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Non-Associative Normed Algebras 2 Volume Hardback Set
 1108679072, 9781108679077

Table of contents :
Non-Associative Normed Algebras, Volume 2: Representation Theory and the Zel’manov Approach
Contents for Volume 2
Contents for Volume 1
Preface
The core of the book revisited
The content of Volume 2
Chapter 5
Chapter 6
Chapter 7
Chapter 8
On the historical notes
Acknowledgements
5 Non-commutative JBW*-algebras, JB*-triples revisited, and a unit-free Vidav–Palmer type non-associative theorem
5.1 Non-commutative JBW*-algebras
5.1.1 The results
5.1.2 Historical notes and comments
5.2 Preliminaries on analytic mappings
5.2.1 Polynomials and higher derivatives
5.2.2 Analytic mappings on Banach spaces
5.2.3 Holomorphic mappings
5.2.4 Historical notes and comments
5.3 Holomorphic automorphisms of a bounded domain
5.3.1 The topology of the local uniform convergence
5.3.2 Holomorphic automorphisms of a bounded domain
5.3.3 The Carathéodory distance on a bounded domain
5.3.4 Historical notes and comments
5.4 Complete holomorphic vector fields
5.4.1 Locally Lipschitz vector fields
5.4.2 Holomorphic vector fields
5.4.3 Complete holomorphic vector fields and one-parameter groups
5.4.4 Historical notes and comments
5.5 Banach Lie structures for aut(Ω) and Aut(Ω)
5.5.1 The real Banach Lie algebra aut(Ω)
5.5.2 The real Banach Lie group Aut(Ω)
5.5.3 Historical notes and comments
5.6 Kaup’s holomorphic characterization of JB*-triples
5.6.1 Bounded circular domains
5.6.2 The symmetric part of a complex Banach space
5.6.5 Historical notes and comments
5.7 JBW*-triples
5.7.1 The bidual of a JB*-triple
5.7.2 The main results
5.7.3 Historical notes and comments
5.8 Operators into the predual of a JBW*-triple
5.8.1 On Pełczyński’s property (V*)
5.8.2 L-embedded spaces have property (V*)
5.8.3 Applications to JB*-triples
5.8.4 Historical notes and comments
5.9 A holomorphic characterization of non-commutative JB*-algebras
5.9.1 Complete normed algebras whose biduals are non-commutative JB*-algebras
5.9.2 The main result
5.9.3 Historical notes and comments
5.10 Complements on non-commutative JB*-algebras and JB*-triples
5.10.1 Selected topics in the theory of non-commutative JBW*-algebras
5.10.2 The strong* topology of a JBW*-triple
5.10.3 Isometries of non-commutative JB*-algebras
5.10.4 Historical notes and comments
6 Representation theory for non-commutative JB*-algebras and alternative C*-algebras
6.1 The main results
6.1.1 Factor representations of non-commutative JB*-algebras
6.1.2 Associativity and commutativity of non-commutative JB*-algebras
6.1.3 JBW*-factors
6.1.4 Classifying prime JB*-algebras: a Zel’manovian approach
6.1.5 Prime non-commutative JB*-algebras are centrally closed
6.1.6 Non-commutative JBW*-factors and alternative W*-factors
6.1.7 Historical notes and comments
6.2 Applications of the representation theory
6.2.1 Alternative C*- and W*-algebras
6.2.2 The strong topology of a non-commutative JBW*-algebra
6.2.3 Prime non-commutative JB*-algebras
6.2.4 Historical notes and comments
6.3 A further application: commutativity of non-commutative JB*-algebras
6.3.1 Le Page’s theorem, and some non-associative variants
6.3.2 The main result
6.3.3 Discussion of results and methods
6.3.4 Historical notes and comments
7 Zel’manov approach
7.1 Classifying prime JB*-triples
7.1.1 Representation theory for JB*-triples
7.1.2 Building prime JB*-triples from prime C*-algebras
7.1.3 The main results
7.1.4 Historical notes and comments
7.2 A survey on the analytic treatment of Zel’manov’s prime theorems
7.2.1 Complete normed J-primitive Jordan algebras
7.2.2 Strong-versus-light normed versions of the Zel’manov prime theorem
7.2.3 The norm extension problem
8 Selected topics in the theory of non-associative normed algebras
8.1 H*-algebras
8.1.1 Preliminaries, and a theorem on power-associative H*-algebras
8.1.2 Structure theory
8.1.3 Topologically simple H*-algebras are ‘very’ prime
8.1.4 Automatic continuity
8.1.5 Isomorphisms and derivations of H*-algebras
8.1.6 Jordan axioms for associative H*-algebras
8.1.7 Real versus complex H*-algebras
8.1.8 Trace-class elements in H*-algebras
8.1.9 Historical notes and comments
8.2 Extending the theory of H*-algebras: generalized annihilator normed algebras
8.2.1 The main result
8.2.2 Generalized annihilator algebras are multiplicatively semiprime
8.2.3 Generalized complemented normed algebras
8.2.4 Historical notes and comments
8.3 Continuing the theory of non-associative normed algebras
8.3.1 Continuity of homomorphisms into normed algebras without topological divisors of zero
8.3.2 Complete normed Jordan algebras with finite J-spectrum
8.3.3 Historical notes and comments
8.3.4 Normed Jordan algebras after Aupetit’s paper [40]: a survey
8.4 The joint spectral radius of a bounded set
8.4.1 Basic notions and results
8.4.2 Topologically nilpotent normed algebras
8.4.3 Involving nearly absolute-valued algebras
8.4.4 Involving tensor products
8.4.5 Historical notes and comments
Bibliography of Volume 1 (updated)
Papers
Books
Additional Bibliography to Volume 2
Papers
Books
Symbol index for Volume 1
Subject index for Volume 1
Symbol index for Volume 2
Subject index for Volume 2

Citation preview

N O N - A S S O C I AT I V E N O R M E D A L G E B R A S Volume 2: Representation Theory and the Zel’manov Approach

This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This second volume revisits JB*-triples, covers Zel’manov’s celebrated work in Jordan theory, proves the unit-free variant of Vidav–Palmer theorem, and develops the representation theory of alternative C*-algebras and non-commutative JB*-algebras. This completes the work begun in the first volume, which introduced these algebras and discussed the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems. This book interweaves pure algebra, geometry of normed spaces, and infinite-dimensional complex analysis. Novel proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, and an extensive bibliography.

Encyclopedia of Mathematics and Its Applications This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and Its Applications cover their subjects comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes are encyclopedic references or manageable guides to major subjects.

Encyclopedia of Mathematics and its Applications All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit www.cambridge.org/mathematics. 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168

M. Deza and M. Dutour Sikiri´c Geometry of Chemical Graphs T. Nishiura Absolute Measurable Spaces M. Prest Purity, Spectra and Localisation S. Khrushchev Orthogonal Polynomials and Continued Fractions H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity F. W. King Hilbert Transforms I F. W. King Hilbert Transforms II O. Calin and D.-C. Chang Sub-Riemannian Geometry M. Grabisch et al. Aggregation Functions L. W. Beineke and R. J. Wilson (eds.) with J. L. Gross and T. W. Tucker Topics in Topological Graph Theory J. Berstel, D. Perrin and C. Reutenauer Codes and Automata T. G. Faticoni Modules over Endomorphism Rings H. Morimoto Stochastic Control and Mathematical Modeling G. Schmidt Relational Mathematics P. Kornerup and D. W. Matula Finite Precision Number Systems and Arithmetic Y. Crama and P. L. Hammer (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering V. Berth´e and M. Rigo (eds.) Combinatorics, Automata and Number Theory A. Krist´aly, V. D. R˘adulescu and C. Varga Variational Principles in Mathematical Physics, Geometry, and Economics J. Berstel and C. Reutenauer Noncommutative Rational Series with Applications B. Courcelle and J. Engelfriet Graph Structure and Monadic Second-Order Logic M. Fiedler Matrices and Graphs in Geometry N. Vakil Real Analysis through Modern Infinitesimals R. B. Paris Hadamard Expansions and Hyperasymptotic Evaluation Y. Crama and P. L. Hammer Boolean Functions A. Arapostathis, V. S. Borkar, and M. K. Ghosh Ergodic Control of Diffusion Processes N. Caspard, B. Leclerc, and B. Monjardet Finite Ordered Sets D. Z. Arov and H. Dym Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations G. Dassios Ellipsoidal Harmonics L. W. Beineke and R. J. Wilson (eds.) with O. R. Oellermann Topics in Structural Graph Theory L. Berlyand, A. G. Kolpakov, and A. Novikov Introduction to the Network Approximation Method for Materials Modeling M. Baake and U. Grimm Aperiodic Order I: A Mathematical Invitation J. Borwein et al. Lattice Sums Then and Now R. Schneider Convex Bodies: The Brunn–Minkowski Theory (Second Edition) G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions (Second Edition) D. Hofmann, G. J. Seal, and W. Tholen (eds.) Monoidal Topology ´ Rodr´ıguez Palacios Non-Associative Normed Algebras I: The M. Cabrera Garc´ıa and A. Vidav–Palmer and Gelfand–Naimark Theorems C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables (Second Edition) L. W. Beineke and R. J. Wilson (eds.) with B. Toft Topics in Chromatic Graph Theory T. Mora Solving Polynomial Equation Systems III: Algebraic Solving T. Mora Solving Polynomial Equation Systems IV: Buchberger Theory and Beyond V. Berth´e and M. Rigo (eds.) Combinatorics, Words and Symbolic Dynamics B. Rubin Introduction to Radon Transforms: With Elements of Fractional Calculus and Harmonic Analysis M. Ghergu and S. D. Taliaferro Isolated Singularities in Partial Differential Inequalities G. Molica Bisci, V. Radulescu, and R. Servadei Variational Methods for Nonlocal Fractional Problems S. Wagon The Banach–Tarski Paradox (Second Edition) K. Broughan Equivalents of the Riemann Hypothesis I: Arithmetic Equivalents K. Broughan Equivalents of the Riemann Hypothesis II: Analytic Equivalents M. Baake and U. Grimm (eds.) Aperiodic Order II: Crystallography and Almost Periodicity ´ Rodr´ıguez Palacios Non-Associative Normed Algebras II: M. Cabrera Garc´ıa and A. Representation Theory and the Zel’manov Approach A. Yu. Khrennikov, S. V. Kozyrev and W. A. Z´un˜ iga-Galindo Ultrametric Pseudodifferential Equations and Applications

Encyclopedia of Mathematics and its Applications

Non-Associative Normed Algebras Volume 2: Representation Theory and the Zel’manov Approach M I G U E L C A B R E R A G A R C ´I A Universidad de Granada

´ N G E L R O D R ´I G U E Z PA L A C I O S A Universidad de Granada

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107043114 DOI: 10.1017/9781107337817 ´ © Miguel Cabrera Garc´ıa and Angel Rodr´ıguez Palacios 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Cabrera Garc´ıa, Miguel. Non-associative normed algebras / Miguel Cabrera Garc´ıa, Universidad de Granada, ´ Angel Rodr´ıguez Palacios, Universidad de Granada. volumes cm. – (Encyclopedia of mathematics and its applications) ISBN 978-1-107-04306-0 (hardback) ´ 1. Banach algebras. 2. Algebra. I. Rodr´ıguez Palacios, Angel. II. Title. QA326.C33 2014 512 .554–dc23 2013045718 ISBN 978-1-107-04306-0 Hardback ISBN - 2 Volume Set 978-1-108-67907-7 Hardback ISBN - Volume I 978-1-107-04306-0 Hardback ISBN - Volume II 978-1-107-04311-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Ana Mar´ıa and In´es

Contents for Volume 2

Contents for Volume 1 Preface 5

page xi xvii

Non-commutative JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer type non-associative theorem 5.1 Non-commutative JBW ∗ -algebras 5.1.1 The results 5.1.2 Historical notes and comments 5.2 Preliminaries on analytic mappings 5.2.1 Polynomials and higher derivatives 5.2.2 Analytic mappings on Banach spaces 5.2.3 Holomorphic mappings 5.2.4 Historical notes and comments 5.3 Holomorphic automorphisms of a bounded domain 5.3.1 The topology of the local uniform convergence 5.3.2 Holomorphic automorphisms of a bounded domain 5.3.3 The Carath´eodory distance on a bounded domain 5.3.4 Historical notes and comments 5.4 Complete holomorphic vector fields 5.4.1 Locally Lipschitz vector fields 5.4.2 Holomorphic vector fields 5.4.3 Complete holomorphic vector fields and one-parameter groups 5.4.4 Historical notes and comments 5.5 Banach Lie structures for aut() and Aut() 5.5.1 The real Banach Lie algebra aut() 5.5.2 The real Banach Lie group Aut() 5.5.3 Historical notes and comments 5.6 Kaup’s holomorphic characterization of JB∗ -triples 5.6.1 Bounded circular domains vii

1 2 3 18 19 20 32 43 53 53 53 61 77 87 88 88 99 126 135 137 137 151 174 174 175

viii

Contents for Volume 2

5.7

5.8

5.9

5.10

6

5.6.2 The symmetric part of a complex Banach space 5.6.3 Numerical ranges revisited 5.6.4 Concluding the proof of Kaup’s theorem 5.6.5 Historical notes and comments JBW ∗ -triples 5.7.1 The bidual of a JB∗ -triple 5.7.2 The main results 5.7.3 Historical notes and comments Operators into the predual of a JBW ∗ -triple 5.8.1 On Pełczy´nski’s property (V ∗ ) 5.8.2 L-embedded spaces have property (V ∗ ) 5.8.3 Applications to JB∗ -triples 5.8.4 Historical notes and comments A holomorphic characterization of non-commutative JB∗ -algebras 5.9.1 Complete normed algebras whose biduals are non-commutative JB∗ -algebras 5.9.2 The main result 5.9.3 Historical notes and comments Complements on non-commutative JB∗ -algebras and JB∗ -triples 5.10.1 Selected topics in the theory of non-commutative JBW ∗ -algebras 5.10.2 The strong∗ topology of a JBW ∗ -triple 5.10.3 Isometries of non-commutative JB∗ -algebras 5.10.4 Historical notes and comments

Representation theory for non-commutative JB∗ -algebras and alternative C∗ -algebras 6.1 The main results 6.1.1 Factor representations of non-commutative JB∗ -algebras 6.1.2 Associativity and commutativity of non-commutative JB∗ -algebras 6.1.3 JBW ∗ -factors 6.1.4 Classifying prime JB∗ -algebras: a Zel’manovian approach 6.1.5 Prime non-commutative JB∗ -algebras are centrally closed 6.1.6 Non-commutative JBW ∗ -factors and alternative W ∗ -factors 6.1.7 Historical notes and comments 6.2 Applications of the representation theory 6.2.1 Alternative C∗ - and W ∗ -algebras

187 193 197 207 211 211 224 236 245 245 253 263 266 268 268 271 272 274 275 298 315 329 347 347 348 352 359 363 367 374 398 411 412

Contents for Volume 2 6.2.2

6.3

7

8

The strong topology of a non-commutative JBW ∗ -algebra 6.2.3 Prime non-commutative JB∗ -algebras 6.2.4 Historical notes and comments A further application: commutativity of non-commutative JB∗ -algebras 6.3.1 Le Page’s theorem, and some non-associative variants 6.3.2 The main result 6.3.3 Discussion of results and methods 6.3.4 Historical notes and comments

ix 416 418 421 422 422 428 432 436

Zel’manov approach 7.1 Classifying prime JB∗ -triples 7.1.1 Representation theory for JB∗ -triples 7.1.2 Building prime JB∗ -triples from prime C∗ -algebras 7.1.3 The main results 7.1.4 Historical notes and comments 7.2 A survey on the analytic treatment of Zel’manov’s prime theorems 7.2.1 Complete normed J-primitive Jordan algebras 7.2.2 Strong-versus-light normed versions of the Zel’manov prime theorem 7.2.3 The norm extension problem

437 437 438 441 444 458

Selected topics in the theory of non-associative normed algebras 8.1 H ∗ -algebras 8.1.1 Preliminaries, and a theorem on power-associative H ∗ -algebras 8.1.2 Structure theory 8.1.3 Topologically simple H ∗ -algebras are ‘very’ prime 8.1.4 Automatic continuity 8.1.5 Isomorphisms and derivations of H ∗ -algebras 8.1.6 Jordan axioms for associative H ∗ -algebras 8.1.7 Real versus complex H ∗ -algebras 8.1.8 Trace-class elements in H ∗ -algebras 8.1.9 Historical notes and comments 8.2 Extending the theory of H ∗ -algebras: generalized annihilator normed algebras 8.2.1 The main result 8.2.2 Generalized annihilator algebras are multiplicatively semiprime 8.2.3 Generalized complemented normed algebras 8.2.4 Historical notes and comments

477 477

461 462 467 470

480 484 489 495 502 508 510 523 545 561 562 571 577 583

x

Contents for Volume 2 8.3

8.4

Continuing the theory of non-associative normed algebras 8.3.1 Continuity of homomorphisms into normed algebras without topological divisors of zero 8.3.2 Complete normed Jordan algebras with finite J-spectrum 8.3.3 Historical notes and comments 8.3.4 Normed Jordan algebras after Aupetit’s paper [40]: a survey The joint spectral radius of a bounded set 8.4.1 Basic notions and results 8.4.2 Topologically nilpotent normed algebras 8.4.3 Involving nearly absolute-valued algebras 8.4.4 Involving tensor products 8.4.5 Historical notes and comments

Bibliography of Volume 1 (updated) References – Papers References – Books Additional Bibliography to Volume 2 References – Papers References – Books Symbol index for Volume 1 Subject index for Volume 1 Symbol index for Volume 2 Subject index for Volume 2

586 586 589 595 597 603 605 620 638 642 650 664 664 691 699 699 712 715 719 725 727

Contents for Volume 1

Preface 1

2

page xi

Foundations 1.1 Rudiments on normed algebras 1.1.1 Basic spectral theory 1.1.2 Rickart’s dense-range-homomorphism theorem 1.1.3 Gelfand’s theory 1.1.4 Topological divisors of zero 1.1.5 The complexification of a normed real algebra 1.1.6 The unital extension and the completion of a normed algebra 1.1.7 Historical notes and comments 1.2 Introducing C∗ -algebras 1.2.1 The results 1.2.2 Historical notes and comments 1.3 The holomorphic functional calculus 1.3.1 The polynomial and rational functional calculuses 1.3.2 The main results 1.3.3 Historical notes and comments 1.4 Compact and weakly compact operators 1.4.1 Operators from a normed space to another 1.4.2 Operators from a normed space to itself 1.4.3 Discussing the inclusion F(X, Y) ⊆ K(X, Y) in the non-complete setting 1.4.4 Historical notes and comments Beginning the proof of the non-associative Vidav–Palmer theorem 2.1 Basic results on numerical ranges 2.1.1 Algebra numerical ranges 2.1.2 Operator numerical ranges 2.1.3 Historical notes and comments xi

1 1 1 16 20 27 30 33 36 38 38 52 53 53 58 68 70 70 75 83 86 94 94 94 104 114

xii

Contents for Volume 1 2.2

2.3

2.4

2.5

2.6

2.7

2.8

An application to Kadison’s isometry theorem 2.2.1 Non-associative results 2.2.2 The Kadison–Paterson–Sinclair theorem 2.2.3 Historical notes and comments The associative Vidav–Palmer theorem, starting from a non-associative germ 2.3.1 Natural involutions of V-algebras are algebra involutions 2.3.2 The associative Vidav–Palmer theorem 2.3.3 Complements on C∗ -algebras 2.3.4 Introducing alternative C∗ -algebras 2.3.5 Historical notes and comments V-algebras are non-commutative Jordan algebras 2.4.1 The main result 2.4.2 Applications to C∗ -algebras 2.4.3 Historical notes and comments The Frobenius–Zorn theorem, and the generalized Gelfand– Mazur–Kaplansky theorem 2.5.1 Introducing quaternions and octonions 2.5.2 The Frobenius–Zorn theorem 2.5.3 The generalized Gelfand–Mazur–Kaplansky theorem 2.5.4 Historical notes and comments Smooth-normed algebras, and absolute-valued unital algebras 2.6.1 Determining smooth-normed algebras and absolute-valued unital algebras 2.6.2 Unit-free characterizations of smooth-normed algebras, and of absolute-valued unital algebras 2.6.3 Historical notes and comments Other Gelfand–Mazur type non-associative theorems 2.7.1 Focusing on complex algebras 2.7.2 Involving real scalars 2.7.3 Discussing the results 2.7.4 Historical notes and comments Complements on absolute-valued algebras and algebraicity 2.8.1 Continuity of algebra homomorphisms into absolute-valued algebras 2.8.2 Absolute values on H ∗ -algebras 2.8.3 Free non-associative algebras are absolute-valued algebras 2.8.4 Complete normed algebraic algebras are of bounded degree

120 120 124 130 131 132 138 143 151 156 160 161 166 171 176 176 177 192 198 203 203 212 216 223 223 227 238 244 249 250 251 257 262

Contents for Volume 1 2.8.5

2.9

3

Absolute-valued algebraic algebras are finite-dimensional 2.8.6 Historical notes and comments Complements on numerical ranges 2.9.1 Involving the upper semicontinuity of the duality mapping 2.9.2 The upper semicontinuity of the pre-duality mapping 2.9.3 Involving the strong subdifferentiability of the norm 2.9.4 Historical notes and comments

xiii

Concluding the proof of the non-associative Vidav–Palmer theorem 3.1 Isometries of JB-algebras 3.1.1 Isometries of unital JB-algebras 3.1.2 Isometries of non-unital JB-algebras 3.1.3 A metric characterization of derivations of JB-algebras 3.1.4 JB-algebras whose Banach spaces are convex-transitive 3.1.5 Historical notes and comments 3.2 The unital non-associative Gelfand–Naimark theorem 3.2.1 The main result 3.2.2 Historical notes and comments 3.3 The non-associative Vidav–Palmer theorem 3.3.1 The main result 3.3.2 A dual version 3.3.3 Historical notes and comments 3.4 Beginning the theory of non-commutative JB∗ -algebras 3.4.1 JB-algebras versus JB∗ -algebras 3.4.2 Isometries of unital non-commutative JB∗ -algebras 3.4.3 An interlude: derivations and automorphisms of normed algebras 3.4.4 The structure theorem of isomorphisms of non-commutative JB∗ -algebras 3.4.5 Historical notes and comments 3.5 The Gelfand–Naimark axiom a∗ a = a∗ a, and the non-unital non-associative Gelfand–Naimark theorem 3.5.1 Quadratic non-commutative JB∗ -algebras 3.5.2 The axiom a∗ a = a∗ a on unital algebras 3.5.3 An interlude: the bidual and the spacial numerical index of a non-commutative JB∗ -algebra 3.5.4 The axiom a∗ a = a∗ a on non-unital algebras 3.5.5 The non-unital non-associative Gelfand–Naimark theorem

270 274 283 284 291 299 310 319 319 319 324 327 332 336 340 340 344 344 345 351 356 359 359 366 370 381 388 392 393 397 404 411 414

xiv

Contents for Volume 1 3.5.6 Vowden’s theorem 3.5.7 Historical notes and comments Jordan axioms for C∗ -algebras 3.6.1 Jacobson’s representation theory: preliminaries 3.6.2 The main result 3.6.3 Jacobson’s representation theory continued 3.6.4 Historical notes and comments

418 421 425 426 432 435 444

Jordan spectral theory 4.1 Involving the Jordan inverse 4.1.1 Basic spectral theory for normed Jordan algebras 4.1.2 Topological J-divisors of zero 4.1.3 Non-commutative JB∗ -algebras are JB∗ -triples 4.1.4 Extending the Jordan spectral theory to Jordan-admissible algebras 4.1.5 The holomorphic functional calculus for complete normed unital non-commutative Jordan complex algebras 4.1.6 A characterization of smooth-normed algebras 4.1.7 Historical notes and comments 4.2 Unitaries in JB∗ -triples and in non-commutative JB∗ -algebras 4.2.1 A commutative Gelfand–Naimark theorem for JB∗ -triples 4.2.2 The main results 4.2.3 Russo–Dye type theorems for non-commutative JB∗ -algebras 4.2.4 A touch of real JB∗ -triples and of real noncommutative JB∗ -algebras 4.2.5 Historical notes and comments 4.3 C∗ - and JB∗ -algebras generated by a non-self-adjoint idempotent 4.3.1 The case of C∗ -algebras 4.3.2 The case of JB∗ -algebras 4.3.3 An application to non-commutative JB∗ -algebras 4.3.4 Historical notes and comments 4.4 Algebra norms on non-commutative JB∗ -algebras 4.4.1 The Johnson–Aupetit–Ransford uniqueness-of-norm theorem 4.4.2 A non-complete variant 4.4.3 The main results 4.4.4 The uniqueness-of-norm theorem for general non-associative algebras 4.4.5 Historical notes and comments

450 450 451 460 463

3.6

4

473

480 487 490 497 498 505 518 521 527 536 536 552 560 562 565 566 571 573 577 592

Contents for Volume 1 4.5

4.6

JB∗ -representations and alternative C∗ -representations of hermitian algebras 4.5.1 Preliminary results 4.5.2 The main results 4.5.3 A conjecture on non-commutative JB∗ -equivalent algebras 4.5.4 Historical notes and comments Domains of closed derivations 4.6.1 Stability under the holomorphic functional calculus 4.6.2 Stability under the geometric functional calculus 4.6.3 Historical notes and comments

References – Papers References – Books Symbol index Subject index

xv 604 605 611 630 632 636 636 644 665 671 696 704 707

Preface

The core of the book revisited In the preface to Volume 1 we proposed as the ‘leitmotiv’ of our work to remove associativity in the abstract characterizations of unital (associative) C∗ -algebras given either by the Gelfand–Naimark theorem or by the Vidav–Palmer theorem, and to study (possibly non-unital) closed ∗-subalgebras of the Gelfand–Naimark or Vidav– Palmer algebras born after removing associativity. To be more precise, for a norm-unital complete normed (possibly non-associative) complex algebra A, we considered the following conditions: (GN) (Gelfand–Naimark axiom). There is a conjugate-linear vector space involution ∗ on A satisfying 1∗ = 1 and a∗ a = a2 for every a in A. (VP) (Vidav–Palmer axiom). A = H(A, 1) + iH(A, 1). In both conditions, 1 denotes the unit of A, whereas, in (VP), H(A, 1) stands for the closed real subspace of A consisting of those elements h ∈ A such that f (h) belongs to R for every bounded linear functional f on A satisfying f  = f (1) = 1. Contrary to what happens in the associative case [696, 725, 787, 930], in the nonassociative setting, (GN) and (VP) are not equivalent conditions. Indeed, as proved in Lemma 2.2.5, it is easily seen that (GN) implies (VP), but, as shown by Example 2.3.65, the converse implication is not true. Therefore, after introducing ‘alternative C∗ -algebras’ and ‘non-commutative JB∗ -algebras’, and realizing that the former are particular cases of the latter, we specified how, by means of Theorems GN and VP which follow, the behaviour of the Gelfand–Naimark and the Vidav–Palmer axioms in the non-associative setting are clarified. Theorem GN Norm-unital complete normed complex algebras fulfilling the Gelfand– Naimark axiom are nothing other than unital alternative C∗ -algebras. Theorem VP Norm-unital complete normed complex algebras fulfilling the Vidav– Palmer axiom are nothing other than unital non-commutative JB∗ -algebras.

xvii

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Then we announced as the main goal of our work to prove Theorems GN and VP, together with their unit-free variants, and to ‘describe’ alternative C∗ -algebras and non-commutative JB∗ -algebras by means of the so-called representation theory. Since Theorems GN and VP and the unit-free variant of Theorem GN were already proved in Theorems 3.2.5, 3.3.11, and 3.5.53, respectively, it remains the main objective of our work to prove the unit-free variant of Theorem VP, and to develop the representation theory of alternative C∗ -algebras and non-commutative JB∗ -algebras. We now do this in Chapters 5 and 6 respectively. Indeed, the unit-free variant of Theorem VP is proved in Theorem 5.9.9, whereas the representation theory of alternative C∗ -algebras and non-commutative JB∗ -algebras can be summarized by means of Corollaries 6.1.11 and 6.1.12, Theorem 6.1.112, and Corollary 6.1.115. The content of Volume 2 As we commented in the preface of Volume 1, the dividing line between the two volumes could be drawn between what can be done before and after involving the holomorphic theory of JB∗ -triples and the structure theory of non-commutative JB∗ algebras. Then the content of Volume 1 was described in some detail, and a tentative content of Volume 2 was outlined. Now we are going to specify with more precision the content of the present second volume. Chapter 5 The main goal of this first chapter of Volume 2 is to prove what can be seen as a unit-free version of the non-associative Vidav–Palmer theorem, namely that noncommutative JB∗ -algebras are precisely those complete normed complex algebras having an approximate unit bounded by one, and whose open unit ball is a homogeneous domain [365] (see Theorem 5.9.9). Some ingredients in the long proof of this result were already established in Volume 1. This is the case of the Bohnenblust– Karlin Corollary 2.1.13, the non-associative Vidav–Palmer theorem proved in Theorem 3.3.11 as well as its dual version shown in Corollary 3.3.26, Proposition 3.5.23 (that every non-commutative JB∗ -algebra has an approximate unit bounded by one), Theorem 4.1.45 (that non-commutative JB∗ -algebras are JB∗ -triples in a natural way), and the equivalence (ii)⇔(vii) in the Braun–Kaup–Upmeier Theorem 4.2.24. ♣ The new relevant ingredients which are proved in the chapter are the following: (i) Edward’s fundamental Fact 5.1.42, which describes how JBW-algebras and JBW ∗ -algebras are mutually determined, and implies, via [738], the uniqueness of the predual of any non-commutative JBW ∗ -algebra (see Theorem 5.1.29(iv)). (ii) The Kaup–Stach´o contractive projection theorem for JB∗ -triples (see Theorem 5.6.59). (iii) Kaup’s holomorphic characterization of JB∗ -triples as those complex Banach spaces whose open unit ball is a homogeneous domain (see Theorem 5.6.68).

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(iv) Dineen’s celebrated result that the bidual of a JB∗ -triple is a JB∗ -triple (see Proposition 5.7.10). (v) The Barton–Horn–Timoney basic theory of JBW ∗ -triples establishing the separate w∗ -continuity of the triple product of a given JBW ∗ -triple (see Theorem 5.7.20) and the uniqueness of the predual (see Theorem 5.7.38). (vi) The Barton–Timoney theorem that the predual of any JBW ∗ -triple is L-embedded (see Theorem 5.7.36). (vii) The Chu–Iochum–Loupias result that bounded linear operators from a JB∗ triple to its dual are weakly compact (see Corollary 5.8.33) or, equivalently, that all continuous products on the Banach space of a JB∗ -triple are Arens regular (see Fact 5.8.39). The original references for the results just listed are [222], [382, 597], [381], [213], [854, 979], [854], and [172], respectively. Our proof of these results are not always the original ones, although sometimes the latter underlie the former. This is the case of results (ii) and (iii), which in our development depend on the foundations of the infinite-dimensional holomorphy done in [710, 751, 814, 837, 1113, 1114, 1124] (see Sections 5.2 to 5.6), on the design of proof suggested in [710, Section 2.5], and, at the end, on numerical range techniques included in Subsection 5.6.3. On the other hand, our proof of result (v) is new, and, contrary to what happens in the original one, it avoids any Banach space result on uniqueness of preduals. Indeed, our proof of Theorem 5.7.20 involves only result (ii) and the Barton–Timoney Theorem 5.7.18, whereas our proof of Theorem 5.7.38 depends only on Theorem 5.7.20 (whose proof has been just remarked on), result (i), and Horn’s Corollary 5.7.28(i)(b). Concerning result (vii), it is noteworthy that a much finer theorem is proved in [172]. Namely, that every bounded linear operator from a JB∗ -triple to its dual factors through a complex Hilbert space. The proof of this more general theorem (a sketch of which can be found in §5.10.151) is very involved, and shall not be completely discussed in our work. As a matter of fact, we re-encounter result (vii) by combining results (iv) and (vi) with Corollary 5.8.19 (asserting that, if Y is a Banach space such that Y  has property (V ∗ ), then every bounded linear operator from Y to Y  is weakly compact) and Theorem 5.8.27 (that L-embedded Banach spaces have property (V ∗ )). Corollary 5.8.19 and Theorem 5.8.27 just reviewed are due to Godefroy–Iochum [957] and Pfitzner [1044], respectively. Nevertheless, the proof of Corollary 5.8.19 in [957] relies heavily on Proposition 5.8.14, whose arguments have been lost in the literature (see §5.8.42). Our proof of Proposition 5.8.14 is taken from Pfitzner’s private communication [1047]. Once the main objective of the chapter is reached in Section 5.9, the chapter concludes with a section devoted to some complements on non-commutative JB∗ algebras and JB∗ -triples. In Subsection 5.10.1 we introduce the strong∗ topology of a non-commutative JBW ∗ -algebra [19] and apply it to build up a functional calculus at each normal element a of a non-commutative JBW ∗ -algebra A, which extends the continuous

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functional calculus (cf. Corollary 4.1.72) and has a sense for all real-valued bounded lower semicontinuous functions on J-sp(A, a). Then we follow [366] to prove a variant for non-commutative JBW ∗ -algebras of Kadison’s isometry theorem for unital C∗ -algebras (cf. Theorem 2.2.29), a consequence of which is that linearly isometric non-commutative JBW ∗ -algebras are Jordan-∗-isomorphic. (We recall that linearly isometric (possibly non-unital) C∗ -algebras are Jordan-∗-isomorphic (a consequence of Theorem 2.2.19), but that linearly isometric (even unital) noncommutative JB∗ -algebras need not be Jordan-∗-isomorphic (cf. Antitheorem 3.4.34).) We also prove the generalization to non-commutative JBW ∗ -algebras of Akemann’s theorem [826] asserting the coincidence of the strong∗ and Mackey topologies on bounded subsets of any W ∗ -algebra. In Subsection 5.10.2 we introduce and study the strong∗ topology of a JBW ∗ triple as done by Barton and Friedman [853, 60], and follow [1061] to prove that, when a non-commutative JBW ∗ -algebra is viewed as a JBW ∗ -triple, its new (triple) strong∗ topology coincides with the (algebra) strong∗ topology introduced in Subsection 5.10.1. We also prove Zizler’s refinement [1137] of Lindenstrauss’s theorem [1001] on norm-density of operators whose transpose attain their norm, and apply it to prove a variant for JBW ∗ -triples of the so-called little Grothendieck’s theorem [853, 964, 1040, 1052]. In Subsection 5.10.3 we provide the reader with a full non-associative discussion of the Kadison–Paterson–Sinclair Theorem 2.2.19 on surjective linear isometries of (possibly non-unital) C∗ -algebras [366]. To this end we introduce the multiplier non-commutative JB∗ -algebra M(A) of a given non-commutative JB∗ -algebra A, and prove that M(A) coincides with the JB∗ -triple of multipliers [873] of the JB∗ triple underlying A. Then we also prove that the Kadison–Paterson–Sinclair theorem remains true verbatim for surjective linear isometries from non-commutative JB∗ algebras to alternative C∗ -algebras, and that no further verbatim generalization is possible.

Chapter 6 Implicitly, the representation theory of JB-algebras underlies our work since, without providing the reader with a proof, we took from the Hanche-Olsen–Stormer book [738] the very deep fact that the closed subalgebra of a JB-algebra generated by two elements is a JC-algebra (cf. Proposition 3.1.3). In that way we were able to develop the basic theory of non-commutative JB∗ -algebras (including the non-associative Vidav–Palmer Theorem 3.3.11 and Wright’s fundamental Fact 3.4.9 which describes how JB-algebras and JB∗ -algebras are mutually determined) without any further implicit or explicit reference to representation theory. In fact, we avoided any dependence on representation theory throughout all of Volume 1, and to the end of Chapter 5 of the present volume. Now, in Chapter 6, we conclude the basic theory of non-commutative JB∗ algebras, and follow [19, 124, 125, 222, 481, 482, 641] to develop in depth the

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representation theory of non-commutative JB∗ -algebras and, in particular, that of alternative C∗ -algebras. To this end, in Subsection 6.1.1 we introduce noncommutative JBW ∗ -factors and non-commutative JBW ∗ -factor representations of a given non-commutative JB∗ -algebra, and prove that every non-commutative JB∗ algebra has a faithful family of type I non-commutative JBW ∗ -factor representations. When these results specialize for classical C∗ -algebras, type I non-commutative JBW ∗ -factors are nothing other than the (associative) W ∗ -factors consisting of all bounded linear operators on some complex Hilbert space [738, Proposition 7.5.2], and, consequently, type I W ∗ -factor representations of a C∗ -algebra A are precisely the irreducible representations of A on complex Hilbert spaces. Subsection 6.1.2 deals with a first application of the representation theory outlined above, which allows us to show that non-commutative JB∗ -algebras are associative and commutative if (and only if) they have no nonzero nilpotent element. As a consequence, we obtain that alternative C∗ -algebras are commutative if and only if they have no nonzero nilpotent element [340]. This generalizes Kaplansky’s associative forerunner [761, Theorem B in Appendix III]. In Subsection 6.1.3, we involve the theory of JB-algebras [738], and invoke result (i) in ♣ to classify all (commutative) JBW ∗ -factors. This classification is applied to prove that i-special JB∗ -algebras are JC∗ -algebras. In Subsection 6.1.4, we combine the result just reviewed with Zel’manovian techniques [437, 662] to prove that, if J is a prime JB∗ -algebra, and if J is neither quadratic (cf. Corollary 3.5.7) nor equal to the unique JB∗ -algebra whose self-adjoint part is H3 (O) (cf. Example 3.1.56 and Theorem 3.4.8), then either there exists a prime C∗ -algebra A such that J is a closed ∗-subalgebra of the JB∗ -algebra M(A)sym containing A, or there exists a prime C∗ -algebra A with a ∗-involution τ such that J is a closed ∗-subalgebra of M(A)sym contained in H(M(A), τ ) and containing H(A, τ ) [255]. In Subsection 6.1.5, we introduce totally prime normed algebras and ultraprime normed algebras, and prove that totally prime normed complex algebras are centrally closed, and that ultraprime normed algebras are totally prime [149]. Then we combine the classification theorem of prime JB∗ -algebras reviewed above with the fact that prime C∗ -algebras are ultraprime [1012] to show that prime non-commutative JB∗ -algebras are ultraprime, and hence centrally closed. In Subsection 6.1.6, we combine the central closedness of prime non-commutative JB∗ -algebras with a topological reading of McCrimmon’s paper [436] to prove that non-commutative JBW ∗ -factors are either commutative or simple quadratic or of the form B(λ) for some (associative) W ∗ -factor B and some 0 ≤ λ ≤ 1. This theorem is originally due to Braun [124]. As a consequence, alternative W ∗ factors are either associative or equal to the alternative C∗ -algebra of complex octonions (cf. Proposition 2.6.8). Now that we have reviewed Section 6.1 in detail, we will explain the content of the remaining sections of Chapter 6. Section 6.2 deals with the main applications of the representation theory, namely the structure of alternative C∗ -algebras [125, 331, 481], the definition and properties of the strong topology of a non-commutative JBW ∗ -algebra [482], and the classification of prime non-commutative JB∗ -algebras

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[363]. Finally, Section 6.3 deals with a rather incidental application. Indeed, we follow [860] to prove a Le Page type theorem for non-commutative JB∗ -algebras, and discuss Le Page’s theorem [999] in a general non-associative and non-star setting.

Chapter 7 This chapter deals with the analytic treatment of Zel’manov’s prime theorems for Jordan structures, thus continuing the approach begun in Subsection 6.1.4. In Subsection 7.1 we follow [448, 449] to prove as the main result that, if X is a prime JB∗ -triple which is neither an exceptional Cartan factor nor a spin triple factor, then either there exist a prime C∗ -algebra A and a self-adjoint idempotent e in the C∗ algebra M(A) of multipliers of A such that X is a closed subtriple of M(A) contained in eM(A)(1 − e) and containing eA(1 − e), or there exist a prime C∗ -algebra A, a selfadjoint idempotent e ∈ M(A), and a ∗-involution τ on A with e + eτ = 1 such that X is a closed subtriple of M(A) contained in H(eM(A)eτ , τ ) and containing H(eAeτ , τ ). Among the many tools involved in the proof of the above classification theorem, we emphasize Horn’s description of Cartan factors [330], the core of the proof of Zel’manov’s prime theorem for Jordan triples [663, 1133, 1134], and the complementary work by D’Amour and McCrimmon on the topic [920, 921]. Proofs of these tools are not discussed in our development. The main results in the Friedman–Russo paper [270], whose proofs are outlined in our development, are also involved. It is noteworthy that, through the description of prime JB∗ -algebras proved in Subsection 6.1.4, Zel’manov’s work underlies again the proof of the classification theorem of prime JB∗ -triples we are dealing with. In Section 7.2 we survey in detail other applications of Zel’manov’s prime theorems on Jordan structures to the study of normed Jordan algebras and triples. In Subsection 7.2.1 we include the general complete normed version [146] of the Anquela–Montaner–Cort´es–Skosyrskii classification theorem of J-primitive Jordan algebras [21, 585], as well as the more precise classification theorem of J-primitive JB∗ -algebras [255, 525]. In Subsection 7.2.2 we include structure theorems for simple normed Jordan algebras [151] (see also [539]) and non-degenerately ultraprime complete normed Jordan complex algebras [152] (see also [428, 855]). This subsection deals also with the limits of normed versions of Zel’manov prime theorems, a question which was first considered in [893], and culminates in the paper of Moreno, Zel’manov, and the authors [147] where it is proved that an associative polynomial p over K is a Jordan polynomial if and only if, for every algebra norm  ·  on the Jordan algebra M∞ (K)sym , the action of p on M∞ (K) is  · -continuous (see also [447, 1082]). Subsection 7.2.3 deals with the so-called norm extension problem, which in its roots is crucially related to normed versions of Zel’manov’s prime theorems. The first significative progress on this problem (reviewed of course in this subsection) is due to Rodr´ıguez, Slinko, and Zel’manov [538], who as the main result prove that,

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if A is a real or complex associative algebra with linear algebra involution ∗, if A is a ‘∗-tight envelope of H(A, ∗)’, if the Jordan algebra H(A, ∗) is semiprime, and if  ·  is a complete algebra norm on H(A, ∗), then there exists an algebra norm on A whose restriction to H(A, ∗) is equivalent to  · . The appropriate versions for Jordan triples of the results of [538], due to Moreno [1025, 1026], are also included. The subsection concludes with a full discussion of results on the norm extension problem in a general non-associative setting. The main reference for this topic is [1029]. Other related results in [1027, 1059, 1064] are also reviewed.

Chapter 8 We devote this concluding chapter to developing some of our favourite parcels of the theory of non-associative normed algebras, not previously included in our work. The first section of the chapter deals with H ∗ -algebras, incidentally introduced in Volume 1 of our work. The reasonably well-behaved co-existence of two structures, namely that of an algebra and that of a Hilbert space, becomes the essence of semiH ∗ -algebras. Indeed, they are complete normed algebras A endowed with a (vector space) conjugate-linear involution ∗, and whose norm derives from an inner product in such a way that, for each a ∈ A, the adjoint of the left multiplication La is precisely La∗ , and the adjoint of the right multiplication Ra is Ra∗ . Since Ambrose’s pioneering paper [20], it is well-known that associative semi-H ∗ -algebras with zero annihilator are H ∗ -algebras, i.e. their involutions are algebra involutions. But this is no longer true in general. We begin Subsection 8.1.1 by recalling those results on semi-H ∗ -algebras, which were already proved in Volume 1 of our work. Then we introduce the classical topologically simple associative complex H ∗ -algebra HS (H) of all Hilbert– Schmidt operators on a nonzero complex Hilbert space H, and show how this algebra allows us to construct natural examples of Jordan and Lie H ∗ -algebras. After showing how the norm of a semi-H ∗ -algebra with zero annihilator determines its involution, we prove that power-associative H ∗ -algebras are non-commutative Jordan algebras [714]. In Subsection 8.1.2, we establish two fundamental structure theorems for a semiH ∗ -algebra A, which, in two successive steps, reduce the general case to the one that A has zero annihilator, and the case that A has zero annihilator to the one that A is topologically simple [199]. According to the structure theory commented in the preceding paragraph, topologically simple semi-H ∗ -algebras merit being studied in depth. This is done in Subsection 8.1.3. To this end we introduce totally multiplicatively prime normed algebras, show that they are totally prime, and prove that topologically simple complex semi-H ∗ -algebras are totally multiplicatively prime [889]. Since, as we already commented in our review of Subsection 6.1.5, totally prime normed complex algebras are centrally closed, it follows that topologically simple complex H ∗ -algebras are centrally closed [148, 149].

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The central closedness of topologically simple complex H ∗ -algebras just reviewed becomes the key tool of Subsection 8.1.4, where we prove that derivations of complex semi-H ∗ -algebras with zero annihilator are continuous [624], and that denserange algebra homomorphisms from complete normed complex algebras to complex H ∗ -algebras with zero annihilator are also continuous [526]. In Subsection 8.1.5 we show that isomorphic complex H ∗ -algebras with zero annihilator are ∗-isomorphic, and that bijective algebra ∗-homomorphisms between topologically simple complex H ∗ -algebras are positive multiples of isometries (hence, essentially, a topologically simple complex H ∗ -algebra has a unique H ∗ -algebra structure) [198]. These results follow from a structure theorem for bijective algebra homomorphisms between complex H ∗ -algebras with zero annihilator, which becomes the appropriate H ∗ -variant of the structure theorem for bijective algebra homomorphisms between non-commutative JB∗ -algebras proved in Theorem 3.4.75. In Subsection 8.1.6, we prove the appropriate H ∗ -variant of the Jordan characterization of C∗ -algebras established in Theorem 3.6.30 [518]. A more than satisfactory H ∗ -variant of Theorem 3.6.25 is also obtained [624]. Subsection 8.1.7 is devoted to providing us with the appropriate tools to transfer results from complex semi-H ∗ -algebras to real ones. The basic tool asserts that the complexification of any real (semi-)H ∗ -algebra becomes a complex (semi-)H ∗ algebra in a natural way. This quite elementary fact already allows to convert many complex results into real ones, all of them involving the assumption that the algebra has zero annihilator. The treatment of topologically simple real (semi-)H ∗ -algebras is more elaborated: there are no topologically simple real (semi-)H ∗ -algebras other than topologically simple complex (semi-)H ∗ -algebras, regarded as real algebras, and the real (semi-)H ∗ -algebras of all fixed points for an involutive conjugate-linear algebra ∗-homomorphism on a topologically simple complex (semi-)H ∗ -algebra [142]. This reduction of topologically simple real (semi-) H ∗ -algebras to complex ones allows us to transfer the remaining results known in the complex setting to the real setting. In particular, we prove that dense-range algebra homomorphisms from H ∗ -algebras with zero annihilator to topologically simple H ∗ -algebras are surjective. Then, after introducing H ∗ -ideals of an arbitrary normed ∗-algebra, we prove that topologically simple normed ∗-algebras have at most one H ∗ -ideal [687]. We begin Subsection 8.1.8 by introducing the complete normed complex ∗-algebra (T C (H),  · τ ) of all trace-class operators on a complex Hilbert space H, as well as the  · τ -continuous trace-form on it. Then we show that (T C (H),  · τ ) can be intrinsically determined into the H ∗ -algebra (HS (H),  · ) of all Hilbert–Schmidt operators on H. This fact allows us to replace HS (H) with an arbitrary real or complex (possibly non-associative) semi-H ∗ -algebra A with zero annihilator, to build an appropriate substitute of (T C (H),  · τ ) into A, denoted by (τ c(A),  · τ ), and to discuss whether or not a  · τ -continuous trace-form on τ c(A) does exist. We prove that, for a semi-H ∗ -algebra A with zero annihilator, τ c(A) is a ∗-invariant ideal of A, (τ c(A),  · τ ) is both a normed algebra and a dual Banach space, and the existence of a  · τ -continuous trace-form on τ c(A) depends on the existence of

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an ‘operator-bounded’ approximate unit in A [424]. This, together with deep results established in Volume 1 (namely Theorem 3.5.53 and Proposition 4.5.36(ii)), allows us to prove that a complex H ∗ -algebra A with zero annihilator is alternative if and only if (A,  · ) has an approximate unit operator-bounded by 1, and the predual of (τ c(A),  · τ ) is a non-associative C∗ -algebra. In the concluding Subsection 8.1.9, we survey the classification theorems of topologically simple H ∗ -algebras in the most familiar classes of algebras. Thus, starting from the well-known fact that there are no topologically simple associative complex H ∗ -algebras other than those of the form HS (H) for a nonzero complex Hilbert space H [20, 374], the corresponding theorems for topologically simple alternative [1042], Jordan and non-commutative Jordan [199, 1118, 1119], Lie [197, 460, 687], Malcev [141], or structurable [140, 144] H ∗ -algebras are established. Section 8.2 deals with generalized annihilator normed algebras, which become non-star generalizations of H ∗ -algebras with zero annihilator. We prove that any generalized annihilator complete normed real or complex algebra with zero weak radical (cf. Definition 4.4.39) is the closure of the direct sum of its minimal closed ideals, which are indeed topologically simple normed algebras [259]. We also show that the weak radical of any real or complex semi-H ∗ -algebra coincides with its annihilator, so that the structure theorem for semi-H ∗ -algebras with zero annihilator proved in Subsection 8.1.2 is rediscovered. We introduce multiplicatively semiprime algebras (i.e. algebras such that both they and their multiplication algebras are semiprime), and show that generalized annihilator normed algebras are multiplicatively semiprime [876]. Even more, we characterize generalized annihilator normed algebras among those normed algebras which are multiplicatively semiprime. We introduce generalized complemented normed algebras, which are particular cases of generalized annihilator normed algebras, and prove that, if A is a generalized complemented complete normed algebra with zero weak radical, and if {Ai }i∈I stands for the family of its minimal closed ideals, then for each a ∈ A there exists a unique summable family {ai }i∈I in A such that ai ∈ Ai  for every i ∈ I, and a = i∈I ai [259, 846]. Section 8.3 deals with other complements into the theory of non-associative normed algebras. In Subsection 8.3.1 we prove that algebra homomorphisms from complete normed complex algebras to complete normed complex algebras with no nonzero two-sided topological divisor of zero are continuous [529]. In Subsection 8.3.2 we show that complete normed J-semisimple non-commutative Jordan complex algebras, each element of which has a finite J-spectrum, are a finite direct sum of closed simple ideals which are either finite-dimensional or quadratic, and derive that complete normed semisimple alternative complex algebras, each element of which has a finite spectrum, are finite-dimensional [91]. After the usual subsection devoted to historical notes and comments, we include a comprehensive survey on the more significant results on normed Jordan algebras which have been not previously developed in our work.

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The concluding Section 8.4 deals with the non-associative discussion done in [452, 453] of the Rota–Strang paper [544] (so in particular of Proposition 4.5.2, cf. p. 632 of Volume 1), and of the theory of topologically nilpotent normed (associative) algebras developed in [927, 928, 929, 1020] (see also [786, pp. 515–7], [1156, Section 11], and §8.4.121). The section discusses also non-associative versions of related results published in [569, 615, 1083] (see also [1030]), and incorporates proofs of most auxiliary results invoked but not proved in [452, 453]. Among these proofs, we emphasize that of Theorem 8.4.76, courtesy of Shulman and Turovskii. In Subsection 8.4.1 we introduce the notion of (joint) spectral radius r(S) of a bounded subset S of any normed algebra A. Then we prove one of the key results in the whole section, namely that, if A is a normed algebra, and if S is a bounded subset of A with r(S) < 1, then the multiplicatively closed subset of A generated by S is bounded, and has the same spectral radius as S. In Subsection 8.4.2, we introduce topologically nilpotent normed algebras as those normed algebras whose closed unit balls have zero spectral radius. Among the results obtained, we emphasize the following: (i) A normed associative algebra A is topologically nilpotent if and only if so is the normed Jordan algebra Asym obtained by symmetrization of its product. (ii) Every non-topologically nilpotent normed algebra can be equivalently algebrarenormed in such a way that the spectral radius of the corresponding closed unit ball is arbitrarily close to 1. (iii) Every topologically nilpotent complete normed algebra is equal to its weak radical. In Subsection 8.4.3, we show that, for every member A in a large class of normed algebras (which contains all commutative C∗ -algebras, all JB-algebras, and all absolute-valued algebras), the conclusion in Proposition 4.5.2 has the following stronger form: for each bounded and multiplicatively closed subset S of A we have that sup{s : s ∈ S} ≤ 1. In Subsection 8.4.4, we involve in our development tensor products of algebras. Thus we prove that the projective tensor product of two normed algebras is topologically nilpotent whenever some of them are topologically nilpotent, and that in fact the converse is true whenever some of them are associative. Moreover, associativity in the above converse cannot be removed. We also prove that a normed algebra A is topologically nilpotent if and only if so is the normed algebra C0 (E, A) for some (equivalently, every) Hausdorff locally compact topological space E. The results obtained about tensor products of normed algebras are then applied to show that most notions introduced in the section can be non-trivially exemplified into a class of algebras almost arbitrarily prefixed.

On the historical notes As in Volume 1, each section of the present volume concludes with a subsection devoted to historical notes and comments. Paraphrasing Dinnen [1155, p. X], in these

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notes ‘we provide information on the history of the subject and references for the material presented. We have tried to be as careful as possible in this regard and take responsibility for the inevitable errors. Accurate and comprehensive records of this kind are not a luxury but essential background information in appreciating and understanding a subject and its evolution’. Errata: A list of errata for Volume 1 can be found in the web page of Volume 2: www.cambridge.org/9781107043114. We hope to continue this for both Volumes. Please send corrections to: [email protected] and/or [email protected]. Acknowledgements We are indebted to many mathematicians whose encouragement, questions, suggestions, and kindly-sent reprints greatly influenced us while we were writing this work: M. D. Acosta, J. Alaminos, C. Aparicio, R. M. Aron, B. A. Barnes, J. Becerra, H. Behncke, G. Benkart, A. Browder, M. J. Burgos, J. C. Cabello, A. J. Cabrera, A. Ca˜nada, C.-H. Chu, M. J. Crabb, J. Cuntz, C. M. Edwards, F. J. Fern´andez Polo, J. E. Gal´e, E. Garc´ıa, G. Godefroy, M. G´omez, A. Y. Helemskii, R. Iordanescu, V. Kadets, O. Loos, G. L´opez, J. Mart´ınez Moreno, M. Mathieu, C. M. McGregor, P. Mellon, A. Morales, J. C. Navarro Pascual, M. M. Neumann, R. Pay´a, J. P´erez Gonz´alez, H. P. Petersson, J. D. Poyato, A. Rochdi, A. Rueda, B. Schreiber, I. P. Shestakov, M. Siles, A. M. Sinclair, A. M. Slinko, R. M. Timoney, A. J. Ure˜na, M. V. Velasco, M. Villegas, A. R. Villena, B. Zalar, and W. Zelazko. Special thanks should be given to D. Beltita, J. A. Cuenca, H. G. Dales, R. S. Doran, A. Fern´andez L´opez, A. Kaidi, W. Kaup (RIP), M. Mart´ın, J. F. Mena, J. Mer´ı, A. Moreno Galindo, A. M. Peralta, H. Pfitzner, V. S. Shulman, Yu. V. Turovskii, H. Upmeier, J. D. M. Wright, and D. Yost, for their substantial contributions to the writting of our work. We would also like to thank the staff of Cambridge University Press for their help and kindness, and particularly R. Astley, C. Dennison, N. Yassar Arafat, R. Munnelly, N. Saxena, and the copy-editor K. Eagan. This work has been partially supported by the Spanish government grant MTM2016-76327-C3-2-P.

5 Non-commutative JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer type non-associative theorem

Non-commutative JB∗ -algebras (cf. Definition 3.3.1) have become central objects in the first volume of our work since, in the unital case, they are the solution to the general non-associative Vidav–Palmer theorem (see Theorem 3.3.11), aned contain alternative C∗ -algebras (cf. §2.3.62) which in their turn become the solution to the general non-associative Gelfand–Naimark theorem (see Theorem 3.5.53). As a concluding main result in the present chapter, we will prove a general non-associative characterization of non-commutative JB∗ -algebras (see Theorem 5.9.9), a germ of which could be the following. Fact 5.0.1 A norm-unital complete normed complex algebra is a non-commutative JB∗ -algebra (for some involution) if and only if it is linearly isometric to a JB∗ -triple. The proof, which only involves results established in the first volume of our work, goes as follows. Proof The ‘only if’ part follows from Theorem 4.1.45. To prove the ‘if’ part, let us recall that, given a complex normed space X and a norm-one element u ∈ X, H(X, u) denotes the set of all hermitian elements of X relative to u (cf. Definition 2.1.12). Now let A be a norm-unital complete normed complex algebra such that there exists a linear isometry φ from A onto some JB∗ -triple J. Then, by Corollary 2.1.13, φ(1) is a vertex of BJ , and hence, by the implication (vii)⇒(ii) in Theorem 4.2.24, J is the underlying Banach space of a JB∗ -algebra with unit φ(1). Therefore, by Lemma 2.2.8(iii), we have J = H(J, φ(1)) + iH(J, φ(1)), and hence A = H(A, 1) + iH(A, 1). Now, by the non-associative Vidav–Palmer theorem (cf. Theorem 3.3.11), A is a noncommutative JB∗ -algebra. 

1

2

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

The actual formulation of Theorem 5.9.9 avoids ‘norm-unital’, and replaces ‘is linearly isometric to a JB∗ -triple’ with ‘has an approximate unit bounded by one and its open unit ball is a homogeneous domain (see Definition 5.3.53)’. To reach Theorem 5.9.9 from Fact 5.0.1, we must go a long way, with most steps having their own interest. Thus in Section 5.1 we introduce non-commutative JBW ∗ algebras (i.e. non-commutative JB∗ -algebras which are dual Banach spaces), and prove Edwards’ results [222] relating non-commutative JBW ∗ -algebras with JBWalgebras (i.e. JB-algebras which are dual Banach spaces). In Sections 5.2, 5.3, 5.4, and 5.5 we study in detail the algebraic and analytic structure of the sets of all biholomorphic automorphisms and of all complete holomorphic vector fields on a bounded domain in a complex Banach space. This study culminates in Section 5.6, where we prove Kaup’s characterization of JB∗ -triples [380, 381] as those complex Banach spaces the open unit balls of which are homogeneous domains. Sections 5.7 and 5.8 are devoted to establishing the basic theory of JBW ∗ -triples (i.e. JB∗ -triples which are dual Banach spaces) and of operators into the predual of a JBW ∗ -triple. These sections contain relevant results originally due Dineen [213], Barton–Timoney [854], Horn [979], and Chu–Iochum–Loupias [172]. It is noteworthy that our proofs of the Barton–Horn–Timoney Theorems 5.7.20 and 5.7.38 (asserting the separate w∗ -continuity of the product and the uniqueness of the predual of a JBW ∗ -triple) are new and avoid any Banach space result on uniqueness of preduals. On the other hand, one of the crucial steps in our proof of the Chu–Iochum–Loupias Theorem 5.8.32 (asserting that bounded linear operators from a JB∗ -triple to the predual of a JBW ∗ -triple are weakly compact) consists of Proposition 5.8.14, a result whose proof is difficult to find in the literature. We include a complete and self-contained proof of this result, which has been communicated to us by Pfitzner [1047]. Section 5.9 contains the (conclusion of) proof of the commented refinement of Fact 5.0.1, namely that non-commutative JB∗ -algebras are precisely those complete normed complex algebras having a bounded approximate unit and whose open unit ball is a homogeneous domain. The chapter concludes with Section 5.10, which contains different complements on non-commutative JB∗ -algebras and JB∗ -triples. Indeed, we study in deep the strong∗ topology of a non-commutative JBW ∗ -algebra and of a JBW ∗ -triple, as well as linear isometries between non-commutative JB∗ -algebras. 5.1 Non-commutative JBW ∗ -algebras Introduction This section is devoted to establishing the basic theory of noncommutative JBW ∗ -algebras. As main results we prove Edwards’ theorems [222] asserting that a non-commutative JB∗ -algebra A is a non-commutative JBW ∗ -algebra if and only if its self-adjoint part H(A, ∗) is a JBW-algebra, and that, if this is the case, then the predual of A is unique, the involution of A is w∗ -continuous, and the product of A is separately w∗ -continuous (see Theorems 5.1.29 and 5.1.38 and Corollary 5.1.30). The proof of the uniqueness of the predual involves deep results of

5.1 Non-commutative JBW ∗ -algebras

3

the theory of JB-algebras (see Theorem 5.1.27) which, as we did in the first volume of our work in similar occasions, are taken from [738] without proof. On the other hand, the proof of the separate w∗ - continuity of the product in the non-commutative case follows an argument in [481]. The section also contains theorems taken from [481] asserting that, in a noncommutative JB∗ -algebra, M-ideals are precisely the closed ideals (Theorem 5.1.22(i)) and that the predual of a non-commutative JBW ∗ -algebra is an L-summand of the dual (Theorem 5.1.32). The section concludes by revisiting real non-commutative JB∗ -algebras in order to prove that c0 is an M-ideal of its bidual (Corollary 5.1.57), a result which will be needed later in the proof of Theorem 5.8.27. 5.1.1 The results Lemma 5.1.1 Let A be an algebra over K with zero annihilator, and let I, J be ideals of A such that A = I ⊕ J. Then I = {a ∈ A : aJ = Ja = 0}. Proof

Put K := {a ∈ A : aJ = Ja = 0}. The inclusion I ⊆ K is clear since IJ + JI ⊆ I ∩ J = 0.

Conversely, let a be in K, and write a = x + y with x ∈ I and y ∈ J. Then, by the inclusion just proved, y = a − x ∈ K ∩ J. But, since J ⊆ {a ∈ A : aI = Ia = 0}, we derive that y ∈ Ann(A) = 0. Therefore a = x ∈ I.  By a direct summand of an algebra A over K we mean any ideal I of A such that there exists another ideal J of A satisfying A = I ⊕ J. As a straightforward consequence of Lemma 5.1.1, we get the following. Fact 5.1.2 Let A be a normed algebra over K with zero annihilator, and let I be a direct summand of A. Then I is closed in A. Definition 5.1.3 Let X be a normed space over K. (i) By an M-projection on X we mean a linear projection P : X → X such that x = max{P(x), x − P(x)} for every x ∈ X. (ii) A subspace Y of X is said to be an M-summand of X if Y is the range of an M-projection on X. Lemma 5.1.4 Let A be a JB∗ -algebra, and let I be a direct summand of A. Then, I is an M-summand of (the Banach space underlying) A. Proof Let J be an ideal of A such that A = I ⊕ J. By Fact 5.1.2, I and J are closed in A, and then, by Proposition 3.4.13, they are ∗-invariant. Therefore, I and J are new JB∗ -algebras, so I × J is a JB∗ -algebra under the sup norm, and the mapping (x, y) → x + y from I × J to A is a bijective algebra ∗-homomorphism. It follows from Proposition 3.4.4 that x + y = max{x, y} for every (x, y) ∈ I × J. Thus the

4

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

projection from A onto I corresponding to the decomposition A = I ⊕ J becomes an M-projection.  §5.1.5 As usual, by a dual Banach space over K we mean (the first component of) a couple (X, Y), where X and Y are Banach spaces over K such that Y  = X. The Banach space Y is called the predual of X, and is usually denoted by X∗ . Definition 5.1.6 By a non-commutative JBW ∗ -algebra (respectively, a JBW ∗ algebra, an alternative W ∗ -algebra, a W ∗ -algebra) we mean a non-commutative JB∗ -algebra (respectively, a JB∗ -algebra, an alternative C∗ -algebra, a C∗ -algebra) which is a dual Banach space. Thus JBW ∗ -algebras are precisely those noncommutative JBW ∗ -algebras which are commutative (cf. Definition 3.3.1), alternative W ∗ -algebras are precisely those non-commutative JBW ∗ -algebras which are alternative (cf. Fact 3.3.2), and therefore W ∗ -algebras are precisely those non-commutative JBW ∗ -algebras which are associative. Non-commutative JBW ∗ algebras (respectively, alternative W ∗ -algebras) were incidentally introduced in the paragraph immediately before Proposition 4.2.71 (respectively, in Remark 3.5.40). We note that, by the Banach–Alaoglu and Krein–Milman theorems, the closed unit ball of a non-commutative JBW ∗ -algebra has extreme points, so that the implication (iv)⇒(i) in Theorem 4.2.36 applies to get the following. Fact 5.1.7 Nonzero non-commutative JBW ∗ -algebras are unital. Lemma 5.1.8 Let A be an algebra over K, and let I be an ideal of A having a unit e. Then e is an idempotent in A and I = eA. Moreover, if A is flexible and powerassociative, then e is central in A. Proof Clearly e is an idempotent in A and we have eA ⊆ I = eI ⊆ eA, hence I = eA. Suppose that A is flexible and power-associative. Let x be in A 1 (e). Then e • x = 12 x, 2

so e • x = e • (e • x) = 12 e • x because e • x ∈ I, and so x = 0. Thus A 1 (e) = 0. 2 Since A 1 (e) = (Asym ) 1 (e), and Asym is power-associative (cf. Corollary 2.4.18), it 2 2 follows from Lemma 3.1.14 that e is central in Asym . By Corollary 4.3.48, e is central in A.  §5.1.9 Given a dual Banach space X over K, any w∗ -closed subspace M of X will be considered canonically as a new dual Banach space. Indeed, by the bipolar theorem, such a subspace M must be the polar in X of its prepolar M◦ in X∗ , and consequently we have M = (M◦ )◦ ≡ (X∗ /M◦ ) . With this convention it becomes clear that the weak∗ topology of M coincides with the restriction to M of the weak∗ topology of X. Fact 5.1.10 Let A be a non-commutative JBW ∗ -algebra, and let I be a w∗ -closed ideal of A. Then: (i) There exists a central idempotent e ∈ A such that I = eA. (ii) I is an M-summand of A.

5.1 Non-commutative JBW ∗ -algebras

5

Proof By Proposition 3.4.13, I is ∗-invariant, and hence it is a new noncommutative JBW ∗ -algebra. Therefore, by Fact 5.1.7, I has a unit e, and the proof of assertion (i) is concluded by applying Lemma 5.1.8. Now it turns out that J := (1 − e)A is an ideal of A and that A = I ⊕ J. Hence I is a direct summand of A, and the proof of assertion (ii) is concluded by invoking Lemma 5.1.4.  The proof we have just given shows that, as happens with the unit of any ∗-algebra, the idempotent e must be self-adjoint. As we note in the next remark, this is not new for us. Remark 5.1.11 Let A be a non-commutative JB∗ -algebra. We already know that central idempotents of A are self-adjoint (cf. Fact 3.3.4, §4.3.38, and either Theorem 4.3.47 or Corollary 4.3.48). Nevertheless, the most natural verification of this result consists of noticing that the centre Z(A) of A is a commutative C∗ -algebra (cf. Proposition 3.4.1(i)), and of applying then to Z(A) the commutative Gelfand– Naimark theorem. Another proof, close to that of Fact 5.1.10, is the following. Let e be a central idempotent of A. Then I := eA is an ideal of A, and is closed in A because I = {a ∈ A : a = ea}. Therefore, by Proposition 3.4.13, I is ∗-invariant. Since e is a unit for I, it follows that e∗ = e. The notions of L-summand and of M-ideal of a normed space were incidentally introduced in Subsection 2.9.4. Now we are going to recall and develop them in a more detailed way. Definition 5.1.12 Let X be a normed space over K. (i) By an L-projection on X we mean a linear projection P : X → X such that x = P(x) + x − P(x) for every x ∈ X. (ii) A subspace Y of X is said to be an L-summand of X if Y is the range of an L-projection on X. (iii) By an M-ideal of X we mean a closed subspace Y of X such that Y ◦ is an L-summand of X  . Some comments on Definitions 5.1.3 and 5.1.12 are in order. §5.1.13 Let X be a normed space over K. There is an obvious duality between L- and M-projections. Indeed, 

P is an

L-projection M-projection



on X if and only if P is an



M-projection L-projection



on X  .

As a consequence, M-summands of X are M-ideals of X. Proposition 5.1.14 Let X be a normed space over K. Then any two L- (respectively, M-)projections on X commute.

6

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Proof

Let P and Q be L-projections on X. Then for x ∈ X we have Q(x) = PQ(x) + (IX − P)Q(x) = QPQ(x) + (IX − Q)PQ(x) + Q(IX − P)Q(x) + (IX − Q)(IX − P)Q(x) = QPQ(x) + Q(x) − QPQ(x) + 2PQ(x) − QPQ(x) ≥ Q(x) + 2PQ(x) − QPQ(x),

so that PQ = QPQ. But likewise we obtain P(IX − Q) = (IX − Q)P(IX − Q) which is equivalent to QP = QPQ. Therefore PQ = QP. Now that we know that any two L-projections commute, the fact that any two M-projections commute follows by invoking §5.1.13.  Corollary 5.1.15 Let X be a normed space over K, and let Y be an L- (respectively, M-)summand of X. Then there is a unique L- (respectively, M-)projection on X whose range is Y. Lemma 5.1.16 Let X be a nonzero complex Banach space, and let P be an L- or M-projection on X. Then P ∈ H(BL(X), IX ). Proof

+ + In both cases there is a function f : R+ 0 × R0 → R0 such that

y + z = f (y, z) for all y ∈ P(X) and z ∈ ker(P). Let r be in R. Then exp(irP) = eir P + IX − P. Therefore for every x ∈ X we have  exp(irP)(x) = eir P(x) + (IX − P)(x) = f (P(x), (IX − P)(x)) = x. Thus  exp(irP) = 1, and Corollary 2.1.9(iii) concludes the proof.



Proposition 5.1.17 Let X be a nonzero complex Banach space, let Y be an M-ideal of X, and let T be a hermitian operator on X. Then T(Y) ⊆ Y. Proof Let P be the L-projection onto the polar Y ◦ of Y in X  . Then the transpose operator T  of T is also a hermitian operator on X  (cf. Corollary 2.1.3) so, for each t ∈ R, exp(itT  ) is a surjective linear isometry on X  (cf. Corollary 2.1.9(iii)). It follows easily that exp(itT  )P exp(−itT  ) is a new L-projection on X  . By Proposition 5.1.14, we realize that [exp(itT  )P exp(−itT  ), P] = 0. Computing the coefficient of t in the power-series development of the left-hand side, we have [[T  , P], P] = 0. Since L-projections are hermitian operators (cf. Lemma 5.1.16), it follows from Corollary  2.4.3 that [T  , P] = 0. Which implies that T  (Y ◦ ) ⊆ Y ◦ , and finally T(Y) ⊆ Y. Fact 5.1.18 Let E and F be topological spaces, let x be in E, and let f : E → F be a function such that f (x) is a cluster point of f (xλ ) whenever xλ is any net in E converging to x. Then f is continuous at x. Proof Assume that f is not continuous at x. Let  stand for the set of all neighbourhoods of x in E ordered by reverse inclusion. Then there exists a neighbourhood

5.1 Non-commutative JBW ∗ -algebras

7

N of f (x) in F such that for every λ ∈  we can find xλ ∈ λ with f (xλ ) ∈ / N. It becomes clear that limλ xλ = x and that f (x) is not a cluster point of the net f (xλ ), contrary to the assumption.  A celebrated theorem of S. Banach (sometimes attributed to M. Krein and ˇ V. Smulyan) asserts that a linear form on the dual X  of a Banach space X is ∗ w -continuous if (and only if) so is its restriction to BX  (see for example [1161, Corollary 3.11.4]). An apparently more general formulation of this result is collected in the following. Fact 5.1.19 Let X be a Banach space over K, let Y be a normed space over K, and let T : X  → Y  be a linear or conjugate-linear mapping whose restriction to BX  is w∗ -continuous. Then T is w∗ -continuous. Proof It is enough to show that for each y ∈ Y the linear form f on X  defined by f (x ) := T(x )(y) (or f (x ) := T(x )(y)) is w∗ -continuous. But this follows from the assumption that T|BX is w∗ -continuous and the Banach theorem quoted immediately above.  Corollary 5.1.20 Let X be a Banach space over K, and let P be a linear projection on X  . Then P is w∗ -continuous if (and only if) P is bounded and both P(X  ) and ker(P) are w∗ -closed in X  . Proof Suppose that P is bounded and that P(X  ) and ker(P) are w∗ -closed in X  . Let x be in BX  , and let xλ be a net in BX  w∗ -convergent to x . Take a cluster point y of the net P(xλ ) in the weak∗ topology. Then x − y is a cluster point of the net xλ − P(xλ ) in the weak∗ topology. Since P(X  ) and ker(P) are w∗ -closed in X  , it follows that y ∈ P(X  ) and x − y ∈ ker(P). Therefore y = P(x ), and hence P(x ) is a cluster point of the net P(xλ ) in the weak∗ topology. Keeping in mind the arbitrariness of x ∈ BX  and of the net xλ w∗ -convergent to x , it follows from Fact 5.1.18 that P|BX is w∗ -continuous. Finally, by Fact 5.1.19, P is w∗ -continuous.  The reader might wonder why we did not introduce the notion of an ‘L-ideal’ of a normed space X over K, meaning a closed subspace of X whose polar is an M-summand of X  . The reason is given by the following. Lemma 5.1.21 Let X be a Banach space over K. Then we have: (i) M-summands of X  are w∗ -closed. (ii) M-projections on X  are w∗ -continuous. (iii) L-summands of X are precisely those closed subspaces of X whose polars are M-summands of X  . Proof To prove assertion (i), let us consider a decomposition X  = Y ⊕∞ Z, and let ˇ us assume to the contrary that Y is not w∗ -closed. In this case, by the Krein–Smulyan  theorem (see for example [778, Corollary 2.7.12]), there exists a net xλ in BY w∗  convergent to some x ∈ Z, x = 0. Then yλ := xλ + xx  defines a net in BX  whose w∗ -limit has norm 1 + x  > 1, a contradiction.

8

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Assertion (ii) follows from assertion (i) and Corollary 5.1.20. Assertion (iii) follows from assertion (ii) and §5.1.13.

Theorem 5.1.22 Let A be a non-commutative

JB∗ -algebra.



Then:

(i) The M-ideals of A are precisely the closed ideals of A. (ii) The M-summands of A are precisely the direct summands of A. Proof Let I be an M-ideal of A. For a ∈ A, let Ta stand for either La or Ra . Let x be in H(A, ∗). Then, by Lemma 3.6.24 and Proposition 5.1.17, we have Tx (I) ⊆ I, i.e. xI, Ix ⊆ I. Since every element a ∈ A can be written as a = x + iy with x, y ∈ H(A, ∗), it follows that I is an ideal of A. Now let I be a closed ideal of A. We know that A is a unital non-commutative JBW ∗ -algebra with separately w∗ -continuous product (cf. Theorem 3.5.34), and that consequently I ◦◦ is an ideal of A . Since I ◦◦ is w∗ -closed, it follows from Fact 5.1.10 that I ◦◦ is an M-summand of A . Therefore, by Lemma 5.1.21(iii), I is an M-ideal of A. This concludes the proof of assertion (i). We already proved in Lemma 5.1.4 that direct summands of A are M-summands. Conversely, let I be an M-summand of A. Let P be the M-projection onto I. Then both I and J := (IA − P)(A) are M-ideals of A with A = I ⊕ J. Therefore, by assertion (i), I and J are ideals of A. Thus I is a direct summand of A. This concludes the proof of assertion (ii).  §5.1.23 Let X be a dual Banach space over K. We say that X∗ is the unique predual of X if, whenever Z is any Banach space over K such that Z  = X, and we see X∗ and Z as subspaces of X  via the corresponding canonical embeddings, we have X∗ = Z. Formulating this notion in a slightly more precise way, as is done in [954, D´efinition 1], the following fact becomes a tautology. Fact 5.1.24 Let X and Y be dual Banach spaces over K having a unique predual. Then surjective linear isometries from X to Y are w∗ -continuous. Now we recall the definition of the order in a JB-algebra, together with some related results. §5.1.25 Let A be a JB-algebra. Since the unital extension A1 of A becomes a unital JB-algebra with n(A1 , 1) = 1 (cf. Corollary 3.1.11 and Proposition 3.1.4(iii)), 1 is a vertex of BA1 , and hence A enjoys of the order induced by the numerical-range order of (A1 , 1) as defined in §2.3.34. Indeed, an element a in A is called positive whenever + V(A1 , 1, a) ⊆ R+ 0 . The set A of all positive elements in A is a closed proper convex cone in A, and the order in A is given by: a ≤ b if and only if b − a ∈ A+ . In the case that A is unital, the passing to the unital extension is unnecessary, i.e. the order in A, as defined above, coincides with the numerical-range order of (A, 1) (cf. §3.1.27). Anyway, if a, b are in A, and if 0 ≤ a ≤ b, then a ≤ b (cf. Fact 2.3.36). Moreover, according to Lemma 3.1.29, we have A+ = {a2 : a ∈ A} (which puts in agreement our definition of the order with the one given in [738, 3.3.3]), and Ua (A+ ) ⊆ A+ for every a ∈ A.

(5.1.1)

5.1 Non-commutative JBW ∗ -algebras

9

Definition 5.1.26 Let A be a JB-algebra. A linear functional f on A is said to be positive if f (a) ≥ 0 whenever a is a positive element of A. The JB-algebra A is said to be monotone complete if each bounded increasing net aλ in A has a least upper bound a in A. Suppose that A is monotone complete. A linear functional f on A is called normal if it is bounded and if f (aλ ) → f (a) for each net aλ as above. Now we involve in our development the following outstanding result whose proof is omitted. Theorem 5.1.27 [738, Theorem 4.4.16] Let A be a JB-algebra. Then the following conditions are equivalent: (i) A is monotone complete and the set of all positive normal linear functionals on A separates the points of A. (ii) A is a dual Banach space. Moreover, if the above conditions are fulfilled, then the predual of A is unique and consists of the normal linear functionals on A. According to the above theorem and the definition of a JBW-algebra in [738, 4.1.1] (as those JB-algebras A satisfying condition (i) above), we introduced JBW-algebras as those JB-algebras which are dual Banach spaces (cf. the paragraph immediately before Proposition 3.1.12). §5.1.28 Let A be a non-commutative JB∗ -algebra. Then H(A, ∗) becomes a JBalgebra in a natural way (cf. Corollary 3.4.3), and hence, as we agreed in §3.4.68, it will be seen endowed with the order remembered in §5.1.25. By the sake of shortness, the order of H(A, ∗) is called the order of A, positive elements of H(A, ∗) are called positive elements of A, and we set A+ := H(A, ∗)+ . Thus, an element a ∈ A is positive if and only if a = h2 for some h ∈ H(A, ∗). As a consequence, we have that a∗ • a ≥ 0 for every a ∈ A.

(5.1.2)

We note that, by Fact 4.1.67(ii) and Proposition 4.5.17(ii), A+ = {h ∈ H(A, ∗) : J-sp(A1 , h) ⊆ R+ 0 }, and that, if A is unital, then we have in fact A+ = {h ∈ H(A, ∗) : J-sp(A, h) ⊆ R+ 0 }. Needless to say that, in the case that A is actually a C∗ -algebra, we find back the usual order on A (cf. §1.2.47 and Proposition 2.3.39(i)). Theorem 5.1.29 Let A be a non-commutative JBW ∗ -algebra. Then we have: (i) H(A, ∗) is w∗ -closed in A, and hence is a JBW-algebra in a natural way (cf. Corollary 3.4.3 and §5.1.9). (ii) The involution of A is w∗ -continuous.

10

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

(iii) A∗ = H(A, ∗)∗ ⊕ iH(A, ∗)∗ , meaning that, for each h∗ ∈ H(A, ∗)∗ , the mapping h + ik → h∗ (h) + ih∗ (k) (h, k ∈ H(A, ∗)) belongs to A∗ , and that, for each a∗ ∈ A∗ , there exist unique functionals h∗ , k∗ ∈ H(A, ∗)∗ such that a∗ (h + ik) = h∗ (h) − k∗ (k) + i(h∗ (k) + k∗ (h)) for all h, k ∈ H(A, ∗). (iv) A has a unique predual. (v) The positive part A+ of A is w∗ -closed in A. (vi) A equals the norm-closed linear hull of the set of its self-adjoint idempotents. Proof We may suppose that A = 0. Then, by Fact 5.1.7, A is unital. ˇ In view of the Krein–Smulyan theorem, to prove assertion (i) it is enough to show ∗ that H(A, ∗) ∩ BA is w -closed in A. We argue by contradiction, so that there exists a net hλ in H(A, ∗) ∩ BA w∗ -convergent to h + ik with h, k ∈ H(A, ∗) and k = 0. Replacing hλ with −hλ if necessary, we may suppose that there is a positive number α in J-sp(A, k) (cf. Fact 4.1.67(i) and Corollary 4.1.72(i)). Take n ∈ N such that n>

1 − α2 . 2α

(5.1.3)

Noticing that, for each λ, the closed subalgebra of A generated by hλ and 1 is a C∗ -algebra (cf. Proposition 3.4.1(ii)), we have 1

1

1

1

hλ + in1 = (hλ + in1)∗ (hλ + in1) 2 = h2λ + n2 1 2 ≤ (h2λ  + n2 ) 2 ≤ (1 + n2 ) 2 . Therefore, since h + ik = w∗ - lim hλ , we conclude that 1

h + ik + in1 ≤ (1 + n2 ) 2 .

(5.1.4)

On the other hand, since H(A, ∗) is a JB-algebra in a natural way (cf. Corollary 3.4.3), and the involution of A is an isometry (cf. Proposition 3.3.13), we have 1

1

k + n1 = (k + n1)2  2 ≤ (k + n1)2 + h2  2 1

= (h + i(k + n1))∗ • (h + i(k + n1)) 2 ≤ h + i(k + n1).

(5.1.5)

Finally, since α + n ≤ k + n1 (because α ∈ J-sp(A, k) and Theorem 4.1.17 applies), it is enough to invoke (5.1.5), (5.1.4), and (5.1.3) to get 1

k + n1 ≤ h + i(k + n1) ≤ (1 + n2 ) 2 < α + n ≤ k + n1, the desired contradiction. Keeping in mind that P := 12 (IA + ∗) is a bounded linear projection on the real Banach space underlying that of A satisfying P(A) = H(A, ∗) and ker(P) = iH(A, ∗), assertion (ii) follows from assertion (i) and Corollary 5.1.20.

5.1 Non-commutative JBW ∗ -algebras

11

In view of assertions (i) and (ii), the direct sum A = H(A, ∗) ⊕ iH(A, ∗) is topological relative to the weak∗ topologies. Therefore, keeping in mind the natural identification between w∗ -continuous real- and complex-linear functionals on A, assertion (iii) follows. Assertion (iv) follows from assertion (iii) and Theorem 5.1.27. The mapping φ : a → 12 (1 + a) from A to A is a w∗ -homeomorphism satisfying φ(BA ∩ H(A, ∗)) = BA ∩ A+ . Therefore, by assertion (i), BA ∩ A+ is w∗ -closed in A. ˇ Then assertion (v) follows from the Krein–Smulyan theorem. Assertion (vi) follows from assertion (i) and Proposition 3.1.12 or [738, Proposition 4.2.3].  Corollary 5.1.30 Let A be a non-commutative JBW ∗ -algebra. Then we have: (i) Surjective linear isometries from A to any non-commutative JBW ∗ -algebra are w∗ -continuous. (ii) Bijective Jordan-∗-homomorphisms (so, in particular, bijective algebra ∗-homomorphisms) from A to any non-commutative JBW ∗ -algebra are w∗ -continuous. (iii) The product of A is separately w∗ -continuous. (iv) The w∗ -closed ideals of A are precisely the sets of the form eA, where e is a central idempotent in A. Proof We may suppose that A = 0. Assertion (i) follows from Theorem 5.1.29(iv) and Fact 5.1.24, whereas assertion (ii) follows from Proposition 3.4.4, Remark 3.4.5, and assertion (i). For a ∈ A, let Ta stand for either La or Ra . Let x be in H(A, ∗). Then, by Lemma 3.6.24 and Corollary 2.1.9(iii), for every r ∈ R, exp(irTx ) : A → A is a surjective linear isometry, and hence, by assertion (i), is w∗ -continuous. Since exp(irTx ) − IA Tx = −i lim r→0 r in the norm topology of BL(A), we conclude that Tx is w∗ -continuous. Since every element a ∈ A can be written as a = x + iy with x, y ∈ H(A, ∗), the proof of assertion (iii) is complete. In view of Fact 5.1.10(i), to prove assertion (iv) it is enough to show that eA is w∗ -closed in A whenever e is a central idempotent of A. But this follows from assertion (iii) since for such an idempotent e we have eA = {x ∈ A : a − ea = 0}.



As a consequence of Theorem 5.1.29(ii) and Corollary 5.1.30(iii), we obtain the following. Corollary 5.1.31 Let A be a non-commutative JBW ∗ -algebra, and let B be any ∗-subalgebra of A. Then the w∗ -closure of B in A is ∗-subalgebra of A, and hence (by §5.1.9) it is a non-commutative JBW ∗ -algebra in a natural way.

12

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

A Banach space is said to be L-embedded if it is an L-summand of its bidual. Theorem 5.1.32 The predual A∗ of a non-commutative JBW ∗ -algebra A is L-embedded. Proof By standard theory of duality, the separate w∗ -continuity of the product of A (cf. Corollary 5.1.30(iii)) is equivalent to (La ) (A∗ ) ⊆ A∗ and (Ra ) (A∗ ) ⊆ A∗ for every a ∈ A. Therefore we have (La ) ((A∗ )◦ ) ⊆ (A∗ )◦ and (Ra ) ((A∗ )◦ ) ⊆ (A∗ )◦ for every a ∈ A, i.e., A(A∗ )◦ ⊆ (A∗ )◦ and (A∗ )◦ A ⊆ (A∗ )◦ in A endowed with the Arens product. Applying the separate w∗ -continuity of the product of A (cf. Theorem 3.5.34) and the w∗ -density of A in A , we deduce that A (A∗ )◦ + (A∗ )◦ A ⊆ (A∗ )◦ . Therefore (A∗ )◦ is a w∗ -closed ideal of A , and hence, by Fact 5.1.10, it is an M-summand of A . Finally, by Lemma 5.1.21(iii), A∗ is an L-summand of A .  Since nonzero JBW-algebras are unital (by the Banach–Alaoglu and Krein– Milman theorems, and Proposition 3.1.9), and the product of any JBW-algebra is separately w∗ -continuous [738, Corollary 4.1.6], we can argue as in the proofs of Fact 5.1.10(i) and Corollary 5.1.30(iv) to derive the following. Fact 5.1.33 [738, Proposition 4.3.6] Let A be a JBW-algebra. Then the w∗ -closed ideals of A are precisely the sets of the form eA, where e is a central idempotent in A. Lemma 5.1.34 The following assertions hold: (i) The range of a w∗ -continuous linear mapping between dual Banach spaces over K is w∗ -closed if (and only if) it is norm-closed. (ii) The range of any algebra ∗-homomorphism (respectively, algebra homomorphism) between non-commutative JB∗ -algebras (respectively, JB-algebras) is norm-closed. (iii) The range of any w∗ -continuous algebra ∗-homomorphism (respectively, algebra homomorphism) between non-commutative JBW ∗ -algebras (respectively, JBW-algebras) is w∗ -closed. Proof Let X and Y be dual Banach spaces over K, and let T : X → Y be a w∗ -continuous linear mapping. Then T(BX ) is w∗ -compact and convex, and T(X) is equal to the linear hull of T(BX ). Therefore, if T(X) is norm closed in Y, then, by [726, Corollary V.5.9], T(X) is w∗ -closed in Y. Thus assertion (i) has been proved. Let A and B be non-commutative JB∗ -algebras (respectively, JB-algebras), and let  : A → B be an algebra ∗-homomorphism (respectively, algebra homomorphism). By Propositions 3.4.4 and 3.4.13 (respectively, Proposition 3.1.4(ii) or [738, Proposition 3.4.3], and [738, Theorem 3.4.2]), A/ ker() is a non-commutative JB∗ -algebra (respectively, a JB-algebra) in a natural way. Since the mapping a + ker() → (a) from A/ ker() to B is an injective algebra ∗-homomorphism (respectively, algebra homomorphism) with range (A), it follows from Proposition 3.4.4 (respectively, Proposition 3.1.4(ii) or [738, Proposition 3.4.3]) that (A) is closed in B. Thus assertion (ii) has been proved. Finally, assertion (iii) follows straightforwardly from assertions (i) and (ii). 

5.1 Non-commutative JBW ∗ -algebras

13

Definition 5.1.35 Let A be a complex ∗-algebra (respectively, a real algebra). By a non-commutative JBW ∗ -representation (respectively, a JBW-representation) of A we mean a w∗ -dense-range algebra ∗-homomorphism (respectively, algebra homomorphism) from A to some non-commutative JBW ∗ -algebra (respectively, to some JBW-algebra). Thus non-commutative JBW ∗ -representations of A are particular types of non-commutative JB∗ -representations of A, as introduced in Definition 4.5.8. Let 1 : A → B1 and 2 : A → B2 be non-commutative JBW ∗ -representations (respectively, JBW-representations) of A. In agreement with Definition 4.5.27, we say that 1 and 2 are equivalent if there exists a bijective algebra ∗-homomorphism (respectively, algebra homomorphism) F : B1 → B2 such that 2 = F ◦ 1 . It is important to recall here that, by Proposition 3.4.4 and Corollary 5.1.30(ii) (respectively, by Proposition 3.1.4(ii), Fact 5.1.24, and Theorem 5.1.27), bijective algebra ∗-homomorphisms (respectively, algebra homomorphisms) between non-commutative JBW ∗ -algebras (respectively, JBW-algebras) are isometric and w∗ -continuous. Now let A be a non-commutative JB∗ -algebra (respectively, a JB-algebra), and recall that, by Theorem 3.5.34 (respectively, by Proposition 3.1.10 or [738, Theorem 4.4.3]), A becomes a non-commutative JBW ∗ -algebra (respectively, a JBW-algebra) containing A as a ∗-subalgebra (respectively, as a subalgebra), and whose product is separately w∗ -continuous. It follows that each central idempotent e in A gives rise to a non-commutative JBW ∗ -representation (respectively, a JBW-representation) of A, namely the mapping a → ea from A to eA . Such a representation will be called the canonical non-commutative JBW ∗ -representation (respectively, the canonical JBW-representation) of A associated to e. Proposition 5.1.36 Let A be a non-commutative JB∗ -algebra (respectively, a JB-algebra). Then every non-commutative JBW ∗ -representation (respectively, JBW-representation) of A is equivalent to the canonical non-commutative JBW ∗ representation (respectively, JBW-representation) of A associated to a suitable central idempotent of A . Proof First we prove the bracket-free version of the proposition. Let  : A → B be a non-commutative JBW ∗ -representation of A. Let π : B → B stand for the Dixmier projection (i.e. the transpose of the inclusion B∗ → B ), and set  := π ◦  : A → B.   is σ (A , A )-σ (B, B∗ ) Since π is σ (B , B )-σ (B, B∗ ) continuous, it follows that     to the σ (A , A )-dense subalgebra A is an algecontinuous. Since the restriction of  bra ∗-homomorphism, it follows from Theorem 5.1.29(ii) and Corollary 5.1.30(iii)  is also an algebra ∗-homomorphism. Since the range of   includes that that  of , which is σ (B, B∗ )-dense in B, it follows from the bracket-free version of ) is a σ (A , A )-closed ideal  maps A onto B. Since ker( Lemma 5.1.34(iii) that   of A , it follows from Fact 5.1.10(i) that there exists a central idempotent e ∈ A ) = (1 − e)A . It is now clear that the restriction of   to eA is a such that ker(  bijective algebra ∗-homomorphism (say F) from eA onto B. Finally, denoting by

14

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

the non-commutative JBW ∗ -representation associated to e, the equality  = F ◦ is obvious. The bracketed version of the proposition is proved in an analogous way. Indeed, it is enough to take ∗ equal to the identity on both A and B, and then to avoid Theorem 5.1.29(ii), and to replace Corollary 5.1.30(iii) with [738, 4.5.6], the bracket-free version of Lemma 5.1.34(iii) with the bracketed version of the same lemma, and Fact 5.1.10(i) with Fact 5.1.33.  Corollary 5.1.37 Let A be a non-commutative JBW ∗ -algebra (respectively, a JBWalgebra). Then A is isometrically and w∗ -bicontinuously ∗-isomorphic (respectively, isomorphic) to some w∗ -closed ideal of A . Proof The identity mapping on A is a non-commutative JBW ∗ -representation (respectively, a JBW-representation) of A, and Proposition 5.1.36 applies.  The following theorem becomes a converse of Theorem 5.1.29(i). Theorem 5.1.38 Let A be a non-commutative JB∗ -algebra such that the JB-algebra H(A, ∗) is in fact a JBW-algebra. Then A is a non-commutative JBW ∗ -algebra. Proof Since A = Asym as Banach spaces (cf. the paragraph immediately after Remark 2.2.6), and Asym is a JB∗ -algebra (cf. Fact 3.3.4), and H(A, ∗) = H(Asym , ∗) as JB-algebras (cf. Corollary 3.4.3), we may suppose that A is commutative. Then, by Lemma 4.2.61(iii), we have H(A, ∗) ≡ H(A , ∗) as JBW-algebras. On the other hand, by the bracketed version of Corollary 5.1.37, the JBW-algebra H(A, ∗) is isomorphic to eH(A, ∗) , for some central idempotent e ∈ H(A, ∗) . It follows that H(A, ∗) is isomorphic to pH(A , ∗), for some central idempotent p ∈ H(A , ∗). Therefore, since A = H(A , ∗) + iH(A , ∗), p is central in A , and, since A = H(A, ∗) + iH(A, ∗), the JB∗ -algebra A is isometrically ∗-isomorphic to pA . Since pA is a non-commutative JBW ∗ -algebra (cf. Corollary 5.1.30(iv)), and bijective algebra ∗-isomorphisms between non-commutative JB∗ -algebras are isometric (cf. Proposition 3.4.4), we see that A is (linearly isometric to) a dual Banach space.  Definition 5.1.39 Let A be a non-commutative JB∗ -algebra. A linear functional f on A is said to be positive if f (a) ∈ R+ 0 whenever a is a positive element of A. Since every element of H(A, ∗) is the difference of two positive elements (cf. Corollary 3.1.5 and Claim 3.1.28), it follows that positive linear functionals on A take real values on H(A, ∗). Much later (see Remark 6.2.19 below), we will realize that the notion of a positive linear functional just given is in agreement with the one in Definition 2.3.30. A is said to be monotone complete if the JB-algebra H(A, ∗) is monotone complete (cf. Definition 5.1.26). Suppose that A is monotone complete. A linear functional f on A is called normal if it is bounded and if f (aλ ) → f (a) whenever aλ is any bounded increasing net in H(A, ∗), and a stands for the least upper bound of the net. Combining Theorems 5.1.27, 5.1.29, and 5.1.38, we obtain the following.

5.1 Non-commutative JBW ∗ -algebras

15

Corollary 5.1.40 Let A be a non-commutative JB∗ -algebra. Then the following conditions are equivalent: (i) A is a non-commutative JBW ∗ -algebra. (ii) The JB-algebra H(A, ∗) is a JBW-algebra. (iii) A is monotone complete and the set of all positive normal linear functionals on A separates the points of A. Moreover, if the above conditions are fulfilled, then the predual of A is unique and consists of the normal linear functionals on A. Combining Theorems 3.4.8 and 5.1.38, we obtain the following. Corollary 5.1.41 Let B be a JBW-algebra. Then there is a unique JBW ∗ -algebra A such that B = H(A, ∗). Now, combining Theorem 5.1.29(i) and Corollary 5.1.41, we derive the following. Fact 5.1.42 Let A denote the category whose objects are the JBW ∗ -algebras, and whose morphisms are the w∗ -continuous algebra ∗-homomorphisms, and let B stand for the category whose objects are the JBW-algebras, and whose morphisms are the w∗ -continuous algebra homomorphisms. Then A → H(A, ∗) establishes a bijective equivalence from A to B. Now that the main results in the section have been proved, we deal with some by-products of them. Corollary 5.1.43 Let A be a non-commutative JBW ∗ -algebra. Then we have: (i) Jordan derivations (so, in particular, derivations) of A are w∗ -continuous. (ii) Bijective Jordan homomorphisms (so, in particular, algebra homomorphisms) from A to any non-commutative JBW ∗ -algebra are w∗ -continuous. Proof In view of Fact 3.3.4, we may suppose that A is a JBW ∗ -algebra, so that Jordan derivations are derivations, and Jordan homomorphisms are algebra homomorphisms. Let D be a ∗-derivation of A. Then, by Lemma 3.4.26, D is norm-continuous, so, for every λ ∈ R, we may consider exp(λD) which is an algebra ∗-automorphism of A (cf. Lemma 2.2.21 and Proposition 3.3.13), and hence a w∗ -continuous mapping (by Corollary 5.1.30(ii)). Since D = lim

λ→0

exp(λD) − IA λ

in the norm topology of BL(A), we conclude that D is w∗ -continuous. Since every derivation of A can be written as D1 + iD2 , with D1 and D2 ∗-derivations of A, assertion (i) follows. Let B be any non-commutative JBW ∗ -algebra, and let F : A → B be a bijective algebra homomorphism. By Theorem 3.4.75, we have F = G ◦ exp(iD) for some ∗-derivation D of A and some bijective algebra ∗-homomorphism G : A → B. But

16

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

exp(iD) and G are w∗ -continuous in view of assertion (i) and Corollary 5.1.30(ii), respectively. Thus assertion (ii) has been proved.  Corollary 5.1.44 Let B be a non-commutative JBW ∗ -algebra (respectively, a JBWalgebra), and let A be a w∗ -dense ∗-subalgebra (respectively, subalgebra) of B. Then BA is w∗ -dense in BB . Proof We may suppose that A is norm-closed in B, and hence that A is a noncommutative JB∗ -algebra (respectively, a JB-algebra). Then the inclusion A → B is a non-commutative JBW ∗ -representation (respectively, a JBW-representation) of A. Therefore the result follows from Proposition 5.1.36 and the fact that, by Goldstine’s  theorem, BA is w∗ -dense in BA . Since alternative C∗ -algebras are non-commutative JB∗ -algebras (cf. Fact 3.3.2), it is enough to invoke Corollary 5.1.40, to derive the following. Corollary 5.1.45 Let A be an alternative C∗ -algebra. Then the following conditions are equivalent: (i) A is an alternative W ∗ -algebra. (ii) The JB-algebra H(A, ∗) is a JBW-algebra. (iii) A is monotone complete and the set of all positive normal linear functionals on A separates the points of A. Moreover, if the above conditions are fulfilled, then the predual of A is unique and consists of the normal linear functionals on A. We conclude this subsection by establishing some results on real non-commutative JB∗ -algebras in order to prove that c0 is an M-ideal of ∞ . As usual in the literature dealing with M-ideals, we say that a Banach space is M-embedded if it is an M-ideal of its bidual. Recalling again that the bidual of a non-commutative JB∗ -algebra is a non-commutative JB∗ -algebra in a natural way (cf. Theorem 3.5.34), and invoking Theorem 5.1.22(i), we are provided with the following. Fact 5.1.46 A non-commutative JB∗ -algebra is M-embedded if and only if it is an ideal of its bidual. Our next goal is to generalize the ‘if’ part of the above fact to the setting of real non-commutative JB∗ -algebras. Recall that by a real non-commutative JB∗ -algebra (respectively, a real JB∗ -algebra, a real alternative C∗ -algebra, or a real C∗ -algebra) we mean a closed real ∗-subalgebra of a (complex) non-commutative JB∗ -algebra (respectively, JB∗ -algebra, alternative C∗ -algebra, or C∗ -algebra). Proposition 5.1.47 Let A and B be real non-commutative JB∗ -algebras, and let F : A → B be an algebra ∗-homomorphism. Then F is contractive. Moreover, if F is injective, then F is an isometry.

5.1 Non-commutative JBW ∗ -algebras

17

Proof Let AC and BC stand for the non-commutative JB∗ -complexifications of A and B (cf. Definition 4.2.56), respectively, and let FC : AC → BC denote the extension of F by complex linearity. Then AC and BC are non-commutative JB∗ -algebras, and FC is an algebra ∗-homomorphism, which is injective whenever F is so. Therefore the result follows from Proposition 3.4.4.  Proposition 5.1.48 Let A be a real non-commutative JB∗ -algebra, and let M be a closed ideal of A. Then M is ∗-invariant. Proof M + iM is a closed ideal of the non-commutative JB∗ -complexification of A, and hence, by Proposition 3.4.13, it is ∗-invariant. Since ∗ commutes with the canonical involution of the complexification (cf. the paragraph immediately before Lemma 1.1.97) the result follows.  Arguing as in the proof of Lemma 5.1.4, with Propositions 5.1.48 and 5.1.47 instead of 3.4.13 and 3.4.4, respectively, we obtain the following. Lemma 5.1.49 Let A be a real JB∗ -algebra, and let I be a direct summand of A. Then I is an M-summand of A. Now consider the following. Definition 5.1.50 By a real non-commutative JBW ∗ -algebra (respectively, a real JBW ∗ -algebra, a real alternative W ∗ -algebra, or a real W ∗ -algebra) we mean a real non-commutative JB∗ -algebra (respectively, real JB∗ -algebra, real alternative C∗ -algebra, or real C∗ -algebra) which is a dual Banach space. By the Banach–Alaoglu and Krein–Milman theorems, the closed unit ball of a real non-commutative JBW ∗ -algebra has extreme points, so that the implication (iii)⇒(i) in Corollary 4.2.58 applies to get the following. Fact 5.1.51 Nonzero real non-commutative JBW ∗ -algebras are unital. Another important fact is the following. Fact 5.1.52 Let A be a real non-commutative JBW ∗ -algebra, and let I be a w∗ -closed ideal of A. Then: (i) There exists a central idempotent e ∈ A such that I = eA. (ii) I is an M-summand of A. Proof Argue as in the proof of Fact 5.1.10, with Proposition 5.1.48, Fact 5.1.51, and Lemma 5.1.49 instead of Proposition 3.4.13, Fact 5.1.7, and Lemma 5.1.4.  Proposition 5.1.53 Let A be a real non-commutative JB∗ -algebra. Then closed ideals of A are M-ideals of A. Proof Argue as in the corresponding part of Theorem 5.1.22(i), with Proposition 4.2.62 and Fact 5.1.52 instead of Theorem 3.5.34 and Fact 5.1.10. 

18

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Recalling the fact, applied in the above proof, that the bidual of a real noncommutative JB∗ -algebra is a real non-commutative JB∗ -algebra in a natural way (cf. Proposition 4.2.62 again), and invoking Proposition 5.1.53, we are provided with the following. Fact 5.1.54 A real non-commutative JB∗ -algebra is M-embedded whenever it is an ideal of its bidual. Lemma 5.1.55 Let X be a normed space over K, let P be an L-projection on X, and let Q be a contractive linear projection on X such that ker(Q) = ker(P). Then P = Q. Proof

Since Q is a contractive linear projection, for x ∈ X and y ∈ ker(Q) we have x − (x − Q(x)) = Q(x) = Q(x − y) ≤ x − y,

and hence x − Q(x) is a best approximant of x in ker(Q). On the other hand, since P is an L-projection, each x ∈ X has a unique best approximant point in ker(P), namely x − P(x). Therefore, since ker(P) = ker(Q), the result follows.  Given a normed space X over K, we denote by πX  the Dixmier projection on X  , i.e., the unique linear projection on X  whose rage is X  and whose kernel is X ◦ (the polar of X in X  ). Proposition 5.1.56 For a Banach space X over K the following conditions are equivalent: (i) X is M-embedded. (ii) πX  is an L-projection on X  . As a consequence, if X is M-embedded, then X  is L-embedded. Proof (i)⇒(ii) By the assumption (i), X ◦ is the kernel of an L-projection P on X  . Since πX  is a contractive linear projection on X  with kernel X ◦ , it follows from Lemma 5.1.55 that P = πX  . (ii)⇒(i) By the assumption (ii), X ◦ is the range of an L-projection on X  , i.e. X is M-embedded.  Finally, combining Facts 5.1.46 and 5.1.54 with Proposition 5.1.56, we get the following. Corollary 5.1.57 The real or complex Banach space c0 is M-embedded. Equivalently, we have 1 = 1 ⊕1 c◦0 . 5.1.2 Historical notes and comments A classical definition of a von Neumann algebra is that of a ∗-algebra of bounded linear operators on a complex Hilbert space, which is closed in the weak operator topology. Since a JB∗ -algebra cannot in general be represented as a Jordan ∗-algebra of bounded linear operators on a complex Hilbert space (see Example 3.1.56 and

5.2 Preliminaries on analytic mappings

19

Theorem 3.4.8), to say that a JB∗ -algebra is ‘closed in the weak operator topology’ makes no sense. However, we recall that non-spatial characterizations on von Neumann algebras are known. Indeed, Kadison [989] (see also [1167, Exercise 7.6.38]) showed that an abstract C∗ -algebra A admits a faithful representation as a von Neumann algebra of operators if and only if A is monotone complete and the set of all positive normal linear functionals on A separates the points of A. Another characterization is that of Sakai [1070] (see also [806, Theorem 1.16.7]) who showed that a C∗ -algebra admits such a representation if and only if it is a dual Banach space. In this way both the definition itself of a non-commutative JBW ∗ algebra (as a non-commutative JB∗ -algebra which is a dual Banach space) and the equivalence (i)⇔(iii) in Corollary 5.1.40 show that non-commutative JBW ∗ -algebras become the reasonable non-associative generalizations of von Neumann algebras. With minor variants, the comments in the present paragraph are taken from Shultz [570, p. 361]. We note that monotone complete C∗ -algebras need not be W ∗ -algebras. This fact, as well as a classification theory of monotone complete C∗ -algebras (using spectroid invariants and a classification semigroup), can be found in Saitˆo and Wright [1183]. As we said in the introduction, Theorems 5.1.29 and 5.1.38 are due to Edwards [222], whereas Theorems 5.1.22 and 5.1.32 are due to Pay´a, P´erez, and Rodr´ıguez [481]. Theorem 5.1.32 can be also found in [19]. Theorem 5.1.27 and the bracketed version of Proposition 5.1.36 are due to Shultz [570]. Proposition 5.1.53 (that closed ideals of real non-commutative JB∗ -algebras are M-ideals) could be new. We do not know whether, conversely, M-ideals of real non-commutative JB∗ -algebras are closed ideals. The answer to this question is affirmative when the real non-commutative JB∗ -algebra is the self-adjoint part of any C∗ -algebra [830, Proposition 6.18]. We guess that, with similar arguments, the answer remains affirmative when the real non-commutative JB∗ -algebra is an arbitrary JB-algebra. Most results on M-ideals and L-summands involved in our development have been taken from the book of Harmand–Werner–Werner [739]. Indeed, Propositions 5.1.14 and 5.1.17, Lemma 5.1.21, the associative forerunners of Theorems 5.1.22 and 5.1.32, Lemma 5.1.55, Proposition 5.1.56, and Corollary 5.1.57 can be found in [739] as Theorem I.1.10(a), Corollary I.1.25, Theorems I.1.9 and V.4.4, Example IV.1.1(b), Propositions I.1.2(b) and III.1.2, and Example III.1.4(a), respectively. The reader is referred to [739, pp. 41, 42, 46, 146, 199, 251, and 255] for a detailed discussion of the paternity of these results. Theorems 5.1.22(i) and 5.1.32 are reviewed in [739, p. 255] without proof. We learned the associative forerunner of Theorem 5.1.22(i) in the Smith–Ward paper [1094].

5.2 Preliminaries on analytic mappings Introduction In this section we develop the basic theory of the analytic mappings between Banach spaces. This theory relies on the concepts of a polynomial and

20

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

of a power series, which allow us to talk about asymptotic developments of finite order and about power series expansions of a mapping at a point. For analytic mappings between complex Banach spaces we prove the Cauchy integral formulae (see Proposition 5.2.50), and derive the Cauchy inequalities (see Corollary 5.2.53), which become one of the key tools in the next sections. As a main result, we prove the equivalence between complex analyticity and holomorphy (see Theorem 5.2.60).

5.2.1 Polynomials and higher derivatives §5.2.1 Let X and Y be normed spaces over K. For each n ∈ N, let BLn (X, Y) stand for the vector space of all continuous n-linear mappings F : X× . n. . ×X → Y, endowed with the norm F = sup F(x1 , . . . , xn ). xk ∈SX

Let BLsn (X, Y) be the vector subspace of BLn (X, Y) consisting of all F ∈ BLn (X, Y) which are symmetric, i.e. such that F(xπ(1) , . . . , xπ(n) ) = F(x1 , . . . , xn ) for all x1 , . . . , xn ∈ X and π ∈ Sn , where Sn is the symmetric group of orden n. Clearly BLsn (X, Y) is a closed subspace of BLn (X, Y). Given F ∈ BLn (X, Y), let Fs be the continuous n-linear mapping defined by Fs (x1 , . . . , xn ) :=

1  F(xπ(1) , . . . , xπ(n) ). n! π∈Sn

Clearly Fs ∈ BLsn (X, Y), and Fs  ≤ F.

(5.2.1)

Hence the mapping F → Fs is a contractive linear projection on BLn (X, Y) with range BLsn (X, Y). In the case that Y is a Banach space, we have that BLn (X, Y), and hence BLsn (X, Y), is a Banach space. §5.2.2 For k ∈ N, a finite sequence α = (α1 , . . . , αk ) ∈ (N ∪ {0})k is called a multiindex of length k. The order of the multi-index α = (α1 , . . . , αk ) is defined by |α| := α1 + · · · + αk . For k ∈ N and n ∈ N ∪ {0} we will consider in what follows the sets A0 (k, n) := {α = (α1 , . . . , αk ) ∈ (N ∪ {0})k : |α| = n} and A(k, n) := {α = (α1 , . . . , αk ) ∈ Nk : |α| = n}.

5.2 Preliminaries on analytic mappings

21

The factorial of the multi-index α = (α1 , . . . , αk ) is defined by α! := α1 ! · · · αk !. Moreover, for each integer n ≥ |α|, the multinomial coefficient is defined by   n! n . := α (n − |α|)! α! Note that, for k = 1, it is reduced to binomial coefficients. §5.2.3 Let X and Y be normed spaces over K, and let F be in BLsn (X, Y). For α ∈ A0 (k, n) and (x1 , . . . , xk ) ∈ X k we will adopt the following confortable notation α

F(x1α1 , . . . , xk k ) := F(x1 , . . . , x1 , . . . , xk , . . . , xk ), α1 times

αk times

and in particular F(xn ) := F(x, . . . , x). n times

Inductive arguments show the binomial formula n    n n−j j F((x1 + x2 )n ) = F(x1 , x2 ), j

(5.2.2)

j=0

and more generally, for k ≥ 2, the multinomial formula  n α n F(x1α1 , . . . , xk k ). F((x1 + · · · + xk ) ) = α

(5.2.3)

α∈A0 (k,n)

Moreover, in the case that α ∈ A0 (k, m) with m < n, for each (x1 , . . . , xk ) ∈ X k , we α will denote by F(x1α1 , . . . , xk k ) the element of BLsn−m (X, Y) defined by α

F(x1α1 , . . . , xk k )(h1 , . . . , hn−m ) := F(x1 , . . . , x1 , . . . , xk , . . . , xk , h1 , . . . , hn−m ). α1 times

αk times

§5.2.4 Let X and Y be normed spaces over K. A (continuous) homogeneous polynomial of degree n from X to Y is a mapping P : X → Y for which there exists F ∈ BLn (X, Y) such that P is the restriction of F to the ‘diagonal’, i.e. P(x) = F(x, . n. ., x) for every x ∈ X. It is clear that Fs also determines P, and, in agreement with our previous notation, P(x) = Fs (xn ) for every x ∈ X. In fact, there is a unique continuous symmetric n-linear mapping P : X× . n. . ×X → Y determining P, which is given by the following polarization formula P(x1 , . . . , xn ) =

1  σ1 . . . σn P(σ1 x1 + · · · + σn xn ). n! 2n σ1 =±1 ··· σn =±1

(5.2.4)

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

We denote by P n (X, Y) the vector space of all homogeneous polynomials of degree n from X to Y, which is a normed space under the norm P  = sup P(x). x∈SX

Moreover, it follows from (5.2.4) that the mapping P → P is a topological isomorphism from P n (X, Y) to BLsn (X, Y) satisfying P  ≤ P  ≤

nn P . n!

(5.2.5)

Clearly, P 1 (X, Y) = BL1 (X, Y) = BL(X, Y). Moreover, we interprete the homogeneous polynomials of degree 0, as well as the 0-linear mappings, as constant mappings, so that P 0 (X, Y) = BL0 (X, Y) = Y. Given X, Y, and Z normed spaces over K, P ∈ P m (X, Y), and Q ∈ P n (Y, Z), the continuous mn-linear mapping F : X mn → Z defined by F(x1 , . . . , xmn ) := Q(P(x1 , . . . , xm ), . . . , P(x(m−1)n+1 , . . . , xmn )) satisfies F(x, .mn . ., x) = (Q ◦ P)(x) for every x ∈ X, and hence Q ◦ P ∈ P mn (X, Z). §5.2.5 Let X and Y be normed spaces over K, let  be a non-empty open subset of X, let x0 be in , and let f :  → Y be a mapping. We recall (see the paragraph immediately before Theorem 1.1.23) that f is said to be (Fr´echet) differentiable at x0 if there exists T ∈ BL(X, Y) such that lim

x→x0 x∈\{x0 }

f (x) − f (x0 ) − T(x − x0 ) = 0. x − x0 

In this case, the operator T is unique, is called the (Fr´echet) derivative of f at x0 , and is denoted by Df (x0 ). The mapping f is said to be differentiable on  if f is differentiable at every point of , and in such a case the mapping Df :  → BL(X, Y) is called the differential mapping of f . When X = K, then it is straightforward to verify that f is differentiable at ζ0 if and only if the derivative f  (ζ0 ) =

lim

ζ →ζ0 ζ ∈\{ζ0 }

f (ζ ) − f (ζ0 ) ∈Y ζ − ζ0

exists, and in this case Df (ζ0 )(ζ ) = ζ f  (ζ0 ). In the case K = C, we can regard  as an open subset of XR and f as a mapping (say fR ) from  to YR . It follows from the natural inclusion BL(X, Y) ⊆ BL(XR , YR ) that f is differentiable at x0 if and only if fR is differentiable at x0 and DfR (x0 ) ∈ BL(X, Y). Moreover, in such a case we have DfR (x0 ) = Df (x0 ). §5.2.6 Let X and Y be normed spaces over K. We recall that there is a natural identification BLn+1 (X, Y) = BL(X, BLn (X, Y)). Indeed, each F ∈ BL(X, BLn (X, Y)) is identified with the continuous (n + 1)-linear mapping (x1 , x2 , . . . , xn+1 ) → F(x1 )(x2 , . . . , xn+1 ),

5.2 Preliminaries on analytic mappings

23

which will be also denoted by F. This identification is usually utilized for the inductive presentation of higher order differentials. Suppose that  is a non-empty open subset of X and f :  → Y is an n times differentiable mapping on , and write Dn f :  → BLn (X, Y) for the nth differential of f . Then f is said to be n + 1 times differentiable at x0 ∈  if the mapping Dn f is differentiable at x0 . In this case, the n + 1 differential of f at x0 is defined by Dn+1 f (x0 ) := D(Dn f )(x0 ) regarded as a member of BLn+1 (X, Y). The mapping f is said to be n + 1 times differentiable on  if f is n + 1 times differentiable at every point of , and in such a case the mapping Dn+1 f :  → BLn+1 (X, Y) is called the n+1 differential mapping of f . If the mapping Dn f is continuous, then f is said to be n times continuously differentiable or of class Cn on . The mapping f is of class C∞ if it is of class Cn for every n ∈ N, and in this case f is also said to be smooth. By the Schwarz symmetric theorem [707, Theorem 7.9], if f is n times differentiable at x0 , then Dn f (x0 ) is symmetric, i.e. Dn f (x0 ) ∈ BLsn (X, Y). As usual, we will identify D0 f ≡ f . Proposition 5.2.7 Let X and Y be normed spaces over K, and let n be in N ∪ {0}. Then every P in P n (X, Y) is of class C∞ . More precisely, for each 0 ≤ k ≤ n, Dk P is the homogeneous polynomial of degree n − k from X to BLsk (X, Y) given by Dk P(x) =

n! P(xn−k ), (n − k)!

whereas Dk P(x) = 0 for each k > n. In particular, . . . , x, h) and Dn P(x)(h1 , . . . , hn ) = n! P(h1 , . . . , hn ). DP(x)(h) = nP(x, n−1 n! P and Dk : P n (X, Y) → P n−k (X, BLsk (X, Y)) is a conMoreover, Dk P = (n−k)! tinuous linear mapping.

Proof Let P ∈ P n (X, Y). Since, for each 0 ≤ k ≤ n, the mapping x → P(xn−k ) is a homogeneous polynomial of degree n − k from X to BLsk (X, Y), it is sufficient deal with the case k = 1 and then apply this case to the successive derivatives of P. It is obvious that DP = 0 if n = 0, otherwise by noting that P = P ◦ d, where d : X → X n is the continuous linear mapping defined by d(x) = (x, . n. ., x), it follows from the chain rule and the symmetry of P that P is differentiable and . . . , x, h), DP(x)(h) = DP(d(x))(d(h)) = DP(x, . n. ., x)(h, . n. ., h) = nP(x, n−1 that is to say DP(x) = nP(xn−1 ). Therefore DP(x1 , . . . , xn−1 )(xn ) = nP(x1 , . . . , xn ), hence DP  = nP , and using (5.2.5) we see that DP  ≤ DP  = nP  ≤

nn+1 P . n!

Thus the linear mapping D : P n (X, Y) → P n−1 (X, BL(X, Y)) is continuous.



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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

§5.2.8 Let X and Y be normed spaces over K. We will denote by P(X, Y) the vector space defined as the algebraic direct sum of the vector spaces P n (X, Y) with n ∈ N ∪ {0}. The elements of P(X, Y) are called polynomials from X to Y. So, a polynomial P from X to Y is a mapping P : X → Y for which there exist (unique) k-homogeneous polynomials Pk (k = 0, 1, . . . , n) such that P(x) = P0 + P1 (x) + · · · + Pn (x) for every x ∈ X. If Pn = 0, then the integer n is called the degree of P. Moreover, P(X, Y) is a normed space under the norm P  = sup P(x), x∈SX

and, for each n ∈ N ∪ {0},

P n (X, Y)

becomes a closed subspace of P(X, Y).

§5.2.9 Let X and Y be normed spaces over K, and let n be in N ∪ {0}. If  is an open subset of X, and if f :  → Y is an n times differentiable mapping at x0 ∈ , then the nth degree Taylor polynomial of f at x0 is defined as the polynomial Tf ,x0 ,n : X → Y given by Tf ,x0 ,n (x) =

n  1 k D f (x0 )((x − x0 )k ). k! k=0

Note that this really is a polynomial of degree at most n. Two consequences of Proposition 5.2.7 are the following. Corollary 5.2.10 Let X and Y be normed spaces over K. If P ∈ P n (X, Y), then TP,x0 ,m = P for all x0 ∈ X and m ≥ n. Proof Given P ∈ P n (X, Y), x0 , x ∈ X, and m ≥ n, it follows from Proposition 5.2.7 and the binomial formula (5.2.2) that m n     1 k n k D P(x0 )((x − x0 ) ) = P(x0n−k , (x − x0 )k ) TP,x0 ,m (x) = k k! k=0

k=0

= P((x0 + (x − x0 ))n ) = P(xn ) = P(x), 

as required.

Corollary 5.2.11 Let X and Y be normed spaces over K, let  be an open subset of X, let x0 be in , and let n be in N. If f :  → Y is a mapping n times differentiable at x0 , then DTf ,x0 ,n = TDf ,x0 ,n−1 . Proof Since Tf ,x0 ,n is the composition of the mapping x → x − x0 from X to X with  the mapping x → nk=0 k!1 Dk f (x0 )(xk ) from X to Y, it follows from Proposition 5.2.7 that n  1 DTf ,x0 ,n (x) = Dk f (x0 )((x − x0 )k−1 ), (k − 1)! k=1

5.2 Preliminaries on analytic mappings

25

and keeping in mind that Dk f (x0 )((x − x0 )k−1 ) = Dk−1 (Df )(x0 )((x − x0 )k−1 ) we realize that DTf ,x0 ,n (x) =

n−1  1 D (Df )(x0 )((x − x0 ) ) = TDf ,x0 ,n−1 (x),

!

=0



as required.

Using the Hahn–Banach theorem, we can give a simple proof of Taylor’s formula for normed spaces. Proposition 5.2.12 Let X and Y be normed spaces over K, let  be an open subset of X, let n be a natural number, and let f :  → Y be a mapping n times differentiable. If a line segment [x0 , x1 ] is entirely contained in , then  f (x1 ) − Tf ,x0 ,n−1 (x1 ) ≤

1 x1 − x0 n sup Dn f (x0 + t(x1 − x0 )). n! 0≤t≤1

Proof Since complex differentiability implies real differentiability, we may suppose that K = R. Then for each y ∈ SY  , we consider the function g : t → y ( f (x0 + t(x1 − x0 ))) from Gx0 ,x1 to R, where Gx0 ,x1 := {t ∈ R : x0 + t(x1 − x0 ) ∈ } is an open subset of R containing the interval [0, 1]. The chain rules give that g is n times differentiable and for each k ∈ {0, 1, . . . , n} g(k) (t) = y (Dk f (x0 + t(x1 − x0 ))((x1 − x0 )k )), and consequently Tg,0,n−1 (1) =

n−1  1  k y (D f (x0 )((x1 − x0 )k )) = y (Tf ,x0 ,n−1 (x1 )). k! k=0

By the classical Taylor formula, |g(1) − Tg,0,n−1 (1)| ≤

1 sup |g(n) (t)|, n! 0≤t≤1

and by noting that g(1) − Tg,0,n−1 (1) = y ( f (x1 )) − y (Tf ,x0 ,n−1 (x1 )) = y ( f (x1 ) − Tf ,x0 ,n−1 (x1 )), and |g(n) (t)| = |y (Dn f (x0 + t(x1 − x0 ))((x1 − x0 )n ))| ≤ Dn f (x0 + t(x1 − x0 ))x1 − x0 n , we deduce that |y ( f (x1 ) − Tf ,x0 ,n−1 (x1 ))| ≤

1 x1 − x0 n sup Dn f (x0 + t(x1 − x0 )), n! 0≤t≤1

and the desired conclusion follows from the Hahn–Banach theorem. In particular, for n = 1, we have the mean value theorem.



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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Fact 5.2.13 Let X and Y be normed spaces over K, let  be an open subset of X, and let f :  → Y be a differentiable mapping. If a line segment [x0 , x1 ] is entirely contained in , then  f (x1 ) − f (x0 ) ≤ x1 − x0  sup Df (x0 + t(x1 − x0 )). 0≤t≤1

As a consequence, we have the following. Corollary 5.2.14 Let X and Y be normed spaces over K, let  be an open subset of X, and let f :  → Y be a mapping of class C1 on . Then f is locally Lipschitz. §5.2.15 Let X and Y be normed spaces over K, let  be an open subset of X, let f :  → Y be a mapping, let x0 be in , and let n be in N ∪ {0}. We say that f has an asymptotic development of order n at x0 if there exists a polynomial P : X → Y of degree at most n, called an n-expansion of f at x0 , satisfying f (x) − P(x) = o(x − x0 n ), i.e., for each ε > 0 there exists δ > 0 such that if x ∈  and x − x0  < δ, then  f (x) − P(x) ≤ εx − x0 n . In such a case, f (x0 ) = P(x0 ) and f is continuous in x0 . Clearly, if P is a polynomial of degree at most n, then P is an n-expansion of itself at any point. The next result is the asymptotic form of the Taylor formula. Proposition 5.2.16 Let X and Y be normed spaces over K, let  be an open subset of X, let x0 be in , and let n be in N. If f :  → Y is a mapping n times differentiable at x0 , then Tf ,x0 ,n is an n-expansion of f at x0 . Proof We will prove that f (x) − Tf ,x0 ,n (x) = o(x − x0 n ) by induction on n. First, by the definition of the differential, it is true for n = 1. We now suppose that it is true up to order n − 1 and consider the case n. Assume that f :  → Y is a mapping which is n − 1 times differentiable in  and n times differentiable at x0 , and consider the mapping φ :  → Y defined by φ(x) = f (x) − Tf ,x0 ,n (x). It follows from Corollary 5.2.11 that Dφ(x) = Df (x) − TDf ,x0 ,n−1 (x), and hence, by hypothesis, Dφ(x) = o(x − x0 n−1 ). Let us fix ε > 0. From what we have just seen, there exists δ > 0 such that Dφ(x) < εx − x0 n−1 if x − x0  < δ. From Fact 5.2.13 we have φ(x) = φ(x) − φ(x0 ) ≤ εx − x0 n , which ends the proof.



Exercise 5.2.17 Let X and Y be normed spaces over K, and let n be in N ∪ {0}. Show that:

5.2 Preliminaries on analytic mappings

27

(i) If P be a polynomial of degree at most n from X to Y such that P(x) = o(xn ), then P = 0. (ii) If  is an open subset of X, and if f :  → Y has an asymptotic development at x0 ∈  of order n, then this development is unique. Solution To prove (i), assume that P is a polynomial of degree at most n from X to Y satisfying the condition P(x) = o(xn ). Note that this condition gives that = 0 for all x ∈ X and k = 0, 1, . . . , n. Write limt→0 P(tx) tk P(x) =

n 

Pk (x) with Pk ∈ P k (X, Y) (0 ≤ k ≤ n).

k=0

n

Then P0 = limt→0 k=0 tk Pk (x) = limt→0 P(tx) = 0. Assume that P0 = P1 = · · · = Pk−1 = 0 for some k with 1 ≤ k ≤ n. Then Pk (x) = lim (Pk (x) + tPk+1 (x) + · · · + tn−k Pn (x)) = lim t→0

t→0

P(tx) = 0. tk

This inductive argument shows that P = 0. Now, assume that  is an open subset of X, that f :  → Y is a mapping, and that P and Q are asymptotic developments of f at x0 ∈  of order n. Then P(x) − Q(x) = ( f (x) − Q(x)) − ( f (x) − P(x)) = o(x − x0 n ), and hence R(x) := P(x0 + x) − Q(x0 + x) is a polynomial of degree at most n from X to Y such that R(x) = o(xn ). It follows from (i) that R = 0, and hence P = Q.  Combining Proposition 5.2.16 with Exercise 5.2.17(ii) we have the following. Corollary 5.2.18 Let X and Y be normed spaces over K, let  be an open subset of X, let f :  → Y be a mapping, let x0 be in , and let n be in N. If f is n times differentiable at x0 , then Tf ,x0 ,n is the unique asymptotic development of f at x0 of order n. In the following result we display the behaviour of asymptotic developments for some operations. Theorem 5.2.19 Let X be a normed space over K, let X be an open of X, let n be in N ∪ {0}, and let x0 ∈ X . We have: (i) If Y is a normed space over K, if the mappings fi : X → Y (1 ≤ i ≤ k) admit  n-expansions Pi at x0 , respectively, and if λi ∈ K (1 ≤ i ≤ k), then ki=1 λi Pi is k the n-expansion of i=1 λi fi at x0 . (ii) If Yi (1 ≤ i ≤ k) are normed spaces over K, and if the mappings fi : X → Yi (1 ≤ i ≤ k) admit n-expansions Pi at x0 , respectively, then (P1 , . . . , Pk ) is the n-expansion of ( f1 , . . . , fk ) at x0 . (iii) If Y and Z are normed spaces over K, if Y is an open of Y, if f : X → Y satisfies f (X ) ⊆ Y and admits at x0 an n-expansion P, and if g : Y → Z

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

admits at f (x0 ) an n-expansion Q, then the truncation to the order n of the polynomial Q ◦ P is the n-expansion of g ◦ f at x0 . (iv) If Yi (1 ≤ i ≤ k) and Z are normed spaces over K, if the mappings fi : X → Yi (1 ≤ i ≤ k) admit n-expansions Pi at x0 , respectively, and if F ∈ BL(Y1 , . . . , Yk ; Z), then the truncation to the orden n of the polynomial F ◦ (P1 , . . . , Pk ) is the n-expansion of F ◦ ( f1 , . . . , fk ) at x0 . Proof Assertions (i) and (ii) are immediately checked.  To prove assertion (iii), write P(x) = nk=0 Pk (x − x0 ) with Pk ∈ P k (X, Y) (0 ≤ k ≤ n). Then we have P0 = f (x0 ), hence f (x) − f (x0 ) =

n 

Pk (x − x0 ) + o(x − x0 n ),

k=1

and so f (x) − f (x0 ) = O(x − x0 ). Therefore (g ◦ f )(x) = g( f (x)) = Q( f (x)) + o( f (x) − f (x0 )n ) = Q( f (x)) + o(x − x0 n ). On the other hand, writing Q(y) =

n 

Qk (y − f (x0 )) with Qk ∈ P k (Y, Z) (0 ≤ k ≤ n),

k=0

we see that Q( f (x)) =

n  k=0

Qk ( f (x) − f (x0 )) =

n 

Qk (P(x) − f (x0 ) + o(x − x0 n )),

k=0

and, invoking the binomial formula (5.2.2), we realize that Q( f (x)) =

n 

Qk (P(x) − f (x0 )) + o(x − x0 n ) = Q(P(x)) + o(x − x0 n ).

k=0

It follows that (g ◦ f )(x) = Q(P(x)) + o(x − x0 n ), and consequently the truncation to the order n of the polynomial Q ◦ P is the n-expansion of g ◦ f at x0 . To prove assertion (iv), note that F is a k-homogeneous polynomial from W := Y1 × · · · × Yk to Z with continuous symmetric k-linear mapping F : W k → Z defined for wi = (yi,1 , . . . , yi,k ) (1 ≤ i ≤ k) by 1  F(w1 , . . . , wk ) = F(yσ (1),1 , . . . , yσ (k),k ). k! σ ∈Sk

Since, by assertion (ii), (P1 , . . . , Pk ) is the n-expansion of f = ( f1 , . . . , fk ), it follows from assertion (iii) that the truncation to the order n of the polynomial F ◦(P1 , . . . , Pk )  is the n-expansion of F ◦ ( f1 , . . . , fk ) at x0 . As a consequence of Theorem 5.2.19(iii)–(iv) we can give explicit formulae for the higher derivatives of composition mappings. The next one is known as the Leibnitz formula.

5.2 Preliminaries on analytic mappings

29

Corollary 5.2.20 Let X, Yi (1 ≤ i ≤ k), and Z be normed spaces over K, let  be an open of X, let n be in N ∪ {0}, and let x0 ∈ . Suppose that fi :  → Yi (1 ≤ i ≤ k) are n times differentiable mappings at x0 , and that F ∈ BL(Y1 , . . . , Yk ; Z). Then, for each h ∈ X, we have Dn(F ◦ ( f1 , . . . , fk ))(x0 )(hn ) =

 n F(Dα1f1 (x0 )(hα1 ), . . . , Dαkfk (x0 )(hαk )). α

α∈A0 (k,n)

Proof For simplicity, write the nth degree Taylor polynomial of fi (1 ≤ i ≤ k) at x0  as Pi (x0 + h) = nj=0 Pi,j (h) with Pi,j ∈ P j (X, Yi ). Then (F ◦ (P1 , . . . , Pk ))(x0 + h) = F(P1 (x0 + h), . . . , Pk (x0 + h))) ⎛ ⎞ n n   = F⎝ P1,α1 (h), . . . , Pk,αk (h)⎠ α1 =0

=

αk =0



(α1 ,...,αk

F(P1,α1 (h), . . . , Pk,αk (h)).

)∈{0,1,...,n}k

Since F ◦ (P1,α1 , . . . , Pk,αk ) ∈ P α1 +···+αk (X, Z), it follows that the n-homogeneous component of (F ◦ (P1 , . . . , Pk ))(x0 + h) is given by (F ◦ (P1 , . . . , Pk ))n (h) =



F(P1,α1 (h), . . . , Pk,αk (h)).

(α1 ,...,αk )∈A0 (k,n)

Now, keeping in mind Theorem 5.2.19(iv), we deduce that 1 n D (F ◦ ( f1 , . . . , fk ))(x0 )(hn ) n!    1 α1 1 αk α1 αk = F D f1 (x0 )(h ), . . . , D fk (x0 )(h ) , α1 ! αk ! α∈A0 (k,n)

and the statement follows.



The next one is known as the Fa`a di Bruno formula. Corollary 5.2.21 Let X, Y, and Z be normed spaces over K, let X and Y be open subsets of X and Y, respectively, let f : X → Y and g : Y → Z be mappings such that f (X ) ⊆ Y , and let n be a natural number. Suppose that f and g are n times differentiable at x0 ∈ X and at f (x0 ), respectively. Then, for each h ∈ X, we have Dn (g ◦ f )(x0 )(hn )   n  1  n Dk g( f (x0 ))(Dα1 f (x0 )(hα1 ), . . . , Dαk f (x0 )(hαk )). = α k! k=1

α∈A(k,n)

30

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Proof Set y0 = f (x0 ) and z0 = g(y0 ). For simplicity, put P and Q to denote the nth degree Taylor polynomials of f and g at x0 and y0 , respectively, and write P(x0 + x) = y0 +

n 

Pi (x) and Q(y0 + y) = z0 +

i=1



(Q ◦ P)(x0 + h) = Q(P(x0 + h)) = Q y0 +

= z0 + = z0 +

Qi (y).

i=1

Then

n 

n 

⎛ Qk ⎝

n 

 Pi (h) = z0 +

i=1 n 

k=1

α1 =1

n 



Pα1 (h), . . . ,

n 



n 

 Qk

k=1

n 

 Pi (h)

i=1

Pαk (h)⎠

αk =1

Qk (Pα1 (h), . . . , Pαk (h)).

k=1 (α1 ,...,αk )∈{1,...,n}k

Since Qk ◦ (P1,α1 , . . . , Pk,αk ) ∈ P α1 +···+αk (X, Z), it follows that the n-homogeneous component of (Q ◦ P)(x0 + h) is given by (Q ◦ P)n (h) =

n 



Qk (P1,α1 (h), . . . , Pk,αk (h)).

k=1 (α1 ,...,αk )∈A(k,n)

Now, keeping in mind Theorem 5.2.19(iii), we deduce that 1 n D (g ◦ f )(x0 )(hn ) n!   n   1 1 α1 1 αk k α1 αk D g( f (x0 )) D f (x0 )(h ), . . . , D f (x0 )(h ) = k! α1 ! αk ! k=1 α∈A(k,n)

n  1  1 = Dk g( f (x0 ))(Dα1f (x0 )(hα1 ), . . . , Dαkf (x0 )(hαk )), k! α1 ! · · · αk ! k=1

α∈A(k,n)



and the statement follows.

To describe the higher derivatives of a power of a mapping we will need the concept of tree-derivatives. Definition 5.2.22 Consider the set S = {{1, . . . , n} : n ∈ N} (totally) ordered by inclusion. Given a natural number n, a tree A of height n is a pair ({Ak }0≤k≤n , {σk }0≤k≤n−1 ), where {Ak }0≤k≤n is an increasing finite sequence in S with A0 = {1} and, for each 0 ≤ k ≤ n − 1, σk : Ak+1 → Ak is an increasing surjective mapping. The cardinality of Ak is called the width of A at the height k, and the width of A at the height n is called the degree of A. The set of all trees of degree m and height n will be denoted by Trees(m, n).

5.2 Preliminaries on analytic mappings

31

The following diagram illustrates the tree of degree 7 and height 2 given by the sets A0 = {1}, A1 = {1, 2, 3}, and A2 = {1, 2, 3, 4, 5, 6, 7}, and the mappings σ0 : A1 → A0 and σ1 : A2 → A1 given by σ0 (1) = σ0 (2) = σ0 (3) = 1 and σ1 (1) = σ1 (2) = 1, σ1 (3) = σ1 (4) = 2, σ1 (5) = σ1 (6) = σ1 (7) = 3. 1

A0

σ0

2

1

A1 σ1 A2

3

1 2 3 4 5 6 7

The proof of the next lemma is quite elementary, and is therefore left to the reader. Lemma 5.2.23 Let m and n be natural numbers. If A = ({Ak }0≤k≤n+1 , {σk }0≤k≤n ) is a tree of degree m and height n + 1, if d is the width of A at the height n, and if αi is the cardinality of σn−1 (i) (1 ≤ i ≤ d), then ψ(A) := (α1 , . . . , αd ) ∈ A(d, m) and ϕ(A) := ({Ak }0≤k≤n , {σk }0≤k≤n−1 ) ∈ Trees(d, n). Moreover, the mapping A → (ψ(A), ϕ(A)) is a bijection from Trees(m, n + 1) onto

m

d=1 [A(d, m) × Trees(d, n)].

§5.2.24 For each natural number m, we will denote by A(m,1) the unique tree of degree m and height 1, which is given by the diagram A0

1

σ0 A1

1

2

... m

Definition 5.2.25 Let X be a normed space over K, let  be an open subset of X, let f :  →  be a C∞ mapping, and let x0 be in . The tree-derivatives of f at x0 are defined inductively, according to their height, as follows. For each m ∈ N, we define PA(m,1) ,f ,x0 ∈ P m (X) by PA(m,1) ,f ,x0 (h) :=

1 m D f (x0 )(hm ). m!

Assume that for some n ≥ 2, the polynomials PB,f ,x are already defined for all trees B of height n − 1 and all points x ∈ . Given A ∈ Trees(m, n), according to Lemma 5.2.23, A is uniquely determined by a natural number d, a multi-index α ∈ A(d, m), and a tree B ∈ Trees(d, n − 1), which allow us to define PA,f ,x0 ∈ P m (X) by   1 αd 1 α1 α1 αd PA,f ,x0 (h) := PB,f ,f (x0 ) D f (x0 )(h ), . . . , D f (x0 )(h ) . α1 ! αd !

32

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Proposition 5.2.26 Let X be a normed space over K, let  be an open subset of X, let f :  →  be a C∞ mapping, let x0 be in , and let n be in N. Then 1 m n D f (x0 )(hm ) = m! Proof



PA,f ,x0 (h) for every m ∈ N.

(5.2.6)

A∈Trees(m,n)

To prove (5.2.6) by induction on n, note that, by Definition 5.2.25, PA(m,1) ,f ,x0 (h) :=

1 m D f (x0 )(hm ) for every m ∈ N, m!

so that (5.2.6) holds for n = 1. Assume that (5.2.6) holds for some natural number n. Then, by Corollary 5.2.21, we have that for any m ∈ N 1 m n 1 m n+1 (x0 )(hm ) = D f D ( f ◦ f )(x0 )(hm ) m! m!   m 1 1  m = Dk f n ( f (x0 ))(Dα1 f (x0 )(hα1 ), . . . , Dαk f (x0 )(hαk )) α m! k! k=1 α∈A(k,m)   m   1 1 α1 1 αk k n α1 αk = D f ( f (x0 )) D f (x0 )(h ), . . . , D f (x0 )(h ) . k! α1 ! αk ! k=1 α∈A(k,m)

On the other hand, the induction hypothesis gives that for any k ∈ N 

1 k n D f ( f (x0 ))(hk ) = k! and hence 1 k n D f ( f (x0 )) = k!

PA, f , f (x0 ) (h),

A∈Trees(k,n)



PA, f , f (x0 ) .

A∈Trees(k,n)

It follows that 1 m n+1 (x0 )(hm ) D f m! m   =



k=1 α∈A(k,m) B∈Trees(k,n)

 PB,f ,f (x0 )

 1 αk 1 α1 D f (x0 )(hα1 ), . . . , D f (x0 )(hαk ) . α1 ! αk !

Now, invoking Lemma 5.2.23, we obtain (5.2.6) for n + 1, and the proof is complete.  5.2.2 Analytic mappings on Banach spaces §5.2.27 Let X and Y be Banach spaces over K. For x0 ∈ X, a power series centred at x0 from X to Y is a series of mappings from X to Y of the form  Pn (x − x0 ), where Pn ∈ P n (X, Y). (5.2.7) n≥0

5.2 Preliminaries on analytic mappings

33

The radius of (uniform) convergence of (5.2.7) is defined as the largest R ≤ +∞  such that the series n≥0 Pn (x − x0 ) converges uniformly for x − x0  ≤ r, whenever r < R. The power series (5.2.7) is called convergent if R > 0. The radius of restricted (uniform) convergence of (5.2.7) is defined as the largest R ≤ +∞ such that the series  Pn (x1 − x0 , . . . , xn − x0 ) n≥0

converges uniformly for every sequence xn with supn∈N xn − x0  ≤ r, whenever r < R.  Lemma 5.2.28 Let X and Y be Banach spaces over K, and let n≥0 Pn (x − x0 ) be a power series from X to Y centred at x0 ∈ X with radius of convergence R and radius of restricted convergence R. Then    n Pn r converges R = sup r ≥ 0 : n≥0

and

 R = sup r ≥ 0 : 



 Pn r converges . n

n≥0

As a consequence, the series n≥0 Pn (x − x0 ) converges absolutely for each x ∈ X  such that x − x0  < R, and the series n≥0 Pn (x1 − x0 , . . . , xn − x0 ) converges absolutely for each sequence xn with supn∈N xn − x0  < R. Proof Since both equalities have an almost identical proof, we  restrict ourselves to  show the second one. Set S = r ≥ 0 : n≥0 Pn rn converges . Given r ∈ S and any sequence xn in X such that supn∈N xn − x0  ≤ r, we see that Pn (x1 − x0 , . . . , xn − x0 ) ≤ Pn rn for every n ∈ N ∪ {0},  hence n≥0 Pn (x1 − x0 , . . . , xn − x0 ) converges uniformly for every sequence xn in X such that supn∈N xn − x0  ≤ r, and so r ≤ R. As a result sup S ≤ R. For the reverse inequality, suppose R > 0, consider 0 ≤ r < R, and fix s such that r < s < R. Then  the series n≥0 Pn (x1 − x0 , . . . , xn − x0 ) converges uniformly for all sequences xn in X such that supn∈N xn − x0  ≤ s, and consequently there exists m ∈ N such that for each n ≥ m we have Pn (x1 − x0 , . . . , xn − x0 ) ≤ 1 for all sequences xn in X such that supn∈N xn −x0  ≤ s. Therefore we have for each n ≥ m that Pn  ≤ s1n , and hence  n Pn rn ≤ rs . Thus the series n≥0 Pn rn converges, and so r ∈ S. It follows from the arbitrariness of r that R ≤ sup S, and so we have proved the equality sup S = R. Finally, given a sequence xn in X such that supn∈N xn − x0  = r < R, we have for each n ∈ N ∪ {0} that Pn (x1 − x0 , . . . , xn − x0 ) ≤ Pn rn , and so the absolute  convergence of the series n≥0 Pn (x1 − x0 , . . . , xn − x0 ) is derived from the conver gence of the series n≥0 Pn rn .  Both radii of convergence are given by the Cauchy–Hadamard formula.

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem  Proposition 5.2.29 Let X and Y be Banach spaces over K, and let n≥0 Pn (x − x0 ) be a power series from X to Y centred at x0 ∈ X with radius of convergence R and radius of restricted convergence R. Then 34

1 1 1 1 = lim sup Pn  n and = lim sup Pn  n , R R n→∞ n→∞

(5.2.8)

and we have R ≤ R ≤ eR. Proof Equalities (5.2.8) follows from Lemma 5.2.28 and the Cauchy–Hadamard formula for power series of real numbers. Taking into account that, by (5.2.5), 1

and that limn→∞

n 1

(n!) n

n

1

Pn  n ≤ Pn  n ≤ = e, we deduce that

1 R

1

Pn  n ,

(n! )

1 n



≤ e R1 , and finally R ≤ R ≤ eR. 

1 R

§5.2.30 Let X be a Banach space over K. For each n ∈ N, we have a natural identification of BLsn (K, X) with X. Indeed, the mapping F → F(1, . n. ., 1) becomes an isometric isomorphism from BLsn (K, X) onto X. As a consequence, P n (K, X) ≡ X. More precisely, each homogeneous polynomial P from K to X of degree n is of the form P(t) = xtn for suitable x ∈ X, and we have that P = P = x. Therefore  any power series from K to X centred at t0 ∈ K is of the form n≥0 xn (t − t0 )n for suitable sequence (xn )n∈N∪{0} in X, and we have as a consequence of (5.2.8) that 1 1 1 = = lim sup xn  n . R R n→∞

Proposition 5.2.31 Let X and Y be Banach spaces over K, let  be a domain in X, and let fn :  → Y be a sequence of differentiable mappings on . Suppose that: (i) There exists one point x0 ∈  such that the sequence fn (x0 ) converges in Y. (ii) For every a ∈ , there is an open ball B(a) of centre a and contained in  such that the sequence Dfn converges uniformly on B(a). Then for each a ∈ , the sequence fn converges uniformly on B(a). Moreover, if, for each x ∈ , we set f (x) = limn→∞ fn (x) and g(x) = limn→∞ Dfn (x), then g(x) = Df (x) for every x ∈ . Proof Let a be in , and let r stand for the radius of B(a). We claim that the existence of a point b ∈ B(a) such that the sequence fn (b) converges implies that the sequence fn converges uniformly on B(a). Given ε > 0, there is by assumption ε a natural number n0 such that, for n, m ≥ n0 , we have Dfn (z) − Dfm (z) ≤ 4r for ε every z ∈ B(a), and moreover  fn (b) − fm (b) ≤ 2 . Then, by Fact 5.2.13 applied to the mapping fn − fm , we have for any x ∈ B(a) that  fn (x) − fm (x) − ( fn (b) − fm (b)) ≤ x − b sup Dfn (z) − Dfm (z) ≤ 2r z∈B(a)

ε ε = , 4r 2

5.2 Preliminaries on analytic mappings

35

and consequently  fn (x) − fm (x) ≤  fn (x) − fm (x) − ( fn (b) − fm (b)) +  fn (b) − fm (b) ≤ ε. As Y is complete, this proves that fn (x) is convergent at every point of B(a), and in fact uniformly convergent on B(a). The above claim first shows that the set U of the points x such that fn (x) is a convergent sequence, is both open and closed in ; as it is not empty by assumption, and  is a domain, we conclude that U = . Secondly, it shows that, for each a ∈ , the sequence fn converges uniformly on B(a). We will finally prove that g is the derivative of f . Fix a ∈ , and let r stand for the radius of B(a). Given ε > 0, there is by assumption a natural number n0 such that, for n, m ≥ n0 , we have Dfn (z) − Dfm (z) ≤ 3ε for every z ∈ B(a), and moreover Dfn (a) − g(a) ≤ 3ε . Applying Fact 5.2.13 to the mapping fn − fm , we have for any x ∈ B(a) that ε  fn (x) − fm (x) − ( fn (a) − fm (a)) ≤ x − a sup Dfn (z) − Dfm (z) ≤ x − a, 3 z∈B(a) and letting m tend to +∞, we see that, for n ≥ n0 and x ∈ B(a), we have ε  fn (x) − fn (a) − ( f (x) − f (a)) ≤ x − a. 3 On the other hand, for any fixed n ≥ n0 , there is r ≤ r such that, for x − a ≤ r , we have  fn (x) − fn (a) − Dfn (a)(x − a) ≤ 3ε x − a. It follows that, for x − a ≤ r , we have  f (x) − f (a) − g(a)(x − a) ≤  f (x) − f (a) − ( fn (x) − fn (a)) +  fn (x) − fn (a) − Dfn (a)(x − a) + (Dfn (a) − g(a))(x − a) ε ≤ 3 x − a = εx − a, 3 which proves that Df (a) exists and is equal to g(a).



§5.2.32 Let X and Y be Banach spaces over K, let  be an open subset of X, and let f :  → Y be a mapping of class C∞ . The Taylor series of f at x0 ∈  is defined as the power series 1 Dn f (x0 )((x − x0 )n ). n! n≥0

The next result shows that the sum mapping f of a convergent power series centred at a point x0 is a C∞ mapping and the power series is uniquely determined by f and x0 . More precisely, the power series becomes the Taylor series of f at x0 .  Proposition 5.2.33 Let X and Y be Banach spaces over K, let n≥0 Pn (x − x0 ) be a convergent power series from X to Y centred at x0 ∈ X with radius of restricted convergence R, and let f : B → Y be the mapping given by the sum of the power series

36

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

on the open ball B of centre x0 and radius R. Then f is a C∞ mapping on B, and, for  n! each k ∈ N, the power series n≥k (n−k)! Pn ((x − x0 )n−k ) from X to BLsk (X, Y) has radius of restricted convergence R and its sum is the mapping Dk f : B → BLsk (X, Y). In particular, we have Dk f (x0 ) = k! Pk for each k ∈ N ∪ {0} ,

(5.2.9)

and consequently f (x) =

∞  1 n D f (x0 )((x − x0 )n ) for every x ∈ B. n! n=0

Proof It is sufficient to deal with the case k = 1 and then apply this case to successive derivatives. Note that, for each n ∈ N, the continuous symmetric n-linear mapping Fn : (x1 , . . . , xn ) → Pn+1 (x1 , . . . , xn ) from X n to BL(X, Y) (determined by the n-homogeneous polynomial x → Pn+1 (xn )) has norm equal to Pn+1 . Since 1 limn→∞ (n + 1) n = 1, it follows that 1

1

lim sup (n + 1)Fn  n = lim sup Pn  n , n→∞

n→∞

and hence, by Proposition 5.2.29, the radius of restricted convergence of the power   series n≥1 nPn ((x − x0 )n−1 ) = n≥0 (n + 1)Pn+1 ((x − x0 )n ) from X to BL(X, Y) is also R. On the other hand, by Proposition 5.2.7, each Pn is differentiable on B and DPn (x) = nPn (xn−1 ) for every x ∈ B. n It follows that the sequence of mappings k=0 Pk (x − x0 ) on B satisfies the conditions in Proposition 5.2.31, so that f is differentiable on B and  n−1) for every x ∈ B. In particular, Df (x ) = P .  Df (x) = ∞ 0 1 n=1 nPn ((x − x0 ) §5.2.34 Let X and Y be Banach spaces over K, let  be an open subset of X, and let f :  → Y be a mapping. The mapping f is said to be analytic (or more precisely, K-analytic) at a point x0 ∈  if it can be expressed as a convergent power series about x0 , which means that there is an open ball B ⊆  with centre x0 and a sequence Pn , where Pn is a homogeneous polynomial of degree n, so that f (x) =

∞ 

Pn (x − x0 ) for every x ∈ B.

(5.2.10)

n=0

The mapping f is said to be analytic whenever f is analytic at each point of . In this case, by the above proposition, f is a smooth mapping on , the power series expansion (5.2.10) about x0 is nothing but the Taylor series of f at x0 , and the derivatives Dk f :  → BLk (X, Y) are again analytic for all k ≥ 0. It is clear that the set of all analytic mappings from  to Y, with operations defined pointwise, becomes a vector space over K. Plainly, polynomials are analytic mappings (cf. Corollary 5.2.10). More generally we have

5.2 Preliminaries on analytic mappings

37 

Proposition 5.2.35 Let X and Y be Banach spaces over K, let n≥0 Pn (x − x0 ) be a convergent power series from X to Y centred at x0 ∈ X with radius of restricted convergence R, and let f : B → Y be the mapping given by the sum of the power series on the open ball B of centre x0 and radius R. Then f is an analytic mapping. Proof that

For simplicity we suppose that x0 = 0. Fix a ∈ B and 0 < r < R − a . Note ∞   ∞   m n=0 m=n

n

Pm a

r =

m−n n

=

m   ∞   m m=0 n=0 ∞ 

n

Pm am−n rn

Pm (a + r)m

m=0

and the last written series converges because of Lemma 5.2.28. From this, we get on the one hand that ∞    m Pm am−n < ∞, n m=n

 m m−n ) defines an element Q ∈ P n (X, Y) for each and hence the series ∞ n m=n ( n ) Pm (a n ∈ N ∪ {0}. On the other hand, it follows that m   ∞ ∞    m f (x) = Pm (x) = Pm (am−n , (x − a)n ) n m=0

=

m=0 n=0

∞  ∞   n=0 m=n





 m Pm (am−n , (x − a)n ) = Qn (x − a) n n=0

uniformly for x ∈ X with x − a < r. Thus f is analytic on B.



In contrast to the case K = C (cf. Theorem 5.2.60 below), in the case K = R there are mappings of class C∞ which are not analytic. As an example, consider the function f : R → R defined by ⎧ ⎨ e− 1x if x > 0 f (x) = ⎩ 0 if x ≤ 0, and note that the Taylor series of f at 0 is the null series. Proposition 5.2.36 Let X and Yi (1 ≤ i ≤ k) be Banach spaces over K, let  be an open subset of X, and let fi :  → Yi (1 ≤ i ≤ k) be analytic mappings. Then ( f1 , . . . , fk ) is an analytic mapping. Moreover, given x0 ∈ , if the power series expan sion of fi about x0 is given by fi (x0 + h) = ∞ n=0 Pi,n (h), and if for each n ∈ N ∪ {0} we consider the homogeneous polynomial Qn ∈ P n (X, Y1 × · · · × Yk ) given by Qn (h) = (P1,n (h), . . . , Pk,n (h)),

38

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

then the power series expansion of ( f1 , . . . , fk ) about x0 is given by ( f1 , . . . , fk )(x0 + h) =

∞ 

Qn (h).

n=0

Proof Let x0 ∈ , let Ri stand for the radius of convergence of the power ∞ series n=0 Pi,n , and set R = min{R1 , . . . , Rk }. By Lemma 5.2.28, the series  n P i,n h (1 ≤ i ≤ k) converges whenever h < Ri , and hence the series  n n≥0  k n≥0 i=1 Pi,n  h converges when h < R. Since we have for each n ∈ N ∪ {0} that  k   Qn (h) ≤ Pi,n  hn , i=1



it follows that the series n≥0 Qn (h) has radius of convergence ≥ R. Finally, since   1 n 1 n 1 n n Qn (h) = D f1 (x0 )(h ), . . . , D fk (x0 )(h ) = Dn ( f1 , . . . , fk )(x0 )(hn ), n! n! n! 

the statement follows from Proposition 5.2.33.

To study the analyticity of the composition of analytic mappings, we need the following result on an absolutely convergent series on K.  Lemma 5.2.37 Let k be a natural number, and let n≥0 ai,n (1 ≤ i ≤ k) be an absolutely convergent series on K. For each n ∈ N ∪ {0}, set  bn := a1,α1 a2,α2 · · · ak,αk . Then the series Proof

α∈A0 (k,n)



n≥0 bn

is absolutely convergent and

∞

n=0 bn

=

k i=1

∞

n=0 ai,n



.

For each n ∈ N ∪ {0} consider Ai,n :=

n 

ai,j (1 ≤ i ≤ k) and Bn :=

j=0

n 

bj .

j=0

Firstly suppose that ai,n ≥ 0 for all i ∈ {1, . . . , k} and n ∈ N ∪ {0}. In this case, for each n ∈ N ∪ {0}, the inclusions n 

A0 (k, j) ⊆ {0, 1, . . . , n} ⊆ k

j=0

and the equality

k

i=1 Ai,n

kn 

A0 (k, j)

j=0

=



α∈{0,1,...,n}k a1,α1 a2,α2 · · · ak,αk

Bn ≤

k 

Ai,n ≤ Bkn .

yield the inequalities (5.2.11)

i=1

Since the sequence (Ai,n )n∈N∪{0} (1 ≤ i ≤ k) is convergent, and so bounded, it follows that the increasing sequence Bn is bounded, and so converges, that is to say the

5.2 Preliminaries on analytic mappings

39



series n≥0 bn is convergent. Moreover, taking limits as n → ∞ in (5.2.11), we conclude that ∞  ∞ k    bn = ai,n . n=0

i=1

n=0

To give a proof in the general case, for each n ∈ N ∪ {0}, set  |a1,α1 ||a2,α2 | · · · |ak,αk |, vn := α∈A0 (k,n)

and consider the real numbers Ui,n :=

n 

|ai,j | (1 ≤ i ≤ k) and Vn :=

j=0

n 

vj ,

j=0

and the set Sn := {α ∈ {0, 1, . . . , n}k : |α| > n}. Then     k   k        a1,α1 · · · ak,αk  ≤ |a1,α1 | · · · |ak,αk | = Ui,n − Vn .  Ai,n − Bn  =      α∈Sn

i=1

α∈Sn

i=1

Since |bn | ≤ vn for every n ∈ N ∪ {0} and, in view of the particular  case, we have  k that the series n≥0 vn is convergent and limn→∞ Vn = 0, we deduce i=1 Ui,n −  k   that the series n≥0 bn is absolutely convergent and limn→∞ i=1 Ai,n − Bn = 0, which concludes the proof.   Corollary 5.2.38 Let k be a natural number, and let n≥0 ai,n zn (1 ≤ i ≤ k) be a power series on K with radius of convergence Ri . For each n ∈ N ∪ {0}, set  a1,α1 a2,α2 · · · ak,αk . bn := α∈A0 (k,n)

Then the power series ∞  n=0

bn z = n



n≥0 bn z

∞ k   i=1

n

has radius of convergence R ≥ min{R1 , . . . , Rk } and 

ai,n z

n

for every z with |z| < min{R1 , . . . , Rk }.

n=0

Proposition 5.2.39 Let X, Yi (1 ≤ i ≤ k), and Z be Banach spaces over K, and let  be an open subset of X. Suppose that fi :  → Yi (1 ≤ i ≤ k) is an analytic mapping, and that F ∈ BL(Y1 , . . . , Yk ; Z). Then F ◦ ( f1 , . . . , fk ) is an analytic mapping. Moreover, given x0 ∈ , if the power series expansion of fi about x0 is given by  fi (x0 + h) = ∞ n=0 Pi,n (h), and if for each n ∈ N ∪ {0} we consider the homogeneous polynomial Qn ∈ P n (X, Z) given by  F(P1,α1 (h), . . . , Pk,αk (h)), Qn (h) = α∈A0 (k,n)

40

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

then the power series expansion of F ◦ ( f1 , . . . , fk ) about x0 is given by (F ◦ ( f1 , . . . , fk ))(x0 + h) =

∞ 

Qn (h).

n=0

Let x0 ∈ , let Ri stand for the radius of convergence of the power series  P , and set R = min{R1 , . . . , Rk }. By Lemma 5.2.28, the series n≥0 Pi,n hn i,n n=0 (1 ≤ i ≤ k) converges whenever h < Ri , and hence, by Corollary 5.2.38, the series   n n≥0 α∈A0 (k,n) P1,α1  · · · Pk,αk h converges when h < R. Since we have for each n ∈ N ∪ {0} that  Qn (h) ≤ F P1,α1  · · · Pk,αk hn ,

Proof ∞

α∈A0 (k,n)

it follows that the series by Corollary 5.2.20,



n≥0 Qn (h)

Qn (h) =

has radius of convergence ≥ R. Finally, since

1 n D (F ◦ ( f1 , . . . , fk ))(x0 )(hn ), n! 

the statement follows from Proposition 5.2.33.

Proposition 5.2.40 Let X, Y, and Z be Banach spaces over K, let X and Y be open subsets of X and Y, respectively, and let f : X → Y and g : Y → Z be analytic mappings such that f (X ) ⊆ Y . Then g ◦ f is an analytic mapping. Moreover, if for x0 ∈ X the power series expansion of f about x0 is given by f (x0 + h) =

∞ 

Pn (h),

n=0

and if for each n ∈ N we consider the homogeneous polynomial Qn ∈ P n (X, Z) given by Qn (h) :=

n  1  Dk g( f (x0 ))(Pα1 (h), . . . , Pαk (h)), k! k=1

α∈A(k,n)

then the power series expansion of g ◦ f about x0 is given by (g ◦ f )(x0 + h) = g( f (x0 )) +

∞ 

Qn (h).

n=1

Proof

According to the definition of Qn we see that n   1 k Qn (h) ≤ D g( f (x0 )) Pα1  · · · Pαk hn , k! k=1

α∈A(k,n)

5.2 Preliminaries on analytic mappings

41

hence we have for each m ∈ N that m 

n m    1 k Qn (h) ≤ Pα1  · · · Pαk hn D g( f (x0 )) k! n=1 n=1 k=1 α∈A(k,n) ⎞ ⎛ m m  1  ⎝ = Pα1  · · · Pαk ⎠ hn , Dk g( f (x0 )) k! n=k

k=1

α∈A(k,n)

and so m  n=1

⎞ ⎛ m m    1 k ⎝ Qn (h) ≤ Pα1  · · · Pαk ⎠ hn . (5.2.12) D g( f (x0 )) k! k=1

n=1

α∈A(k,n)

 Let Rf be the radius of convergence of the power series n≥0 Pn . According to Lemma 5.2.28 we can consider the function σ : [0, Rf [→ R defined by  n σ (r) = ∞ n=1 Pn r . Note that, as a consequence of Corollary 5.2.38, the inequality (5.2.12) gives that m  n=1

m  1 k Qn (h) ≤ D g( f (x0 ))σ (h)k when h < Rf . k!

(5.2.13)

k=1

Let Rg be the radius of restricted convergence of the power series expansion of g about f (x0 ). Since σ is continuous and σ (0) = 0, there exists r0 ∈]0, Rf [ such that  1 σ (r0 ) ∈ [0, Rg [, and hence, by Lemma 5.2.28, the series n≥0 n! Dn g( f (x0 ))σ (r0 )n  converges. It follows from (5.2.13) that the power series n≥1 Qn (h) has radius of 1 n D (g ◦ f )(x0 )(hn ), convergence ≥ r0 . Finally, since by Corollary 5.2.21, Qn (h) = n! the statement follows from Proposition 5.2.33.  Now we deal with the powers of an analytic mapping. Proposition 5.2.41 Let X be a Banach space over K, let  be an open subset of X, let f :  →  be an analytic mapping, and let n be in N. Then f n is analytic, and the power series expansion of f about x0 ∈  is given by f n (x0 + h) = f n (x0 ) +

∞ 



PA,f ,x0 (h).

(5.2.14)

m=1 A∈Trees(m,n)

Proof By Proposition 5.2.40, f n is analytic. Now, the statement follows from the Taylor expansion of f n at x0 , by keeping in mind Proposition 5.2.26.  We also recall the concept of directional derivative. §5.2.42 Let X and Y be Banach spaces over K, let  be an open subset of X, and let f :  → Y be a mapping. Given x0 ∈  and x ∈ X, the Y-valued function fx : t → f (x0 + tx) is defined on an open interval of R containing 0, and f is said to be differentiable at x0 in the direction x whenever fx is differentiable at 0. In this case, the derivative of f at x0 in the direction x, denoted by Df (x0 ; x) is defined as

42

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

the derivative of fx at 0. If f is differentiable at x0 , then the directional derivative Df (x0 ; x) is defined for any direction x ∈ X and Df (x0 ; x) = Df (x0 )(x). An important global property of analytic mappings is specified by the so-called principle of analytic continuation. Proposition 5.2.43 Let X and Y be Banach spaces over K, let  be a domain in X, let f :  → Y be an analytic mapping, and let Z be a real subspace of X such that KZ is dense in X. Suppose there exists a non-empty open subset U of  ∩ Z such that f (U) = 0. Then f = 0. Proof Since U is an open subset of Z and f (U) = 0, we have for any (x0 , x) ∈ U × Z that the derivative of f at x0 in the direction x, and hence Df (x0 )(x), vanishes. By the arbitrariness of (x0 , x) in U × Z, and the density of KZ in X, we deduce that Df (U) = 0. Now, an induction argument gives that Dn f (U) = 0 for every n ∈ N, and hence U is contained in the set A := {x ∈  : Dn f (x) = 0 for every n ∈ N ∪ {0}}. It is clear that A is a closed subset of . Moreover, by Proposition 5.2.33, f vanishes in a neighbourhood of any point x0 in A, and so A is open. Therefore A = , and in particular f = 0.  As a first consequence we have the next corollary. Corollary 5.2.44 Let X and Y be complex Banach spaces, let  be a domain in X, and let f :  → Y be a real analytic mapping. If there exists an open ball B contained in  such that f|B is holomorphic, then f is holomorphic in . Proof Let x0 ∈ X be given. It is clear that the valuation mappings F → F(x0 ) and F → F(ix0 ) from BL(XR , YR ) to Y, as well as the mapping y → iy from Y to Y, are continuous real-linear mappings. Since the mapping Df :  → BL(XR , YR ) is real analytic (cf. §5.2.34), it follows from Proposition 5.2.40 that the mappings x → Df (x)(ix0 ) and x → iDf (x)(x0 ) from  to Y are real analytic, and hence the mapping h : x → Df (x)(ix0 ) − iDf (x)(x0 ) from  to Y is so. Now, since f|B is holomorphic, we have h = 0 on B, and hence, by the principle of analytic continuation, we have in fact that h = 0 on , i.e. Df (x)(ix0 ) = iDf (x)(x0 ) for every x ∈ . The arbitrariness of x0 ∈ X gives that Df (x) ∈ BL(X, Y) for every x ∈ , and hence f is holomorphic in  as required.  §5.2.45 Let X be a real Banach space. Recall that, by Lemma 1.1.97, the (projective) normed complexification XC = X ⊕ iX of X is a complex Banach space such that the natural embedding X → XC and the canonical involution  of XC are isometries, and consequently max{x, y} ≤ x + iy ≤ x + y for all x, y ∈ X.

(5.2.15)

5.2 Preliminaries on analytic mappings

43

If P is an n-homogeneous polynomial from X to any real Banach space Y, then, according to the binomial formula (5.2.2), the mapping PC : XC → YC defined by n    n m PC (u + iv) = (5.2.16) i P(un−m , vm ) m m=0

becomes the unique n-homogeneous polynomial from XC to YC extending P. Note that P ≤ PC  ≤ 2n P.

(5.2.17)

Indeed, the first inequality is a consequence of the fact that the natural embeddings X → XC and Y → YC are isometric, and that (PC )|X = P. The second one is a consequence of (5.2.15) and (5.2.16) because n    n PC (u + iv) ≤ Pun−m vm = P(u + v)n ≤ 2n Pu + ivn m m=0

for every u + iv ∈ XC .

 Lemma 5.2.46 Let X and Y be real Banach spaces. If the power series n≥0 Pn from X to Y has radius of convergence R, then the radius of convergence RC of the  1 complexified power series n≥0 (Pn )C from XC to YC satisfies 2e R ≤ RC ≤ R. Proof By (5.2.17), Pn  ≤ (Pn )C  ≤ 2n Pn  for every n ∈ N ∪ {0}, hence, by Proposition 5.2.29, R1 ≤ R1C ≤ 2 R1 ≤ 2e R1 , and the statement follows.  Combining Lemma 5.2.46 with Proposition 5.2.43 we derive the following fact, which allows to deduce many results about R-analytic mappings from the corresponding results for C-analytic mappings. Fact 5.2.47 Let X and Y be real Banach spaces, let  be an open subset of X, and let f :  → Y be an analytic mapping. Then there exist an open subset C of XC and an analytic mapping fC : C → YC such that  ⊆ C and ( fC )| = f . 5.2.3 Holomorphic mappings Holomorphic mappings from an open subset  of a complex Banach space X into a complex Banach space Y has been widely discussed in Volume 1, mainly in the development of the holomorphic functional calculus at an element of a complete normed unital associative complex algebra (cf. Section 1.3), and, more generally, of a complete normed unital non-commutative Jordan complex algebra (cf. Section 4.1). We know that the C-analytic mappings are holomorphic (cf. §5.2.34). One of the main objectives of this subsection will be to state the converse (cf. Theorem 5.2.60 below). §5.2.48 Let X and Y be complex Banach spaces, let  be an open subset of X, and let f :  → Y be a mapping. The mapping f is said to be G-analytic (or more precisely, Gˆateaux analytic) if for all x0 ∈  and x ∈ X the Y-valued mapping

44

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem fx0 ,x : ζ → f (x0 + ζ x)

is analytic on the open set x0 ,x := {ζ ∈ C : x0 + ζ x ∈ }. The mapping f is said to be weakly analytic (respectively, weakly G-analytic) if y ◦ f is analytic (respectively, G-analytic) for every y ∈ Y  . It follows from Proposition 5.2.40 that these notions are related by means of the following diagram: f analytic

⇒

⇓ f weakly analytic

f G-analytic ⇓

⇒

f weakly G-analytic

Lemma 5.2.49 Let  be an open subset of C, let Y be a complex Banach space, and let f :  → Y be a weakly analytic function. Then f is continuous. Proof Given λ0 ∈ , let us fix r > 0 such that λ0 + 2r ⊆ . Let y ∈ Y  , and let λ ∈ λ0 + r with λ = λ0 . Using the Cauchy integral formula for holomorphic functions of one complex variable [1149, Proposition IV.2.6], we can write     1 (y ◦ f )(ζ ) (y ◦ f )(ζ )   − dζ (y ◦ f )(λ) − (y ◦ f )(λ0 ) = 2πi |ζ −λ0 |=2r ζ −λ ζ − λ0  λ − λ0 (y ◦ f )(ζ ) = dζ , 2πi |ζ −λ0 |=2r (ζ − λ)(ζ − λ0 ) and hence

     f (λ) − f (λ0 )  y  ≤ 1 M 4πr = M ,   2π 2r2 λ − λ0 r

where M := sup{|(y ◦ f )(ζ )| : |ζ − λ0 | = 2r}. By the uniform boundedness principle, there exists K > 0 such that f (λ) − f (λ0 ) ≤ K for every λ ∈ λ0 + r with λ = λ0 . λ − λ0 This shows that f is continuous at λ0 .



One of the most important distinctions between the theories of real and complex analytic mappings is that the complex analytic mappings have the Cauchy integral representation. Proposition 5.2.50 Let X and Y be complex Banach spaces, let  be an open subset of X, and let f :  → Y be a weakly G-analytic mapping. Let x0 ∈ , x ∈ X, and r > 0 be such that x0 + ζ x ∈  for every ζ ∈ C with |ζ | ≤ r. Then for each λ ∈ C such that |λ| < r we have:

5.2 Preliminaries on analytic mappings

45

(i) The Cauchy integral formula 1 f (x0 + λx) = 2πi



f (x0 + ζ x) dζ . ζ −λ |ζ |=r

(5.2.18)

(ii) The series expansion f (x0 + λx) =

∞ 

cn λn , where cn =

n=0

1 2πi



f (x0 + ζ x) dζ , (5.2.19) ζ n+1 |ζ |=r

and this series has radius of convergence ≥ r. (iii) The Cauchy inequalities cn  ≤

1 sup f (x0 + ζ x) for every n ∈ N ∪ {0} . rn |ζ |=r

(5.2.20)

Proof By assumption, rBC ⊆ x0 ,x and the function fx0 ,x : x0 ,x → Y is weakly analytic. Therefore, by Lemma 5.2.49, fx0 ,x is continuous, and hence we can consider for each λ with |λ| < r  f (x0 + ζ x) dζ ∈ Y. ζ −λ |ζ |=r Fix y ∈ SY  , and consider the complex function g : x0 ,x → C defined by g(ζ ) := y ( f (x0 + ζ x)). Since g is holomorphic on an open neighbourhood of rBC , by the Cauchy integral formula for holomorphic functions of one complex variable, we have for each λ with |λ| < r that   1 g(ζ ) y ( f (x0 + ζ x)) 1 y ( f (x0 + λx)) = g(λ) = dζ = dζ 2πi |ζ |=r ζ − λ 2πi |ζ |=r ζ −λ    f (x0 + ζ x) 1 dζ . = y 2πi |ζ |=r ζ − λ Since SY  separates the points of Y, the equality (5.2.18) follows.  1 λn = ∞ For λ and ζ with |λ| < |ζ | = r we have ζ −λ n=0 ζ n+1 , and hence ∞

f (x0 + ζ x)  f (x0 + ζ x) n = λ . ζ −λ ζ n+1 n=0

Since f is bounded on the set {x0 + ζ x : |ζ | = r}, the series converges absolutely and uniformly for |ζ | = r and |λ| ≤ s < r, and hence we can integrate this series term by term to obtain   ∞   f (x0 + ζ x) f (x0 + ζ x) dζ λn dζ = ζ −λ ζ n+1 |ζ |=r |ζ |=r n=0

46

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

and this last series converges absolutely and uniformly for |λ| ≤ s, so that it has radius of convergence ≥ r. Now, an application of (5.2.18) completes the proof of assertion (ii). Finally, for each n ∈ N ∪ {0}, the Cauchy inequality (5.2.20) is a direct consequence of the basic property of boundedness of the integral and the description of cn provided by (5.2.19).  Corollary 5.2.51 Let X and Y be complex Banach spaces, let  be an open subset of X, and let f :  → Y be a mapping. Then f is G-analytic if (and only if) f is weakly G-analytic. Proof Suppose that f is weakly G-analytic. Given x0 ∈ , x ∈ X, and λ0 ∈ x0 ,x , take r > 0 such that x0 + (λ0 + ζ )x ∈  for every ζ ∈ C with |ζ | ≤ r. Applying Proposition 5.2.50(ii) to the function fx0 +λ0 x,x , we obtain a series expansion  n f (x0 + λx) = ∞ n=0 cn (λ − λ0 ) with radius of convergence greater than or equal to r. This shows that f is G-analytic.  §5.2.52 Let X be a complex Banach space. We denote by X the open unit ball of X. Now let Y be another complex Banach spaces, let  be an open subset of X, let f :  → Y be an analytic mapping, and let x0 ∈ . The radius of boundedness rb ( f , x0 ) of f at x0 is defined as the supremum of all r > 0 such that x0 + rBX ⊆  and f is bounded on x0 + rBX . Corollary 5.2.53 Let X and Y be complex Banach spaces, let  be an open subset  of X, let f :  → Y be an analytic mapping, let f (x0 + h) = ∞ n=0 Pn (h) be the power series expansion of f at x0 ∈ , and let 0 < r < rb ( f , x0 ). Then for each n ∈ N ∪ {0} we have the Cauchy integral formula for the n-differential  1 f (x0 + ζ x) dζ for every x ∈ BX , (5.2.21) Pn (x) = 2πi |ζ |=r ζ n+1 and the Cauchy inequalities for the n-differential 1  f B , rn ! n "n Dn f (x0 ) ≤  f B , r Pn  ≤

(5.2.22) (5.2.23)

and Dn f (x0 ) ≤ n!

! e "n r

 f B ,

(5.2.24)

where B := x0 + rX ⊆ , and f B := supx∈B f (x). Proof Given x ∈ BX , by the convergence of the power series expansion of f at x0 , there exists ε > 0 such that f (x0 + λx) =

∞  n=0

Pn (λx) =

∞  n=0

Pn (x)λn

5.2 Preliminaries on analytic mappings

47

for |λ| ≤ ε. Now, comparing this series expansion with the one given by (5.2.19), we deduce (5.2.21). Given x ∈# SX and 0 < s < r, by (5.2.21), we have for each n ∈ N ∪ {0} that f (x0 +ζ x) 1 1 Pn (x) = 2πi |ζ |=s ζ n+1 dζ , and hence Pn (x) ≤ sn sup|ζ |=s f (x0 + ζ x). Now, (5.2.22) follows from the arbitrariness of x in SX and s in ]0, r[. It follows from (5.2.9), (5.2.5), and (5.2.22) that ! n "n Dn f (x0 ) = n! Pn  ≤ nn Pn  ≤  f B , r proving (5.2.23). Finally, (5.2.24) follows from (5.2.23) since as a consequence of Stirling formula n  we have nn! ≤ en . Proposition 5.2.54 Let X and Y be complex Banach spaces, let  be an open subset of X, let f :  → Y be an analytic mapping, and let x0 ∈ . If  − x0 is balanced, then the Taylor series of f at x0 converges to f uniformly on a suitable neighbourhood of each compact subset of . Proof

Let K be a compact subset of . Consider the continuous mapping ϕ : C ×  → X given by ϕ(ζ , x) = x0 + ζ (x − x0 ).

Since  − x0 is balanced, it follows that ϕ(BC × ) ⊆ , and so ϕ(BC × K) is a compact subset of . Fix M > 0 such that f (ϕ(BC ×)) ⊆ MY . Then ϕ −1 ( f −1 (MY )) is an open neighbourhood of BC × K in C × , and hence there exist r > 1 and V open neighbourhood of K in  such that ϕ(rBC × V) ⊆ f −1 (MY ). Therefore the set B := {x0 + ζ (x − x0 ) : ζ ∈ rBC , x ∈ V} is contained in , and f is bounded on B. Hence we can write ∞

f (x0 + ζ (x − x0 ))  f (x0 + ζ (x − x0 )) = ζ −1 ζ n+1 n=0

and this series converges absolutely and uniformly for x ∈ V and |ζ | = r. After integrating over the circle |ζ | = r and applying the Cauchy integral formulae (5.2.18)  and (5.2.21) we conclude that f (x) = ∞ n=0 Pn (x − x0 ) and this series converges absolutely and uniformly for x ∈ V.  Theorem 5.2.55 Let X and Y be complex Banach spaces, let  be an open subset of X, let f :  → Y be an analytic mapping, and let x0 ∈ . Then rb ( f , x0 ) = min{rc ( f , x0 ), d(x0 , X \ )}, where rc ( f , x0 ) denotes the radius of convergence of the Taylor series of f at x0 , and d(x0 , X \ ) denotes the distance from x0 to the boundary of . As a consequence, if X is finite dimensional, then rb ( f , x0 ) = d(x0 , X \ ) and rc ( f , x0 ) ≥ d(x0 , X \ ).

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem  Proof Let f (x) = ∞ n=0 Pn (x − x0 ) be the power series expansion of f about x0 . Since clearly rb ( f , x0 ) ≤ d(x0 , X \ ), in order to show the inequality 48

rb ( f , x0 ) ≤ min{rc ( f , x0 ), d(x0 , X \ )},

(5.2.25)

it suffices to show that rb ( f , x0 ) ≤ rc ( f , x0 ). Let 0 ≤ r < rb ( f , x0 ). Then x0 + rBX ⊆  and f is bounded on x0 + rBX (say by M). It follows from the Cauchy inequalities (5.2.22) that Pn  ≤ M rn for every n ∈ N ∪ {0}, and an application of the Cauchy– Hadamard formula (cf. Proposition 5.2.29) gives that r ≤ rc ( f , x0 ). Letting r tend to rb ( f , x0 ), we get that rb ( f , x0 ) ≤ rc ( f , x0 ), and (5.2.25) follows. In order to prove the reverse inequality rb ( f , x0 ) ≥ min{rc ( f , x0 ), d(x0 , X \ )},

(5.2.26)

let us fix r and s with 0 ≤ r < s < min{rc ( f , x0 ), d(x0 , X \ )}. Since s < d(x0 , X \ ), it follows that x0 + sX ⊆ , and then, by Proposition 5.2.54 applied to each point in x0 + sX , we get f (x) =

∞ 

Pn (x − x0 ) for every x ∈ x0 + sX .

(5.2.27)

n=0

On the other hand, it follows from the Cauchy–Hadamard formula (cf. Proposition 1 5.2.29) that lim supn→∞ Pn  n = rc ( f1,x0 ) < 1s , and hence there exists a constant K ≥ 1 such that K Pn  < n for every n ∈ N ∪ {0}. (5.2.28) s It follows from (5.2.27) and (5.2.28) that  f (x) ≤

∞ ! r "n  K for every x ∈ x0 + rBX . s n=0

Hence rb ( f , x0 ) ≥ r, and so (5.2.26) follows. Finally, if X is finite dimensional, then each closed ball with finite radius is compact, and whence it follows that rb ( f , x0 ) = d(x0 , X \ ), and as a consequence rc ( f , x0 ) ≥ d(x0 , X \ ).  §5.2.56 A polydisc in Cn is a product of discs. Given a = (a1 , . . . , an ) ∈ Cn and r = (r1 , . . . , rn ) ∈ (R+ )n , the open and closed polydiscs with centre a and polyradius r are defined by n (a, r) := {z ∈ Cn : |zj − aj | < rj for j = 1, . . . , n} and n

 (a, r) := {z ∈ Cn : |zj − aj | ≤ rj for j = 1, . . . , n}, respectively. The torus with centre a and polyradius r is defined by T(a, r) := C(a1 , r1 ) × · · · × C(an , rn ),

5.2 Preliminaries on analytic mappings

49

where the circles C(aj , rj ) have a positive orientation when we consider the integral of a function along T(a, r). Now we can state the Cauchy integral formula for a polydisc which extends Proposition 5.2.50. Proposition 5.2.57 Let X and Y be complex Banach spaces, let  be an open subset of X, let f :  → Y be a G-analytic mapping such that the restriction of f to any finite dimensional subspace of X is continuous, and let x0 ∈ , x1 , . . . , xn ∈ X, and n r1 , . . . , rn ∈ R+ be such that x0 + ζ1 x1 + · · · + ζn xn ∈  for every ζ ∈  (0, r). Then for each λ ∈ n (0, r) we have: (i) The Cauchy integral formula f (x0 + λ1 x1 + · · · + λn xn ) =

1 (2πi)n



f (x0 + ζ1 x1 + · · · + ζn xn ) d(ζ1 , . . . , ζn ). T(0,r) (ζ1 − λ1 ) · · · (ζn − λn )

(ii) The multiple series expansion f (x0 + λ1 x1 + · · · + λn xn ) =



cα λα1 1 · · · λαn n ,

α∈(N∪{0})n

where cα =

1 (2πi)n



f (x0 + ζ1 x1 + · · · + ζn xn ) T(0,r)

ζ1α1 +1 · · · ζnαn +1

d(ζ1 , . . . , ζn ). n

This multiple series converges absolutely and uniformly for λ ∈  (0, s) provided 0 ≤ sj < rj for every j. n

Proof Since the polydisc  (0, r) is compact, we can find R1 > r1 , . . . , Rn > rn such that x0 +ζ1 x1 +· · ·+ζn xn ∈  for every ζ ∈ n (0, R). If y ∈ Y  , then the function g : n (0, R) → C given by g(ζ1 , . . . , ζn ) := y ( f (x0 + ζ1 x1 + · · · + ζn xn )) is separately holomorphic in each of the variables ζ1 , . . . , ζn when the other variables are held fixed. Then repeated applications of the Cauchy integral formula for holomorphic functions of one variable lead to the iterated integral    1 dζ1 dζ2 g(ζ1 , . . . , ζn ) ··· dζn g(λ1 , . . . , λn ) = n (2πi) |ζ1 |=r1 ζ1 − λ1 |ζ2 |=r2 ζ2 − λ2 ζn − λn |ζn |=rn for every λ ∈ n (0, r). Since the function (ζ1 , . . . , ζn ) →

g(ζ1 , . . . , ζn ) (ζ1 − λ1 )(ζ2 − λ2 ) · · · (ζn − λn )

is continuous on the compact set T(0, r), Fubini’s theorem allows us to replace the iterated integral by a multiple integral. Since g = y ◦ f and Y  separates points, we conclude the proof of assertion (i).

50

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

If |λj | < |ζj | = rj for j = 1, . . . , n, then we can write f (x0 + ζ1 x1 + · · · + ζn xn ) = (ζ1 − λ1 ) · · · (ζn − λn )



f (x0 + ζ1 x1 + · · · + ζn xn )

α∈(N∪{0})n

ζ1α1 +1 · · · ζnαn +1

λα1 1 · · · λαn n ,

and this multiple series converges absolutely and uniformly for |ζj | = rj and |λj | ≤ sj < rj . After integrating this series term to term, assertion (ii) follows from assertion (i).  Proposition 5.2.58 Let X and Y be complex Banach spaces, let  be an open subset of X, and let f :  → Y be a mapping. Then the following conditions are equivalent: (i) f is analytic. (ii) f is continuous and G-analytic. (iii) f is continuous and f|∩M is analytic for each finite dimensional subspace M of X. Proof The implication (i)⇒(ii) is clear. Suppose that f is continuous and G-analytic. Let M be a finite dimensional subspace of X, let x0 ∈ ∩M, and let e1 , . . . , en be a basis for M. Then, by Proposition 5.2.57, we have a series expansion of the form  cα λα1 1 · · · λαn n , f (x0 + λ1 e1 + · · · + λn en ) = α∈(N∪{0})n

where this multiple series converges absolutely and uniformly on a suitable polydisc n (0, r). If for each m ∈ N ∪ {0} we define Pm ∈ P m (M, Y) by  cα λα1 1 · · · λαn n , Pm (λ1 e1 + · · · + λn en ) = α∈A0 (n,m)

then we have a power series expansion f (x0 + λ1 e1 + · · · + λn en ) =

∞ 

Pm (λ1 e1 + · · · + λn en )

m=0

with uniform convergence on n (0, r). This shows (ii)⇒(iii). Now, suppose that f is continuous and that f|∩M is analytic for each finite dimensional subspace M of X. Fix x0 ∈  and 0 < r < d(x0 , X \ ), and let B stand for the open ball of centre x0 and radius r. If M is a finite dimensional subspace of X containing x0 , then, by hypothesis, f|∩M is analytic, and then, by Theorem 5.2.55,  the Taylor series of f|∩M at x0 , say n≥0 PM n (x − x0 ), has radius of convergence > r, and in particular f (x) =

∞ 

PM n (x − x0 ) for every x ∈ B ∩ M.

n=0

If M and N are two finite dimensional subspaces of X containing x0 , then the uniqueN ness of the Taylor series expansion gives that PM n (x) = Pn (x) for all x ∈ M ∩ N

5.2 Preliminaries on analytic mappings

51

and n ∈ N ∪ {0}. Let Pn : X → Y be defined by Pn (x) = PM n (x) if M is any finite dimensional subspace of X containing x0 and x. Therefore f (x) =

∞ 

Pn (x − x0 ) for every x ∈ B.

n=0

Now, since f is continuous, we can find s < r and a positive constant K such that  f (x) ≤ K for every x ∈ x0 + sBX . Given x ∈ BX , let M be any finite dimensional subspace of X containing x0 and x. Then by the Cauchy inequality (5.2.22) we get that K Pn (x) = PM n (x) ≤ n . s Note that Pn (x) = Pn (x, . n. ., x) for every x ∈ X, where Pn : X× . n. . ×X → Y is the symmetric n-linear mapping given by Pn via the polarization formula (5.2.4), hence Pn is continuous, and so Pn ∈ P n (X, Y) and Pn  ≤ sKn . Now, by Cauchy–Hadamard  formula (5.2.8), we deduce that the power series n≥0 Pn (x − x0 ) has a radius of convergence greater than or equal to s. Thus we have proved the implication (iii)⇒(i), and the proof of the statement is complete.  Proposition 5.2.59 Let X and Y be complex Banach spaces, let  be an open subset of X, and let f :  → Y be a G-analytic mapping. Then f is continuous if (and only if) f is locally bounded. Proof Suppose that f is locally bounded. Given x0 ∈ , choose constants r > 0 and K > 0 such that x0 + rX ⊆  and  f (x) < K for every x ∈ x0 + rX . Given x ∈ X with x = r, it is clear that x ⊆ rX , and so  ⊆ x0 ,x . Since f is G-analytic, it follows that the function g :  → Y defined by g(ζ ) = f (x0 + ζ x) − f (x0 ) is analytic. Moreover, it is clear that g(ζ ) < 2K for every ζ ∈ . Given y ∈ SY  , we have as a consequence that the function h:ζ →

1  y ( f (x0 + ζ x) − f (x0 )) 2K

is a holomorphic function from  to  such that h(0) = 0. By the classical Schwarz’s lemma (see Lemma 5.3.42 below), we deduce that |h(ζ )| ≤ |ζ | for every ζ ∈ . Therefore we have for each ζ ∈  that |y ( f (x0 + ζ x) − f (x0 ))| ≤ 2K|ζ |, and an application of the Hahn–Banach theorem gives that  f (x0 + ζ x) − f (x0 ) ≤ 2K|ζ |. As a consequence, for each x ∈ X \ {0} with x < r we obtain that   r x − f (x0 ) ≤ 2K|ζ | for every ζ ∈ , f x0 + ζ x 2K and taking ζ = x r we conclude that  f (x0 + x) − f (x0 ) ≤ r x, proving that f is continuous at x0 . Since x0 is arbitrary in , we conclude that f is continuous. 

As we have already remarked, analytic mappings are of class C∞ . The next result tells us the converse when K = C.

52

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Theorem 5.2.60 Let X and Y be complex Banach spaces, let  be an open subset of X, and let f :  → Y be a mapping. Then f is holomorphic if and only if it is analytic. Proof Suppose that f is holomorphic. Then, for each x0 ∈ , x ∈ X, and y ∈ Y  , the set x0 ,x = {ζ ∈ C : x0 + ζ x ∈ } is open in C, and, by the chain rule, the function g : x0 ,x → C defined by g(ζ ) = y ( f (x0 + ζ x)) is holomorphic, and hence analytic by the corresponding result for functions of one complex variable. Therefore f is weakly G-analytic, and hence, by Corollary 5.2.51, f is G-analytic. Since clearly f is continuous, it follows from Proposition 5.2.58 that f is analytic. The reverse implication was established in Proposition 5.2.33.  From here on, we will not distinguish between the terms analytic and holomorphic; we usually speak of holomorphic mapping. We also extend this convention to the notions introduced in §5.2.48. For complete the information about these notions we will include the famous Dunford theorem. Proposition 5.2.61 Let X and Y be complex Banach spaces, let  be an open subset of X, and let f :  → Y be a mapping. Then f is holomorphic if (and only if) f is weakly holomorphic. Proof Suppose that f is weakly holomorphic. Then f is weakly G-holomorphic, and hence G-holomorphic by Corollary 5.2.51. We claim that f is locally bounded. To show this, let K be a compact subset of . Since f is weakly holomorphic, the set f (K) is weakly bounded, and hence norm bounded by the uniform boundedness principle. Thus f is bounded on each compact subset of , and consequently f is locally bounded. Indeed, if there exists some point x0 in , such that f is not bounded on any neighbourhood of x0 , then for each n ∈ N there exists xn ∈ (x0 + 1n X ) ∩  such that  f (xn ) > n, and hence f is not bounded on the compact set {xn : n ∈ N ∪ {0}}, a contradiction. Now that the claim is proved, by Proposition 5.2.59, f is continuous, and, invoking Proposition 5.2.58, we conclude that f is holomorphic.  Now, we derive Liouville’s theorem as a consequence of (5.2.22). Fact 5.2.62 Let X and Y be complex Banach spaces, and let f : X → Y be a mapping. Suppose that f is holomorphic on the entire space X and bounded. Then f is constant. Proof Let M be a positive constant such that  f (x) ≤ M for every x ∈ X, and let  f (x) = ∞ n=0 Pn (x) be the power series expansion of f about 0. For each n ∈ N and δ ∈ R+ , it follows from (5.2.22) that Pn  ≤ δ1n M. Hence Pn  = 0 for every n ∈ N, and thus, f = P0 is constant.  Finally, to close this subsection, we include the inverse mapping theorem, referring the reader to [814, Theorem 1.23] for a proof.

5.3 Holomorphic automorphisms of a bounded domain

53

Theorem 5.2.63 Let X and Y be Banach spaces over K, let  be an open subset of X, let x0 be in , and let f :  → Y be an analytic mapping such that Df (x0 ) is a continuous bijective linear mapping from X to Y. Then there exist open neighbourhoods U ⊆  of x0 and V of f (x0 ) such that f : U → V is bianalytic. 5.2.4 Historical notes and comments The material in this section is basic in the classical theory of analytic mappings between Banach spaces, and has been elaborated mainly from the books of Cartan [1146], Chae [707], Chaperon [1147], Coleman [1148], Dieudonn´e [1152], Dineen [1154, 1155], Franzoni and Vesentini [1159], Isidro and Stach´o [751], Lelong-Ferrand and Arnaudi`es [1172], Mujica [1175], and Upmeier [814, 815]. For the properties of holomorphic mappings between locally convex spaces the reader is referred to the books of Dineen [1154, 1155] and Herv´e [1160]. We also encourage the reader to consult the final notes of the chapters of the book [1155] to get extensive information about the history of the infinite dimensional holomorphy. 5.3 Holomorphic automorphisms of a bounded domain Introduction In this section we begin the study of the set of all holomorphic automorphisms of a bounded domain in a complex Banach space, which becomes a metrizable topological group for the topology of the local uniform convergence. This topology appears as a natural generalization to infinite dimension of the compact open topology, and is studied in Subsection 5.3.1. Subsection 5.3.2 revolves around the Cartan linearity theorem (Proposition 5.3.24) and the Cartan uniqueness theorem (Proposition 5.3.25), providing several topological and metric improvements of these results. Subsection 5.3.3 deals with the Carath´eodory distance on a bounded domain, which becomes a natural generalization of the Poincar´e distance on the open unit disc in the complex plane. For any complex Banach space X, the Carath´eodory distance on X is complete (Proposition 5.3.60). Finally, the completeness of the group of automorphisms of a bounded domain is discussed in Theorem 5.3.61 and Examples 5.3.62 and 5.3.63. 5.3.1 The topology of the local uniform convergence §5.3.1 Let E be a topological space and let Y be a normed space over K. In agreement with the notation in Example 1.1.4(d), we will denote by C(E, Y) the vector space over K of all continuous mappings from E to Y. We recall that the compact open topology or the topology of the uniform convergence on compact sets is the locally convex topology τc on C(E, Y) which is generated by the seminorms of the form  f K := sup  f (x), x∈K

54

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

where K varies among all the compact subsets of E. The subspace Cb (E, Y) of C(E, Y), consisting of all bounded continuous functions from E to Y, becomes a normed space over K under the sup norm  f E := sup  f (x). x∈E

Proposition 5.3.2 Let Y be a Banach space over K. We have: (i) If E is a topological space, then (Cb (E, Y),  · E ) is a Banach space. (ii) If E is a metric space, then (C(E, Y), τc ) is a complete locally convex space. Proof Suppose that E is a topological space, and that fn is a Cauchy sequence in (Cb (E, Y),  · E ). Then, for each x ∈ E, the sequence fn (x) is a Cauchy sequence in Y. As Y is complete, we can consider the function f : E → Y defined by f (x) = lim fn (x). n→∞

Given ε > 0, there is by assumption a natural number n0 such that, for n, m ≥ n0 , we have fn (x) − fm (x) ≤ ε for every x ∈ E. Letting m tend to +∞, we see that, for n ≥ n0 and x ∈ E, we have  fn (x) − f (x) ≤ ε. As a consequence  f (x) ≤  fn0 E + ε for every x ∈ E, and hence f is bounded. On the other hand, for any fixed x0 ∈ E, it follows from the continuity of fn0 that there exists a neighbourhood V of x0 in E such that  fn0 (x0 ) − fn0 (x) ≤ ε for every x ∈ V. Therefore, for each x ∈ V, we have  f (x0 ) − f (x) ≤  f (x0 ) − fn0 (x0 ) +  fn0 (x0 ) − fn0 (x) +  fn0 (x) − f (x) ≤ 3ε. It follows from the arbitrariness of ε > 0 and x0 in E that f ∈ Cb (E, Y), and so fn  · E -converges to f . Suppose that E is a metric space, and fλ is a Cauchy net in (C(E, Y), τc ). Then we have that the net ( fλ )|K is  · K -Cauchy for each compact K ⊆ E. Since, by assertion (i), (C(K, Y),  · K ) is complete, a continuous uniform limit fK : K → Y exists for each K. It is easily seen that if K1 ⊆ K2 , then ( fK2 )|K1 = fK1 , and from this it follows that the function f : E → Y defined by f (x) = fK (x) for x ∈ K, is well defined. Given x ∈ E and a sequence xn in E converging to x, by considering the compact set K = {xn : n ∈ N} ∪ {x}, the continuity of fK gives that f (xn ) converges to f (x). Thus f is continuous in E, and fλ converges to f uniformly on each compact subset K of E.  §5.3.3 Let X and Y be complex Banach spaces, and let  be an open subset of X. We will denote by H (, Y) the subspace of C(, Y) consisting of all holomorphic mappings from  to Y, and we will define Hb (, Y) := H (, Y) ∩ Cb (, Y). Proposition 5.3.4 Let X and Y be complex Banach spaces, and let  be an open subset of X. Then:

5.3 Holomorphic automorphisms of a bounded domain

55

(i) H (, Y) is a closed subspace of (C(, Y), τc ). In particular (H (, Y), τc ) is a complete locally convex space. (ii) Hb (, Y) is a closed subspace of (Cb (, Y),  ·  ). In particular (Hb (, Y),  ·  ) is a Banach space. Proof Let fλ be a net in H (, Y) which τc -converges to a mapping f ∈ C(, Y). Given x0 ∈ , x ∈ X, and y ∈ Y  , set hλ (ζ ) = y ( fλ (x0 + ζ x)) and h(ζ ) = y ( f (x0 + ζ x)) for every ζ ∈ x0 ,x , where x0 ,x := {ζ ∈ C : x0 + ζ x ∈ }. Then hλ is holomorphic on the open subset x0 ,x of C and the net hλ converges to h uniformly on compact sets of x0 ,x . By the theorem of Weierstrass for holomorphic functions of one complex variable [1149, Theorem 2.1], the function h is holomorphic on x0 ,x . Since x0 is arbitrary in , we have proved that f is weakly G-holomorphic. Now, it follows from Corollary 5.2.51 and Proposition 5.2.58 that f ∈ H (, Y). Thus H (, Y) is a closed subspace of (C(, Y), τc ). The last assertion in (i) follows from Proposition 5.3.2(ii). Now, let fλ be a net in Hb (, Y) which  ·  -converges to f ∈ Cb (, Y). It is clear that fλ τc -converges to f , and hence, by (i), f ∈ Hb (, Y). Thus Hb (, Y) is a closed subspace of (Cb (, Y),  ·  ). The last assertion in (ii) follows from Proposition 5.3.2(i).  In complex context, Proposition 5.2.35 can be improved as follows.  Corollary 5.3.5 Let X and Y be complex Banach spaces, let n≥0 Pn (x − x0 ) be a convergent power series from X to Y centred at x0 ∈ X with radius of convergence R, and let f : B → Y be the mapping given by the sum of the power series on the open ball B of centre x0 and radius R. Then f is a holomorphic mapping.  Proof Since the power series n≥0 Pn (x − x0 ) converges uniformly for x − x0  < r whenever r < R, the mapping f is continuous. Since the mappings x → Pn (x − x0 ) lie in H (B, Y), assertion (i) in the above proposition implies that f ∈ H (B, Y).  §5.3.6 Let X be a complex Banach space, and let  be an open subset of X. We will say that a non-empty subset K of  lies strictly inside  or is completely interior to , a fact that will be denoted by K  , whenever there exists δ > 0 such that Kδ := K + δX ⊆ , where (according to §5.2.52) X denotes the open unit ball of X. §5.3.7 Let X and Y be complex Banach spaces, and let  be an open subset of X. Then each open ball B contained in  induces a seminorm .B defined by  f B := sup f (x) x∈B

of H (, Y) consisting of all holomorphic mappings on the subspace from  to Y which are bounded on B. If  is a domain in X, then, in light of the Hb (B, Y)

56

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

principle of analytic continuation (Proposition 5.2.43),  · B is in fact a norm on Hb (B, Y), which can be seen naturally as a subspace of the Banach space Hb (B, Y) (cf. Proposition 5.3.4(ii)). Let H0 (, Y) be the subspace of H (, Y) consisting of all f ∈ H (, Y) which are bounded on each open ball B strictly inside , i.e. satisfying rb ( f , x) = d(x, X \ ) for every x ∈  (cf. §5.2.52), or equivalently, in light of Theorem 5.2.55, rc ( f , x) ≥ d(x, X \ ) for every x ∈ . The space H0 (, Y) becomes a locally convex space when it is endowed with the topology of local uniform convergence (or briefly, T-topology): the topology defined by the family of seminorms ·B , where B ranges over all open balls strictly inside . Note that P(X, Y) can be seen in a natural way as a subspace of H0 (, Y), and that the T-topology on P(X, Y) is nothing other than the norm topology (cf. §5.2.8). In a more general sense, we will say that a net fλ in H (, Y) T-converges to f ∈ H (, Y) if for each open ball B   there exists λ0 such that fλ − f ∈ Hb (B, Y) for every λ ≥ λ0 and lim  fλ − f B = 0. λ≥λ0

We write in this case f = T- limλ fλ . Since each compact subset of  is contained in a finite union of open balls strictly inside , it follows that the T-topology is stronger than the compact open topology. Moreover, both topologies coincide if and only if dim(X) < ∞. As a straightforward consequence of Proposition 5.3.4 we have the following. Corollary 5.3.8 Let X and Y be complex Banach spaces, and let  be an open subset of X. Then H0 (, Y) is complete with respect to the T-topology. Proof Let fλ be a T-Cauchy net in H0 (, Y). Then fλ is a τc -Cauchy net in H (, Y), and, by Proposition 5.3.4(i), there exists f ∈ H (, Y) such that fλ τc -converges to f . Moreover, given any open ball B  , it follows from Proposition 5.3.4(ii) that ( fλ )|B  · B -converges to f|B ∈ Hb (B, Y). Thus f ∈ H0 (, Y), and fλ T-converges to f .  Cauchy inequalities allow relate T-convergence with pointwise convergence for the n-differentials. Fact 5.3.9 Let X and Y be complex Banach spaces, let  be an open subset of X, let fλ be a net in H (, Y), and let f be in H (, Y). If fλ T-converges to f , then for each n ∈ N ∪ {0} the net Dn fλ pointwise converges to Dn f . Proof Fix x ∈ , choose r with 0 < r < d(x, X \ ), and set B := x + rX . If the net fλ T-converges to f , then there exists λ0 such that fλ − f ∈ Hb (B, Y) for every λ ≥ λ0 and limλ≥λ0  fλ − f B = 0. For any n ∈ N ∪ {0}, it follows from (5.2.23) that

5.3 Holomorphic automorphisms of a bounded domain 57 ! n "n Dn fλ (x) − Dn f (x) = Dn ( fλ − f )(x) ≤  fλ − f B for every λ ≥ λ0 , r  and consequently limλ Dn fλ (x) − Dn f (x) = 0, as required. Proposition 5.3.10 Let X and Y be complex Banach spaces, let  be an open subset of X, and let B be an open ball strictly inside  with centre x0 . Suppose that fλ is a T-bounded net in H0 (, Y), and that f ∈ H0 (, Y). Then the following assertions are equivalent: (i) fλ converges uniformly to f on B. (ii) For each n ∈ N ∪ {0}, the net Dn fλ (x0 ) converges to Dn f (x0 ) in BLn (X, Y). Proof The implication (i)⇒(ii) follows from Fact 5.3.9 applied to the net ( fλ )|B and the mapping f|B . To prove the implication (ii)⇒(i), consider the power series expansion of the mappings fλ and f about x0 : fλ (x0 + h) =

∞ 

Pλ,n (h) and f (x0 + h) =

n=0

∞ 

Pn (h).

n=0

If r stands for the radius of B, then each one of these power series has radius of convergence > r (by Theorem 5.2.55), and we have for each λ and each h ∈ X with h < r that  fλ (x0 + h) − f (x0 + h) =

∞  (Pλ,n − Pn )(h) n=0



∞ 

Pλ,n − Pn hn ≤

n=0

Fix δ > r such that

B

∞ 

Pλ,n − Pn rn .

n=0

:= x0 + δX  . By (5.2.22) we have for each λ that

Pλ,n − Pn  ≤

1  δ n  fλ − f B

for every n ∈ N ∪ {0},

and consequently ∞ 

Pλ,n − Pn rn ≤  fλ − f B

n=0

∞ ! "  r n n=0

δ

.

Given ε > 0, since fλ is a T-bounded net in H0 (, Y) and f ∈ H0 (, Y), the set { fλ − f B : λ} is bounded, and so we can find n0 ∈ N such that  fλ − f 

B

∞ ! "  r n ε < for every λ. δ 2

n=n0 +1

On the other hand, by (ii), we can also find λ0 such that for every λ ≥ λ0 we have n0  n=0

Pλ,n − Pn rn ≤

n0  1 ε Dn fλ (x0 ) − Dn f (x0 )rn < . n! 2 n=0

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

It follows that  fλ (x0 + h) − f (x0 + h) < ε for every λ ≥ λ0 . Now, from the arbitrariness of h ∈ X with h < r, we conclude that  fλ − f B ≤ ε.  Let X and Y be complex Banach spaces, and let  be a domain in X. Our next goal is to show that the topology induced on T-bounded subsets of H0 (, Y) by any two open balls strictly inside  is the same. To this end we need the following lemma. Lemma 5.3.11 Let X be a complex Banach space, let  be a domain in X, and let x and y be in . Then there exist r > 0 and x = x0 , x1 , . . . , xn = y in  such that xk+1 ∈ xk + rX   for every k ∈ {0, 1, . . . , n − 1}. Proof Since  is arcwise connected, there is a continuous function γ : [0, 1] →  such that γ (0) = x and γ (1) = y. By compactness of γ ([0, 1]), we can find ρ > 0 such that γ ([0, 1]) + ρX ⊆ . Since γ is uniformly continuous, there exists δ > 0 such that ρ for all s, t ∈ [0, 1] with |s − t| < δ. γ (s) − γ (t) < 2 Now, fix a partition P = {0 = t0 < t1 < . . . < tn = 1} of [0, 1] such that tk −tk−1 < δ for every k ∈ {1, . . . , n}. Then by taking xk := γ (tk ) (0 ≤ k ≤ n) and r := ρ2 , the statement follows.  Proposition 5.3.12 Let X and Y be complex Banach spaces, let  be a domain in X, let F be a T-bounded subset of H0 (, Y), and let B and C be two open balls strictly inside . Then for every ε > 0 there exists δ > 0 such that f C ≤ ε whenever f ∈ F with f B ≤ δ.

(5.3.1)

Proof Suppose first that the centre u of C belongs to B. If r is the radius of C, then we can choose r1 with 0 < r1 < r such that u + r1 X ⊆ B. Fix d > 0 such that C +dX ⊆ , and put r2 := r + d2 . Given f ∈ F , v ∈ X , and y ∈ SY  , by Hadamard’s three circles theorem [1149, Theorem VI.3.13] applied to the holomorphic function ζ → y ( f (u + ζ v)) from (r + d) to C, we obtain that r 1

log r2

M(r)

r2

≤ M(r1 )log r M(r2 )

log rr

1

,

where, for each s with r1 ≤ s ≤ r2 , M(s) := sup{|y ( f (u + ζ v))| : |ζ | = s}. It is clear that M(r1 ) ≤ sup{f (u + ζ v) : |ζ | = r1 } ≤ f u+r1 X ≤ f B . Moreover, if K > 0 is such that hu+r2 X ≤ K for every h ∈ F , then we have M(r2 ) ≤ sup{f (u + ζ v) : |ζ | = r2 } ≤ f u+r2 X ≤ K.

5.3 Holomorphic automorphisms of a bounded domain

59

Therefore p

M(r) ≤ f B K q , where p =

log rr1 log rr2 and q = . log rr21 log rr21

Now, given x ∈ C, taking v = 1r (x − u) and y ∈ SY  such that f (x) = |y ( f (x))| (by p the Hahn–Banach theorem), it follows that f (x) ≤ f B K q . Therefore p

 f C ≤ f B K q , and consequently (5.3.1) holds if B contains the centre u of C. In the general case, by Lemma 5.3.11, there exist r > 0 and x0 , x1 , . . . , xn in  such that x0 is the centre of B, xn is the centre of C, and xk+1 ∈ xk + rX   for every k ∈ {0, 1, . . . , n − 1}. Then by taking B0 := B, Bk := xk−1 + rX (1 ≤ k ≤ n + 1), Bn+2 = C we obtain a chain of open balls strictly inside  starting in B and ending in C, and with the property that each one of them contains the centre of the next one. Thus the statement follows from the above considered particular case.  Corollary 5.3.13 Let X and Y be complex Banach spaces, let  be a domain in X, and let B be an open ball strictly inside . Then we have: (i) If fλ is a T-bounded net in H0 (, Y), which is ·B -Cauchy, then fλ is T-Cauchy. (ii) If F be a T-bounded subset of H0 (, Y), then the topology on F induced by the T-topology is metrizable. More precisely, the mapping ( f , g) →  f − gB is a distance on F whose associated topology coincides with the one induced by the T-topology. Proof Let fλ be a T-bounded net in H0 (, Y), which is  · B -Cauchy. Then the set {fλ − fμ : λ, μ running over the index set} is T-bounded in H0 (, Y), and an application of Proposition 5.3.12 gives that fλ is  · C -Cauchy for any open ball C strictly inside , that is fλ is T-Cauchy. Let F be a T-bounded subset of H0 (, Y). If fλ is a net in F that  · B -converges to f ∈ F , then, by (i), fλ is T-Cauchy in H0 (, Y), and hence, by Corollary 5.3.8, fλ T-converges to f . Thus the topologies induced on F by the  · B -topology and the T-topology agree.  Combining Proposition 5.3.10 and Corollary 5.3.13(ii), we derive the following infinite dimensional version of Vitali’s theorem. Theorem 5.3.14 Let X and Y be complex Banach spaces, let  be a domain in X, let fλ be a T-bounded net in H0 (, Y), and let f ∈ H0 (, Y). Then the following assertions are equivalent: (i) fλ converges to f in the T-topology. (ii) There exists an open ball B   such that fλ converges to f uniformly on B.

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

(iii) There exists x0 ∈  such that, for each n ∈ N ∪ {0}, the net Dn fλ (x0 ) converges to Dn f (x0 ) in BLn (X, Y). (iv) For each x0 ∈  and for each n ∈ N ∪ {0}, the net Dn fλ (x0 ) converges to Dn f (x0 ) in BLn (X, Y). As a consequence of Corollary 5.2.53 and the mean value theorem (Fact 5.2.13) we have the following. Proposition 5.3.15 Let X and Y be complex Banach spaces, let  be an open subset of X, let x0 , x1 ∈  and δ > 0 be such that [x0 , x1 ] + δX ⊆ , and let f :  → Y be a bounded holomorphic mapping. Then, for each n ∈ N ∪ {0}, we have   n + 1 n+1 Dn f (x1 ) − Dn f (x0 ) ≤  f  x1 − x0 . (5.3.2) δ Proof Let us fix n ∈ N ∪ {0}. By the mean value theorem applied to the mapping Dn f :  → BLn (X, Y) we have Dn f (x1 ) − Dn f (x0 ) ≤ x1 − x0  sup Dn+1 f (x0 + t(x1 − x0 )). 0≤t≤1

For each t ∈ [0, 1], it follows from (5.2.23) that for any 0 < r < δ   n + 1 n+1 n+1 D f (x0 + t(x1 − x0 )) ≤  f  , r and hence we have D

 n+1

f (x0 + t(x1 − x0 )) ≤

Now, we can realize that (5.3.2) holds.

n+1 δ

n+1  f  . 

Corollary 5.3.16 Let X and Y be complex Banach spaces, let  be an open subset of X, let A be a convex set strictly inside , and let δ > 0 be such that Aδ ⊆ . Suppose that f :  → Y is a bounded holomorphic mapping. Then, for each n ∈ N ∪ {0}, the ! "n+1 mapping Dn f is Lipschitz on A with Lipschitz constant n+1  f  . δ Lemma 5.3.17 Let X and Y be complex Banach spaces, let  be a domain in X, let xλ be a net in  converging to x0 ∈ , let fλ be a T-bounded net in H0 (, Y), and let f ∈ H (, Y). If for some n ∈ N ∪ {0} the net Dn fλ (x0 ) converges to Dn f (x0 ), then the net Dn fλ (xλ ) converges to Dn f (x0 ). In particular, if T- lim fλ = f , then Dn fλ (xλ ) converges to Dn f (x0 ) for every n ∈ N ∪ {0}. Proof Suppose that n ∈ N ∪ {0} is such that the net Dn fλ (x0 ) converges to Dn f (x0 ). Fix δ > 0 such that x0 + δX  , and take M > 0 such that  fλ x0 +δX ≤ M for every λ. For each x ∈ B := x0 + 2δ X , we have that [x0 , x] + 2δ X ⊆ x0 + δX , and hence, by Proposition 5.3.15, we have for each λ that   2(n + 1) n+1 Dn fλ (x) − Dn fλ (x0 ) ≤ Mx − x0 . δ

5.3 Holomorphic automorphisms of a bounded domain

61

Since xλ converges to x0 , there exists λ0 such that xλ ∈ B for every λ ≥ λ0 , and we have the inequalities Dn fλ (xλ ) − Dn f (x0 ) ≤ Dn fλ (xλ ) − Dn fλ (x0 ) + Dn fλ (x0 ) − Dn f (x0 )   2(n + 1) n+1 Mxλ − x0  + Dn fλ (x0 ) − Dn f (x0 ), ≤ δ which allow us to conclude that Dn fλ (xλ ) converges to Dn f (x0 ). In the case that T- lim fλ = f , invoking Fact 5.3.9, it follows from the above that Dn fλ (xλ ) converges to Dn f (x0 ) for every n ∈ N ∪ {0}.  Proposition 5.3.18 Let X, Y, and Z be complex Banach spaces, let X and Y be domains in X and Y, respectively, and let fλ and gλ be T-bounded nets in H0 (X , Y) and H0 (Y , Z), respectively, such that T- lim fλ = f ∈ H0 (X , Y) and T- lim gλ = g ∈ H0 (Y , Z). If fλ (X ) ⊆ Y for every λ, if the net gλ ◦ fλ is T-bounded in H0 (X , Z), if f (X ) ⊆ Y , and if g ◦ f ∈ H0 (X , Z) then T- lim(gλ ◦ fλ ) = g ◦ f . Proof Suppose that fλ (X ) ⊆ Y for every λ, that the net gλ ◦ fλ is T-bounded in H0 (X , Z), that f (X ) ⊆ Y , and that g ◦ f ∈ H0 (X , Z). Let us fix x0 ∈ X arbitrarily, and set y0 := f (x0 ) and yλ := fλ (x0 ) for every λ. By Fact 5.3.9, we find that Dn fλ (x0 ) → Dn f (x0 ) and Dn gλ (y0 ) → Dn g(y0 ) for every n ∈ N ∪ {0}, and applying Lemma 5.3.17 we get that Dn gλ (yλ ) → Dn g(y0 ) for every n ∈ N ∪ {0}. Thus, for n = 0, we have (gλ ◦ fλ )(x0 ) → (g ◦ f )(x0 ), and, for each n ∈ N, keeping in mind Corollary 5.2.21, we realize that Dn (gλ ◦ fλ )(x0 ) → Dn (g ◦ f )(x0 ). Finally, invoking Theorem 5.3.14 we conclude that gλ ◦ fλ converges to g ◦ f in the T-topology.  5.3.2 Holomorphic automorphisms of a bounded domain §5.3.19 Let X be a complex Banach space, and let  be a domain in X, we denote by Aut() the group ( for the composition) of all biholomorphic mappings from  onto . Fact 5.3.20 Let X be a complex Banach space, and let  be a bounded domain in X. Then Aut() is a T-bounded subset of H0 (, X) and, for any open ball B strictly inside , the topology induced on Aut() by the distance associated to  · B is precisely the topology on Aut() induced by the T-topology.

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Proof Since  is bounded, it follows that Aut() ⊆ BHb (B,X) for every open ball B  , and hence Aut() is a T-bounded subset of H0 (, X). Now, given an open ball B  , Corollary 5.3.13(ii) gives that the topology on Aut() induced by the distance associated to  · B agrees with the one induced by the T-topology.  Lemma 5.3.21 Let X be a complex Banach space, let  be a bounded domain in X, and let B be an open ball strictly inside . Then there exists k > 0 such that  f − gB ≤ kg−1 ◦ f − I B for all f , g ∈ Aut(). Proof Let f and g be in Aut(), and let d be the distance from B to the boundary of . Consider any x ∈ B, and set y := g−1 ( f (x)). If y lies in B d , then [x, y] + d2 X ⊆ , 2 and hence, by Proposition 5.3.15, we have 2  f (x) − g(x) = g(y) − g(x) ≤ g y − x d 2 2 ≤ y − x ≤ g−1 ◦ f − I B . d d On the other hand, if y ∈ / B d , then obviously y − x ≥ 2

d 2

whence

2 2  f (x) − g(x) ≤ diam() ≤ diam()y − x ≤ diam()g−1 ◦ f − I B . d d Thus, by the arbitrariness of x in B, we obtain  f − gB ≤

2 max{, diam()}g−1 ◦ f − I B , d

and the statement is proved.



Theorem 5.3.22 Let X be complex Banach space, and let  be a bounded domain in X. Then Aut() is a topological group for the T-topology acting continuously on . Proof The composition operation is continuous in Aut() with regard to the T-topology because of Proposition 5.3.18. In order to prove the T-continuity of the operation f → f −1 , fix an open ball B strictly inside  and take a sequence fn in Aut()  · B -converging to f ∈ Aut(). By Lemma 5.3.21, there exists a positive constant k > 0 such that  f −1 − fn−1 B ≤ k fn ◦ f −1 − I B for every n ∈ N. Since the T-continuity of the composition operation gives that the sequence fn ◦ f −1  · B -converges to I , it follows that fn−1  · B -converges to f −1 . Now, invoking Fact 5.3.20, we conclude that the mapping f → f −1 is continuous in Aut() with regard to the T-topology. Finally, the continuity of the mapping ( f , x) → f (x) from Aut() ×  to  follows from Lemma 5.3.17.  Lemma 5.3.23 Let X be a normed space over K, let  be an open subset of X, let f :  →  be an m times differentiable mapping on  for some m ∈ N with m ≥ 2, and let x0 be in . Suppose that

5.3 Holomorphic automorphisms of a bounded domain

63

(11 ) f (x0 ) = x0 , (21 ) Df (x0 ) = IX , and (31 ) Dk f (x0 ) = 0 for k = 2, . . . , m − 1. Then for each n ∈ N we have (1n ) f n (x0 ) = x0 , (2n ) Df n (x0 ) = IX , (3n ) Dk f n (x0 ) = 0 for k = 2, . . . , m − 1, and (4n ) Dm f n (x0 ) = nDm f (x0 ). Proof We proceed by induction on n. By assumption, the result is true for n = 1. Assume that the result holds for some natural number n. By (11 ) and (1n ), we have f n+1 (x0 ) = f ( f n (x0 )) = f (x0 ) = x0 , and hence (1n+1 ) holds. Using the chain rule, we derive from (1n ), (21 ) and (2n ) that Df n+1 (x0 ) = D( f ◦ f n )(x0 )) = Df (x0 ) ◦ Df n (x0 ) = Ix ◦ IX = IX , and hence (2n+1 ) holds. In order to prove (3n+1 ) and (4n+1 ) we will apply the Fa`a di Bruno formula (Corollary 5.2.21). Given k with 2 ≤ k ≤ m − 1, we have for each h ∈ X that Dk f n+1 (x0 )(hk ) = Dk ( f ◦ f n )(x0 )(hk )   k  1  k = Dj f ( f n (x0 ))(Dα1 f n (x0 )(hα1 ), . . . , Dαj f n (x0 )(hαj )). α j! j=1

α∈A(j,k)

It follows from (1n ) and (31 ) that Dj f ( f n (x0 )) = 0 for each j with 2 ≤ j ≤ k, and hence all corresponding summands are zero. Since A(1, k) = {(k)}, the summand corresponding to j = 1 is nothing but Df ( f n (x0 ))(Dk f n (x0 )(hk )), which is also equal to zero because of (3n ). Thus Dk f n+1 (x0 ) = 0, and hence (3n+1 ) holds. Finally, we have for each h ∈ X that Dm f n+1(x0 )(hm) = Dm ( f ◦ f n )(x0 )(hm )   m  1  m j n = D f ( f (x0 ))(Dα1 f n(x0 )(hα1), . . . , Dαj f n (x0 )(hαj )). α j! j=1

α∈A(j,m)

As above, it follows from (1n ) and (31 ) that all summands corresponding to j with 2 ≤ j ≤ m − 1 are zero. Since A(1, m) = {(m)}, the summand corresponding to j = 1 is nothing but Df ( f n (x0 ))(Dm f n (x0 )(hm )), which is equal to nDm f (x0 )(hm ) because of (1n ), (21 ), and (4n ). Since A(m, m) = {(1, .m. ., 1)}, the summand corresponding to j = m is no other than Dm f ( f n (x0 ))(Df n (x0 )(h), .m. ., Df n (x0 )(h)), which is equal to Dm f (x0 )(hm ) because of (1n ) and (2n ). Thus Dm f n+1 (x0 ) = (n + 1)Dm f (x0 ), and hence (4n+1 ) holds.  The next result is know as Cartan’s linearity theorem, and can be seen as a generalized classical Schwartz lemma (see Lemma 5.3.42 below).

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Proposition 5.3.24 Let X be a complex Banach space, let  be a bounded domain in X, let f :  →  be a holomorphic mapping, and let x0 be in . Suppose that f (x0 ) = x0 and Df (x0 ) = IX . Then f = I . Proof Without loss of generality we may suppose that  is a neighbourhood of 0 and x0 = 0. Suppose f = I . Since, by assumption f (0) = 0 and Df (0) = IX , there exists a natural number m ≥ 2 such that D2 f (0) = · · · = Dm−1 f (0) = 0 and Dm f (0) = 0. Therefore the Taylor series of f about 0 is  f (x) = x + Pm (x) + Pk (x), k>m

where Pk ∈ P k (X, X) for k ≥ m and Pm = 0. It follows from Lemma 5.3.23 that, for each n ∈ N, the Taylor series of f n about 0 is given by  Qn,k (x) f n (x) = x + nPm (x) + k>m

for suitable Qn,k ∈ we have

P k (X, X)

for k > m. Now, by the Cauchy’s inequalities (5.2.22),

1 n f  for every n ∈ N, dm where d is the distance from 0 to the boundary of . Since  is bounded, we get nPm  ≤

 for every n ∈ N. ndm Letting n → +∞ we see that Pm = 0. This contradiction completes the proof. Pm  ≤



The next consequence of Cartan’s linearity theorem is know as Cartan’s uniqueness theorem for Aut(). Proposition 5.3.25 Let X be a complex Banach space, let  be a bounded domain in X, and let x0 be in . Suppose that f , g ∈ Aut() satisfy f (x0 ) = g(x0 ) and Df (x0 ) = Dg(x0 ). Then f = g. Proof

Define h := g−1 ◦ f . Then h(x0 ) = g−1 ( f (x0 )) = g−1 (g(x0 )) = x0

and Dh(x0 ) = Dg−1 ( f (x0 )) ◦ Df (x0 ) = Dg−1 (g(x0 )) ◦ Dg(x0 ) = IX . Thus, from Proposition 5.3.24, we have h = I , which implies f = g.



Cartan’s uniqueness theorem states that, given a point x0 in a bounded domain  of a complex Banach space X, the mapping Tx0 : Aut() → X ⊕ BL(X) defined by Tx0 ( f ) := ( f (x0 ), Df (x0 )) is injective. Our next goal will be to study the topological properties of this mapping (cf. Theorem 5.3.30 below).

5.3 Holomorphic automorphisms of a bounded domain

65

Lemma 5.3.26 Let A = ({Ak }0≤k≤n , {σk }0≤k≤n−1 ) be a tree of degree m and height n. Put dk for the cardinality of Ak (0 ≤ k ≤ n − 1) and αk,j for the cardinality of σk−1 (j) (0 ≤ k ≤ n − 1, 1 ≤ j ≤ dk ). Then there exists a positive constant K such that for each complex Banach space X, each domain  in X, each holomorphic mapping f :  → , and each x0 ∈  we have PA,f ,x0  ≤ K

dk n−1 

Dαk,j f ( f n−(k+1) (x0 )).

k=0 j=1

Proof By Lemma 5.2.23, A is uniquely determined by the multi-index and the tree given by α = (αn−1,1 , . . . , αn−1,dn−1 ) and B = ({Ak }0≤k≤n−1 , {σk }0≤k≤n−2 ). Keeping in mind Definition 5.2.25 and the inequalities (5.2.5), we realize that there exists a positive constant K1 , which only depends on A, such that 

dn−1

PA,f ,x0  ≤ K1 PB,f ,f (x0 ) 

Dαn−1,j f (x0 ).

j=1



An inductive argument concludes the proof.

§5.3.27 For each n ∈ N, we will denote by A(1,n) the unique tree of degree 1 and height n. Note that, for n ≥ 2, according to Lemma 5.2.23, the multi-index α = (1) and the tree A(1,n−1) determine the tree A(1,n) , and consequently, for f and x0 as in Definition 5.2.25, we have PA(1,n) ,f ,x0 = PA(1,n−1) ,f ,f (x0 ) ◦ Df (x0 ), and so, by a finite induction PA(1,n) ,f ,x0 = Df ( f n−1 (x0 )) ◦ Df ( f n−2 (x0 )) ◦ · · · ◦ Df ( f (x0 )) ◦ Df (x0 ). Given k, m, n ∈ N with 1 ≤ k ≤ n, we will denote by A(k;m,n) the unique tree of degree m and height n satisfying A0 = A1 = · · · = Ak−1 ⊆ Ak = Ak+1 = · · · = An . ...

A(1; m, n)

A(2; m, n)

1

1

1

1

1

A0

A(n; m, n)

σ0 1

2

...

m

A2

1

2

...

m

1

2

...

m

1

An−1

1

2

...

m

1

2

...

m

1

1

2

...

m

1

2

...

m

A1 σ1

σn−1 An

1

2

...

m

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Note that, for n ≥ 2, according to Lemma 5.2.23, A(k;m,n) is determined by the multi-index and the tree given by $ α = (1, .m. ., 1) and A(k;m,n−1) if k < n, α = (m)

and A(1,n−1)

if k = n.

Therefore, for f and x0 as in Definition 5.2.25, we have $ PA(k;m,n−1) ,f ,f (x0 ) ◦ Df (x0 ) PA(k;m,n) ,f ,x0 = PA(1,n−1) ,f ,f (x0 ) ◦ PA(m,1) ,f ,x0

if k < n, if k = n,

and so, by a finite induction PA(k;m,n) ,f ,x0 = Df ( f n−1 (x0 )) ◦ Df ( f n−2 (x0 )) ◦ · · · ◦ Df ( f n−(k−1) (x0 )) ◦ PA(m,1) ,f ,f n−k (x0 ) ◦ Df ( f n−(k+1) (x0 )) ◦ · · · ◦ Df ( f (x0 )) ◦ Df (x0 ). Next result can be seen as an approximate version of Lemma 5.3.23. Proposition 5.3.28 Let X be a complex Banach space, let  be a bounded domain in X, let fλ :  →  be a net of holomorphic mappings, let x0 be in , and let m be in N with m ≥ 2. Suppose that (i) fλ (x0 ) → x0 , (ii) Dfλ (x0 ) → IX , and (iii) Dk fλ (x0 ) → 0 for k = 2, . . . , m − 1. Then Dm fλn (x0 ) − nDm fλ (x0 ) → 0 for every n ∈ N. Proof We will begin by noticing that, in light of Lemma 5.3.17, the assumptions (i)–(iii) imply the truthfulness of the conditions obtained once that x0 is replaced in p (i)–(iii) by fλ (x0 ) with p ∈ N ∪ {0}. Thus, as a first application of Lemma 5.3.17 (by taking there xλ = fλ (x0 ), f the constantly x0 function, and n = 0), it follows from (i) that fλ2 (x0 ) → x0 , and an inductive argument gives that in fact p

fλ (x0 ) → x0 for every p ∈ N ∪ {0}.

(5.3.3) p

Moreover, as a second application of Lemma 5.3.17 (now, taking there xλ = fλ (x0 ), f = I , and n = 1), it follows from (ii) and (5.3.3) that p

Dfλ ( fλ (x0 )) → IX for every p ∈ N ∪ {0}.

(5.3.4) p

Finally, as a third application of Lemma 5.3.17 (in this case, taking there xλ = fλ (x0 ), f = I , and n = k), we derive from (iii) and (5.3.3) that p

Dk fλ ( fλ (x0 )) → 0 for all p ∈ N ∪ {0} and 2 ≤ k ≤ m − 1.

(5.3.5)

Fix n ∈ N, and A ∈ Trees(m, n). Keeping the notation introduced in the statement of Lemma 5.3.26, suppose that there exist k0 , j0 with 0 ≤ k0 ≤ n − 1 and 1 ≤ j0 ≤ dk0 such that 2 ≤ αk0 ,j0 ≤ m − 1. Given δ ∈]0, 1] such that x0 + δX ⊆ , by (5.3.3), there

5.3 Holomorphic automorphisms of a bounded domain

67

p

exists λ0 such that, for all λ ≥ λ0 and p = 1, . . . , m, we have fλ (x0 ) ∈ x0 + 2δ X , hence p fλ (x0 ) + 2δ X ⊆ , and so, by (5.2.23),  q 2q p  for every q ∈ N. (5.3.6) Dq fλ ( fλ (x0 )) ≤ δ Since the bounds in (5.3.6) are independent of λ, it follows from (5.3.5) and Lemma 5.3.26 that PA,fλ ,x0 → 0.

(5.3.7)

Now, suppose that αk,j = 1 or m for all 0 ≤ k ≤ n − 1 and 1 ≤ j ≤ dk . Then, there exists k with 1 ≤ k ≤ n such that A = A(k;m,n) , and, by §5.3.27, we have for every λ that PA(k;m,n) ,fλ ,x0 = Dfλ ( fλn−1 (x0 )) ◦ · · · ◦ Dfλ ( fλn−(k−1) (x0 )) ◦ PA

n−k (m,1) ,fλ ,fλ (x0 )

n−(k+1)

◦ Dfλ ( fλ

(x0 )) ◦ · · · ◦ Dfλ (x0 ),

and hence, by (5.3.4), lim(PA(k;m,n) ,fλ ,x0 − PA

n−k (m,1) ,fλ ,fλ (x0 )

λ

) = 0.

(5.3.8)

Moreover, given δ ∈]0, 1] such that x0 + δX ⊆ , by (5.3.3), there exists λ0 such p that, for all λ ≥ λ0 and p = 0, . . . , n − 1, we have fλ (x0 ) ∈ x0 + 2δ X , hence the line p p segment [x0 , fλ (x0 )] verifies [x0 , fλ (x0 )] + 2δ X ⊆ , and so, by Proposition 5.3.15, 

p

Dm fλ ( fλ (x0 )) − Dm fλ (x0 ) ≤

2(m + 1) δ

m+1

p

 fλ (x0 ) − x0 .

  p Therefore, by (5.3.3), limλ Dm fλ ( fλ (x0 )) − Dm fλ (x0 ) = 0, hence lim(PA

n−k (m,1) ,fλ ,fλ (x0 )

λ

− PA(m,1) ,fλ ,x0 ) = 0 for every k with 1 ≤ k ≤ n,

so, by (5.3.8), lim(PA(k;m,n) ,fλ ,x0 − PA(m,1) ,fλ ,x0 ) = 0 for every k with 1 ≤ k ≤ n, λ

and finally lim λ

& 1 % m n D fλ (x0 ) − nDm fλ (x0 ) = lim[PA(m,1) ,fλn ,x0 − nPA(m,1) ,fλ ,x0 ] λ m! ' ( n  PA(k;m,n) ,fλ ,x0 . = lim PA(m,1) ,fλn ,x0 − λ

k=1

Since, by Proposition 5.2.41, PA(m,1) fλn ,x0 =

 A∈Trees(m,n)

PA,fλ ,x0 ,

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

it follows from (5.3.7) that lim λ

& 1 % m n D fλ (x0 ) − nDm fλ (x0 ) = lim λ m!



PA,fλ ,x0 = 0,

A∈Trees(m,n)\{A(k;m,n) :1≤k≤n}



and the proof is complete.

Corollary 5.3.29 Let X be a complex Banach space, let  be a bounded domain in X, let fλ :  →  be a net of holomorphic mappings, and let x0 be in . If fλ (x0 ) → x0 , and Dfλ (x0 ) → IX , then fλ converges to I in the T-topology. Proof Suppose that fλ (x0 ) → x0 , Dfλ (x0 ) → IX , and fλ not converge to I in the T-topology. Then, by Theorem 5.3.14, there exists m ∈ N with m ≥ 2 such that Dk fλ (x0 ) → 0 for k = 2, . . . , m − 1, and Dm fλ (x0 )  0. By passing to a subnet if necessary, we may suppose that there exists ε > 0 such that Dm fλ (x0 ) ≥ ε for every λ. Given n ∈ N, by Proposition 5.3.28, there exists λ such that Dm fλn (x0 ) − nDm fλ (x0 ) < n 2ε . Therefore ε n ≤ nDm fλ (x0 ) − Dm fλn (x0 ) − nDm fλ (x0 ) ≤ Dm fλn (x0 ). 2 Now, fix δ > 0 so that the open ball B of centre x0 and radius δ is strictly inside , m and note that the Cauchy estimate (5.2.24) give that n 2ε ≤ m! δe . Given the arbitrariness of n, we find a contradiction, and the proof is complete.  Theorem 5.3.30 Let X be a complex Banach space, let  be a bounded domain in X, and let x0 be in . Then the mapping Tx0 : Aut() → X ⊕ BL(X) defined by Tx0 ( f ) := ( f (x0 ), Df (x0 )) is a homeomorphism from Aut(), endowed with the T-topology, into X ⊕ BL(X) endowed with the topology underlying its Banach space structure. Proof The mapping Tx0 is injective because of Proposition 5.3.25. Keeping in mind Corollary 5.3.13(ii), in order to prove that Tx0 is bicontinuous, we must show that, for fn sequence in Aut() and f ∈ Aut(), the following conditions are equivalent: (i) fn converges to f in the T-topology. (ii) fn (x0 ) → f (x0 ), and Dfn (x0 ) → Df (x0 ). The implication (i)⇒(ii) follows from Fact 5.3.9. Suppose that assertion (ii) holds. Then ( f −1 ◦ fn )(x0 ) = f −1 ( fn (x0 )) → f −1 ( f (x0 )) = x0 , and the chain rule gives that D( f −1 ◦ fn )(x0 ) = Df −1 ( fn (x0 )) ◦ Dfn (x0 ) converges to Df −1 ( f (x0 )) ◦ Df (x0 ) = D( f −1 ◦ f )(x0 ) = DI (x0 ) = IX . Therefore, by Corollary 5.3.29, f −1 ◦ fn converge to I in the T-topology. Finally, by Theorem 5.3.22, fn = f ◦ ( f −1 ◦ fn ) converge to f in the T-topology. 

5.3 Holomorphic automorphisms of a bounded domain

69

Given a bounded domain  in a complex Banach space, a point x0 ∈ , and an open ball B  , it follows from Theorem 5.3.30 and Corollary 5.3.13 that, for each ε > 0 there exists δ > 0 such that for any f ∈ Aut() we have max{ f (x0 ) − x0 , Df (x0 ) − IX } < δ ⇒  f − I B < ε. Our next goal is to provide a stronger result (cf. Proposition 5.3.35 below). Lemma 5.3.31 Let X be a complex Banach space, let  be a bounded domain in X, let f :  →  be a holomorphic mapping, let B be an open ball strictly inside , and let δ and d be so that 0 < δ < d ≤ d(B, X \ ). Suppose that x ∈ B is such that f (x) ∈ Bδ . Then for each natural number p we have  1  f k − I Bd .  f (x) − x d−δ

(5.3.9)

p−1  f (x) − x max  f k − I Bd . 1≤k≤p−1 d−δ

(5.3.10)

p−1

 f p (x) − x − p( f (x) − x) ≤

k=1

In particular,  f p (x) − x − p( f (x) − x) ≤

Proof Since f (x) ∈ Bδ , it follows that [x, f (x)] + (d − δ)X ⊆ Bd . By Proposition 5.3.15, we have for any k ∈ N that  f k+1 (x) − f (x) − ( f k (x) − x) = ( f k − I )( f (x)) − ( f k − I )(x) 1  f k − I Bd  f (x) − x. ≤ d−δ By noticing that for any natural number p f p (x) − x − p( f (x) − x) =

p−1 ) *  f k+1 (x) − f (x) − ( f k (x) − x) , k=1

the inequality (5.3.9) follows from the triangle inequality, while the inequality (5.3.10) is a straightforward consequence of the one (5.3.9).  Lemma 5.3.32 Let X be a complex Banach space, let  be a bounded domain in X, let B be an open ball strictly inside , and let d be so that 0 < d ≤ d(B, X \ ). Then there exists a positive constant K such that for any holomorphic mapping f :  → , any natural number p, and any x ∈ B, we have  f p (x) − x − p( f (x) − x) ≤ Kp f (x) − x max  f k − I Bd . 1≤k≤p−1

(5.3.11)

Proof Let f :  →  be a holomorphic mapping, let p be in N, and let x be in B. Choose δ so that 0 < δ < d. In the case that f (x) ∈ Bδ , by the inequality (5.3.10) in Lemma 5.3.31, we have  f p (x) − x − p( f (x) − x) ≤

1 p f (x) − x max  f k − I Bd . 1≤k≤p−1 d−δ

(5.3.12)

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Suppose that f (x) ∈ / Bδ . Then  f (x) − x ≥ δ, hence  f (x) − x max  f k − I Bd ≥  f (x) − x f − I Bd ≥  f (x) − x2 ≥ δ 2 , 1≤k≤p−1

and so 1  f (x) − x max  f k − I Bd ≥ 1. 1≤k≤p−1 δ2 On the other hand,  f p (x) − x − p( f (x) − x) ≤  f p − I  + p f − I  ≤ (p + 1)diam() ≤ 2p diam(). It follows that 2p diam()  f (x) − x max  f k − I Bd . (5.3.13) 1≤k≤p−1 δ2 + , 1 2 diam() Now, taking K = max d−δ , δ2 , the inequality (5.3.11) follows from the ones (5.3.12) and (5.3.13).   f p (x) − x − p( f (x) − x) ≤

Proposition 5.3.33 Let X be a complex Banach space, let  be a bounded domain in X, let B be an open ball strictly inside , let d be so that 0 < d ≤ d(B, X \), and let u and v be real numbers so that 0 < u < 1 < v. Then there exists a positive constant α such that for any holomorphic mapping f :  →  and any natural number p satisfying  f k − I Bd < α for every k with 0 ≤ k ≤ p − 1,

(5.3.14)

we have u f p (x) − x ≤ p f (x) − x ≤ v f p (x) − x for every x ∈ B. Proof Let K be a positive constant satisfying the conclusion in Lemma 5.3.32, and set   1 1 1 α := min 1 − , − 1 . K v u For any holomorphic mapping f :  →  and any natural number p satisfying (5.3.14), and for any x ∈ B we see that  f p (x) − x − p( f (x) − x) ≤ Kαp f (x) − x, hence  f p (x) − x ≤  f p (x) − x − p( f (x) − x) + p f (x) − x ≤ (1 + Kα)p f (x) − x 1 ≤ p f (x) − x u

5.3 Holomorphic automorphisms of a bounded domain

71

and  f p (x) − x ≥ p f (x) − x −  f p (x) − x − p( f (x) − x) ≥ (1 − Kα)p f (x) − x 1 ≥ p f (x) − x, v 

as required.

Corollary 5.3.34 Let X be a complex Banach space, let  be a bounded domain in X, and let C be an open ball contained in . Then there exists a positive constant β with the following property: If f :  →  is a holomorphic mapping satisfying  f p − I C < β for every p ∈ N, then f = I . Proof If C has centre x0 and radius r, then we consider the open ball B centred in x0 with radius 2r . Fix real numbers u and v so that 0 < u < 1 < v, and consider the positive constant α provided by Proposition 5.3.33 for B, d = 2r , u, and v. If f :  →  is a holomorphic mapping satisfying  f p − I C < α for every p ∈ N, then we have p f − I B ≤ v f p − I B ≤ v f p − I C < vα for every p ∈ N, hence  f − I B = 0, and so f = I by the principle of analytic continuation (Proposition 5.2.43).  Proposition 5.3.35 Let X be a complex Banach space, let  be a bounded domain in X, and let x0 ∈ . Then, for each open ball B   centred at x0 , there exists a constant K such that for any holomorphic mapping f :  →  we have  f − I B ≤ K max{ f (x0 ) − x0 , Df (x0 ) − IX }. Proof We proceed by contradiction. Assume that there exists an open ball B   centred at x0 and a sequence fn of holomorphic mappings from  to  such that  fn − I B > n max{ fn (x0 ) − x0 , Dfn (x0 ) − IX }.

(5.3.15)

Since  fn − I B ≤ diam(), we have max{ fn (x0 ) − x0 , Dfn (x0 ) − IX } → 0, hence, by Corollary 5.3.29, T-lim fn = I , and so  fn − I B → 0. Let us fix d, u and v such that 0 < d ≤ d(B, X \ ) and 0 < u < 1 < v. By Proposition 5.3.33 and Corollary 5.3.34, there exists a positive constant α satisfying: (i) For any holomorphic mapping f :  →  and any natural number p such that  f k − I Bd < α for every k with 0 ≤ k ≤ p − 1, we have u f p (x) − x ≤ p f (x) − x ≤ v f p (x) − x for every x ∈ B.

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

(ii) If f :  →  is a holomorphic mapping satisfying  f p − I Bd < α for every p ∈ N, then f = I . Since, by (5.3.15), fn = I for each n ∈ N, according to (ii), we can define   pn := min p ∈ N :  fnp − I Bd ≥ α . p

p

By definition of pn , we have  fn n − I Bd ≥ α, and hence fn n  I with respect to the norm  · Bd . However, we shall show that p

p

fn n (x0 ) → x0 and Dfn n (x0 ) → IX which is a contradiction by Corollary 5.3.29. Fix n, p ∈ N with p ≤ pn . As an application of (i) we obtain that u fnp − I B ≤ p fn − I B ≤ v fnp − I B ,

(5.3.16)

and hence p

p≤v

diam()  fn − I B ≤v .  fn − I B  fn − I B

On the other hand, as a new application of (i) we obtain that u fnp (x0 ) − x0  ≤ p fn (x0 ) − x0  ≤ v fnp (x0 ) − x0 , and hence p  fnp (x0 ) − x0  ≤  fn (x0 ) − x0 . u It follows that  fnp (x0 ) − x0  ≤

v diam()  fn (x0 ) − x0  . u  fn − I B

Since, by (5.3.15),  fn (x0 ) − x0  ≤ 1n  fn − I B , we conclude that  fnp (x0 ) − x0  ≤

v diam() 1 . u n

(5.3.17)

p

In particular,  fn n (x0 ) − x0  → 0. p Now, let us consider the sequence Dfn n (x0 ). By the chain rule we have for each n ∈ N that Dfnpn (x0 ) = Dfn ( fnpn −1 (x0 )) ◦ · · · ◦ Dfn (x0 ) =

pn 

(IX + Hn,k ),

k=1

where Hn,k = Dfn ( fnk−1 (x0 )) − IX for k = 1, . . . , pn .

5.3 Holomorphic automorphisms of a bounded domain

73

Therefore Dfnpn (x0 ) − IX  =

pn 



Hn,k

∅=J⊆{1,...,pn } k∈J

k=1





(IX + Hn,k ) − IX = 



Hn,k  =

∅=J⊆{1,...,pn } k∈J

pn 

(1 + Hn,k ) − 1.

k=1

We must prove that the right-hand side tends to 0. To do this, let us denote by r the radius of B and choose n0 ∈ N such that v diam() 1 r < for every n ≥ n0 . u n 2 Then, by (5.3.17), we have for all n ≥ n0 and k = 1, . . . , pn that fnk−1 (x0 ) lies in B := x0 + 2r X , and so [x0 , fnk−1 (x0 )] + 2r X ⊆ B. Therefore, by Proposition 5.3.15, we see that Hn,k  = Dfn ( fnk−1 (x0 )) − IX  = D( fn − I )( fnk−1 (x0 )) − D( fn − I )(x0 ) + D( fn − I )(x0 ) ≤ D( fn − I )( fnk−1 (x0 )) − D( fn − I )(x0 ) + D( fn − I )(x0 )  2 4 ≤  fn − I B  fnk−1 (x0 ) − x0  + Dfn (x0 ) − IX , r and hence, by (5.3.17) and (5.3.15),  2 4 v diam() 1 1  fn − I B +  fn − I B . Hn,k  ≤ r u n n ! "2 Setting γ = 4r v diam() + 1 (a constant independent of n and k), we have u Hn,k  ≤ γ

 fn − I B for all n ≥ n0 and k = 1, . . . , pn . n

Therefore pn  k=1



 fn − I B (1 + Hn,k ) ≤ 1 + γ n

 pn

for every n ≥ n0 .

Note that, by (5.3.16), p

pn

 fn − I B v fn n − I B v diam() ≤ ≤ → 0. n n n

= 1, it follows that Since limx→0 log(1+x) x    fn − I B  fn − I B lim pn log 1 + γ = lim pn γ = 0, n→∞ n→∞ n n

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

and hence

" !    f −I   fn − I B pn pn log 1+γ n n B = lim e = 1. lim 1 + γ n→∞ n→∞ μn

Now, the inequalities 0≤ show that

 pn

pn  k=1

   fn − I B pn (1 + Hn,k ) − 1 ≤ 1 + γ −1 n

k=1 (1 + Hn,k ) − 1 → 0,

and the proof is complete.



Theorem 5.3.36 Let X be a complex Banach space, let  be a bounded domain in X, let x0 be a point in , and let B be an open ball strictly inside . Then there are an open neighbourhood U of x0 in  and a positive constant K such that for any holomorphic mapping f :  →  and any point a in U we have  f − I B ≤ K max{ f (a) − a, Df (a) − IX }. Proof By simplicity, for any holomorphic mapping f :  →  and any point x in , we will set mx ( f ) := max{ f (x) − x, Df (x) − IX }. Suppose at first that B is centred at x0 and has radius r. By Proposition 5.3.35, there exists a positive constant K0 such that for any holomorphic mapping f :  →  we have  f − I B ≤ K0 mx0 ( f ).

(5.3.18)

Fix δ with 0 < δ < r, and set C := x0 + δX . Let f :  →  be a holomorphic mapping. Since C + (r − δ)X ⊆ B, it follows from Corollary 5.3.16 that the mappings 1 f − I and D( f − I ) are Lipschitz on C with Lipschitz constants r−δ  f − I B  2 2 and r−δ  f − I B , respectively, which, in view of (5.3.18), can be replaced by K0 4K0 r−δ mx0 ( f ) and (r−δ)2 mx0 ( f ). Therefore we have for each a ∈ C that  f (a) − a ≥  f (x0 ) − x0  −

K0 mx ( f )x0 − a r−δ 0

and Df (a) − IX  ≥ Df (x0 ) − IX  −

4K0 mx ( f )x0 − a, (r − δ)2 0

and hence

 .  K0 4K0 ma ( f ) ≥ mx0 ( f ) 1 − − a . + x 0 r − δ (r − δ)2  K0 4K0  1 + (r−δ) Fix ρ with 0 < ρ < δ such that r−δ 2 ρ < 2 , and let U stand for the open ball with centre x0 and radius ρ. Then we have for each a ∈ U that ma ( f ) ≥ 12 mx0 ( f ), and, by (5.3.18),  f − I B ≤ 2K0 ma ( f ). It follows from the arbitrariness of the

5.3 Holomorphic automorphisms of a bounded domain

75

holomorphic mapping f :  →  that the statement is proved in the case in which B is centred at x0 . Now suppose that B is an arbitrary open ball strictly inside , and put y0 for the centre of B. By Lemma 5.3.11, there exist r > 0 and x0 , x1 , . . . , xn = y0 such that the open balls Bj (0 ≤ j ≤ n + 1) given by Bj := xj + rX (0 ≤ j ≤ n) and Bn+1 := B are strictly inside  and satisfy xj+1 ∈ Bj (0 ≤ j ≤ n). By the above paragraph, there exist a neighbourhood U of x0 in  and a positive constant α such that for any holomorphic mapping f :  →  and any point a in U we have  f − I B0 ≤ αma ( f ).

(5.3.19)

For each j ∈ {1, . . . , n}, by Proposition 5.3.35, there exists Kj > 0 such that we have for any holomorphic mapping f :  →  that  f − I Bj ≤ Kj mxj ( f ). Moreover, since xj ∈ Bj−1 , it follows from the inequalities (5.2.23) that there exists Mj > 0 such that we have for any holomorphic mapping f :  →  that mxj ( f ) ≤ Mj  f − I Bj−1 . Therefore, for any holomorphic mapping f :  → ,  f − I Bj ≤ Kj Mj  f − I Bj−1 ,

(5.3.20)

and it follows from (5.3.19) and (5.3.20) that for any point a in U we have  f − I B ≤ αK1 · · · Kn+1 M1 · · · Mn+1 ma ( f ), 

as required.

Corollary 5.3.37 Let X be a complex Banach space, let  be a bounded domain in X, and let B1 and B2 be open balls strictly inside . Then there exists a positive constant K such that  f − I B2 ≤ K f − I B1 for every holomorphic mapping f :  → . Proof Set x1 and r1 for the centre and the radius of B1 . By Theorem 5.3.36, there exist δ with 0 < δ < r1 and K > 0 such that  f − I B2 ≤ K max{ f (a) − a, Df (a) − IX } for all a ∈ B0 := x1 + δX and f :  →  holomorphic. Since B0 + (r1 − δ)X ⊆ B1 , the Cauchy inequalities (5.2.23) provide a positive constant M such that max{ f (a) − a, Df (a) − IX } ≤ M f − I B1 for all a ∈ B0 and f :  →  holomorphic. As a result  f − I B2 ≤ KM f − I B1 for every holomorphic mapping f :  → , and the proof is complete.



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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Proposition 5.3.38 Let X be a complex Banach space, let  be a bounded domain in X, let x0 ∈ , let A   be a convex subset, and let B   be an open ball. Then there exist an open neighbourhood U of x0 contained in  and a constant K > 0 such that g−1 ◦ f − I B ≤ K max{ f (x) − g(x), Df (x) − Dg(x)} for all x ∈ U and f , g ∈ Aut() satisfying f (x0 ) ∈ A and g(x0 ) ∈ A. Proof Given an open ball C   centred at x0 , by Corollary 5.3.16, there exists a positive constant k such that all the automorphisms of  are k-Lipschitz on C. Fix ε > 0 so that Aε   and fix δ > 0 such that δ < εk and U := x0 + δX ⊆ C. Note that, for each x ∈ U and h ∈ Aut() such that h(x0 ) ∈ A, we have that h(x) − h(x0 ) ≤ kx − x0  ≤ kδ < ε, and hence h(x) = h(x0 ) + (h(x) − h(x0 )) ∈ Aε . Moreover, note that Corollary 5.3.16 provides the existence of a positive constant k0 such that all the automorphisms of  are k0 -Lipschitz on Aε . Fix x ∈ U and f , g ∈ Aut() satisfying f (x0 ) ∈ A and g(x0 ) ∈ A. It follows that g−1 ( f (x)) − x = g−1 ( f (x)) − g−1 (g(x)) ≤ k0  f (x) − g(x).

(5.3.21)

On the other hand, we see that D(g−1 ◦ f )(x) − IX  = Dg−1( f (x)) ◦ Df (x) − Dg−1(g(x)) ◦ Dg(x) ≤ [Dg−1 ( f (x)) − Dg−1 (g(x))] ◦ Dg(x) + Dg−1 ( f (x)) ◦ [Df (x) − Dg(x)] ≤ Dg−1 ( f (x)) − Dg−1 (g(x))Dg(x) + Dg−1 ( f (x))Df (x) − Dg(x). By the Cauchy inequality (5.2.23), there exists a positive constant M such that for each h ∈ Aut() and z ∈ U ∪ Aε we have Dh(z) ≤ M. Moreover, Corollary 5.3.16 provides the existence of a positive constant k1 such that, for each h ∈ Aut(), we have that Dh is k1 -Lipschitz on Aε . It follows that D(g−1 ◦ f )(x) − IX  ≤ M(k1 + 1) max{ f (x) − g(x), Df (x) − Dg(x)}. (5.3.22) Now, setting K := max{k0 , M(k1 + 1)}, we deduce from (5.3.21) and (5.3.22) that max{(g−1 ◦ f )(x) − x, D(g−1 ◦ f )(x) − IX } ≤ K max{ f (x) − g(x), Df (x) − Dg(x)} for all x ∈ U and f , g ∈ Aut() satisfying f (x0 ) ∈ A and g(x0 ) ∈ A. Finally, invoking Theorem 5.3.36, the proof concludes.  Combining Lemma 5.3.21 with Proposition 5.3.38 we have the following. Theorem 5.3.39 Let X be a complex Banach space, let  be a bounded domain in X, let x0 ∈ , let A   be a convex subset, and let B   be an open ball.

5.3 Holomorphic automorphisms of a bounded domain

77

Then there exist an open neighbourhood U of x0 contained in  and a constant K > 0 such that  f − gB ≤ K max{ f (x) − g(x), Df (x) − Dg(x)} for all x ∈ U and f , g ∈ Aut() satisfying f (x0 ) ∈ A and g(x0 ) ∈ A. Proposition 5.3.40 Let X be a complex Banach space, let  be a bounded domain in X, and let tλ and fλ be nets in R and H (, ), respectively. Set hλ := tλ ( fλ − I ) for every λ, and suppose the existence of an open ball B0   and of a mapping h ∈ H (B0 , X) such that hλ − hB0 → 0. Then there exists a mapping / h ∈ H0 (, X) such that T / h. h|B0 = h and hλ −→ /

Proof Shrinking B0 if necessary, we may suppose that h is bounded on B0 , and hence there exist M > 0 and λ0 such that hλ B0 ≤ M for every λ ≥ λ0 . Let B be an open ball strictly inside . By Lemma 5.3.11, we can find a finite sequence B1 , . . . , Bn of open balls such that U := ∪nj=1 Bj is a domain strictly inside  such that B0 ∪B  U. By Corollary 5.3.37, for each j ∈ {1, . . . , n}, there exists Kj > 0 such that  fλ − I Bj ≤ Kj  fλ − I B0 for every λ, hence hλ Bj ≤ Kj hλ B0 ≤ Kj M for every λ ≥ λ0 , and so {hλ }λ≥λ0 is uniformly bounded on U. Since {hλ }λ≥λ0 is  · B0 -Cauchy, it follows from Corollary 5.3.13(i) that it is T-Cauchy in H0 (U, X). Now, by Corollary 5.3.8, there exists a mapping T / hU in H0 (U, X). Finally, the arbitrariness of hU ∈ H0 (U, X) such that {hλ }λ≥λ0 −→ /

B   and the principle of analytic continuation (Proposition 5.2.43) allow us to conclude the proof. 

5.3.3 The Carath´eodory distance on a bounded domain / of complex Banach spaces X and Y, respec§5.3.41 Given open subsets  and  / tively, we will denote by H (, ) the subset of H (, Y) consisting of all holomor/. phic mappings f from  to Y satisfying f () ⊆  We begin by recalling the classical Schwarz lemma which play a crucial role in geometric function theory. We recall that the open unit disc in the complex plane is simply denoted by . Lemma 5.3.42 (Classical Schwarz lemma) If f ∈ H (, ) and f (0) = 0, then: (i) | f (z)| ≤ |z| for every z ∈ . (ii) | f  (0)| ≤ 1. Moreover, if |f (z)| = |z| for some nonzero z ∈  or | f  (0)| = 1, then f (z) = eiθ z for some θ ∈ R and every z ∈ .

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

§5.3.43 For a ∈  let us define φa as the restriction to  of the M¨obius transformation defined by a−z φa (z) = . 1 − az It is easy to see that for each z ∈  we have 1 − |φa (z)|2 =

(1 − |a|2 )(1 − |z|2 ) > 0, |1 − az|2

and consequently, |φa (z)| < 1. Thus φa ∈ H (, ). Moreover, it is routine to verify that 1 − ab φφb (a) for all a, b ∈ . φa ◦ φb = − 1 − ab Since φa (a) = 0 and φ0 = −I , we have in particular that φa2 = I , and so φa ∈ Aut() and φa−1 = φa . It is also clear that, for each θ ∈ R, the restriction to  of the rotation rθ defined by rθ (z) = eiθ z lies in Aut() and rθ−1 = r−θ . Note also that rθ ◦ φa = φrθ (a) ◦ rθ . If f ∈ Aut(), then ψ := φf (0) ◦ f ∈ Aut() and ψ(0) = 0. By the Schwarz lemma |ψ  (0)| ≤ 1. Since ψ −1 ∈ Aut() we also have 1 |ψ  (0)|

= |(ψ −1 ) (0)| ≤ 1.

Hence |ψ  (0)| = 1 and by the Schwarz lemma ψ(z) = eiθ z for some θ ∈ R and every z ∈ . Hence f = φf (0) ◦ rθ = rθ ◦ φr−1 ( f (0)) , θ

and so we have proven the following fact. Fact 5.3.44 Aut() = {rθ ◦ φa : θ ∈ R, a ∈ }. From this fact it is easy to see that the following result is a generalization of the classical Schwarz lemma. Lemma 5.3.45 (Schwarz–Pick lemma) If f ∈ H (, ), then:       z−w   (i)  f (z)−f (w)  ≤  1−zw  for all z, w ∈ . 1−f (z)f (w)

(ii)

|f  (z)| 1−|f (z)|2



1 1−|z|2

for every z ∈ .

We have equality in (i) and (ii) if f ∈ Aut(). If equality holds in (i) for one pair of points z = w or if equality holds in (ii) at one point z, then f ∈ Aut().

5.3 Holomorphic automorphisms of a bounded domain

79

§5.3.46 We recall that the hyperbolic tangent function tanh z =

e2z − 1 e2z + 1

is a biholomorphic transformation from the open horizontal strip {z ∈ C : |(z)| < π2 } onto , and its inverse is given by tanh−1 z =

1 1+z log for every z ∈ . 2 1−z

§5.3.47 We say that a distance d on a domain  in a complex Banach space X is Aut()-invariant if we have d( f (x), f (y)) = d(x, y) for every f ∈ Aut() and all x, y ∈ . From the point of view of geometric function theory on , the Euclidean distance on  is inappropriate. In the next statement we include the definition and properties of the so called Poincar´e (hiperbolic) distance on . We refer to [721, Section 3.2] or [1180, Section 3.1] for details. Theorem 5.3.48 The mapping ρ :  ×  → R defined by     −1  z − w  ρ(z, w) := tanh  1 − zw  is a distance on  satisfying the properties: (i) ρ is invariant under Aut(). (ii) ρ(0, t) = ρ(0, s) + ρ(s, t) for all s, t ∈]0, 1[ with s < t. (iii) limt→0+

ρ(0,t) t

= 1.

Properties (i)–(iii) characterize ρ in the set of distances on . Moreover, we have: (iv) ρ( f (z), f (w)) ≤ ρ(z, w) for every f ∈ H (, ) and all z, w ∈ . So, f ∈ H (, ) is isometric with respect to ρ if and only if f ∈ Aut(). (v) (, ρ) is a complete unbounded metric space and ρ induces on  the Euclidean topology. Lemma 5.3.49 Let X be a complex Banach space, let  be a domain in X, and let x and y be in . Then the set {ρ( f (x), f (y)) : f ∈ H (, )}   is bounded. Moreover, tanh−1 1r y − x is a bound for this set in the case that r is a positive number such that y ∈ x + rX ⊆ . Proof We may suppose that x = y. Firstly suppose in addition the existence of a positive number r such that y ∈ x + rX ⊆ . Then consider the holomorphic function φ :  →  defined by φ(ζ ) := x + ζ r

y−x . y − x

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Given f ∈ H (, ), by Theorem 5.3.48(iv) applied to f ◦ φ ∈ H (, ) we have       1 1 y − x ≤ ρ 0, y − x , ρ f (φ(0)), f φ r r i.e. ρ( f (x), f (y)) ≤ tanh−1



 1 y − x , r

(5.3.23)

and the statement is proved in this case. If x and y are arbitrary points in  then, by Lemma 5.3.11, we can find r > 0 and a chain x0 , x1 , . . . , xn of points of  such that x0 = x, xn = y, and xk+1 ∈ xk + rX ⊆  for k = 0, 1, . . . , n − 1. By (5.3.23), we have for each f ∈ H (, ) that   n  −1 1 tanh ρ( f (x), f (y)) ≤ xk − xk−1  , r k=1

and the proof is complete.



§5.3.50 We are now in a position to define the Carath´eodory semidistance on an arbitrary domain  in a complex Banach space X as the mapping d :  ×  → R given by d (x, y) := sup{ρ( f (x), f (y)) : f ∈ H (, )}. It is easy to check that d is a semidistance on . Proposition 5.3.51 The following assertions hold: (i) The Carath´eodory semidistance assigned to  is the Poincar´e distance. (ii) Let 1 and 2 be domains in complex Banach spaces X1 and X2 , respectively. If f ∈ H (1 , 2 ), then d2 ( f (x), f (y)) ≤ d1 (x, y) for all x, y ∈ 1 . As a consequence, if f is a biholomorphic mapping from 1 onto 2 , then f is an isometry with respect to the Carath´eodory semidistances. Proof Since holomorphic functions from  into itself are nonexpansive with respect to the Poincar´e distance (cf. Theorem 5.3.48(iv)), we have d (z, w) ≤ ρ(z, w) for all z, w ∈ . On the other hand, using I we see that d (z, w) ≥ ρ(z, w) for all z, w ∈ . Thus d = ρ, and assertion (i) is proved. Suppose that 1 and 2 are domains in complex Banach spaces X1 and X2 , respectively, and that f ∈ H (1 , 2 ). If h ∈ H (2 , ), then h ◦ f ∈ H (1 , ) and so, for any x, y ∈ 1 , we have ρ(h( f (x)), h( f (y))) ≤ d1 (x, y). It follows from the arbitrariness of h in H (2 , ) that d2 ( f (x), f (y)) ≤ d1 (x, y) for all x, y ∈ 1 .  Corollary 5.3.52 Let X be a complex Banach space, let x0 be in X, and let r > 0. Set B for the open ball in X with centre x0 and radius r. Then   1 dB (x0 , x) = ρ 0, x − x0  for every x ∈ B. r

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81

Proof Since h : x → 1r (x − x0 ) is a biholomorphic mapping from B to X such that h(x0 ) = 0, in view of assertion (ii) in the above proposition, it is sufficient to prove that dX (0, x) = ρ(0, x) for every x ∈ X . To this end, we may suppose that x ∈ X \ {0}. Given f ∈ H (X, ), consider the holomorphic function ψ :  →   x defined by ψ(ζ ) := (φf (0) ◦ f ) ζ x , and note that, again by assertion (ii) in the above proposition, we have ρ( f (0), f (x)) = ρ(φf (0) ( f (0)), φf (0) ( f (x))) = ρ(ψ(0), ψ(x)) ≤ ρ(0, x). Hence dX (0, x) ≤ ρ(0, x). On the other hand, taking ϕ ∈ BX  such that ϕ(x) = x,  we see that ρ(0, x) = ρ(ϕ(0), ϕ(x)) ≤ dX (0, x), and the proof is complete. Definition 5.3.53 A domain  in a complex Banach space X is said to be homogeneous if the automorphism group Aut() acts transitively on , that is to say, for each x and y in , there exists ϕ ∈ Aut() such that ϕ(x) = y. It follows from Fact 5.3.44 that the open unit disc  in the complex plane is a homogeneous domain. Therefore, as a consequence of Riemann’s mapping theorem, every simply connected domain in C is homogeneous. Corollary 5.3.54 Let X be a complex Banach space such that X is a homogeneous domain. If for x ∈ X , Fx denotes an authomorphism of X such that Fx (x) = 0, then dX (x, y) = tanh−1 (Fx (y)) for every y ∈ X . Proof If x, y ∈ X and if Fx ∈ Aut(X ) satisfies Fx (x) = 0, then, by Proposition 5.3.51(ii) and Corollary 5.3.52, we have dX (x, y) = dX (0, Fx (y)) = ρ(0, Fx (y)) = tanh−1 (Fx (y)).



A further obvious but useful consequence of assertion (ii) in Proposition 5.3.51 is the following fact. Fact 5.3.55 If 1 and 2 are domains in a complex Banach space X such that 1 ⊆ 2 , then d2 (x, y) ≤ d1 (x, y) for all x, y ∈ 1 . Proposition 5.3.56 Let X be a complex Banach space, and let  be a bounded domain in X. Then the norm topology and the topology induced by the Carath´eodory semidistance coincide on . (In particular, d is also a distance, not just a semidistance.) More precisely, for each x, y ∈  we have:  1  (i) tanh−1 R(x) x − y ≤ d (x, y), where R(x) := sup{x − z : z ∈ }. In particular, 1 x − y ≤ d (x, y). diam()  1  (ii) d (x, y) ≤ tanh−1 r(x) x − y whenever x − y < r(x), where r(x) denotes the norm-distance of x to the boundary of .

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem In particular, d (x, y) ≤ tanh−1

  2 r(x) 1 x − y whenever x − y < . 2 r(x) 2

Proof Given x ∈ , it is clear that  is contained in the open ball B with centre x and radius R(x). Therefore, by Fact 5.3.55 and Corollary 5.3.52, for each y ∈  we have     1 1 x − y = tanh−1 x − y . d (x, y) ≥ dB (x, y) = ρ 0, R(x) R(x) Moreover, since tanh−1 is convex on [0, 1[ and its derivative at 0 is equal to 1, we have tanh−1 (t) ≥ t for every t ∈ [0, 1[, so that d (x, y) ≥

1 1 x − y ≥ x − y, R(x) diam()

and assertion (i) is proved. The first part of assertion (ii) was proved in Lemma 5.3.49. Now, keeping in mind that tanh−1 (0) = 0 and tanh−1 is convex on [0, 1[, we realize that .   1 1 tanh−1 (t) ≤ 2 tanh−1 t for every t ∈ 0, , 2 2 and hence for any x, y ∈  with x − y < 12 r(x) we have     2 1 −1 −1 1 x − y ≤ tanh x − y, d (x, y) ≤ tanh r(x) 2 r(x) 

as required.

Proposition 5.3.57 Let X be a complex Banach space, let  be a bounded domain in X, let G be a subgroup of Aut(), and let x0 be in . If the orbit G(x0 ) of x0 by G is a neighbourhood of x0 , then G(x0 ) = , and the subgroup G acts transitively on . Proof First we show that G(x0 ) is an open subset of . As G(x0 ) is supposed to be a neighbourhood of x0 , there exists some open subset W ⊆  such that x0 ∈ W ⊆ G(x0 ). Given x ∈ G(x0 ), fix g ∈ G such that g(x0 ) = x, and note that x = g(x0 ) ∈ g(W) ⊆ g(G(x0 )) = (g ◦ G)(x0 ) = G(x0 ). Since g is a homeomorphism, g(W) is open, and so the arbitrariness of x gives that G(x0 ) is open. Now we show that G(x0 ) is a closed subset of . Let x ∈  be any point of the closure of G(x0 ) in . Take a sequence xn in G(x0 ) converging to x, and for each n ∈ N fix gn in G such that gn (x0 ) = xn . Since we have supposed that G(x0 ) is a neighbourhood of x0 and, by Proposition 5.3.56, d induces the norm topology on , there exists ε > 0 such that {y ∈  : d (y, x0 ) < ε} ⊆ G(x0 ).

(5.3.24)

5.3 Holomorphic automorphisms of a bounded domain

83

Moreover, as xn → x there exists n0 ∈ N such that d (gn (x0 ), x) = d (xn , x) < ε for all n ≥ n0 .

(5.3.25)

Since d is G-invariant (cf. Proposition 5.3.51), it follows from (5.3.25) that d (x0 , g−1 n0 (x)) < ε, so that, by (5.3.24), g−1 n0 (x) ∈ G(x0 ), and therefore x ∈ gn0 (G(x0 )) = (gn0 ◦ G)(x0 ) = G(x0 ). Thus G(x0 ) is closed in . Since  is connected, we have G(x0 ) = , and consequently  is homogeneous under the action of G.  Our next goal is to include two results about the completeness of the Carath´eodory distance. To this end, we need the following lemma. Lemma 5.3.58 Let X be a complex Banach space, and let  be a bounded domain in X. If xλ is a d -Cauchy net strictly inside , then xλ converges to a point in . Proof Suppose that {xλ }λ∈ is a d -Cauchy net strictly inside , and fix δ > 0 such that {xλ : λ ∈ }δ  . Choose λ0 ∈  such that d (xλ , xμ )
0 such that Kδ  . Then r(x) > δ for every x ∈ K, where r(x) stands for the distance of x to the boundary of . By Proposition 5.3.56(ii), we have   δ −1 1 for all x, y ∈ K satisfying x − y < . (5.3.26) d (x, y) ≤ tanh 2 2 Fix n ∈ N such that diam(K) < 2δ n. Given arbitrary x, y ∈ K, putting xk := x + nk (y − x) (0 ≤ k ≤ n), we get a chain of points in [x, y] ⊆ K such that x0 = x, xn = y, and xk − xk+1  < 2δ (0 ≤ k ≤ n − 1). Therefore, by the triangle inequality and (5.3.26), we have   n−1  1 d (x, y) ≤ d (xk , xk+1 ) ≤ n tanh−1 . 2 k=0

Thus K is d -bounded. Keeping in mind Proposition 5.3.51(ii), to prove the ‘if’ part, we may suppose without loss of generality that 0 ∈ , and it is suffices to show that, for any r > 0, the set Br := {x ∈  : d (0, x) < r} lies strictly inside . Let ϕ ∈ X  be such that sup (ϕ(x)) ≤ 1,

(5.3.27)

x∈ ϕ(x) and consider the mapping f :  → C defined by f (x) = 2−ϕ(x) . Then f (0) = 0 and |f (x)| < 1 for every x ∈ . Hence f ∈ H (, ). Now if x ∈ Br , then

tanh−1 (| f (x)|) = ρ(0, f (x)) = ρ( f (0), f (x)) ≤ d (0, x) < r. Hence | f (x)| < tanh(r) < 1. Thus  f Br < 1 and there exists t0 ∈]0, 1[ such that (ϕ(x)) ≤ t0 for every x ∈ Br . Let α > 0 be chosen so that αX ⊆ . If ϕ ∈ X  satisfies (5.3.27), then (ϕ(z)) ≤ α1 for every z ∈ X , and so (ϕ(y + α(1 − t0 )z)) ≤ 1 for all y ∈ Br and z ∈ X . Hence, by the Hahn–Banach separation theorem, we have that Br +α(1−t0 )X ⊆ . Thus Br lies strictly inside .

5.3 Holomorphic automorphisms of a bounded domain

85

Finally, given a d -Cauchy sequence xn in , we have that K := {xn : n ∈ N} is d -bounded, and invoking Lemma 5.3.58 and the ‘if’ part in the proof, we realize that xn converges to a point in .  Now we discuss the completeness of the group of automorphisms of a bounded domain. Theorem 5.3.61 Let X be a complex Banach space, and let  be a bounded domain in X. If fλ is a T-Cauchy net in Aut(), and if there exists x0 ∈  such that fλ (x0 ) converges in , then there exists f ∈ Aut() such that T- lim fλ = f . Proof Suppose that {fλ }λ∈ is a T-Cauchy net in Aut(), and that there exists a point x0 ∈  such that fλ (x0 ) converges to y0 ∈ . By Corollary 5.3.8, there exists f ∈ H0 (, X) such that f = T- lim fλ , and consequently f (x0 ) = y0 . Let us choose any open balls B, B  , centred respectively at x0 and y0 such that f (B) ⊆ B , and denote by r(B ) and r(u) the distances of B and u ∈ B to the boundary of , respectively. It is clear that r(B ) < r(u) for each u ∈ B , and hence, by Proposition 5.3.56(ii),   2 1 −1 1 tanh u − v for all u ∈ B , v ∈  with u − v < r(B ). d (u, v) ≤  r(B ) 2 2 Since fλ converges to f uniformly in B, we can choose λ0 ∈  such that for any λ ≥ λ0 and x ∈ B we have  fλ (x) − f (x) < 12 r(B ), and consequently     2 2 −1 1 −1 1 d ( fλ (x), f (x)) ≤ tanh  fλ (x) − f (x) ≤ tanh  fλ − f B . r(B ) 2 r(B ) 2 Thus there is no loss of generality in assuming the existence of a positive constant K such that d ( fλ (x), f (x)) ≤ K fλ − f B for all λ ∈  and x ∈ B.

(5.3.28)

By Proposition 5.3.56(i), we can fix δ > 0 so that the open d -ball C := {x ∈  : d (x, x0 ) < δ} is contained in B. Let us introduce also the open d -ball   1 C := y ∈  : d (y, y0 ) < δ , 2 and set yλ := fλ (x0 ). Since yλ converges to y0 , in light of Proposition 5.3.56(ii), we may also suppose without loss of generality that yλ ∈ C for every λ ∈ . It follows from Proposition 5.3.51(ii) and the triangle inequality that, for any y ∈ C , d ( fλ−1 (y), x0 ) = d ( fλ ( fλ−1 (y)), fλ (x0 )) = d (y, yλ ) 1 1 ≤ d (y, y0 ) + d (y0 , yλ ) < δ + δ = δ, 2 2 and so fλ−1 (y) ∈ C for all λ ∈  and y ∈ C .

(5.3.29)

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Given λ, μ ∈ , again by Proposition 5.3.51(ii), for any y ∈ C we have d ( fλ−1 (y), fμ−1 (y)) = d ( fλ ( fλ−1 (y)), fλ ( fμ−1 (y))) = d ( fμ ( fμ−1 (y)), fλ ( fμ−1 (y))) ≤ d ( fμ ( fμ−1 (y)), f ( fμ−1 (y))) + d ( f ( fμ−1 (y)), fλ ( fμ−1 (y))). Since, by (5.3.29), fμ−1 (y) ∈ C ⊆ B, it follows from (5.3.28) that d ( fλ−1 (y), fμ−1 (y)) ≤ K( fμ − f B +  fλ − f B ) for all λ, μ ∈  and y ∈ C , (5.3.30) Now, invoking Proposition 5.3.56(i), we realize that  fλ−1 − fμ−1 C ≤ diam()K( fμ − f B +  fλ − f B ) for all λ, μ ∈ , and hence the net fλ−1 is  · C -Cauchy. Since, by Proposition 5.3.56(ii), C contains some open ball, invoking Corollary 5.3.13(i) we obtain that fλ−1 is a T-Cauchy net.

(5.3.31)

Again, by Corollary 5.3.8, there exists g ∈ H0 (, X) such that g = T- lim fλ−1 . Consider  ×  as a directed set with the relation ≤ given by declaring that (λ, μ) ≤ (λ , μ ) whenever λ ≤ λ and μ ≤ μ , and take the ×-indexed net fμ ◦ fλ−1 in Aut(). Note that, via Proposition 5.3.51(ii), (5.3.30) can be read as d (( fμ ◦ fλ−1 )(y), y) ≤ K( fμ − f B +  fλ − f B ) for all λ, μ ∈  and y ∈ C , and that as a consequence of Proposition 5.3.56(i) we have  fμ ◦ fλ−1 − I C ≤ diam()K( fμ − f B +  fλ − f B ) for all λ, μ ∈ . Therefore fμ ◦ fλ−1  · C -converges to I , and hence fμ ◦ fλ−1 converges to I uniformly in any open ball contained in C . It follows from Theorem 5.3.14 that T- lim( fμ ◦ fλ−1 ) = I . Given x ∈ , we have in particular that ( fμ ◦ fλ−1 )(x) → x so that, given any ball B   centred at x, there is λ0 ∈  such that ( fλ0 ◦ fλ−1 )(x) ∈ B for every λ ≥ λ0 . Keeping in mind Proposition 5.3.51(ii), for all λ, μ ≥ λ0 we have d (( fλ0 ◦ fλ−1 )(x), ( fλ0 ◦ fμ−1 )(x)) = d ( fλ−1 (x), fμ−1 (x)) = d (( fμ ◦ fλ−1 )(x), x), hence d (( fλ0 ◦ fλ−1 )(x), ( fλ0 ◦ fμ−1 )(x)) → d (x, x) = 0, and so (( fλ0 ◦ fλ−1 )(x))λ≥λ0 is a d -Cauchy net. It follows from Lemma 5.3.58 that there exists y ∈ B ⊆  such that (( fλ0 ◦ fλ−1 )(x))λ≥λ0 → y. Therefore we have (lim( fλ0 ◦ fλ−1 )(x)) = fλ−1 (y) ∈ . g(x) = lim fλ−1 (x) = fλ−1 0 0 It follows from the arbitrariness of x in  that g() ⊆ , so that f ◦ g is defined in , and Proposition 5.3.18 gives that f ◦ g = I . Note also that, by Propositions 5.3.51(ii) and 5.3.56, the condition fλ (x0 ) → y0 gives that d ( fλ−1 (y0 ), x0 ) = d (y0 , fλ (x0 )) → 0,

5.3 Holomorphic automorphisms of a bounded domain

87

and hence fλ−1 (y0 ) → x0 . This fact together with assertion (5.3.31) allows us to apply all our results to the net fλ−1 instead of fλ , and conclude the equality g ◦ f = I . Thus f ∈ Aut(), as required.  In the following examples we analyze the assumptions in Theorem 5.3.61. The former justifies the assumption of the existence of a point x0 ∈  such that the net fλ (x0 ) converges to y0 ∈ . Example 5.3.62 In the open unit disc  in the complex plane, consider the sequence of automorphisms −φ− tanh(n) , where for each n ∈ N −φ− tanh(n) (ζ ) =

tanh(n) + ζ 1 + tanh(n)ζ

(ζ ∈ )

(cf. Fact 5.3.44). Given 0 < r < 1, note that for each ζ with |ζ | < r we have    (1 − ζ )(1 − tanh(n))  (1 + r)(1 − tanh(n))  < |1 + φ− tanh(n) (ζ )| =  → 0,  1 + tanh(n)ζ 1 − tanh(n)r and consequently T- lim −φ− tanh(n) = f , where f is the mapping constantly 1, and so f∈ / Aut(). The second example shows that the boundedness of  is also of a crucial importance for Theorem 5.3.61. Example 5.3.63 In a complex Banach space X consider the sequence fn in Aut(X) given by 1 fn (x) := x for all n ∈ N and x ∈ X. n Then fn is T-convergent to the constant mapping 0 which clearly is not in Aut(X). 5.3.4 Historical notes and comments The material in this section has been elaborated mainly from the survey article of Arazy [837] and the books of Dineen [721], Franzoni and Vesentini [1159], Isidro and Stach´o [751], Mujica [1175], Reich and Shoikhet [1180], Upmeier [814, 815], Vigu´e [1187], and Willard [1188]. Other sources are quoted in what follows. The study of the automorphisms of a bounded domain in Cn was carried out by H. Cartan [899, 1145]. Propositions 5.3.24 and 5.3.25 were showed for bounded domains in C2 in [899], while their proofs for a bounded domain in a complex Banach space remain without substantial change. An elementary exposition of the works of H. Cartan is given in Narasimhan [1176]. Theorem 5.3.39 is due to Vigu´e [1187, 1114], and it seems that this result had no finite-dimensional forerunners. We refer the reader to the books [721, 1180] for a proof of the classical Schwarz lemma 5.3.42 and the Schwarz–Pick lemma 5.3.45, as well as for the properties of the Poincar´e distance on  included in Theorem 5.3.48. Moreover, these books

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contain various generalizations of the Schwarz lemma, a detailed treatment of several semidistances on an arbitrary domain, and precise historical notes. All results in Subsection 5.3.3 are taken from [721, 751, 1180]. 5.4 Complete holomorphic vector fields Introduction In Subsection 5.4.1 we study locally Lipschitz vector fields on an open subset of a complex Banach space. As the main result we prove the existence of a flow for such vector fields and determine its properties (Theorem 5.4.9). In Subsection 5.4.2 we focus on holomorphic vector fields and prove that their flows are in fact analytic mappings (Theorem 5.4.22). Moreover, the consideration of holomorphic vector fields on an open subset  of a complex Banach space X as ‘differential operators’ on spaces of holomorphic mappings allows us to show that H (, X) becomes a Lie complex algebra in a natural way, and to provide a description of the exponential of those holomorphic vector fields subjected to suitable conditions of boundedness on open balls strictly inside  (Fact 5.4.40). Finally, in Subsection 5.4.3, we establish the close relationship between complete holomorphic vector fields on a bounded domain  and analytic one-parameter groups of biholomorphic automorphisms of  (Proposition 5.4.48). 5.4.1 Locally Lipschitz vector fields §5.4.1 Let X be a Banach space over K, and let  be an open set of X. By a vector field on  we simply mean a mapping  :  → X, which we interpret as assigning a vector to each point of . Given a point x0 ∈ , a local flow or an integral curve of  at x0 is a differentiable function ϕ : I → , where I is an open interval of R containing 0, which is a solution of the initial value problem $  x (t) = (x(t)) (t ∈ I) (5.4.1) x(0) = x0 . Fact 5.4.2 Let X be a Banach space over K, let  be a continuous vector field on an open set  of X, let x0 be in , let I be an open interval of R containing 0, and let ϕ : I →  be a continuous function. Then ϕ is a local flow of  at x0 if and only if  t (ϕ(s))ds for every t ∈ I. (5.4.2) ϕ(t) = x0 + 0

Proof Suppose that ϕ is a local flow of  at x0 . Consider the function g : I → X given by  t (ϕ(s))ds for every t ∈ I. g(t) := x0 + 0

Clearly g(0) = x0 . Moreover, by the fundamental theorem of calculus, g is differentiable in I and g (t) = (ϕ(t)) = ϕ  (t) for every t ∈ I. Now, as a consequence of the mean value theorem (Fact 5.2.13) we deduce that ϕ = g, and so ϕ satisfies (5.4.2).

5.4 Complete holomorphic vector fields

89

Conversely, if ϕ satisfies (5.4.2), then clearly ϕ(0) = x0 , and, by the fundamental theorem of calculus, ϕ is differentiable in I and ϕ  (t) = (ϕ(t)) for every t ∈ I. Thus ϕ is a local flow of  at x0 .  The next result is the well-known Gronwall inequality. Proposition 5.4.3 Let f , g : [a, b] → R be continuous and non-negative functions, and let c be in [a, b]. For each t ∈ [a, b], set It for the closed interval of R determined by c and t. Suppose the existence of a non-negative constant α such that  f (t) ≤ α + f (s)g(s)ds for every t ∈ [a, b]. It

Then

 f (t) ≤ α exp

 g(s)ds for every t ∈ [a, b].

It

Proof

Consider the function h : [a, b] → R defined by  h(t) := α + f (s)g(s)ds. It

The assumptions give that 0 ≤ α ≤ h and f ≤ h. Moreover, by the fundamental theorem of calculus, h is a continuous function such that $ −f (t)g(t) if a ≤ t < c  h (t) = f (t)g(t) if c < t ≤ b. If we now consider the continuous function k : [a, b] → R defined by    k(t) := h(t) exp − g(s)ds , It

then we obtain

⎧  #  ⎨ (h (t) + h(t)g(t)) exp − I g(s)ds if a ≤ t < c t k (t) = ⎩ (h (t) − h(t)g(t)) exp  − # g(s)ds if c < t ≤ b, It

and hence, in light of the inequality f (t)g(t) ≤ h(t)g(t), we realize that k (t) ≥ 0 if t ∈ [a, c[ and k (t) ≤ 0 if t ∈]c, b]. Therefore k(t) ≤ k(c) = h(c) = α, and so     g(s)ds ≤ α exp g(s)ds . f (t) ≤ h(t) = k(t) exp It

This finishes the proof.

It



We recall that, according to §5.2.52, X denotes the open unit ball of a given Banach space X. Moreover, our notation ϕx is in agreement with the customary one for separate mappings of a mapping with two arguments ϕ(t, x).

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Lemma 5.4.4 Let X be a Banach space over K, and let  : X → X be a Lipschitz vector field. Then there exist a positive constant R and a continuous mapping 1 ϕ : [−R, R] × X → X 2 such that, for each r ∈]0, R] and x ∈ 12 X , the mapping (ϕx )|]−r,r[ is the unique ] − r, r[-defined local flow of  at x. Moreover, ϕ is uniformly Lipschitz in the x-variable, i.e. there exists a positive constant M such that 1 ϕ(t, x) − ϕ(t, y) ≤ Mx − y for all t ∈ [−R, R] and x, y ∈ X . 2 Proof Let K be a positive constant such that (x) − (y) ≤ Kx − y for all x, y ∈ X . Note that  is bounded. Indded, for each x ∈ X we have (x) ≤ (x) − (0) + (0) ≤ Kx + (0) ≤ K + (0). 1 Set R := 2(K+(0)) and fix x0 ∈ 12 X . For each r ∈]0, R], consider the Banach space C([−r, r], X) with the sup norm, and note that the set

C([−r, r], X ) := {ψ ∈ C([−r, r], X) : ψ([−r, r]) ⊆ X } is open in C([−r, r], X). For each ψ ∈ C([−r, r], X ), consider the mapping Sr (ψ) : [−r, r] → X defined by

 Sr (ψ)(t) := x0 +

t

(ψ(s))ds. 0

By the fundamental theorem of calculus, Sr (ψ) is continuous. Moreover, for each t ∈ [−r, r] we see that  t   t    Sr (ψ)(t) − x0  = (ψ(s))ds ≤  (ψ(s))ds 0

0

1 ≤ r(K + (0)) ≤ R(K + (0)) = , 2 and hence 1 (5.4.3) Sr (ψ)(t) ≤ x0  + . 2 Therefore Sr (ψ) ∈ C([−r, r], X ). Moreover, given ψ, φ ∈ C([−r, r], X ), we have for each t ∈ [−r, r] that  t ((ψ(s)) − (φ(s)))ds Sr (ψ)(t) − Sr (φ)(t) = 0

≤ r sup (ψ(s)) − (φ(s)) s∈[−r,r]

≤ rK sup ψ(s) − φ(s) s∈[−r,r]

= rKψ − φ,

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91

and hence Sr (ψ) − Sr (φ) ≤ rKψ − φ. Since rK ≤ RK < 2R(K + (0)) = 1, it follows that Sr : C([−r, r], X ) → C([−r, r], X ) becomes a contractive mapping. Set α0 := x0  + 12 , and, for each α with α0 ≤ α < 1, consider the set Er,α := {ψ ∈ C([−r, r], X) : ψ(0) = x0 and ψ([−r, r]) ⊆ αBX }. It is clear that Er,α is a closed subset of C([−r, r], X), and hence Er,α becomes a complete metric space. Since Er,α ⊆ C([−r, r], X ) and Er,α is Sr -invariant (in fact, by (5.4.3), Sr (Er,α ) ⊆ Er,α0 ), it follows from the contraction mapping principle that there exists a unique ϕr,α ∈ Er,α such that Sr (ϕr,α ) = ϕr,α . Note that, for any α1 , α2 with α0 ≤ α1 ≤ α2 < 1, we have Er,α1 ⊆ Er,α2 , and consequently ϕr,α1 = ϕr,α2 . On the other hand, it is clear from the definition of Sr that, for 0 < r1 ≤ r2 ≤ R, we have (Sr1 )(ψ|[−r1 ,r1 ] ) = (Sr2 (ψ))|[−r1 ,r1 ] for every ψ ∈ C([−r2 , r2 ], X ). Therefore, if we put ϕx0 := ϕR,α0 , then we realize that ϕr,α = (ϕx0 )|[−r,r] for every (r, α) ∈]0, R] × [α0 , 1[. It follows from Fact 5.4.2 that, for each r ∈]0, R], (ϕx0 )|]−r,r[ is a local flow of  at x0 . Suppose that φ is a ] − r, r[-defined local flow of  at x0 for some r ∈]0, R]. Given r ∈]0, r[, we can fix α ∈ [α0 , 1[ such that φ|[−r ,r ] ∈ Er ,α , and, keeping in mind Fact 5.4.2, we deduce from the above that φ|[−r ,r ] = (ϕx0 )|[−r ,r ] . As a consequence, φ = (ϕx0 )|]−r,r[ . Summarizing, we have proven the existence of ϕx0 ∈ C([−R, R], X ) satisfying that, for every r ∈]0, R], (ϕx0 )|]−r,r[ is the unique ] − r, r[-defined local flow of  at x0 . Now, according with the above paragraph, consider the mapping 1 ϕ : [−R, R] × X → X 2 defined by ϕ(t, x) := ϕx (t) where, for each x ∈ 12 X , ϕx : [−R, R] → X is the continuous function satisfying for each r ∈]0, R] that (ϕx )|]−r,r[ is the unique ] − r, r[-defined local flow of  at x. Given t ∈ [−R, R] and x, y ∈ 12 X with x = y, we have    t  t (ϕx (s))ds − y + (ϕy (s))ds ϕ(t, x) − ϕ(t, y) = x + 0



= x−y+ 0

0 t

((ϕx (s)) − (ϕy (s)))ds



≤ x − y +

t

((ϕx (s)) − (ϕy (s)))ds .

0

Therefore, by putting It for the interval determined by 0 and t, we have  ϕ(t, x) − ϕ(t, y) ≤ x − y + (ϕx (s)) − (ϕy (s))ds 

It

Kϕx (s) − ϕy (s)ds.

≤ x − y + It

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

By the Gronwall inequality (Proposition 5.4.3) applied to the functions f (s) := ϕx (s) − ϕy (s) and g(s) := K we deduce that



 Kds x − y = exp(K|t|)x − y ≤ exp(KR)x − y.

ϕ(t, x) − ϕ(t, y) ≤ exp It

Thus ϕ is uniformly Lipschitz in the second variable. Finally, if (tn , xn ) is a sequence in [−R, R] × 12 X converging to (t, x) ∈ [−R, R] × 12 X , then ϕ(tn , xn ) − ϕ(t, x) ≤ ϕ(tn , xn ) − ϕ(tn , x) + ϕ(tn , x) − ϕ(t, x) ≤ exp(KR)xn − x + ϕx (tn ) − ϕx (t), and as a result ϕ(tn , xn ) converges to ϕ(t, x). Thus ϕ is continuous, and the proof is complete.  Proposition 5.4.5 Let X be a Banach space over K, let  be an open set of X, let x0 be in , and let  :  → X be a locally Lipschitz vector field. Then there exist ρ, δ > 0, and a continuous mapping ϕ : [−ρ, ρ] × (x0 + 2δ X ) → x0 + δX such that x0 + δX ⊆  and, for each r ∈]0, ρ] and x ∈ x0 + 2δ X , the function (ϕx )|]−r,r[ is the unique ] − r, r[-defined and (x0 + δX )-valued local flow of  at x. Moreover, if I is an open interval of R containing 0, if S is a subset of  containing x0 , and if ψ : I ×S →  is a continuous mapping such that, for each x ∈ S, ψx : I →  is a local flow of  at x, then there exists a neighbourhood W of (0, x0 ) in R × X contained in [−ρ, ρ] × (x0 + 2δ X ) such that ϕ and ψ are equal on W ∩ (I × S). Proof Fix δ > 0 such that x0 + δX ⊆  and |(x0 +δX ) is Lipschitz. Consider the Lipschitz vector field ϒ : X → X defined by ϒ(x) := (x0 + δx). By Lemma 5.4.4, there exist R > 0 and a continuous mapping ψ : [−R, R] × 12 X → X such that, for each r ∈]0, R] and x ∈ 12 X , the function (ψx )|]−r,r[ is the unique ] − r, r[-defined local flow of ϒ at x. Set ρ := Rδ, and consider the continuous mapping   δ ϕ : [−ρ, ρ] × x0 + X → x0 + δX 2 defined by ϕ(t, x) := x0 + δψ( 1δ t, 1δ (x − x0 )). For each x ∈ x0 + 2δ X and t ∈] − ρ, ρ[ we have   1 1 ϕ(0, x) = x0 + δψ 0, (x − x0 ) = x0 + δ (x − x0 ) = x δ δ and

 .  . d d d 1 1 ϕx (t) = x0 + δψ 1 (x−x0 ) t =δ ψ 1 (x−x0 ) t δ δ dt dt δ dt δ       1 1 1 = δ ϒ ψ 1 (x−x0 ) t =  x0 + δψ 1 (x−x0 ) t = (ϕx (t)). δ δ δ δ δ

Therefore (ϕx )|]−ρ,ρ[ is a local flow of  at x.

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93

Now, suppose that r ∈]0, ρ] and x ∈ x0 + 2δ X , and that φ :] − r, r[→ x0 + δX is a local flow of  at x. Then γ :] − δr , δr [→ X defined by γ (t) := 1δ (φ(δt) − x0 ) satisfies γ (0) = 1δ (x − x0 ) and   1   γ (t) = φ (δt) = (φ(δt)) = ϒ (φ(δt) − x0 )) = ϒ(γ (t) . δ Thus γ is a local flow of ϒ at 1δ (x − x0 ). Since δr ∈]0, R] and 1δ (x − x0 ) ∈ 12 X , it follows from Lemma 5.4.4 that γ = (ψ 1 (x−x0 ) )|]− δr , δr [ , and consequently δ φ = (ϕx )|]−r,r[ . This concludes the proof of the first assertion in the statement. Now, suppose that I is an open interval of R containing 0, S is a subset of  containing x0 , and ψ : I × S →  is a continuous mapping such that, for each x ∈ S, ψx : I →  is a local flow of  at x. Then ψ(0, x0 ) = ψx0 (0) = x0 , and hence, by continuity, we can fix ρ  , δ  > 0 such that δ ρ  ≤ ρ, ] − ρ  , ρ  [⊆ I, δ  ≤ , and ψ(W ∩ (I × S)) ⊆ x0 + δX , 2 where W :=] − ρ  , ρ  [×(x0 + δ  X ). Since, for each x ∈ S ∩ (x0 + δ  X ), (ψx )|]−ρ  ,ρ  [ is a (x0 +δX )-valued local flow of  at x, we derive that (ψx )|]−ρ  ,ρ  [ = (ϕx )|]−ρ  ,ρ  [ , and consequently ψ and ϕ are equal on W ∩ (I × S).  Lemma 5.4.6 Let X be a Banach space over K, let  be an open of X, let x0 be in , and let  :  → X be a locally Lipschitz vector field. We have: (i) There exist local flows of  at x0 . (ii) If I and J are open intervals of R containing 0, and if ψ : I →  and φ : J →  are local flows of  at x0 , then ψ and φ are equal on I ∩ J. (iii) There exists a unique maximal local flow of  at x0 . (iv) If ϕ : I →  is the maximal local flow of  at x0 , and if t0 ∈ I, then the function ϕ t0 : I − t0 →  defined by ϕ t0 (s) := ϕ(s + t0 ) is the maximal local flow of  at ϕ(t0 ). Proof Assertion (i) is a straightforward consequence of the first assertion in Proposition 5.4.5. Suppose that I and J are open intervals of R containing 0, and that ψ : I →  and φ : J →  are local flows of  at x0 . We begin noticing that, by the second assertion in Proposition 5.4.5, ψ and φ are equal on some neighbourhood of 0 contained in I ∩ J. Now, consider the set K := {t ∈ I ∩ J : ψ(t) = φ(t)}. It is clear that K is a closed subset of I ∩ J. Let t be in K and define ψ t and φ t near 0 by ψ t (s) := ψ(t + s) and φ t (s) := φ(t + s). Then ψ t and φ t are local flows of  at the point ψ(t) = φ(t), so they agree on some neighbourhood of 0. Thus some neighbourhood of t lies in K, and so K is open. Since I ∩ J is connected, we conclude that K = I ∩ J, and the proof of assertion (ii) is complete.

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Assertion (iii) follows from assertion (ii) by considering the family F of all local flows of  at x0 , the interval I obtained as the union of the domains of all members of F , and the function ϕ : I →  defined by ϕ(t) := ψ(t) for any ψ in F such that t lies in the domain of ψ. Finally, suppose that ϕ : I →  is the maximal local flow of  at x0 , that t0 ∈ I, and that ψ : J →  is the maximal local flow of  at ϕ(t0 ). Since ϕ t0 : I − t0 →  defined by ϕ t0 (s) := ϕ(s + t0 ) is a local flow of  at ϕ(t0 ), we deduce that I − t0 ⊆ J and ϕ t0 = ψ |I−t0 . By interchanging roles, we have also that J + t0 ⊆ I and ψ −t0 = ϕ|J+t0 . Therefore J = I − t0 and ψ = ϕ t0 . Thus assertion (iv) is proved.  Corollary 5.4.7 Let X be a Banach space over K, let  be an open of X, let  :  → X be a locally Lipschitz vector field, and let ψ : I →  and φ : J →  be maximal local flows of . Then either ψ(I) ∩ φ(J) = ∅ or there exists t0 ∈ I such that J = I − t0 and φ(t − t0 ) = ψ(t) for every t ∈ I. Proof Suppose that ψ(I) ∩ φ(J) = ∅, and that ψ(t1 ) = φ(t2 ) for suitable t1 ∈ I and t2 ∈ J. By Lemma 5.4.6(iv), ψ t1 and φ t2 are the maximal local flows of  at ψ(t1 ) and φ(t2 ), respectively. Since ψ(t1 ) = φ(t2 ), it follows that ψ t1 = φ t2 , and in particular I − t1 = J − t2 . Therefore J = I − t0 , where t0 := t1 − t2 ∈ I. Moreover, φ(t − t0 ) = φ(t − t1 + t2 ) = φ t2 (t − t1 ) = ψ t1 (t − t1 ) = ψ(t) for every t ∈ I.  §5.4.8 Given a locally Lipschitz vector field  in an open set  of a Banach space X, for each x ∈ , the maximal local flow of  at x will be denoted by ϕx , the domain of definition of ϕx will be denoted by I(x), and t− (x) (respectively, t+ (x)) will denote the inf (respectively, sup) of I(x) and will be called the negative (respectively, positive) lifetime of x. Theorem 5.4.9 Let X be a Banach space over K, let  be an open of X, and let  :  → X be a locally Lipschitz vector field. Then there exists an open subset D of R ×  containing {0} × , and a continuous mapping ϕ from D to  which is a solution of the Cauchy problem ⎧ ∂ ⎨ f (t, x) = ( f (t, x)) ((t, x) ∈ D) ∂t ⎩ f (0, x) = x (x ∈ ), and satisfies the following properties: (i) If for each x ∈  we consider the set Dx := {t ∈ R : (t, x) ∈ D} and the function ϕx : Dx →  defined by ϕx (t) := ϕ(t, x), then Dx = I(x) and ϕx is the maximal local flow of  at x. (ii) For s ∈ R and (t, x) ∈ D, we have that (s, ϕ(t, x)) ∈ D if and only if (s + t, x) ∈ D. In this case, ϕ(s + t, x) = ϕ(s, ϕ(t, x)).

5.4 Complete holomorphic vector fields

95

(iii) If for each t ∈ R we consider the set Dt := {x ∈  : (t, x) ∈ D} and the mapping ϕt : Dt →  defined by ϕt (x) := ϕ(t, x), then Dt is open in X and, in the case that Dt = ∅, the mapping ϕt is a homeomorphism from Dt onto D−t , and ϕt−1 = ϕ−t . (iv) The lifetime functions t− (·) and t+ (·) are upper and lower semicontinuous respectively. (v) If x ∈  is such that ϕx is bounded, and if t− (x) (respectively, t+ (x)) is finite, then limt→t− (x) ϕx (t) (respectively, limt→t+ (x) ϕx (t)) exists and belongs to the boundary of . Proof According to §5.4.8, for each x ∈ , ϕx : I(x) →  will denote the maximal local flow of  at x. Let D be the set of all points (t, x) in R ×  such that t lies in I(x), and consider the mapping ϕ : D →  defined by ϕ(t, x) := ϕx (t). Clearly D contains {0} × . To prove that D is open and ϕ is continuous, we will show the following claim: For each (t0 , x0 ) ∈ D there exist ρ, δ > 0 such that ]t0 − ρ, t0 + ρ[×(x0 + δX ) ⊆ D and ϕ|]t0 −ρ,t0 +ρ[×(x0 +δX ) is continuous. Fix x0 ∈ . By the first assertion in Proposition 5.4.5, the claim is true for t0 = 0. In order to prove the claim for all t0 ∈]0, t+ (x0 )[, we consider the set Q consisting of all r ∈]0, t+ (x0 )[ such that for each t with 0 < t < r there exist ρ, δ > 0 such that ]t − ρ, t + ρ[×(x0 + δX ) ⊆ D and ϕ|]t−ρ,t+ρ[×(x0 +δX ) is continuous. Then Q is an interval and, by the first assertion in Proposition 5.4.5, there exists ρ > 0 such that ]0, ρ[⊆ Q. We must prove that Q =]0, t+ (x0 )[. This is clear whenever Q is not bounded from above. If Q is bounded from above, we let b be its least upper bound. We must prove that b = t+ (x0 ). Suppose that this is not the case, and put x1 = ϕ(b, x0 ). By Proposition 5.4.5, there exist ρ1 , δ1 > 0 and a unique continuous mapping   δ1 ψ : [−ρ1 , ρ1 ] × x1 + X → x1 + δ1 X 2 such that x1 + δ1 X ⊆  and, for each x ∈ x1 + δ21 X , the function (ψx )|]−ρ1 ,ρ1 [ is a local flow of  at x. By continuity of ϕx0 , we can find ρ with 0 < ρ < min{b, ρ1 } and so small that whenever b − ρ < t < b we have ϕ(t, x0 ) ∈ x1 + δ21 X . Select a point t0 such that b − ρ < t0 < b. By the hypothesis on b, there exist ρ0 , δ0 > 0 such that ]t0 − ρ0 , t0 + ρ0 [×(x0 + δ0 X ) ⊆ D and the mapping ϕ is continuous on ]t0 − ρ0 , t0 + ρ0 [×(x0 + δ0 X ). Moreover, by the continuity of ϕ at (t0 , x0 ), we can suppose that ϕ(]t0 − ρ0 , t0 + ρ0 [×(x0 + δ0 X )) ⊆ x1 + δ21 X . We define φ :]t0 − ρ1 , t0 + ρ1 [×(x0 + δ0 X ) →  by φ(t, x) := ψ(t − t0 , ϕ(t0 , x)). It is clear that φ is a continuous mapping. Moreover, we have for t ∈]t0 − ρ1 , t0 + ρ1 [ and x ∈ x0 + δ0 X that φ(t0 , x) = ψ(0, ϕ(t0 , x)) = ϕ(t0 , x)

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JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

and ∂ ∂ φ(t, x) = ψ(t − t0 , ϕ(t0 , x)) = (ψ(t − t0 , ϕ(t0 , x))) = (φ(t, x)). ∂t ∂t t

t

Hence both φx0 and ϕx0 are local flows of  with the same value at 0. They coincide on any interval on which they are defined by Lemma 5.4.6(ii), and hence ϕ|]t0 −ρ1 ,t0 +ρ1 [×(x0 +δ0 X ) is continuous. Since ρ < ρ1 , we see that b ∈]t0 − ρ1 , t0 + ρ1 [, which is a contradiction. Therefore b is the positive lifetime of x0 . Similarly, one proves the claim for all t0 ∈]t− (x0 ), 0[, and we therefore see that D is open in R × X and ϕ is continuous on D. Now, assertion (i) follows from the very definition of D and ϕ. Given (t, x) ∈ D, by Lemma 5.4.6(iv), we have that I(ϕ(t, x)) = I(x) − t and ϕϕ(t,x) (s) = ϕx (t + s) for every s ∈ I(ϕ(t, x)), and so assertion (ii) follows. Given t ∈ R, the mapping x → (t, x) from X to R × X is continuous, and hence the set Dt is an open in X because D is an open in R × X. Moreover, in the case that Dt = ∅, the function ϕt is continuous because ϕ is continuous. Suppose that Dt = ∅, and fix x ∈ Dt . Then (t, x) ∈ D and, since (0, x) ∈ D, it follows from assertion (ii) that (−t, ϕ(t, x)) ∈ D and ϕ(−t, ϕ(t, x)) = ϕ(0, x). Therefore ϕt (Dt ) ⊆ D−t and (ϕ−t ◦ ϕt )(x) = ϕ0 (x) = x for every x ∈ Dt . Now, replacing t by −t, we obtain that ϕ−t (D−t ) ⊆ Dt and (ϕt ◦ ϕ−t )(x) = ϕ0 (x) = x for every x ∈ D−t . Thus we have that ϕt is a homeomorphism from Dt onto D−t , and ϕt−1 = ϕ−t , and assertion (iii) is proved. Let us fix α ∈ R, and consider the set Sα := {x ∈  : t− (x) < α}. Given x0 ∈ Sα , select t such that t− (x0 ) < t < α. Since D is open, there exist an open interval J containing t and an open set V in X containing x0 such that J × V ⊆ D, and consequently, by assertion (i), t ∈ I(x) for every x ∈ V. Therefore t− (x) < α for every x ∈ V, and hence V ⊆ Sα . It follows from the arbitrariness of x0 in Sα that Sα is open, and hence t− (·) is upper semicontinuous. Analogously, one can prove that t+ (·) is lower semicontinuous, and the proof of assertion (iv) is complete. Suppose that x0 ∈  is such that I(x0 ) = R and there exists M > 0 satisfying ϕx 0 (t) ≤ M for every t ∈ I(x0 ), and suppose for example that t− (x0 ) > −∞. Given a sequence tn in I(x0 ) converging to t− (x0 ), in view of Fact 5.4.2 and the basic property of boundedness of the integral we have for all natural numbers p and q that  tp  tp (ϕx0 (t))dt = ϕx 0 (t)dt ≤ M|tp − tq |, ϕx0 (tp ) − ϕx0 (tq ) = tq

tq

hence ϕx0 (tn ) is a Cauchy sequence in X, and so converges. From this it follows the existence of y ∈  such that y=

lim ϕx0 (t).

t→t− (x0 )

(5.4.4)

Suppose that y ∈ . It follows from assertion (ii) that I(ϕx0 (t)) = I(x0 ) − t, and hence t− (ϕx0 (t)) = t− (x0 ) − t for every t ∈ I(x0 ). Therefore limt→t− (x0 ) t− (ϕx0 (t)) = 0, and, by (5.4.4), we realize that

5.4 Complete holomorphic vector fields

97

lim sup t− (x) ≥ 0. x→y

Since, by assertion (iv), the negative lifetime function t− (·) is upper semicontinuous, we obtain the contradiction t− (y) ≥ 0. Thus y belongs to the boundary of , and assertion (v) is proved.  §5.4.10 Given a locally Lipschitz vector field  on an open subset  of a Banach space X, the function ϕ : D →  in the statement of Theorem 5.4.9 is called the flow or the integral of . The vector field  is said to be complete whenever D = R × , that is to say t− (x) = −∞ and t+ (x) = +∞ for every x ∈ . Corollary 5.4.11 Let X be a Banach space over K, let  be an open subset of X, let  :  → X be a locally Lipschitz vector field, let ϕ : D →  be the flow of , and let X0 be a closed subspace of X. Suppose that 0 :=  ∩ X0 is not empty and (0 ) ⊆ X0 , regard |0 from 0 to X0 , and set D0 := D ∩ (R × 0 ). Then ϕ(D0 ) ⊆ 0 , and ϕ|D0 regarded from D0 to 0 is the flow of |0 . Consequently, if  is complete, then so is |0 . Proof It is clear that each local flow of |0 is a local flow of . Therefore, if we denote by t0− (·) and t0+ (·) the negative and positive lifetime functions for |0 , then we have for every x ∈ 0 that t− (x) ≤ t0− (x) < t0+ (x) ≤ t+ (x), and (ϕx )|]t− (x),t+ (x)[ is 0

0

the maximal flow of |0 at x. Suppose that t− (x0 ) < t0− (x0 ) for some x0 ∈ 0 . Then ϕ(t0− (x0 ), x0 ) ∈ , and since X0 is closed in X we have also that ϕ(t0− (x0 ), x0 ) =

lim ϕ(t, x0 ) ∈ X0 .

t→t0− (x0 ) t>t0− (x0 )

Therefore ϕ(t0− (x0 ), x0 ) ∈ 0 , a contradiction. Thus t− (x) = t0− (x) for every x ∈ 0 . Analogously t+ (x) = t0+ (x) for every x ∈ 0 . Hence D0 is the domain of the flow of |0 , and ϕ|D0 is the flow of |0 .  Since clearly each local flow of a bounded vector field has bounded derivative, we have the following consequence of Theorem 5.4.9(v). Corollary 5.4.12 Let X be a Banach space over K, let  be an open subset of X, let  :  → X be a bounded locally Lipschitz vector field, and let 0 be an open subset of X such that 0 ⊆ . If all local flows of  at points of 0 are contained in 0 , then |0 is a complete vector field. The next fact is elementary. Fact 5.4.13 Let X be a Banach space over K, let  be an open subset of X, let  be a locally Lipschitz vector field on , let ϕ : D →  be the flow of , and let t0 ∈ R. Then the vector field t0  is locally Lipschitz and its flow is the mapping ψ from E := {(t, x) ∈ R ×  : (t0 t, x) ∈ D} to  defined by ψ(t, x) := ϕ(t0 t, x).

98

JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem

Proof It is clear that t0  is locally Lipschitz and E is an open subset of R ×  containing {0} × . The chain rule gives for each (t, x) ∈ E that d ∂ ψ(t, x) = ψx (t) = ϕx (t0 t) = t0 ϕx (t0 t) = t0 (ϕ(t0 t, x)) = t0 (ψ(t, x)). ∂t dt Moreover, we have for each x ∈  that ψ(0, x) = ϕ(0, x) = x. Therefore ψ is the  : E →  of t0 . If t0 = 0, then E = R × , so ψ = ψ , restriction to E of the flow ψ and in this case the proof is complete. Suppose that t0 = 0. Then, interchanging the roles of  and t0 , and keeping in mind that D = {(t, x) ∈ R ×  : (t0−1 t, x) ∈ E }, it , and the proof is finished. follows that E = E and ψ = ψ  §5.4.14 Let X be a Banach space over K, let  be an open subset of X, let  be a locally Lipschitz vector field on , and let ϕ : D →  be the flow of . Following the standard terminology in the theory of vector fields, the mapping ϕ1 : D1 →  is called the exponential of , and is denoted by exp(). Note that, according to the above fact, we have exp(t) := ϕt for every t ∈ R. To close this subsection we consider a class of vector fields which is particularly easy to study. These are the ones determined by a bounded linear operator on a Banach space, which are called linear vector fields. For these vector fields, the exponential has the same meaning in both operator theory and the theory of vector fields. Exercise 5.4.15 Let X be a Banach space over K, and let F be in BL(X). Prove that: (i) The linear vector field F is complete and its flow is the mapping ϕ : R × X → X given by ϕ(t, x) := exp(tF)(x) (in the meaning of operator theory). (ii) If K = R, then F|X is complete if and only if V(BL(X), IX , F) = 0. (iii) If K = C, then iF|X is complete if and only if F is hermitian. Solution Consider for each t ∈ R the exponential exp(tF) of the operator tF in BL(X), fix x ∈ X, and set ϕ(t, x) := exp(tF)(x). Since exp(0F) = IX , it follows that ϕ(0, x) = x. Moreover, since dtd exp(tF) = F ◦ exp(tF), and the evaluation at x is a continuous linear mapping from BL(X) to X, we see that   ∂ d ∂ ϕ(t, x) = (exp(tF)(x)) = exp(tF) (x) = F(exp(tF)(x)) = F(ϕ(t, x)). ∂t ∂t dt Now, the arbitrariness of x in X gives that the mapping ϕ : R × X → X is the flow of F, and consequently F is complete. Now that (i) is proved, we deduce as a consequence that F|X is complete if and only if ϕ(R × X ) ⊆ X , that is to say  exp(tF)(x) < 1 for every (t, x) ∈ R × X , which is equivalent to  exp(tF) ≤ 1 for every t ∈ R, which in turn is equivalent to  exp(tF) = 1 for every t ∈ R. Now, (ii) and (iii) follow from the assertions (ii) and (iii) in Corollary 2.1.9 applied to F and iF, respectively. 

5.4 Complete holomorphic vector fields

99

5.4.2 Holomorphic vector fields In this subsection we will prove that the flow of any holomorphic vector field is a real analytic mapping, and we will obtain an explicit description of the flow of any holomorphic vector field on a domain under suitable conditions of boundedness. We begin with the next fact, which is a consequence of the mean value theorem. Fact 5.4.16 Let X and Y be normed spaces over K, let  be an open subset of X, and let  :  → Y be a differentiable mapping. If a line segment [x0 , x1 ] is entirely contained in , and if z lies is [x0 , x1 ], then (x1 )−(x0 )−D(z)(x1 −x0 ) ≤ x1 −x0  sup D(x0 +t(x1 −x0 ))−D(z). 0≤t≤1

Proof Apply the mean value theorem (Fact 5.2.13) to the mapping f :  → Y given by f (x) = (x) − D(z)(x).  Lemma 5.4.17 Let X and Y be complex Banach spaces, and let  : X → Y be a bounded holomorphic mapping. Then the mapping ∗ : Hb (,X) → Hb (, Y) given by ∗ ( f ) =  ◦ f is holomorphic, and D∗ ( f )(g)(ζ ) = D( f (ζ ))(g(ζ )) for all f ∈ Hb (,X) , g ∈ Hb (, X), and ζ ∈ . Proof Let f ∈ Hb (,X) . For each g ∈ Hb (, X), consider the function T(g) :  → Y given by T(g)(ζ ) = D( f (ζ ))(g(ζ )). Note that, by the chain rule, T(g) is a holomorphic function because of the following facts: – The function ζ → ( f (ζ ), g(ζ )) from  to X ×X is holomorphic since its component functions are holomorphic. – The mapping (x, z) → (D(x), z) from X × X → BL(X, Y) × X is holomorphic since its component mappings are holomorphic. – The mapping (F, x) → F(x) from BL(X, Y)×X → Y is a bounded bilinear mapping, and hence is holomorphic. Fix δ such that 0 < δ < 1 −  f . Then, it is clear that f () + δX ⊆ X , and hence, by the inequality (5.2.23), we have D( f (ζ )) ≤ 1δ  for every ζ ∈ . Therefore 1 T(g)(ζ ) = D( f (ζ ))(g(ζ )) ≤ D( f (ζ ))g(ζ ) ≤ g δ for every ζ ∈ , and hence T(g) is bounded and T(g) ≤ 1δ g. Moreover, since the mapping T : g → T(g) from Hb (, X) to Hb (, Y) is linear, we conclude that T is a bounded linear operator and T ≤ 1δ . On the other hand, for each g ∈ Hb (, X) with g < 2δ and each ζ ∈ , we see that δ [ f (ζ ), f (ζ ) + g(ζ )] + X ⊆ X , 2

100 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem and using Fact 5.4.16 and Proposition 5.3.15 we realize that ∗ ( f + g)(ζ ) − ∗ ( f )(ζ ) − T(g)(ζ ) = ( f (ζ ) + g(ζ )) − ( f (ζ )) − D( f (ζ ))(g(ζ )) ≤ g(ζ ) sup D( f (ζ ) + tg(ζ )) − D( f (ζ )) 0≤t≤1  2

≤ g(ζ ) ≤

4 δ

g(ζ )

 2 4 g2 δ

for every ζ ∈ , and hence ∗ ( f + g) − ∗ ( f ) − T(g) ≤

 2 4 g2 . δ

It follows that ∗ is differentiable at f and D∗ ( f ) = T, as required.



Lemma 5.4.18 Let X and Y be complex Banach spaces, let B be an open ball in X, and let E : Hb (B, Y) × B → Y be the mapping defined by E( f , x) = f (x). Then E is holomorphic, and DE( f0 , x0 )( f , x) = Df0 (x0 )(x) + f (x0 ) for all ( f0 , x0 ) ∈ Hb (B, Y) × B and ( f , x) ∈ Hb (B, Y) × X. Proof We may suppose that B = X . Let ( f0 , x0 ) ∈ Hb (X , Y) × X . Consider the linear mapping T : Hb (X , Y) × X → Y defined by T( f , x) = Df0 (x0 )(x) + f (x0 ). Since T( f , x) = Df0 (x0 )(x) + f (x0 ) ≤ Df0 (x0 )x +  f (x0 ) ≤ (Df0 (x0 ) + 1)( f X + x), it follows that T is a bounded linear operator. Fix δ with 0 < δ < 1 − x0  and pick ( f , x) ∈ Hb (X , Y) × X such that ( f , x) < 2δ . Then, it is clear that δ [x0 , x0 + x] + X ⊆ X , 2 and hence, by the mean value theorem (Fact 5.2.13) and the inequality (5.2.23), we have 2 1  f (x0 + x) − f (x0 ) ≤ x sup Df (x0 + tx) ≤ x f X ≤ ( f X + x)2 . δ δ 0≤t≤1 Therefore E(( f0 , x0 ) + ( f , x)) − E( f0 , x0 ) − T( f , x) =  f0 (x0 + x) − f0 (x0 ) − Df0 (x0 )(x) + f (x0 + x) − f (x0 ) 1 ≤  f0 (x0 + x) − f0 (x0 ) − Df0 (x0 )(x) + ( f X + x)2 , δ

5.4 Complete holomorphic vector fields

101

and hence E(( f0 , x0 ) + ( f , x)) − E( f0 , x0 ) − T( f , x) = 0. ( f ,x)→(0,0)  f X + x lim

Thus E is differentiable at ( f0 , x0 ) and DE( f0 , x0 ) = T, as required.



§5.4.19 Let X be a complex Banach space. Via natural identifications (cf. §5.2.30), the Taylor series at 0 of any holomorphic function f from  to X is of the form  n n≥0 xn ζ for suitable sequence (xn )n∈N∪{0} in X. Since the radius of convergence  of this power series is ≥ 1 (cf. Theorem 5.2.55), and the power series n≥0 xn ζ n  and n≥1 n1 xn−1 ζ n have the same radius of convergence (cf. Proposition 5.2.29), we deduce that f has a primitive. Indeed, the function F :  → X defined by F(ζ ) =

∞  1 n=1

n

xn−1 ζ n

is the primitive of f vanishing at 0. Arguing as in the case of complex functions of one complex variable [1149, Proposition IV.2.15], we see that F admits the following integral representation  ζ F(ζ ) = f (ν)dν, 0



where 0 f is the integral of f along any piecewise continuously differentiable curve in  with initial and end points 0 and ζ respectively. Note that whenever f ∈ Hb (, X), it follows from the basic property of boundedness of the integral that   1 F(ζ ) = f (ν)dν = f (tζ )ζ dt ≤ |ζ | f  for every ζ ∈ , [0,ζ ]

0

and consequently F ∈ Hb (, X) and F ≤  f . For the next results, recall that any mapping of class C1 is locally Lipschitz (cf. Corollary 5.2.14), and therefore any holomorphic vector field in an open set of a complex Banach space has a flow (cf. Theorem 5.4.9). Proposition 5.4.20 Let X be a complex Banach space, let  : X → X be a bounded holomorphic vector field, and let ϕ : D → X be the flow of . Then there exist positive numbers ε and δ and a holomorphic mapping ψ : ε×δX → X such that ψ(0, x) = x for every x ∈ δX , and ∂ ψ(ζ , x) = (ψ(ζ , x)) for every (ζ , x) ∈ ε × δX . ∂ζ Therefore ] − ε, ε[×δX ⊆ D, and ψ = ϕ on ] − ε, ε[×δX . Proof Note that for x ∈ 12 X and f ∈ 12 Hb (,X) , the function ν → (x + f (ν)) is a well-defined bounded holomorphic function from  to X, and hence its primitive

102 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem #ζ vanishing at 0, namely the function ζ → 0 (x + f (ν))dν, is also holomorphic and bounded. Therefore, we can consider the mapping 1 1  : C × X × Hb (,X) → Hb (, X) 2 2 defined by



ζ

(ω, x, f )(ζ ) = f (ζ ) − ω

(x + f (ν))dν for every ζ ∈ .

0

Now, note also the following facts: – The mappings π1 : (ω, x, f ) → ω, π2,3 : (ω, x, f ) → (x, f ), and π3 : (ω, x, f ) → f are bounded linear operators from C × X × Hb (, X) to C, X × Hb (, X), and Hb (, X), respectively, and hence π1 , π2,3 , and π3 are holomorphic and Dπ1 (ω, x, f ) = π1 , Dπ2,3 (ω, x, f ) = π2,3 , and Dπ3 (ω, x, f ) = π3 for every (ω, x, f ) ∈ C × X × Hb (, X). – The mapping α : (x, f ) → x + f is a bounded linear operator from X × Hb (, X) to Hb (, X), hence α is holomorphic and Dα(x, f ) = α for every (x, f ) ∈ X × Hb (, X). – By Lemma 5.4.17, the mapping ∗ : f →  ◦ f from Hb (,X) to Hb (, X) is holomorphic and D∗ ( f )(g)(ζ ) = D( f (ζ ))(g(ζ )) for all f ∈ Hb (,X) , g ∈ Hb (, X), ζ ∈ . #ζ – The mapping β : Hb (, X) → Hb (, X) determined by β( f )(ζ ) = 0 f (ν)dν is a bounded linear operator, hence β is holomorphic and Dβ( f ) = β for every f ∈ Hb (, X). – The mapping γ : (ω, f ) → ωf is a bounded bilinear operator from C × Hb (, X) to Hb (, X), hence γ is holomorphic and Dγ (ω, f )(ν, g) = ωg + νf for all (ω, f ), (ν, g) ∈ C × Hb (, X). Since  = π3 − γ ◦ (π1 , β ◦ ∗ ◦ α ◦ π2,3 ), where it is understood that π1 , π2,3 and π3 are restricted to C × 12 X × 12 Hb (,X) , it follows that  is holomorphic. Moreover, the rules of differentiation give that  ζ D(ω, x, f )(ν, y, g)(ζ ) = g(ζ ) − ω D(x + f (μ))(y + g(μ))dμ  −ν

0 ζ

(x + f (μ))dμ

0

for all (ω, x, f ) ∈ C × 12 X × 12 Hb (,X) , (ν, y, g) ∈ C × X × Hb (, X), and ζ ∈ . Therefore the partial derivative  ζ ∂ D(x + f (μ))(g(μ))dμ (ω, x, f )(g)(ζ ) = g(ζ ) − ω ∂f 0

5.4 Complete holomorphic vector fields

103

for all (ω, x, f ) ∈ C × 12 X × 12 Hb (,X) , g ∈ Hb (, X), and ζ ∈ , and in particular ∂ ∂ ∂f (0, 0, 0) = IHb (,X) . Since (0, 0, 0) = 0 and ∂f (0, 0, 0) is an isomorphism from Hb (, X) onto itself, by the implicit function theorem, there exist positive numbers ε and δ and a holomorphic mapping φ : ε × δX → Hb (, X) such that (ω, x, φ(ω, x)) ∈ C × 12 X × 12 Hb (,X) and (ω, x, φ(ω, x)) = 0

(5.4.5)

for every (ω, x) ∈ ε × δX . Therefore for each (ω, x) ∈ ε × δX and ζ ∈  we have that  ζ φ(ω, x)(ζ ) = ω (x + φ(ω, x)(μ))dμ, (5.4.6) 0

and consequently ∂ φ(ω, x)(ζ ) = ω(x + φ(ω, x)(ζ )). ∂ζ

(5.4.7)

Note as a consequence of the first assertion in (5.4.5) we have δ ≤ 12 , and  ε also  that 1 φ 2 , x ∈ 2 Hb (,X) for every x ∈ δX . Therefore we have for each ζ ∈ 2ε  and    x ∈ δX that 2ζε ∈ , x ∈ 12 X , and x + φ 2ε , x 2ζε ∈ X . Hence we can consider the mapping ε ψ :  × δX → X 2    defined by ψ(ζ , x) = x + φ 2ε , x 2ζε . Taking into account that the constant mappings and the projections π1 and π2 of the product space C × X are holomorphic; the mapping σ : (x, y) → x + y from X × X to X is a bounded linear operator, and hence holomorphic; and the mapping E : ( f , ζ ) → f (ζ ) from Hb (, X) ×  to X is holomorphic (by Lemma 5.4.18), it follows from the equality  !  " 2  ε ψ = σ ◦ π2 , E ◦ φ ◦ , π2 , π1 2 ε that ψ is holomorphic. Moreover, it follows from (5.4.6) and (5.4.7) that for each ζ ∈ 2ε  and x ∈ δX we have ψ(0, x) = x and

∂ ψ(ζ , x) = (ψ(ζ , x)). ∂ζ

Therefore ] − 2ε , 2ε [×δX ⊆ D and ψ|]− 2ε , 2ε [×δX = ϕ|]− 2ε , 2ε [×δX , and the proof is complete.  Corollary 5.4.21 Let X be a complex Banach space, let  be an open of X, let  :  → X be a holomorphic vector field, and let ϕ : D →  be the flow of . Then for each x0 in  there exist positive numbers ε and δ and a holomorphic mapping ψ : ε × (x0 + δX ) →  such that for each ζ ∈ ε and x ∈ x0 + δX we have ψ(0, x) = x and

∂ ψ(ζ , x) = (ψ(ζ , x)), ∂ζ

and consequently ] − ε, ε[×(x0 + δX ) ⊆ D and ψ = ϕ on ] − ε, ε[×(x0 + δX ).

104 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof

Let x0 ∈ . Since  is open and  is continuous, we can fix ρ > 0 such that x0 + ρX ⊆  and (x0 + ρX ) ⊆ (x0 ) + X .

Consider the bounded holomorphic vector field ϒ : X → X defined by ϒ(x) = (x0 + ρx). By Proposition 5.4.20, there exists positive numbers ε and δ and a holomorphic mapping φ : ε × δX → X such that for each ζ ∈ ε and x ∈ δX we have φ(0, x) = x and

∂ φ(ζ , x) = ϒ(φ(ζ , x)), ∂ζ

and consequently ] − ε, ε[×δX ⊆ D(ϒ). Consider the holomorphic mapping ψ : ερ × (x0 + δρX ) → x0 + ρX defined by



 1 1 ζ , (x − x0 ) , ψ(ζ , x) = x0 + ρφ ρ ρ

and note that

  1 1 ψ(0, x) = x0 + ρφ 0, (x − x0 ) = x0 + ρ (x − x0 ) = x for every x ∈ x0 + δρX , ρ ρ

and

     ∂ 1 1 ∂ 1 1 ψ(ζ , x) = ρ φ ζ , (x − x0 ) = ϒ φ ζ , (x − x0 ) ∂ζ ∂ζ ρ ρ ρ ρ    1 1 = (ψ(ζ , x)) ζ , (x − x0 ) =  x0 + ρφ ρ ρ

for every (ζ , x) ∈ ερ × (x0 + δρX ). It follows from the uniqueness of the flows having the same initial condition that ] − ερ, ερ[×(x0 + δρX ) ⊆ D() and ψ = ϕ on ] − ερ, ερ[×(x0 + δρX ), as required.  Now, we can adapt the arguments in the first paragraph in the proof of Theorem 5.4.9 to prove the following result. Theorem 5.4.22 Let X be a complex Banach space, let  be an open set of X, let  :  → X be a holomorphic vector field, and let ϕ : D →  be the flow of . Then for each (t0 , x0 ) ∈ D there exist positive numbers ε and δ and a holomorphic mapping φ : (t0 + ε) × (x0 + δX ) → X such that ]t0 − ε, t0 + ε[×(x0 + δX ) ⊆ D and φ = ϕ on ]t0 − ε, t0 + ε[×(x0 + δX ). Therefore ϕ is real analytic on D and, for each t ∈ R such that Dt = ∅, the mapping exp(t) : Dt → D−t is biholomorphic.

5.4 Complete holomorphic vector fields

105

Proof Fix x0 ∈ . By Corollary 5.4.21, the statement holds for t0 = 0. In order to verify the statement for all t0 ∈]0, t+ (x0 )[, consider the set Q of all r ∈]0, t+ (x0 )[ such that for each t ∈]0, r[ there exist positive numbers ε and δ and a holomorphic function φ defined on the open set (t + ε) × (x0 + δX ) such that ]t − ε, t + ε[×(x0 + δX ) ⊆ D and φ = ϕ on ]t − ε, t + ε[×(x0 + δX ). Then Q is an interval, and again by Corollary 5.4.21 there exists ε > 0 such that ]0, ε[⊆ Q. We must prove that Q =]0, t+ (x0 )[. This is clear whenever Q is not bounded from above. Suppose that Q is bounded from above, and let b be its least upper bound. We must prove that b = t+ (x0 ). Suppose that this is not the case, and put x1 = ϕ(b, x0 ). By Corollary 5.4.21, there exist ε1 , δ1 > 0 and a holomorphic mapping ψ1 : ε1  × (x1 + δ1 X ) →  such that we have for ζ ∈ ε1  and x ∈ x1 + δ1 X that ψ1 (0, x) = x and

∂ ψ1 (ζ , x) = (ψ1 (ζ , x)), ∂ζ

and consequently ] − ε1 , ε1 [×(x1 + δ1 X ) ⊆ D and ψ1 = ϕ on ] − ε1 , ε1 [×(x1 + δ1 X ). By continuity of ϕ, we can find a positive number ρ with ρ < min{b, ε1 } and so small that whenever t ∈]b − ρ, b[ we have ϕ(t, x0 ) ∈ x1 + δ1 X . Select a point t0 ∈]b − ρ, b[. Then x1 + δ1 X is an open neighbourhood of ϕ(t0 , x0 ) contained in  and, by the hypothesis on b, there exist ε0 , δ0 > 0 and a holomorphic mapping ψ0 : (t0 + ε0 ) × (x0 + δ0 X ) → x1 + δ1 X such that ]t0 − ε0 , t0 + ε0 [×(x0 + δ0 X ) ⊆ D and ψ0 = ϕ on ]t0 − ε0 , t0 + ε0 [×(x0 + δ0 X ). We define φ : (t0 +ε1 )×(x0 +δ0 X ) →  by φ(ζ , x) := ψ1 (ζ −t0 , ψ0 (t0 , x)). Then φ is a holomorphic mapping and we have for t ∈]t0 − ε1 , t0 + ε1 [ and x ∈ x0 + δ0 X that φ(t, x) = ψ1 (t − t0 , ψ0 (t0 , x)) = ϕ(t − t0 , ϕ(t0 , x)) = ϕ(t, x). Hence ]t0 − ε1 , t0 + ε1 [×(x0 + δ0 X ) ⊆ D and φ = ϕ on ]t0 − ε1 , t0 + ε1 [×(x0 + δ0 X ), and so φ is a holomorphic extension of ϕ|]t0 −ε1 ,t0 +ε1 [×(x0 +δ0 X ) . Since ρ < ε1 , we see that b ∈]t0 − ε1 , t0 + ε1 [, and consequently t0 + ε1 ∈ Q, which is a contradiction. Thus b cannot be strictly smaller than the positive lifetime of x0 . Similarly, one proves the statement for all t0 ∈]t− (x0 ), 0[, and the proof is complete.  §5.4.23 Let X and Y be complex Banach spaces, let U ⊆  be open subsets of X, and let  be a holomorphic vector field on . Then  can be regarded as a

106 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem ‘differential operator’ on the space H (U, Y) by considering for each f ∈ H (U, Y) the holomorphic mapping ( f ) : U → Y defined by ( f )(x) := Df (x)((x)) for every x ∈ U.

(5.4.8)

Indeed, given f ∈ H (U, Y), the mapping F : U → BL(X, Y) × X defined by F(x) := (Df (x), (x)) is holomorphic (because its component mappings (Df , ) are holomorphic), and DF(x) = (D(Df )(x), D(x)). On the other hand, the continuous bilinear mapping E : BL(X, Y) × X → Y defined by E(F, x) := F(x) is holomorphic and we have DE(F, x)(G, z) = E(F, z) + E(G, x) = F(z) + G(x). Since ( f ) = E ◦ F, it follows from the chain rule that ( f ) is holomorphic and D(( f ))(x)(z) = Df (x)(D(x)(z)) + D2 f (x)((x), z)

(5.4.9)

for all x ∈ U and z ∈ X. Note also that the law (, f ) → ( f ) is a bilinear mapping from H (, X) × H (U, Y) to H (U, Y). ∂ According to (5.4.8), it is frequent in the literature represent by the symbol  ∂x to the holomorphic vector field  when one wishes to emphasize that  is being seen ∂ ’ notation for convenience. as a differential operator, however we will drop the ‘ ∂x With this abuse of notation in mind, in the case X = Y, it should be noted that (x) = (IU )(x) for every x ∈ U.

(5.4.10)

In the next statement we will involve the associative complex algebra L(H (, Y)) of all linear mappings from H (, Y) into itself, as well as its associated Lie algebra (H (, Y))ant . In the particular case Y = C, this Lie algebra can be replaced by the Lie algebra Der(H (, C)) of all derivations of the associative and commutative algebra H (, C). Proposition 5.4.24 Let X be a complex Banach space, and let  be an open subset of X. Then H (, X) is a Lie complex algebra for the Lie bracket defined by [, ](x) := D(x)((x)) − D(x)((x))

(5.4.11)

for all ,  ∈ H (, X) and x ∈ . Moreover, given a complex Banach space Y, for each  ∈ H (, X), the mapping () : f → ( f ) becomes a linear operator on H (, Y), and the mapping  : H (, X) → (L(H (, Y)))ant is an injective Lie homomorphism. In the particular case that Y = C, for each  ∈ H (, X), the mapping () is a derivation of the associative algebra H (, C), and the mapping  : H (, X) → Der(H (, C)) is an injective Lie homomorphism. Proof Given a complex Banach space Y, it is clear that () ∈ L(H (, Y)) for every  ∈ H (, X), and that  is a linear mapping from H (, X) to L(H (, Y)).

5.4 Complete holomorphic vector fields

107

Moreover, given norm-one elements y in Y and x in X  , we see that f : x → x (x)y is a holomorphic mapping from  to Y such that, for each  ∈ H (, X), ()( f )(x) = ( f )(x) = Df (x)((x)) = x ((x))y for every x ∈ . Therefore  is injective as a consequence of the Hahn–Banach theorem. Given ,  ∈ H (, X), by (5.4.9), we have for all f ∈ H (, Y) and x ∈  that [(), ()]( f )(x) = (() ◦ ())( f )(x) − (() ◦ ())( f )(x) = (( f ))(x) − (( f ))(x) = D(( f ))(x)((x)) − D(( f ))(x)((x)) = Df (x)(D(x)((x))) + D2 f (x)((x), (x)) − Df (x)(D(x)((x))) − D2 f (x)((x), (x)), and keeping in mind the symmetry of D2 f (x) we conclude that [(), ()]( f )(x) = Df (x)([, ](x)) = [, ]( f )(x) = ([, ])( f )(x). Therefore ([, ]) = [(), ()] for all ,  ∈ H (, X). Thus, via , we can regard H (, X) with the product (5.4.11) as a subalgebra of the Lie complex algebra (L(H (, Y)))ant , and consequently H (, X) with the product (5.4.11) becomes a Lie complex algebra. Finally, suppose that Y = C. Since D( fg)(x)(y) = Df (x)(y)g(x) + f (x)Dg(x)(y) for all f , g ∈ H (, C), x ∈ , and y ∈ X, it follows that, for each  ∈ H (, X), the linear mapping () : H (, C) → H (, C) is in fact a derivation. Thus  becomes an injective Lie homomorphism from H (, X) to Der(H (, C)), and the proof is complete.  Lemma 5.4.25 Let X be a complex Banach space, let  be an open subset of X, let  be in H (, X), and let ϕ : D →  be the flow of . Then we have: % & ∂ ∂ ∂ ∂ (i) ∂t ∂x ϕ(t, x) = D(ϕ(t, x)) ◦ ∂x ϕ(t, x), and ∂x ϕ(0, x) = IX . (ii) If  ∈ H (, X), then for each (t0 , x0 ) ∈ D we have  d  Dϕ−t (ϕ(t, x0 ))((ϕ(t, x0 ))) = Dϕ−t0 (ϕ(t0 , x0 ))([, ](ϕ(t0 , x0 ))). dt t=t0 Proof

Since ϕ is real analytic in D (cf. Theorem 5.4.22), we have . . ∂ ∂ ∂ ∂ ∂ [(ϕ(t, x))] = D( ◦ ϕt )(x) ϕ(t, x) = ϕ(t, x) = ∂t ∂x ∂x ∂t ∂x = D(ϕt (x)) ◦ Dϕt (x) = D(ϕ(t, x)) ◦

Moreover,

∂ ∂x ϕ(0, x) = Dϕ0 (x) = DI (x) = IX .

∂ ϕ(t, x). ∂x

Thus assertion (i) is proved.

108 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem ∂ In what follows, for simplicity, we will set F = ∂x ϕ, that is F : D → BL(X) is the ∂ real analytic mapping defined by F(t, x) := ∂x ϕ(t, x) for every (t, x) ∈ D, and hence its partial derivative mappings

∂ ∂ F : D → BL(R, BL(X)) and F : D → BL(X, BL(X)) ∂t ∂x are also real analytic. As usual, we identify the spaces BL(R, BL(X)) ≡ BL(X), so that ∂ ∂ ∂t F(t, x) ≡ ∂t F(t, x)(1), and consequently DF : D → BL(R × X, BL(X)) is given by ∂ ∂ F(t, x) + F(t, x)(y) for all (t, x) ∈ D and (s, y) ∈ R × X. ∂t ∂x In order to prove assertion (ii), let us fix x0 ∈ . Keeping in mind Theorem 5.4.9(ii), we can consider the functions DF(t, x)(s, y) = s

β : dom(ϕx0 ) → D and α : dom(ϕx0 ) → BL(X) defined by β(t) := (−t, ϕ(t, x0 )) and α(t) := F(β(t)). Since β is differentiable and β  (t) = (−1, (ϕ(t, x0 ))), the chain rule guarantees that α is differentiable and α  (t) = (F ◦ β) (t) = D(F ◦ β)(t)(1) = DF(β(t))(Dβ(t)(1)) = DF(β(t))(β  (t)) = DF(β(t))(−1, (ϕ(t, x0 ))) = −

∂ ∂ F(β(t)) + F(β(t))((ϕ(t, x0 ))), ∂t ∂x

and keeping in mind (i) ∂ F(−t, ϕ(t, x0 ))((ϕ(t, x0 ))). ∂x On the other hand, given  ∈ H (, X), the function γ : dom(ϕx0 ) → X defined by γ (t) := (ϕ(t, x0 )) is differentiable and α  (t) = −D(ϕ(−t, ϕ(t, x0 ))) ◦ F(−t, ϕ(t, x0 )) +

γ  (t) = Dγ (t)(1) = D( ◦ ϕx0 )(t)(1) = (D(ϕx0 (t)) ◦ Dϕx0 (t))(1) = D(ϕx0 (t))(ϕx 0 (t)) = D(ϕ(t, x0 ))((ϕ(t, x0 ))). It follows that the function ζ : dom(ϕx0 ) → BL(X) × X defined by ζ (t) := (α(t), γ (t)) is differentiable and ζ  (t) = (α  (t), γ  (t)). Taking into account that the valuation mapping E : (T, x) → T(x) from BL(X) × X to X is bilinear and continuous, it follows that the function η : dom(ϕx0 ) → X defined by η(t) := α(t)(γ (t)) is differentiable and η (t) = Dη(t)(1) = D(E ◦ ζ )(t)(1) = (DE(ζ (t)) ◦ Dζ (t))(1) = DE(ζ (t))(ζ  (t)) = DE(α(t), γ (t))(α  (t), γ  (t)) = α(t)(γ  (t)) + α  (t)(γ (t)) = F(−t, ϕ(t, x0 ))[D(ϕ(t, x0 ))((ϕ(t, x0 )))] − D(ϕ(−t, ϕ(t, x0 )))[F(−t, ϕ(t, x0 ))((ϕ(t, x0 )))] . ∂ F(−t, ϕ(t, x0 ))((ϕ(t, x0 ))) ((ϕ(t, x0 ))). + ∂x

5.4 Complete holomorphic vector fields Since, by (i), F0 (x) = IX for every x ∈ , and consequently it follows that

109

∂ ∂x F(0, x0 ) = DF0 (x0 ) = 0,

η (0) = D(x0 )((x0 )) − D(x0 )((x0 )) = [, ](x0 ).

(5.4.12)

Given t0 ∈ dom(ϕx0 ), set x1 := ϕ(t0 , x0 ), and consider the functions α1 : dom(ϕx1 ) → BL(X), γ1 : dom(ϕx1 ) → X, and η1 : dom(ϕx1 ) → X defined by α1 (t) :=

∂ ϕ(−t, ϕ(t, x1 )), γ1 (t) := (ϕ(t, x1 )), and η1 (t) := α1 (t)(γ1 (t)). ∂x

Moreover, fix ε > 0 such that ] − ε, ε[⊆ dom(ϕx1 ) ∩ [dom(ϕx0 ) − t0 ], and note that for each t with |t| < ε we have ∂ ϕ(−(t0 + t), ϕ(t0 + t, x0 )) = Dϕ−(t0 +t) (ϕ(t0 + t, x0 )) ∂x = D(ϕ−t0 ◦ ϕ−t )(ϕ(t, ϕ(t0 , x0 ))) = Dϕ−t0 (ϕ(t0 , x0 )) ◦ Dϕ−t (ϕ(t, x1 ))

α(t0 + t) =

= α(t0 ) ◦ α1 (t) and γ (t0 + t) = (ϕ(t0 + t, x0 )) = (ϕ(t, ϕ(t0 , x0 ))) = (ϕ(t, x1 )) = γ1 (t), and hence η(t0 + t) = α(t0 + t)(γ (t0 + t)) = (α(t0 ) ◦ α1 (t))(γ1 (t)) = α(t0 )(η1 (t)). Since α(t0 ) ∈ BL(X), it follows that      d  d  d  η(t) =  η(t0 + t) =  [α(t0 )(η1 (t))] = α(t0 ) η1 (0) . dt t=t0 dt t=0 dt t=0 The proof is now concluded by applying (5.4.12) with η1 instead of η.



Proposition 5.4.26 Let X be a complex Banach space, let  be an open subset of X, let ,  be in H (, X), and let ϕ : D →  and ψ : D  →  be the flows of  and , respectively. Then the following conditions are equivalent: (i) [, ] = 0. (ii)  ◦  =  ◦  (whenever  and  are regarded as differential operators on H (, X)). (iii)  is invariant under the flow of , that is Dϕt (x)((x)) = (ϕ(t, x)) for every (t, x) ∈ D . (iv)  is invariant under the flow of , that is Dψs (x)((x)) = (ψ(s, x)) for every (s, x) ∈ D  .

110 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof

Given f ∈ H (, X) and x ∈ , using (5.4.9), we see that ( ◦ )( f )(x) = (( f ))(x) = D(( f ))(x)((x)) = Df (x)(D(x)((x))) + D2 f (x)((x), (x)),

and analogously ( ◦ )( f )(x) = Df (x)(D(x)((x))) + D2 f (x)((x), (x)). Since D2 f (x) is symmetric, it follows that ( ◦  −  ◦ )( f )(x) = Df (x)(D(x)((x)) − D(x)((x))) = Df (x)([, ](x)), which gives the implication (i)⇒(ii). Moreover, by taking f = I , we deduce the converse implication. Thus we have proven the equivalence (i)⇔(ii). Suppose that [, ] = 0. Given x0 ∈ , consider the function η : dom(ϕx0 ) → X defined by η(t) := Dϕ−t (ϕ(t, x0 ))((ϕ(t, x0 ))). It follows from Lemma 5.4.25(ii) that η = 0, and hence η is constant. Since η(0) = DI (x0 )((x0 )) = IX ((x0 )) = (x0 ), we have η(t) = (x0 ) for every t ∈ dom(ϕx0 ). Therefore Dϕt (x0 )((x0 )) = Dϕt (x0 )(η(t)) = D(ϕt ◦ ϕ−t )(ϕt (x0 ))((ϕ(t, x0 ))) = IX ((ϕ(t, x0 ))) = (ϕ(t, x0 )) for every t ∈ dom(ϕx0 ), and we have proven the implication (i)⇒(iii). Now, to prove the converse implication, suppose that  is invariant under ϕ. Given x0 ∈  and t ∈ dom(ϕx0 ), applying Dϕ−t (ϕt (x0 )) to both sides in the equality Dϕt (x0 )((x0 )) = (ϕt (x0 )), we conclude that (x0 ) = Dϕ−t (ϕt (x0 ))((ϕt (x0 ))). Therefore  d 0 =  Dϕ−t (ϕt (x0 ))((ϕt (x0 ))), dt t=t0 and hence, by Lemma 5.4.25(ii), [, ](x0 ) = 0. The arbitrariness of x0 in  gives that [, ] = 0. Thus we have proven the equivalence (i)⇔(iii). Finally, the anticommutativity of the Lie bracket and the equivalence (i)⇔(iii) yield to the equivalence (i)⇔(iv).  Corollary 5.4.27 Let X be a complex Banach space, let  be an open subset of X, let ,  be in H (, X) such that [, ] = 0, and let ϕ, ψ, and φ be the flows of , , and  + , respectively. If x0 ∈ , and if I and J are open intervals containing [0, 1] such that ϕ(t, ψ(s, x0 )) is defined for every (t, s) ∈ I × J, then φ(1, x0 ) is defined and φ(1, x0 ) = ϕ(1, ψ(1, x0 )).

5.4 Complete holomorphic vector fields

111

Proof Suppose that x0 ∈ , and I and J are open intervals containing [0, 1] such that ϕ(t, ψ(s, x0 )) is defined for every (t, s) ∈ I × J. Set K := I ∩ J, and consider the functions α : K → K ×  and γ : K →  defined by α(t) := (t, ψ(t, x0 )) and γ (t) := ϕ(α(t)). Since α is differentiable and α  (t) = (1, ψx 0 (t)), the chain rule gives that γ is differentiable and γ  (t) = Dϕ(α(t))(α  (t)) = Dϕ(α(t))(1, ψx 0 (t)) = D1 ϕ(α(t)) + D2 ϕ(α(t))(ψx 0 (t)). Note that D1 ϕ(α(t)) = (ϕ(α(t))) = (γ (t)) and, keeping in mind that  is invariant under ϕ (cf. Proposition 5.4.26), D2 ϕ(α(t))(ψx 0 (t)) = D2 ϕ(t, ψ(t, x0 ))((ψ(t, x0 ))) = (ϕ(t, ψ(t, x0 ))) = (γ (t)). Therefore γ  (t) = ( + )(γ (t)) for every t ∈ K, hence γ is a local flow of  +  at the point γ (0) = x0 , and so (1, x0 ) ∈ dom(φ) and φ(1, x0 ) = γ (1) = ϕ(1, ψ(1, x0 )).  Proposition 5.4.28 Let X be a complex Banach space, let  be an open subset of X, let ,  be in H (, X), and let ϕ : D →  and ψ : D  →  be the flows of  and , respectively. Then the following conditions are equivalent: (i) [, ] = 0. (ii) The flows of  and  commute, that is: If x ∈ , and if I and J are open intervals containing 0 such that one of the expressions ϕ(t, ψ(s, x)) or ψ(s, ϕ(t, x)) is defined for every (t, s) ∈ I × J, then both are defined and they are equal. Proof Suppose that  and  commute, and that x0 ∈  and I and J are open intervals containing 0 such that (ϕt ◦ ψs )(x0 ) is defined for every (t, s) ∈ I × J. (The same proof with  and  reversed works under the assumption that the other expression is defined on such rectangle.) By Proposition 5.4.26, the hypotheses implies that  is invariant under ϕ. Fix any t ∈ I, and consider the curve γ : J →  defined by γ (s) := ϕ(t, ψ(s, x0 )). This curve satisfies γ (0) = ϕ(t, x0 ) and γ  (s) = (ϕt ◦ ψx0 ) (s) = D(ϕt ◦ ψx0 )(s)(1) = Dϕt (ψ(s, x0 ))(Dψx0 (s)(1)) = Dϕt (ψ(s, x0 ))(ψx 0 (s)) = Dϕt (ψ(s, x0 ))((ψ(s, x0 ))) = (ϕ(t, ψ(s, x0 ))) = (γ (s)). Therefore γ is a local flow of  at ϕ(t, x0 ), and consequently J ⊆ dom(ψϕ(t,x0 ) ) and ψ(s, ϕ(t, x0 )) = γ (s) for every s ∈ J. Now, the arbitrariness of t in I gives that (ψs ◦ ϕt )(x0 ) is defined for every (t, s) ∈ I × J and (ψs ◦ ϕt )(x0 ) = (ϕt ◦ ψs )(x0 ). Thus ϕ and ψ commute. Conversely, suppose that the flows of  and  commute, and fix x0 ∈ . If ε > 0 is chosen small enough that (ψs ◦ ϕt )(x0 ) is defined whenever t, s ∈] − ε, ε[, then the

112 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem hypothesis guarantees that (ψs ◦ ϕt )(x0 ) = (ϕt ◦ ψs )(x0 ) for all such t and s. This can be rewritten in the form ψϕt (x0 ) (s) = (ϕt ◦ ψx0 )(s). Differentiating this relation with respect to s, we get (ψ(s, ϕt (x0 ))) = Dϕt (ψ(s, x0 ))((ψ(s, x0 ))), and in particular for s = 0 we have (ϕt (x0 )) = Dϕt (x0 )((x0 )). Applying Dϕ−t (ϕ(t, x0 )) to both sides, we conclude Dϕ−t (ϕ(t, x0 ))((ϕ(t, x0 ))) = (x0 ). Now, differentiating with respect to t at t = 0 and invoking Lemma 5.4.25(ii) we obtain that [, ](x0 ) = 0. The arbitrariness of x0 in  shows that [, ] = 0, and the proof is complete.  Proposition 5.4.29 Let X be a complex Banach space, let  be an open subset of X, let x0 be in , and let ,  be in H (, X) such that [, ] = 0. If I and J are open intervals containing 0 such that exp(t)(exp(s)(x0 )) is defined for every (t, s) ∈ I × J, then exp(t + s)(x0 ) and exp(s)(exp(t)(x0 )) are defined and exp(t + s)(x0 ) = exp(t)(exp(s)(x0 )) = exp(s)(exp(t)(x0 ))

(5.4.13)

for every (t, s) ∈ I × J. Proof By Proposition 5.4.28, exp(s)(exp(t)(x0 )) is defined and the second equality in (5.4.13) holds for every (t, s) ∈ I × J. In order to prove the remaining part of the conclusion, let us fix (t0 , s0 ) ∈ I × J with t0 = 0 and s0 = 0. It is clearly that t0 , s0  ∈ H (, X) satisfy [t0 , s0 ] = 0, and exp(tt0 )(exp(ss0 )(x0 )) is defined for every (t, s) ∈ t10 I × s10 J. Since t10 I and s10 J are open intervals containing [0, 1], it follows from Corollary 5.4.27 that exp(t0  + s0 )(x0 ) is defined and exp(t0  + s0 )(x0 ) = exp(t0 )(exp(s0 )(x0 )).

(5.4.14)

The proof concludes keeping in mind the arbitrariness of (t0 , s0 ) in I × J with t0 = 0 and s0 = 0, and the fact that clearly exp(t0  + s0 )(x0 ) is defined and verifies (5.4.14) whenever (t0 , s0 ) in I × J and either t0 = 0 or s0 = 0.  §5.4.30 Let X and Y be complex Banach spaces, let X and Y be open subsets of X and Y, respectively, and let g : X → Y be a biholomorphic mapping. For each open subset U of X contained in X , we can consider the mappings g : H (U, X ) → H (g(U), Y ) and g♦ : H (U, X) → H (g(U), Y) defined respectively by g ( f ) := g ◦ f ◦ (g−1 )|g(U) and g♦ () := (g|U ) ◦ (g−1 )|g(U) . Exercise 5.4.31 Let X and Y be complex Banach spaces, let X and Y be open subsets of X and Y, respectively, let g : X → Y be a biholomorphic mapping, and let U be an open subset of X contained in X . Prove that: (i) g is a bijection from H (U, X ) onto H (g(U), Y ) such that (g )−1 = (g−1 ) . (ii) g♦ is a Lie algebra isomorphism from H (U, X) onto H (g(U), Y) such that (g♦ )−1 = (g−1 )♦ .

5.4 Complete holomorphic vector fields

113

(iii) For each  ∈ H (U, X), we have Dg♦ () = {(t, g(x)) : (t, x) ∈ D } and g (exp(t))(y) = exp(tg♦ ())(y) for every (t, y) ∈ Dg♦ () . Solution It is easy to verify assertion (i), as well as g♦ is a linear bijection from H (U, X) onto H (g(U), Y) with inverse (g−1 )♦ . Given y ∈ g(U), keeping in mind (5.4.9) and setting x := g−1 (y), note that for each  ∈ H (U, X) we have D(g♦ ())(y) = D((g) ◦ g−1 )(y) = D((g))(x) ◦ Dg−1 (y) = [Dg(x) ◦ D(x) + D(Dg)(x)((x))] ◦ Dg−1 (y). Hence, for any 1 , 2 ∈ H (U, X), we have D(g♦ (2 ))(y)(g♦ (1 )(y)) = D(g♦ (2 ))(y)((1 (g) ◦ g−1 )(y)) = D(g♦ (2 ))(y)(Dg(x)(1 (x))) = [Dg(x) ◦ D2 (x) + D(Dg)(x)(2 (x))](1 (x)) = Dg(x)(D2 (x)(1 (x))) + D2 g(x)(2 (x), 1 (x)). Interchanging the role of 1 and 2 , and subtracting, we obtain [g♦ (1 ), g♦ (2 )](y) = D(g♦ (2 ))(y)(g♦ (1 )(y)) − D(g♦ (1 ))(y)(g♦ (2 )(y)) = Dg(x)[D2 (x)(1 (x)) − D1 (x)(2 (x))] = Dg(x)([1 , 2 ](x)) = [1 , 2 ](g)(x) = g♦ ([1 , 2 ])(y). So g♦ is a homomorphism of Lie algebras, and the proof of (ii) is finished. Let  ∈ H (U, X), and let ϕ : D → U stand for the flow of . Consider the set D = {(t, g(x)) : (t, x) ∈ D } and mapping ψ : D → g(U) defined by ψ(t, y) := g(ϕ(t, g−1 (y))), and note that ψ(0, y) = g(ϕ(0, g−1 (y))) = g(g−1 (y)) = y and

  ∂ ∂ ψ(t, y) = Dg(ϕ(t, g−1 (y))) ϕ(t, g−1 (y)) ∂t ∂t = Dg(ϕ(t, g−1 (y)))((ϕ(t, g−1 (y)))) = (g)(ϕ(t, g−1 (y))) = ((g) ◦ g−1 )(g(ϕ(t, g−1 (y)))) = g♦ ()(ψ(t, y)).

Therefore ψ is the restriction to D of the flow of g♦ (). But, interchanging the roles of (g, ) and (g−1 , g♦ ()), we see that in fact D = Dg♦ () , and so ψ is the flow of

114 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem g♦ (). Moreover, keeping in mind the relationship between the flow ϕ of  and the flow ψ of g♦ (), we realize that for each (t, y) ∈ Dg♦ () we have (g ◦ ϕt ◦ g−1 )(y) = g(ϕ(t, g−1 (y))) = ψ(t, y) = ψt (y), and hence (for the mappings g and g♦ relative to the open set dom(exp(t))) we find that g (exp(t))(y) = exp(tg♦ ())(y), 

as required.

Lemma 5.4.32 Let X be a complex Banach space, and let  be a bounded domain in X. Suppose that tλ and gλ are nets in R \ {0} and Aut(), respectively, such that tλ → 0, and

1 T (gλ − I ) −→  ∈ H (, X). tλ

We have: 1 −1 (i)  ∈ H0 (, X), gλ −→ I , g−1 λ −→ I , and tλ (gλ − I ) −→ −. (ii) If B is an open ball strictly inside , and if δ > 0 is such that Bδ  , then there exists λ0 such that for each λ ≥ λ0 we have B ⊆ gλ (B δ ) ∩ g−1 λ (B δ ). T

T

T

2

Proof

For each λ we have

2

1 tλ (gλ − I ) 

≤ |t1λ | diam(), and hence t1λ (gλ − I ) is T assumption, t1λ (gλ − I ) −→ , it follows

a net in Hb (, X) ⊆ H0 (, X). Since, by from Corollary 5.3.8 that  ∈ H0 (, X). Moreover, for any open ball C   we have   1 1 (gλ − I ) ≤ |tλ | (gλ − I ) −  + C −→ 0, gλ − I C = |tλ | tλ tλ C C

and hence T-limλ gλ = I . Since Aut() is a topological group for the T-topology (Theorem 5.3.22), we have also that g−1 λ −→ I . T

T

Let B   be an open ball, and let δ > 0 be so that Bδ  . Since gλ −→ I and δ g−1 λ −→ I , there exists λ0 such that for each λ ≥ λ0 we have gλ − I B < 2 and δ g−1 λ − I B < 2 , hence we have for each x ∈ B that gλ (x) = x + (gλ (x) − x) ∈ B δ T

2

−1 −1 and g−1 λ (x) = x + (gλ (x) − x) ∈ B δ , and so gλ (B) ∪ gλ (B) ⊆ B δ , that is to say 2

2

B ⊆ gλ (B δ ) ∩ g−1 λ (B 2δ ). Thus, we have proven assertion (ii). Now, for each λ ≥ λ0 2 and x ∈ B, set . 1 −1 (x)) − (g (x)) and βλ (x) := (x) − (g−1 αλ (x) := − (gλ − I ) (g−1 λ λ λ (x)), tλ and note that

" 1 ! −1 gλ − I (x) + (x) = αλ (x) + βλ (x). tλ

Taking into account that g−1 λ (x) ∈ B δ , we have αλ (x) ≤ 2

hence αλ B ≤

1 tλ

(gλ − I )−



2

1 tλ

(5.4.15) (gλ − I ) − 



2

,

, and so αλ B → 0. Moreover, keeping in mind

5.4 Complete holomorphic vector fields

115

that [x, gλ (x)] ⊆ B δ , Bδ  , and  is bounded on Bδ , the mean value theorem (Fact 2 5.2.13) and the Cauchy inequality (5.2.24) give −1 −1 βλ (x) ≤ g−1 λ (x) − x sup D(x + t(gλ (x) − x)) ≤ gλ − I B 0≤t≤1

2e Bδ , δ

2e hence βλ B ≤ g−1 λ − I B δ Bδ , and so βλ B → 0. Now, by (5.4.15) and the triangle inequality, we see that  t1λ (g−1 λ − I ) + B ≤ αλ B + βλ B , and we find that 1 −1 ·B (gλ − I ) −→ −. tλ

Finally, keeping in mind the arbitrariness of B  , we conclude that 1 −1 T (g − I ) −→ −, tλ λ 

and the proof of assertion (i) is also complete.

For a good understanding of the formulation of the next proposition, recall the notation introduced in §5.3.7. Proposition 5.4.33 Let X be a complex Banach space, let  be a bounded domain in X, let U ⊆  be an open subset of X, let B  U be an open ball, and let δ > 0 be so that Bδ  U. Suppose that tλ , gλ , and fλ are nets in R \ {0}, Aut(), and HbU (Bδ , X), respectively, such that tλ → 0,

·Bδ 1 T (gλ − I ) −→  ∈ H (, X), and fλ −→ f ∈ HbU (Bδ , X). tλ

Then there exists λ0 such that, for each λ ≥ λ0 , the mappings −1 1 tλ ( fλ ◦ gλ − fλ )

1 tλ ( fλ ◦ gλ − fλ )

and

are defined on an open subset of X containing B, and

1 1 ·B ·B ( fλ ◦ gλ − fλ ) −→ ( f ), and ( fλ ◦ g−1 λ − fλ ) −→ −( f ). tλ tλ Proof By Lemma 5.4.32(ii), there exists λ0 such that, for each λ ≥ λ0 , we have −1 B ⊆ gλ (B δ ) ∩ g−1 λ (B 2δ ), hence gλ (U) ∩ gλ (U) ∩ U is an open subset of X containing 2 B, in which the mappings 1 1 ( fλ ◦ gλ − fλ ) and ( fλ ◦ g−1 λ − fλ ) tλ tλ are defined. For each λ ≥ λ0 and x ∈ gλ (U) ∩ g−1 λ (U) ∩ U, set & 1% fλ (gλ (x)) − fλ (x) − Dfλ (x)(gλ (x) − x) , tλ   1 βλ (x) := Dfλ (x) (gλ (x) − x) − (x) , and γλ (x) := D( fλ − f )(x) ((x)) , tλ

αλ (x) :=

116 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem and note that 1 ( fλ ◦ gλ − fλ )(x) − ( f )(x) = αλ (x) + βλ (x) + γλ (x). tλ

(5.4.16)

For any x ∈ B, keeping in mind the inclusions [x, gλ (x)] ⊆ B δ ⊆ Bδ ⊆ U, and the fact 2 that fλ is bounded on Bδ , we deduce from the Taylor formula (Proposition 5.2.12) and the Cauchy inequality (5.2.24) that 1 1 gλ (x) − x2 sup D2 fλ (x + t(gλ (x) − x)) |tλ | 2 0≤t≤1  2 2e 1  fλ Bδ gλ (x) − x2 ≤ |tλ | δ  2 1 2e ≤ (gλ − I ) gλ − I B  fλ Bδ . tλ δ B

αλ (x) ≤

Since  t1λ (gλ − I )B → B , gλ − I B → 0, and  fλ Bδ →  f Bδ , we find that αλ B → 0. On the other hand, the Cauchy inequality (5.2.24) gives βλ (x) ≤ Dfλ (x) ≤

2e  fλ Bδ δ

1 (gλ (x) − x) − (x) tλ 1 (gλ − I ) −  , tλ B

and we find also that βλ B → 0. Analogously, 2e  fλ − f Bδ B , δ and hence γλ B → 0. Now, by (5.4.16) and the triangle inequality, we see that  t1λ ( fλ ◦ gλ − fλ ) − ( f )B ≤ αλ B + βλ B + γλ B , and conclude that γλ (x) ≤ D( fλ − f )(x)(x) ≤

1 ·B ( fλ ◦ gλ − fλ ) −→ ( f ). tλ T 1 −1 tλ (gλ − I ) −→ − (Lemma ·B that t1λ ( fλ ◦ g−1 λ − fλ ) −→ −( f ).

Finally, keeping in mind that changing roles, we have also

5.4.32(i)), by inter

§5.4.34 Let X and Y be complex Banach spaces, and let U and  be open subsets of X such that U ⊆ . The consideration of holomorphic vector fields on  as differential operators on H (U, Y) (cf. §5.4.23) provide us permission to speak of ‘composition’. Accordingly, given a holomorphic vector field  on , for n ∈ N ∪ {0}, the nth power n of  (as a differential operator on H (U, Y)) is defined inductively by 0 ( f ) = f and n+1 ( f ) = (n ( f )) for every f ∈ H (U, Y). Note that, if as usual ϕ : D →  stands for the flow of , then for each f ∈ H (U, Y) and (t, x) ∈ D with ϕ(t, x) ∈ U, the chain rule gives that d ( f ◦ ϕx )(t) = Df (ϕx (t))(ϕx (t)) = Df (ϕx (t))((ϕx (t))) = ( f )(ϕx (t)), dt

5.4 Complete holomorphic vector fields that is to say

d dt ( f

117

◦ ϕx ) = ( f ) ◦ ϕx , and arguing by induction we see that dn ( f ◦ ϕx ) = n ( f ) ◦ ϕx for every n ∈ N ∪ {0}. dtn

Since ϕx (0) = x, we obtain in particular for each x ∈ U that   dn ( f ◦ ϕx ) = n ( f )(x) for every n ∈ N ∪ {0}. dtn t=0 In the particular case that Y = X, taking f = I we get for each x ∈  that  dn  ϕx = n (I )(x) for every n ∈ N ∪ {0}. dtn t=0

(5.4.17)

(5.4.18)

(5.4.19)

Now, keeping in mind that ϕx is an analytic function (cf. Theorem 5.4.22), we deduce from the equalities (5.4.19) that ϕx (t) =

∞ n  t n=0

n!

n (I )(x) whenever |t| is small enough.

(5.4.20)

Lemma 5.4.35 Let X and Y be complex Banach spaces, let C be an open ball in X, and let ε be a positive number. If  ∈ H (Cε , X) is bounded on C, and if g ∈ Hb (Cε , Y), then (g) ∈ H (Cε , Y) is bounded on C and 1 (g)C ≤ C gCε . ε

(5.4.21)

Proof Suppose that  ∈ H (Cε , X) is bounded on C, and that g ∈ Hb (Cε , Y). By §5.4.23, we have that (g) ∈ H (Cε , Y). For each x ∈ C, we have x + εX  Cε , hence ε < rb (g, x), and invoking (5.2.23) we realize that 1 (g)(x) = Dg(x)((x)) ≤ Dg(x)(x) ≤ gCε C . ε Therefore (g) is bounded on C and (5.4.21) holds.



Lemma 5.4.36 Let X and Y be complex Banach spaces, let B be an open ball in X, let δ be a positive number, and let n be a natural number. If 1 , . . . , n ∈ Hb (Bδ , X) and f ∈ Hb (Bδ , Y), then (n ◦ · · · ◦ 1 )( f ) ∈ H (Bδ , Y) is bounded on B and ! n "n n Bδ · · · 1 Bδ  f Bδ . (5.4.22) (n ◦ · · · ◦ 1 )( f )B ≤ δ Proof Suppose that 1 , . . . , n ∈ Hb (Bδ , X) and f ∈ Hb (Bδ , Y). For n = 1 the statement follows from Lemma 5.4.35 with  = 1 , C = B, ε = δ, and g = f . Assume that n > 1. It follows from Lemma 5.4.35 for  = 1 , C = B (n−1)δ , ε = nδ , and g = f n that 1 ( f ) is bounded on B (n−1)δ and n

n 1 ( f )B (n−1)δ ≤ 1 B (n−1)δ  f Bδ . δ n n

118 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Arguing by induction on k with 2 ≤ k ≤ n − 1 and applying Lemma 5.4.35 for  = k , C = B (n−k)δ , ε = nδ , and g = (k−1 ◦ · · · ◦ 1 )( f ) we obtain that n (k ◦ · · · ◦ 1 )( f ) is bounded on B (n−k)δ and n

(k ◦ · · · ◦ 1 )( f )B (n−k)δ n

n ≤ k B (n−k)δ (k−1 ◦ · · · ◦ 1 )( f )B (n−k+1)δ . δ n n

Now, a new application of Lemma 5.4.35, in this case for  = n , C = B, ε = nδ , and g = (n−1 ◦ · · · ◦ 1 )( f ), gives that (n ◦ · · · ◦ 1 )( f ) is bounded on B and n (n ◦ · · · ◦ 1 )( f )B ≤ n B (n−1 ◦ · · · ◦ 1 )( f )B δ . δ n Linking these n inequalities, we get at the end of the process that ! n "n n B n−1 B δ · · · 1 B (n−1)δ  f Bδ . (n ◦ · · · ◦ 1 )( f )B ≤ δ n n Finally, since B ⊆ B δ ⊆ . . . ⊆ B (n−1)δ ⊆ Bδ , and consequently n

n

B ≤ B δ ≤ · · · ≤ B (n−1)δ ≤ Bδ n

n

for any holomorphic mapping  bounded on Bδ , the inequality (5.4.22) follows.



Corollary 5.4.37 Let X and Y be complex Banach spaces, let B be an open ball in X, let δ > 0, and let n ∈ N ∪ {0}. If  ∈ Hb (Bδ , X) and f ∈ Hb (Bδ , Y), then n ( f ) ∈ H (Bδ , Y) is bounded on B and ! n "n nBδ  f Bδ . (5.4.23) n ( f )B ≤ δ Proof The inequality (5.4.23) holds for n = 0 because 0 = IH (Bδ ,Y) and clearly  f B ≤  f Bδ . For each n ∈ N, the inequality (5.4.23) follows from (5.4.22) by taking k =  for every k ∈ {1, . . . , n}.  Proposition 5.4.38 Let X and Y be complex Banach spaces, let B be an open ball in X, and let δ > 0. Regarding each holomorphic vector field on Bδ as a differential operator on H (Bδ , Y), we have:  1 n (i) n≥0 n!  is a power series from Hb (Bδ , X) to BL(Hb (Bδ , Y), Hb (B, Y)) with radius of convergence ≥ δe , and hence defines a holomorphic mapping from δ Y), Hb (B, Y)). e Hb (Bδ ,X) to BL(Hb (Bδ , 1 n (ii) For each f ∈ Hb (Bδ , Y), n≥0 n!  ( f ) is a power series from Hb (Bδ , X) to Hb (B, Y) with radius of convergence ≥ δe , and hence defines a holomorphic mapping from δe Hb (Bδ ,X) to Hb (B, Y). Moreover, for each  ∈ Hb (Bδ , X) with  1 n Bδ < δe , the series n≥0 n!  ( f ) is absolutely convergent in the Banach space Hb (B, Y). Proof Set W := BL(Hb (Bδ , Y), Hb (B, Y)). For n = 0, consider the constant mapping F0 :  → IHb (Bδ ,Y) from Hb (Bδ , X) to W, and, for each n ∈ N, consider the mapping

5.4 Complete holomorphic vector fields

119

Fn : Hb (Bδ , X)× . n. . ×Hb (Bδ , X) → W 1 defined by Fn (1 , . . . , n ) := n! (1 ◦ · · · ◦ n ). It follows from Lemma 5.4.36 that, for each n ∈ N ∪ {0}, Fn is a continuous n-linear mapping with associated 1 n n-homogeneous polynomial Pn : Hb (Bδ , X) → W given by Pn () := n!  . There 1 n fore n≥0 n!  is a power series centred at 0 from Hb (Bδ , X) to W. Since, by    n 1 1 n n (5.4.23), we have Pn  ≤ n! ≤ δe , and hence lim supn→∞ Pn  n ≤ δe , it δ follows from Proposition 5.2.29 that the radius of convergence of the power series  δ 1 n n≥0 n!  is ≥ e . Now, by Corollary 5.3.5, the mapping given by the sum of the

power series on δe Hb (Bδ ,X) is holomorphic, and assertion (i) is proved. Given f ∈ Hb (Bδ , Y), it is clear that the valuation mapping Ef : F → F( f ) is a continuous linear mapping from W to Hb (B, Y) and Ef  ≤  f Bδ , and hence, for each n ∈ N ∪ {0}, Ef ◦ Pn is a n-homogeneous polynomial from Hb (Bδ , X) to  1 n Hb (B, Y) with Ef ◦ Pn  ≤ Pn  f Bδ . Therefore, the power series n≥0 n!  (f ) δ from Hb (Bδ , X) to Hb (B, Y) has radius of convergence ≥ e , and hence defines a holomorphic mapping from δe Hb (Bδ ,X) to Hb (B, Y). Moreover, by Lemma 5.2.28,  1 n for each  ∈ Hb (Bδ , X) with Bδ < δe , the series n≥0 n!  ( f ) is absolutely convergent in Hb (B, Y).  Proposition 5.4.39 Let X and Y be complex Banach spaces, let  be a domain in X, let B be an open ball strictly inside , and let δ > 0 be such that Bδ ⊆ . Then there exists R with 0 < R < δe such that for each holomorphic vector field  on  which is bounded on Bδ with Bδ < R we have: (i) B ⊆ dom(exp()) and exp()(B) ⊆ Bδ .  1 n (ii) For each f ∈ Hb (Bδ , Y), f ◦ exp()|B = ∞ n=0 n!  ( f ) in Hb (B, Y). Proof First, we consider the particular case Y = X. By Proposition 5.4.38, the  1 n  (I ) from Hb (Bδ , X) to Hb (B, X) has radius of convergence power series n≥0 n! δ ≥ e , and hence defines a holomorphic mapping from δe Hb (Bδ ,X) to Hb (B, X). As  δ 1 n a consequence the mapping F :  → ∞ n=1 n!  (I ) from e Hb (Bδ ,X) to Hb (B, X) is holomorphic. Since F(0) = 0, if follows that there exists 0 < R < δe such that F()B < δ for every  ∈ Hb (Bδ , X) with Bδ < R. Therefore, for each  ∈ Hb (Bδ , X) with Bδ < R we have that ∞ ∞   1 n 1 n  (I )(x) = x +  (I )(x) ∈ Bδ for every x ∈ B. n! n! n=0

n=1

Now, given any holomorphic vector field  on  bounded on Bδ with Bδ < R,  tn n we have that the power series n≥0 n!  (I ) from R to Hb (B, X) has radius of R convergence ≥ τ := B > 1 and the mapping φ :] − τ , τ [×B → X defined by δ

φ(t, x) :=

∞ n  t n=0

n!

n (I )(x)

120 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem is in fact Bδ -valued. It follows from the principle of analytic continuation (Proposition 5.2.43) and (5.4.20) that, for each x ∈ B, φx is a local flow of  at x. Therefore, if ϕ : D →  is the flow of , then we have that ] − τ , τ [×B ⊆ D and φ = ϕ on ] − τ , τ [×B, and in particular φ1 = exp()|B . Hence B ⊆ dom(exp()), exp()(B) ⊆ Bδ , and exp()|B =

∞  1 n  (I ) in Hb (B, X). n! n=0

Now we deal with an arbitrary complex Banach space Y. Given f ∈ Hb (Bδ , Y), keeping in mind Proposition 5.4.38, we find that, for each x ∈ B, the power series  tn n n≥0 n!  ( f )(x) from R to Y has radius of convergence ≥ τ , and hence determines an analytic function ψx :] − τ , τ [→ Y defined by ψx (t) =

∞ n  t n=0

n!

n ( f )(x).

Since the local flow ϕx :] − τ , τ [→  of  at x is in fact Bδ -valued, we can consider the analytic function f ◦ (ϕx )|]−τ ,τ [ , which in view of (5.4.18) satisfies ( f ◦ ϕx )(n) (0) = n ( f )(x) for every n ∈ N ∪ {0}. Therefore f ◦ ϕx = ψx , and hence ∞  1 n ( f ◦ exp())(x) = ( f ◦ ϕx )(1) = ψx (1) =  ( f )(x). n! n=0

Finally, the arbitrariness of x in B allows us to derive that f ◦exp()|B = in Hb (B, Y), as desired.

∞

1 n n=0 n!  ( f )



Using the notation introduced in §5.4.14, we can state the next fact as a consequence of the above two propositions. Fact 5.4.40 Let X be a complex Banach space, let  be a domain in X, let B be an open ball strictly inside , and let δ be a positive number such that Bδ ⊆ . Then the power series centred at the origin from Hb (Bδ , X) to Hb (B, X) given by 1 n (I ) n!

(5.4.24)

n≥0

is convergent. Moreover, there exists R > 0 such that for each  ∈ RH  (Bδ ,X) we b have B ⊆ dom(exp()), exp()(B) ⊆ Bδ , and exp()|B =

∞  1 n  (I ). n! n=0

As a consequence, the mapping exp(·)|B : RH  (Bδ ,X) → Hb (B, X) is holomorphic b and, for each n ∈ N, the n differential of exp(·)|B at 0 is determined by Dn (exp(·)|B )(0)(, . n. ., ) = n (I )|B for every  ∈ Hb (Bδ , X).

5.4 Complete holomorphic vector fields

121

Moreover, given x0 ∈ B, we have for each  ∈ RH  (Bδ ,X) that b

exp()(x0 ) =

∞  n=0

1 n  (I )(x0 ), n!

and the mapping x0 : RH  (Bδ ,X) → X given by x0 () := exp()(x0 ) is holob morphic and Dn x0 (0)(, . n. ., ) = n (I )(x0 ) for all n ∈ N and  ∈ Hb (Bδ , X). Proof It follows from Proposition 5.4.39 that there exists R > 0 such that, for each  ∈ RH  (Bδ ,X) , we have that b

B ⊆ dom(exp()), exp()(B) ⊆ Bδ , and exp()|B =

∞  1 n  (I ) in Hb (B, X), n! n=0

and hence the mapping exp(·)|B : RH  (Bδ ,X) → Hb (B, X) is holomorphic. Now, b invoking Proposition 5.2.33, we realize that, for each n ∈ N, Dn (exp(·)|B )(0)(, . n. ., ) = n (I )|B for every  ∈ Hb (Bδ , X). Given x0 ∈ B, the valuation of elements of Hb (B, X) at x0 becomes a continuous lin 1 n ear mapping Ex0 from Hb (B, X) to X, and hence exp()(x0 ) = ∞ n=0 n!  (I )(x0 ) for each  ∈ RH  (Bδ ,X) . Finally, the chain rule gives that x0 := Ex0 ◦ exp(·)|B is b a holomorphic mapping from RH  (Bδ ,X) to X, and for each n ∈ N b

Dn x0 (0) = Dn (Ex0 ◦ exp(·)|B )(0) = Ex0 ◦ Dn (exp(·)|B )(0), and hence Dn x0 (0)(, . n. ., ) = n (I )(x0 ) for every  ∈ Hb (Bδ , X).



Lemma 5.4.41 Let X be a Banach space over K, and let (xm,n )(m,n)∈N×N be a double sequence in X such that for each m ∈ N the sequence (xm,n )n∈N converges to some xm ∈ X. Suppose in addition that there exists a sequence (am )m∈N of non-negative  real numbers such that the series m∈N am converges and xm,n  ≤ am for every   (m, n) ∈ N × N. Then the series m∈N xm and m∈N xm,n , for each n ∈ N, are absolutely convergent, and ∞  m=1

xm = lim

n→∞

∞ 

xm,n .

m=1

Proof By the assumptions, it is clear that xm  ≤ am for each m ∈ N. Therefore   the series m∈N xm,n , for each n ∈ N, and m∈N xm are in fact absolutely conver ε gent. Given ε > 0, there exists M ∈ N such that ∞ m=M+1 am < 4 . Moreover, since ε limn→∞ xm,n = xm for each m ∈ N, there exists n0 ∈ N such that xm,n − xm  < 2M for all n ≥ n0 and 1 ≤ m ≤ M. Therefore we have for each n ≥ n0 that

122 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem ∞  m=1

xm,n −

∞ 

xm =

m=1



∞ 

(xm,n − xm ) ≤

m=1 M 

xm,n − xm  +

m=1

∞ 

xm,n − xm 

m=1 ∞ 

2am ≤

m=M+1

M  ε ε + 2 = ε, 2M 4

m=1



and the proof is complete.

Proposition 5.4.42 Let X be a complex Banach space, let  be a bounded domain in X, let fn be a sequence in Aut(), and let tn be a null sequence of positive real numbers. Suppose that  ∈ H (, X) is such that 1 ( fn − I ) = , n→∞ tn

T- lim

(5.4.25)

that B open ball strictly inside  and δ > 0 are such that Bδ  , and that K > 0. Then there exists τ > 0 such that for each t ∈ [0, τ [, each f ∈ H (, X) such that [t]

 f Bδ ≤ K, and each n ∈ N, the mappings f ◦ fn tn and n ( f ) are bounded on B. Moreover, for any t ∈ [0, τ [ and for any f ∈ H (, X) such that  f Bδ ≤ K, in the Banach space (Hb (B, X),  · B ), we have:   tn n tn n (i) The series n≥0 n!  ( f ) is absolutely convergent, and ∞ n=1 n!  ( f )B < δ. t  [ ] tn n (ii) The sequence f ◦ fn tn converges to ∞ n=0 n!  ( f ). Proof For each n ∈ N, set hn := t1n ( fn − I ). Note that hn ∈ Hb (, X) because  is bounded, and hence  ∈ H0 (, X) (by Corollary 5.3.8). Moreover, for each n ∈ N, consider the linear mapping Hn on H (, X) defined by Hn (h) :=

1 (h ◦ fn − h) for every h ∈ H (, X). tn

We will break the proof into four steps. Step 1: Let U ⊆  be an open subset of X such that |U is bounded, let g be in H (, X) such that g|U is bounded, and let gn be a sequence in H (, X) converging to g uniformly on each open ball strictly inside U. Then Hn (gn ) converges to (g) uniformly on each open ball strictly inside U. Fix an open ball C  U and d > 0 such that Cd  U. Since hn converges to  uniformly on C and |C is bounded, tn is a null sequence, and gn converges to g uniformly on Cd and g|Cd is bounded, there exists m ∈ N such that d hn C ≤ 1 + C , tn hn C < , and gn Cd ≤ 1 + gCd for every n ≥ m. 2 Fix x ∈ C and n ≥ m. Then [x, x + tn hn (x)] + d2 X ⊆ U and, by Fact 5.4.16 and Proposition 5.3.15, we have

5.4 Complete holomorphic vector fields

123

gn (x + tn hn (x)) − gn (x) − Dgn (x)(tn hn (x)) ≤ tn hn (x) sup Dgn (x + stn hn (x)) − Dgn (x) 0≤s≤1

 2 4 gn Cd tn hn (x) ≤ tn hn (x) d  2 2 4 gn Cd hn 2C ≤ tn d  2 4 ≤ tn2 (1 + gCd )(1 + C )2 . d On the other hand, using the inequality (5.2.23), we realize that Dgn (x)(hn (x)) − Dg(x)((x)) = Dgn (x)(hn (x)) − Dg(x)(hn (x)) + Dg(x)(hn (x)) − Dg(x)((x)) ≤ D(gn − g)(x)(hn (x)) + Dg(x)(hn (x) − (x)) ≤ D(gn − g)(x)hn (x) + Dg(x)hn (x) − (x) 1 1 ≤ gn − gCd hn C + gCd hn − C d d & 1% ≤ (1 + C )gn − gCd + gCd hn − C . d Since 1 [gn ( fn (x)) − gn (x)] − Dg(x)((x)) tn 1 = [gn (x + tn hn (x)) − gn (x) − Dgn (x)(tn hn (x))] tn + Dgn (x)(hn (x)) − Dg(x)((x)),

Hn (gn )(x) − (g)(x) =

it follows that

 2 4 Hn (gn )(x) − (g)(x) ≤ tn (1 + gCd )(1 + C )2 d & 1% + (1 + C )gn − gCd + gCd hn − C , d

and so Hn (gn ) converges to (g) uniformly on C. Step 2: Suppose that C is an open ball concentric with B such that C2ε ⊆ Bδ for some ε > 0. If n ∈ N is such that  fn − I Bδ < ε, and if g ∈ H (, X) is such that g|C2ε is bounded, then Hn (g)|C is bounded and 1 Hn (g)C ≤ gC2ε hn C . ε Let n ∈ N be such that  fn − I Bδ < ε. Then [x, fn (x)] + εX ⊆ C2ε for every x ∈ C.

124 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Therefore, for any g ∈ H (, X) with gC2ε < ∞, by Proposition 5.3.15 we have Hn (g)(x) =

1 11 1 g( fn (x)) − g(x) ≤ gC2ε  fn (x) − x ≤ gC2ε hn C tn tn ε ε

for every x ∈ C, and the proof of Step 2 is complete. Since |Bδ is bounded, and, by (5.4.25), hn converges to  uniformly on Bδ , it follows that the hn ’s are uniformly bounded on Bδ . From now on, we set M := sup hn Bδ < ∞. n∈N

Step 3: Let f be in H (, X) such that f|Bδ is bounded, and let m be in N ∪ {0}. δ Then, for each n ∈ N such that mtn < 2M , the mapping Hnm ( f ) is bounded on B and   2mM m Hnm ( f )B ≤  f Bδ . (5.4.26) δ Moreover, the sequence Hnm ( f ) converges to m ( f ) uniformly on B. Consequently m ( f ) is bounded on B and   2mM m m ( f )B ≤  f Bδ . (5.4.27) δ We can suppose that m = 0. Set ε = that, for each n ∈ N such that mtn
mn } is finite. For each n ∈ N, the inequalities mn ≤ ttn < mn + 1 give that 0 ≤ t − mn tn < tn , hence mn tn → t as n → ∞, and more generally, for each k ∈ N ∪ {0}, (mn − k)tn → t as n → ∞. Therefore,   1 tm mn m (mn tn )[(mn − 1)tn ] · · · [(mn − (m − 1))tn ] → . tn = m m! m! On the other hand, for each n ∈ N such that m ≤ mn we have δ , 2M and invoking Step 3 we realize that Hnm ( f ) and m ( f ) are bounded on B and Hnm ( f ) converges to m ( f ) in the Banach space (Hb (B, X),  · B ). Therefore the sequence tm m (m,n )n∈N converges to m := m!  ( f ) in the Banach space (Hb (B, X),  · B ). Moreover, it follows from (5.4.26) that for each n ∈ N such that m ≤ mn      m   2teM m mn m m mn m 2mM m,n B =  f Bδ ≤  f Bδ , t H ( f )B ≤ t m n n m n δ δ 0 ≤ mtn ≤ mn tn ≤ t < τ
0 such that ht B ≤ M for every t ∈ R \ {0}. (ii) If {ft }t∈R is T-differentiable, then its generator lies in H0 (, X). Proof Suppose that {ft }t∈R is T-continuous, and fix an open ball B   and d > 0 such that Bd  . Since limt→0  ft − I Bd = 0, we can find τ > 0 so that we have d sup  ft − I Bd < . 3 |t|≤τ % & τ Fix t ∈] − τ2 , τ2 [\{0} arbitrarily, and set n := |t| . Then k|t| ≤ τ for every k with 1 ≤ k ≤ n, and hence d max  ftk − I Bd = max  fkt − I Bd < . 1≤k≤n 1≤k≤n 3 Using the inequality (5.3.10) in Lemma 5.3.31 with ( ft , d3 , d, n) instead of ( f , δ, d, p) we obtain that n  fnt − I − n( ft − I )B =  ftn − I − n( ft − I )B ≤  ft − I B , 2 whence by the triangle inequality we have  fnt − I B = n( ft − I ) + [fnt − I − n( ft − I )]B ≥ n ft − I B −  fnt − I − n( ft − I )B n ≥ n ft − I B −  ft − I B 2 n =  ft − I B , 2 and consequently ht B =

1 2  ft − I B ≤  fnt − I B . |t| n|t|

Since n|t| ≤ τ , we have  fnt − I Bd < d3 , and in particular  fnt − I B < d3 . Moreτ over, since |t| < n + 1 and |t| < τ2 we see that τ < n|t| + |t| < n|t| + τ2 , and hence τ 2 < n|t|. As a result, ht B ≤

2 d3 τ 2

=

4d . 3τ

On the other hand, in the case |t| ≥ τ2 , we have for each x ∈ B that ht (x) =

1 2  ft (x) − x ≤ diam(), |t| τ

and the proof of assertion (i) is finished. Assertion (ii) follows from the first assertion in Lemma 5.4.32(i).



130 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Theorem 5.4.50 Let X be a complex Banach space, let  be a bounded domain in X, and let {ft }t∈R be a one-parameter group in Aut(). Then the following conditions are equivalent: (i) { ft }t∈R is T-continuous. (ii) { ft }t∈R is T-differentiable. (iii) { ft }t∈R is analytic. Proof To prove the implication (i)⇒(ii), suppose that {ft }t∈R is T-continuous, and set 1 ht := ( ft − I ) for every t = 0. t First, we show that h 1 is a T-convergent sequence in H0 (, X). Let B   be an n open ball, and suppose that ε > 0 has been given. By Lemma 5.4.49(i), there exists M > 0 such that ht B ≤ M for every t ∈ R \ {0}. Fix d > 0 such that Bd  , and δ ε choose 0 < δ < d small enough to have 0 < d−δ ≤ 2M . By assumption we can find a number τ > 0 such that  ft − I Bd < δ for every t ∈] − τ , τ [. Let m, n ∈ N be large enough to have m, n > τ1 . Observe that for k ≤ max{m, n} we k have mn < τ , and hence f k − I B < δ. Therefore, (5.3.10) in Lemma 5.3.31 can d mn be applied to the automorphism f 1 and the natural number m to obtain mn

m − I − m( f f mn

1 mn

− I )

B

= f m1 − I − m( f mn

1 mn

− I )

B



ε m f 1 − I mn 2M

B

or, multiplying by n, h1 −h n

1 mn

In a similar way we obtain h 1 − h m

B 1 mn

≤ B

ε h1 2M mn

ε ≤ . 2

≤ 2ε , and by the triangle inequality we get

h1 −h 1 n

B

m

B

≤ ε.

Therefore the sequence h 1 is  · B -Cauchy in Hb (B, X). It follows from the arbin trariness of B   that h 1 is T-Cauchy in H0 (, X), and, invoking Corollary 5.3.8, n we can confirm the existence of  ∈ H0 (, X) such that h 1 T-converges to . n Second, we will prove that T- limt→0+ ht = . To this end, fix an open ball B   and d > 0 so that Bd  , and, invoking Lemma 5.4.49(i), take M > 0 so that ht B ≤ M for every t ∈ R \ {0}. Fix ε with 0 < %ε 0 is such that Bδ  , then  is a topological embedding from (H0 (, X),  · B ) into the normed space BL((H0 (, Y),  · Bδ ), (H0 (, Y),  · B )). More precisely, we have 1 1 B ≤ () ≤ B for every  ∈ H0 (, X). Bδ  δ Proof

Taking into account that the Lie product in H (, X) is given by [, ] = () − () for all ,  ∈ H (, X),

assertion (i) follows from Lemma 5.5.1 for Y = X. Moreover, by that lemma, we have for each  ∈ H0 (, X) that () maps H0 (, Y) into H0 (, Y), and hence assertion (ii) follows from Proposition 5.4.24. Given an open ball B strictly inside  and δ > 0 such that Bδ  , by Lemma 5.4.35, for each  ∈ H0 (, X) we have in fact that ( f )B ≤ 1δ B  f Bδ for every f ∈ H0 (, Y). Therefore () : (H0 (, Y),  · Bδ ) → (H0 (, Y),  · B ) is a bounded linear operator with () ≤ 1δ B , and hence also the linear mapping  : (H0 (, X),  · B ) → BL((H0 (, Y),  · Bδ ), (H0 (, Y),  · B )) is a bounded linear operator with  ≤ 1δ . On the other hand, given norm-one elements y ∈ Y and x ∈ X  , it is clear that the mapping f :  → Y given by f (x) := x (x)y lies in H0 (, Y) and we have for each  ∈ H0 (, X) that ()( f )(x) = ( f )(x) = Df (x)((x)) = x ((x))y for every x ∈ . Hence, for  ∈ H0 (, X) and x ∈ B, we see that x ((x))y = ()( f )(x) ≤ ()( f )B ≤ () f Bδ ≤ ()Bδ , and the Hahn–Banach theorem gives that (x) ≤ ()Bδ . As a consequence B ≤ ()Bδ , and the proof is complete.  From now on in this subsection we will focus on the study of the set aut() of all complete holomorphic vector fields on a bounded domain  in a complex Banach space. Proposition 5.5.3 Let X be a complex Banach space, let  be a domain in X, and let ,  be in aut() such that [, ] = 0. Then we have: (i) exp( + ) = exp() ◦ exp() = exp() ◦ exp(). (ii) The mapping h : R2 ×  →  defined by h(t, s, x) := exp(t + s)(x) is real analytic, and we have for each x ∈  that Dhx (0, 0)(t, s) = t(x) + s(x) and Dhx (t, s) = Dhh(t,s,x) (0, 0) for every (t, s) ∈ R2 .

5.5 Banach Lie structures for aut() and Aut()

139

Proof Proposition 5.4.29 gives assertion (i) straightforwardly, as well as the mapping h in assertion (ii) is defined by h(t, s, x) = ϕ(t, ψ(s, x)) = ψ(s, ϕ(t, x)) on whole R2 × , where ϕ and ψ denote the flows of  and , respectively. The first equality can be rewritten as h = ϕ ◦ (πR , ψ) ◦ (πR , πR× ), and hence, by Propositions 5.2.36 and 5.2.40 and Theorem 5.4.22, we obtain that h is real analytic. Finally, note that for any (t, s, x) ∈ R2 ×  we have ∂ ∂ hx (t, s) = ϕt (ψs (x)) = (ϕt (ψs (x))) = (h(t, s, x)) ∂t ∂t and ∂ ∂ hx (t, s) = ψs (ϕt (x)) = (ψs (ϕt (x))) = (h(t, s, x)). ∂s ∂s In particular, we have ∂ ∂ hx (0, 0) = (x) and hx (0, 0) = (x), ∂t ∂s and we derive that Dhx (0, 0)(t, s) = t(x)+s(x) and Dhx (t, s) = Dhh(t,s,x) (0, 0).



In general, for an unbounded domain , aut() is not closed under addition. However, for bounded domains we have the next result. Proposition 5.5.4 Let X be a complex Banach space, and let  be a bounded domain in X. Then we have: (i) aut() is a real subspace of H0 (, X). (ii)  · B1 and  · B2 are equivalent norms on aut(), for any open balls B1 and B2 strictly inside . (iii) (aut(),  · B ) is a Banach space, for any open ball B strictly inside . Proof By Proposition 5.4.48, we have the inclusion aut() ⊆ H0 (, X). Moreover, if  and  are in aut(), and if ϕ and ψ are its respective flows, then 1 1 T-lim (ϕt − I ) =  and T-lim (ψt − I ) = . t→0 t t→0 t Since T-limt→0 ϕt = I , it follows from Proposition 5.4.33 that 1 T-lim (ϕt ◦ ψt − ϕt ) = (I ) = . t→0 t Writing, for any t = 0, 1 1 1 (ϕt ◦ ψt − I ) = (ϕt − I ) + (ϕt ◦ ψt − ϕt ), t t t we realize that 1 T-lim (ϕt ◦ ψt − I ) =  + , t→0 t and invoking Theorem 5.4.51 we conclude that + ∈ aut(). Given a nonzero real 1 (ϕαt − I ) = α, and number α, it is clear that T-limt→0 1t (ϕαt − I ) = T-limt→0 α αt

140 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem invoking Theorem 5.4.51 again we obtain that α ∈ aut(). Thus aut() is a real subspace of H0 (, X), and assertion (i) is proved. Let B1 and B2 be open balls strictly inside . By Corollary 5.3.37, there exists a positive constant K such that  f − I B2 ≤ K f − I B1 for every f ∈ Aut(). Let  be in aut(), and let ϕ stand for the flow of . Keeping in mind Proposition 5.4.48, we see that ϕt − I B2 ≤ Kϕt − I B1 for every t ∈ R, hence 1 (ϕt − I ) t

≤K B2

1 (ϕt − I ) t

for every t ∈ R \ {0}, B1

and taking limits as t → 0 we conclude that B2 ≤ KB1 . It follows from the arbitrariness of  in aut() that  · B2 ≤ K · B1 on aut(). By interchanging roles, we also find a positive constant K  such that  · B1 ≤ K   · B2 on aut(). Thus  · B1 and  · B2 are equivalent norms on aut(), and assertion (ii) is proved. Let B be an open ball strictly inside . Suppose that n is a ·B -Cauchy sequence in aut(). Then, by (i) and (ii), n is a T-Cauchy sequence in H0 (, X), and hence, T

by Corollary 5.3.8, there exists  ∈ H0 (, X) such that n −→ . We must show that  lies in aut(). Clearly we may suppose that  = 0. Fix an open ball C strictly inside  and a positive number δ so that C2δ  . For each n ∈ N, let ϕn denote the flow of n , and set   1 fn := ϕn , · ∈ Aut(). n Since n Cδ → Cδ > 0, there exists n0 ∈ N such that n Cδ < r := 32 Cδ for every n ≥ n0 , and, by Proposition 5.4.39, there exists τ > 0 such that ϕn (t, x) ∈ Cδ for all n ≥ n0 and (t, x) ∈] − τ , τ [×C. Fix n1 ≥ n0 such that n11 < τ . Given n ≥ n1 and x ∈ C, using the Taylor formula (Proposition 5.2.12), we see that   1 1 1 fn (x) − x − n (x) = ϕn , x − ϕn (0, x) − n (ϕn (0, x)) n n n     1 1 , x − ϕn (0, x) − D(ϕn (·, x))(0) = ϕn n n   1 1 1 2 sup (ϕ (·, x)) t D ≤ n 2 n2 0≤t≤1 n 1 1 sup D2 (ϕn (·, x))(t). = 2 n2 0≤t≤ 1 n

5.5 Banach Lie structures for aut() and Aut()

141

By (5.4.17), we have for each t ∈ R that d2 (ϕn (·, x))(t) = (2n (I ) ◦ ϕn (·, x))(t) = 2n (I )(ϕn (t, x)) dt2 = n (n )(ϕn (t, x)) = Dn (ϕn (t, x))(n (ϕn (t, x))), 2

and hence, keeping in mind that  dtd 2 (ϕn (·, x))(t) = D2 (ϕn (·, x))(t), we derive that D2 (ϕn (·, x))(t) ≤ Dn (ϕn (t, x))n (ϕn (t, x)). Since ϕn (t, x) ∈ Cδ for each t ∈ [0, 1n ], it follows from the Cauchy inequality (5.2.24) that e D2 (ϕn (·, x))(t) ≤ Mn for every t ∈ [0, 1n ], where Mn = n C2δ n Cδ . δ Therefore  fn (x) − x − 1n n (x) ≤ 12 n12 Mn , hence n( fn (x) − x) − n (x) ≤ and the arbitrariness of x in C gives that n( fn − I ) − n C ≤ 12 n1 Mn . Since

11 2 n Mn ,

n( fn − I ) − C ≤ n( fn − I ) − n C + n − C ·C

and the sequence Mn is bounded, it follows that n( fn − I ) −→ , and the arbitrariness of C yields to T

n( fn − I ) −→ . Finally, by Theorem 5.4.51, we conclude that  ∈ aut(), and the proof is complete.  We remark that, according to Proposition 5.5.4, for any bounded domain  in a complex Banach space, we will always consider aut() as a real Banach space with respect to the norm  · B for any open ball B strictly inside  (which in many cases will not be specified). Cartan uniqueness theorem for Aut() (Proposition 5.3.25) allow us to give an elementary proof of the so-called Cartan’s uniqueness theorem for aut(). Like there, this result show that, for bounded domains, complete holomorphic vector fields are determined by their Taylor coefficients of order ≤ 1 at a single point in . Proposition 5.5.5 Let X be a complex Banach space, let  be a bounded domain in X, and let x0 be in . Suppose that ,  ∈ aut() satisfy (x0 ) = (x0 ) and D(x0 ) = D(x0 ). Then  = . Proof Consider ϒ :=  − . Then ϒ ∈ aut() (cf. Proposition 5.5.4(i)) satisfy ϒ(x0 ) = 0 and Dϒ(x0 ) = 0. To prove the statement, it is enough to show that ϒ = 0. Let ϕ : R ×  →  be the flow of ϒ. Since ϒ(x0 ) = 0, it follows that the constant

142 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem function ψ : R →  defined by ψ(t) := x0 is a local flow of ϒ at x0 . Therefore, by Lemma 5.4.6(ii), ϕx0 = ψ, i.e. ϕ(t, x0 ) = x0 for every t ∈ R.

(5.5.1)

Keeping in mind the first assertion in Lemma 5.4.25(i), and using (5.5.1) and the fact that Dϒ(x0 ) = 0, we see that     ∂ ∂ ∂ ∂ ϕ(t, x) = Dϒ(ϕ(t, x0 )) ◦ ϕ(t, x0 ) = Dϒ(x0 ) ◦ ϕ(t, x0 ) = 0. ∂t ∂x ∂x ∂x x=x0  ∂ So the function t → ∂x ϕ(t, x)x=x is constant, hence 0     ∂ ∂  = = Dϕ0 (x0 ) = DI (x0 ) = IX . ϕ(t, x) ϕ(0, x) ∂x ∂x x=x0 x=x0 Therefore, for any fixed t ∈ R, the mapping ϕt ∈ Aut() satisfies ϕt (x0 ) = x0 and Dϕt (x0 ) = IX , implying by Proposition 5.3.25 that ϕt = I . Hence we have for each x ∈  that ∂ ∂ ϒ(x) = ϒ(ϕ(t, x)) = ϕ(t, x) = x = 0. ∂t ∂t So ϒ = 0, and the proof is complete.



With the help of Theorem 5.3.36 we can provide the topological version of Cartan’s uniqueness theorem immediately above (compare with Theorem 5.3.30). Theorem 5.5.6 Let X be a complex Banach space, let  be a bounded domain in X, let B be an open ball strictly inside , and let x0 be in . Then the mapping Tx0 : (aut(),  · B ) → (X ⊕ BL(X), max{ · X ,  · BL(X) }) defined by Tx0 () := ((x0 ), D(x0 )) becomes a bicontinuous R-linear bijection from aut() onto its range. In particular, the range of Tx0 is a closed real subspace of the complex Banach space X ⊕ BL(X). Proof It is clear that the mapping Tx0 is R-linear, and it follows from Proposition 5.5.4(ii) and Fact 5.3.9 that Tx0 is continuous. By Theorem 5.3.36, there exists a positive constant K such that  f − I B ≤ K max{ f (x0 ) − x0 , Df (x0 ) − IX } for every holomorphic mapping f :  → . Given  ∈ aut(), setting ϕ : R× →  for the flow of , we derive from the above inequality that   1 1 1 (5.5.2) (ϕt − I ) ≤ K max (ϕt (x0 ) − x0 ) , (Dϕt (x0 ) − IX ) t t t B

5.5 Banach Lie structures for aut() and Aut()

143

for every t ∈ R \ {0}. Since T-limt→0 1t (ϕt − I ) =  (cf. Proposition 5.4.48), it is clear that 1 ·B (ϕt − I ) −→ , t and it follows from Fact 5.3.9 that ·BL(X) 1 1 ·X (ϕt (x0 ) − x0 ) −→ (x0 ) and (Dϕt (x0 ) − IX ) −→ D(x0 ). t t

Now, taking limits as t → 0 in the inequality (5.5.2) we obtain B ≤ K max{(x0 ), D(x0 )}. This inequality shows that Tx0 is injective and T−1 x0 is continuous. Thus Tx0 is a bicontinuous R-linear bijection from aut() onto its range, and hence, by Proposition 5.5.4(iii), it is a closed real subspace of X ⊕ BL(X).  §5.5.7 Let X be a complex Banach space, and let  be a domain in X. According to §5.4.14, we can consider the exponential mapping exp : aut() → Aut() defined by exp() := (ϕ )1 , where, for each  ∈ aut(), ϕ : R ×  →  denote the flow of . Proposition 5.5.8 Let X be a complex Banach space, let  be a bounded domain in X, and let B be an open ball strictly inside . Regard Aut() inside the Banach space (Hb (B, X),  · B ). Then there exists an open neighbourhood M of the origin in aut() satisfying:  1 n  (I ) converges to exp() ∈ Aut() in the (i) For any  ∈ M the series n≥0 n! Banach space (Hb (B, X),  · B ). (ii) The mapping  → exp() from M to (Hb (B, X),  · B ) is real analytic. Proof Fix δ > 0 such that Bδ  , and regard aut() as a closed real subspace of the complex Banach space (Hb (Bδ , X),  · Bδ ). It follows from Proposition 5.4.38(ii)  1 n that n≥0 n!  (I ) is a power series from aut() to Hb (B, X) with radius of convergence ≥ δe , and hence defines a real analytic mapping from δe aut() to Hb (B, X). Moreover, by Proposition 5.4.39, there exists R with 0 < R < δe such that for each  1 n  ∈ aut() with Bδ < R we have that exp() = ∞ n=0 n!  (I ) in the Banach space (Hb (B, X),  · B ). Thus the result follows by taking M := { ∈ aut() : Bδ < R}.



For a good understanding of the formulation of the next theorem, recall the notation introduced in §§5.3.7 and 5.4.30.

144 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Theorem 5.5.9 Let X be a complex Banach space, let  be a bounded domain in X, let U ⊆  be an open subset of X, let B  U be an open ball, and let δ > 0 be so that Bδ  U. Suppose that tλ , gλ , fλ , and λ are nets in R \ {0}, Aut(), H (U, ), and HbU (Bδ , X), respectively, such that tλ → 0,

·Bδ 1 T (gλ − I ) −→  ∈ H (, X), fλ −→ f ∈ H (U, ) satisfying tλ ·Bδ

f (Bδ ) ⊆ C for suitable open ball C  , and λ −→  ∈ HbU (Bδ , X). Then there exists λ0 such that for each λ ≥ λ0 the mappings 1 ♦ tλ (gλ (λ ) − λ )

1  tλ (gλ ( fλ ) − fλ )

and

are defined on an open subset of X containing B, and

1  1 ♦ ·B ·B (g ( fλ ) − fλ ) −→  ◦ f − ( f ), and (g (λ ) − λ ) −→ [, |U ]. tλ λ tλ λ Proof According to Lemma 5.4.32(ii), there exists λ0 such that for each λ ≥ λ0 we have B ⊆ gλ (B δ ) ⊆ gλ (U), and so B is contained in gλ (U) ∩ U, which is the domain 2 of definition of the mappings 1  1 ♦ (gλ ( fλ ) − fλ ) and (g (λ ) − λ ). tλ tλ λ ·Bδ

Fix ε > 0 such that Cε  . Since fλ −→ f , there exists λ1 ≥ λ0 such that for each λ ≥ λ1 we have  fλ − f Bδ < 2ε , and hence we have for each x ∈ Bδ that fλ (x) = f (x) + ( fλ − f )(x) ∈ C 2ε . Therefore (gλ ◦ fλ − f )(x) ≤ (gλ ( fλ (x)) − fλ (x) +  fλ (x) − f (x) ≤ gλ − I C ε +  fλ − f Bδ , 2

hence gλ ◦ fλ − f Bδ ≤ gλ − I C ε +  fλ − f Bδ , and so gλ ◦ fλ − f Bδ → 0, that ·Bδ

2

is to say gλ ◦ fλ −→ f . Now, by Proposition 5.4.33, we obtain that  ·B 1 gλ ◦ fλ ◦ g−1 λ − gλ ◦ fλ −→ −( f ). tλ

(5.5.3)

On the other hand, for each λ ≥ λ1 and x ∈ Bδ , the triangle inequality, the mean value theorem (Fact 5.2.13) and the Cauchy inequality (5.2.24) give 1 (gλ ◦ fλ − fλ ) (x) − ( ◦ f )(x) tλ   1 (gλ − I ) −  ( fλ (x)) + ( fλ (x)) − ( f (x)) ≤ tλ 2e 1 +  fλ − f Bδ Cε , ≤ (gλ − I ) −  tλ ε Cε 2

5.5 Banach Lie structures for aut() and Aut() hence

1 tλ

(gλ ◦ fλ − fλ ) −  ◦ f





and so

1 tλ

(gλ ◦ fλ − fλ ) −  ◦ f



→ 0, that is to say

1 tλ

(gλ − I ) − 



2

145

+  fλ − f Bδ 2e ε Cε ,

·Bδ 1 (gλ ◦ fλ − fλ ) −→  ◦ f . tλ

(5.5.4)

It follows from (5.5.3) and (5.5.4), and the equality 1 1  1 (gλ ( fλ ) − fλ ) = (gλ ◦ fλ − fλ ) + (gλ ◦ fλ ◦ g−1 λ − gλ ◦ fλ ) tλ tλ tλ ·B



that t1λ (gλ ( fλ ) − fλ ) −→  ◦ f − ( f ), and so the first conclusion in the statement is proved. In order to prove the remaining part of the conclusion, note that for each λ ≥ λ0 and each x ∈ gλ (U) ∩ U we have 1 1 ♦ −1 (g (λ ) − λ )(x) = [Dgλ (g−1 λ (x))(λ (gλ (x))) − λ (x)] tλ λ tλ = aλ (x) + bλ (x) + cλ (x) + dλ (x) + eλ (x), where 1 −1 D(gλ − I )(g−1 λ (x))(λ (gλ (x)) − λ (x)), tλ 1 bλ (x) := D(gλ − I )(g−1 λ (x))(λ (x) − (x)), tλ   1 cλ (x) := D (gλ − I ) −  (g−1 λ (x))((x)), tλ aλ (x) :=

dλ (x) := D(g−1 λ (x))((x)), 1 eλ (x) := (λ ◦ g−1 λ − λ )(x). tλ For any λ ≥ λ0 and x ∈ B, we have that g−1 λ (x) ∈ B 2δ , and the mean value theorem (Fact 5.2.13) and the Cauchy inequality (5.2.24) give 1 −1 D(gλ − I )(g−1 λ (x))λ (gλ (x)) − λ (x) |tλ | 1 2e −1 ≤ gλ − I Bδ g−1 λ (x) − x sup Dλ (x + t(gλ (x) − x)) |tλ | δ 0≤t≤1  2 1 2e (gλ − I ) g−1 ≤ λ − I B λ Bδ . δ tλ Bδ

aλ (x) ≤

Since, by assumption,

1 tλ (gλ

− I )



→ Bδ and λ Bδ → Bδ , and, by ·B

Lemma 5.4.32(i), g−1 λ − I B → 0, we deduce that aλ −→ 0. Again using the Cauchy inequality (5.2.24), we see that

146 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem 1 D(gλ − I )(g−1 λ (x))(λ − )(x) |tλ | 1 2e ≤ gλ − I Bδ λ − B |tλ | δ 2e 1 ≤ (gλ − I ) λ − B , δ tλ Bδ

bλ (x) ≤

·B

and we obtain that bλ −→ 0. Analogously,   1 (gλ − I ) −  (g−1 cλ (x) ≤ D λ (x)) (x) tλ 2e 1 (gλ − I ) −  B , ≤ δ tλ Bδ ·B

and we get that cλ −→ 0. Once again the mean value theorem (Fact 5.2.13) and the Cauchy inequality (5.2.24) give dλ (x) − D(x)((x)) ≤ D(g−1 λ (x)) − D(x)(x) −1 2 ≤ g−1 λ (x) − x sup D (x + t(gλ (x) − x))B

 ≤2

2e δ

2

0≤t≤1

g−1 λ − I B Bδ B ,

·B

·B

and so dλ −→ (|U ). Finally, by Proposition 5.4.33, we find that eλ −→ −(). As a result 1 ♦ ·B (gλ (λ ) − λ ) −→ (|U ) − () = [, |U ].  tλ The above theorem gives as a consequence the next corollary. Corollary 5.5.10 Let X be a complex Banach space, and let  be a bounded domain in X. Suppose that tλ is a net in R \ {0} such that tλ → 0, and that fλ and gλ are nets in Aut() satisfying 1 1 T T ( fλ − I ) −→  ∈ H0 (, X) and (gλ − I ) −→  ∈ H0 (, X). tλ tλ Then 1 T −1 (gλ ◦ fλ ◦ g−1 λ ◦ fλ − I ) −→ [, ]. 2 tλ Proof

Note that for each λ we have 1 −1 (gλ ◦ fλ ◦ g−1 λ ◦ fλ − I ) = aλ + bλ + cλ , tλ2

5.5 Banach Lie structures for aut() and Aut() where aλ :=

147

* 1) −1 −1 −1 ◦ f ◦ g − f ) ◦ f − (g ◦ f ◦ g − f ) , (g λ λ λ λ λ λ λ λ λ tλ2

& 1% (gλ − I ) ◦ fλ − (gλ − I ) ◦ g−1 λ , 2 tλ * 1) cλ := 2 (fλ − I ) ◦ g−1 λ − ( f λ − I ) . tλ

bλ :=

T

Since fλ −→ I (Lemma 5.4.32(i)), it follows from the first conclusion of Theorem T −1 1 tλ (gλ ◦ fλ ◦ gλ − fλ ) −→  ◦ I − (I ) = 0, and hence, by Proposition T we have aλ −→ −(0) = 0. Moreover, by Proposition 5.4.33, we have that

5.5.9 that 5.4.33,

dλ :=

& T 1% (gλ − I ) ◦ fλ − (gλ − I ) −→ (). 2 tλ

(5.5.5)

Let B   be an open ball, and let δ > 0 be so that Bδ  . By Lemma 5.4.32(ii), there exists λ0 such that for each λ ≥ λ0 we have that g−1 λ (B) ⊆ B 2δ , and hence, for each x ∈ B, the mean value theorem (Fact 5.2.13) and the Cauchy inequality (5.2.24) give −1 dλ (g−1 λ (x)) − ()(x) ≤ dλ (gλ (x)) − dλ (x) + dλ (x) − ()(x) −1 ≤ g−1 λ (x) − x sup Ddλ (x + t(gλ (x) − x)) + dλ − ()B 0≤t≤1

2e ≤ g−1 dλ Bδ + dλ − ()B . λ − I B δ Since, by Lemma 5.4.32(i), g−1 λ − I B → 0, and, by (5.5.5), dλ Bδ → ()Bδ ·B

and dλ − ()B → 0, it follows that bλ = dλ ◦ g−1 λ −→ (). In view of the T

arbitrariness of B, we conclude that bλ −→ (). Finally, by Proposition 5.4.33, we T

have that cλ −→ −(). As a result, 1 T −1 (gλ ◦ fλ ◦ g−1 λ ◦ fλ − I ) −→ () − () = [, ]. tλ2



Theorem 5.5.11 Let X be a complex Banach space, and let  be a bounded domain in X. Then aut() is a real subalgebra of the Lie complex algebra H0 (, X), and it is a Banach Lie real algebra for the norm  · B for any open ball B strictly inside . Further, the Lie algebra aut() is purely real in the sense that aut() ∩ i aut() = 0. Proof By Proposition 5.5.4(i), aut() is a real subspace of H0 (, X). Moreover, if  and  are in aut(), and if ϕ and ψ are its respective flows, then 1 1 T-lim (ϕt − I ) =  and T-lim (ψt − I ) = , t→0 t t→0 t

148 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem and hence, by Corollary 5.5.10, we have 1 (ψt ◦ ϕt ◦ ψt−1 ◦ ϕt−1 − I ) = [, ]. t→0 t2

T-lim

Now, invoking Theorem 5.4.51, we conclude that [, ] ∈ aut(). Thus aut() is a real subalgebra of the Lie complex algebra H0 (, X). Recall that, by Proposition 5.5.4(iii), (aut(),  · B ) is a real Banach space for any open ball B strictly inside . In order to prove the continuity of the Lie product on aut(), let us fix an open ball B strictly inside  and δ > 0 such that Bδ  . By Proposition 5.5.4(ii), we know that there exists a positive constant M such that  · Bδ ≤ M · B on aut(). Therefore, given ,  ∈ aut(), by using the inequality (5.2.23), we have for each x ∈ B that [, ](x) = D(x)((x)) − D(x)((x)) ≤ D(x)(x) + D(x)(x) 1 2M 1 B B , ≤ Bδ B + Bδ B ≤ δ δ δ hence [, ]B ≤ 2M δ B B , and so [·, ·] is  · B -continuous. Finally, suppose that  ∈ aut() ∩ i aut(). Since [, i] = 0, it follows from Proposition 5.5.3(ii) that the mapping h : R2 ×  →  defined by h(t, s, x) := exp(t + s(i))(x) is real analytic, and Dhx (0, 0)(t, s) = (t + is)(x) and Dhx (t, s) = Dhh(t,s,x) (0, 0) for all x ∈  and (t, s) ∈ R2 . Both equalities highlight the fact that the real-linear mappings Dhx (t, s) are in fact complex-linear mappings once that we identify R2 with C, and hence hx regarded as a mapping from C to X becomes holomorphic. Therefore hx is an entire function on C with range in , which is bounded. By Liouville’s theorem (Fact 5.2.62), hx is a constant. Thus 0 = hx (0) ≡ Dhx (0, 0)(1, 0) = (x). Since x ∈  is arbitrary, we conclude that  = 0, and so the proof is complete.



To close this subsection we deal with some relevant auxiliary results that will be useful later. We begin with a variant of Lemma 5.4.18, which is proposed as an exercise. Exercise 5.5.12 Let X be a complex Banach space, let B be an open ball in X, and let δ > 0. Set U := {g ∈ Hb (B, X) : g − IB B < 2δ }. Prove that: (i) For each g ∈ U we have g(B) ⊆ B δ , and hence f ◦ g ∈ Hb (B, X) for every 2 f ∈ Hb (Bδ , X). (ii) Given ( f , g) ∈ Hb (Bδ , X) × U and (h, k) ∈ Hb (Bδ , X) × Hb (B, X), the mapping T( f ,g) (h, k) : x → Df (g(x))(k(x)) + h(g(x)) lies in Hb (B, X), and the linear mapping T( f ,g) : Hb (Bδ , X) × Hb (B, X) → Hb (B, X) is continuous.

5.5 Banach Lie structures for aut() and Aut()

149

(iii) The mapping  : ( f , g) → f ◦ g from Hb (Bδ , X) × U to Hb (B, X) is holomorphic, and D( f , g) = T( f ,g) for every ( f , g) ∈ Hb (Bδ , X) × U . Solution Given g ∈ U , we have for each x ∈ B that g(x) − x < 2δ , and hence g(x) = x + (g(x) − x) ∈ B δ . Therefore g(B) ⊆ B δ , and hence f ◦ g ∈ Hb (B, X) for 2 2 every f ∈ Hb (Bδ , X). Thus assertion (i) is proved. Given ( f , g) ∈ Hb (Bδ , X) × U and (h, k) ∈ Hb (Bδ , X) × Hb (B, X), note that T( f ,g) (h, k) = E ◦ α ◦ (g, k) + h ◦ g, where α : Bδ × X → BL(X) × X is defined by α(x, y) := (Df (x), y) and E : BL(X) × X → X is defined by E(F, x) := F(x). Therefore T( f ,g) (h, k) : B → X is holomorphic. Moreover, for each x ∈ B, using the inequality (5.2.23), we see that T( f ,g) (h, k)(x) ≤ Df (g(x))k(x) + h(g(x)) 2 ≤  f Bδ kB + hBδ δ     2 ≤  f Bδ + 1 hBδ + kB . δ Therefore T( f ,g) (h, k) ∈ Hb (B, X), and     2 T( f ,g) (h, k)B ≤  f Bδ + 1 hBδ + kB . δ

(5.5.6)

It is immediate to verify that T( f ,g) is a linear mapping, and hence (5.5.6) gives that it is continuous. Now, the proof of assertion (ii) is complete. Given ( f , g) ∈ Hb (Bδ , X) × U , for any (h, k) ∈ Hb (Bδ , X) × Hb (B, X) such that kB < 4δ and for any x ∈ B, we see that [g(x), g(x) + k(x)] ⊆ B 3δ , and hence 4

(( f , g) + (h, k))(x) = (( f + h) ◦ (g + k))(x) is defined. Using the Taylor formula (Proposition 5.2.12) and the Cauchy inequality (5.2.23), we have ((( f , g) + (h, k)) − ( f , g) − T( f ,g) (h, k))(x) =  f (g(x) + k(x)) + h(g(x) + k(x)) − f (g(x)) − Df (g(x))(k(x)) − h(g(x)) ≤  f (g(x) + k(x)) − f (g(x)) − Df (g(x))(k(x)) + h(g(x) + k(x)) − h(g(x)) 1 ≤ k(x)2 sup D2 f (g(x) + tk(x)) + k(x) sup Dh(g(x) + tk(x)) 2 0≤t≤1 0≤t≤1  2 1 4 4 ≤ k2B  f Bδ + kB hBδ 2 δ δ     2 4 ≤ kB hBδ + kB  f Bδ + 1 . δ δ

150 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Therefore (( f , g) + (h, k)) − ( f , g) − T( f ,g) (h, k)B     2 4 ≤ kB hBδ + kB  f Bδ + 1 . δ δ This inequality shows that  is differentiable at ( f , g) and D( f , g) = T( f ,g) , as required.  Next result becomes a variant of Lemma 5.4.25(ii). Proposition 5.5.13 Let X be a complex Banach space, let  be a bounded domain in X, let U ⊆  be an open subset of X, let B  U be an open ball, let δ > 0 be so that Bδ  U, and let  ∈ aut(). If ϕ : R ×  →  stand for the flow of , then there exists ρ > 0 such that, for each  ∈ HbU (Bδ , X), the mapping t → ϕt♦ () is defined from ] − ρ, ρ[ to the Banach space Hb (B, X) and is real analytic. Moreover, d ♦ (5.5.7) (ϕ ()) = [ϕt♦ (), ] forevery t ∈] − ρ, ρ[, dt t n d ϕ0♦ () = , and n (ϕt♦ ())(0) = (Tn ◦ · · · ◦ T2 ◦ T1 )() forevery n ∈ N, dt where, for each n ∈ N, Tn : Hb (B δn , X) → Hb (B 2

Tn (ϒ) := [ϒ, ].

δ 2n+1

, X) is defined by

Proof Let ε > 0 be such that Bδ+ε  U. By Proposition 5.5.8, there exist τ > 0 such that the mapping t → ϕt from ] − τ , τ [ to Hb (Bδ+ε , X) is analytic. Given  in HbU (Bδ , X), by Lemma 5.4.35, the linear mapping g → (g) from Hb (Bδ+ε , X) to Hb (Bδ , X) is continuous, and hence the mapping t → (ϕt ) from ] − τ , τ [ to Hb (Bδ , X) is analytic. On the other hand, T-limt→0 1t (ϕt − I ) =  (cf. Proposition 5.4.48), hence T-limt→0 ϕt = I , and in particular limt→0 ϕt − I B = 0, and so there exists ρ with 0 < ρ ≤ τ such that ϕt −I B < 2δ (i.e. ϕt ∈ U in the terminology introduced in Exercise 5.5.12) for every t ∈] − ρ, ρ[. It follows that the mapping t → ((ϕt ), ϕ−t ) from ] − ρ, ρ[ to Hb (Bδ , X) × Hb (B, X) is analytic (cf. Proposition 5.2.36) and Hb (Bδ , X) × U -valued. Now, invoking Exercise 5.5.12, we derive that the mapping t → ϕt♦ () = (ϕt ) ◦ ϕ−t from ] − ρ, ρ[ to Hb (B, X) is analytic. Note also that, for each t ∈] − ρ, ρ[, ϕt♦ () is defined on the open set ϕt−1 (U), and we have B ⊆ ϕt−1 (B δ ) ⊆ ϕt−1 (U) because ϕt ∈ U . 2 Replacing in the above paragraph the open ball B with B δ , there exists ρ1 > 0 2

such that the mapping t → ϕt♦ () from ] − ρ1 , ρ1 [ to Hb (B δ , X) is analytic and, 2

for each t ∈] − ρ1 , ρ1 [, ϕt♦ () is defined on the open set ϕt−1 (U), and we have B δ ⊆ ϕt−1 (B 3δ ) ⊆ ϕt−1 (U). It is clear that, for each ε with 0 ≤ ε < 2δ , we have 2 4 Bε  B δ (where we mean that B0 = B) and the linear mapping f → f|Bε from 2

Hb (B δ , X) to Hb (Bε , X) is continuous. Therefore the mapping t → ϕt♦ () from 2 ] − ρ1 , ρ1 [ to Hb (Bε , X) is also analytic. Now, for a fixed t in ] − ρ1 , ρ1 [, invoking the last conclusion in Theorem 5.5.9 we realize that

5.5 Banach Lie structures for aut() and Aut()

151

·Bε 1 ♦ ♦ (ϕs (ϕt ()) − ϕt♦ ()) −→ [ϕt♦ (), ] as s → 0. s By noticing that, for each s ∈] − ρ1 , ρ1 [ such that s + t ∈] − ρ1 , ρ1 [, we have the ♦ equality ϕs+t () = ϕs♦ (ϕt♦ ()) on Bε , we derive that

d ♦ (5.5.8) (ϕ ()) = [ϕt♦ (), ] in the Banach space Hb (Bε , X), dt t and in particular we have (5.5.7). Given n ∈ N, using Lemma 5.4.35, we have for each ϒ ∈ Hb (B δn , X) that 2

[ϒ, ]B

δ 2n+1

= ϒ() − (ϒ)B ≤

δ 2n+1

≤ ϒ()B

δ 2n+1

+ (ϒ)B

δ 2n+1

2n+2 ϒB δ B δ δ 2n 2n

and hence the linear mapping Tn : Hb (B δn , X) → Hb (B 2

Tn (ϒ) := [ϒ, ] is continuous. Taking ε =

δ , 22

δ 2n+1

, X) given by

we derive from (5.5.8) that

d ♦ (ϕ ()) = T1 (ϕt♦ ()). dt t Analogously, taking ε = hence

δ , 23

we derive from (5.5.8) that

♦ ♦ d dt (ϕt ()) = T2 (ϕt ()),

  d2 ♦ d d ♦ ♦ (ϕ ()) = (ϕ ())) = T ()) = T2 (T1 (ϕt♦ ())), (T (ϕ 2 t 2 dt dt t dt2 t

and so

d2 (ϕ ♦ ()) = (T2 ◦ T1 )(ϕt♦ ()). Arguing by induction, we find that dt2 t dn ♦ (ϕ ()) = (Tn ◦ · · · ◦ T2 ◦ T1 )(ϕt♦ ()) for every n ∈ N. dtn t

Finally, keeping in mind that ϕ0♦ () = , we obtain that dn ♦ (ϕ ())(0) = (Tn ◦ · · · ◦ T2 ◦ T1 )() for every n ∈ N. dtn t



5.5.2 The real Banach Lie group Aut() We begin by dealing with a variant of Exercise 5.4.31. Proposition 5.5.14 Let X and Y be complex Banach spaces, let X and Y be bounded domains in X and Y, respectively, and let g : X → Y be a biholomorphic mapping. Then we have: (i) The law g : f → g ◦ f ◦ g−1 is a bicontinuous isomorphism from the topological group Aut(X ) onto the topological group Aut(Y ) such that (g )−1 = (g−1 ) . (ii) The law g♦ :  → (g) ◦ g−1 is a bicontinuous isomorphism from the Banach Lie real algebra aut(X ) onto the Banach Lie real algebra aut(Y ) such that (g♦ )−1 = (g−1 )♦ . (iii) g (exp(t)) = exp(tg♦ ()) for all  ∈ aut(X ) and t ∈ R.

152 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof It is clear that g ( f ) := g ◦ f ◦ g−1 ∈ Aut(Y ) for every f ∈ Aut(X ), and it is routine to verify that g : Aut(X ) → Aut(Y ) is an isomorphism of groups with inverse (g−1 ) . The T-continuity of g follows from Proposition 5.3.18. Replacing g with g−1 , we realize that g is a homeomorphism for the T-topologies, and assertion (i) has been proved. In light of Exercise 5.4.31, we realize that g♦ is a Lie algebra isomorphism from aut(X ) onto aut(Y ) such that (g♦ )−1 = (g−1 )♦ , as well as assertion (iii) holds. In order to prove the continuity of g♦ , let us fix open balls B  X and C  Y such that C ⊆ g(B), and a positive number δ such that Bδ  X , and consider the norms  · B and  · C in the spaces aut(X ) and aut(Y ), respectively (cf. Proposition 5.5.4). Then we have for each y ∈ C that g♦ ()(y) = (g)(g−1 (y)) = Dg(g−1 (y))((g−1 (y))) 1 ≤ Dg(g−1 (y))(g−1 (y)) ≤ gBδ B , δ where the last inequality is a consequence of (5.2.23). Therefore 1 g♦ ()C ≤ gBδ B , δ and hence g♦ is continuous. Now, replacing g with g−1 , we obtain that g♦ is bicontinuous. Thus assertion (ii) is proved, and the proof is complete.  Corollary 5.5.15 Let X be a complex Banach space, let  be a bounded domain in X, and let  ∈ aut(). Then exp()♦ = exp(ad )−1 in BL(aut()). Proof Let ϕ : R ×  →  stand for the flow of . By Proposition 5.5.14(ii), for each t ∈ R, ϕt♦ is a bicontinuous automorphism of the Banach Lie real algebra ♦ aut() with inverse ϕ−t . On the other hand, by Proposition 5.5.13, for any open ball B  , there exists ρ > 0 such that for all t ∈] − ρ, ρ[ and  ∈ aut() we have that ♦ d ♦ dt ϕt () = [ϕt (), ] in the Banach space Hb (B, X), and hence in the Banach space aut(). Therefore γ :] − ρ, ρ[→ aut() defined by γ (t) := ϕt♦ () is a local flow at γ (0) =  of the linear vector field −ad in the Banach space aut(), and hence, by Exercise 5.4.15, ϕt♦ () = exp(−tad )() = exp(tad )−1 () for every t ∈] − ρ, ρ[. The arbitrariness of  gives that ϕt♦ = exp(tad )−1 for every t ∈] − ρ, ρ[. Now, given t ∈ R, fix n ∈ N such that nt ∈] − ρ, ρ[, and note that    −1 n  n −1 ! "♦ ! "n t t ♦ ♦ n ϕt = ϕ t = ϕ t = exp = exp = exp (tad )−1 . ad ad n n n n In particular, we have ϕ1♦ = exp(ad )−1 , as required.



Corollary 5.5.16 Let X be a complex Banach space, let  be a bounded domain in X, and let φ be a continuous automorphism of the Banach Lie real algebra aut(). Then φ(exp()♦ ()) = exp(φ())♦ (φ()) for all ,  ∈ aut().

5.5 Banach Lie structures for aut() and Aut() Proof

153

Given ,  ∈ aut(), we have φ(ad ()) = φ([, ]) = [φ(), φ()] = adφ() (φ()),

and reiterating the argument we obtain φ(adn ()) = adnφ() (φ()) for every n ∈ N. Now, the continuity of φ gives that ∞  ∞  (−1)n  (−1)n n n φ ad () = adφ() (φ()), n! n! n=0

n=0

)−1 ())

= exp(adφ() )−1 (φ()). Finally, invoking Corollary that is φ(exp(ad 5.5.15 we conclude that φ(exp()♦ ()) = exp(φ())♦ (φ()), as required.  For a given bounded domain  in a complex Banach space, the exponential mapping exp : aut() → Aut() is in general not injective. But it becomes so when restricted to an appropriate neighbourhood of the origin. Lemma 5.5.17 Let X be a complex Banach space, and let  be a bounded domain in X. Then there exists a symmetric open neighbourhood B of the origin in aut(), contained in the open neighbourhood M of the origin given in Proposition 5.5.8, such that the mapping exp : B → Aut() is injective. Proof Fix an open ball B strictly inside . According to Proposition 5.5.8, the mapping exp : M → Hb (B, X) is real analytic, and so the mapping F : M → Hb (B, X) given by F() := exp() − IB −  is as well. Since D exp(0) ∈ BL(aut(), Hb (B, X)) is given by D exp(0)() = (I ) =  for every  ∈ aut(), we deduce that DF(0) = 0. It follows from the continuity of DF that there exists ρ > 0 such that 1 for every  ∈ ρ(aut(),·B ) . 2 Then, for any 1 , 2 ∈ ρ(aut(),·B ) satisfying exp(1 ) = exp(2 ), we have that F(1 ) − F(2 ) = 2 − 1 , and using the mean value theorem (Fact 5.2.13) we see that ρ(aut(),·B ) ⊆ M and DF() ≤

1 − 2 B = F(1 ) − F(2 )B ≤ 1 − 2 B

DF()

sup ∈ρ(aut(),·

B)

1 ≤ 1 − 2 B , 2 and we deduce that 1 = 2 . Thus the exponential mapping is injective on the open ball ρ(aut(),·B ) . 

154 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Next we summarize some well-known results of the theory of free Lie algebras (see for example [687, Chapter 1], [1142], [1143, Chapter II], or [1181]). §5.5.18 Let X be a non-empty set of indeterminates. An associative word with characters in X is a finite sequence of elements of X, including the empty sequence, called the empty word and denoted by 1. With the concatenation product, the set of all associative words over X gives rise to a monoid M(X) with neutral element 1, called the free monoid on X. The free vector space over K generated by M(X), with product equal to the extension by bilinearity of the product of M(X), is called the unital free associative algebra over K generated by X, and is denoted by A (X). The subalgebra L (X) of (A (X))ant generated by X is called the free Lie algebra over K generated by X. The elements in A (X) are called associative polynomials and the elements in L (X) are called Lie polynomials. Of course, as the reader will suspect, these algebras enjoy the corresponding universality properties. Moreover, it is proved that A (X) becomes the universal enveloping associative algebra of the Lie algebra L (X). We will consider the larger algebras of formal series: ⎧ ⎫ ⎨ ⎬ A(X) := pn : pn is a homogeneous associative polynomial of degre n ⎩ ⎭ n≥0

and L(X) :=

⎧ ⎨ ⎩

n≥1

⎫ ⎬ pn : pn is a homogeneous Lie polynomial of degre n . ⎭

A(X) is a unital associative algebra with product defined by ⎞⎛ ⎞ ⎛  n      ⎝ pn ⎠ ⎝ qn ⎠ = pk qn−k , n≥0

A0 (X) :=





n≥0

n≥0 pn ∈ A (X) : p0  (A (X))ant contained

n≥0

k=0

 = 0 is an ideal of A(X), and L(X) is a subin A0 (X). In this setting, we can introduce the

algebra of exponential and logarithm mappings

exp : A0 (X) → 1 + A0 (X) and log : 1 + A0 (X) → A0 (X) defined by exp(a) :=

 (−1)n−1 1 an and log(b) := (b − 1)n . n! n n≥0

n≥1

These mappings are bijections inverse to one another. The famous Baker–Campbell– Hausdorff formula highlights that, for x, y ∈ X, the series c(x, y) := log(exp(x) exp(y))

5.5 Banach Lie structures for aut() and Aut()

155

is a Lie series, i.e. belongs to L(X). Therefore ⎛ ⎞⎛ ⎞  1 1 xm ⎠ ⎝ yn ⎠ , exp(c(x, y)) = exp(x) exp(y) = ⎝ m! n! m≥0

(5.5.9)

n≥0

and hence the term Ej (x, y) of total homogeneity j in (x, y) of the associative series  1 exp(c(x, y)) = n≥0 n! (c(x, y))n is given by  xm yn . (5.5.10) Ej (x, y) = m! n! m+n=j

The next classical result about the convergence of the Baker–Campbell–Hausdorff series in Banach Lie algebras can be seen in [687, Proposition 1.33], [1142, Theorem 5.30], or [1143, Proposition II.7.1]. Proposition 5.5.19 Let L be a Banach Lie algebra over K such that [x, y] ≤ xy for all x, y ∈ L, and set D := {(x, y) ∈ L × L : x + y < log 2}. Then, for any (x, y) ∈ D, the series c(x, y) in L (obtained from the Lie series c(x, y) by substituting x by x and y by y, and replacing Lie brackets in L (X) by Lie brackets in L) converges absolutely, and the mapping c : D → L is K-analytic. Theorem 5.5.20 Let X be a complex Banach space, and let  be a bounded domain in X. Then there are an open neighbourhood M0 of the origin in aut() and a real analytic mapping C : M0 × M0 → aut() such that exp(C(1 , 2 )) = exp(1 ) ◦ exp(2 ) for all 1 , 2 ∈ M0 .

(5.5.11)

Proof For the sake of simplicity we will write H0 instead of H0 (, X). By Proposition 5.5.2 we know that H0 is a Lie complex algebra, that the mapping  from H0 to (L(H0 ))ant becomes an injective Lie homomorphism, and that, given a pair (B, δ), where B is an open ball strictly inside  and δ is a positive number such that Bδ  , we have that for each  ∈ H0 the mapping () : f → ( f ) is a bounded operator from (H0 ,  · Bδ ) to (H0 ,  · B ), as well as that  becomes a topological embedding from (H0 ,  · B ) to BL((H0 ,  · Bδ ), (H0 ,  · B )). More precisely, if we denote by  · (B,δ) the operator norm in the space BL((H0 ,  · Bδ ), (H0 ,  · B )), then we have 1 1 B ≤ ()(B,δ) ≤ B for every  ∈ H0 . Bδ  δ In what follows, when we refer to a pair (B, δ) we will always understand that B is an open ball strictly inside  and that δ is a positive number such that Bδ  . Consider the set L := {T ∈ L(H0 ) : T ∈ BL((H0 ,  · Bδ ), (H0 ,  · B )) for every pair (B, δ)}. It is clear that L is a subspace of L(H0 ) containing IH0 . Moreover, given S, T ∈ L and any pair (B, δ), we have that for every f ∈ H0 (S ◦ T)( f )B = S(T( f ))B ≤ S(B, δ ) T( f )B δ ≤ S(B, δ ) T(B δ , δ ) f Bδ , 2

2

2

2

2

156 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem and so S ◦ T ∈ BL((H0 ,  · Bδ ), (H0 ,  · B )) and S ◦ T(B,δ) ≤ S(B, δ ) T(B δ , δ ) . 2

2

(5.5.12)

2

In view of the arbitrariness of the pair (B, δ), we conclude that S ◦ T ∈ L. Thus L is a unital subalgebra of L(H0 ). Moreover,  is an injective Lie homomorphism from H0 to Lant , and, for any pair (B, δ),  is a topological embedding from (H0 ,  · B ) to (L,  · (B,δ) ). Note that (5.5.12) gives that the bilinear mapping (S, T) → S ◦ T from (L,  · (B, δ ) ) × (L,  · (B δ , δ ) ) to (L,  · (B,δ) ) is continuous, and hence it can 2

2

2

be extended to a continuous bilinear mapping F from the Banach space product of the completions of the normed spaces (L,  · (B, δ ) ) and (L,  · (B δ , δ ) ) to the Banach 2

2

2

space completion of the normed space (L,  · (B,δ) ). Note also that given two pairs (B, δ) and (C, ε) such that B ⊆ C and Cε ⊆ Bδ we have that for any T ∈ L T( f )B ≤ T( f )C ≤ T(C,ε)  f Cε ≤ T(C,ε)  f Bδ for every f ∈ H0 , and hence T(B,δ) ≤ T(C,ε) .

(5.5.13)

On the other hand, by Theorem 5.5.11, aut() is a real subalgebra of the Lie complex algebra H0 , and (aut(),  · B ) is a Banach Lie real algebra for any open ball B strictly inside . Therefore, by §1.1.3 and Proposition 5.5.19, the Baker– Campbell–Hausdorff series determines a real analytic mapping c : U × U → aut() for a suitable open neighbourhood U of the origin in aut(). Moreover, since for any open balls B, C strictly inside  we have that  · B and  · C are equivalent norms on aut() (cf. Proposition 5.5.4(ii)), it follows from Proposition 5.4.38(i) that, for any pair (B, δ) fixed, there exists an open neighbourhood V of the origin in aut() such  1 ()n from aut() to L defines an analytic mapping that the power series n≥0 n! from V to the completion of L for the norms  · (B, δ ) and  · (B δ , δ ) (and hence, by 2

2

2

(5.5.13), for the norm  · (B,δ) ). Moreover, by Proposition 5.4.39, we may suppose that for each  ∈ U and for each f ∈ H0 we have that ∞  1 ()n ( f ) = f ◦ exp() n!

(5.5.14)

n=0

in the Banach spaces Hb (B, X) and Hb (B δ , X). Since c(0, 0) = 0, shrinking U 2 if necessary, we may suppose that U ⊆ V and c(U × U ) ⊆ V . Moreover, by Proposition 5.2.39, we may suppose that the power series from U × U to the completion of (L,  · (B,δ) ) given by ∞  ∞ 1  1 n n (, ) → F () , () n! n! n=0

n=0

converges and its homogeneous terms lie in L. Since, via , L can be seen as an envelope associative of aut(), it follows from (5.5.9) and (5.5.10) that both power

5.5 Banach Lie structures for aut() and Aut() 157 ! "  ∞ 1 ∞ 1 1 n n n have the same series ∞ n=0 n! (c(, )) and F n=0 n! () , n=0 n! () homogeneous terms, hence ∞  ∞ ∞  1  1 1 (c(, ))n = F ()n , ()n , n! n! n! n=0

n=0

n=0

and so, by (5.5.14), ∞  ∞ ∞  1  1 1 n n n exp(c(, )) = (c(, )) (I ) = F () , () (I ) n! n! n! n=0 n=0 n=0  ∞  ∞ 1 1 ()n ()n (I ) = n! n! n=0

n=0

∞  1 = ()n (exp()) = exp() ◦ exp(). n! n=0

Finally, since the mapping (, ) → (, ) becomes a linear homeomorphism on the Banach space aut() × aut(), the result follows by considering the mapping C from U × U to aut() defined by C(, ) := c(, ).  Theorem 5.5.21 Let X be a complex Banach space, and let  be a bounded domain in X. Suppose that B is the symmetric open neighbourhood of the origin in aut() given in Lemma 5.5.17, and consider     1 B := exp B :n∈N . n Then we have: (i) B is a fundamental system of neighbourhoods of I for a unique Hausdorff topology Ta on Aut() which is compatible with the group structure. (ii) Ta is finer that T. Proof From the general theory of topological groups (see, for example, [689, Theorems 3.3 and 3.4]), in order to prove assertion (i) it suffices to show that B satisfies the following properties:   (1) I ∈ exp 1n B for every n ∈ N. (2) If n1 , n2 ∈ N, then there exists m ∈ N such that   3    1 1 1 B exp B . B ⊆ exp exp m n1 n2 (3) If n ∈ N, then there exists m ∈ N such that       1 1 1 exp B ◦ exp B ⊆ exp B . m m n

158 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem (4) If n ∈ N, then there exists m ∈ N such that   −1  1 1 . B ⊆ exp B exp m n (5) If n ∈ N and g ∈ Aut(), then there exists m ∈ N such that     1 1 B ⊆ g ◦ exp B ◦ g−1 . exp m n   4 (6) n∈N exp 1n B = {I }.   Property (1) is clear because I = exp(0) ∈ exp 1n B for every n ∈ N.   To verify property (2) it suffices to consider m = max{n1 , n2 }. Since 1n B : n ∈ N is a fundamental system of neighbourhoods of 0 in aut(), and the real analytic mapping C : M0 × M0 → aut() given in Theorem 5.5.20 satisfies C(0, 0) = 0, it follows  that for any given n ∈ N we can find m ∈ N such that m1 B ⊆ M0 and C m1 B, m1 B ⊆ 1n B, hence, by (5.5.11),          1 1 1 1 1 B ◦ exp B = exp C B, B ⊆ exp B , exp m m m m n and property (3) holds. Moreover, since B = −B, we have for each n ∈ N that  exp

    −1 1 1 1 , B = exp − B = exp B n n n

and so property (4) is proved. Let n ∈ N and g ∈ Aut() be given. By Proposition 5.5.14(ii), the mapping g♦ : aut()  → aut() is an automorphism of the Banach Lie real algebra aut(), hence g♦ 1n B is a neighbourhood of 0 and we can find some m ∈ N such that m1 B ⊆ g♦ 1n B , and consequently, by Proposition 5.5.14(iii),           1 1 1 ♦ 1  exp B ⊆ exp g B = g exp B = g ◦ exp B ◦ g−1 , m n n n and property (5) has been proved. Finally, given f ∈ exp(B) \ {I }, there is some  ∈ B \ {0} such that f = exp(), and we can find some m ∈ N such that / m1 B.  1  ∈ Since the exponential / exp m B , and so   mapping is injective on B, we have that f ∈ 4 f∈ / n∈N exp 1n B . Thus property (6) is satisfied by B. Keeping in mind Theorem 5.3.22, to show that Ta is finer than T it suffices to prove that every T-neighbourhood of I contains a Ta -neighbourhood of I . To this end, note that in view of Fact 5.3.20, once an open ball B   has been fixed, a fundamental system of T-neighbourhoods of I in Aut() is given by the family {Vε : ε > 0}, where Vε := {f ∈ Aut() :  f − I B < ε}. Since the mapping exp : B → Hb (B, X) is in fact Aut()-valuated, and it is continuous  1  at 0, it follows that for each ε > 0 there exists some n ∈ N such that exp n B ⊆ Vε . Therefore T ⊆ Ta . 

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159

§5.5.22 The topology Ta introduced in Theorem 5.5.21 immediately above on Aut() is called the analytic topology on Aut(). This topology is in general different from the topology T of local uniform convergence (see [1114, Theorem 2.4.8]). Corollary 5.5.23 Let X be a complex Banach space, and let  be a bounded domain in X. We have: (i) If λ is a net in aut() such that λ converges to 0, then exp(λ ) Ta -converges to I . (ii) If λ is a net in the symmetric open neighbourhood B of the origin in aut() given in Lemma 5.5.17 such that exp(λ ) Ta -converges to I , then λ converges to 0. (iii) For any  in aut(), the mapping t → exp(t) from R to (Aut(), Ta ) is continuous. Proof Assertions (i) and (ii) follows straightforwardly from the definition of the Ta -topology. In order to prove assertion (iii), let  be in aut(), and suppose that tλ is a net in R converging to t0 ∈ R. Then we have that (tλ − t0 ) converges to 0, hence, by (i), exp((tλ − t0 )) Ta -converges to I , and so exp(tλ ) = exp((tλ − t0 )) ◦ exp(t0 ) Ta -converges to I ◦ exp(t0 ) = exp(t0 ).



§5.5.24 Let M be a Hausdorff topological space, and let X be a Banach space over K. A chart of M (at the point p ∈ M) over X is a pair (U, ϕ), where U is an open subset of M (containing p) and ϕ is a homeomorphism from U onto an open subset of X. An atlas of M over X is a family {(Ui , ϕi ) : i ∈ I} of charts of M over X satisfying the following conditions: (i) The family {Ui : i ∈ I} is an open cover of M. (ii) For each i, j ∈ I such that Ui ∩ Uj = ∅, the charts (Ui , ϕi ) and (Uj , ϕj ) are analytically compatible, i.e. the transition homeomorphism ϕj ◦ (ϕi−1 )|ϕi (Ui ∩Uj ) : ϕi (Ui ∩ Uj ) → ϕj (Ui ∩ Uj ) is analytic. Note that every atlas of M over X can be extended in a unique way to a maximal atlas. A K-analytic structure of M over X is a maximal atlas of M over X. We say that M is a K-analytic Banach manifold modelled on X if M is endowed with a K-analytic structure over X. Example 5.5.25 (a) Let X be a Banach space over K. Note that each nonempty open subset U of X becomes a K-analytic Banach manifold modelled on X for the K-analytic structure determined by the chart (U, IU ). More generally, if M is a K-analytic Banach manifold modelled on X, if {(Ui , ϕi ) : i ∈ I} is an atlas determining its K-analytic structure, and if U is a nonempty open subset of M, then U becomes a

160 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem K-analytic Banach manifold modelled   on X for the K-analytic structure determined by the atlas Ui ∩ U, (ϕi )|Ui ∩U : i ∈ I . (b) Let M and N be K-analytic Banach manifolds modelled on the Banach spaces X and Y, respectively. If (U, ϕ) and (V, ψ) are charts of M and N, respectively, then the mapping ϕ × ψ : U × V → X × Y given by (ϕ × ψ)(x, y) := (ϕ(x), ψ(y)) provides a chart of M × N over X × Y. Then M × N endowed with the K-analytic structure determined by charts so constructed becomes a K-analytic Banach manifold modelled on X × Y. (c) Let M be a complex analytic Banach manifold modelled on the complex Banach space X. Any chart (U, ϕ) of M over X is a chart over the real Banach space XR underlying X, and the family of these charts endows M with a real analytic Banach manifold structure modelled on XR . In light of Example 5.5.25(a), we may extend the notion of analytic mapping between Banach spaces (cf. §5.2.34) to the more general setting of analytic Banach manifolds. §5.5.26 Let M and N be K-analytic Banach manifolds modelled on the Banach spaces X and Y, respectively. A mapping f : M → N is called analytic if, for each p ∈ M, there are charts (U, ϕ) of M at p and (V, ψ) of N at f (p) such that f (U) ⊆ V and the mapping ψ ◦ f ◦ ϕ −1 : ϕ(U) → Y is analytic. A bijective mapping f from M onto N is called bianalytic if f and f −1 are analytic mappings. Note that the existence of a bianalytic mapping between M and N implies that X and Y are topologically isomorphic. It is clear that every analytic mapping is continuous and, by Proposition 5.2.40, the composition of analytic mappings is analytic. It follows from Example 5.5.25(a) that f : U → V is analytic whenever f : M → N is analytic and U and V are nonempty open subsets of M and N, respectively, such that f (U) ⊆ V. Note also that the charts in any analytic Banach manifold are bianalytic mappings. §5.5.27 Let M be a K-analytic Banach manifold modelled on the Banach space X, and let p be a point of M. Consider the set C of all pairs [(U, ϕ), x], where (U, ϕ) is a chart of M at p, and x ∈ X. It is straightforward to verify that the relation ∼ = defined on C by [(U1 , ϕ1 ), x1 ] ∼ = [(U2 , ϕ2 ), x2 ] if and only if D(ϕ2 ◦ ϕ1−1 )(ϕ1 (p))(x1 ) = x2 is an equivalence  relation. The equivalence class of C containing [(U, ϕ), x] is usually ∂  denoted by x ∂ϕ p , and is called a tangent vector to M at p. Let us fix any chart (U, ϕ)  ∂  of M at p. Then the mapping x → x ∂ϕ is a bijection from X to C / ∼ = which allow p us to transfer the Banach space structure of X to C / ∼ =. We say that C / ∼ = endowed with this Banach space structure is the tangent space of M at p, and is denoted by Tp (M).

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161

Note that in the case in which we consider an open subset  of a Banach space X as an analytic Banach manifold, then we have for each x0 ∈  that Tx0 () ≡ X via the identification x ∂I∂ x ≡ x. 0

§5.5.28 Let M and N be K-analytic Banach manifolds modelled on the Banach spaces X and Y, respectively, let f : M → N be an analytic mapping, and let p be a point of M. Fix (U, ϕ) and (V, ψ) charts of M at p and of N at f (p), respectively, such that f (U) ⊆ V. Then the mapping ψ ◦ f ◦ ϕ −1 : ϕ(U) → Y is analytic, and consequently f induces a mapping df (p) : Tp (M) → Tf (p) (N) defined by

   ∂  ∂  −1 , := D(ψ ◦ f ◦ ϕ )(ϕ(p))(x) df (p) x ∂ϕ p ∂ψ f (p) 

which becomes a continuous linear mapping. It is easy to see that the definition of df (p) does not depend on the charts (U, ϕ) and (V, ψ) we have chosen. The mapping df (p) is called the differential of f at p. In the case in which M and N are open subsets X and Y of Banach spaces X and Y, respectively, and f : X → Y is an analytic mapping, identifying, for each x ∈ X , the tangent spaces Tx (X ) and Tf (x) (Y ) with X and Y (cf. §5.5.27), we realize that the differential df (x) : Tx (X ) → Tf (x) (Y ) is nothing but the derivative Df (x) : X → Y. For any K-analytic Banach manifold M, it is clear that dIM (p) = ITp (M) for every p ∈ M. Moreover, the chain rule can be extended to analytic mappings between analytic Banach manifolds: If M, N, and W are K-analytic Banach manifolds, and if the mappings f : M → N and g : N → W are analytic, then d(g ◦ f )(p) = dg( f (p)) ◦ df (p) for every p ∈ M.

(5.5.15)

The following inverse mapping theorem for analytic Banach manifolds becomes an extension of the inverse mapping theorem for Banach spaces (Theorem 5.2.63). Theorem 5.5.29 Let M and N be K-analytic Banach manifolds, let f : M → N be an analytic mapping, and let p be a point of M. If df (p) : Tp (M) → Tf (p) (N) is bijective, then f is bianalytic from an open neighbourhood of p onto an open neighbourhood of f (p). Proof Suppose that M and N are modelled on the Banach spaces X and Y, respectively, and that df (p) is bijective. Choose charts (U, ϕ) of M at p and (V, ψ) of N at f (p) such that f (U) ⊆ V. Then ψ ◦ f ◦ ϕ −1 : ϕ(U) → Y is an analytic mapping such that D(ψ ◦ f ◦ ϕ −1 )(ϕ(p)) is a continuous bijective linear mapping from X to Y. Therefore, by Theorem 5.2.63, there exist open neighbourhoods B ⊆ ϕ(U) of ϕ(p)

162 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem and C ⊆ ψ(V) of ψ( f (p)) such that (ψ ◦ f ◦ ϕ −1 )|B : B → C is bianalytic. As a consequence, ϕ −1 (B) ⊆ U and ψ −1 (C) ⊆ V are open neighbourhoods of p and f (p), respectively, and f|ϕ −1 (B) = ψ −1 ◦ (ψ ◦ f ◦ ϕ −1 ) ◦ ϕ|ϕ −1 (B) : ϕ −1 (B) → ψ −1 (C) is bianalytic.  §5.5.30 Let M be a K-analytic Banach manifold modelled on the Banach space X. The tangent bundle of M is defined as the disjoint union  Tp (M), T(M) := p∈M

and the mapping πM : T(M) → M determined by the condition πM (Tp (M)) = p is called the bundle projection of M. The tangent bundle T(M) is a K-analytic Banach manifold modelled on X × X with an analytic structure induced from that of M. Indeed, given an atlas {(Ui , ϕi ) : i ∈ I} determining the analytic structure of M, for  each i ∈ I, set T(Ui ) = p∈Ui Tp (M) and consider the mapping T(ϕi ) : T(Ui ) → ϕi (Ui ) × X defined by



  ∂  := (ϕi (p), x) for all p ∈ Ui and x ∈ X. T(ϕi ) x ∂ϕi p

Now, define a topology on T(M) in which the open sets O are those such that, for each i ∈ I, T(ϕi )(O ∩ T(Ui )) is open in X × X. Then T(M) is a K-analytic Banach manifold modelled on the Banach space X × X for the analytic structure determined by the atlas {(T(Ui ), T(ϕi )) : i ∈ I}. To immediately verify that, for any chart (U, ϕ) on M, we have πM (T(U)) ⊆ U and ϕ ◦πM ◦T(ϕ)−1 = (π1 )|T(ϕ)(T(U)) , where the mapping π1 : X × X → X is the projection given by π1 (x1 , x2 ) = x1 . Thus, πM becomes an analytic mapping from T(M) to M. If M and N are K-analytic Banach manifolds modelled on the Banach spaces X and Y, respectively, and if f : M → N is an analytic mapping, then we can consider the mapping T( f ) : T(M) → T(N) determined by the condition that T( f )|Tp (M) acts as df (p) for every p ∈ M. Given charts (U, ϕ) of M and (V, ψ) of N such that f (U) ⊆ V, we have the inclusion T( f )(T(U)) ⊆ T(V). Moreover, we have for each p ∈ U and x ∈ X that    ∂  −1 (T(ψ) ◦ T( f ) ◦ T(ϕ) )(ϕ(p), x) = (T(ψ) ◦ T( f )) x ∂ϕ p     ∂  = T(ψ) D(ψ ◦ f ◦ ϕ −1 )(ϕ(p))(x) ∂ψ f (p) = (ψ( f (p)), D(ψ ◦ f ◦ ϕ −1 )(ϕ(p))(x)). Since  : BL(X, Y) × X → Y given by (F, x) = F(x), ψ ◦ f ◦ ϕ −1 : ϕ(U) → Y (and so D(ψ ◦ f ◦ ϕ −1 ) : ϕ(U) → BL(X, Y)), and the projections πi : X × X → X (i = 1, 2), are analytic, it follows from Propositions 5.2.36, 5.2.39, and 5.2.40 that the mappings

5.5 Banach Lie structures for aut() and Aut() 163   ψ ◦ f ◦ ϕ −1 ◦ (π1 )|ϕ(U)×X and  ◦ D(ψ ◦ f ◦ ϕ −1 ) ◦ (π1 )|ϕ(U)×X , (π2 )|ϕ(U)×X . are analytic, and hence T(ψ) ◦ T( f ) ◦ T(ϕ)−1 is analytic. Thus, T( f ) becomes an analytic mapping from T(M) to T(N). This mapping is usually denoted by df and is called the differential of f . §5.5.31 Let M be a K-analytic Banach manifold modelled on the Banach space X. By an analytic vector field on M we simply mean an analytic mapping  : M → T(M) such that πM ◦  = IM , i.e. (p) ∈ Tp (M) for every p ∈ M. Given an analytic vector field  on M, for each chart (U, ϕ) of M, there exists a unique analytic mapping / : ϕ(U) → X such that   ∂  /(ϕ(p)) (p) =  for every p ∈ U. (5.5.16) ∂ϕ p Indeed, given a chart (U, ϕ) of M, consider the chart (T(U), T(ϕ)) of T(M), and note that (U) ⊆ T(U). Therefore we can consider the analytic mapping / := π2 ◦ T(ϕ) ◦  ◦ ϕ −1 ,  where π2 : X × X → X is the projection given by π2 (x1 , x2) = x2 . If for each p ∈ U ∂  we consider the vector xp ∈ X determined by (p) = xp ∂ϕ , then we have that p    ∂  / (ϕ(p)) = (π2 ◦ T(ϕ) ◦ )(p) = (π2 ◦ T(ϕ)) xp = π2 (ϕ(p), xp ) = xp , ∂ϕ p / satisfies (5.5.16). The uniqueness of  / is clear. and hence  The vector space of all analytic vector fields on M, under pointwise addition and scalar multiplication, will be denoted by T(M). If  is an open subset of a Banach space X, then T() ≡  × X with the bundle projection π given by π (x, v) = x. Therefore analytic vector fields on  are of /(x)), where  / :  → X is an analytic mapping, and we may the form (x) = (x,  thus identify T() with the space of all analytic mappings from  to X, which is in agreement with our introduction of the notion of a vector field on  (cf. §5.4.1). The next result is an extension of Proposition 5.4.24 to the more general setting of analytic Banach manifolds. Proposition 5.5.32 Let M be a K-analytic Banach manifold modelled on the Banach space X. We have: (i) If ,  ∈ T(M), and if for each chart (U, ϕ) of M we define

    & ∂   / / / / [, ](p) := D (ϕ(p)) (ϕ(p)) − D(ϕ(p))  (ϕ(p)) ∂ϕ p %

(5.5.17)

/ and / for every p ∈ U, where   are the analytic mappings from ϕ(U) to X satisfying (5.5.16) for  and , respectively, then the law p → [, ](p) from M to T(M) becomes a well-defined analytic vector field on M. (ii) T(M) is a Lie algebra over K for the product [·, ·].

164 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof Let  and  be in T(M). Suppose that (U, ϕ) and (V, ψ) are charts of M such that U ∩ V = ∅, and consider analytic mappings /, / ,    : ϕ(U) → X and   : ψ(V) → X satisfying (5.5.16) for ,  in relation to the charts (U, ϕ) and (V, ψ), respectively. Then we have for each p ∈ U ∩ V that     ∂  ∂  ∂  ∂   /  / = (ψ(p)) and  (ϕ(p)) =  (ψ(p)) , (ϕ(p)) ∂ϕ p ∂ψ p ∂ϕ p ∂ψ p that is to say /(ϕ(p))) =  (ψ(p)) D(ψ ◦ ϕ −1 )(ϕ(p))(

(5.5.18)

D(ψ ◦ ϕ −1 )(ϕ(p))(/  (ϕ(p))) =   (ψ(p)).

(5.5.19)

and

Writing (5.5.18) as follows /(ϕ(p))) = (  ◦ ψ ◦ ϕ −1 )(ϕ(p)), D(ψ ◦ ϕ −1 )(ϕ(p))( and differentiating this relation (keeping in mind (5.4.9)), we obtain /(ϕ(p))(x)) + D2 (ψ ◦ ϕ −1 )(ϕ(p))( /(ϕ(p)), x) D(ψ ◦ ϕ −1 )(ϕ(p))(D (ψ(p))(D(ψ ◦ ϕ −1 )(ϕ(p))(x)) = D for all p ∈ U ∩ V and x ∈ X. Now, if for each p ∈ U ∩ V we take x = /  (ϕ(p)), then it follows from (5.5.19) that /(ϕ(p))(/ D(ψ ◦ ϕ −1 )(ϕ(p))(D  (ϕ(p)))) /(ϕ(p)), / (ψ(p))(  (ϕ(p))) = D  (ψ(p))). + D2 (ψ ◦ ϕ −1 )(ϕ(p))( By interchanging the roles of  and , we have also that /(ϕ(p)))) D(ψ ◦ ϕ −1 )(ϕ(p))(D/  (ϕ(p))( /(ϕ(p))) = D (ψ(p))).  (ϕ(p)),   (ψ(p))( + D2 (ψ ◦ ϕ −1 )(ϕ(p))(/ Subtracting and keeping in mind the symmetry of D2 (ψ ◦ ϕ −1 )(ϕ(p)) we conclude that /(ϕ(p))) − D /(ϕ(p))(/ D(ψ ◦ ϕ −1 )(ϕ(p))[D/  (ϕ(p))(  (ϕ(p)))]     = D (ψ(p))((ψ(p))) − D(ψ(p))( (ψ(p))), that is to say /(ϕ(p))) − D /(ϕ(p))(/ [D/  (ϕ(p))(  (ϕ(p)))]

 ∂  ∂ϕ p

(ψ(p))) − D (ψ(p))( = [D  (ψ(p))(  (ψ(p)))]

 ∂  . ∂ψ p

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165

Thus, for each p ∈ M, the definition of [, ](p) does not depend on the chosen chart at p, and hence [, ] is a well-defined mapping from M to T(M) satisfying that [, ](p) ∈ Tp (M) for every p ∈ M. Moreover, given a chart (U, ϕ) of M, if we consider the chart (T(U), T(ϕ)) of T(M), then we have [, ](U) ⊆ T(U) and, for each p ∈ U, /(ϕ(p))) − D /(ϕ(p))(/ (T(ϕ) ◦ [, ] ◦ ϕ −1 )(ϕ(p)) = (ϕ(p), D/  (ϕ(p))(  (ϕ(p)))). Since the component mappings of the mapping T(ϕ) ◦ [, ] ◦ ϕ −1 : ϕ(U) → X × X are analytic (cf. §5.2.34 and Proposition 5.2.39), it follows from Proposition 5.2.36 that T(ϕ) ◦ [, ] ◦ ϕ −1 is analytic. Thus [, ] is an analytic vector field on M, and assertion (i) is proved. Let {(Ui , ϕi ) : i ∈ I} be an atlas of M over X. For each  ∈ T(M) and i ∈ I, denote by ρi () the analytic mapping from ϕi (Ui ) to X satisfying (5.5.16) for  in relation to the chart (Ui , ϕi ). It follows from (5.5.17) that ρi ([, ]) = [ρi (), ρi ()] for all ,  ∈ T(M), hence the mapping ρi becomes an algebra homomorphism from (T(M), [·, ·]) to (T(ϕi (Ui )), [·, ·]), and so the mapping ρ :  → (ρi ())i∈I defines an injective algebra homomorphism from (T(M), [·, ·]) to the direct product algebra  i∈I (T(ϕi (Ui )), [·, ·]). By Proposition 5.4.24 (replacing C with K, and holomorphy with analitycity), we know that each (T(ϕi (Ui )), [·, ·]) is a Lie algebra, and  hence i∈I (T(ϕi (Ui )), [·, ·]) is also a Lie algebra. Thus (T(M), [·, ·]) is a Lie algebra because it can be seen, via ρ, as a subalgebra of a Lie algebra, and the proof is complete.  §5.5.33 A Banach Lie group over K is a set G where we have a group structure together with a K-analytic structure in such a way that the following conditions are satisfied: (i) The product mapping ( f , g) → fg from G × G to G is analytic. (ii) The inversion mapping f → f −1 from G to G is analytic. Fact 5.5.34 Let G be a set endowed with a group structure and a K-analytic structure. Then G is a Banach Lie group if and only if the mapping ( f , g) → fg−1 from G × G to G is analytic. Proof

If the mappings f → f −1 and ( f , g) → fg are analytic, then so is ( f , g) → ( f , g−1 ) → fg−1 .

Conversely, if the mapping ( f , g) → fg−1 is analytic, then so are the mappings f → (e, f ) → ef −1 and ( f , g) → ( f , g−1 ) → f (g−1 )−1 .



In practice, the next result is very useful. Proposition 5.5.35 Let G be a set endowed with a group structure and a K-analytic structure. Then G is a Banach Lie group if (and only if) the following conditions are satisfied:

166 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem (i) For each g ∈ G, the mapping f → gf from G to G is analytic. (ii) For each g ∈ G, the mapping f → gfg−1 from G to G is analytic in an open neighbourhood of the unit element e of G. (iii) The mapping ( f , g) → fg−1 from G × G to G is analytic in an open neighbourhood of (e, e). Proof

Given ( f0 , g0 ) ∈ G × G, we have % −1 −1 −1 & −1 fg−1 = ( f0 g−1 g0 for all f , g ∈ G. 0 )g0 ( f0 f )(g0 g)

Thus the mapping ( f , g) → fg−1 from G × G to G can be represented in a neighbourhood of ( f0 , g0 ) as a composition of mappings of the types mentioned in conditions (i)–(iii), and hence it is analytic on such a neighbourhood. Now, the result follows  from the arbitrariness of ( f0 , g0 ) in G × G and Fact 5.5.34. §5.5.36 Clearly every Banach Lie group G is a topological group, and hence its topology is completely determined by any system of neighbourhoods of the unit element e. In fact, its analytic structure is also determined by a system of charts at e. Indeed, for any element f in G, it follows from (i) in §5.5.33 that the left translation Lf : g → fg becomes a bianalytic mapping from G to G, and hence ( fU, ϕ ◦ (Lf −1 )|fU ) is a chart of G at f whenever (U, ϕ) is a chart of G at e. Moreover, if X is the Banach space modelling G, then we have      ∂  ∂  dLf (g) x (5.5.20) = x  for all g ∈ U and x ∈ X, ∂ϕ g ∂(ϕ ◦ (Lf −1 )|fU )  fg

because D((ϕ ◦ Lf −1 ) ◦ Lf ◦ ϕ −1 )(ϕ(g)) = DIϕ(U) (ϕ(g)) = IX . §5.5.37 Let G be a Banach Lie group with unit element e. An analytic vector field  : G → T(G) is called left invariant if ( fg) = dLf (g)((g)) for all f , g ∈ G. Any left invariant analytic vector field is determined entirely by its value at e. Indeed, keeping in mind (5.5.15), we realize that an analytic vector field  on G is left invariant if and only if ( f ) = dLf (e)((e)) for every f ∈ G.

(5.5.21)

On the other hand, for any w ∈ Te (G), the mapping w : f → dLf (e)(w) is a left invariant analytic vector field on G satisfying w (e) = w. Indeed, fix a chart (U, ϕ) ∂  of G at e, write w = x ∂ϕ for suitable x in the Banach space modelling G, and, for e each f ∈ G, consider the charts ( fU, ϕ ◦ (Lf −1 )|fU ) and (T( fU), T(ϕ ◦ (Lf −1 )|fU )) of G at f and of T(G) at w ( f ), respectively. For each g ∈ U, it follows from (5.5.20) that   ∂  w ( fg) = x  , ∂(ϕ ◦ (L( fg)−1 )|fgU )  fg

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167

and hence (T(ϕ ◦ Lf −1 ) ◦ w ◦ (ϕ ◦ Lf −1 )−1 )((ϕ ◦ Lf −1 )( fg)) = ((ϕ ◦ Lf −1 )( fg), x). Therefore the mapping T(ϕ ◦ Lf −1 ) ◦ w ◦ (ϕ ◦ Lf −1 )−1 is analytic because its component mappings so are, and hence w is analytic. Moreover, we have w (e) = dLe (e)(w) = dIG (e)(w) = w, hence w satisfies the condition (5.5.21), and so it is left invariant. The set L (G) of all left invariant analytic vector fields on G is clearly a subspace of the vector space T(G). It follows from the above paragraphs that the mapping  → (e) is a linear bijection from L (G) onto Te (G) with inverse w → w . This linear bijection allow us to transfer the Banach space structure of Te (G) ≡ X (cf. §5.5.27) to L (G). Proposition 5.5.38 Let G be a Banach Lie group. Then L (G) is a subalgebra of the Lie algebra T(G). Proof Let (U, ϕ) be a chart of G at the unit element e, and let X be the Banach space modelling G. According to §5.5.31, for each  ∈ L (G) there is an analytic / from ϕ(U) to X such that mapping   ∂  /(ϕ(g)) for every g ∈ U. (g) =  ∂ϕ g Given f ∈ G, we have for each g ∈ fU that f −1 g ∈ U, and hence 

   ∂  /(ϕ( f −1 g)) (g) = ( ff −1 g) = dLf ( f −1 g)(( f −1 g)) = dLf ( f −1 g)  . ∂ϕ f −1 g

Therefore, by (5.5.20), we have

  ∂  /(ϕ( f −1 g)) (g) =   for every g ∈ fU. ∂(ϕ ◦ (Lf −1 )|fU ) 

(5.5.22)

g

Given  and  in L (G), it follows from Proposition 5.5.32 and the equalities (5.5.20) and (5.5.22), that, for each f ∈ G, both dLf (e)([, ](e)) and [, ]( f ) are equal to   ∂  /(ϕ(e))) − D /(ϕ(e))(/ [D/  (ϕ(e))(  (ϕ(e)))]  , ∂(ϕ ◦ (Lf −1 )|fU )  f

and so [, ]( f ) = dLf (e)([, ](e)). Finally, the arbitrariness of f in G gives that [, ] ∈ L (G).  Proposition 5.5.39 Let G be a Banach Lie group. Then the Lie product on L (G) is continuous.

168 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof Set X for the Banach space modelling G, and fix a chart (V, ϕ) of G at e such that ϕ(e) = 0. Since the product of G is continuous, we can also fix an open neighbourhood U of e such that UU ⊆ V, and we have in particular that U ⊆ V. Then the set  := ϕ(U) × ϕ(U) is an open neighbourhood of (0, 0) in X × X, and the mapping  :  → X defined by (x, y) := ϕ(ϕ −1 (x)ϕ −1 (y)) is analytic. Note that (0, x) = x for every x ∈ ϕ(U), i.e.  ◦ α|ϕ(U) = Iϕ(U) , where α is the continuous linear mapping from X to X × X defined by α(x) := (0, x). It follows from the chain rule that D(0, x)(0, w) = w for all x ∈ ϕ(U) and w ∈ X, and in particular D(0, 0)(0, w) = w for every w ∈ X.

(5.5.23)

From now on in this  proof, by simplicity, we identify each vector w ∈ X with the ∂  ∈ Te (G). Given w ∈ X, we have for each f ∈ U that tangent vector w ∂ϕ e  ∂  w ( f ) = dLf (e)(w) = D(ϕ ◦ Lf ◦ ϕ −1 )(ϕ(e))(w) ∂ϕ f  ∂  = D(ϕ ◦ Lϕ −1 (ϕ( f )) ◦ ϕ −1 )(0)(w) , ∂ϕ  f

5w : ϕ(U) → X associated to w is given by and hence the analytic mapping  5w (x) = D(x, 0)(0, w).  5w = E(0,w) ◦ D ◦ β, where β : X → X × X and E(0,w) : BL(X × X, X) → X Writing  are the continuous linear mappings defined by β(x) = (x, 0) and E(0,w) (F) = F(0, w), 5w (x)(w ) = D2 (x, 0)((w , 0), (0, w)), and in particular the chain rule gives that D 5w (0)(w ) = D2 (0, 0)((w , 0), (0, w)) for every w ∈ X. D

(5.5.24)

Now, given u, v ∈ X, we deduce from (5.5.23) and (5.5.24) that 5v (0)) = D2 (0, 0)((v, 0), (0, u)), 5u (0)( D 5v ](0) = (u, v), where : X × X → X is the continuous bilinear map5u ,  hence [ ping defined by (u, v) := D2 (0, 0)((u, 0), (0, v)) − D2 (0, 0)((v, 0), (0, u)). Therefore [u , v ](e) = (u, v), and hence [u , v ] =  (u,v) . Now, the continuity of the Lie product on L (G) follows from the continuity of .  §5.5.40 Let G be a Banach Lie group with unit element e. As commented in §5.5.37, the linear bijection  → (e) from L (G) to Te (G) allows us to transfer the Banach structure of Te (G) to L (G). Likewise, keeping in mind Propositions 5.5.38 and 5.5.39, this linear bijection allows us to see Te (G) as a Banach Lie algebra for the product defined by [(e), (e)] := [, ](e) for all ,  ∈ L (G). This Banach Lie algebra is called the Banach Lie algebra of G.

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169

The basic examples of Banach Lie groups are provided by complete normed unital associative algebras. Proposition 5.5.41 Let A be a complete normed unital associative algebra over K. Then Inv(A) endowed with its canonical structures of group and K-analytic Banach manifold is a Banach Lie group over K whose Banach Lie algebra is Aant . Proof By Theorem 1.1.23, the group Inv(A) is open in A, and hence it becomes naturally a K-analytic Banach manifold modelled on A (cf. Example 5.5.25(a)). The product in Inv(A) is analytic because it is the restriction of a continuous bilinear mapping in A. Since every real algebra is a full real subalgebra of its complexification, in order to prove the analyticity of the mapping x → x−1 we may suppose that K = C, in which case it is followed from Theorems 1.1.23 and 5.2.60. Thus Inv(A) is a Banach Lie group over K, and  T1 (Inv(A)) is identical to the Banach space underlying A by ∂  identifying a ∂IInv(A) 1 ≡ a. Given a ∈ A, we have for each x ∈ Inv(A) that     ∂  ∂  −1 a (x) = dLx (1) a = D(IInv(A) ◦ Lx ◦ IInv(A) )(1)(a) ∂IInv(A) 1 ∂IInv(A) x    ∂  ∂  ∂  = DLx (1)(a) = Lx (a) = xa ,   ∂IInv(A) x ∂IInv(A) x ∂IInv(A) x 5a = (Ra )|Inv(A) . Therefore and hence 

 ∂  [a , b ](1) = [DRb (1)(Ra (1)) − DRa (1)(Rb (1))] ∂IInv(A) 1   ∂  ∂  = (Rb (a) − Ra (b)) = [a, b] ∂IInv(A) 1 ∂IInv(A) 1

for all a, b ∈ A, and so Aant can be regarded as the Banach Lie algebra of Inv(A).  §5.5.42 Let X be a complex Banach space, and let  be a bounded domain in X. Our next goal is to prove that the exponential mapping exp : aut() → Aut() induces an analytic structure on (Aut(), Ta ) making Aut() into a real Banach Lie group with Banach Lie algebra aut(), acting analytically on . To this end, keeping in mind Proposition 5.5.8, Lemma 5.5.17, and Theorem 5.5.20, we can consider a symmetric open neighbourhood B0 of the origin in aut() such that the mapping exp : B0 → Aut() is injective and real analytic, and additionally there is a real analytic mapping C : B0 × B0 → aut() satisfying (5.5.11). Moreover, taking advantage of the continuity of the mapping C, we fix symmetric open neighbourhoods B1 and B2 of the origin in aut() satisfying B2 ⊆ B1 ⊆ B0 , C(B1 , B1 ) ⊆ B0 , and C(B2 , B2 ) ⊆ B1 .

(5.5.25)

Lemma 5.5.43 Let X be a complex Banach space, let  be a bounded domain in X, and let B0 , B1 , B2 , and C be as in §5.5.42 immediately above. Then we have: (i) C(, 0) = C(0, ) =  and C(, −) = C(−, ) = 0 for every  ∈ B1 .

170 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem (ii) C(C(1 , 2 ), 3 ) = C(1 , C(2 , 3 )) for all 1 , 2 , 3 ∈ B2 . (iii) exp : B2 → exp(B2 ) is a homeomorphism for the topologies induced on B2 and exp(B2 ) by aut() and (Aut(), Ta ), respectively. Proof

By (5.5.11), we have for any  ∈ B1 that exp(C(, 0)) = exp() ◦ exp(0) = exp() ◦ I = exp()

and exp(C(, −)) = exp() ◦ exp(−) = exp() ◦ exp()−1 = I = exp(0). Now, the second condition in (5.5.25) and the injectivity of exp on B0 give that C(, 0) =  and C(, −) = 0. A similar argument yields to C(0, ) =  and C(−, ) = 0, and the proof of assertion (i) is complete. Given 1 , 2 , 3 ∈ B2 , it follows from (5.5.25) that C(1 , 2 ) and C(2 , 3 ) lie in B1 , and hence C(C(1 , 2 ), 3 ) and C(1 , C(2 , 3 )) lie in B0 . Since, by (5.5.11), we have that exp(C(C(1 , 2 ), 3 )) = exp(1 ) ◦ exp(2 ) ◦ exp(3 ) = exp(C(1 , C(2 , 3 ))), assertion (ii) follows from the injectivity of the exponential on B0 . It follows from Corollary 5.5.23(i)–(ii) that, for any net λ in B0 , we have λ converges to 0 if and only if exp(λ ) Ta -converges to I .

(5.5.26)

Let λ be a net in B2 , and let  be in B2 . If λ converges to , then C(λ , −) converges to C(, −) = 0, hence exp(λ ) ◦ exp()−1 = exp(C(λ , −)) Ta converges to I (by (5.5.26)), and so exp(λ ) Ta -converges to exp(). Conversely, if exp(λ ) Ta -converges to exp(), then exp(C(λ , −)) = exp(λ ) ◦ exp()−1 Ta -converges to I , hence C(λ , −) converges to 0 (by (5.5.26)), and so C(C(λ , −), ) converges to C(0, ) = . Note that, by (i) and (ii), we have C(C(λ , −), ) = C(λ , C(−, )) = C(λ , 0) = λ , and consequently λ converges to . Summarizing, we have proved that λ converges to  if and only if exp(λ ) Ta -converges to exp(). Thus the proof is finished.  §5.5.44 Let X be a complex Banach space, let  be a bounded domain in X, and let B0 , B1 , and B2 be as in §5.5.42 above. We will denote by log : exp(B0 ) → B0 to the inverse mapping of exp : B0 → exp(B0 ), and we will consider the family F := {V : V is an open subset of aut() such that 0 ∈ V ⊆ B2 } . It follows from Theorem 5.5.21 that the family {exp(V) : V ∈ F } is a fundamental system of neighbourhoods of I for Ta .

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171

Theorem 5.5.45 Let X be a complex Banach space, and let  be a bounded domain in X. Then we have: (i) There exists a unique real analytic Banach manifold structure on (Aut(), Ta ) for which the family {(exp(V), log| exp(V) ) : V ∈ F } is a system of charts at I . (ii) The manifold (Aut(), Ta ) is a real Banach Lie group whose Banach Lie algebra is aut(). (iii) The mapping : ( f , x) → f (x) from Aut() ×  to  is real analytic. Proof By Lemma 5.5.43(iii), the family of pairs {(exp(V), log| exp(V) ) : V ∈ F } becomes a system of charts at I . Taking into account that, for each g ∈ Aut(), the left translation Lg : f → g ◦ f is an automorphism of the topological group (Aut(), Ta ) whose inverse is Lg−1 , we can define a system of charts at g as the family of pairs {(g ◦ exp(V), log ◦(Lg−1 )|g◦exp(V) ) : V ∈ F }. It is clear that {g ◦ exp(V) : g ∈ Aut(), V ∈ F } is an open cover of Aut() for the topology Ta . Suppose that (g1 ◦ exp(V1 )) ∩ (g2 ◦ exp(V2 )) = ∅ for some g1 , g2 ∈ Aut() and V1 , V2 ∈ F . Given f ∈ (g1 ◦ exp(V1 )) ∩ (g2 ◦ exp(V2 )), there are 1 ∈ V1 and 2 ∈ V2 such that g1 ◦ exp(1 ) = f = g2 ◦ exp(2 ), −1 = exp( ) ◦ exp(− ). Since V and V and hence g−1 2 1 1 2 2 ◦ g1 = exp(2 ) ◦ exp(1 ) are contained in B2 , by (5.5.25) and (5.5.11), we have that

3 := C(2 , −1 ) ∈ B0 and g−1 2 ◦ g1 = exp(3 ). For any  ∈ (log ◦Lg−1 )((g1 ◦ exp(V1 )) ∩ (g2 ◦ exp(V2 ))) we see that 1

[log ◦Lg−1 ◦ (log ◦Lg−1 )−1 ]() = (log ◦Lg−1 )(g1 ◦ exp()) 2

1

2

= log(g−1 2 ◦ g1 ◦ exp()) = log(exp(3 ) ◦ exp()), and, keeping in mind that  ∈ V1 , again using (5.5.11) and (5.5.25), we obtain that [log ◦Lg−1 ◦ (log ◦Lg−1 )−1 ]() = C(3 , ). 2

1

Now, the analyticity of the mapping C and of the mapping  → (3 , ) from aut() to aut() × aut() gives that the transition homeomorphism corresponding to the charts (g1 ◦ exp(V1 ), log ◦(Lg−1 )|g1 ◦exp(V1 ) ) and (g2 ◦ exp(V2 ), log ◦(Lg−1 )|g2 ◦exp(V2 ) ) 1

is analytic, and so assertion (i) is proved.

2

172 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem In order to prove the first assertion in (ii), we will show that (Aut(), Ta ) satisfies the three conditions in Proposition 5.5.35. (a) For each g ∈ Aut(), the left translation Lg : f → g ◦ f is analytic. Indeed, given g ∈ Aut(), we have for any f ∈ Aut() and V ∈ F that ( f ◦ exp(V), log ◦(Lf −1 )|f ◦exp(V) ) and (g ◦ f ◦ exp(V), log ◦(L(g◦ f )−1 )|g◦ f ◦exp(V) ) are charts of (Aut(), Ta ) at f and g ◦ f , respectively. It is clear that Lg ( f ◦ exp(V)) ⊆ g ◦ f ◦ exp(V) and, for any  ∈ V, [log ◦L(g◦ f )−1 ◦ Lg ◦ (log ◦Lf −1 )−1 ]() = (log ◦L(g◦ f )−1 ◦ Lg )(f ◦ exp()) = (log ◦L(g◦ f )−1 )(g ◦ f ◦ exp()) = , hence log ◦(L(g◦ f )−1 )|g◦ f ◦exp(V) ◦ Lg ◦ (log ◦(Lf −1 )|f ◦exp(V) )−1 = IV is analytic, and so Lg is analytic. (b) For each g ∈ Aut(), the inner automorphism g : f → g ◦ f ◦ g−1 is analytic. Let g be in Aut(). Given V ∈ F , the continuity of g♦ (cf. Proposition 5.5.14(ii)) gives the existence of an element V1 ∈ F such that g♦ (V1 ) ⊆ V, and we will consider, for any f ∈ Aut(), the charts of (Aut(), Ta ) at f and g ( f ) given by ( f ◦ exp(V1 ), log ◦(Lf −1 )|f ◦exp(V1 )) and (g ( f ) ◦ exp(V), log ◦(Lg ( f )−1 )|g ( f )◦exp(V)). Keeping in mind Proposition 5.5.14(iii), for any  ∈ V1 , we see that g ( f ◦ exp()) = g ( f ) ◦ g (exp()) = g ( f ) ◦ exp(g♦ ()), and hence g ( f ◦ exp(V1 )) ⊆ g ( f ) ◦ exp(V). Moreover, we have for any  ∈ V1 that [log ◦Lg ( f )−1 ◦ g ◦ (log ◦Lf −1 )−1 ]() = (log ◦Lg ( f )−1 ◦ g )(f ◦ exp()) = (log ◦Lg ( f )−1 )(g ( f ) ◦ exp(g♦ ())) = g♦ (), and hence log ◦(Lg ( f )−1 )|g ( f )◦exp(V) ◦ g ◦ (log ◦(Lf −1 )|f ◦exp(V1 ) )−1 = (g♦ )|V1 . Since g♦ is a continuous linear mapping (cf. Proposition 5.5.14(ii)), and hence analytic, we conclude that g is analytic. (c) The mapping F : ( f , g) → f ◦ g−1 is analytic in an open neighbourhood of (I , I ). Indeed, consider a symmetric open neighbourhood B3 of the origin in aut() such that B3 ⊆ B2 and C(B3 , B3 ) ⊆ B2 , and note that (exp(B3 ) × exp(B3 ), log| exp(B3 ) × log| exp(B3 ) ) and (exp(B2 ), log| exp(B2 ) )

5.5 Banach Lie structures for aut() and Aut()

173

are charts of (Aut()×Aut(), Ta ×Ta ) at (I , I ) and of (Aut(), Ta ) at I , respectively. For each ,  ∈ B3 , we have F(exp(), exp()) = exp() ◦ exp()−1 = exp() ◦ exp(−) = exp(C(, −)), and hence F(exp(B3 ) × exp(B3 )) ⊆ exp(B2 ). Moreover, [log ◦F ◦ (log × log)−1 ](, ) = (log ◦F) (exp(), exp()) = log(exp(C(, −))) = C(, −). −1 Therefore log| exp(B2 ) ◦F ◦ log| exp(B3 ) × log| exp(B3 ) = C ◦ (IB3 × (−IB3 )) is analytic, and we conclude that F is analytic at exp(B3 ) × exp(B3 ). Now, invoking Proposition 5.5.35, we can confirm that (Aut(), Ta ) is a real Banach Lie group. On the other hand, for any f ∈ Aut(), it is immediate to verify that the mapping dLf (I ) : TI (Aut()) → Tf (Aut()) is given by      ∂  ∂ dLf (I )  ∂ log  =  ∂[log ◦(L −1 )|f ◦exp(V) ]  for every V ∈ F . 

I

f

f

Therefore, for any fixed  ∈ aut(), the left invariant analytic vector field determined  : Aut() → T(Aut()) is given by by  (according to §5.5.37)    ∂  ( f ) =  ∂[log ◦(L −1 )|f ◦exp(V) ]  for every f ∈ Aut(), f

f

 can be identified with , and finally aut() can be regarded as the Banach Lie so  algebra of Aut(). In order to prove assertion (iii), it suffices to show that, for any x0 ∈ , there are open neighbourhoods U ⊆  of x0 and U of I in (Aut(), Ta ) such that the mapping is analytic at U × U. To this end, given x0 ∈ , fix any open ball B   centred at x0 and any positive number δ so that Bδ  , fix also an open ball B centred at the origin  in (aut(),  · Bδ ) such that B ⊆ B2 , and consider the chart exp(B), log| exp(B) × (B, IB ) of Aut() ×  at (I , x0 ). Since the mapping exp : B → Aut() ⊆ Hb (B, X) is real analytic, we have that the mapping (, x) → (exp(), x) from B × B to Hb (B, X) × B is real analytic. On the other hand, by Lemma 5.4.18, the mapping ( f , x) → f (x) from Hb (B, X) × B to X is holomorphic, and in particular real analytic. It follows that the mapping  : B × B → X defined by (, x) = exp()(x) is real analytic. Now, keeping in mind that the chart log| exp(B) ×IB : exp(B) × B → B × B

  is real analytic (cf. §5.5.26), we conclude that | exp(B)×B =  ◦ log| exp(B) ×IB is real analytic, and the proof is complete. 

174 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Since the charts in any analytic Banach manifold are bianalytic mappings, assertion (i) in the above theorem gives the following corollary. Corollary 5.5.46 Let X be a complex Banach space, and let  be a bounded domain in X. Then the exponential mapping is bianalytic from an open neighbourhood of the origin in aut() onto an open neighbourhood of I in (Aut(), Ta ). We close this subsection with the following tautologic result. Proposition 5.5.47 Let X and Y be complex Banach spaces, let X and Y be bounded domains in X and Y, respectively, and let g : X → Y be a biholomorphic mapping. Then g : Aut(X ) → Aut(Y ) is an isomorphism of Banach Lie groups (i.e. g is a bicontinuous isomorphism of topological groups and a bianalytic mapping of real analytic Banach manifolds). 5.5.3 Historical notes and comments The material in this section has been elaborated mainly from the survey article of Arazy [837] and the books of Beltita [687], Bonfiglioli and Fulci [1142], Bourbaki [1143], Chu [710], Isidro and Stach´o [751], Reutenauer [1181], Upmeier [814, 815], and Vigu´e [1187]. Other sources are quoted in what follows. Theorem 5.5.6 is based on Vigu´e [1187]. Our proof of Lemma 5.5.17 follows the arguments in the proof of [814, Lemma 7.3]. Theorems 5.5.11, 5.5.21, and 5.5.45 become an infinite dimensional version of a famous theorem by H. Cartan on groups of holomorphic transformations, and were independently proved in 1976 by Upmeier [1186, 1112] and Vigu´e [1187, 1114]. The fact that, for any bounded domain  in a complex Banach space, the group Aut() carries the structure of a real Banach Lie group acting analytically on , is no longer true for unbounded domains in general, even in the finite-dimensional case. We have had serious difficulties in finding a proof of Theorem 5.5.20 (the key tool in the proof of Theorem 5.5.21), which could fit well in the framework of our book. Actually, Theorem 5.5.20 is invoked at the beginning of the proof of [814, Theorem 7.4], and is formulated in [751, Theorem 6.50]. Nevertheles, in both cases, the authors of [751, 814] limit themselves to refer the reader to Bourbaki’s monograph [1143] for a proof. Certainly [1143] contains the information summarized in §5.5.18, as well as a proof of Proposition 5.5.19, but, at least explicitly, it does not contain Theorem 5.5.20. Therefore we have asked for advice from Upmeier, who has kindly written a note [1113], which completes the proof of Theorem 5.5.20, and which has been incorporated into our development. We are very grateful to Upmeier for having written [1113] specifically to be included in our work. 5.6 Kaup’s holomorphic characterization of JB∗ -triples Introduction In this section we consider bounded circular domains in complex Banach spaces, pay special attention to the open unit balls of complex Banach spaces,

5.6 Kaup’s holomorphic characterization of JB∗ -triples

175

and, as the main result in the section, prove Kaup’s holomorphic characterization of JB∗ -triples as those complex Banach spaces whose open unit balls are homogeneous domains (see Theorem 5.6.68). In Subsection 5.6.1 we describe the complete holomorphic vector fields on a bounded circular domain  of a complex Banach space X (Proposition 5.6.6 and §§5.6.7 and 5.6.9), and show that  induces a ‘partial Banach Jordan ∗-triple structure’ on X (Proposition 5.6.11). As the main result we prove that, in the case that  is actually a bounded balanced domain in X, we have the equalities Aut()(0) = Aut0 ()(0) = A =  ∩ aut()(0), where A denotes the set of all points x ∈  such that Aut0 ()(x) is an analytic subset of  (Theorem 5.6.24). In Subsection 5.6.2 we particularize our study to the case that the domains under consideration are precisely the open unit balls of complex Banach spaces. As the main result we prove that the open unit balls of two complex Banach spaces are biholomorphically equivalent (if and) only if the spaces are linearly isometric (Theorem 5.6.35). We also include the fact that complete holomorphic vector fields on the open unit ball of a complex Banach space are characterized by tangency to the unit sphere (Fact 5.6.38), and prove a ‘partial’ version of the Kaup–Stach´o contractive projection Theorem 5.6.59 (see Proposition 5.6.39). Subsection 5.6.3 contains some auxiliary tools on numerical ranges, needed in our approach to the (conclusion of) proof of Kaup’s holomorphic characterization of JB∗ -triples. The main results in the subsection are Theorems 5.6.45 and 5.6.51. Subsection 5.6.4 contains the last steps in the proof of Kaup’s Theorem 5.6.68, as well as the most immediate consequences of this theorem. Among them we emphasize the fact that open unit balls of non-commutative JB∗ -algebras are homogeneous domains (Corollary 5.6.69).

5.6.1 Bounded circular domains §5.6.1 A subset S of complex vector space X is called circular if 0 ∈ S and TS ⊆ S, i.e. S remains invariant under the rotation mappings rt : X → X (t ∈ R) defined by rt (x) := eit x. If  is a circular domain in a complex Banach space X, then it is clear that the set {rt : t ∈ R} can be seen as a subgroup of the group Aut(), and it is called the circular subgroup or the rotation subgroup. / be Proposition 5.6.2 Let X and Y be complex Banach spaces, and let  and  bounded circular domains in X and Y, respectively. If f is a biholomorphic mapping / such that f (0) = 0, then f is the restriction to  of a continuous between  and  linear bijection from X onto Y. More precisely, Df (0) is bijective and f = Df (0)| . / are circular, for each real number t, we can regard the Proof Since both  and  /. rotation mappings rt and/ rt on X and Y, respectively, as automorphisms on  and 

176 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Consider the mapping gt := r−t ◦ f −1 ◦/ rt ◦ f . Clearly gt is biholomorphic on . Moreover, gt (0) = e−it f −1 (eit f (0)) = e−it f −1 (0) = 0 and, by use of the chain rule, Dgt (0) = (e−it Df −1 (0)) ◦ (eit Df (0)) = e−it eit Df −1 (0) ◦ Df (0) = D( f −1 ◦ f )(0) = IX . Now it follows from Proposition 5.3.24 that gt = I , i.e. eit f (x) = f (eit x) for every x ∈ . Since f (0) = 0, the Taylor expansion of f around 0 is of the form  n f (x) = ∞ n=1 Pn (x), where Pn ∈ P (X, Y). It follows that eit

∞ 

Pn (x) = eit f (x) = f (eit x) =

n=1

∞  n=1

Pn (eit x) =

∞ 

eint Pn (x)

n=1

for any real number t. From the uniqueness of the power series expansion, we infer that Pn = 0 for n ≥ 2, i.e. f = (P1 )| . Finally, note that P1 = Df (0) is bijective /. because P1 () =   §5.6.3 Let X be a complex Banach space. The linear vector field iIX is a complete holomorphic vector field on X whose flow ϕ : R × X → X is given by ϕ(t, x) = exp(itIX )(x) = eit x (cf. Exercise 5.4.15). If  is a circular domain in X, then clearly ϕ(R × ) ⊆ , and hence  := (iIX )| is a complete holomorphic vector field on , which will be called in the following the circular vector field on . Let (A, [·, ·]) be a Lie algebra over K. As usual, for each a ∈ A we denote by ada the multiplication operator by a on A defined by ada (x) := [a, x]. Accordingly the multiplication algebra of A, that is the subalgebra of L(A) generated by IA and all the operators ada (a ∈ A), is usually denoted by adA and called the adjoint algebra of A. It is clear that the mapping a → ada from A to adA is linear, and the mapping (F, a) → F(a) from adA × A to A provides a left adA -module structure for A. Thus, for each p(x) ∈ K[x] and for each a, b ∈ A, the meaning of p(ada )(b) is not in doubt. Lemma 5.6.4 Let X be a complex Banach space, let  be a bounded circular domain in X, let p(x) ∈ C[x], and let f ∈ H0 (, X). If  denotes the circular vector field on , if ad is the multiplication operator by  in the complex Lie algebra H0 (, X),  and if f (x) = ∞ n=0 Pn (x) is the power series expansion about 0 of f , then the power series expansion about 0 of p(ad )( f ) is given by p(ad )( f )(x) =

∞  n=0

p(i(n − 1))Pn (x).

5.6 Kaup’s holomorphic characterization of JB∗ -triples

177

Proof Let P be in P n (X, X). For each x ∈  we have that D(x) = iIX and, by Proposition 5.2.7, DP(x) = nP(xn−1 ), hence [, P](x) = DP(x)((x)) − D(x)(P(x)) = nP(xn−1 )(ix) − iIX (P(x)) = niP(x) − iP(x) = i(n − 1)P(x), and so [, P] = i(n − 1)P.

(5.6.1)

 Let B be an open ball centred at 0 and contained in  such that f (x) = ∞ n=0 Pn (x) uniformly on B. It follows from the T-continuity of the Lie product of H0 (B, X) (cf. Proposition 5.5.2(i)) that [, f ](x) =

∞ 

[, Pn ](x) uniformly on

n=0

1 B, 2

 and hence, by (5.6.1), that [, f ](x) = ∞ n=0 i(n − 1)Pn (x) is the power series expansion about 0 of [, f ]. Arguing inductively, we find for each k ∈ N that adk ( f )(x) =

∞ 

(i(n − 1))k Pn (x)

n=0

is the power series expansion about 0 of adk ( f ).  k Finally, if p(x) = m k=0 λk x , then  m  ∞  m m     k k k λk ad ( f ) = λk ad ( f ) = λk (i(n − 1)) Pn p(ad )( f ) = =

k=0  m ∞   n=0

k=0

k=0



λk (i(n − 1))k Pn =

k=0

∞ 

n=0

p(i(n − 1))Pn ,

n=0



as required.

Corollary 5.6.5 Let X be a complex Banach space, let  be a bounded circular domain in X, let p(x) ∈ R[x], and let  ∈ aut(). If  denotes the circular vector field on , if ad is the multiplication operator by  in the real Lie algebra aut(), and  if (x) = ∞ n=0 Pn (x) is the power series expansion about 0 of , then p(ad )() lies in aut(), and its power series expansion about 0 is given by p(ad )()(x) =

∞ 

p(i(n − 1))Pn (x).

n=0

Proof Since aut() is a real subalgebra of the complex Lie algebra H0 (, X), we () can consider for any  ∈ aut() the multiplication operators adaut ∈ adaut() and  H (,X)

∈ adH0 (,X) determined by  in both Lie algebras. It is clear that, for any aut polinomial q(x) with real coefficients, we have that q(ad () ) ∈ adaut() can be ad 0

178 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem H (,X)

seen as the restriction to aut() of q(ad 0 follows immediately from Lemma 5.6.4.

) ∈ adH0 (,X) . Thus the statement 

The following simple result is the key for the analysis of the structure of aut(X ) and is the basis for the connection between the holomorphic structure of X and the structure of the complex Banach space X. Proposition 5.6.6 Let X be a complex Banach space, let  be a bounded circular domain in X, and let  be in aut(). Then we have: (i)  is quadratic in the sense that its Taylor series at 0 is  = 0 + 1 + 2 with n homogeneous polynomial of degree n. (ii) 1 and 0 + 2 belong to aut(). (iii) 0 = 0 if and only if 2 = 0. (iv) i(0 − 2 ) ∈ aut().  Proof Let  be the circular vector field on , and let (x) = ∞ n=0 n (x) be the Taylor series of  at 0. Taking p(x) = x(1 + x2 ) we have p(i(n − 1)) = −in(n − 1) (n − 2), and consequently, by Corollary 5.6.5, we get that p(ad )() ∈ aut() and p(ad )() = −i

∞ 

n(n − 1)(n − 2)n .

n=0

Since p(ad )() vanishes at 0 and has vanishing derivative there, by Proposition 5.5.5, p(ad )() = 0, and so n = 0 for n ≥ 3, proving assertion (i). Next, take the polynomial p(x) = 1 + x2 . Then the same argument yields p(ad )() =

2 

n(2 − n)n = 1 ∈ aut().

n=0

Also, 0 + 2 =  − 1 ∈ aut(), proving assertion (ii). If 0 = 0 then 2 =  − 1 ∈ aut() and we have 2 (0) = (0) − 1 (0) = 0 = 0 and D2 (0) = D(0) − D1 (0) = 1 − 1 = 0. Hence 2 = 0 by Proposition 5.5.5. Conversely, if 2 = 0 then 0 ∈ aut(). But the flows of a constant vector field are line segments, so 0 is incomplete unless it is zero. Thus, assertion (iii) is proved. Finally, by considering the polynomial p(x) = x, and by applying Corollary 5.6.5, we have 2  i(n − 1)n = −i(0 − 2 ), ad () = n=0

hence i(0 − 2 ) ∈ aut(), proving assertion (iv).



§5.6.7 Let X be a complex Banach space, and let  be a bounded circular domain in X. Regarding P(X, X) inside H0 (, X), Proposition 5.6.6 tells us that aut() ⊆ P 0 (X, X) ⊕ P 1 (X, X) ⊕ P 2 (X, X),

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179

and if we set aut0 () := aut() ∩ P 1 (X, X) and aut0 () := aut() ∩ (P 0 (X, X) ⊕ P 2 (X, X)), then we have the direct sum decomposition aut() = aut0 () ⊕ aut0 ().

(5.6.2)

Since, by Proposition 5.2.7, D(0) =  for every  ∈ aut0 () and D(0) = 0 for every  ∈ aut0 (), it follows that aut0 () = { ∈ aut() : (0) = 0} and aut0 () = { ∈ aut() : D(0) = 0}. −1 Therefore aut0 () = T−1 0 ({0} ⊕ BL(X)) and aut0 () = T0 (X ⊕ {0}), where

T0 : aut() → X ⊕ BL(X) is given by T0 () := ((0), D(0)). Hence, by Theorem 5.5.6, aut0 () and aut0 () are closed subspaces of aut(), and so the direct sum decomposition (5.6.2) is topological. We will also consider the orbit of zero under aut() X0 := aut()(0) = {(0) :  ∈ aut()}. Proposition 5.6.8 Let X be a complex Banach space, and let  be a bounded circular domain in X. Then we have: (i) X0 is a closed complex subspace of X. (ii) The mapping E0 : aut() → X0 given by E0 () = (0) is a continuous surjective R-linear mapping with kernel aut0 (), and (E0 )|aut0 () is bicontinuous. (iii) For each y ∈ X0 there exists a unique continuous quadratic mapping qy : X → X such that the vector field x → y−qy (x) on  belongs to aut(), and the mapping y → qy from X0 to P 2 (X, X) is conjugate-linear and continuous. Proof The fact that X0 is a closed real subspace of X follows easily from Theorem 5.5.6, by noticing that X0 can be seen as PX (T0 (aut())). By Proposition  5.6.6(iv), we have that i(0 −2 ) ∈ aut() for every  = 2n=0 n ∈ aut(), hence i(0) = i0 = i(0 − 2 )(0) ∈ X0 , and so X0 is a complex subspace of X. It is clear that the mapping E0 is R-linear and surjective with kernel aut0 (). Moreover, given an open ball B0   centred at 0, the obvious inequality (0) ≤ B0 for every  ∈ aut(), together with Proposition 5.5.4(ii), give that E0 is continuous. Now, the Banach isomorphism theorem implies that (E0 )|aut0 () is bicontinuous, and assertion (ii) is proved. Given y ∈ X0 , it follows from the above that there exists a unique 2 ∈ P 2 (X, X) such that y + 2 ∈ aut(), and so qy := −2 is the unique continuous quadratic mapping on X such that the vector field y : x → y − qy (x) on  belongs to aut(). It is clear that the mapping y → qy from X0 to P 2 (X, X) is R-linear. Moreover, by Proposition 5.6.6(iv), qiy = −iqy , and consequently this mapping is conjugatelinear. Finally, fix r > 0 such that the open ball B0 with centre 0 and radius r is

180 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem strictly inside , and according to assertion (ii) consider a constant M > 0 such that y B0 ≤ My for every y ∈ X0 . Then, for each x ∈ X with x < 1 we have 1 1 1 qy (rx) = 2 y − y (rx) ≤ 2 (y + y (rx)) r2 r r 1 1 ≤ 2 (y + y B0 ) ≤ 2 (1 + M)y, r r

qy (x) =

hence qy  ≤ r12 (1 + M)y, and so the mapping y → qy from X0 to P 2 (X, X) is continuous. Thus, the proof is complete.  §5.6.9 Let X be a complex Banach space, and let  be a bounded circular domain in X. According to assertions (ii) and (iii) in Proposition 5.6.8 immediately above, we have aut0 () = {y : y ∈ X0 }, where for each y ∈ X0 we are denoting by y the complete holomorphic vector field on  given by y (x) := y − qy (x). Note that y is the unique element of aut() satisfying y (0) = y and Dy (0) = 0 (cf. Propositions 5.2.7 and 5.5.5). Moreover, the mapping y → y is a homeomorphism from X0 onto aut0 (). Indeed, this mapping is precisely the inverse of the mapping (E0 )|aut0 () in Proposition 5.6.8(ii). In what follows, contrary to our convention in §5.2.4, instead of qy , we will set qy (·, ·) : X × X → X for the unique continuous symmetric bilinear mapping such that qy (x, x) = qy (x) for every x ∈ X. Proposition 5.6.10 Let X be a complex Banach space, and let  be a bounded circular domain in X. Then we have: (i) aut0 () is a closed subalgebra of the Banach Lie algebra aut(), and the inclusion aut0 () → BL(X) becomes an algebraic-topological embedding from the opposite algebra of aut0 () into BL(X)ant . More precisely, [F, G] = G ◦ F − F ◦ G for all F, G ∈ aut0 () (i.e. the products induced on aut0 () by the Lie algebras aut() and BL(X)ant are opposite). (ii) aut0 ()(X0 ) ⊆ X0 and [aut0 (), aut0 ()] ⊆ aut0 (). More precisely, for all y ∈ X0 and F ∈ aut0 () we have F(y) ∈ X0 , qF(y) (x) = F(qy (x)) − 2qy (F(x), x), and [y , F] = F(y) . (iii) aut0 ()(X0 ) ⊆ X0 and [aut0 (), aut0 ()] ⊆ aut0 (). More precisely, for all y1 , y2 ∈ X0 and x ∈  we have y1 (y2 ) ∈ X0 , qy1 (qy2 (x), x) = qy2 (qy1 (x), x), and [y1 , y2 ] = 2(qy1 (y2 , ·) − qy2 (y1 , ·)) ∈ aut0 ().

(5.6.3)

5.6 Kaup’s holomorphic characterization of JB∗ -triples

181

As a consequence, iqy (y, ·) ∈ aut0 () for every y ∈ X0 . (iv) aut()(X0 ) = X0 . Proof By Theorem 5.5.6, the mapping aut0 () → BL(X) is a bicontinuous R-linear bijection from aut0 () onto its range in BL(X), and aut0 () is closed in BL(X). Moreover, given F, G ∈ aut0 (), if [·, ·] denotes the product in aut(), then we have for each x ∈  that [F, G](x) = DG(x)(F(x)) − DF(x)(G(x)) = G(F(x)) − F(G(x)), hence [F, G] = G ◦ F − F ◦ G, and (i) is proved. Given y ∈ X0 and F ∈ aut0 (), we see that for each x ∈  [y , F](x) = DF(x)(y (x)) − Dy (x)(F(x)) = F(y (x)) + 2qy (x, F(x)) = F(y) − F(qy (x)) + 2qy (x, F(x)). Since the mapping x → F(qy (x)) − 2qy (x, F(x)) is an element of P 2 (X, X), it follows that F(y) ∈ X0 , qF(y) (x) = F(qy (x)) − 2qy (F(x), x), and [y , F] = F(y) , proving assertion (ii). Given y1 , y2 ∈ X0 , we have for each x ∈  that [y1 , y2 ](x) = Dy2 (x)(y1 (x)) − Dy1 (x)(y2 (x)) = −2qy2 (x, y1 (x)) + 2qy1 (x, y2 (x)) = −2qy2 (x, y1 ) + 2qy2 (x, qy1 (x)) + 2qy1 (x, y2 ) − 2qy1 (x, qy2 (x)) & % & % = 2 qy1 (y2 , x) − qy2 (y1 , x) − 2 qy1 (qy2 (x), x) − qy2 (qy1 (x), x) . Since the mapping x → qy1 (qy2 (x), x) − qy2 (qy1 (x), x) is an element of P 3 (X, X), by Proposition 5.6.6(i), it must be null, and hence qy1 (qy2 (x), x) = qy2 (qy1 (x), x), and we find (5.6.3). Now, given y ∈ X0 , keeping in mind assertions (i) and (iii) in Proposition 5.6.8 and taking y1 = y and y2 = iy, we deduce that 1 1 [y , iy ] = (qy (iy, ·) − qiy (y, ·)) = iqy (y, ·) 4 2 lies in aut0 (). As a consequence, by (ii), we have that qy (y, X0 ) ⊆ X0 . Finally, given y1 , y2 ∈ X0 , by (5.6.3), 1 qy1 (y2 ) = qy2 (y2 , y1 ) + [y1 , y2 ](y2 ) ∈ X0 , 2 and hence y1 (y2 ) = y1 − qy1 (y2 ) ∈ X0 . Thus the proof of assertion (iii) is complete. Finally, the inclusion aut()(X0 ) ⊆ X0 follows from the decomposition (5.6.2) and the first assertions in (ii) and (iii). The converse inclusion is obvious.  Now we will prove that each bounded circular domain in a complex Banach space X induces a ‘partial Banach Jordan ∗-triple structure’ of X (cf. Definitions 4.1.32 and 4.1.37).

182 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proposition 5.6.11 Let X be a complex Banach space, and let  be a bounded circular domain in X. Then the mapping {· · · } : X × X0 × X → X defined by {xyz} := qy (x, z) for all x, z ∈ X and y ∈ X0 is linear in the outer variables, conjugate-linear in the middle variable, and satisfies the ‘commutative’ condition {xyz} = {zyx} for all x, z ∈ X and y ∈ X0 and the ‘partial’ Jordan triple identity {uv{xyz}} = {{uvx}yz} − {x{vuy}z} + {xy{uvz}} for all u, v, y ∈ X0 and x, z ∈ X. Moreover {· · · } is continuous and {X0 X0 X0 } ⊆ X0 . Proof It is clear from the definition that {· · · } is linear in the outer variables, conjugate-linear in the middle variable, and satisfies the commutative condition. Given u ∈ X0 , by Proposition 5.6.10(ii)–(iii), F := iqu (u, ·) ∈ aut0 (), and for all y ∈ X0 and x ∈ X we have that F(y) ∈ X0 and F(qy (x)) = qF(y) (x) + 2qy (F(x), x), hence i{uu{xyx}} = {x, i{uuy}, x} + 2{i{uux}, y, x}, and so {uu{xyx}} = −{x{uuy}x} + 2{{uux}yx}. By polarization law (4.2.2) we obtain that {uv{xyx}} = −{x{vuy}x} + 2{{uvx}yx} for all u, v, y ∈ X0 and x ∈ X, and finally linearizing in the variable x we derive the partial Jordan triple identity. Since, by Proposition 5.6.8(iii), the conjugate-linear mapping y → qy from X0 to P 2 (X, X) is continuous, it follows from (5.2.5) that the conjugate-linear mapping y → qy (·, ·) from X0 to BLs2 (X, X) is continuous. Therefore there exists a positive constant M such that qy (·, ·) ≤ My for every y ∈ X0 , and so we have {xyz} = qy (x, z) ≤ qy (·, ·)xz ≤ Mxyz for all x, z ∈ X and y ∈ X0 . Thus {· · · } is continuous. For any y ∈ X0 , by Proposition 5.6.10(iii), we have y (X0 ) ⊆ X0 , hence qy (X0 ) ⊆ X0 , and so qy (X0 , X0 ) ⊆ X0 . The arbitrariness of y in X0 gives that {X0 X0 X0 } ⊆ X0 , as required.  In the remaining of this subsection we will deal with the Banach Lie group Aut() for a bounded circled domain . Lemma 5.6.12 Let X be a complex Banach space, let  be a bounded circular domain in X, and let  be in aut(). Then exp()( ∩ X0 ) ⊆  ∩ X0 . Proof Since X0 is a closed subspace of X (Proposition 5.6.8(i)) and (X0 ) ⊆ X0 (Proposition 5.6.10(iv)), the result follows from Corollary 5.4.11. 

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183

§5.6.13 Let G be a Hausdorff topological group, and let W be a symmetric (i.e. W = W −1 ) connected neighbourhood of the unit of G. For each n ∈ N we set W n := {g1 g2 · · · gn : gk ∈ W (1 ≤ k ≤ n)}. It is well-known in the general theory of topological groups (see, for example, [689, Exercises 4.11 and 4.14]) that H := ∪n∈N W n is a closed normal subgroup of G and that H is the connected component of the unit of G. Thus H does not depend on the choice of W. As a particularization of this general fact we have the following. Fact 5.6.14 Let X be a complex Banach space, and let  be a bounded domain in X. Let us denote by Aut0 () the connected component of I in (Aut(), Ta ). Then Aut0 () is a closed normal subgroup of (Aut(), Ta ) and Aut0 () = {exp(1 ) ◦ · · · ◦ exp(n ) : n ∈ N, k ∈ aut() (1 ≤ k ≤ n)}.

(5.6.4)

Proof For any  ∈ aut() given, it follows from Corollary 5.5.23(iii) that the function γ : [0, 1] → Aut() defined by γ (t) = exp(t) is a continuous path which connects I and exp() in the topological space (Aut(), Ta ). Therefore exp(aut()) is a connected subset of (Aut(), Ta ). Since exp(aut()) is a neighbourhood of I in (Aut(), Ta ) (by Theorem 5.5.21), and exp(aut()) is clearly symmetric, the result follows from §5.6.13.  §5.6.15 Let X be a complex Banach space, and let  be a bounded circular domain in X. Since X0 is a closed complex subspace of X, it is clear that  ∩ X0 is a bounded circular open subset of the complex Banach space X0 . However,  ∩ X0 may fail to be connected. Proposition 5.6.16 Let X be a complex Banach space, and let  be a bounded circular domain in X. We have: (i) Aut0 ()( ∩ X0 ) ⊆  ∩ X0 . If in addition  ∩ X0 is connected, then: (ii) For each f ∈ Aut0 (), we have f ( ∩ X0 ) =  ∩ X0 , and hence f regarded as a mapping from  ∩ X0 to  ∩ X0 becomes an element in Aut( ∩ X0 ). In this way, {f|∩X0 : f ∈ Aut0 ()} can be regarded as a subgroup of Aut( ∩ X0 ). (iii) Aut0 ()(0) =  ∩ X0 . Proof Assertion (i) is a consequence of the equality (5.6.4) and of Lemma 5.6.12. Now suppose that ∩X0 is connected, and so ∩X0 is a bounded circular domain in the complex Banach space X0 . Given f in Aut0 (), by (i) we have f ( ∩ X0 ) ⊆  ∩ X0 and f −1 ( ∩ X0 ) ⊆  ∩ X0 . Therefore  ∩ X0 = f ( f −1 ( ∩ X0 )) ⊆ f ( ∩ X0 ) ⊆  ∩ X0 , hence f ( ∩ X0 ) =  ∩ X0 ,

184 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem and so f|∩X0 (regarded as a mapping from  ∩ X0 to  ∩ X0 ) becomes an element in Aut( ∩ X0 ). Thus (ii) is proved. In order to prove (iii), fix an open ball B centred at 0 and a positive number δ so that Bδ  , and consider the neighbourhood B of the origin in aut() given in Lemma 5.5.17. Since the mapping η : y → y from X0 to (aut(),  · Bδ ) is continuous and real-linear, we can find an open neighbourhood U of 0 in X0 such that U ⊆  and x ∈ B for every x ∈ U. By Proposition 5.5.8, the mapping exp : B → Aut0 () ⊆ (Hb (B, X),  · B ) is real analytic and D exp(0)() = (I ) =  for every  ∈ aut(). On the other hand, as a consequence of Lemma 5.4.18, the mapping E0 : (Hb (B, X),  · B ) → X given by E0 ( f ) = f (0) is holomorphic and DE0 ( f )(g) = Df (0)(0) + g(0) for all f , g ∈ Hb (B, X). It follows that the mapping h := E0 ◦ exp ◦η|U is real analytic on U with range contained in Aut0 ()(0) ⊆ X0 (by assertion (i)). Moreover, the chain rule gives that, for each x ∈ X0 , Dh(0)(x) = DE0 (IB )(D exp(0)(Dη(0)(x))) = DE0 (IB )(D exp(0)(x )) = DE0 (IB )(x ) = DIB (0)(0) + x (0) = x, and hence Dh(0) = IX0 . Since h(0) = 0, the inverse mapping theorem 5.2.63 guarantees the existence of open neighbourhoods U0 and V0 of 0 in X0 with U0 ⊆ U such that h : U0 → V0 is real bianalytic. Since V0 = h(U0 ) ⊆ h(U) ⊆ Aut0 ()(0), it follows that Aut0 ()(0) is a neighbourhood of 0 in X0 . Now, keeping in mind that Aut0 () can be seen as a subgroup of Aut( ∩ X0 ) (by assertion (ii)), and invoking Proposition 5.3.57, we conclude that Aut0 ()(0) =  ∩ X0 .  Let U be an open subset of a complex Banach space. For any holomorphic function f : U → C we set Z( f ) := {x ∈ U : f (x) = 0}, and for any family F of holomorphic functions from U to C we define Z(F ) := ∩f ∈F Z( f ). Definition 5.6.17 Let X be a complex Banach space, and let  be an open subset of X. An subset A of  is said to be (complex-)analytic in  if, for every x ∈ , there exist an open neighbourhood U of x and a family Fx of holomorphic functions from U to C such that A ∩ U = Z(Fx ). It is immediate that, if A is an analytic set in , then A is closed in . Fact 5.6.18 Let X1 and X2 be complex Banach spaces, let 1 and 2 be open subsets of X1 and X2 , respectively, and let h : 1 → 2 be a holomorphic mapping. If A is an analytic set in 2 , then h−1 (A) is an analytic set in 1 . Proof Let A be an analytic set in 2 . Given x ∈ 1 , consider an open neighbourhood V of h(x) in 2 and a family Gh(x) of holomorphic functions from V to C such that A ∩ V = Z(Gh(x) ). It is immediate to verify that U := h−1 (V) is an open neighbourhood of x in 1 and Fx := {g ◦ h|U : g ∈ Gh(x) } is a family of holomorphic functions from U to C such that h−1 (A) ∩ U = Z(Fx ). Thus h−1 (A) is an analytic set in 1 .  The following consequences are straightforward.

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185

Corollary 5.6.19 Let X be a complex Banach space, let  be an open subset of X, and let A be an analytic set in . We have: (i) If 0 is an open subset of X contained in , then A ∩ 0 is an analytic set in 0 . (ii) If Y is a complex Banach space, and if h :  → Y is a biholomorphic bijection from  to h(), then h(A) is an analytic set in h(). Proposition 5.6.20 Let X be a complex Banach space, let  be a domain in X, and let A be an analytic set in . We have: (i) If A has non-empty interior, then A = . (ii)  \ A is connected. Proof Suppose that the interior B of A is non-empty. Given x in the closure of B in , choose a connected open neighbourhood U of x in  and a family Fx of holomorphic functions from U to C such that A ∩ U = Z(Fx ). Therefore each f ∈ Fx vanishes on B ∩ U, hence by the principle of analytic continuation (Proposition 5.2.43) f = 0 on U, and so U ⊆ A. Therefore U ⊆ B, and in particular x ∈ B. As a result B is closed in . Since B is open and  is connected, we conclude that B = , and hence A = , as required. By (i), in order to prove assertion (ii), we may suppose that A has empty interior. We claim that each x ∈  has a connected open neighbourhood U in  such that U \ A is connected. Indeed, given x ∈ , choose a convex open neighbourhood U of x in  and a family Fx of holomorphic functions from U to C such that A∩U = Z(Fx ). For any fixed x0 , x1 ∈ U \ A consider the sets V := {ζ ∈ C : ζ x0 + (1 − ζ )x1 ∈ U} and B := {ζ ∈ V : ζ x0 + (1 − ζ )x1 ∈ A}, and for each f ∈ Fx consider the function  f : V → C defined by  f (ζ ) := f (ζ x0 + (1 − ζ )x1 ). It is immediate to verify that V is a convex open subset of C such that 0, 1 ∈ V, that each  f is a holomorphic function,  and that 0, 1 ∈ / B = ∩f ∈Fx Z(f ). It follows from the principle of isolated zeroes [1149, Theorem 3.7] that B is discrete in V, and hence V \ B is connected. If γ : [0, 1] → V is a curve joining 0 and 1 in V \ B, then it is clear that / γ : [0, 1] → U defined by γ (t) := γ (t)x0 + (1 − γ (t))x1 is a curve joining x0 and x1 in U \ A. Thus U \ A is / connected. Now that the claim is proved, to obtain a contradiction, assume that  \ A is not connected, and write  \ A = 1 ∪ 2 where 1 and 2 are disjoint non-empty open sets. Since A has empty interior, we have  =  \ A = 1 ∪ 2 = 1 ∪ 2 , where denotes closure in , and hence 1 ∩ 2 = ∅ because  is connected. Given x ∈ 1 ∩ 2 , by the claim, there is a connected open neighbourhood U of x in  such that U \ A is connected. But U \ A = U ∩( \ A) = U∩(1∪2 ) = (U∩1 )∪(U∩2 ), and consequently either U ∩ 1 = ∅ or U ∩ 2 = ∅, a contradiction with the fact that x ∈ 1 ∩ 2 .  §5.6.21 Let X be a complex Banach space, and let  be a bounded circular domain in X. We define A := {x ∈  : Aut0 ()(x) is analytic in }.

186 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Lemma 5.6.22 Let X be a complex Banach space, and let  be a bounded circular domain in X such that  ∩ X0 is connected. Then 0 ∈ A and Aut()(A ) = A . Proof Since X0 is a closed complex subspace of X (cf. Proposition 5.6.8(i)), it follows that the polar (X0 )◦ of X0 in X  is a family of holomorphic functions from X to C such that X0 ∩  = {x ∈  : f (x) = 0 for every f ∈ (X0 )◦ }, and hence X0 ∩  is analytic in . Keeping in mind that, by Proposition 5.6.16(iii), Aut0 ()(0) = ∩X0 , we conclude that 0 ∈ A . Now, we remark that the normality of Aut0 () in Aut() (cf. Fact 5.6.14) implies that A is Aut()-invariant. Indeed, given h ∈ Aut() and x ∈ A , we have that Aut0 ()(x) is analytic in , and hence, by Corollary 5.6.19(ii), h(Aut0 ()(x)) is also analytic in . But h(Aut0 ()(x)) = (h ◦ Aut0 () ◦ h−1 )(h(x)) = Aut0 ()(h(x)). So h(x) ∈ A . Therefore Aut()(A ) ⊆ A . The reverse inclusion is clear, completing the proof.  We recall that a subset S of complex vector space X is called balanced if BC S ⊆ S. It is clear that every balanced subset of X is circled. Lemma 5.6.23 Let X be a complex Banach space, let  be a bounded balanced domain in X, and let x ∈ A . Then Cx ∩  ⊆ Aut0 ()(x). Proof The conclusion is straightforward for x = 0 because I ∈ Aut0 (). So we shall suppose that x ∈ A \ {0}, and we set V := {ζ ∈ C : ζ x ∈ }. Since  is bounded and balanced, it is clear that there exists r > 1 such that V = r. Moreover, since Aut0 ()(x) is analytic in , and the function ζ → ζ x from V to  is holomorphic, it follows from Fact 5.6.18 that the set A := {ζ ∈ V : ζ x ∈ Aut0 ()(x)} is analytic in V. As  is circular, Aut0 () contains the rotation subgroup, hence Aut0 ()(x) contains eiθ x for every θ ∈ R, and so A contains T. Therefore V \ A = [(V \ A) ∩ ] ∪ [(V \ A) ∩ (V \ BC )]. Since, by Proposition 5.6.20(ii), V \ A is connected, we see that  ⊆ A or V \ BC ⊆ A. In any case, A has non-empty interior, and hence, by Proposition 5.6.20(i), V = A.  So Cx ∩  ⊆ Aut0 ()(x), as desired. Note that, if  is a bounded balanced domain in a complex Banach space X, then clearly  ∩ X0 is also balanced, hence connected. Theorem 5.6.24 Let X be a complex Banach space, and let  be a bounded balanced domain in X. Then Aut()(0) = Aut0 ()(0) = A =  ∩ X0 . Proof

It follows from Lemma 5.6.22 that Aut()(0) ⊆ A .

(5.6.5)

Given x ∈ A , by Lemma 5.6.23, 0 ∈ Aut0 ()(x), and so x ∈ Aut0 ()(0). Therefore A ⊆ Aut0 ()(0).

(5.6.6)

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Combining the inclusions (5.6.5) and (5.6.6) with the obvious inclusion Aut0 ()(0) ⊆ Aut()(0), we find Aut()(0) = Aut0 ()(0) = A . Finally, invoking Proposition 5.6.16(iii), the proof is complete.  §5.6.25 Let X be a complex Banach space, and let  be a bounded circular domain in X. We set Aut0 () := Aut() ∩ P 1 (X, X). In view of Proposition 5.6.2 we have Aut0 () = {f ∈ Aut() : f (0) = 0} and Aut0 () ⊆ Inv(BL(X)).

(5.6.7)

Since T-convergence implies pointwise convergence (cf. Fact 5.3.9), it is clear from the first assertion in (5.6.7) that Aut0 () is a closed subgroup of Aut() for the T-topology (and hence for the Ta -topology). Moreover, if fn is a sequence in Aut0 () that  · -converges to F ∈ Inv(BL(X)), then we see that F() is an open subset of X contained in , and hence F() ⊆ . Since fn−1  · -converges to F −1 , we have also that F −1 () ⊆ . It follows that F() = , and hence F ∈ Aut0 (). Thus Aut0 () is a closed subgroup of Inv(BL(X)). The group Aut0 () is called the isotropy subgroup of Aut() at the origin. Corollary 5.6.26 Let X be a complex Banach space, and let  be a bounded balanced domain in X. Then Aut() = Aut0 () ◦ Aut0 () = Aut0 () ◦ Aut0 (). Proof Let f ∈ Aut(). By Theorem 5.6.24, there exists g ∈ Aut0 () such that f (0) = g(0). Therefore h := g−1 ◦ f ∈ Aut() verifies h(0) = 0, and so h ∈ Aut0 () (cf. §5.6.25). Hence f = g ◦ h ∈ Aut0 () ◦ Aut0 (). This shows that Aut() = Aut0 () ◦ Aut0 (). The normality of Aut0 () in Aut() yields now that Aut() = Aut0 () ◦ Aut0 ().  5.6.2 The symmetric part of a complex Banach space In this subsection we will deal with special bounded circular domains: the open unit balls of complex Banach spaces. Recall that, for any bounded circular domain  in a complex Banach space X, the orbit of zero under aut() was denoted by X0 in §5.6.7. §5.6.27 Let X be a complex Banach space. The orbit of zero under aut(X ) is called the symmetric part of X, and is denoted by Xs . Thus Xs := {(0) :  ∈ aut(X )}.

188 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem The following facts summarize some properties of the symmetric part of a complex Banach space, which are particularizations of general results in the preceding subsection. Fact 5.6.28 Let X be a complex Banach space. Then Xs is a closed complex subspace of X with open unit ball Xs = Aut(X )(0) = Aut0 (X )(0), and hence both Aut(X ) and Aut0 (X ) act on it transitively. Proof The fact that Xs is a closed complex subspace of X follows from Proposition 5.6.8(i), and the equalities Xs = Aut(X )(0) = Aut0 (X )(0) are given by Theorem 5.6.24.  Fact 5.6.29 Let X be a complex Banach space. Then we have: (i) For each y ∈ Xs there exists a unique continuous quadratic mapping qy : X → X such that (the restriction to X of) the vector field y : x → y − qy (x) on X belongs to aut(X ), and the mapping y → qy from Xs to P 2 (X, X) is conjugatelinear and continuous. (ii) The so-called (intrinsic) partial triple product of X is the mapping · · ·  from X × Xs × X to X defined by xyz := qy (x, z) for all x, z ∈ X and y ∈ Xs ,

(5.6.8)

where qy (·, ·) : X × X → X stands for the unique continuous symmetric bilinear mapping such that qy (x, x) = qy (x) for every x ∈ X. Moreover, · · ·  is symmetric bilinear in the outer variables, conjugate-linear in the middle variable, and verifies the partial Jordan triple identity uvxyz = uvxyz − xvuyz + xyuvz for all u, v, y ∈ Xs and x, z ∈ X. Furthermore · · ·  is continuous and Xs Xs Xs  ⊆ Xs . Proof Assertion (i) is a particularization of Proposition 5.6.8(iii), and assertion (ii) is a particularization of Proposition 5.6.11.  Another remarkable fact concerning the structure of the Banach Lie real algebra aut(X ) for a complex Banach space X is the following. Fact 5.6.30 Let X be a complex Banach space. Then we have: (i) aut0 (X ) = { ∈ aut(X ) : (0) = 0} and aut0 (X ) = { ∈ aut(X ) : D(0) = 0} are closed subspaces of aut(X ), and aut(X ) = aut0 (X ) ⊕ aut0 (X ) is a topological direct sum decomposition. Moreover, aut0 (X ) = {iF|X : F ∈ H(BL(X), IX )} and aut0 (X ) = {y : y ∈ Xs }.

5.6 Kaup’s holomorphic characterization of JB∗ -triples

189

(ii) [aut0 (X ), aut0 (X )] ⊆ aut0 (X ) (i.e. aut0 (X ) is a subalgebra of the Banach Lie real algebra aut(X )), and [aut0 (X ), aut0 (X )] ⊆ aut0 (X ) and [aut0 (X ), aut0 (X )] ⊆ aut0 (X ). Explicitly, for x, y ∈ Xs , z ∈ X, and  ∈ aut0 (X ), [x , y ](z) = 2yxz − 2xyz and [x , ] = (x) .

(5.6.9)

(iii) For any x ∈ Xs , the operator L(x, x) := xx · is hermitian. Proof

With exception of the equality aut0 (X ) = {iF|X : F ∈ H(BL(X), IX )},

(5.6.10)

which follows from Exercise 5.4.15(iii), the remaining part of assertion (i) is contained in §§5.6.7 and 5.6.9. Assertion (ii) is contained in Proposition 5.6.10(i)–(iii). Given x ∈ Xs , it follows from (5.6.9) that 4iL(x, x) = [x , ix ] ∈ aut0 (X ), and hence, by (5.6.10), L(x, x) ∈ H(BL(X), IX ).  Since both Xs and the partial triple product of a complex Banach space X are intrinsically defined, we have the following. Proposition 5.6.31 Let X and Y be complex Banach spaces, and let F : X → Y be a surjective linear isometry. Then F(Xs ) = Ys and F(xyzX ) = F(x)F(y)F(z)Y for every (x, y, z) ∈ X × Xs × X. Now we show that biholomorphic mappings between the open unit balls of complex Banach spaces preserve the open unit balls of their symmetric parts. Theorem 5.6.32 Let X and Y be complex Banach spaces, and let f : X → Y be a biholomorphic bijection. Then f (Xs ) = Ys . Proof By Proposition 5.5.47, the mapping f  : ϕ → f ◦ ϕ ◦ f −1 is a bicontinuous isomorphism between the real Banach Lie groups Aut(X ) and Aut(Y ), and hence it preserves the connected components of the identity automorphisms, i.e. we have f  (Aut0 (X )) = Aut0 (Y ) (cf. Fact 5.6.14). As a consequence, for each x ∈ X we have f (Aut0 (X )(x)) = f  (Aut0 (X ))( f (x)) = Aut0 (Y )( f (x)). Keeping in mind §5.6.21 and Theorem 5.6.24, we realize that, for each x ∈ Xs , Aut0 (X )(x) is analytic in X , hence Aut0 (Y )( f (x)) = f (Aut0 (X )(x)) is analytic in Y (by Corollary 5.6.19(ii)), and so f (x) ∈ Ys . Thus f (Xs ) ⊆ Ys . Applying the same argument with f −1 we get f (Xs ) = Ys as desired.  A direct consequence of Proposition 5.6.2 is the following.

190 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Corollary 5.6.33 Let X and Y be complex Banach spaces, and let f be a biholomorphic mapping between X and Y . If f (0) = 0, then f is the restriction to X of a surjective linear isometry from X to Y. More precisely, Df (0) is a surjective linear isometry from X to Y, and f = Df (0)|X . Proof The result follows from Proposition 5.6.2 by noticing that, if F : X → Y is a linear mapping inducing a bijection from X onto Y , then F is a linear isometry of X onto Y.  §5.6.34 Let X be a complex Banach space. In view of the above corollary, the isotropy subgroup of Aut(X ) at the origin (cf. §5.6.25), Aut0 (X ) = {f ∈ Aut(X ) : f (0) = 0} consists of the restriction to X of all surjective linear isometries of X to itself. Thus the group G (X) of all surjective linear isometries on X is identified naturally with Aut0 (X ). Theorem 5.6.35 Let X and Y be complex Banach spaces. Then X and Y are biholomorphically equivalent if and only if X and Y are linearly isometric. Proof If F : X → Y is a surjective linear isometry, then clearly F(X ) = Y and F, regarded as a mapping from X to Y , is a biholomorphic bijection. Conversely, suppose that f : X → Y is a biholomorphic bijection. By Theorem 5.6.32 and Fact 5.6.28, there exist ϕ ∈ Aut(Y ) such that f (0) = ϕ(0). Thus ψ := ϕ −1 ◦ f : X → Y is biholomorphic and ψ(0) = 0. By Corollary 5.6.33, ψ = Dψ(0)|X and Dψ(0) is a surjective linear isometry from X to Y.  Lemma 5.6.36 Let X be a Banach space over K, let  be an open subset of X, let  :  → X be a locally Lipschitz vector field, and let ϕ : I →  be a local flow of . Then the function n : I → R defined by n(t) := ϕ(t) satisfies       ϕ(t) ϕ(t) n− (t) = min  V X, and n+ (t) = max  V X, , ϕ  (t) , ϕ  (t) ϕ(t) ϕ(t) for every t ∈ I such that ϕ(t) = 0. Proof Fix t0 ∈ I such that ϕ(t0 ) = 0, and consider the X-valued function f defined +rϕ(t0 )) ϕ(t0 ) in an open interval about 0 by f (r) := ϕ(t0ϕ(t . Note that f (0) = ϕ(t and 0 ) 0 )   f (0) = ϕ (t0 ). Keeping in mind Corollary 2.1.6, we realize that ϕ(t0 +t) ϕ(t0 ) − 1 ϕ(t0 + t) − ϕ(t0 ) f (−r) − 1 lim = − lim = lim t t r t→0− t→0− r→0+ ϕ(t0 )       ϕ(t0 ) ϕ(t0 )   = − max  V X, = min  V X, , −ϕ (t0 ) , ϕ (t0 ) ϕ(t0 ) ϕ(t0 )

n− (t0 ) =

5.6 Kaup’s holomorphic characterization of JB∗ -triples

191

and ϕ(t0 +t) ϕ(t0 ) − 1 ϕ(t0 + t) − ϕ(t0 ) lim = lim t t t→0+ t→0+ ϕ(t0 )    f (r) − 1 ϕ(t0 ) = lim = max  V X, , ϕ  (t0 ) . r ϕ(t0 ) r→0+

n+ (t0 ) =



Lemma 5.6.37 Let X be a Banach space over K, let  be an open subset of X containing BX , let  :  → X be a locally Lipschitz vector field satisfying (V(X, x, (x))) = 0 for every x ∈ SX ,

(5.6.11)

let a be in SX , and let ϕ : I →  be the maximal local flow of  at a. Then ϕ(I) ⊆ SX . Proof

Note that the mapping : X \ {0} → X defined by  x :=

Lipschitz. Indeed, given x ∈ X \ {0}, for y ∈ X such that x − y
x − x = x, 2 2 and hence for all y, z ∈ X \ {0} such that x − y < 12 x and x − z < 12 x we have y z 1  zy − yz  − = y z yz 1 =  z(y − z) + (z − y)z  yz 1 ≤ (z y − z + | z − y | z) yz 2 4 ≤ y − z < y − z. y x

 y − z =

 : X \ {0} → X defined by  (x) := ( As a consequence, the vector field  x) is locally  Lipschitz. Let ψ : J → X \ {0} be the maximal local flow of  at a, and consider the function n : J → R defined by n(t) := ψ(t). By (5.6.11), we have for each t ∈ J that    ψ(t)  6 6 6  , ψ (t) , 0 = (V(X, ψ(t), (ψ(t)))) = (V(X, ψ(t), (ψ(t)))) =  V X, ψ(t) and hence, by Lemma 5.6.36, n (t) = 0. Therefore n(t) = n(0) = a = 1 for every t ∈ J, that is to say ψ(J) ⊆ SX . Hence ψ is a local flow of  at a, and consequently J ⊆ I and ψ = ϕ|J . If inf(I) < inf(J) = α, then ϕ(α) ∈ SX , and the above reasoning with ϕ(α) instead of a allows us to find a prolongation of ψ, which is a contradiction. Analogously, sup(J) < sup(I) yields a contradiction. Therefore J = I, ψ = ϕ, and so  ϕ(I) ⊆ SX . The following statement expresses the fact that complete holomorphic vector fields on the open unit ball of a complex Banach space are characterized by tangency to the unit sphere.

192 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Fact 5.6.38 Let X be a complex Banach space, and let  be a holomorphic vector field on X . Then  is complete if and only if it has a bounded holomorphic extension ) to a neighbourhood of BX satisfying (say  (x)) ⊆ iR for every x ∈ SX . V(X, x,  Proof Suppose that  is complete. By Proposition 5.6.6,  = 0 + 1 + 2 , with n homogeneous polinomial of degree n. Therefore,  has a bounded holomorphic  → 2X stand for the flow of  . Since  is complete,  to 2X . Let  extension  ϕ:D  it follows that R × X ⊆ D and  ϕ (R × X ) = X . Therefore, given x ∈ SX , we  at x is [(2X ) \ X ]-valued (cf. Corollary realize that the maximal local flow  ϕx of  5.4.7), that is,  ϕx (t) ≥ 1 for every t ∈]t− (x), t+ (x)[. On the other hand, for each n (t), t∈]t− (x), t+ (x)[, it follows from the continuity of  ϕ that  ϕx (t) = limn→∞  ϕ n+1 x − + and hence  ϕx (t) ≤ 1. Therefore, for each x ∈ SX ,  ϕx (]t (x), t (x)[) ⊆ SX , and (x))). hence, by Lemma 5.6.36, 0 = (V(X,  ϕx (0),  ϕx (0))) = (V(X, x,   to some open Conversely, suppose that  has a bounded holomorphic extension    containing BX and satisfying (V(X, x, (x))) = 0 for every x ∈ SX . By Lemma  at a norm-one vector is contained in SX , and 5.6.37, each maximal local flow of   is bounded, by Corollary consequently the flow of  is contained in X . Since  5.4.12, we conclude that  is complete.  Proposition 5.6.39 Let X be a complex Banach space, let π : X → X be a contractive linear projection, and write Y := π(X). Then: (i) π(Xs ) ⊆ Ys , and we have xπ(y)zY = π(xyzX ) for every (x, y, z) ∈ Y × Xs × Y. (ii) Y ∩ Xs ⊆ Ys , and we have xyzY = π(xyzX ) for every (x, y, z) ∈ Y × (Y ∩ Xs ) × Y. (iii) π(xyzX ) = π(xπ(y)zX ) for every (x, y, z) ∈ Y × (Xs ∩ π −1 (Xs )) × Y. Proof

Let y be in Xs , and let x be in SY . By Facts 5.6.29(i) and 5.6.38, we have V(X, x, y − xyxX ) ⊆ iR.

Since π is a contractive linear mapping with π(x) = x, it follows from Corollary 2.1.2 that V(Y, x, π(y) − π(xyxX )) ⊆ iR. Since x is arbitrary in SY , we can apply again Fact 5.6.38 to derive that (the restriction to Y of) the mapping  : z → π(y) − π(zyzX ) from Y to Y is a complete holomorphic vector field on Y . Therefore, by §5.6.27, π(y) = (0) ∈ Ys , and then, by Fact 5.6.29(i) again, we have zπ(y)zY = π(zyzX ) for every z ∈ Y. Since y is arbitrary in Xs , the proof of assertion (i) is complete. Assertion (ii) follows straightforwardly from assertion (i).

5.6 Kaup’s holomorphic characterization of JB∗ -triples

193

Let (x, y, z) be in Y × (Xs ∩ π −1 (Xs )) × Y. Then, by assertion (i), π(y) ∈ Ys and xπ(y)zY = π(xyzX ). But, on the other hand, π(y) ∈ Y ∩ Xs , which implies in view of assertion (ii) (with π(y) instead of y) that xπ(y)zY = π(xπ(y)zX ). It follows that π(xyzX ) = π(xπ(y)zX ).  5.6.3 Numerical ranges revisited The next fact is a slight variant of Fact 2.9.63. Fact 5.6.40 Let X be a normed space over K, and let u be a norm-one element in X. Then for every x ∈ X we have 3 {φ(x) : φ ∈ BX  , |φ(u) − 1| < δ}− . V(X, u, x) = δ>0

Proof

Let x be in X. Then, by Fact 2.9.63, we have   3 − V(X, u, x) = {φ(x) : φ ∈ SX  , |φ(u) − 1| < δ} ⊆

3

δ>0

{φ(x) : φ ∈ BX  , |φ(u) − 1| < δ}− .

δ>0

Conversely, for each positive number δ consider the set

and let μ be in

4

Bδ := {φ(x) : φ ∈ BX  , |φ(u) − 1| < δ}, δ>0 Bδ .

Then for each δ > 0 there is φδ ∈ BX  with

|φδ (u) − 1| < δ and |φδ (x) − μ| < δ. Therefore for every λ ∈ K we have |μ − λ| = |μ − φδ (x) + φδ (x) − λφδ (u) + λ(φδ (u) − 1)| ≤ δ + x − λu + |λ|δ, and hence, by letting δ → 0, we get |μ − λ| ≤ x − λu. Thus, by Proposition 2.1.1, 4  we obtain that μ ∈ λ∈K BK (λ, x − λu) = V(X, u, x). According to §2.1.30, given a normed space X over K, we denote by (X) the subset of X × X  defined by (X) := {(x, φ) : x ∈ SX , φ ∈ D(X, x)}, and by π1 the natural projection from X ×X  onto X. The next lemma was formulated in Proposition 2.9.64 without discussing its proof. Lemma 5.6.41 Let X be a normed space over K, let u be in SX , and let  be any subset of (X) such that π1 () is dense in SX . Then for every x ∈ X we have 3 co{φ(x) : (y, φ) ∈ , y − u < δ}. (5.6.12) V(X, u, x) = δ>0

Proof Let δ > 0, and let (y, φ) be in  such that y − u < δ. Then φ lies in BX  and |φ(u) − 1|(= |φ(u − y)|) < δ. Therefore for every x ∈ X we have {φ(x) : (y, φ) ∈ , y − u < δ} ⊆ {φ(x) : φ ∈ BX  , |φ(u) − 1| < δ}.

194 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Since the right-hand side of the above inclusion is convex, the inclusion ⊇ in the equality (5.6.12) follows from Fact 5.6.40 above. To prove the converse inclusion we argue by contradiction. Assume that there exist x ∈ X and δ > 0 such that V(X, u, x)  co{φ(x) : (y, φ) ∈ , y − u < δ}. Then there is α ∈ K such that max (αV(X, u, x)) > sup (α{φ(x) : (y, φ) ∈ , y − u < δ}). Replacing x with αx, we actually may suppose that max (V(X, u, x)) > K := sup ({φ(x) : (y, φ) ∈ , y − u < δ}).

(5.6.13)

Now, since in the case K = C we have D(XR , y) = { ◦ φ : φ ∈ D(X, y)} for every y ∈ SX , we may also suppose that K = R. Write B := {y ∈ SX : y − u < δ}. Then, for each y ∈ B, there exists φ y ∈ D(X, y) such that φ y (x) ≤ K. (Indeed, since π1 () is dense in SX , there exists a sequence (yn , φn ) in  with yn ∈ B for every n and y = limn yn , and then, since φn (x) ≤ K for every n, it is enough to take φ y equal to any cluster point of the sequence φn in the w∗ -topology of X  .) Now let / X denote / stand for the restriction of φ to / the linear hull of {u, x} in X and, for φ ∈ X  , let φ X. u u / / / / Since φ ∈ D(X , u) and φ (x) ≤ K < max V(X, u, x) = max V(X , u, x), it follows that / X is not smooth at u, and as a consequence / X is two-dimensional. Thus D(/ X , u) is a \ {u, −u} has exactly two connected components nontrivial closed segment, and S/ X / (the ‘two sides of u on S/ X ’). By taking a sequence zn in B ∩ X on either side of u on S/ X with u = limn zn , and passing to a subsequence if necessary, we may suppose zn is convergent in / zn ∈ D( that the sequence φ5 X  , and then (since φ5 X , zn ) for every n) z we realize that the sequence φ5n actually converges to either one or other of the two zn (x) ≤ K for every n, it follows that ψ(x) ≤ K for extreme points of D(/ X , u). Since φ5 every ψ ∈ D(/ X , u). By taking ψ = / η, where η is any element of D(X, u) such that η(x) = max V(X, u, x), we are in a contradiction with (5.6.13).  Lemma 5.6.42 Let X be a normed space over K, and let  be a subset of (X) such that π1 () is dense in SX . Then for every g ∈ B(π1 (), X) we have 3 co{φ(g(y)) : (x, φ) ∈ , y ∈ π1 (), x − y < δ}, V(B(π1 (), X), 1 , g) = δ>0

where 1 stands for the natural embedding π1 () → X. Proof Set Y := B(π1 (), X) and, for (y, φ) ∈ π1 () × X  , denote by y ⊗ φ the element of Y  defined by (y ⊗ φ)(h) := φ(h(y)) for every h ∈ Y. Now, consider the set /  of all couples (h, ψ) ∈ SY × Y  such that there exists (y, φ) ∈ π1 () × X  satisfying that (h(y), φ) belongs to  and that ψ = y ⊗ φ. Clearly /  is a subset of / (Y). We claim that π1 ( ) is dense in SY . Let f be in SY and 0 < ε < 1. There exists y ∈ π1 () with f (y) > 1 − ε, and, by the density of π1 () in SX , there exists (x, φ) ∈  with x − ff (y) (y)  < ε. Consider the element h of Y defined by h(y) = x  and and h(z) = f (z) whenever z belongs to π1 () \ {y}. Then (h, y ⊗ φ) belongs to / h − f  = x − f (y) < ε(2−ε) . Now that the claim has been proved, we fix g ∈ Y, 1−ε 4 and apply Lemma 5.6.41 to obtain V(Y, 1 , g) = δ>0 co(Aδ ), where  , h − 1  < δ}. Aδ := {ψ(g) : (h, ψ) ∈ /

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Thus, to conclude the proof it is enough to show that Aδ = Bδ , where Bδ := {φ(g(y)) : (x, φ) ∈ , y ∈ π1 (), x − y < δ}. Let λ be in Aδ . Then there exists (h, ψ) ∈ /  such that h − 1  < δ and ψ(g) = λ. By the definition of /  , there exists (y, φ) ∈ π1 () × X  such that (h(y), φ) ∈  and ψ = y ⊗ φ. It follows that, writing x := h(y), we have λ = φ(g(y)) with (x, φ) ∈ , y ∈ π1 (), and x − y < δ, i.e. λ belongs to Bδ . Conversely, suppose that λ is in Bδ . Then there exist (x, φ) ∈  and y ∈ π1 () such that x − y < δ and φ(g(y)) = λ. Considering the element h of Y defined by h(y) = x and h(z) = z whenever z belongs  and to π1 () \ {y}, and writing ψ := y ⊗ φ, we have λ = ψ(g) with (h, ψ) ∈ /  h − 1  < δ, i.e. λ belongs to Aδ . For the sake of shortness, given a normed space X over K and a bounded function f : SX → X, we write V( f ) := V(B(SX , X), IX , f ), where here IX stands for the inclusion SX → X. With this convention, the choice  = (X) in Lemma 5.6.42 yields the following. Corollary 5.6.43 Let X be a normed space over K, and let f be a bounded function from SX to X. Then we have 3 V( f ) = co{φ( f (y)) : (x, φ) ∈ (X), y ∈ SX , x − y < δ}. δ>0

Corollary 5.6.44 Let X be a normed space over K, let  be a subset of (X) such that π1 () is dense in SX , and let f be a bounded continuous function from SX to X. Then we have 3 V( f ) = co{φ( f (y)) : (x, φ) ∈ , y ∈ π1 (), x − y < δ} δ>0

=

3

co{φ( f (y)) : (x, φ) ∈ , y ∈ SX , x − y < δ}.

δ>0

Proof Denoting by Z the subspace of B(SX , X) consisting of all bounded continuous functions from SX to X, and writing Y := B(π1 (), X), the mapping f → f|π1 () from Z to Y becomes a linear isometry sending 1 to 1 . Therefore, for every f ∈ Z we have V( f ) = V(Z, 1, f ) = V(Y, 1 , f|π1 () ). Now, the first equality in the present corollary follows from Lemma 5.6.42. The second equality follows from the former, obvious inclusions, and Corollary 5.6.43.  The next result was formulated in Theorem 2.1.50 without discussing its proof. Theorem 5.6.45 Let X be a normed space over K, let  be a subset of (X) such that π1 () is dense in SX , and let f : SX → X be any bounded and uniformly continuous function. Then V( f ) = co{φ( f (x)) : (x, φ) ∈ }. Proof Let ε > 0. Take δ > 0 such that f (x) − f (y) < ε whenever x, y are in SX and x − y < δ. Then we have (φ( f (y))) < (φ( f (x))) + ε whenever (x, φ) is in , y is in SX , and x − y < δ. Since

196 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem V( f ) ⊆ co{φ( f (y)) : (x, φ) ∈ , y ∈ SX , x − y < δ} (by Corollary 5.6.44), we deduce max (V( f )) ≤ max (C) + ε, where C := co{φ( f (x)) : (x, φ) ∈ }. Now, the arbitrariness of ε yields max (V( f )) ≤ max (C). Finally, for μ ∈ SK , the last inequality, applied to μf instead f , gives max (μV( f )) ≤ max (μC), which implies that V( f ) ⊆ C because C is a compact and convex subset of K.  Arguing as in the proof of Corollary 2.1.34, with Theorem 5.6.45 above instead of Proposition 2.1.31, we obtain the following. Corollary 5.6.46 Let X be a Banach space over K, and let f : SX  → X  be any bounded and uniformly continuous function. Then V(B(SX  , X  ), IX  , f ) = co{F( f )(x) : (x, f ) ∈

(X)}.

According to Mazur’s theorem (see for example [800, Proposition 9.4.3]), if X is a separable Banach space over K, then the set {x ∈ SX : X is smooth at x} is dense in SX . Therefore the next corollary follows from Theorem 5.6.45. Corollary 5.6.47 Let X be a separable Banach space over K. Then, for every bounded and uniformly continuous function f : SX → X, we have V( f ) = co{φ( f (x)) : (x, φ) ∈

(X) and X is smooth at x}.

Lemma 5.6.48 Let X be a complex Banach space, let T be in H(BL(X), IX ), let Y be a closed subspace of XR invariant under iT, and let p be a norm-one element in Y. Then V(Y, p, iT(p)) = 0. Proof Since the inclusion BL(X)R → BL(XR ) is a linear isometry, it follows from Corollary 2.1.2(ii) and Proposition 2.1.4 that V(BL(XR ), IX , iT)[= V(BL(X)R , IX , iT) = (V(BL(X), IX , iT))] = 0. On the other hand, (iT)|Y (regarded as a mapping form Y to Y) lies in BL(Y). Therefore, by lemma 2.2.24, we have that V(BL(Y), IY , (iT)|Y ) = 0. Finally, since the mapping F → F(p) from BL(Y) to Y is a linear contraction carrying IY to p, the result follows from Corollary 2.1.2(i).  In what follows, the reader should keep in mind that, if A is a norm-unital normed complex algebra, then i[H(A, 1), H(A, 1)] ⊆ H(A, 1) (cf. Lemma 2.3.1). Proposition 5.6.49 Let A be a norm-unital complete normed complex algebra, let p be a norm-one element in H(A, 1), and let y be in H(A, 1). Then V(H(A, 1), p, i[y, p]) = 0. Proof Let T ∈ BL(A) be defined by T(a) := [y, a] for every a ∈ A. Then iT leaves H(A, 1) invariant. On the other hand, since T = Ly − Ry , it follows from Lemma

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197

2.1.10 that T lies in H(BL(A), IA ). Therefore the result follows from Lemma 5.6.48 above (with X = A and Y = H(A, 1)).  Lemma 5.6.50 Let A be a norm-unital normed complex algebra, and let p be a normone element in H(A, 1) such that 1 − p < 2. Then D(A, p) ⊆ D(A, 1). Proof

Let η be in D(A, p). Then  ◦ η ∈ D(AR , p), hence ( ◦ η)|H(A,1) ∈ D(H(A, 1), p).

On the other hand, by §2.3.37 and Fact 2.9.62, we have that D(H(A, 1), p) ⊆ D(H(A, 1), 1). It follows that ( ◦ η)|H(A,1) ∈ D(H(A, 1), 1). As a consequence, (η(1)) = 1, and hence, since η = 1, we obtain that η(1) = 1, and finally η ∈ D(A, 1).  Theorem 5.6.51 Let A be a norm-unital complete normed complex algebra, write Y := H(A, 1) ⊕ iH(A, 1) (cf. Corollary 2.1.13), and let p be a norm-one element in H(A, 1) such that 1 − p < 2. Then V(A, p, [y, p]) = 0 for every y ∈ Y. Proof Let y be in Y, and let φ be in D(A, p). Write y = h + ik with h, k ∈ H(A, 1), and note that, for every x ∈ H(A, 1), the equality (φ([x, p])) = 0 holds because [x, p] ∈ iH(A, 1) and Lemma 5.6.50 applies. Then, by Proposition 2.1.4, Corollary 2.1.2, and Proposition 5.6.49, we have φ(i[k, p]) ∈ (V(A, p, i[k, p])) = V(AR , p, i[k, p]) = V(H(A, 1), p, i[k, p]) = 0. Therefore, since (φ([h, p])) = 0, we conclude that (φ([y, p])) = 0. Since Y is a complex subspace of A, and y is an arbitrary element of Y, we can replace y with iy in the above equality to finally obtain φ([y, p]) = 0.  Corollary 5.6.52 Let A be a norm-unital complete normed complex algebra, and let p be in SA such that V(A, 1, p) ⊆ R+ 0 . Then (i) D(A, p) ⊆ D(A, 1). (ii) V(A, p, [y, p]) = 0 for every y ∈ H(A, 1). Proof In view of Lemma 5.6.50 and Theorem 5.6.51, it is enough to show that 1 − p < 2. But, since V(A, 1, p) ⊆ [0, 1], and n(H(A, 1)) = 1, and V(A, 1, p)(= (V(A, 1, p)) = V(AR , 1, p)) = V(H(A, 1), 1, p), we actually have that 1 − p ≤ 1. 5.6.4 Concluding the proof of Kaup’s theorem Lemma 5.6.53 Let X be a JB∗ -triple. Then for all x, y ∈ X we have v(BL(X), IX , L(x, y)) ≤ xy.



198 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof Let φ be in D(BL(X), IX ). Then, according to Definition 4.1.37 and §4.1.39, the mapping (x, y) → φ(L(x, y)) becomes a non-negative hermitian sesquilinear form on X, and hence, by the Cauchy–Schwarz inequality, for x, y ∈ X we have 7 7 |φ(L(x, y))| ≤ φ(L(x, x))φ(L(y, y)) ≤ L(x, x)L(y, y) = xy, the equality above being true thanks to Corollary 4.1.51. Therefore v(BL(X), IX , L(x, y)) = sup{|φ(L(x, y))| : φ ∈ D(BL(X), IX )} ≤ xy.



The next lemma is a variant of Corollary 2.1.2(i). Lemma 5.6.54 Let (X, u) and (Y, v) be numerical-range complex spaces, and let F : X → Y be a conjugate-linear contraction such that F(u) = v. Then V(Y, v, F(x)) ⊆ {λ : λ ∈ V(X, u, x)} for every x ∈ X. Proof Let x be in X and let μ be in V(Y, v, F(x)). Then there exists φ ∈ D(Y, v) such that μ = φ(F(x)). Since clearly the mapping z → φ(F(z)) (from X to C) lies in D(X, u), it follows that μ = λ, where λ := φ(F(x)) ∈ V(X, u, x).  Lemma 5.6.55 Let X be a JB∗ -triple, let x be in SX , and let x be in D(X, x). We have: (i) The linear functional a → x ({xax}) on X belongs to D(X, x). (ii) If X is smooth at x (cf. §2.6.1), then x (a − {xax}) ∈ iR for every a ∈ X. Proof For T ∈ BL(X) we write ||| T ||| := v(BL(X), IX , T), and note that ||| · ||| is a norm on BL(X) equivalent to the operator norm (cf. Proposition 2.1.11). Then, according to Lemma 5.6.53, the mapping F : a → L(x, a) from X to (BL(X), ||| · |||) becomes a conjugate-linear contraction satisfying F(x) = L(x, x). Therefore, noticing that ||| L(x, x) ||| = 1 (by Corollary 4.1.51 and Proposition 2.3.4), it follows from Lemma 5.6.54 that for every a ∈ X we have that V((BL(X), ||| · |||), L(x, x), L(x, a)) ⊆ {λ : λ ∈ V(X, x, a)}.

(5.6.14)

Now let φ stand for the linear functional on BL(X) defined by φ(T) := x (T(x)) for every T ∈ BL(X). Then, by Corollary 4.2.12, we have φ(L(x, x))(= x ({xxx})) = 1. On the other hand, since clearly φ belongs to D(BL(X), IX ), for T ∈ BL(X) we have that φ(T) ∈ V(BL(X), IX , T), so |φ(T)| ≤ ||| T |||, and hence ||| φ ||| ≤ 1. It follows that φ lies in D((BL(X), ||| · |||), L(x, x)). Therefore, by (5.6.14), for a ∈ X we have x ({xax})(= φ(L(x, a))) ∈ V(X, x, a), and assertion (i) follows straightforwardly. Suppose that X is smooth at x, so that x is the unique element of D(X, x). Thus, by (i), for every a ∈ X we have x ({xax}) = x (a), hence (x ({xax})) = (x (a)), and finally x (a − {xax}) ∈ iR, as asserted in (ii).  Let X be a complex Banach space such that X = Xs (cf. §5.6.27). Then the partial triple product of X (cf. Fact 5.6.29) works on the whole X × X × X, and consequently it will be called the intrinsic triple product of X.

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199

Proposition 5.6.56 Let X be a JB∗ -triple. Then: (i) For every a ∈ X the holomorphic vector field a : x → a − {xax} is complete on the open unit ball X of X. (ii) X = Xs , and the triple product of X as a JB∗ -triple coincides with the intrinsic triple product of X as a complex Banach space. Proof

Let a be in X. In view of Fact 5.6.38, to prove (i) it is enough to show that V(X, x, a (x)) ⊆ iR for every x ∈ SX .

Suppose at first that X is separable. Then, by Lemma 5.6.55(ii), for (x, φ) ∈ (X) such that X is smooth at x we have that φ(a (x)) ∈ iR. It follows from Corollary 5.6.47 that V((a )|SX ) ⊆ iR. Now, if x is any element of SX , the mapping g → g(x) from B(SX , X) to X is a linear contraction taking IX to x, and hence, by Corollary 2.1.2(i), we have V(X, x, a (x)) ⊆ V((a )|SX ) ⊆ iR, as desired. Now, remove the assumption that X is separable. Then, given x ∈ SX , the closed subtriple of X generated by a and x (say Y) is a separable JB∗ -triple, and therefore, by Corollary 2.1.2 and the above, we have that V(X, x, a (x)) = V(Y, x, a (x)) ⊆ iR. Keeping in mind assertion (i) just proved, for every a ∈ X we have that  a = a (0) ∈ Xs . Therefore assertion (ii) follows from Fact 5.6.29. Keeping in mind Proposition 5.6.56(ii), the next theorem follows straightforwardly from Proposition 5.6.31. Theorem 5.6.57 Surjective linear isometries between JB∗ -triples preserve triple products. Putting together Proposition 4.1.52 and Theorem 5.6.57, we are provided with a complete proof of Theorem 2.2.28 (stated there without a proof) that surjective linear isometries between JB∗ -triples are precisely those bijective linear mappings which preserve triple products. Corollary 5.6.58 Let X be a JB∗ -triple, and let T be in H(BL(X), IX ). Then T({xyz}) = {T(x)yz} − {xT(y)z} + {xyT(z)} for all x, y, z ∈ X. Proof Let r be in R. By Corollary 2.1.9(iii), exp(irT) : X → X is a surjective linear isometry. Therefore, by Theorem 5.6.57, we have exp(irT)({xyz}) = {exp(irT)(x) exp(irT)(y) exp(irT)(z)} for all x, y, z ∈ X. Now take derivative at r = 0.



Theorem 5.6.59 Suppose that X is a JB∗ -triple, let π : X → X be a contractive linear projection, and set Y := π(X). Then we have: (i) Y becomes a JB∗ -triple under the triple product {· · ·}π defined by {xyz}π := π({xyz}) for all x, y, z ∈ Y. (ii) π({xyz}) = π({xπ(y)z}) for every y ∈ X and all x, z ∈ Y.

200 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof By Proposition 5.6.56(ii), X = Xs and the triple product of X as a JB∗ -triple coincides with the intrinsic triple product of X. Then, by Proposition 5.6.39(ii) and Fact 5.6.29(ii), Y = Ys and the intrinsic triple product of Y is nothing other than {· · ·}π . It follows from Fact 5.6.29 that Y, endowed with the triple product {· · ·}π , satisfies the Jordan triple identity. Now, for T ∈ BL(X), let us regard π ◦ T|Y as a mapping from Y to Y. Then the mapping T → π ◦ T|Y from BL(X) to BL(Y) becomes a linear contraction carrying IX to IY , and therefore, by Corollary + 2.1.2(i), V(BL(Y), IY , π ◦ T|Y ) ⊆ R+ 0 whenever V(BL(X), IX , T) ⊆ R0 . By taking π T = L(x, x) (x ∈ X), we realize that (Y, {· · ·} ) becomes a positive hermitian Banach Jordan ∗-triple satisfying {yyy}π  ≤ y3 for every y ∈ Y. To prove the converse inequality we may assume that y ∈ SY . Then, taking φ ∈ D(X, y), we see that φ ◦ π ∈ D(X, y), and hence, by Corollary 4.2.12, φ({yyy}π )[= φ(π({yyy}))] = 1, which implies {yyy}π  ≥ 1 = y3 . Now the proof of assertion (i) is complete. Since X = Xs , and the triple product of X coincides with its intrinsic triple product, assertion (ii) follows from Proposition 5.6.39(iii).  Theorem 5.6.59 was already formulated without proof (cf. Theorem 2.3.74). Lemma 5.6.60 Let X be a hermitian Banach Jordan ∗-triple. Then the equality L(x(3) , x(3) ) = L(x, x)3 holds for every x ∈ X. Proof Assume at first that X is abelian. Then, by Proposition 4.2.5, the closed linear hull of L(X, X) ∪ {IX } in BL(X) is a subalgebra of BL(X), and is in fact a C∗ -algebra for the involution determined by L(x, y)∗ = L(y, x). Therefore for x ∈ X we have that L(x, x)3  = L(x, x)3 . Since L(x(3) , x(3) ) = L(x, x)3 for every x ∈ X (because X is abelian), the result follows in this particular case. Now remove the assumption that X is abelian. Then, by Lemma 4.1.38, all closed subtriples of X are hermitian Banach Jordan ∗-triples. Let x be in X, and let M stand for the closed subtriple of X generated by x. Then, as a consequence of Lemma 4.1.42, we have r(LM (z, z)) = r(LX (z, z)) for every z ∈ M. Therefore, Proposition 2.3.22 applies to get LM (z, z) = r(LM (z, z)) = r(LX (z, z)) = LX (z, z) for every z ∈ M. In particular, we have that LM (x, x) = LX (x, x) and LM (x(3) , x(3) ) = LX (x(3) , x(3) ). Finally, since LM (x(3) , x(3) ) = LM (x, x)3 because M is abelian (cf. Lemma 4.1.49) and the first paragraph of the proof applies, we conclude that LX (x(3) , x(3) ) = LX (x, x)3 .



Let X be a hermitian Banach Jordan ∗-triple, and let u, v be in X. Then, clearly, and L(u,v)−L(v,u) lie in H(BL(X), IX ), and both L(u,v)+L(v,u) 2 2i L(u, v) =

L(u, v) − L(v, u) L(u, v) + L(v, u) +i . 2 2i

(5.6.15)

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Suppose that the Banach Jordan ∗-triple X is actually positive hermitian, and let x be in X such that L(x, x) = 1. Then, as a consequence of Corollary 5.6.52(i), we have that H(BL(X), IX ) ⊆ H(BL(X), L(x, x)), and therefore L(u, v) + L(v, u) L(u, v) − L(v, u) , ∈ H(BL(X), L(x, x)). 2 2i

(5.6.16)

As a straightforward consequence of (5.6.15) and (5.6.16), we are provided with the following. Lemma 5.6.61 Let X be a positive hermitian Banach Jordan ∗-triple, let x be in X such that L(x, x) = 1, and let u, v be in X. Then: (i) [V(BL(X), L(x, x), L(u, v) + L(v, u))] = 2[V(BL(X), L(x, x), L(u, v))]. (ii) If V(BL(X), L(x, x), L(u, v) − L(v, u)) = 0, then V(BL(X), L(x, x), L(u, v)) ⊆ R. Theorem 5.6.62 Let X be a positive hermitian Banach Jordan ∗-triple. We have: √ (i) The mapping x → x∞ := L(x, x) is a continuous seminorm on X satisfying {xxx}∞ = x3∞ for every x ∈ X. (ii) If there is M > 0 such that Mx2 ≤ L(x, x) for every x ∈ X, then  · ∞ is an equivalent norm on X, and (X,  · ∞ ) becomes a JB∗ -triple under the triple product of X. Proof Let φ be in D(BL(X), IX ). Then the mapping (x, y) → φ(L(x, y)) becomes a non-negative hermitian sesquilinear form on X, and hence, by Minkowski’s inequality, for x, y ∈ X we have 7 7 7 φ(L(x + y, x + y)) ≤ φ(L(x, x)) + φ(L(y, y)) ≤ x∞ + y∞ . Therefore, by Proposition 2.3.4, 7 7 x + y∞ = L(x + y, x + y) = v(BL(X), IX , L(x + y, x + y)) 7 = sup{φ(L(x + y, x + y)) : φ ∈ D(BL(X), IX )} ≤ x∞ + y∞ . √ Thus  · ∞ is a seminorm on X. Since for x ∈ X we have x∞ ≤ Kx (where K stands for the norm of the triple product of X), the seminorm  · ∞ is continuous. Now the proof of (i) is concluded by noticing that the equality {xxx}∞ = x3∞ (x ∈ X) follows from Lemma 5.6.60. 2 √ Suppose that there is M > 0 such that Mx ≤ L(x, x) for every x ∈ X. Then M· ≤ ·∞ on X, and hence, since ·∞ is continuous, it is in fact an equivalent norm on X. Let x be in X with x∞ = 1 and let y be in X. Then for r ∈ R we have x + ry2∞ = L(x + ry, x + ry) = L(x, x) + r(L(y, x) + L(x, y)) + r2 L(y, y), and hence, for 0 = r ∈ R, (x + ry∞ + 1)

x + ry∞ − 1 L(x, x) + r(L(y, x) + L(x, y)) + r2 L(y, y) − 1 = . r r

202 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Therefore, noticing that L(x, x) = 1, it follows from Corollary 2.1.6 that 2 max [V((X,  · ∞ ), x, y)] = max [V(BL(X), L(x, x), L(y, x) + L(x, y))], which, in view of Lemma 5.6.61(i), can be reformulated as max [V((X,  · ∞ ), x, y)] = max [V(BL(X), L(x, x), L(y, x))].

(5.6.17)

Since y is arbitrary in X, we can replace y with λy (λ ∈ SC ) in (5.6.17) to obtain that V((X,  · ∞ ), x, y) = V(BL(X), L(x, x), L(y, x)).

(5.6.18)

Now let a be in X. It follows from Corollary 5.6.52(ii) that V(BL(X), L(x, x), [L(a, a), L(x, x)]) = 0, which, in view of the identity (4.2.11) in p. 506 of Volume 1, reads as V[BL(X), L(x, x), L({aax}, x) − L(x, {aax})] = 0. By Lemma 5.6.61(ii), the last equality can be reformulated as V[BL(X), L(x, x), L({aax}, x)] ⊆ R, and hence, taking y = {aax} in (5.6.18), we obtain that V((X,  · ∞ ), x, {aax}) ⊆ R.  Since V((BL(X),  · ∞ ), IX , L(a, a)) = co {V((X,  · ∞ ), x, {aax}) : x ∈ S(X,·∞ ) } (cf. Proposition 2.1.31), we realize that V((BL(X),  · ∞ ), IX , L(a, a)) ⊆ R. Thus, since a is arbitrary in X, we have proved that (X,  · ∞ ) is a hermitian Banach Jordan ∗-triple under the triple product of X. Since the positivity of a hermitian Banach Jordan ∗-triple can be spectrally checked, and the spectrum of a bounded linear operator on X is the same in BL(X) and in (BL(X),  · ∞ ), and X is positive, we see that (X,  · ∞ ) is positive. Finally, since the equality {xxx}∞ = x3∞ holds for every x ∈ X (by assertion (i) already proved), (X,  · ∞ ) is a JB∗ -triple under the triple product of X.  Corollary 5.6.63 A positive hermitian Banach Jordan ∗-triple X can be equivalently renormed as a JB∗ -triple if (and only if) there is M > 0 such that Mx2 ≤ L(x, x) for every x ∈ X. Lemma 5.6.64 Let X be a nonzero hermitian Banach Jordan ∗-triple such that there is a ∈ X satisfying sp(BL(X), L(a, a)) ∩ R− = ∅. Then there exists 0 = b ∈ X such that sp(BL(X), L(b, b)) ⊆ R− 0. Proof Let M denote the closed subtriple of X generated by a. In view of Lemma 4.1.38, M is a hermitian Banach Jordan ∗-triple and, thanks to Lemma 4.1.42, we have that sp(BL(M), LM (a, a)) ∩ R− = ∅, and that sp(BL(X), L(y, y)) ⊆ R− 0 for every − M y ∈ M such that sp(BL(M), L (y, y)) ⊆ R0 . Therefore we can suppose that X is generated by a as a Banach Jordan ∗-triple. Then the a-homotope Jordan algebra X (a) (cf. Definition 4.1.36) is associative (cf. Lemma 4.1.49). Moreover, an easy induction argument shows that for every j ∈ N we have that aj = a(2j−1) , where the

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left-hand side of the equality is seen in X (a) whereas the right-hand side is seen in X. Moreover, clearly, L(a, a)(an ) = an+1 (n ∈ N), and hence L(a, a)m (an ) = am+n for all n, m ∈ N.

(5.6.19)

On the other hand, the equality {a(2j−1) a(2k−1) a(2n−1) } = a(2(j+k+n−1)−1) for all j, k, n ∈ N (cf. (4.1.29)) can be read as {aj ak an } = aj+k+n−1 . It follows that L(aj , ak )(an ) = L(a, a)j+k−1 (an ) for all j, k, n ∈ N.  j Now let p(x) = m p(x) := xp(x) ∈ R[x]. Then, since j=0 λj x be in R[x], and consider (a)  p(0) = 0, we can think about  p(a) in X , and we have that  n L( p(a), p(a))(a ) = λj λk L(aj+1 , ak+1 )(an ) 0≤j,k≤m

=



λj λk L(a, a)j+k+1 (an )

0≤j,k≤m





= ⎣L(a, a) ◦ ⎝

m 

⎞  ⎤ m  λj L(a, a)j ⎠ ◦ λk L(a, a)k ⎦ (an )

j=0

k=0

= (L(a, a) ◦ p(L(a, a)) )(a ). 2

n

Since X = lin{an : n ∈ N} (cf. Lemma 4.1.49 again), we obtain that L( p(a), p(a)) = L(a, a) ◦ p(L(a, a))2 .

(5.6.20)

Note also that, as a consequence of (5.6.19) with n = 1, we have that p(L(a, a))(a) =  p(a).

(5.6.21)

Now recall that, since X is abelian (cf. Lemma 4.1.49 once more), it follows from Proposition 4.2.5 that the closed linear hull of L(X, X) ∪ {IX } in BL(X) (say A) is a commutative subalgebra of BL(X), and is in fact a unital C∗ -algebra for the involution determined by L(x, y)∗ = L(y, x). As a consequence, for x ∈ X we have that sp(A, L(x, x)) ⊆ R (cf. Proposition 1.2.20(ii)), and hence, by Proposition 1.1.93(iii), sp(BL(X), L(x, x)) = sp(A, L(x, x)).

(5.6.22)

Write K := sp(A, L(a, a)), and let f ∈ C C (K) be defined by f (t) = 0 if t ∈ K ∩ R+ 0 and − 2 3 f (t) = t if t ∈ K ∩ R . Then, since tf (t) = f (t) for every t ∈ K, it follows from the continuous functional calculus of A at L(a, a) (cf. Theorem 1.2.28 and §1.2.29) and the spectral mapping theorem (cf. Proposition 1.2.34) that sp[A, L(a, a) ◦ ( f (L(a, a))2 ] = {f (t)3 : t ∈ K} ⊆ R− 0.

(5.6.23)

Now, since f is a real-valued continuous function on K, it follows from the Stone– Weierstrass theorem (cf. Theorem 1.2.10) the existence of a sequence pn in R[x] such

204 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem that pn (t) → f (t) uniformly on K. Since pn (L(a, a))(a) =  pn (a) (by (5.6.21)), and pn (L(a, a)) converges to f (L(a, a)) in A (and hence in BL(X)), we realize that  pn (a) converges to b := f (L(a, a))(a), and so L( pn (a), pn (a)) converges to L(b, b). Then, keeping in mind (5.6.20), we realize that L(b, b) = L(a, a) ◦ ( f (L(a, a))2 . Finally, since K ∩ R− ⊇ sp(BL(X), L(a, a)) ∩ R− = ∅, it follows from (5.6.22) and (5.6.23) that b = 0 and that sp(BL(X), L(b, b)) ⊆ R−  0. Lemma 5.6.65 Let X be a complex Banach space such that X = Xs , let a be a normone element in X, and regard X as a Banach Jordan ∗-triple under its intrinsic triple product (cf. Fact 5.6.29). Then the operator IX − L(a, a) is not invertible in BL(X). Proof Suppose, to obtain a contradiction, that IX − L(a, a) is invertible in BL(X). For any x ∈ X, we know, by §5.6.9, that x ∈ aut(X ), and hence, by Theorem 5.5.11, x := 12 [x , a ] + x ∈ aut(X ). Moreover, by (5.6.3) in Proposition 5.6.10, we have that x (·) = qx (a, ·) − qa (x, ·) + x − qx (·),

(5.6.24)

and in particular x (a) = (IX − L(a, a))(x). By applying Fact 5.4.40 with  = X and B = δX for a prefixed δ > 1, there exists a positive number r such that for every polynomial P in X with P2δX < r we have that B ⊆ dom(exp(P)) and the mapping a : {P ∈ P(X, X) : P2δX < r} → X given by a (P) := exp(P)(a) is holomorphic and D a (0)(P) = P(a). On the other hand, it is clear from (5.6.24) that H : x → x is a continuous R-linear mapping from X to P(X, X), and hence DH(x) = H for every x ∈ X. Now, fix an open neighbourhood V of 0 in X such that x 2δX < r for every x ∈ V, and consider the mapping F = a ◦ H : V → X. By the chain rule we have DF(0)(x) = D( a ◦ H)(0)(x) = D a (H(0))(DH(0)(x)) = D a (0)(H(x)) = D a (0)(x ) = x (a) = (IX − L(a, a))(x), and hence DF(0) = IX − L(a, a) ∈ Inv(BL(X)). Therefore, by the inverse mapping theorem 5.2.63, there exists an open neighbourhood U of 0 in X contained in V such that F(U) is an open neighbourhood of F(0) = exp(0 )(a) = exp(0)(a) = IX (a) = a, and the mapping F : U → F(U) is R-bianalytic. But, for each x ∈ U, exp(x ) is a continuous function on B such that exp(x )(X ) = X , hence exp(x )(a) ∈ SX , and so F(U) ⊆ BX . This completes the contradiction and the proof.  Corollary 5.6.66 Let X be a complex Banach space such that X = Xs , and regard X as a Banach Jordan ∗-triple under its intrinsic triple product. Then {a ∈ X : sp(BL(X), L(a, a)) ⊆] − ∞, 1[} ⊆ X . As a consequence, if for a ∈ X we have sp(BL(X), L(a, a)) ⊆ R− 0 , then a = 0. Proof Suppose that a ∈ X \ {0} satisfies sp(BL(X), L(a, a)) ⊆] − ∞, 1[. Since, by a a Lemma 5.6.65, 1 ∈ sp(BL(X), L( a , a )), it follows that    a a a2 ∈ a2 sp BL(X), L , = sp(BL(X), L(a, a)) ⊆] − ∞, 1[, a a and so a < 1.

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Finally, if a ∈ X is such that sp(BL(X), L(a, a)) ⊆ R− 0 , then for each t ∈ R we − 2 see that sp(BL(X), L(ta, ta)) = t sp(BL(X), L(a, a)) ⊆ R0 , hence, by the foregoing, ta ∈ X for every t ∈ R, and consequently a = 0.  Fact 5.6.67 Let X be a complex Banach space such that X = Xs , and regard X as a Banach Jordan ∗-triple under its intrinsic triple product. Then: (i) X is a positive hermitian Banach Jordan ∗-triple satisfying x2 ≤ L(x, x) for every x ∈ X, and hence (by Theorem 5.6.62) the mapping 7 x → x∞ := L(x, x) is an equivalent norm on X converting X into a JB∗ -triple. (ii) Surjective linear isometries on X become isometries for the norm  · ∞ . (iii) H(BL(X), IX ) ⊆ H((BL(X),  · ∞ ), IX ). Proof By Fact 5.6.30(iii), X is hermitian. Assume that there exists a ∈ X such that sp(BL(X), L(a, a)) ∩ R− = ∅. Then, by Lemma 5.6.64, there exists also a nonzero element b ∈ X such that sp(BL(X), L(b, b)) ⊆ R− 0 , which contradicts the consequence in Corollary 5.6.66. Therefore X is positive hermitian. To conclude the proof of (i) note that, for x ∈ X we have that x2 ∈ sp(BL(X), L(x, x)) (thanks to Lemma 5.6.65), and hence the inequality x2 ≤ L(x, x) holds. Let F : X → X be a surjective linear isometry. Then, by Proposition 5.6.31, F preserves the triple product of X. Therefore, since (X,  · ∞ ) is a JB∗ -triple (by assertion (i) already proved), it follows from Proposition 4.1.52 that F is an isometry for the norm  · ∞ . This proves (ii). Assertion (iii) follows from (ii) by invoking Corollary 2.1.9(iii).  Theorem 5.6.68 Let X be a complex Banach space. The following conditions are equivalent: (i) X is a homogeneous domain. (ii) Xs = X. (iii) X is a JB∗ -triple for some triple product. Moreover, if the above conditions are fulfilled, then the triple product of X as a JB∗ -triple coincides with the intrinsic triple product of X as a complex Banach space. Proof The equivalence (i)⇔(ii) follows straightforwardly from the equality Xs = Aut(X )(0) in Fact 5.6.28. Moreover, Proposition 5.6.56(ii) gives the implication (iii)⇒(ii), as well as the fact that, whenever X satisfies (iii), the triple product of X as a JB∗ -triple coincides with the intrinsic triple product of X as a complex Banach space. Thus it only remains to prove the implication (ii)⇒(iii). Suppose that Xs = X, and regard X as a Banach Jordan ∗-triple under its intrinsic triple product. By Fact 5.6.67(i), we know that X is positive hermitian, that  ·  ≤  · ∞ , and that  · ∞ is an equivalent norm on X converting X into a JB∗ -triple. Let  be in aut(X ). Then, according to Fact 5.6.30(i), we can write  = iT + a for suitable T ∈ H(BL(X), IX ) and a ∈ X. Moreover, keeping in mind Fact 5.6.67(iii) and Proposition 5.6.56(i), we

206 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem have that T ∈ H((BL(X),  · ∞ ), IX ) and a ∈ aut((X,·∞ ) ), and hence, by Fact 5.6.30(i) again, we obtain that  ∈ aut((X,·∞ ) ). Thus aut(X ) ⊆ aut((X,·∞ ) ). Since (X,·∞ ) ⊆ X , it follows that exp()|(X,·∞ ) , regarded as a mapping from (X,·∞ ) to itself, belongs to Aut((X,·∞ ) ). On the other hand, given x ∈ X , by Facts 5.6.14 and 5.6.28, we have that x = [exp(1 ) ◦ · · · ◦ exp(n )](0) for suitable 1 , . . . , n ∈ aut(X ). It follows that x ∈ (X,·∞ ) . Therefore X = (X,·∞ ) , i.e.  ·  =  · ∞ , and hence X is a JB∗ -triple.  Corollary 5.6.69 Let A be a non-commutative JB∗ -algebra. Then the open unit ball of A is a homogeneous domain. Proof Combine Theorem 4.1.45 with the implication (iii)⇒(i) in Theorem 5.6.68 above.  Corollary 5.6.70 The bidual X  of a complex Banach space X is a JB∗ -triple if and only if the symmetric part of X  contains X. Proof The ‘only if’ part follows from Proposition 5.6.56(ii). Suppose that (X  )s ⊇ X. Then, by Fact 5.6.29(i), for each x ∈ X there exists a unique continuous quadratic mapping qx : X  → X  such that (the restriction to X  of) the function x : β → x − qx (β) from X  to X  becomes a complete holomorphic vector field on X  . Moreover, the mapping x → qx , from X to the Banach space of all X  -valued continuous quadratic functions on X  , is conjugate-linear and continuous. For α ∈ X  consider the continuous conjugate-linear mapping Fα : y → qy (α) from X to X  . Denote by Gα the unique w∗ -continuous conjugate-linear mapping from X  to X  which extends Fα (i.e. the composition of Fα : X  → X  with the Dixmier projection π : X  → X  ), and consider the function α : β → α − Gβ (α). Note that the definition of α just given is consistent with the one previously introduced in the particular case that α = x ∈ X. Now let x and (φ, β) be elements of X and (X  ), respectively. Since x is a complete holomorphic vector field on X  , and (β, φ) belongs to (X  ), it follows from Fact 5.6.38 that (x − Gβ (x))(φ)[= (x − Fβ (x))(φ) = x (β)(φ)] ∈ iR. Since x is arbitrary in X, it follows from the w∗ -denseness of X in X  that α (β)(φ)[= (α − Gβ (α))(φ)] ∈ iR for every α ∈ X  . Note now that α is a holomorphic mapping on X  bounded on 2BX  . Actually it follows easily from the continuity of the mapping x → qx , and the way of defining Gα , that γ → Gγ (α) is a continuous quadratic mapping from X  to X  . Since (φ, β) is arbitrary in (X  ), it follows from Corollary 5.6.46 that V(B(SX  , X  ), IX  , (α )|SX ) ⊆ iR, and hence, since for γ ∈ SX  the mapping f → f (γ ) from B(SX  , X  ) to X  is a linear contraction, we derive from Corollary 2.1.2(i) that V(X  , γ , α (γ )) ⊆ iR. Therefore, by Fact 5.6.38, α is a complete homomorphic vector field on X  , and hence

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207

α = α (0) belongs to (X  )s . Since α is arbitrary in X  , we conclude that X  = (X  )s . Finally, by the implication (ii)⇒(iii) in Theorem 5.6.68, X  is a JB∗ -triple.  5.6.5 Historical notes and comments The material in this section has been elaborated mainly from the survey article of Arazy [837] and the books of Chu [710], Friedmann and Scarr [732], Isidro and Stach´o [751], Narasimhan [1176], and Upmeier [814, 815]. Other sources are quoted in what follows. Most results in Subsection 5.6.1 are due to Braun, Kaup, and Upmeier [867]. Concerning them, our development follows that of [751, Chapter 7], where the approach to the fundamental Theorem 5.6.24 relies on complex analytic sets (see [751, Section 7.3] for details) instead of on complex submanifolds (as done in [837]). Proposition 5.6.20 is taken from [1176, Proposition 4.1]. The symmetric part of a complex Banach space, defined in §5.6.27, was introduced by Kaup and Upmeier [997], who proved results from Fact 5.6.28 to Theorem 5.6.32, as well as Theorem 5.6.35. In our approach, most of these results follow easily from more general ones published later but previously proved by us in Subsection 5.6.1. Corollary 5.6.33 is due to Harris [972] and Phillips [1049]. Several different proofs of this result, relying on the Cartan linearity theorem (Proposition 5.3.24), appear in the literature (see [837, Lemma 2.1] or [721, Proposition 1.1]). As a consequence of Corollary 5.6.33, if f is an automorphism of the open unit ball X of a complex Banach space X satisfying f (0) = 0, then f is the restriction to X of a surjective linear isometry from X to X (more precisely, f = Df (0)|X ). As commented in [721, p. 9], the converse holds: if f ∈ H (X , X ), and if Df (0) is a surjective linear isometry from X to X, then f ∈ Aut(X ). This result, when f (0) = 0, is a consequence of the following continuous form of the Schwarz lemma which is also due to Harris [973]. Proposition 5.6.71 Let X be a complex Banach space, and let f be in H (X , X ) such that f (0) = 0. Then  f (x) − Df (0)(x) ≤

8x2 d(Df (0), G ) for every x ∈ X , (1 − x)2

where G is the group of all surjective linear isometries from X to X and d is the distance in the operator norm. Fact 5.6.38 seems to be folklore in the theory. Our formulation and proof is close to those of Upmeier [815, Lemma 4.4] (see also [25, Proposition 1.3], [26, Proposition 2.5], [382, 597], [837, p. 139], and [1097, 1098]). Arazy’s survey paper [837] becomes the best possible guide for the results in Subsection 5.6.2. In particular, in [837, Proposition 5.19], the reader can find a previously unpublished example of Vigu´e showing that the symmetric part Xs of a complex Banach space X need not be complemented in X.

208 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Lemma 5.6.41 is due to Lumer and Phillips [409, Lemma 5.1]. Results from Lemma 5.6.42 to Corollary 5.6.46 are due to Rodr´ıguez [534]. Altough straightforwardly derivable from previously known results, Corollary 5.6.47 has been not pointed out in the literature. Results from Lemma 5.6.48 to Corollary 5.6.52 are new. To obtain an interesting unit-free variant of Theorem 5.6.51 (see Proposition 5.6.74 below), we need some auxiliary results. Let A be an algebra over K. As usual in our work, for a ∈ A we denote by La• the operator of multiplication by a in Asym . Following [808, p. 20], the Lie multiplication algebra of A is defined as the subalgebra of L(A)ant generated by LA ∪ RA , and is denoted by LM (A). Fact 5.6.72 Let A be a non-commutative Jordan algebra over K. We have: (i) If A is unital, then every operator T ∈ LM (A) such that T(1) = 0 is a Jordan derivation of A. (ii) If a, b are in A, then the mappings [a, ·] and [a, ·, b] are Jordan derivations of A. Proof Let D denote the subalgebra of L(A)ant consisting of all Jordan derivations of A, and write L := LA• + D. Then, keeping in mind Lemma 2.4.15, for a ∈ A we have that La = La• + 12 (La − Ra ) ∈ LA• + D = L and 1 Ra = La• − (La − Ra ) ∈ LA• + D = L , 2 and therefore LA ∪ RA ⊆ L . On the other hand, by Fact 2.4.7 and Lemma 3.1.23, for a, b ∈ A and D, E ∈ D we have that • [La• + D, Lb• + E] = LD(b)−E(a) + ([La• , Lb• ] + [D, E]) ∈ LA• + D = L ,

and hence L is a subalgebra of L(A)ant . It follows that LM (A) ⊆ L . Now, let T be in LM (A). Then, by the inclusion just proved, we have that T = La• + D for suitable a ∈ A and D ∈ D. Therefore, if A is unital, and if T(1) = 0, then a = (La• + D)(1) = 0, and so, as asserted in (i), T = D is a Jordan derivation of A. Since the unital extension of A is a non-commutative Jordan algebra, to prove (ii) we can suppose that A is unital. Then, since for a, b, x ∈ A the equalities [a, x] = (La − Ra )(x), [a, x, b] = [Rb , La ](x), and [a, 1] = 0 = [a, 1, b] hold, assertion (ii) follows from (i).



The next result follows from the fact that every non-commutative JB∗ -algebra has an approximate unit bounded by one (cf. Proposition 3.5.23). Fact 5.6.73 Let A be a non-commutative JB∗ -algebra. Then the mappings x → Lx and x → Rx from A to BL(A) are linear isometries. We recall that, according to Fact 3.3.4 and Lemma 3.4.26, Jordan derivations of non-commutative JB∗ -algebras are automatically continuous.

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209

Proposition 5.6.74 Let A be a (possibly non-unital) non-commutative JB∗ -algebra, and let p be a norm-one self-adjoint element of A such that IA − Lp•  < 2 (which happens for example if p is a norm-one positive element of A, or if A is unital and p is a norm-one self-adjoint element of A such that 1 − p < 2). Then for every Jordan derivation D of A we have that V(A, p, D(p)) = 0. As a consequence, for all a, b ∈ A the equalities V(A, p, [a, p]) = 0 = V(A, p, [a, p, b]) hold. Proof Let D be a Jordan derivation of A. Then, by Lemma 3.4.27 applied to the JB∗ -algebra Asym , D belongs to H(BL(A), IA ) ⊕ iH(BL(A), IA ). On the other hand, we know that Lp• is a norm-one element in H(BL(A), IA ) (cf. Lemma 3.6.24). Therefore, since IA − Lp•  < 2, it follows from Fact 2.4.7 and Theorem 5.6.51 that • )[= V(BL(A), Lp , [D, Lp• ])] = 0. V(BL(A), Lp• , LD(p)

Since the mapping x → Lx• from A to BL(A) is a linear isometry (cf. Fact 5.6.73), it follows from Corollary 2.1.2(ii) that V(A, p, D(p)) = 0. The consequence follows from Fact 5.6.72(ii).    0 . Then p is Remark 5.6.75 Let A stand for the C∗ -algebra M2 (C), and set p := 10 −1 2 a self-adjoint element of A such that p = 1. Therefore p = 1 and more precisely, by the implication (i)⇒(ii) in Theorem 2.1.27, p is a vertex of BA . Since p is not central, it follows that V(A, p, [a, p]) = 0 for some a ∈ A. This shows that the requirement IA − Lp•  < 2 (equivalent to 1 − p < 2 in the unital case) in Proposition 5.6.74 above cannot be removed. Corollaries 5.6.46, 5.6.47, and 5.6.52, become the results in Subsection 5.6.3 which are crucial in our subsequent development (see the proofs of Proposition 5.6.56, of Lemmas 5.6.60 and 5.6.61, of Theorem 5.6.62, and of Corollary 5.6.70). Fact 5.6.38 and Subsection 5.6.3, together with their later applications mentioned above, close our work concerning numerical ranges. In this regard, we note that numerical ranges underlie the recent monograph by Kadets, Mart´ın, Mer´ı, and P´erez [1166] (see [1166, Section 1.4] for details). Subsection 5.6.4 contains the last steps in the proof of Kaup’s Theorem 5.6.68, as well as the most immediate consequences of this theorem. Implicitly or explicitly, all arguments are due to Kaup [380, 381]. In particular, Lemma 5.6.53 and Theorem 5.6.62 have been taken from [381] almost verbatim. It is noteworthy that other arguments in [380, 381], like Corollary 4.1.51 and the whole Subsection 4.2.1, were already proved in Volume 1. Kaup’s argument also underlies our approach through Chu’s design of proof in [710, Section 2.5]. Indeed, Lemmas 5.6.64 and 5.6.65, and Corollary 5.6.66 are taken from (a corrected version of) the proof of Proposition 2.5.22 in [710], and Lemmas 2.5.18 and 2.5.19 in [710], respectively. Nevertheless, as a whole, our (conclusion of) proof of Kaup’s Theorem 5.6.68 is new. Indeed, as mentioned in the above paragraph, through the proofs of Proposition 5.6.56, of Lemmas 5.6.60 and 5.6.61, and of Theorem 5.6.62, our approach essentially involves the numerical range results established in Subsection 5.6.3.

210 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem The first proof that the open unit ball of a JB∗ -triple is a homogeneous domain appears in [380, Proposition 5.2]. Later, Kaup himself [381] gives a more intrinsic proof of this result. Indeed, he proves that, given a JB∗ -triple X and an arbitrary element x ∈ X , the spectrum of the Bergmann operator B(x, x) (cf. (4.2.39)) consists only of positive numbers, IX + L(z, x) is a bijective operator on X for each z ∈ X (a consequence of Lemmas 5.6.53, 2.3.21, and 1.1.20), and the ‘M¨obius transformation’ defined by 1

gx (z) := x + B(x, x) 2 (IX + L(z, x))−1 (z) for every z ∈ X , is a biholomorphic automorphism of X which satisfies gx (0) = x, g−1 x = g−x , and 1 Dgx (0) = B(x, x) 2 (defined in terms of the holomorphic functional calculus, cf. Proposition 1.3.25(i)). This nice result is carefully reviewed in [710, 995, 996, 1006, 1173]. The homogeneity of open unit balls of JB∗ -triples, proved by Kaup, had been shown earlier by Harris [314] for those JB∗ -triples which can be seen as closed subtriples of C∗ -algebras. Theorem 5.6.57 and Corollary 5.6.58 are due to Kaup (see [380, Proposition 5.4] and [381, Proposition 5.5]). As we commented immediately before Theorem 2.3.74, the contractive projection Theorem 5.6.59 (the core of the proof of which was established in Proposition 5.6.39) is due to Kaup [382] and Stach´o [597]. An additional bibliography on the contractive projection problem can be found in [136, 229, 507, 515, 874, 951]. Corollary 5.6.69 is due to Youngson [655]. Its unital forerunner is due to Braun, Kaup, and Upmeier [126]. Corollary 5.6.70 is due to Becerra, Rodr´ıguez, and Wood [80]. In relation to this corollary it is noteworthy that, as we will prove in Proposition 5.7.10, the bidual of any JB∗ -triple is a JB∗ -triple but that, as shown in [71, Example 3.10], complex Banach spaces whose biduals are JB∗ -triples need not be JB∗ -triples. By a symmetric domain in a complex Banach space X we mean a domain  ⊆ X such that, for each x0 ∈ , there exists S ∈ Aut() such that S2 = I and x0 is an isolated fixed point for S. It is easily seen that every circular homogeneous domain is symmetric (see for example [710, Lemma 2.5.11]). It was known to E. Cartan [898] that all bounded symmetric domains in Cn are homogeneous, and that all homogeneous bounded domains of dimension ≤ 3 are symmetric, and he raised the question whether it is so in higher dimension. The answer was given by Pyatetskii-Shapiro in [1055] providing the first examples of non-symmetric homogeneous bounded domains in C4 and C5 . According to Kaup’s holomorphic characterization of JB∗ -triples we are dealing with, the open unit ball of any JB∗ -triple is a bounded symmetric domain. Actually, Kaup’s concluding theorem in [381] is the following. Theorem 5.6.76 A bounded domain in a complex Banach space is symmetric (if and) only if it is biholomorphically equivalent to the open unit ball of a JB∗ -triple.

5.7 JBW ∗ -triples

211

Theorem 5.6.76 above contains Vigu´e’s celebrated forerunner [1187, 1114] asserting that every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to a balanced bounded (symmetric) domain. For additional information about symmetric domains the reader is referred to [173, 314, 380, 381, 385, 721, 751, 772, 814, 815, 993, 994, 1115, 1169, 1173]. 5.7 JBW ∗ -triples Introduction In Subsection 5.7.1 we establish the classical version of the principle of local reflexivity for Banach spaces, without proving it, and follow Heinrich’s survey [318] to prove the reformulation of this principle in terms of ultrapowers (see Proposition 5.7.8). Then we prove the Barton–Timoney refinement [854] of Dineen’s forerunner [213] asserting that the bidual of any JB∗ -triple X becomes a JB∗ -triple under a suitable triple product which is separately w∗ -continuous and extends the triple product of X (see Proposition 5.7.10 and Theorem 5.7.18). In Subsection 5.7.2 we give new and very short proofs of the Barton–Horn– Timoney theorems [854, 979] asserting the separate w∗ -continuity of the product and the uniqueness of the predual of a JBW ∗ -triple (see Theorems 5.7.20 and 5.7.38, respectively). The subsection contains also theorems taken from [854] asserting that, in a JB∗ -triple, M-ideals are precisely the closed triple ideals (Theorem 5.7.34(i)) and that the predual of a JBW ∗ -triple is an L-summand of the dual (Theorem 5.7.36). The subsection concludes with the Barton–Dang–Horn variant for JB∗ -triples [852] of Proposition 5.1.36 (see Proposition 5.7.45). 5.7.1 The bidual of a JB∗ -triple The next lemma becomes a variant of Lemma 2.2.24. Lemma 5.7.1 Let X be a normed space, let S be in BL(X), and let M be an S-invariant closed subspace of X, then we have V(BL(X/M), IX/M , SM ) ⊆ V(BL(X), IX , S), where SM denotes the element of BL(X/M) defined by SM (x + M) := S(x) + M. Proof The set A of those operators T ∈ BL(X) such that T(M) ⊆ M is a subspace of BL(X) containing IX , and the mapping T → T M from A to BL(X/M) is a linear contraction taking IX to IX/M . Therefore, by Corollary 2.1.2, we have V(BL(X/M), IX/M , SM ) ⊆ V(A, IX , S) = V(BL(X), IX , S).



By a triple ideal of a Jordan ∗-triple X we mean a subspace M of X such that {MXX} + {XMX} ⊆ M. It is clear that, if M is a (closed) triple ideal of a (Banach) Jordan ∗-triple X, then X/M becomes naturally a (Banach) Jordan ∗-triple. Now we are provided with the following variant of Lemma 4.1.38.

212 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Lemma 5.7.2 Let X be a hermitian (respectively, positive hermitian) Banach Jordan ∗-triple, and let M be a closed triple ideal of X. Then the Banach Jordan ∗-triple X/M is hermitian (respectively, positive hermitian). Proof Let x be in X. Then M is invariant under L(x, x) and, with the notation in Lemma 5.7.1, we have L(x + M, x + M) = L(x, x)M . Therefore, by that lemma, V(BL(X/M), IX/M , L(x + M, x + M)) ⊆ R whenever X is hermitian (respectively, V(BL(X/M), IX/M , L(x + M, x + M)) ⊆ R+ 0 

whenever X is positive hermitian).

In Corollary 4.1.114, we formulated without a complete proof that, for elements x, y, z in a JB∗ -triple, the inequality {xyz} ≤ xyz holds. We can now prove the following less accurate result. Fact 5.7.3 Let X be a JB∗ -triple. Then {xyz} ≤ 4xyz for all x, y, z ∈ X. Proof It is enough to show that, for x, y ∈ BX , we have L(x, y) ≤ 4. But, by the polarization law (4.2.2) (already applied in the proof of Proposition 4.2.5), for x, y ∈ X we have 4L(x, y) = L(x + y, x + y) − L(x − y, x − y) + i (L(x + iy, x + iy) − L(x − iy, x − iy)) , and hence, by Corollary 4.1.51, 4L(x, y) ≤ x + y2 + x − y2 + x + iy2 + x − iy2 ≤ 16 

whenever x, y lie in BX . Now we can prove the following generalization of Fact 4.2.29.

Lemma 5.7.4 Let {Xi }i∈I be any family of JB∗ -triples, and let X stand for the ∞ -sum of the family {Xi }i∈I . Then {{xi yi zi }} lies in X whenever {xi }, {yi }, and {zi } are in X, and X becomes a JB∗ -triple under the triple product {{xi }{yi }{zi }} := {{xi yi zi }}.

(5.7.1)

Proof The first conclusion follows from Fact 5.7.3. Then we straightforwardly realize that X becomes a Banach Jordan ∗-triple under the triple product defined in (5.7.1). Moreover, for {xi } ∈ X we have {{xi }{xi }{xi }} = {{xi xi xi }} = sup{{xi xi xi } : i ∈ I} = sup{xi 3 : i ∈ I} = {xi }3 . Therefore it only remains to show that X is positive hermitian. To realize this, let A stand for the ∞ -sum of the family {BL(Xi )}i∈I , define a mapping φ : A → BL(X) by φ({Ti })({xi }) := {Ti (xi )} for all {Ti } ∈ A and {xi } ∈ X, and note that φ becomes

5.7 JBW ∗ -triples

213

a linear isometry satisfying φ({IXi }) = IX and φ({L(xi , xi )}) = L({xi }, {xi }) for every {xi } ∈ X. Then, by Corollaries 2.1.2 and 2.9.51, for every {xi } ∈ X we have V(BL(X), IX , L({xi }, {xi })) = V(A, {IXi }, {L(xi , xi )}) = co [∪i∈I V(BL(Xi ), IXi , L(xi , xi ))] ⊆ R+ 0.



For the formulation and proof of the next result, the reader is invited to recall the notion of ultraproduct of a family of Banach spaces, and the notation used in this setting (cf. §2.8.58). Proposition 5.7.5 Let {Xi }i∈I be a family of JB∗ -triples, and let U be an ultrafilter on I. Then the ultraproduct (Xi )U becomes a JB∗ -triple under the (well-defined) triple product {(xi )(yi )(zi )} := ({xi yi zi }). 0. By a (1 + ε)-isometry from X to Y we mean a linear mapping T : X → Y satisfying (1 + ε)−1 x ≤ T(x) ≤ (1 + ε)x for every x ∈ X. Now we need to invoke the so-called local reflexivity principle, which reads as follows. Proposition 5.7.7 [671, Theorem 11.2.4] Let X be a Banach space over K, let M and N be finite-dimensional subspaces of X  and X  , respectively, and let ε > 0. Then there exists a (1 + ε)-isometry T : M → X satisfying: (i) T(x) = x for every x ∈ M ∩ X. (ii) x (T(x )) = x (x ) for every (x , x ) ∈ N × M. Proposition 5.7.8 Let X be a Banach space over K. Then there exist an ultrafilter U on a suitable set I, a linear isometry φ : X  → XU , and a contractive linear mapping Q : XU → X  such that:

214 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem (i) The restriction of φ to X coincides with the canonical inclusion X ⊆ XU . (ii) The restriction of Q to X coincides with the canonical inclusion X ⊆ X  . (iii) Q ◦ φ is the identity mapping on X  . As a consequence, π := φ ◦ Q : XU → XU is a contractive linear projection with range φ(X  ), and its restriction to X coincides with the canonical inclusion X ⊆ XU . Moreover, we have Q((xi )) = w∗ - limU xi for every (xi ) ∈ XU . Proof Consider the set I consisting of all triplets (M, N, ε), where M and N are finite-dimensional subspaces of X  and X  , respectively, and 0 < ε ≤ 1. Then I becomes a directed set under the order defined by (M1 , N1 , ε1 ) ≤ (M2 , N2 , ε2 ) if and only if M1 ⊆ M2 , N1 ⊆ N2 , and ε1 ≥ ε2 . For i ∈ I, write Ai := {j ∈ I : j ≥ i}, and set A := {Ai : i ∈ I}. Then A becomes a filter base on I, and therefore we may choose an ultrafilter U on I with U ⊇ A . On the other hand, according to Proposition 5.7.7, for each i = (Mi , Ni , εi ) ∈ I we may fix a (1 + εi )-isometry Ti : Mi → X satisfying Ti (x) = x for every x ∈ Mi ∩ X and x (Ti (x )) = x (x ) for every (x , x ) ∈ Ni × Mi . Now, denote by B(I, X) the Banach space of all bounded functions from I to X, define a mapping ξ : X  → B(I, X) by ξ(x )(i) = Ti (x ) if x ∈ Mi , and ξ(x )(i) = 0 otherwise, and set φ := q ◦ ξ , where q stand for the quotient mapping B(I, X) → XU . We claim that φ is a linear isometry. Indeed, it is clear that φ(λx ) = λφ(x ) for all λ ∈ K and x ∈ X  . Moreover, given x , y ∈ X  , denoting by M0 the linear hull of {x , y }, and writing i0 := (M0 , 0, 1) ∈ I, we see that Ai0 ⊆ {i ∈ I : ξ(x + y )(i) − ξ(x )(i) − ξ(y )(i) = 0}, hence the right-hand set above belongs to U , and so lim ξ(x + y )(i) − ξ(x )(i) − ξ(y )(i) = 0 U

or, equivalently, φ(x + y ) = φ(x ) + φ(y ). On the other hand, given x ∈ X  and ε > 0, and writing i0 := (Kx , 0, ε0 ) ∈ I, where 0 < ε0 ≤ 1 and ε0 x  < ε, we see that for each i = (Mi , Ni , εi ) ∈ I such that i ≥ i0 we have x  − ε ≤ (1 − ε0 )x  ≤ (1 + ε0 )−1 x  ≤ (1 + εi )−1 x  ≤ Ti (x ) ≤ (1 + εi )x  ≤ (1 + ε0 )x  ≤ x  + ε. Since for i ≥ i0 we have Ti (x ) = ξ(x )(i), we realize that Ai0 ⊆ {i ∈ I : |ξ(x )(i) − x | ≤ ε}, and so the right-hand set above belongs to U . It follows from the arbitrariness of ε that limU ξ(x )(i) = x  or, equivalently, φ(x ) = x .

5.7 JBW ∗ -triples

215

Since for each i = (Mi , Ni , εi ) ∈ I and each x ∈ Mi ∩ X we have Ti (x) = x, we easily realize that the restriction of φ to X coincides with the canonical embedding X → XU . Given (x , x ) ∈ X  × X  , it is enough to set i0 := (Kx , Kx , 1) ∈ I to obtain x [ξ(x )(i)] = x (Ti (x )) = x (x ) for every i ≥ i0 . As a consequence, we have lim x [ξ(x )(i)] = x (x ) for every (x , x ) ∈ X  × X  . U

(5.7.2)

Now we define a contractive linear mapping Q : XU → X  by the formula Q((xi )) := w∗ - lim xi . U

w∗ -compactness

This limit exists by the of closed balls of X  . The definitions of Q and φ, together with (5.7.2), imply that [Q(φ(x ))](x ) = lim x [ξ(x )(i)] = x (x ) for every (x , x ) ∈ X  × X  . U

This shows that Q ◦ φ is the identity mapping on X  . Therefore Q is surjective. Moreover, keeping in mind that Q(x) = x for every x ∈ X, and setting π := φ ◦ Q : XU → XU , we get the contractive linear projection satisfying the properties asserted in the statement.  Lemma 5.7.9 Let X be a JB∗ -triple. Suppose that there exist an ultrafilter U on a set I, a linear isometry φ : X  → XU , and a contractive linear mapping Q : XU → X  satisfying properties (i), (ii), and (iii) in Proposition 5.7.8. Denote by {· · ·}XU the triple product of the JB∗ -triple XU (cf. Corollary 5.7.6). Then X  becomes a JB∗ -triple under the triple product {x y z }X  := Q({φ(x )φ(y )φ(z )}XU ), which extends indeed that of X. Proof

As noticed in the statement of Proposition 5.7.8, the mapping π := φ ◦ Q : XU → XU

is a contractive linear projection with range φ(X  ). Then, by Theorem 5.6.59, φ(X  ) becomes a JB∗ -triple under the triple product {φ(x )φ(y )φ(z )}π := π({φ(x )φ(y )φ(z )}XU ). Therefore X  is a JB∗ -triple under the triple product {x y z }X  := φ −1 (π({φ(x )φ(y )φ(z )}XU )) = Q({φ(x )φ(y )φ(z )}XU ). Since the triple product {· · ·}XU extends that of X, and φ and Q become the identity on X, it becomes clear that {· · ·}X  extends the triple product of X. 

216 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Putting together Proposition 5.7.8 and Lemma 5.7.9 we get the following result. Proposition 5.7.10 The bidual X  of any JB∗ -triple X becomes a JB∗ -triple under a triple product which extends that of X. For the formulation and proof of the following results, the reader should recall the Peirce decomposition of a Jordan ∗-triple X relative to a tripotent e (cf. Fact 4.2.14 and Lemma 4.2.20). Since no confusion arises, for k ∈ {1, 12 , 0}, we will simply write Xk for the Peirce subspace Xk (e) and Pk for the Peirce projection Pk (e). Lemma 5.7.11 Let X be a JB∗ -triple, and let e be a tripotent in X. We have: (i) (ii) (iii) (iv)

P1 (x) + P0 (x) = max{P1 (x), P0 (x)} for every x ∈ X. (P1 ) (x ) + (P0 ) (x ) = (P1 ) (x ) + (P0 ) (x ) for every x ∈ X  . If x ∈ X  satisfies (P1 ) (x ) = x , then (P0 ) (x ) = 0. If x ∈ X  satisfies (P0 ) (x ) = x , then (P1 ) (x ) = 0.

Proof Assertion (i) is a reformulation of Corollary 4.2.30(iii)(a), whereas assertion (ii) follows from (i). Since P1 and P0 are contractive (cf. the inequality (4.2.6) in Proposition 4.2.15), assertion (i) implies P1 + P0  ≤ 1, and hence (P1 ) (x ) + (P0 ) (x ) ≤ x  for every x ∈ X  . Now, keeping in mind (ii), assertions (iii) and (iv) follow straightforwardly.



We recall that, given a JB∗ -triple X and an element x ∈ X, the operators L(x, x) : X → X and Qx : X → X are defined by L(x, x)(y) := {xxy} and Qx (y) := {xyx} for every y ∈ X. For the sake of shortness, we will write Dx := L(x, x). Lemma 5.7.12 Let X be a JB∗ -triple, and let n be in N. Then there exist unique mappings αkn : X × X → X, for k ∈ N ∪ {0} with k ≤ 3n , such that n

(3n )

(x + ty)

=

3 

tk αkn (x, y) forall x, y ∈ X and t ∈ R.

(5.7.3)

k=0

Moreover, we have: (i) There exist constants Ckn ≥ 0 such that αkn (x, y) ≤ Ckn x3

n −k

(ii) The mappings α0n and α1n are given by n α0n (x, y) = x(3 )

and

α1n (x, y) =

yk for all x, y ∈ X. ' n−1  m=0

( (2Dx(3m ) + Qx(3m ) ) (y).

5.7 JBW ∗ -triples

217

(iii) If e is a tripotent in X, if x is in X1 ∪ X0 , and if y is in X 1 , then 2

α0n (x, y) ∈ X1 ∪ X0

and

α1n (x, y) = 2n

' n−1 

( (y) ∈ X 1 .

D

m x(3 )

2

m=0

Proof The uniqueness of the mappings αkn follows from the fact that a polynomial mapping is identically zero if and only if all its coefficients are zero. To prove the remaining assertions in the statement we proceed by induction on n. For all x, y ∈ X and t ∈ R we have (x + ty)(3) = {x + ty, x + ty, x + ty} = {xxx} + t(2{xxy} + {xyx}) + t2 (2{xyy} + {yxy}) + t3 {yyy}, and hence we find that α01 (x, y) = {xxx} = x(3) , α11 (x, y) = 2{xxy} + {xyx} = (2Dx + Qx )(y), α21 (x, y) = 2{xyy} + {yxy}, and α31 (x, y) = {yyy}. Note that the existence of constants Ck1 (0 ≤ k ≤ 3), as required in (i) for n = 1, follows from Fact 5.7.3. Moreover, if e is a tripotent in X, if x is in X1 ∪ X0 , and if y is in X 1 , then, by Proposition 4.2.22, α01 (x, y) = x(3) ∈ X1 ∪ X0 , {xyx} = 0, and 2

α11 (x, y) = 2Dx (y) ∈ X 1 . Thus we have proved the whole statement for n = 1. 2 Now, assume that the statement holds for some natural number n. Then, by Lemma 4.1.49, for all x, y ∈ X and t ∈ R we have (x + ty)(3

n+1 )

n

n

n

= {(x + ty)(3 ) , (x + ty)(3 ) , (x + ty)(3 ) } ⎧ ⎫ 3n 3n 3n ⎨ ⎬   = tp αpn (x, y), tq αqn (x, y), tr αrn (x, y) ⎩ ⎭ p=0

=

q=0



r=0

tp+q+r {αpn (x, y), αqn (x, y), αrn (x, y)}

0≤p,q,r≤3n

=

n+1 3

k=0

hence we find that αkn+1 (x, y) =

tk



{αpn (x, y), αqn (x, y), αrn (x, y)},

p+q+r=k



{αpn (x, y), αqn (x, y), αrn (x, y)}

p+q+r=k

and in particular n

n

n

α0n+1 (x, y) = {α0n (x, y), α0n (x, y), α0n (x, y)} = {x(3 ) x(3 ) x(3 ) } = x(3

n+1 )

218 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem and α1n+1 (x, y) = 2{α0n (x, y), α0n (x, y), α1n (x, y)} + {α0n (x, y), α1n (x, y), α0n (x, y)} ' n−1 (  = (2Dx(3n ) + Qx(3n ) ) (2Dx(3m ) + Qx(3m ) ) (y) ' =

m=0 n 

(

(2Dx(3m ) + Qx(3m ) ) (y).

m=0

Moreover, it follows from Fact 5.7.3 that  αkn+1 (x, y) ≤ 4 αpn (x, y)αqn (x, y)αrn (x, y) p+q+r=k







≤ 4⎝

Cpn Cqn Crn ⎠ x3

n −k

yk ,

p+q+r=k

 and hence we can take Ckn+1 = 4 p+q+r=k Cpn Cqn Crn . Finally, if e is a tripotent in X, if x is in X1 ∪ X0 , and if y is in X 1 , then, by Proposition 4.2.22, 2

α0n+1 (x, y) = x

(3n+1 )

and

∈ X1 ∪ X0 , {α0n (x, y), α1n (x, y), α0n (x, y)} = 0,

 α1n+1 (x, y) = 2Dx(3n )

n−1

2

n−1 





Dx(3m ) (y) = 2

n

m=0

n 

 Dx(3m ) (y) ∈ X 1 . 2

m=0



Proposition 5.7.13 Let X be a JB∗ -triple, let e be a tripotent in X, and let x be in X  such that (P1 ) (x ) = x . Then (P1 ) (x ) = x . Proof By Lemma 5.7.11(iii), (P0 ) (x ) = 0. Since IX  = (P1 ) + (P 1 ) + (P0 ) , it 2 remains to prove that (P 1 ) (x ) = 0. To this end let y ∈ X 1 . We are to prove that 2 2 x (y) = 0. We may propose that x  = 1, x (y) ≥ 0, and y ≤ 1. Then, by assumption, (P1 ) (x ) = 1, and hence, for ε > 0 we may choose x ∈ X1 with x = 1 and x (x) ≥ 1 − ε. Then for t ∈ R, x + ty ≥ x (x + ty) = x (x) + tx (y) ≥ 1 − ε + tx (y), and hence, for each n ∈ N we have (1 − ε + tx (y))3 ≤ x + ty3 = (x + ty)(3 ) . n

n

n

Since, by Lemma 5.7.12 n

(3n )

(x + ty)

 ≤ x

(3n )

3n −1

+ |t|2 x n

y +

3  k=2

≤ 1 + |t|2n +

3n  k=2

|t|k Ckn ,

|t|k Ckn x3

n −k

yk

5.7 JBW ∗ -triples writing fn (t) :=

3 n

k=2 t

k−2 Cn , k

219

we realize that

(1 − ε + tx (y))3 ≤ 1 + |t|2n + t2 fn (|t|) n

and so, letting ε → 0, we obtain (1 + tx (y))3 ≤ 1 + |t|2n + t2 fn (|t|). n

Now, writing (1+tx (y))3 = 1+t3n x (y)+t2 gn (t) for a suitable polynomial mapping gn : R → R, we see that n

t3n x (y) + t2 gn (t) ≤ t2n + t2 fn (t) for every t ∈ R+ , and dividing by t3n results in  n  n  n 1 2 1  gn (t) ≤ +t fn (t) for every t ∈ R+ . x (y) + t 3 3 3  n Letting t → 0, we obtain that x (y) ≤ 23 . Finally, letting n → ∞, we conclude that x (y) = 0.  We recall that JBW ∗ -triples were defined as those JB∗ -triples which are dual Banach spaces (cf. the paragraph immediately before Proposition 4.2.65). Lemma 5.7.14 Let X be a JBW ∗ -triple whose triple product is w∗ -continuous in the middle variable. Then the triple product of X is separately w∗ -continuous. Proof By polarization and symmetry in the outer variables it suffices to show that, for every y ∈ X, the mapping Dy : x → {yyx} from X to X is σ (X, X∗ )-continuous. We first show that Dy is ‘locally’ σ (X, X∗ )-continuous. Fix any tripotent e ∈ X, and note that, by Lemma 4.2.20, Q2e is the projection onto the Peirce 1-eigenspace X1 (e) relative to e. Let t be in R, set Ut := exp(itDy ), and let  et := Ut (e). By Corollary 2.1.9(iii), Ut is a surjective linear isometry, and hence a bijective triple homomorphism (cf. Theorem 5.6.57). Therefore  et is a tripotent and Ut Q2e (X) = Q2et (X). Write A := Q2e (X) and Bt := Q2et (X). Then, by Corollary 4.2.30(iii)(b), A and Bt are JB∗ -algebras for suitable products and involutions. Moreover, since Q2e and Q2et are σ (X, X∗ )-continuous, and A = ker(IX − Q2e ), and Bt = ker(IX − Q2et ), it is enough to invoke §5.1.9 to realize that A and Bt become canonically dual Banach spaces in such a way that σ (A, A∗ ) = σ (X, X∗ )|A and σ (Bt , (Bt )∗ ) = σ (X, X∗ )|Bt . Since (Ut Q2e )|A : A → Bt is a surjective linear isometry, it follows from Theorem 5.1.29(iv) and Fact 5.1.24 that (Ut Q2e )|A : A → X is σ (A, A∗ )σ (X, X∗ ) continuous. Since Q2e : X → A is σ (X, X∗ )-σ (A, A∗ ) continuous, it follows that Ut Q2e = ((Ut Q2e )|A ) ◦ Q2e : X → X is σ (X, X∗ )-σ (X, X∗ ) continuous. Since (exp(itDy ) − IX )Q2e Ut Q2e − Q2e = −i lim t→0 t→0 t t

Dy Q2e = −i lim

in the norm topology of BL(X), we conclude that Dy Q2e is σ (X, X∗ )-continuous, and so Dy Qe = Dy Q2e Qe is, too. It follows from the identity (4.2.21) before Proposition 4.2.16 that Qe Dy = 2Q{yye},e − Dy Qe . Thus Qe Dy is the difference of two σ (X, X∗ )continuous operators and so is itself σ (X, X∗ )-continuous.

220 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Finally, let x∗ ∈ SX∗ . It remains that x∗ ◦ Dy is σ (X, X∗ )-continuous. Since the set D(X∗ , x∗ ) := {x ∈ BX : x(x∗ ) = 1} is a σ (X, X∗ )-closed face of BX , it follows from the Krein–Milman theorem that D(X∗ , x∗ ) contains an extreme point e of BX . By Theorem 4.2.34, e is a tripotent. Thus 1 = x∗ (e) = x∗ Q2e (e) ≤ x∗ Q2e  = (Q2e ) (x∗ ) ≤ Q2e  ≤ 1, where for the last inequality we have applied the inequality (4.2.6) in Proposition 4.2.15. Therefore x∗ = (Q2e ) (x∗ ) = x∗ Q2e by Proposition 5.7.13. Then x∗ Dy = x∗ Q2e Dy = (x∗ Qe ) ◦ (Qe Dy ) is a composition of σ (X, X∗ )-continuous operators, and the proof is complete.



Proposition 5.7.15 Let E be a uniformizable topological space, let D be a dense subset of E, let F be a compact Hausdorff topological space, and let f : E → F be a function which satisfies the following condition: (∗) If x is in E, and if yλ is any net in D converging to x, then f (x) is a cluster point of the net f (yλ ). Then f is continuous. Proof We begin by noticing that the hypothesis (∗) implies that lim f (yλ ) = f (x), whenever yλ is a net in D converging to x. Indeed, if there would exist a net yλ in D converging to x and such that the net f (yλ ) does not converge to f (x), then, by the compactness of F, f (yλ ) would have a cluster point z ∈ F with z = f (x). By taking a subnet f (yλ(μ) ) convergent to z (see for example [778, Proposition 2.1.35]), we would be provided with a net yλ(μ) in D convergent to x and such that f (x) is not a cluster point of f (yλ(μ) ), contrary to the assumption (∗). Let x be in E, and let (xλ )λ∈ be any net in E converging to x. Let  be any uniformity on E generating the topology of E, order  by reverse inclusion, and order  ×  component-wise, that is, (λ1 , γ1 ) ≤ (λ2 , γ2 ) if λ1 ≤ λ2 and γ1 ≤ γ2 . Then  ×  becomes a directed set. Since D is dense in E, for every (λ, γ ) ∈  ×  we can choose y(λ,γ ) ∈ γ (xλ ) ∩ D. We claim that lim

(λ,γ )∈×

y(λ,γ ) = x.

Indeed, for each neighbourhood V of x in E there exist γ1 ∈  with γ1 (x) ⊆ V, γ0 ∈  with γ0 ◦ γ0 ⊆ γ1 , and λ0 ∈  such that xλ ∈ γ0 (x) whenever λ ≥ λ0 ; in this way, for all (λ, γ ) ≥ (λ0 , γ0 ) we have y(λ,γ ) ∈ γ (xλ ) ⊆ γ0 (xλ ) and xλ ∈ γ0 (x), hence y(λ,γ ) ∈ (γ0 ◦ γ0 )(x) ⊆ γ1 (x) ⊆ V. Now that the claim has been proved, it follows from the first paragraph of the proof that lim

(λ,γ )∈×

f (y(λ,γ ) ) = f (x).

Let N be any closed neighbourhood of f (x) in F. Then there is (λ0 , γ0 ) ∈  ×  with f (y(λ,γ ) ) ∈ N for all (λ, γ ) ≥ (λ0 , γ0 ). Fix any λ ≥ λ0 . Since xλ = limγ ∈ y(λ,γ )

5.7 JBW ∗ -triples

221

(because y(λ,γ ) ∈ γ (xλ ) for every γ ∈  and the set {γ (xλ ) : γ ∈ } is a basis of neighbourhoods of xλ ), by the first paragraph of the proof we have f (xλ ) = limγ ∈ f (y(λ,γ ) ). Hence f (xλ ) ∈ N since N is closed in F and f (y(λ,γ ) ) ∈ N whenever γ ≥ γ0 . Thus f (x) = limλ∈ f (xλ ). Therefore f is continuous.  Corollary 5.7.16 Let X and Y be normed spaces over K, and let T : X  → Y  be a bounded linear or conjugate-linear mapping which satisfies the following condition: (∗∗) If x is in BX  , and if xλ is any net in BX converging to x in the weak∗ topology of X  , then T(x ) is a cluster point of the net T(xλ ) in the weak∗ topology of Y  . Then T is w∗ -continuous. Proof As any subset of a topological vector space, (BX  , w∗ ) becomes a uniformizable topological space. Therefore, taking in Proposition 5.7.15 E := (BX  , w∗ ), D := BX , F := (TBY  , w∗ ), and f := T|E (regarded as a mapping from E to F), condition (∗∗) in the present corollary becomes condition (∗) in that proposition, and hence the restriction of T to BX  is w∗ -continuous. By Fact 5.1.19, T is w∗ -continuous.  Lemma 5.7.17 Let U and V be ultrafilters on the sets I and J, respectively, and  consider the collection U " V of those subsets of I × J of the form j∈B f (j) × {j}, where B ∈ V and f is a function from B to U . Then we have: (i) U " V is a filter base on I × J, and A × B belongs to U " V whenever A and B are in U and V , respectively. (ii) For every ultrafilter W on I × J with W ⊇ U " V , for every compact Hausdorff topological space E, and for every function k : I × J → E, we have lim k(i, j) = lim lim k(i, j). W

V

U

Proof Let B and C be in V , let f : B → U and g : C → U be functions, and consider the mapping h : B ∩ C → U defined by h(j) := f (j) ∩ g(j) for every j ∈ B ∩ C. Then we have ⎞ ⎛ ⎞ ⎛  3   ⎝ f (j) × {j}⎠ ⎝ g(j) × {j}⎠ = h(j) × {j}. j∈B

j∈C

j∈B∩C

This proves the nontrivial parts of assertion (i). Let W be an ultrafilter on I × J with W ⊇ U " V , let E be a compact Hausdorff topological space, and let k : I × J → E be a function. Put x := lim lim k(i, j) V

U

and let N be any neighbourhood of x in E. Then   B := j ∈ J : lim k(i, j) ∈ N ∈ V , U

222 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem and for each j ∈ B the set f (j) := {i ∈ I : k(i, j) ∈ N} ∈ U because N is a neighbourhood of limU k(i, j) in E. Since  f (j) × {j} ⊆ {(i, j) ∈ I × J : k(i, j) ∈ N} j∈B

and the left-hand set above is in U " V ⊆ W , the right-hand set is also in W . Since N is an arbitrary neighbourhood of x in E, this proves that x = limW k(i, j).  Theorem 5.7.18 The bidual X  of any JB∗ -triple X becomes a JB∗ -triple under a triple product which extends that of X and is separately w∗ -continuous. Proof Let I, U , φ : X  → XU , and Q : XU → X  be the set, the ultrafilter, the linear isometry, and the contractive linear mapping given by Proposition 5.7.8. By Lemma 5.7.9, X  is a JB∗ -triple under the triple product {x y z }X  := Q({φ(x )φ(y )φ(z )}XU )

(5.7.4)

which extends indeed that of X. We are going to show that {· · ·}X  is separately w∗ -continuous. To this end, in view of Corollary 5.7.16 and Lemma 5.7.14, it is enough to show that, for every (x , y , z ) ∈ X  × BX  × X  and every net (yλ )λ∈ in BX w∗ -convergent to y , {x y z }X  is a w∗ -cluster point of the net ({x yλ z }X  )λ∈ . Let y be in BX  , and let (yλ )λ∈ be a net in BX w∗ -convergent to y . For λ ∈ , write Bλ := {λ ∈  : λ ≥ λ}, and set B := {Bλ : λ ∈ }. Then B becomes a filter base on , and therefore we may choose an ultrafilter V on  with V ⊇ B. Now, according to Lemma 5.7.17(i), we may choose an ultrafilter W on I ×  containing the filter base U " V . Given a bounded family {xi }i∈I of elements of X, we consider the family {x(i,λ) }(i,λ)∈I× defined by x(i,λ) := xi for every (i, λ) ∈ I × , and we note the following fact: (*) If E is any set, if f : X → E is any mapping, and if S is a subset of E such that {i ∈ I : f (xi ) ∈ S} ∈ U , then {(i, λ) ∈ I ×  : f (x(i,λ) ) ∈ S} ∈ W . Indeed, since {i ∈ I : f (xi ) ∈ S} ×  = {(i, λ) ∈ I ×  : f (x(i,λ) ) ∈ S}, this follows because A ×  lies in U " V whenever A is in U . As a first consequence of (*), taking E = X and f = IX , and letting S run over the set of all neighbourhoods of zero in X, we see that lim xi = 0 ⇒ lim x(i,λ) = 0. U

W

Therefore, we can define a mapping ϕ : XU → XW by ϕ((xi )) := (x(i,λ) ). It is clear that ϕ is linear. Moreover, as a second application of (*), taking E = R and f equal to

5.7 JBW ∗ -triples

223

the norm of X, and letting S run over the set of all neighbourhoods of a suitable real number, we derive that (xi ) = lim xi  = lim x(i,λ)  = ϕ((xi )) U

W

for every (xi ) ∈ XU . Thus ϕ is an isometry. Now consider the mappings  : X  → XW and  Q : XW → X  φ  := ϕ ◦ φ and  defined by φ Q((x(i,λ) )) := w∗ - limW x(i,λ) . For later application in the present proof, we note that, by the definition of ϕ : XU → XW , for every x ∈ X  , we (x ) = (x(i,λ) ) for some bounded family {x(i,λ) }(i,λ)∈I× with x(i,λ) = xi can write φ for every (i, λ) ∈ I × . Since φ is a linear isometry extending the canonical inclu is a linear isometry extending the canonical sion X ⊆ XU , it becomes clear that φ inclusion X ⊆ XW . It is also clear that  Q is a contractive linear mapping extending the canonical inclusion X ⊆ X  . As a last application of (*), taking E = X  and f equal to the canonical inclusion X ⊆ X  , and letting S run over the set of all neighbourhoods of a suitable element of X  , we obtain that w∗ - lim xi = w∗ - lim x(i,λ) , U

W

i.e. Q((xi )) =  Q(ϕ((xi ))). Therefore Q =  Q ◦ ϕ, and hence  =  Q◦φ Q ◦ ϕ ◦ φ = Q ◦ φ = IX  . ,  Now we are in a position to apply Lemma 5.7.9, with (I × , W , φ Q) instead of  ∗ (I, U , φ, Q), to derive that X becomes a JB -triple under the triple product (u )φ (v )φ (w )}XW ), Q({φ (u , v , w ) →  and hence, by Theorem 5.6.57, we have (u )φ (v )φ (w )}XW ) for all u , v , w ∈ X  . Q({φ {u v w }X  = 

(5.7.5)

y(i,λ) ) of XW . Now, for (i, λ) ∈ I × , set / y(i,λ) := yλ , and consider the element (/ Then, by Lemma 5.7.17(ii), we have  y(i,λ) = w∗ - lim w∗ - lim/ y(i,λ) = w∗ - lim yλ = y . Q((/ y(i,λ) )) = w∗ - lim/ W

V

U

V

(5.7.6)

◦  Consider the mapping  π := φ Q : XW → XW , which becomes a contractive linear  (X ), and note that projection with range φ  ◦  Q= Q. Q◦ π = Q◦φ Q = IX  ◦ 

(5.7.7)

Then, for all x , z ∈ X  we have (x )φ (y )φ (z )}XW ) Q({φ {x y z }X  =  ( (z )}XW ) (x )φ Q((/ y(i,λ) )))φ = Q({φ (z )}XW ) (x ) π ((/ y(i,λ) ))φ = Q({φ  (z )}XW ). (x ) π ((/ y(i,λ) ))φ = Q π ({φ 



(by (5.7.5)) (by (5.7.6)) (by definition of  π) (by (5.7.7)).

224 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Writing (x ) = (x(i,λ) ) ∈ XW and φ (z ) = (z(i,λ) ) ∈ XW φ

(5.7.8)

x(i,λ) = xi and z(i,λ) = zi for every (i, λ) ∈ I × ,

(5.7.9)

(xi ) = φ(x ) and (zi ) = φ(z ),

(5.7.10)

with

where

we have (x )(/ (z )}XW ) Q π ({φ y(i,λ) )φ {x y z }X  =  (z )}XW ) (x )(/ y(i,λ) )φ = Q({φ y(i,λ) )(z(i,λ) )}XW ) = Q({(x(i,λ) )(/  y(i,λ) z(i,λ) }X )) = Q(({x(i,λ)/

= Q(({xi yλ zi }X )) ∗

= w - lim{xi yλ zi }X W

= w∗ - lim w∗ - lim{xi yλ zi }X V

U

(by Theorem 5.6.59(ii)) (by (5.7.7)) (by (5.7.8)) (by Corollary 5.7.6) (by (5.7.9)) (by definition of  Q) (by Lemma 5.7.17(ii))

= w∗ - lim Q(({xi yλ zi }X ))

(by definition of Q)

= w∗ - lim Q({(xi )(yλ )i (zi )}XU )

(by Corollary 5.7.6)

V V

= w∗ - lim Q({φ(x )φ(yλ )φ(z )}XU )

(by (5.7.10))

= w∗ - lim{x yλ z }X 

(by (5.7.4)).

V V

Therefore, since B ⊆ V , it follows that, for every w∗ -neighbourhood N of {x y z }X  in X  and for every λ0 ∈ , the set Bλ0 ∩{λ ∈  : {x yλ z }X  ∈ N} lies in V , and hence is non-empty. Thus {x y z }X  is a w∗ -cluster point of the net {x yλ z }X  .  5.7.2 The main results §5.7.19 We recall that every non-commutative JB∗ -algebra becomes a JB∗ -triple under its own norm and the triple product {abc} := Ua,c (b∗ ) (cf. Theorem 4.1.45). Hence non-commutative JBW ∗ -algebras are JBW ∗ -triples in a natural manner. Moreover, in view of Theorem 5.1.29(ii) and Corollary 5.1.30(iii), the triple product of any non-commutative JBW ∗ -algebra is separately w∗ -continuous. This is not casual, since we have in fact the following. Theorem 5.7.20 Let X be a JBW ∗ -triple. Then the triple product of X is separately w∗ -continuous. Proof We know that it is enough to show the w∗ -continuity of the triple product of X in the middle variable (cf. Lemma 5.7.14). We know also that X  is a JB∗ triple under a triple product which extends that of X and is separately σ (X  , X  )continuous (cf. Theorem 5.7.18). Let x, z be in X, let y be in BX , and let yλ be a net

5.7 JBW ∗ -triples

225

in BX σ (X, X∗ )-converging to y. Let π : X  → X stand for the Dixmier projection (i.e. the transpose of the inclusion X∗ → X  ), and note that π is σ (X  , X  )-σ (X, X∗ )continuous. Take a cluster point y ∈ X  of the net yλ in the σ (X  , X  )-topology. Then π(y ) is a cluster point of the net π(yλ ) = yλ in the σ (X, X∗ )-topology, and hence π(y ) = y. On the other hand, {xy z} is a cluster point of the net {xyλ z} in the σ (X  , X  )-topology, and hence π({xy z}) is a cluster point of the net π({xyλ z}) = {xyλ z} in the σ (X, X∗ )-topology. Since π({xy z}) = π({xπ(y )z}) = {xπ(y )z} (cf. Theorem 5.6.59(ii)) and π(y ) = y, it follows that {xyz} is a cluster point of the net {xyλ z} in the σ (X, X∗ )-topology. Keeping in mind the arbitrariness of y ∈ BX and of the net yλ σ (X, X∗ )-convergent to y, it follows from Fact 5.1.18 that the restriction to BX of the function {x, ·, z} : X → X is σ (X, X∗ )-continuous. Finally, by Fact 5.1.19, {x, ·, z} is σ (X, X∗ )-continuous.  As a first consequence of the above theorem, we have the following. Corollary 5.7.21 Let X be a JBW ∗ -triple, let e be a tripotent in X, and let k ∈ {1, 12 , 0}. Then Xk (e) is a w∗ -closed subtriple of X. Proof Since Xk (e) = {x ∈ X : {eex} = kx} (cf. Fact 4.2.14(i)), it follows from Theorem 5.7.20 that Xk (e) is w∗ -closed in X. Moreover, by Corollary 4.2.30(i), Xk (e) is a subtriple of X.  Given a Jordan ∗-triple X over K, the annihilator of X, denoted by Ann(X), is defined by the equality Ann(X) := {x ∈ X : {xXX} = {XxX} = 0}. Arguing as in the proof of Lemma 5.1.1, we get the following. Lemma 5.7.22 Let X be a Jordan ∗-triple over K with zero annihilator, and let I, J be triple ideals of X such that X = I ⊕ J. Then I = {x ∈ X : {xXJ} = {JxX} = {xJX} = 0}. By a direct summand of a Jordan ∗-triple X over K we mean any triple ideal I of X such that there exists another triple ideal J of X satisfying X = I ⊕ J. Keeping in mind Theorem 5.7.20 for the bracketed version, the following fact follows straightforwardly from Lemma 5.7.22. Fact 5.7.23 Let X be a Banach Jordan ∗-triple over K with zero annihilator (respectively, a JBW ∗ -triple), and let I be a direct summand of X. Then I is closed (respectively, w∗ -closed) in X. Lemma 5.7.24 Let X be a JB∗ -triple, and let I be a direct summand of X. Then I is an M-summand of X. Proof Let J be a triple ideal of X such that X = I ⊕ J. By Fact 5.7.23, I and J are closed in X. Therefore I and J are new JB∗ -triples (cf. Fact 4.1.40), so I × J is a JB∗ -triple under the sup norm (cf. Fact 4.2.29), and the mapping (x, y) → x + y from I × J to A is a bijective triple homomorphism. It follows from Proposition 4.1.52 that

226 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem x + y = max{x, y} for every (x, y) ∈ I × J. Thus the projection from X onto I corresponding to the decomposition X = I ⊕ J becomes an M-projection.  Keeping in mind Theorem 5.7.20 and arguing as in the last paragraph of the proof of Lemma 5.7.14, we obtain the following. Fact 5.7.25 Let X be a JBW ∗ -triple, and ϕ be in SX∗ . Then there exists a complete tripotent e ∈ X such that ϕ(e) = 1. As a consequence, X has a complete tripotent. Recall that by an alternative W ∗ -algebra (respectively, a W ∗ -algebra) we mean an alternative C∗ -algebra (respectively, a C∗ -algebra) which is a dual Banach space. Thus W ∗ -algebras are precisely those alternative W ∗ -algebras which are associative. As any non-commutative JB∗ -algebra, every C∗ -algebra becomes a JB∗ -triple under its own norm and the triple product {xyz} := Ux,z (y∗ ), which in this case can be expressed as 12 (xy∗ z + zy∗ x) (cf. Fact 4.1.41). Hence W ∗ -algebras are JBW ∗ -triples in a natural manner and, as a consequence, commutative W ∗ -algebras are abelian JBW ∗ -triples. Conversely, we have the following. Proposition 5.7.26 Let X be an abelian JBW ∗ -triple. Then X is the JBW ∗ -triple underlying a commutative W ∗ -algebra. Proof By Fact 5.7.25, X contains a complete tripotent e. Because X is abelian one has {ee{eex}} = {{eee}ex} = {eex} for every x ∈ X, and hence L(e, e)2 = L(e, e). Therefore, by Fact 4.2.14, we have P1 (e) = L(e, e) and P 1 (e) = 0. But, since P0 (e) = 0, we have in fact L(e, e) = IX , i.e. e is a 2 unitary element of X (cf. Definition 4.1.53). Keeping in mind that X is abelian, it follows from Theorem 4.1.55 that X is the JBW ∗ -triple underlying an associative JBW ∗ -algebra. But, by Fact 3.3.2, associative JBW ∗ -algebras and commutative W ∗ -algebras are the same.  §5.7.27 Let X be a JBW ∗ -triple, and let S be a non-empty subset of X. Since the intersection of any family of w∗ -closed subtriples of X is again a w∗ -closed subtriple of X, it follows that the intersection of all w∗ -closed subtriples of X containing S is the smallest w∗ -closed subtriple of X containing S. This subtriple is called the w∗ -closed subtriple of X generated by S. Invoking Theorem 5.7.20, the following assertions are easily verified: (i) If S is a subtriple of X, then the w∗ -closure of S in X is a subtriple of X. (ii) The w∗ -closed subtriple of X generated by S coincides with the w∗ -closure in X of the subtriple of X generated by S (cf. the paragraph immediately before Corollary 4.1.44). (iii) If S is an abelian subtriple of X, then the w∗ -closed subtriple of X generated by S is abelian.

5.7 JBW ∗ -triples

227

Corollary 5.7.28 Let X be a JBW ∗ -triple. We have: (i) If x is in X, then (a) the w∗ -closed subtriple of X generated by x is the JBW ∗ -triple underlying a commutative W ∗ -algebra; (b) there exists a tripotent e ∈ X such that x = Q2e (x). (ii) X equals the norm-closed linear hull of the set of its tripotents. Proof We note that, thanks to Fact 4.1.40 and §5.1.9, w∗ -closed subtriples of X are JBW ∗ -triples in a natural way. Therefore, since single-generated subtriples of Jordan ∗-triples are abelian (cf. Lemma 4.1.49), it is enough to invoke Proposition 5.7.26 to get assertion (i)(a). Assertion (i)(b) follows from assertion (i)(a) by invoking Fact 5.1.7. Finally, keeping in mind that self-adjoint idempotents in a C∗ -algebra become tripotents in its underlying JB∗ -triple, assertion (ii) follows from assertion (i)(a) and Theorem 5.1.29(vi).  Let X be a Jordan ∗-triple over K. By an inner ideal of X we mean any subspace M of X satisfying {MXM} ⊆ M. Lemma 5.7.29 Let X be a Jordan ∗-triple over K, and let e ∈ X be a tripotent. We have: (i) X1 (e) is an inner ideal of X. (ii) X1 (e) = Qe (X). (iii) If f is a tripotent of X with f ∈ X1 (e), then X1 ( f ) ⊆ X1 (e). Given x, z ∈ X1 (e) and y ∈ X, writing y = y1 + y 1 + y0 , where 2   yk ∈ Xk (e) for k ∈ 1, 12 , 0 , it follows from Proposition 4.2.22 that {xy1 z} ∈ X1 (e) and {xy 1 z} = {xy0 z} = 0, and consequently {xyz} = {xy1 z} ∈ X1 (e). Thus

Proof

2

{X1 (e), X, X1 (e)} ⊆ X1 (e), and assertion (i) is proved. Since P1 (e) = Q2e , it follows that X1 (e) = P1 (e)(X) = Q2e (X) ⊆ Qe (X). On the other hand, by assertion (i), Qe (X) ⊆ X1 (e). Thus assertion (ii) is proved. Propose that f is a tripotent of X with f ∈ X1 (e). Then, by assertions (ii) and (i), X1 ( f ) = Qf (X) = {fXf } ⊆ X1 (e), and the proof is complete.  §5.7.30 Let X be a Jordan ∗-triple over K. Two tripotents e and f in X are said to be orthogonal if {eef } = 0. Let e and f be orthogonal tripotents in X. Note that then f ∈ X0 (e), and hence, as a consequence of the second line of (4.2.41) in Proposition 4.2.22, we have L(e, f ) = L( f , e) = 0, so in particular {ffe} = 0. Thus e + f is a tripotent in X. It follows from the Jordan triple identity (4.2.11) that L(e, e) and L( f , f ) commute. Since the Peirce projections Pk (e) (k ∈ {1, 12 , 0}) are polynomials over the integer in L(e, e) (cf. Fact 4.2.14(ii)), it follows that e and f are compatible in the sense that Pj (e) and Pk ( f ) commute for all j, k ∈ {1, 12 , 0}. Moreover, it follows from the Jordan triple identity (4.1.13) that for each x ∈ X we have

228 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem {xf {efe}} = {{xfe}fe} − {e{fxf }e} + {ef {xfe}}, hence {e{fxf }e} = 0, and so Qe Qf = 0. Since P1 (e) = Q2e and P1 ( f ) = Q2f , we realize that P1 (e)P1 ( f ) = 0, hence P1 (e) = P1 (e)IX = P1 (e)(P1 ( f ) + P 1 ( f ) + P0 ( f )) = P1 (e)(P 1 ( f ) + P0 ( f )), 2

2

and analogously P1 ( f ) = (P 1 (e) + P0 (e))P1 ( f ). Thus writing 2

IX = (P1 (e) + P 1 (e) + P0 (e))(P1 ( f ) + P 1 ( f ) + P0 ( f )) 2

2

we derive the following decomposition of IX as a sum of orthogonal projections IX = P1 (e) + P1 ( f ) + P 1 (e)P 1 ( f ) + P 1 (e)P0 ( f ) + P0 (e)P 1 ( f ) + P0 (e)P0 ( f ). 2

2

2

2

Therefore, we have the decomposition = Xjk = X00 ⊕ X01 ⊕ X02 ⊕ X11 ⊕ X12 ⊕ X22 , X= 0≤j≤k≤2

where the Peirce spaces Xjk (0 ≤ j ≤ k ≤ 2) with respect to the pair (e, f ) are defined by X00 := X0 (e) ∩ X0 ( f ), X11 := X1 (e),

X01 := X 1 (e) ∩ X0 ( f ), 2

X12 := X 1 (e) ∩ X 1 ( f ), 2

2

X02 := X0 (e) ∩ X 1 ( f ), 2

and

X22 := X1 ( f ).

Putting X10 := X01 , X20 := X02 , and X21 := X12 it follows from Proposition 4.2.22 that {Xij Xjk Xk } ⊆ Xi for all i, j, k, ∈ {0, 1, 2}, and the triple products

(5.7.11)

of Peirce spaces which cannot be written in this form vanish. Let X be a Jordan ∗-triple over K, and let e be a tripotent in X. Since e becomes a unitary element of the Jordan ∗-triple X1 (e), it follows from Proposition 4.1.54 that X1 (e) is a Jordan ∗-algebra over K with unit e in a natural manner. We note that selfadjoint idempotents in the Jordan ∗-algebra X1 (e) become tripotents in X. Moreover, given a self-adjoint idempotent z in the Jordan ∗-algebra X1 (e), we see that for each x ∈ X1 (e) we have {zzx} = Uz,x (z) = (Lz Lx + Lx Lz − Lzx )(z) = z(xz) + xz2 − (zx)z = xz.

(5.7.12)

Lemma 5.7.31 Let X be a Jordan ∗-triple over K, let e be a complete tripotent in X, let z be a self-adjoint central idempotent in the Jordan ∗-algebra X1 (e), and set w := e − z. We have:

5.7 JBW ∗ -triples

229

(i) =

X1 (e)



X 1 (e)

 X0 (w) =

 X1 (z)



 X 1 (z)

⊕ X0 (z)

⊕ ⊕ X1 (w) ⊕ X 1 (w)

X

(ii) If X is in fact a summand of X.

JBW ∗ -triple,

=

2

2

2

then X0 (w) is a triple ideal of X, and hence a direct

Proof Clearly, z and w are orthogonal tripotents in X, so that we can consider the Peirce decomposition of X with respect to the pair (z, w). Moreover, it follows from (5.7.12) that, for each x ∈ X 1 (z) ∩ X1 (e) we have x = 2xz, hence 2

xz = 2(xz)z = 2xz2 = 2xz, so xz = 0, and finally x = 0. Therefore X 1 (z) ∩ X1 (e) = 0. 2

(5.7.13)

Moreover, since e is a complete tripotent in X, we have X = X1 (e) ⊕ X 1 (e). 2

(5.7.14)

On the other hand, since z and w are orthogonal tripotents, they are compatible and we have L(z, w) = L(w, z) = 0. Therefore L(e, e) = L(z, z) + L(w, w), and e and z are also compatible. It follows from (5.7.13) and (5.7.14) that X 1 (z) ⊆ X 1 (e). 2

2

(5.7.15)

As a consequence of (5.7.15), for each x ∈ X 1 (z), we have 2

1 1 x = {eex} = {zzx} + {wwx} = x + {wwx}, 2 2 hence {wwx} = 0, and so x ∈ X0 (w). Therefore X 1 (z) ⊆ X0 (w),

(5.7.16)

X 1 (z) ∩ X 1 (w) = 0.

(5.7.17)

2

and in particular 2

2

By interchanging the roles of z and w, it follows from (5.7.16) that X 1 (w) ⊆ X0 (z). 2

(5.7.18)

Moreover, since the inclusion X0 (z) ∩ X0 (w) ⊆ X0 (e) is clear, we have X0 (z) ∩ X0 (w) = 0.

(5.7.19)

230 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem It follows from (5.7.16), (5.7.17), (5.7.18), and (5.7.19) that X00 = 0,

X01 = X10 = X 1 (z), 2

X11 = X1 (z),

X12 = X21 = 0,

X02 = X20 = X 1 (w), 2

and

X22 = X1 (w),

and consequently X = X 1 (z) ⊕ X 1 (w) ⊕ X1 (z) ⊕ X1 (w). 2

(5.7.20)

2

Moreover, it follows from the compatibility of z and w that X0 (w) = X1 (z) ⊕ X 1 (z) and X0 (z) = X1 (w) ⊕ X 1 (w). 2

2

Thus X = X0 (w) ⊕ X0 (z). Since, by Lemma 5.7.29(iii), X1 (z) ⊆ X1 (e) and X1 (w) ⊆ X1 (e), we have the inclusion X1 (z) ⊕ X1 (w) ⊆ X1 (e). Moreover, it follows from (5.7.15) and the analogous inclusion for w that X 1 (z) ⊕ X 1 (w) ⊆ X 1 (e). Now, the decompositions (5.7.14) and 2 2 2 (5.7.20) give that X1 (e) = X1 (z) ⊕ X1 (w) and X 1 (e) = X 1 (z) ⊕ X 1 (w). Thus asser2 2 2 tion (i) has been proved. Now propose that X is a JBW ∗ -triple. Keeping in mind assertion (i) and the multiplication rules (5.7.11), to prove that X0 (w) is a triple ideal of X it is enough to show that L(a, b) = L(b, a) = 0 whenever a and b are in X 1 (z) and X 1 (w), respectively. 2 2 Then, by symmetry and Corollaries 5.7.21 and 5.7.28(ii), it suffices to show that L(a, b) = 0 whenever a is a tripotent in X 1 (z) and b is arbitrary in X 1 (w). Let a be 2 2 a tripotent in X 1 (z), and let b be in X 1 (w). Since a ∈ X 1 (z) ⊆ X0 (w), it follows that 2 2 2 {wwa} = 0, hence w and a are orthogonal tripotents, and so c := a + w is a tripotent. Write b = b1 + b 1 + b0 , where bk ∈ Xk (a) for k ∈ {1, 12 , 0}. Since a ∈ X 1 (z) ⊆ X0 (w) 2 2 and b ∈ X 1 (w), it follows that {aba} = 0, and consequently 2

b1 = P1 (a)(b) = Q2a (b) = 0. Thus b = b 1 + b0 , and as a consequence 2

b 1 = 2{aab 1 } = 2({aab 1 } + {aab0 }) = 2{aab}. 2

2

2

From this, taking again into account that a ∈ X 1 (z) ⊆ X0 (w) and b ∈ X 1 (w), we 2

2

see that b 1 ∈ X 1 (w), hence {ccb 1 } = {aab 1 } + {wwb 1 } = 12 b 1 + 12 b 1 = b 1 , and so 2 2 2 2 2 2 2 2 b 1 ∈ X1 (c). Summarizing b 1 ∈ X 1 (w) ∩ X1 (c). Suppose, to obtain a contradiction, 2 2 2 that L(a, b) = 0. Since a ∈ X1 (a) and b0 ∈ X0 (a), it follows that L(a, b0 ) = 0, hence L(a, b) = L(a, b 1 + b0 ) = L(a, b 1 ), and so b 1 = 0. But, b 1 ∈ X1 (c), and consequently 2

2

2

b 1 = Q2c (b 1 ). Therefore Qc (b 1 ) = 0. On the other hand 2

2

2

2

Qc (b 1 ) = {cb 1 c} = {(a + w)b 1 (a + w)} = {ab 1 a} + 2{ab 1 w} + {wb 1 w}. 2

2

2

2

2

2

5.7 JBW ∗ -triples

231

Keeping in mind that a ∈ X1 (a), w ∈ X0 (a), and b 1 ∈ X 1 (a) we obtain that 2

2

{ab 1 a} = 0 and {wb 1 w} = 0, 2

2

and on account that a ∈ X 1 (z) ⊆ X0 (w), b 1 ∈ X 1 (w) ⊆ X0 (z), and w ∈ X1 (w) ⊆ X0 (z) 2 2 2 we realize that {ab 1 w} ∈ X 1 (z) ∩ X 1 (w) = 0. 2

2

2

Thus Qc (b 1 ) = 0. This contradiction concludes the proof.



2

Lemma 5.7.31(i) should be kept in mind for the proof of the following. Theorem 5.7.32 Every w∗ -closed triple ideal of a JBW ∗ -triple X is a direct summand of X. Proof Fix a complete tripotent e ∈ X (cf. Fact 5.7.25) and note that, by Corollary 5.7.21, X1 (e) is w∗ -closed in X. Let J be a w∗ -closed triple ideal of X, and set I := J ∩ X1 (e). Then I is a w∗ -closed ideal of the JBW ∗ -algebra X1 (e) (cf. Corollary 4.2.30(iii)(b) and §5.1.9). By Fact 5.1.10(i), there exists a central idempotent z in X1 (e) such that I = zX1 (e). For x ∈ X1 (e) we have that zx ∈ X1 (e), hence, by (5.7.13), we see that {zz(zx)} = z(zx) = z2 x = zx, and so zx ∈ X1 (z). Thus I ⊆ X1 (z). Conversely, it follows from (5.7.12) and the inclusion X1 (z) ⊆ X1 (e) that for each x ∈ X1 (z) we have x = {zzx} = zx ∈ I, and hence X1 (z) ⊆ I. Thus I = X1 (z). Putting w := e − z, from the inclusion X1 (w) ⊆ X1 (e), we deduce that X1 (w) ∩ J ⊆ X1 (e) ∩ J = I = X1 (z), and the fact that X1 (z) ∩ X1 (w) = 0 allows us to conclude that X1 (w) ∩ J = 0.

(5.7.21)

Since J is a triple ideal of X, it is clear that X1 (z) + X 1 (z) is contained in J. Assume 2 that X1 (z) + X 1 (z) = J. Then X0 (z) ∩ J = 0. Take a nonzero element x ∈ X0 (z) ∩ J, 2 and write x = x1 + x 1 with xk ∈ Xk (w). Since J is a triple ideal we see that 2 x1 = 2{wwx} − x ∈ J, and hence x1 = 0 because of (5.7.21). Thus x = x 1 ∈ X 1 (w) ∩ J. 2 2 This implies X1 (w) ∩ J = 0 by Proposition 4.2.32(iii), which is a contradiction because of (5.7.21). Therefore J = X1 (z)+X 1 (z) = X0 (w), and the proof is concluded 2 by applying Lemma 5.7.31(ii).  Putting together Theorem 5.7.32 and Lemma 5.7.24, we obtain the following. Fact 5.7.33 Let X be a JBW ∗ -triple, and let I be a w∗ -closed triple ideal of X. Then I is an M-summand of X. Theorem 5.7.34 Let X be a JB∗ -triple. Then: (i) The M-ideals of X are precisely the closed triple ideals of X. (ii) The M-summands of X are precisely the direct summands of X. Proof Let I be an M-ideal of X. By Proposition 5.1.17, for every x ∈ X we have L(x, x)(I) ⊆ I, and hence, by polarization, L(x, y)(I) ⊆ I for all x, y ∈ X.

232 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem We know that X  is a JBW ∗ -triple under a triple product which extends that of X (cf. Proposition 5.7.10). Clearly I ◦◦ is an M-summand of X  , so there is an M-projection P from X  onto I ◦◦ . Fix x ∈ X and y ∈ I. Write {xyx} = u + v , where u ∈ I ◦◦ and P(v ) = 0. Using Lemma 5.1.16 and Corollary 5.6.58, we obtain u = P({xyx}) = 2{P(x)yx} − {xyx} = 2{P(x)yx} − (u + v ). Thus v = 2{P(x)yx} − 2u ∈ I ◦◦ , since {P(x)yx} ∈ I ◦◦ by the first paragraph of the proof (with (X  , I ◦◦ ) instead of (X, I)). Consequently v = 0. Hence {xyx} ∈ I ◦◦ ∩ X = I, so I is a triple ideal in X. Now let I be a closed triple ideal of X. We know that X  is a JBW ∗ -triple under a triple product which extends that of X and is separately w∗ -continuous (cf. Theorem 5.7.18). Consequently, since I is w∗ -dense in I ◦◦ , and X is w∗ -dense in X  , we see that I ◦◦ is a triple ideal of X  . Since I ◦◦ is w∗ -closed, it follows from Fact 5.7.33 that I ◦◦ is an M-summand of X  . Therefore, by Lemma 5.1.21(iii), I is an M-ideal of X. This concludes the proof of assertion (i). We already proved in Lemma 5.7.24 that direct summands of X are M-summands. Conversely, let I be an M-summand of X. Let P be the M-projection onto I. Then both I and J := (IX − P)(X) are M-ideals of X with X = I ⊕ J. Therefore, by assertion (i), I and J are triple ideals of X. Thus I is a direct summand of X. This concludes the proof of assertion (ii).  Corollary 5.7.35 Let X be a JBW ∗ -triple, and let S be a subset of X. Then the following conditions are equivalent: (i) S is an M-summand of X. (ii) S is a direct summand of X. (iii) S is a w∗ -closed triple ideal of X. Proof (i)⇒(ii) By Theorem 5.7.34(ii). (ii)⇒(iii) By the bracketed version of Fact 5.7.23. (iii)⇒(i) By Fact 5.7.33.



Given complex normed spaces X and Y, and a bounded conjugate-linear operator T : X → Y, the transpose T  of T is defined as the bounded conjugate-linear operator from Y  to X  given by T  (y )(x) := y (T(x)) for all y ∈ Y  and x ∈ X. Theorem 5.7.36 The predual X∗ of a JBW ∗ -triple X is L-embedded. Proof

For x, y ∈ X, let L(x, y) and Qx,y denote the operators on X defined by L(x, y)(z) := {xyz} and Qx,y (z) := {xzy},

respectively. By standard theory of duality, the separate w∗ -continuity of the triple product of X (cf. Theorem 5.7.20) is equivalent to the inclusions (L(x, y)) (X∗ ) ⊆ X∗ and (Qx,y ) (X∗ ) ⊆ X∗ for all x, y ∈ X. Therefore we have (L(x, y)) ((X∗ )◦ ) ⊆ (X∗ )◦ and

5.7 JBW ∗ -triples

233

(Qx,y ) ((X∗ )◦ ) ⊆ (X∗ )◦ for all x, y ∈ X. Keeping in mind that X  is a JB∗ -triple under a triple product which extends that of X (cf. Proposition 5.7.10), the above inclusions read as {XX(X∗ )◦ } ⊆ (X∗ )◦ and {X(X∗ )◦ X} ⊆ (X∗ )◦ in X  . Applying the separate w∗ -continuity of the triple product of X  (cf. Theorem 5.7.18) and the w∗ -density of X in X  , we deduce {X  X  (X∗ )◦ } + {X  (X∗ )◦ X  } ⊆ (X∗ )◦ . Therefore (X∗ )◦ is a w∗ -closed triple ideal of X  , and hence, by Fact 5.7.33, it is an M-summand of X  . Finally, by Lemma 5.1.21(iii), X∗ is an L-summand of X  .  Since non-commutative JBW ∗ -algebras are JBW ∗ -triples in a natural way, Theorem 5.7.36 above contains Theorem 5.1.32. On the other hand, combining Theorems 5.1.22 and 5.7.34, we get the following. Corollary 5.7.37 Let A be a non-commutative JB∗ -algebra, and let X stand for the JB∗ -triple underlying A. Then: (i) Closed ideals of A and closed triple ideals of X coincide. (ii) Direct summands of A and direct summands of X coincide. The next theorem generalizes the fact, proved in Theorem 5.1.29(iv), that noncommutative JBW ∗ -algebras have unique predual. Theorem 5.7.38 JBW ∗ -triples have unique predual. Proof Let X be a JBW ∗ -triple, and let Y, Z be preduals of X. It is enough to show that the identity mapping IX : (X, σ (X, Y)) → (X, σ (X, Z)) is a homeomorphism. By symmetry, it suffices to show that IX : (X, σ (X, Y)) → (X, σ (X, Z)) is continuous. Let x be in BX , and let xλ be a net in BX σ (X, Y)-convergent to x. Take a σ (X, Z)-cluster point  x ∈ X of the net xλ . Let e be any tripotent in X, note that, by Fact 4.2.14(ii) and Theorem 5.7.20, P1 (e) : X → X is σ (X, Y) − σ (X, Y) and σ (X, Z) − σ (X, Z) continuous, and set A := P1 (e)(X). Since A = ker(IX − P1 (e)), it follows that A is both σ (X, Y)- and σ (X, Z)-closed in X. But, by Corollary 4.2.30(iii)(b), A is a JB∗ -algebra for suitable product and involution. Therefore, by §5.1.9, A is a JBW ∗ -algebra in two manners, with corresponding weak∗ topologies equal to σ (X, Y)|A and σ (X, Z)|A . Since JBW ∗ -algebras have unique predual (cf. Theorem 5.1.29(iv)), σ (X, Y) and σ (X, Z) coincide on A. Now P1 (e)(xλ ) is a net in A σ (X, Y)converging to P1 (e)(x) ∈ A, and P1 (e)( x) ∈ A is a σ (X, Z)-cluster point of the net x) = P1 (e)(x). By choosing e adequately (cf. Corollary P1 (e)(xλ ). Therefore P1 (e)( 5.7.28(i)(b)), we obtain  x − x = P1 (e)( x − x) = 0, hence x (=  x) is a σ (X, Z)cluster point of the net xλ . Keeping in mind the arbitrariness of x ∈ BX and of the net xλ σ (X, Y)-convergent to x, it follows from Fact 5.1.18 that the restriction to BX of IX : (X, σ (X, Y)) → (X, σ (X, Z)) is continuous. Finally, by Fact 5.1.19, IX : (X, σ (X, Y)) → (X, σ (X, Z)) is continuous.  Keeping in mind Proposition 4.1.52 and Fact 5.1.24, Theorem 5.7.38 above implies the following.

234 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Corollary 5.7.39 Bijective triple homomorphisms between JBW ∗ -triples are w∗ -continuous. Let X be a (Banach) Jordan ∗-triple over K, and let I be a (closed) triple ideal of X. Then X/I becomes a (Banach) Jordan ∗-triple under the triple product defined by {x + I, y + I, z + I} := {xyz} + I. Proposition 5.7.40 Let X be a JB∗ -triple, and let I be a closed triple ideal of X. Then the Banach Jordan ∗-triple X/I is a JB∗ -triple. Proof Since I is w∗ -dense in I ◦◦ , it follows from Theorem 5.7.18 that I ◦◦ becomes a w∗ -closed triple ideal of X  . Then, by Corollary 5.7.35, we have X  = I ◦◦ ⊕∞ J for some w∗ -closed triple ideal J of X  . Therefore, denoting by π : X  → X  /I ◦◦ the quotient mapping, π|J : J → X  /I ◦◦ becomes a surjective isometric triple homomorphism. As a result, X  /I ◦◦ is a JBW ∗ -triple in a natural way. Finally, the mapping x + I → x + I ◦◦ from X/I to X  /I ◦◦ becomes an isometric triple homomorphism, hence X/I is a JB∗ -triple.  Proposition 5.7.41 Let X and Y be JB∗ -triples, and let F : X → Y be a triple homomorphism. We have: (i) F is contractive. (ii) If F is injective, then F is isometric. (iii) F has closed range. Proof Let x be in X. To prove that F(x) ≤ x, we may consider the closed subtriple of X generated by x (say Z), and the closure in Y of F(Z), and we may see F as a triple homomorphism from Z to F(Z). Since Z is abelian (cf. Lemma 4.1.49), so is F(Z), and hence, F(Z) can be seen as a closed subtriple of C0C (E), for some locally compact Hausdorff topological space E (cf. Theorem 4.2.7). Therefore it is better to see F as a triple homomorphism from Z to C0C (E). Let t be in E, and let us denote by δt : C0C (E) → C the valuation at t. Then δt ◦ F : Z → C is a triple homomorphism, and hence, by Lemmas 4.1.48(i) and 4.2.2, it is continuous with δt ◦ F ≤ 1, so F(x)(t) ≤ x. By taking maximum in t ∈ E, we get F(x) ≤ x, as desired. Propose that F is injective, and let x be in X. To prove that F(x) = x, we may consider the closed subtriples of X and Y generated by x and F(x), respectively, which are the JB∗ -triples underlying suitable commutative C∗ -algebras A and B, respectively (cf. Theorem 4.2.9), we may apply assertion (i) proved above to realize that F(A) ⊆ B, and, consequently, we may see F as a triple homomorphism from A to B with dense range. Let 1 denote the unit of A (cf. Theorem 2.2.15 and §2.3.53), and set u := F  (1) ∈ B . Then we have {uuF(a)} = F(a) for every a ∈ A. Since F(A) is norm-dense in B, and B is w∗ -dense in B , we see that {uub } = b for every b ∈ B , i.e., u is a unitary element of the JB∗ -triple underlying B (cf. Definition 4.1.53), and hence is a unitary element of B (cf. the implication (vi)⇒(i) in Theorem 4.2.28). Since B is commutative, it follows that the mapping  : A → B defined by

5.7 JBW ∗ -triples

235

(a) := F(a)u∗ is an injective algebra ∗-homomorphism. Therefore, by Corollary 1.2.52,  (and hence F) is an isometry. Thus F(x) = x, as desired. Assertion (iii) follows by arguing as in the proof of the bracket-free version of Lemma 5.1.34(ii). Indeed, after doing the appropriate terminological changes, it is enough to replace the first application of Proposition 3.4.4 with the current assertion (i), Proposition 3.4.13 with Proposition 5.7.40, and the second application of Proposition 3.4.4 with the current assertion (ii).  The next corollary follows from Lemma 5.1.34(i) and Proposition 5.7.41(iii) above. Corollary 5.7.42 The range of any w∗ -continuous triple homomorphism between JBW ∗ -triples is w∗ -closed. Definition 5.7.43 Let X be a complex Jordan ∗-triple. By a JBW ∗ -representation of X we mean a w∗ -dense-range triple homomorphism from X to some JBW ∗ -triple. Let 1 : X → Y1 and 2 : X → Y2 be JBW ∗ -representations of X. We say that 1 and 2 are equivalent if there exists a bijective triple homomorphism F from Y1 to Y2 such that 2 = F ◦ 1 . It is important to recall here that, by Proposition 4.1.52 and Corollary 5.7.39, bijective triple homomorphisms between JBW ∗ -triples are isometric and w∗ -continuous. Definition 5.7.44 Let X be a JBW ∗ -triple. According to Corollary 5.7.35, w∗ -closed ideals of X are the sets of the form P(X), when P runs over the set of all M-projections on X. It follows then that each M-projection on X becomes a w∗ -continuous triple homomorphism (cf. Corollary 5.1.20). Now let X be any JB∗ -triple. Then, by Proposition 5.7.10, X  becomes a JBW ∗ -triple containing X as a subtriple. Therefore each M-projection P on X  gives rise to a w∗ -dense-range triple homomorphism from X to a certain JBW ∗ -triple, namely the mapping x → P(x) from X to P(X  ). Such a representation will be called the canonical JBW ∗ -representation of X associated to P. Proposition 5.7.45 Let X be a JB∗ -triple. Then every JBW ∗ -representation of X is equivalent to the canonical JBW ∗ -representation of X associated to a suitable M-projection on X  . Proof The result follows by arguing as in the proof of the bracket-free version of Proposition 5.1.36. Indeed, after doing the appropriate terminological changes, it is enough to forget Theorem 5.1.29(ii), and to replace Corollary 5.1.30(iii) with Theorem 5.7.20, Lemma 5.1.34(iii) with Corollary 5.7.42, and Fact 5.1.10(i) with Fact 5.7.33.  Corollary 5.7.46 Let X be a JBW ∗ -triple. Then X is isometrically and w∗ bicontinuously triple-isomorphic to a w∗ -closed triple ideal of X  . Proof The identity mapping on X is a JBW ∗ -representation of X, and Proposition 5.7.45 applies. 

236 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Corollary 5.7.47 Let Y be a JBW ∗ -triple, and let X be a w∗ -dense subtriple of Y. Then BX is w∗ -dense in BY . Proof After doing the appropriate terminological changes, the result follows by arguing as in the proof of Corollary 5.1.44, with Proposition 5.7.45 instead of Proposition 5.1.36.  Proposition 5.7.45 and Corollaries 5.7.46 and 5.7.47 become the appropriate variants for JB∗ -triples of Proposition 5.1.36 and Corollaries 5.1.37 and 5.1.44, respectively. 5.7.3 Historical notes and comments In [1002], Lindenstrauss and Rosenthal discovered a striking result which shows that the bidual of any Banach space is ‘finitely representable’ in the original space. A slightly stronger version of this result, just the one formulated in Proposition 5.7.7 under the name of local reflexivity principle, was found by Johnson, Rosenthal, and Zippin [988]. The reformulation of the local reflexivity principle in terms of ultrapowers, established in Proposition 5.7.8, is due to Henson, Moore, Jr. [977], and Stern [1101]. The proof we have given is that of Heinrich [318], although we have incorporated some clarifications taken from the proof of [671, Proposition 11.1.12]. For additional historical information about the local reflexivity principle and its relation with ultraproducts, the reader is referred to [790, Subsection 6.1.3]. Proposition 5.7.10 is due to Dineen [923]. The proof we have given is also due to him [213]. Results from Lemma 5.7.11 to Proposition 5.7.13 are due to Friedman and Russo [269]. Results from Lemma 5.7.14 to Theorem 5.7.18 are due to Barton and Timoney [854]. Our proofs are essentially the original ones. Nevertheless, we have made a slight change in the proof of Theorem 5.7.18. To understand the reason of this change, the reader is invited to browse in both ours and the original proofs, and to follow our notation. Then the reader can see that, in the Barton–Timoney proof, the net yλ in BX w∗ -convergent to y ∈ BX  is proposed to be indexed by the set (say N ) of all w∗ -neighbourhoods of 0 ∈ X  directed by reverse inclusion, and that, arguing as in our proof, they derive that {x y z }X  is a w∗ -cluster point of the net {x yλ z }X  whenever x , z are in X  . The difficulty appears when, to conclude the proof, they say: ‘Since any net in X  which is w∗ -convergent to y has a subnet indexed by N , Lemma 1.2 (equal to our Corollary 5.7.16) implies that {x , ·, z }X  is w∗ -continuous’. As a matter of fact, we have been unable to verify that any net in X  which is w∗ -convergent to y has a subnet indexed by N . Therefore, in our proof, the net yλ is indexed by an arbitrary directed set , and the ultrafilter refining the ultrafilter of the local reflexivity is not fixed, but depends on . Anyway, the slight change we have introduced should not be understood as a demerit of the whole original proof of Theorem 5.7.18, which consists also of the results from Lemma 5.7.14 to Lemma 5.7.17, and which becomes one of the deepest and cleverest arguments included in our work.

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With the exception of Proposition 5.7.26, Corollary 5.7.28, Lemma 5.7.31, and Theorem 5.7.32 (which are due to Horn [979]), results from Theorem 5.7.20 to Corollary 5.7.39 are due to Barton and Timoney [854]. Our proofs of the fundamental Theorems 5.7.20 and 5.7.38 (asserting the separate w∗ -continuity of the triple product and the uniqueness of the predual of a JBW ∗ -triple) are new. The original proofs depend heavily on results of Godefroy [955], which are explained in what follows. Let X be a Banach space over K. For any subset S of X, let / S stand for the w∗   closure of S in X , and denote by Bu the subset of X consisting of those elements x ∈ X  such that, for every closed convex bounded subset C of X, the mapping x → x (x ) from (/ C, w∗ ) to K has at least one continuity point. It is proved in [955, Lemme 13] that Bu is a norm-closed subspace of X  . The universal frame γu (X) of X in X  is defined as the prepolar of Bu ∩ X ◦ in X  , where X ◦ stands for the polar of X in X  . It is clear that X ⊆ γu (X). The Banach space X is said to be well-framed if γu (X) = X, i.e., if Bu ∩ X ◦ is w∗ -dense in X ◦ . As one of the main results in [955], Godefroy proves the following. Theorem 5.7.48 [955, Theorems 15 and 16] Let X be a well-framed Banach space over K. Then we have: (i) X is the unique predual of X  (in the sense of §5.1.23). (ii) Every closed subspace of X is well-framed. With the aid of Theorem 5.7.48(i) above, Horn [979] proves that Theorems 5.7.20 and 5.7.38 are ‘equivalent’. More precisely, he establishes the following. Fact 5.7.49 For a JBW ∗ -triple X, the following conditions are equivalent: (i) The triple product of X is separately w∗ -continuous. (ii) X∗ is well-framed. (iii) The predual of X is unique. Indeed, the implications (i)⇒(ii) and (iii)⇒(i) are shown in [979, Proposition 3.20] and [979, Proposition 3.24], respectively, whereas the implication (ii)⇒(iii) follows from Theorem 5.7.48(i). Finally, Barton and Timoney [854] combine their Theorem 5.7.18 with Horn’s Fact 5.7.49 above to show that all conditions in Fact 5.7.49 are automatically fulfilled, thus proving in particular Theorems 5.7.20 and 5.7.38. Their argument is not, but could have been, the following. Original proof of Theorems 5.7.20 and 5.7.38 Let X be a JBW ∗ -triple. Then, by Theorem 5.7.18, X  is a JBW ∗ -triple whose triple product is separately w∗ continuous. Therefore, by the implication (i)⇒(ii) in Fact 5.7.49, X  is well-framed. Now, by Theorem 5.7.48(ii), X∗ is well-framed, and hence, by Theorem 5.7.48(i), the predual of X is unique (the conclusion in Theorem 5.7.38). Finally, by the implication (iii)⇒(i) in Fact 5.7.49, the triple product of X is separately w∗ -continuous (the conclusion in Theorem 5.7.20). 

238 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem §5.7.50 Curiously, in his survey paper [956] (on uniqueness of preduals), Godefroy does not include Theorem 5.7.48. Even more, in [956, Application VII.11(1)], Godefroy says ‘. . . Horn [979, Theorem 3.21] has shown (by using Theorem V.3) that JBW ∗ -triples have a unique predual’. As a matter of fact, if one goes to [956, Theorem V.3], then one finds the Godefroy–Talagrand theorem [960, 961] (that Banach spaces satisfying the so-called ‘property (X)’ are unique preduals of their duals) instead of Theorem 5.7.48 invoked by Horn. Keeping in mind the ‘Erratum’ in [961, p. 32], this reference is correct but, since (to the best of our knowledge) the details have not been treated in the literature, we follow Pfitzner [1048] to provide the reader with the necessary modifications.  Let X be a Banach space over K. A generalized series α∈I xα in X is called wuC (weakly unconditionally Cauchy) if the family {|x (xα )|}α∈I is summable in R for every x ∈ X  . If I = N, then we encounter the wuC-series as defined in §5.8.1 below. Arguing as in the proof of the implication (i)⇒(ii) in Proposition 5.8.2 below, we  see that, if α∈I xα is a wuC generalized series in X, then there is a constant C ≥ 0 such that  tα xα ≤ C sup |tα | (5.7.22) α∈F

α∈F

for every (tα ) ∈ ∞ (I) and every finite subset F of I. As a first consequence of  (5.7.22), if α∈I xα is a wuC generalized series in X  , then the family {xα }α∈I is w∗  summable in X  , i.e. there exists x = w∗ - limF α∈F xα ∈ X  where F runs through the family of all finite subsets of I ordered by inclusion. In this case we use the notation   x = w∗ - α∈I xα or, to be more precise, the notation x = σ (X  , X)- α∈I xα . Definition 5.7.51 Let X be a Banach space over K. We say that X has property (X0 ) if, for any x ∈ X  such that the equality      ∗  xα = x (xα ) (5.7.23) x w α∈I



α∈I  α∈I xα in

holds for every wuC generalized series X  , we have x ∈ X. Now let κ be an infinite cardinal number. We say that X has property (X) of level κ  if for any x ∈ X  such that (5.7.23) holds for every wuC generalized series α∈I xα in X  with |I| ≤ κ, we have x ∈ X. Finally, we say that X has property (X) if X has property (X) of level ℵ0 . Remark 5.7.52 (a) It is immediate that property (X) implies property (X) of level κ for every infinite cardinal κ, and that property (X) of level κ for some infinite cardinal κ implies property (X0 ). (b) Property (X) was introduced by Godefroy and Talagrand [960, 961], whereas property (X) of level κ was introduced by Neufang [1031]. Actually, Neufang has shown that property (X) of level κ is equivalent to property (X) if and only if κ is not measurable [1031, Corollary 3.15 and Remark 3.11(i)]. Property (X0 ) has been introduced by Pfitzner in the note [1048] we are reproducing here.

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Concerning our interest, the following result (essentially due to Horn [979]) becomes relevant. Fact 5.7.53 The following assertions hold: (i) If X is the predual of a JBW ∗ -triple, then: (a) Every family of pairwise orthogonal tripotents in X  is w∗ -summable in X  .   (b) If x ∈ X  is such that x (w∗ - α∈I xα ) = α∈I x (xα ) for every family {xα }α∈I of pairwise orthogonal tripotents in X  , then x ∈ X. (ii) If Y is any JB∗ -triple, and if {yα }α∈I is a family of pairwise orthogonal tripotents  in Y, then the generalized series α∈I yα is wuC in Y. (iii) If X is the predual of a JBW ∗ -triple, then X has property (X0 ). Proof Assertion (i)(a) follows from the Barton–Timoney Theorem 5.7.20 and [979, Corollary 3.13]. (We note that the separate w∗ -continuity of the triple product of X, assured by Theorem 5.7.20, is taken as an axiom in [979].) Then assertion (i)(b) is nothing other than [979, Proposition 3.19]. Let Y be a JB∗ -triple, and let {yα }α∈I be a family of pairwise orthogonal tripotents in Y. Then, by Proposition 5.7.10, {yα }α∈I can be seen as a family of pairwise orthogonal tripotents in the JBW ∗ -triple Y  . Therefore, by assertion (i)(a) with X = Y  , {yα }α∈I is σ (Y  , Y  )-summable in Y  . But this implies that the generalized  series α∈I yα is wuC in Y. This proves assertion (ii). (In the case that Y is in fact a JBW ∗ -triple, the conclusion follows straightforwardly from (i)(a) and Banach space results; indeed, since the family {yα }α∈I is w∗ -summable in Y, the generalized series  ∗ α∈I yα is ‘w uC’, and then, as pointed out in [1031, Remark 3.9], we can argue as in the proof of [1184, Lemma 15.1] or [1151, Corollary 11 in p. 49] to realize that,  as desired, α∈I yα is wuC.) Keeping in mind assertions (i) and (ii) already proved, assertion (iii) follows  straightforwardly from the definition itself of property (X0 ). In [961, Theorem 6(1)] it is shown that a Banach space X having property (X) is the unique predual of its dual X  . In what follows we show the same with (X0 ) instead of (X). First we show a lemma which corresponds to [961, Lemma 2] and displays the crucial fact that the weak∗ -limit of a wuC generalized series in a dual Banach space does not depend on the predual.  Lemma 5.7.54 Let X be a Banach space, and let α∈I xα be a wuC generalized series in X  . If Z is a Banach space such that Z  = X  , then   σ (X  , X)xα = σ (X  , Z)xα . α∈I

α∈I

 and z = σ (X  , Z)- α∈I xα . We claim that,  if y ∈ X  and C ≥ 0 are such that y + α∈F xα  ≤ C for every finite subset F of  I, then y + x − z + α∈F xα  ≤ C for every finite subset F of I.

Proof

Set x = σ (X  , X)-



 α∈I xα

240 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem  Indeed, propose that y ∈ X  and C ≥ 0 are such that y + α∈F xα  ≤ C for every finite subset F of I. Then for all finite subsets F, G, H of I we have that y +





xα +

α∈F

xα = y +

α∈H\(F∪G)



xα ≤ C.

α∈F∪(H\G)



If H runs through the finite subsets of I, then α∈H\(F∪G) xα σ (X  , X)-converges to  x − α∈F∪G xα , and by σ (X  , X)-lower semicontinuity of the norm of X  we have y +





xα + x −

α∈F

xα ≤ C for all finite subsets F, G of I,

α∈F∪G

that is to say y + x −



xα ≤ C for all finite subsets F, G of I.

α∈G\F

   Similarly, if G runs through the finite subsets of I, then α∈G\F xα σ (X , Z)    converges to z − α∈F xα , and by σ (X , Z)-lower semicontinuity of the norm of X  we have y + x − z +



xα ≤ C for every finite subset F of I.

α∈F

Now that the claim is proved, keeping in mind that, by (5.7.22), there exists a constant  C ≥ 0 such that  α∈F xα  ≤ C for every finite subset F of I, applying inductively  the claim, we get that n(x − z ) + α∈F xα  ≤ C for every n ∈ N and every finite subset F of I. In particular x − z  ≤ Cn for every n ∈ N. Thus x = z .  Theorem 5.7.55 A Banach space X satisfying property (X0 ) is the unique predual of its dual X  (in the sense of §5.1.23). Proof Suppose Z is a Banach space such that Z  = X  , and consider X and Z as subspaces of X  via the canonical embeddings. Take z ∈ Z ⊆ X  . Then, by Lemma  5.7.54, for every wuC generalized series xα in X  we have that ! ! !  "  "  "  xα = z σ (X  , Z)xα = z σ (Z  , Z)xα = z(xα ), z σ (X  , X)which entails that z ∈ X by property (X0 ). This shows Z ⊆ X. In fact, equality holds by the Hahn–Banach theorem because Z is 1-norming for X  .  The proofs of Lemma 5.7.54 and of Theorem 5.7.55 above are practically the same as those of Lemma 2 and Theorem 6(1) in [961]. This corresponds, at least with respect to uniqueness of predual, to the comment in the ‘Erratum’ of [961] asserting that ‘L’ensemble des d´emonstrations et des r´esultats de cet article s’adapte identiquement, aux notations pr`es, a` ce nouveau cadre’. But, as we have already said, as far as we know such an adaptation has not been done explicitly in the

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literature. Now Fact 5.7.53(iii) and Theorem 5.7.55 show that, as asserted in §5.7.50, Godefroy’s reference in [956] to the proof of Horn’s implication (i)⇒(iii) in Fact 5.7.49 is correct. We are very grateful to Pfitzner for having written the note [1048] specifically to be included in our work. Theorem 5.7.32 is only a part of a deep result of Horn [979, Theorem 4.2] asserting that, fixed any complete tripotent e in a JBW ∗ -triple X, the mapping assigning to each w∗ -closed ideal I of the JBW ∗ -algebra X1 (e) the w∗ -closed triple ideal of X generated by I becomes a bijection from the set of all w∗ -closed ideals of X1 (e) onto the set of all w∗ -closed triple ideals of X, with inverse J → J ∩ X1 (e). In [1045], Pfitzner applies the Godefroy–Talagrand theorem quoted above [961, Theorem 6] (see also [956, Theorem V.3(i) and Corollary VII.2]) to prove that every separable L-embedded Banach space over K is the unique predual of its dual. If separability could be removed in Pfitzner’s theorem, then Theorem 5.7.38 (that JBW ∗ -triples have unique predual) would become a Banach space consequence of Theorem 5.7.36 (that the predual of any JBW ∗ -triple is L-embedded). Note that our proof of Theorem 5.7.38 does not involve Theorem 5.7.36. Indeed, Theorem 5.7.38 could have been proved immediately after Corollary 5.7.28. In Godefroy’s survey paper [956] already quoted, the reader can find interesting results about JBW ∗ -triples, such as the following. Theorem 5.7.56 [956, Corollary VII.9] Let X be a JBW ∗ -triple, let Y be any dual complex Banach space, and let T : X → Y be a bounded linear operator. Then T is w∗ -continuous if and only if both T(BX ) and ker(T) are w∗ -closed. Although deep, Proposition 5.7.40 seems to be folklore in the theory. As pointed out by Dineen [213, p. 67], it follows by combining Kaup’s characterization of bounded homogeneous domains in complex Banach spaces [380, 381] (cf. Theorem 5.6.68) with Proposition 2.4 in Vigu´e’s paper [1116]. (Incidentally, we note that there is a misprint in [213] when [1116, Proposition 2.3] is invoked instead of the correct [1116, Proposition 2.4].) Our proof of Proposition 5.7.40 is inspired by that of Proposition 3 of [852] (which is nothing other than our Corollary 5.7.42). Results from Proposition 5.7.41 to Corollary 5.7.47 are due to Barton, Dang, and Horn [852]. A slightly more elementary proof of Proposition 5.7.41(ii) can be given, by applying the following result of Bouhya and Fern´andez. Proposition 5.7.57 [121, Proposition 13] Let E be a locally compact Hausdorff topological space, and let ||| · ||| be a vector space norm on C0C (E) satisfying ||| x ||| ≤ x and ||| xy∗ z ||| ≤ ||| x |||||| y |||||| z ||| for all x, y, z ∈ C0C (E). Then ||| · ||| =  ·  on C0C (E). Proof Let x and y be in C0C (E). Given ε > 0, there exists a compact subset K of E such that if t ∈ E \ K then |x(t)y(t)| ≤ ε. Since E is locally compact, we can take an open subset U and a compact subset C of E such that K ⊆ U ⊆ C. Let  E be the compactification of E. Then K and  E \ U are disjoint closed subsets of  E. By

242 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Urysohn’s lemma, there exists a continuous function v from  E to [0, 1] such that  v(t) = 1 for t ∈ K and v(E \ U) = 0. Then the restriction u of v to E is an element of C0C (E) satisfying u∗ = u and u = 1. Now we have |x(t)y(t) − x(t)u(t)y(t)| = |x(t)y(t)||1 − u(t)| = 0 for every t ∈ K, and |x(t)y(t) − x(t)u(t)y(t)| = |x(t)y(t)||1 − u(t)| ≤ ε for every t ∈ E \ K. Then xy − xuy ≤ ε. Hence ||| xy ||| ≤ ||| xuy ||| + ||| xy − xuy ||| ≤ ||| xuy ||| + xy − xuy ≤ ||| xuy ||| + ε ≤ ||| x |||||| u |||||| y ||| + ε ≤ ||| x |||||| y ||| + ε, which proves that ||| xy ||| ≤ ||| x |||||| y |||. Now ||| · ||| is an algebra norm in the C∗ -algebra C0C (E), and hence, by Proposition 1.2.51, ||| · ||| =  · .  Alternative proof of Proposition 5.7.41(ii) Let X and Y be JB∗ -triples, and let F : X → Y be an injective triple homomorphism. Let x be in X. To prove that F(x) = x, we consider the closed subtriple of X generated by x (which, by Theorem 4.2.9, is the JB∗ -triple underlying C0C (E) for some locally compact Hausdorff topological space E), we denote by Z the closure of F(C0C (E)) in Y (which is an abelian JB∗ -triple), and we see F as a triple homomorphism from C0C (E) to Z. Then, the mapping x → ||| x ||| := F(x) becomes a norm on C0C (E) satisfying ||| · ||| ≤  ·  on C0C (E) (by Proposition 5.7.41(i)) and ||| uv∗ w ||| ≤ ||| u |||||| v |||||| w ||| for all u, v, w ∈ C0C (E) (by Lemma 4.1.48(i)). Therefore, by Proposition 5.7.57 above, we have ||| · ||| =  ·  on C0C (E), which implies F(x) = x, as desired.  Most results on JB∗ -triples proved in this section remain true in the more general setting of real JB∗ -triples (cf. Definition 4.2.50). Indeed, involving the brackedfree version of Proposition 4.2.54, §4.2.60, and Theorem 5.7.18, Isidro, Kaup, and Rodr´ıguez proved the following generalization of Theorem 5.7.18. Proposition 5.7.58 [341, Lemma 4.2] The bidual X  of any real JB∗ -triple X becomes a real JB∗ -triple under a triple product which extends that of X and is separately w∗ -continuous. Let us define real JBW ∗ -triples as those real JB∗ -triples which are dual Banach spaces. Invoking Proposition 5.7.58, the following fact was proved also in [341]. Fact 5.7.59 [341, Definition 4.1 and Theorem 4.4] For a real JB∗ -triple X the following conditions are equivalent: (i) The JB∗ -triple complexification of X (cf. the bracket-free version of Definition 4.2.56) is a JBW ∗ -triple. (ii) X is a w∗ -closed real subtriple of some JBW ∗ -triple. (iii) X is a real JBW ∗ -triple whose triple product is separately w∗ -continuous.

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243

Later, Mart´ınez and Peralta [426] showed that, as conjectured in [341, p. 329], in condition (iii) of Fact 5.7.59 the separate w∗ -continuity of the triple product can be dropped. More precisely, they invoke Fact 4.2.14, Theorems 4.2.57 and 5.7.48, and Proposition 5.7.58, to prove the following generalization of Theorems 5.7.20 and 5.7.38. Theorem 5.7.60 Let X be a real JBW ∗ -triple. Then we have: (i) The predual of X is unique. (ii) The triple product of X is separately w∗ -continuous. Contrary to what happens with the complex forerunner of the preceding theorem (namely Theorems 5.7.20 and 5.7.38), we do not know any proof of Theorem 5.7.60 avoiding the application of Godefroy’s Theorem 5.7.48. With Theorem 5.7.60(ii) in mind, Lemma 4.3 of [341] (1995) becomes the generalization of Horn’s Theorem 5.7.32 to the setting of real JBW ∗ -triples. Much later the following generalization of Theorem 5.7.36 is proved in [66] (2004). Proposition 5.7.61 The predual of any real JBW ∗ -triple is L-embedded. At the end of Subsection 5.1.1 we proved a few results on real non-commutative JBW ∗ -algebras. Now we can realize that, in fact, most results on (complex) noncommutative JBW ∗ -algebras, established in that subsection, remain true in the more general setting of their real counterparts. Indeed, we are provided with the following. Proposition 5.7.62 Let A be a nonzero real non-commutative JB∗ -algebra. Then the following conditions are equivalent: (i) The non-commutative JB∗ -complexification of A (cf. the bracketed version of Definition 4.2.56) is a non-commutative JBW ∗ -algebra. (ii) A is a w∗ -closed real ∗-subalgebra of some non-commutative JBW ∗ -algebra. (iii) A is a real non-commutative JBW ∗ -algebra. Moreover, if the above conditions are fulfilled, then we have: (iv) (v) (vi) (vii)

The predual of A is unique, and is L-embedded. The involution of A is w∗ -continuous. Jordan derivations (so, in particular, derivations) of A are w∗ -continuous. Bijective Jordan homomorphisms (so, in particular, bijective algebra homomorphisms) from A to any real non-commutative JBW ∗ -algebra are w∗ -continuous. (viii) The product of A is separately w∗ -continuous. Proof Let AC stand for the non-commutative JB∗ -complexification of A. (i)⇒(ii) Since the canonical involution of AC (say ) is a conjugate-linear bijective algebra ∗-homomorphism, the assumption (i) and a conjugate-linear version

244 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem of Corollary 5.1.30(ii) imply that  is w∗ -continuous. Therefore A = H(AC , ) is a w∗ -closed real ∗-subalgebra of the non-commutative JBW ∗ -algebra AC . (ii)⇒(iii) By §5.1.9. Now recall that A becomes a real JB∗ -triple, and that AC becomes a JB∗ -triple under the triple product {xyz} = Ux,z (y∗ ) (see Example 4.2.51(b) and Theorem 4.1.45, respectively). (iii)⇒(i) By the assumption (iii), the real JB∗ -triple underlying A is a JBW ∗ triple. Therefore, by Theorem 5.7.60(ii) and the implication (iii)⇒(i) in Fact 5.7.59, the JB∗ -triple complexification of the real JB∗ -triple underlying A is a JBW ∗ -triple. Since the JB∗ -triple complexification of the real JB∗ -triple underlying A coincides with the JB∗ -triple underlying AC (by the uniqueness of the norm in Fact 4.2.55), condition (i) follows. Now propose that conditions (i) to (iii) are fulfilled. Then assertion (iv) follows from condition (iii), Theorem 5.7.60(i), and Proposition 5.7.61. Assertion (v) follows from condition (iii), Fact 5.1.51, the equality x∗ = {1x1}, and Theorem 5.7.60(ii). Assertion (vi) follows by keeping in mind condition (i), the fact that Jordan derivations of A extend by complex-linearity to Jordan derivations of AC , and Corollary 5.1.43(i). Assertion (vii) is proved analogously by replacing ‘Jordan derivations’ with ‘bijective Jordan homomorphisms’ and Corollary 5.1.43(i) with Corollary 5.1.43(ii). The separate w∗ -continuity of the product of Asym follows from condition (iii), Theorem 5.7.60(ii), and the equality x • y = Ux,y (1). Finally, to conclude the proof of assertion (viii), note that the separate w∗ -continuity of the mapping (x, y) → [x, y] follows from assertion (vi) already proved and the fact that for each x ∈ A the mapping y → [x, y] is a Jordan derivation of A (cf. Lemma 2.4.15).  As an immediate consequence, we have the following. Corollary 5.7.63 Let A be a nonzero real alternative C∗ -algebra. Then the following conditions are equivalent: (i) The alternative C∗ -complexification of A (see again the bracketed version of Definition 4.2.56) is an alternative W ∗ -algebra. (ii) A is a w∗ -closed real ∗-subalgebra of some alternative W ∗ -algebra. (iii) A is a real alternative W ∗ -algebra. Moreover, if the above conditions are fulfilled, then we have: (iv) (v) (vi) (vii)

The predual of A is unique, and is L-embedded. The involution of A is w∗ -continuous. Jordan derivations (so, in particular, derivations) of A are w∗ -continuous. Bijective Jordan homomorphisms (so, in particular, bijective algebra homomorphisms) from A to any real alternative W ∗ -algebra are w∗ -continuous. (viii) The product of A is separately w∗ -continuous. Proposition 5.7.62, and even Corollary 5.7.63, could be new. The associative forerunner of Corollary 5.7.63 is essentially known in [343].

5.8 Operators into the predual of a JBW ∗ -triple

245

5.8 Operators into the predual of a JBW ∗ -triple Introduction In Subsection 5.8.1 we introduce Pełczy´nski’s property (V ∗ ) for Banach spaces, and prove the Godefroy–Iochum result [957] asserting that, if the dual X  of a Banach space X has property (V ∗ ), then every bounded linear operator from X to X  is weakly compact (see Corollary 5.8.19). The core of the proof of this result consists of a rather technical characterization of property (V ∗ ) (see Proposition 5.8.14), announced by Godefroy and Saab [958], and whose proof has been communicated to us by Pfitzner [1047]. As a main result of Subsection 5.8.2, we prove Pfitzner’s theorem [1044] asserting that L-embedded Banach spaces enjoy property (V ∗ ) (see Theorem 5.8.27). In Subsection 5.8.3 we combine Proposition 5.7.10 (that the dual of a JB∗ -triple is the predual of a JBW ∗ -triple) and Theorem 5.7.36 (that preduals of JBW ∗ -triples are L-embedded) with Corollary 5.8.19 and Theorem 5.8.27 quoted above to straightforwardly re-encounter the Chu–Iochum–Loupias theorem [172] asserting that bounded linear operators from a JB∗ -triple to its dual are weakly compact (see Corollary 5.8.33). Through a folklore discussion relating weak compactness of operators with Arens regularity of bilinear mappings, the Chu–Iochum–Loupias theorem can be equivalently reformulated by saying that all continuous products on a JB∗ -triple are Arens regular (see Fact 5.8.39). This result will become one of the crucial ingredients in the proof of Theorem 5.9.9. 5.8.1 On Pełczynski’s ´ property (V ∗ )  §5.8.1 Let X be a Banach space over K. A series xn in X is called weakly uncon ditionally Cauchy (in short, wuC) if, given any permutation π of N, ( nk=1 xπ(k) ) is ∞  a weakly Cauchy sequence; equivalently if n=1 |x (xn )| < ∞ for every x ∈ X  , see  [1151, p. 44]. Trivially, if xn is a wuC-series in X, and if (tn ) is an element in ∞ ,   then tn xn is a wuC-series in X. In particular, if xn is a wuC-series in X, then so   is xnk for any subsequence (xnk ). It is also clear that, if xn is a wuC-series in X, then the sequence (xn ) converges weakly to 0. Proposition 5.8.2 Let X be a Banach space over K, and let (xn ) be a sequence in X. Then the following conditions are equivalent:  (i) xn is wuC. (ii) There is a constant C > 0 such that for any (tn ) ∈ ∞ sup

n 

tk xk ≤ C sup |tn |.

n∈N k=1

n∈N

 (iii) For any (tn ) ∈ c0 , the series tn xn converges. (iv) There is a constant C > 0 such that for any finite subset A of N and any λn ∈ SK (n ∈ A) we have  λn xn ≤ C. n∈A

246 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof (i)⇒(ii) Define T : X  → 1 by T(x ) = (x (xn )). T is a well-defined linear mapping with a closed graph; therefore, T is bounded. From this we see that for any (tn ) ∈ B ∞ and any x ∈ BX  ,    n   n           tk xk  =  tk x (xk ) = |(t1 , . . . , tn , 0, 0, . . .)(T(x ))| ≤ T, x     k=1

k=1

and (ii) follows. (ii)⇒(iii) Let (tn ) ∈ c0 . Then keeping m < n and letting both go off to ∞, we have n 

tk xk ≤ C sup |tk | → 0 m≤k≤n

k=m

from which (iii) follows easily.  (iii)⇒(iv) If (iii) holds, then the sequence Tn : (tk ) → nk=1 tk xk of bounded linear operators from c0 to X converges pointwise to a mapping T : c0 → X defined  by T(tn ) = ∞ n=1 tn xn . Therefore, by the Banach–Steinhaus closure theorem (cf. Proposition 1.4.16), T is a bounded linear operator. The values of T on Bc0 are  bounded. In particular, vectors of the form n∈A λn xn , where A ranges over the finite subsets of N and λn ∈ SK for each n in such a A, are among the values of T on Bc0 and that is assertion (iv). (iv)⇒(i) If (iv) is in effect, then for any x ∈ BX  we have                λn x (xn ) = x λn xn  ≤ λn xn ≤ C      n∈A

n∈A

n∈A

for any finite subset A of N and any choice of λn ∈ SK . By choosing λn ∈ SK such that   λn x (xn ) = |x (xn )| we get n∈A |x (xn )| ≤ C. That n |x (xn )| < ∞ follows directly from this, and along with it we get assertion (i).   Corollary 5.8.3 Let X and Y be Banach spaces over K, and let xn be a wuC-series in X. We have  (i) If X is isomorphically embedded in Y, then xn is a wuC-series in Y.  (ii) If F : Y → X  is a bounded linear operator, then F  (xn ) is a wuC-series in Y  . Proof Assertion (i) follows immediately from the equivalence (i)⇔(ii) in Proposition 5.8.2. Now let F : Y → X  be a bounded linear operator. Then for each y ∈ Y  and n ∈ N  we have y (F  (xn )) = F  (y )(xn ). Since F  (y ) ∈ X  and, by assertion (i), xn    (F  (x ))| < ∞. Thus  (x ) is a is a wuC-series in X  , we realize that ∞ |y F n n n=1 wuC-series in Y  .  Lemma 5.8.4 Let X be a Banach space over K, and let (xn ) be a weak Cauchy sequence in X. Then   xi wuC-series in X  . (5.8.1) lim sup |xi (xn )| = 0 for every i→∞ n∈N

5.8 Operators into the predual of a JBW ∗ -triple

247



Proof Let xi be a wuC-series in X  . Then T : x → (xi (x)) defines a linear operator from X to 1 , which is bounded (by the closed graph theorem). Therefore (T(xn )) is a weak Cauchy sequence in 1 , and hence, by the Schur property of 1 (see for example [729, Theorem 5.36]), (T(xn )) is a norm-convergent sequence in 1 . That is to say there is l = (l(i)) ∈ 1 such that, setting an,i = |xi (xn ) − l(i)|, we have ∞ 

n

an,i = T(xn ) − l → 0.

(5.8.2)

i=1

Now for each n ∈ N we have that lim an,i = 0 and sup an,i ≤

i→∞

i∈N

∞ 

an,i .

(5.8.3)

i=1

Fix an arbitrary ε > 0. It follows from (5.8.2) and the second assertion in (5.8.3) that there exists n0 ∈ N such that an,i ≤ ε for all n ≥ n0 and i ∈ N. Moreover, for each k = 1, . . . , n0 − 1, it follows from the first assertion in (5.8.3) that there exists ik ∈ N such that ak,i ≤ ε for every i ≥ ik . Setting i0 := max{i1 , . . . , in0 −1 }, it follows that an,i ≤ ε for all n ∈ N and i ≥ i0 , and as a result supn∈N an,i ≤ ε for every i ≥ i0 . Hence lim sup an,i = 0.

(5.8.4)

i→∞ n∈N

Since we have for all n, i ∈ N that |xi (xn )| ≤ an,i + |l(i)|, and consequently we have for each i ∈ N that sup |xi (xn )| ≤ sup an,i + |l(i)|,

n∈N

n∈N

i

(5.8.1) follows from (5.8.4) and the fact that |l(i)| → 0.



ˇ Recall that, via the Eberlein–Smulyan theorem, a subset K of a Banach space X is weakly relatively compact if and only if every sequence in K has a subsequence which is weakly convergent in X. Corollary 5.8.5 Let X be a Banach space over K, and let K be a weakly relatively compact subset of X. Then   lim sup |xi (x)| = 0 for every xi wuC-series in X  . (5.8.5) i→∞ x∈K

Proof We argue by contradiction, so we assume the existence of a wuC-series in X  such that



xi

lim sup |xi (x)| = 0.

i→∞ x∈K

Then there are ε > 0 and a subsequence (xin ) of (xi ) such that supx∈K |xin (x)| > ε    for every n ∈ N. Since n xin is wuC-series in X  , we may replace xi with n xin , and suppose without loss of generality that supx∈K |xi (x)| > ε for every i ∈ N. Therefore for each i ∈ N we can choose xi ∈ K satisfying |xi (xi )| > ε. Now,

248 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem given any subsequence (xik ) of (xi ), we have |xik (xik )| > ε for every k ∈ N, hence supj∈N |xik (xij )| > ε, and so lim sup |xik (xij )| = 0.

k→∞ j∈N

 Since k xik is a wuC-series in X  , it follows from Lemma 5.8.4 that the sequence (xik ) is not weakly Cauchy, and hence is not weakly convergent. Therefore the sequence (xi ) lies in K and has no weakly convergent subsequence, the desired contradiction.  Remark 5.8.6 (a) Condition (5.8.5) is not sufficient for a subset K of a Banach space X to be weakly relatively compact. Indeed, take X = c0 and K = BX . Then K is not weakly relatively compact. However, by the Schur property of 1 , for every wuC series xi in X  , the sequence (xi ) norm-converges to 0, and hence sup |xi (x)| = xi  → 0. x∈K

(b) If a subset K of a Banach space X satisfies (5.8.5), then it is bounded. For, if K is not bounded, then, for each i ∈ N, one can choose xi ∈ SX  and xi ∈ K such that 1  xi (xi ) > 2i , so that x is a wuC-series in X  and (5.8.5) cannot hold. 2i i Corollary 5.8.5 and Remark 5.8.6(a) suggest the following definition. Definition 5.8.7 We say that a Banach space X over K has property (V ∗ ) if its weakly relatively compact subsets are precisely those subsets K ⊆ X satisfying (5.8.5). Since the weakly relatively compact subsets of a reflexive Banach space are precisely the bounded subsets, Corollary 5.8.5 and Remark 5.8.6(b) yield the following. Fact 5.8.8 Reflexive Banach spaces over K have property (V ∗ ). As another application of Lemma 5.8.4, we obtain the following result. Corollary 5.8.9 A Banach space X over K having property (V ∗ ) is weakly sequentially complete. Proof

Let (xn ) be a weakly Cauchy sequence in X. By Lemma 5.8.4,   xi wuC-series in X  . lim sup |xi (xn )| = 0 for every i→∞ n∈N

Since X has the property (V ∗ ), we can assert that the set K := {xn : n ∈ N} is weakly relatively compact. Therefore, there exists a subsequence (xnk ) of (xn ) which is weakly convergent. Since (xn ) is weakly Cauchy and (xnk ) is weakly convergent, we conclude that (xn ) is weakly convergent.  Let X be a Banach space over K. We say that a sequence (xn ) in X is an

1 -sequence if it is bounded and satisfies

5.8 Operators into the predual of a JBW ∗ -triple n  k=1

αk xk ≥ r

n 

249

|αk | for some r > 0 and all n ∈ N and α1 , . . . , αn ∈ K,

k=1

 equivalently, if it is bounded and the mapping l → ∞ i=1 l(i)xi from 1 to X becomes an isomorphism onto its range. Since any subsequence of an 1 -sequence is an

1 -sequence, and the canonical basis of 1 is not weakly Cauchy (indeed, apply to it the element ((−1)i ) ∈ ∞ ), it follows that an 1 -sequence in X has no weakly Cauchy subsequence. We recall that, according to the Rosenthal 1 -theorem, every bounded sequence in an infinite-dimensional Banach space contains a subsequence which is either weakly Cauchy or an 1 -sequence (see for example [729, Theorem 5.37]). As a consequence we have the following result. Lemma 5.8.10 Let X be a weakly sequentially complete Banach space over K, and let K be a bounded subset of X. Then the following conditions are equivalent: (i) K is not weakly relatively compact. (ii) K contains an 1 -sequence. Proof (i)⇒(ii) If K is not weakly relatively compact, then K contains a sequence (xn ) which has no weakly convergent subsequence. Since X is weakly sequentially complete, it follows that (xn ) has no weakly Cauchy subsequence. Now, by Rosenthal’s 1 -theorem, (xn ) contains an 1 -subsequence, and hence K contains an

1 -sequence. (ii)⇒(i) If (xn ) is an 1 -sequence contained in K, then the set {xn : n ∈ N}, and hence K, is not weakly relatively compact.  Corollary 5.8.11 Let X be a weakly sequentially complete Banach space over K, and let Y be a non-reflexive closed subspace of X. Then Y contains an isomorphic copy of 1 . Proof Since Y is weakly sequentially complete, and BY is not a weakly relatively compact subset of Y, the result follows from Lemma 5.8.10.  Proposition 5.8.12 Let X be a Banach space over K. Then the following conditions are equivalent: (i) X has property (V ∗ ). (ii) X is weakly sequentially complete and contains no 1 -sequence (xn ) satisfying (5.8.1). Proof (i)⇒(ii) That property (V ∗ ) implies weak sequential completeness was already noted in Corollary 5.8.9. Let (xn ) be an 1 -sequence in X. Then the set {xn : n ∈ N} is not weakly relatively compact. Since X has (V ∗ ), it follows that (xn ) does not satisfy (5.8.1). (ii)⇒(i) Let K ⊆ X be not weakly relatively compact. We may suppose that K is bounded by Remark 5.8.6(b). Since X is weakly sequentially complete, it follows

250 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem from Lemma 5.8.10 that K contains an 1 -sequence (xn ). Then, since (xn ) does not satisfy (5.8.1), it follows that K does not satisfy (5.8.5). Thus X has (V ∗ ).  w

Lemma 5.8.13 Let Z be a Banach space over K, let zi → 0 in Z, and let (zn ) be bounded in Z  . Then, given ε > 0, there is a strictly increasing sequence (nj ) in N such that ε sup |znk (znj )| ≤ j for every j ∈ N. (5.8.6) 2 k=j Proof Since only the evaluations of the zn on the zi are concerned we may suppose without loss of generality that Z is the closed linear hull of the zi , and hence that Z is separable. Then, by passing to a subsequence if necessary, we may also suppose that the weak∗ -limit z = w∗ - lim zn exists by weak∗ -metrizability of the unit ball of Z  . Let ε > 0. It is enough to construct two sequences (nj ) and (mj ) in N by induction over j ∈ N such that nj+1 > nj , nj+1 ≥ mj for every j ∈ N; ε |znk (znj )| < j for all j, k ∈ N with k < j; 2 ε  |zn (znj )| < j for every n ≥ mj . 2

(5.8.7) (5.8.8) (5.8.9)

For j = 1 we choose n1 such that |z (zn1 )| < 2ε whence the existence of m1 such that (5.8.9) holds for j = 1. For the induction step j → j + 1 we choose nj+1 such that ε (5.8.7) holds for j and (5.8.8) holds for j + 1, and such that |z (znj+1 )| < 2j+1 holds. From the latter we obtain mj+1 fulfilling (5.8.9) for j + 1. Now, to obtain (5.8.6) note that if k < j then condition (5.8.8) gives directly that |znk (znj )| < 2εj , whereas if k > j then (5.8.7) yields nk ≥ nj+1 ≥ mj , and hence (5.8.9) for n = nk gives |znk (znj )| < 2εj .  Proposition 5.8.14 For a Banach space X over K the following conditions are equivalent. (i) X has (V ∗ ). (ii) Each bounded non weakly relatively compact set K ⊆ X contains an 1 -sequence such that the 1 -copy spanned by this sequence is complemented in X. (iii) For each Banach space Y over K, if a bounded linear operator T : Y → X is not weakly compact then there is a subspace V ⊆ Y isomorphic to 1 such that the restriction T|V is an isomorphism and T(V) is complemented in X. Proof (i)⇒(ii) Let K ⊆ X be bounded but not weakly relatively compact. By the assumption (i), Lemma 5.8.10, and Proposition 5.8.12, K contains an 1 -sequence (xn ) which does not satisfy (5.8.1). Therefore, by passing to a subsequence of (xn ) if  necessary, there exists a wuC-series i xi in X  such that α := inf |xn (xn )| > 0. n∈N

(5.8.10)

5.8 Operators into the predual of a JBW ∗ -triple 251 n n Let r > 0 be such that  m=1 αm xm  ≥ r m=1 |αm | for all n ∈ N and α1 , . . . , αn ∈ K. We apply Lemma 5.8.13 for ε = 1 with Z = X  , zi = xi and zn = xn . Therefore, there is a strictly increasing sequence (nj ) in N such that sup |xn j (xnk )| ≤ k=j

1 for every j ∈ N. 2j

(5.8.11)

For each j, let γj denote the linear functional on the linear hull of (xnk ) determined by $ 0 if k = j xnk → xn j (xnk ) if k = j. Then, since   m    m m     1  1   αk xnk  ≤ |αk | sup |γj (xnk )| ≤ αk xnk j γj   r 2 1≤k≤m k=1

k=1

k=1

for all m ∈ N and α1 , . . . , αm ∈ K (by (5.8.11)), we realize that γj is bounded with γj  ≤ r21 j . Now define βj ∈ X  as a Hahn–Banach extension of γj , and set yj :=

1 (x − βj ). xn j (xnj ) nj

Then yj (xnk ) =



if k = j if k = j.

1 0



yj is wuC because ⎞ ⎛ ∞ ∞ ∞    1 |x (yj )| ≤ ⎝ |x (xn j )| + x  βj ⎠ < ∞ for every x ∈ X  . α

The series

j=1



j=1

j=1

yj

is wuC and the sequence (xnk ) is bounded, the series Since the series is absolutely convergent for every x ∈ X, so that, by construction, π :x→

∞ 



 k yk (x)xnk

yk (x)xnk

k=1

defines the desired projection from X onto the 1 -copy spanned by (xnk ). (ii)⇒(iii) Suppose that Y is a Banach space over K and that T : Y → X is a bounded linear operator not weakly compact. Then Y contains a bounded sequence (yn ) such that the set K := {xn : n ∈ N} (where xn = T(yn )) is not weakly relatively compact. By (ii), we (may pass to an appropriate subsequence and) suppose that (xn ) is an ·

1 -sequence such that U := lin (xn ) is complemented in X. But now (yn ) is an

1 -sequence because (xn ) is so and   m m m   1 1  αn yn ≥ αn yn = αn xn T T T n=1

n=1

n=1

252 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem for all m ∈ N and α1 , . . . , αm ∈ K. Therefore T acts as an isomorphism on · V = lin (yn ). (iii)⇒(i) Let K ⊆ X be not weakly relatively compact. We may suppose by Remark 5.8.6(b) that K is bounded. Then K contains a (bounded) sequence (xn ) which has no weakly convergent subsequence. Hence the operator T : 1 → X given by en → xn , where (en ) is the canonical basis of 1 , is not weakly compact. By (iii), Y = 1 contains an 1 -copy V with 1 -basis (vn ) such that T acts as an isomorphism on V and U := T(V) (with basis (un ), where un = T(vn )) is complemented in X by a projection π : X → X. Define an isomorphism S : U → 1 by un → en , and consider the bounded linear operator S1 := S ◦ π from X to 1 . Set xi = S1 (ei ), where (en ) is  the canonical basis of c0 (⊆ ∞ ). Since clearly ei is a wuC-series in c0 , it follows   from Corollary 5.8.3(ii) that xi is a wuC-series in X  . Set yi = T  (xi ). Then yi (vi ) = T  (xi )(vi ) = xi (T(vi )) = xi (ui ) = S1 (ei )(ui ) = ei (S1 (ui )) = ei (S(ui )) = ei (ei ) = 1. Write vi =

∞

 in 1 . We have vi  = ∞ n=1 |λn,i | and ∞        λn,i yi (en ) ≤ vi  sup |yi (en )|. 1 = |yi (vi )| =    n∈N

n=1 λn,i en

n=1

Hence for each i ∈ N there is ni such that

1 2

< vi |yi (eni )|, and so

|xi (xni )| = |xi (T(eni ))| = |T  (xi )(eni )| = |yi (eni )| > and (5.8.5) cannot hold. This concludes the proof.

1 2vi  

Remark 5.8.15 The above proof shows that in condition (iii) it is enough to take Y = 1 . Corollary 5.8.16 Let X be a non-reflexive weakly sequentially complete Banach space over K. Then the following conditions are equivalent: (i) X has property (V ∗ ). (ii) X contains 1 -sequences, and each 1 -sequence in X contains a subsequence spanning a complemented subspace in X isomorphic to 1 . Proof (i)⇒(ii) By Corollary 5.8.11, X contains 1 -sequences. Moreover, by the implication (i)⇒(ii) in Proposition 5.8.14, each 1 -sequence in X contains a subsequence spanning a complemented 1 -copy in X. (ii)⇒(i) Let K be a bounded non weakly relatively compact subset of X. By (ii) and Lemma 5.8.10, K contains an 1 -sequence spanning a complemented 1 -copy in X. Now, by the implication (ii)⇒(i) in Proposition 5.8.14, we realize that X has  property (V ∗ ).

5.8 Operators into the predual of a JBW ∗ -triple

253

The fact that a Banach space contains a complemented copy of 1 if and only if its dual contains a copy of c0 (see, for example, [1151, Theorem V.10]) allows us to state the following consequence. Corollary 5.8.17 If X is a non-reflexive Banach space over K with property (V ∗ ), then X  contains copies of c0 . Proposition 5.8.18 Let X and Y be Banach spaces over K such that X has property (V ∗ ) and Y  is weakly sequentially complete. Then every bounded linear operator from Y to X is weakly compact. Proof If a bounded linear operator T : Y → X is not weakly compact then, by the implication (i)⇒(iii) in Proposition 5.8.14, there is an 1 -copy V ⊆ Y such that U = T(V) is complemented in X by a projection π. Then, with j = (T|V )−1 the isomorphism from U onto V, we get a surjection S = j◦π ◦T from Y onto the 1 -copy V whose adjoint S is an isomorphic embedding of ∞ into Y  . But, by assumption, Y  is weakly sequentially complete, and therefore it cannot contain ∞ .  Combining Corollary 5.8.9 and Proposition 5.8.18, we get the following. Corollary 5.8.19 Let X and Y be Banach spaces over K such that both X and Y  have property (V ∗ ). Then every bounded linear operator from Y to X is weakly compact. 5.8.2 L-embedded spaces have property (V ∗ ) Lemma 5.8.20 Let X be an L-embedded Banach space over K, let P be the L-projection on X  whose range is X, let (xn ) be a sequence in X, let (un ) be a sequence in Xs := ker(P), and let x and u be in X  . Suppose that there exist K, L ≥ 0 such that for every η > 0, every z , t ∈ X  , and every n0 ∈ N there is n ≥ n0 with |z (xn ) − x (z )| ≤ Kz  + η and |un (t ) − u (t )| ≤ Lt  + η.

(5.8.12)

Furthermore, write x = x0 + xs and u = u0 + us , with x0 , u0 ∈ X and xs , us ∈ Xs , and let x , u ∈ SX  be such that x (x0 ) = 0 and us (u ) = 0. Finally, let 0 < ε < 1 and let (εn )  be a sequence of positive numbers decreasing to zero such that ∞ n=1 (1+εn ) < 1+ε. Then there is a strictly increasing sequence (nk ) in N and there are two wuC-series    xk and uk in X  satisfying for every k ∈ N the conditions unk (xk ) = 0 and uk (xnk ) = 0,

(5.8.13)

and |xk (xnk ) − xs (x )| ≤ K(1 + ε) + εk and |unk (uk ) − u (u0 )| ≤ L(1 + ε) + εk . (5.8.14) Proof We shall construct by induction over k ∈ N, using the local reflexivity principle (cf. Proposition 5.7.8), a strictly increasing sequence (nk ) of indices and four

254 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem sequences (xk ), (yk ), (uk ), and (vk ) in X  such that the following conditions hold for every k ∈ N: ⎫ ⎞ ⎛ k k ⎪   ⎪ ⎪ ⎪ αj xj ≤ ⎝ (1 + εj )⎠ max |αj | and α0 yk + ⎪ ⎪ ⎪ 0≤j≤k ⎬ j=1 j=1 (5.8.15) ⎞ ⎛ ⎪ k k ⎪   ⎪ ⎪ αj uj ≤ ⎝ (1 + εj )⎠ max |αj | for all α0 , . . . , αk ∈ K, ⎪ α0 vk + ⎪ ⎪ ⎭ 0≤j≤k j=1

j=1

unk (xk ) = 0

and uk (xnk ) = 0,

(5.8.16)

yk (x0 ) = 0

and us (vk ) = 0,

(5.8.17)

xs (yk ) = xs (x )

and



k−1 

|xk (xnk ) − xs (x )| ≤ K ⎝

⎞ (1 + εj )⎠ + εk

j=1

⎛ |unk (uk ) − u (u0 )| ≤ L ⎝

vk (u0 ) = u (u0 ),

k−1 

⎞ (1 + εj )⎠ + εk .

j=1

(5.8.18)

⎫ ⎪ ⎪ ⎪ and ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(5.8.19)

For the following it is useful to recall some properties of P : The restriction of P to X  is an isometric isomorphism from X  onto Xs◦ with (P (y ))|X = y for every y ∈ X  , Q := IX  − P is a contractive projection, and X  = Xs◦ ⊕∞ X ◦ (where X ◦ is the polar of X in X  ). We use (5.8.12) with η = ε1 , z = x , t = u , and n0 = 1 in order to find n1 ∈ N such that |x (xn1 ) − x (x )| ≤ K + ε1 and |un1 (u ) − u (u )| ≤ L + ε1 .

(5.8.20)

Consider the finite-dimensional subspaces M1 = lin({x , u , P (x ), P (u )}) ⊆ X  and N1 = lin({x0 , xs , u0 , us , xn1 , un1 }) ⊆ X  . By the local reflexivity principle, there is a bounded linear operator R1 : M → X  with R1  ≤ 1 + ε1 satisfying f  (R1 (e )) = e ( f  ) for all e ∈ M1 and f  ∈ N1 .

(5.8.21)

Note that clearly Q (x ) and Q (u ) lie in M1 . We define x1 = R1 (P (x )), y1 = R1 (Q (x )), u1 = R1 (Q (u )) and v1 = R1 (P (u )). In light of the analogy between (K, x0 , xs , x , xn1 , x1 , y1 ) and (L, us , u0 , u , un1 , u1 , v1 ), in order to prove the conditions (5.8.15)-(5.8.19) for k = 1 it is enough to verify the first assertion in each one of these conditions. For each α0 , α1 ∈ K, we have

5.8 Operators into the predual of a JBW ∗ -triple

255

α0 y1 + α1 x1  = R1 (Q (α0 x ) + P (α1 x )) ≤ (1 + ε1 )Q (α0 x ) + P (α1 x ) = (1 + ε1 ) max{Q (α0 x ), P (α1 x )} ≤ (1 + ε1 ) max{α0 x , α1 x } = (1 + ε1 ) max{|α0 |, |α1 |}, and hence (5.8.15) holds for k = 1. The conditions (5.8.16), (5.8.17), and (5.8.18) for k = 1 are easy to verify because P(un1 ) = 0, Q(x0 ) = 0, and Q(xs ) = xs and thus, by (5.8.21), we have un1 (x1 ) = un1 (R1 (P (x ))) = P (x )(un1 ) = P(un1 )(x ) = 0, y1 (x0 ) = R1 (Q (x ))(x0 ) = x0 (R1 (Q (x ))) = Q (x )(x0 ) = Q(x0 )(x ) = 0, and xs (y1 ) = xs (R1 (Q (x ))) = Q (x )(xs ) = Q(xs )(x ) = xs (x ). Now, note that x1 (xn1 ) = R1 (P (x ))(xn1 ) = P (x )(xn1 ) = x (P(xn1 )) = x (xn1 ), and hence (5.8.19) for k = 1 follows from (5.8.20) and the fact that x (x0 ) = 0. For the induction step k → k + 1 suppose now that n1 < · · · < nk and x1 , . . . , xk ,  y1 , . . . , yk , u1 , . . . , uk , v1 , . . . , vk have been constructed and satisfy conditions (5.8.15)– (5.8.19). By (5.8.12) with η = εk+1 , z = yk , t = vk , and n0 = nk + 1, there is an index nk+1 > nk such that |yk (xnk+1 ) − x (yk )| ≤ Kyk  + εk+1 and |vk (unk+1 ) − u (vk )| ≤ Lvk  + εk+1 . (5.8.22) Set A = {x1 , . . . , xk }, B = {u1 , . . . , uk }, and C = {yk , vk }, and consider the finitedimensional subspaces Mk+1 = lin(A ∪ B ∪ C ∪ P (A ∪ B ∪ C)) ⊆ X  and Nk+1 = lin({x0 , xs , u0 , us , xnk+1 , unk+1 }) ⊆ X  . By the local reflexivity principle, there is a bounded linear operator Rk+1 : Mk+1 → X  with Rk+1  ≤ 1 + εk+1 such that f  (Rk+1 (e )) = e ( f  ) for all e ∈ Mk+1 and f  ∈ N,

(5.8.23)

and (Rk+1 )|Mk+1 ∩X  = IMk+1 ∩X  . Clearly Q (xj ), Q (yk ), Q (uj ), Q (vk ) ∈ Mk+1 for 0 ≤ j ≤ k. We define  = Rk+1 (P (yk )), yk+1 = Rk+1 (Q (yk )), xk+1

uk+1 = Rk+1 (Q (vk )), and vk+1 = Rk+1 (P (vk )).

(5.8.24)

256 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem To prove the conditions (5.8.15)–(5.8.19) for k + 1, it is again enough to verify the first assertion in these conditions because of the analogy between  (K, x0 , xs , x , xnk+1 , xk+1 , yk+1 ) and (L, us , u0 , u , unk+1 , uk+1 , vk+1 ).

For all α0 , . . . , αk+1 ∈ K, by (5.8.24), we have ⎛ ⎛ ⎞ ⎛ ⎞⎞ k+1 k k    α0 yk+1 + αj xj = Rk+1 ⎝Q ⎝α0 yk + αj xj ⎠ + P ⎝αk+1 yk + αj xj ⎠⎠ , j=1

j=1

j=1

and hence α0 yk+1 +

k+1  j=1

αj xj ⎛

≤ (1 + εk+1 ) Q ⎝α0 yk +

= (1 + εk+1 ) max

≤ (1 + εk+1 ) max ⎛

k+1 

≤⎝



⎩ ⎧ ⎨ ⎩







αj xj ⎠ + P ⎝αk+1 yk +

j=1

Q ⎝α0 yk +

k 



k 



⎞ αj xj ⎠

j=1

αj xj ⎠ , P ⎝αk+1 yk +

j=1

α0 yk +

k 

αj xj , αk+1 yk +

j=1



k  j=1

αj xj

⎫ ⎬

k  j=1

⎞ ⎫ ⎬ αj xj ⎠ ⎭



  (1 + εj )⎠ max max |αj |, max |αj | 0≤j≤k

j=1

k+1 

=⎝

⎧ ⎨

k 

1≤j≤k+1

⎞ (1 + εj )⎠ max |αj |,

j=1

0≤j≤k+1

where the last inequality comes from (5.8.15). Thus (5.8.15) ( for k + 1 instead of k) is proved. The conditions (5.8.16), (5.8.17), and (5.8.18) ( for k + 1 instead of k) are easy to verify because P(unk+1 ) = 0, Q(x0 ) = 0 and Q(xs ) = xs thus, by (5.8.23)  unk+1 (xk+1 ) = unk+1 (Rk+1 (P (yk ))) = P (yk )(unk+1 ) = P(unk+1 )(yk ) = 0,

yk+1 (x0 ) = Rk+1 (Q (yk ))(x0 ) = x0 (Rk+1 (Q (yk ))) = Q (yk )(x0 ) = Q(x0 )(yk ) = 0, and xs (yk+1 ) = xs (Rk+1 (Q (yk ))) = Q (yk )(xs ) = Q(xs )(yk ) = xs (yk ) = xs (x ). Finally, we have  xk+1 (xnk+1 ) = Rk+1 (P (yk ))(xnk+1 ) = xnk+1 (Rk+1 (P (yk )))

= P (yk )(xnk+1 ) = yk (P(xnk+1 )) = yk (xnk+1 ),

5.8 Operators into the predual of a JBW ∗ -triple

257

and hence, keeping in mind (5.8.17) and (5.8.18), we deduce from (5.8.22) that  (xnk+1 ) − xs (x )| = |yk (xnk+1 ) − xs (yk )| ≤ Kyk  + εk+1 . |xk+1  Since yk  ≤ kj=1 (1 + εj ) as a consequence of (5.8.15), it follows that (5.8.19) holds for k + 1. This ends the induction. Now, the statement follows by noticing that (5.8.13) is nothing other than (5.8.16), that (5.8.14) is a consequence of (5.8.19),     uk are wuC as a consequence of (5.8.15) and the and that the series xk and implication (ii)⇒(i) in Proposition 5.8.2. 

Corollary 5.8.21 Let X be an L-embedded Banach space over K, let P be the L-projection on X  whose range is X, let (xn ) be a sequence in X, and let (un ) be a sequence in Xs := ker(P). Furthermore, suppose that (x , u ) is a weak∗ -cluster point of the sequence (xn , un ) in X  ×X  , and write x = x0 +xs and u = u0 +us with x0 , u0 ∈ X, xs , us ∈ Xs . Finally, let x , u ∈ SX  be such that x (x0 ) = 0 and us (u ) = 0. Then there is a strictly increasing sequence (nk ) in N and there are two wuC-series    xk and uk in X  such that unk (xk ) = 0 and uk (xnk ) = 0 for every k ∈ N,

(5.8.25)

lim xk (xnk ) = xs (x ) and lim unk (uk ) = u (u0 ).

(5.8.26)

and k→∞

k→∞

Proof Since (x , u ) is a weak∗ -cluster point of the sequence (xn , un ) in X  × X  , we realize that the assumption on (x , u ) in Lemma 5.8.20 holds with K = L = 0. Therefore, taking ε = 12 in that lemma, we find a sequence (εn ) of positive numbers  converging to zero, a strictly increasing sequence (nk ) in N, and two wuC-series xk   and uk in X  satisfying (5.8.25) and |xk (xnk ) − xs (x )| ≤ εk and |unk (uk ) − u (u0 )| ≤ εk for every k ∈ N, and so (5.8.26). 



Fact 5.8.22 Let X be a Banach space over K, and let xn be a wuC-series in X  .   Then for each (tn ) ∈ ∞ the series tn xn weak∗ converges (with sum ∗ tn xn , say)  in X  , and the mapping T : (tn ) → ∗ tn xn becomes a bounded linear operator from

∞ to X  . Proof By the implication (i)⇒(ii) in Proposition 5.8.2, there exists C ≥ 0 such that, for any (tn ) ∈ ∞ we have sup

n 

n∈N k=1

tk xk ≤ C sup |tn |.

(5.8.27)

n∈N

 Since for each (tn ) ∈ ∞ , the series tn xn is also wuC in X  , it follows from the  weakly∗ completeness of closed balls in X  that the series tn xn weak∗ converges in   X  . Let ∗ tn xn stand for the weak∗ sum of the series tn xn in X  . It is clear that the

258 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem  mapping T : (tn ) → ∗ tn xn is a linear operator from ∞ to X  . Moreover, keeping in mind (5.8.27), for (tn ) ∈ ∞ we realize that  n  n    ∗  ∗  tn xn = w - lim tk xk tk xk ≤ C sup |tn |, ≤ lim inf n→∞

k=1

n→∞

k=1

n∈N



and hence T is bounded.

The dual space of ∞ can be identified in a natural way with a normed space whose underlying set consists of all the bounded finitely additive K-valued set functions on the subsets of N. To be more precise, each x ∈ ∞ can be identified with the bounded finitely additive K-valued measure μ on 2N defined by μ(A) = x (χA ) for every A ⊆ N, where χA denotes the characteristic function of A. See [1151] for an  excellent discussion of ∞ . Note that, if xn is a wuC-series in the dual of a Banach space X, if we consider the bounded linear operator T : ∞ → X  associated to the  series xn via Fact 5.8.22, and if x ∈ X  , then μ := T  (x ) is the element in ∞ defined by  ∗    μ(A) = x xk for every A ⊆ N. (5.8.28) k∈A

We also recall Phillips’ original lemma asserting that for any sequence μn in ∞ sat isfying limn μn (A) = 0 for each A ⊆ N, we have limn j∈N |μn ({j})| = 0 [1151, p. 83]. Theorem 5.8.23 Let X be an L-embedded Banach space over K, and let P be the L-projection on X  whose range is X. Then P is weak∗ sequentially continuous. Therefore P, regarded as a mapping from X  onto X, is weak∗ -weak sequentially continuous. Proof Suppose that the sequence xn in X  is weak∗ -null and that xn = xn + un with xn = P(xn ). Then the sequence (xn , un ) is bounded in X  × X  , and hence admits a weak∗ -cluster point, say (x , u ) ∈ X  × X  . Therefore x + u is a weak∗ -cluster w∗

point of xn + un = xn → 0, hence x + u = 0, and so u = −x . Write x = x0 + xs with x0 ∈ X and xs ∈ Xs (so that u = −x0 − xs ), and let x , u ∈ SX  be such that x (x0 ) = 0 and xs (u ) = 0. We apply Corollary 5.8.21 to obtain a strictly increasing   sequence (nk ) in N and two wuC-series xk and uk in X  such that unk (xk ) = 0 and uk (xnk ) = 0 for every k ∈ N,

(5.8.29)

and lim xk (xnk ) = xs (x ) and

k→∞

lim unk (uk ) = −u (x0 ).

k→∞

(5.8.30)

For each n ∈ N, consider the mesures μn and νn in ∞ associated to xn via the wuC    series xk and uk , respectively. It follows from (5.8.28) that μn (A) → 0 and νn (A) → 0 for every A ⊆ N. By Phillips’ lemma we get

5.8 Operators into the predual of a JBW ∗ -triple   k (5.8.29) |xnk (xj )| = |μnk ({j})| −→ 0 |xk (xnk )| = |xnk (xk )| ≤ j∈N

and |unk (uk )|

(5.8.29)

=

|xnk (uk )| ≤

 j∈N

259

j∈N

|xnk (uj )| =



k

|νnk ({j})| −→ 0.

j∈N

Therefore, by (5.8.30), xs (x ) = 0 and u (x0 ) = 0. Since x and u were arbitrary in the unit sphere of X  satisfying x (x0 ) = 0 and xs (u ) = 0, we deduce that ker(x0 ) = ker(xs ), and as a result x0 = xs = 0. It follows that (0, 0) is the unique weak∗ -cluster point of the sequence (xn , un ) in X  × X  . As a consequence, the sequence (xn ) is weak∗ null in X  (and hence weakly null in X). This proves the theorem.  Corollary 5.8.24 Every L-embedded Banach space over K is weakly sequentially complete. Proof As a consequence of Theorem 5.8.23, X is weak∗ sequentially closed in X  . Therefore, since X  is weak∗ sequentially complete (by the w∗ -compactness of closed balls in X  ), and the restriction of the weak∗ topology of X  to X coincides with the weak topology of X, the result follows.  In the next lemma, we shall consider the canonical inclusions

1 → 1 and c0 → ∞ = 1 , and c◦0 will denote the polar of c0 in 1 . Lemma 5.8.25 Let (en ) denote the usual basis of the real or complex Banach space

1 . Then all w∗ -cluster points of the sequence en in 1 lie in Sc◦0 . Proof Let x = (tn ) ∈ c0 , and take ε > 0. Since en (x) = tn → 0, there exists n0 ∈ N such that |en (x)| < ε for every n ≥ n0 . Now let μ be a w∗ -cluster point of the sequence en in 1 . Then there exists a natural number m ≥ n0 such that |μ(x) − em (x)| < ε, and consequently |μ(x)| ≤ |μ(x) − em (x)| + |em (x)| < 2ε. Thus μ(x) = 0 because of the arbitrariness of ε > 0. Now, by the arbitrariness of x ∈ c0 , we obtain that μ ∈ c◦0 . Finally, consider the norm-one element u = {1, 1, 1, . . .} in ∞ = 1 , and note that K := {x ∈ 1 : x (u) = 1} is a w∗ -closed subset of 1 containing {en : n ∈ N}. Therefore μ ∈ K, and hence μ ≥ 1. Since the reverse inequality is clear, we  conclude that μ ∈ Sc◦0 , as desired. Lemma 5.8.26 Let X be an L-embedded Banach space over K, and let P be the L-projection on X  whose range is X. Let the closed subspace Y ⊆ X be an almost L-summand in its bidual in the sense that there is a number 0 < ε < 14 such that Y  = Y ⊕ Ys and y + ys  ≥ (1 − ε)(y + ys ) for all y ∈ Y and ys ∈ Ys .

260 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem 1

Then P|Y ◦◦ − Q ≤ 3ε 2 , where Y  and Y ◦◦ ⊆ X  are identified and Q denotes the projection from Y ◦◦ onto Y corresponding to the decomposition Y ◦◦ = Y ⊕ Ys . Proof By hypothesis there is a closed subspace Z ⊆ X  such that Y  ≡ Y ◦◦ = Y ⊕ Z with y + z ≥ (1 − ε)(y + z). Let y ∈ Y ◦◦ , and write y = y + z with y ∈ Y and z ∈ Z. Because P(y ) − Q(y ) = P(y + z) − Q(y + z) = P(z) 1

and because (ε 2 + 2ε)z ≤

1

ε 2 +2ε 1−ε y + z

1

≤ 3ε 2 y + z for any y ∈ Y, z ∈ Z, it is

1

enough to show that P(z) ≤ (ε 2 + 2ε)z for each z ∈ Z. Decompose z = x + xs 1 in X  = X ⊕1 Xs . Since we are done if x = P(z) ≤ ε 2 z, we suppose that 1 x > ε 2 z from now on. We obtain y + x = (y + x) + xs  − xs  = y + z − xs  ≥ (1 − ε)(y + z) − xs  = (1 − ε)(y + x + xs ) − xs  = (1 − ε)(y + x) − εxs  ≥ (1 − ε)(y + x) − εz 1

≥ (1 − ε)(y + x) − ε 2 x, hence 1

y + x ≥ (1 − 2ε 2 )(y + x)

(5.8.31)

for every y ∈ Y, which extends to all y ∈ Y ◦◦ , as will be shown in a moment: 1

y + x ≥ (1 − 2ε 2 )(y  + x).

(5.8.32)

For the time being we take (5.8.32) for granted and have in particular that, for z ∈ Y ◦◦ , 1

1

1

xs  =  − z + x ≥ (1 − 2ε 2 )(z + x) ≥ (1 − 2ε 2 )(z + ε 2 z), and finally 1

1

1

P(z) = x = z − xs  ≤ z − (1 − 2ε 2 )(1 + ε 2 )z = (ε 2 + 2ε)z. It remains to prove inequality (5.8.32). We first observe that x ∈ / Y because otherwise 1

0 =  − x + x ≥ (1 − 2ε 2 )( − x + x) > 0 1

by our standing assumption that x > ε 2 z. Thus it makes sense to consider the direct sum G = Y ⊕ Kx ⊆ X, and we let ι denote the identity from G onto 1 / G = Y ⊕1 Kx. Therefore / G ≡ Y ◦◦ ⊕1 Kx and ι ≤ (1−2ε 2 )−1 by (5.8.31). Inequality (5.8.32) follows now with y + x ∈ G◦◦ from 1

y  + x = ι (y + x) ≤ (1 − 2ε 2 )−1 y + x. Theorem 5.8.27 Every L-embedded Banach space X over K has property (V ∗ ).



5.8 Operators into the predual of a JBW ∗ -triple

261

Proof It is enough to show that X satisfies condition (ii) in Proposition 5.8.12. We already know from Corollary 5.8.24 that X is weakly sequentially complete. Let (xn ) be an 1 -sequence in X. We may suppose that c

n 

|αk | ≤

k=1

n 

αk xk ≤

k=1

n 

|αk |

k=1

for suitable c > 0 and all n ∈ N and α1 , . . . , αn ∈ K. Fix numbers ε and δ such that 2 0 < ε < 14 and 0 < δ < 9ε2 . By James’ distortion theorem [982, Lemma 2.1] (see also [769, Proposition 2.e.3]) there are pairwise disjoint finite sets Ak ⊆ N and a sequence  λn in K such that the sequence yk = n∈Ak λn xn satisfies (1 − δ)

n  k=1

|αk | ≤

n 

αk yk ≤

k=1

n 

|αk | for all n ∈ N and α1 , . . . , αn ∈ K,

k=1

(5.8.33) and



|λn |
0, every x ∈ X  , and every k0 ∈ N there is k ≥ k0 with 1

|x (yk ) − xs (x )| ≤ 3δ 2 x  + η.

(5.8.36)

To this end, denote the usual basis of 1 by (en ), denote by π 1 the canonical projection from 1 = 1 ⊕1 c◦0 onto 1 (cf. Corollary 5.1.57), take a w∗ -cluster point μ of the sequence en in 1 , and note that, according to Lemma 5.8.25, we have μ ∈ Sc◦0 . We will map μ into Xs . Set Y = lin S : Y → 1 satisfies

·

(yk ). In view of (5.8.33), the canonical isomorphism

y  ≤ S (y ) ≤

1  1−δ y 

for every y ∈ Y  .

In particular 1 − δ ≤ zs  ≤ 1 for zs = (S )−1 (μ). Consider zs ∈ X  via the identification of Y  and Y ◦◦ ⊆ X  . Denote by Q the canonical projection from Y ◦◦ onto Y (i.e. Q = S−1 ◦ π 1 ◦ S ). Then zs ∈ Ys = ker(Q) because μ ∈ ker(π 1 ), and zs is a w∗ -cluster point in X  of the set {yk : k ∈ N} in Ys . Finally set xs := (IX  − P)(zs ) ∈ ker(P) = Xs , where P is as usual the L-projection on X  = X ⊕1 Xs whose range is X. For the decomposition y = y + ys in Y ◦◦ = Y ⊕ Ys of any element y ∈ Y ◦◦ we have

262 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem y + ys  ≥ (1 − δ)S (y) + S (ys ) = (1 − δ)(S (y) + S (ys )) ≥ (1 − δ)(y + ys ). Since δ < result that

1 4

the assumptions of Lemma 5.8.26 are satisfied. We deduce from this 1

zs − xs  = P(zs ) = P(zs ) − Q(zs ) ≤ 3δ 2 zs .

(5.8.37)

Hence 1

1

xs  = zs − P(zs ) = zs  − P(zs ) ≥ (1 − 3δ 2 )zs  ≥ 1 − 4δ 2 , and so we have proved (5.8.35). We use the fact that zs is a w∗ -cluster point of the sequence yk , together with (5.8.37), to find (5.8.36). Indeed, given η > 0, x ∈ X  , and k0 ∈ N, there is k ≥ k0 such that |x (yk ) − zs (x )| ≤ η, and hence 1

|x (yk ) − xs (x )| ≤ |zs (x ) − xs (x )| + |x (yk ) − zs (x )| ≤ 3δ 2 x  + η. This ends the proof of the claim. Now that the claim is proved, we apply Lemma 5.8.20 with xk = yk and 1 uk = 0 for every k ∈ N, x = xs , u = 0, K = 3δ 2 , L = 0, x , u ∈ SX  such that |xs (x )| > 34 xs , and any sequence of positive numbers (εn ) decreasing to zero such  that ∞ < 1 + ε. Then there exist a strictly increasing sequence (kn ) in N n=1 (1 + εn ) and a wuC-series yn in X  such that 1

|yn (ykn ) − xs (x )| ≤ 3δ 2 (1 + ε) + ε for every n ∈ N. 1

Since 0 < δ 2 < (x )|

> since |xs (5.8.38) that

ε 9 and 3 4 xs ,

1

(5.8.38)

0 < ε < 14 , it follows that 3δ 2 (1 + ε) + ε < we deduce from (5.8.35) that |xs |yn (ykn )| ≥

5 16

(x )|

for every n ∈ N.



2 3.

17 48 .

Moreover,

It follows from (5.8.39)

Note that, by (5.8.34), for each i ∈ N there is ni ∈ Aki such that |yi (xni )| > because otherwise we would have  1 5 5 |λn ||yi (xn )| < |yi (yki )| ≤ c= , c 16 16

5 16 c,

n∈Aki

a contradiction with (5.8.39). Thus (xn ) fails to satisfy (5.8.1). Since (xn ) is an arbitrary 1 -sequence in X, the proof that X fulfils condition (ii) in Proposition 5.8.12 is concluded.  Combining Proposition 5.8.18 with Theorem 5.8.27, we obtain the following. Corollary 5.8.28 Let X and Y be Banach spaces over K such that X is L-embedded and Y  is weakly sequentially complete. Then every bounded linear operator from Y to X is weakly compact.

5.8 Operators into the predual of a JBW ∗ -triple

263

5.8.3 Applications to JB∗ -triples Since preduals of JBW ∗ -triples are L-embedded (cf. Theorem 5.7.36), Corollaries 5.8.29 and 5.8.30 follow from Corollaries 5.8.24 and 5.8.28, respectively. Corollary 5.8.29 The predual of any JBW ∗ -triple is weakly sequentially complete. Corollary 5.8.30 Let X be a JBW ∗ -triple, and let Y be a complex Banach space such that Y  is weakly sequentially complete. Then every bounded linear operator from Y to X∗ is weakly compact. Taking Y = X∗ in the above corollary, we obtain the following. Proposition 5.8.31 Let X be a weakly sequentially complete JBW ∗ -triple. Then the Banach space of X is reflexive. Now recall that, by Proposition 5.7.10, the dual of any JB∗ -triple is the predual of a JBW ∗ -triple. Therefore it is enough to invoke Corollaries 5.8.29 and 5.8.30 to obtain the following. Theorem 5.8.32 Let X be a JBW ∗ -triple, and let Y be a JB∗ -triple. Then every bounded linear operator from Y to X∗ is weakly compact. Invoking again Proposition 5.7.10, we obtain the following. Corollary 5.8.33 Let X and Y be JB∗ -triples. Then every bounded linear operator from Y to X  is weakly compact. Remark 5.8.34 Keeping in mind Dineen’s classical Proposition 5.7.10, applied to derive Corollary 5.8.33 from Theorem 5.8.32, these last two results are ‘equivalent’. For, if T is a bounded linear operator from the JB∗ -triple Y to the the predual of a JBW ∗ -triple X, then, via the canonical inclusion X∗ → X  , we can see T as a bounded linear operator from Y to X  , so that, by Corollary 5.8.33 and Fact 1.4.1(iv), T is weakly compact. Now we are going to discuss the relationship between weak compactness of operators and Arens regularity of continuous bilinear mappings. We recall that, given normed spaces X, Y, Z over K, and a bounded bilinear mapping m : X ×Y → Z, the bounded bilinear mappings mr : Y ×X → Z (the flip mapping of m) and m : Z  × X → Y  (the adjoint operation of m) are defined by mr (y, x) := m(x, y) and m (z , x)(y) := z (m(x, y)) for every (x, y, z ) ∈ X × Y × Z  (cf. Definition 2.2.14 and §2.2.11, respectively). Lemma 5.8.35 Let X, Y, Z be normed spaces over K, and let m : X × Y → Z be a bounded bilinear mapping. Then mr |Z  ×Y = mr .

264 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proof

For (x, y, z ) ∈ X × Y × Z  , we have mr (z , y)(x) = m (y, z )(x) = y(m (z , x)) = m (z , x)(y) = z (m(x, y)) = z (mr (y, x)) = mr (z , y)(x),

and hence mr (z , y) = mr (z , y).



Let X, Y, Z and m be as above. We recall that the bounded bilinear mapping m : X  × Y  → Z  is called the first Arens extension of m and is denoted by mt , and that m is said to be Arens regular if mrt = mtr (cf. again Definition 2.2.14). Moreover, since the mappings mt (x, ·) and mt (·, y ) are w∗ -continuous for every (x, y ) ∈ X × Y  (cf. Lemma 2.2.12(i)-(ii)), we realize that mt , and analogously mrtr (the second Arens extension of m), are determined by mt (x , y ) = w∗ - lim [w∗ - lim m(xα , yβ )]

(5.8.40)

mrtr (x , y ) = w∗ - lim [w∗ - lim m(xα , yβ )],

(5.8.41)

α

β

and β

α

whenever (x , y ) is in X  × Y  , and xα and yβ are nets in X and Y which converge to x and y in the w∗ -topologies of X  and Y  , respectively. Proposition 5.8.36 Let X, Y, Z be normed spaces over K, and let m : X × Y → Z be a bounded bilinear mapping. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

m is Arens regular. m t = mrtr |Z  ×X  . m t (Z  × X  ) ⊆ Y  . For every z ∈ Z  , the operator m (z , ·) from X to Y  is weakly compact.

Proof (i)⇒(ii) By Lemma 5.8.35, we have mrtr |Z  ×X  = mrtr . Therefore, if m is Arens regular, then mrtr |Z  ×X  = m t . Let (x , y , z ) ∈ X  × Y  × Z  be arbitrary. (ii)⇒(iii) Since mrt = mr = mrt , and mrt : X  × Z  → Y  is an extension of mr : X  × Z  → Y  , it is enough to invoke the assumption (ii) to obtain m t (z , x ) = mrt (x , z ) = mr (x , z ) ∈ Y  . (iii)⇔(iv) Put T := m (z , ·) : X → Y  . Then, by Lemma 2.2.12(iii), the operator : X  → Y  is w∗ -continuous and extends T. Therefore mt (z , ·) = T  . But, by Proposition 1.4.6, T is weakly compact if and only if T  (X  ) ⊆ Y  . (iii)⇒(i) Let xα and yβ be nets in X and Y which converge to x and y in the ∗ w -topologies of X  and Y  , respectively. Then, applying at the appropriate time the assumption (iii) and the equality (5.8.41), we see that mt (z , ·)

m t (x , y )(z ) = z (m t (x , y )) = m t (z , x )(y ) = y (m t (z , x )) = lim yβ (m t (z , x )) = lim m t (z , x )(yβ ) = lim z (m t (x , yβ )) β

β

β

5.8 Operators into the predual of a JBW ∗ -triple

265

= lim m t (x , yβ )(z ) = lim x (m (yβ , z )) = lim lim xα (m (yβ , z )) β

β



β

α



α

= lim lim m (yβ , z )(xα) = lim lim yβ (m (z , xα)) = lim lim m (z , xα)(yβ) β





α

β



β

α

= lim lim z (m(xα , yβ )) = lim lim m(xα , yβ )(z ) = m (x , y )(z ), β

α



β

rtr

α

and hence m t = mrtr , i.e., m is Arens regular.



Corollary 5.8.37 Let X, Y be normed spaces over K. Then the following conditions are equivalent: (i) Every bounded linear operator from X to Y  is weakly compact. (ii) For every normed space Z over K, all bounded bilinear mappings from X × Y to Z are Arens regular. (iii) All bounded bilinear forms on X × Y are Arens regular. (iv) There exists a nonzero normed space Z over K such that all bounded bilinear mappings from X × Y to Z are Arens regular. Proof The implication (i)⇒(ii) follows from the implication (iv)⇒(i) in Proposition 5.8.36, whereas the ones (ii)⇒(iii)⇒(iv) are clear. (iv)⇒(i) Suppose that condition (iv) is fulfilled. Let T : X → Y  be any bounded linear operator. Then we can choose a nonzero element z ∈ Z, consider the bounded bilinear mapping m : X × Y → Z defined by m(x, y) := [T(x)(y)]z, and observe that m (z , x)(y) = z (z)T(x)(y) for every (x, y, z ) ∈ X × Y × Z  . By taking z ∈ Z  such that z (z) = 1 we obtain that m (z , ·) = T(·). Therefore, since m is Arens regular, it follows from the implication (i)⇒(iv) in Proposition 5.8.36 that T is a weakly compact operator.  By a product on a vector space X over K we mean a bilinear mapping from X × X to X. The next corollary follows straightforwardly from the one immediately above. Corollary 5.8.38 Let X be a normed space over K. Then all bounded linear operators from X to X  are weakly compact if and only if all continuous products on X are Arens regular. Keeping in mind the above corollary, and taking Y = X in Corollary 5.8.33, we obtain the following. Fact 5.8.39 All continuous products on a JB∗ -triple are Arens regular. Keeping in mind the equivalence (ii)⇔(iv) in Lemma 2.3.51 and the equivalence (i)⇔(iii) in Corollary 5.8.37, Corollary 5.8.33 can be reformulated as follows. Corollary 5.8.40 Let X and Y be JB∗ -triples. Then every bounded bilinear form on X × Y has a separately w∗ -continuous extension to a bilinear form on X  × Y  . Since non-commutative JB∗ - and JBW ∗ -algebras are JB∗ - and JBW ∗ -triples, respectively (cf. Theorem 4.1.45), the following corollary follows from results previously proved in this subsection.

266 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Corollary 5.8.41 Let A be a non-commutative JBW ∗ -algebra, let X be a complex Banach space, and let B, C be non-commutative JB∗ -algebras. We have: (i) A∗ is weakly sequentially complete. (ii) If X  is weakly sequentially complete, then all bounded linear operators from X to A∗ are weakly compact. (iii) If A is weakly sequentially complete, then the Banach space of A is reflexive. (iv) Bounded linear operators from B to A∗ are weakly compact. (v) Bounded linear operators from B to C are weakly compact. (vi) All continuous products on B are Arens regular. (vii) Every bounded bilinear form on B × C has a separately w∗ -continuous extension to a bilinear form on B × C . 5.8.4 Historical notes and comments Property (V ∗ ) was introduced by Pełczy´nski [1036] in connection with his study of Banach spaces on which every unconditionally converging operator (see [1151, p. 54] for definition) is weakly compact. Among other results, Corollary 5.8.9 is proved in [1036]. §5.8.42 Proposition 5.8.14 is due to Godefroy and Saab [958, Proposition III.1]. Since the paper [958] is essentially an announcement of results, its authors limit themselves to comment that ‘L’implication la moins facile [in Proposition 5.8.14] est (i)⇒(ii); sa d´emonstration suit la construction faite par G. A. Edgard dans [932, p. 94]; l’implication (ii)⇒(i) a e´ t´e observ´ee par G. Emmanuele [937].’ According to [958, p. 503], ‘Les r´esultats pr´esent´es dans cette Note feront l’object d’un article plus d´etaill´e.’ However, we have found only a later paper of Godefroy and Saab, namely [959], where all results in [958] are proved save for Proposition 5.8.14. This proposition can be found as Theorem 3.3.B(b) of the Harmand–Werner–Werner book [739], but a complete proof of it is missing there. In fact, Emmanuele’s paper [937], cited as a preprint in [958] and ignored in [739], contains a proof of the equivalence (i)⇔(ii). Another proof of this equivalence can be found in Bombal’s paper [863]. Actually it seems to us that, in the (end of the) 80s, Proposition 5.8.14 was ‘in the air’ and known to experts. The proof of Proposition 5.8.14 we have given (including Lemmas 5.8.4 and 5.8.13) is due to Pfitzner [1047] (2014). According to him, Lemma 5.8.13 resembles a nice useful extraction lemma of Simons [1089]. Our proof of the implication (ii)⇒(i) in Proposition 5.8.14 passes through condition (iii). A direct proof of this implication, taken from [1047] (see also [937]), is the following. Let K ⊆ X be not weakly relatively compact. We may suppose by Remark 5.8.6(b) that K is bounded. Then, by (ii), K contains an 1 -sequence (xn ) spanning a · complemented 1 -copy U = lin (xn ) in X. Let (en ) and (en ) be the canonical basis of c0 and of 1 , respectively. Denote by π the projection from X onto U,

5.8 Operators into the predual of a JBW ∗ -triple

267

denote by S : U → 1 the isomorphism determined by S(xn ) = en , and consider  the operator T = S ◦ π from X to 1 . Set xi = T  (ei ). Since clearly ei is a wuC  series in c0 , it follows from Corollary 5.8.3(ii) that xi is a wuC-series in X  . Moreover, for all i, n ∈ N we have  1 if n = i    xi (xn ) = T (ei )(xn ) = ei (T(xn )) = ei (S(xn )) = ei (en ) = 0 if n = i. Thus (5.8.5) cannot hold. This shows (i).



Corollary 5.8.16 was noted by Randrianantoanina [1057]. Proposition 5.8.18 is the result of a careful reading of the proof of Proposition I.1 in the Godefroy–Iochum paper [957] (1988), where the case X = Y  of Corollary 5.8.19 is established. It is noteworthy that the original Godefroy–Iochum proof, like ours, involves Proposition 5.8.14. We are very grateful to Pfitzner for having written the note [1047] specifically to be included in our work. Theorem 5.8.23 is due to Pfitzner [1046]. Corollary 5.8.24 (which follows almost straightforwardly from Theorem 5.8.23) is earlier. It is originally due to Godefroy [955], who has given several different proofs of it in other papers (see [739, p. 201] for details). As said in the introduction, Theorem 5.8.27 is due to Pfitzner [1044]. Corollary 5.8.24 and Theorem 5.8.27 can be found in [739, Theorem IV.2.2] and [739, Theorem IV.2.7], respectively. Most material in Subsections 5.8.1 and 5.8.2, whose paternity has not been discussed previously, has been taken from Diestel’s book [1151]. We thank J. F. Mena for his collaboration in the writing of these subsections. As we commented in Remark 5.8.34, Theorem 5.8.32 and Corollary 5.8.33 are ‘equivalent’. Thus these results are originally due to Chu, Iochum, and Loupias [172] (1989), who actually prove the following much finer theorem. Theorem 5.8.43 Let X and Y be JB∗ -triples, and let T : Y → X  be a bounded linear operator. Then T factors through a complex Hilbert space. More precisely, there exist a complex Hilbert space H and bounded linear operators S : Y → H, R : H → X  √ satisfying T = R ◦ S and RS ≤ 2(1 + 2 3)T. The proof of the above theorem is very involved. A sketch of it will be discussed later (see §5.10.151). Today, the equivalence (i)⇔(iv) in Proposition 5.8.36 seems to be folklore (see for example [717, Corollary in p. 12]). We learned it by reading the proof of Theorem 6.1 of the Alvermann–Janssen paper [19], and published shortly later a note [1060] generalizing the Alvermann–Janssen result and simplifying their argument. Actually, the equivalence (i)⇔(iv) in Proposition 5.8.36 was known much earlier, and seems to go back to Pym [1056, Theorem 4.2], who proved it in a very general situation, which covers the case that X = Y = Z equals a normed associative algebra A, and m equals the product of A (see also [715, Theorem 2.6.17]). The much greater generality of

268 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Pym’s theorem, and the fact that the notions involved in its formulation have to be searched in a previous paper, could have produced that the particularization just commented had remained unknown for many people, and had been rediscovered several times. Thus, in the papers [1060, 1109] (published almost at the same time), the equivalence (i)⇔(iv) in Proposition 5.8.36 is established in its actual setting, and Corollaries 5.8.37 and 5.8.38 are first pointed out. The actual formulation of Proposition 5.8.36, as well as the proof we have given, are taken from the paper of Mohammadzadeh and Vishki [1022]. A curious forerunner of Fact 5.8.39 is due to Gulick [966], who proved that all continuous associative and commutative products on a unital commutative C∗ -algebra are Arens regular. Corollary 5.8.40 and Theorem 5.8.43 were claimed in [853] (1987). Nevertheless, as pointed out in [1040, p. 607], the proofs of these results in [853] have severe gaps. The associative forerunners of assertions (i) to (v) in Corollary 5.8.41 are due to Akemann [826]. Those of assertions (vi) and (vii) can be found in [1060, 1109]. The current versions of assertions (i) and (ii) in Corollary 5.8.41 were noted in [19] (see §5.10.127). That of assertion (vi) was derived in [1060]. 5.9 A holomorphic characterization of non-commutative JB∗ -algebras Introduction We prove a converse to Theorem 3.5.34, namely that a complete normed complex algebra is a non-commutative JB∗ -algebra (for its own norm and some involution) whenever its bidual (endowed with the Arens product) is so (see Theorem 5.9.7). Then we apply this fact, together with many other results in the chapter (including Fact 5.0.1), to prove that non-commutative JB∗ -algebras are precisely those complete normed complex algebras having a bounded approximate unit and whose open unit ball is a homogeneous domain (see Theorem 5.9.9). 5.9.1 Complete normed algebras whose biduals are non-commutative JB∗ -algebras We recall that JC∗ -algebras were defined as those JB∗ -algebras which can be seen as closed ∗-subalgebras of Asym for some C∗ -algebra A (cf. the paragraph immediately before Lemma 3.3.5). Lemma 5.9.1 Let A be a non-commutative JB∗ -algebra, and let e be an idempotent in A such that e = 1. Then e∗ = e. Proof By Fact 3.3.4, we may suppose that A is commutative. Then, by Proposition 3.4.6, the closed subalgebra of A generated by {e, e∗ } is a JC∗ -algebra. Therefore e can be regarded as a norm-one idempotent in a C∗ -algebra, so that, by Corollary 1.2.50, we have e∗ = e, as required.  Lemma 5.9.2 Let A be a power-associative flexible algebra over K, and let e ∈ A be an idempotent. Then Ue (A) is a subalgebra of A. Actually Ue (A) is the largest subalgebra of A having e as a unit.

5.9 A holomorphic characterization of non-commutative JB∗ -algebras

269

Proof According to Fact 3.3.3 and Lemma 2.5.3, Ue (A) is a subalgebra of Asym , and the following equalities (which should be omnipresent through this proof) hold: Ue (A) = {x ∈ A : e • x = x} = {x ∈ A : ex = xe = x}.

(5.9.1)

Let x and y be in Ue (A). Then, by Lemma 2.4.15, we have [x, y] = [x, e • y] = [x, e] • y + e • [x, y] = e • [x, y], and consequently [x, y] ∈ Ue (A). Therefore xy = 12 [x, y] + x • y ∈ Ue (A).



From now on, the bidual A of any normed algebra A will be regarded without notice as a normed algebra relative to the Arens product (cf. §2.2.11). The next result refines Proposition 3.4.78. Proposition 5.9.3 Let A and B be non-commutative JB∗ -algebras, and let  : A → B be a contractive algebra homomorphism. Then  is a ∗-mapping. Proof Suppose at first that A and B are unital, and that  preseves units. Then, by Corollary 3.3.17(a),  is a ∗-mapping, as desired. Now remove any additional assumption on A, B, and . To prove that  is a ∗-mapping we may suppose that  = 0. Then A and B are unital non-commutative JB∗ -algebras whose involutions extend those of A and B, respectively (cf. Theorem 3.5.34),  : A → B is a contractive algebra homomorphism (cf. Lemma 3.1.17), and e := (1) is a norm-one idempotent in B . Therefore, by Lemmas 5.9.1 and 5.9.2, Ue (B ) is a closed ∗-subalgebra of B (hence a unital non-commutative JB∗ algebra) containing  (A ). Then  , regarded as a mapping from A to Ue (B ), becomes a unit-preserving contractive algebra homomorphism. By the first para graph of the proof,  (and hence ) is a ∗-mapping. Corollary 5.9.4 Let B be a non-commutative JB∗ -algebra, and let A be a closed subalgebra of B which is a non-commutative JB∗ -algebra for the norm of B and some involution . Then  = ∗ on A, and therefore A is ∗-invariant. Proof Apply Proposition 5.9.3 to the isometric algebra homomorphism  := (A, ) → (B, ∗).  In the proof of the following lemma, for a subset S of a complex vector space, the symbol linR (S) will stand for the real linear hull of S. Lemma 5.9.5 Let A be a unital complete normed complex algebra such that A is a non-commutative JB∗ -algebra for some involution ∗. Then A is a ∗-invariant subset of A , and hence is a non-commutative JB∗ -algebra. Proof First note that the unit 1 of A is a unit for A (cf. Corollary 2.2.13), and therefore we have 1 = 1. Let f be in linR D(A, 1) ∩ i linR D(A, 1). Then there are g, h, r, s in D(A, 1) and α, β, γ , δ in R+ 0 satisfying f = αg − βh = i(γ r − δs).

270 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Denoting by j the canonical injection of A into A , j(g), j(h), j(r), j(s) are elements of D(A , 1), and the equalities j( f ) = αj(g) − βj(h) = i(γ j(r) − δj(s)) hold. Therefore j( f ) belongs to linR D(A , 1) ∩ i linR D(A , 1). Since A is a noncommutative JB∗ -algebra, we can apply the ‘only if’ part of Corollary 3.3.26 to conclude that j( f ) = 0, and therefore f = 0. Now, we have linR D(A, 1) ∩ i linR D(A, 1) = 0, and hence, by the ‘if’ part of Corollary 3.3.26, A is a non-commutative JB∗ -algebra for its own norm and some involution. Therefore, by Corollary 5.9.4, A is a ∗ invariant subset of A . The next fact was already proved in Lemma 2.3.52. For the sake of convenience, we repeat it here. Fact 5.9.6 Let B be a normed algebra over K, and let A be a subalgebra of B. Then the bipolar A◦◦ of A in B is a subalgebra of B , and the natural Banach space identification of A◦◦ with A becomes a bijective algebra homomorphism. Now the main result in this subsection reads as follows. Theorem 5.9.7 Let A be a complete normed complex algebra such that A , endowed with a suitable involution ∗, is a non-commutative JB∗ -algebra. Then A is a ∗-invariant subset of A , and hence is a non-commutative JB∗ -algebra. Proof We may suppose that A = 0. Then, in view of Lemma 5.9.5, we may also suppose that A is not unital. However, since A is a non-commutative JBW ∗ -algebra, A certainly has a unit 1 (cf. Fact 5.1.7). Put B := A (actually, we can take B equal to any closed ∗-subalgebra of A containing 1 and A), and consider the closed subalgebra C := C1 + A of B. Taking bipolars in B , we obtain C◦◦ = C1 + A◦◦ . Since A is a non-commutative JB∗ -algebra, it follows from Fact 5.9.6 that A◦◦ is a non-commutative JB∗ -algebra for some involution. Therefore, since B is a non-commutative JB∗ -algebra in a natural way (cf. Theorem 3.5.34), it follows from Corollary 5.9.4 that A◦◦ is a ∗-invariant subset of B . Now, keeping in mind Fact 5.9.6 again, C is a unital complete normed complex algebra whose bidual C ≡ C◦◦ = C1 + A◦◦ is a non-commutative JB∗ -algebra. Therefore, by Lemma 5.9.5, C is ∗-invariant, and hence is a non-commutative JB∗ -algebra. Since A is a closed ideal of C = C1 + A, it follows from Proposition 3.4.13 that A is also ∗-invariant.  Keeping in mind Fact 3.3.2, the following corollary follows straightforwardly from Theorem 5.9.7. Corollary 5.9.8 Let A be a complete normed complex algebra such that A , endowed with a suitable involution ∗, is a C∗ -algebra (respectively, an alternative C∗ -algebra

5.9 A holomorphic characterization of non-commutative JB∗ -algebras

271

or a JB∗ -algebra). Then A is a ∗-invariant subset of A , and hence is a C∗ -algebra (respectively, an alternative C∗ -algebra or a JB∗ -algebra).

5.9.2 The main result In the first volume of our work we proved that non-commutative JB∗ -algebras have approximate units bounded by one (cf. Proposition 3.5.23). Now, in the second volume, we have proved that open unit balls of non-commutative JB∗ -algebras are homogeneous domains (cf. Corollary 5.6.69). Thus the ‘only if’ part of the following theorem holds. Theorem 5.9.9 A complete normed complex algebra A is a non-commutative JB∗ -algebra (for some involution) if and only if A has an approximate unit bounded by one and the open unit ball of A is a homogeneous domain. Proof Let A be a nonzero complete normed complex algebra having an approximate unit bounded by one and such that A is a homogeneous domain. By Kaup’s Theorem 5.6.68, A is linearly isometric to a JB∗ -triple. As a first consequence, since every product on a JB∗ -triple is Arens regular (by Fact 5.8.39), A is an Arens regular normed algebra, and therefore, by Lemma 3.5.24(ii) and its proof, A has a unit 1 with 1 = 1. On the other hand, by Proposition 5.7.10, A is linearly isometric to a JB∗ -triple. Now, since A is a norm-unital complete normed complex algebra, it follows from Fact 5.0.1 that A is a non-commutative JB∗ -algebra. Finally, by Theorem 5.9.7, A is a non-commutative JB∗ -algebra.  Remark 5.9.10 The condition in the above theorem that A has an approximate unit bounded by one cannot be omitted. Indeed, take A equal to the Banach space of an arbitrary nonzero JB∗ -triple (cf. Theorem 5.6.68), endowed with the zero product. More illuminating examples of complete normed complex algebras A with homogeneous open unit balls, and which are not non-commutative JB∗ -algebras, are listed in what follows. (i) The normed associative algebra A whose normed space is that of an arbitrary JB∗ -triple J of dimension ≥ 2, and whose product is given by xy := f (x)y, where f is a norm-one element of J  such that f (e) = 1 for some norm-one element e of J. In this case e becomes a norm-one left unit for A. (ii) The normed associative and commutative algebra A whose normed space is that of an arbitrary nonzero JB∗ -triple J, and whose product is given by 1 xy := ( f (x)y + f (y)x − f (x)f (y)e), 3 where f and e are chosen as in the first-listed example. In this case 3e is a unit for A. (iii) The normed non-commutative Jordan algebra A obtained from an arbitrary nonzero non-commutative JB∗ -algebra by replacing its natural product xy with

272 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Mxy, where 0 < M < 1. In this case A is bicontinuously isomorphic to a non-commutative JB∗ -algebra, and hence has a bounded approximate unit. (iv) The normed associative algebra A of all Hilbert-Schmidt operators on a complex Hilbert space H of dimension ≥ 2 (see Example 8.1.3). Then, since A is a Hilbert space, A is linearly isometric to a JB∗ -triple (cf. Remark 4.2.38). If H is finitedimensional, then A has a unit. Otherwise, A has an unbounded approximate unit. As a straightforward consequence of Theorem 5.9.9, we obtain the following. Corollary 5.9.11 A normed complex algebra is a non-commutative JB∗ -algebra if and only if it is linearly isometric to a non-commutative JB∗ -algebra and has an approximate unit bounded by one. Now, keeping in mind Fact 3.3.2, the following corollaries follow from Theorem 5.9.9. Corollary 5.9.12 A complete normed associative complex algebra is a C∗ -algebra if and only if it has an approximate unit bounded by one and its open unit ball is a homogeneous domain. Corollary 5.9.13 A complete normed alternative complex algebra is an alternative C∗ -algebra if and only if it has an approximate unit bounded by one and its open unit ball is a homogeneous domain. Corollary 5.9.14 A normed alternative complex algebra is an alternative C∗ -algebra if and only if it is linearly isometric to a non-commutative JB∗ -algebra and has an approximate unit bounded by one. Corollary 5.9.15 A normed alternative complex algebra is an alternative C∗ -algebra if and only if it is linearly isometric to an alternative C∗ -algebra and has an approximate unit bounded by one. Corollary 5.9.16 A normed associative complex algebra is a C∗ -algebra if and only if it is linearly isometric to a non-commutative JB∗ -algebra and has an approximate unit bounded by one. Corollary 5.9.17 A normed associative complex algebra is a C∗ -algebra if and only if it is linearly isometric to a C∗ -algebra and has an approximate unit bounded by one. Concerning Corollary 5.9.11, it is noteworthy that linearly isometric noncommutative JB∗ -algebras need not be Jordan-∗-isomorphic (cf. Antitheorem 3.4.34). 5.9.3 Historical notes and comments Most results and arguments in this section (including Lemma 5.9.1) are due to Kaidi, Morales, and Rodr´ıguez [365]. Other sources are quoted as adequate. It is noteworthy

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that Theorems 5.9.7 and 5.9.9 had no associative forerunners in the literature. Indeed, their associative consequences, formulated in the bracket-free version of Corollary 5.9.8 and in Corollary 5.9.12, were unknown before the publication of [365]. Lemma 5.9.2 is due to Albert [12, Theorem I.5]. Its specialization to the particular case of non-commutative Jordan algebras can be found in [436, p. 5]. This specialization (enough for our purposes) is the one invoked by the authors of [365] in their proof of Theorem 5.9.7. The fact (asserted in Corollary 3.3.17(a) and applied in the proof of Proposition 5.9.3) that unit-preserving linear contractions between unital non-commutative JB∗ algebras are ∗-mappings is folklore in the theory. Of course, its associative forerunner is also folklore (see for example the last sentence of the proof of [862, Theorem 1.7] and the last paragraph of [788, p. 88]). Through its Corollary 5.9.4, Proposition 5.9.3 (whose paternity is discussed in §5.9.21) has allowed us to significantly simplify the original proof in [365] of Theorem 5.9.7. Proposition 5.9.3 can be generalized as follows. Corollary 5.9.18 Let A and B be non-commutative JB∗ -algebras, and let  : A → B be a contractive Jordan homomorphism. Then  is a ∗-mapping. Proof By Fact 3.3.4, Asym and Bsym are JB∗ -algebras in a natural way. Therefore, regarding  as an algebra homomorphism from Asym to Bsym , the result follows from the commutative version of Proposition 5.9.3.  Now, invoking Fact 3.3.2, we derive the following. Corollary 5.9.19 Let A and B be alternative C∗ -algebras, and let  : A → B be a contractive Jordan homomorphism. Then  is a ∗-mapping. In [938], Essaleh, Peralta, and Ram´ırez prove the following. Theorem 5.9.20 Let X, Y be JB∗ -triples, and let , be linear contractions from X to Y such that the equalities ({abc}) = {(a) (b)(c)} and ({abc}) = { (a)(b) (c)}

(5.9.2)

hold for all a, b, c ∈ X. Then  = , and hence  is a triple homomorphism. §5.9.21 Now let A and B be non-commutative JB∗ -algebras, and let  : A → B be a contractive algebra homomorphism. Then, by Proposition 3.3.13, ∗ := ∗ ◦  ◦ ∗ : A → B is a linear contraction. Moreover, regarding A and B as JB∗ -triples (cf. Theorem 4.1.45) and writing := ∗ , the equalities (5.9.2) hold for all a, b, c ∈ A. Therefore, by Theorem 5.9.20,  = ∗ is a ∗-mapping. In this way, Theorem 5.9.20 contains Proposition 5.9.3. We are very grateful to Peralta for providing us with a copy of [938] and for asking us whether Proposition 5.9.3 was known. Our answer to Peralta’s question was that Proposition 5.9.3 was not known, but that it is easily

274 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem derivable from previously known results. Then we wrote our own proof of that proposition, we communicated it to him, and he and his co-authors had the deference to reproduce it at the end of [938]. Corollary 5.9.17 is originally due to Rodr´ıguez [528]. 5.10 Complements on non-commutative JB∗ -algebras and JB∗ -triples Introduction In Subsection 5.10.1 we introduce the strong∗ topology of a noncommutative JBW ∗ -algebra and study its basic properties (see Definition 5.10.3 and Proposition 5.10.5). As a first application, we build up a functional calculus at each normal element a of a non-commutative JBW ∗ -algebra A, which extends the continuous functional calculus (cf. Corollary 4.1.72) and has a sense for all real-valued bounded lower semicontinuous functions on J-sp(A, a) (see Proposition 5.10.7). This allow us to prove a variant for non-commutative JBW ∗ -algebras (see Theorem 5.10.9) of Kadison’s isometry theorem for unital C∗ -algebras (cf. Theorem 2.2.29), a consequence of which is that linearly isometric non-commutative JBW ∗ -algebras are Jordan-∗-isomorphic (Corollary 5.10.10). (We recall that linearly isometric unital alternative C∗ -algebras are Jordan-∗-isomorphic (cf. Corollary 3.4.33), that linearly isometric (possibly non-unital) C∗ -algebras are Jordan∗-isomorphic (a consequence of Theorem 2.2.19), but that linearly isometric (even unital) non-commutative JB∗ -algebras need not be Jordan-∗-isomorphic (cf. Antitheorem 3.4.34).) We introduce the support idempotent of a weak∗ -continuous positive linear functional on a non-commutative JBW ∗ -algebra (see Definition 5.10.18) as a tool to prove the generalization to non-commutative JBW ∗ -algebras of Akemann’s theorem [826] asserting the coincidence of the strong∗ and Mackey topologies on bounded subsets of any W ∗ -algebra (see Theorem 5.10.33). The technology in the proof of Theorem 5.10.33 allows us to prove the generalization to non-commutative JB∗ -algebras of Jarchow’s characterization [985] of weakly compact operators from a C∗ -algebra to a complex Banach space (see Theorem 5.10.37). Subsection 5.10.1 concludes by proving Sakai’s theorem [1071] asserting that C∗ -algebras whose Banach spaces are reflexive are in fact finite-dimensional (see Theorem 5.10.43), and by following [66] to establish a later outstanding refinement originally obtained in [67] (Corollary 5.10.51). In Subsection 5.10.2 we follow the Friedman–Russo paper [269] to introduce the support tripotent of a w∗ -continuous linear functional on a JBW ∗ -triple (see Definition 5.10.58) and to establish its basic properties. Then we introduce and study the strong∗ topology of a JBW ∗ -triple as done by Barton and Friedman [853, 60] (see Definition 5.10.61 and Theorem 5.10.63), and prove that, when a non-commutative JBW ∗ -algebra is viewed as a JBW ∗ -triple, its new (triple) strong∗ topology coincides with the (algebra) strong∗ topology introduced in Subsection 5.10.1 [1061] (see Proposition 5.10.62). We prove Zizler’s refinement [1137] of Lindenstrauss theorem [1001] on norm-density of operators whose transpose attain their norm (see Proposition 5.10.69). Then, following [1061, 1040], we combine Proposition

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5.10.69 with Proposition 5.10.67 (due to Barton and Friedman [853]) to prove a variant for JBW ∗ -triples of the so-called little Grothendieck’s theorem [964] (see Corollary 5.10.72), and to derive a Banach space characterization of the strong∗ topology of a JBW ∗ -triple (see Theorem 5.10.77). This is applied to realize that the strong∗ topology of a JBW ∗ -triple and the strong∗ topology of a w∗ -closed subtriple coincide on bounded subsets of the subtriple (see Proposition 5.10.83). In Subsection 5.10.3 we follow [366] to provide the reader with a full nonassociative discussion of the Kadison–Paterson–Sinclair Theorem 2.2.19 on surjective linear isometries of (possibly non-unital) C∗ -algebras. To this end we introduce the multiplier non-commutative JB∗ -algebra M(A) of a given non-commutative JB∗ algebra A, which turns out to be a closed ∗-subalgebra of A containing the unit of A and containing A as an essential ideal (see Theorem 5.10.90). Moreover, according to Proposition 5.10.96, M(A) coincides with the JB∗ -triple of multipliers [873] of the JB∗ -triple underlying A. We prove that the Kadison–Paterson–Sinclair theorem remains true verbatim for surjective linear isometries from non-commutative JB∗ algebras to alternative C∗ -algebras (see Theorem 5.10.102), and that no further verbatim generalization is possible (see Proposition 5.10.108). Nevertheless the consequence that non-commutative JB∗ -algebras linearly isometric to an alternative C∗ -algebra A are Jordan-∗-isomorphic to A becomes a relevant property of A, which is not fulfilled by all choices of A in the class of non-commutative JB∗ -algebras (recall again Antitheorem 3.4.34) but is enjoyed by certain choices of A outside the class of alternative C∗ -algebras (indeed, as we commented above, this is the case when A is any non-commutative JBW ∗ -algebra). Intrinsic characterizations of this property are obtained in Corollary 5.10.115. 5.10.1 Selected topics in the theory of non-commutative JBW ∗ -algebras Fact 5.10.1 Let A be a unital non-commutative JB∗ -algebra. An element a ∈ A is positive if and only if V(A, 1, a) ⊆ R+ 0. Proof

By Corollary 2.1.2, Proposition 2.1.4, and Lemma 2.2.5, we have

V(H(A, ∗), 1, h) = V(AR , 1, h) = (V(A, 1, h)) = V(A, 1, h) for every h ∈ H(A, ∗). (5.10.1) Now, let a be a positive element of A. Then, by definition (cf. §§5.1.25 and 5.1.28), + a ∈ H(A, ∗) and V(H(A, ∗), 1, a) ⊆ R+ 0 . Therefore, by (5.10.1), V(A, 1, a) ⊆ R0 . + Conversely, let a be in A such that V(A, 1, a) ⊆ R0 . By Lemma 2.2.5, a ∈ H(A, ∗), and then, by (5.10.1) again, V(H(A, ∗), 1, a) ⊆ R+ 0 . Thus a is a positive element of A.  Lemma 5.10.2 Let A be a unital non-commutative JB∗ -algebra, and let f be a linear functional on A. Then the following conditions are equivalent: (i) f is positive. (ii) There are ϕ ∈ D(A, 1) and r ∈ R+ 0 such that f = rϕ.

276 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Moreover, if the above conditions are fulfilled, then: (iii) |f (a)|2 ≤ f (a∗ • a)f (1) for every a ∈ A. √ (iv) The mapping a → f (a∗ • a) is a seminorm on A. Proof By Fact 5.10.1, every element of D(A, 1) is a positive linear functional on A. This proves the implication (ii)⇒(i). Let f be a positive linear functional on A, and let a be in A. Since, by (5.1.2), 0 ≤ a∗ • a ≤ a2 1, we have 0 ≤ f (a∗ • a) ≤ a2 f (1). As a consequence, the mapping (x, y) → f (y∗ • x) becomes a non-negative hermitian sesquilinear form on A, and hence, by the Cauchy– Schwarz inequality, we have 7 7 |f (a)| = |f (1∗ • a)| ≤ f (a∗ • a)f (1∗ • 1) = f (a∗ • a)f (1) ≤ a f (1). This shows that f is bounded with  f  = f (1). Therefore, if f = 0, then it is enough to set ϕ :=  ff  and r :=  f  to have ϕ ∈ D(A, 1), r ≥ 0, and f = rϕ. Thus the implication (i)⇒(ii) and assertion (iii) have been proved. Assertion (iv) follows from Minkowski’s inequality for non-negative hermitian sesquilinear forms.  Definition 5.10.3 Let A be a unital non-commutative JB∗ -algebra. According to √ Lemma 5.10.2, for each ϕ ∈ D(A, 1), the mapping pϕ : a → ϕ(a∗ • a) is a seminorm on A. Now suppose that A is in fact a nonzero non-commutative JBW ∗ -algebra. The strong∗ topology of A is defined as the locally convex topology on A generated by the family of seminorms {pϕ : ϕ ∈ D(A, 1) ∩ A∗ }, and is denoted by s∗ = s∗ (A, A∗ ). §5.10.4 The proof of the next proposition will involve the fact that, for elements a, b in a Jordan algebra A over K, we have (Ua (b))2 = Ua (Ub (a2 )).

(5.10.2)

Indeed, thinking about the unital extension of A, it is enough to invoke the fundamental formula (cf. Proposition 3.4.15) to derive (Ua (b))2 = UUa (b) (1) = Ua (Ub (Ua (1))) = Ua (Ub (a2 )), as desired. Proposition 5.10.5 Let A be a nonzero non-commutative JBW ∗ -algebra. We have: (i) The involution of A is strong∗ -continuous. (ii) The strong∗ topology of A is stronger than the weak∗ topology. (iii) Every bounded increasing net aλ in H(A, ∗) is strong∗ -convergent to its least upper bound. (iv) The product of Asym is separately strong∗ -continuous. (v) The product of Asym is jointly strong∗ -continuous on bounded subsets of A. Proof Since pϕ (a∗ ) = pϕ (a) for all a ∈ A and ϕ ∈ D(A, 1) ∩ A∗ , assertion (i) becomes obvious.

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Let aλ be a net in A s∗ -convergent to a ∈ A. Then, by Lemma 5.10.2, for every λ and every ϕ ∈ D(A, 1) ∩ A∗ we have |ϕ(a − aλ )| ≤ pϕ (a − aλ ), and hence limλ ϕ(a − aλ ) = 0 for every ϕ ∈ D(A, 1) ∩ A∗ . Since A∗ is the linear hull of D(A, 1) ∩ A∗ (cf. Corollary 2.9.32), we derive that limλ ϕ(a − aλ ) = 0 for every ϕ ∈ A∗ , i.e. aλ w∗ -converges to a. Thus assertion (ii) has been proved. Let aλ be a bounded increasing net in H(A, ∗). It follows from Corollary 5.1.40 that aλ has a least upper bound a ∈ H(A, ∗), and that a = w∗ - limλ aλ . Let M be in R+ such that aλ  ≤ M for every λ. Then we have (a − aλ )2 ≤ a − aλ (a − aλ ) ≤ (a + M)(a − aλ ) for every λ, and hence [pϕ (a − aλ )]2 = ϕ((a − aλ )2 ) ≤ (a + M)ϕ(a − aλ ) for every λ and every ϕ ∈ D(A, 1) ∩ A∗ . Therefore limλ pϕ (a − aλ ) = 0 for every ϕ ∈ D(A, 1) ∩ A∗ . This proves assertion (iii). Let a be in A, and let La• denote the operator of multiplication by a on Asym . Since 1 La• = (U1+a − Ua − IA ), 2 and La• = Lh• + iLk• whenever a = h + ik with h, k ∈ H(A, ∗), to prove assertion (iv) it suffices to show the s∗ -continuity of the operator Uh for each h ∈ H(A, ∗). Then, in view of assertion (i), it actually suffices to show that, for each h ∈ H(A, ∗), the restriction of Uh to H(A, ∗) is s∗ -continuous. Let h be in H(A, ∗), and let kλ be a net in H(A, ∗) s∗ -convergent to 0. Then ψ(kλ2 ) = [pψ (kλ )]2 → 0 for every ψ ∈ D(A, 1) ∩ A∗ .

(5.10.3)

Now, let ϕ be in D(A, 1) ∩ A∗ . Then, by Corollary 5.1.30(iii), ϕ ◦ Uh lies in A∗ and, by the inclusion (5.1.1) in §5.1.25, ϕ ◦ Uh is a positive linear functional on A. It follows from Lemma 5.10.2 that there are ψ ∈ D(A, 1)∩A∗ and r ∈ R+ 0 such that ϕ ◦Uh = rψ. 2 Therefore, by (5.10.3), we have ϕ(Uh (kλ )) → 0. But, by (5.10.2) and (5.1.1) again, for every λ we have (Uh (kλ ))2 = Uh (Ukλ (h2 )) ≤ h2 Uh (Ukλ (1)) = h2 Uh (kλ2 ). Thus ϕ((Uh (kλ ))2 ) → 0. By the arbitrariness of ϕ ∈ D(A, 1) ∩ A∗ , we see that Uh (kλ ) s∗ -converges to 0. In view of assertion (i), to prove assertion (v) it is enough to show that aλ • bλ s∗ -converges to a • b whenever a and b are in H(A, ∗) and aλ and bλ are bounded nets in H(A, ∗) s∗ -convergent to a and b, respectively. Suppose first that aλ s∗ -converges to 0. From the inequalities 0 ≤ (a2λ )2 ≤ a2λ a2λ we find that a2λ s∗ -converges to 0. If furthermore bλ s∗ -converges to 0, then it follows from the equality 1 a • b = [(a + b)2 − a2 − b2 ] 2

278 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem that aλ • bλ s∗ -converges to 0. In the general case, consider the equality a • b − aλ • bλ = (a − aλ ) • b + (aλ − a) • (b − bλ ) + a • (b − bλ ). From the above the middle term s∗ -converges to 0, while by assertion (iv) the other two terms s∗ -converge to 0.  As usual, for λ ∈ R we write λ+ := max{λ, 0}. Since the mapping λ → λ+ is continuous, for each self-adjoint element x in a unital C∗ -algebra we can consider the positive element x+ in the sense of the continuous functional calculus. Lemma 5.10.6 Let B be a unital commutative C∗ -algebra, and let x, y be in H(B, ∗). Then (x+ − y+ )2 ≤ (x − y)2 . Proof Keeping in mind the commutative Gelfand–Naimark theorem (cf. Theorem 1.2.23), it is enough to show that, for λ, μ ∈ R, we have (λ+ − μ+ )2 ≤ (λ − μ)2 . But this is of straightforward verification.  For the formulation and proof of the next result, the reader should keep in mind the continuous functional calculus at a normal element of a non-commutative JB∗ -algebra, as stated in Corollary 4.1.72. Proposition 5.10.7 Let A be a nonzero non-commutative JBW ∗ -algebra, let a be a normal element of A, and set E := J-sp(A, a). Consider the set C of those complexvalued functions f on E such that there exists a bounded sequence fn in CC (E) in such a way that fn converges pointwise to f and fn (a) s∗ -converges to some element of A. Then: (i) C becomes a ∗-algebra under the operations defined pointwise, and contains CC (E). (ii) The correspondence f → f (a) := s∗ - limn fn (a), for fn as above, becomes a welldefined algebra ∗-homomorphism from C to A, which extends the continuous functional calculus at a, and preserves natural orders (i.e. f (a) lies in A+ whenever f is in C and f (t) ≥ 0 for every t ∈ E). (iii)  f (a) ≤  f ∞ for every f ∈ C . (iv) H(C , ∗) contains all real-valued bounded lower semicontinuous functions on E. (v) If f is a nonzero bounded lower semicontinuous function on E such that f (t) ≥ 0 for every t ∈ E, then f (a) = 0. Proof Let us see C as a subset of the ∗-algebra F C (E) of all complex-valued functions on E, with operations defined pointwise. The inclusion CC (E) ⊆ C is clear. Therefore, to prove assertion (i) it is enough to show that C is a ∗-subalgebra of F C (E). The fact that C is a vector subspace of F C (E) is clear, whereas the ∗-invariance of C follows from Proposition 5.10.5(i). Let f , g be in C . Then f = lim fn and g = lim gn pointwise, for suitable bounded sequences fn , gn in CC (E) such that s∗

s∗

fn (a) → x and gn (a) → y for some x, y ∈ A.

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Therefore fg = lim fn gn pointwise and, by Proposition 5.10.5(v), s∗

( fn gn )(a) = fn (a)gn (a) = fn (a) • gn (a) → x • y. Thus fg lies in C . Now we prove that the correspondence f → f (a) in assertion (ii) is a mapping. To this end it is enough to show that, whenever fn is a bounded sequence in CC (E) s∗

with fn → 0 pointwise and fn (a) → x for some x ∈ A, we have x = 0. Let fn be such a sequence. Since fn → 0 pointwise, it follows from [1151, Theorem VII.1] that fn → 0 in the weak topology of the Banach space CC (E), so fn (a) → 0 in the weak topology of the Banach space A (because norm-continuous linear mappings between normed w∗

s∗

spaces are weakly continuous), and so fn (a) → 0. On the other hand, since fn (a) → x, w∗

it follows from Proposition 5.10.5(ii) that fn (a) → x. Therefore x = 0, as desired. Clearly, the mapping f → f (a) in assertion (ii) is linear. Moreover, it is a ∗-mapping, thanks to Proposition 5.10.5(i). Let f , g be in C . Then f = lim fn and g = lim gn pointwise, for suitable bounded sequences fn , gn in CC (E) such that s∗

s∗

fn (a) → f (a) and gn (a) → g(a),

(5.10.4)

and, as we showed in the first paragraph of the proof, we have ( fg)(a) = f (a) • g(a).

(5.10.5)

Now, invoking (5.10.4), Proposition 5.10.5(ii), and Corollary 5.1.30(iii), for m ∈ N we have g(a)fm (a) = w∗ - lim[gn (a)fm (a)] = w∗ - lim[fm (a)gn (a)] = fm (a)g(a), n

n

and then g(a)f (a) = w∗ - lim[g(a)fm (a)] = w∗ - lim[fm (a)g(a)] = f (a)g(a). m

m

Therefore, keeping in mind (5.10.5), we obtain ( fg)(a) = f (a)g(a). Thus the mapping f → f (a) in assertion (ii) is an algebra ∗-homomorphism. Moreover, clearly, it extends the continuous functional calculus at a. To conclude the proof of assertion (ii), let us prove that the mapping f → f (a) preserves natural orders. First of all, we note that, by the preceding paragraph, the set {f (a) : f ∈ C } becomes a unital associative and commutative ∗-subalgebra of A, and hence, by Proposition 3.4.1(i), its norm-closure in A becomes a unital commutative C∗ -algebra (say B). Let f be in C such that f (t) ≥ 0 for every t ∈ E. Then there exists a bounded sequence fn in CC (E) in such a way that fn converges pointwise to f and fn (a) s∗ -converges to f (a) ∈ H(A, ∗). Replacing fn with 12 ( fn + fn∗ ) if necessary, and keeping in mind Proposition 5.10.5, we may suppose that fn ∈ CR (E). Then applying Lemma 5.10.6, in the C∗ -algebra B above we have [( fn (a))+ − ( f (a))+ ]2 ≤ ( fn (a) − f (a))2

280 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem for every n ∈ N, which implies that the sequence ( fn (a))+ s∗ -converges to ( f (a))+ . But, for each n ∈ N, we have fn (a)+ = fn+ (a), where fn+ ∈ CR (E) denotes the function t → ( fn (t))+ (cf. Corollary 4.1.72(iii)), and the sequence fn+ converges pointwise to f . It follows that f (a) = ( f (a))+ ∈ A+ . Now we prove assertion (iii). Let f be in C . Then we have f ∗ (t)f (t) ≤  f 2∞ for every t ∈ E. Therefore, since the algebra ∗-homomorphism f → f (a) preserves natural orders, we obtain f (a)∗ f (a) ≤  f 2∞ 1. Therefore, thinking about the C∗ -algebra B in the preceding paragraph, we get  f (a) ≤  f ∞ . Let f : E → R be a bounded lower semicontinuous function. Then there is an (automatically bounded) increasing sequence fn in CR (E) converging pointwise to f (see for example [1188, 7K.4]). Since the continuous functional calculus at a preserve orders, fn (a) is a bounded increasing sequence in H(A, ∗), and hence, by Proposition 5.10.5(iii), fn (a) is s∗ -convergent in A. Therefore f lies in C . Thus assertion (iv) has been proved. Now let f be a nonzero bounded lower semicontinuous function on E such that f (t) ≥ 0 for every t ∈ E. Then, as in the preceding paragraph, there is an increasing sequence fn in CR (E) converging pointwise to f , and for any choice of such a s∗

sequence we have fn (a) → f (a). But, since f (t) ≥ 0 for every t ∈ E, the sequence fn above can be chosen in such a way that fn (t) ≥ 0 for all n ∈ N and t ∈ E. Moreover, since f = 0, there exists m ∈ N such that fm = 0. Therefore, since fm (a) = 0 (because the continuous functional calculus at a is injective), and 0 ≤ fm (a) ≤ f (a) (by assertion (ii)), we deduce that f (a) = 0. Thus assertion (v) has been proved.  Corollary 5.10.8 Let A be a nonzero non-commutative JBW ∗ -algebra, and let a be a J-unitary element of A (cf. Definition 4.2.25). Then there exists a J-unitary element b of A such that b2 = a. Proof We know that a is normal (cf. Fact 4.2.26(v)). Let B denote the norm-closed subalgebra of A generated by {1, a, a∗ }. Then, by Fact 3.4.22, B is a C∗ -algebra. Since a becomes a unitary element of B, it follows from Proposition 1.2.20(i) that J-sp(A, a) ⊆ sp(B, a) ⊆ T := {z ∈ C : |z| = 1}. Consider the lower semicontinuous function s : R → R defined by s(r) = 1 if r > 0 and s(r) = −1 otherwise. Then the mapping z → s((z)) from J-sp(A, a) to R is lower semicontinuous, and hence belongs to C in view of Proposition 5.10.7(iv). In what follows, the remaining parts ? 5.10.7 will be applied without ? of Proposition notice. Since the mappings z → 1+(z) and z → 1−(z) from J-sp(A, a) to R are 2 2 continuous, it follows that the mapping f : J-sp(A, a) → C defined by @ @ 1 + (z) 1 − (z) f (z) := + is((z)) 2 2

also belongs to C . It is clear that f (z)2 = z for every z ∈ J-sp(A, a), and consequently it is enough to set b := f (a) to have b2 = a. On the other hand, it is also clear that

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f • f ∗ = 1 and f 2 • f ∗ = f , which implies that b • b∗ = 1 and b2 • b∗ = b, i.e. b is J-unitary in A.  Now we can prove the following variant of the Kadison isometry theorem (cf. Theorem 2.2.29). Theorem 5.10.9 Let A and B be non-commutative JBW ∗ -algebras, and let :A→B be a mapping. Then the following conditions are equivalent: (i) There are a J-unitary element u of B and a bijective Jordan-∗-homomorphism F : A → B such that  = Uu ◦ F. (ii)  is a surjective linear isometry. Proof The implication (i)⇒(ii) follows from Proposition 3.4.4, Remark 3.4.5, and the implication (i)⇒(vii) in Theorem 4.2.28. (ii)⇒(i) Suppose that  is a surjective linear isometry. Set v := (1). Since 1 is a J-unitary element of A, and J-unitary elements of unital non-commutative JB∗ -algebras are characterized in terms of their underlying Banach spaces (cf. the equivalences (i)⇔(iv) or (i)⇔(v) in Theorem 4.2.28), we see that v is a J-unitary element of B. Therefore, by Corollary 5.10.8, there exists a J-unitary element u of B such that u2 = v. Write F := Uu∗ ◦  : A → B. Since u∗ is a J-unitary element of B, it follows from the implication (i)⇒(vii) in Theorem 4.2.28 that F is a surjective linear isometry, and clearly we have F(1) = 1. Therefore, by the implication (ii)⇒(i) in Proposition 3.4.25, F is a bijective Jordan-∗-homomorphism. Finally, the equality   = Uu ◦ F is clear. Corollary 5.10.10 Linearly isometric non-commutative JBW ∗ -algebras are Jordan∗-isomorphic. We note that linearly isometric non-commutative JB∗ -algebras need not be Jordan∗-isomorphic, nor even if they are in fact unital JC∗ -algebras (cf. Antitheorem 3.4.34). Corollary 5.10.11 Let A and B be non-commutative JBW ∗ -algebras. Then we have: (i) Unit-preserving w∗ -continuous Jordan-∗-homomorphisms from A to B are s∗ -continuous. (ii) Surjective linear isometries from A to B are s∗ -continuous. Proof Let  : A → B be a unit-preserving w∗ -continuous Jordan-∗-homomorphism. Then, since  is contractive (cf. Proposition 3.4.4 and Remark 3.4.5), for every ϕ ∈ D(B, 1) ∩ B∗ , the functional ψ := ϕ ◦  lies in D(A, 1) ∩ A∗ , and we have pϕ ◦  = pψ . This proves assertion (i). Now let  : A → B be a surjective linear isometry. Then, by Theorem 5.10.9, there are a J-unitary element u of B and a bijective Jordan-∗-homomorphism

282 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem F : A → B such that  = Uu ◦ F. Keeping in mind that F is w∗ -continuous (cf. Corollary 5.1.30(ii)), the s∗ -continuity of F follows from assertion (i). Finally, since Uu = 2(Lu• )2 − Lu•2 , the s∗ -continuity of Uu follows from Proposition 5.10.5(iv).  Now we deal with some by-products of the results proved above. Lemma 5.10.12 A linear functional on a non-commutative JBW ∗ -algebra is s∗ continuous if and only if it is w∗ -continuous. Proof The ‘if’ part follows from Proposition 5.10.5(ii). Let A be a non-commutative JBW ∗ -algebra, and let f be a s∗ -continuous linear functional on A. Keeping in mind Corollary 5.1.40, to prove the ‘only if’ part it is enough to show that f is a normal functional. But, in view of Proposition 5.10.5(iii), for every bounded increasing net aλ in H(A, ∗) with least upper bound a, we have f (aλ ) → f (a). Therefore f is a normal functional, as desired.  Proposition 5.10.13 Let B be a non-commutative JBW ∗ -algebra, and let A be a w∗ -dense ∗-subalgebra of B. Then BA is s∗ -dense in BB . Proof By Lemma 5.10.12, the closed convex subsets of B are the same for both the weak∗ and strong∗ topologies. Therefore the result follows from Corollary 5.1.44.  Now recall that, according to Theorem 3.5.34, the bidual of any non-commutative JB∗ -algebra A becomes naturally a non-commutative JBW ∗ -algebra containing A as a ∗-subalgebra. Lemma 5.10.14 Let A be a non-commutative JB∗ -algebra. We have: (i) A+ ∩ BA is s∗ (A , A )-dense in (A )+ ∩ BA . (ii) Each norm-continuous positive linear functional on A remains positive when it is regarded as a σ (A , A )-continuous linear functional on A . (iii) If f and g are norm-continuous positive linear functionals on A, then  f + g =  f  + g. Proof By Proposition 5.10.13, BA is s∗ (A , A )-dense in BA . Let a be in (A )+ ∩ BA , and write a = b2 with b ∈ (A )+ ∩ BA . It follows that there exists a net bλ in BA s∗ (A , A )-convergent to b. Therefore, by assertions (i) and (iv) in Proposition 5.10.5, the net b∗λ • bλ s∗ (A , A )-converges to b2 = a. Since b∗λ • bλ ∈ A+ ∩ BA for every λ, assertion (i) in the lemma follows. Assertion (ii) follows from (i) since, by Proposition 5.10.5(ii), all functionals in A become s∗ (A , A )-continuous when they are regarded as σ (A , A )-continuous linear functionals on A . Let f and g be norm-continuous positive linear functionals on A. Then, by assertion (ii) and Lemma 5.10.2, we have  f + g = ( f + g)(1) = f (1) + g(1) =  f  + g.



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§5.10.15 Let A be a non-commutative Jordan algebra over K. By an inner ideal of A we mean any subspace B of A such that Ub (A) ⊆ B for every b ∈ B. We note that, by Fact 3.3.3, this definition in agreement with the one given in §4.4.71 for Jordan algebras. We note also that, if A has a unit 1, then inner ideals of A are subalgebras of Asym , i.e. they are strict inner ideals of Asym (cf. §4.4.71 again). For, if B is an inner ideal of A, and if b is in B, then b2 = Ub (1) ∈ B. Standard examples of inner ideals of A are the sets of the form Ua (A), where a is any element of A. Indeed, this follows straightforwardly from the fundamental formula UUa (b) = Ua Ub Ua . The particular case of inner ideals of the form Ue (A), for some idempotent e ∈ A, is specially relevant in our development, and is discussed in the next lemma. Fact 5.10.16 Let A be a non-commutative JBW ∗ -algebra. Then e → Ue (A) becomes a bijection from the set of all self-adjoint idempotents of A onto the set of all w∗ -closed ∗-invariant inner ideals of A. Proof Let e be a self-adjoint idempotent of A. Then Ue (A) is an inner ideal of A (cf. §5.10.15), is ∗-invariant, and is w∗ -closed thanks to Corollary 5.1.30(iii) and the equalities (5.9.1) in the proof of Lemma 5.9.2. Conversely, let B be a w∗ -closed ∗-invariant inner ideal of A. Since B is a w∗ -closed ∗-subalgebra of the JBW ∗ -algebra Asym (cf. §5.10.15 again), B is a JBW ∗ -algebra (cf. §5.1.9), and hence, by Fact 5.1.7, B has a unit e, which is a self-adjoint idempotent of A. The inclusion Ue (A) ⊆ B follows because e ∈ B and B is an inner ideal of A, whereas the reverse inclusion follows from the equalities (5.9.1) and the fact that e is the unit of B. Thus B = Ue (A). To conclude the proof, we must show that e = z whenever e and z are self-adjoint idempotents in A such that Ue (A) = Uz (A). Let e, z be such idempotents. Then z ∈ Ue (A), and hence, by the equalities (5.9.1), we have e • z = z. By symmetry, z • e = e. Therefore e = z, as desired.  Proposition 5.10.17 Let A be a non-commutative JB∗ -algebra with a unit 1, and let f be a positive linear functional on A. We have: (i) The set S := {a ∈ A : f (a∗ • a) = 0} is a ∗-invariant inner ideal of A. (ii) If A is in fact a non-commutative JBW ∗ -algebra, and if f is w∗ -continuous, then the set S above is w∗ -closed in A. Proof By Lemma 5.10.2(iv), S is a subspace of A. On the other hand, the ∗-invariance of S is clear. Therefore, to prove that S is an inner ideal of A it is enough to show that Uh (H(A, ∗)) ⊆ S whenever h is in S ∩ H(A, ∗). Let h be in S ∩ H(A, ∗), and let k be in H(A, ∗). Then, by the inclusion (5.1.1) in §5.1.25 and the equality (5.10.2) in §5.10.4, we have (Uh (k))2 = Uh (Uk (h2 )) ≤ Uk (h2 )Uh (1) = Uk (h2 )h2 , and hence Uh (k) ∈ S. Thus assertion (i) has been proved.

284 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Suppose that A is a non-commutative JBW ∗ -algebra and that f is w∗ -continuous. Then, by assertions (i), (ii), and (v) in Proposition 5.10.5, the mapping a → f (a∗ • a) is s∗ -continuous on BA . Therefore the set S ∩ BA is s∗ -closed in A. By Lemma 5.10.12, S ∩ BA is w∗ -closed in A, hence S is w∗ -closed in A thanks to the Krein– ˇ Smulyan theorem.  Definition 5.10.18 Let A be a non-commutative JBW ∗ -algebra, let f be a w∗ continuous positive linear functional on A, and let 1 denote the unit of A. Combining Fact 5.10.16 and Proposition 5.10.17, we realize that {a ∈ A : f (a∗ • a) = 0} = U1−e (A) for a unique self-adjoint idempotent e ∈ A. Such an idempotent is called the support idempotent of f . Lemma 5.10.19 Let A be a non-commutative JB∗ -algebra with a unit 1, let f be a positive linear functional on A, and let e ∈ A be a self-adjoint idempotent such that f (e) = f (1). Then f ◦ Ue = f . Proof

By the Cauchy–Schwarz inequality, for every b ∈ A we have 7 |f ((1 − e) • b)| ≤ f (1 − e)f (b∗ • b) = 0,

and hence f (e • b) = f (b). Applying this fact several times (with b • a instead of b and then with a instead of b), and using that Ue = 2(Le• )2 − Le• , we find f (Ue (a)) = f (a) for every a ∈ A.  Let A be a non-commutative JB∗ -algebra. A positive linear functional f on A is said to be faithful if f (a) > 0 whenever a is a nonzero positive element of A. Proposition 5.10.20 Let A be a non-commutative JBW ∗ -algebra, let f be a w∗ -continuous positive linear functional on A, and let e denote the support idempotent of f . Then we have: (i) Ue (A) is a w∗ -closed ∗-subalgebra of A, and the restriction of f to Ue (A) is faithful. (ii) f (e) = f (1) =  f . (iii) If a ∈ A satisfies f (a) =  f a, then Ue (a) = ae. (iv) e is equal to the unit of A if and only if f is faithful. Proof By Lemma 5.9.2 and Fact 5.10.16, Ue (A) is a w∗ -closed ∗-subalgebra of A. Let a be in A+ ∩ Ue (A) such that f (a) = 0. Since a ∈ Ue (A), it follows from the equalities (5.9.1) in the proof of Lemma 5.9.2 that a = e • a. On the other hand, since a ≥ 0 (which implies that a2 ≤ aa) and f (a) = 0, we have f (a2 ) = 0, and hence, again by the definition of the support idempotent, a ∈ U1−e (A), which reads as e • a = 0 again by (5.9.1). Therefore a = 0. Since 1 − e ∈ U1−e (A), it follows from the definition of the support idempotent that f (e) = f (1). But f (1) =  f  thanks to the implication (i)⇒(ii) in Lemma 5.10.2.

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To prove assertion (iii), we may suppose that  f  = 1, and then it suffices to show that Ue (a) = e whenever a ∈ A satisfies a = f (a) = 1. Let a be such an element of A, and set b := Ue (a), so that we must show b = e. Since f (b) = f (Ue (a)) = f (a) = 1 (cf. Lemma 5.10.19), it follows from assertion (ii) that f (e − b) = 1 − 1 = 0. Suppose at first that a = a∗ . Then b = b∗ , and hence e − b ∈ Ue (A)+ since b ≤ 1 because Ue is contractive (cf. Proposition 3.4.17). Therefore, since f (e − b) = 0 and f is faithful on Ue (A), we get that b = e, as desired. Now remove the assumption that a = a∗ . Then   1 1 1 1 ∗ ∗ (a + a ) = f (a) + f (a) = 1, 1 ≥ (a + a ) ≥ f 2 2 2 2 and hence  12 (a + a∗ ) = f ( 12 (a + a∗ )) = 1. Therefore 12 (b + b∗ ) = e by the above case (with 12 (a + a∗ ) instead of a). Since e is an extreme point of the unit ball of Ue (A) (cf. Corollary 2.1.13 and Lemma 2.1.25), we conclude that b = e. The ‘only if’ part of assertion (iv) follows from assertion (i). To prove the ‘if’ part, suppose that f is faithful. Let a ∈ A be such that f (a∗ • a) = 0. Then a∗ • a = 0 by faithfulness of f , and hence, by Lemma 3.4.65, a = 0. Therefore, by the definition of the support idempotent, we have e = 1.  Proposition 5.10.21 Let A be a unital non-commutative JB∗ -algebra, and let e and f be self-adjoint idempotents in A. Then the following conditions are equivalent: (i) (ii) (iii) (iv) (v) (vi)

e ≤ f. U1−f (e) = 0. f • e = e. fe = ef = e. Uf Ue Uf = Ue . Ue (A) ⊆ Uf (A).

Proof (i)⇒(ii) The assumption 0 ≤ e ≤ f , together with the inclusion (5.1.1) in §5.1.25, yields 0 ≤ U1−f (e) ≤ U1−f ( f ) = 0, and hence U1−f (e) = 0. (ii)⇒(iii) By the assumption (ii) and the equality (5.10.2) in §5.10.4, we have (Ue (1 − f ))2 = 0, and hence Ue (1 − f ) = 0. Now note that, as a consequence of the Shirshov–Cohn theorem (cf. Theorem 3.1.55), for all x, y ∈ A we have 4(x • y)2 = 2x • Uy (x) + Ux (y2 ) + Uy (x2 ). It follows that ((1 − f ) • e)2 = 0, so (1 − f ) • e = 0, and so f • e = e. (iii)⇒(iv) By the equalities (5.9.1) in the proof of Lemma 5.9.2. (iv)⇒(v) By the fundamental formula (cf. Proposition 3.4.15), we have Uf Ue Uf = UUf (e) . But, by the assumption (iv), Uf (e) = e. (v)⇒(vi) This is clear. (vi)⇒(i) Since e ∈ Ue (A), the assumption (vi) implies that e ∈ Uf (A). Therefore e ≤ f because Uf (A) is a subalgebra of A with unit f . 

286 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Remark 5.10.22 Let A be a non-commutative JBW ∗ -algebra, let S denote the set of all self-adjoint idempotents in A, endowed with the order induced from that of A, and let F stand for the family of all w∗ -closed ∗-invariant inner ideals of A, ordered by inclusion. It follows from Fact 5.10.16 and the equivalence (i)⇔(vi) in Proposition 5.10.21 that e → Ue (A) becomes an order isomorphism from S onto F . Therefore, since the intersection of any subfamily of F lies in F , it follows that every non-empty subset of S has a greatest lower bound in S. But, since the mapping e → 1 − e from S to S is an order anti-isomorphism, we see that S is a complete lattice. Lemma 5.10.23 Let A be a non-commutative JBW ∗ -algebra, let ψ be a w∗ continuous faithful positive linear functional on A, and consider the norm ||| · ||| √ on A defined by ||| a ||| := ψ(a∗ • a). Then the topology of ||| · ||| and the strong∗ topology coincide on bounded subsets of A. Proof Let τ denote the topology of ||| · |||. Then the inequality τ ≤ s∗ is clear. Now let ϕ be in D(A, 1) ∩ A∗ , and let 0 < ε < 1. Then, by Theorem 5.1.29(v), E := {x ∈ A+ : ϕ(x) ≥ ε2 } ∩ BA is a non-empty w∗ -compact subset of A. Therefore there exists δ := min ψ(E), and δ > 0 by the faithfulness of ψ. It follows that ϕ(x) < ε2 whenever x is in A+ ∩ BA with ψ(x) < δ. As a consequence, √ (5.10.6) pϕ (a) < ε whenever a is in BA with ||| a ||| < δ. Now let S be any bounded subset of A, and let aλ be any net in S τ -convergent 1 to some a ∈ S. Take √ M > sups∈S s. Then 2M (a − aλ ) ∈ BA for every λ, and 1 ||| 2M (a − aλ ) ||| < δ for λ big enough. Therefore, by (5.10.6), we have pϕ (a − aλ ) < 2Mε for λ big enough. It follows from the arbitrariness of ε ∈]0, 1[ that limλ pϕ (a − aλ ) = 0. Then, since ϕ is arbitrary in D(A, 1) ∩ A∗ , we realize that the net aλ s∗ -converges to a. The above shows that s∗ ≤ τ on bounded subsets of A.  Lemma 5.10.24 Let A be a unital non-commutative JB∗ -algebra, let e ∈ A be a selfadjoint idempotent, and let (x, y) be in Ue (A) × U1−e (A). Then x + y = max{x, y}. Proof See e as a tripotent of the JB∗ -triple underlying A (cf. Theorem 4.1.45), and apply Corollary 4.2.30(iii)(a).  Let A be a JBW-algebra, or a non-commutative JBW ∗ -algebra. We denote by A+ ∗ the cone of all w∗ -continuous positive linear functionals on A. The following result was incidentally invoked immediately after Proposition 3.1.57. Proposition 5.10.25 [738, Proposition 4.5.3] Let A be a JBW-algebra, and let ϕ be in A∗ . Then there are ϕ1 , ϕ2 ∈ A+ ∗ such that ϕ = ϕ1 − ϕ2 and ϕ = ϕ1  + ϕ2 .

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Now let A be a non-commutative JBW ∗ -algebra. According to Theorem 5.1.29, H(A, ∗) is a JBW-algebra, the involution of A is the transpose of a conjugate-linear involution on A∗ (which will be denoted also by ∗), and H(A, ∗)∗ identifies with H(A∗ , ∗) just as the real subspace of A∗ consisting of those ϕ ∈ A∗ such that ϕ(h) ∈ R for every h ∈ H(A, ∗). With these ideas in mind, the next corollary follows straightforwardly from Proposition 5.10.25. Corollary 5.10.26 Let A be a non-commutative JBW ∗ -algebra, and let ϕ be in H(A∗ , ∗). Then there are ϕ1 , ϕ2 ∈ A+ ∗ such that ϕ = ϕ1 − ϕ2 and ϕ = ϕ1  + ϕ2 . Now, given a non-commutative JBW ∗ -algebra A, for each functional ϕ ∈ A∗ there exist ϕ1 , ϕ2 , ϕ3 , ϕ4 ∈ A+ ∗ satisfying ϕ = ϕ1 − ϕ2 + i(ϕ3 − ϕ4 ), ϕ1 − ϕ2  = ϕ1  + ϕ2 , and ϕ3 − ϕ4  = ϕ3  + ϕ4 . We suppose that, for each ϕ ∈ A∗ , a choice of ϕ1 , ϕ2 , ϕ3 , ϕ4 ∈ A+ ∗ as above has been made. Then we write [ϕ] := ϕ1 + ϕ2 + ϕ3 + ϕ4 , and note that [ϕ] ≤ 2ϕ. Linearizing the identity (5.10.2) (cf. §5.10.4) in the variable b, we get the following. Fact 5.10.27 For elements a, b, c in a non-commutative Jordan algebra over K, we have Ua (b) • Ua (c) = Ua (Ub,c (a2 )). Fact 5.10.28 Let A be a non-commutative JB∗ -algebra, and let a be in A. Then Ua∗ ,a (A+ ) ⊆ A+ . Proof Write a = h + ik with h, k ∈ H(A, ∗). Then Ua∗ ,a = Uh + Uk . Therefore the result follows from the inclusion (5.1.1) in §5.1.25.  Lemma 5.10.29 Let A be a non-commutative JBW ∗ -algebra, let an be a sequence in H(A, ∗) ∩ BA s∗ -convergent to 0, and let δ > 0. Then there exists a sequence en of self-adjoint idempotents of A s∗ -converging to 1 and satisfying max{Uen (an ), Uen ,1−en (an )} ≤ δ for every n ∈ N.

(5.10.7)

Proof Let ξ : R → R denote the characteristic function of the interval ] − δ, δ], and let n be in N. Since ξ is the difference of two lower semicontinuous functions, and J-sp(A, an ) ⊆ R (cf. Fact 4.1.67(i)), we can define en := ξ(an ) ∈ A in the sense of Proposition 5.10.7. Then en is a self-adjoint idempotent and we have δ −2 a2n ≥ 1 − en = (1 − en )2 .

(5.10.8)

Since s∗ -lim an = 0, it follows from (5.10.8) that 1 = s∗ -lim en . On the other hand, it is immediate that an en  ≤ δ for each n ∈ N, which implies (5.10.7) since 1, an , and en lie in a common associative subalgebra of A. 

288 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Lemma 5.10.30 Let A be a non-commutative JBW ∗ -algebra, let E be a weakly relatively compact subset of A∗ , and let an be a sequence in BA s∗ -converging to 0. Then an → 0 uniformly on E as n → ∞. Proof We may suppose that E ⊆ BA∗ . On the other hand, by Proposition 5.10.5(i), the self-adjoint and skew-adjoint parts of the sequence an both s∗ -converge to 0, so we may additionally suppose that each an is self-adjoint. Suppose the lemma is false. Then there exists ε > 0 and sequences ϕn in E and xn in BA , where xn is a subsequence of an such that |ϕn (xn )| > ε for every n ∈ N.

(5.10.9)

ˇ By the Eberlein–Smulyan theorem (see for example [729, Theorem 3.109]), there exists a subsequence of ϕn which is weakly convergent. For notational simplicity we suppose that the sequence ϕn converges weakly to ϕ0 . Let α > 0 be given. By Lemma 5.10.29, there exists a sequence en of self-adjoint idempotents of A s∗ -converging to 1 and satisfying max{Uen (an ), Uen ,1−en (an )} ≤ α6 for every n ∈ N. Then, setting gj := ϕj − ϕ0 ∈ 2BA∗ , and keeping in mind that the equality IA = Ux + 2Ux,1−x + U1−x holds for any x ∈ A, we derive that |gj (xn )| ≤ |gj (Uen (xn ))| + 2|gj (Uen ,1−en (xn ))| + |gj (U1−en (xn ))| ≤ α + |gj (U1−en (xn ))| for all j, n ∈ N.

(5.10.10)

The sequence ϕj converges pointwise to ϕ0 on BA , which is w∗ -compact. By the Osgood theorem [1168, Theorem 9.5, p. 86], there is x0 ∈ BA such that the family {ϕj : j ∈ N ∪ {0}} is equicontinuous in x0 when restricted to BA . As a consequence, we can find a w∗ -neighbourhood N of 0 in A such that |ϕj (x) − ϕj (x0 )| < α6 for every j ∈ N ∪ {0} whenever x is in V := (x0 + N ) ∩ BA . Now choose j0 such that j ≥ j0 implies |gj (x0 )| < α6 . It follows that |gj (x)|
0. Then there exists δ > 0 and a finite subset F of E such that

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|ψ(a)| < ε for every ψ ∈ E whenever a is in BA and [ψ](a∗ • a) < δ for every ψ ∈ F. Proof Suppose the lemma is false for some ε > 0. Then by induction we can construct sequences ϕn in E and an in BA such that |ϕn+1 (an )| ≥ ε and [ϕn ](a∗m • am ) < 21m for every m ≥ n. Set ϕ :=

∞  1 [ϕn ] ∈ A+ ∗, 2n n=1

and let e denote the support idempotent of ϕ. We claim that ϕn (a) = ϕn (Ue (a)) for any a ∈ A and each n ∈ N. Indeed, writing ϕn = ϕ1,n − ϕ2,n + i(ϕ3,n − ϕ4,n ), with ϕ1,n , ϕ2,n , ϕ3,n , ϕ4,n ∈ A+ ∗ such that [ϕn ] := ϕ1,n + ϕ2,n + ϕ3,n + ϕ4,n , for i = 1, 2, 3, 4 we have {a ∈ A : ϕ(a∗ • a) = 0} ⊆ {a ∈ A : ϕi,n (a∗ • a) = 0}, and hence, by Definition 5.10.18 and the implication (vi)⇒(i) in Proposition 5.10.21, we have e ≥ ei,n , where ei,n denotes the support idempotent of ϕi,n ; this implies ϕi,n (e) = ϕi,n , so that, by Lemma 5.10.19, we obtain ϕi,n = ϕi,n ◦ Ue , and the claim follows. Moreover, for m ∈ N we have ϕ(a∗m • am ) =

∞  1 [ϕn ](a∗m • am ) 2n n=1

m ∞   1 1 ∗ [ϕ ](a • a ) + sup ψ ≤ n m m n n−1 2 2 ψ∈E n=1

n=m+1

m 1  1 1 ≤ m + sup ψ m−1 2 2n ψ∈E 2 n=1

1 1 ≤ m + sup ψ m−1 , 2 2 ψ∈E and hence ϕ(a∗m • am ) → 0 as m → ∞. Since for m ∈ N we have (Ue (am ))∗ • Ue (am ) = Ue (a∗m ) • Ue (am ) = Ue (Ua∗m ,am (e)) (by Fact 5.10.27) and Ua∗m ,am (e) ≤ Ua∗m ,am (1) = a∗m • am (by Fact 5.10.28), we derive that ϕ[(Ue (am ))∗ • Ue (am )] = ϕ[Ue (Ua∗m ,am (e))] = ϕ(Ua∗m ,am (e)) ≤ ϕ(a∗m • am ), and hence ϕ[(Ue (am ))∗ • Ue (am )] → 0 as m → ∞. Since ϕ is faithful on Ue (A), it follows from Lemma 5.10.23 that the sequence Ue (an ) s∗ -converges to 0. By Lemma 5.10.30, this sequence then converges to 0 uniformly on E. However  |ϕn+1 (Ue (an ))| = |ϕn+1 (an )| ≥ ε, a contradiction, and the proof is complete.

290 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proposition 5.10.32 Let A be a non-commutative JBW ∗ -algebra, and let E be a weakly relatively compact subset of A∗ . Then there is ψ ∈ A+ ∗ with the property that for any ε > 0 there exists δ > 0 such that |ϕ(a)| < ε for every ϕ ∈ E whenever a is in BA and ψ(a∗ • a) < δ. Proof Let εn = 1n and choose δn and Fn = {ϕ1n , . . . , ϕmn n } for each n ∈ N according    1 mn 1 n to Lemma 5.10.31. Set ψ := ∞ n=1 2n i=1 2i [ϕi ] . It is clear that ψ is a desired element of A+  ∗. Given a dual Banach space X over K, we denote by m(X, X∗ ) the Mackey topology on X relative to its duality with X∗ , i.e. the topology of uniform convergence on absolutely convex and weakly compact subsets of A∗ (see for example [1161, pp. 203–6]). Theorem 5.10.33 Let A be a non-commutative JBW ∗ -algebra. Then m(A, A∗ ) and s∗ (A, A∗ ) coincide on bounded subsets of A. Proof By Lemma 5.10.12 and the Mackey–Arens theorem, we have m(A, A∗ ) ≥ s∗ (A, A∗ ). Thus we need only show that the converse inequality holds on bounded subsets of A. Now let S be any bounded subset of A, and let aλ be any net in S s∗ (A, A∗ )convergent to some a ∈ S. Let E be a weakly relatively compact subset of A∗ , and let ε > 0. Take M > sups∈S s, and let ψ ∈ A+ ∗ and δ > 0 be given by Proposition 1 5.10.32. Since 2M (a − aλ ) ∈ BA for every λ, and ∗  . - 1 1 (a − aλ ) • (a − aλ ) < δ ψ 2M 2M for λ big enough, it follows that |ϕ(a − aλ )| < 2Mε for every ϕ ∈ E and λ big enough. By the arbitrariness of ε ∈ R+ , we see that the net aλ converges to a uniformly on E. Since E is an arbitrary weakly relatively compact subset of A∗ , we finally realize that aλ m(A, A∗ )-converges to a.  Now that Theorem 5.10.33 has been proved, we are going to deal with other outstanding applications of Proposition 5.10.32. Lemma 5.10.34 Let X be a dual Banach space over K, and let ·, · be a separately w∗ -continuous non-negative hermitian sesquilinear form on X. Then there exists a w∗ -continuous linear mapping T from X to a Hilbert space over K such that √ x, x = T(x) for every x ∈ X. (5.10.12) Proof By the closed graph theorem, ·, · is separately norm-continuous, and hence, by Lemma 1.1.8, it is jointly norm-continuous. Consider the Hilbert space H over K completion of the pre-Hilbert space (X/N, (·|·)), where N := {x ∈ X : x, x = 0} and (x + N|y + N) := x, y for all x, y ∈ X,

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and the linear operator T : X → H given by T(x) := x + N. Then T is normcontinuous, and (5.10.12) holds. To prove that T is w∗ -continuous it is enough to show that, for each h ∈ H, the mapping  h : x → (T(x)|h) from X to K is w∗ -continuous, a fact which is clearly true if h = y + N ∈ X/N for some y ∈ X. (Indeed, consider in this case the equality (T(x)|h) = x, y and apply the separate w∗ -continuity of ·, ·.) The norm-continuity of T, together with the norm-density of X/N in H, gives us that, for arbitrary h ∈ H, the functional  h equals the uniform h is limit on BX of a sequence hn with hn ∈ X/N for every n ∈ N. Therefore  w∗ -continuous on BX , so that, by Fact 5.1.19,  h is w∗ -continuous, as desired.  Lemma 5.10.35 Let X and Y be Banach spaces over K, and let F : X → Y be a linear mapping such that there exists a bounded linear operator G from X to a reflexive Banach space Z over K, and a function N : R+ → R+ , in such a way that F(x) ≤ N(ε)G(x) + εx for all x ∈ X and ε > 0. Then F is weakly compact. Proof

Clearly, F is bounded. Let us fix ε > 0, and for x ∈ X set xε := N(ε)G(x) + εx.

Then  · ε is an equivalent norm on X, the mapping T : x → (N(ε)G(x), εx) from (X,  · ε ) to Z ⊕1 X is a linear isometry, and F : (X,  · ε ) → Y is a linear contraction. Since (Z ⊕1 X) =, Z  ⊕1 X  , and T  (x ) = (N(ε)G (x ), εx ), and F  : (X  ,  · ε ) → Y  is a linear contraction (where now  · ε denotes the bidual norm on X  of  · ε on X), it follows that the mapping x → (N(ε)G (x ), εx ) from (X  ,  · ε ) to Z  ⊕1 X  is a linear isometry, i.e. x ε := N(ε)G (x ) + εx  for every x ∈ X  , and then that F  (x ) ≤ N(ε)G (x ) + εx  for every x ∈ X  .

(5.10.13)

On the other hand, since G(BX ) is w∗ -dense in G (BX  ), and Z is reflexive, G(BX ) is w-dense (so norm-dense) in G (BX  ). Let x be in BX  , and take a sequence xn in BX such that the sequence G(xn ) norm-converges to G (x ). Since F  (x ) − F(xn ) ≤ N(ε)G (x ) − G(xn ) + 2ε for every n ∈ N (by (5.10.13)), we have lim supn F  (x ) − F(xn ) ≤ 2ε. By letting ε → 0, and keeping in mind the arbitrariness of x ∈ BX  , we realize that F  (BX  ) ⊆ Y. Thus F(X  ) ⊆ Y, and F is weakly compact in view of Proposition 1.4.6.  Proposition 5.10.36 Let X be a complex Banach space, let A be a non-commutative JBW ∗ -algebra, and let F : X → A∗ be a bounded linear operator. Then the following conditions are equivalent:

292 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem (i) F is weakly compact. + + (ii) There exists ψ ∈ A+ ∗ , and a function N : R → R , such that 1

F  (a) ≤ N(ε) (ψ(a • a∗ )) 2 + εa for all a ∈ A and ε > 0.

(5.10.14)

(iii) There exists a w∗ -continuous linear operator G from A to a complex Hilbert space, and a function N : R+ → R+ , such that F  (a) ≤ N(ε)G(a) + εa for all a ∈ A and ε > 0.

(5.10.15)

(iv) There exists a bounded linear operator G from A to a complex Hilbert space, and a function N : R+ → R+ , such that (5.10.15) holds. Proof (i)⇒(ii) By the assumption (i), E := F(BX ) is a weakly relatively compact subset of A∗ . Let ψ ∈ A+ ∗ be given by Proposition 5.10.32, and take ε > 0. It follows that there exists δ = δ(ε) > 0 with the property that F  (a) ≤ ε whenever a is in BA with ψ(a∗ • a) < δ. Now let a be in A \ {0}, and set a b := ? . ∗ 2ψ(a •a) + a δ Then b ∈ BA and ψ(b∗ • b) < δ. Therefore F  (b) ≤ ε, i.e. @ 2ψ(a∗ • a)  F (a) ≤ ε + a. δ

? 2 Now to conclude that condition (ii) is fulfilled it is enough to take N(ε) := ε δ(ε) . (ii)⇒(iii) Suppose that condition (ii) is fulfilled. Then, by Theorem 5.1.29(ii) and Corollary 5.1.30(iii), the mapping (a, b) → ψ(b∗ • a) is a separately w∗ -continuous non-negative hermitian sesquilinear form on A. Therefore, by Lemma 5.10.34, condition (iii) holds. (iii)⇒(iv) This is clear. (iv)⇒(i) By the assumption (iv) and Lemma 5.10.35, F  is weakly compact, and hence, by Theorem 1.4.45(iii), so is F.  Keeping in mind that the dual of a non-commutative JB∗ -algebra is the predual of a non-commutative JBW ∗ -algebra (cf. Theorem 3.5.34), and that a bounded linear operator between Banach spaces is weakly compact if and only if so is its transpose (cf. Theorem 1.4.45), the following theorem follows from Proposition 5.10.36. Theorem 5.10.37 Let A be a non-commutative JB∗ -algebra, let Y be a complex Banach space, and let T : A → Y be a bounded linear operator. Then the following conditions are equivalent: (i) T is weakly compact. (ii) There exists a positive linear functional ψ ∈ A , and a function N : R+ → R+ , such that 1

T(a) ≤ N(ε) (ψ(a • a∗ )) 2 + εa for all a ∈ A and ε > 0.

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(iii) There exists a bounded linear operator G from A to a complex Hilbert space, and a function N : R+ → R+ , such that T(a) ≤ N(ε)G(a) + εa for all a ∈ A and ε > 0. Proposition 5.10.38 For a non-commutative JB∗ -algebra A, the following conditions are equivalent: (i) The Banach space of A is reflexive. (ii) There exists a positive linear functional ϕ ∈ A satisfying a2 ≤ ϕ(a∗ • a) for every a ∈ A. (iii) The Banach space of A is isomorphic to a complex Hilbert space. Proof (i)⇒(ii) By the assumption (i), A is a non-commutative JBW ∗ -algebra and the identity mapping on A∗ is weakly compact. Therefore we may apply the implication (i)⇒(ii) in Proposition 5.10.36, and take ε = 12 in (5.10.14), to get that   1 1  ψ(a • a∗ ) 2 a ≤ 2N 2 1 2 for some ψ ∈ A+ ∗ and every a ∈ A. By setting ϕ := 4(N( 2 )) ψ, we are provided with the desired positive linear functional in condition (ii). √ (ii)⇒(iii) Suppose that condition (ii) is fulfilled. For a in A, set ||| a ||| := ϕ(a∗ • a). √ Then we have a ≤ ||| a ||| ≤ ϕa for every a ∈ A. Therefore ||| · ||| becomes an equivalent norm on A converting A into a complex Hilbert space. (iii)⇒(i) This is clear. 

According to Theorem 3.5.5, all quadratic non-commutative JB∗ -algebras are examples of non-commutative JB∗ -algebras whose Banach spaces are isomorphic to Hilbert spaces. Thus, by Corollary 3.5.7, there are non-commutative JB∗ -algebras with reflexive Banach space of arbitrary density character. Now we are proving that this cannot happens in the particular case of C∗ -algebras (see Theorem 5.10.43). Let A be a unital non-commutative JB∗ -algebra, and let e be a self-adjoint idempotent in A. We say that e is minimal if e = 0 and there is no nonzero self-adjoint idempotent in A less than or equal to e other than e. According to the equivalence (i)⇔(iv) in Proposition 5.10.21, e is not minimal if and only if either e = 0 or there are nonzero orthogonal self-adjoint idempotentes f , g ∈ A such that e = f + g. Corollary 5.10.39 Let A be a non-commutative JB∗ -algebra whose Banach space is reflexive. Then every nonzero self-adjoint idempotent of A is a finite sum of pairwise orthogonal minimal self-adjoint idempotents. Proof Let ϕ ∈ A be the positive linear functional on A given by the implication (i)⇒(ii) in Proposition 5.10.38, and let ||| · ||| denote the norm on A defined by 7 ||| a ||| := ϕϕ(a∗ • a).

294 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem √ Then ϕ ·  ≤ ||| · ||| ≤ ϕ ·  on A, and hence for a, b ∈ A we have ||| ab ||| ≤ ϕab ≤ ϕab ≤ ||| a |||||| b |||. As a consequence, we have ||| e ||| ≥ 1 whenever e ∈ A is a nonzero idempotent. Moreover, the equality ||| a + b |||2 = ||| a |||2 + ||| b |||2 holds whenever a, b are in A with a∗ • b = 0. Now, if e ∈ A is a nonzero self-adjoint idempotent with ||| e |||2 < 2, then e is minimal. For, otherwise, we could write e = f + g with f , g ∈ A orthogonal nonzero self-adjoint idempotents, and consequently we would have ||| e |||2 = ||| f |||2 + ||| g |||2 ≥ 2, a contradiction. Assume inductively that, for some n ∈ N, every nonzero self-adjoint idempotent such that ||| e |||2 < n is a finite sum of pairwise orthogonal minimal self-adjoint idempotents. If e ∈ A is a nonzero non-minimal self-adjoint idempotent with ||| e |||2 < n + 1, then e = f + g with f , g ∈ A orthogonal nonzero self-adjoint idempotents, so ||| f |||2 = ||| e |||2 − ||| g |||2 < (n + 1) − 1 = n ||| g |||2 = ||| e |||2 − ||| f |||2 < (n + 1) − 1 = n, hence, by the induction hypothesis, e is a finite sum of pairwise orthogonal minimal self-adjoint idempotents.  Fact 5.10.40 Let A be a non-commutative JBW ∗ -algebra, and let e ∈ A be a minimal self-adjoint idempotent. Then Ue (A) = Ce. Proof By Lemma 5.9.2 and Fact 5.10.16, Ue (A) is a w∗ -closed ∗-subalgebra of A, and hence it is a non-commutative JBW ∗ -algebra. But the minimality of e in A implies that e is the unique nonzero self-adjoint idempotent in Ue (A). Therefore the result follows from Theorem 5.1.29(vi).  Lemma 5.10.41 Let A be an associative semiprime algebra over K, and let L be a left ideal of A such that LL = 0. Then L = 0.  Proof The set I := { ni=1 xi ai : n ∈ N, xi ∈ L, ai ∈ A} becomes an ideal of A satisfying II = 0. Therefore I = 0 by semiprimeness of A. But this reads as that L ⊆ Ann (A) := {a ∈ A : aA = 0}. Since Ann (A) is an ideal of A with Ann (A) Ann (A) = 0, we have Ann (A) = 0, again by semiprimeness of A. Therefore L = 0.  When convenient, left ideals of a given associative algebra A will be seen as left A-modules (cf. Example 3.6.34). Fact 5.10.42 Let A be an associative algebra over K, and let e and f be nonzero idempotents in A. We have:

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(i) For each a ∈ eAf the mapping ma : x → xa from Ae to Af is a module homomorphism, and the mapping a → ma becomes a linear bijection from eAf to the vector space over K of all module homomorphisms from Ae to Af . (ii) If A is semiprime, and if eAe = Ke and fAf = Kf , then dim(eAf ) ≤ 1. Proof Certainly, for each a ∈ eAf , the mapping ma : Ae → Af , as defined in (i), is a module homomorphism, and the mapping a → ma is linear. Let a be in eAf such that ma = 0. Then a = ma (e) = 0. Therefore the mapping a → ma is injective. Now let F : Ae → Af be any module homomorphism, and set a := F(e) ∈ Af . Since F(e) = F(e2 ) = eF(e), we realize that a ∈ eAf . Finally, for every x ∈ Ae we have F(x) = F(xe) = xF(e) = xa = ma (x). Suppose that eAe = Ke. Then, as a consequence of assertion (i), the space of all module homomorphisms from Ae to Ae reduces to the scalar multiples of the identity operator on Ae. Now suppose additionally that A is semiprime. Let L be a nonzero submodule of Ae. Then L is a nonzero left ideal of A, and hence, by Lemma 5.10.41, there are x, y ∈ L such that xy = 0. Since xe = x and ye = y, it follows that eye = 0. Therefore, since eAe = Ke, there exists 0 = λ ∈ K such that ey = eye = λe. This implies that e ∈ L since ey ∈ L, and hence that L = Ae. Thus Ae is an irreducible left A-module (cf. Definition 3.6.35). Finally suppose additionally that fAf = Kf and that eAf = 0. Then, by assertion (i), there exists a nonzero module homomorphism F : Ae → Af . On the other hand, by the preceding paragraph, Ae and Af are irreducible left A-modules. Therefore F is bijective (indeed, ker(F) is a submodule of Ae different from Ae, and F(Ae) is a nonzero submodule of Af ). Now, if G : Ae → Af is any module homomorphism, then F −1 ◦ G : Ae → Ae is a module homomorphism, and hence, by the second paragraph in the proof, we have F −1 ◦ G ∈ KIAe , so G ∈ KF. Thus the space of all module homomorphisms from Ae to Af is one-dimensional. By assertion (i) again, dim(eAf ) = 1.  Theorem 5.10.43 Let A be a C∗ -algebra whose Banach space is reflexive. Then A is finite-dimensional. Proof We suppose that A = 0, and note that A is a W ∗ -algebra with A∗ = A . By Corollary 5.10.39, there exist minimal self-adjoint idempotents e1 , . . . , en ∈ A such  that 1 = ni=1 ei . By Fact 5.10.40, we have ei Aei = Cei for every i = 1, . . . , n, so that, by Fact 5.10.42(ii), dim(ei Aej ) ≤ 1 for all i, j = 1, . . . , n. Therefore A = 1A1 =   1≤i,j≤n ei Aej is finite-dimensional. Corollary 5.10.44 Let A be a C∗ -algebra, let B be a normed complex algebra, and let F : A → B be a weakly compact Jordan homomorphism. Then the range of F is finite-dimensional. Proof By Proposition 3.6.11(i), ker(F) is a closed ideal of A, and hence, by Proposition 2.3.43, ker(F) is ∗-invariant, and A/ ker(F) is a C∗ -algebra. Moreover,

296 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem the induced mapping a + ker(F) → F(a) from A/ ker(F) to B becomes a weakly compact Jordan homomorphism with the same range as that of F. Therefore we may suppose that F is injective. Then the mapping a → F(a) becomes a continuous algebra norm on the JB∗ -algebra Asym . Therefore, by Lemma 4.4.28(ii) applied to Asym , there exists M > 0 such that MF(a) ≤ a ≤ Fa for every a ∈ A. Thus F is a topological embedding, hence F(A) is a Banach space and is closed in B. It follows from Fact 1.4.1(iv) and Corollary 1.4.3 that F(A) is a reflexive Banach space. Therefore A is a C∗ -algebra whose Banach space is reflexive, and hence, by Theorem 5.10.43, is finite-dimensional. As a result, F(A) is finite-dimensional.  Now we are going to refine Theorem 5.10.43 by obtaining the same conclusion from much weaker assumptions. Since C∗ -algebras are JB∗ -triples in a natural way (cf. Fact 4.1.41), our first result in this line follows straightforwardly from Corollary 5.8.41(iii) and Theorem 5.10.43. Corollary 5.10.45 Let A be a weakly sequentially complete W ∗ -algebra. Then A is finite-dimensional. §5.10.46 To motivate our next result, let us recall some well-known facts in the theory of Banach spaces. Indeed, for a Banach space X, each of the conditions (i) to (x) which follow implies the subsequent one: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

Finite-dimensionality Hilbertizability Superreflexivity Reflexivity The Radon–Nikodym property ‘Abundance’ of denting points of BX Existence of denting points of BX Existence of slices of BX of arbitrarily small diameter Existence of non-empty relatively w-open subsets of BX of arbitrarily small diameter (x) Existence of non-empty relatively w-open subsets of BX with diameter less than 2.

The implication (viii)⇒(ix) above follows because, in fact, slices of BX are non-empty relatively w-open subsets of BX . We note in addition that denting points of BX are points of w − n continuity of the identity mapping on BX , and that the mere existence of a point of w − n continuity of the identity on BX implies condition (ix) above. We recall that a slice of the closed unit ball of the Banach space X is a set of the form S(X, f , α) := {x ∈ BX : ( f (x)) > 1 − α} for some f in SX  and α > 0, that a denting point of BX is an element of BX such that there are slices of BX of arbitrarily small diameter containing it, and that condition (viii) above is usually called dentability of BX .

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Proposition 5.10.47 Let X be a Banach space over K such that X  is L-embedded. If there exists a non-empty relatively w-open subset of BX with diameter less than 2, then X is reflexive. Proof We have X  = X  ⊕1 N for some subspace N of X  , and hence X  = (X  )◦ ⊕∞ N ◦ . Suppose that there is a non-empty relatively w-open subset U of BX with diam(U) < 2. Then U contains a set V of the form {x ∈ BX : |fi (x − x0 )| < 1 for every i = 1, . . . , n}, for suitable x0 ∈ BX , n ∈ N, and f1 , . . . , fn ∈ X  . Put V  := {z ∈ BX  : |fi (z − x0 )| < 1 for every i = 1, . . . , n}. Since V  is relatively w∗ -open in BX  , and BX is w∗ -dense in BX  , the set V (= V  ∩ BX ) is w∗ -dense in V  . Therefore V − V is w∗ -dense in V  − V  , and consequently, by the lower w∗ -semicontinuity of the norm of X  , we have diam(V  ) = diam(V) ≤ diam(U) < 2. In the same way, the set V  := {β ∈ BX  : |fi (β − x0 )| < 1 for every i = 1, . . . , n} has diameter less than 2. Write x0 = u + v with (u, v) ∈ (X  )◦ × N ◦ . We claim that B(X  )◦ + v is contained in V  . Indeed, for α ∈ B(X  )◦ , α + v belongs to BX  because X  = (X  )◦ ⊕∞ N ◦ , and, on the other hand, for every i = 1, . . . , n we have fi (α + v − x0 ) = fi (α − u) = 0 because (α − u, fi ) belongs to (X  )◦ × X  . Keeping in mind that diam(V  ) < 2, it follows from the claim just shown that diam(B(X  )◦ ) = diam(B(X  )◦ + v) < 2. Therefore X is reflexive.  Now, combining Proposition 5.7.10 and Theorem 5.7.36 with Proposition 5.10.47, we obtain the following. Corollary 5.10.48 Let X be a JB∗ -triple. If there exists a non-empty relatively w-open subset of BX with diameter less than 2, then X is reflexive. Since non-commutative JB∗ -algebras are JB∗ -triples in a natural way (cf. Theorem 4.1.45), it is enough to invoke Proposition 5.10.38 and Corollary 5.10.48 to get the following. Corollary 5.10.49 Let A be a non-commutative JB∗ -algebra. If there exists a non-empty relatively w-open subset of BA with diameter less than 2, then the Banach space of A is isomorphic to a complex Hilbert space. Remark 5.10.50 If one wishes a JB∗ -triple-free proof of the above corollary, then it is enough to invoke Theorem 5.1.32 and Propositions 5.10.47 and 5.10.38. Finally, keeping in mind Theorem 5.10.43, the above corollary yields the following.

298 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Corollary 5.10.51 Let A be a C∗ -algebra. If there exists a non-empty relatively w-open subset of BA with diameter less than 2, then A is finite-dimensional. 5.10.2 The strong∗ topology of a JBW ∗ -triple We recall that, if X is a Jordan ∗-triple over K, and if e is a tripotent in X, then e becomes a unitary element of the Jordan ∗-triple X1 (e), and hence, by Proposition 4.1.54, X1 (e) is a Jordan ∗-algebra over K with unit e in a natural manner. Lemma 5.10.52 Let X be a Jordan ∗-triple over K, let e be a tripotent in X, and let z be a self-adjoint idempotent in the Jordan ∗-algebra X1 (e). Then Uz (X1 (e)) = X1 (z). Proof Clearly, z is tripotent in X. Moreover, by Lemma 5.7.29(iii), we have X1 (z) ⊆ X1 (e). On the other hand, for a ∈ X1 (e) we have za = {zea} = {zza} + {z, e − z, a} = {zza}. Keeping in mind the equalities (5.9.1) in the proof of Lemma 5.9.2, it follows that Uz (X1 (e)) = X1 (z).  Fact 5.10.53 Let X be a JB∗ -triple, let e be a nonzero tripotent in X, and let x be in D(X, e). Then x = x ◦ P1 (e). Proof Since P1 (e) is contractive (cf. inequality (4.2.6) in Proposition 4.2.15), we have 1 = x (e) = x (P1 (e)(e)) ≤ x ◦ P1 (e) ≤ 1, and hence x ◦ P1 (e) = 1. The result now follows from Proposition 5.7.13.  Lemma 5.10.54 Let X be a JB∗ -triple, let e be a nonzero tripotent in X, and let x be in X satisfying x = 1 and P1 (e)(x) = e. Then P 1 (e)(x) = 0. 2

Proof Let x = x1 + x 1 + x0 be the Peirce decomposition of x with respect to e. 2 By assumption x1 = e and we wish to prove that x 1 = 0. Set y := e − ix 1 − x0 and 2

2

z := 12 (x + y). By the equality (4.2.5) in Proposition 4.2.15, we have y = x = 1, hence z ≤ 1, and so {zzz} = z3 ≤ 1. Note that z = e + λx 1 with λ = 12 (1 − i), 2 and that the Peirce arithmetic (cf. Proposition 4.2.22) gives P1 (e)({zzz}) = e + 2|λ|2 {x 1 x 1 e} = e + {x 1 x 1 e}. 2

2

2

2

Now, by the inequality (4.2.6) in Proposition 4.2.15, we have e + {x 1 x 1 e} ≤ {zzz} ≤ 1. 2

2

But, by Proposition 4.2.32(ii), we have 1 + {x 1 x 1 e} = e + {x 1 x 1 e}. Therefore 2 2 2 2 {x 1 x 1 e} = 0, and hence, by Proposition 4.2.32(iii), x 1 = 0.  2

2

2

§5.10.55 Let X be a Jordan ∗-triple over K, and let T denote the set of all tripotents in X. We can define an order relation on T by e ≤ f if and only if f − e is a tripotent orthogonal to e (cf. §5.7.30). We note that, reflexivity of the relation ≤ being obvious, only antisymmetry and transitivity need to be checked.

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Suppose that e, f are tripotents in X such that e ≤ f and f ≤ e. Then f − e is a tripotent and L(e, f − e) = L( f , f − e) = 0, so f − e = {f − e, f − e, f − e} = (L( f , f − e) − L(e, f − e))( f − e) = 0, hence e = f . Now suppose that e, f , g are tripotents in X such that e ≤ f and f ≤ g. Then f − e and g − f are tripotents and L(e, f − e) = L( f − e, e) = L( f , g − f ) = L(g − f , f ) = 0. Therefore {f − e, f − e, g − f } = {f − e, f , g − f } − {f − e, e, g − f } = L(g − f , f )( f − e) − L( f − e, e)(g − f ) = 0, hence f − e and g − f are orthogonal, and so g − e = (g − f ) + ( f − e) is a tripotent. Moreover, we have {e, e, g − e} = {e, e, g − f + f − e} = {e, e, g − f } = {e, f − ( f − e), g − f } = {e, f − e, g − f } = 0, and hence e and g − e are orthogonal. Thus e ≤ g. Corollary 5.10.56 Let X be a JB∗ -triple, and let e, f be tripotents in X. Then e ≤ f if and only if P1 (e)( f ) = e. Proof

Suppose that e ≤ f . Then L(e, f − e) = 0, so P1 (e)( f − e) = Q2e ( f − e) = Qe ({e, f − e, e}) = Qe L(e, f − e)(e) = 0,

hence P1 (e)( f ) = P1 (e)(e) + P1 (e)( f − e) = e. Conversely, suppose that P1 (e)( f ) = e. To prove that e ≤ f , we may suppose that f = 0. Then, by Lemma 5.10.54, we have P 1 (e)( f ) = 0. Therefore 2

x0 := f − e = f − P1 (e)( f ) = P0 (e)( f ) ∈ X0 (e), i.e. {eex0 } = 0. Now it only remains to show that x0 is a tripotent. To this end, note that, by Peirce arithmetic, we have {ex0 x0 } = 0 = {fex0 } and {ffe} = {e + x0 , e + x0 , e} = e. Therefore {x0 x0 x0 } = {f − e, x0 , x0 } = {fx0 x0 } = {f , f − e, x0 } = {ffx0 } = {f , f , f − e} = f − e = x0 , as desired.



Let X be a JBW ∗ -triple, and let e be a tripotent in X. We recall that, in view of Corollaries 4.2.30(iii)(b) and 5.7.21, X1 (e) becomes a JBW ∗ -algebra with unit e in a natural way.

300 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proposition 5.10.57 Let X be a JBW ∗ -triple, and let ϕ be in X∗ . Then there exists a unique tripotent e ∈ X such that ϕ = ϕ ◦ P1 (e) and ϕ|X1 (e) is a faithful w∗ -continuous positive linear functional on the JBW ∗ -algebra X1 (e). Moreover, the tripotent e above satisfies the following properties: (i) ϕ|X1 (e)  = ϕ(e) = ϕ. (ii) e is the support idempotent of ϕ|X1 (e) in the JBW ∗ -algebra X1 (e). Proof Throughout the proof, we may and shall suppose that ϕ = 1. Let us prove the existence of e. By Fact 5.7.25, there is a tripotent f ∈ X such that ϕ ∈ D(X, f ). Therefore, by Fact 5.10.53, we have ϕ = ϕ ◦ P1 ( f ). Set ψ := ϕ|X1 ( f ) . Then, clearly ψ is w∗ -continuous. Moreover, since 1 = ϕ ≥ ψ ≥ ψ( f ) = ϕ( f ) = 1, we realize that ψ lies in D(X1 ( f ), f ), and hence ψ is positive in view of the implication (ii)⇒(i) in Lemma 5.10.2. Let e be the support idempotent of ψ in the JBW ∗ algebra X1 ( f ) (cf. Definition 5.10.18). Then, clearly, e is a tripotent in X. Moreover, by Proposition 5.10.20(ii) and Fact 5.10.53, we have ϕ = ϕ ◦ P1 (e), and ϕ|X1 (e) is a faithful positive w∗ -continuous linear functional (cf. Proposition 5.10.20(i) and Lemma 5.10.52). Now, let e ∈ X be a tripotent satisfying the conditions in the first conclusion of the proposition, and set φ := ϕ|X1 (e) . Since e is the unit of X1 (e), it follows from Lemma 5.10.2 that φ(e) = φ. Moreover, since P1 (e) is contractive (cf. the inequality (4.2.6) in Proposition 4.2.15), for x ∈ X we have |ϕ(x)| = |ϕ(P1 (e)(x))| = |φ(P1 (e)(x))| ≤ φx, and hence 1 = ϕ ≤ φ = φ(e) = ϕ(e) ≤ 1. Thus e fulfils property (i) in the statement. Also e fulfils property (ii) thanks to Proposition 5.10.20(iv). To conclude the proof, let us show the uniqueness of e. Suppose that e1 and e2 are tripotents in X such that both satisfy the conditions in the first conclusion of the proposition. Let φk := ϕ|X1 (ek ) for k = 1, 2. Then P1 (e1 )(e2 ) ∈ X1 (e1 ), and, by property (i) applied to both e1 and e2 , we have φ1 (P1 (e1 )(e2 )) = ϕ(P1 (e1 )(e2 )) = ϕ(e2 ) = 1 = φ1 , which implies that P1 (e1 )(e2 ) = 1. Therefore, by property (ii) (applied to e1 ) and Proposition 5.10.20(iii), P1 (e1 )(e2 ) = e1 , so e1 ≤ e2 by Corollary 5.10.56. By symmetry e2 ≤ e1 , and thus e1 = e2 .  Definition 5.10.58 Let X be a JBW ∗ -triple, and let ϕ be in X∗ . The unique tripotent e ∈ X given by Proposition 5.10.57 will be called the support tripotent of ϕ in X. Lemma 5.10.59 Let X be a Jordan ∗-triple over K, and let e ∈ X be a tripotent. We have: (i) If z0 is in X0 (e), then for every x ∈ X we have P1 (e)({x, x, z0 }) = 0.

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(ii) If ϕ is a linear functional on X such that ϕ = ϕ ◦ P1 (e), then for every x ∈ X we have ϕ(x) = ϕ({eex}).  Proof Let z0 be in X0 (e), and let x be in X. Write x = k∈{1, 1 ,0} xk with xk ∈ Xk (e). 2 Then, by Peirce arithmetic, we have {xxz0 } = u + v, where u := {x1 x 1 z0 } + {x 1 x0 z0 } ∈ X 1 (e) and v := {x 1 x 1 z0 } + {x0 x0 z0 } ∈ X0 (e). 2

2

2

2

2

Therefore, P1 (e)({x, x, z0 }) = 0. Let ϕ be a linear functional on X such that ϕ = ϕ ◦P1 (e), and let x be in X. Then, by Fact 4.2.14(i), we have P1 (e)(x) = {e, e, P1 (e)(x)}. Therefore, since P1 (e) commutes with L(e, e) (cf. Fact 4.2.14(ii)), we have ϕ(x) = ϕ(P1 (e)(x)) = ϕ({e, e, P1 (e)(x)}) = ϕ(P1 (e)({eex})) = ϕ({eex}).



Proposition 5.10.60 Let X be a JBW ∗ -triple. Then we have: (i) For each ϕ ∈ SX∗ and each z ∈ D(X∗ , ϕ), the mapping (x, y) → ϕ({xyz}) becomes a non-negative hermitian sesquilinear form on X, and hence the mapping √ x → xϕ := ϕ({xxz}) is a seminorm on X. (ii) For each x ∈ X and each ϕ ∈ SX∗ , the symbol xϕ above is appropriate because the number ϕ({xxz}) does not depend on the point of support z ∈ D(X∗ , ϕ). (iii) For each x ∈ X and each ϕ ∈ SX∗ , we have |ϕ(x)| ≤ xϕ . (iv) For every x ∈ X, we have sup{xϕ : ϕ ∈ SX∗ } = x. Proof Assertion (i) was already proved in the paragraph immediately before Proposition 4.2.65. Let ϕ be in SX∗ , and let z be in D(X∗ , ϕ). Consider the support tripotent e of ϕ. Then, by the equality ϕ = ϕ ◦ P1 (e) in Proposition 5.10.57, we have 1 = ϕ(z) = ϕ(P1 (e)(z)) ≤ P1 (e)(z) ≤ z = 1, so P1 (e)(z) = 1 and ϕ(P1 (e)(z)) = P1 (e)(z). Therefore, by Propositions 5.10.57(ii) and 5.10.20(iii), we have P1 (e)(z) = e, and hence, by Lemma 5.10.54, P 1 (e)(z) = 0. Now, by Lemma 5.10.59(i), we have 2

P1 (e)({xxz}) = P1 (e)({xxe}) + P1 (e)({x, x, P0 (e)(z)}) = P1 (e)({xxe}) for every x ∈ X. Consequently, ϕ({xxz}) = ϕ(P1 (e)({xxz})) = ϕ(P1 (e)({xxe})) = ϕ({xxe}), so xϕ does not depend on z. Thus assertion (ii) has been proved. Let x be in X, and let ϕ be in SX∗ . As in the preceding paragraph, let e denote the support tripotent of ϕ. Then, by the Cauchy–Schwarz inequality, we have √ |ϕ({xye})| ≤ ϕ({xxe})ϕ({yye}) for every y ∈ X. Taking y = e, and invoking Lemma 5.10.59(ii), the left-hand side of the above inequality converts into |ϕ(x)|, whereas the right-hand side converts into xϕ . Thus assertion (iii) has been proved. Assertion (iv) is nothing other than Proposition 4.2.65. 

302 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Definition 5.10.61 Let X be a JBW ∗ -triple. According to Proposition 5.10.60, each functional ϕ ∈ SX∗ gives rise naturally to a seminorm  · ϕ on X. The strong∗ topology of X is defined as the locally convex topology on X generated by the family of seminorms { · ϕ : ϕ ∈ SX∗ }, and is denoted by s∗ = s∗ (X, X∗ ). Our next result puts in agreement the strong∗ topology of a JBW ∗ -triple, just defined, with the strong∗ topology of a non-commutative JBW ∗ -algebra, introduced in Definition 5.10.3. Indeed, we have the following. Proposition 5.10.62 Let A be a non-commutative JBW ∗ -algebra, and let X denote the JBW ∗ -triple underlying A (cf. §5.7.19). Then the strong∗ topology of A coincides with the strong∗ topology of X. Proof Let ϕ be in D(A, 1) ∩ A∗ . Then ϕ ∈ SX∗ and 1 ∈ D(X∗ , ϕ). Therefore, by Proposition 5.10.60(i)-(ii), for every x ∈ X we have x2ϕ = ϕ({xx1}). But, since {xx1} = x∗ • x, we realize that xϕ = pϕ (x) for every x ∈ A (cf. Definition 5.10.3 again for the notation). Therefore, since ϕ is arbitrary in D(A, 1) ∩ A∗ , we see that the the strong∗ topology of A is weaker than the strong∗ topology of X. Conversely, let ϕ be in SX∗ , choose z ∈ D(X∗ , ϕ), and let ψ denote the mapping x → ϕ(z • x) from A to C. Clearly ψ ∈ A∗ , ψ ≤ 1 and ψ(1) = 1, hence ψ ∈ D(A, 1) ∩ A∗ . Moreover, from the equality {xxz} + {x∗ x∗ z} = 2z • (x∗ • x), we obtain x2ϕ + x∗ 2ϕ = 2ψ(x∗ • x) = 2pψ (x)2 , √ and hence xϕ ≤ 2pψ (x) for every x ∈ A. Since ϕ is arbitrary in SX∗ , we see that the strong∗ topology of A is stronger than the strong∗ topology of X.  Now the next theorem generalizes Proposition 5.10.5(ii) and Lemma 5.10.12. Theorem 5.10.63 Let X be a JBW ∗ -triple. Then we have: (i) The strong∗ topology of X is stronger than the weak∗ topology. (ii) A linear functional on X is s∗ -continuous if and only if it is w∗ -continuous. Proof Let xλ be a net in X s∗ -convergent to x ∈ X. Then, by Proposition 5.10.60(iii), for every λ and every ϕ ∈ SX∗ we have |ϕ(x − xλ )| ≤ x − xλ ϕ , and hence lim ϕ(x − xλ ) = 0 for every ϕ ∈ SX∗ . λ

Since X∗ = CSX∗ , we derive that limλ ϕ(x − xλ ) = 0 for every ϕ ∈ X∗ , i.e. xλ w∗ -converges to x. Thus assertion (i) has been proved. The ‘if’ part of assertion (ii) follows from assertion (i) just proved. Let ϕ be in SX∗ , and let e denote the support tripotent of ϕ. For each x ∈ X, define a linear mapping {ϕ, x, e} : X → C by {ϕ, x, e}(y) := ϕ({yxe}) for every y ∈ X.

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Then {ϕ, x, e} ∈ X∗ since the triple product of X is separately w∗ -continuous (cf. Theorem 5.7.20). Let xλ be a net in X w∗ -convergent to some x ∈ X. Then for every y ∈ X, we have {ϕ, xλ , e}(y) = ϕ({yxλ e}) → ϕ({yxe}) = {ϕ, x, e}(y) since the triple product of X is separately w∗ -continuous. Thus, the conjugatelinear mapping x → {ϕ, x, e} from X to X∗ is w∗ -w continuous. In particular, the set {ϕ, BX , e} is w-compact and absolutely convex. Now let τ denote the Mackey topology on X relative to its duality with X∗ , and note that the closed convex subsets of X are the same for both τ and w∗ . Let f be an s∗ -continuous linear functional on X, and let yλ be a net in ker( f ) ∩ BX τ -converging to some y ∈ X. Then ψ(y − yλ ) → 0 uniformly for ψ ∈ 2{ϕ, BX , e} by definition of the Mackey topology. Since y − yλ ∈ 2BX for every λ, we deduce that y − yλ 2ϕ = {ϕ, y − yλ , e}(y − yλ ) → 0. By the arbitrariness of ϕ ∈ SX∗ , we realize that the net yλ s∗ -converges to y. Then, since f is s∗ -continuous, and yλ ∈ ker( f ) for every λ, we have y ∈ ker( f ). In this way we have shown that ker( f ) ∩ BX is τ -closed (so w∗ -closed) in X. By the Krein– ˇ Smulyan theorem, ker( f ) is w∗ -closed in X, and hence f is w∗ -continuous. Thus the proof of the ‘only if’ part of assertion (ii) is concluded.  The next corollary generalizes Proposition 5.10.13. Corollary 5.10.64 Let Y be a JBW ∗ -triple, and let X be a w∗ -dense subtriple of Y. Then BX is s∗ -dense in BY . Proof By Theorem 5.10.63(ii), the closed convex subsets of Y are the same for  both w∗ and s∗ . Therefore the result follows from Corollary 5.7.47. Now we are going to describe the strong∗ topology of a JBW ∗ -triple in terms which have a sense for any dual Banach space. Lemma 5.10.65 Let X be a Jordan ∗-triple over K, and let e ∈ X be a tripotent. Then e is complete if and only if the operator L(e, e) is bijective. Proof If L(e, e) is bijective, then X0 (e) = 0 because X0 (e) = ker(L(e, e)). Conversely, if X0 (e) = 0, then, by Fact 4.2.14(ii), we have (L(e, e) − IX )(2L(e, e) − IX ) = 0, hence IX = L(e, e)(3IX − 2L(e, e)), which shows that L(e, e) is invertible in L(X).



Lemma 5.10.66 Let X be a JBW ∗ -triple, let Y be a complex Banach space, and let T : X → Y be a w∗ -w-continuous linear operator which attains its norm at some element of BX . Then T attains its norm at a complete tripotent of X. Proof We may suppose that T = 1. Let x ∈ BX be such that T(x) = 1, and take y ∈ SY  satisfying y (T(x)) = 1. Then y ◦ T ∈ X∗ and y ◦ T = 1. Therefore, by

304 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Fact 5.7.25, there exists a complete tripotent e ∈ X such that y (T(e)) = 1. Now, clearly, T(e) = 1.  Proposition 5.10.67 Let X be a JBW ∗ -triple, let H be a complex Hilbert space, and let T : X → H be a w∗ -continuous linear operator which attains √ its norm at some element of BX . Then there exists ϕ ∈ SX∗ such that T(x) ≤ 2Txϕ for every x ∈ X. Proof By Lemma 5.10.66, T attains its norm at some complete tripotent e ∈ X. We suppose that T = 1. Define ϕ(x) = (T(x)|T(e)) for every x ∈ X, and note that 1 = ϕ = ϕ(e). Fix a ∈ X. Since 1 L(e, a) + L(a, e) = (L(e + a, e + a) − L(e − a, e − a)) ∈ H(BL(X), IX ) 2 and e = 1, it follows from Corollary 2.1.9(iii) that, for every t ∈ R with |t| small, we have   2 1 2 2 1 ≥ T(exp[it(L(e, a) + L(a, e))](e)) = T e + italin − t asq + O(|t|3 ), 2 (5.10.16) where alin := {eae} + {aee} and asq := {ea{eae}} + {ea{aee}} + {ae{eae}} + {ae{aee}}. By (5.10.16) and the parallelogram law we have   2 1 2 2 2 T(e) − t T(asq ) + itT(alin ) ≤ 2 + O(|t|3 ) 2 for every t ∈ R with |t| small. Now, 1 T(e) − t2 T(asq ) 2

2

 2   1 ≥  T(e) − t2 T(asq )|T(e)  2  2   1 2  = 1 − t ϕ(asq ) 2

(5.10.17)

(5.10.18)

= 1 − t2 ϕ(asq ) + O(|t|4 ). As t → 0, (5.10.17) and (5.10.18) yield T(alin )2 ≤ ϕ(asq ).

(5.10.19)

We pause to calculate ϕ(asq ). Write ak = Pk (e)(a), for k = 1, 12 . By identity (4.1.15) in Definition 4.1.32, {ea{eae}} = {e{aea}e}. Consequently, using Proposition 5.10.60(i), Fact 5.10.53, and Peirce arithmetic, we have ϕ({ea{eae}}) = ϕ({{aea}ee}) = ϕ({{a1 ea1 }ee}) = ϕ({a1 ea1 }) = ϕ({aea}).

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By Fact 5.10.53 and Peirce arithmetic, we also have ϕ({ae{aee}}) = ϕ({aea}). Thus, ϕ({ae{aee}} + {ea{eae}}) = 2(ϕ({aea})).

(5.10.20)

Along similar lines one can establish that 1 ϕ({ea{aee}} + {ae{eae}}) = 2ϕ({a1 a1 e}) + ϕ({a 1 a 1 e}) 2 2 2 = 2ϕ({{eea}{eea}e})

(5.10.21)

= 2(ϕ({{eea}{eea}e})). By (5.10.19) to (5.10.21) we have T({aee}) + T({eae})2 ≤ 2(ϕ({{eea}{eea}e} + {aea})).

(5.10.22)

Writing (5.10.22) with ia instead of a, we obtain T({aee}) − T({eae})2 ≤ 2(ϕ({{eea}{eea}e} − {aea})).

(5.10.23)

Summing up inequalities (5.10.22) and (5.10.23), and applying the parallelogram law again, we get T({aee}2 + T({eae}2 ≤ 2(ϕ({{eea}{eea}e})), and hence T({aee})2 ≤ 2(ϕ({{eea}{eea}e})). Now let x = {eea}. Then T(x)2 ≤ 2(ϕ({xxe})). This holds for all x ∈ X because, by Lemma 5.10.65, the operator L(e, e) is bijective. Since ϕ(e) = 1 implies that x2ϕ = ϕ({xxe}) (cf. Proposition 5.10.60), this completes the proof.  Lemma 5.10.68 Let X and Y be dual Banach spaces over K, and let F : X → Y be a w∗ -continuous linear operator such that there exist sequences xn in SX and ϕn in SY∗ satisfying |ϕm (F(xn ))| ≥ F − m1 for all n, m ∈ N with m ≤ n. Then F attains its norm at some element of BX . Proof Take a w∗ -cluster point x ∈ BX of the sequence (xn )n∈N , and let us fix m ∈ N. Then, since F is w∗ -continuous, |ϕm (F(x))| is a cluster point of the sequence (|ϕm (F(xn ))|)n∈N . Therefore, since |ϕm (F(xn ))| ≥ F − m1 for all n ≥ m we derive that |ϕm (F(x))| ≥ F − m1 . But F ≥ F(x) ≥ |ϕm (F(x))|, and m is arbitrary in N. Therefore F(x) = F.  Proposition 5.10.69 Let X and Y be dual Banach spaces over K, let T : X → Y be a w∗ -continuous linear operator, and let ε > 0. Then there is a w∗ -continuous linear operator F : X → Y which attains its norm at some element of BX and such that T − F is compact with T − F ≤ ε. Proof We may suppose that T = 1 and 0 < ε < 13 . Choose a decreasing sequence εn of positive numbers such that

306 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem 2

∞ 

εm < εn2 for every n ∈ N, and 2

m=n+1

∞ 

εm < ε.

(5.10.24)

m=1

Next choose inductively sequences Tn in BL(X, Y), xn in SX , and ϕn in SY∗ such that T1 = T

(5.10.25)

Tn (xn ) ≥ Tn  − εn2 for every n ∈ N,

(5.10.26)

ϕn (Tn (xn )) ≥ Tn (xn ) − εn2 for every n ∈ N,

(5.10.27)

Tn+1 (x) = Tn (x) + εn ϕn (Tn (x))Tn (xn ) for all x ∈ X and n ∈ N.

(5.10.28)

Having chosen these sequences we verify that the following hold. Tn − Tm  ≤ 2

n−1 

εk for all n, m ∈ N with m < n, (5.10.29)

k=m

4 2 ≤ Tn  ≤ for every n ∈ N; 3 3 Tn+1  ≥ Tn  + εn Tn 2 − 4εn2 for every n ∈ N;

(5.10.30)

Tn  ≥ Tm  ≥ 1 for all n, m ∈ N with m < n;

(5.10.31)

and

|ϕm (Tm (xn ))| ≥ Tm  − 6εm for all n, m ∈ N with m < n.

(5.10.32)

Assertion (5.10.29) is proved by using induction on n, as follows. Since we suppose that T = 1, it follows from (5.10.25) that T1  = 1, and hence 23 ≤ T1  ≤ 43 . Moreover, by (5.10.28), we see that for every x ∈ X (T2 − T1 )(x) = ε1 ϕ1 (T1 (x))T1 (x1 ) ≤ 2ε1 x, and hence T2 − T1  ≤ 2ε1 . Then, as a consequence of the second assertion in (5.10.24) we obtain that 4 T2  ≤ T2 − T1  + T1  ≤ 2ε1 + 1 ≤ ε + 1 ≤ , 3 and 2 T2  ≥ T1  − T2 − T1  ≥ 1 − 2ε1 ≥ 1 − ε ≥ . 3 Now, suppose that n ≥ 2 is such that (5.10.29) holds for every k ≤ n. Given m < n+1, by (5.10.28), we see that for every x ∈ X (Tn+1 − Tm )(x) = =

n  k=m n 

(Tk+1 − Tk )(x) ≤

n 

εk ϕk (Tk (x))Tk (xk ) ≤

k=m

(Tk+1 − Tk )(x)

k=m n  k=m

εk Tk 2 x ≤ 2x

n  k=m

εk ,

5.10 Complements on non-commutative JB∗ -algebras and JB∗ -triples 307  and hence Tn+1 − Tm  ≤ 2 nk=m εk . As a consequence of the second assertion in (5.10.24) we obtain that Tn+1  ≤ Tn+1 − T1  + T1  ≤ 2

n  k=1

4 εk + 1 ≤ ε + 1 ≤ , 3

and Tn+1  ≥ T1  − Tn+1 − T1  ≥ 1 − 2

n  k=1

2 εk ≥ 1 − ε ≥ , 3

and the proof of the induction is complete. For the remaining part of the proof, we note that 1 1 for every n ≥ 2. ε1 ≤ , and εn ≤ 6 10n

(5.10.33)

Indeed, it follows from the second assertion in (5.10.24) that ε1 ≤ 12 ε ≤ 16 , and it 1 1 follows from the first assertion in (5.10.24) that ε2 ≤ 12 ε12 ≤ 12 36 ≤ 10.2 . Now, it 1 is suffice to observe that, if n ≥ 2 is such that εn ≤ 10n , then the first assertion in 1 (5.10.24) gives that εn+1 ≤ 12 εn2 ≤ 12 1012 n2 ≤ 10(n+1) . To prove (5.10.30), first of all note that, by (5.10.26), (5.10.27), and (5.10.29), for every n ∈ N we have ϕn (Tn (xn )) ≥ Tn (xn ) − εn2 ≥ Tn  − 2εn2 ≥

2 2 4 − = > 0, 3 9 9

and hence, by (5.10.26), (5.10.27), and (5.10.28), we realize that Tn+1  ≥ Tn+1 (xn ) = [1 + εn ϕn (Tn (xn ))]Tn (xn ) ≥ (1 + εn Tn  − 2εn3 )(Tn  − εn2 ) ≥ Tn  + εn Tn 2 − εn2 (3εn Tn  + 1). Relation (5.10.30) follows easily from this inequality, since, as a consequence of (5.10.33), we have εn Tn  ≤ 1 for every n ∈ N . Now, (5.10.31) is an immediate consequence of (5.10.25) and (5.10.30) since, again by (5.10.33), we have 4εn ≤ Tn 2 for every n ∈ N. Finally, we verify (5.10.32). By the triangle inequality, (5.10.26), (5.10.29), (5.10.31), and (5.10.24), for m < n we have Tm+1 (xn ) ≥ Tn (xn ) − Tn − Tm+1  ≥ Tn  − εn2 − 2

n−1 

2 εk ≥ Tm+1  − 2εm .

k=m+1

Hence, by (5.10.28) and (5.10.30), 2 , εm |ϕm (Tm (xn ))|Tm  + Tm  ≥ Tm+1 (xn ) ≥ Tm  + εm Tm 2 − 6εm

so that |ϕm (Tm (xn ))| ≥ Tm  − 6εm , and this proves (5.10.32).  Now let δ > 0. According to (5.10.24), take p ∈ N such that 2 ∞ m=p εm < δ. It follows from (5.10.29) that Tm − Tn  < δ whenever n, m ≥ p. Therefore Tn is a

308 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Cauchy sequence in BL(X, Y), and hence converges in norm to some F ∈ BL(X, Y). By (5.10.24), (5.10.25), and (5.10.29) we have T − Tn  = T1 − Tn  ≤ 2

n−1 

εk < ε for every n ∈ N

k=1

and Tm+1 − Tn  ≤ 2

n−1 

2 εk < εm for all m < n,

k=m+1

and letting n → ∞ we have 2 for every m ∈ N. T − F ≤ ε and Tm+1 − F ≤ εm

(5.10.34)

On the other hand, writing (5.10.28) as (Tn+1 − T)(x) = (Tn − T)(x) + εn [ϕn ((Tn − T)(x))Tn (xn ) + ϕn (T(x))Tn (xn )], and keeping in mind (5.10.25) and that T is w∗ -continuous, an easy induction argument shows that, for every n ∈ N, Tn − T is a finite-rank and w∗ -continuous operator. Therefore F − T is compact and w∗ -continuous. Now, to conclude the proof it is enough to show that F satisfies the requirements in Lemma 5.10.68. Indeed, clearly, F = (F − T) + T is w∗ -continuous. Moreover, by (5.10.32) and (5.10.34), for 2 ≤ m ≤ n we have 2 |ϕm (F(xn ))| ≥ |ϕm (Tm (xn ))| − Tm − F ≥ Tm  − 6εm − εm−1 2 2 ≥ F − F − Tm  − 6εm − εm−1 ≥ F − 6εm − 2εm−1 ≥ F −

where the last inequality is consequence of (5.10.33).

1 , m 

Lemma 5.10.70 Let X and Y be dual Banach spaces over K, and let T : X → Y be a compact and w∗ -continuous linear operator. Then T attains its norm at some element of BX . Proof We may suppose that T = 1. Then, since T is compact, there is a sequence xn in BX in such a way that T(xn ) → 1 and that the sequence T(xn ) converges to some y ∈ BY in the norm topology of Y. As a consequence, y = 1 and the sequence T(xn ) converges to y in the weak∗ topology of Y. Now take a w∗ -cluster point x ∈ BX of the sequence xn . Since T is w∗ -continuous, T(x) is a w∗ -cluster point of the sequence T(xn ). Therefore T(x) = y, and hence T(x) = y = 1.  Proposition 5.10.71 Let X be a JBW ∗ -triple, let H be a complex Hilbert space, and √ let T : X → H be a w∗ -continuous linear operator. Let K > 2 and ε > 0. Then there exist ϕ1 , ϕ2 ∈ SX∗ such that  1 T(x) ≤ K T x2ϕ2 + ε2 x2ϕ1 2 for every x ∈ X.

5.10 Complements on non-commutative JB∗ -algebras and JB∗ -triples

309

Proof Without loss of generality we can suppose T = 1. Take δ > 0 such that 7 2 δ ≤ ε and 2((1 + δ)2 + δ) ≤ K. By Proposition 5.10.69, there is a w∗ -continuous linear operator F : X → Y which attains its norm at some element of BX and such that T − F is compact with T − F ≤ δ. Then, by Lemma 5.10.70, T − F also attains its norm. Therefore, by Proposition 5.10.67, there exist ϕ1 , ϕ2 ∈ SX∗ such that √ √ (T − F)(x) ≤ 2T − Fxϕ1 and F(x) ≤ 2Fxϕ2 for every x ∈ X. Therefore for x ∈ X we have √ √ T(x) ≤ F(x) + (T − F)(x) ≤ 2Fxϕ2 + 2T − Fxϕ1 √ √ √ ≤ 2(1 + δ)xϕ2 + 2δ δxϕ1 ! ! "1 "1 7 2 2 ≤ 2((1 + δ)2 + δ) x2ϕ2 + δx2ϕ1 ≤ K x2ϕ2 + ε2 x2ϕ1 .



Given a JBW ∗ -triple X and functionals ϕ1 , ϕ2 ∈ SX∗ , we denote by  · ϕ1 ,ϕ2 the seminorm on X defined by  1 xϕ1 ,ϕ2 := x2ϕ2 + x2ϕ1 2 for every x ∈ X. Taking K = 2 and ε = 1 in Proposition 5.10.71, we get the following. Corollary 5.10.72 Let X be a JBW ∗ -triple, and let T be a w∗ -continuous linear operator from X to a complex Hilbert space. Then there exist ϕ1 , ϕ2 ∈ SX∗ such that T(x) ≤ 2Txϕ1 ,ϕ2 for every x ∈ X. Corollary 5.10.73 Let A be a non-commutative JBW ∗ -algebra, let H be a complex Hilbert space, let T : A → H be a w∗ -continuous linear operator, and let M > 2. Then there exists ψ ∈ D(A, 1) ∩ A∗ such that 1

T(x) ≤ MT (ψ(x • x∗ )) 2 for every x ∈ A. ? √ Proof Taking K := M and ε := M−2 in Proposition 5.10.71, we find 2 ϕ1 , ϕ2 ∈ SA∗ such that 1  T(x) ≤ KT x2ϕ2 + ε2 x2ϕ1 2 for every x ∈ A. Let x be in A, and let i = 1, 2. As we saw in the proof of Proposition 5.10.62, there exist ψi ∈ D(A, 1) ∩ A∗ such that x2ϕi ≤ 2ψi (x • x∗ ). Therefore T(x) ≤ Finally, writing ψ := T(x) ≤



1 (ψ2 + ε2 ψ1 ), 1+ε2

7

1

2KT(ψ2 (x • x∗ ) + ε2 ψ1 (x • x∗ )) 2 . we have ψ ∈ D(A, 1) ∩ A∗ and 1

1

2(1 + ε2 )KT(ψ(x • x∗ )) 2 = MT(ψ(x • x∗ )) 2 .



310 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Corollary 5.10.74 Let A be a non-commutative JB∗ -algebra, and let T be a bounded linear operator from A to a complex Hilbert space. Then there exists a positive linear functional ψ ∈ SA satisfying 1

T(x) ≤ 2T (ψ(x • x∗ )) 2 for every x ∈ A. Proof T  becomes a w∗ -continuous linear operator from the non-commutative JBW ∗ -algebra A (cf. Theorem 3.5.34) to H. Therefore, by Corollary 5.10.73, for each n ∈ N there is a positive linear functional ψn ∈ SA satisfying   1 1 T(ψn (x • x∗ )) 2 T(x) ≤ 2 + n for every x ∈ A. Take in A a w∗ -cluster point η of the sequence ψn . Then η is a positive linear functional with η ≤ 1, and the inequality 1

T(x) ≤ 2T(η(x • x∗ )) 2 holds for every x ∈ A. If η = 0, then T = 0 and nothing has to be proved. Otherwise 1 take ψ := η η.  For a good understanding of the following corollaries, we note that, since the bidual of a JB∗ -triple X is a JBW ∗ -triple whose triple product extends that of X (cf. Proposition 5.7.10), for each ϕ ∈ SX  , the seminorm  · ϕ on X  can and will be restricted to X. Corollary 5.10.75 Let X be a JB∗ -triple, let H be a√complex Hilbert space, and let T : X → H be a bounded linear operator. Let K > 2 and ε > 0. Then there exist ϕ1 , ϕ2 ∈ SX  such that  1 T(x) ≤ K T x2ϕ2 + ε2 x2ϕ1 2 for every x ∈ X. Proof T  becomes a w∗ -continuous linear operator from the JBW ∗ -triple X  to H. Now apply Proposition 5.10.71.  Corollary 5.10.76 Let X be a JB∗ -triple, and let T be a bounded linear operator from X to a complex Hilbert space. Then there exist ϕ1 , ϕ2 ∈ SX  such that T(x) ≤ 2Txϕ1 ,ϕ2 for every x ∈ X. Proof

Take K = 2 and ε = 1 in Corollary 5.10.75.



Theorem 5.10.77 Let X be a dual Banach space over K. Then the following topologies coincide in X: (i) The topology on X generated by the family of seminorms of the form 7 x → x, x, where ., . is any separately w∗ -continuous non-negative hermitian sesquilinear form on X.

5.10 Complements on non-commutative JB∗ -algebras and JB∗ -triples

311

(ii) The topology on X generated by the family of seminorms x → T(x), when T runs over all w∗ -continuous linear operators from X to arbitrary Hilbert spaces over K. Moreover, if X is a JBW ∗ -triple, then the above topologies coincide with the strong∗ topology of X. Proof Let us denote by τ1 and τ2 the topologies arising in paragraphs (i) and (ii), respectively. Let T be any w∗ -continuous linear operator from X to a complex Hilbert space. Then the mapping (x, y) → x, y := (T(x)|T(y)) from X × X to K becomes a separately w∗ -continuous non-negative hermitian sesquilinear form on X satisfying √ T(x) = x, x for every x ∈ X. Keeping in mind the arbitrariness of T, this shows that τ2 ≤ τ1 . Conversely, let ·, · be any separately w∗ -continuous non-negative hermitian sesquilinear form on X. Then, by Lemma 5.10.34, there exists a w∗ -continuous √ linear operator T from X to a Hilbert space over K such that x, x = T(x) for every x ∈ X. Keeping in mind the arbitrariness of ·, ·, this shows that τ1 ≤ τ2 . Now suppose that X is a JBW ∗ -triple. Then, by Corollary 5.10.72, we have τ2 ≤ s∗ . Let ϕ ∈ SX∗ . Take z ∈ D(X∗ , ϕ). Then, according to Theorem 5.7.20 and Proposition 5.10.60, the mapping (x, y) → x, y := ϕ({xyz}) from X × X to C becomes a separately w∗ -continuous non-negative hermitian sesquilinear form on X satisfying x2ϕ = x, x for every x ∈ X. Keeping in mind the arbitrariness of ϕ ∈ SX∗ , this shows that s∗ ≤ τ1 .  Definition 5.10.78 Let X be a dual Banach space over K. In agreement with Theorem 5.10.77, the strong∗ topology of X is defined as the topology on X generated by the family of seminorms x → T(x), where T is any w∗ -continuous linear operator from X to a Hilbert space over K. This topology will be denoted by s∗ = s∗ (X, X∗ ). Now we can generalize Theorem 5.10.63 as follows. Proposition 5.10.79 Let X be a dual Banach space over K. Then we have: (i) The strong∗ topology of X is stronger than the weak∗ topology. (ii) A linear functional on X is s∗ -continuous if and only if it is w∗ -continuous. Proof Assertion (i) follows since each element of X∗ becomes a w∗ -continuous linear operator from X to the Hilbert space K. Let τ denote the Mackey topology on X relative to its duality with X∗ . In view of assertion (i) and the Mackey–Arens theorem, to prove (ii) it is enough to show that s∗ ≤ τ . Let T be a w∗ -continuous linear operator from X to a Hilbert space H over K. Then T = S for some S ∈ BL(H∗ , X∗ ) and, by reflexivity of H∗ , D := S(BH∗ ) becomes a w-compact and absolutely convex subset of X∗ satisfying T(x) = sup |ϕ(x)| for every x ∈ X. ϕ∈D

312 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Keeping in mind the arbitrariness of T and the definition of the Mackey topology, this shows that s∗ ≤ τ , as desired.  Corollary 5.10.80 Let X and Y be dual Banach spaces over K, and let F : X → Y be a linear mapping. Then F is s∗ -continuous if and only if it is w∗ -continuous. Proof Suppose that F is w∗ -continuous. Let T be any w∗ -continuous linear operator from Y to a Hilbert space over K. Then / T := T ◦ F is a w∗ -continuous linear operator from X to a Hilbert space over K satisfying T(F(x)) = / T (x) for every x ∈ X. Keeping in mind the arbitrariness of T and the definition of the strong∗ topologies on both X and Y (cf. Definition 5.10.78), we realize that F is s∗ -continuous. Now suppose that F is s∗ -continuous. To prove that F is w∗ -continuous it is enough to show that ϕ ◦ F : X → K is w∗ -continuous whenever ϕ is an arbitrary functional in Y∗ . Let ϕ be such a functional. By the supposed s∗ -continuity of F and Proposition 5.10.79(i), ϕ ◦ F is s∗ -continuous. Therefore, by Proposition 5.10.79(ii), ϕ ◦ F is w∗ -continuous, as desired.  We note that, via Corollary 5.1.30(i), Proposition 5.10.62, Theorem 5.10.77, and Definition 5.10.78, Corollary 5.10.11 is fully contained in Corollary 5.10.80. Along the same lines, it is enough to combine Corollaries 5.1.43 and 5.10.80 to get the following. Corollary 5.10.81 Let A be a non-commutative JBW ∗ -algebra. Then we have: (i) Derivations of A are s∗ -continuous. (ii) Bijective algebra homomorphisms from A to any non-commutative JBW ∗ algebra are s∗ -continuous. Corollary 5.10.82 Let X be a dual Banach space over K, and let Y be a w∗ -closed subspace of X. Then s∗ (X, X∗ )|Y ≤ s∗ (Y, Y∗ ). Proof Since the inclusion Y → X is w∗ -continuous, it follows from Corollary 5.10.80 that it is s∗ (Y, Y∗ )-s∗ (X, X∗ )-continuous, so that the result follows.  Proposition 5.10.83 Let X be a JBW ∗ -triple, and let Y be a w∗ -closed subtriple of X. Then s∗ (X, X∗ ) and s∗ (Y, Y∗ ) coincide on bounded subsets of Y. Proof Let g be in SY∗ , and let n be in N. Since Y∗ = X∗ /Z, where Z is the unique closed subspace of X∗ such that Y equals the polar of Z in the duality (X∗ , X) (cf. §5.1.9), there exists en ∈ X∗ with (en )|Y = g and 1 ≤ en  < 1 + 4n12 . Now take y ∈ D(Y∗ , g) ⊆ SY ⊆ SX , and set dn := en −1 en . Then we have |1 − dn (y)| < 4n12 so, by the Bishop–Phelps–Bollob´as theorem (cf. Theorem 2.9.7), there are fn ∈ SX∗ and yn ∈ D(X∗ , fn ) such that  fn − dn  < 1n and yn − y < 1n . Now, for any b ∈ Y we have |b2g − en b2fn | = |g({bby}) − en  fn ({bbyn })| = |en ({bby}) − en  fn ({bbyn })|

5.10 Complements on non-commutative JB∗ -algebras and JB∗ -triples

313

= en  |dn ({bby}) − fn ({bbyn })| = en  |dn ({b, b, y − yn }) + (dn − fn )({bbyn })|   1 2 ≤ 1+ 2 b2 . 4n n Since n is arbitrary in N, it follows that  · 2g is the uniform limit on bounded subsets of Y of a sequence of restrictions to Y of functions (namely en  · 2fn ) which are s∗ (X, X∗ )-continuous. Therefore  · g is s∗ (X, X∗ )-continuous on bounded subsets of Y. Since g is arbitrary in SY∗ , this shows that s∗ (Y, Y∗ ) ≤ s∗ (X, X∗ ) on bounded subsets of Y. The converse inclusion follows from Corollary 5.10.82.  As we have seen in Subsection 5.10.4, Proposition 5.10.83 becomes one of the key tools to prove that the triple product of a JBW ∗ -triple is jointly s∗ -continuous on bounded sets (compare Proposition 5.10.5(v)). Keeping in mind Definition 5.10.78, Proposition 5.10.79 is better understood if we are aware of the following result. Proposition 5.10.84 Let X be a dual Banach space over K. Then the Mackey topology m(X, X∗ ) coincides with the topology on X generated by the family of seminorms x → T(x), where T is any w∗ -continuous linear operator from X to a reflexive Banach space over K. Proof Let us denote by τ the second topology arising in the statement. As in the proof of Proposition 5.10.79, if T is any w∗ -continuous linear operator from X to a reflexive Banach space over K, then there exists a w-compact and absolutely convex subset D of X∗ such that the equality T(x) = supϕ∈D |ϕ(x)| holds for every x ∈ X. This shows that τ ≤ m(X, X∗ ). Conversely, let D be a w-compact and absolutely convex subset of X∗ . Consider the Banach space 1 (D) and the bounded linear operator F : 1 (D) → X∗ given by  F({λϕ }ϕ∈D ) := λϕ ϕ. ϕ∈D

Then we have F(B 1 (D) ) = D, and hence F is weakly compact. By the Davis–Figiel– Johnson–Pełczy´nski theorem (cf. Theorem 1.4.46), there exists a reflexive Banach space Y, together with bounded linear operators S : 1 (D) → Y and R : Y → X∗ , such that F = R ◦ S. Then, for x ∈ X, we have sup |ϕ(x)| = sup |x(F(z))| = sup |x(R(S(z)))| ϕ∈D

z∈B 1 (D)

z∈B 1 (D)

≤ S sup |x(R(y))| = SR (x). y∈BY

Since D is an arbitrary w-compact and absolutely convex subset of X∗ , and R is a w∗ -continuous linear operator from X to the reflexive Banach space Y  , the inequality m(X, X∗ ) ≤ τ follows. 

314 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Theorem 5.10.33 suggests to studying those dual Banach spaces X which satisfy the conclusion in that theorem (cf. Theorem 5.10.77 and Definition 5.10.78). This is done in the next result. Proposition 5.10.85 Let X be a dual Banach space over K. Then the following conditions are equivalent: (i) The topologies m(X, X∗ ) and s∗ (X, X∗ ) coincide on bounded subsets of X. (ii) For every w∗ -continuous linear operator F from X to a reflexive Banach space over K, there exists a w∗ -continuous linear operator G from X to a Hilbert space over K satisfying F(x) ≤ G(x) + x for every x ∈ X. (iii) For every w∗ -continuous linear operator F from X to a reflexive Banach space over K, there exist a w∗ -continuous linear operator G from X to a Hilbert space over K and a mapping N : R+ → R+ satisfying F(x) ≤ N(ε)G(x) + εx for all x ∈ X and ε > 0. Proof (i)⇒(ii) Let F be a w∗ -continuous linear operator from X to a reflexive Banach space over K. Then, by Proposition 5.10.84 N := {y ∈ BX : F(y) ≤ 1} is a m(X, X∗ )|BX -neighbourhood of zero in BX . By assumption, there exist Hilbert spaces H1 , . . . , Hn over K and w∗ -continuous linear operators Gi : X → Hi (i : 1, . . . , n) 4 1ε ). Then G is w∗ -continuous. Indeed, given y = {hn } ∈ H, for each n ∈ N, the mapping αn : x → (Gn (x)|hn ) is an element of X∗ , and we have

5.10 Complements on non-commutative JB∗ -algebras and JB∗ -triples A A B∞ B∞ ∞ ∞    B 1 hn  B αn 2 C ≤ hn  C < ∞. ≤ nGn  n n2 n=1

n=1

n=1

315

n=1

∞

αn Therefore α := n=1 n G is an element of X∗ which, in view of (5.10.36), satisfies n (G(x)|y) = α(x) for every x ∈ X. Since y is arbitrary in H, the w∗ -continuity of G follows. Moreover, keeping in mind (5.10.35), (5.10.36) and the definitions of n(·) and N(·), for all ε > 0 and x ∈ X we have

F(x) ≤

1 1 Gn(ε) (x) + x n(ε) n(ε)

≤ Gn(ε) G(x) +

1 x ≤ N(ε)G(x) + εx. n(ε)

(iii)⇒(i) Let xλ be a net in BX converging to zero in the topology s∗ (X, X∗ ). Let F be a w∗ -continuous linear operator from X to a reflexive Banach space over K, and let ε > 0. By assumption, there exist a w∗ -continuous linear operator G from X to a Hilbert space over K and a mapping N : R+ → R+ satisfying !ε" ε G(x) + x for every x ∈ X. F(x) ≤ N 2 2 ε Take λ0 such that G(xλ ) ≤ 2 N( ε whenever λ ≥ λ0 . Then we have F(xλ ) ≤ ε 2) for every λ ≥ λ0 . By Proposition 5.10.84, xλ m(X, X∗ )-converges to zero. 

5.10.3 Isometries of non-commutative JB∗ -algebras As usual in our work, the bidual A of any normed algebra A will be seen without notice as a normed algebra relative to the Arens product. Now let A be a non-commutative JB∗ -algebra. Once more, we recall that A is in fact a non-commutative JBW ∗ -algebra with a unit, that the involution of A extends that of A and is w∗ -continuous, and that the product of A is separately w∗ -continuous (cf. Lemma 2.3.51 and Theorem 3.5.34). We define the set of multipliers, M(A), of A as the subset of A given by the equality M(A) := {x ∈ A : xA + Ax ⊆ A}. In the particular case that A is a C∗ -algebra, we know that A is an ideal of M(A) (cf. §2.2.16) and that, in a categorical sense, M(A) is the largest C∗ -algebra containing A as an essential ideal (cf. pp. 149–50 of Volume 1). In the general case, it is clear that M(A) is a closed ∗-invariant subspace of A containing A and the unit of A . In this way, the equality M(A) = A holds if and only if A has a unit (cf. Corollary 2.2.13). It is also clear that, if M(A) were a subalgebra of A , then A would be an ideal of M(A). Actually, we are showing in what follows that M(A) is in fact a subalgebra of A (and hence a non-commutative JB∗ -algebra) which contains A as an essential ideal. We also will show that M(A) is the largest non-commutative JB∗ -algebra containing A as a closed essential ideal.

316 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Our argument begins by recalling that the bidual B of any JB-algebra B is a JBW-algebra with a unit, and that the product of B is separately w∗ -continuous (cf. Proposition 3.1.10). Then we invoke Proposition 3.1.18, which is formulated here as follows. Lemma 5.10.86 Let B be a JB-algebra. Then M(B) := {x ∈ B : xB ⊆ B} is a subalgebra of B . Now let X be a complex Banach space. By a conjugation on X we mean an involutive conjugate-linear isometry on X. Conjugations τ on a complex Banach space X give rise by natural transposition to conjugations (also denoted by τ ) on X  . As we commented in §4.2.60, given a conjugation τ on X, we have a natural identification H(X, τ ) ≡ H(X  , τ ) as dual Banach spaces. On the other hand, we recall that, given any non-commutative JB∗ -algebra A, the self-adjoint part of A, regarded as a real subalgebra of the JB∗ -algebra Asym , is a JB-algebra (cf. Corollary 3.4.3), and that the equality (Asym ) = (A )sym holds (thanks to the Arens regularity of A). Therefore it is enough to invoke Lemma 4.2.61(iii) to obtain the following. Corollary 5.10.87 Let A be a non-commutative JB∗ -algebra. Then the Banach space identification H(A, ∗) ≡ H(A , ∗) is also a JB-algebra identification. Now, combining Lemma 5.10.86 and Corollary 5.10.87, the next result follows. Corollary 5.10.88 Let A be a (commutative) JB∗ -algebra. Then M(A) is a subalgebra of A . Now, to obtain the non-commutative generalization of the above corollary we only need a single new fact, which is proved in the next lemma. We recall that every derivation of a JB∗ -algebra is automatically continuous (cf. Lemma 3.4.26). Lemma 5.10.89 Let A be a JB∗ -algebra, and let D be a derivation of A. Then M(A) is D -invariant. Proof Let a, x be in A and M(A), respectively. Since D is a derivation of A , we have D (x)a = D(xa) − xD(a) ∈ A . Therefore, since a is arbitrary in A, it follows that D (x) lies in M(A).



Theorem 5.10.90 Let A be a non-commutative JB∗ -algebra. Then M(A) is a normclosed ∗-subalgebra of A containing A as an essential ideal. Moreover, if B is any non-commutative JB∗ -algebra containing A as a closed essential ideal, then B can be seen as a closed ∗-subalgebra of M(A) containing A. In addition we have M(A) = M(Asym ). Proof Keeping in mind the equality (A )sym = (Asym ) , the inclusion M(A) ⊆ M(Asym ) is clear. Let b be in A. Then the mapping D : a → [b, a] from

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A to A is a derivation of Asym (cf. Lemma 2.4.15), and we have D (x) = [b, x] for every x in A . It follows from Lemma 5.10.89 that D (M(Asym )) ⊆ M(Asym ), or equivalently [b, x] ∈ M(Asym ) for every x in M(Asym ). Therefore, for x in M(Asym ) we have [b2 , x] = 2b • [b, x] ∈ A • M(Asym ) ⊆ A . Since b is arbitrary in A, and A is the linear hull of the set of squares of its elements, we deduce that [A, x] ⊆ A for every x in M(Asym ). It follows M(Asym )A + AM(Asym ) ⊆ A , and hence M(Asym ) ⊆ M(A). Then the equality M(Asym ) = M(A) is proved. Now, for x, y ∈ M(A) and a ∈ A we have [x, y] • a = [x, y • a] − [x, a] • y ∈ [M(A), A] + A • M(A) ⊆ A , and hence [M(A), M(A)] ⊆ M(Asym ). On the other hand, Corollary 5.10.88 applies to Asym giving M(A) • M(A) ⊆ M(Asym ) • M(Asym ) ⊆ M(Asym ). It follows from the first paragraph of the proof that M(A) is a subalgebra of A . Suppose that P is an ideal of M(A) with P ∩ A = 0. Then, since A is an ideal of M(A), we actually have AP = 0, so A P = 0, and so P = 0. Therefore A is an essential ideal of M(A). Let B be a non-commutative JB∗ -algebra, and let ϕ : A → B be an injective (automatically isometric) algebra ∗-homomorphism such that ϕ(A) is an essential ideal of B. Then ϕ  is an injective algebra ∗-homomorphism from A to B whose range is a w∗ -closed ideal of B . By Fact 5.1.10(i), we have ϕ  (A ) = B e for a suitable central (automatically self-adjoint) idempotent e in B . Now (B (1 − e)) ∩ B is an ideal of B whose intersection with ϕ(A) is zero, and hence (B (1 − e)) ∩ B = 0 because ϕ(A) is an essential ideal of B. Therefore the mapping ψ : b → be from B to ϕ  (A ) is an injective algebra ∗-homomorphism. Then η := (ϕ  )−1 ◦ ψ is an injective algebra ∗-homomorphism from B to A satisfying η(ϕ(a)) = a for every a in A. In this way we can see B as a closed ∗-invariant subalgebra of A containing A as an ideal. In this regarding we have clearly B ⊆ M(A).  Remark 5.10.91 Let A be a non-commutative JB∗ -algebra. The above theorem allows us to say that M(A) is the non-commutative JB∗ -algebra of multipliers of A. The equality M(A) = M(Asym ) in the theorem can be understood in the sense that, if we consider the JB-algebra H(A, ∗), then the JB-algebra of multipliers of H(A, ∗) (in the sense of Lemma 5.10.86) is nothing other than H(M(A), ∗). Now we are going to discuss multipliers on JB∗ -triples. To this end, we begin by proving the following. Proposition 5.10.92 Let X be a JB∗ -triple, and let I be a closed subspace of X. Then I is a triple ideal of X if (and only if) {uxy} ∈ I whenever u ∈ I and x, y ∈ X. Proof

From the Jordan triple identity (4.1.13), for u ∈ I and x, y, z, v ∈ X, {x{vuy}z} = {{uvx}yz} + {xy{uvz}} − {uv{xyz}}.

318 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem By hypothesis, the right-hand side belongs to I. To deduce that {vuy} ∈ I we need the simple fact that, for each w ∈ X, there is a sequence xn ∈ X such that w = limn→∞ {xn wxn }. This can be checked using the facts that the closed subtriple of X generated by w is (isometrically) triple-isomorphic to C0C (E) for some subset E of R+ such that E ∪ {0} is compact, and that, in this identification, w converts into the inclusion mapping E → C (cf. Theorem 4.2.9). Thus it is sufficient to prove the result for the case that X = C0C (E), for E as above, and that w(t) = t for every t ∈ E. In this case, it is enough to define the desired sequence xn by xn (t) = 1 if t ≥ 1n , and xn (t) = nt otherwise.  The next fact generalizes Corollary 4.2.11. Given an element x in a JB∗ -triple X, we write Xx for the closed subtriple of X generated by x. Fact 5.10.93 Let X be a JB∗ -triple, let x be in X, and let n be a natural number. Then there exists a unique y ∈ X such that y(2n+1) = x. Moreover, the element y above lies in Xx . Proof Suppose at first that X is equal to (the JB∗ -triple underlying) C0C (E) for some locally compact Hausdorff topological space E. Then, for all z ∈ X and m ∈ N, we have z(2m+1) (t) = |z(t)|2m z(t) for every t ∈ E. Therefore the mapping y : E → C, defined by y(t) = 2n+1√x(t) 2n if x(t) = 0 and y(t) = 0 otherwise, is the unique element |x(t)|

in X such that y(2n+1) = x. Now let X be an arbitrary JB∗ -triple, and let x be in X \ {0}. Then, according to Theorem 4.2.9 and a half of the preceding paragraph there exists y ∈ Xx such that y(2n+1) = x. Let z be any element of X satisfying z(2n+1) = x. Then Xx ⊆ Xz , and hence y, z ∈ Xz . It follows from Theorem 4.2.9 again and the other half of the preceding paragraph that z = y.  Lemma 5.10.94 Let X be a JB∗ -triple, let x be in X, and let n be a natural number. Then Xx = Xx(2n+1) . Proof Since Xx ⊇ Xx(2n+1) , it is enough to show the converse inclusion. But it follows from Fact 5.10.93 (taking there x(2n+1) instead of x) that x ∈ Xx(2n+1) . Therefore Xx ⊆ Xx(2n+1) , as desired.  We recall that the bidual X  of any JB∗ -triple X becomes a JB∗ -triple under a triple product extending that of X (cf. Proposition 5.7.10). Proposition 5.10.95 Let X be a JB∗ -triple. Then the set Mult(X) := {x ∈ X  : {xXX} ⊆ X} is a closed subtriple of X  (hence a JB∗ -triple). It is the largest closed subtriple of X  which contains X as a triple ideal. Proof Since Mult(X) is clearly norm closed, the second statement will follow from the first, by Proposition 5.10.92. Let x ∈ Mult(X) and a ∈ X. Then, by (4.1.15),

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we have {axa(2n+3) } = {a{xaa(2n+1) }a} ∈ X for every n ≥ 0. This implies that {axXa(3) } ⊆ X. Therefore, by Lemma 5.10.94, we have that {axa} ∈ X, and hence that {XxX} ⊆ X, upon linearizing. In turn, this shows that {xx{aaa}} = 2{a{xaa}x} − {{axa}ax} ∈ X where we have used (4.2.15). Therefore {xx{XXX}} ⊆ X (cf. the polarization formula in §4.2.76) and {xxX} ⊆ X (cf. Corollary 4.2.11). In addition, the Jordan triple identity (4.1.13) and then (4.2.13) gives {x{aaa}x} = 2{{aax}ax} − {aa{xax}} = 2{{aax}ax} − {a{axa}x} ∈ X which implies that {xXx} ⊆ X. Consequently, {{xxx}aa} = 2{xx{xaa}} − {x{aax}x} ∈ X from which we deduce that {{xxx}XX} ⊆ X, so that {xxx} ∈ Mult(X). Hence Mult(X) is a subtriple of X  .  According to the above proposition, given a JB∗ -triple X, the set Mult(X) defined above is called the JB∗ -triple of multipliers of X. Now, given a given noncommutative JB∗ -algebra A, we can consider the non-commutative JB∗ -algebra of multipliers of A, M(A), and the JB∗ -triple of multipliers of the JB∗ -triple underlying A (cf. Theorem 4.1.45). Actually, the following result holds. Proposition 5.10.96 Let A be a non-commutative JB∗ -algebra. Then we have M(A) = Mult(A). Proof The inclusion M(A) ⊆ Mult(A) is clear. To prove the converse inclusion, we start by noticing that, clearly, the JB∗ -triples underlying A and Asym coincide, and that, by Theorem 5.10.90, the equality M(A) = M(Asym ) holds, so that we may suppose that A is commutative. We note also that, since the equality {xyz}∗ = {x∗ y∗ z∗ } is true for all x, y, z in A , and A is a ∗-invariant subset of A , Mult(A) is ∗-invariant too, and therefore it is enough to show that ax lies in A whenever a and x are selfadjoint elements of A and Mult(A), respectively. But, for such a and x, we can find a self-adjoint element b in A satisfying b3 = a, and apply the Shirshov–Cohn theorem (cf. Theorem 3.1.55) to obtain that xa = xb3 = b(2Ux,b (b) − Ub (x)) = b(2{xbb} − {bxb}) belongs to A.



In Theorem 4.1.55, we determined those JB∗ -triples which underlie unital noncommutative JB∗ -algebras. Indeed, keeping in mind that the JB∗ -triple underlying a given non-commutative JB∗ -algebra A is the same as the one underlying the JB∗ -algebra Asym , and summarizing something, Theorem 4.1.55 asserts that a nonzero JB∗ -triple underlies a unital non-commutative JB∗ -algebra if and only if it has a unitary element (cf. Definition 4.1.53). Now we can prove the unit-free version of this result.

320 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Proposition 5.10.97 A JB∗ -triple X underlies a non-commutative JB∗ -algebra if and only if the JB∗ -triple of multipliers of X has a unitary element. Proof Suppose that X underlies a non-commutative JB∗ -algebra A. Then, keeping in mind that A = X  as Banach spaces, we easily realize that the unit 1 of the noncommutative JB∗ -algebra A becomes a unitary element of the JB∗ -triple X  . Since 1 ∈ M(A), it is enough to invoke Proposition 5.10.96 to obtain that 1 is a unitary element of Mult(X). Conversely, suppose that Mult(X) has a unitary element e. By Theorem 4.1.55, Mult(X), endowed with the product xy := {xey} and the involution x∗ := {exe}, becomes a JB∗ -algebra (say B). Since X is a triple ideal of Mult(X), we realize that X is a (closed) ∗-subalgebra of B, and hence it is a JB∗ -algebra (say A). Finally, that X is the JB∗ -triple underlying A follows from the last conclusion in Theorem 4.1.55 (applied to Mult(X)).  Now that multipliers on non-commutative JB∗ -algebras have been studied, we are going to do a non-associative discussion of the classical Kadison–Paterson–Sinclair theorem on isometries of C∗ -algebras (cf. Theorem 2.2.19), which we formulate as follows. Theorem 5.10.98 Let A and B be nonzero C∗ -algebras, and let F be a mapping from B to A. Then F is a surjective linear isometry (if and) only if there exists a surjective Jordan-∗-homomorphism G from B onto A, and a unitary element u in the C∗ -algebra of multipliers of A satisfying F(b) = uG(b) for every b in B. Concerning concepts involved in the statement, the general formulation of Theorem 5.10.98 could have a sense in the more general setting of nonzero noncommutative JB∗ -algebras, by replacing ‘unitary element’ with ‘J-unitary element’ (cf. Definition 4.2.25) and ‘the C∗ -algebra of multipliers’ with ‘the non-commutative JB∗ -algebra of multipliers’. Nevertheless, even the ‘if’ part of Theorem 5.10.98 does not remain true in the setting of non-commutative JB∗ -algebras. Indeed, surjective Jordan-∗-homomorphisms between non-commutative JB∗ -algebras are isometries but, as a matter of fact, left multiplications by J-unitary elements of a unital noncommutative JB∗ -algebra need not be isometries. This handicap becomes more than an anecdote in view of the following result, which in fact characterizes alternative C∗ -algebras among non-commutative JB∗ -algebras. Given an element x in the multiplier non-commutative JB∗ -algebra of a non-commutative JB∗ -algebra A, we denote by Tx : A → A the linear operator defined by Tx (a) := xa for every a ∈ A. We recall also that J-unitaries and unitaries of an alternative ∗-algebra are the same (cf. §4.1.1 and Definitions 4.1.2 and 4.1.56). Proposition 5.10.99 For a nonzero non-commutative JB∗ -algebra A, the following conditions are equivalent: (i) A is an alternative C∗ -algebra. (ii) For every J-unitary element u of M(A), Tu is a surjective isometry. (iii) For every J-unitary element u of M(A), Tu is an isometry.

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Proof (i)⇒(ii) Suppose that A is an alternative C∗ -algebra. Then it follows from Corollary 3.5.35 and Theorem 5.10.90 that M(A) is a unital alternative C∗ -algebra. Therefore, by Lemma 3.4.30, left multiplications on M(A) by unitary elements of M(A) are surjective linear isometries. Since A is invariant under these isometries and their inverses, it follows that Tu : A → A is a surjective isometry whenever u is a unitary element in M(A). (ii)⇒(iii) This is clear.  (iii)⇒(i) For x in A , denote by LxA the operator of left multiplication by x on A .  We remark that LxA = (Tx ) whenever x belongs to M(A) (indeed, both sides of the equality are w∗ -continuous operators on A coinciding on A). Let h be in H(M(A), ∗), and let r be a real number. Then exp(irh) is a J-unitary element of M(A), and thereA = (Texp(irh) ) just established, fore, by the assumption (iii) and the equality Lexp(irh) 





A A A Lexp(irh) is an isometry on A . Now set Gr := Lexp(irh) ◦ Lexp(−irh) . Then Gr is an   isometry on A preserving the unit of A . Since G0 = IA and the mapping r → Gr is continuous, there exists a positive number k such that Gr is surjective whenever |r| < k. It follows from Proposition 3.4.25 that, for |r| < k, Gr is a Jordan 1 n ∗-automorphism of A . If n≥0 n! r Fn is the power series development of Gr ,   then we easily obtain F0 = IA , F1 = 0, and F2 = 2((LhA )2 − LhA2 ). By Corollary 3.2.2 and Lemma 3.2.4, F2 is a Jordan ∗-derivation of A . Therefore, by Lemma 3.4.27, we have

F2 ∈ iH(BL(A ), IA ),

(5.10.37)

and then, by Lemma 2.3.21, sp(F2 ) ⊆ iR. Now we argue as in the conclusion of the  proof of Theorem 3.2.5: by Lemmas 2.1.10 and 2.3.21, we have also sp(LhA ) ⊆ R    and sp(LhA2 ) ⊆ R, so that, since LhA and LhA2 commute (by Corollary 3.2.2(ii)), Corollary 1.1.81(i) gives 











sp(F2 ) = sp((LhA )2 − LhA2 ) ⊆ sp((LhA )2 ) − sp(LhA2 ) ⊆ (sp(LhA ))2 − sp(LhA2 ) ⊆ R. It follows that sp(F2 ) = {0}. Then, invoking (5.10.37) and Proposition 2.3.22, we   have 0 = F2 = (LhA )2 − LhA2 . In particular, for x in M(A) we have h(hx) = h2 x. Since h is an arbitrary element of H(M(A), ∗), an easy linearization argument gives y(yx) = y2 x for all x, y in M(A). By applying the involution of M(A) to both sides of the above equality, it follows that M(A) (and hence A) is the alternative.  Remark 5.10.100 As we have pointed out at the beginning of the above proof, given an alternative C∗ -algebra A, the non-commutative JB∗ -algebra of multipliers of A is in fact an alternative C∗ -algebra. Therefore it will be called the alternative C∗ -algebra of multipliers of A. Now that we know that alternative C∗ -algebras are the unique non-commutative JB∗ -algebras which can play the role of A in a verbatim non-associative generalization of the ‘if’ part of Theorem 5.10.98, we proceed to prove that they are in fact ‘good’ for the non-associative generalization of the ‘only if’ part of that theorem. Let X and Y be JB∗ -triples, and let F : X → Y be a surjective linear isometry. It follows easily from Theorem 5.6.57 that F  (Mult(X)) = Mult(Y). In the particular

322 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem case of non-commutative JB∗ -algebras (where an autonomous proof of Theorem 5.6.57 was provided in Proposition 4.2.44), we can apply Proposition 5.10.96 to arrive in the following non-associative generalization of Corollary 2.2.20. Lemma 5.10.101 Let A and B be non-commutative JB∗ -algebras, and let F : B → A be a surjective linear isometry. Then we have F (M(B)) = M(A). In particular, bijective Jordan-∗-homomorphisms from B to A extend uniquely to bijective Jordan∗-homomorphisms from M(B) to M(A). Proof In view of the previous comments, we only must prove the uniqueness of bijective Jordan-∗-homomorphisms from M(B) to M(A) extending a given bijective Jordan-∗-homomorphism (say G) from B to A. But, if R and S are bijective Jordan∗-homorphisms from M(B) to M(A) extending G, then for b in B and x in M(B) we have (R(x) − S(x)) • G(b) = R(x) • R(b) − S(x) • S(b) = R(x • b) − S(x • b) = G(x • b) − G(x • b) = 0.



Theorem 5.10.102 Let A be a nonzero alternative C∗ -algebra, let B be a noncommutative JB∗ -algebra, and let F be a mapping from B to A. Then F is a surjective linear isometry if and only if there exists a bijective Jordan-∗-homomorphism G : B → A, and a unitary element u in the alternative C∗ -algebra of multipliers of A satisfying F = Tu ◦ G. Proof The ‘if’ part follows from Proposition 3.4.4 (see also Remark 3.4.5) and the implication (i)⇒(ii) in Proposition 5.10.99. Suppose that F is a surjective linear isometry. Then, as a consequence of Lemma 5.10.101, F extends to a surjective linear isometry H : M(B) → M(A). Since M(A) is a unital alternative C∗ -algebra, Corollary 3.4.32 applies, so that there exists a bijective Jordan-∗-homomorphism T : M(B) → M(A), and a unitary element u in M(A) ◦ T. Therefore, since H extends F, for every b ∈ B M(A) such that H = Lu we have F(b) = uT(b), and hence T(b) = u∗ F(b) ∈ M(A)A ⊆ A. It follows that G := T|B , regarded as an operator from B to A, is a bijective Jordan-∗-homomorphism satisfying F = Tu ◦ G.  Since every non-commutative JB∗ -algebra has an approximate unit bounded by 1 (cf. Proposition 3.5.23), the following corollary is apparently better than the theorem. Corollary 5.10.103 Let A be a nonzero alternative C∗ -algebra, let B be a normed complex algebra having an approximate unit bounded by 1, and let F : B → A be a surjective linear isometry. Then there exists a bijective Jordan homomorphism G : B → A, and a unitary element u ∈ M(A) satisfying F = Tu ◦ G. Proof Since B has an approximate unit bounded by 1, and is linearly isometric to an alternative C∗ -algebra, and alternative C∗ -algebras are non-commutative JB∗ algebras, it follows from Corollary 5.9.11 that B is a non-commutative JB∗ -algebra for some involution. Now apply Theorem 5.10.102. 

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As promised in p. 157 of Volume 1, the above corollary becomes a unit-free version of Proposition 2.3.20. Through Corollary 5.9.11, the proof we have just given involves the main result in the chapter, namely Theorem 5.9.9. Now we are going to provide the reader with a second proof of Corollary 5.10.103, which somehow avoids Theorem 5.9.9, and has it own interest. To this end, we begin by proving the following generalization of Lemma 2.2.17(ii). Lemma 5.10.104 Let X be a Jordan-∗-triple over K, let Y be a subtriple of X, and let u be a unitary element of X. Suppose that Qy (u) and Qu (y) belong to Y for every y ∈ Y. Then {uyz} lies in Y for all y, z ∈ Y. Proof

Linearizing the assumption Qy (u) ∈ Y for every y ∈ Y, we get {yuz} ∈ Y for all y, z ∈ Y.

(5.10.38)

On the other hand, for each y, z ∈ Y, the Jordan triple identity (4.1.13) gives {uy{uuz}} = {{uyu}uz} − {u{yuu}z} + {uu{uyz}}, and hence, keeping in mind that u is unitary, we derive that {uyz} = {{uyu}uz}. Since, by assumption, {uyu} ∈ Y, the result follows from (5.10.38).  For the proof of the next proposition, we recall that the product of any JBW ∗ algebra is separately w∗ -continuous (cf. Corollary 5.1.30(iii)), and that the triple product of the bidual of any JB∗ -triple is separately w∗ -continuous (cf. Theorem 5.7.18). Proposition 5.10.105 Let X be a JB∗ -triple, and let u be a unitary element of X  such that Qx (u) belongs to X whenever x is in X. Then u lies in the JB∗ -triple of multipliers of X. Proof According to Theorem 4.1.55, X  , endowed with the product xy := {xuy} and the involution x∗ := Qu (x), becomes a JBW ∗ -algebra (say B). Then, by the assumption on u, X can be seen as a subalgebra of B (say A). Since B is equal to A endowed with the Arens product (as both algebras have the same Banach spaces, and the products of both algebras are separately w∗ -continuous and coincide on A), it follows from Theorem 5.9.7 that A is ∗-invariant. This means that Qu (x) belongs to X for every x ∈ X. Now invoke the assumption on u again, and apply Lemma 5.10.104.  Combining the implication (i)⇒(vi) in Theorem 4.2.28, and Propositions 5.10.96 and 5.10.105, we obtain the following. Corollary 5.10.106 Let A be a nonzero non-commutative JB∗ -algebra, and let u be a J-unitary element in A such that Ua (u∗ ) belongs to A whenever a is in A. Then u lies in the non-commutative JB∗ -algebra of multipliers of A. Keeping in mind Corollary 3.5.35 (cf. also Remark 5.10.100), Corollary 5.10.106 yields the following.

324 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Corollary 5.10.107 Let A be a nonzero alternative C∗ -algebra, and let u be a unitary element in A such that au∗ a belongs to A whenever a is in A. Then u lies in the alternative C∗ -algebra of multipliers of A. As a straightforward consequence of the above corollary, if A is a C∗ -algebra and if u is a unitary element in A such that au∗ a ∈ A for every a ∈ A, then ua∗ u ∈ A for every a ∈ A, a fact whose proof was promised in Remark 2.2.18. Second proof of Corollary 5.10.103 Let A, B, and F be as in Corollary 5.10.103. Then, since B has an approximate unit bounded by 1, and is linearly isometric to an alternative C∗ -algebra, and every product on an alternative C∗ -algebra is Arens regular (a consequence of Fact 5.8.39), we realize that B is Arens regular, and hence, by Lemma 3.5.24(ii) and its proof, B is norm-unital. Since A is unital, we are in a position to apply Corollary 3.4.32 to obtain the existence of a bijective Jordan∗-homomorphism H : B → A together with a unitary element u ∈ A satisfying F  (b ) = uH(b ) for every b ∈ B . Now, for an arbitrary element a ∈ A, we can write a = F(b) for a suitable b in B, and using Proposition 2.5.24(ii) we see that au∗ a = F(b)u∗ F(b) = (uH(b))u∗ (uH(b)) = uH(b)H(b) = uH(b2 ) = F(b2 ) ∈ A. By Corollary 5.10.107, u belongs to the alternative C∗ -algebra of multipliers of A. Therefore H maps B onto A, and the mapping G : b → H(b) from B onto A is a bijective Jordan homomorphism.  To conclude the discussion about verbatim non-associative versions of Theorem 5.10.98, we show that alternative C∗ -algebras are also the unique non-commutative JB∗ -algebras which can play the role of A in the ‘only if’ part of such versions. Let A be a non-commutative JB∗ -algebra, and let u be a J-unitary element of M(A). By applying Lemma 4.2.41 to the JB∗ -algebra (A )sym , we realize that the Banach space of A with product •u and involution ∗u defined by a •u b := Ua,b (u∗ ) and a∗u := Uu (a∗ ), respectively, becomes a JB∗ -algebra. Such a JB∗ -algebra will be denoted by A(u). Proposition 5.10.108 Let A be a non-commutative JB∗ -algebra which is not an alternative C∗ -algebra. Then there exists a non-commutative JB∗ -algebra B, and a surjective linear isometry F : B → A which cannot be written as Tu ◦ G with u a J-unitary element in M(A) and G a bijective Jordan-∗-homomorphism from B to A. Proof By Proposition 5.10.99, there is a J-unitary element v in M(A) such that Tv is not an isometry on A. Take B equal to A(v), and F : B → A equal to the identity mapping. Suppose that F = Tu ◦ G for some J-unitary u in M(A) and some bijective Jordan-∗-homomorphism G from B to A. Noticing that the JB∗ -algebra B is nothing but A (v) (by the w∗ -continuity of the involutions and the separate w∗ -continuity of the products on biduals of non-commutative JB∗ -algebras), we have G (v) = 1 (because v is the unit of A (v) and G is a bijective Jordan-∗-homomorphism), and hence F  (v) = u. Therefore v = u (because F is the identity mapping). Finally, the equality F = Tv ◦ G implies that Tv is an isometry, contrary to the choice of v. 

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Now that the verbatim non-associative variant of Theorem 5.10.98 has been altogether discussed, we pass to consider the consequence of that theorem, namely that linearly isometric C∗ -algebras are Jordan-∗-isomorphic. We already know that the natural non-associative variant of such an assertion is not true. Indeed, according to Antitheorem 3.4.34, linearly isometric non-commutative JB∗ -algebras need not be Jordan-∗-isomorphic, the counterexample being given by two unital JC∗ -algebras. Therefore, for a non-commutative JB∗ -algebra A, we consider the property (P) which follows. (P) Non-commutative JB∗ -algebras which are linearly isometric to A are in fact Jordan-∗-isomorphic to A. As a consequence of Theorem 5.10.102, all alternative C∗ -algebras fulfil Property (P). But we already know that the class of those non-commutative JB∗ -algebras A satisfying Property (P) is much larger than that of alternative C∗ -algebras, since it contains all non-commutative JBW ∗ -algebras (cf. Corollary 5.10.10), and therefore (passing to the bidual) every non-commutative JB∗ -algebra can be enlarged to a noncommutative JB∗ -algebra fulfilling Property (P). Another way of constructing noncommutative JB∗ -algebras which are not alternative, but have Property (P), is that of considering appropriate ‘mutations’ of alternative C∗ -algebras. §5.10.109 Let A be an algebra over K, and let λ be in K. The λ-mutation of A, denoted by A(λ) , is defined as the algebra whose vector space is that of A, and whose product (say ) is defined by a b := λab + (1 − λ)ba. Without enjoying its name, the above notion was already involved in Example 4.5.43 and subsequent discussions (cf. Remarks 4.5.44 and 4.5.53(b)). We note that (A(λ) )sym = Asym , and that flexibility passes from A to A(λ) . Therefore, if A is a non-commutative Jordan algebra, then so is A(λ) . We note also that, if A is a normed algebra, and if λ lies in the closed real interval [0, 1], then A(λ) becomes a normed algebra under the norm of A. It follows from Fact 3.3.4 and the above considerations that, if A is a non-commutative JB∗ -algebra, then so is A(λ) for every λ ∈ [0, 1]. Now let A be a non-commutative JB∗ -algebra, and let λ be in [0, 1]. Since Property (P) means the same for A and Asym , and (A(λ) )sym = Asym , it follows that (A(λ) )sym satisfies Property (P) whenever A does. Therefore it is enough to invoke Theorem 5.10.102 to derive the following. Corollary 5.10.110 Let B be an alternative C∗ -algebra, let λ be in [0, 1], and set A := B(λ) . Then A fulfils Property (P). Other sufficient conditions for a non-commutaive JB∗ -algebra to satisfy Property (P) are obtained in what follows. They rely on the roots of the fact already emphasized that non-commutative JBW ∗ -algebras have Property (P) (cf. Corollary 5.10.8 and Theorem 5.10.9). We remark that, if u is a J-unitary element in the noncommutative JB∗ -algebra of multipliers of a non-commutative JB∗ -algebra A, then the operator Uu , acting on A, is a surjective linear isometry. Indeed, Uu , acting on

326 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem A , is a surjective linear isometry (cf. Theorem 4.2.28), and A is invariant under Uu and its inverse Uu∗ . Theorem 5.10.111 Let A be a nonzero non-commutative JB∗ -algebra. Then the following conditions are equivalent: (i) For every non-commutative JB∗ -algebra B, and every surjective linear isometry F : B → A, there exists a bijective Jordan-∗-homomorphism G : B → A, and a J-unitary element u ∈ M(A) satisfying F = Uu ◦ G. (ii) For each J-unitary element v ∈ M(A) there is a J-unitary element u ∈ M(A) such that u2 = v. As a consequence, if A satisfies condition (ii) above, then A fulfils Property (P). Proof (i)⇒(ii) Let v be a J-unitary element of M(A). Take B equal to A(v), and F : B → A equal to the identity mapping. By the assumption (i), we have F = Uu ◦ G for some J-unitary element u ∈ M(A) and some bijective Jordan-∗-homomorphism G : B → A. Arguing as in the proof of Proposition 5.10.108, we find G (v) = 1, and hence F  (v) = u2 . Therefore v = u2 (because F is the identity mapping). (ii)⇒(i) Let B be a non-commutative JB∗ -algebra, and let F : B → A a surjective linear isometry. Put v := F  (1). By Theorem 4.2.28, v is a J-unitary element of A , and, by Lemma 5.10.101, v belongs to M(A). By the assumption (ii), there is a J-unitary element u in M(A) with u2 = v. Write G := Uu∗ ◦ F. Then G is a bijective Jordan-∗-homomorphism from B to A because it is a surjective linear isometry satisfying G (1) = 1, and Proposition 3.4.25 applies. On the other hand, the equality F = Uu ◦ G is clear.  The next result is a variant of Theorem 5.10.111. For a good understanding of its formulation and proof, we note that, if u, v are J-unitary elements of a unital non-commutative Jordan ∗-algebra A over K, then Uu (v) is J-unitary. Indeed, by Proposition 3.4.15 and Theorem 4.1.3, Uu (v) is J-invertible with −1 (Uu (v))−1 = UU Uu (v) = Uu−1 Uv−1 Uu−1 Uu (v) = Uu−1 (v−1 ) = Uu∗ (v∗ ) = (Uu (v))∗ . u (v)

Thus the set of all J-unitary elements of A is invariant under Uu and its inverse Uu∗ , and hence Uu acts naturally as a bijection in that set. Theorem 5.10.112 Let A be a nonzero non-commutative JB∗ -algebra, let JU stand for the set of all J-unitary elements of M(A), and let G denote the subgroup of Inv(L(M(A))) generated by the set {Uu : u ∈ JU}. Then the following conditions are equivalent: (i) For every non-commutative JB∗ -algebra B, and every surjective linear isometry F : B → A, there exists a bijective Jordan-∗-homomorphism G : B → A, and an operator T ∈ G , satisfying F = T ◦ G. (ii) G acts transitively on JU. As a consequence, if A satisfies condition (ii) above, then A fulfils Property (P).

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Proof We note that for every u ∈ JU we have u∗ ∈ JU and Uu−1 = Uu∗ , and that this implies that G = {Uu1 ◦ · · · ◦ Uun : n ∈ N; u1 , . . . , un ∈ JU}. Then condition (ii) becomes equivalent to the following. (iii) For each v ∈ JU, there is n ∈ N, and u1 , u2 , . . . , un ∈ JU such that Uu1 Uu2 · · · Uun (1) = v. Note also that condition (ii) in Theorem 5.10.111 is nothing other than the case n = 1 of condition (iii) above. Now, keeping in mind these remarks, the current proof is similar to that of Theorem 5.10.111, and therefore is left to the reader.  Remark 5.10.113 As we have pointed out in the above proof, condition (ii) in Theorem 5.10.111 is stronger than condition (ii) in Theorem 5.10.112, which is already sufficient for Property (P). Nevertheless, the weakest of these conditions is not necessary for Property (P). Indeed, in the associative and commutative case, both conditions are equivalent, so that the commutative C∗ -algebra A := CC (SC ) do not satisfy condition (ii) in Theorem 5.10.112 (as there is no unitary element v ∈ A = M(A) such that v2 equals the inclusion SC → C), but has Property (P) (by Corollary 5.10.110 with λ = 1). Intrinsic characterizations of Property (P) are obtained in Corollary 5.10.115. Proposition 5.10.114 Let A and B be nonzero non-commutative JB∗ -algebras, and let F : B → A be a mapping. Then the following conditions are equivalent: (i) F is a surjective linear isometry. (ii) There exists a J-unitary element v ∈ M(A) such that F, regarded as a mapping from B to the JB∗ -algebra A(v), becomes a bijective Jordan-∗-homomorphism. Proof Only the implication (i)⇒(ii) merits a proof. Suppose that (i) holds. Then, as in the proof of the implication (ii)⇒(i) in Theorem 5.10.111, v := F  (1) is a J-unitary element of M(A). Since F  (1) = v and v is the unit of A (v), it follows from Proposition 3.4.25 that F  : B → A (v) is a Jordan-∗-homomorphism. Therefore F : B → A(v) is a Jordan-∗-homomorphism.  Let A and B be non-commutative JB∗ -algebras. According to Lemma 5.10.101, each surjective linear isometry F : B → A extends canonically to a surjective linear isometry  F : M(B) → M(A), which is a Jordan-∗-isomorphism whenever F is so. Moreover, in the case B = A, the passing F →  F respects the composition of mappings. Furthermore, by the equivalence (i)⇔(v) in Theorem 4.2.28,  F maps the set of all J-unitary elements of M(B) onto the set of all J-unitary elements of M(A). In what follows, we will not distinguish between F and  F. Corollary 5.10.115 Let A be a nonzero non-commutative JB∗ -algebra. Then the following conditions are equivalent:

328 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem (i) A satisfies Property (P). (ii) A is Jordan-∗-isomorphic to A(v) for every J-unitary element v ∈ M(A). (iii) The group of all surjective linear isometries on A acts transitively on the set of all J-unitary elements of M(A). Proof (i)⇒(ii) Let v be any J-unitary element in M(A). Since A and A(v) are linearly isometric, it follows from the assumption (i) that A is Jordan-∗-isomorphic to A(v). (ii)⇒(iii) Let v ∈ M(A) be any J-unitary element. Then, by the assumption (ii), there is a bijective Jordan-∗-homomorphism F : A → A(v). Since v is the unit of M(A(v)), we have F(1) = v. (iii)⇒(i) Let B be any non-commutative JB∗ -algebra linearly isometric to A. Then, by Proposition 5.10.114, B is Jordan-∗-isomorphic to A(v) for some J-unitary element v ∈ M(A). Now, by the assumption (iii), there exists a surjective linear isometry F : A → A such that F(1) = v. Since v is the unit of M(A(v)), it follows from Proposition 3.4.25 that F : M(A) → M(A(v)) (and hence F : A → A(v)) is a bijective Jordan-∗-homomorphism. Therefore A and B are Jordan-∗-isomorphic.  Now we are going to determine the hermitian operators on a non-commutative JB∗ -algebra. Our determination generalizes and unifies those of Propositions 2.2.26 and 3.4.28, which deal with the (possibly non-unital) associative case and the unital non-associative case, respectively. Theorem 5.10.116 Let A be a nonzero non-commutative JB∗ -algebra, and let R be a bounded linear operator on A. Then R is hermitian if and only if it can be written as Tx + iD for some self-adjoint element x ∈ M(A) and some Jordan ∗-derivation D of A. Proof Let x be in H(M(A), ∗). Since the mapping y → Ty from M(A) to BL(A) is a linear isometry sending 1 to IA , and the equality H(M(A), ∗) = H(M(A), 1) holds, we obtain that Tx belongs to H(BL(A), IA ), i.e., Tx is an hermitian operator on A. Now let D be a Jordan ∗-derivation of A. Then, by Lemma 3.4.27 applied to Asym , iD is a hermitian operator on A. Conversely, let R be an hermitian operator on A. Then, for λ in R, exp(iλR) is a surjective linear isometry on A, so, by Lemma 5.10.101, we have exp(iλR )(M(A)) = (exp(iλR)) (M(A)) = M(A), and so exp(iλR )(1) − 1 λ→0 iλ

x := R (1) = lim

lies in M(A). On the other hand, since R is a hermitian operator on A , and the mapping S → S(1) from BL(A ) to A is a linear contraction sending IA to 1, we deduce that x belongs to H(A , 1). It follows that x lies in H(M(A), ∗). Put D := i(Tx − R). By the first paragraph of the proof, iD is a hermitian operator on A.

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Now iD is a hermitian operator on A and D (1) = 0, so that, by Proposition 3.4.28, D is a Jordan ∗-derivation of A . Therefore D is a Jordan ∗-derivation of A. Since the equality R = Tx + iD is obvious, the proof is concluded.  To motivate our concluding outstanding result in the subsection (namely Theorem 5.10.119), let us formulate the following straightforward consequence of Corollaries 3.1.22 and 3.4.3, and Proposition 3.4.4 (see also Remark 3.4.5). Fact 5.10.117 Two non-commutative JB∗ -algebras are Jordan-∗-isomorphic if and only if their self-adjoint parts are linearly isometric. With Antitheorem 3.4.34, the above fact yields the following. Corollary 5.10.118 There are linearly isometric non-commutative JB∗ -algebras (even unital JC∗ -algebras) whose self-adjoint parts are not linearly isometric. Now Fact 5.10.117 can be specified as follows. Theorem 5.10.119 Let A and B be nonzero non-commutative JB∗ -algebras, let F : H(A, ∗) → H(B, ∗) be a linear mapping, and let FC : A → B denote the extension of F by complex linearity. Then the following conditions are equivalent: (i) F is a surjective linear isometry. (ii) There exists a central self-adjoint element u ∈ M(B) with u2 = 1, and a bijective Jordan-∗-homomorphism G : A → B, such that FC (a) = uG(a) for every a ∈ A. (iii) FC is a surjective linear isometry. Proof (i)⇒(ii) By the assumption (i) and Theorem 3.1.21, there exists a central symmetry u ∈ M(H(B, ∗)), and a bijective algebra homomorphism  : H(A, ∗) → H(B, ∗) such that F(h) = u • (h) for every h ∈ H(A, ∗). Now note that, by Remark 5.10.91, u is a self-adjoint element of M(B) satisfying u2 = 1, and that u is central in M(B) because it is obviously central in M(B)sym and Theorem 4.3.47 applies. It follows that, denoting by G : A → B the extension of  by complex linearity, we have FC (a) = uG(a) for every a ∈ A. At this time, the implications (ii)⇒(iii)⇒(i) are clear. 

5.10.4 Historical notes and comments W ∗ -algebra.

Let A be a Since we know that a∗ a ≥ 0 for every a ∈ A (cf. Proposition ∗ 2.3.39(i)), the strong topology of A is usually defined as the locally convex topology √ √ on A generated by the seminorms a → ϕ(a∗ a) and a → ϕ(aa∗ ), where ϕ ranges over D(A, 1) ∩ A∗ [806, Definition 1.8.7]. Since for every a ∈ A we have 2a∗ • a = a∗ a + aa∗ , 2a∗ • a ≥ a∗ a, and 2a∗ • a ≥ aa∗ ,

330 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem it becomes clear that the strong∗ topology thus defined coincides with the strong∗ topology introduced in Definition 5.10.3. Now let A be a non-commutative JBW ∗ -algebra. It follows from the definition itself of the strong∗ topology that the strong∗ topology of A and that of the JBW ∗ -algebra Asym coincide. Therefore those results and arguments on the strong∗ topology, which do not involve the non-commutative product of A, can be reduced to the case that A is commutative. Actually, thanks to Theorem 5.1.29(i), the study of the strong∗ topology of A can be reduced in most cases to the study of its restriction to the JBW-algebra H(A, ∗) (called the strong topology in [738]). Thus Propositions 5.10.5 and 5.10.13, Corollary 5.10.11, and Lemma 5.10.12 are tributaries of the results in [738]. Somehow, our approach to the extended functional calculus at a normal element of a non-commutative JBW ∗ -algebra, as formulated in Proposition 5.10.7, is inspired by Section 106 of the Riesz–Nagy book [796]. As in the particular case of a (possibly non-commutative) W ∗ -algebra, this functional calculus can be reduced to the even more particular case of a commutative W ∗ -algebra. To realize this, let A be a non-commutative JBW ∗ -algebra, and let a ∈ A be a normal element. Then, by Corollary 5.1.31, the w∗ -closure in A of the subalgebra of A generated by {1, a, a∗ } is a w∗ -closed associative and commutative ∗-subalgebra of A, and hence a commutative W ∗ -algebra (say B) which contains the norm-closed subalgebra of A generated by {1, a, a∗ } (equal to the range of the continuous functional calculus at a). Moreover, by Theorem 4.1.71(ii), we have J-sp(A, a) = sp(B, a), and, by Proposition 5.10.5(ii), the range of the extended functional calculus at a is contained in B. Further, by Propositions 5.10.62 and 5.10.83, the ∗-algebra of complex-valued functions C = C (A, a) in Proposition 5.10.7 coincides with the corresponding one C (B, a) relative to B. Again let A be a non-commutative JBW ∗ -algebra, and let a ∈ A be a normal element. Then the ∗-algebra C (of those complex-valued functions on J-sp(A, a) for which the extended functional calculus given by Proposition 5.10.7 has a sense) consists of bounded Borel functions [789, Proposition 6.2.7]. Actually, keeping in mind the reduction in the preceding paragraph, and invoking the classical literature on W ∗ -algebras, a still more extended functional calculus can be considered, which has a sense for all bounded Borel complex-valued functions on J-sp(A, a) (see, for example, [781, p. 72] for details). As an application, we are provided with the following result which refines Corollary 5.10.8 and whose associative forerunner can be seen in [781, Theorem 2.5.8] or [758, Theorem 5.2.5]. Theorem 5.10.120 Let A be a nonzero non-commutative JBW ∗ -algebra, and let a be a J-unitary element of A. Then there exists h ∈ H(A, ∗) such that a = exp(ih). Theorem 5.10.9 is due to Kaidi, Morales, and Rodr´ıguez [366]. The results leading to the definition and properties of the support idempotent, of a w∗ -continuous positive linear functional on a non-commutative JBW ∗ -algebra, are scattered in the literature. Via Corollary 5.1.40, most of these results can be

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reduced to the theory of JBW-algebras. Thus, for example, a definition (equivalent to ours) of the support idempotent, of a w∗ -continuous positive linear functional on a JBW-algebra, can be found in [673, Definition 5.2], whereas Lemma 5.10.19 is nothing other than the non-commutative JB∗ -algebra version of [673, Proposition 1.41(1.55)]. Our approach follows Sakai’s classical one for W ∗ -algebras [806, Definition 1.14.2], by Jordanizing it suitably. For the sake of completeness, we recall here the classical definition of the support idempotent of a w∗ -continuous positive linear functional on a W ∗ -algebra. The argument begins with the following straightforward consequence of the equivalence (i)⇔(iv) in Proposition 5.10.21. Lemma 5.10.121 Let A be a C∗ -algebra, and let e and f be self-adjoint idempotents in A. Then e ≤ f if and only if Ae ⊆ Af . Now we are provided with the following associative variant of Fact 5.10.16. Fact 5.10.122 [806, Proposition 1.10.1] Let A be a W ∗ -algebra. Then e → Ae becomes a bijection from the set of all self-adjoint idempotents of A onto the set of all w∗ -closed left ideals of A. Proof Let e be a self-adjoint idempotent of A. Then the left ideal Ae is w∗ -closed because Ae = {a ∈ A : a − ae = 0} and the product of A is separately w∗ -continuous (cf. Corollary 5.1.30(iii)). Conversely, let L be a w∗ -closed left ideal of A. Then, by Theorem 5.1.29(ii), L ∩ L∗ is a w∗ -closed ∗-subalgebra of A, and hence, by Fact 5.1.7, L ∩ L∗ has a unit e, which is a self-adjoint idempotent of A. The inclusion Ae ⊆ L follows because e ∈ L and L is a left ideal of A. But, if x is in L, then x∗ x ∈ L ∩ L∗ , so [x(1 − e)]∗ [x(1 − e)] = (1 − e)x∗ x(1 − e) = 0, hence x = xe ∈ Ae. Thus L = Ae. To conclude the proof, we must show that e = f whenever e and f are self-adjoint idempotents in A such that Ae = Af . But this follows from Lemma 5.10.121.  Now let A be a W ∗ -algebra, and let ϕ ∈ A∗ be a positive linear functional. Then it is easily realized that the set T := {a ∈ A : ϕ(a∗ a) = 0} is a w∗ -closed left ideal of A, so that, by Fact 5.10.122, we have T = A(1 − e) for a unique self-adjoint idempotent e ∈ A. Such an idempotent is called the support idempotent of ϕ. We leave the reader to realize that, when the W ∗ -algebra A is regarded as a non-commutative JBW ∗ -algebra, this notion of support idempotent is in agreement with Definition 5.10.18. Assertion (iii) in Proposition 5.10.20 is reducible to the (commutative) JBW ∗ algebra case, but not to the JBW-algebra case. Its specialization to the particular case that A is a w∗ -closed ∗-subalgebra of Bsym , for some W ∗ -algebra B, is proved by Friedman and Russo [951, Lemma 2.4], who know that the current version of

332 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem assertion (iii) in Proposition 5.10.20 is true [269, proof of Proposition 2]. Most arguments in the proof of Proposition 5.10.21 are taken from [738, Lemmas 4.1.13 and 4.2.2]. The associative forerunners (in short, a.f.’s) of Lemmas 5.10.29 and 5.10.31, of Proposition 5.10.32, and of Theorem 5.10.33 are due to Akemann [826] (1967), and can be found as Lemmas II.4.12 and III.5.6, implication (i)⇒(iv) in Theorem III.5.4, and Theorem III.5.7 of Takesaki’s book [1185], where Akemann’s paper is fully discussed. On the other hand, the a.f. of Lemma 5.10.23 can be found in [1185, Proposition III.5.3], whereas the a.f. of Lemma 5.10.30 is due to Aarnes [823] (1968), who also reproves the a.f.’s of Lemma 5.10.29, of Proposition 5.10.32, and of Theorem 5.10.33. Our proof of Lemma 5.10.23 is different from that of its a.f. in [1185]. On the other hand, as in the associative case, we are provided with the following additional information. Lemma 5.10.123 Let A, ψ, and ||| · ||| be as in Lemma 5.10.23. Then the closed unit ball of A is a complete metric space relative to the distance d(a, b) := ||| a − b |||. Proof

By Theorem 5.1.29(ii) and Corollary 5.1.30(iii), the mapping (a, b) → ψ(b∗ • a)

is a separately w∗ -continuous non-negative hermitian sesquilinear form on A. Therefore, by Lemma 5.10.34, there exists a complex Hilbert space H and a w∗ -continuous linear mapping T : A → H such that ||| a ||| = T(a) for every a ∈ A. Now the restriction of T to BA is an isometry from (BA , d) onto T(BA ), and therefore it is enough to show that T(BA ) is norm-closed in H. But this is indeed true, since T(BA ) is w∗ -compact because of the w∗ -continuity of T.  In Akemann’s original proof of Theorem 5.10.33, Aarnes’ a.f. of Lemma 5.10.30 was replaced with the a.f. of the following (see also [1185, Lemma III.5.5]). Lemma 5.10.124 Let A be a non-commutative JBW ∗ -algebra, let ϕn be a sequence in A∗ converging weakly to some ϕ0 in A∗ . Suppose an is a sequence in BA s∗ -converging to 0. Then limn ϕi (an ) = 0 uniformly for i ∈ N ∪ {0}. A proof avoiding Lemma 5.10.123 is the following. Proof We first make a number of reductions to simplify the proof. Since the sequence ϕn is bounded, we may suppose that ϕn ∈ BA∗ for every n ∈ N ∪ {0}. Set ϕ :=

∞  1 [ϕn ] ∈ A+ ∗. 2n n=0

Let e be the support idempotent of ϕ. Arguing as in the proof of Lemma 5.10.31, we realize that ϕn (a) = ϕn (Ue (a)) for any a ∈ A and each n ∈ N ∪ {0}. Therefore we may restrict attention to the algebra Ue (A), or instead merely suppose (as we shall

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do) that ϕ is faithful on A. By Proposition 5.10.5(i), the self-adjoint and skew-adjoint parts of the sequence an both s∗ -converge to 0, so we may suppose that each an is self-adjoint. 1 Define a new norm ||| · ||| on A by ||| a ||| = ϕ(a∗ a) 2 for every a ∈ A. Then, by Lemma 5.10.23, the topology of ||| · ||| and the strong∗ topology coincide on BA . Let α > 0 be given and define Hi = {a ∈ BA : |ϕj (a) − ϕ0 (a)| ≤ α for every j ≥ i}. Clearly each Hi is w∗ -closed in BA since the ϕn are w∗ -continuous on BA . Also  BA = i∈N Hi , since the sequence ϕn converges weakly to ϕ0 . Thus we may apply the Baire category theorem for compact spaces to get the existence of a0 ∈ BA , a w∗ -neighbourhood N of a0 in A, and j0 ∈ N such that N ∩ BA ⊆ Hj0 . Since w∗ ≤ s∗ (cf. Proposition 5.10.5(ii)), and the topology of ||| · ||| and the strong∗ topology coincide on BA , N ∩ BA is a ||| · |||-neighbourhood of a0 in BA . Therefore there exists β ∈ R+ such that, if a is in BA and ||| a − a0 ||| ≤ β, then |ϕj (a)−ϕ0 (a)| ≤ α for every j ≥ j0 . We now apply Lemma 5.10.29 to get the existence of a sequence en of self-adjoint idempotents of A s∗ -converging to 1 and satisfying max{Uen (an ), Uen ,1−en (an )} ≤ α6 for every n ∈ N. Therefore, setting gj := ϕj − ϕ0 ∈ 2BA , we derive that |gj (an )| ≤ |gj (Uen (an ))| + 2|gj (Uen ,1−en (an ))| + |gj (U1−en (an ))| ≤ α + |gj (U1−en (an ))| for every j ∈ N.

(5.10.39)

Now set bn := Uen (a0 ) + U1−en (an ). Then, by Proposition 3.4.17 and Lemma 5.10.24, bn is in BA . Moreover, in view of Proposition 3.4.17, the inclusion (5.1.1) in §5.1.25, and the equality (5.10.2) in §5.10.4, for x ∈ H(A, ∗) ∩ BA we have (U1−en (x))2 = U1−en (Ux (1 − en )) ≤ Ux (1 − en )U1−en (1) ≤ Ux (1 − en )(1 − en ) ≤ 1 − en , and hence ||| U1−en (x) ||| ≤ ||| 1 − en |||. Therefore we have ||| bn − a0 ||| = ||| 2Uen ,1−en (a0 ) + U1−en (a0 ) − U1−en (an ) ||| ≤ 2||| Uen ,1−en (a0 ) ||| + 2||| 1 − en |||. Since the sequence 1 − en s∗ -converges to 0, and the product of Asym is jointly on BA , there exists n0 such that n > n0 implies

s∗ -continuous

β β and ||| 1 − en ||| ≤ . 4 4 Then for n > n0 we have that ||| bn − a0 ||| ≤ β so ||| Uen ,1−en (a0 ) ||| ≤

|gj (bn )| = |gj (Uen (a0 )) + gj (U1−en (an ))| ≤ α for j > j0 and n > n0 .

(5.10.40)

Similarly, ||| Uen (a0 ) − a0 ||| ≤ 2||| Uen ,1−en (a0 ) ||| + ||| 1 − en ||| ≤

3β ≤ β for n > n0 , 4

334 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem hence |gj (Uen (a0 ))| ≤ α for j > j0 and n > n0 .

(5.10.41)

Therefore, keeping in mind (5.10.40) and (5.10.41), we have |gj (U1−en (an ))| ≤ 2α for j > j0 and n > n0 . Thus, invoking (5.10.39), we get |gj (an )| ≤ 3α for j > j0 and n > n0 . Since α > 0 was arbitrary, we obtain that limn (gi )(an ) = 0 uniformly for i ∈ N ∪ {0}. Finally, since |ϕi (an )| ≤ |gi (an )| + |ϕ0 (an )|, and ϕ0 (an ) → 0, we are done.



§5.10.125 In a first attempt, we were unable to prove Lemma 5.10.123. Therefore we renounced to prove Lemma 5.10.124 (since the proof of its a.f. seemed to involve the a.f. of Lemma 5.10.123 in an essential way), and replaced it with Lemma 5.10.30. Later we were able to modify the proof of the a.f. of Lemma 5.10.124 to get the proof of its non-associative generalization given above, which avoids Lemma 5.10.123. Finally, much later, we have found a proof of Lemma 5.10.123. The a.f. of Proposition 5.10.32 is only a part of the following. Theorem 5.10.126 [1185, Theorem III.5.4] For a subset E of the predual A∗ of a W ∗ -algebra A, the following conditions are equivalent: (i) E is a weakly relatively compact subset of A∗ . (ii) The restriction E|B of E to each maximal commutative ∗-subalgebra B of A is a weakly relatively compact subset of B∗ . (iii) E is bounded and for any decreasing sequence en of self-adjoint idempotents in A with inf en = 0, we have limn→∞ ϕ(en ) = 0 uniformly for ϕ ∈ E. (iv) E is bounded and there is ψ ∈ A+ ∗ with the property that for any ε > 0 there exists δ > 0 such that |ϕ(a)| < ε for every ϕ ∈ E whenever a is in BA and ψ(a∗ • a) < δ. (v) E is bounded and for any increasing net eλ of self-adjoint idempotents in A, limλ ϕ(eλ ) exists uniformly for ϕ ∈ E. According to [1185, p. 180], ‘Theorem 5.10.126 is a combination of results due to several mathematicians: A. Grothendieck [965], S. Sakai [1072], M. Takesaki [1103], H. Umegaki [1110], and, finally, C. Akemann [826]’. §5.10.127 In [19, pp. 108-109] Alvermann and Janssen assert that ‘there is no difficulty to modify the proofs of [Akemann in] [826] to obtain the following’: (1) The predual of any non-commutative JBW ∗ -algebra is weakly sequentially complete (i.e. assertion (i) in Corollary 5.8.41 holds). (2) The appropriate variant of Theorem 5.10.126 for non-commutative JBW ∗ algebras holds. (Indeed, Theorem 5.10.126 is true with ‘non-commutative

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JBW ∗ -algebra’ instead of ‘W ∗ -algebra’, and ‘commutative and associative’ instead of ‘commutative’ in condition (ii).) (3) Theorem 5.10.33 holds. (4) All bounded linear mappings from a complex Banach space, with a weakly sequentially complete dual, to a non-commutative JBW ∗ -algebra are weakly compact (i.e., assertion (ii) in Corollary 5.8.41 is true). We have given complete proofs of the Alvermann–Janssen claims (1), (3), and (4) above, and of a part of claim (2) (namely Proposition 5.10.32). We feel that Alvermann and Janssen are right and that, therefore, the remaining parts of claim (2) are true. Nevertheless, we encourage the interested reader to do the appropriate modifications in [826, 1185] to conclude the proof of claim (2). Thus he/she will enjoy overcoming the obstacles that could appear, as was the case for us in similar situations (cf. §5.10.125). Anyway, the reader can consult Peralta’s paper [1039], where the appropriate variant of Theorem 5.10.126 for JBW ∗ -triples is established, and claim (2) is derived. (Curiously, no reference to the Alvermann–Janssen paper [19] is done there.) We will go back to deal with [1039] later (see the comments before Proposition 5.10.139 and after Theorem 5.10.141). It is noteworthy how the authors of [19] apply their claims (1) and (4) to show that non-commutative JB∗ -algebras are Arens regular, and then to provide a new proof of Theorem 3.5.34. Lemma 5.10.34 keeps the core of the proof of [1061, Corollary 1]. Lemma 5.10.35 and the associative forerunner of Theorem 5.10.37 are due to Jarchow (see [1163, Theorem 20.7.3] and [985, Thorem 1.3], respectively). The current version of Theorem 5.10.37 is due to Chu and Iochum [906, Theorem 9]. According to them, ‘Jarchow’s . . . [associative forerunner of Theorem 5.10.37] . . . can easily be extended to [non-commutative] JB∗ -algebras. . . . The proof is verbatim, the only modification is to use the result of [19] in the proof of (i) implies (ii)’. Actually, ‘the result of [19]’ invoked by Chu and Iochum is precisely the implication (i)⇒(iv) in [19, Theorem 5.12], which is nothing other than our Proposition 5.10.32. The argument in the proof of Corollary 5.10.39 is taken from [1153, (15.8.6)(i)]. A part of the argument in the proof of Fact 5.10.42(ii) is a simplification of that of [795, Lemma 2.1.8]. We showed in that proof that, if A is a semiprime associative algebra over K, and if e ∈ A is a nonzero idempotent such that eAe = Ke, then Ae is a minimal ideal of A (a consequence of [795, Corollary 2.1.9]). Actually, the following better result holds (see Lemma 2.1.5 and Corollary 2.1.9 of [795]). Fact 5.10.128 Let A be a semiprime associative algebra over K. Then minimal left ideals of A are precisely the sets of the form Ae, where e is a nonzero idempotent of A such that eAe is a division algebra. In the case that the semiprime associative algebra A above is normed and complex, to say that eAe is a division algebra is the same as to affirm that eAe = Ce (cf. Corollary 1.1.43). Moreover, if A is actually a C∗ -algebra, then for each

336 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem idempotent e ∈ A there is a self-adjoint idempotent f ∈ A such that ef = e and fe = f (cf. Corollary 4.3.17), hence Af ⊆ Ae, and f = 0 whenever e = 0. Therefore, invoking Fact 5.10.128, we derive the following. Proposition 5.10.129 Let A be a C∗ -algebra. Then minimal left ideals of A are precisely the sets of the form Af , where f is a nonzero self-adjoint idempotent of A such that fAf = Cf . A more classical argument (avoiding Corollary 4.3.17) is given in the following. Proof

Let L be a minimal left ideal of A. Take 0 = x ∈ L. Then 0 = y := x∗ x ∈ L ∩ H(A, ∗).

Now, since Ay2 is a nonzero left ideal of A contained in L, it follows from the minimality of L that L = Ay2 . Therefore, since y ∈ L, there must exist a ∈ A such that y = ay2 . Set f := ya∗ = 0. Then f ∗ f = ay2 a∗ = ya∗ = f . Therefore f is a nonzero self-adjoint idempotent of A, and f = f ∗ = ay ∈ L. Then, by minimality of L as a left ideal, we have L = Af .  Theorem 5.10.43 is originally due to Sakai. In fact he proves the following refinement of Corollary 5.10.45. Theorem 5.10.130 [1071, Proposition 2] Weakly sequentially complete C∗ -algebras are finite-dimensional. Our proof of Theorem 5.10.43 (by deriving it from Proposition 5.10.38) follows the lines of [798, Section III.1]. Corollary 5.10.44 and the proof we have given are due to P´erez, Rico, and Rodr´ıguez [488, Corollary 13] (1994). The consequence (that weakly compact homomorphisms from C∗ -algebras to normed complex algebras have finite-dimensional ranges) is earlier. Indeed, this is proved by Gal´e, Ransford, and White [276, Theorem 3.1] (1992) as a consequence of a more general result. A very simple proof (such as that of Corollary 5.10.44, already commented) had been published by Mathieu [1011] (1989), although it seems to us that it is coeval to [276] (see the remark at the end of p. 820 of [276]). Proposition 5.10.47 is due to Becerra, L´opez, Peralta, and Rodr´ıguez [66]. Corollaries 5.10.48 to 5.10.51 had been proved in [67] with other methods. A description of JB∗ -triples whose Banach space is reflexive can be found in [67, Proposition 2.4]. We note that, in view of §5.10.46, Corollary 5.10.48 contains the result, previously proved by Chu and Iochum [907, Theorem 6], that JB∗ -triples enjoying the Radon–Nikodym property are reflexive. According to Yeadon [1122], the strong∗ topology of a W ∗ -algebra A coincides with the Mackey topology on the whole algebra (if and) only if A is finite-dimensional. Keeping in mind Theorem 5.10.43 and Corollary 3.5.7, a non-associative generalization of the above result would consist of proving that the strong∗ topology of a

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non-commutative JBW ∗ -algebra A coincides with the Mackey topology on the whole algebra if and only if the Banach space of A is reflexive (cf. Proposition 5.10.38). Results from Lemma 5.10.52 to Proposition 5.10.57, which lead to the definition of the support tripotent of a w∗ -continuous linear functional on a JBW ∗ -triple, are due to Friedman and Russo [269]. The order in the set of tripotents of a Jordan ∗-triple, introduced in §5.10.55, is due to Loos [772, §5]. Results from Lemma 5.10.59 to Proposition 5.10.67 are due to Barton and Friedman [853, 60]. Concerning Proposition 5.10.62, they only prove its associative forerunner, and that the ‘triple’ strong∗ topology of any non-commutative JBW ∗ algebra is greater than the ‘algebra’ strong∗ topology. The converse inequality, which completes Proposition 5.10.62, is due to Rodr´ıguez [1061]. In Proposition 5.10.67, we have formulated the fruit of the correct part of the argument in the proof of [853, Theorem 1.3] (1987). Actually, in the original argument, the fact that the w∗ -continuous linear operator T : X → H attains its norm does not appear as an assumption, but as a conclusion. As noticed by Peralta [1037] (2001), this becomes a severe gap in the original argument. The counter-example is easy. Indeed, take X equal to the complex Hilbert space 2 (which, as we saw in Remark 4.2.38, is a JBW ∗ -triple), set H := X, and let T : X → H denote the (automatically w∗ -continuous) bounded linear operator whose associated matrix is ⎛1 ⎞ 0 ... 0 ... ⎜2 ⎟ ⎜ ⎟ 2 ⎜0 ⎟ . . . 0 . . . ⎜ ⎟ 3 ⎜ ⎟. ⎜. . . . . . . . . ⎟ . . . . . . ⎜ ⎟ n ⎜0 0 ... . . .⎟ ⎝ ⎠ n+1 ... ... ... ... ... Then T does not attain its norm. It is noteworthy that the above is a counter-example to the argument, but not to the formulation, of the gapped version of Proposition 5.10.67 (i.e. Proposition 5.10.67 when the assumption that T attains its norm is removed). Indeed, when a complex Hilbert space H is regarded as a JBW ∗ -triple, for every ϕ ∈ SH∗ we have  · ϕ =  ·  on H. According to the above comments, the following problem, whose affirmative answer is claimed in [853], seems to remain open. √ Problem 5.10.131 Is there a universal constant M (possibly equal to 2) such that, for each JBW ∗ -triple X and each w∗ -continuous linear operator from X to a complex Hilbert space, there exists ϕ ∈ SX∗ satisfying T(x) ≤ MTxϕ for every x ∈ X? Lemma 5.10.68 and Proposition 5.10.69 are due to Zizler [1137] (1973), who acknowledges that his arguments follow with minor changes those of Lindenstrauss [1001] (1963). Given Banach spaces X and Y over K, denote by NA(X, Y) the set of

338 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem those bounded linear operators from X to Y which attain their norms. Lindenstrauss proves that, for each T ∈ BL(X, Y) there is a sequence Tn in BL(X, Y) norm-convergent to T in such a way that Tn  ∈ NA(X  , Y  ) and T − Tn is compact for every n ∈ N, whereas Zizler’s Proposition 5.10.69 can be rephrased as that, for each T ∈ BL(X, Y) there is a sequence Tn in BL(X, Y) norm-convergent to T in such a way that Tn  ∈ NA(Y  , X  ) and T − Tn is compact for every n ∈ N. We note that, if a bounded linear operator attains its norm, then so does its transpose. We also note that, in general, NA(X, Y) is not dense in BL(X, Y) (even with X = Y) [1001, Section 3]. Actually, as proved recently by Mart´ın [1009, Theorem 8], there is a Banach space X and a compact operator T : X → X which is not in the closure of NA(X, X) in BL(X). For additional information about norm-attaining operators the reader is referred to Section 7.5 of the Fabian–Habala–H´ajek–Montesinos–Zizler book [729] and to Acosta’s survey paper [825]. With the exception of Corollary 5.10.81, which could be new, results from Proposition 5.10.71 to Proposition 5.10.83 are due to Rodr´ıguez [1061] and Peralta– Rodr´ıguez [1040]. Germinally, most of these results appear already in [1061] (1991) (see also [525, Proposition D.17] (1994)), but their validity is affected by the absence of an affirmative answer to Problem 5.10.131. Fortunately, as Bunce comments in [872] (2001), ‘the remarkable recent article [1040] [(2001)] provides antidotes to some subtle difficulties in [853] and subsequent works [such as [1061]]’. This is the case of Proposition 5.10.71, or merely its Corollary 5.10.72, which, in the absence of an affirmative answer to Problem 5.10.131, restores full validity to the results in [1061]. The original proof of Proposition 5.10.71 does not invoke Zizler’s Proposition 5.10.69, but a variant, due to Poliquin and Zizler [1053, Corollary 2], asserting that, given a dual Banach space X over K, a Banach space Y over K, and a w∗ -w-continuous linear operator T : X → Y, there is a sequence Tn : X → Y of w∗ -w-continuous linear operators which converges to T in the norm topology of BL(X, Y) and satisfies that Tn ∈ NA(X, Y) and T − Tn is of rank one for every n ∈ N. The proof of the Poliquin–Zizler theorem, just formulated, is much more difficult than that of Proposition 5.10.69. Therefore we have preferred to prove and invoke this proposition, paying a small tribute consisting of the elementary Lemma 5.10.70. In [60, Question 4], Barton and Friedman raise the question whether the triple product of a JBW ∗ -triple is jointly s∗ -continuous on bounded sets. An affirmative answer to this question is given in [1061]. The unique ingredient in the proof not previously discussed is the following. Fact 5.10.132 Every JBW ∗ -triple can be seen as a w∗ -closed subtriple of a JBW ∗ algebra.

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Proof Let X be a JBW ∗ -triple. By Theorem 4.1.113 (see also Corollary 7.1.6), X can be seen as a norm-closed subtriple of a JB∗ -algebra A. Then, by Proposition 3.5.26, A is a JBW ∗ -algebra with w∗ -continuous involution and separately w∗ continuous product. Therefore, by bitransposing the inclusion X → A, X  becomes a w∗ -closed subtriple of A . Since X can be regarded as a w∗ -closed triple ideal of X  (cf. Corollary 5.7.46), the result follows.  Now, as announced above, we can prove the following. Theorem 5.10.133 Let X be a JBW ∗ -triple. Then the triple product of X is jointly s∗ -continuous on bounded subsets of X. Proof According to Fact 5.10.132, there exists a JBW ∗ -algebra A such that X is a w∗ -closed subtriple of A. By assertions (i) and (v) in Proposition 5.10.5, the triple product of A is jointly s∗ -continuous on bounded subsets of A. Therefore the result follows from Propositions 5.10.62 and 5.10.83.  Answering affirmatively a question raised in [1061], Bunce [872] proves the following. Theorem 5.10.134 Let X be a JBW ∗ -triple, and let Y be a w∗ -closed subtriple of X. Then each w∗ -continuous linear functional on Y has a norm-preserving extension to a w∗ -continuous linear functional on X. The proof of the above theorem is really tough. Indeed, by means of Propositions 5.7.13 and 5.10.57, Bunce reduces the proof to the case that X is a JBW ∗ -algebra and Y is a w∗ -closed ∗-subalgebra containing the unit of X. Then he proves the statement by applying a jordanization of its associative forerunner [806, Proposition 1.24.5], and invoking, via Corollary 5.1.40, the structure theory of JBW-algebras (see [738, Theorem 7.2.6]), as well as deep results of Haagerup and Hanche-Olsen [969], Pedersen and Størmer [1035], and Størmer [602, 603]. Now, as conjectured in [1061] and proved in [872], Proposition 5.10.83 can be refined, even with a shorter proof. Indeed, we have the following. Proposition 5.10.135 Let X be a JBW ∗ -triple, and let Y be a w∗ -closed subtriple of X. Then s∗ (X, X∗ )|Y = s∗ (Y, Y∗ ). Proof Let g be in SY∗ . By Theorem 5.10.134, there exists f ∈ SX∗ such that f|Y = g. Taking y ∈ D(Y∗ , g) ⊆ SY ⊆ SX , we realize that for every b ∈ Y we have b2g = g({bby}) = f ({bby}) = b2f . Since g is arbitrary in SY∗ , this shows that s∗ (Y, Y∗ ) ≤ s∗ (X, X∗ )|Y . The converse inclusion follows from Corollary 5.10.82.  In relation to Theorem 5.10.134, the following earlier result of Edwards and R¨uttimann [933] becomes illuminating.

340 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Theorem 5.10.136 Let X be a JBW ∗ -triple, and let Y be a w∗ -closed subtriple of X. Then Y is an inner ideal in X if and only if every w∗ -continuous linear functional on Y has a unique norm-preserving extension to a w∗ -continuous linear functional on X. Using the Horn–Neher structure theory for JBW ∗ -triples [330, 980], Chu and Iochum [906, Proposition 2] prove that every JBW ∗ -triple X can be regarded as a subtriple of a suitable JB∗ -algebra A in such a way that X becomes the range of a contractive linear projection on A. Then, as shown in [525], an easy argument, involving Proposition 3.5.26 and Corollaries 5.7.35 and 5.7.46, allows to derive the following refinement of Fact 5.10.132. Proposition 5.10.137 [525, Theorem D.20] Every JBW ∗ -triple X can be regarded as a subtriple of a suitable JBW ∗ -algebra A in such a way that X becomes the range of a w∗ -continuous contractive linear projection on A. Now, combining Theorem 5.10.33 with Propositions 5.10.62, 5.10.83, and 5.10.137, we obtain the following. Theorem 5.10.138 [525, Theorem D.21] Let X be a JBW ∗ -triple. Then m(X, X∗ ) and s∗ (X, X∗ ) coincide on bounded subsets of X. Propositions 5.10.84 and 5.10.85 are taken from [1040]. The following result, due to Peralta [1039], becomes an appropriate variant of Proposition 5.10.32. Proposition 5.10.139 Let X be a nonzero JBW*-triple, and let E be a weakly relatively compact subset of X∗ . Then there are norm-one elements ϕ1 , ϕ2 ∈ SX∗ with the property that for any ε > 0 there exists δ > 0 such that |φ(x)| < ε for every φ ∈ E whenever x is in BX and xϕ1 ,ϕ2 < δ. Proof Let D denote the weakly closed absolutely convex hull of E in X∗ . Then D is w-compact (see for example [778, Theorem 2.8.14]) and absolutely convex. Therefore, arguing as in the third paragraph of the proof of Proposition 5.10.84, there exists a reflexive complex Banach space Y and a bounded linear operator R : Y → X∗ such that sup |φ(x)| ≤ R (x) for every x ∈ X. φ∈D

Moreover, by Theorem 5.10.138 and the implication (i)⇒(iii) in Proposition 5.10.85, there exists a w∗ -continuous linear operator G from X to a complex Hilbert space, and a mapping N : R+ → R+ satisfying R (x) ≤ N(η)G(x) + ηx for all x ∈ X and η > 0. Furthermore, by Corollary 5.10.72, there exist ϕ1 , ϕ2 ∈ SX∗ such that G(x) ≤ 2Gxϕ1 ,ϕ2 for every x ∈ X.

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It follows that sup |φ(x)| ≤ 2N(η)Gxϕ1 ,ϕ2 + ηx for all x ∈ X and η > 0.

φ∈D

Finally, given ε > 0, it is enough to take η = 3ε and δ = 6N( εε)G to realize that 3 |φ(x)| < ε for every φ ∈ E whenever x is in BX and xϕ1 ,ϕ2 < δ.  Proposition 5.10.140 Let X be a complex Banach space, let Y be a nonzero JBW ∗ triple, and let F : X → Y∗ be a bounded linear operator. Then the following conditions are equivalent: (i) F is weakly compact. (ii) There exist ϕ1 , ϕ2 ∈ SY∗ , and a function N : R+ → R+ , such that F  (y) ≤ N(ε)yϕ1 ,ϕ2 + εy for all y ∈ Y and ε > 0. (iii) There exists a w∗ -continuous linear operator G from Y to a complex Hilbert space, and a function N : R+ → R+ , such that F  (y) ≤ N(ε)G(y) + εy for all y ∈ Y and ε > 0.

(5.10.42)

(iv) There exists a bounded linear operator G from Y to a complex Hilbert space, and a function N : R+ → R+ , such that (5.10.42) holds. Proof Argue as in the proof of Proposition 5.10.36, replacing Proposition 5.10.32 with Proposition 5.10.139, and Theorem 5.1.29(ii) and Corollary 5.1.30(iii) with Theorem 5.7.20.  Keeping in mind that the dual of a JB∗ -triple is the predual of a JBW ∗ -triple (cf. Proposition 5.7.10), the following theorem follows from Proposition 5.10.139. Theorem 5.10.141 [1040] Let X be a JB∗ -triple, let Y be a complex Banach space, and let T : X → Y be a bounded linear operator. Then the following conditions are equivalent: (i) T is weakly compact. (ii) There exist ϕ1 , ϕ2 ∈ SX  , and a function N : R+ → R+ , such that T(x) ≤ N(ε)xϕ1 ,ϕ2 + εx for all x ∈ X and ε > 0. (iii) There exists a bounded linear operator G from X to a complex Hilbert space, and a function N : R+ → R+ , such that T(x) ≤ N(ε)G(x) + εx for all x ∈ X and ε > 0. The above theorem is established in [906, Theorem 11], with .ϕ1 ,ϕ2 in condition (ii) replaced with .ϕ for a single functional ϕ ∈ SX  . Since this refinement depends on an affirmative answer to Problem 5.10.131, it should remain in doubt. In Peralta’s original proof of Proposition 5.10.139, that proposition was derived from Theorem 5.10.141. It seems to us that, actually, what he had in mind is not Theorem 5.10.141 but Proposition 5.10.140, which, although not explicitly formulated in

342 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem [1040] (or anywhere else outside of our work), could have been easily deduced from the results proved in [1040]. Arguing as in the proof of Proposition 5.10.38, with Proposition 5.10.137 instead of Proposition 5.10.36, we get the following. Proposition 5.10.142 For a JB∗ -triple X, the following conditions are equivalent: (i) The Banach space of X is reflexive. (ii) There are ϕ1 , ϕ2 ∈ SX∗ , and a positive constant K > 0, such that x ≤ Kxϕ1 ,ϕ2 for every x ∈ X. (iii) The Banach space of X is isomorphic to a complex Hilbert space. Now, applying Corollary 5.10.48, we obtain the following generalization of Corollary 5.10.49. Corollary 5.10.143 Let X be a JB∗ -triple. If there exists a non-empty relatively w-open subset of BX with diameter less than 2, then the Banach space of X is isomorphic to a Hilbert space. Proposition 5.10.71 (or its Corollary 5.10.72) and Corollary 5.10.73 become versions for JBW ∗ -triples and non-commutative JBW ∗ -algebras, respectively, of the so-called little Grothendieck’s theorem. Appropriate version for non-commutative JB∗ -algebras and JB∗ -triples have been established in Corollaries 5.10.74, 5.10.75, and 5.10.76. The original little Grothendieck’s theorem [964] (see also [1052, Theorem 5.2(i)]) reads as follows. Theorem 5.10.144 Let E be a compact Hausdorff topological space, let H be a Hilbert space over K, and let T : CK (E) → H be a bounded linear operator. Then there is a probability measure λ on E such that  T(x) ≤ MK T where MR =

?

π 2

and MC =

?

4 π.

1 |x| dλ 2

2

for every x ∈ CK (E),

Moreover, the constant MK is sharp.

According to the comments immediately before [1052, Theorem 8.1], the next complex non-commutative variant of Theorem 5.10.144 ‘was first proved [by Pisier] in [1051] with a larger constant. Haagerup [967] obtained the constant 1, that was shown to be optimal in [970] (see [1052, §11] for details)’. Theorem 5.10.145 Let A be a C∗ -algebra, let H be a complex Hilbert space, and let T : A → H be a bounded linear operator. Then there are positive linear functionals ϕ1 , ϕ2 ∈ SA such that 1

T(x) ≤ T(ϕ1 (x∗ x) + ϕ2 (xx∗ )) 2 for every x ∈ A.

5.10 Complements on non-commutative JB∗ -algebras and JB∗ -triples

343

Haagerup’s Theorem 5.10.145 contains the associative forerunner of Corollary 5.10.74. Indeed, since for x ∈ A we have x∗ x ≤ 2x∗ • x and xx∗ ≤ 2x∗ • x, it is enough to take ψ := 12 (ϕ1 + ϕ2 ), and invoke Lemma 5.10.14(iii), to get the following. Corollary 5.10.146 Let A be a C∗ -algebra, and let T be a bounded linear operator from A to a complex Hilbert space. Then there exists a positive linear functional ψ ∈ SA satisfying 1

T(x) ≤ 2T (ψ(x • x∗ )) 2 for every x ∈ A. Now let us comment on the so-called (big) Grothendieck’s theorem [964] (see also [1052, Theorem 2.1]), whose original formulation reads as follows. Theorem 5.10.147 Let E, F be compact Hausdorff topological spaces, and let V : CK (E) × CK (F) → K be a bounded bilinear form. Then there are probability measures λ and μ, respectively on E and F, such that  |V(x, y)| ≤ KV

 1  |x|2 dλ

1

2

|y|2 dμ

2

for all x ∈ CK (E) and y ∈ CK (F),

where K is a universal constant. According to the comments immediately before [1052, Theorem 7.1], the next complex non-commutative variant of Theorem 5.10.147 ‘was first proved [by Pisier] in [1051] with an additional approximation assumption, and [by Haagerup] in [968] in general. Actually, a posteriori, by [968] (see also [990]), the assumption needed in [1051] always holds’. Theorem 5.10.148 Let A and B be C∗ -algebras, and let V : A × B → C be a bounded bilinear form. Then there are positive linear functionals ϕ1 , ϕ2 ∈ SA and ψ1 , ψ2 ∈ SB such that 1

1

|V(x, y)| ≤ V(ϕ1 (x∗ x) + ϕ2 (xx∗ )) 2 (ψ1 (y∗ y) + ψ2 (yy∗ )) 2 for all x ∈ A and y ∈ B. In [172], Chu, Iochum, and Loupias provide the reader with a detailed discussion on the modifications needed in Haagerup’s proof of the above theorem in order to obtain an appropriate non-associative generalization. (Among the tools involved in these modifications, the authors of [172] invoke Proposition 3.4.6 and Corollary 3.4.7.) Thus they formulate the following. Theorem 5.10.149 Let A and B be non-commutative JB∗ -algebras, and let V : A × B → C be a bounded bilinear form. Then there are positive linear functionals ϕ ∈ SA and ψ ∈ SB such that √ 1 1 |V(x, y)| ≤ 2(1 + 2 3)V(ϕ(x • x∗ )) 2 (ψ(y • y∗ )) 2 for all x ∈ A and y ∈ B. As noted in [172], the above theorem implies the following particular case of Theorem 5.8.43.

344 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem Corollary 5.10.150 Let A and B be non-commutative JB∗ -algebras, and let T be a bounded linear operator from A to B . Then T factors through a complex Hilbert space. More precisely, there exist a complex Hilbert space H and bounded √ linear operators S : A → H, R : H → B satisfying T = R ◦ S and RS ≤ 2(1 + 2 3)T. Hint Consider the bounded bilinear form V : A × B → C defined by V(x, y) := T(x)(y), and apply Theorem 5.10.149.  §5.10.151 The authors of [172] combine Corollary 5.10.150 with the Chu–Iochum proposition commented in the paragraph before Proposition 5.10.137 to prove Theorem 5.8.43. Therefore Theorem 5.10.149 and the Horn–Neher structure theory for JBW ∗ -triples [330, 980] become the main ingredients in the proof of Theorem 5.8.43. Now we can state and prove an appropriate version of Grothendieck’s theorem for JBW ∗ -triples in the line of Haagerup’s associative forerunner [968, Proposition 2.3]. √ Theorem 5.10.152 [1040] Let M > 4(1 + 2 3) and ε > 0. For every couple (X, Y) of JBW ∗ -triples and every separately weak∗ -continuous bilinear form V on X × Y, there exist norm-one functionals ϕ1 , ϕ2 ∈ X∗ , and ψ1 , ψ2 ∈ Y∗ satisfying  1  1 |V(x, y)| ≤ M V x2ϕ2 + ε2 x2ϕ1 2 y2ψ2 + ε2 y2ψ1 2 for every (x, y) ∈ X × Y. Proof We begin by noticing that a bilinear form U on X × Y is separately weak∗ continuous if and only if there exists a weak∗ -to-weak-continuous linear operator FU : X → Y∗ such that the equality U(x, y) = FU (x)(y) holds for every (x, y) ∈ X × Y. Put T := FV : X → Y∗ in the sense of the preceding paragraph. Regarding T as an operator from X to Y  , it follows from Theorem 5.8.43 that there exist a complex  Hilbert space H and bounded linear √ operators S : X → H, R : H → Y satisfying T = R ◦ S and RS ≤ 2(1 + 2 3)T. Replacing H with the closure of S(X) in H (say H1 ) and R with R|H1 if necessary, we can suppose that R takes its values into Y∗ . Therefore from now on we will see R as an operator from H to Y∗ . On the other hand, we also may suppose that R is injective. Indeed, take H0 equals to the orthogonal complement of ker(R) in H, R0 := R|H0 and S0 := πH0 ◦ S, where πH0 is the orthogonal projection √ from H onto H0 , to have T = R0 ◦ S0 with R0 injective and R0 S0  ≤ 2(1 + 2 3)T. Let x be in BX , and let xλ be a net in BX weak∗ -convergent to x. Take a cluster point h of S(xλ ) in the weak topology of H. Then R(h) is a cluster point of T(xλ ) = R(S(xλ )) in the weak topology of Y∗ . On the other hand, since T is weak∗ -to-weak-continuous, we have T(xλ ) → T(x) weakly. It follows that R(h) = T(x), and hence h = S(x) by the injectivity of R. Therefore S(x) is a cluster point of S(xλ ) in the weak topology of H. Keeping in mind the arbitrariness of x ∈ BX and of the net xλ weak∗ -convergent to x, it follows from Fact 5.1.18 that S|BX is weak∗ -continuous. By Fact 5.1.19, S is weak∗ -continuous.

5.10 Complements on non-commutative JB∗ -algebras and JB∗ -triples

345

Now that we know S is weak∗ -continuous, we apply Proposition ? that the operator √ M√ 5.10.71 with K = > 2 to find norm-one functionals ϕ1 , ϕ2 ∈ X∗ , and 2(1+2 3)

ψ1 , ψ2 ∈ Y∗ satisfying

 1  1 S(x) ≤ KS x2ϕ2 + ε2 x2ϕ1 2 and R (y) ≤ KR y2ψ2 + ε2 y2ψ1 2 for all x ∈ X and y ∈ Y. Therefore |V(x, y)| = |T(x)(y)| = |R(S(x))(y)| = |R (y)(S(x))|  1  1 M ≤ √ RS x2ϕ2 + ε2 x2ϕ1 2 y2ψ2 + ε2 y2ψ1 2 2(1 + 2 3)  1  1 ≤ MV x2ϕ2 + ε2 x2ϕ1 2 y2ψ2 + ε2 y2ψ1 2 for every (x, y) ∈ X × Y.



In the same way as Corollary 5.10.73 was derived from Proposition 5.10.71, we √ can obtain from Theorem 5.10.152 that, given M > 8 (1 + 2 3), non-commutative JBW ∗ -algebras A, B, and a separately weak∗ -continuous bilinear form V on A × B, there exist norm-one positive functionals ϕ ∈ A∗ and ψ ∈ B∗ satisfying 1

1

|V(x, y)| ≤ MV (ϕ(x • x∗ )) 2 (ψ(y • y∗ )) 2 for every (x, y) ∈ A × B. Since biduals of JB∗ -triples are JBW ∗ -triples, the following corollary follows from Corollary 5.8.40 and Theorem 5.10.152. √ Corollary 5.10.153 Let M > 4(1 + 2 3) and ε > 0. Then for every couple (X, Y) of JB∗ -triples and every bounded bilinear form V on X × Y there exist norm-one functionals ϕ1 , ϕ2 ∈ X ∗ and ψ1 , ψ2 ∈ Y ∗ satisfying 1  1  |V(x, y)| ≤ MV x2ϕ2 + ε2 x2ϕ1 2 y2ψ2 + ε2 y2ψ1 2 for every (x, y) ∈ X × Y. Taking M = 18 and ε = 1 in the above corollary, we find the following. Fact 5.10.154 Let X, Y be JB∗ -triples, and let V be a bounded bilinear form on X ×Y. Then there exist norm-one functionals ϕ1 , ϕ2 ∈ X ∗ and ψ1 , ψ2 ∈ Y ∗ satisfying |V(x, y)| ≤ 18Vxϕ1 ,ϕ2 yψ1 ,ψ2 for every (x, y) ∈ X × Y. In [853, Theorem 1.4], Barton and Friedman claim that for every pair of complex JB∗ -triples X, Y, and every bounded bilinear form V on X × Y, there exist norm-one functionals ϕ ∈ X ∗ and ψ ∈ Y ∗ such that the inequality √ (5.10.43) |V(x, y)| ≤ (3 + 2 3) V xϕ yψ holds for every (x, y) ∈ X × Y. However, among other gaps (see [1040, p. 607]), the beginning of the Barton–Friedman proof assumes as true that, for X, Y and V as

346 JBW ∗ -algebras, JB∗ -triples revisited, and a unit-free Vidav–Palmer theorem above, every separately weak∗ -continuous bilinear extension of V to X  × Y  attains its norm. Unfortunately, as the next example shows, the above assumption is not right. Example 5.10.155 Take X and Y equal to the complex 2 space, and consider the bounded bilinear form on X × Y defined by V(x, y) := (S(x)|σ (y)) where S is the bounded linear operator on 2 whose associated matrix is ⎛1 ⎞ 0 ... 0 ... ⎜2 ⎟ ⎜ ⎟ 2 ⎜0 ... 0 . . .⎟ ⎜ ⎟ 3 ⎜ ⎟, ⎜. . . . . . . . . ⎟ . . . . . . ⎜ ⎟ n ⎜0 0 ... . . .⎟ ⎝ ⎠ n+1 ... ... ... ... ... and σ is the conjugation on 2 fixing the elements of the canonical basis. Then V does not attain its norm. It is noteworthy that, although the bilinear form V above does not attain its norm, it satisfies inequality (5.10.43) for all x, y ∈ 2 and all norm-one elements ϕ, ψ ∈ ∗2 . Therefore it does not become a counterexample to the Barton–Friedman claim. In fact we do not know if Theorem 1.4 of [853] is true. Another alleged claim [853, Theorem 1.4] (with √ √ proof of the Barton–Friedman constant 3 + 2 3 replaced with 4(1 + 2 3)) appears in the Chu–Iochum–Loupias paper already quoted (see [172, Theorem 6]). However such a proof (whose clever argument has been corrected and refined in the proof of Theorem 5.10.152) depends on an affirmative answer to Problem 5.10.131. By the way, we refer the reader to Peralta’s paper [1038] for partial affirmative answers to that problem. Most of the results in Subsection 5.10.3 are due to Kaidi, Morales, and Rodr´ıguez [366, Sections 5 and 6, and §7.7]. Other sources are quoted in what follows. Proposition 5.10.92 is due to Dineen and Timoney [924], whereas Lemma 5.10.94 and Proposition 5.10.95 are due to Bunce and Chu [873]. The particular cases of Corollaries 5.10.103 and 5.10.107, with ‘C∗ -algebra’ instead of ‘alternative C∗ algebra’, are originally due to Rodr´ıguez [528, Theorem A] and Akemann–Pedersen [3, Proposition 4.4], respectively. Actually, as pointed out in [366, §7.7], once Corollary 5.10.107 has been proved, the second proof of Corollary 5.10.103 mimics that of its forerunner in [528]. Facts 5.10.93 and 5.10.117, Propositions 5.10.97 and 5.10.114, Corollaries 5.10.115 and 5.10.118, Theorem 5.10.119, and the proof of Lemma 5.10.94 are new. Most notions and results on JB∗ - and JBW ∗ -triples we have dealt with in this section have their appropriate variants for real JB∗ - and real JBW ∗ -triples (see [872, 908, 1037, 1040, 1041]). For additional information on the matter, see [983, 984], Russo’s survey paper [1069], and references therein.

6 Representation theory for non-commutative JB∗ -algebras and alternative C∗ -algebras

Implicitly, representation theory of JB-algebras underlies our work since, without providing the reader with a proof, we took from the Hanche-Olsen–Stormer book [738] the very deep fact that the closed subalgebra of a JB-algebra generated by two elements is a JC-algebra (cf. Proposition 3.1.3). In that way we were able to develop the basic theory of non-commutative JB∗ -algebras (including the non-associative Vidav–Palmer Theorem 3.3.11 and Wright’s fundamental Fact 3.4.9 [641], which describes how JB-algebras and JB∗ -algebras are mutually determined) without any further implicit or explicit reference to representation theory. In fact, we avoided any dependence on representation theory throughout all of Volume 1, and to the end of Chapter 5 of the present volume. To conclude the basic theory of non-commutative JB∗ -algebras, we need to develop in detail the representation theory of these objects. Thus we devote Section 6.1 of this chapter to explaining how the theory of non-commutative JB∗ -algebras can be reduced to the knowledge of the so-called non-commutative JBW ∗ -factors, and to describeing in detail these last objects. Section 6.2 deals with the main applications of the representation theory, namely the structure of alternative C∗ algebras, the definition and properties of the strong topology of a non-commutative JBW ∗ -algebra, and the classification of prime non-commutative JB∗ -algebras. Finally, Section 6.3 deals with a rather incidental application. Indeed, a Le Page type theorem for non-commutative JB∗ -algebras is proved, and a general non-associative non-∗ discussion of Le Page’s theorem [999] is included. 6.1 The main results Introduction In Subsection 6.1.1, we introduce non-commutative JBW ∗ -factors and non-commutative JBW ∗ -factor representations of a given non-commutative JB∗ -algebra. As the main result, we prove that every non-commutative JB∗ -algebra has a faithful family of non-commutative JBW ∗ -factor representations (see Corollary 6.1.11). Subsection 6.1.2 deals with a first application of the representation theory, which allows us to show as the main result that non-commutative JB∗ -algebras are 347

348

Representation theory for non-commutative JB∗ -algebras

associative and commutative if (and only if) they have no nonzero nilpotent element (see Theorem 6.1.17). As a consequence, we obtain that alternative C∗ -algebras are commutative if and only if they have no nonzero nilpotent element (see Corollary 6.1.19). In Subsection 6.1.3, once more in our development we involve the theory of JB-algebras [738], and invoke Fact 5.1.42 to classify all (commutative) JBW ∗ factors (see Proposition 6.1.41). This classification is applied to prove that i-special JB∗ -algebras are JC∗ -algebras (see Theorem 6.1.44). In Subsection 6.1.4, we combine Theorem 6.1.44 just reviewed with Zel’manovian techniques [437, 662] to prove an appropriate version for JB∗ -algebras of Zel’manov’s prime theorem (see Theorem 6.1.57 and §6.1.132). In Subsection 6.1.5, we introduce totally prime normed algebras and ultraprime normed algebras, and prove that totally prime normed complex algebras are centrally closed, and that ultraprime normed algebras are totally prime (see Theorems 6.1.60 and 6.1.63). Then we combine Theorem 6.1.57 with the fact that prime C∗ -algebras are ultraprime (see Corollary 6.1.70) to show as main result that prime non-commutative JB∗ -algebras are ultraprime (see Theorem 6.1.78). Therefore, as asserted in Corollary 6.1.79, prime non-commutative JB∗ -algebras are centrally closed. Finally, in Subsection 6.1.6, we combine Corollaries 6.1.19 and 6.1.79, already reviewed, with a topological reading of McCrimmon’s paper [436] to prove as a main result that non-commutative JBW ∗ -factors are either commutative or simple quadratic or of the form B(λ) for some (associative) W ∗ -factor B and some 0 ≤ λ ≤ 1 (see Theorem 6.1.112). As a consequence, alternative W ∗ -factors are either associative or equal to the alternative C∗ -algebra of complex octonions (see Proposition 2.6.8 and Corollary 6.1.115). 6.1.1 Factor representations of non-commutative JB∗ -algebras By a JBW-factor (respectively, a non-commutative JBW ∗ -factor, a JBW ∗ -factor, an alternative W ∗ -factor, or a W ∗ -factor) we mean a prime JBW-algebra (respectively, non-commutative JBW ∗ -algebra, JBW ∗ -algebra, alternative W ∗ -algebra, or W ∗ -algebra). Fact 6.1.1 Let A be a nonzero non-commutative JBW ∗ -algebra. Then the following conditions are equivalent: (i) (ii) (iii) (iv) (v)

A is a non-commutative JBW ∗ -factor. A has no w∗ -closed ideal other than zero and A. The centre of A equals C1. The JBW ∗ -algebra Asym is a JBW ∗ -factor. The JBW-algebra H(A, ∗) (cf. Theorem 5.1.29) is a JBW-factor.

Proof (i)⇒(ii) Since prime algebras have no nonzero proper direct summand, this implication follows from Fact 5.1.10(ii).

6.1 The main results

349

(ii)⇒(i) Let B, C be ideals of A such that B = 0 and BC = 0. Then the separate

w∗ -continuity of the product of A (cf. Corollary 5.1.30(iii)) yields that the w∗ -closure

of B in A (say D) is a nonzero w∗ -closed ideal of A such that DC = 0. Therefore, since D = A (by the assumption (ii)), we have that AC = 0, and hence C = 0. Thus A is prime. (ii)⇒(iii) By the assumption (ii) and Corollary 5.1.30(iv), 1 is the unique nonzero idempotent in Z(A). Therefore, since Z(A) is a W ∗ -algebra (by Corollary 5.1.30(iii)), condition (iii) holds thanks to Theorem 5.1.29(vi). (iii)⇒(ii) By Fact 5.1.10(i). (iii)⇔(iv) Since Z(A) = Z(Asym ) (cf. Theorem 4.3.47), this equivalence follows from the one (i)⇔(iii) already proved (applied to both A and Asym ). (iv)⇔(v) Since H(A, ∗) = H(Asym , ∗) as JBW-algebras, this equivalence follows from Fact 5.1.42.  Lemma 6.1.2 Every quadratic algebra over K of dimension = 2 and whose algebraic norm function is nondegenerate is simple. Proof

Let A be a quadratic algebra over K. Then for each a ∈ A we have that a2 − t(a)a + n(a)1 = 0,

(6.1.1)

where t and n are the trace function and the algebraic norm function on A (cf. Proposition 2.5.12). Suppose that n is nondegenerate, and that A contains a nonzero proper ideal I. Given x ∈ I \ {0}, it follows from (6.1.1) that n(x)1 = t(x)x − x2 ∈ I, which forces n(x) = 0 because 1 ∈ / I. Therefore, if we denote by n(·, ·) the unique symmetric bilinear form on A such that n(a, a) = n(a) for every a ∈ A, then we derive from the arbitrariness of x in I \ {0} that n(x, y) = 0 for all x, y ∈ I.

(6.1.2)

Note that linearizing (6.1.1), we have 2a • b − t(a)b − t(b)a + 2n(a, b)1 = 0 for all a, b ∈ A, and in particular 2a • x − t(a)x − t(x)a + 2n(a, x)1 = 0 for all a ∈ A and x ∈ I.

(6.1.3)

Since n is nondegenerate, it follows from (6.1.2) that, for any x ∈ I \ {0}, there exists a ∈ A \ I such that n(a, x) = 0. Then, since 2a • x − t(a)x ∈ I and 1 ∈ / I, we deduce from (6.1.3) that t(x) = 0, hence the trace function is injective on I, and so dim(I) = 1. Moreover, if x ∈ I is such that t(x) = 1, then (6.1.3) gives that, for each a ∈ A, a = 2a • x − t(a)x + 2n(a, x)1 ∈ I + K1, hence dim(A) = 2, and so the proof finishes.



Corollary 6.1.3 Let A be a quadratic non-commutative JB∗ -algebra. Then: (i) A is a non-commutative JBW ∗ -algebra. (ii) A is a non-commutative JBW ∗ -factor if and only if it is simple, if and only if dim(A) = 2.

350

Representation theory for non-commutative JB∗ -algebras

Proof By Theorem 3.5.5, the algebraic norm function on A is nondegenerate, and the Banach space of A is isomorphic to a complex Hilbert space. Therefore, the Banach space of A is reflexive so, in particular, A is a non-commutative JBW ∗ algebra, and (i) is proved. Now, since a quadratic non-commutative JB∗ -algebra of dimension 2 is (isometrically) isomorphic to the C∗ -algebra C2 (which is not prime), the proof of (ii) is concluded by invoking Lemma 6.1.2.  Let A be a non-commutative JB∗ -algebra. In agreement with Definition 5.1.35, by a (non-commutative JBW ∗ -)factor representation of A we mean a w∗ -dense-range algebra ∗-homomorphism from A to some non-commutative JBW ∗ -factor. Proposition 6.1.4 Let A be a non-commutative JB∗ -algebra. Then every factor representation of A is equivalent to one of the form a → ea from A to eA , where e is a minimal central idempotent in A . Proof Let π be a factor representation of A. Then, by Proposition 5.1.36, there exists a central idempotent e ∈ A such that π is equivalent to the factor representation a → ea from A to eA ⊆ A . But, since eA is a factor, it follows from the implication (i)⇒(iii) in Fact 6.1.1 that Z(eA ) = Ce. Therefore, since eZ(A ) = Z(eA ), we realize that e is a minimal central idempotent in A .  Let X be a Banach space over K. Then, as a consequence of Proposition 5.1.14 (see [1140, p. 37] for details), the closure of the sum of any family of M-ideals of X is an M-ideal of X. Therefore, every closed subspace Y of X contains a largest M-ideal of X, namely the closure of the sum of all M-ideals of X contained in Y. Given x ∈ X  we denote by Mx the largest M-ideal of X contained in ker(x ). We denote by J the canonical imbedding of X into X  . The fact (assured by Theorem 5.1.22) that M-ideals and closed ideals of a noncommutative JB∗ -algebra coincide should be omnipresent in what follows. Proposition 6.1.5 Let A be a non-commutative JB∗ -algebra, and let a be in A . Then J(Ma ) = J(A) ∩ MJ(a ) . Proof Ma◦◦ is an M-ideal of A obviously contained in ker(J(a )), so Ma◦◦ ⊆ MJ(a ) . Hence J(Ma ) = J(A) ∩ Ma◦◦ ⊆ J(A) ∩ MJ(a ) . To prove the reverse inclusion, set L := {a ∈ A : J(a) ∈ MJ(a ) }. Since MJ(a ) is a closed ideal of A , and the mapping J is a continuous algebra homomorphism, it follows that L is a closed ideal of A obviously contained in ker(a ). Therefore, J(A) ∩ MJ(a ) = J(L) ⊆ J(Ma ).  Lemma 6.1.6 Let X be a nonzero normed space over K, let Y and Z be complementary L-summands of X, and let x be in X. Then x is an extreme point of BX if and only if either x is an extreme point of BY or x is an extreme point of BZ .

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351

Proof Let x be in SX such that x ∈ / Y ∪ Z. Then x = y + z with y and z nonzero y z elements of Y and Z, respectively, and hence we have that x = y y + z z . y z Since y + z = 1 and y , z ∈ BX , we realize that x is not an extreme point of BX . Therefore, extreme points of BX lie in either Y or Z. Now, let y be in BY such that y is not an extreme point of BX . Then there exist u, v ∈ BX such that y = 12 (u + v). Writing u = y1 + z1 and v = y2 + z2 with y1 , y2 ∈ Y and z1 , z2 ∈ Z, we see that y = 12 (y1 + y2 + z1 + z2 ), which implies that y = 12 (y1 + y2 ). Since y1 , y2 ∈ BY (because L-projections are contractive), it follows that y is not an extreme point of BY . Therefore, extreme points of BY are extreme points of BX .  Let X be a Banach space over K. According to [1140, p. 73], an M-ideal Y of X is said to be primitive if there exists an extreme point x of BX  such that Y = Mx . As established in [1140, p. 73], we are provided with the following. Fact 6.1.7 Let X be a Banach space over K, let Y be a primitive M-ideal of X, and let Z, T be M-ideals of X such that Z ∩ T ⊆ Y. Then either Z ⊆ Y or T ⊆ Y. A non-commutative JBW ∗ -factor is said to be of type I if the closed unit ball of its predual has extreme points. Lemma 6.1.8 Let A be a nonzero non-commutative JBW ∗ -algebra, and let u∗ be an extreme point of BA∗ . Then MJ(u∗ ) is an M-summand of A, and its complementary M-summand is a non-commutative JBW ∗ -factor of type I. Proof By Theorem 5.1.32, J(A∗ ) is an L-summand of A . But then, by Lemma 6.1.6, J(u∗ ) is an extreme point of BA . Because of the w∗ -continuity of J(u∗ ) and the separate w∗ -continuity of the product of A, MJ(u∗ ) is a w∗ -closed ideal of A. Therefore, by Fact 5.1.10, MJ(u∗ ) is an M-summand of A, and there exists a central idempotent e ∈ A such that MJ(u∗ ) = eA (note that e = 1). Let B stand for the complementary M-summand of MJ(u∗ ) . Then B = (1 − e)A. Let f be a central idempotent in B and set g := 1 − e − f . Then f B and gB are w∗ -closed ideals of B (and hence of A) satisfying B = f B ⊕ gB. Since f B ∩ gB = {0} ⊆ MJ(u∗ ) and MJ(u∗ ) is a primitive M-ideal of A, it follows from Fact 6.1.7 that either f B ⊆ MJ(u∗ ) or gB ⊆ MJ(u∗ ) , i.e. either f = 0 or g = 0, which proves that 1 − e (the unit of B) is a minimal central idempotent of B. Therefore, by Fact 5.10.40, Z(B) = C(1 − e), and hence B is a non-commutative JBW ∗ -factor (by the implication (iii)⇒(i) in Fact 6.1.1). Let P : A → A be the M-projection from A onto B. Then, by Lemma 5.1.21(ii), P is ∗ w -continuous, and hence P = Q for some L-projection Q : A∗ → A∗ . Since the prepolar B◦ of B in A∗ is equal to ker Q, and Q(A∗ ) ≡ A∗ / ker(Q) (where ≡ means to be linearly isometric), and A∗ /B0 ≡ B∗ (cf. §5.1.9), it follows that B∗ ≡ Q(A∗ ). On the other hand, since Q(A∗ ) and ker(Q) are complementary L-summands of A∗ , Lemma 6.1.6 yields that either u∗ ∈ Q(A∗ ) or u∗ ∈ ker(Q). But the possibility u∗ ∈ ker(Q) cannot happen for, otherwise, we would have B(u∗ ) = P(A)(u∗ ) = A(Q(u∗ )) = 0, so B would be an M-ideal of A contained in ker(J(u∗ )), hence B would be contained in MJ(u∗ ) , and finally 1 − e would belong to eA ∩ B = 0, a contradiction. Therefore u∗

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is an extreme point of BQ(A∗ ) . Then, since B∗ ≡ Q(A∗ ), we see that BB∗ has extreme points, i.e., the non-commutative JBW ∗ -factor B is of type I.  Now we can conclude the proof of the main result in this subsection, namely the following. Theorem 6.1.9 Let A be a nonzero non-commutative JB∗ -algebra. Then every primitive M-ideal of A is the kernel of a type I factor representation of A. Proof Let u be an extreme point of BA . Then, by Lemma 6.1.8 (with A and u instead of A and u∗ , respectively), MJ(u ) is an M-summand of A whose complementary M-summand (say B) is a non-commutative JBW ∗ -factor of type I. Now, writing B = pA with p a suitable central idempotent of A , it is easy to see that the mapping π : A → B defined by π(a) = pJ(a) is a non-commutative JBW ∗ -representation of A satisfying ker(π) = J −1 ((1 − p)J(A)) = J −1 (J(A) ∩ MJ(u ) ). Therefore, by Proposition 6.1.5, Mu is the kernel of the type I factor representation π of A.  Fact 6.1.10 Let X be a Banach space over K. Then the intersection of all primitive M-ideals of X reduces to zero. Proof As a consequence of the Banach–Alaoglu and Krein–Milman theorems, the intersection of the kernels of extreme points of BX  reduces to zero, and therefore the result follows straightforwardly.  Combining Theorem 6.1.9 with Fact 6.1.10, we obtain the following. Corollary 6.1.11 Every non-commutative JB∗ -algebra has a faithful family of type I non-commutative JBW ∗ -factor representations. Let A be a non-commutative JB∗ -algebra, and let π : A → F be a non-commutative JBW ∗ -factor representation. Since π(A) is w∗ -dense in F, and the product of F is separately w∗ -continuous, we realize that F satisfies all multilinear identities satisfied by A. Therefore, since JB∗ -algebras (respectively, alternative C∗ -algebras) are precisely those non-commutative JB∗ -algebras which are commutative (respectively, alternative), Corollary 6.1.11 yields the following. Corollary 6.1.12 Every JB∗ -algebra (respectively, alternative C∗ -algebra) has a faithful family of type I JBW ∗ -factor (respectively, alternative W ∗ -factor) representations. 6.1.2 Associativity and commutativity of non-commutative JB∗ -algebras We recall that an element a of a ∗-algebra A over K is said to be normal if the subalgebra of A generated by a and a∗ is associative and commutative (cf. Definition 3.4.20). Proposition 6.1.13 Let A be a non-commutative JB∗ -algebra, and let B be a subalgebra of A consisting only of normal elements of A. Then B is associative and commutative.

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Proof Let b be in B. Then, by Fact 3.4.22, the closed subalgebra of A generated by b and b∗ is a commutative C∗ -algebra. Therefore, by Lemma 1.2.12, we have b = r(b). Since b is arbitrary in B, the result follows from Lemma 3.6.27.  Proposition 6.1.14 The following assertions hold: (i) A non-commutative JB∗ -algebra A is associative and commutative if and only if the JB∗ -algebra Asym is associative. (ii) A JB∗ -algebra is associative if and only if all its JC∗ -subalgebras are associative. Proof Concerning assertion (i), only the ‘if’ part needs a proof. Let A be a noncommutative JB∗ -algebra such that Asym is associative. Then, by Fact 3.3.2, Asym is a commutative C∗ -algebra and, by Lemma 2.4.15, for any fixed a ∈ A, the mapping x → [a, x] is a derivation of Asym . Since a commutative C∗ -algebra has no nonzero derivation (cf. Corollary 3.4.51), A is commutative. Therefore A(= Asym ) is associative. Concerning assertion (ii), also only the ‘if’ part needs a proof. Let A be a JB∗ algebra all JC∗ -subalgebras of which are associative. Let a be in A. Since the closed subalgebra of A generated by a and a∗ is a JC∗ -algebra (cf. Proposition 3.4.6), and this JC∗ -algebra is associative by assumption, a is a normal element of A. Since a is arbitrary in A, it follows from Proposition 6.1.13 that A is associative.  Fact 6.1.15 Let A be a unital non-commutative JB∗ -algebra, and let π : A → F be a non-commutative JBW ∗ -factor representation. Then π(1) is the unit of F. Proof For a ∈ A we have π(a)π(1) = π(a) = π(1)π(a). Therefore, since π(A) is w∗ -dense in F, and the product of F is separately w∗ -continuous, we derive that xπ(1) = x = π(1)x for every x ∈ F.  Lemma 6.1.16 Let A be a JC∗ -algebra with no nonzero nilpotent elements. If a and b are in A with a ≥ 0 and ab = ba = 0, then aAb = 0. Proof We may suppose that b lies in H(A, ∗) and also it is enough to prove that aH(A, ∗)b = 0. So let c be in H(A, ∗) and write √ √ x := acb, s := x + x∗ , and t := x∗ x − xx∗ . Since x + x∗ = 2Ua,b (c), x∗ x = Ub Uc (a2 ) and xx∗ = Ua Uc (b2 ) (all the U-operators are taken in the sense of the Jordan algebra ∗). From √ √A), s and t lie2 in H(A, √ the 2 2 ∗ ∗ ∗ assumption ab √ √ = 0, we obtain x = 0, x x xx = 0 and s2 = t . Since x x x = ∗ ∗ xx x and x xx = 0, we also have s • t = 0. Now (s + it) = 0, so s + it = 0 by assumption. In particular, s = acb + bca = 0. Replacing in the above argument a by √ √ √ √ a (note that we also have ab = 0) we obtain acb + bc a = 0 and, multiplying √ on the left by a, we have acb = 0, as required.  Non-commutative JB∗ -algebras which are associative and commutative are nothing other than commutative C∗ -algebras, and hence, by the commutative Gelfand–Naimark theorem, they have no nonzero nilpotent element. Conversely, we have the following.

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Theorem 6.1.17 Let A be a non-commutative JB∗ -algebra with no nonzero nilpotent element. Then A is associative and commutative. Proof By Proposition 6.1.14 we may suppose that A is actually a JC∗ -algebra and that A has a unit. Assume that there is a JBW ∗ -factor representation π : A → F with F = C1. Then there exists a ∈ H(A, ∗) such that J-sp(π(a)) contains at least two points. (Indeed, otherwise we would have π(H(A, ∗)) = R1, so π(A) = C1, and hence F = C1, by the w∗ -density of π(A) in F, contrary to the assumption.) Moreover, since the unit of F lies in the range of π (cf. Fact 6.1.15), we may suppose that the J-spectrum of π(a) contains 1 and −1. Let a = a+ − a− be the orthogonal decomposition of a, and set H := {b ∈ B : a+ b = ba+ = 0}. Then, by Lemma 6.1.16, H is an ideal of A. Let M denote the w∗ -closure of π(H) in F. Then M is an ideal of F, π(a− ) belongs to M, and J-sp(π(a− )) = J-sp((π(a))− ) ⊇ {1}. Thus M is a nonzero w∗ -closed ideal of the factor F and therefore M = F. Moreover, the obvious equality π(a+ )π(H) = 0 implies π(a+ )M = π(a+ )F = 0, so π(a+ ) = 0. But 1 ∈ J-sp(π(a+ )), a contradiction. Thus we have proved that every factor representation of A is one-dimensional, which implies the associativity of A by the bracket-free version of Corollary 6.1.12.  As we established in Remark 3.4.29, alternative algebras satisfy the identity [xy, z] − x[y, z] − [x, z]y = 3[x, y, z].

(6.1.4)

Therefore we have the following. Fact 6.1.18 Alternative commutative algebras over K are associative. Keeping in mind the above fact, the next corollary follows straightforwardly from Theorem 6.1.17. Corollary 6.1.19 An alternative C∗ -algebra is commutative if and only if it has no nonzero nilpotent element. Theorem 6.1.20 Let A be an alternative √ C∗ -algebra, let a be a nonzero element of A 2 such that a = 0, and write E := sp(A1 , a∗ a) \ {0}. Then the closed ∗-subalgebra of A generated by a is ∗-isomorphic to C0 (E, M2 (C)). More precisely, we have: (i) There exists a unique algebra ∗-homomorphism F :C0 (E, M2 (C)) → A such that F(u) = a, where u stands for the function t → 0t 00 . (ii) The algebra ∗-homomorphism F above is isometric, and its range coincides with the closed ∗-subalgebra of A generated by a. Proof Since the closed ∗-subalgebra generated by a single element of an alternative C∗ -algebra is associative (a consequence of Artin’s Theorem 2.3.61), it follows

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from Corollary 3.5.36 and Theorem 4.1.71(ii) that we can suppose that A is associative. Then, according to Lemma 4.3.5, there exists a unique continuous triple homomorphism  : C0C (E) → A taking the inclusion mapping v : E → C into a, and moreover,  is an isometry, and the range of  equals the closed subtriple of A generated by a. For i, j ∈ {l, 2}, consider the linear isometry i j : C0C (E) → A defined as follows: √ √ 11 ( f ) = f ( a∗ a), 22 ( f ) = f ( aa∗ ), 21 ( f ) = ( f ), and 12 ( f ) = [(f )]∗ . It is easily realized that the isometries i j above satisfy i j ( f )k (g) = 0 if j = k and i j ( f )k (g) = i ( fg) if j = k,

(6.1.5)

for all f , g ∈ C0C (E). Indeed, the equality ij ( f )kl (g) = il ( fg) for j = k follows from Lemma 4.3.5(iii). To realize that ij ( f )kl (g) = 0 if j = k, keep in mind that a2 = 0, take f (respectively, g) of the form p(v2 ) for some p ∈ C[x] with p(0) = 0, if i = j (respectively, k = l), and of the form vp(v2 ) for some p ∈ C[x], otherwise, and apply a Stone–Weierstrass denseness argument. Now that the multiplication laws (6.1.5) have been established, it follows from them that the mapping   f11 f12 F: → 11 ( f11 ) + 22 ( f22 ) + 21 ( f21 ) + 12 ( f12 ) f21 f22 from M2 (C0C (E)) (= C0 (E, M2 (C)) in a natural way) to A is an algebra homomorphism. Moreover, keeping in mind the properties of  and the definition of the ∗-homomorphism and that isometries ij , we realize that F is actually an  algebra  F(u) = a, where u stands for the function t → 0t 00 . Let G : C0 (E, M2 (C)) → A be any other algebra ∗-homomorphism such that G(u) = a. Then, by Corollary 1.2.14, the set {w ∈ C0 (E, M2 (C)) : G(w) = F(w)} becomes a closed ∗-subalgebra of C0 (E, M2 (C)) containing u, and hence is equal to the whole algebra C0 (E, M2 (C)) because, as an easy consequence of the Stone–Weierstrass theorem, u generates C0 (E, M2 (C)) as a normed ∗-algebra. Therefore G = F, and assertion (i) is proved. In view of Corollary 1.2.52, to prove that F is isometric, it is enough to show that it is injective. To this end, let fi j be in C0C (E) ( for i, j ∈ {1, 2}) such that  i,j i j ( fi j ) = 0. Multiplying this equality on the right and on the left by j0 i0 ( fi0 j0 ) (where i0 , j0 are arbitrary but fixed values of the indices i, j) and using (6.1.5), we obtain j0 i0 ( fi30 j0 ) = 0. Therefore fi0 j0 = 0, as desired. Now that we know that F is isometric, let C denote the closed ∗-subalgebra of A generated by a. Then the range of F is a closed ∗-subalgebra of A containing a, and hence it contains C. But, by the properties of  and the definition of the isometries ij and that of F, the range of F is contained in C. Therefore the range of F equals C, which concludes the proof of assertion (ii). 

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Corollary 6.1.21 Let A be an alternative C∗ -algebra. Then A is not commutative (if and) only if it contains (as a closed ∗-subalgebra) a copy of either M2 (C) or C0 (]0, 1], M2 (C)). Proof Suppose that A is not commutative. Then, by Corollary 6.1.19, there exists a norm-one element a ∈ A such that a2 = 0. Therefore, by Theorem 6.1.20, A contains C0 (E, M2 (C)) for a suitable locally compact subset E of R such that E ∪ {0} is compact and 1 ∈ E ⊆]0, 1]. Write K := E ∪ {0}, and identify C0 (E, M2 (C)) with the C∗ -algebra B := {w ∈ C(K, M2 (C)) : w(0) = 0}. Suppose that E =]0, 1]. Then K = [0, 1], so K is not connected (because {0, 1} ⊆ K ⊆ [0, 1]), hence K = K0 ∪ K1 for suitable non-empty disjoint closed sets K0 , K1 with 0 ∈ K0 . Now consider the set function χE1 ∈ CC (K) and for α ∈ M2 (C) define  α ∈ B by  α (t) := χE1 (t)α for every t ∈ K. Then the mapping α →  α becomes an (automatically isometric) injective algebra ∗-homomorphism from M2 (C) to B. Thus M2 (C) ⊆ B ⊆ A, as required.  §6.1.22 We remark that the C∗ -algebra C0 (]0, 1], M2 (C)) does not contain any copy of M2 (C). For, if it does, then the unit of M2 (C) is a nonzero self-adjoint idempotent (say e) in C0 (]0, 1], M2 (C)), and the range of the function f : t → e(t) from ]0, 1] to R (being connected and contained in {0, 1}) has to be reduced to {1}, which is impossible because f ∈ C0R (]0, 1]). Corollary 6.1.23 An alternative C∗ -algebra A is not commutative (if and) only if there exist elements a, b ∈ A such that ab = 0 and ba = 0. Proof In view of Corollary 6.1.21, it is enough to prove the result in the cases that A = M2 (C) or  A = C0 (]0, 1], M2 (C)). In the first case, the choice a := 00 10 and b := 10 00 is adequate. In the second case, the functions α : t → ta and β : t → tb ( from ]0, 1] to M2 (C))) lie in C0 (]0, 1], M2 (C)), and satisfy αβ = 0 and βα = 0.  Proposition 6.1.24 An alternative W ∗ -algebra A is not commutative (if and) only if it contains M2 (C). Proof Keeping in mind Corollaries 5.1.31 and 6.1.21, it is enough to show that, if A contains C := C0 (]0, 1], M2 (C)), then the w∗ -closure of C in A (say B) contains M2 (C). So let us suppose that A contains C. Then, by Proposition 5.1.36, we have that B ≡ C p (as W ∗ -algebras) for a suitable central self-adjoint idempotent p in the W ∗ -algebra C . On the other hand, since M2 (C) is finite-dimensional, it follows from [717, Example 4.2(2) and Proposition 4.3(1)] that C is linearly isometric to the injective tensor product C0C (]0, 1]) ⊗ε M2 (C), hence, by [717, Theorem 6.4], we have that C ≡ (C0C (]0, 1])) ⊗π (M2 (C)) , and finally, by [717, Proposition 3.2 and Example 4.2(1)],

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C ≡ BL[(C0C (]0, 1])) , M2 (C)] = F[(C0C (]0, 1])) , M2 (C)] ≡ D ⊗ε M2 (C), where D stands for the commutative W ∗ -algebra (C0C (]0, 1])) . Let  : C → D ⊗ε M2 (C) be the surjective linear isometry given by the above identification. One easily realizes that |C is an algebra ∗-homomorphism. Therefore, since  is w∗ -continuous (cf. Corollary 5.1.30(i)), and involutions of W ∗ -algebras are w∗ -continuous (cf. Theorem 5.1.29(ii)), and products of W ∗ -algebras are separately w∗ -continuous (cf. Corollary 5.1.30(iii)), we obtain that  is a bijective algebra ∗-homomorphism. Now consider the self-adjoint idempotent p ∈ Z(C ) obtained in the first paragraph of the proof. Since B ≡ C p (as ∗-algebras), and C ≡ D ⊗ M2 (C) (also as ∗-algebras), it follows that Z(C ) ≡ D ⊗ C1, p = e ⊗ 1 for some self-adjoint idempotent e ∈ D, and B ≡ De ⊗ M2 (C). Therefore e ⊗ M2 (C) is a copy of M2 (C) contained in B, as desired.  For the formulation and proof of the following result, we recall that, as we did in p. 176 of Volume 1, the three-dimensional spin factor C3 can be introduced as the ∗-subalgebra of the JB∗ -algebra M2 (C)sym consisting of the fixed points for the involutive algebra ∗-antiautomorphism ϑ of M2 (C) given by     λ11 λ12 λ22 λ12 ϑ := . λ21 λ22 λ21 λ11 Therefore C3 is a JB∗ -algebra. We also recall that every non-commutative JB∗ -algebra becomes a JB∗ -triple in a natural way (cf. Theorem 4.1.45) and that, according to Definition 4.2.10, each nonzero element x of a JB∗ -triple X has a triple spectrum σ (x) ⊆ R+ determined by Theorem 4.2.9. We note that σ (x) does not change when we replace X with any closed subtriple of X containing x, and that σ (x) ∪ {0} is compact. Theorem 6.1.25 Let B be a JB∗ -algebra, let b be a nonzero element of B such that b2 = 0, and write E := σ (b). Then the closed ∗-subalgebra of B generated by b is ∗-isomorphic to C0 (E, C3 ). More precisely, we have: (i) There exists a unique algebra ∗-homomorphism 0 G  : C0 (E, C3 )) → B such that 0 G(u) = b, where u stands for the function t → t 0 . (ii) The algebra ∗-homomorphism G above is isometric, and its range coincides with the closed ∗-subalgebra of B generated by b. Proof Let D denote the closed ∗-subalgebra of B generated by b. Since D is a JC∗ -algebra (cf. Proposition 3.4.6), there is a C∗ -algebra A such that D becomes a closed ∗-subalgebra of Asym . Therefore, by Corollary 4.3.28 and Theorem 6.1.20, there exists an isometric algebra ∗-homomorphism  F: C0 (E, M2 (C)) → A such that F(u) = b, where u stands for the function t → 0t 00 . Now note that the inclusion C3 ⊆ M2 (C) induces a natural embedding C0 (E, C3 ) ⊆ C0 (E, M2 (C)). It follows

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that, by restricting F to C0 (E, C3 ), we are provided with an isometric algebra ∗-homomorphism G : C0 (E, C3 )) → Asym such that G(u) = b. Now, since clearly D is the closed ∗-subalgebra of Asym generated by b, to prove that the range of G is equal to D it is enough to show that the JB∗ -algebra C0 (E, C3 ) is generated by u as a normed ∗-algebra. Let R denote the closed ∗-subalgebra of C0 (E, C3 ) generated by u. Let p(x) = λ0 + λ1 x + · · · + λn xn be in C[x]. Then up(u∗ u) = λ0 u + λ1 Uu (u∗ ) + λ2 Uu Uu∗ (u) + · · · lies in R, and a routine calculation shows that up(u∗ u) is nothing other than ! " the function t → tp(t0 2 ) 00 . By the Stone–Weierstrass theorem, the function " ! " ! 0 0 0 f (t) by the self-adjointness of R) lies in R for t → f (t) 0 (and hence t → 0 0 every f ∈ C0 (E). Moreover, since p(2u • u!∗ ) lies in "R for every p ∈ C[x] with 2 p(0) = 0, and p(2u • u∗ ) is the function t → p(t ) 02 , it follows that the function 0 p(t ) ! " f (t) 0 t → 0 f (t) lies in R for every f ∈ C0 (E). Now, for all f , g, h ∈ C0 (E) the function " ! f (t) g(t) t → h(t) f (t) is in R, which proves that C0 (E, C3 ) ⊆ R. Thus, C0 (E, C3 ) = R, as desired. Now that we know that the JB∗ -algebra C0 (E, C3 ) is generated by u as a normed ∗-algebra, the uniqueness of G under the requirements in assertion (i) follows from the automatic continuity of algebra ∗-homomorphisms between  JB∗ -algebras (cf. Proposition 3.4.4). This completes the proof of the theorem. Corollary 6.1.26 A JB∗ -algebra is not associative if and only if it contains (as a closed ∗-subalgebra) a copy of either C3 or C0 (]0, 1], C3 ). Proof Argue as in the proof of Corollary 6.1.21 with Theorems 6.1.17 and 6.1.25 instead of Corollary 6.1.19 and Theorem 6.1.20, respectively.  Arguing as in §6.1.22, we realize that the JB∗ -algebra C0 (]0, 1], C3 ) does not contain any copy of C3 . Proposition 6.1.27 A JBW ∗ -algebra is not associative if and only if it contains C3 . Proof Argue as in the proof of Proposition 6.1.24 with Corollary 6.1.26 instead of Corollary 6.1.21.  The self-adjoint part of the three-dimensional (complex) spin factor C3 is the unique three-dimensional quadratic JB-algebra. It is called the three-dimensional real spin factor, and is denoted by S3 . Via Fact 3.4.9, Corollary 6.1.26 is equivalent to the following. Corollary 6.1.28 A JB-algebra is not associative if and only if it contains (as a closed subalgebra) a copy of either S3 or C0 (]0, 1], S3 ). Via Fact 5.1.42, Proposition 6.1.27 is equivalent to the following.

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Corollary 6.1.29 A JBW-algebra is not associative if and only if it contains the three-dimensional real spin factor. 6.1.3 JBW ∗ -factors We begin by recalling that in Subsection 2.5.1, the division real algebras H of quaternions and O of octonions were introduced by iterating the so called Cayley–Dickson doubling process starting from the Cayley algebra R (with standard involution equal to the identity mapping). In fact, this procedure is more general as shown in §6.1.30. §6.1.30 From now on, we suppose that F is a field of characteristic not two. By a Cayley algebra over F we mean a unital algebra A over F endowed with a linear algebra involution (called the standard involution of A) satisfying a + a ∈ F1 and aa ∈ F1 for every a ∈ A.

(6.1.6)

Every Cayley algebra over F is a quadratic algebra with trace function t : A → F and algebraic norm function n : A → F given by t(a)1 = a + a and n(a)1 = aa = aa for every a ∈ A (cf. Proposition 2.5.20). Given a Cayley algebra (A, ) over F and 0 = μ ∈ F, the algebra over F consisting of the vector space A × A and the product defined by (a1 , a2 )(a3 , a4 ) := (a1 a3 + μa4 a2 , a1 a4 + a3 a2 ) becomes a new Cayley algebra with unit element 1 = (1, 0) and standard involution given by (a1 , a2 ) := (a1 , −a2 ). This algebra will be denoted by C D(A, μ), and will be called a Cayley–Dickson doubling of A. The mapping a → (a, 0) allows us to regard A as a subalgebra of C D(A, μ), and the passing from A to C D(A, μ) is known as the Cayley–Dickson doubling process. Starting from the one-dimensional Cayley algebra A0 = F with standard involution IF , and iterating the Cayley–Dickson doubling process for nonzero scalars μi (1 ≤ i ≤ n) we find 2n -dimensional Cayley algebras over F for every n ∈ N. In Subsection 2.5.1, we dealt with the particular case F = R, we remarked that C = C D(R, −1), and we introduced H = C D(C, −1), and O = C D(H, −1). In the general case, starting from A0 = F with standard involution IF and any 0 = μ1 , μ2 , μ3 ∈ F, we obtain a two-dimensional algebra A1 := C D(A0 , μ1 ), called a binarion algebra over F, then a four-dimensional associative non-commutative algebra A2 := C D(A1 , μ2 ), called a quaternion algebra over F, and finally an eight-dimensional alternative non-associative algebra A3 := C D(A2 , μ3 ), called an octonion algebra over F. Now, we can formulate the celebrated generalized Hurwitz’s theorem, completely describing the unital composition algebras. Theorem 6.1.31 Any unital algebra A over F endowed with a nondegenerate quadratic form n : A → F admitting composition (i.e., satisfying n(ab) = n(a)n(b)

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for all a, b ∈ A) has dimension 1, 2, 4, or 8, and is one of the following: the ground field A0 = F, a binarion algebra A1 = C D(A0 , μ1 ), a quaternion algebra A2 = C D(A1 , μ2 ), or an octonion algebra A3 = C D(A2 , μ3 ). Theorem 6.1.31 is due to Kaplansky [377] and is included in [822, Theorem 1 in p. 32]. Definition 6.1.32 Let A be any algebra. If A is prime and has a nonzero centre Z, then Z has no nonzero zero divisor in itself, and as a consequence one can consider the central localization (Z \ {0})−1 A which is a prime algebra over (Z \ {0})−1 Z (the field of fractions of Z) [822, p. 185]. The algebra A is said to be a central order in an algebra B if A is prime with nonzero centre, and its central localization is isomorphic to B. The algebra A is said to be a Cayley–Dickson ring if it is a central order in an octonion algebra over the field (Z \ {0})−1 Z. The octonion algebras play a relevant role in the structure theory of alternative algebras, as shows the following Slater’s theorem included in [822, p. 194]. Theorem 6.1.33 Let A be a prime alternative algebra over a field F of characteristic different from 3, and suppose that A is not associative. Then A is a Cayley–Dickson ring. Any unital composition algebra over F is either a division algebra or has divisors of zero (cf. Proposition 2.5.26). The composition algebras having zero divisors are called split. §6.1.34 Over any field F, there exists a unique split octonion algebra, denoted by C(F) [808, Corollary III.3.24]. Moreover, this algebra can be described in terms of 2 × 2 matrices whose diagonal entries are scalars in F and whose off-diagonal entries are vectors in F3 , via the so-called Zorn’s vector matrices (cf. §2.5.1 for a particular case). To be precise, C(F) can be introduced as the algebra over F whose vector space is    α u : α, β ∈ F, u, v ∈ F3 , v β and whose product is defined ‘in a natural way’ by     α1 u1 α2 u2 α1 α2 + (u1 , v2 ) := v1 β1 v2 β2 α2 v1 + β1 v2 + u1 × u2

 α1 u2 + β2 u1 − v1 × v2 , β1 β2 + (v1 , u2 )

where, for vectors u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ) ∈ F3 , (u, v) := u1 v1 + u2 v2 + u3 v3

(the ‘scalar product’)

and u × v := (u2 v3 − u3 v2 , u3 v1 − u1 v3 , u1 v2 − u2 v1 )

(the ‘vector product’),

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and the standard involution on C(F) is given by     β −u α u a= −→ a := −v α v β (see, for example, [822, p. 46] or [777, Example II.2.4.6]). If F is an algebraically closed field, then every unital composition algebra over F of dimension > 1 is split, and is isomorphic to F ⊕ F (with coordinate-wise multiplication), M2 (F), or C(F) [822, pp. 44–7]. In particular, the unital composition complex algebras are C, C ⊕ C, M2 (C), and C(C). However, as we commented incidentally in p. 218 of Volume 1, the unital composition real algebras are R, C, R ⊕ R, H, M2 (R), O, and C(R). We recall that O is an absolute valued algebra (p. 176 of Volume 1), and that C(C)(= C ⊗π O) becomes an alternative C∗ -algebra (cf. Proposition 2.6.8) in an essentially unique way (cf. Corollary 3.4.76). Moreover, the split real octonion algebra C(R) becomes a real alternative C∗ -algebra because C(R) can be regarded as the real subalgebra of C(C) consisting of all fixed points for the conjugate-linear automorphism of C(C) obtained by composing the C∗ -algebra involution with the standard involution. §6.1.35 Given an octonion algebra C over F and γ1 , γ2 , γ3 nonzero elements in F, consider the algebra M3 (C) of all 3 × 3 matrices with entries in C (with the usual matrix product) and the diagonal matrix  := diag{γ1 , γ2 , γ3 }. If we consider the involution ∗ on M3 (C) given by X −→ X ∗ :=  −1 X  t

t

where X := (xji ) when X = (xij ), then the subspace H3 (C, ) := H(M3 (C), ∗) of all ∗-invariant elements in M3 (C), endowed with the symmetrized product, becomes a 27-dimensional central simple Jordan algebra over F [808, Theorem 4.8]. When  is equal to the identity matrix 1 = diag{1, 1, 1}, we simply write H3 (C) := H3 (C, 1). The following theorem is of special relevance in our development. Theorem 6.1.36 [755, Theorem 2.5.3] The Jordan algebra H3 (C, ) is not i-special (cf. p. 425 of Volume 1 for definition). If either F is algebraically closed or F = R, then the construction in §6.1.35 becomes specially relevant in view of the next theorem. We recall that a Jordan algebra is called exceptional if it is not special. Theorem 6.1.37 [828, Theorems 4 and 10] (i) If F is algebraically closed, then H3 (C(F)) is the unique finite-dimensional exceptional simple Jordan algebra over F. (ii) There are exactly three non-isomorphic finite-dimensional exceptional central simple Jordan algebras over R, namely H3 (C(R)), H3 (O), and H3 (O, diag{1, −1, 1}).

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§6.1.38 We recall that there is a unique norm on H3 (O) converting it into a JB-algebra (cf. Example 3.1.56 and Proposition 3.1.4(ii)). Therefore, it follows from Theorem 3.4.8 that H3 (C(C))(= C ⊗ H3 (O)) can be structured as a JB∗ -algebra in an essentially unique way (cf. Corollary 3.4.76). Definition 6.1.39 Let A be a ∗-algebra over K. In what follows we will consider linear algebra ∗-involutions on A (cf. p. 39 of Volume 1), which, for the sake of brevity, will be called simply ∗-involutions on A. We note that, if A is a C∗ -algebra with a ∗-involution τ , then H(A, τ ) is a closed ∗-subalgebra of the JB∗ -algebra Asym , so it is a JB∗ -algebra, and therefore H(A, ∗) ∩ H(A, τ ) is a JB-algebra. The following classification of JBW-factors follows from [738, Theorems 5.3.8, 5.3.9, and 7.3.5, and Proposition 7.3.3]. Theorem 6.1.40 The JBW-factors are the following: (i) (ii) (iii) (iv)

The JB-algebra H3 (O). The simple quadratic JB-algebras. The JB-algebras of the form H(A, ∗), where A is a W ∗ -factor. The JB-algebras of the form H(A, ∗) ∩ H(A, τ ), where A is a W ∗ -factor with a ∗-involution τ .

Keeping in mind Fact 5.1.42 and the equivalence (i)⇔(v) in Fact 6.1.1, Theorem 6.1.40 implies the following. Proposition 6.1.41 The JBW ∗ -factors are the following: (i) (ii) (iii) (iv)

The JB∗ -algebra H3 (C(C)). The simple quadratic JB∗ -algebras. The JB∗ -algebras of the form Asym , where A is a W ∗ -factor. The JB∗ -algebras of the form H(A, τ ), where A is a W ∗ -factor with a ∗-involution τ .

On the other hand, keeping in mind Fact 3.4.9, the next result follows from [738, Section 6.2]. Fact 6.1.42 Simple quadratic JB∗ -algebras are JC∗ -algebras. §6.1.43 Keeping in mind that the involution of any non-commutative JB∗ -algebra is isometric (cf. Proposition 3.3.13), it becomes clear that the ∞ -sum of any family of non-commutative JB∗ -algebras (respectively, JB∗ -algebras or C∗ -algebras), endowed with the product and involution defined coordinate-wise, is a noncommutative JB∗ -algebra (respectively, JB∗ -algebra or C∗ -algebra). Theorem 6.1.44 Let J be an i-special JB∗ -algebra. Then J is a JC∗ -algebra. Proof Since algebra ∗-homomorphisms between JB∗ -algebras are contractive and in fact isometric if in addition they are injective (cf. Proposition 3.4.4), it follows from Corollary 6.1.12 that every JB∗ -algebra can be seen as a norm-closed

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∗-subalgebra of a suitable ∞ -sum of JBW ∗ -factors, namely the JBW ∗ -factors of valuation of its factor representations. But, since J is i-special, and H3 (C(C)) is not (by Theorem 6.1.36), H3 (C(C)) cannot appear among the JBW ∗ -factors of valuation of factor representations of J. Therefore, by Proposition 6.1.41 and Fact 6.1.42, J is a closed ∗-subalgebra of a suitable ∞ -sum of JC∗ -algebras, and hence is a JC∗ -algebra.  The above theorem will be applied later in the following form. Corollary 6.1.45 Let J be an i-special JB∗ -algebra. Then there exists a C∗ -algebra A with ∗-involution τ such that J is a closed ∗-subalgebra of Asym contained in H(A, τ ). Proof By Theorem 6.1.44, there exists a C∗ -algebra B such that J is a closed ∗-subalgebra of Bsym . Now, by considering the C∗ -algebra A := B ⊕∞ B(0) (where B(0) denotes the opposite algebra of B), the exchange involution τ on A, and the injective algebra ∗-homomorphism x → x ⊕ x from J to Asym , the result follows.  6.1.4 Classifying prime JB∗ -algebras: a Zel’manovian approach We recall that closed ideals of a non-commutative JB∗ -algebra are ∗-invariant (cf. Proposition 3.4.13), and we refer the reader to §2.2.16 (a particular case of Theorem 5.10.90) for the notion of the C∗ -algebra of multipliers M(A) of a given C∗ -algebra A. Proposition 6.1.46 Let J be a JB∗ -algebra containing a closed essential ideal which, regarded as a JB∗ -algebra, is of the form Asym for a suitable C∗ -algebra A. Then J can be regarded as a closed ∗-subalgebra of M(A)sym containing A. Proof By assumption, there exists an isometric Jordan-∗-homomorphism φ : A →J such that φ(A) is an essential ideal of J. Since the biduals A and J  are, respectively, a C∗ -algebra and a JB∗ -algebra (cf. Theorem 2.2.15 and Proposition 3.5.26) with separately w∗ -continuous products (by the implication (iv)⇒(ii) in Lemma 2.3.51) and w∗ -continuous involutions, the bitranspose φ  of φ is an isometric Jordan∗-homomorphism from A into J  whose range is a w∗ -closed ideal of J  . By Corollary 5.1.30(iv), we have φ  (A ) = eJ  for a suitable central idempotent e ∈ J  . Now ((1 − e)J  ) ∩ J is an ideal of J whose intersection with φ(A) is zero, and hence, since φ(A) is essential in J, we have ((1 − e)J  ) ∩ J = 0. Therefore the mapping ψ : x → ex from J to φ  (A ) is an injective (hence isometric) algebra ∗-homomorphism. Then the mapping η := (φ  )−1 ψ is an isometric Jordan-∗-homomorphism from J into A , and routinely we see that η ◦ φ is equal to the canonical embedding A → A . Therefore, we can regard J as a closed ∗-subalgebra of (A )sym containing A as an ideal. Finally, by Lemma 2.2.17(i), J is in fact contained in M(A).  §6.1.47 Let A be a C∗ -algebra with a ∗-involution τ . Since A is a C∗ -algebra with separately w∗ -continuous product and w∗ -continuous involution ∗, τ  becomes a ∗-involution on A which, clearly, leaves M(A) invariant. In this way, the restriction of τ  to M(A), regarded as a mapping from M(A) to itself, is a ∗-involution on M(A) which extends τ and will be denoted with the same symbol.

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Representation theory for non-commutative JB∗ -algebras

Proposition 6.1.48 Let J be a JB∗ -algebra containing a closed essential ideal of the form H(A, τ ) for a suitable C∗ -algebra A with ∗-involution τ such that A is generated as a normed algebra by H(A, τ ). Then J can be regarded as a closed ∗-subalgebra of (M(A))sym contained in H(M(A), τ ) and containing H(A, τ ). Proof The proof is very similar to that of the preceding proposition, so we give only an outline of it. Using additionally the natural identification H(A, τ ) = H(A , τ  ), we get as above that J can be regarded as a closed ∗-subalgebra of (A )sym contained in H(A , τ  ) and containing H(A, τ ) as an ideal. Let x and a be in J and H(A, τ ), respectively. Then, [x, a2 ] = 2[x • a, a] ∈ [H(A, τ ), H(A, τ )] ⊆ A. Therefore, since H(A, τ ) is a JB∗ -algebra, and every JB∗ -algebra is the linear hull of the set of squares of its elements, we see that [J, H(A, τ )] ⊆ A, which together with J • H(A, τ ) ⊆ H(A, τ ) implies that both JH(A, τ ) and H(A, τ )J are contained in A. Now the set {a ∈ A : aJ ⊆ A and Ja ⊆ A} is a closed ∗-subalgebra of A containing H(A, τ ) so, by the generation of A by H(A, τ ) (this is the only point in the proof where this assumption is used), we have that JA and AJ are contained in A, i.e. J is contained in M(A). Finally, observe that M(A) ∩ H(A , τ  ) = H(M(A), τ ) because M(A) is invariant under any linear algebra involution on A leaving A invariant, and τ extends the given involution τ on A.  Definition 6.1.49 (a) A non-commutative Jordan algebra A is said to be nondegenerate if Ua = 0 implies a = 0. As a direct consequence of the axiom Ua (a∗ ) = a3 we have that every non-commutative JB∗ -algebra is nondegenerate. (b) By an Albert ring we mean any algebra A which is a central order in H3 (C, ) (cf. Definition 6.1.32 and §6.1.35) for some octonion algebra C over the field (Z \ {0})−1 Z, where Z denotes the centre of A. Proposition 6.1.50 (‘prime dichotomy theorem’, see the concluding theorem in the Introduction of [662], or [777, Theorem III.9.2.1]). Every prime nondegenerate Jordan algebra is either i-special or an Albert ring. Remark 6.1.51 From Zel’manov’s structure theorem for prime nondegenerate Jordan algebras [662, Theorem 3] it follows that prime nondegenerate Jordan algebras are actually either special or Albert rings. However, since we are not able to deduce the main result in this subsection only from the statement of Zel’manov’s structure theorem and pre-Zel’manovian JB∗ -theory, we are interested in involving the ‘minimum’ of the algebraic ingredients needed for our purposes, and this is the reason why Proposition 6.1.50 has been formulated in its present form. In this respect, we must say that there exists an earlier weaker version of Proposition 6.1.50 asserting the same result under the additional assumption that the Jordan algebra has no nonzero nil ideal [1131]. Since JB∗ -algebras have no nonzero nil ideal, even this weaker version is enough for us. §6.1.52 Let A be a quadratic commutative algebra over K. Then, according to Proposition 2.5.13, there exists a vector space V over K, and a symmetric bilinear

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form f on V, in such a way that A = K1 ⊕ V as vector spaces, and the product of A is determined by (α1 + x)(β1 + y) = (αβ + f (x, y))1 + (αy + βx). By this reason, quadratic commutative algebras are usually called Jordan algebras of a symmetric bilinear form (on a vector space). Proposition 6.1.53 [662, 437]. Let A be an associative algebra with linear algebra involution τ , let J be a nondegenerate prime subalgebra of Asym contained in H(A, τ ), and suppose that J is not a central order in a Jordan algebra of a symmetric bilinear form. Then there exists a τ -invariant subalgebra A0 of A such that H(A0 , τ ) is a nonzero ideal of J. Moreover, we can choose the subalgebra A0 generated by H(A0 , τ ) as an algebra. Remark 6.1.54 Because the proof of this proposition is one of the most impressive parts of Zel’manov’s work, we give here a brief guide to it. Since J is a prime nondegenerate Jordan algebra which is not a central order in the Jordan algebra of a symmetric bilinear form, by [662, Theorem 1] J does not satisfy a suitable polynomial identity, the so-called f of [662], and by [662, Section l] f is an ‘hermitian polynomial’ in the sense of [437, p. 144], so the proposition follows from assertion (1.3) in [437, p. 145]. As remarked in [437], the existence of such Jordan polynomials as f above, which are at the same time hermitian and ‘Clifford’ (see [437, p. 167] for a definition), is one of the most important novelties in Zel’manov’s work. The following fact follows from the commutative Gelfand–Naimark theorem (cf. Theorem 1.2.23). Fact 6.1.55 Let A be a unital commutative C∗ -algebra with no nonzero divisor of zero. Then A = C1. Definition 6.1.56 An algebra A with algebra involution τ is said to be τ -prime if whenever P, Q are τ -invariant ideals of A with PQ = 0 we have either P = 0 or Q = 0. Now we can conclude the proof of the main result in this subsection, namely the following. Theorem 6.1.57 The prime JB∗ -algebras are the following: (i) The JB∗ -algebra H3 (C(C)) (cf. §6.1.38). (ii) The simple quadratic JB∗ -algebras (cf. Corollaries 3.5.7 and 6.1.3). (iii) The closed ∗-subalgebras of M(A)sym containing A, where A is any prime C∗ -algebra. (iv) The closed ∗-subalgebras of M(A)sym contained in H(M(A), τ ) and containing H(A, τ ), where A is any prime C∗ -algebra with ∗-involution τ (in this case A can be chosen in such a way that it is generated by H(A, τ ) as a normed algebra).

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Representation theory for non-commutative JB∗ -algebras

Proof As we have already pointed out, every JB∗ -algebra is a nondegenerate Jordan algebra. Note also that the centre of a prime JB∗ -algebra is either Cl in the unital case or 0 otherwise. Indeed, keeping in mind that the centre of a prime JB∗ -algebra is a commutative C∗ -algebra (cf. Proposition 3.4.1(i)) with no nonzero divisor of zero, this follows from Fact 6.1.55. Therefore, if a prime JB∗ -algebra J is either an Albert ring or a central order in the Jordan algebra of a symmetric bilinear form, then the passing to the central localization in the definition of these concepts trivializes, thus giving J = H3 (C(C)) in the first possibility (and we are in case (i) of our statement), while for the second possibility we have that either J = C (and we are trivially in both cases (iii) and (iv) with A = C), or J is a simple quadratic JB∗ -algebra, being then in case (ii). If none of the assumptions considered above is true for the prime JB∗ -algebra J, we can apply Corollary 6.1.45 and Propositions 6.1.50 and 6.1.53, to get a C∗ -algebra B with ∗-involution τ such that J is contained in H(B, τ ) (in the best possible sense) and a τ -invariant subalgebra B0 of B such that H(B0 , τ ) is a nonzero ideal of J. Denoting by A and I the norm closures of B0 and H(B0 , τ ) in B, respectively, we have clearly that I = H(A, τ ) and that I is a closed (so ∗-invariant, by Proposition 3.4.13) ideal of J. Since we may choose B0 generated by H(B0 , τ ) as an algebra, it follows that A is generated as a normed algebra by the ∗- invariant set I, so A is a closed ∗-subalgebra of B, and hence a C∗ -algebra. The primeness of J implies that the closed ideal I = H(A, τ ) is essential, so by Proposition 6.1.48 we have H(A, τ ) ⊆ J ⊆ H(M(A), τ ) (again in the best possible sense) and so, if the C∗ -algebra A is prime, we are in case (iv) of our statement. But in any case we claim that A is τ -prime. Since H(A, τ ) is a nonzero ideal of the prime nondegenerate Jordan algebra J, it is also a prime Jordan algebra [662, Lemma 19] and so, by semiprimeness of C∗ -algebras, we are in the following situation: A is a semiprime associative algebra with linear algebra involution τ such that H(A, τ ) is a prime Jordan algebra. But this implies that A is τ -prime, thus concluding the proof of the ‘claim’: if P and Q are τ -invariant ideals of A with P ∩ Q = 0, at least one of the intersections P ∩ H(A, τ ) or Q ∩ H(A, τ ) must be zero, say P ∩ H(A, τ ) = 0, so that P is an algebra with involution τ satisfying H(P, τ ) = 0 (equivalently, P is anticommutative and τ = −IA on P). This implies that P = 0 by semiprimeness of A. Now we return to the fundamental line of the proof by considering the unique remaining possible case for the prime JB∗ -algebra J, namely the one that H(A, τ ) ⊆ J ⊆ H(M(A), τ ), where τ is a ∗-involution on a non-prime τ -prime C∗ -algebra A. Using the fact that A is semiprime but not prime, we can find nonzero closed ideals C and D of A with C ∩ D = 0. By τ -primeness, and without loss of generality, we may suppose that C ∩ τ (C) = 0. Now the mapping x → x + τ (x) is an injective Jordan-∗-homomorphism (so isometric, and so with closed range) from the C∗ -algebra C into J (actually with values in H(A, τ )). It is routine to see that its range is an ideal of J (use that, by [758, Corollary 4.2.10] or Corollary 7.1.2, C is an ideal in M(A)). Therefore J contains a closed nonzero (so essential, by primeness of J) ideal of the form Csym with C a C∗ -algebra, and it is enough to apply Proposition 6.1.46 to obtain C ⊆ J ⊆ M(C)sym . Since Csym

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is an ideal of J, Csym is prime by [662, Lemma 19] again, and hence C is a prime associative algebra. Then we are in case (iii) of our statement. Conversely, all the JB∗ -algebras listed in the theorem are prime. Only the cases (iii) and (iv) require some explanation. First note that if A is prime then M(A) is prime (because A is an essential ideal of M(A)). Now, by [1016], we have that both M(A)sym and H(M(A), τ ) are prime Jordan algebras. Hence, in any case, J is prime because it contains a nonzero closed ideal which is essential and a prime Jordan algebra.  Corollary 6.1.58 The topologically simple JB∗ -algebras are the following: (i) The JB∗ -algebra H3 (C(C)). (ii) The simple quadratic JB∗ -algebras. (iii) The JB∗ -algebras of the form Asym , where A is any topologically simple C∗ -algebra. (iv) The JB∗ -algebras of the form H(A, τ ), where A is any topologically simple C∗ -algebra with ∗-involution τ . Proof Since topologically simple normed algebras are prime, it is enough to select among the JB∗ -algebras listed in Theorem 6.1.57 those which are actually topologically simple. Obviously H3 (C(C)) and simple quadratic JB∗ -algebras should be selected. If the prime JB∗ -algebra J is in the situation A ⊆ J ⊆ (M(A))sym , where A is a prime C∗ -algebra, then, since A is a nonzero closed ideal of J, applying in a first instance the eventual topological simplicity of J, we obtain that J = Asym , and then we should verify that J is topologically simple if and only if A is, a fact which follows from Proposition 3.6.11(i). Let us finally suppose that H(A, τ ) ⊆ J ⊆ H(M(A), τ ), where A is a prime C∗ -algebra with ∗-involution τ . If J is topogically simple, then, since H(A, τ ) is a nonzero closed ideal of J, we have that J = H(A, τ ), and then, since we may choose A generated by H(A, τ ) as a normed algebra, we realize that A is topologically simple. (Indeed, if P is a nonzero closed ideal of A, then P ∩ τ (P) = 0 by primeness of A, and hence P ∩ H(A, τ ) = 0 since otherwise P ∩ τ (P) would be a nonzero anticommutative C∗ -algebra; therefore P∩H(A, τ ) = H(A, τ ) by topological simplicity of H(A, τ ), so P = A because A is generated by H(A, τ ) as a normed algebra.) To conclude the proof we note that, for any topologically simple C∗ -algebra A with ∗-involution τ , the JB∗ -algebra H(A, τ ) is topologically simple, a fact which follows with minor topological changes from the arguments in the proof of [743, Theorem 2.1(1)].  6.1.5 Prime non-commutative JB∗ -algebras are centrally closed Let A be an algebra over K. We define the (unital) multiplication algebra, M (A), of A as the subalgebra of L(A) generated by the identity operator IA on A and all left and right multiplication operators by elements of A. This notion was already considered in the particular setting of Jordan algebras (cf. the case B = A in §3.3.36). For each a, b ∈ A, we denote by Na,b , the bilinear mapping from M (A) × M (A) into A defined

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by Na,b (F, G) = F(a)G(b) for all F, G in M (A). Keeping in mind that the ideal generated by any element a ∈ A agrees with M (A)(a), we realize that A is a prime algebra if and only if it is nonzero and, for a, b ∈ A, Na,b = 0 implies either a = 0 or b = 0. Therefore, for a normed algebra A over K, a reasonable strengthening of primeness is to require that A = 0 and that there exists a positive number L such that Na,b  ≥ Lab for all a, b in A,

(6.1.7)

where, of course, for the computation of Na,b , we consider M (A) as a normed space under the operator norm. Normed algebras satisfying the above requirements will be called totally prime. §6.1.59 Let A be a prime algebra over K. For the notion and properties of the extended centroid, CA , of A, the reader is referred to §2.5.41, Lemma 2.5.42, Definition 2.5.43, and Proposition 2.5.44. In this last result we proved that CA is a field extension of K. We say that A is centrally closed whenever CA = K. Theorem 6.1.60 Every totally prime normed complex algebra is centrally closed. Proof In view of Lemma 2.5.46, it is enough to show that partially defined centralizers on any totally prime normed algebra are continuous. Let A be a totally prime normed algebra, and let L be a positive constant such that (6.1.7) holds. Let f be a partially defined centralizer on A and let x, y be in dom( f ). Then F(x)G(y) belongs to dom( f ) for all F, G ∈ M (A), hence F( f (x))G(y) = f (F(x)G(y)) = F(x)G( f (y)), so Nf (x),y = Nx,f (y) , and so Lf (x)y ≤ Nf (x),y  = Nx,f (y)  ≤ xf (y). Therefore f is continuous with f  ≤

1 inf{f (y) : y ∈ Sdom( f ) }. L



Remark 6.1.61 The proof of Theorem 6.1.60 shows that the extended centroid of a totally prime normed real algebra is either R or C. In §2.8.58 we introduced the ultraproduct of a family of Banach spaces {Xi }i∈I (with respect to a given ultrafilter on the set of indices I), and in particular the ultrapower of a Banach space X (with respect to an ultrafilter on an arbitrary set). Now, for the sake of convenience, we remark that completeness of the spaces Xi in the notion of ultraproduct, and in particular that of X in the notion of ultrapower, is irrelevant. Therefore, we can and will consider the (normed) ultraproduct of a given family of normed spaces, as well as the particular case of the (normed) ultrapower of a given normed space. As pointed out in §2.8.61 for the complete case, ultraproducts of a given family of normed algebras become normed algebras in a natural way. In particular, any ultrapower of a normed algebra becomes a normed algebra.

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An ultrafilter U on a set I is said to be countably incomplete if it is not closed under countable intersections of their elements. It is routine to verify that this condition is equivalent to the existence of a strictly decreasing sequence {Uk }k∈N in U such that U1 = I and ∩k∈N Uk = ∅. We say that a normed algebra A over K is ultraprime if some ultrapower AU , with respect to a countably incomplete ultrafilter U , is a prime algebra. A nice characterization of the ultraprimeness for normed associative algebras is given by the equivalence (i)⇔(ii) in the following. Proposition 6.1.62 For a normed associative algebra A the following conditions are equivalent: (i) A is ultraprime. (ii) There exists a positive number K such that Ma,b  ≥ Kab for all a, b in A. (Here Ma,b denotes the operator on A given by Ma,b (x) = axb.) (iii) Every ultrapower of A is prime. Proof Assume that condition (ii) does not hold. Then for each k ∈ N there exist norm-one elements ak , bk ∈ A such that Mak ,bk  ≤ 1k . Let U be any countably incomplete ultrafilter on an arbitrary (infinite) set I, and let {Uk }k∈N be a strictly decreasing sequence in U such that U1 = I and ∩k∈N Uk = ∅. Then, for each i ∈ I, there is precisely one k ∈ N such that i belongs to Uk \ Uk+1 , and so we can consider the elements (ai ), (bi ) of AU determined by ai = ak and bi = bk if i lies in Uk \ Uk+1 . Since ai  = bi  = 1 for every i ∈ I, we have that (ai ) and (bi ) are nonzero elements of AU . Fixed ε > 0, let k be in N such that 1k < ε. Then for each i ∈ Uk we have Mai ,bi  ≤ 1k < ε. Therefore, the set {i ∈ I : Mai ,bi  < ε} contains Uk and thus it is an element of U . It follows from the arbitrariness of ε that limU Mai ,bi = 0 in BL(A). This implies (ai )AU (bi ) = 0, hence AU is not prime. The arbitrariness of U now shows that A is not ultraprime. In this way we have proved that (i) implies (ii). Since condition (ii) implies primeness, to prove that (ii) implies (iii) it is enough to show that (ii) passes to ultrapower. Suppose that condition (ii) is fulfilled. Let U be any ultrafilter on an arbitrary set I, let (ai ), (bi ) be in AU , and let 0 < L < K. Then, for each i ∈ I, there is a norm-one element xi ∈ A such that Mai ,bi (xi ) ≥ Lai bi . Therefore, M(ai ),(bi )  ≥ M(ai ),(bi ) ((xi )) = (Mai ,bi (xi )) = lim Mai ,bi (xi ) ≥ lim Lai bi  = L(ai )(bi ). U

U

By letting L → K, we realize that (ii) holds with AU instead of A (and even with the same positive constant K). The implication (iii)⇒(i) is clear.  In the non-associative setting, the most natural examples of ultraprime normed algebras are provided by nearly absolute-valued algebras. Indeed, nearly absolutevalued algebras are prime, and, as pointed out in §2.8.61 (for the particular case of complete absolute-valued algebras), ultrapowers of (nearly) absolute-valued algebras are (nearly) absolute-valued algebras.

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Representation theory for non-commutative JB∗ -algebras

We are going to show that ultraprimeness implies total primeness. To this end, we note that it is easy to verify that if A is a normed algebra over K, and if U is any ultrafilter on a set I, then the correspondence ϕ from M (A)U into BL(AU ), defined by ϕ((Fi ))(ai ) = (Fi (ai )) for all (Fi ) in M (A)U and all (ai ) in AU , is a well-defined isometric algebra homomorphism from M (A)U to BL(AU ). Since ϕ((Lai )) = L(ai ) , ϕ((Rai )) = R(ai ) , and ϕ((IA )) = IAU , we have that ϕ(M (A)U ) contains M (AU ), so M (AU ) can and will be seen in a canonical way as a subalgebra of the algebra M (A)U . Theorem 6.1.63 Every ultraprime normed algebra over K is totally prime. Proof Let A be a normed algebra which is not totally prime. Then, for every k ∈ N there exist norm-one elements ak , bk ∈ A such that Nak ,bk  < 1k . Let U be any countably incomplete ultrafilter on an arbitrary set I, and let {Uk }k∈N be a strictly decreasing sequence in U such that U1 = I and ∩k∈N Uk = ∅. Then, for each i ∈ I, there is precisely one k ∈ N such that i belongs to Uk \ Uk+1 , and so we can consider the elements (ai ), (bi ) of AU determined by ai = ak and bi = bk if i lies in Uk \ Uk+1 . Since ai  = bi  = 1 for every i ∈ I, we have that (ai ) and (bi ) are nonzero elements of AU , and therefore M (AU )(ai ) and M (AU )(bi ) are nonzero ideals of AU . The proof will be concluded by showing that M (AU )(ai )M (AU )(bi ) = 0. Recalling the natural embedding M (AU ) → M (A)U , it is enough to show that (Fi (ai )Gi (bi )) = 0 for all (Fi ), (Gi ) ∈ M (A)U . So let us fix (Fi ) and (Gi ) in M (A)U , and let M1 , M2 be positive numbers such that Fi  ≤ M1 and Gi  ≤ M2 for all i ∈ I. Fixed ε > 0, let k be in N such that M1 M2 < kε. Then for each i ∈ Uk we have 1 Fi (ai )Gi (bi ) = Nai ,bi (Fi , Gi ) ≤ Nai ,bi Fi Gi  < M1 M2 < ε. k Therefore, the set {i ∈ I : Fi (ai )Gi (bi ) < ε} contains Uk and thus it is an element of U . This fact proves that (Fi (ai )Gi (bi )) is zero in AU , as desired.  Corollary 6.1.64 Let A be a nearly absolute-valued algebra over K. Then the extended centroid of A is isomorphic to R or C if K = R, and to C if K = C. Proof Since nearly absolute-valued algebras are ultraprime, the result follows from Theorems 6.1.60 and 6.1.63, and Remark 6.1.61.  As a straightforward consequence of Theorems 6.1.60 and 6.1.63 we obtain the following. Corollary 6.1.65 Ultraprime normed complex algebras are centrally closed. §6.1.66 Let A be an associative algebra. For every subset S of A, the left and right annihilators of S in A are defined by Ann (S) := {a ∈ A : ab = 0 for every b ∈ S} and Annr (S) := {a ∈ A : ba = 0 for every b ∈ S}.

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It is clear that Ann (S) is a left ideal of A and Annr (S) is a right ideal of A. For every ideal I of A, the annihilator of I in A is defined by Ann(I) := {a ∈ A : ax = xa = 0 for every x ∈ I}. Clearly, Ann(I) = Ann (I) ∩ Annr (I), and Ann(I) is an ideal of A. Lemma 6.1.67 Let A be a semiprime associative algebra, and let I be an ideal of A. Then we have: (i) Ann(I) = Ann (I) = Annr (I). (ii) I is an essential ideal of A if and only if Ann(I) = 0. (iii) I ∩ Ann(I) = 0 and I ⊕ Ann(I) is an essential ideal of A. Proof To prove assertion (i), it is enough to show that Ann (I) = Annr (I). Note that, since I is an ideal of A, both Ann (I) and Annr (I) are ideals of A. Now, by semiprimenes, the equalities Ann (I)I = 0 and I Annr (I) = 0 yield I Ann (I) = 0 and Annr (I)I = 0, and consequently Ann (I) = Annr (I). The remaining assertions follow easily from assertion (i).  It is clear that every prime algebra is semiprime, and that every nonzero ideal of a prime algebra is an essential ideal. Proposition 6.1.68 Let A be an associative algebra and let D be an essential ideal of A. Then the following conditions are equivalent: (i) D is a prime algebra. (ii) If aDb = 0, for a, b ∈ A, then a = 0 or b = 0. (iii) A is a prime algebra. Proof (i)⇒(ii) Let a, b ∈ A be such that aDb = 0, and let I and J stand for the ideals of A generated by a and b, respectively. Note that ID and DJ are ideals of D satisfying (ID)(DJ) = 0, and consequently ID = 0 or DJ = 0. If ID = 0, then (I ∩D)2 = 0, hence I ∩ D = 0, so I = 0, and so a = 0. Analogously, DJ = 0 leads to b = 0. (ii)⇒(iii) Let I and J be ideals of A such that IJ = 0. Then IDJ = 0, and consequently I = 0 or J = 0. (iii)⇒(i) Let I and J be ideals of D such that IJ = 0. Then DID and DJD are ideals of A such that (DID)(DJD) = 0, and consequently DID = 0 or DJD = 0. Now, in view of Lemma 6.1.67, we conclude that I = 0 or J = 0.  Theorem 6.1.69 Let A be a prime C∗ -algebra. Then for all a, b ∈ A we have that Ma,b  = ab. Proof Let a, b be elements in A. Suppose first that a = b = 1 and that a and b are positive elements in the C∗ -algebra M(A) of multipliers of A. Given 0 < α < 1, + consider an R+ 0 -valued continuous function f defined on R0 such that f (t) = 0 if t ∈ [0, α] and f (1) = 1. Then, by continuous functional calculus at a, we have that c := f (a) is a nonzero positive element in M(A) such that αc ≤ ac. In the same way,

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we find a nonzero positive element d in M(A) such that αd ≤ bd. Since A is an essential ideal of M(A) (cf. Proposition 2.3.56(i)), it follows from Proposition 6.1.68 that there exists u ∈ A such that x := cud = 0. Thus, by Proposition 2.3.39, we obtain Ma,b (x)2 = acudb2 = bdu∗ ca2 cudb ≥ α 2 bdu∗ c2 udb = α 2 cudb2 = α 2 cudb2 du∗ c ≥ α 4 cud2 u∗ c = α 4 cud2 = α 4 x2 . Now, the arbitrariness of α in ]0, 1[ gives that Ma,b  ≥ 1. The reverse inequality is clear, and hence Ma,b  = 1. To conclude the proof, we remove the assumptions in the above paragraph that a = b = 1 and that a and b are positive elements in M(A). Then we have a2 b2 = a∗ abb∗  = Ma∗ a,bb∗  = Ma∗ ,b∗ Ma,b  ≤ Ma∗ ,b∗ Ma,b  ≤ abMa,b , and we conclude that Ma,b  = ab.



Combining Proposition 6.1.62 with Theorem 6.1.69, we obtain the following. Corollary 6.1.70 Prime C∗ -algebras are ultraprime. §6.1.71 According to §2.5.11 and Proposition 2.5.12, given a quadratic algebra A over K, there are a linear form t (‘the trace function’) and a quadratic form n (‘the algebraic norm function’) on A such that a2 − t(a)a + n(a)1 = 0 for every a ∈ A. Lemma 6.1.72 Every ultraproduct of a family of quadratic normed complex algebras is a quadratic algebra. Proof Let A be a quadratic normed complex algebra. Then, for a ∈ A, the spectrum of a relative to the (associative) subalgebra of A generated by a and 1 consists of the roots (say λ1 , λ2 ) of the complex polynomial λ2 − t(a)λ + n(a), so t(a) = λ1 + λ2 and n(a) = λ1 λ2 , and hence |t(a)| ≤ 2a and |n(a)| ≤ a2 Now let {Ai }i∈I be a family of quadratic normed complex algebras, and let U be an ultrafilter on I. Then, since each Ai is unital (with unit 1i say), (Ai )U is unital with unit 1 = (1i ). For i ∈ I, let ti and ni denote the trace function and the algebraic norm function, respectively, on Ai . By the first paragraph in this proof, for (ai ) ∈ (Ai )U , {ti (ai )}i∈I and {ni (ai )}i∈I are bounded families of complex numbers, and therefore t : (ai ) → lim ti (ai ) and n : (ai ) → lim ni (ai ) U

U

become well-defined mappings from (Ai )U to C satisfying (ai )2 − t((ai ))(ai ) + n((ai ))1 = 0 for every (ai ) ∈ (Ai )U . Thus (Ai )U is a quadratic algebra.



§6.1.73 It follows straightforwardly from §6.1.43 that, if U is an ultrafilter on a nonempty set I, and if {Ai }i∈I is a family of non-commutative JB∗ -algebras (respectively, JB∗ -algebras or C∗ -algebras), then the ultraproduct (Ai )U (cf. §2.8.61) is a

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non-commutative JB∗ -algebra (respectively, JB∗ -algebra or C∗ -algebra) in a natural way. Proposition 6.1.74 Prime JB∗ -algebras are ultraprime. Proof Let J be a prime JB∗ -algebra, and let U be any ultrafilter on a set I. It is enough to show that the JB∗ -algebra JU is prime. To this end, we invoke Theorem 6.1.57, and discuss the matter case by case. If J is finite-dimensional ( for example, if J = H3 (C(C))), then we have JU = J (cf. Lemma 2.8.59), so JU is prime. If J is quadratic and infinite-dimensional, then, by Lemma 6.1.72, JU is a quadratic JB∗ -algebra with dim(JU ) ≥ dim(J) > 2, so JU is prime. For the study of the remaining cases note that, if A is a prime C∗ -algebra, we have a canonical imbedding M(A)U ⊆ M(AU ). Indeed, by Proposition 2.3.56(i) and the implication (i)⇒(iii) in Proposition 6.1.68, M(A) is a prime C∗ -algebra, so M(A)U is a prime C∗ -algebra (cf. Corollary 6.1.70). On the other hand, the inclusion A → M(A) as an ideal induces canonically an inclusion AU → M(A)U as an ideal, so AU is a nonzero (hence essential) ideal of the prime C∗ -algebra M(A)U , and so the inclusion M(A)U ⊆ M(AU ) follows from Proposition 2.3.56(ii). If our prime JB∗ -algebra J is in the situation A ⊆ J ⊆ M(A)sym with A a prime C∗ -algebra, then we have AU ⊆ JU ⊆ (M(A)U )sym ⊆ M(AU )sym , and setting B := AU we are in the situation B ⊆ JU ⊆ M(B)sym with B a prime C∗ -algebra. Hence JU is prime by (the trivial part of) Theorem 6.1.57. Analogously, if H(A, τ ) ⊆ J ⊆ H(M(A), τ ) for some prime C∗ -algebra A with ∗-involution τ , denoting by τU the natural extension of τ to the corresponding ultrapowers, we have H(AU ,τU ) = (H(A,τ ))U ⊆ JU ⊆ (H(M(A),τ ))U = H(M(A)U ,τU ) ⊆ H(M(AU ),τU ), so that we are in the situation H(B, τ ) ⊆ JU ⊆ H(M(B), τ ), where B is the prime C∗ -algebra AU and τ is the ∗-involution τU . Again JU is prime by the trivial part of Theorem 6.1.57.  Fact 6.1.75 Let A be a semiprime algebra over K, and let P, Q be ideals of A. Then the conditions PQ = 0, P ∩ Q = 0, and QP = 0 are equivalent. Proof If PQ = 0, then P ∩ Q is an ideal of A satisfying (P ∩ Q)(P ∩ Q) = 0, hence P ∩ Q = 0 by semiprimeness of A, and then QP ⊆ P ∩ Q = 0. By interchanging the roles of P and Q, the result follows.  Lemma 6.1.76 Let A be an algebra over K. We have: (i) If Asym is semiprime, then so is A. (ii) If Asym is prime, then so is A. Proof Suppose that Asym is semiprime. Let P be an ideal of A with PP = 0. Then P is an ideal of Asym with P • P = 0. Therefore P = 0 by semiprimeness of Asym . This proves assertion (i).

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Finally suppose that Asym is prime. Let P, Q be ideals of A with PQ = 0. Since Asym is semiprime, it follows from (i) that A is semiprime, and hence, by Fact 6.1.75, we have QP = 0, so P • Q ⊆ PQ + QP = 0. Since Asym is in fact prime, and P, Q are ideals of Asym , we deduce that either P = 0 or Q = 0. This proves assertion (ii).  In the proof of the next lemma we will apply that non-commutative JB∗ algebras are semiprime. For, if P is an ideal of a non-commutative JB∗ -algebra with PP = 0, then for x ∈ P we have Ux (x∗ ) ∈ PP = 0, and hence x = 0 by the definition itself of a non-commutative JB∗ -algebra. (Alternatively, note that every nondegenerate non-commutative Jordan algebra is semiprime, and recall the comment in Definition 6.1.49(a).) Note also that the ‘if’ part of the lemma follows from Lemma 6.1.76(ii). Lemma 6.1.77 A non-commutative JB∗ -algebra A is prime (if and) only if Asym is prime. Proof Suppose that A is prime. Let P, Q be ideals of Asym such that P • Q = 0. We want to show that either P = 0 or Q = 0. To this end, we may suppose that P and Q are closed in A. Then, by Proposition 3.6.11(i), P and Q are ideals of A. On the other hand, since Asym is a JB∗ -algebra (cf. Fact 3.3.4), and JB∗ -algebras are semiprime, Fact 6.1.75 applies to get that P ∩ Q = 0. It follows from the primeness of A that either P = 0 or Q = 0, as desired.  Now we can unify Corollary 6.1.70 and Proposition 6.1.74 by means of the following. Theorem 6.1.78 Prime non-commutative JB∗ -algebras are ultraprime. Proof Let A be a prime non-commutative JB∗ -algebra. Then, by Fact 3.3.4, Lemma 6.1.77, and Proposition 6.1.74, Asym is ultraprime, and hence there is an uncountable incomplete ultrafilter U such that (Asym )U is prime. Since (Asym )U = (AU )sym , it follows that (AU )sym is prime. Therefore, by Lemma 6.1.76(ii), AU is prime. Thus A is ultraprime.  Combining Corollary 6.1.65 with Theorem 6.1.78, we obtain the following. Corollary 6.1.79 Prime non-commutative JB∗ -algebras are centrally closed. 6.1.6 Non-commutative JBW ∗ -factors and alternative W ∗ -factors We recall that, linearizing the flexible identity [a, b, a] = 0, we get [a, b, c] + [c, b, a] = 0, and hence the equality Lab − La Lb = Rba − Ra Rb

(6.1.8)

holds for all elements a, b in any flexible algebra (cf. Corollary 3.2.2(i)). We recall also that, for each element a in a non-commutative Jordan algebra A, {La , Ra , La2 , Ra2 } is a commutative subset of L(A)

(6.1.9)

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375

(cf. Corollary 3.2.2(ii)). As a consequence of (6.1.8) and (6.1.9), the U-operator in a non-commutative Jordan algebra can be written indistinctly as follows Ua = La (La + Ra ) − La2 = Ra (La + Ra ) − Ra2 .

(6.1.10)

Since non-commutative Jordan algebras are power-associative (cf. Proposition 2.4.19), they enjoy a Peirce decomposition relative to any idempotent (cf. Lemma 2.5.3). Next we give a refined version of Lemma 2.5.3 in the setting of noncommutative Jordan algebras. Lemma 6.1.80 Let A be a non-commutative Jordan algebra over K, and let e be an idempotent in A. Then A is the direct sum A = A1 ⊕ A 1 ⊕ A0 of Peirce subspaces 2

1 Ak := {x ∈ A : e • x = kx} for k = 1, , 0, 2 and we have that Ak = {x ∈ A : ex = xe = kx} for k = 1, 0. These subspaces satisfy the following Peirce relations: (PR1) (PR2) (PR3) (PR4)

Ak Ak ⊆ Ak for k = 1, 0. Ak A 1 + A 1 Ak ⊆ A 1 for k = 1, 0. 2 2 2 A 1 • A 1 ⊆ A1 + A0 . 2 2 A1 A0 = A0 A1 = 0.

Moreover, the projections Pk from A onto Ak (k = 1, 12 , 0) corresponding to the decomposition A = A1 ⊕ A 1 ⊕ A0 are given by 2

P1 = Ue , P 1 = 2Ue,1−e , 2

and

P0 = U1−e .

(6.1.11)

Proof With the exception of the relations (PR1), (PR2), and (PR4), and the new computations of the Peirce projections, all assertions in the present lemma are already known in Lemma 2.5.3. Linearizing the identities [Lx , Rx ] = 0 and [Lx , Rx2 ] = 0, we obtain the identities [Lx , Ry ] + [Ly , Rx ] = 0 and [Lx , Ry2 ] + 2[Ly , Rx•y ] = 0, and taking x ∈ Ak (k = 1, 0) and y = e we derive that 0 = [Lx , Re ] + 2[Le , Rx•e ] = [Lx , Re ] + 2k[Le , Rx ] = (1 − 2k)[Lx , Re ], and hence [Lx , Re ] = 0. Arguing in the same way with the identities [Rx , Lx2 ] = 0, [Rx , Rx2 ] = 0, and [Lx , Lx2 ] = 0, we obtain that [Rx , Le ] = 0, [Rx , Re ] = 0, and [Lx , Le ] = 0, respectively. As a consequence, multiplications by elements of A1 or A0 commute with Le• . Since, by Lemma 2.5.3, the Peirce projections are given by P1 = Le• (2Le• − IA ), P 1 = 4Le• (IA − Le• ), and P0 = (Le• − IA )(2Le• − IA ), 2 (6.1.12)

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376

it follows that the multiplications by elements of A1 or A0 commute with the Peirce projections, and so leave the subspaces Ak (k = 1, 12 , 0) invariant. This establishes the relations (PR1) and (PR2), and shows that A1 A0 ⊆ A1 ∩ A0 = 0 and A0 A1 ⊆ A0 ∩ A1 = 0, whence (PR4). Finally, keeping in mind the flexible identity and its linearization (6.1.8), we deduce that [Le , Re ] = 0 and Le − Le2 = Re − R2e , and then (6.1.11) follows straightforwardly from (6.1.12).  As a by-product of the above lemma and its proof, we derive the following. Corollary 6.1.81 Let A be a non-commutative Jordan algebra over K, and let e be an idempotent in A. Then [F, G] = 0 for all F ∈ LAk ∪ RAk (k = 1, 0) and G ∈ {Le , Re }. As a first consequence of the Peirce relations we have the following. Corollary 6.1.82 Let A be a non-commutative Jordan algebra over K, and let e be an idempotent in A. If xk ∈ Ak (k = 1, 0), and if x 1 , y 1 ∈ A 1 , then 2

2

2

xk Pk (x 1 • y 1 ) = xk (x 1 • y 1 ) and Pk (x 1 • y 1 )xk = (x 1 • y 1 )xk , 2

2

2

2

2

2

2

2

and hence xk • Pk (x 1 • y 1 ) = xk • (x 1 • y 1 ). 2

2

2

2

Proof By (PR3), x 1 • y 1 = P1 (x 1 • y 1 ) + P0 (x 1 • y 1 ) for all x 1 , y 1 ∈ A 1 . Now, the 2 2 2 2 2 2 2 2 2 statement follows from (PR4).  Note that in the description of the Peirce projections in (6.1.11) we have involved the unital extension A1 of the algebra A. This is clearly unnecessary in the case that A has a unit element. Anyway, note also that in fact the statement of Lemma 6.1.80 remains true whenever e is an idempotent in A1 , which allows us to display the symmetry of the Peirce decomposition. Fact 6.1.83 Let A be a non-commutative Jordan algebra over K, and let e be an (e) (e) (e) (1 −e) (1 −e) (1 −e) idempotent in A1 . If A = A1 ⊕ A 1 ⊕ A0 and A = A1 ⊕A1 ⊕ A0 are the 2

2

Peirce decompositions of A relative to e and 1 − e, respectively, then (1 −e) (e) (1 −e) (e) (1 −e) A(e) , A1 = A1 , and A0 = A1 . 1 = A0 2

2

(e)

Moreover, A 1 = {x ∈ A : ex = x(1 − e)} = {x ∈ A : xe = (1 − e)x}, and hence (by the 2

flexible identity and (PR2)) we have ex 1 e = (1 − e)x 1 (1 − e) ∈ A(e) for every x 1 ∈ A(e) 1 1 . 2

2

2

2

2

Peirce relation (PR1) can be also derived from (6.1.11), Lemma 5.9.2, and Fact 6.1.83 The next result displays some formulae which allow us to recover certain products in the Peirce decomposition of a non-commutative Jordan algebra A relative to an

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idempotent e from products in Asym together with multiplications by e. These formulae allow us to provide an improvement of Corollary 6.1.81 that will be useful later. Lemma 6.1.84 Let A be a non-commutative Jordan algebra over K, and let e be an idempotent in A. Then we have: (i) For x1 ∈ A1 , x 1 ∈ A 1 , and x0 ∈ A0 , we have 2

2

2e(x 1 • x0 ) = 2(ex 1 ) • x0 = x 1 x0 , 2(x 1 • x0 )e = 2(x 1 e) • x0 = x0 x 1 , 2

2

2

2

2

2

2e(x 1 • x1 ) = 2(ex 1 ) • x1 = x1 x 1 , 2(x 1 • x1 )e = 2(x 1 e) • x1 = x 1 x1 . 2

2

2

2

2

2

(ii) [F, G] = 0 for all F ∈ LA1 ∪ RA1 and G ∈ LA0 ∪ RA0 . Proof

It follows from Corollary 6.1.81 that 2e(x 1 • x0 ) = Le (Lx0 + Rx0 )(x 1 ) = (Lx0 + Rx0 )Le (x 1 ) = 2(ex 1 ) • x0 . 2

2

2

2

On the other hand, replacing a by e and b by x0 in (6.1.8) we find that Le Lx0 = Re Rx0 , and hence e(x0 x 1 ) = (x 1 x0 )e. Since, by (PR2), x 1 x0 ∈ A 1 , and hence 2

2

2

2

e(x 1 x0 ) + (x 1 x0 )e = x 1 x0 , 2

2

2

we deduce that 2e(x 1 • x0 ) = x 1 x0 . Thus we have proved the first expression in (i). 2 2 The second one follows in the same way, and the remaining ones follow by symmetry (cf. Fact 6.1.83). Let F be in LA1 ∪ RA1 , and let G be in LA0 ∪ RA0 . It follows from (PR1) and (PR4) that [F, G](A1 + A0 ) = 0. In order to prove that [F, G](A 1 ) = 0, note that, by (i), for 2 each x1 ∈ A1 and x0 ∈ A0 we have that Lx1 = 2Lx•1 Le , Rx1 = 2Lx•1 Re , Lx0 = 2Lx•0 Re , and Rx0 = 2Lx•0 Le on A 1 , 2

and hence, by Corollary 6.1.81, [F, G] = 4[Lx•1 , Lx•0 ]H on A 1 , for suitable H ∈ {Le2 , Le Re , R2e }. 2

At this time, remembering the identity (4.1.1) for Jordan algebras in p. 452 of Volume 1, we have that • , L• ] + [L• , L• ] + [L• , L• ] = 0 for all a, c, d ∈ A, [La•c d a•d c c•d a

and replacing a with x1 , c with e, and d with x0 we obtain that [Lx•1 , Lx•0 ] = 0, hence [F, G] = 0, and the proof is complete.  Lemma 6.1.85 Let A be a non-commutative Jordan algebra over K, and let e be an idempotent in A. Then: (i) For any x 1 , y 1 ∈ A 1 we have 2

2

2

2P1 ((ex 1 ) • y 1 ) = 2P1 (x 1 • (y 1 e)) = P1 (x 1 y 1 ) and 2

2

2

2

2

2

2P0 (x 1 • (ey 1 )) = 2P0 ((x 1 e) • y 1 ) = P0 (x 1 y 1 ). 2

2

2

2

2

2

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(ii) lin(Pk (A 1 A 1 )) = lin(Pk (A 1 • A 1 )) for k = 1, 0. 2

2

2

2

(iii) UAi ,Aj (Ak ) ⊆ Ai+j−k for i, j, k ∈ {1, 12 , 0}, where A := 0 whenever ∈ / {1, 12 , 0}. (iv) For any yk ∈ Ak (k = 1, 0) and x 1 ∈ A 1 , we have 2

2

Ux 1 (yk ) = 2P1−k (x 1 (yk • x 1 )) = 2P1−k ((x 1 • yk )x 1 ). 2

2

2

2

2

(v) For any yk ∈ Ak (k = 1, 0) and x 1 ∈ A 1 , we have 2

2

yk x21 = 2Pk ((yk • x 1 )x 1 ) and x21 yk = 2Pk (x 1 (x 1 • yk )) , 2

2

2

2

2

2

and hence x21 • yk = 2Pk (x 1 • (x 1 • yk )). 2

2

Proof

2

Note that for any x 1 , y 1 ∈ A 1 we have 2

2

2

2(ex 1 ) • y 1 = (ex 1 )y 1 + y 1 (ex 1 ) = (ex 1 )y 1 + 2y 1 (e • x 1 ) − y 1 (x 1 e) 2

2

2

2

2

2

2

2

2

2

2

2

= y 1 x 1 + (Lex 1 − Rx 1 e )(y 1 ), 2

2

2

2

2

and then, by (6.1.8), 2(ex 1 ) • y 1 = y 1 x 1 + (Le Lx 1 − Re Rx 1 )(y 1 ) = (IA − Re )(y 1 x 1 ) + Le (x 1 y 1 ). 2

2

2

2

2

2

2

2

2

2

2

Now, keeping in mind that Le and Re commute with P1 and P0 , and that Le P1 = Re P1 = P1 and Le P0 = Re P0 = 0, we deduce that 2P1 ((ex 1 ) • y 1 ) = P1 (x 1 y 1 ) and 2P0 ((ex 1 ) • y 1 ) = P0 (y 1 x 1 ). 2

2

2

2

2

2

2

2

Analogously we see that 2(x 1 e) • y 1 = (IA − Le )(x 1 y 1 ) + Re (y 1 x 1 ), and then we 2 2 2 2 2 2 deduce that 2P1 ((x 1 e) • y 1 ) = P1 (y 1 x 1 ) and 2P0 ((x 1 e) • y 1 ) = P0 (x 1 y 1 ). 2

2

2

2

2

2

2

2

Thus assertion (i) is proved. The inclusion lin(Pk (A 1 • A 1 )) ⊆ lin(Pk (A 1 A 1 )) is obvious. The converse inclu2 2 2 2 sion follows from assertion (i) keeping in mind that eA 1 + A 1 e ⊆ A 1 . Thus assertion 2 2 2 (ii) is proved. In order to prove assertion (iii), consider for each t ∈ R \ {0} the element et := te + 1 − e in the unital extension A1 of A, and note that, as a consequence of (6.1.11), the operator Uet on A is given by Uet = t2 P1 + tP 1 + P0 . It follows that 2

1 Ak = {x ∈ A : Uet (x) = t2k x for every t ∈ R \ {0}} for k = 1, , 0, 2 as well as that Uet Ues = Uets for all t, s ∈ R\{0}, and in particular Uet Uet−1 = IA . Now, given i, j, k ∈ {1, 12 , 0} and xi ∈ Ai , xj ∈ Aj , xk ∈ Ak , keeping in mind the fundamental formula (3.4.3) (cf. Proposition 3.4.15) we realize that, for each t ∈ R \ {0},

6.1 The main results

379

Uet Uxi ,xj (xk ) = Uet Uxi ,xj Uet Uet−1 (xk ) = UUet (xi ),Uet (xj ) Uet−1 (xk ) = Ut2i xi ,t2j xj (t−2k xk ) = t2(i+j−k) Uxi ,xj (xk ), and consequently Uxi ,xj (xk ) ∈ Ai+j−k . Finally, given x 1 ∈ A 1 and yk ∈ Ak (k = 1, 0), by (6.1.10), we have 2

2

Ux 1 (yk ) = 2x 1 (x 1 • yk ) − x21 yk = 2(x 1 • yk )x 1 − yk x21 . 2

2

2

2

2

2

(6.1.13)

2

It follows from assertion (iii) that Ux 1 (yk ) ∈ A1−k , and it follows from (PR3), (PR1), 2

and (PR4) that yk x21 and x21 yk lie in Ak . Now, applying P1−k and Pk in the equalities 2

2



(6.1.13) we derive assertions (iv) and (v), respectively.

Lemma 6.1.86 Let A be a non-commutative Jordan algebra over K, and let e ∈ A be a nonzero idempotent. Write B 1 := A 1 and Bk := lin(Pk (A 1 • A 1 )) (k = 1, 0). 2

2

2

2

Then we have: (i) Bk is an ideal of Ak for k = 1, 0. (ii) B := B1 + B 1 + B0 is an ideal of A, which coincides with the ideal of Asym 2 generated by A 1 . 2

Proof For x 1 , y 1 ∈ A 1 and xk ∈ Ak , the linearization of the equalities in Lemma 2 2 2 6.1.85(v), together with (PR2) and Lemma 6.1.85(ii), gives that both xk (x 1 • y 1 ) and 2 2 (x 1 • y 1 )xk lie in Bk . Since, by Corollary 6.1.82, 2

2

xk Pk (x 1 • y 1 ) = xk (x 1 • y 1 ) and Pk (x 1 • y 1 )xk = (x 1 • y 1 )xk , 2

2

2

2

2

2

2

2

we realize that xk Pk (x 1 • y 1 ) and Pk (x 1 • y 1 )xk belong to Bk . Thus Bk is an ideal of 2 2 2 2 Ak for k = 1, 0. Now that assertion (i) has been proved, the verification that B is an ideal of A reduces to that of the inclusions Bk A 1 + A 1 Bk ⊆ B, B 1 A 1 + A 1 B 1 ⊆ B, and Bk A1−k + A1−k Bk ⊆ B. 2

2

2

2

2

2

The former and the latter hold because, as a consequence of (PR2) and (PR4), we have in fact Bk A 1 + A 1 Bk ⊆ B 1 and Bk A1−k + A1−k Bk = 0, respectively. In 2 2 2 order to prove the middle inclusion, note that, by Lemma 6.1.85(i), we have for all x 1 , y 1 ∈ A 1 that 2

2

2

x 1 y 1 = P1 (x 1 y 1 ) + P 1 (x 1 y 1 ) + P0 (x 1 y 1 ) 2

2

2

2

2

2

2

2

2

= 2P1 ((ex 1 ) • y 1 ) + P 1 (x 1 y 1 ) + 2P0 (x 1 • (ey 1 )). 2

2

2

2

2

2

2

Since e ∈ A1 , and hence, by (PR2), eA 1 ⊆ A 1 , we derive that x 1 y 1 belongs to 2 2 2 2 B, and so B 1 A 1 + A 1 B 1 ⊆ B. Thus B is an ideal of A, and hence of Asym . Since 2 2 2 2 clearly lin(A 1 • A 1 ) is contained in the ideal of Asym generated by A 1 , and since, by 2

2

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Representation theory for non-commutative JB∗ -algebras

380

Fact 3.3.3, P1 = Ue = Ue• and P0 = U1−e = U1• −e , it follows that B is contained in the ideal of Asym generated by A 1 , and so B coincides with the ideal of Asym generated 2 by A 1 .  2

Next, we show another formulae for certain products in a Peirce decomposition that will be useful later. Lemma 6.1.87 Let A be a non-commutative Jordan algebra over K, and let e be an idempotent in A. Then, for zk ∈ Ak (k = 1, 0) and x 1 , y 1 ∈ A 1 , we have 2

2

2

zk • P 1 (x 1 y 1 ) = P 1 (x 1 (zk • y 1 )) = P 1 ((zk • x 1 )y 1 ). 2

2

2

2

2

2

2

2

(6.1.14)

2

Proof Set x 1 y 1 = a1 + a 1 + a0 and y 1 x 1 = b1 + b 1 + b0 , and note that, as a 2 2 2 2 2 2 consequence of (PR3), we have that b 1 = −a 1 . The linearization of the flexible 2 2 identity gives that [x 1 , y 1 , zk ] + [zk , y 1 , x 1 ] = 0, hence 2

2

2

2

0 = P 1 ([x 1 , y 1 , zk ] + [zk , y 1 , x 1 ]) 2

2

2

2

2

= P 1 ((x 1 y 1 )zk − x 1 (y 1 zk ) + (zk y 1 )x 1 − zk (y 1 x 1 )) 2

2

2

2

2

2

2

2

2

= P 1 (a 1 zk − x 1 (y 1 zk ) + 2(zk • y 1 )x 1 − (y 1 zk )x 1 + zk a 1 ) (by (PR1) and (PR4)) 2

2

2

2

2

2

2

2

2

= 2P 1 (a 1 • zk − x 1 • (y 1 zk ) + (zk • y 1 )x 1 ) 2

2

2

2

2

2

= 2a 1 • zk + 2P 1 ((zk • y 1 )x 1 )

(by (PR2) and (PR3))

= 2a 1 • zk − 2P 1 (x 1 (zk • y 1 ))

(by (PR2) and (PR3)),

2

2

2

2

2

2

2

2

and so zk • P 1 (x 1 y 1 ) = zk • a 1 = P 1 (x 1 (zk • y 1 )). This gives the first equality in 2 2 2 2 2 2 2 (6.1.14). For the second one, note that zk • P 1 (x 1 y 1 ) = zk • a 1 = −zk • b 1 = −zk • P 1 (y 1 x 1 ), 2

2

2

2

2

2

2

2

and that the first equality in (i) (with the roles of x 1 and y 1 exchanged) leads to 2 2 zk • P 1 (x 1 y 1 ) = −P 1 (y 1 (zk • x 1 )). Finally, keeping in mind (PR2) and (PR3), we 2 2 2 2 2 2  conclude that zk • P 1 (x 1 y 1 ) = P 1 ((zk • x 1 )y 1 ), and the proof is complete. 2

2

2

2

2

2

Now we introduce multiple Peirce decompositions. Proposition 6.1.88 Let A be a unital non-commutative Jordan algebra over K, and  let e1 , e2 , . . . , en be pairwise orthogonal idempotents in A with nk=1 ek = 1. Set Pi i := Uei (1 ≤ i ≤ n) and Pij := 2Uei ,ej (1 ≤ i < j ≤ n).

(6.1.15)

Then {Pij }1≤i≤ j≤n form a supplementary family of projections on A: IA =

 1≤i≤ j≤n

Pij and Pij Pk = δik δj Pij (1 ≤ i ≤ j ≤ n, 1 ≤ k ≤ ≤ n),

6.1 The main results

381

and therefore the space A breaks up as the direct sum of the ranges: we have the Peirce decomposition of A into Peirce subspaces = A= Aij for Aij := Pij (A). (6.1.16) 1≤i≤ j≤n

Moreover, if for 1 ≤ i < j ≤ n we set Aj i := Aij , and if for each 1 ≤ i ≤ n we consider (e ) (e ) (e ) the Peirce decomposition A = A1 i ⊕ A 1 i ⊕ A0 i relative to the single idempotent ei , 2

then we have i) A1(ei ) = Ai i , A(e = 1 2



i) Aij , and A(e 0 =

j=i



Ajj +

j=i



Ak .

(6.1.17)

k, =i k=

As a consequence, for 1 ≤ i = j ≤ n, 3 (ej ) 3 (e ) (e ) (e ) Ai i ⊆ A0 and Aij = A 1 i ∩ A 1 j ⊆ A0 k , 2

j=i

2

k=i,j

and we have Ai i = {x ∈ A : ei x = xei = x} and Aij = {x ∈ A : ei x + xei = ej x + xej = x}. Proof

Note that

IA = U1 = Uni=1 ei =



Uei ,ej =

1≤i,j≤n

n  i=1



Uei +

2Uei ,ej =

1≤i 0 satisfying yx ≤ kxy for all x, y ∈ A. If Ann(A) = 0, then A is commutative. Proof By Corollary 6.3.2, it is enough to show that the conditions AA ⊆ Z(A) and Ann(A) = 0 imply that A is commutative. But, applying the first condition, for x, y, z, t in A we have 0 = [xyz, t] = xy[z, t] + [xy, t]z = xy[z, t]. Therefore, again by the first condition, we have [A, A]AA = A[A, A]A = AA[A, A] = 0, and hence both [A, A]A and A[A, A] are contained in Ann(A). Now, applying the second condition twice, it follows that [A, A] = 0.  The following celebrated theorem of Le Page [999] follows straightforwardly from anyone of Corollaries 6.3.3 or 6.3.4. Theorem 6.3.5 Let A be a normed unital associative complex algebra such that there exists k > 0 satisfying yx ≤ kxy for all x, y ∈ A. Then A is commutative. Let E be a vector space. As usual in our work, by a product on E we mean any bilinear mapping (x, y) → x y from E × E into E. Given a product on E, and an element u ∈ E, we say that is right u-admissible if the equality x u = x holds for every x in E. The next result generalizes Theorem 6.3.5. Fact 6.3.6 Let A be a normed associative complex algebra with a right unit e, and let be a right e-admissible (possibly non-associative) product on (the vector space of)

424

Representation theory for non-commutative JB∗ -algebras

A such that there exists k > 0 satisfying x coincides with the canonical product of A.

y ≤ kxy for all x, y ∈ A. Then

Proof By Proposition 6.3.1, with E = A and h(x, y) = x x y = x (ye) = (xy) e = xy.

y, for all x, y ∈ A we have 

Remark 6.3.7 (a) Since the unital non-commutative associative real algebra H of Hamilton’s quaternions is an absolute valued algebra (cf. Subsection 2.5.1), results from Corollary 6.3.2 to Fact 6.3.6 need not remain true with ‘real’ instead of ‘complex’. Since Corollary 6.3.2 could be derived from Proposition 6.3.1 without any reference to the base field, also Proposition 6.3.1 need not remain true with ‘real’ instead of ‘complex’. (b) Let A denote the class of all associative complex algebras, and let N stand for the class of all normed complex algebras. Now let C denote the class of those members of A which are commutative, let Z denote the class of those members A of A such that AA ⊆ Z(A), and let L stand for the class of those members A of A ∩ N such that there exists k > 0 satisfying yx ≤ kxy for all x, y ∈ A. Then, according to Corollary 6.3.2, we have C ∩N ⊆ L ⊆ Z ∩N . It is noteworthy that the two inclusions above are strict. To realize this, let us fix λ ∈ C, and consider the normed complex algebra A whose vector space is C3 , whose product and norm are defined by (x1 , x2 , x3 )(y1 , y2 , y3 ) := (0, 0, x1 y2 + λx2 y1 ) and (x1 , x2 , x3 ) := max{1, |λ|} max{|x1 |, |x2 |, |x3 |}, respectively. Since (AA)A = 0 = A(AA), certainly A is associative and the inclusion AA ⊆ Z(A) holds, i.e. A is a member of Z ∩ N . Suppose that A is a member of L . Then, for all x1 , x2 , y1 , y2 ∈ C we have |y1 x2 + λy2 x1 | ≤ k|x1 y2 + λx2 y1 |, hence, taking x1 = x2 = y1 = 1 and y2 = −λ, we obtain 1 − λ2 = 0, and therefore λ = ±1. Thus, for λ = ±1, A lies in (Z ∩ N ) \ L . If λ = −1, then A becomes the easiest example of a normed associative anticommutative complex algebra with nonzero product. Such an algebra A lies in L \ (C ∩ N ). After the discussion just done, it seems to us interesting the question whether an algebraically defined class P of associative complex algebras can be found in such a way that L = P ∩ N . Now we are going to deal with non-associative variants of Theorem 6.3.5. We begin by considering the case of alternative algebras. Proposition 6.3.8 Let A be a normed alternative complex algebra such that there exists k > 0 satisfying yx ≤ kxy for all x, y ∈ A. Then [A • A, A] = 0. Proof Let a, b be in A, and let B denote the subalgebra of A generated by a and b. By Artin’s Theorem 2.3.61, B is associative. Therefore, by Corollary 6.3.2, we have [a2 , b] = 0. Since a, b are arbitrary elements of A, we can linearize the above equality in the variable a to get [a • c, b] = 0 for all a, b, c ∈ A.  The next corollary follows straightforwardly from Proposition 6.3.8.

6.3 Commutativity of non-commutative JB∗ -algebras

425

Corollary 6.3.9 Let A be a normed alternative complex algebra such that there exists k > 0 satisfying yx ≤ kxy for all x, y ∈ A. If lin(A • A) is dense in A (for example, if A is unital), then A is commutative. Remark 6.3.10 (a) If the alternative algebra A in Proposition 6.3.8 is in fact associative, then, by Corollary 6.3.2, we have actually [AA, A] = 0. As the next example shows, this need not remain true in the general case. Let A be the complex algebra whose vector space is C7 and whose product is defined by (x1 , x2 , x3 , x4 , x5 , x6 , x7 )(y1 , y2 , y3 , y4 , y5 , y6 , y7 ) := (0, 0, 0, x1 y2 − x2 y1 , x1 y3 − x3 y1 , x2 y3 − x3 y2 , x1 y6 − x6 y1 + x5 y2 − x2 y5 + x3 y4 − x4 y3 ). Then A is alternative and anticommutative. Therefore, since (0, 0, 1, 0, 0, 0, 0)[(1, 0, 0, 0, 0, 0, 0)(0, 1, 0, 0, 0, 0, 0)] = (0, 0, 0, 0, 0, 0, 1), it follows from the anticommutativity of A that [AA, A] = 0. (b) The algebra in the above example also shows that in Corollary 6.3.4 the associativity of A cannot be relaxed to the alternativity. Indeed, for every algebra norm  ·  on the alternative anticommutative algebra A of the example, and all x, y ∈ A, we have xy = yx. However it is easy to realize that the equality Ann(A) = 0 holds. Now we deal with general non-associative algebras. Given a vector space X over K, and a linear mapping T : X → X, we denote by sp(T) the spectrum of T relative to L(X). We remark that, if X is in fact a Banach space, and if T is continuous, then sp(T) coincides with the spectrum of T relative to BL(X) (cf. Example 1.1.32(d)). Theorem 6.3.11 Let A be a complete normed complex algebra with a right unit e, let E be a complex normed space, let h : A × A → E be a bilinear mapping such that there exists k > 0 satisfying h(x, y) ≤ kxy for all x, y ∈ A, and let z be in A such that sp(Rz ) is countable. Then, for every x ∈ A, we have h(x, z) = h(xz, e). Proof Replacing z by z − αe, for a suitable α ∈ C, we can suppose that 0 ∈ / sp(Rz ). Then K := {μ−1 : μ ∈ sp(Rz )} is a countable compact subset of C. Writing  := C \ K, and considering the analytic mapping ϕ :  → A given by ϕ(λ) := e − λz, we realize that, for every λ in , the operator Rϕ(λ) is bijective. For y ∈ A, denote by Ty : A → E the continuous linear mapping defined by Ty (a) := h(a, y). Then the assumption on h leads to the inequality Tϕ(λ) (a) ≤ kRϕ(λ) (a) for all λ ∈  and a ∈ A. Equivalently, we have Tϕ(λ) [R−1 ϕ(λ) (x)] ≤ kx for all λ ∈  and x ∈ A, and

hence Tϕ(λ) ◦ R−1 ϕ(λ)  ≤ k for every λ ∈ . Now, let us fix an arbitrary continuous linear form f on the complex Banach space BL(A, E) of all bounded linear mappings from A to E. Then the function : λ → f (Tϕ(λ) ◦ R−1 ϕ(λ) ) from  to C is analytic and bounded. Since  is the complement in C of a countable compact set, it follows from an extended version of Liouville’s theorem (see, for example, [803, Exercise 10(a) in p. 324]) that is constant. As a consequence, we have f (Te ◦ Rz − Tz ) =  (0) = 0.

426

Representation theory for non-commutative JB∗ -algebras

Since f is arbitrary in the dual of BL(A, E), the Hahn–Banach theorem yields the equality Tz = Te ◦ Rz , that is, h(x, z) = h(xz, e) for every x ∈ A.  The next result is a straightforward consequence of Theorem 6.3.11. Corollary 6.3.12 Let A be a complete normed complex algebra with a right unit e, and let be a right e-admissible product on (the vector space of) A such that there exists k > 0 satisfying x y ≤ kxy for all x, y ∈ A. If the linear hull of the set {z ∈ A : sp(Rz ) is countable} is dense in A (for example, if A is finite-dimensional), then coincides with the canonical product of A. In the case that e is in fact a (two-sided) unit for the algebra A, the product on A defined by x y := yx is right e-admissible, hence Corollary 6.3.12 applies to obtain the following variant of Le Page’s theorem. Corollary 6.3.13 Let A be a complete normed unital complex algebra. If the linear hull of the set {z ∈ A : sp(Rz ) is countable} is dense in A, and if there exists k > 0 such that yx ≤ kxy for all x, y ∈ A, then A is commutative. As usual, given an element z in an algebra A over K, we denote by R•z the mapping x → 12 (zx + xz) from A to A. With this notation, the next result follows from Theorem 4.1.10 and the convention done in Definition 4.1.65. Fact 6.3.14 Let A be a complete normed unital Jordan-admissible complex algebra, and let z be in A. Then we have 1 sp(R•z ) ⊆ (J-sp(A, z) + J-sp(A, z)). 2 Keeping in mind the above fact, Theorem 6.3.11 and Corollary 6.3.12 yield Corollaries 6.3.15 and 6.3.16 which follow. Corollary 6.3.15 Let A be a complete normed unital Jordan complex algebra, let E be a normed complex space, let h : A × A → E be a bilinear mapping such that there exists k > 0 satisfying h(x, y) ≤ kxy for all x, y ∈ A, and let z be in A such that J-sp(A, z) is countable. Then, for every x ∈ A, we have h(x, z) = h(xz, 1). Corollary 6.3.16 Let A be a complete normed unital Jordan complex algebra, and let be a right 1-admissible product on (the vector space of) A such that there exists k > 0 satisfying x y ≤ kxy for all x, y ∈ A. If the linear hull of the set {z ∈ A : J-sp(A, z) is countable} is dense in A (for example, if A is finite-dimensional), then coincides with the canonical product of A. Fact 6.3.17 Let A be a complete normed unital complex algebra such that there exists k > 0 satisfying yx ≤ kxy for all x, y ∈ A, and let z be in A. We have: (i) The boundary of sp(Rz ) in C is contained in sp(R•z ). (ii) If sp(R•z ) is countable, then so is sp(Rz ).

6.3 Commutativity of non-commutative JB∗ -algebras

427

Proof Let λ be in the boundary of sp(Rz ) in C. Then, by Corollary 1.1.95, there exists a sequence xn in SA such that limn λxn − Rz (xn ) = 0. Since 1 λxn − R•z (xn ) = λxn − (xn z + zxn ) 2 1 1 1 ≤ (λxn − xn z) + (λxn − zxn ) ≤ (1 + k)λxn − Rz (xn ), 2 2 2 it follows that limn λxn − R•z (xn ) = 0. Since xn  = 1 for every n ∈ N, we deduce that λ belongs to sp(R•z ). Thus assertion (i) has been proved. Assertion (ii) follows from (i).  Now, keeping in mind Facts 6.3.17(ii) and 6.3.14, Corollary 6.3.13 yields the following. Corollary 6.3.18 Let A be a complete normed unital Jordan-admissible complex algebra. If the linear hull of the set {z ∈ A : J-sp(A, z) is countable} is dense in A, and if there exists k > 0 such that yx ≤ kxy for all x, y ∈ A, then A is commutative. Since every quadratic algebra over K is a unital Jordan-admissible algebra (cf. Lemma 2.6.3) all elements of which have a finite J-spectrum, and the completion of a normed quadratic algebra over K is a quadratic algebra (a consequence of Proposition 3.5.4), the following corollary follows from the above one. Corollary 6.3.19 Let A be a normed quadratic complex algebra such that there exists k > 0 satisfying yx ≤ kxy for all x, y ∈ A. Then A is commutative. Proposition 6.3.20 Let A be a normed complex algebra of degree 1, and suppose that there exists k > 0 satisfying λx + μy + xy ≤ kλx + μy + yx for all λ, μ ∈ C and x, y ∈ A. Then A is commutative. Proof Since A is of degree 1, for each a ∈ A we have a2 ∈ Ca. As a consequence, the normed unital extension A1 = C1 ⊕ A of A (cf. Proposition 1.1.107) becomes a normed quadratic complex algebra. Let λ, μ be in C, and let x, y be in A. Then (μ1 + x)(λ1 + y) = λμ1 + λx + μy + xy, (λ1 + y)(μ1 + x) = λμ1 + λx + μy + yx. Therefore, by the assumption, (μ1 + x)(λ1 + y) = |λμ| + λx + μy + xy ≤ k(|λμ| + λx + μy + yx) = k(λ1 + y)(μ1 + x). Keeping in mind the arbitrariness of λ, μ ∈ C and x, y ∈ A, it follows from Corollary  6.3.19 that A1 (and hence A) is commutative. Since nonzero anticommutative algebras over K are of degree 1 over K, the next result follows from Proposition 6.3.20.

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Representation theory for non-commutative JB∗ -algebras

Corollary 6.3.21 Let A be a normed anticommutative complex algebra, and suppose that there exists k > 0 satisfying λx + μy + xy ≤ kλx + μy − xy for all λ, μ ∈ C and x, y ∈ A. Then A has zero product. Remark 6.3.22 According to Definition 2.6.4, a pre-H-algebra is a real pre-Hilbert space E endowed with an anticommutative product ∧ satisfying x ∧ y ≤ xy and (x ∧ y|z) = (x|y ∧ z) for all x, y, z ∈ E. Let E be a pre-H-algebra. By taking z = y in the equality above, we obtain (x ∧ y|y) = 0 and (x ∧ y|x) = 0 for all x, y ∈ E. Therefore E is a normed anticommutative real algebra satisfying λx + μy + x ∧ y2 = (λx + μy2 + x ∧ y2 =)λx + μy − x ∧ y2 for all λ, μ ∈ R and x, y ∈ E. Nevertheless, E need not have zero product. Indeed, according to Theorem 2.6.9 and Fact 2.6.16, the absolute-valued unital real algebra H of Hamilton’s quaternions is the flexible quadratic algebra of some pre-H-algebra E (cf. the paragraph immediately before Proposition 2.6.5 for definition), and such a pre-H-algebra must have nonzero product because H is not commutative. The discussion just done shows that neither Proposition 6.3.20 nor Corollary 6.3.21 remain true with ‘real’ instead of ‘complex’. We conclude this subsection with the following fact and theorem. Fact 6.3.23 Let A be a complete normed unital Jordan-admissible complex algebra, and let z be in A such that J-sp(A, z) is countable. Suppose that there exists k > 0 satisfying yx ≤ kxy for all x, y ∈ A. Then z commutes with every element of A. Proof By Facts 6.3.17(ii) and 6.3.14, sp(Rz ) is countable. Therefore the result follows by applying Theorem 6.3.11 with E := A and h(x, y) := yx for all x, y ∈ A.  Theorem 6.3.24 Let A be a complete normed unital non-commutative Jordan complex algebra. If the subalgebra of A generated by the set S := {z ∈ A : J-sp(A, z) is countable} is dense in A, and if there exists k > 0 such that yx ≤ kxy for all x, y in A, then A is commutative. Proof Since A is Jordan-admissible, it follows from Fact 6.3.23 that S is a commutative subset of A. Therefore, since A is flexible, it follows from Corollary 2.4.16 that the subalgebra of A generated by S is commutative. Thus, if this subalgebra is dense in A, then clearly A is commutative.  6.3.2 The main result As a motivating result, we begin this subsection with the next fact, which follows straightforwardly from Corollary 6.1.23.

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429

Fact 6.3.25 Let A be an alternative C∗ -algebra such that there exists k > 0 satisfying yx ≤ kxy for all x, y in A. Then A is commutative. The main goal in this subsection is to prove that the conclusion in Fact 6.3.25 remains true if we relax the assumption that A is an alternative C∗ -algebra to the one that A is a non-commutative JB∗ -algebra. Lemma 6.3.26 Let A be a non-commutative JB∗ -algebra, let M be a closed ideal of A, and let λ be in R with 0 ≤ λ ≤ 1. Then, for x, y ∈ A, we have λxy + (1 − λ)yx + M = inf{λ(x + m)y + (1 − λ)y(x + m) : m ∈ M}. Proof

Let x, y be in A. Then, clearly, the inequality λxy + (1 − λ)yx + M ≤ inf{λ(x + m)y + (1 − λ)y(x + m) : m ∈ M}

holds. To prove the converse inequality, let us fix ε > 0 and, for elements u, v in any complex algebra containing A as a subalgebra, write uv := λuv + (1 − λ)vu. Then we can choose q ∈ M such that xy + q ≤ xy + M + ε. We claim that there exists a positive element p ∈ M satisfying p ≤ 1 and q − pq + p(xy) − (px)y < ε. Since M is ∗-invariant (cf. Proposition 3.4.13), it is a non-commutative JB∗ -algebra, and hence we can apply Proposition 3.5.23 to get a net {eλ }λ∈ of positive elements of M such that eλ  ≤ 1 for every λ ∈ , and limλ eλ m = m = limλ meλ for every m ∈ M. On the other hand, the bidual A of A can be regarded as a non-commutative JB∗ -algebra, which enlarges A (cf. Theorem 3.5.34) and whose product becomes separately w∗ -continuous (cf. Corollary 5.1.30(iii)). Then the bipolar M ◦◦ of M in A is a w∗ -closed ideal of A , and hence we have M ◦◦ = A e for some central idempotent e ∈ A (cf. Fact 5.1.10(i)). Since e is a unit for M, it follows from the separate w∗ -continuity of the product of A and the w∗ -density of M in M ◦◦ that e is the unique possible w∗ -cluster point of the net {eλ } in A . Since the closed unit ball of A is w∗ -compact, we actually have that w∗ -lim{eλ } = e. Now, note that the product  on A is separately w∗ -continuous, and regard the space A × A as the bidual of the Banach space A × A with the sum norm. Then in A × A we have w∗ -lim[(q − eλ q, eλ (xy) − (eλ x)y)] λ

= (q − eq, e(xy) − (ex)y) = (0, 0), where the last equality holds because e is a unit for (M ◦◦ , ) and a central element of (A , ). Since (q − eλ q, eλ (xy) − (eλ x)y) belongs to A × A for every λ ∈ , it follows that the net {(q − eλ q, eλ (xy) − (eλ x)y)}λ∈ converges to (0, 0) in

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the weak topology of A × A, and therefore, for a suitable element p in the convex hull of the set {eλ : λ ∈ }, we have q − pq + p(xy) − (px)y < ε. Clearly, such a p lies in M, is positive, and satisfies p ≤ 1. Now that the claim has been proved, recall that A has a unit 1 (cf. Fact 5.1.7) which is also a unit for (A , ), so that we can write inf{(x + m)y : m ∈ M} ≤ (x − px)y ≤ xy − p(xy) + p(xy) − (px)y = (1 − p)(xy + q) − (1 − p)q + p(xy) − (px)y ≤ xy + q + q − pq + p(xy) − (px)y ≤ xy + M + 2ε. By letting ε → 0, we obtain inf{(x + m)y : m ∈ M} ≤ (|λ| + |1 − λ|)xy + M.



Fact 6.3.27 Let A be a non-commutative JB∗ -algebra, and let M be a closed ideal of A. Then the following assertion holds: (i) For all x, y ∈ A we have xy + M = inf{(x + m)y : m ∈ M} = inf{x(y + m) : m ∈ M}. (ii) If λ is in R with 0 ≤ λ ≤ 1, and if there exists k > 0 such that λyx + (1 − λ)xy ≤ kλxy + (1 − λ)yx for all x, y ∈ A, then we have λβα + (1 − λ)αβ ≤ kλαβ + (1 − λ)βα for all α, β ∈ A/M. Proof By taking λ = 0, 1 in Lemma 6.3.26, assertion (i) follows. Let λ and k be as supposed in assertion (ii). Then, invoking Lemma 6.3.26 again, for all x, y ∈ A we have λyx + (1 − λ)xy + M ≤ inf{λy(x + m) + (1 − λ)(x + m)y : m ∈ M} ≤ k inf{λ(x + m)y + (1 − λ)y(x + m) : m ∈ M} = kλxy + (1 − λ)yx + M. In other words, the conclusion in assertion (ii) is true.



Lemma 6.3.28 Let A be an alternative C∗ -algebra such that there exist λ ∈ R \ { 12 } with 0 ≤ λ ≤ 1, and k > 0, satisfying λyx + (1 − λ)xy ≤ kλxy + (1 − λ)yx for all x, y ∈ A. Then A is commutative.

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Proof Suppose at first that A is unital. Let B denote the complex algebra consisting of the vector space of A and the product  defined by xy := λxy + (1 − λ)yx. Then the unit of A is a unit for B, and, endowed with the norm of A, B becomes a complete normed algebra. Moreover, since Bsym = Asym , B is Jordan admissible and for x ∈ A we have sp(A, x) = J-sp(B, x). Now take z ∈ A such that z2 = 0. Since J-sp(B, z) = {0}, and yx ≤ kxy for all x, y ∈ B, it follows from Fact 6.3.23 that z -commutes with every element of B. Applying that λ = 12 , we realize that z commutes (in the usual sense) with every element of A, in particular with z∗ . Therefore, keeping in mind Artin’s theorem, we have z4 = z∗ z2 = (z∗ z)2  = z2 (z∗ )2  = 0, hence z = 0. Therefore, by Corollary 6.1.19, A is commutative, thus concluding the proof in the particular unital case. Now remove the assumption that A is unital. Since the condition λyx + (1 − λ)xy ≤ kλxy + (1 − λ)yx for all x, y ∈ A is inherited by any subalgebra of A, it follows from the preceding paragraph that every closed unital ∗-subalgebra of A is commutative. As a consequence, A cannot contain a copy of M2 (C) as a ∗-subalgebra. Therefore, according to Corollary 6.1.21, to conclude the proof it is enough to show that A also cannot contain a copy of C0 (]0, 1], M2 (C)) as a closed ∗-subalgebra. Suppose the contrary. Then the inequality λyx + (1−λ)xy ≤ kλxy+(1− λ)yx would be true for all x, y ∈ C0 (]0, 1], M2 (C)). Write M := {x ∈ C0 (]0, 1], M2 (C)) : x(1) = 0}. Since M is a closed ideal of C0 (]0, 1], M2 (C)), it would follow from Fact 6.3.27(ii) that λβα + (1 − λ)αβ ≤ kλαβ + (1 − λ)βα for all α, β ∈ C0 (]0, 1], M2 (C))/M. But, since C0 (]0, 1], M2 (C))/M is isometrically isomorphic to M2 (C), the above cannot happen in view of the first paragraph of the proof.  Theorem 6.3.29 Let A be a non-commutative JB∗ -algebra such that there exists k > 0 satisfying yx ≤ kxy for all x, y ∈ A. Then A is commutative. Proof Let M be any closed ideal of A. Taking λ = 1 in Fact 6.3.27(ii), we realize that, for all α, β ∈ A/M, we have βα ≤ kαβ. Therefore, if A/M is quadratic or of the form C(λ) for some C∗ -algebra C and some 0 ≤ λ ≤ 1, then A/M is commutative (cf. Corollary 6.3.19 or Lemma 6.3.28, respectively). Now take a faithful family {πi : A → Fi }i∈I of non-commutative JBW ∗ -factor representations of A (cf. Corollary 6.1.11), for i ∈ I write Mi := ker(πi ) and Ai := πi (A), and note that Ai is a norm-closed (cf. Corollary 4.4.30) and w∗ -dense ∗-subalgebra of the non-commutative JBW ∗ -factor Fi , and that moreover, Ai is isometrically ∗-isomorphic to the non-commutative JB∗ -algebra A/Mi (cf. Propositions 3.4.13 and 3.4.4). It follows from Corollary 6.1.113 and the first paragraph of the proof that A/Mi is commutative for every i ∈ I. Since ∩i∈I Mi = 0, we deduce that A is commutative. 

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Remark 6.3.30 If A is a nonzero non-commutative JBW ∗ -algebra, then A is unital (cf. Fact 5.1.7) and is equal to the norm-closed linear hull of its ∗-invariant idempotents (cf. Theorem 5.1.29(vi)). Since idempotents of any unital Jordanadmissible algebra have a finite J-spectrum, we realize that, when A is actually a non-commutative JBW ∗ -algebra, Theorem 6.3.29 follows straightforwardly from Fact 6.3.23. It is also noteworthy that, if A is a unital (commutative) JBW ∗ -algebra, then, by Corollary 6.3.16, every right 1-admissible product on A satisfying a x y ≤ kxy for some positive constant k and all x, y ∈ A must coincide with the canonical product of A. 6.3.3 Discussion of results and methods In Subsection 6.3.1, we have obtained some partial affirmative answers to the following. Problem 6.3.31 Let A be a complete normed unital complex algebra, and suppose that there exists a positive constant k satisfying yx ≤ kxy for all x, y ∈ A. Must A be commutative? More ambitious questions are the following. Problem 6.3.32 Let A be a complete normed complex algebra with a right unit e, and let be a right e-admissible product on A satisfying x y ≤ kxy for some positive constant k and all x, y ∈ A. Does coincide with the canonical product of A? Problem 6.3.33 Let A be a complete normed complex algebra with a right unit e, let E be a complex Banach space, and let h : A × A → E be a bilinear mapping satisfying h(x, y) ≤ kxy for some positive constant k and all x, y ∈ A. Does the equality h(x, y) = h(xy, e) holds for every x, y ∈ A? According to Proposition 6.3.1 and Theorem 6.3.11, the answer to Problem 6.3.33 (and hence also to Problems 6.3.32 and 6.3.31) is affirmative if A is either associative or finite-dimensional. Therefore, most probably, the answer must remain affirmative without any additional requirement. Actually, Le Page’s argument and our techniques share a common idea, which we explain in what follows. Let A be a complete normed complex algebra A with a right unit e. For z ∈ A, consider the following property: (Pz ) There exists a couple (, ϕ), where  is the complement in C of a countable compact set such that 0 ∈ , and ϕ :  → A is an analytic mapping satisfying: (a) ϕ(0) = e. (b) ϕ  (0) = z. (c) The operator Rϕ(λ) is bijective for every λ ∈ . Then, looking at the proof of Theorem 6.3.11, we realize that (Pz ) holds whenever sp(Rz ) is countable and 0 ∈ / sp(Rz ), and that Problem 6.3.33 has an affirmative answer whenever A is the closed linear hull of the set {z ∈ A : z satisfies (Pz )}. Thus, Problem 6.3.33 answers affirmatively in the finite-dimensional case. If A is

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associative, then Problem 6.3.33 answers affirmatively because every element z ∈ A satisfies the refined version of (Pz ) given by (Pz∗ ) There exists an analytic function ϕ : C → A such that ϕ(0) = e, ϕ  (0) = z, and the operator Rϕ(λ) is bijective for every λ in C. Indeed, if A is associative, and if z is in A, then the analytic mapping ϕ : C → A defined by ∞ n  λ n z ϕ(λ) := e + n! n=1

(Pz∗ ).

satisfies all the requirements in (Note that, thanks to the equality Rϕ(λ) = exp(λRz ), Rϕ(λ) is certainly a bijective operator for every λ ∈ C.) As we see in the sequel, a similar privilege situation happens in the more general case that A is right alternative (i.e. the equality yx2 = (yx)x holds for all x, y ∈ A). For x in such an algebra A, the right alternative identity reads as Rx2 = (Rx )2 , so that, after linearization, we obtain Rx•y = Rx • Ry for all x, y ∈ A. Now take z ∈ A, and define a sequence zn in A by z1 := z and zn+1 := z • zn . It follows from an elementary induction that Rzn = (Rz )n for every n ∈ N, and hence the analytic mapping ϕ : C → A defined by ϕ(λ) := e +

∞ n  λ n=1

n!

zn

satisfies ϕ(0) = e, ϕ  (0) = z, and Rϕ(λ) = exp(λRz ). Therefore we can formulate the result which follows. Proposition 6.3.34 Problem 6.3.33 has an affirmative answer whenever A is right alternative. The following example shows that the privilege situation for the property (Pz ) occurring in the right alternative setting cannot be expected in general. Example 6.3.35 Let A be the unital complex algebra whose vector space is C3 and whose product is defined by (x1 , x2 , x3 )(y1 , y2 , y3 ) := (x1 y1 + x2 y2 , x1 y2 + x2 y1 , x1 y3 + x3 y1 ). Then A is a Jordan algebra. Moreover, a straightforward calculation shows that, for x = (x1 , x2 , x3 ) ∈ A, the equality det(Rx ) = x1 (x12 − x22 ) holds. Let us fix z = (z1 , z2 , z3 ) ∈ A satisfying (Pz∗ ), so that there are complex valued entire functions ϕ1 , ϕ2 , ϕ3 satisfying ϕ1 (0) = 1, ϕ2 (0) = ϕ3 (0) = 0, ϕi (0) = zi for i = 1, 2, 3, and ϕ1 (λ)(ϕ1 (λ)2 − ϕ2 (λ)2 ) = 0 for every λ ∈ C. Since ϕ1 (λ) = 0 for every λ ∈ C, and the mapping λ → ϕϕ21 (λ) (λ) is an entire function which does not take the values 1 and −1, it follows from Picard’s theorem (see for example [803, Theorem 16.22]) that there exists c ∈ C such that ϕ2 (λ) = cϕ1 (λ) for every λ ∈ C. Now, we have c = cϕ1 (0) = ϕ2 (0) = 0, so ϕ2 = 0, and so z2 = ϕ2 (0) = 0. In this way, the (closed) linear hull of the set {z ∈ A : z satisfies (Pz∗ )} is not the whole algebra A.

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According to Example 6.3.35, the refinement of Le Page’s argument used in our development (by considering Property (Pz ) instead of (Pz∗ )) becomes crucial when we want to remove associativity. Building on the discussion so far, let us consider the following. Claim 6.3.36 Let A be a complete normed complex algebra with a right unit e, let be a right e-admissible product on A such that the inequality x y ≤ kxy holds for some positive constant k and all x, y ∈ A, and let z be in A satisfying (Pz ) (respectively (Pz∗ )) relative to the product . Then z satisfies (Pz ) (respectively (Pz∗ )) relative to the canonical product of A. Proof Choose a couple (, ϕ), where  is the complement in C of a countable compact (eventually empty) set satisfying 0 ∈ , and ϕ :  → A is an analytic mapping such that ϕ(0) = e, ϕ  (0) = z, and the operator Rϕ(λ) is bijective for every λ in . Then, for all x ∈ A and λ ∈ , we have x ≤ (Rϕ(λ) )−1  Rϕ(λ) (x) ≤ k(Rϕ(λ) )−1  Rϕ(λ) (x). Therefore, for every λ ∈ , the operator Rϕ(λ) is bounded below. On the other hand, P := {Rϕ(λ) : λ ∈ } is a connected subset of BL(A), and contains some bijective operator (indeed, Rϕ(0) is the identity mapping on A). It follows from Lemma 2.7.18 that P consists only of bijective operators, i.e. Rϕ(λ) is bijective for every λ ∈ .  The claim just proved, together with the previous discussion about the proof of Theorem 6.3.11, leads to the next result. Proposition 6.3.37 Problem 6.3.32 has an affirmative answer whenever the product is right alternative. Corollary 6.3.38 Problem 6.3.31 has an affirmative answer if A has a right alternative mutation (for example, if Asym is associative). Proof By assumption, there exists λ ∈ C such that the product on A defined by x y := λxy + (1 − λ)yx is right alternative. If λ = 1, then the result follows from Proposition 6.3.34. Otherwise, since for x, y ∈ A we have x

y ≤ (|λ| + k|1 − λ|)xy,

the result follows from Proposition 6.3.37.



Corollary 6.3.39 Let A be a complete normed unital right alternative complex algebra such that there exist λ ∈ C \ { 12 } and k > 0 satisfying λyx + (1 − λ)xy ≤ kλxy + (1 − λ)yx for all x, y ∈ A. Then A is commutative. Proof Denote by the product of A(λ) . Then, up to multiplication of the norm of A by a suitable positive number, A(λ) becomes a complete normed unital complex algebra satisfying y x ≤ kx y for all x, y ∈ A(λ) . Moreover, since λ = 12 , it

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follows from (6.1.44) that A(λ) has a right alternative mutation. Therefore the result follows from Corollary 6.3.38.  Now, let us point out that Le Page’s original technique can be easily adapted to provide further interesting developments in the non-associative setting. A first sample of this procedure is shown in the following. Proposition 6.3.40 Let A be a complete normed unital power-associative complex algebra. Then A is associative if (and only if) there exists k > 0 satisfying x(yz) ≤ k(xy)z for all x, y, z ∈ A. Proof

Let λ be in C, and let x, y, z be in A. Then we have  exp(λx)[(exp(−λx)y)z] ≤ k[(exp(λx)(exp(−λx)y)]z ≤ k exp(λx)(exp(−λx)y) z ≤ k2 y z.

Therefore the analytic mapping λ → exp(λx)[(exp(−λx)y)z)] = yz + λ[x(yz) − (xy)z)] + · · · from C to A is bounded, and hence constant. It follows x(yz) − (xy)z = 0.



Remark 6.3.41 The complete normed absolute-valued unital real algebra O of Cayley numbers (cf. Subsection 2.5.1) is power-associative, is not associative, and nevertheless it clearly satisfies x(yz) = (xy)z for all x, y, z ∈ O. This shows that Proposition 6.3.40 is not true with ‘real’ instead of ‘complex’. Other non-associative applications of Le Page’s technique follow from the next general result. Proposition 6.3.42 Let A be a complete normed unital non-commutative Jordan complex algebra, and let P : A → BL(A) be a quadratic mapping such that P1 = IA and Px (y) ≤ kUx (y) for some k > 0 and all x, y ∈ A. Then P = U. Proof Let λ be in C and let x be in A. Then exp(λx) is a J-invertible element of A, and hence, by Proposition 4.1.58, Uexp(λx) is a bijective operator. Therefore, −1 by the assumed inequality, we have Pexp(λx) ◦ Uexp(λx)  ≤ k. Thus the mapping

−1 λ → Pexp(λx) ◦ Uexp(λx) from C to BL(A) is analytic and bounded, and hence the equality Pexp(λx) = Uexp(λx) holds for every λ ∈ C (since P1 = IA ). Now, computing first and second derivatives at λ = 0, and combining the two resulting equalities, we  find Px = Ux .

By taking in Proposition 6.3.42 Px (y) := xyx, Px (y) := x2 y, and Px (y) := 2x(xy) − x2 y, and applying well-known identities in non-commutative Jordan and alternative algebras, we get assertions (i), (ii), and (iii), respectively, in the following.

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Corollary 6.3.43 Let A be a complete normed unital non-commutative Jordan complex algebra. Then we have: (i) A is alternative if (and only if ) there exists k > 0 satisfying xyx ≤ kUx (y) for all x, y ∈ A. (ii) A is associative and commutative if (and only if ) there exists k > 0 satisfying x2 y ≤ kUx (y) for all x, y ∈ A. (iii) A is commutative if (and only if ) there exists k > 0 satisfying 2x(yx) − x2 y ≤ kUx (y) for all x, y ∈ A. 6.3.4 Historical notes and comments Most results in this section are due to Benslimane, Mesmoudi, and Rodr´ıguez [860]. Some results included in our development were known before the publication of [860]. Thus Corollary 6.3.2 is due to Oudadess [1034] (see also [1021, 934]), Theorem 6.3.5 is due to Le Page [999], Fact 6.3.6 is a variant of a theorem by Baker and Pym [848] which is included in the Bonsall–Duncan book [696, Proposition 15.5], and Corollary 6.3.19 is due to Benslimane and Mesmoudi [859]. Propositions 6.3.8 and 6.3.20, Corollaries 6.3.9 and 6.3.21, and Remark 6.3.22 could be new. As pointed out by Aupetit [682, p. 43], in the complete normed associative unital case, Lemma 3.6.27 can be derived from Le Page’s Theorem 6.3.5 by invoking Fact 1.1.33(ii) and Theorem 1.1.46. As pointed out in [860], Skosyrskii’s Theorem 6.2.31, together with Corollaries 6.3.19, 6.3.38, and 6.3.39, implies the following. Corollary 6.3.44 Let A be a complete normed unital centrally closed prime nondegenerate non-commutative Jordan complex algebra, and suppose that there exists a positive constant k satisfying yx ≤ kxy for all x, y ∈ A. Then A is commutative.

7 Zel’manov approach

In a series of papers (see [662, 663, 1131, 1133, 1134]) Zel’manov provided the mathematical community with his unexpected classification theorems for prime nondegenerate Jordan algebras and triples. Zel’manov’s prime theorems, though received enthusiastically by algebraists since its appearance, took a relatively long time to be assimilated by analysts to obtain new structure theorems for normed prime nondegenerate Jordan algebras and triples, results which could not be approached by the familiar technique of the existence of a nonzero socle or by duality methods in JBW ∗ -theory. The reason could have been that the statements of Zel’manov’s prime theorems, to attain a nice simple form, perhaps conceal in their formulations some crucial information that is needed in the applications. This meant the analyst needed to find out the deep and very difficult proofs of Zel’manov’s theorems, a fact which took time. After this time was taken, the production on the analytic treatment of Zel’manov’s prime theorems had a flourishing development, which began with the Fern´andez–Garc´ıa–Rodr´ıguez paper [255] (where the classification of prime JB∗ -algebras was obtained), and even involved Zel’manov himself [147, 538]. Since the main result in [255] was already established in Subsection 6.1.4, we devote this chapter to establishing the classification of prime JB∗ -triples (see Section 7.1) and to surveying the rest of the topic (see Section 7.2).

7.1 Classifying prime JB∗ -triples Introduction As the main result in this section, we prove that, if X is a prime JB∗ triple which is neither an exceptional Cartan factor nor a spin triple factor, then either there exist a prime C∗ -algebra A and a self-adjoint idempotent e in the C∗ -algebra M(A) of multipliers of A such that X is a closed subtriple of M(A) contained in eM(A)(1 − e) and containing eA(1 − e), or there exist a prime C∗ -algebra A, a selfadjoint idempotent e ∈ M(A), and a ∗-involution τ on A with e + eτ = 1 such that X is a closed subtriple of M(A) contained in H(eM(A)eτ , τ ) and containing H(eAeτ , τ ) (see Theorem 7.1.41). 437

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Zel’manov approach

Among the many tools involved in the proof of Theorem 7.1.41 just quoted, we emphasize Horn’s classification of Cartan factors [330], the core of the proof of Zel’manov’s classification theorem for prime nondegenerate Jordan triples [663, 1133, 1134], and the complementary work by D’Amour and McCrimmon on the topic [920, 921]. Proofs of these tools are not discussed in our development. It is noteworthy that Zel’manov’s work is also involved through the description of prime JB∗ -algebras given by Theorem 6.1.57 (see the proof of Corollary 7.1.10). Other tools involved in the proof of Theorem 7.1.41 could have been proved much earlier, and are shown immediately below. Fact 7.1.1 Let X be a JB∗ -triple, let J be a closed triple ideal of X, and let K be a closed triple ideal of J. Then K is a triple ideal of X. Proof Take a, b ∈ X and y ∈ K. By Corollary 4.2.11, there is z ∈ K such that {zzz} = y. Using the Jordan triple identity (cf. (4.1.13)), we obtain {aby} = {ab{zzz}} = 2{{abz}zz} − {z{baz}z} ∈ {JKK} + {KJK} ⊆ K. Therefore, by Proposition 5.10.92, K is a triple ideal of X.



Fact 7.1.1 is the natural variant for JB∗ -triples of the result for non-commutative JB∗ -algebras which follows. Corollary 7.1.2 Each closed ideal K of a non-commutative JB∗ -algebra A is ∗-invariant. A closed ideal I of K is an ideal of A. Proof The first assertion was proved in Proposition 3.4.13. The second assertion follows from the first one, Corollary 5.7.37(i), and Fact 7.1.1.  A Jordan ∗-triple X is said to be prime if whenever P, Q are triple ideals of X with {PXQ} = 0 we have either P = 0 or Q = 0. The following result follows directly from Fact 7.1.1. Corollary 7.1.3 Every closed triple ideal of a prime JB∗ -triple is a prime JB∗ -triple. 7.1.1 Representation theory for JB∗ -triples We recall that every non-commutative JB∗ -algebra becomes a JB∗ -triple under the triple product defined by {abc} := Ua,c (b∗ ) (cf. Theorem 4.1.45). In particular, JB∗ -algebras and C∗ -algebras are JB∗ -triples in a natural way. §7.1.4 Certain prime JB∗ -triples (the so-called Cartan factors) are well understood in the literature. Prime JBW ∗ -triples are called JBW ∗ -triple factors. A Cartan factor is a JBW ∗ -triple factor such that the closed unit ball of its predual has some extreme point. According to Horn [330], Cartan factors come in the following six types. In,m := BL(H, K) (the Banach space of all bounded linear mappings from H to K), where H, K are complex Hilbert spaces of hilbertian dimension n, m, respectively, with 1 ≤ n ≤ m, and the triple product is defined by

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1 {xyz} := (xy∗ z + zy∗ x) 2 for all x, y, z ∈ BL(H, K). IIn := {x ∈ BL(H) : σ x∗ σ = −x} as closed subtriple of BL(H) (:= BL(H, H)), where H is a complex Hilbert space of hilbertian dimension n ≥ 5 and σ is a conjugation on H. IIIn := {x ∈ BL(H) : σ x∗ σ = x} as closed subtriple of BL(H), where H is a complex Hilbert space of hilbertian dimension n ≥ 2 and σ is a conjugation on H. IVn := H, where H is a complex Hilbert space of hilbertian dimension n ≥ 3, σ is a conjugation on H, and the triple product and the norm are given by {xyz} := (x|y)z + (z|y)x − (x|σ (z))σ (y) and

? x2 := (x|x) + (x|x)2 − |(x|σ (x))|2 ,

respectively, for all x, y, z ∈ H. V := M12 (C(C)) the 1 × 2-matrices over the complex Cayley numbers C(C), regarded as a closed subtriple of the type VI immediately below. VI := H3 (C(C)) the JB∗ -algebra of all hermitian 3 × 3-matrices over C(C), regarded as a JB∗ -triple. Type IVn Cartan factors are usually called spin triple factors. In an agreement with Definition 5.7.43, by a (JBW ∗ -)factor representation of a JB∗ -triple X we mean a w∗ -dense-range triple homomorphism from X to some JBW ∗ -triple factor. In Subsection 6.1.1 we showed that every primitive M-ideal of a non-commutative JB∗ -algebra A is the kernel of a type I factor representation of A (cf. Theorem 6.1.9). Essentially, the proof of this theorem involved only that the bidual of any non-commutative JB∗ -algebra is a non-commutative JBW ∗ -algebra in a natural way, that the product of any non-commutative JBW ∗ -algebra is separately w∗ -continuous, that M-ideals and closed ideals of a non-commutative JB∗ -algebra are the same, that minimal M-summands of a non-commutative JBW ∗ -algebra are non-commutative JBW ∗ -factors, that the predual of a non-commutative JBW ∗ -algebra is L-embedded, and that injective algebra ∗-homomorphisms between non-commutative JB∗ algebras are isometric. Since all results just listed have their JB∗ -triple counterparts (cf. Theorems 5.7.18, 5.7.20, and 5.7.34(i), Corollary 5.7.35, Theorem 5.7.36, and Proposition 5.7.41(ii), respectively), we easily realized that every primitive M-ideal of a JB∗ -triple X is the kernel of a Cartan factor representation of X. Therefore, invoking Fact 6.1.10, we obtain the following. Theorem 7.1.5 Every JB∗ -triple has a faithful family of Cartan factor representations. Corollary 7.1.6 Let X be a JB∗ -triple. Then there exists a JB∗ -algebra A containing X as a closed subtriple.

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Proof Since triple homomorphisms between JB∗ -triples are contractive and in fact isometric if in addition they are injective (cf. Proposition 5.7.41), it follows from Theorem 7.1.5 that every JB∗ -triple can be seen as a norm-closed subtriple of a suitable ∞ -sum of Cartan factors. Therefore it is enough to show that every Cartan factor is a closed subtriple of a JB∗ -algebra. But this follows from Corollary 3.5.7 and §7.1.4.  Corollary 7.1.7 Let X be a JB∗ -triple. Then, for all x, y, z ∈ X, the inequality {xyz} ≤ xyz holds. Proof

Combine Corollary 7.1.6 with Propositions 3.3.13 and 3.4.17.



Corollaries 7.1.6 and 7.1.7 were reviewed in Theorem 4.1.113 and Corollary 4.1.114, respectively. Proposition 7.1.8 Let X be a JB∗ -triple. Then there exists a JB∗ -algebra A containing X as a closed subtriple and with the property that every nonzero closed ideal of A has a nonzero intersection with X. Proof By Corollary 7.1.6, there exists a JB∗ -algebra B containing X as a closed subtriple. Consider the family F of all closed ideals of B which have zero intersection with X, ordered by inclusion. Let C be a totally ordered subset of F . Then the closure of the union of the members of C is a closed ideal of B (say P). If I is in C , then B/I is a JB∗ -algebra in a natural manner (cf. Proposition 3.4.13), and the mapping x → x + I from X to B/I is an injective triple homomorphism, hence, by Proposition 5.7.41, the equality x + I = x holds for every x ∈ X. Therefore we have x − y ≥ x for all x ∈ X and y ∈ P, and hence P is an upper bound of C in F . Now that we know that F is inductive, we may take a maximal element Q in F , and consider the JB∗ -algebra A := B/Q together with the embedding x → x + Q from X to A.  Proposition 7.1.8 leads directly to the following corollary. Corollary 7.1.9 Let J be a prime real JB∗ -triple. Then there exists a prime JB∗ algebra A containing J as a closed subtriple. Closed subtriples of C∗ -algebras are known under the name of JC∗ -triples. Corollary 7.1.10 Let X be a prime JB∗ -triple which is not a JC∗ -triple. Then X is finite-dimensional. Proof Since X is prime, we can apply Corollary 7.1.9 to find a prime JB∗ -algebra A containing X as a closed subtriple. Since X is not a JC∗ -triple, A cannot be a JC∗ -algebra (cf. the paragraph immediately before Lemma 3.3.5 for definition). Now, looking at the list of prime JB∗ -algebras provided by Theorem 6.1.57, and keeping in mind Fact 6.1.42, we realize that the unique prime JB∗ -algebra which

7.1 Classifying prime JB∗ -triples

441

is not a JC∗ -algebra is the 27-dimensional type VI Cartan factor H3 (C(C)) of all hermitian 3 × 3-matrices over the complex Cayley algebra C(C).  After Corollary 7.1.10, the next theorem follows from the work of Loos on finitedimensional JB∗ -triples [772]. In fact, via [772, 2.9], finite-dimensional JB∗ -triples can be recognized in Loos’ book as the so-called complex Jordan pairs with a positive hermitian involution. Since every finite-dimensional JB∗ -triple is a direct sum of simple ideals [772, 4.10, 4.11, 11.4], it follows that finite-dimensional prime JB∗ -triples are simple. Now, keeping in mind Corollary 7.1.10 and looking at the list of finite-dimensional simple JB∗ -triples provided in [772, 4.14], we get the following. Theorem 7.1.11 The prime JB∗ -triples which are not JC∗ -triples are the type V Cartan factor M1,2 (C(C)) of all 1 × 2-matrices over the complex Cayley algebra C(C), and the type VI Cartan factor H3 (C(C)) of all hermitian 3 × 3-matrices over C(C). 7.1.2 Building prime JB∗ -triples from prime C∗ -algebras By a matricial decomposition of a C∗ -algebra A we mean a family {Aij }i,j∈{1,2} of closed subspaces of A satisfying < Aij , and Aij Akl ⊆ δjk Ail A∗ij = Aji , A = i,j∈{1,2}

for all i, j, k, l ∈ {1, 2}. A matricially decomposed C∗ -algebra will be a C∗ -algebra endowed with a matricial decomposition. From now on, for every C∗ -algebra A, M(A) will denote the C∗ -algebra of multipliers of A (cf. §2.2.16, Theorem 5.10.90, and Remark 5.10.91). The following proposition will be applied without notice in the sequel. Proposition 7.1.12 Let A be a matricially decomposed C∗ -algebra. Then the matricial decomposition of A can be uniquely extended to a matricial decomposition of the C∗ -algebra M(A) of multipliers of A. Proof Let i, j be in {1, 2}. Denote by pij the (continuous) linear projection on A  with range Aij and kernel (k,l)=(i,j) Akl . For α, β ∈ A we have pij (β)α = (pi1 + pi2 )[β(pj1 + pj2 )(α)] and αpij (β) = (p1j + p2j )[(p1i + p2i )(α)β]. Then, for x ∈ M(A) and a ∈ A we obtain pij (x)a = (pi1 + pi2 )[x(pj1 + pj2 )(a)] and apij (x) = (p1j + p2j )[(p1i + p2i )(a)x],

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hence pij (x) belongs to M(A). Therefore M(A) inherits the natural matricial decomposition of A , which of course extends that of A. If {qij }i,j∈{1,2} is the family of projections on M(A) corresponding to any matricial decomposition of M(A) extending that of A, then, for x ∈ M(A), a ∈ A and i, j ∈ {1, 2} we have qij (x)a = (pi1 + pi2 )[x(pj1 + pj2 )(a)], so [pij (x) − qij (x)]A = 0, and so qij = pij

|M(A)

.



We note that, if A is a matricially decomposed C∗ -algebra, then A12 is a closed subtriple of A. Proposition 7.1.13 Let A be a matricially decomposed prime C∗ -algebra. Then every closed subtriple of M(A) contained in M(A)12 and containing A12 is a prime JB∗ -triple. Proof Let X be a closed subtriple of M(A) contained in M(A)12 and containing A12 . It is enough to show that the conditions x, y ∈ X and {xXy} = 0 imply either x = 0 or y = 0. Let x, y be in X such that {xXy} = 0. Then, by the multiplication rules of matricial decompositions, we have {xAy} = {xA12 y} ⊆ {xXy} = 0. Therefore, for every a ∈ A the equality xay = −yax holds. Then for a, b ∈ A we have xaybxay = (xay)bxay = −(yax)bxay = −(y(axb)x)ay = (x(axb)y)ay = xa(xby)ay = −xa(ybx)ay = −xaybxay, and hence (xay)A(xay) = 0. Since a is arbitrary in A and A is prime, we deduce that xAy = 0. Since x and y belong to M(A), again the primeness of A gives us that either x = 0 or y = 0.  Let A be a matricially decomposed C∗ -algebra, and let τ be a ∗-involution (cf. Definition 6.1.39) on A. We say that τ is even-swapping whenever the equalities τ (A11 ) = A22 and τ (A12 ) = A12 hold. If this is the case, then H(A12 , τ ) becomes a closed subtriple of A. We recall that τ extends uniquely to a ∗-involution (which will be denoted by the same symbol τ ) on M(A) (cf. §6.1.47). If τ is even-swapping, then the extension of τ to M(A) is even-swapping too. Proposition 7.1.14 Let A be a matricially decomposed prime C∗ -algebra with an even-swapping ∗-involution τ . Then every closed subtriple of M(A) contained in H(M(A)12 , τ ) and containing H(A12 , τ ) is a prime JB∗ -triple. Proof

It is enough to prove that the conditions x, y ∈ H(M(A)12 , τ ) and {xH(A12 , τ )y} = 0

imply either x = 0 or y = 0. In a first step we prove that if x and y are in H(M(A)12 , τ ) and if xH(A21 , τ )y = 0, then either x = 0 or y = 0. Let x, y be in H(M(A)12 , τ ) such that xH(A21 , τ )y = 0. For a, b ∈ A21 we have

7.1 Classifying prime JB∗ -triples

443

(xay)τ b(xay) = yaτ xb[x(a + aτ )y] − yaτ [x(bxaτ + axbτ )y] + [x(aτ xa)y]τ bτ y = 0, so (xay)τ A(xay) = (xay)τ A21 (xay) = 0 (by the multiplication rules of matricial decompositions), and so xay = 0 (by the primeness of A). Since a is arbitrary in A21 and the equality xAy = xA21 y holds, we actually have xAy = 0, so that again the primeness of A gives either x = 0 or y = 0. Now, let x, y be in H(M(A)12 , τ ) such that {xH(A12 , τ )y} = 0. If h and k are in H(A21 , τ ), then h∗ and k∗ belong to H(A12 , τ ), so that we have (xhx)k(yhy) = xh{xk∗ y}hx − {xh∗ y}kxhy − {xk∗ y}hxhx + {x(hxk + kxh)∗ y}hy − xk{x(hyh)∗ y} + xkyh{xh∗ y} = 0. Therefore, for h ∈ H(A21 , τ ) we obtain (xhx)H(A21 , τ )(yhy) = 0, and hence, by the first step in the proof, we have either xhx = 0 or yhy = 0. Even more, we actually have either xH(A21 , τ )x = 0 or yH(A21 , τ )y = 0. (Indeed, if there were h1 , h2 ∈ H(A21 , τ ) satisfying xh1 x = 0 and yh2 y = 0, then yh1 y = xh2 x = 0, so that would have h := h1 + h2 ∈ H(A21 , τ ), xhx = 0, and yhy = 0, which is not possible.) Again by the first step in the proof we obtain either x = 0 or y = 0.  Remark 7.1.15 If A is a C∗ -algebra, if e is a self-adjoint idempotent in M(A), and if we set e1 := e and e2 := 1 − e, then {ei Aej }i,j∈{1,2} becomes a matricial decomposition of A. Our next result asserts that every matricial decomposition of a given C∗ -algebra A comes from a self-adjoint idempotent e ∈ M(A) by the procedure described in the above remark. Proposition 7.1.16 Let A be a matricially decomposed C∗ -algebra. Then there exists a self-adjoint idempotent e ∈ M(A) such that, by writing e1 := e and e2 := 1 − e, we have Aij = ei Aej for i, j ∈ {1, 2}. Proof First suppose that A is actually a W ∗ -algebra and that the sum A = ⊕i,j∈{1,2} Aij is w∗ -topological. Then A11 + A12 is a w∗ -closed right ideal of A, and therefore there exists a self-adjoint idempotent e ∈ A such that A11 + A12 = eA (cf. Fact 5.10.122). By taking adjoints, we obtain A11 + A21 = Ae, hence A11 ⊆ eA ∩ Ae = eAe = e1 Ae1 . On the other hand, since (A12 + A22 )(A11 + A12 ) = 0 and e belongs to A11 + A12 , we deduce A12 + A22 ⊆ A(1 − e). By taking adjoints again, we obtain A21 + A22 ⊆ (1 − e)A, and hence A22 ⊆ (1 − e)A ∩ A(1 − e) = (1 − e)A(1 − e) = e2 Ae2 .

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Zel’manov approach

Moreover, we clearly have A12 ⊆ eA ∩ A(1 − e) = eA(1 − e) = e1 Ae2 , and hence, A21 ⊆ e2 Ae1 . Now, since A = ⊕i,j∈{1,2} Aij , and A = ⊕i,j∈{1,2} ei Aej , and Aij ⊆ ei Aej for i, j ∈ {1, 2}, we conclude Aij = ei Aej for i, j ∈ {1, 2}. Now let A be an arbitrary matricially decomposed C∗ -algebra. Then A is a W ∗ algebra and, denoting by pij the (continuous) linear projection on A with range  Aij and kernel (k,l)=(i,j) Akl , the family {pij (A )}i,j∈{1,2} is a matricial decomposition of A . Moreover, the sum A = ⊕i,j∈{1,2} pij (A ) is w∗ -topological. By applying the first paragraph in the proof we find a self-adjoint idempotent e ∈ A such that pij (A ) = ei A ej , where e1 := e and e2 := 1 − e. Then for a ∈ A we have ea = eae + ea(1 − e) = (p11 + p12 )(a) ∈ A. 

Therefore e belongs to M(A).

Remark 7.1.17 Let A be a C∗ -algebra, let e be a self-adjoint idempotent in M(A), and let τ be a ∗-involution on A. It is straightforward that τ is even-swapping relative to the matricial decomposition {ei Aej }i,j∈{1,2} (where e1 := e and e2 := 1 − e) if and only if e + eτ = 1. Now the next theorem follows from Propositions 7.1.13 and 7.1.14, and Remarks 7.1.15 and 7.1.17. Theorem 7.1.18 Let A be a prime C∗ -algebra, and let e ∈ M(A) be a self-adjoint idempotent. Then: (i) Every closed subtriple of M(A) contained in eM(A)(1 − e) and containing eA(1 − e) is a prime JB∗ -triple. (ii) If τ is any ∗-involution on A such that e + eτ = 1, then every closed subtriple of M(A) contained in H(eM(A)eτ , τ ) and containing H(eAeτ , τ ) is a prime JB∗ -triple. 7.1.3 The main results Now we pass to explain those Zel’manovian techniques which are needed in our development. Definition 7.1.19 Let F be a field of characteristic different from 2 and 3. By a Jordan triple over F we mean a vector space T over F together with a trilinear triple product {· · · } : T × T × T −→ T satisfying {ab{xyz}} = {{abx}yz} − {x{bay}z} + {xy{abz}} for all a, b, x, y, z ∈ T. According to Loos [772, 2.8], ‘this definition may be extended to arbitrary base fields but then the identity above turns out to be too weak to develop a satisfactory theory, and has to be replaced by more complicated identities

7.1 Classifying prime JB∗ -triples

445

(cf. [771, 1.2]). As long as the base field has characteristic different from 2 and 3, however, the identity above is sufficient (cf. [771, 2.2])’. Forgetting the analytic conditions, and restricting the scalars, every JB∗ -triple can and will be seen as a Jordan triple over R. Any associative algebra A becomes a Jordan triple under the triple product {abc} := 12 (abc + cba). If moreover, A has two commuting linear algebra involutions ∗, τ , then H(A, ∗) ∩ S(A, τ ) is a Jordan subtriple of A. (Here S(A, τ ) stands for the set of those elements a ∈ A such that τ (a) = −a.) A Jordan triple is called special whenever it is (isomorphic to) a Jordan subtriple of some associative algebra, and i-special if it is the homomorphic image of a special Jordan triple. By a real JC∗ -triple we mean a closed real subtriple of a JC∗ -triple or, equivalently, a closed real subtriple of a C∗ -algebra. If A is a C∗ -algebra, then H(A, ∗) is both a real JC∗ -triple and a special Jordan triple over R under the same triple product, namely 1 {xyz} := (xyz + zyx). 2 The above is more than an example since we have the following. Fact 7.1.20 Let X be a real JC∗ -triple. Then there exists a C∗ -algebra A such that X can be seen as a closed subtriple of the JB-algebra H(A, ∗). Therefore X is a special Jordan triple over R. Proof Let B be any C∗ -algebra containing X as a closed real subtriple, and let A!stand" for the C∗ -algebra M2 (B) (cf. Proposition 2.4.22). Then the mapping x → x0∗ 0x from X to H(A, ∗) becomes an isometric triple homomorphism.  Since JC∗ -triples are real JC∗ -triples, it follows from the above fact that JC∗ triples are special Jordan triples over R. From now on, let X denote an infinite set of indeterminates. We consider the nonunital free associative algebra A (X) over F. A (X) has two natural linear algebra involutions, namely the one ∗ leaving the elements of the set X fixed, and the one τ which maps each element x in X into −x. The Jordan subtriple of A (X) generated by X is called the free special Jordan triple on the set of free generators X, and is denoted by S T = S T (X). Clearly, the inclusion S T ⊆ H(A (X), ∗) ∩ S(A (X), τ ) holds. It follows from the universal property of A (X) that, if A is an associative algebra with two commuting linear algebra involutions ∗, τ , and if T is a Jordan subtriple of A contained in H(A, ∗) ∩ S(A, τ ) then every map φ : X → T extends to a unique ˆ algebra ∗-τ -homomorphism φˆ : A (X) → A such that φ(S T ) ⊆ T. Since every special Jordan triple can be regarded as a Jordan subtriple of a suitable associative algebra with two commuting linear algebra involutions (A, ∗, τ ) contained in H(A, ∗) ∩ S(A, τ ), it follows that, given a special Jordan triple T, every mapping from X to T extends uniquely to a triple homomorphism from S T to T. Keeping in mind

446

Zel’manov approach

the definition of i-special Jordan triples, the above universal property remains true in the more general case that T is an i-special Jordan triple. In this way, we can consider valuations of elements of S T in any i-special Jordan triple. For elements a1 , . . . , an in an associative algebra A, we define the n-tad {a1 · · · an }n as the element of A given by 1 {a1 · · · an }n = (a1 · · · an + an · · · a1 ). 2 An ideal I of S T is called formal if p(x1 , . . . , xn ) ∈ I ⇒ p(σ (x1 ), . . . , σ (xn )) ∈ I for all permutations σ of X, and hermitian if it is formal and n-tad closed in A (X) for all odd n ≥ 5, i.e. {I · · · I }n ⊆ I for all odd n ≥ 5 . Now, if A is an associative algebra with two commuting linear algebra involutions ∗ and τ , if T is a Jordan subtriple of A contained in H(A, ∗) ∩ S(A, τ ), and if I is a hermitian ideal of S T , then {a1 · · · an }n lies in I (T) whenever n is any odd natural number and a1 , . . . , an are in I (T). Lemma 7.1.21 Let B be an associative algebra with two commuting linear algebra involutions ∗, τ , and let T be a Jordan subtriple of B contained in H(B, ∗) ∩ S(B, τ ). If I is a hermitian ideal of S T satisfying I (T) = 0, then the subalgebra C of B generated by I (T) is ∗-τ -invariant and we have I (T) = H(C, ∗) ∩ S(C, τ ). Proof By the previous comments, I (T) is an n-tad closed ideal of T for all odd n ≥ 5, and the inclusion I (T) ⊆ H(B, ∗) ∩ S(B, τ ) holds. From that inclusion we deduce that the subalgebra C of B generated by I (T) is ∗-τ -invariant and that I (T) is contained in H(C, ∗) ∩ S(C, τ ). Let z be in H(C, ∗) ∩ S(C, τ ). Since z is in C,  z = kn=1 b1n · · · binn for suitable k ∈ N and bin ∈ I (T) (n = 1, . . . , k, i = 1, . . . , in ). Moreover, we have  k ∗ k   1 in ∗ τ ∗ in in 1 bn · · · bn = z = z = (−z ) = − (−1) bn · · · bn n=1

n=1

=−

k 

(−1)in b1n · · · binn ,

n=1

hence

k

in 1 n=1 bn · · · bn in even

z=

= 0, and so z =

k

in 1 n=1 bn · · · bn . in odd

It follows

k k   1 1 {b1n · · · binn }in ∈ I (T). (bn · · · binn + binn · · · b1n ) = 2

n=1 in odd

n=1 in odd



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447

§7.1.22 One of the key tools in Zel’manov’s work is the discovery of a precise hermitian ideal G (see [1134, p. 730]) in S T with the property that the behaviour of i-special prime ‘nondegenerate’ Jordan triples drastically differs depending on whether or not G vanishes on them. By a Jordan triple of hermitian type we mean an i-special Jordan triple T satisfying G (T) = 0. Lemma 7.1.23 Let T be an i-special complex Jordan triple. Then G (T) is invariant under every conjugate-linear automorphism of T. Proof If A (X) is the free complex associative algebra on the infinite set X of indeterminates, then the vector space of A (X) is the free complex vector space on the set of all associative words with letters in X, so that we can define a mapping     : A (X) → A (X) by ( i λi wi ) := i λi wi (here i λi wi is any finite linear combination of associative words wi ). Clearly,  is an involutive conjugate-linear algebra automorphism of A (X) which fixes the elements of X. Therefore the free special complex Jordan triple S T (X) is invariant under . Since the Zel’manovian complex ideal G of S T (X) is generated (as ideal) by a set of elements of S T (X) whose expressions as linear combinations of associative words only involve real scalars, it follows that G is -invariant. Now, if φ is a conjugate-linear automorphism  of T, if p(x1 , . . . , xn ) is in G , and if p = k λk qk where λk are complex numbers  and qk are Jordan triple monomials, then (p) = k λk qk lies in G , and hence, for t1 , . . . tn ∈ T,   φ[p(t1 , . . . , tn )] = λk φ[qk (t1 , . . . , tn )] = λk qk (φ(t1 ), . . . , φ(tn )) k

belongs to G (T).

k



Lemma 7.1.24 Let X be a JC∗ -triple. Then there exists a matricially decomposed C∗ -algebra B with an even-swapping ∗-involution τ such that X can be seen as a closed subtriple of B contained in H(B, τ ) ∩ B12 . Proof Let A be any C∗ -algebra containing X as a closed subtriple. Consider the opposite algebra A(0) of A, and the algebra direct product A × A(0) , which becomes a C∗ -algebra under the involution defined coordinate-wise and the sup norm. Now, consider the C∗ -algebra B! := M2 (A × A(0)") (cf. ! Proposition 2.4.22) " (u ,v ) (u ,v ) (v ,u ) (v ,u ) and the ∗-involution τ on B given by τ (u11 ,v11 ) (u12 ,v12 ) = (v22 ,u22 ) (v12 ,u12 ) , and 21 21 22 22 21 21 11 11 note that τ is even-swapping relative to the natural matricial decomposition of B. (Regarding B = M2 (C) ⊗ (A × A(0) ), τ is nothing other than the tensor product of the symplectic involution on the C∗ -algebra M2 (C) ! by the"exchange involution on the (0,0) (x,x) ∗ (0) C -algebra A × A ). Then the mapping x → (0,0) (0,0) from X to H(B, τ ) becomes an isometric triple homomorphism.  Proposition 7.1.25 Let X be a JC∗ -triple of hermitian type. Then X contains a nonzero closed triple ideal of the form H(A, τ ) ∩ A12 , where A is a matricially

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decomposed C∗ -algebra, τ is an even-swapping ∗-involution on A, and A is generated as C∗ -algebra by H(A, τ ) ∩ A12 . Proof Let B and τ be the matricially decomposed C∗ -algebra and the evenswapping ∗-involution on B, respectively, given by Lemma 7.1.24, so that we have X ⊆ H(B, τ ) ∩ B12 .



For b = bij with bij ∈ Bij , we write π(b) = τ (b11 + b22 ) − τ (b12 + b21 ), so that π becomes an even-swapping ∗-involution on B commuting with τ , and we have H(B, τ ) ∩ S(B, π) = H(B, τ ) ∩ B12 + H(B, τ ) ∩ B21 . Put T

:= X + X ∗ ,

and note that T ⊆ H(B, τ ) ∩ S(B, π ).

Then T is a closed subtriple of B, but for the moment this is not relevant for our argument. At this point, we emphasize that T is a Jordan subtriple of B (i.e. T is a subspace of B closed under the triple product [abc] := 12 (abc + cba)). In fact, for x1 , y1 , x2 , y2 , x3 , y3 ∈ X, we have [(x1 + y∗1 )(x2 + y∗2 )(x3 + y∗3 )] = {x1 y2 x3 } + {y1 x2 y3 }∗ , where {· · · } is the triple product of the JB∗ -triple X. Since the set {x + x∗ : x ∈ X} is a copy of XR contained in T, and XR is of hermitian type, T (regarded as a Jordan triple) is also of hermitian type. By Lemma 7.1.21, the subalgebra C of B generated by G (T) is invariant under τ and π, and we have G (T) = H(C, τ ) ∩ S(C, π ). Moreover, since T is a ∗-invariant subset of B, and the restriction of ∗ to T is a conjugate-linear Jordan triple automorphism of T, it follows from Lemma 7.1.23 that G (T) is ∗-invariant. Since C is generated by G (T), we conclude that C is ∗-invariant. On the other hand, the decomposition T = X ⊕ X ∗ exhibits T as a ‘polarized’ Jordan triple in the sense [920, p. 229, (5.1)], and therefore G (T) inherits the polarization (see [920, p. 231]). This means that G (T) = G (T) ∩ X + G (T) ∩ X ∗ .

 Now, G (T) is contained in B12 ∩ C + B21 ∩ C and hence in i,j∈{1,2} Bij ∩ C. Since the last sum is a subalgebra of B, and C is the subalgebra of B generated by G (T), it follows that  C= Cij , i,j∈{1,2}

where Cij := Bij ∩ C.  To conclude the proof, we keep in mind that the sum B = i,j∈{1,2} Bij is topological to obtain that all properties proved for C pass to the closure of C (say A) in B.

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Therefore, A is a closed ∗-subalgebra of B, and inherits the matricial decomposition of B. Moreover, denoting by K the closure of G (T) in B, K is an ideal of the Jordan triple T satisfying K = H(A, τ ) ∩ S(A, π) = H(A, τ ) ∩ A12 + H(A, τ ) ∩ A21 . Then, clearly, H(A, τ ) ∩ A12 is a closed triple ideal of X.



The structure theorem for prime JB∗ -triples of hermitian type will follow from the above proposition and some general results on matricially decomposed C∗ -algebras. Proposition 7.1.26 For a matricially decomposed C∗ -algebra A the following assertions hold: (i) All closed ideals of A inherit the matricial decomposition. (ii) If A is generated as a closed ideal by A12 , then every nonzero closed ideal of A meets A12 . (iii) If A has an even-swapping ∗-involution τ , and if A is generated as a closed ideal by H(A, τ ) ∩ A12 , then every nonzero τ -invariant closed ideal of A meets H(A, τ ) ∩ A12 . Proof Let i, j be in {1, 2}. Let pij be the (continuous) linear projection on A with  range Aij and kernel (k,l)=(i,j) Akl , and notice that, for a, b, c ∈ A, the equality pij (apij (b)c) = pii (a)bpjj (c) follows straightforwardly from the multiplication rules of the matricial decomposition. Now, if P is a closed ideal of A, and if x is in P, it is enough to take an approximate unit {eλ }λ∈ for A, to conclude that pij (x) = lim lim pij (eλ pij (x)eμ ) = lim lim pii (eλ )xpjj (eμ ) λ

μ

λ

μ

lies in P. Thus assertion (i) is proved. Suppose that A is generated as a closed ideal by A12 . Let P be a closed ideal of A satisfying P ∩ A12 = 0. By the ∗-invariance of P, we have also that P ∩ A21 = 0. Hence, keeping in mind assertion (i), we deduce p12 (P) = p21 (P) = 0, and therefore A12 p22 (P) = 0. Since obviously A12 p11 (P) = 0, we obtain that A12 P = 0. Now, A12 is contained in the left annihilator Ann (P) of P in A (cf. §6.1.66). Since A is generated as a closed ideal by A12 and Ann (P) is a closed ideal of A, we conclude that A ⊆ Ann (P) and therefore P = 0. Thus assertion (ii) is proved. Now suppose that A has an even-swapping ∗-involution τ , and that A is generated as a closed ideal by H(A, τ ) ∩ A12 . Let P be a closed τ -invariant ideal of A satisfying P ∩ H(A, τ ) ∩ A12 = 0. Let P◦◦ be the bipolar of P in A . Since τ and p12 commute (because τ is even-swapping), the equality P ∩ H(A, τ ) ∩ A12 = 0 can be read as (IA + τ )p12 (P) = 0, and hence we have (IA + τ  )p12 (P◦◦ ) = 0. Now, write P◦◦ = A e for a suitable central idempotent e ∈ A . Since the family {pij (A )}i,j∈{1,2} is a matricial decomposition of A , and A e and A (1 − e) are closed ideals of A , we can apply assertion (i) to obtain that, for i, j ∈ {1, 2} and α ∈ A , we have

450

Zel’manov approach pij (αe) = pij (αe)e and pij (α(1 − e)) = pij (α(1 − e))(1 − e),

so pij Re = Re pij Re and pij (IA − Re ) = (IA − Re )pij (IA − Re ), and so Re and pij commute. It follows Re (H(A , τ  ) ∩ A12 ) = Re (IA + τ  )p12 (A ) = (IA + τ  )p12 Re (A ) = (IA + τ  )p12 (P◦◦ ) = 0, hence H(A , τ  ) ∩ A12 ⊆ ker(Re ). Since ker(Re ) is a w∗ -closed ideal of A , and A is generated by H(A , τ  ) ∩ A12 as a w∗ -closed ideal (because A is generated by H(A, τ ) ∩ A12 as a norm-closed ideal), we actually have P ⊆ P◦◦ = Re (A ) = 0, and so P = 0. Now, the proof of assertion (iii) is complete.  A triple ideal I of a Jordan ∗-triple X is said to be essential if I ∩ J = 0 for every nonzero triple ideal J of X. We recall (cf. Proposition 5.7.10) that the bidual X  of any JB∗ -triple X becomes a JB∗ -triple under a triple product which extends that of X. Proposition 7.1.27 Let X be a JB∗ -triple. Then the set Mult(X) := {x ∈ X  : {xXX} ⊆ X} becomes a closed subtriple of X  containing X as a closed essential triple ideal. Moreover, if K is any JB∗ -triple containing X as a closed essential triple ideal, then K can be seen as a closed subtriple of Mult(X) containing X. Proof That Mult(X) is a closed subtriple of X  containing X as a closed triple ideal was proved in Proposition 5.10.95. The remaining part of the proposition can be proved by arguing as in the corresponding part of the proof of Theorem 5.10.90 by keeping in mind Theorem 5.7.18 and by replacing Fact 5.1.10 with Theorem 5.7.32 (see also the comments in Definition 5.7.44). Alternatively, the reader can go to [873, Lemmas 2.2 and 2.3].  Proposition 7.1.28 For a matricially decomposed C∗ -algebra A the following assertions hold: (i) If A has an even-swapping ∗-involution τ , and if A is generated by H(A, τ ) ∩ A12 as a normed ∗-algebra, then Mult(H(A, τ ) ∩ A12 ) = H(M(A), τ ) ∩ M(A)12 . (ii) If A is generated by A12 as a normed ∗-algebra, then Mult(A12 ) = M(A)12 .

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Proof Suppose that A has an even-swapping ∗-involution τ , and that A is generated by H(A, τ ) ∩ A12 as a normed ∗-algebra. Put X := H(A, τ ) ∩ A12 . Through the natural identification of X  with X ◦◦ , we can regard Mult(X) as a closed subtriple of A contained in X ◦◦ . Then, the inclusion Mult(X) ⊇ H(M(A), τ ) ∩ M(A)12 is clear. Let x be in Mult(X) and z be in X. Since τ and p12 commute, we have X ◦◦ = H(A , τ  ) ∩ (A )12 , and hence, clearly, xz = 0 = zx. Moreover, by Corollary 4.2.11, there exists y ∈ X such that z = {yyy}, and consequently, from the equalities xz∗ = x{yyy}∗ = 2{xyy}y∗ − y{yxy}∗ and z∗ x = {yyy}∗ x = 2y∗ {xyy} − {yxy}∗ y it follows that xz∗ , z∗ x belong to A. Now, the set B := {a ∈ A : xa, xa∗ , ax, a∗ x ∈ A} is a closed ∗-subalgebra of A containing X, so B = A (because A is generated by X as a normed ∗-algebra), and so, x lies in M(A). Since x is an arbitrary element of Mult(X), we deduce Mult(X) ⊆ H(M(A), τ ) ∩ M(A)12 , and the proof of assertion (i) is complete. Keeping in mind that (A12 )◦◦ = (A )12 , the proof of assertion (ii) is similar to that of assertion (i), and is left to the reader.  Now we are ready to formulate and prove the following. Theorem 7.1.29 Let X be a prime JB∗ -triple of hermitian type. Then one of the following assertions is true for X: (i) There exists a matricially decomposed prime C∗ -algebra A such that X can be regarded as a closed subtriple of the C∗ -algebra M(A) contained in M(A)12 and containing A12 . (ii) There exists a matricially decomposed prime C∗ -algebra A with an evenswapping ∗-involution τ such that X can be regarded as a closed subtriple of the matricially decomposed C∗ -algebra M(A) contained in H(M(A), τ ) ∩ M(A)12 and containing H(A, τ ) ∩ A12 . Proof Since X is i-special and all JB∗ -triples listed in Theorem 7.1.11 are non i-special, it follows from that theorem that X is a JC∗ -triple. Then, by Proposition 7.1.25, X contains a nonzero closed triple ideal of the form H(A, τ )∩A12 , where A is a matricially decomposed C∗ -algebra, τ is an even-swapping ∗-involution on A, and A is generated by H(A, τ ) ∩ A12 as a normed ∗-algebra. Since X is prime, H(A, τ ) ∩ A12 is an essential triple ideal of X, so that Propositions 7.1.27 and 7.1.28(i) allow us to

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see X as a closed subtriple of the matricially decomposed C∗ -algebra M(A) contained in H(M(A), τ ) ∩ M(A)12 and containing H(A, τ ) ∩ A12 . If A is prime, then we are in case (ii). Suppose from now on that A is nonprime. Since A is τ -prime (by Corollary 7.1.3 and Proposition 7.1.26(iii)), we can find a nonzero closed ideal P of A such that P ∩ τ (P) = 0. But P is a prime algebra because, if R, S are closed ideals of P with RS = 0, then R + τ (R), S + τ (S) are τ -invariant ideals of A (by Corollary 7.1.2) satisfying (R+τ (R))(S+τ (S)) = 0 and therefore either R+τ (R) = 0 or S+τ (S) = 0. By Proposition 7.1.26(i), P inherits the matricial decomposition of A. Now, P is a matricially decomposed C∗ -algebra, and the mapping φ : x → x + τ (x) from P12 to H(A, τ ) ∩ A12 is an injective triple homomorphism whose range is a triple ideal of H(A, τ ) ∩ A12 . Let us denote by A the closed ∗-subalgebra of P generated by  P12 . Since i,j∈{1,2} Pij ∩ A is a closed ∗-subalgebra of P containing P12 we deduce  A = i,j∈{1,2} Pij ∩ A, and hence A inherits the matricial decomposition of P. In this way, A is a matricially decomposed C∗ -algebra satisfying A12 = P12 . Moreover, keeping in mind Proposition 5.7.41, we can forget the embedding φ above and regard A12 as a closed triple ideal of H(A, τ )∩A12 . Since H(A, τ )∩A12 is a closed triple ideal of X, it follows from Fact 7.1.1 that A12 can be seen as a nonzero closed triple ideal of the prime JB∗ -triple X. By Corollary 7.1.3, A12 is a prime JB∗ -triple, and hence, in view of Proposition 7.1.26(ii), A is a prime C∗ -algebra. By Propositions 7.1.27 and 7.1.28(ii), X can be regarded as a closed subtriple of the matricially decomposed C∗ algebra M(A) contained in M(A)12 and containing A12 . Therefore X is in case (i).  §7.1.30 A Jordan triple T is said to be of Clifford type if it is i-special and satisfies G (T) = 0, where G is the Zel’manovian ideal of S T introduced in §7.1.22. §7.1.31 Now we establish the foundations for the earlier determination of all prime JB∗ -triple of Clifford type. More precisely, we obtain here the list of Cartan factors which are of Clifford type. This list will be obtained by considering their associated Jordan pairs and then by applying the determination of prime nondegenerate Jordan pairs of Clifford type provided by D’Amour and McCrimmon [921]. A Jordan pair over a field F of characteristic different from 2 and 3 is a couple (V + , V − ) of vector spaces over F together with trilinear mappings [· · · ]ε : V ε × V −ε × V ε −→ V ε (ε = ±) satisfying [ab[xyz]ε ]ε = [[abx]ε yz]ε − [x[bay]−ε z]ε + [xy[abz]ε ]ε for all a, x, z ∈ V ε and b, y ∈ V −ε . Every Jordan triple T over F has an associated Jordan pair over F, namely the pair (V + , V − ), where V ε = T and [· · · ]ε = {· · · } (ε = ±). In [921], a suitable notion of Jordan pair of Clifford type is given in such a way that a Jordan triple is of Clifford type (in the sense introduced in §7.1.30) if and only if its associated Jordan pair is of Clifford type. Although JB∗ -triples are only real Jordan triples, the real Jordan pair associated to a JB∗ -triple actually is the realification of a complex Jordan pair. Indeed, if X is a JB∗ -triple, and if X denotes

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the complex vector space obtained from that of X by replacing the complex structure with the conjugate one, then the realifications of X and X coincide, and, taking V + = X and V − = X, the mappings (x, y, z) → [xyz]ε = {xyz} from V ε × V −ε × V ε to V ε (ε = ±) become complex-linear in each of their variables. Proposition 7.1.32 The Cartan factors of Clifford type are the following: (i) (ii) (iii) (iv)

Those of type In,m for n = 1, 2 and n ≤ m, The one of type II5 , The one of type III2 , All type IVn Cartan factors, for n ≥ 5.

Proof First we show that Cartan factors not listed in the proposition are not of Clifford type. Let X = BL(H, K) be a type In,m Cartan factor with 3 ≤ n ≤ m. Take 3-dimensional subspaces H1 and K1 of H and K, respectively. Then, denoting by H1⊥ the orthogonal complement of H1 in H, the set X1 := {x ∈ BL(H, K) : x(H1⊥ ) = 0, x(H) ⊆ K1 } is a closed subtriple of X isomorphic to the type I3,3 Cartan factor. But it easily seen that the complex Jordan pair associated to the type I3,3 Cartan factor is isomorphic to (M3 (C), M3 (C)) (where Mn (C) denotes n × n-complex matrices) with trilinear mappings [xyz]ε = 12 (xyz + zyx). Since this Jordan pair is not of Clifford type [921], it follows that X1 (and hence X) is not of Clifford type. Let X = {x ∈ BL(H) : σ x∗ σ = −x} be a type IIn Cartan factor with n ≥ 6. Take a σ -invariant 6-dimensional subspace H1 of H. Then the set X1 := {x ∈ X : x(H1⊥ ) = 0, x(H) ⊆ H1 } is a closed subtriple of X isomorphic to the type II6 Cartan factor. Since the complex Jordan pair associated to X1 is isomorphic to the pair (A6 (C),A6 (C)) (where An (C) := {x ∈ Mn (C) : xt = −x}, t the transpose involution) with trilinear mappings [xyz]ε = − 12 (xyz + zyx), and this Jordan pair is not of Clifford type [921], we obtain that X is not of Clifford type. Let X = {x ∈ BL(H) : σ x∗ σ = x} be a type IIIn Cartan factor with n ≥ 3. Arguing as in the previous case, we realize that X contains as a closed subtriple a copy of the type III3 Cartan factor, whose associated complex Jordan pair is isomorphic to (H3 (C), H3 (C)) (where Hn (C) := {x ∈ Mn (C) : xt = x}) with trilinear mappings [xyz]ε = 12 (xyz + zyx). As above, the fact that X is not of Clifford type follows from [921]. Since the type V and VI Cartan factors are not i-special, the first part of the proof is concluded. Now, we prove that all Cartan factors listed in the proposition are of Clifford type. Let X = BL(H, K) be a type In,m Cartan factor with n = 1, 2 and n ≤ m. Let p(x1 , . . . , xr ) be in the Zel’manovian ideal G , and y1 , . . . , yr be elements of X. Take

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a subspace K1 of K containing y1 (H) + · · · + yr (H) and having finite dimension q ≥ n. Then X1 := {x ∈ BL(H, K) : x(H) ⊆ K1 } is a closed subtriple of X isomorphic to the type In,q Cartan factor and contains {y1 , . . . , yr }. Since the complex Jordan pair associated to the type In,q Cartan factor is isomorphic to (Mn,q (C), Mq,n (C)) (where Mi,j (C) denotes i × j-complex matrices) with trilinear mappings [xyz]ε = 12 (xyz + zyx) and this Jordan pair is of Clifford type [921], it follows that X1 is of Clifford type, and therefore p(y1 , . . . , yr ) = 0. Since the complex Jordan pair associated to the type II5 Cartan factor is isomorphic to (A5 (C), A5 (C)), and this Jordan pair is of Clifford type [921], such a Cartan factor is of Clifford type. The remaining cases listed in the proposition can be treated in a unified way because, according to [772, 4.18], the type III2 Cartan factor is isomorphic to the type IV3 Cartan factor. If X is a type IVn Cartan factor (n ≥ 3) then the complex Jordan pair associated to X is isomorphic to (H, H), where H is a complex Hilbert space of hilbertian dimension n, with trilinear mappings [xyz]ε := (x|σ (y))z + (z|σ (y))x − (x|σ (z))y, where σ is a conjugation on H. Since the mapping (x, y) → (x|σ (y)) from H × H to C is a nondegenerate symmetric bilinear form, the above Jordan pair is of Clifford type [921], hence X is of Clifford type.  We say that a Banach space E is hilbertizable if there are positive constants m, M, and an inner product (·|·) on E such that mx2 ≤ (x|x) ≤ Mx2 for every x ∈ E. A family {Ei }i∈I of Banach spaces is said to be uniformly hilbertizable if there are positive constants m, M, and inner products (·|·)i on Ei (i ∈ I) satisfying mxi 2 ≤ (xi |xi )i ≤ Mxi 2 for all i ∈ I and xi ∈ Ei . Corollary 7.1.33 Every family of Cartan factors of Clifford type is uniformly hilbertizable. Proof Let X = BL(H, K) be a type In,m Cartan factor with n ≤ m and n finite. Take an orthonormal basis {η1 , . . . , ηn } of H. Then the mapping 1 (x(ηi )|y(ηi )) n n

(x, y) → (x|y) :=

i=1

from X × X to C is an inner product on X satisfying (x|x) ≤ x2 for every x ∈ X.  Moreover, for x ∈ X and η = ni=1 λi ηi ∈ H, we have  n 2  n  n     2 2 2 x(η) ≤ |λi | x(ηi ) ≤ |λi | x(ηi ) = nη2 (x|x), i=1

and hence

1 2 n x

≤ (x|x).

i=1

i=1

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Let X be a type IVn Cartan factor with n ≥ 3. Then there exist a complex Hilbert space (H, (·|·)) of hilbertian dimension n and a conjugation σ on H such that X = H as complex vector spaces and the equality ? x2 := (x|x) + (x|x)2 − |(x|σ (x))|2 holds for every x ∈ H. Therefore we have 12 x2 ≤ (x|x) ≤ x2 for every x ∈ X. It follows from this and Proposition 7.1.32 that every Cartan factor X of Clifford type has an inner product (·|·) satisfying 15 x2 ≤ (x|x) ≤ x2 for every x ∈ X.  Now we conclude the determination of prime JB∗ -triples of Clifford type. Such a determination will be obtained by combining Proposition 7.1.32 and Corollary 7.1.33 with the technology of Banach ultraproducts (cf. §2.8.58). According to Lemma 5.7.4, the ∞ -sum of any family of JB∗ -triples, endowed with the triple product defined point-wise, becomes a JB∗ -triple. Then the following lemma is a direct consequence of Proposition 5.7.41. Lemma 7.1.34 Let X be a JB∗ -triple, let I be a non-empty set, and, for each i ∈ I, let φi be a triple homomorphism from X to a JB∗ -triple Xi . If ∩i∈I ker(φi ) = 0, then x = sup{φi (x) : i ∈ I} for every x ∈ X. We note that, if U is an ultrafilter on a non-empty set I, and if {Ei }i∈I is a uniformly hilbertizable family of Banach spaces, then the ultraproduct (Ei )U is a hilbertizable Banach space. Indeed, if m, M are positive constants and (·|·)i is an inner product on Ei (i ∈ I) satisfying mxi 2 ≤ (xi |xi )i ≤ Mxi 2 for all i ∈ I and xi ∈ Ei , then ((xi ), (yi )) → ((xi )|(yi )) := limU (xi |yi ) is a well-defined inner product on (Ei )U satisfying m(xi )2 ≤ ((xi )|(xi )) ≤ M(xi )2 for every (xi ) ∈ (Ei )U . We note also that, if for every i ∈ I, Ei is a JB∗ -triple, then (Ei )U is a JB∗ -triple under the triple product {(xi )(yi )(zi )} := ({xi yi zi }) (cf. Proposition 5.7.5). Lemma 7.1.35 Let X be a prime JB∗ -triple, let I be a non-empty set, and, for each i ∈ I, let φi be a triple homomorphism from X to a JB∗ -triple Xi . Suppose that ∩i∈I ker(φi ) = 0. Then there exists an ultrafilter U on I such that the triple homomorphism φ : x → (φi (x)) from X to (Xi )U is injective. Proof Argue as in the proof of Lemma 6.2.25 with Lemma 7.1.34 instead of Lemma 6.2.24.  Proposition 7.1.36 Every prime JB∗ -triple of Clifford type is a Cartan factor. Proof First we prove that, if X is a JB∗ -triple of Clifford type, and if φ is a representation of X into a JBW ∗ -triple factor Z, then Z is also of Clifford type. To this end, we invoke the strong∗ topology of an arbitrary JBW ∗ -triple Z (cf. Definition 5.10.61), and recall that the strong∗ topology is a Hausdorff vector space topology, that, if Y is a w∗ -dense closed subtriple of Z, then the closed unit ball of Y is strong∗ dense in the closed unit ball of Z (Corollary 5.10.64), and that the triple product of Z is jointly strong∗ -continuous on bounded subsets of Z (Theorem 5.10.133).

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Let X be a JB∗ -triple of Clifford type, and let φ be a representation of X into a JBW ∗ -triple factor Z. By Proposition 5.7.41, φ(X) is a closed subtriple of Z and clearly φ(X) is of Clifford type. If Z were not i-special, then it would be finitedimensional (by Corollary 7.1.10), so φ(X) = Z would be not i-special, which is a contradiction. Let p(x1 , . . . , xr ) be in the Zel’manovian ideal G , and z1 , . . . , zr be elements of Z. Put M := max {z1 , . . . , zr }. Then, there exist nets {yλi }λ∈ in φ(X), with yλi  ≤ M for every λ ∈ , strong∗ -convergent to zi (i ∈ {1, 2, . . . , r}). Since the triple product is jointly strong∗ -continuous on bounded sets, we have that {p(yλ1 , . . . , yλr )}λ strong∗ -converges to p(z1 , . . . , zr ). Since p(yλ1 , . . . , yλr ) = 0 for every λ ∈ , we obtain p(z1 , . . . , zr ) = 0. Therefore Z is of Clifford type. Now, let X be a prime JB∗ -triple of Clifford type. Then, by Theorem 7.1.5, there exists a faithful family of Cartan factor representations of X (say, {φi : X → Xi }i∈I ). By the preceding paragraph, for each i ∈ I, Xi is a Cartan factor of Clifford type. Let U be the ultrafilter on I whose existence is provided by Lemma 7.1.35. It follows from Corollary 7.1.33 that (Xi )U is a hilbertizable Banach space. Since, by Lemma 7.1.35 and Proposition 5.7.41, the mapping x → (φi (x)) from X to (Xi )U is a linear isometry, X is also a hilbertizable Banach space. As a consequence, the Banach space of X is reflexive, hence it is a dual Banach space whose predual has extreme points in its closed unit ball. Since X is a prime JB∗ -triple, it follows that X is a Cartan factor.  The following theorem follows from Propositions 7.1.32 and 7.1.36. Theorem 7.1.37 The prime JB∗ -triples of Clifford type are the type In,m (n = 1, 2, n ≤ m), II5 , III2 , and IVn (n ≥ 5) Cartan factors. In what follows we summarize the results previously obtained for prime JB∗ triples in a single statement. Simultaneously, in the formulation of such a summarized version of the Zel’manovian classification of prime JB∗ -triples, we will avoid any reference to the technical classification of prime Jordan triples in the cases exceptional, hermitian, and Clifford. Lemma 7.1.38 If X is a type In,m (1 ≤ n ≤ m) Cartan factor, then there is a matricially decomposed W ∗ -factor A such that X = A12 . If X is either a type IIn (n ≥ 2) or type IIIn (n ≥ 1) Cartan factor, then there is a matricially decomposed W ∗ -factor A with an even-swapping ∗-involution τ such that X = H(A, τ ) ∩ A12 . Proof For X = BL(H, K) a type In,m Cartan factor, take A = BL(H ⊕ K) with matricial decomposition given by A11 := {x ∈ A : x(K) ⊆ K, x(H) = 0}, A12 := {x ∈ A : x(H) ⊆ K, x(K) = 0}, A21 := {x ∈ A : x(H) = 0, x(K) ⊆ H}, A22 := {x ∈ A : x(K) = 0, x(H) ⊆ H}, to obtain X = A12 . For X = {x ∈ BL(H) : σ x∗ σ = ∓x} a type IIn or IIIn Cartan factor, take A = M2 (C) ⊗ BL(H), the matricial decomposition of A given by Aij := uij ⊗ BL(H)

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(where the" uij ! are the usual matrix units for M2 (C)), and τ = τ∓ ⊗ t (where ! " λ11 λ12 λ22 ∓λ12 τ∓ λ λ  = ∓λ λ , and t(x) = σ x∗ σ ), to get X = H(A, τ ) ∩ A12 . 21

22

21

11

The following Zel’manovian classification theorem for prime JB∗ -triples follows directly from the preceding lemma and Theorems 7.1.11, 7.1.29, and 7.1.37. Theorem 7.1.39 If X is a prime JB∗ -triple, then one of the following assertions holds for X: (i) X is either the type V or the type VI Cartan factor. (ii) X is a spin triple factor. (iii) There exists a matricially decomposed prime C∗ -algebra A such that X can be regarded as a closed subtriple of the C∗ -algebra M(A) contained in M(A)12 and containing A12 . (iv) There exists a matricially decomposed prime C∗ -algebra A with an evenswapping ∗-involution τ such that X can be regarded as a closed subtriple of the matricially decomposed C∗ -algebra M(A) contained in H(M(A), τ ) ∩ M(A)12 and containing H(A, τ ) ∩ A12 . A Banach Jordan ∗-triple is said to be topologically simple if it has nonzero product and has no nonzero proper closed triple ideal. Clearly, topologically simple Banach Jordan ∗-triples are prime. Corollary 7.1.40 If X is a topologically simple JB∗ -triple, then one of the following assertions holds for X: (i) X is either the type V or the type VI Cartan factor. (ii) X is a spin triple factor. (iii) There exists a matricially decomposed topologically simple C∗ -algebra A such that X = A12 . (iv) There exists a matricially decomposed topologically simple C∗ -algebra A with an even-swapping ∗-involution τ such that X = H(A, τ ) ∩ A12 . Proof If X is not a JC∗ -triple, then, by Theorem 7.1.11, we are in case (i). Suppose that X is of Clifford type. Then, by Theorem 7.1.37, either X is a spin triple factor (and we are in case (ii)) or X is a type In,m (n = 1, 2, n ≤ m) or II5 Cartan factor. In the case that X is a type II5 Cartan factor, by Lemma 7.1.38 and its proof, we have X = H(A, τ ) ∩ A12 for some matricially decomposed algebraically (hence topologically) simple C∗ -algebra A with an even-swapping ∗-involution τ , and therefore we are in case (iv). In the case that X = BL(H, K) is a type In,m (n = 1, 2, n ≤ m) Cartan factor, the matricially decomposed W ∗ -factor A = BL(H ⊕ K) given by Lemma 7.1.38 and its proof, which satisfies X = A12 , need not be topologically simple. However, the topologically simple C∗ -algebra A := K(H ⊕ K) of all compact operators on H ⊕ K (cf. Corollary 1.4.33) inherits the matricial decomposition of A, and, by the finite dimensionality of H, the equality A12 = A12 holds, which leads to case (iii).

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Finally, suppose that X is of hermitian type. By the topological simplicity of X and Theorem 7.1.29, we have the following two possibilities for X: (1) There exists a matricially decomposed prime C∗ -algebra A such that X = A12 . (2) There exists a matricially decomposed prime C∗ -algebra A with an evenswapping ∗-involution τ such that X = H(A, τ ) ∩ A12 . Moreover, by the proof of Theorem 7.1.29, in both situations A is generated as C∗ algebra by X. Now, to conclude the proof it is enough to show that the topological simplicity of X implies that of A. In the case that X is in the situation (1), this follows from Proposition 7.1.26(ii). If X is in the situation (2), and if P is a nonzero closed ideal of A, then P ∩ τ (P) is a nonzero (by primeness of A) τ -invariant closed ideal of A, so P ∩ τ (P) = A (by Proposition 7.1.26(iii)), and so P = A.  Now, with the help of Proposition 7.1.16 and Remark 7.1.17 we can avoid any mention to matricial decompositions of C∗ -algebras, and reformulate Theorem 7.1.39 and Corollary 7.1.40 as follows. Theorem 7.1.41 If X is a prime JB∗ -triple, then one of the following assertions holds for X: (i) X is either the type V or the type VI Cartan factor. (ii) X is a spin triple factor. (iii) There exist a prime C∗ -algebra A and a self-adjoint idempotent e ∈ M(A) such that X can be regarded as a closed subtriple of the C∗ -algebra M(A) contained in eM(A)(1 − e) and containing eA(1 − e). (iv) There exist a prime C∗ -algebra A, a self-adjoint idempotent e ∈ M(A), and a ∗-involution τ on A with e + eτ = 1 such that X can be regarded as a closed subtriple of the C∗ -algebra M(A) contained in H(eM(A)eτ , τ ) and containing H(eAeτ , τ ). Corollary 7.1.42 If X is a topologically simple JB∗ -triple, then one of the following assertions holds for X: (i) X is either the type V or the type VI Cartan factor. (ii) X is a spin triple factor. (iii) There exist a topologically simple C∗ -algebra A and a self-adjoint idempotent e ∈ M(A) such that X = eA(1 − e). (iv) There exist a topologically simple C∗ -algebra A, a self-adjoint idempotent e ∈ M(A), and a ∗-involution τ on A with e + eτ = 1 such that X = H(eAeτ , τ ). We note that, in view of Theorem 7.1.18, all JB∗ -triples listed in Theorem 7.1.41 are prime. 7.1.4 Historical notes and comments Most results in this section are due to Moreno and Rodr´ıguez [448, 449]. Other sources are quoted in what follows.

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Corollary 7.1.2 could be new. Its associative forerunner (which actually is what applies in our development) can be found in [758, Corollary 4.2.10]. Theorem 7.1.5 and Corollaries 7.1.6 and 7.1.7 are originally due to Friedman–Russo [270]. Fact 7.1.20 is due to Harris [975], and can be found in [341, Corollary 2.4]. The nonzero hermitian ideal of the free special Jordan triple, discovered by Zel’manov [1134] and reviewed in §7.1.22, becomes crucial in our development, as was the case for the original Zel’manov’s classification of general prime nondegenerate Jordan triples [1133, 1134, 663]. Proposition 7.1.27 is due to Bunce and Chu [873]. As the reader can realize by looking at the proof of Proposition 7.1.32, that proposition depends heavily on the D’Amour–McCrimmon classification of Jordan pairs of Clifford type [921]. In our approach to the structure of prime JB∗ -triples of hermitian type, the JC∗ -triples of the form A12 , where A is a matricially decomposed C∗ -algebra, have become crucial. It is noteworthy that such JC∗ -triples are ‘more’ than closed subtriples of C∗ -algebras. Actually, if A is a matricially decomposed C∗ -algebra, then A12 is a ‘ternary ring of operators’, in the sense of Zettl [665]. This means that A12 is a norm-closed subspace of a C∗ -algebra closed under the associative triple product of the second kind xy∗ z. Of course, the JB∗ -triple structure of a ternary ring of operators is obtained by symmetrizing its associative triple product in the outer variables. Keeping in mind this fact, it turns out that, philosophically, Theorem 7.1.29 is close to the Zel’manov-type theorem for Jordan triples of hermitian type proved by D’Amour [920, Theorem 4.1]. However, we note that associative triple products arising in D’Amour’s theorem are of first kind and linear in the middle variable, whereas associative triple products of ternary rings of operators are of second kind and conjugate-linear in the middle variable. In the already quoted paper [449], Moreno and Rodr´ıguez prove relevant specializations of Theorem 7.1.29 when the hermitian prime JB∗ -triple X in that theorem is in fact a JBW ∗ -triple factor. Consequently, they obtain a classification theorem for JBW ∗ -triple factors, which, in its setting, refine Theorem 7.1.41. The precise formulation of this result is the following. Theorem 7.1.43 If X is a JBW ∗ -triple factor, then one of the following assertions holds for X: (i) X is either the type V or the type VI Cartan factor. (ii) X is a spin triple factor. (iii) There exist a W ∗ -factor A and a self-adjoint idempotent e ∈ A such that X = eA(1 − e). (iv) There exist a W ∗ -factor A, a self-adjoint idempotent e ∈ A, and a ∗-involution τ on A with e + eτ = 1 such that X = H(eAeτ , τ ). It is noteworthy that an apparently different classification of JBW ∗ -triple factors follows from the structure theory for general JBW ∗ -triples developed by Horn and Neher (see [330, 980]). According to that theory, every JBW ∗ -triple factor X which

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is not in cases (i) and (ii) of the above theorem must satisfy one of the following three assertions: (a) There exist a W ∗ -factor B and a self-adjoint idempotent p ∈ B such that X = pB. (b) There exists a W ∗ -factor B with ∗-involution π such that X = H(B, π ). (c) There exists a W ∗ -factor B with ∗-involution π such that X = S(B, π ). It is shown in [449] that the classification of JBW ∗ -triple factors given by Theorem 7.1.43 is in agreement with the one just reviewed. Indeed the authors of [449] prove that case (iii) in Theorem 7.1.43 leads to case (a), and that case (iv) in Theorem 7.1.43 leads to cases (b) or (c). The main results proved in [448, 449] for (complex) JB∗ -triples have been applied by Steptoe [1100] to show that a JC∗ -triple whose Cartan factor representations all have rank greater than two has a composition series in which successive quotients are isomorphic to inner ideals in a universally reversible JC∗ -algebra. Real Cartan factors are defined as those real JB∗ -triples whose JB∗ -triple complexification (cf. Definition 4.2.56) is a (complex) Cartan factor. They are real JBW ∗ triple factors (i.e. prime real JBW ∗ -triples). Their classification is due to Loos [772] in the finite dimensional case, and to Kaup [383] in the general case. They come in 12 different types: H C R H R H r,s O C(R) IR , VIO , VIC(R) . n,m , I2p,2q , In,n , IIn , II2p , IIIn , III2p , IVn , V , V

The notation has the property that, erasing the superscripts, we obtain the JB∗ -triple complexification of the given real Cartan factor. We note that the type VIO and VIC(R) real Cartan factors are nothing other than the JB-algebra H3 (O) (cf. Example 3.1.56) and the real JB∗ -algebra H3 (C(R)) ⊆ H3 (C(C)), respectively, regarded as real JB∗ -triples (cf. Example 4.2.51(b)). The type VO and VC(R) real Cartan factors are M12 (O) and M12 (C(R)) regarded as subtriples of the type VIO and VIC(R) real Cartan factors, respectively. By a generalized real Cartan factor we mean either a Cartan factor (regarded as a real JBW ∗ -triple) or a real Cartan factor. Generalized real Cartan factors can be intrinsically characterized as those real JBW ∗ -triple factors X such that BX∗ has extreme points (see [383, Lemma 4.5]). By a real spin triple factor we mean a JB∗ -triple whose JB∗ -triple complexification is a spin triple factor. Consequently, by a generalized real spin triple factor we mean either a spin triple factor (regarded as a real JBW ∗ -triple) or a real spin triple factor. In [448], the Zel’manovian classification of prime real JB∗ -triples is also obtained. The arguments are similar to those used for the complex case. Even the treatment of prime real JB∗ -triples of hermitian type is easier than the corresponding one for the complex case. (This should not surprise because real JB∗ -triples are real Jordan triples, and Zel’manov’s techniques apply verbatim.) As a result, the authors of [448] prove Theorem 7.1.44 in the next paragraph. Given a real C∗ -algebra A, the real C∗ -algebra of multipliers of A, M(A), is defined as in the complex case (see §2.2.16) and has similar properties (indeed, keep in mind Proposition 4.2.62 instead of Theorem 2.2.15).

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Theorem 7.1.44 If X is a prime real JB∗ -triple, then one of the following assertions holds for X: (i) X is the type V, VI, VO , VC(R) , VIO , or VIC(R) generalized real Cartan factor. (ii) X is a generalized real spin triple factor. (iii) There exists a prime real C∗ -algebra A such that X can be regarded as a closed subtriple of the real C∗ -algebra M(A) contained in H(M(A), ∗) and containing H(A, ∗). (iv) There exists a prime real C∗ -algebra A with ∗-involution τ such that X can be regarded as a closed subtriple of the real C∗ -algebra M(A) contained in S(M(A), τ ) ∩ H(M(A), ∗) and containing S(A, τ ) ∩ H(A, ∗). The classification of topologically simple real JB∗ -triples, as well as that of real JBW ∗ -triple factors, is also obtained in [448, 449]. Indeed, we have the following. Theorem 7.1.45 [448] If X is a topologically simple real JB∗ -triple, then one of the following assertions holds for X: (i) (ii) (iii) (iv)

X is the type V, VI, VO , VC(R) , VIO , or VIC(R) generalized real Cartan factor. X is a generalized real spin triple factor. There exists a topologically simple real C∗ -algebra A such that X = H(A, ∗). There exists a topologically simple real C∗ -algebra A with ∗-involution τ such that X = S(A, τ ) ∩ H(A, ∗).

Theorem 7.1.46 [449] If X is a real JBW ∗ -triple factor, then one of the following assertions holds for X: (i) (ii) (iii) (iv)

X is the type V, VI, VO , VC(R) , VIO , or VIC(R) generalized real Cartan factor. X is a generalized real spin triple factor. There exists a real W ∗ -factor A such that X = H(A, ∗). There exists a real W ∗ -factor A with ∗-involution τ such that X = S(A, τ ) ∩ H(A, ∗).

Of course, by a real W ∗ -factor we mean a prime real W ∗ -algebra. We note that, in its setting, the above theorem refines Theorem 7.1.44. To conclude this subsection, let us say that, with the help of Zel’manov’s prime theorem for Jordan triples [663] (see also [920]), Bouhya and Fern´andez [121] classified prime Banach Jordan-∗-triples with nonzero socle and without nilpotent elements. As a consequence, the authors of [121] rediscovered the Bunce–Chu structure theorem for compact JB∗ -triples [873]. 7.2 A survey on the analytic treatment of Zel’manov’s prime theorems Introduction In Subsections 6.1.4, 6.1.5, and 6.1.7 we have developed in detail the applications of Zel’manov’s prime theorem for Jordan algebras (cf. §6.1.132) to the classification of prime JB∗ -algebras (cf. Theorem 6.1.57), and, as a by-product, to the behaviour of prime non-commutative JB∗ -algebras (cf. Theorem 6.1.78).

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Later, in Section 7.1, we have applied the techniques in the proof of Zel’manov’s prime theorem for Jordan triples to obtain the classification of prime JB∗ -triples (cf. Theorem 7.1.41). In this section we survey in detail the remaining known applications of Zel’manov’s prime theorems on Jordan structures to the study of normed Jordan algebras and triples. The material we are reviewing has been partially surveyed in other places (see [145, 384, 450, 525, 532, 895, 939, 1027, 1028, 1062, 1063]). Now we are going to combine and update these references to offer a complete panoramic view of such material. In reviewing the results, we have not respected the chronology of their appearance. In fact we have preferred to assemble the different results according to their nature. Thus, we include in Subsections 7.2.1 and 7.2.2 structure theorems for complete normed J-primitive Jordan algebras (Theorem 7.2.8), J-primitive JB∗ algebras (Theorem 7.2.10), simple normed Jordan algebras (Theorem 7.2.11), and nondegenerately ultraprime complete normed Jordan complex algebras (Theorem 7.2.12). Subsection 7.2.3 deals with the so-called norm extension problem, which in its roots is crucially related to normed versions of Zel’manov’s prime theorems for Jordan structures. 7.2.1 Complete normed J-primitive Jordan algebras Along this section F will denote a field of characteristic different from two. §7.2.1 Let J be a Jordan algebra over F. In agreement with §4.4.71, a vector subspace M of J such that Um (J) ⊆ M for every m ∈ M is called an inner ideal of J. If in addition M is also a subalgebra of J, then it is called a strict inner ideal of J. Given x ∈ J, a strict inner ideal M of J is called x-modular when the following three conditions are satisfied: (i) U1−x (J) ⊆ M. (ii) U1−x,m (z) ∈ M for all z ∈ J1 and m ∈ M. (iii) x2 − x ∈ M. If there exists x ∈ J such that M is a maximal element (relative to the inclusion) of the family of all proper x-modular strict inner ideals of J, then we say that M is a maximal modular inner ideal of J. Now, in agreement with §4.4.73, J-primitive ideals of J are defined as the cores in J of maximal modular inner ideals of J, and J is said to be a J-primitive algebra if zero is a J-primitive ideal of J. The structure of J-primitive Jordan algebras is given by the next variant of Zel’manov’s prime theorem for Jordan algebras, already reviewed in §6.1.132. Such a variant has been obtained independently by Anquela, Montaner, and Cort´es [21], and by Skosyrskii [585]. Theorem 7.2.2 The J-primitive Jordan algebras over F are the following: (i) The finite-dimensional central simple exceptional Jordan algebras over a field extension of F (cf. Theorem 6.1.37 for the case F = R or C).

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(ii) The Jordan algebras of a nondegenerate symmetric bilinear form on a vector space X over a field extension  of F with dim (X) ≥ 2 (cf. §6.1.52). (iii) The subalgebras of Q(A)sym containing A as an ideal, where A is a primitive associative algebra over F, and Q(A) stands for the Martindale algebra of symmetric quotients of A (cf. §6.1.126). (iv) The subalgebras of Q(A)sym contained in H(Q(A), ∗) and containing H(A, ∗) as an ideal, where A is a primitive associative algebra over F with a linear algebra involution ∗. The structure of complete normed J-primitive Jordan algebras, obtained by Moreno and the authors [145, 146], and which will be reviewed in what follows, consists of a case-by-case Banach treatment of the above theorem. We begin by considering cases (i), (ii), and (iii) in Theorem 7.2.2, whose normed study does not need any Zel’manovian technique. Actually, the normed treatment of case (i) in Theorem 7.2.2 reduces to conbining Theorem 6.1.37 with the next immediate consequence of the Gelfand–Mazur theorem. Proposition 7.2.3 Let J be a unital algebra over a field extension F of K. If J is a normed algebra over K, then F = C if K = C, and F = R or C if K = R. Now we pass to deal with the normed treatment of Jordan algebras of a bilinear form. If (X,  · ) is a normed space over K and if f is a continuous symmetric bilinear form on X with f  ≤ 1, then the Jordan algebra J(X, f ) = K1 ⊕ X (cf. §6.1.52) with norm  ·  defined by α1 + x := |α| + x becomes a normed algebra called the normed Jordan algebra of the continuous symmetric bilinear form f on the normed vector space (X,  · ). Clearly, such a normed algebra is complete if and only if X is a Banach space. The following result (which is nothing other than a reformulation of the commutative particularization of Proposition 3.5.4) asserts that, up to a topological isomorphism, these algebras are the unique Jordan algebras of a symmetric bilinear form which are normed algebras. Proposition 7.2.4 Let (J,  · ) be a normed Jordan algebra over K, and suppose that J = J(X, f ) for a symmetric bilinear form f on a vector space X over a field extension F of K. Then F = C if K = C, and F = R or C if K = R. Moreover, there exists a norm ||| · ||| on X such that X is a normed space over F, f is continuous with ||| f ||| ≤ 1, and the norm  ·  of J is equivalent to the norm given by α1 + x → |α| + ||| x |||. The Banach treatment of case (iii) in Theorem 7.2.2 was made in [892] and [145] for F = C and F = R, respectively, giving rise to the next proposition. Proposition 7.2.5 Let (J,  · ) a complete normed Jordan algebra over K, and suppose that there exists a primitive associative algebra A over K such that J is a subalgebra of Q(A)sym containing A as an ideal. Then there exists a Banach space X over K and an injective algebra homomorphism  from Q(A) into the complete normed associative algebra BL(X) such that (A) acts irreducibly on X and the restriction of  to J is continuous.

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The Banach treatment of case (iv) of Theorem 7.2.2 needs deep Zel’manovian techniques, which are summarized in what follows. By an involutive algebra over F we mean a (possibly non-associative) algebra over F endowed with a linear algebra involution. From now on, X will stand for a countably infinite set of indeterminates. We denote by A (X) the (non-unital) free associative algebra over F on X, and by J (X) the free special Jordan algebra over F on X, namely the subalgebra of A (X)sym generated by X. Intuitively, the elements of J (X), called Jordan polynomials, are those elements in A (X) which can be obtained from that of X by a finite process of taking sums, Jordan products a • b, and products by elements of F. If ∗ denotes the unique linear algebra involution on A (X) fixing the elements of X, we clearly have J (X) ⊆ H(A (X), ∗). For every element a in any involutive algebra (A, ∗) over F, write {a} := 12 (a + a∗ ) . Following [437], we say that a Jordan polynomial p (involving m indeterminates, say x1 , ..., xm ) is an imbedded pentad eater if there exist a natural number k and Jordan polynomials pij (1 ≤ i ≤ k, 1 ≤ j ≤ 3) involving m + 4 indeterminates such that, for all natural numbers r and s and all z1 , . . . , zr , y1 , . . . , y4 , w1 , . . . , ws in X, we have in A (X) {z1 . . . zr y1 . . . y4 pw1 . . . ws } =

k  {z1 . . . zr pi1 pi2 pi3 w1 . . . ws }, i=1

where, to be brief, in the right-hand side of the above equality we have written pij instead of pij (y1 , . . . , y4 , x1 , . . . , xm ). The set of all imbedded pentad eaters is a subspace of J (X), and, in fact, an ideal of J (X) [834, Theorem 2.7], which is denoted by I5 . For any special Jordan algebra J, I5 (J) will mean the ideal of J of all valuations on J of the polynomials in I5 . The celebrated Zel’manov prime theorem for Jordan algebras (see §6.1.132) asserts that, if J is a prime nondegenerate Jordan algebra over F, and if J is neither an Albert ring nor a central order in a Jordan algebra of a bilinear form, then there exists a ∗-prime involutive associative algebra (A, ∗) over F, which is generated by H(A, ∗) as an algebra, such that J can be seen as a subalgebra of Q(A)sym contained in H(Q(A), ∗) and containing H(A, ∗) as an ideal. The proof of Zel’manov’s theorem shows that the involutive algebra (A, ∗) above can be chosen with the additional property that I5 (J) = H(A, ∗) (see [437] for details). Precisely, thanks to the above equality, the authors of [146] were able to prove the following germinal normed version of the Zel’manov prime theorem. Theorem 7.2.6 Let (J,  · ) be a prime nondegenerate normed Jordan algebra over K, and suppose that J is neither an Albert ring nor a central order in the Jordan algebra of a bilinear form. Then there exists a normed ∗-prime involutive associative algebra (A, ∗, ||| · |||) over K, which is generated by H(A, ∗) as an algebra, such that J can be seen as a subalgebra of Q(A)sym contained in H(Q(A), ∗) and containing H(A, ∗) as an ideal, and the following properties are satisfied: (i) h ≤ ||| h ||| for every h ∈ H(A, ∗).

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(ii) If a is in A, and satisfies aJ + Ja ⊆ A, then the mappings x → ax and x → xa from (J,  · ) into (A, ||| · |||) are continuous, (iii) ||| a∗ ||| = ||| a ||| for every a ∈ A. ˆ ∗) denotes the completion of (A, ∗) relative to the norm ||| · |||, then every (iv) If (A, ˆ ∗). nonzero ∗-invariant ideal of Aˆ meets H(A, As stated in Theorem 7.2.7, if the norm  ·  on the Jordan algebra J above is complete, then a better result holds. The proof of such a result given in [146] involves Theorem 7.2.6 and a ‘very Zel’manovian’ theorem on extensions of Jordan homomorphisms due to McCrimmon [1018, Theorem 2.2]. Theorem 7.2.7 Let (J,  · ) be a prime nondegenerate complete normed Jordan algebra over K, and suppose that J is neither an Albert ring nor a central order in the Jordan algebra of a bilinear form. Then there exists a normed ∗-prime involutive associative algebra (A, ∗, ||| · |||) over K, which is generated by H(A, ∗) as an algebra, such that J can be seen as a subalgebra of Q(A)sym contained in H(Q(A), ∗) and containing H(A, ∗) as an ideal, and the following properties are satisfied: (i) If a is in A, and satisfies aJ + Ja ⊆ A, then the mappings x → ax and x → xa from (J,  · ) into (A, ||| · |||) are continuous. (ii) ||| a∗ ||| = ||| a ||| for every a ∈ A. ˆ ∗, ||| · |||) denotes the completion of (A, ∗, ||| · |||), then the inclusion (iii) If (A, A ⊆ Q(A) extends in a unique way to an injective algebra ∗-homomorphism ˆ ∗) into J and satisfying h ≤ ||| h ||| for all h in Aˆ → Q(A) mapping H(A, ˆ ∗). H(A, (iv) Every nonzero ∗-ideal of Aˆ meets H(A, ∗). Theorems 7.2.6 and 7.2.7 make no direct reference to J-primitive Jordan algebras. However, as matter of fact, J-primitive Jordan algebras are particular examples of prime nondegenerate Jordan algebras, and the authors of [145, 146] were able to obtain the structure of complete normed J-primitive Jordan algebras (as we are presenting here) only by passing through the germinal complete normed version of Zel’manov’s prime theorem provided by Theorem 7.2.7. All the remaining material previously included in this subsection is also involved in the proof of the following. Theorem 7.2.8 A complete normed Jordan algebra J over K is J-primitive (if and) only if one of the following assertions holds: (i) J = H3 (C(C)) if K = C, and J = H3 (C(C)), J = H3 (C(R)), J = H3 (O), or J = H3 (O, diag{1, −1, 1}) if K = R. (ii) J is the complete normed Jordan algebra of a continuous nondegenerate symmetric bilinear form on a Banach space over F with dimF (X) ≥ 2, where F = C if K = C, and F = R or C if K = R. (iii) There exist a Banach space X over K and an associative subalgebra A of BL(X) acting irreducibly on X such that J can be seen as a subalgebra of BL(X)sym containing A as an ideal, and the inclusion J → BL(X) is continuous.

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(iv) There exist a Banach space X over K and an associative subalgebra A of BL(X) acting irreducibly on X such that J can be seen as a subalgebra of BL(X)sym , the inclusion J → BL(X) is continuous, the identity mapping on J extends to a linear algebra involution ∗ on the subalgebra B of BL(X) generated by J, A is a ∗-invariant subset of B, H(A, ∗) is an ideal of J, and A is generated by H(A, ∗) as an algebra. In relation to the germinal normed versions of Zel’manov’s prime theorem for Jordan algebras given by Theorems 7.2.6 and 7.2.7, it would be interesting to try a description of those (complete) normed prime nondegenerate Jordan algebras which are either Albert rings or central orders in Jordan algebras of bilinear forms. The tensor product D ⊗ H3 (C(C)), where D denotes the disc algebra (cf. Example 2.9.67), becomes an example of an algebra in such a (complete) situation. Remark 7.2.9 (a) As we already commented in p. 338 of Volume 1, Theorem 7.2.8 has been successfully applied by Villena [625] to extend to the setting of complete normed Jordan algebras the celebrated Johnson–Sinclair theorem [355] asserting the automatic continuity of derivations on semisimple complete normed associative algebras. Villena’s result has been generalized by Boudi, Fern´andez, Marhnine, and Zarhouti in the setting of (linear) derivations and additive derivations of complete normed Jordan triples and pairs (see [945, 865]). (b) Let J be a complete normed Jordan complex algebra, and let D : J → J be a (possibly discontinuous) derivation on J. According to Mathieu [1014], the following three theorems are the main results of the Bresar–Villena paper [869]. Theorem 3.1: Suppose that [D(a), b, c] = 0 for some a ∈ J and all b, c ∈ J; then D(a) is quasinilpotent. Theorem 4.6: Suppose that [D(a), D(b), D(c)] = 0 for all a, b, c ∈ J; then D2 (a) is quasi-nilpotent for every a ∈ J. Theorem 4.14: Suppose that [D(a), b, a] = 0 for all a, b ∈ J; then D(J) is contained in J-Rad(J). The latter result is the hardest to obtain; in fact, it relies on Theorem 7.2.8. Theorem 4.14 generalizes the corresponding result for derivations of complete normed associative algebras obtained by Mathieu and Runde [1015]. The authors of [869] also formulate an analogue of the well-known non-commutative (associative) Singer–Wermer conjecture (see p. 391 of Volume 1) for derivations of complete normed Jordan algebras, and obtain a number of equivalent reformulations. The Zel’manovian classification of prime JB∗ -algebras, given by Theorem 6.1.57, suggests the possibility of describing ‘geometrically primitive JB∗ -algebras’ (i.e. JB∗ -algebras having an injective type I factor representation) by selecting among the prime JB∗ -algebras listed there those which are geometrically primitive. With the help of (the associative forerunner of) Theorem 6.2.30, this careful selection has been done by Fern´andez, Garc´ıa, and Rodr´ıguez [255]. But, looking at the list of geometrically primitive JB∗ -algebras provided by [255, Theorem 4.2], and regarding Theorem 6.1.57 under the light of the tools developed in [21] for the proof of Theorem 7.2.2, we realize that, as asserted in the note added in proof to

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[255], geometrically primitive JB∗ -algebras are no other thing than J-primitive JB∗ algebras. Therefore we obtain the next result, first formulated in [525, Theorem F.9]. Theorem 7.2.10 The J-primitive JB∗ -algebras are the following: (i) The JB∗ -algebra H3 (C(C)). (ii) The simple quadratic JB∗ -algebras. (iii) The closed ∗-subalgebras of M(A)sym containing A, where A is any primitive C∗ -algebra. (iv) The closed ∗-subalgebras of M(A)sym contained in H(M(A), τ ) and containing H(A, τ ), where A is any primitive C∗ -algebra with ∗-involution τ . With the well-known result of Dixmier [926] that separable prime C∗ -algebras are primitive, the above theorem implies easily that separable prime JB∗ -algebras are J-primitive (see [255, Corollary 4.3] for details). Since Dixmier’s result just quoted, the question whether every prime C∗ -algebra is primitive remained open for many years. This question was negatively answered by Weaver [1120] (see also Crabb [910] for a simplified construction). Of course, if A is a prime non-primitive C∗ -algebra, then Asym is a prime non-J-primitive JB∗ -algebra.

7.2.2 Strong-versus-light normed versions of the Zel’manov prime theorem The classification theorem for complete normed J-primitive Jordan algebras we have just reviewed becomes an example of the so-called ‘light’ normed versions of Zel’manov’s prime theorem for Jordan algebras (cf. §6.1.132 again). This means that the topology of the norm of the normed Jordan algebra J in cases (iii) and (iv) of the theorem does not arise as the restriction to J of the topology of some algebra norm on its natural associative envelope (in our present case, the subalgebra of BL(X) generated by J). There are in the literature examples of ‘strong’ (i.e. free of the pathology just described) versions of the Zel’manov prime theorem, like the following. Theorem 7.2.11 [151] Up to bicontinuous isomorphisms, the unital simple (complete) normed Jordan algebras over K are the following: (i) H3 (C(C)) if K = C, and H3 (C(C)), H3 (C(R)), H3 (O), and H3 (O, diag{1, −1, 1}) if K = R. (ii) The (complete) normed Jordan algebras J(X, f ) of a continuous nondegenerate symmetric bilinear form f on a (complete) normed vector space X over F with dimF (X) ≥ 2, where F = C if K = C, and F = R or C if K = R. (iii) The Jordan algebras of the form Asym , where A is a unital simple (complete) normed associative algebra over K. (iv) The Jordan algebras of the form H(A, ∗), where A is a unital simple (complete) normed associative algebra over K, and ∗ is an isometric linear algebra involution on A.

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In fact Theorem 7.2.11 was proved by the authors in [151] only for the case K = C, but, with the help of Theorem 6.1.37(ii) and Propositions 7.2.3 and 7.2.4, the arguments in the proof remain valid for the case K = R. Another strong normed version of the Zel’manov prime theorem is the one proved by the authors [152] for nondegenerately ultraprime complete normed Jordan algebras. A normed Jordan algebra J is said to be nondegenerately ultraprime if there exists a countably incomplete ultrafilter U on a suitable set such that the corresponding normed ultrapower JU is prime and nondegenerate. Since JB∗ -algebras are nondegenerate Jordan algebras, and every ultrapower of a JB∗ -algebra is a JB∗ -algebra (cf. §6.1.73), and prime JB∗ -algebras are ultraprime (cf. Proposition 6.1.74), prime JB∗ -algebras become examples of nondegenerately ultraprime normed Jordan algebras. With the Beidar–Mikhalev–Slin’ko characterization of prime nondegenerate Jordan algebras [855], it can be proved easily that a normed Jordan algebra J is nondegenerately ultraprime if and only if there exists k > 0 such that Ux,y  ≥ kxy for all x, y in J. As a consequence, all normed ultrapowers of a nondegenerately ultraprime normed Jordan algebra are prime and nondegenerate. Following ideas by Mathieu in [428] and [1013], in [152] we introduced ultraτ -prime normed associative algebras with continuous linear algebra involution τ , and characterized them (without any reference to ultrapowers) as those normed associative algebras A with continuous linear algebra involution τ satisfying max{Ma,b , Maτ ,b } ≥ ka b for some fixed k > 0 and all a, b ∈ A, where as usual Ma,b (c) := acb for every c ∈ A. For such an ultra-τ -prime normed associative algebra (A, τ ), a large τ -invariant subalgebra Qb (A) of its symmetric Martindale algebra of quotients Q(A) (cf. Proposition 6.1.130) can be converted in an ultra-τ -prime normed algebra in such a way that the natural inclusion A → Qb (A) becomes a topological embedding. Then subalgebras of Qb (A)sym contained in H(Qb (A), τ ) and containing H(A, τ ) are examples of nondegenerately ultraprime normed Jordan algebras. Now the main result in [152] reads as follows. Theorem 7.2.12 Up to bicontinous algebra isomorphisms, the nondegenerately ultraprime complete normed Jordan complex algebras are the following: (i) H3 (C(C)). (ii) The complete normed Jordan algebras J(X, f ) of a continuous nondegenerate symmetric bilinear form f on a complex Banach space X with dim(X) ≥ 2 and such that the natural embedding x → f (·, x) from X into its dual is topological. (iii) The closed subalgebras of Qb (A)sym contained in H(Qb (A), τ ) and containing H(A, τ ) as an ideal, where A is an ultra-τ -prime complete normed complex algebra with continuous linear algebra involution τ such that H(A, τ ) generates A as a normed algebra. The proof of the above theorem is very long and difficult. Indeed, it uses Corollary 6.1.65 and Theorem 6.1.132, as well as Theorem 7.2.19 and Proposition 7.2.21,

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which will be reviewed in the next subsection. The proof of Theorem 7.2.12 involves also the next three propositions. Proposition 7.2.13 [824] Let X, Y, and Z be normed spaces over K, and let F : X → Y and G : X → Z be bounded linear operators. Then 1 sup{F(x)G(x) : x ∈ BX } ≥ FG. 4 Proposition 7.2.14 [152] Let A be a unital complete normed prime associative complex algebra with a (possibly discontinuous) linear algebra involution τ . Then for every s ∈ S(A, τ ) satisfying 1 − s2  < 1 we have  2 ? 2 M1+s,1−s  ≥ 1 + 1 − 1 − s  . Proposition 7.2.15 [894] The centre of a nondegenerate Jordan algebra J coincides with the set {x ∈ J : 2Ux•y = Ux Uy + Uy Ux for every y ∈ J}. Proposition 7.2.15 was proved by the authors involving strong structural methods but, according to an old private communication of Y. A. Medvedev, the proof can be considerably liberated of its structural nature. To conclude our review of Theorem 7.2.12, let us remark that it would be interesting to know if the algebras arising in case (iii) of that theorem fall in one of the following two cases: (1) The closed subalgebras of Qb (A)sym containing A as an ideal, where A is an ultraprime complete normed associative complex algebra. (Here Qb (A) denotes the Mathieu’s symmetric algebra of bounded quotients of such an algebra A [1013].) (2) The closed subalgebras of Qb (A)sym contained in H(Qb (A), τ ) and containing H(A, τ ) as an ideal, where A is an ultraprime complete normed associative complex algebra with continuous linear algebra involution τ . This problem seems to be an essentially associative problem, namely if every ultraτ -prime complete normed associative prime complex algebra A, with continuous involution τ , is ultraprime. Even a very particular case of this question is also open, namely, if every ultra-τ -prime complete normed associative simple complex algebra, with continuous involution τ , is ultraprime. Despite the examples of strong normed versions of Zel’manov’s prime theorem provided by Theorems 7.2.11 and 7.2.12 in some particular settings, a general strong normed version of Zel’manov’s theorem cannot be expected. Nor even for normed simple Jordan algebras without a unit (see [893, Remark 4] for details). This is a consequence of the next theorem, proved by Moreno and the authors [893]. We denote by M∞ (K) the simple associative algebra of all countably infinite matrices over K with a finite number of nonzero entries.

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Theorem 7.2.16 There exists an algebra norm · on the Jordan algebra M∞ (K)sym which is not equivalent to any algebra norm on M∞ (K). More precisely, there exists a linear algebra involution ∗ on M∞ (K) such that there is no algebra norm ||| · ||| on M∞ (K) such that the restrictions of  ·  and ||| · ||| to H(M∞ (K), ∗) are equivalent. It is shown in [893, Section 3] that, for a suitable choice of the pathological norm  ·  above, the completions of the normed Jordan algebras (M∞ (K)sym ,  · ) and (H(M∞ (K), ∗),  · ) are complete normed J-primitive Jordan algebras over K whose topologies cannot be obtained by restricting to them the topology of any algebra norm in their natural associative envelopes. In other words, a strong normed version of Theorem 7.2.8 cannot be expected. Results from Proposition 7.2.5 to Theorem 7.2.8, as well as Theorem 7.2.16, can be found in Moreno’s PhD thesis [1174], written under the advise of the authors. Strong normed versions of the Zel’manov prime theorem depend heavily on the so-called ‘norm extension problem’, which will be considered in some detail in the next subsection. As a matter of fact, one of the main results in [538] (see Theorem 7.2.18) essentially links the norm extension problem with the continuity of a typical non-Jordan polynomial (namely, the ‘tetrad’). This invited Moreno, Zel’manov, and the authors [147] to apply the techniques in the proof of Theorem 7.2.16 to obtain analytical characterizations of Jordan polynomials, a question previously tried out by Arens and Goldberg [838]. In this direction the authors of [147] proved the following result. Theorem 7.2.17 An associative polynomial p over K is a Jordan polynomial if and only if, for every algebra norm  ·  on the Jordan algebra M∞ (K)sym , the action of p on M∞ (K) is  · -continuous. Actually a better result holds. Indeed, according to Moreno [447], there exists an algebra norm  ·  on the Jordan algebra M∞ (K)sym such that Jordan polynomials over K are precisely those associative polynomials which act  · -continuously on M∞ (K). In [1082] Shestakov introduces the notion of a normed special Jordan algebra over K as a Jordan algebra J over K which is isomorphic and homeomorphic to a subalgebra of Asym for some normed associative algebra A over K, and suggests that Moreno’s result reviewed above could give some evidence that there exist (algebraically) special normed Jordan algebras over K which are not normed special. 7.2.3 The norm extension problem Throughout this subsection (C, ∗) will stand for an involutive algebra over K. As usual, we will consider H(C, ∗) as a subalgebra of Csym . Obviously, if  ·  is an algebra norm on C, then its restriction to H(C, ∗) is an algebra norm on H(C, ∗). The so-called norm extension problem (NEP) is the following. NEP Given an involutive algebra (C, ∗) over K, and an algebra norm  ·  on H(C, ∗), is there an algebra norm on C whose restriction to H(C, ∗) is equivalent to  · ?

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It is easy to see that, when we are able to answer affirmatively the above question, then actually we can choose the algebra norm on C extending the topology of the norm  ·  on H(C, ∗) in such a way that ∗ becomes continuous. It is also not difficult to realize that, in studying the NEP, the additional assumption that the algebra C is a ∗-tight envelope of H(C, ∗) (i.e. C is generated by H(C, ∗) as an algebra, and every nonzero ∗-invariant ideal of C has nonzero intersection with H(C, ∗)) is not too restrictive. Indeed, in any case one can find an involutive algebra (D, τ ) such that (D, τ ) is a τ -tight envelope of H(D, τ ), and the algebras H(D, τ ) and H(C, ∗) are isomorphic. Since, as we commented in the previous subsection, strong normed versions of Zel’manov’s prime theorem crucially depends on the norm extension problem, in the present subsection we are reviewing in detail the main results about that problem. Suppose that the algebra C is associative and that the NEP has an affirmative answer. Then, clearly, the tetrad mapping 1 (x, y, z, t) → {xyzt} := (xyzt + tzyx) 2 from H(C, ∗) × H(C, ∗) × H(C, ∗) × H(C, ∗) to H(C, ∗) is  · -continuous. The following partial converse of the fact just quoted was proved by Rodr´ıguez, Slinko, and Zel’manov in [538, Corollary 1]. Theorem 7.2.18 Suppose that C is associative and a ∗-tight envelope of H(C, ∗). Then the NEP has an affirmative answer if (and only if) the tetrad mapping is  · continuous. The two following positive results on the norm extension problem involve in their proof the general criterium given by the above theorem. Theorem 7.2.19 [538, Theorem 2] Suppose that C is associative and a ∗-tight envelope of H(C, ∗), that H(C, ∗) is semiprime, and that the norm  ·  on H(C, ∗) is complete. Then the NEP has an affirmative answer. Theorem 7.2.20 [893, Theorem 4] Suppose that C is associative and a ∗-tight envelope of H(C, ∗), and that H(C, ∗) is unital and simple. Then the NEP has an affirmative answer. The last theorem is an example of a ‘global’ affirmative answer to the NEP. This means that the NEP answers affirmatively without any condition on the algebra norm  ·  on H(C, ∗), and hence, under suitable algebraic assumptions (for instance, if C is associative and a ∗-tight envelope of H(C, ∗), and H(C, ∗) is unital and simple), every algebra norm on H(C, ∗) can be extended to an algebra norm on C. According to Theorem 7.2.16, for C = M∞ (K) with a suitable linear algebra involution ∗ there exists an algebra norm  ·  on H(C, ∗) such that the NEP has a negative answer. This shows that neither the assumption of completeness of the norm  ·  in Theorem 7.2.19 nor that the existence of a unit for H(C, ∗) in Theorem 7.2.20 can be removed. Another affirmative answer to the NEP is provided by the following.

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Proposition 7.2.21 [538] Suppose that C is associative and a ∗-tight envelope of H(C, ∗), and that there exists K > 0 such that the algebra norm  ·  of H(C, ∗) satisfies s2  ≤ KDs 2 for those s ∈ S(C, ∗) for which the mapping Ds : h → [s, h] from H(C, ∗) to H(C, ∗) is  · -continuous. Then the NEP has an affirmative answer. More precisely, Ds is  · -continuous for every s ∈ S(C, ∗), and there exists ρ > 0 (depending only on K) such that, by defining ||| h + s ||| := ρ(h + Ds ) for all h ∈ H(C, ∗) and s ∈ S(C, ∗), ||| · ||| becomes an algebra norm on C. Although, in relation to the analytic treatment of Zel’manov’s prime theorem, the norm extension problem is only interesting in the case that the algebra C is associative, in some positive results on that problem the associativity of C is not needed. This happens by the first time in the paper of Moreno and Rodr´ıguez [1029]. Let (B, ∗) be a unital involutive algebra over K. If b1 , . . . , bn are ∗-invariant invertible elements in the nucleus of B (cf. Definition 6.2.7), and if we set d := diag{b1 , . . . , bn }, then the operator ∗ on Mn (B) given by (bij )∗ := d−1 (b∗ji )d is a linear algebra involution on Mn (B). Involutions on Mn (B) defined in this way are called canonical involutions. The standard involution on Mn (B) is nothing other than the canonical involution corresponding to the identity diagonal matrix. In its easiest form, the main result of [1029] reads as follows. Theorem 7.2.22 [1029, Theorem 3.3] The NEP has an affirmative answer if (C, ∗) is of the form (Mn (B), ∗), for some natural number n ≥ 3 and some (possibly nonassociative) unital involutive algebra (B, ∗) over K, and if the linear algebra involution ∗ on Mn (B) is a canonical involution. Moreover, for (C, ∗) as above, the following two assertions hold: (i) Two algebra norms on C are equivalent whenever they make ∗ continuous and their restriction to H(C, ∗) are equivalents (‘uniqueness of the extended norm topology’). (ii) If the algebra norm  ·  on H(C, ∗) is complete, then the essentially unique algebra norm on C making ∗ continuous and generating on H(C, ∗) the topology of  ·  is complete too (‘complete-to-complete extension property’). Let (A, ∗) be a finite dimensional ∗-simple associative complex involutive algebra, and let n denote the degree of H(A, ∗) (cf. §2.6.28). Then, according to [754, pp. 208–9], (A, ∗) is isomorphic to (Mn (D), ∗), where D is a complex composition associative algebra and ∗ on Mn (D) denotes the standard involution relative to the Cayley involution on D (cf. Theorem 6.1.31). Now, let us consider in addition a unital involutive complex algebra (B, ∗). Then we have A ⊗ B = Mn (D) ⊗ B = Mn (C) ⊗ D ⊗ B = Mn (D ⊗ B), and the tensor involution on A ⊗ B is nothing other than the standard involution on Mn (D ⊗ B) relative to the tensor involution on D ⊗ B. It follows from Theorem 7.2.22 (with D ⊗ B instead of B) that, if n ≥ 3, then the NEP has a global affirmative answer if C is of the form A ⊗ B, for (A, ∗) and (B, ∗) involutive algebras satisfying the

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properties just required, and if the linear algebra involution on C is the tensor involution. With some additional effort (see the proof of [1029, Theorem 3.5] for details), the result we have just shown for the case K = C remains essentially true when K = R. In this way we have the following abstract version of Theorem 7.2.22 (note that, if (A, ∗) is a unital ∗-simple involutive associative algebra over K, then H(A, ∗) is a unital simple Jordan algebra, and therefore the centre of H(A, ∗) is a field). Theorem 7.2.23 [1029, Theorem 3.5] The NEP has an affirmative answer whenever (C, ∗) is of the form (A, ∗) ⊗ (B, ∗), where (A, ∗) is a finite dimensional ∗-simple involutive associative algebra over K whose hermitian part H(A, ∗) is of degree ≥ 3 over its centre, (B, ∗) is a (possibly non-associative) unital involutive algebra over K, and the linear algebra involution on C is the tensor involution. Moreover, for such an involutive algebra (C, ∗), we enjoy the uniqueness of the extended norm topology and the complete-to-complete extension property. We note that the affirmative answers to the NEP given by Theorems 7.2.22 and 7.2.23 are of global type. The necessity of the assumptions in Theorems 7.2.22 and 7.2.23 are fully discussed in [1029]. Actually, concerning the hypothesis n ≥ 3 in Theorem 7.2.22 we have the next anti-theorem (see [1029, Theorem 4.3] or [1028]). Theorem 7.2.24 Let X be an arbitrary infinite-dimensional normed space over K. Then there exists a unital involutive associative algebra (B, ∗) over K satisfying: (i) X = H(M2 (B), ∗), as vector spaces. (ii) Up to multiplication by a suitable positive number if necessary, the norm of X becomes an algebra norm on H(M2 (B), ∗). (iii) There is no algebra norm on M2 (B) whose restriction to H(M2 (B), ∗) is equivalent to the norm of X. (iv) M2 (B) is a ∗-tight envelope of H(M2 (B), ∗). We note that the above anti-theorem shows in addition that the assumption of semiprimeness of H(C, ∗) in Theorem 7.2.19 cannot be removed. One of the tools in the proof of Theorem 7.2.24 is the following. Lemma 7.2.25 On every infinite-dimensional normed space over K there is a discontinuous anticommutative associative product. As a consequence, every infinite-dimensional normed space (X,  · ) over K can be converted into an associative algebra (say A) in such a way that  ·  becomes an algebra norm on the Jordan algebra Asym , but  ·  cannot be equivalent to any algebra norm on A. Lemma 7.2.25 shows that the assumption of semiprimeness cannot be removed in the following. Theorem 7.2.26 Let A be an alternative semiprime algebra over K, and let  ·  be a complete algebra norm on the Jordan algebra Asym . Then  ·  is equivalent to an algebra norm on A.

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Lemma 7.2.25 and Theorem 7.2.26 are due to Rodr´ıguez [1059]. More available references for the proof of the lemma (respectively, of the theorem) are [1029, Lemma 4.2] or [1028] (respectively, [518, Proposition 3]). We note that, if C = A ⊕ A(0) for some algebra A over K, and if ∗ stands for the exchange involution on C, then H(C, ∗) ≡ Asym as algebras over K, and therefore the NEP converts into the question whether algebra norms on Asym are equivalent to algebra norms on A. In this way Lemma 7.2.25 and Theorem 7.2.26 become respectively negative and affirmative answers to the NEP. As carefully reviewed in [1027, Section 4], other results closely related to Theorem 7.2.26 are Theorems 3 and 6, Corollary 1, and Proposition 1 of [893], as well as Corollaries 6.6, 6.8, and 6.11 of [1029]. A relevant part of Theorem 7.2.23 remains true if we relax the assumption of finite dimensionality for the ∗-simple involutive associative algebra (A, ∗) to the mere existence of a unit for A, but we assume that the algebra B is associative. This is proved in [1029, Theorem 5.5], by applying Zel’manovian techniques, and precisely reads as follows. Theorem 7.2.27 The NEP has a global affirmative answer whenever (C, ∗) is of the form (A, ∗) ⊗ (B, ∗), where (A, ∗) is a unital ∗-simple involutive associative algebra over K whose degree over its centre is ≥ 3, (B, ∗) is a unital involutive associative algebra over K, and the linear algebra involution on C is the tensor involution. In obtaining the above theorem, the following purely algebraic fact becomes crucial. If (A, ∗) is a ∗-simple involutive associative algebra whose hermitian part is of degree ≥ 2 over its centroid, and if (B, ∗) is a unital involutive algebra, then the involutive algebra (A, ∗) ⊗ (B, ∗) is a ∗-tight envelope of its hermitian part [1029, Proposition 5.1]. The uniqueness of the extended norm topology and the complete-to-complete extension property, first discovered in the setting of Theorem 7.2.22, was later systematically considered by Rodr´ıguez and Velasco [539, 1064]. Suppose that the involutive algebra (C, ∗) is associative and a ∗-tight envelope of H(C, ∗), that H(C, ∗) is unital and simple, and that the algebra norm  ·  on H(C, ∗) is complete. Then, according to either Theorem 7.2.19 or Theorem 7.2.20, the NEP has an affirmative answer. As the main result, it is shown in [539] that, under the above assumptions, the topologies of all algebra norms on C making ∗ continuous and extending the topology of  ·  on H(C, ∗) coincide, and are complete. This result is applied in [539, Section 3] to simplify the original proof of the complete case of Theorem 7.2.11. Now suppose that (C, ∗) is of the form (A, ∗) ⊗ (B, ∗), where (A, ∗) is a finite dimensional ∗-simple involutive associative algebra over K whose hermitian part H(A, ∗) is of degree ≥ 2 over its centre, and (B, ∗) is a unital involutive (possibly non-associative) algebra over K. Then, according to Theorem 7.2.24, the NEP need not have a global affirmative answer. However, if for some particular algebra norm  ·  on H(C, ∗) the NEP has an affirmative answer, then, as in Theorem 7.2.23, we

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enjoy the uniqueness of the extended norm topology and the complete-to-complete extension property [1064]. The uniqueness of the extended norm topology is applied in [539, 1064] (see also [1029, Theorem 5.3]) to derive the continuity of algebra homomorphisms and derivations from certain normed involutive algebras into arbitrary normed algebras and modules, respectively, from the continuity of such isomorphisms and derivations on the hermitian part. As a sample, we have the following. Theorem 7.2.28 [1064] Let (C, ∗) be a normed involutive algebra over K which, algebraically regarded, is of the form (A, ∗) ⊗ (B, ∗) for some central simple finitedimensional involutive algebra (A, ∗) over K whose involution is different from ±IA and some unital involutive algebra (B, ∗) over K. Then we have: (i) Algebra homomorphisms from C to arbitrary normed algebras over K are continuous whenever they are continuous on H(C, ∗). (ii) Derivations from C to arbitrary normed C-bimodules over K (cf. §4.6.3) are continuous whenever they are continuous on H(C, ∗). For some particular choices of the involutive algebra (A, ∗) above, there exists a positive number K (only depending on (C, ∗)) such that, for every continuous algebra homomorphism ϕ from C to a normed algebra over K, and for every continuous derivation δ from C to a normed C-bimodule over K, we have ϕ ≤ Kϕ|H(C,∗) 3 and δ ≤ Kδ|H(C,∗)  (see [1064, Corollary 4]). Now, let us consider the NEP in the case that H(C, ∗) is finite dimensional. If C is associative, and if H(C, ∗) generates C as an algebra, then it is well know that the equality C = H(C, ∗)+ lin(H(C, ∗)H(C, ∗))+ lin(H(C, ∗)H(C, ∗)H(C, ∗)) holds. Therefore, if in addition dim H(C, ∗) < ∞, then dim C < ∞, and hence the NEP has an obvious and global affirmative answer. If C is not associative, a similar result cannot be expected. This is a consequence of the following anti-theorem, due to Moreno. Theorem 7.2.29 [1027, Theorem 3] Let X be an arbitrary vector space over K with dim X ≥ 2. Then there exists an involutive algebra (C, ∗) over K satisfying the following conditions: (i) X = H(C, ∗) as vector spaces. (ii) C is a ∗-tight envelope of H(C, ∗). (iii) The product of H(C, ∗) is zero (and hence every norm on X is an algebra norm on H(C, ∗)). (iv) There is no algebra norm on C. To conclude this subsection, let us summarize the work of Moreno [1025, 1026] on the norm extension problem for Jordan triples. Let A be an associative algebra with two commuting linear algebra involutions τ and π, and consider the Jordan triple

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T := H(A, τ ) ∩ S(A, π) under the triple product {xyz} := 12 (xyz + zyx). As we already we have shown in Section 7.1, Jordan triples as the one T above play an important role in Zel’manov’s prime theorem for Jordan triples. In fact they play a role similar to that played by the Jordan algebra H(A, τ ) in Zel’manov’s prime Theorem 6.1.132 for Jordan algebras. Thus, a ‘triple-norm extension problem’ merits consideration in relation to eventual future normed versions of Zel’manov’s prime theorem for Jordan triples. Let A and T be as above. If  ·  is an algebra norm on A, then, clearly, the restriction of  ·  to T is a triple norm on T (i.e. a norm on the vector space T making the triple product of T continuous). The converse question, called the triple norm extension problem, is the following: given a triple norm ||| · ||| on T, is there an algebra norm on A whose restriction to T is equivalent to ||| · |||? Suppose from now on that A is a τ -π-tight envelope of T (which means that A is generated by T as an algebra and that every τ -π-invariant ideal of A meets T). Then the triple norm extension problem has an affirmative answer if (and only if) the pentad mapping {· · · · ·}5 is ||| · |||-continuous, where {· · · · ·}5 is the function from T × T × T × T × T to T defined by 1 {t1 t2 t3 t4 t5 }5 := (t1 t2 t3 t4 t5 + t5 t4 t3 t2 t1 ) 2 [1025, Theorem 1.2] (compare Theorem 7.2.18). Moreover, if T is nondegenerate (i.e. the conditions x ∈ T and {xTx} = 0 imply x = 0), and if the triple norm ||| · ||| on T is complete then the triple norm extension problem has an affirmative answer [1026, Theorem 2] (compare Theorem 7.2.19). Moreno’s paper [1025] just reviewed motivated the following theorem, due to Cabrera and Mohammed [887]. Theorem 7.2.30 Let A be a unital associative finite-dimensional algebra over F with two commuting linear algebra involutions τ and π. Then there exists a natural number n and an injective algebra τ -π -homomorphism from A to M4n (F), where τ and π on M4n (F) stand for the symmetric and symplectic involutions.

8 Selected topics in the theory of non-associative normed algebras

We devote this chapter to developing some of our favourite aspects of the theory of normed algebras, not previously included in our work. The first section deals with H ∗ -algebras, incidentally introduced in Volume 1 of our work. A second section deals with generalized annihilator normed algebras, which become non-star generalizations of H ∗ -algebras with zero annihilator. The third section is a miscellany: automatic continuity of algebra homomorphisms into complete normed algebras with no nonzero divisor of zero is discussed; the structure of complete normed J-semisimple non-commutative Jordan algebras, all elements of which have finite J-spectra, is obtained; and a comprehensive survey on normed Jordan algebras is included. In the concluding fourth section, the notions of spectral radius of a bounded subset of a normed algebra and of a topologically nilpotent normed algebra are studied in depth. 8.1 H∗ -algebras Introduction The reasonably well-behaved coexistence of two structures, namely that of an algebra and that of a Hilbert space, becomes the essence of semi-H ∗ algebras. Indeed, they are complete normed algebras A endowed with a (vector space) conjugate-linear involution ∗, and whose norm derives from an inner product in such a way that, for each a ∈ A, the adjoint of the left multiplication La is precisely La∗ , and the adjoint of the right multiplication Ra is Ra∗ . Since Ambrose’s pioneering paper [20], it is well-known that associative semi-H ∗ -algebras with zero annihilator are H ∗ -algebras, i.e. their involutions are algebra involutions. But this is no longer true in general. Actually there exist even simple complex semi-H ∗ -algebras which are not H ∗ -algebras (see Example 8.1.5). Thus we devote this section to discuss in depth the theory of semi-H ∗ -algebras, paying special attention to H ∗ -algebras. We begin Subsection 8.1.1 by recalling those results on semi-H ∗ -algebras, which were already proved in Volume 1 of our work. Then we introduce the classical topologically simple associative complex H ∗ -algebra HS (H) of all Hilbert–Schmidt operators on a nonzero complex Hilbert space H, and show how natural examples of 477

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Jordan and Lie H ∗ -algebras can be constructed from it. After showing in Proposition 8.1.6 how the norm of a semi-H ∗ -algebra with zero annihilator determines its involution, we prove that power-associative H ∗ -algebras are non-commutative Jordan algebras (see Theorem 8.1.9). As main results in Subsection 8.1.2, we establish two fundamental structure theorems for a semi-H ∗ -algebra A, which, in two successive steps, reduce the general case to the one that A has zero annihilator, and the case that A has zero annihilator to the one that A is topologically simple (see Theorems 8.1.10 and 8.1.16). It is noteworthy that the involution of a semi-H ∗ -algebra A with zero annihilator is automatically continuous, and is isometric if and only if A is an H ∗ -algebra (see Corollary 8.1.12). We conclude the subsection by applying the above result to prove that alternative semi-H ∗ -algebras with zero annihilator are in fact H ∗ -algebras. According to the structure theory discussed in the preceding paragraph, topologically simple semi-H ∗ -algebras are worth further study. Since topologically simple complex H ∗ -algebras need not be ultraprime (see Proposition 8.1.36), we could wonder whether at least they are totally prime (cf. Theorem 6.1.63). We answer this question affirmatively by introducing totally multiplicatively prime normed algebras, showing that they are totally prime (see Proposition 8.1.27), and proving, as the main result in Subsection 8.1.3, that topologically simple complex semiH ∗ -algebras are totally multiplicatively prime (see Theorem 8.1.32). Since totally prime normed complex algebras are centrally closed (cf. Theorem 6.1.60), it follows that, as asserted in Corollary 8.1.34, topologically simple complex H ∗ -algebras are centrally closed. Corollary 8.1.34 just reviewed becomes the key tool of Subsection 8.1.4, where we prove that derivations of complex semi-H ∗ -algebras with zero annihilator are continuous (see Theorem 8.1.41), that dense-range algebra homomorphisms from complete normed complex algebras to complex H ∗ -algebras with zero annihilator are continuous (see Theorem 8.1.52), and that dense-range algebra homomorphisms from complete normed complex algebras to complex semi-H ∗ -algebras with zero annihilator are continuous whenever their ranges are ∗-invariant (see Theorem 8.1.53). In Subsection 8.1.5 we study derivations and bijective algebra homomorphisms of complex H ∗ -algebras. As main results, we show that isomorphic complex H ∗ -algebras with zero annihilator are ∗-isomorphic, and that bijective algebra ∗-homomorphisms between topologically simple complex H ∗ -algebras are positive multiples of isometries (hence, essentially, a topologically simple complex H ∗ -algebra has a unique H ∗ -algebra structure). These results follow from Theorem 8.1.64 and Corollary 8.1.67, respectively. Keeping in mind Corollary 8.1.58, Theorem 8.1.64 just quoted becomes the appropriate H ∗ -variant of the structure theorem for isomorphisms of non-commutative JB∗ -algebras proved in Theorem 3.4.75. As the main result in Subsection 8.1.6, we prove the appropriate H ∗ -variant of the Jordan characterization of C∗ -algebras established in Theorem 3.6.30 (see Theorem 8.1.76). A more than satisfactory H ∗ -variant of Theorem 3.6.25 is also obtained in Proposition 8.1.71.

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Subsection 8.1.7 is devoted to providing us with the appropriate tools to transfer results from complex semi-H ∗ -algebras to real ones. The basic tool is given by Proposition 8.1.77, which asserts that the complexification of any real (semi-)H ∗ algebra becomes a complex (semi-)H ∗ -algebra in a natural way. This quite elementary fact already allows to convert many complex results into real ones, all of them involving the assumption that the algebra has zero annihilator (see Corollaries 8.1.79 to 8.1.86 and 8.1.91). The treatment of topologically simple real (semi-)H ∗ -algebras is more elaborate. Indeed, according to Theorem 8.1.88, there are no topologically simple real (semi-)H ∗ -algebras other than topologically simple complex (semi-)H ∗ algebras, regarded as real algebras, and the real (semi-)H ∗ -algebras of all fixed points for an involutive conjugate-linear algebra ∗-homomorphism on a topologically simple complex (semi-)H ∗ -algebra. This reduction of topologically simple real (semi-) H ∗ -algebras to complex ones allows to transfer the remaining results known in the complex setting to the real one (see Theorems 8.1.92 and 8.1.95, and Corollaries 8.1.96 and 8.1.97). Special mention should be made of Corollary 8.1.98, asserting that dense-range algebra homomorphism from H ∗ -algebras with zero annihilator to topologically simple H ∗ -algebras are surjective. We introduce H ∗ -ideals of an arbitrary normed ∗-algebra, and apply Corollary 8.1.98 just reviewed to prove that topologically simple normed ∗-algebras have at most one H ∗ -ideal (see Theorem 8.1.101). On the other hand we show that Theorem 8.1.32 (that topologically simple complex semi-H ∗ -algebras are totally multiplicatively prime) and Corollary 8.1.33 remain true verbatim in the real setting (see Theorem 8.1.107 and Corollary 8.1.108, respectively). To conclude this subsection we show that, if the complexification of a real algebra with zero annihilator can be structured as an H ∗ -algebra, then the algebra itself can be structured as an H ∗ -algebra (see Theorem 8.1.109). We begin Subsection 8.1.8 by introducing the complete normed complex ∗-algebra (T C (H),  · τ ) of all trace-class operators on a complex Hilbert space H, as well as the  · τ -continuous trace-form on it (see Theorem 8.1.111). Then we show that (T C (H),  · τ ) can be structurally determined into the H ∗ -algebra (HS (H),  · ) of all Hilbert–Schmidt operators on H (see Fact 8.1.110 and §8.1.112). Such an intrinsic characterization of (T C (H), ·τ ) into (HS (H), ·) allows us to replace HS (H) with an arbitrary real or complex ( possibly non-associative) semi-H ∗ algebra A with zero annihilator, to build an appropriate substitute of (T C (H),  · τ ) into A, denoted by (τ c(A),  · τ ), and to discuss whether or not a  · τ -continuous trace-form on τ c(A) does exist. As we could expect, for a semi-H ∗ -algebra A with zero annihilator, τ c(A) is a ∗-invariant ideal of A, (τ c(A),  · τ ) is both a normed algebra and a dual Banach space (see Theorem 8.1.116), and the existence of a  · τ -continuous trace-form on τ c(A) depends on the existence of an ‘operatorbounded’ approximate unit in A (see Corollary 8.1.137). This, together with deep results established in Volume 1 (namely Theorem 3.5.53 and Proposition 4.5.36(ii)), allows us to prove that a complex H ∗ -algebra A with zero annihilator is alternative if and only if (A,  · ) has an approximate unit operator-bounded by 1, and the predual of (τ c(A),  · τ ) is a non-associative C∗ -algebra (see Corollary 8.1.141).

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To conclude the review of Subsection 8.1.8 we note that the complete normed ∗-algebra (T C (H),  · τ ) (for a complex Hilbert space H) was already incidentally involved in the proof of Theorem 2.9.70. In the concluding Subsection 8.1.9, we try to settle the paternity of the material reviewed earlier, and establish some complementary results, most times referring the reader to the original papers for a proof when these results are known. Among them, we emphasize the classification theorems of topologically simple complex H ∗ -algebras in the most familiar classes of algebras. The starting result in this direction is Theorem 8.1.151, asserting that there are no topologically simple associative complex H ∗ -algebras other than those of the form HS (H) for a nonzero complex Hilbert space H. Then the corresponding theorems for topologically simple alternative, Jordan, non-commutative Jordan, Lie, Malcev, or structurable complex H ∗ -algebras can be formulated in terms involving Theorem 8.1.151 just reviewed (see Facts 8.1.152 and 8.1.154, Propositions 8.1.156 and 8.1.157, and Theorems 8.1.155, 8.1.158, 8.1.159, and 8.1.161). 8.1.1 Preliminaries, and a theorem on power-associative H∗ -algebras In p. 254 of Volume 1 we introduced semi-H ∗ -algebras over K as those algebras A over K which are also Hilbert spaces and are endowed with a conjugate-linear vector space involution ∗ satisfying (ab|c) = (b|a∗ c) = (a|cb∗ ) for all a, b, c ∈ A.

(8.1.1)

We also recall that H ∗ -algebras over K had been previously introduced in Remark 2.6.54. Indeed, they are those semi-H ∗ -algebras over K whose involution ∗ is an algebra involution. We proved in Lemma 2.8.12 the following. Fact 8.1.1 Let A be a semi-H ∗ -algebra over K. We have: (i) Up to the multiplication of the Hilbertian norm of A by a suitable positive number, A becomes a complete normed algebra. (ii) If the involution of A is isometric, then A is in fact an H ∗ -algebra. Assertion (i) in the above fact will be applied without notice through this section. §8.1.2 In Volume 1 we also proved the following: (i) Jordan complex semi-H ∗ -algebras with zero annihilator are H ∗ -algebras (cf. Corollary 4.1.104). (ii) Associative real or complex semi-H ∗ -algebras with zero annihilator are H ∗ -algebras (cf. pp. 492–3 of Volume 1). (iii) Complex H ∗ -algebras with zero annihilator have a unique complete algebra norm topology (cf. Remark 4.4.48(f)) To conclude our review of what we know about semi-H ∗ -algebras since Volume 1, let us recall that we devoted the whole Subsection 2.8.2 and a part of Subsection 2.8.6

8.1 H ∗ -algebras

481

to a full discussion of absolute values on H ∗ -algebras [200], being Proposition 2.8.13 and Theorem 2.8.88 the main results concerning this topic. Now we introduce the fundamental example of an associative H ∗ -algebra. Example 8.1.3 Let H be a complex Hilbert space. Let {uλ }λ∈ and {vγ }γ ∈ be orthonormal bases in H. According to Schatten [809, Section II.1] (see also [795, A.1.3] and [1182, Section 2.4]), for each bounded linear operator F on H, the families {F(uλ )2 }λ∈ , {|(F(uλ )|vγ )|2 }λ∈,γ ∈ , and {F ∗ (vγ )2 }γ ∈ are simultaneously summable in R or not. Whenever they are summable, their sum is the same, independent of {uλ }λ∈ and {vγ }γ ∈ . Those operators F ∈ BL(H) for which the above families are summable are called Hilbert–Schmidt operators on H. The class HS (H) of all Hilbert–Schmidt operators on H becomes a ∗-invariant ideal of BL(H) (so an associative complex ∗-algebra) contained in the algebra K(H) (of all compact operators on H) and containing the algebra F(H) (of all finite-rank operators on H). Moreover, for F, G ∈ HS (H) and for any orthonormal basis {uλ }λ∈ in H, the family {(F(uλ )|G(uλ ))}λ∈ is summable in C with sum independent of the chosen orthonormal basis, and HS (H) becomes a complex Hilbert space under the inner product defined by  (F(uλ )|G(uλ )). (F|G) := λ∈

In this way HS (H) turns out to be an H ∗ -algebra, which is topologically simple whenever H = 0. §8.1.4 As usual in our work, given an algebra A over K, we will denote by Asym , Aant , and A(0) the algebras with the same vector space as that of A, and products given respectively by 1 (x, y) → x • y := (xy + yx) 2 (x, y) → [x, y] := xy − yx (x, y) → yx. If A is a (semi-)H ∗ -algebra over K, then Asym , Aant , and A(0) can and will be seen also as (semi-)H ∗ -algebras under the same inner product and involution that those of A. Thus, starting from a complex Hilbert space H, taking A = HS (H), and thinking about Asym (respectively, Aant ), we are provided with natural examples of Jordan (respectively, Lie) complex H ∗ -algebras. Now we exhibit an example of a semi-H ∗ -algebra which is not an H ∗ -algebra. Example 8.1.5 Let A be the two-dimensional complex algebra with basis {h, k}, and multiplication table given by h2 = ih + 2k, hk = kh = 2h − ik, and k2 = 0,

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endowed with the unique conjugate-linear vector space involution ∗ satisfying h∗ = h and k∗ = k, and the inner product (·|·) defined by (λ1 h + λ2 k|μ1 h + μ2 k) := (λ1 + iλ2 )(μ1 + iμ2 ) + 3(λ1 − iλ2 )(μ1 − iμ2 ), becomes a simple commutative complex semi-H ∗ -algebra, which is not an H ∗ algebra. Indeed, the equalities (8.1.1) follow by checking that (h2 |h) = 0 = (h|h2 ), (h2 |k) = 6 = (h|hk), (k2 |k) = 0 = (k|k2 ), (k2 |h) = 0 = (k|hk), and the fact that the involution ∗ is not an algebra involution follows from (h2 )∗ = −ih + 2k = ih + 2k = (h∗ )2 . The simplicity of A is left to the reader. Now we show how the norm of a semi-H ∗ -algebra with zero annihilator determines its involution. Proposition 8.1.6 Let A and B be semi-H ∗ -algebras over K with zero annihilator, and let F : A → B be a bijective algebra homomorphism or antihomomorphism. Suppose that F is a positive multiple of an isometry. Then F is a ∗-mapping. Proof Replacing B with its opposite algebra (in the case that F is an algebra antihomomorphism), we may suppose that F is an algebra homomorphism. Moreover, replacing the inner product of B with a suitable positive number, we may suppose in addition that F is an isometry. Then for all x, y, z ∈ A we have (x|zy∗ ) = (xy|z) = (F(xy)|F(z)) = (F(x)F(y)|F(z)) = (F(x)|F(z)F(y)∗ ) = (x|z[F −1 (F(y)∗ )]), hence zy∗ = z[F −1 (F(y)∗ )] and analogously y∗ z = [F −1 (F(y)∗ )]z, so that y∗ − F −1 (F(y)∗ ) ∈ Ann(A) = 0. Therefore F(y∗ ) = F(y)∗ for every y ∈ A.



In the case of topologically simple H ∗ -algebras, also their involutions determine their norms. But this will be proved much later (see Corollary 8.1.97). A bilinear form ·, · on an algebra B over K is said to be associative if the equalities xy, z = x, yz = y, zx hold for all x, y, z ∈ B. We recall that, according to Proposition 2.4.19, non-commutative Jordan algebras are power-associative. Proposition 8.1.7 Let A be a power-associative algebra over K, and suppose that there exists a nondegenerate symmetric associative bilinear form ·, · on A. Then A is a non-commutative Jordan algebra. Proof As pointed out in §2.5.2, the linearization of the obvious identities [a2 , a] = 0 and [a2 , a, a] = 0 yields the identities 2[a • b, a] + [a2 , b] = 0

(8.1.2)

8.1 H ∗ -algebras

483

and 2[a • b, a, a] + [a2 , b, a] + [a2 , a, b] = 0.

(8.1.3)

Now note that ·, · becomes an associative bilinear form on both Asym and Aant , i.e. we have x • y, z = x, y • z and [x, y], z = x, [y, z] for all x, y, z ∈ A. It follows from (8.1.2) that 0 = 2[x • y, x] + [x2 , y], z = 2[x • y, x], z + [x2 , y], z = 2x • y, [x, z] + x2 , [y, z], and hence 2x • y, [x, z] = x2 , [z, y]. An interchange of the roles of y and z gives that 2x • z, [x, y] = x2 , [y, z]. Comparing these equalities, we see that 2x • y, [x, z] = 2x • z, [y, x], and consequently 2[x • y, x], z = 2x • [y, x], z. Since z is arbitrary in A and ·, · is nondegenerate, we deduce that [x • y, x] = x • [y, x], and hence, by Lemma 2.4.14, A is flexible. To conclude the proof it only remains to show that A is Jordan-admissible. To this end, in view of Lemma 2.4.18 we may replace A with Asym if necessary, and suppose in consequence that A is commutative. It follows from (8.1.3) that 0 = 2[xy, x, x] + [x2 , y, x] + [x2 , x, y] = 2((xy)x)x − 2(xy)x2 + (x2 y)x − x2 (yx) + x3 y − x2 (xy) = 2((xy)x)x − 4(xy)x2 + (x2 y)x + x3 y, and hence 2((xy)x)x + x3 y = 4(xy)x2 − (x2 y)x. Therefore, for each z ∈ A, we have that 2((xy)x)x, z + x3 y, z = 4(xy)x2 , z − (x2 y)x, z. Note that both summands in the left-hand side of this equality are symmetric in y and z, and hence the right-hand side must be symmetric in y and z too, so that 4(xy)x2 , z − (x2 y)x, z = 4(xz)x2 , y − (x2 z)x, y. Since (xy)x2 , z = (x2 z)x, y and (xz)x2 , y = (x2 y)x, z, we derive that 5(x2 z)x, y = 5(x2 y)x, z, hence z, (xy)x2  = (x2 y)x, z. Since z is arbitrary in A and ·, · is nondegenerate, we deduce that (xy)x2 = (x2 y)x, and hence A is a Jordan algebra.  Now the following straightforward fact becomes crucial. Fact 8.1.8 Let A be an H ∗ -algebra over K. Then the mapping (x, y) → x, y := (x|y∗ ) + (y|x∗ ) becomes a nondegenerate symmetric associative bilinear form on A. Combining Proposition 8.1.7 with Fact 8.1.8, we obtain the following.

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Theorem 8.1.9 Power-associative H ∗ -algebras over K are non-commutative Jordan algebras. 8.1.2 Structure theory Theorem 8.1.10 becomes the first structure theorem for semi-H ∗ -algebras. Someway, it reduces the general theory to the case of semi-H ∗ -algebras with zero annihilator. Theorem 8.1.10 Let A be a semi-H ∗ -algebra over K, and let B stand for the closed linear hull of AA. Then we have: Ann(A) is a ∗-invariant ideal of A. A equals the orthogonal sum B ⊕ Ann(A). B is an ideal of A with zero annihilator in itself. B becomes a semi-H ∗ -algebra under the involution ' defined by b' := π(b∗ ), where π stands for the orthogonal projection from A onto B. (v) Ann(A) = {a ∈ A : aA = 0} = {a ∈ A : Aa = 0}.

(i) (ii) (iii) (iv)

Proof Since the annihilator of any algebra is an ideal of the algebra, assertion (i) follows from the equalities 8.1.1 in the definition of semi-H ∗ -algebras. To prove assertion (ii) it is enough to show that the orthogonal (AA)⊥ of AA in A equals Ann(A). But, invoking again the equalities 8.1.1, for a ∈ A we have aA = 0 ⇔ (aA|A) = 0 ⇔ (a|AA) = 0 ⇔ a ∈ (AA)⊥ , Aa = 0 ⇔ (Aa|A) = 0 ⇔ (a|AA) = 0 ⇔ a ∈ (AA)⊥ , and hence a ∈ Ann(A) ⇔ aA = 0 ⇔ Aa = 0 ⇔ a ∈ (AA)⊥ , as desired. Note that, along the way, we have proved assertion (v). Since the linear hull of AA is clearly an ideal of A, the fact that B is an ideal of A follows from Exercise 1.1.48. To conclude the proof of assertion (iii), note that the fact that B has zero annihilator in itself is a straightforward consequence of assertion (ii) previously proved. According to assertion (ii) and the definition of ', given b ∈ B, b' is the unique element of B such that b∗ − b' lies in Ann(A). But then, by assertion (i), (b' )∗ − b remains in Ann(A), and hence (b' )' = b. Thus ' is a (conjugate-linear vector space) involution on B. Now let x, y, z be in B. Write x∗ = x' + u and y∗ = y' + v with u, v ∈ Ann(A). Then we have (xy|z) = (y|x∗ z) = (y|(x' + u)z) = (y|x' z) and (xy|z) = (x|zy∗ ) = (x|z(y' + v)) = (x|zy' ), which concludes the proof of assertion (iv).



Remark 8.1.11 Let A be an H ∗ -algebra over K, and let B stand for the closed linear hull of AA. Then B becomes in fact an H ∗ -algebra under the involution ' defined by b' := π(b∗ ), where π stands for the orthogonal projection from A onto B. Indeed,

8.1 H ∗ -algebras

485

in view of assertion (iv) in Theorem 8.1.10, it is enough to show that (xy)' = y' x' for all x, y ∈ B. To this end note that, for x, y ∈ A, both xy and y∗ x∗ lie in B and that, since A is an H ∗ -algebra, (xy)∗ − y∗ x∗ = 0 ∈ Ann(A), hence (xy)' = y∗ x∗ . Therefore, if x, y are actually in B, then it is enough to write x∗ = x' + u and y∗ = y' + v with u, v ∈ Ann(A) to derive that (xy)' = y' x' , as desired. Corollary 8.1.12 Let A be a semi-H ∗ -algebra over K with zero annihilator. Then we have: (i) The involution of A is continuous. (ii) A equals the closed linear hull of AA. (iii) A is an H ∗ -algebra if and only if the involution of A is isometric. Proof Assertion (i) follows from the equalities 8.1.1 and the closed graph theorem, whereas assertion (ii) follows from Theorem 8.1.10(ii). The ‘if’ part of assertion (iii) is a consequence of Fact 8.1.1(ii). Suppose that A is in fact an H ∗ -algebra. Then for a, b, c ∈ A we have ((ab)∗ |c) = (b∗ a∗ |c) = (a∗ |bc) = (a∗ c∗ |b) = (c∗ |ab). Keeping in mind assertions (i) and (ii) proved previously, we deduce that the equality (d∗ |c) = (c∗ |d) holds for all c, d ∈ A. By taking d = c∗ , we finally get c = c∗ , i.e. the involution of A is isometric.  Assertion (i) in the above corollary will be applied without notice through this section. Let A be a normed algebra over K. By a minimal closed ideal of A we mean a nonzero closed ideal M of A which does not contain any nonzero closed ideal of A other than M. Proposition 8.1.13 Let A be a semi-H ∗ -algebra over K, and let M be a closed ideal of A. We have: (i) M is a direct summand of A. (More precisely, M ⊥ is an ideal of A.) (ii) Closed ideals of M are precisely the closed ideals of A contained in M. Moreover, if A has zero annihilator, then: (iii) M has zero annihilator in itself. (iv) M is a topologically simple normed algebra if and only if it is a minimal closed ideal of A. (v) M is ∗-invariant. (Hence M is a semi-H ∗ -algebra, and is an H ∗ -algebra if A is so.) Proof Assertion (i) follows from the equalities (8.1.1), whereas (ii) follows straightforwardly from (i). Suppose that A has zero annihilator. Then (iii) follows from (i), and (iv) follows from (ii) and (iii). Now note that (M ⊥ M ∗ |A) = (M ⊥ |AM) ⊆ (M ⊥ |M) = 0.

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Therefore, M ⊥ M ∗ = 0. Analogously, M ∗ M ⊥ = 0. It follows from (i) and Lemma 5.1.1 that M ∗ ⊆ M. Thus assertion (v) has been proved.  Remark 8.1.14 Let A be a semi-H ∗ -algebra over K. It follows from Proposition 8.1.13 that A is semiprime if (and only if) A has zero annihilator, and that A is prime (if and) only if A is topologically simple. Exercise 8.1.15 Let H be a Hilbert space over K, let M be a closed subspace of H, and let F, G be bounded linear operators on H such that F(M) ⊆ M, G(M ⊥ ) ⊆ M ⊥ , and F(M ⊥ ) = 0 = G(M). Prove that F + G = max{F, G}. Hint Take x ∈ H and compute (F + G)(x) after writing x = y + z with y ∈ M and  z ∈ M⊥ . Theorem 8.1.16, which follows, becomes the second and last structure theorem for semi-H ∗ -algebras. According to its reformulation in Corollary 8.1.18, it completely reduces the theory of semi-H ∗ -algebras with zero annihilator to the topologically simple case. Theorem 8.1.16 Let A be a (semi-)H ∗ -algebra over K with zero annihilator. Then A is the closure of the orthogonal sum of its minimal closed ideals, and these are topologically simple (semi-)H ∗ -algebras. Proof We may suppose that A = 0. The fact that minimal closed ideals of A are topologically simple (semi-)H ∗ -algebras follows from Proposition 8.1.13(iv)–(v). Moreover, two different minimal closed ideals M, N of A are orthogonal. Indeed, we have M ∩ N = 0 by minimality, so MN = 0 = NM, and so N ⊆ M ⊥ by Proposition 8.1.13(i) and Lemma 5.1.1. Let X denote the normed space over K consisting of the vector space of A and the norm ||| · ||| defined by ||| x ||| := Lx . (The fact that ||| x ||| = 0 implies x = 0 follows from Theorem 8.1.10(v).) Then, given any closed ideal M of A and elements x ∈ M and y ∈ M ⊥ , we have Lx (M) ⊆ M and, by Proposition 8.1.13(i), also Ly (M ⊥ ) ⊆ M ⊥ and Lx (M ⊥ ) = 0 = Ly (M). Therefore, by Exercise 8.1.15, we get ||| x + y ||| = Lx+y  = Lx + Ly  = max{Lx , Ly } = max{||| x |||, ||| y |||}. Thus M and M ⊥ are complementary M-summands of X. Therefore, by §5.1.13, M ◦ := {f ∈ X  : f (M) = 0} and (M ⊥ )◦ := {f ∈ X  : f (M ⊥ ) = 0} become complementary L-summands of the dual space X  of X, hence, by Lemma 6.1.6, for every extreme point f of BX  , we have that either f ∈ M ◦ or f ∈ (M ⊥ )◦ . Now let f be in X  . Then for every x ∈ X we have |f (x)| ≤ ||| f |||||| x ||| = ||| f |||Lx  ≤ ||| f |||ρx, where ρ denotes the norm of the product of A. Therefore, by the Riesz–Fr´echet representation theorem, there exists  f ∈ A satisfying f (x) = (x| f ) for every x ∈ X. Suppose that f is an extreme point of BX  . We claim that the closed ideal of A generated by  f (say N) is a minimal closed ideal of A. To this end, let M be a

8.1 H ∗ -algebras

487

closed ideal of A contained in N. It follows from the preceding paragraph that either (M| f ) = 0 or (M ⊥ | f ) = 0, i.e. either  f ∈ M ⊥ or  f ∈ M. Keeping in mind Proposition 8.1.13(i), the first possibility gives M ⊆ N ⊆ M ⊥ , hence M = 0, whereas the second implies M = N. This proves the claim. To conclude the proof it only remains to show that the sum of all minimal closed ideals of A is dense in A. But, if this were not true, there would exist a nonzero element x0 ∈ A satisfying (x0 |N) = 0 for every minimal closed ideal N of A so, according to the preceding paragraph, we would have in particular (x0 | f ) = 0 for every extreme point f of BX  . This would mean that f (x0 ) = 0 for every extreme point f of BX  , which is not possible because, as a consequence of the Banach– Alaoglu and Krein–Milman theorems, the linear hull of the set of all extreme points  of BX  is w∗ -dense in X  . For a better understanding of the above theorem we include here the following. Fact 8.1.17 Let {Aα } be a family of (semi-)H ∗ -algebras over K, and suppose that there are positive constants K, L such that aα bα  ≤ Kaα bα  and a∗α  ≤ Laα  for every α and all aα , bα ∈ Aα . Then the Hilbert space 2 -sum of the family {Aα }, with product and involution defined coordinate-wise, becomes a (semi-)H ∗ -algebra, which is called the (semi-)H ∗ -algebra 2 -sum of the family of (semi-)H ∗ -algebras {Aα }. Proof



Straightforward. (semi-)H ∗ -algebras

We note that the assumptions on the family of {Aα } in the above fact are automatically fulfilled if all the Aα are closed ∗-invariant subalgebras of a common (semi-)H ∗ -algebra with zero annihilator. This is the case when {Aα } is the family of all minimal closed ideals of a given (semi-)H ∗ -algebra A with zero annihilator. Therefore it is enough to invoke Theorem 8.1.16 to get the following. Corollary 8.1.18 Let A be a (semi-)H ∗ -algebra over K with zero annihilator. Then minimal closed ideals of A are topologically simple (semi-)H ∗ -algebras, and A is totally isomorphic to the (semi-)H ∗ -algebra 2 -sum of the family of its minimal closed ideals. The concluding part of this subsection is devoted to derive some by-products of Theorem 8.1.10 and Corollary 8.1.12. Fact 8.1.19 Let A be a semi-H ∗ -algebra over K, and suppose that the symmetrized algebra Asym has zero annihilator and that the involution of A is an algebra involution on Asym . Then A is an H ∗ -algebra. Proof By the assumption, Asym , endowed with the inner product and the involution ∗ of A, is an H ∗ -algebra with zero annihilator. Therefore, by Corollary 8.1.12(iii), ∗ is isometric. Finally, by Fact 8.1.1(ii), A is an H ∗ -algebra. 

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As a first consequence of Fact 8.1.19, we can generalize assertion (i) in §8.1.2 as follows. Corollary 8.1.20 Let A be a Jordan-admissible complex semi-H ∗ -algebra, and suppose that Asym has zero annihilator. Then A is an H ∗ -algebra. Proof



Combine §8.1.2(i) with Fact 8.1.19.

Since the annihilator of an algebra is invariant under all derivations of the algebra, it follows from Lemma 2.4.15 that, for every flexible algebra A, the annihilator of Asym is an ideal of A. Therefore, if a topologically simple normed flexible algebra A over K is not anticommutative, then Asym has zero annihilator. Hence, invoking Fact 8.1.19 and Corollary 8.1.20, we get Corollaries 8.1.21 and 8.1.22, respectively. Corollary 8.1.21 Let A be a topologically simple flexible semi-H ∗ -algebra over K which is not anticommutative, and suppose that the involution of A is an algebra involution on Asym . Then A is an H ∗ -algebra. Corollary 8.1.22 Let A be a topologically simple non-commutative Jordan complex semi-H ∗ -algebra which is not anticommutative. Then A is an H ∗ -algebra. In relation to Corollaries 8.1.21 and 8.1.22, we note that anticommutative algebras are non-commutative Jordan algebras. Let A and B be semi-H ∗ -algebras over K, and let F : A → B be a linear mapping. According to §3.4.71, we denote by F ∗ the linear mapping from A into B defined by F ∗ (a) := (F(a∗ ))∗ for every a ∈ A. Therefore, since we cannot use the same symbol for different things, when F is continuous we denote by F • : B → A the adjoint operator of F determined by (F(a)|b) = (a|F • (b)) for all a ∈ A and b ∈ B. With the above conventions, the equalities (8.1.1) in the definition of a semi-H ∗ algebra A can be read as (La )• = La∗ and (Ra )• = Ra∗ for every a ∈ A,

(8.1.4)

whereas, if A is in fact an H ∗ -algebra, then we have (La )∗ = Ra∗ and (Ra )∗ = La∗ for every a ∈ A.

(8.1.5)

The next proposition generalizes assertion (ii) in §8.1.2. In the proof we will apply that, given an alternative algebra A, the mapping a → La is a Jordan homomorphism from A to the algebra L(A) of all linear operators on A. Proposition 8.1.23 Let A be an alternative semi-H ∗ -algebra over K. We have: (i) Ann(A) = Ann(Asym ) = {a ∈ A : a∗ • a = 0}. (ii) If A has zero annihilator, then A is an H ∗ -algebra.

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Proof The inclusions Ann(A) ⊆ Ann(Asym ) ⊆ {a ∈ A : a∗ • a = 0} are clear. Let a ∈ A be such that a∗ • a = 0. Then we have (La )• • La = La∗ • La = La∗ •a = 0. Therefore, for every b ∈ A we have La (b)2 + (La )• (b)2 = ((La )• La (b)|b) + (La (La )• (b)|b)   = 2 [(La )• • La ](b)|b = 0, and hence ab = 0. By Theorem 8.1.10(v), a lies in Ann(A). Thus assertion (i) is proved. Suppose that A has zero annihilator. Then for a, b ∈ A we have L(a•b)∗ = (La•b )• = (La • Lb )• = (Lb )• • (La )• = Lb∗ • La∗ = Lb∗ •a∗ , hence, again by Theorem 8.1.10(v), ∗ is an algebra involution on Asym . Since Asym has zero annihilator (by assertion (i) already proved), it follows from Fact 8.1.19 that A is an H ∗ -algebra. Now the proof is complete.  The complex case of assertion (ii) in Proposition 8.1.23 was reviewed in p. 493 of Volume 1. 8.1.3 Topologically simple H∗ -algebras are ‘very’ prime We introduced and studied totally prime normed algebras in Subsection 6.1.5. Now we are going to show that this class of normed algebras contains a relevant subclass, namely the one of totally multiplicatively prime normed algebras, which in turn contains that of all topologically simple complex semi-H ∗ -algebras. An algebra A over K is said to be multiplicatively prime whenever both A and M (A) are prime algebras. Examples of multiplicatively prime algebras are all simple algebras. Indeed, if A is a simple algebra, and if F, G are in M (A) with FM (A)G = 0 and G = 0, then F = 0 because the linear hull of M (A)G(A) is a nonzero ideal of A and hence it is equal to A. In general, prime algebras need not be multiplicatively prime. Indeed, we have the following. Example 8.1.24 Consider the three-dimensional unital algebra A over K with basis {1, u, v} and multiplication table given by u2 = 1, uv = v2 = v, vu = 0. This algebra was already discussed in the proof of Lemma 4.4.83(iii), where we showed that it is prime. Also we pointed out there that Kv is an ideal of A. Since Lv Ru (A) = Kv, it follows that Lv Ru M (A)Lv Ru = 0. Since Lv Ru = 0, we realize that M (A) is not prime (nor even semiprime). Let A be an algebra over K. For F ∈ M (A) and a ∈ A, we denote by WF,a the linear operator from M (A) to A defined by WF,a (T) := FT(a) for every T ∈ M (A). An useful ideal-free characterization of the multiplicative primeness is provided by the following. Proposition 8.1.25 Let A be an algebra over K. Then the following conditions are equivalent:

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(i) A is multiplicatively prime. (ii) A has nonzero product and, whenever F ∈ M (A) and a ∈ A satisfy WF,a = 0, we have either F = 0 or a = 0. (iii) A has zero annihilator and M (A) is prime. (iv) A is semiprime and M (A) is prime. Proof Suppose that A is multiplicatively prime. Then the primeness of A implies that Ann(A) = 0, and in particular A has nonzero product. Now, suppose that F ∈ M (A) \ {0} and a ∈ A satisfy WF,a = 0. Then FM (A)(a) = 0, and consequently FM (A)La (x) = FM (A)Rx (a) = 0 and FM (A)Ra (x) = FM (A)Lx (a) = 0 for every x ∈ A, hence FM (A)La = FM (A)Ra = 0, and the primeness of M (A) gives that La = Ra = 0, that is a ∈ Ann(A), and so a = 0. Thus the implication (i)⇒(ii) is proved. Now, suppose that A has nonzero product and that WF,a = 0, with F ∈ M (A) and a ∈ A, implies either F = 0 or a = 0. Given ideals P and Q of M (A) such that PQ = 0, we see that WF,G(a) = 0 for all F ∈ P, G ∈ Q, and a ∈ A, hence either P = 0 or Q(A) = 0, that is Q = 0. Thus M (A) is prime. On the other hand, the fact that A has nonzero product allows us to ensure the existence of an element a ∈ A such that La = 0. Since WLa ,x = 0 for every x ∈ Ann(A), we derive that Ann(A) = 0, and the proof of the implication (ii)⇒(iii) finishes. Suppose that A has zero annihilator and that M (A) is prime. Let I be an ideal of A, and consider the following ideals of M (A) I ann := {F ∈ M (A) : F(I) = 0} and [I : A] := {F ∈ M (A) : F(A) ⊆ I}. Since I ann [I : A] = 0 and M (A) is prime, it follows that either I ann = 0 or [I : A] = 0. Now suppose that II = 0. Then for every x ∈ I we have Lx , Rx ∈ I ann ∩ [I : A], hence Lx = Rx = 0, i.e. I ⊆ Ann(A), and so I = 0. Thus A is semiprime, and we have proved the implication (iii)⇒(iv). Finally, suppose that A is semiprime and that M (A) is prime. Suppose that I and J are ideals of A such that IJ = 0. Then Fact 6.1.75 yields I ∩ J = 0, and keeping in mind that [I : A][J : A](A) ⊆ I ∩ J, we derive that [I : A][J : A] = 0, and hence, by the primeness of M (A), either [I : A] = 0 or [J : A] = 0. It follows from the semiprimeness of A and the inclusions II ⊆ [I : A](I) and JJ ⊆ [J : A](J) that either I = 0 or J = 0. Thus A is prime, and we have proved the implication (iv)⇒(i).  We already know that simple algebras are multiplicatively prime. In the normed case, we have the following better result. Corollary 8.1.26 Topologically simple normed algebras over K are multiplicatively prime.

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Proof Let A be a topologically simple normed algebra over K. Let F and a be in M (A) and A, respectively, such that WF,a = 0 (i.e. FM (A)(a) = 0). If a = 0, then, by the topological simplicity of A and the continuity of F, we have F(A) = F(M (A)(a)) ⊆ FM (A)(a) = 0, hence F = 0. Now apply the implication (ii)⇒(i) in Proposition 8.1.25.



In view of the equivalence (i)⇔(ii) in Proposition 8.1.25, for a normed algebra A over K, a reasonable strengthening of the multiplicative primeness is to require that A has nonzero product and that there exists a positive number K such that KFa ≤ WF,a  for all F ∈ M (A) and a ∈ A.

(8.1.6)

Normed algebras satisfying the above requirements will be called totally multiplicatively prime. Proposition 8.1.27 Let A be a totally multiplicatively prime normed algebra over K. Then A is totally prime. More precisely, if the normed algebra A has nonzero product and satisfies (8.1.6) for some positive constant K, then A satisfies Na,b  ≥ K 2 ρab for all a, b in A, where ρ denotes the norm of the product of A; i.e. A satisfies (6.1.7) with L = K 2 ρ. Proof Let K be a positive constant such that (8.1.6) holds. We claim that the inequality Kρa ≤ sup{RF(a)  : F ∈ M (A) with F = 1} holds for every a ∈ A, where ρ denotes the norm of the product of A, that is ρ = sup{xy : x, y ∈ A with x = y = 1}. Fix an element a in A and an ε ∈]0, 1[. Choose x in A of norm one such that Lx  ≥ ερ. Since for every F ∈ M (A) we have WLx ,a (F) = xF(a) ≤ RF(a) , it follows that WLx ,a  ≤ sup{RF(a)  : F ∈ M (A) with F = 1}. It follows from this and the inequalities Kερa ≤ KLx a ≤ WLx ,a  that Kερa ≤ sup{RF(a)  : F ∈ M (A) with F = 1}. Since ε is arbitrary in ]0, 1[ the claim follows. Finally, fix a, b ∈ A and note that for F, G ∈ M (A) with F = G = 1 we have WRF(b) ,a (G) = G(a)F(b) = Na,b (G, F) ≤ Na,b , hence WRF(b) ,a  ≤ Na,b , and so KRF(b) a ≤ Na,b . Keeping in mind the above claim one obtains immediately that K 2 ρab ≤ Na,b . 

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Remark 8.1.28 It follows from the compactness of the unit sphere of finitedimensional normed spaces that prime (respectively, multiplicatively prime) finitedimensional algebras over K, endowed with any algebra norm (cf. Proposition 1.1.7), are totally prime (respectively, totally multiplicatively prime). Since there are prime finite-dimensional algebras which are not multiplicatively prime (cf. Example 8.1.24), we are provided with totally prime normed algebras which are not totally multiplicatively prime. To prove (as announced at the beginning of this subsection) that topologically simple complex semi-H ∗ -algebras are totally multiplicatively prime, we need some auxiliary results. Lemma 8.1.29 Let A be a topologically simple complex semi-H ∗ -algebra. Then the centroid, A , of A (cf. Definition 1.1.10) is equal to CIA . Proof By Proposition 1.1.11(ii), we have A ⊆ BL(A), where, as usual, BL(A) stands for the normed algebra of all bounded linear operators on A. Now, since A is the commutator of LA ∪ RA in BL(A), and LA ∪ RA is a self-adjoint subset of BL(A) (cf. (8.1.4)), it follows from Proposition 1.1.11(i) that A is a commutative C∗ algebra of operators on the Hilbert space of A containing IA . If f , g are nonzero elements in A with fg = 0, then ker( f ) becomes a nonzero closed proper ideal of A, contrary to the topological simplicity of A. Therefore the unital commutative C∗ algebra A has no nonzero divisor of zero, and hence, by Fact 6.1.55, we have that A = CIA .  Let X be a normed space over K. The strong operator topology of BL(X) is defined as the topology of pointwise convergence in BL(X) when X is endowed with the norm topology, and will be denoted by SOT. The proofs of the following two theorems will not be discussed here. Theorem 8.1.30 (von Neumann’s bicommutant theorem) [723, Corollary I.3.4.1] Let H be a complex Hilbert space, and let B be a self-adjoint subalgebra of the C∗ -algebra BL(H) containing IH . Then the bicommutant of B in BL(H) coincides with the SOT-closure of B in BL(H). Theorem 8.1.31 (Kaplansky’s density theorem) [723, Theorem I.3.5.3] Let H be a complex Hilbert space, and let B be an SOT-dense self-adjoint subalgebra of BL(H). Then the closed unit ball of B is SOT-dense in the closed unit ball of BL(H). For the proof of the following theorem and later applications, we note that, as a consequence of (8.1.4), the multiplication algebra of a semi-H ∗ -algebra A is a selfadjoint subalgebra of the C∗ -algebra BL(A). Theorem 8.1.32 Let A be a topologically simple complex semi-H ∗ -algebra. Then A is totally multiplicatively prime. More precisely, we have WF,a  = Fa for all F ∈ M (A) and a ∈ A.

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Proof It suffices to show that WF,a  = 1 whenever F ∈ M (A) and a ∈ A have norm one. The inequality WF,a  ≤ 1 is clear. To prove the converse inequality, note that by Lemma 8.1.29 we have M (A)c = CIA , where c denotes commutant in BL(A), and therefore M (A)cc = BL(A). Now, by Theorems 8.1.30 and 8.1.31, BM (A) is SOT-dense in BBL(A) . Since the mappings T → T(a) from (BL(A), SOT) into A, x → F(x) from A into A, and x → x from A into R are continuous it follows that the mapping h from (BL(A), SOT) into R defined by h(T) = FT(a) is continuous. This, together with the clear fact that h(BM (A) ) ⊆ [0, WF,a ] and the SOT-density of BM (A) in BBL(A) allows us to obtain that h(BBL(A) ) ⊆ [0, WF,a ]. For each x in A consider the operator Tx on A defined by Tx (y) = (y|a)x for every y in A. It is clear that Tx (a) = x and Tx is bounded with Tx  = x. Therefore, for x in A with x = 1 we have F(x) = FTx (a) = h(Tx ) ≤ WF,a . From this we obtain 1 ≤ WF,a , as required.



Corollary 8.1.33 Let A be a topologically simple complex semi-H ∗ -algebra. Then A is totally prime. More precisely, we have Na,b  = ρab for all a, b ∈ A, where ρ denotes the norm of the product of A. Proof The inequality Na,b  ≤ ρab is clear. The converse inequality follows from Proposition 8.1.27 and Theorem 8.1.32.  The following corollary follows from Theorem 6.1.60 and Corollary 8.1.33. Corollary 8.1.34 Every topologically simple complex semi-H ∗ -algebra is centrally closed. Remark 8.1.35 Since finite-dimensional prime algebras over K, endowed with any algebra norm, are ultraprime (a consequence of Lemma 2.8.59), the existence of finite-dimensional prime algebras which are not multiplicatively prime (assured by Example 8.1.24) shows that there exist ultraprime normed algebras which are not multiplicatively prime (much less, totally multiplicatively prime). On the other hand, by Theorem 8.1.32 and Proposition 8.1.36, totally multiplicatively prime normed algebras need not be ultraprime. Proposition 8.1.36 Let H be a nonzero complex Hilbert space, and let ||| · ||| denote the norm of H as well as the corresponding operator norm on BL(H). Let A stand for the topologically simple complex H ∗ -algebra HS (H) (cf. Example 8.1.3), and let  ·  denote the norm of A as well as the corresponding operator norm on BL(A). Then we have Ma,b  = ||| a |||||| b ||| for all a, b ∈ A. As a consequence, A is ultraprime (if and) only if H is finite-dimensional.

(8.1.7)

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Proof It is easily seen that A is a norm ideal on H in the sense of Definition 6.1.141 (see [809, Lemma II.1.3(iv)–(v)]). Therefore (8.1.7) follows from assertion (6.1.50) in Proposition 6.1.142. If H were infinite-dimensional and if A were ultraprime, then, by Proposition 6.1.142(a),  ·  and ||| · ||| would be equivalent norms on A. But this is certainly impossible. For, given an arbitrary natural number n, we can find pairwise orthogonal norm-one elements x1 , . . . , xn ∈ H, and hence, by [809, Theorems I.1.1 and II.1.2], we have   n n    √   x x xk xk = n.  = 1 and  k k   k=1

k=1

As the next proposition shows, total multiplicative primeness of a normed algebra and ultraprimeness of the multiplication algebra are closely related. Proposition 8.1.37 Let A be a normed algebra over K. We have: (i) If A is a totally multiplicatively prime, then M (A) is ultraprime. (ii) If M (A) is ultraprime, and if there exists a positive constant K such that Ka ≤ max{La , Ra } for every a ∈ A,

(8.1.8)

then A is totally multiplicatively prime. (iii) If both A and M (A) are ultraprime, then A is totally multiplicatively prime. Proof Suppose that A is totally multiplicatively prime. Let K be a positive number such that (8.1.6) holds. Then for F, G, T ∈ M (A) and a ∈ A we have WF,G(a) (T) = FTG(a) ≤ FTGa = MF,G (T)a ≤ MF,G Ta, so WF,G(a)  ≤ MF,G a, and so KFG(a) ≤ MF,G a. In this way, for F, G ∈ M (A) we have KFG ≤ MF,G , and consequently, by Proposition 6.1.62, M (A) is ultraprime. This proves (i). Now suppose that M (A) is ultraprime and that there exists a positive constant K1 such that K1 a ≤ max{La , Ra } for every a ∈ A. Let K2 be a positive constant such that K2 FG ≤ MF,G  for all F, G ∈ M (A). Then for F, T ∈ M (A) and a, x ∈ A we have MF,La (T)(x) = FTLa (x) = FT(ax) = FTRx (a) = WF,a (TRx ) ≤ WF,a Tx, hence MF,La  ≤ WF,a , and so K2 FLa  ≤ WF,a . Analogously we can obtain K2 FRa  ≤ WF,a , and consequently K1 K2 Fa ≤ WF,a . Thus A satisfies (8.1.6) with K = K1 K2 . This proves (ii). In view of (ii), to prove (iii) it is enough to show that, if A is ultraprime, then (8.1.8) holds for some K > 0. Assume by the contrary that there is no K > 0 such that (8.1.8) is fulfilled. Then, arguing as in the first paragraph of the proof of Proposition 6.1.62, we realize that, for every countably incomplete ultrafilter U , we have Ann(AU ) = 0, hence AU is not prime, and finally, keeping in mind the arbitrariness of U , A is not ultraprime, contrary to the assumption. 

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8.1.4 Automatic continuity The first goal in this subsection is to prove the automatic continuity of derivations of complex semi-H ∗ -algebras with zero annihilator. The starting auxiliary result, formulated in the next proposition, is an easy consequence of Theorem 3.1 in the Erickson–Martindale–Osborn paper [246], where it is proved a result for centrally closed prime algebras, which remembers Jacobson density theorem (cf. Theorem 3.6.57) and asserts that, given linearly independent elements x1 , . . . , xp in a centrally closed prime algebra A over K, there exists T ∈ M (A) such that T(x1 ) = · · · = T(xp−1 ) = 0 and T(xp ) = 0. In fact we will need only the case p = 2 of this statement, whose proof is then almost straightforward: if T(x2 ) = 0 whenever T lies in M (A) with T(x1 ) = 0, then S(x1 ) → S(x2 ) is a well-defined partially defined centralizer on A (say f ) with dom( f ) = M (A)(x1 ), so x1 and x2 are linearly dependent because f (x1 ) = x2 and A is centrally closed. For the sake of convenience, given a vector space X over K, x ∈ X, and linear mappings T1 , . . . , Tn : X → X, we will write T1 · · · Tn x to denote (T1 ◦ · · · ◦ Tn )(x). Proposition 8.1.38 Let A be a centrally closed prime algebra over K such that dim(T(A)) > 1 for every nonzero T in the multiplication algebra M (A) of A. Then there exist sequences bn in A and Tn in M (A) such that Tn · · · T1 bn = 0 and Tn+1 Tn · · · T1 bn = 0 for every n ∈ N. Proof Let b1 ∈ A and T1 ∈ M (A) be such that T1 b1 = 0 and suppose inductively that b1 , . . . , bk and T1 , . . . , Tk have been chosen so that Tj · · · T1 bj−1 = 0 and Tj · · · T1 bj = 0 for j = 2, . . . , k. Since dim(Tk · · · T1 (A)) > 1, there exists bk+1 ∈ A such that Tk · · · T1 bk and Tk · · · T1 bk+1 are linearly independent, and so there exists Tk+1 ∈ M (A) such that Tk+1 Tk · · · T1 bk = 0 and Tk+1 Tk · · · T1 bk+1 = 0. The sequences bn and Tn constructed in this way satisfy the requirements in the conclusion of the proposition.  Now we are going to realize that the possibility dim(T(A)) = 1 for some T in the multiplication algebra of a topologically simple complete normed algebra A over K implies the continuity of derivations on A. The proof is very simple, involving only the well-known easy fact that the separating space for a derivation of a normed algebra is a closed ideal (cf. Lemma 1.1.57) and the following purely algebraic observation already pointed out in Fact 2.4.7. If D is a derivation of an algebra A, then we have DLa − La D = LD(a) and DRa − Ra D = RD(a) for every a ∈ A, hence for every T ∈ M (A) the operator d(T) = DT − TD lies in M (A). Proposition 8.1.39 Let D be a derivation of a topologically simple complete normed algebra A over K, and suppose that there exists a nonzero T ∈ M (A) with finitedimensional range. Then D is continuous. Proof Since T is continuous with finite-dimensional range, DT is continuous, so also TD is continuous because TD = DT − d(T) and d(T) lies in M (A). Thus the separating space S(D) of D is contained in ker(T). Since S(D) is a closed ideal of

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A, and T is nonzero, we obtain that S(D) = 0, and the result follows from the closed graph theorem (cf. Fact 1.1.56).  Lemma 8.1.40 Let A be an algebra over K with zero annihilator. Then direct summands of A are invariant under any derivation of A. Proof Let I be a direct summand of A, let D be a derivation of A, and let x be in I. Write A = I ⊕ J for an ideal J of A. Then for every y ∈ J we have xy = 0, so 0 = D(xy) = xD(y) + D(x)y, hence xD(y) = 0 because xD(y) ∈ I and D(x)y ∈ J. Analogously D(y)x = 0 for every y ∈ J. It follows from Lemma 5.1.1 that D(x) lies in I.  Now we can conclude the proof of our first main result in this subsection. To this end, we recall that, for a semi-H ∗ -algebra A, M (A) is a self-adjoint subalgebra of the C∗ -algebra BL(A). Theorem 8.1.41 Let A be a complex semi-H ∗ -algebra with zero annihilator, and let D be a derivation of A. Then D is continuous. Proof We suppose first that A is topologically simple, and we argue by contradiction. If D is not continuous, then, by Corollary 8.1.34 and Propositions 8.1.38 and 8.1.39, there exist sequences bn in A and Tn in M (A) such that Tn · · · T1 bn = 0 and Tn+1 · · · T1 bn = 0 for all n in N, and clearly we may suppose bn  = Tn  = 1 for every n ∈ N. On the other hand, the discontinuity of D together with the closed graph theorem implies the existence of an element b ∈ A such that the linear functional x → (D(x)|b) from A to C is not continuous. But it is straightforward to show that the set I = {y ∈ A : x → (D(x)|y) is continuous} is an ideal of A which is closed thanks to the classical Banach–Steinhaus theorem (cf. Proposition 1.4.16). It follows from the topological simplicity of A that I = 0, that is: the functional x → (D(x)|y) is discontinuous for every nonzero element y ∈ A. Now, using this fact, we can construct inductively a sequence an in A with the property that for every n ∈ N we have (i) an  ≤ 2−n , and  • • • • (ii) |(D(an )|Tn · · · T1 bn )| ≥ n + | n−1 j=1 (D(T1 · · · Tj aj )|bn )| + d(T1 · · · Tn ) + • • d(T1 · · · Tn+1 ).  • • Now we consider the element a ∈ A defined by a := ∞ j=1 T1 · · · Tj aj , and for n ∈ N ∞ • • we write cn := an+1 + j=n+2 Tn+2 · · · Tj aj . Then we have: (D(a)|bn ) =

n−1 

(D(T1• · · · Tj• aj )|bn ) + (D(T1• · · · Tn• an )|bn )

j=1

⎞ ⎞   + ⎝D ⎝ T1• · · · Tj• aj ⎠ bn ⎠  j=n+1 ⎛ ⎛

∞ 

8.1 H ∗ -algebras =

n−1 

497

(D(T1• · · · Tj• aj )|bn ) + (d(T1• · · · Tn• )(an ) + T1• · · · Tn• D(an )|bn )

j=1 • + (D(T1• · · · Tn+1 cn )|bn )

=

n−1 

(D(T1• · · · Tj• aj )|bn ) + (d(T1• · · · Tn• )(an )|bn ) + (D(an )|Tn · · · T1 bn )

j=1 • + (d(T1• · · · Tn+1 )(cn )|bn ) + (D(cn )|Tn+1 · · · T1 bn )

= (D(an )|Tn · · · T1 bn ) +

n−1  (D(T1• · · · Tj• aj )|bn ) + (d(T1• · · · Tn• )(an )|bn ) j=1

• + (d(T1• · · · Tn+1 )(cn )|bn ),

where for the last equality we have used that Tn+1 · · · T1 bn = 0. Therefore, since cn  ≤ 1, we obtain    n−1    D(a) ≥ |(D(a)|bn )| ≥ |(D(an )|Tn · · · T1 bn )| −  (D(T1• · · · Tj• aj )|bn )  j=1  • )(cn )|bn )| − |(d(T1• · · · Tn• )(an )|bn )| − |(d(T1• · · · Tn+1      n−1  ≥ |(D(an )|Tn · · · T1 bn )| −  (D(T1• · · · Tj• aj )|bn )  j=1  • ) ≥ n. − d(T1• · · · Tn• ) − d(T1• · · · Tn+1

The contradiction D(a) ≥ n for every n ∈ N shows that D is continuous if A is topologically simple. The general case can be reduced to the above proved one as follows. Let M be any minimal closed ideal of A. Then, by Lemma 8.1.40, D(M) is contained in M. Since M is a topologically simple H ∗ -algebra (cf. Proposition 8.1.13(iv)–(v)), it follows from the first part of the proof that D is continuous on M. Now let xn be a sequence in A such that lim xn = 0 and lim D(xn ) = x for some x ∈ A. Then for every y ∈ M we have that 0 = lim D(xn y) = lim(D(xn )y + xn D(y)) = xy, and in the same way yx = 0. Thus S(D)M = MS(D) = 0 for every minimal closed ideal M of A, so, by Theorem 8.1.16, S(D)A = AS(D) = 0 and thus S(D) = 0. By the closed graph theorem, D is continuous.  Now we begin the proof of the automatic continuity of dense-range algebra homomorphisms from complete normed complex algebras to complex H ∗ -algebras with zero annihilator. Given a vector space X over K and a linear mapping T : X → X, we denote by sp(T) the spectrum of T relative to L(X). We remark that, if X is in fact a Banach

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space, and if T is continuous, then sp(T) coincides with the spectrum of T relative to BL(X) (cf. Example 1.1.32(d)). Lemma 8.1.42 Let X and Y be normed spaces over K, let  : X → Y be a denserange linear mapping, let F be a bounded linear operator on X, let G be a compact operator on Y, and suppose that F = G. Then sp(G) ∪ {0} ⊆ sp(F) ∪ {0}. Proof It is enough to show that 1 ∈ C \ sp(F) implies 1 ∈ C \ sp(G). But, if 1 is not in the spectrum of F, then IX − F is a surjective operator on X and, since (IX − F) = (IY − G) and  has dense range in Y, it follows that IY − G has dense range in Y. Since G is a compact operator on Y, the Riesz–Schauder theory (cf. Theorem 1.4.27(ii)) gives us that IY − G is actually a bijective operator on Y, so 1 is not in the spectrum of G, as desired.  With Newburgh’s theorem asserting the continuity of the spectrum at elements of complete normed associative complex algebras with totally discontinuous spectrum (see [682, Corollary 7 in p. 8]), Lemma 8.1.42 yields directly to the next one. Lemma 8.1.43 Let X and Y be complex Banach spaces, let  : X → Y be a denserange linear mapping, let {Fn } be a sequence of bounded linear operators on X norm-convergent to zero, let {Gn } be a sequence of compact operators on Y normconvergent to some (compact) linear operator G on Y, and suppose that the equality Fn = Gn  holds for every n ∈ N. Then sp(G) = {0}. The next lemma is folklore. Lemma 8.1.44 Let X be a complex vector space, and let F : X → X be a linear mapping with finite-dimensional range and satisfying sp(F) = {0}. Then F n+1 = 0, where n denotes the dimension of F(X). Proof Since F(X) is an F-invariant subspace of X of dimension n, there exist n ≥ m ∈ N and p ∈ C[x] such that [IX − Fp(F)]F m = 0 on F(X), and therefore [IX − Fp(F)]F m+1 = 0 on X. But, by Proposition 1.3.4(ii), we have sp(Fp(F)) = {0}, and hence IX − Fp(F) is bijective. It follows that F m+1 = 0.  Lemma 8.1.45 Let B be an algebra over K, and suppose the existence of a nondegenerate symmetric associative bilinear form ·, · on B. Then we have: (i) There exists a unique linear algebra involution # on M (B) satisfying Lb# = Rb and R#b = Lb for every b ∈ B. (ii) For x, y ∈ B and T ∈ M (B), the equality T(x), y = x, T # (y) holds. Moreover, if in addition we suppose that the algebra B is prime, then: (iii) I, b = 0, with b ∈ B and I an ideal of B, implies either I = 0 or b = 0. (iv) B is multiplicatively prime. Proof The first two assertions are clear from the observation that the set of linear operators on B having an adjoint with respect to ·, · is an algebra of operators on B,

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499

and that the passing to the operator adjoint is a linear algebra involution on this algebra of operators. Suppose that B is prime, and let I and J be ideals of B with I, J = 0. Then 0 = IB, J = B, JI, so JI = 0 (since ·, · is nondegenerate), and so either I = 0 or J = 0 (since B is prime). Since the set {b ∈ B : I, b = 0} is an ideal of B, (iii) is proved. Let P and Q be ideals of M (B) with PQ = 0. Then 0 = (PQ)(B), B = Q(B), P# (B), hence either P = 0 or Q = 0 because Q(B) is an ideal of B and (iii) applies. Therefore M (B) is a prime algebra, and (iv) is proved.  Lemma 8.1.46 Every dense subalgebra of a normed centrally closed prime algebra over K is a centrally closed prime algebra. Proof Let B be a normed centrally closed prime algebra, and let C be any dense subalgebra of B. The primeness of C follows easily from that of B and the density of C in B. To prove that C is centrally closed, let f be a nonzero partially defined centralizer on C, and consider the set Z of those x ∈ B such that there exists a sequence yn in dom( f ) with 0 = lim yn and x = lim f (yn ). Since Z is nothing other than the separating subspace of f regarded as a linear mapping from dom( f ) to B, it follows from Lemma 1.1.57 that Z is closed in B. This information, together with the behaviour of f and the density of C in B, gives easily that Z is an ideal of B. On the other hand, for y ∈ dom( f ) and x ∈ Z (x = lim f (yn ) for a suitable sequence yn in dom( f ) with 0 = lim yn ), we have xy = lim f (yn )y = lim yn f (y) = 0. In this way we obtain that Zdom( f ) = 0, hence also ZI = 0, where I denotes the closure of dom( f ) in B. Since dom( f ) is a nonzero ideal of C, and C is dense in B, I is a nonzero ideal of B. It follows from the primeness of B that Z = 0, i.e. f , regarded as a partially defined linear operator on B, is closeable (cf. p. 651 of Volume 1 for definition). Now it is routine to verify that the closure f of f (cf. again p. 651 of Volume 1) is a partially defined centralizer on B. Since B is centrally closed, we have f (x) = λx for some λ ∈ K and every x ∈ dom(f ), hence f (y) = λy for every y ∈ dom( f ).  The next result is one of the main lemmas in the general theory of automatic continuity. Its proof will not be discussed here. Lemma 8.1.47 [810, Lemma 1.6] Let X and Y be Banach spaces over K, and let Fn and Gn be sequences of bounded linear operators on X and Y respectively. If  is a linear mapping from X to Y satisfying Fn = Gn  for every n ∈ N, then there is a natural number N such that (G1 · · · Gn S())− = (G1 · · · GN S())− for every n ≥ N, where − denotes closure in X. Notation 8.1.48 From now on to Proposition 8.1.51, A and B will be complete normed complex algebras,  : A → B will be a dense-range algebra homomorphism,

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and C will denote the range of . If D is any of the algebras A, B, or C, then as usual, for d in D, LdD and RD d will denote the operators of left and right, respectively, multiplication by d on D. Keeping in mind that M (D) contains only bounded operators and that the equalities B  and RAa = RB(a)  LaA = L(a)

hold for every a ∈ A, it follows easily that for each F ∈ M (A) there exists a (F)) in M (B) such that F =  (F), and that the mapping unique element (say  :F→ (F) is a dense-range algebra homomorphism from M (A) to M (B) (where  of course M (A) and M (B) are viewed as normed algebras under the operator norm). Moreover, observing that, for c in C, LcC and RC c are nothing other than the restrictions to C of LcB and RBc , respectively, and denoting by T the only bounded linear operator on B extending a given bounded linear operator T on C, it follows that the mapping . These T → T is a bijective algebra homomorphism from M (C) onto the range of  facts will be used in what follows without notice. Proposition 8.1.49 Suppose that B is multiplicatively prime and that there exists a nonzero element T ∈ M (C) with finite-dimensional range. Then  is continuous. Proof Arguing by contradiction, assume that  is not continuous, so that, by the closed graph theorem, the separating space S() is not zero. It is easy to see ), so that, since B has zero that, for b ∈ S(), the operators LbB and RBb lie in S(   is a dense-range algebra annihilator, we have also S() = 0. Moreover, since   homomorphism from M (A) to M (B), S() is an ideal of M (B) (cf. Lemma ), so that G = lim  (Fn ) for a suitable 1.1.58). Let G be an arbitrary element in S( sequence Fn in M (A) converging to zero, and let T denote the continuous extension to B of the nonzero element T in M (C) with finite-dimensional range, whose (S) for some S in M (A), hence existence has been assumed. We know that T =  we have (Fn )T = lim  (Fn S), 0 = lim Fn S and GT = lim  and we may apply Lemma 8.1.43 to obtain sp(GT) = {0}. (Note that, for every (Fn )T has finite-dimensional range, hence it is (Fn S) =  n ∈ N, the operator  compact.) Denoting by p the dimension of the range of T, the dimension of the range of GT is less than or equal to p, a fact that, together with sp(GT) = {0}, ), and implies (GT)p+1 = 0 (cf. Lemma 8.1.44). Now, since G is arbitrary in S( ) is an ideal of M (B), S( )T is a left ideal (hence a subalgebra) of M (B) S( satisfying the assumptions of the Nagata–Higman theorem reviewed in §2.8.42 (see [753, Appendix C in p. 271] for a proof). By applying this theorem, we have that )T is a nilpotent left ideal of M (B). Finally use the assumed primeness of S( ) is a nonzero ideal of )T = 0, and also T = 0 (since S( M (B) to obtain S( M (B)). Therefore T = 0, a contradiction.  Putting together Corollary 8.1.26 and Proposition 8.1.49, we obtain the following.

8.1 H ∗ -algebras

501

Corollary 8.1.50 Suppose that B is topologically simple and that there exists a nonzero element in M (C) with finite-dimensional range. Then  is continuous. Proposition 8.1.51 Suppose that B is prime and centrally closed, and that there exists a continuous nondegenerate symmetric associative bilinear form ·, · on B. Then  is continuous. Proof If there exists a nonzero element in M (C) with finite-dimensional range, then, by Lemma 8.1.45(iv) and Proposition 8.1.49,  is continuous. Otherwise, by Lemma 8.1.46 and Proposition 8.1.38, there are sequences cn in C and Tn in M (C) such that Tn · · · T1 cn = 0 and Tn+1 Tn · · · T1 cn = 0 for every n ∈ N. We know that, for n ∈ N, the continuous extension to B of Tn , (denoted as in Notation 8.1.48 by Tn ) lies  : M (A) → M (B). Recalling in the range of the induced algebra homomorphism  that this range is generated by {LcB , RBc : c ∈ C}, it follows that it is invariant under the involution # on M (B) given by Lemma 8.1.45(i). Therefore, for n ∈ N, there exists # (Fn ). Since Fn = T #n  for every n ∈ N, we may apply Fn ∈ M (A) with T n =  Lemma 8.1.47 to obtain the existence of a natural number N such that (T 1 · · · T n S())− = (T 1 · · · T N S())− #

#

#

#

for every n ≥ N. Therefore, since TN+1 ...T1 cN = 0 and ·, · is continuous, it follows from Lemma 8.1.45(ii) that #

#

0 = S(), TN+1 · · · T1 cN  = S(), T N+1 · · · T 1 cN  = T 1 · · · T N+1 S(), cN  = (T 1 · · · T N+1 S())− , cN  = (T 1 · · · T N S())− , cN  = S(), TN · · · T1 cN . #

#

#

#

Since TN · · · T1 cN = 0 and S() is an ideal of B, Lemma 8.1.45(iii) gives S() = 0. Hence  is continuous by the closed graph theorem.  Theorem 8.1.52 Let A be a complete normed complex algebra, let B be a complex H ∗ -algebra with zero annihilator, and let  : A → B be a dense-range algebra homomorphism. Then  is continuous. Proof Suppose at first that B is topologically simple. Then, by Fact 8.1.8, Corollaries 8.1.12(i) and 8.1.34, and Proposition 8.1.51,  is continuous. The general case can be reduced to the above proved one as follows. Consider the compositions π (where π is the orthogonal projection onto any minimal closed ideal of B), which, by Proposition 8.1.13, are dense-range algebra homomorphisms from A to topologically simple H ∗ -algebras. By the preceding paragraph, they are continuous. Now apply Theorem 8.1.16 together with the closed graph theorem.  We do not know if the above theorem remains true with ‘semi-H ∗ -algebra’ instead of ‘H ∗ -algebra’. Anyway, we are provided with the following variant. Theorem 8.1.53 Let A be a complete normed complex algebra, let B be a complex semi-H ∗ -algebra with zero annihilator, and let  : A → B be an algebra homomorphism whose range is ∗-invariant and dense in B. Then  is continuous.

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Proof Suppose at first that B is topologically simple. Let C denote the range of , so that we are covered by Notation 8.1.48. If there exists a nonzero element in M (C) with finite-dimensional range, then, by Corollary 8.1.50,  is continuous. Otherwise, by Lemma 8.1.46 and Proposition 8.1.38, there are sequences cn in C and Tn in M (C) such that Tn · · · T1 cn = 0 and Tn+1 Tn · · · T1 cn = 0 for every n ∈ N. We know that, for n ∈ N, the continuous extension to B of Tn (denoted by Tn ) lies  : M (A) → M (B). Since B in the range of the induced algebra homomorphism  ∗ is a semi-H -algebra and C is a ∗-invariant subset of B, the set {LcB , RBc : c ∈ C} is a (M (A)) is generated self-adjoint subset of the C∗ -algebra BL(B). Therefore, since  as an algebra by {LcB , RBc : c ∈ C}, it is a ∗-subalgebra of BL(B), and hence for n in • (Fn ). Since Fn = T •n  for every n ∈ N, we N there exists Fn ∈ M (A) with T n =  may apply Lemma 8.1.47 to obtain the existence of a natural number N such that •







(T 1 · · · T n S())− = (T 1 · · · T N S())− for every n ≥ N. Therefore, since TN+1 · · · T1 cN = 0, we obtain 0 = (S()|TN+1 · · · T1 cN ) = (S()|T N+1 · · · T 1 cN ) •



= (T 1 · · · T N+1 S()|cN )

• • • • = ((T 1 · · · T N+1 S())− |cN ) = ((T 1 · · · T N S())− |cN )

= (S()|TN · · · T1 cN ). Since TN ...T1 cN = 0, and S() is an ideal of B, and B is topologically simple, we deduce that S() = 0. Hence  is continuous by the closed graph theorem. The reduction of the general case to the above considered one is done as in the last paragraph of the proof of Theorem 8.1.52 since references applied there for  H ∗ -algebras cover the case of semi-H ∗ -algebras. Corollary 8.1.54 Surjective algebra homomorphisms from complete normed complex algebras to complex semi-H ∗ -algebras with zero annihilator are continuous. Therefore every complex semi-H ∗ -algebra with zero annihilator has a unique complete algebra norm topology. 8.1.5 Isomorphisms and derivations of H∗ -algebras We recall that, by Theorem 8.1.41, derivations of complex semi-H ∗ -algebras with zero annihilator are continuous. In relation to assertions (iv) to (vii) in the next lemma (which in the present subsection will be only applied in the complex case), we also note that, as we will prove in Corollary 8.1.82, the above result remains true with ‘real’ instead of ‘complex’. Let A be an algebra over K. By a quasi-centralizer on A we mean a linear mapping F : A → A such that F(a)b = aF(b) for all a, b ∈ A.

(8.1.9)

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503

Arguing as in the proof of Proposition 1.1.11(ii), we realize that if A is a complete normed algebra over K with zero annihilator, then quasi-centralizers on A are continuous. Lemma 8.1.55 Let A be a semi-H ∗ -algebra over K with zero annihilator, and let F be a quasi-centralizer on A. Then: (i) For all a, b ∈ A we have that F• (ab) = F ∗ (a)b = aF ∗ (b). (ii) F ∗ is a quasi-centralizer on A, and we have that (F ∗ )• (ab) = F(a)b = aF(b) for all a, b ∈ A. (iii) F is a centralizer if and only if F∗ = F • . Now let D be a derivation of A. Then: (iv) For all a, b ∈ A we have that D• (ab) = aD• (b) − D∗ (a)b and D• (ab) = D• (a)b − aD∗ (b).

(8.1.10)

(v) D• + D∗ is a quasi-centralizer on A. (vi) D• + D∗ is a centralizer if and only if (D• )∗ = (D∗ )• . (vii) If A is in fact an H ∗ -algebra, then D• + D∗ is a centralizer on A. Proof Let a, b be in A. Then condition (8.1.9) yields La F = LF(a) and Rb F = RF(b) . Therefore, taking adjoints, we obtain F • La∗ = LF(a)∗ and F • Rb∗ = RF(b)∗ . By replacing a with a∗ and b with b∗ , respectively, and applying the resulting equalities to b and a, respectively, assertion (i) follows. Assertion (ii) follows straightforwardly from (i), whereas assertion (iii) follows from (ii) and Corollary 8.1.12(ii). Let a, b be in A. Then we have that LD(a) = [D, La ], and taking adjoints we obtain that LD(a)∗ = [La∗ , D• ]. Replacing a with a∗ , and applying the resulting equality to b, we obtain the first equality in (8.1.10). The second equality is obtained in a similar way, thus concluding the proof of (iv). Again let a, b be in A. It follows from (iv) that aD• (b) − D∗ (a)b = D• (a)b − aD∗ (b), or equivalently a((D• + D∗ )(b)) = ((D• + D∗ )(a))b. This proves assertion (v). Assertion (vi) follows from (v) and (iii). Suppose that A is an H ∗ -algebra. Then, since ∗ is a surjective conjugatelinear isometry on (the Hilbert space of) A (cf. Corollary 8.1.12(iii)), we have (G• )∗ = (G∗ )• for every G ∈ BL(A). Therefore assertion (vii) follows from (vi).  Theorem 8.1.56 For every derivation D of a complex H ∗ -algebra A with zero annihilator we have that D• = −D∗ . Proof By Theorem 8.1.16 and Lemma 8.1.40, we can suppose that A is topologically simple. Then, by Lemmas 8.1.55(vii) and 8.1.29, we have D• + D∗ = λIA for suitable λ ∈ C. Now suppose that D = D∗ . Then λ belongs to R because D• + D

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is a self-adjoint operator on the Hilbert space of A. Write G := 12 i(D• − D), then G is a self-adjoint operator and D = 12 λIA + iG, so sp(D) ⊆ 12 λ + iR, which together with Corollary 3.4.44(i) gives λ = 0. Thus we have proved that D• = −D under the additional assumption that D = D∗ . But for arbitrary D, since the involution of A is an algebra involution, it follows that D1 := 12 (D + D∗ ) and D2 := 12 i(D∗ − D) are also derivations of A satisfying D1 = D∗1 , D2 = D∗2 and D = D1 + iD2 . Therefore D• = −D1 + iD2 = −D∗ , as desired.  Corollary 8.1.57 The spectrum of any derivation of a complex H ∗ -algebra A with zero annihilator is invariant under the mapping z → −z from C to C. Proof

The mapping F → (F ∗ )• is an algebra antiautomorphism of BL(A), so sp(F) = sp((F ∗ )• ) for every F ∈ BL(A).

Now apply the Theorem 8.1.56.



From now on we will use without notice the fact that bijective algebra homomorphisms from a complex H ∗ -algebra with zero annihilator to a complete normed non-associative complex algebra are automatically continuous (a reformulation of §8.1.2(iii)). Corollary 8.1.58 Let A be a complex H ∗ -algebra with zero annihilator. Then the mapping D → exp(D) is a homeomorphism from the set of all derivations D of A with D∗ = −D onto the set of all algebra automorphisms F of A with F = (F −1 )∗ and sp(F) ⊆ R+ . Proof The proof of this corollary is very similar to that of Lemma 3.4.72. Anyway, for the sake of completeness, we give here the details. By the holomorphic functional calculus, the mapping F → exp(F) is a homeomorphism from the set of those elements in BL(A) whose spectrum is included in the complex band {z ∈ C : −π < (z) < π } onto the set of those elements in BL(A) whose spectrum is included in C \ R− 0 . Now, let D be a derivation of A with D∗ = −D. In view of Theorem 8.1.56, we have that sp(D) ⊆ R, and hence, by the spectral mapping theorem, sp(exp(D)) ⊆ R+ . Since clearly [(exp(D))−1 ]∗ = exp(D), we have in fact that F := exp(D) is an algebra automorphism of A with F = (F −1 )∗ and sp(F) ⊆ R+ . Conversely, let F be an algebra automorphism of A with F = (F −1 )∗ and sp(F) ⊆ R+ . Then there exists a unique element D ∈ BL(A) such that sp(D) ⊆ R and F = exp(D). Moreover, since exp(D∗ ) = F ∗ = F −1 = exp(−D), and sp(D∗ ) ⊆ R (by the equality (3.4.28) in §3.4.71, and Example 1.1.32(d)), we get D∗ = −D. Finally, by Theorem 3.4.49, D is a derivation of A.  A straightforward consequence of the preceding corollary is the following.

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Corollary 8.1.59 Let A be a complex H ∗ -algebra with zero annihilator and denote by P the set of all algebra automorphisms F of A with F = (F −1 )∗ and sp(F) ⊆√R+ . Then for each F in P there is a unique G in P such that G2 = F (we write G = F). Corollary 8.1.60 Let A and P be as in the above corollary. Then for F in P we have that F • = F. Proof Write F = exp(D) with D a derivation of A such that D∗ = −D (Corollary 8.1.58). Then D• = D (Theorem 8.1.56). Now F • = (exp(D))• = exp(D• ) = exp(D) = F.



Lemma 8.1.61 Let A and B be semi-H ∗ -algebras over K, and let F : A → B be a continuous algebra homomorphism. Then for all y ∈ A and z ∈ B we have that F • (z)y = F • (zF ∗ (y)). Proof

Let x, y be in A, and let z be in B. Then (F(xy)|z) = (xy|F • (z)) = (x|F • (z)y∗ )

and (F(xy)|z) = (F(x)F(y)|z) = (F(x)|z(F(y))∗ ) = (x|F • (z(F(y))∗ )). Therefore F • (z)y∗ = F • (z(F(y))∗ ). Replacing y with y∗ , the result follows.



We recall that, by Theorem 8.1.52, dense-range algebra homomorphisms from a complete normed complex algebra to a complex H ∗ -algebra with zero annihilator are continuous. Theorem 8.1.62 Let A and B be complex H ∗ -algebras, and let F : A → B be a denserange algebra homomorphism. Suppose that A has zero annihilator and that B is topologically simple. Then there is a positive number λ such that F ∗ F • = λIB . As a consequence, F is surjective. Proof Since the involutions of A and B are algebra involutions, it follows that F ∗ is an algebra homomorphism. Therefore, by Lemma 8.1.61, for y ∈ A and z ∈ B we have F ∗ F • (z)F ∗ (y) = F ∗ F • (zF ∗ (y)). Thus F ∗ F • (z)w = F ∗ F • (zw) for all z ∈ B and w ∈ F ∗ (A).

(8.1.11)

Since F ∗ (A)(= F(A)∗ ) is dense in B, we have also that (8.1.11) is valid for all w in B. An analogous argument shows that wF ∗ F • (z) = F ∗ F • (wz) and so F ∗ F • lies in the centroid of B and therefore F ∗ F • = λIB with suitable λ ∈ C (cf. Lemma 8.1.29). The complex number λ must be different from zero. For otherwise we would have F • (B) ⊆ ker(F ∗ ) = (ker(F))∗ = ker(F)

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(the last equality because ker(F) is a closed two-sided ideal of A so a self-adjoint subset) hence FF • = 0 and F = 0, which is a contradiction. It follows from F ∗ F • = λIB with λ = 0 that F is surjective. Now it only remains to show that λ lies in R+ . To this end, consider the equality A = ker(F) ⊕ (ker(F))⊥ and keep in mind that (ker(F))⊥ is an ideal of A (cf. Proposition 8.1.13(i)) and that the restriction of F to (ker(F))⊥ is a bijective algebra homomorphism from (ker(F))⊥ to B. Then, since (ker(F))⊥ is a new H ∗ -algebra with zero annihilator (cf. Proposition 8.1.13(iii)–(v)), replacing A with (ker(F))⊥ and F with its restriction to (ker(F))⊥ we can suppose that F is a bijective algebra homomorphism from A onto B. Then (F ∗ )−1 = λ−1 F • ⇒ (F ∗ )−1 F = λ−1 F • F ⇒ sp((F ∗ )−1 F) ⊆ λ−1 R+ . Now (F ∗ )−1 F is an algebra automorphism of A whose spectrum lies in λ−1 R+ . Therefore, as desired, λ ∈ R+ as a consequence of Theorem 3.4.42(ii).  Lemma 8.1.63 Let A and B be nonzero complex H ∗ -algebras with zero annihilator, and let F : A → B be a bijective algebra homomorphism. Write G := (F ∗ )−1 F. Then G is an algebra automorphism of A satisfying (G−1 )∗ = G and sp(G) ⊆ R+ . Proof It is clear that G is an algebra automorphism of A with (G−1 )∗ = G. In order to prove that sp(G) ⊆ R+ consider the family {Mα } of all minimal closed ideals of A. For each α write Nα := F(Mα ). Clearly Nα is a minimal closed ideal of B and the mapping Fα : xα → F(xα ) is a bijective algebra homomorphism from the topologically simple H ∗ -algebra Mα to the topologically simple H ∗ -algebra Nα so • (Fα∗ )−1 Fα = λ−1 α Fα Fα

for suitable λα ∈ R+ (Theorem 8.1.62). But, if we consider A as the 2 -sum of the family of H ∗ -algebras {Mα } (cf. Corollary 8.1.18), then with standard notation like • that in [723, p. 19] G is the operator ((Fα∗ )−1 Fα ) = (λ−1 α Fα Fα ) which is obviously a + positive operator on the Hilbert space of A. So sp(G) ⊆ R because 0 ∈ / sp(G).  Theorem 8.1.64 Let A and B be complex H ∗ -algebras with zero annihilator, and let F : A → B be a bijective algebra homomorphism. Then F can be written in a unique way as F = F2 F1

(8.1.12)

with F2 : A → B a bijective algebra ∗-homomorphism and F1 an algebra automorphism of A satisfying (F1−1 )∗ = F1 and sp(F1 ) ⊆ R+ . Proof Assume that there is a decomposition like that in (8.1.12) and, according to Corollary 8.1.58, write F1 = exp(D) with D a derivation of A such that D∗ = −D. Then we have (F ∗ )−1 F = (F1∗ )−1 (F2∗ )−1 F2 F1 = (F1∗ )−1 F1 = exp(2D).

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Since 2D is a derivation of A with (2D)∗ = −2D, it follows from a new application of Corollary 8.1.58 that D (and hence F1 ) is uniquely determined by F. Since F2 = F exp(−D), the uniqueness of the decomposition (8.1.12) follows. To show the existence of a decomposition like that in (8.1.12), write G := (F ∗ )−1 F. Then G is an algebra automorphism of A with (G−1√)∗ = G and sp(G) ⊆ R+ (Lemma 8.1.63). Consider the algebra automorphism F1 := G of A, in the sense of Corollary 8.1.59, which satisfies also (F1−1 )∗ = F1 and sp(F1 ) ⊆ R+ , and let F2 : A → B be the bijective algebra homomorphism defined by F2 := FF1−1 . Of course F = F2 F1 . Now it is easily deduced that F2 is a bijective algebra ∗-homomorphism.  Corollary 8.1.65 Let A and B be topologically simple complex H ∗ -algebras. Then there is a positive number K (K = 1 if A = B) such that for every bijective algebra homomorphism F : A → B we have that F• = K(F ∗ )−1 . Proof For every bijective algebra homomorphism G from a topologically simple complex H ∗ -algebra X to another Y there is a positive number λ(G) such that G• = λ(G)(G∗ )−1 (Theorem 8.1.62). Also the equality λ(HG) = λ(H)λ(G) is easily verified if H is another bijective algebra homomorphism from Y to a (topologically simple) H ∗ -algebra Z. Now our corollary states that λ(F) = λ(E) for any two bijective algebra homomorphisms F and E from A to B. To prove this, suppose first that A = B so F is an algebra automorphism of A. Write F = F2 F1 as in Theorem 8.1.64. Then F1 belongs to P (notation as in Corollary 8.1.59), so by Corollary 8.1.60 F1• = F1 = (F1∗ )−1 and λ(F1 ) = 1. Since F2 is an algebra ∗-automorphism of A we have F2• = λ(F2 )F2−1 so (λ(F2 ))− 2 F2 is a unitary operator on the Hilbert space of A and so 1

1

sp((λ(F2 ))− 2 F2 ) ⊆ {z ∈ C : |z| = 1}. By a new application of Theorem 3.4.42(ii), we obtain that λ(F2 ) = 1 and so λ(F)(= λ(F2 )λ(F1 )) = 1 for every algebra automorphism F of A. Now let F and E be two bijective algebra homomorphisms from A to B. Then F −1 E is an algebra automorphism of A, so λ(F −1 E) = 1 and therefore λ(E) = λ(FF −1 E) = λ(F)λ(F −1 E) = λ(F), as required.



Corollary 8.1.66 The spectrum of any algebra automorphism of a topologically simple complex H ∗ -algebra A is invariant under the mapping z → 1z from C \ {0} to C \ {0}. Proof Argue as in the proof of Corollary 8.1.57 with Corollary 8.1.65 instead of Theorem 8.1.56. 

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Corollary 8.1.67 Let A and B be topologically simple complex H ∗ -algebras. Then there is a positive number L (L = 1 if A = B) such that, for every bijective algebra ∗-homomorphism F : A → B, the mapping LF is isometric. Proof

1

Let K be as in Corollary 8.1.65 and write L := K − 2 .



Corollary 8.1.68 Let A be a topologically simple complex H ∗ -algebra, and let τ be a ∗-involution on A. Then τ is isometric. Proof The opposite algebra A(0) of A is a complex H ∗ -algebra in a natural way (cf. §8.1.4), and τ : A → A(0) becomes a bijective algebra ∗-homomorphism. Therefore, by Corollary 8.1.67, there is a positive number L such that Lτ is an isometry. Since  τ 2 = IA , we deduce that L = 1. §8.1.69 Complex H ∗ -algebras with zero annihilator have a certain essential uniqueness of the H ∗ -algebra structure. Indeed, as a consequence of Theorem 8.1.64, isomorphic complex H ∗ -algebras with zero annihilator are ∗-isomorphic. The situation improves in the topologically simple particular case because, by Corollary 8.1.67, bijective algebra ∗-homomorphisms between topologically simple complex H ∗ -algebras are positive multiples of isometries. No more can be said because, by multiplying the inner product of a given (semi-)H ∗ -algebra by a positive number different from 1, one obtains a new (semi-)H ∗ -algebra which is obviously ∗-isomorphic to the given one, but is not isometrically ∗-isomorphic. Remark 8.1.70 Some of the H ∗ -algebras appearing in the statements of this subsection have been supposed to be topologically simple. It must be noticed that this assumption cannot be replaced by the weaker one that the H ∗ -algebra has zero annihilator. Let A (respectively, B) be the complex H ∗ -algebra whose Hilbert space is the familiar space 2 of all sequences of complex numbers {λn }  such that {λn }2 = |λn |2 < ∞, with involution {λn }∗ = {λn } and product {λn }{μn } = {n−1 λn μn } (respectively, {λn }{μn } = {λn μn }). Then A and B are (associative and commutative) H ∗ -algebras with zero annihilator, and the mapping F : {λn } → {n−1 λn } is a non surjective dense-range algebra ∗-homomorphism from A to B (compare with Theorem 8.1.62). Now let A be the algebra C × C with involution (λ, μ)∗ := (λ, μ) and inner product ((λ1 , μ1 )|(λ2 , μ2 )) := λ1 λ2 + 2μ1 μ2 . Then A is an (associative and commutative) H ∗ -algebra with zero annihilator and the mapping F : (λ, μ) → (μ, λ) is an algebra ∗-automorphism of A which is not a scalar multiple of an isometry (compare with Corollaries 8.1.65 and 8.1.67). 8.1.6 Jordan axioms for associative H∗ -algebras The next result resembles Theorem 3.6.25. Proposition 8.1.71 Let A be a flexible complex ∗-algebra, and suppose that the algebra Asym , obtained by symmetrization of the product of A, is an H ∗ -algebra with

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zero annihilator for some inner product and the given involution. Then A with the same inner product and involution is an H ∗ -algebra. Proof By Lemma 2.4.15, for any a ∈ A, the mapping Da : b → ab − ba is a derivation of Asym , and hence, by Theorem 8.1.56, we have D•a = −D∗a = Da∗ . This means that (ab − ba|c) = (b|a∗ c − ca∗ ) for all a, b, c ∈ A. Since Asym is an H ∗ -algebra, we have that (ab + ba|c) = (b|a∗ c + ca∗ ), and so (ab|c) = (b|a∗ c) and (ab|c) = (a|cb∗ ) for all a, b, c ∈ A.  Lemma 8.1.72 Let A be an associative semiprime algebra over K. Then Asym has zero annihilator. sym

Proof Let x be in Ann(Asym ). Then we have 0 = UxA semiprimeness of A.

(A) = xAx, hence x = 0 by 

Lemma 8.1.73 Let A be a C∗ -algebra, and let B be a subalgebra of A consisting only of normal elements of A. Then B is commutative. Proof In view of Lemma 1.2.12, we have b = r(b) for every b ∈ B. Therefore the result follows from Lemma 3.6.27.  The above lemma is a straightforward consequence of Proposition 6.1.13. Nevertheless the autonomous proof we have just given has its own interest, since it avoids the notion of a normal element of a ( possibly non-associative) ∗-algebra over K (cf. Definition 3.4.20), as well as the characterization of normal elements of a noncommutative JB∗ -algebra proved in Fact 3.4.22. Lemma 8.1.74 Let A be an associative semiprime complex algebra endowed with a conjugate-linear vector space involution ∗ satisfying (ab)∗ = a∗ b∗ for all a, b ∈ A, and suppose that Asym is an H ∗ -algebra for some inner product and the given involution ∗. Then A is commutative. Proof Keeping in mind Lemma 8.1.72, the notation and arguments at the beginning of the proof of Proposition 8.1.71 now give that D•a = −D∗a = −Da∗ . Since Asym is an H ∗ -algebra, this implies that (ab|c) = (b|ca∗ ) for all a, b.c ∈ A. This means that, for every a ∈ A, the adjoint operator of La is Ra∗ . Therefore, since A is associative, La is a normal element in the C∗ -algebra BL(A) for every a ∈ A. But, again by associativity of A, LA is a subalgebra of BL(A). It follows from Lemma 8.1.73 that LA is a commutative algebra. Finally, since A is an associative semiprime algebra, the mapping a → La is an injective algebra homomorphism, hence A is commutative, as desired.  Lemma 8.1.75 Let A be a flexible algebra over K such that Asym has zero annihilator. Then direct summands of Asym are ideals of A. Proof By Lemmas 2.4.15 and 8.1.40, direct summands of Asym are invariant under the mappings Da : b → ab − ba from A to A, when a runs over A. 

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Now we can conclude the proof of the main result in this subsection, namely the following. Theorem 8.1.76 Let A be an associative semiprime complex algebra, and suppose that Asym is an H ∗ -algebra for a suitable inner product and a suitable involution ∗. Then A, endowed with the same inner product and involution, becomes an H ∗ -algebra. Proof By Lemma 8.1.72, Asym has zero annihilator. Therefore, in view of Proposition 8.1.71, it is enough to show that ∗ is an algebra involution on A. Suppose first that A is prime. Then, since ∗ is a conjugate-linear algebra involution on Asym , the mapping x → x∗ from AR to AR is a surjective Jordan homomorphism. Therefore, by Proposition 3.6.29 (see [743, Theorem 3.1] for a proof), it is either an algebra homomorphism or an algebra antihomomorphism. In both cases, keeping in mind Lemma 8.1.74, ∗ becomes an algebra involution on A, as desired. Now remove the assumption that A is prime. Let M be a minimal closed ideal of Asym . Then, by Proposition 8.1.13(v), M is ∗-invariant. On the other hand, by Proposition 8.1.13(i), M is a direct summand of Asym , and hence, by Lemma 8.1.75, M is an ideal (so a subalgebra) of A. Moreover, M, considered as an associative algebra, is prime because M sym is topologically simple (hence prime) and Lemma 6.1.76(ii) applies. By the preceding paragraph, ∗ is an algebra involution on M.  Finally, by Theorem 8.1.16 applied to Asym , ∗ is an algebra involution on A. The above theorem becomes a reasonable H ∗ -variant of Theorem 3.6.30. 8.1.7 Real versus complex H∗ -algebras Let B be a complex (semi-)H ∗ -algebra. Then the underlying real algebra BR , obtained by restriction of the base field, can and will be considered as a real (semi-) H ∗ -algebra (called the (semi-)H ∗ -algebra realification of B) under the involution ∗ of B and the real inner product (·|·). Clearly closed ∗-invariant subalgebras of the real (semi-)H ∗ -algebra BR are new examples of real (semi-)H ∗ -algebras. But we are interested in the following apparently particular case of this last situation. Let  be any continuous involutive conjugate-linear algebra ∗-automorphism of B. Then H(B, ) is a closed ∗-invariant subalgebra of BR , so a real (semi-)H ∗ -algebra. We remark that, as a consequence of Corollary 8.1.54, the assumption of continuity for  is superfluous if B has zero annihilator. Actually, if B is in fact a topologically simple H ∗ -algebra, then  is an isometry. (Indeed,  ◦ ∗ is a ∗-involution on B, and Corollaries 8.1.12(iii) and 8.1.68 apply.) We note that if  is isometric, then for x, y ∈ H(B, ) we have x2 + y2 − 2[i(x|y)] = x + iy2 = x − iy2 = x2 + y2 + 2[i(x|y)], hence (x|y) is a real number, and the passing to (·|·) in the real (semi-)H ∗ -algebra H(B, ) is unnecessary. Proposition 8.1.77 Let A be a real (semi-)H ∗ -algebra. Then there exists an essentially unique couple (B, ), where B is a complex (semi-)H ∗ -algebra and  is

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an isometric involutive conjugate-linear algebra ∗-automorphism of B, satisfying A = H(B, ). Moreover, we have Ann(B) = Ann(A) ⊕ i Ann(A). Proof Suppose that some couple (B, ) satisfies the requirements in the statement. Then, as a complex algebra, B = A ⊕ iA cannot be other than the complexification of A (cf. pp. 31–2 of Volume 1). On the other hand, by the comments immediately before, the (real-valued) inner product of A is the restriction to A of the (complexvalued) inner product of B, and hence the inner product of B must coincide with the extension to B by sesquilinearity of the inner product of A. Moreover, clearly, the involution ∗ of B must coincide with the extension to B by conjugate linearity of the involution of A. This shows the essential uniqueness of the couple (B, ). To prove the existence of such a couple we follow the reverse way. Indeed, it is straightforward to realize that the complexification AC of A becomes naturally a complex (semi-)H ∗ algebra with involution ∗ equal to the extension by conjugate-linearity of the one of A, and inner product equal to the extension by sesquilinearity of the one of A. By taking B equal to the complex (semi-)H ∗ -algebra just described, and  equal to the canonical involution of B = AC (cf. Lemma 1.1.97), the couple (B, ) satisfies the requirements in the statement. Finally, the equality Ann(B) = Ann(A) ⊕ i Ann(A) is of straightforward verification.  Let A be a real (semi-)H ∗ -algebra. The complex (semi-)H ∗ -algebra B in the above proposition will be called the (semi-)H ∗ -algebra complexification of A, and the mapping  will be called the canonical involutive conjugate-linear algebra ∗-automorphism of this last (semi-)H ∗ -algebra. Corollary 8.1.78 Let B be a complex (semi-)H ∗ -algebra, and let  be a continuous involutive conjugate-linear algebra ∗-automorphism of B. Then B is bicontinuously ∗-isomorphic to a suitable complex (semi-)H ∗ -algebra in such a way that  converts into an isometry. Proof Write A := H(B, ). Then A is a real (semi-)H ∗ -algebra whose (semi-)H ∗ algebra complexification is bicontinuously ∗-isomorphic to B in such a way that  converts into the canonical involutive conjugate-linear algebra ∗-automorphism of this last algebra, which is isometric thanks to Proposition 8.1.77.  The above corollary is only a bi-product of Proposition 8.1.77 with no special significance. The actual interest of the proposition is that it allows us to transfer to real (semi-)H ∗ -algebras many results known to now only for complex (semi-)H ∗ algebras. As a first application, we can prove the definitive version of assertion (i) in §8.1.2. Corollary 8.1.79 Jordan semi-H ∗ -algebras over K with zero annihilator are H ∗ algebras. Proof We may suppose that K = R. So let A be a Jordan real semi-H ∗ -algebra with zero annihilator. Then the semi-H ∗ -algebra complexification of A (say B) is a

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Jordan complex semi-H ∗ -algebra with zero annihilator, and hence an H ∗ -algebra by the previously known complex case. Since A is a closed ∗-invariant subalgebra of  BR , A is indeed an H ∗ -algebra. Now we can derive the definitive version of Corollary 8.1.20. Corollary 8.1.80 Let A be a Jordan-admissible semi-H ∗ -algebra over K, and suppose that Asym has zero annihilator. Then A is an H ∗ -algebra. Proof

Combine the above corollary with Fact 8.1.19.



The following definitive version of Corollary 8.1.22 is derived from Corollary 8.1.80 in the same way as Corollary 8.1.22 was derived from Corollary 8.1.20. Corollary 8.1.81 Let A be a topologically simple non-commutative Jordan semi-H ∗ algebra over K which is not anticommutative. Then A is an H ∗ -algebra. Now, thinking about more outstanding results, we can derive the definitive version of Theorems 8.1.41, 8.1.52, and 8.1.53. Corollary 8.1.82 Let A be a semi-H ∗ -algebra over K with zero annihilator, and let D be a derivation of A. Then D is continuous. Proof We may suppose that K = R. Then the semi-H ∗ -algebra complexification of A (say B) is a complex semi-H ∗ -algebra with zero annihilator. On the other hand, the extension of D to B by complex linearity is a derivation of B, and hence is continuous by the previously known result for the complex case. Therefore D is continuous, as desired.  Corollary 8.1.83 Let A be a complete normed algebra over K, let B be an H ∗ algebra over K with zero annihilator, and let  : A → B be a dense-range algebra homomorphism. Then  is continuous. Proof We may suppose that K = R. Let AC stand for the normed complexification of A (cf. Proposition 1.1.98), and let BC stand for the H ∗ -algebra complexification of B. Then AC is a complete normed complex algebra, and BC is a complex H ∗ algebra with zero annihilator. On the other hand, the extension of  to AC by complex linearity becomes a dense-range algebra homomorphism from AC to BC , and hence is continuous by the previously known result for the complex case. Therefore  is continuous, as desired.  Corollary 8.1.84 Let A be a complete normed algebra over K, let B be a semi-H ∗ algebra over K with zero annihilator, and let  : A → B be an algebra homomorphism whose range is ∗-invariant and dense in B. Then  is continuous. The proof of the above corollary is left to the reader because, with minor changes, it is similar to that of Corollary 8.1.83. Now the definitive version of Corollary 8.1.54 is immediate.

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Corollary 8.1.85 Surjective algebra homomorphisms from complete normed algebras over K to semi-H ∗ -algebras over K with zero annihilator are continuous. Therefore every semi-H ∗ -algebra over K with zero annihilator has a unique complete algebra norm topology. Let A be a real algebra. Since both flexibility and associativity of A pass to its complexification AC , and (AC )sym = (Asym )C , it is enough to apply Proposition 8.1.77 to Asym to derive the following definitive versions of Proposition 8.1.71 and of Theorem 8.1.76. Corollary 8.1.86 For an algebra A over K, the following assertions hold: (i) If A is flexible and has a conjugate-linear algebra involution, and if Asym is an H ∗ -algebra for some inner product and the given involution, then A, endowed with the same inner product and involution, is an H ∗ -algebra. (ii) If A is associative and semiprime, and if Asym is an H ∗ -algebra for some inner product and some involution, then A, endowed with the same inner product and involution, is an H ∗ -algebra. Now we are going to deal with more elaborate consequences of Proposition 8.1.77. Proposition 8.1.87 Let B be a topologically simple complex (semi-)H ∗ -algebra. Then the (semi-)H ∗ -algebra realification of B is a topologically simple real (semi-) H ∗ -algebra. Moreover, if  is any (automatically isometric) involutive conjugatelinear algebra ∗-automorphism of B, then H(B, ) is a topologically simple real (semi-)H ∗ -algebra. Proof Let M be a closed ideal of BR . Then M ∩ iM is a closed ideal of B so, by the topological simplicity of B, we have either M ∩ iM = B, in which case M = BR , or M ∩ iM = 0. From this last equality we deduce MB = M(iB) = i(MB) ⊆ M ∩ iM = 0 and, analogously, BM = 0 so M ⊆ Ann(B) = 0. Therefore BR is topologically simple. Now let  be an involutive conjugate-linear algebra ∗-automorphism of B, and write A := H(B, ). We know that A is a real (semi-)H ∗ -algebra. Let M be a closed ideal of A. Since B = A ⊕ iA, and this direct sum is topological, it becomes clear that M + iM is a closed ideal of B, hence either M + iM = B (and so M = A) or M + iM = 0 (and so M = 0). Therefore A is topologically simple.  Our next result will show that the topologically simple real (semi-)H ∗ -algebras obtained in Proposition 8.1.87 are the unique topologically simple real (semi-) H ∗ -algebras. Theorem 8.1.88 Let A be a topologically simple real (semi-)H ∗ -algebra. Then either A is the (semi-)H ∗ -algebra realification of a topologically simple complex (semi-) H ∗ -algebra, or there exist a topologically simple complex (semi-)H ∗ -algebra B and an involutive conjugate-linear algebra ∗-automorphism  of B such that A = H(B, ).

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Proof Let (B, ) be the couple given by Proposition 8.1.77, so that A = H(B, ). If the complex (semi-)H ∗ -algebra B is topologically simple, then we are in the last situation of the statement. Otherwise B has a nonzero closed proper ideal M. If M were -invariant, then M ∩ A would be a nonzero closed proper ideal of A, contrary to the topological simplicity of A. Therefore such an ideal M is not -invariant, and hence M ∩ M  = 0, so M and M  are mutually orthogonal (by Proposition 8.1.13 and Lemma 5.1.1), and then B = M ⊕ M  because M ⊕ M  is a nonzero closed -invariant ideal of B. Now from the fact that M is an arbitrary nonzero closed proper ideal of B it follows easily that M and M  are the unique nonzero closed proper ideals of B and, as a consequence, that M is a topologically simple complex (semi-)H ∗ -algebra. Now routine verification shows that the mapping m → m + m is a total isomorphism (isometric bijective algebra ∗-homomorphism) from the (semi-)H ∗ -algebra realification of M (once its inner product has been multiplied previously by two) onto A.  The proof of the above theorem shows that the two possibilities in the statement for the topologically simple real (semi-)H ∗ -algebra A are mutually exclusive because they depend on whether or not the (semi-)H ∗ -algebra complexification of A is topologically simple. The following corollary exhibits more explicitly this fact. Corollary 8.1.89 Let A be a topologically simple real (semi-)H ∗ -algebra. Then the centroid of A is isomorphic to C or R depending on whether A is the realification of a topologically simple complex (semi-)H ∗ -algebra or there is a topologically simple complex (semi-)H ∗ -algebra B and an involutive conjugate-linear algebra ∗-automorphism  of B such that A = H(B, ). The proofs of the above corollary and of Lemma 8.1.93 need some new notions and an auxiliary lemma. Let A be an algebra over K. The subspace of L(A) consisting of all quasicentralizers on A will be called the quasi-centroid of A, and will be denoted by q-A . By a ring centralizer (respectively, ring quasi-centralizer) on A we mean an additive mapping F : A → A such that F(ab) = F(a)b = aF(b) (respectively, F(a)b = aF(b)) for all a, b ∈ A. Now let B be another algebra over K. By a ring homomorphism from A to B we mean an additive mapping G : A → B such that G(ab) = G(a)G(b). Lemma 8.1.90 Let A and B be algebras over K with zero annihilator. Then we have: (i) Ring (quasi-)centralizers on A are (quasi-)centralizers. (ii) For each bijective ring homomorphism G : A → B there is a bijective additive mapping  G : q-A → q-B such that G(F(a)) =  G(F)(G(a)) for all a ∈ A and F ∈ q-A . G Moreover, we have  G(A ) = B and, regarded as a mapping from A to B ,  becomes a bijective ring homomorphism. (iii) If K = C, then q-AR = q-A and AR = A .

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Proof To prove (i) it is enough to show that all ring quasi-centralizers on A are linear. Let F be a quasi-centralizer on A. Then for a, b ∈ A and λ ∈ K we have (λF(a))b = λaF(b) = F(λa)b and analogously b(λF(a)) = bF(λa), hence λF(a) − F(λa) ∈ Ann(A) = 0. Assertion (ii) is a formalization of (i). Let G : A → B be a bijective ring homomorphism. Then, given a mapping F : A → A, we realize that GFG−1 is a ring (quasi-) centralizer on B if and only if F is a ring (quasi-)centralizer on A. Therefore, writing  G(F) := GFG−1 for F ∈ q-A , and invoking (i) for both A and B, assertion (ii) follows straightforwardly. Suppose that K = C. Then, since ring (quasi-)centralizers on A and on AR are the same, assertion (iii) follows from (i).  Proof of Corollary 8.1.89 Let A be a topologically simple real (semi-)H ∗ -algebra. If A is the realification of a topologically simple complex (semi-)H ∗ -algebra, then by Lemmas 8.1.29 and 8.1.90(iii), we have A = C. Otherwise, by Theorem 8.1.88, A = H(B, ) for a suitable topologically simple complex (semi-)H ∗ -algebra B and an involutive conjugate-linear algebra ∗-automorphism  of B. Since B = A ⊕ iA, every element of A can be extended by complex linearity to an element of B . Therefore A = R because, again by Lemma 8.1.29, B = C.  Now we are going to show the real versions of the main results in Subsection 8.1.5. We note that the formulations of Corollary 8.1.91 and of Theorem 8.1.92 involve Corollary 8.1.82. Corollary 8.1.91 For every derivation D of an H ∗ -algebra A over K with zero annihilator we have that D• = −D∗ . Proof Argue as in the proof of Corollary 8.1.82, with Theorem 8.1.56 instead of Theorem 8.1.41.  We note that the above corollary refines assertion (vii) in Lemma 8.1.55 in a substantial way. Now we are going to prove that algebraically isomorphic topologically simple real H ∗ -algebras are actually totally isomorphic. The first step to this end is the following theorem on decomposition of isomorphisms between real H ∗ -algebras with zero annihilator, which implies that isomorphic real H ∗ -algebras with zero annihilator are ∗-isomorphic. Theorem 8.1.92 Let A and B be H ∗ -algebras over K with zero annihilator, and let F : A → B be a bijective algebra homomorphism. Then F can be written in a unique way as F = G exp(D) with G : A → B a bijective algebra ∗-homomorphism and D a derivation of A such that D∗ = −D. Proof If K = C, then the result follows from Corollary 8.1.58 and Theorem 8.1.64. Suppose that K = R. Consider the H ∗ -algebra complexifications AC and BC of A and B, respectively, and the mapping FC : AC → BC obtained by extending F

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by complex linearity. Then FC is a bijective algebra homomorphism, and hence, by the above commented complex case, FC can be written in a unique way as FC = G0 exp(D0 ) where G0 : AC → BC is a bijective algebra ∗-homomorphism and D0 is a derivation of AC with D∗0 = −D0 . Now, let  denote indistinctly the canonical involutive conjugate-linear algebra ∗-automorphisms of AC and of BC . Then, with    the notation in §3.4.71, we have FC = FC = (G0 exp(D0 )) = G0 exp(D0 ). Since    G0 : AC → BC is a bijective algebra homomorphism with (G0 )∗ = (G∗0 ) = G0 , and   ∗  D0 is a derivation of AC with (D0 ) = (D∗0 ) = −D0 , it follows from the uniqueness   of the decomposition for FC that G0 = G0 and D0 = D0 . (Note that the equality (T  )∗ = (T ∗ ) , applied above with T = G0 and T = D0 , follows from the fact that the involutions  on both AC and BC are ∗-mappings.) Therefore G0 (A) = B and D0 (A) ⊆ A. Now, if we consider the mappings G : A → B and D : A → A defined by G(x) := G0 (x) and D(x) := D0 (x) for every x ∈ A, then they satisfy the requirements in the statement of the theorem. The uniqueness of the decomposition for F follows  easily from the uniqueness of the decomposition for FC . Our next purpose is to show that bijective algebra ∗-homomorphisms between given topologically simple real H ∗ -algebras are constant positive multiples of isometries. The proof depends on Theorem 8.1.88 and the following two lemmas. Actually Lemma 8.1.94 is a small refinement of Corollary 8.1.65. Lemma 8.1.93 Let A and B be topologically simple complex semi-H ∗ -algebras. Then any bijective algebra homomorphism from AR to BR is, as a mapping from A to B, either linear or conjugate-linear. Proof Let F : AR → BR be a bijective algebra homomorphism. Since F : A → B is a bijective ring homomorphism, Lemma 8.1.90(ii) applies, so that F induces a bijective ring homomorphism u : A → B such that F(λx) = u(λ)F(x) for all x ∈ A and λ ∈ A . Since A = C = B (cf. Lemma 8.1.29), the proof is concluded by observing that in our case u is real-linear.  We note that the above lemma remains true with ‘central complex algebras’ instead of ‘topologically simple complex semi-H ∗ -algebras’. The formulation of the following lemma involves assertion (iii) in §8.1.2. Lemma 8.1.94 Let A and B be topologically simple complex H ∗ -algebras. Then there is a positive number K (K = 1 if A = B) such that for every bijective algebra homomorphism or antihomomorphism F : A → B we have that F• = K(F ∗ )−1 . Proof By Corollary 8.1.65, there is a positive number K (K = 1 if A = B) such that F • = K(F ∗ )−1 for every bijective algebra homomorphism F : A → B, and also, by regarding each bijective algebra antihomomorphism from A to B as a bijective algebra homomorphism from A to the H ∗ -algebra B(0) , there is a positive number L such that G• = L(G∗ )−1 for every bijective algebra antihomomorphism G : A → B. Therefore we must show that if A and B are at the same time isomorphic

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517

and antiisomorphic, then K = L. First suppose that A = B, and let F be an algebra antiautomorphism of A. Then, since F 2 is an automorphism, we have (F 2 )• = ((F 2 )∗ )−1 and also clearly (F 2 )• = L2 ((F 2 )∗ )−1 so L = 1 in this particular case. In general let G : A → B be a bijective algebra antihomomorphism, and let H be an algebra antiautomorphism of A. Then GH : A → B is a bijective algebra homomorphism, so we have K((GH)∗ )−1 = (GH)• = H • G• = L(H ∗ )−1 (G∗ )−1 = L((GH)∗ )−1 , and so K = L as required.



The formulation of the following theorem involves Corollary 8.1.85. Theorem 8.1.95 Let A and B be topologically simple H ∗ -algebras over K. Then there is a positive number K (K = 1 if A = B) such that for every bijective algebra homomorphism F : A → B we have that F• = K(F ∗ )−1 . Proof In view of Corollary 8.1.65, we may suppose that K = R, and obviously we may also suppose that A and B are isomorphic. If A and B are the H ∗ -algebra realifications of suitable complex H ∗ -algebras, then the equality H • = K(H ∗ )−1 is true for some positive number K and every complex-linear bijective algebra homomorphism or antihomomorphism H : A → B (cf. Lemma 8.1.94). But if F : A → B denotes any real-linear bijective algebra homomorphism, then, by Lemma 8.1.93, F is either complex-linear or conjugate-linear, in which case the mapping x → F(x)∗ is a complex-linear bijective algebra antihomomorphism. In both cases, keeping in mind for the second one that the H ∗ -algebra involution of B is isometric, we have clearly F • = K(F ∗ )−1 . In view of Theorem 8.1.88 and Corollary 8.1.89, it only remains to consider the case of the existence of topologically simple complex H ∗ -algebras C and D with involutive conjugate-linear algebra ∗-automorphisms, both denoted by , such that A = H(C, ) and B = H(D, ). Now, since C = A ⊕ iA and D = B ⊕ iB, each bijective algebra homomorphism from A to B extends in a unique way to a bijective algebra homomorphism from C to D, so in this case the statement of our theorem follows straightforwardly from the analogous one for complex algebras (cf. Lemma 8.1.94 again).  Now, arguing as in the proof of Lemma 8.1.94, with Theorem 8.1.95 instead of Corollary 8.1.65, we get the following. Corollary 8.1.96 Let A and B be topologically simple H ∗ -algebras over K. Then there is a positive number K (K = 1 if A = B) such that for every bijective algebra homomorphism or antihomomorphism F : A → B we have that F• = K(F ∗ )−1 . Corollary 8.1.97 Let A and B be topologically simple H ∗ -algebras over K. Then there is a positive number L (L = 1 if A = B) such that, for every bijective algebra ∗-homomorphism or ∗-antihomomorphism F : A → B, the mapping LF is isometric. Now the comments in §8.1.69, on the essential uniqueness of the H ∗ -algebra structure of complex H ∗ -algebras with zero annihilator and, in particular, of topologically

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simple complex H ∗ -algebras, remain valid verbatim replacing ‘complex’ with ‘real’. Indeed, it is enough to replace Theorem 8.1.64 and Corollary 8.1.67 with Theorem 8.1.92 and Corollary 8.1.97, respectively, in the quoted comments. To conclude the discussion about the real versions of the main results in Subsection 8.1.5, let us prove the definitive version of Theorem 8.1.62, the formulation of which involves Corollary 8.1.83. Corollary 8.1.98 Let A and B be H ∗ -algebras over K, and let F : A → B be a denserange algebra homomorphism. Suppose that A has zero annihilator and that B is topologically simple. Then there exists a positive number λ such that F ∗ F • = λIB . As a consequence, F is surjective. Proof We may suppose that K = R. Then, arguing as in the proof of Theorem 8.1.62, we realize that there exists μ ∈ B \{0} such that F ∗ F • = μ. Since B is a field (cf. Theorem 8.1.88 and Corollary 8.1.89), μ is a bijective mapping, and therefore F ∗ (and hence F) is surjective. Now write C := (ker(F))⊥ and G := F|C . Then C is an H ∗ -algebra over K (cf. assertions (i) and (v) in Proposition 8.1.13) and G : C → B is a bijective algebra homomorphism. Therefore, by Theorem 8.1.95, there is a positive number λ such that G∗ G• = λIB . Moreover, denoting by P the orthogonal projection of A onto C regarded as a mapping from A to C, we have PP• = IC , P∗ = P (because C and C⊥ are ∗-invariant subsets of A), and F = GP. Therefore F ∗ F • = G∗ P∗ P• G• = G∗ PP• G• = G∗ IC G• = G∗ G• = λIB .



Let A be a normed ∗-algebra over K. By an H ∗ -ideal of A we mean any ideal I of A such that there exists an H ∗ -algebra B over K and a continuous algebra ∗-homomorphism φ : B → A satisfying φ(B) = I. Lemma 8.1.99 Let A be a normed ∗-algebra over K, and let I, J be H ∗ -ideals of A. Then I ∩ J is an H ∗ -ideal of A. Proof Take H ∗ -algebras B, C over K, and continuous algebra ∗-homomorphisms φ : B → A, ψ : C → A such that φ(B) = I and ψ(C) = J. Then B ⊕ 2 C becomes an H ∗ -algebra over K in a natural way, the set D := {(b, c) ∈ B ⊕ 2 C : φ(b) = ψ(c)} is a closed ∗-subalgebra of B ⊕ 2 C (hence an H ∗ -algebra over K), and the mapping η : D → A defined by η(b, c) := φ(b) becomes a continuous algebra ∗-homomorphism satisfying η(D) = I ∩ J.  Exercise 8.1.100 Let A be an algebra over K, let B be a topologically simple normed algebra over K, let I be a nonzero ideal of A, and let φ : A → B be an injective algebra homomorphism such that φ(A) is an ideal of B. Prove that φ(I) is dense in B. Theorem 8.1.101 Let A be a topologically simple normed ∗-algebra over K. Then A has at most one nonzero H ∗ -ideal. Proof Let I be any nonzero H ∗ -ideal of A. Take an H ∗ -algebra B over K, and a continuous algebra ∗-homomorphisms φ : B → A such that φ(B) = I. Then for every

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519

b ∈ Ann(B) we have φ(b)I = 0 = Iφ(b), and hence φ(b) ∈ Ann(A) = 0 because A is topologically simple and I is dense in A. Therefore Ann(B) ⊆ ker(φ), and consequently [ker(φ)]⊥ ⊆ [Ann(B)]⊥ . But, by Theorem 8.1.10 and Remark 8.1.11, [Ann(B)]⊥ is an H ∗ -algebra with zero annihilator for a suitable involution, in such a way that the restriction of φ to [Ann(B)]⊥ remains an algebra ∗-homomorphism. Since [ker(φ)]⊥ is a closed ideal of [Ann(B)]⊥ , it follows that [ker(φ)]⊥ is an H ∗ algebra in a natural way (cf. Proposition 8.1.13(v)) and that the restriction of φ to [ker(φ)]⊥ becomes an injective algebra ∗-homomorphism whose range equals I. Therefore, replacing B with [ker(φ)]⊥ , and φ with the restriction of φ to [ker(φ)]⊥ , we may and will suppose that φ is injective. Then B must be topologically simple since otherwise, by Proposition 8.1.13(i), there would exist two non-trivial complementary direct summands M and N of B, we would have φ(M)φ(N) = 0, and, by Exercise 8.1.100, we would conclude AA = 0, a contradiction. Now let J be any other nonzero H ∗ -ideal of A. Then we have I ∩ J = 0 because A is prime. Therefore, in view of Lemma 8.1.99, to prove as desired that J = I we may suppose that J ⊆ I. According to the preceding paragraph, take an H ∗ -algebra C over K, and a continuous injective algebra ∗-homomorphisms ψ : C → A such that ψ(C) = J. Then the chain of algebra homomorphisms ψ

φ −1

C −→ J → I −→ B, where the first and last arrows are bijective, provides us with an algebra homomorphism F : C → B whose range contains φ −1 (J). Since φ −1 (J) is dense in B (because B is topologically simple and Exercise 8.1.100 applies), it follows from Corollary 8.1.98 that F is surjective. Therefore the inclusion J → I is surjective, i.e., J = I.  Corollary 8.1.102 Let H be a nonzero complex Hilbert space. Then HS (H) is the unique nonzero H ∗ -ideal of K(H). Thus the C∗ -algebra K(H), of all compact operators on H, determines the H ∗ -algebra HS (H), of all Hilbert–Schmidt operators on H, in a non-spacial way. Proof Since HS (H) is obviously a nonzero H ∗ -ideal of K(H) (cf. Example 8.1.3), and K(H) is topologically simple (cf. Corollary 1.4.33), the result follows from Theorem 8.1.101.  We note that topologically simple complete normed complex ∗-algebras need not have nonzero H ∗ -ideals (see Remark 8.1.126(a)). Let A be an algebra over K. For an element a ∈ A, Ea will denote the mapping from M (A) into A defined by Ea (F) = F(a) for every F ∈ M (A). Given a mapping h and a subset S of its domain, hS will denote the restriction of h to S. Now we are going to prove the real version of Theorem 8.1.32. To this end some auxiliary results have to be shown previously. Lemma 8.1.103 Let A be a totally multiplicatively prime algebra over K, and let K be a positive number such that (8.1.6) holds for A. Then:

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(i) For every a ∈ A and every nonzero ideal P of M (A) we have Ka ≤ EaP  ≤ a.

(8.1.13)

(ii) For all F ∈ M (A) and a ∈ A, and every nonzero ideal P of M (A), we have P K 2 Fa ≤ WF,a .

(8.1.14)

Proof (i) Let a be in A and let P be a nonzero ideal of M (A). Take F ∈ P with F = 1. Then for every G ∈ M (A) with G = 1 we have WF,a (G) = FG(a) = EaP (FG) ≤ EaP FG ≤ EaP , hence WF,a  ≤ EaP , and therefore Ka ≤ EaP . The inequality EaP  ≤ a is obvious. (ii) Let F and a be in M (A) and A, respectively. and let P be a nonzero ideal of M (A). Take G in M (A) and T in P with G = T = 1. It follows from P (GT) that W P the equality WF,T(a) (G) = FGT(a) = WF,a F,T(a)  ≤ WF,a , hence P P KFT(a) ≤ WF,a , and therefore KFEa (T) ≤ WF,a . Now (8.1.14) follows from (8.1.13).  According to §3.6.53, given an algebra A over K, we denote by M  (A) the multiplication ideal of A, i.e. the subalgebra of L(A) generated by LA ∪ RA . Since clearly M (A) = KIA + M  (A), it follows that M  (A) is an ideal of M (A). Lemma 8.1.104 Let X be a complex vector space, and let S be a subset of L(X) such that CS ⊆ S. Then the subalgebras of L(X) and of L(XR ) generated by S coincide. As a consequence, for every complex algebra A we have that M  (A) = M  (AR ). Proof The proof of the first conclusion is left to the reader as an exercise. The  second one follows from the former by taking S = LA ∪ RA . Let X be a complex normed space, and let ' be a conjugation (i.e. a conjugatelinear involutive isometry) on X. According to the notation and facts in §3.4.71, by defining T ' := ' ◦ T ◦ ' for T ∈ BL(X), we are provided with an isometric involutive conjugate-linear algebra automorphism (denoted also by ') of BL(X) such that the normed real algebras BL(H(X, ')) and H(BL(X), ') are bicontinuously isomorphic in a natural way. Lemma 8.1.105 Let X be a complex pre-Hilbert space, let ' be a conjugation on X, let φ : BL(H(X, ')) → H(BL(X), ') be the natural bijective algebra homomorphism, and let Y and Z be '-invariant subspaces of BL(X) with Z ⊆ Y. Then the following assertions hold: (i) φ is an isometry. (ii) Z is SOT-dense in Y if and only if φ −1 (H(Z, ')) is SOT-dense in φ −1 (H(Y, ')). (iii) If BZ is SOT-dense in BY , then Bφ −1 (H(Z,')) is SOT-dense in Bφ −1 (H(Y,')) .

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Proof We note that, for F ∈ BL(H(X, ')), φ(F) is the unique element in BL(X) such that φ(F)(h) = F(h) for every h ∈ H(X, '). Consequently, for T ∈ H(BL(X), '), we have T(h) = φ −1 (T)(h) for every h ∈ H(X, '). Let F be in BL(H(X, ')). Then for h, k ∈ H(X, ') we have φ(F)(h + ik)2 = F(h) + iF(k)2 = F(h)2 + F(k)2 ≤ F2 (h2 + k2 ) = F2 h + ik2 , and hence φ(F) ≤ F. Since the converse inequality is clear, assertion (i) follows. Suppose that Z is SOT-dense in Y. Let F be in φ −1 (H(Y, ')). Then, since φ(F) ∈ H(Y, ') ⊆ Y, there is a net Tλ in Z such that Tλ (x) is norm-convergent to φ(F)(x) for every x ∈ X. As a consequence, Tλ (h) is norm-convergent to ' φ(F)(h) = F(h) for every h ∈ H(X, '). Now 12 (Tλ + Tλ ) is a net in H(Z, ') such that ' φ −1 ( 12 (Tλ + Tλ )) is SOT-convergent to F in BL(H(X, ')). Therefore φ −1 (H(Z, ')) is SOT-dense in φ −1 (H(Y, ')). In this way we have proved the ‘only if’ part of assertion (ii). Conversely, suppose that φ −1 (H(Z, ')) is SOT-dense in φ −1 (H(Y, ')). Let T be in Y, and write T = R + iS with R, S ∈ H(Y, '). Then since φ −1 (H(Z, ')) × φ −1 (H(Z, ')) is SOT × SOT-dense in φ −1 (H(Y, ')) × φ −1 (H(Y, ')), there is a net (Fλ , Gλ ) in φ −1 (H(Z, ')) × φ −1 (H(Z, ')) such that, for every h ∈ H(X, '), the nets Fλ (h) and Gλ (h) are norm-convergent to φ −1 (R)(h) = R(h) and φ −1 (S)(h) = S(h), respectively. Now φ(Fλ ) + iφ(Gλ ) is a net in Z such that, for all h, k ∈ H(X, '), the net [φ(Fλ ) + iφ(Gλ )](h + ik) norm-converges to T(h + ik), and hence φ(Fλ ) + iφ(Gλ ) SOT-converges to T in BL(X). Therefore Z is SOT-dense in Y. In this way we have proved the ‘if’ part of assertion (ii). With minor changes, the proof of (iii) is similar to that of the ‘only if’ part of assertion (ii). Indeed, it is enough to keep in mind that, if F belongs to BBL(H(X,')) , then, by (i), φ(F) lies in BBL(X) , and that, if T belongs to BBL(X) , then, again by (i), φ −1 ( 12 (T + T ' )) lies in BBL(H(X,')) .  Let H be a real Hilbert space, let X denote the Hilbert complexification of H, and let ' stand for the canonical involution of X. Then H(X, ') = H and, according to Definition 4.2.45, H(BL(X), ') becomes a real C∗ -algebra. Since the natural bijective algebra homomorphism φ : BL(H) → H(BL(X), ') is clearly a ∗-mapping, it follows from Lemma 8.1.105(i) that BL(H) is a real C∗ -algebra and that BL(X) is (a materialization of) the C∗ -complexification of BL(H) (cf. Definition 4.2.56). The following proposition contains the real versions of von Neumann’s bicommutant and Kaplansky’s density theorems. Proposition 8.1.106 Let H be a Hilbert space over K, and let B be a self-adjoint subalgebra of BL(H). We have: (i) If IH belongs to B, then the bicommutant of B in BL(H) coincides with the SOT-closure of B in BL(H). (ii) If B is SOT-dense in BL(H), then BB is SOT-dense in BBL(H) .

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Proof We may suppose that K = R. Let X denote the Hilbert complexification of H, and let ' stand for the canonical involution of X, so that we have H(X, ') = H. With the notation in the above lemma, we realize that Z := φ(B) + iφ(B) is a '-invariant self-adjoint subalgebra of the C∗ -algebra BL(X). Moreover, we have Z c = φ(Bc ) + iφ(Bc ) and then Z cc = φ(Bcc ) + iφ(Bcc ), where the commutants in the left- (respectively, right-) hand sides of these equalities are taken in BL(X) (respectively, in BL(H)). On the other hand, if IH ∈ B, then IX ∈ Z, and hence, by Theorem 8.1.30, Z cc coincides with the SOT-closure of Z in BL(X). Since H(Z, ') = φ(B) and H(Z cc , ') = φ(Bcc ), it follows from the ‘only if’ part of assertion (ii) in Lemma 8.1.105 that, if IH ∈ B, then Bcc is the SOT-closure of B in BL(H), which proves (i). Suppose that B is SOT-dense in BL(H). Then, by the ‘if’ part of assertion (ii) in Lemma 8.1.105, Z is SOT-dense in BL(X). Therefore, by Theorem 8.1.31, BZ is SOT-dense in BBL(X) . Finally, by Lemma 8.1.105(iii), BB is SOT-dense in BBL(H) . Now the proof of the proposition is complete.  Theorem 8.1.107 Let A be a topologically simple semi-H ∗ -algebra over K. Then A is totally multiplicatively prime. More precisely, we have WF,a  = Fa for all F ∈ M (A) and a ∈ A.

(8.1.15)

Proof We may suppose that K = R. Let us first suppose that there exists a topologically simple complex H ∗ -algebra B such that A is the H ∗ -algebra realification of B. Let F and a be in M (A) and A, respectively. Then, since M (A) = RIA + M # (A) ⊆ CIB + M # (B) = M (B), M (A)

WF,a could have two meaning, namely WF,a

M (B)

or WF,a

, depending on whether M (B)

WF,a acts on M (A) or M (B). But, by Theorem 8.1.32, we have WF,a Therefore, invoking Lemmas 8.1.104 and 8.1.103, it follows that M (A) M WF,a  ≥ WF,a

 (A)

M  = WF,a

 (B)

 = Fa.

 ≥ Fa.

M (A)

Since the inequality WF,a  ≤ Fa is clear, (8.1.15) follows in this case. Now, in view of Theorem 8.1.88, it only remains to consider the case of the existence of a topologically simple semi-H ∗ -algebra B and of an involutive conjugatelinear algebra ∗-automorphism ' of B such that A = H(B, '). Then, by Corollary 8.1.89, we have that A = RIA , and therefore M (A)cc = Ac = BL(A). Therefore, by Proposition 8.1.106, M (A) is SOT-dense in BL(A). Now, arguing as in the proof of Theorem 8.1.32, we realize that (8.1.15) also holds in this last case.  The following definitive version of Corollary 8.1.33 follows from Theorem 8.1.107 and Proposition 8.1.27. Corollary 8.1.108 Let A be a topologically simple semi-H ∗ -algebra over K. Then A is totally prime. More precisely, we have Na,b  = ρab for all a, b ∈ A, where ρ denotes the norm of the product of A.

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According to Proposition 8.1.77, the complexification of any real (semi-) becomes naturally a complex (semi-)H ∗ -algebra. We conclude this subsection by proving the following partial converse.

H ∗ -algebra

Theorem 8.1.109 Let A be a real algebra with zero annihilator and suppose that its complexification AC can be structured as an H ∗ -algebra. Then A can also be structured as an H ∗ -algebra. Proof Let us denote by τ : x + iy → x − iy the canonical involution of AC , and recall that τ is a conjugate-linear algebra automorphism of AC = A ⊕ iA. Let P stand for the set of all (linear) algebra automorphisms F of A such that F = (F −1 )∗ ∗ and sp(F) ⊆ R+ . Denoting by A C the H -algebra having the same structure of AC except for the product by complex numbers (defined by λx := λx) and the inner product (defined by (x|y) := (y|x)), and regarding τ as a (linear) bijective algebra homomorphism from AC to A C , we can apply Theorem 8.1.64 to obtain the existence of a unique conjugate-linear algebra ∗-automorphism T of AC and a unique ψ ∈ P such that τ = Tψ. Since the mapping G → TGT −1 is a conjugate-linear algebra automorphism of the complex algebra BL(AC ), for any G ∈ BL(AC ) the equality sp(TGT −1 ) = sp(G) holds. On the other hand, given any algebra automorphism η of AC , TηT −1 is an algebra automorphism of AC , and η−1 lies in P whenever η belongs to P. It follows that TPT −1 ⊆ P and that Tψ −1 T −1 lies in P. Now, since τ = Tψ and τ = τ −1 , we have that τ = ψ −1 T −1 = T −1 (Tψ −1 T −1 ). Since T −1 is a conjugate-linear algebra ∗-automorphism of AC , and Tψ −1 T −1 belongs to P, it follows from the uniqueness of the decomposition for τ that T = T −1 (so T is actually an involutive conjugate-linear algebra ∗-automorphism of AC ) and then that ψ = Tψ −1 T. By Corollary 8.1.59, there exists a unique ϑ ∈ P such that ϑ 2 = ψ. But Tϑ −1 T belongs to P and (Tϑ −1 T)2 = Tϑ −1 TTϑ −1 T = Tϑ −2 T = Tψ −1 T = ψ. Therefore Tϑ −1 T = ϑ, and hence τ = Tψ = Tϑ 2 = TTϑ −1 Tϑ = ϑ −1 Tϑ. It follows from the equality τ = ϑ −1 Tϑ just obtained that the real algebras H(AC , τ ), and H(AC , T) are isomorphic. The proof is concluded by observing that A = H(AC , τ ), whereas H(AC , T) is a closed ∗-invariant real subalgebra of the complex H ∗ -algebra AC , hence a real H ∗ -algebra.  8.1.8 Trace-class elements in H∗ -algebras The starting point for this subsection is the following. Fact 8.1.110 Let H be a nonzero complex Hilbert space, and let ||| · ||| denote the norm of H as well as the corresponding operator norm on BL(H). Let A stand for the complex H ∗ -algebra HS (H) (cf. Example 8.1.3), and let  ·  denote the norm of A as well as the corresponding operator norm on BL(A). Then we have ||| a ||| = La  = Ra  for every a ∈ A.

524

Selected topics in the theory of non-associative normed algebras

Proof As we already know since the proof of Proposition 8.1.36, HS (H) is a norm ideal on H. Therefore the result follows from Proposition 6.1.142(b).  With the above notation, Fact 8.1.110 shows that the norm of an element of A, regarded as a bounded linear operator on H, has an intrinsic (i.e. non-spacial) meaning relative to A. To build on this, let us recall the celebrated Schatten–von Neumann theorem (see [795, Appendix A.1.4], [809, Sections III.1 and IV.1], and [1182, Section 2.2]), which, again with the notation in Fact 8.1.110, reads as follows. Theorem 8.1.111 Let H be a nonzero complex Hilbert space, and consider the set T C (H) := AA ⊆ A (whose elements are called ‘trace-class operators’ on H). Then: (i) T C (H) is a ∗-invariant ideal of BL(H) containing F(H). (ii) There exists a unique linear form τ on T C (H) (called the ‘trace-form’) such that τ (ab) = (a|b∗ ) for all a, b ∈ A; the trace-form is commutative, i.e. the equality τ (ab) = τ (ba) holds for all a, b ∈ A, and satisfies τ ∗ = τ . (iii) There exists a complete algebra norm  · τ on T C (H) (called the ‘tracenorm’) making τ continuous and satisfying abcτ ≤ ||| a |||bτ ||| c ||| for every b ∈ T C (H) and all a, c ∈ BL(H); moreover,  ·  ≤  · τ on T C (H), b∗ τ = bτ for every b ∈ T C (H), and for a ∈ A we have that aa∗ τ = a2 . (iv) The normed algebra (T C (H),  · τ ) is topologically simple. (b)(c) := τ (cb∗ )  : (T C (H), ·τ ) → (K(H), ||| · |||) , defined by φ (v) The mapping φ for all c ∈ K(H) and b ∈ T C (H), becomes a bijective conjugate-linear isometry. (vi) The mapping ψ : (BL(H), ||| · |||) → (T C (H), ·τ ), defined by ψ(d)(b) := τ (d∗ b) for all d ∈ BL(H) and b ∈ T C (H), becomes a bijective conjugate-linear isometry. §8.1.112 According to assertions (v) and (vi) in the above theorem, regarded as Banach spaces, (T C (H),  · τ ) is the dual of (K(H), ||| · |||), as well as the unique predual of (BL(H), ||| · |||) (cf. Theorem 5.1.29(iv)). Since A is ||| · |||-dense in K(H) (by Corollary 1.4.33), it follows that (A, ||| · |||) becomes a ( possibly incomplete) natural predual of (T C (H),  · τ ), which is intrinsically determined by (A,  · ) thanks to Fact 8.1.110. Moreover, combining assertions (ii) and (v) in Theorem 8.1.111, we realize that, in this duality, operators in T C (H) ⊆ A identify with those elements b ∈ A such that the linear form ( · |b) on A is ||| · |||-continuous, and consequently the trace-norm  · τ of T C (H) converts into the dual norm of (A, ||| · |||) . This intrinsic characterization of (T C (H),  · τ ) into (A,  · ) will allow us in what follows to replace A (equal to HS (H) in our present discussion) with an arbitrary real or complex ( possibly non-associative) semi-H ∗ -algebra B with zero annihilator, to build an appropriate substitute of (T C (H),  · τ ) into B, which will

8.1 H ∗ -algebras

525

be denoted by (τ c(B),  · τ ) (see Theorem 8.1.116), and to discuss whether or not a  · τ -continuous ‘trace-form’ on τ c(B) does exist (see Corollary 8.1.137). Lemma 8.1.113 Let A be a normed algebra over K, and for a ∈ A write ||| a ||| := max{La , Ra }. Then for all a, b ∈ A we have ||| ab ||| ≤ a||| b ||| and ||| ba ||| ≤ a||| b |||. Therefore ||| · ||| is a (vector space) seminorm on A making the product of A separately continuous. Proof

Let a, b, c be in A. Then we have Lab (c) = (ab)c ≤ abc ≤ Rb ac ≤ ||| b |||ac,

hence Lab  ≤ ||| b |||a. Analogously, Rab (c) = c(ab) ≤ cab ≤ cRb a ≤ c||| b |||a, hence Rab  ≤ ||| b |||a. It follows that ||| ab ||| ≤ a||| b |||. By symmetry, we have also that ||| ba ||| ≤ a||| b |||.  The following lemma underlies the proof of Theorem 8.1.16. Lemma 8.1.114 Let H be a Hilbert space over K, let ||| · ||| be a continuous norm on H (say ||| · ||| ≤ K ·  on H for some positive constant K), let Y be the subspace of H consisting of those y ∈ H such that the linear form φ(y) := ( · |y) on H is ||| · |||-continuous, and for y ∈ Y write || y || := ||| φ(y) |||. Then (Y, || · ||) becomes a dual Banach space over K (cf. §5.1.5) with predual equal to the completion of (H, ||| · |||), and the inequality y ≤ K || y || holds for every y ∈ Y. Proof Clearly, the mapping φ : y → φ(y) from (Y, || · ||) to the dual space (H, ||| · |||) of (H, ||| · |||) is conjugate-linear and isometric. Moreover, since ||| · |||-continuous linear forms on H are  · -continuous, it follows from the Riesz–Fr´echet representation theorem that φ is surjective. Therefore, up to the identification (Y, || · ||) ≡ (H, ||| · |||) just described, (Y, || · ||) becomes a dual Banach space over K with (complete) predual equal to the completion of (H, ||| · |||). On the other hand, for y ∈ Y we have y2 = (y|y) = φ(y)(y) ≤ ||| φ(y) |||||| y ||| = || y || ||| y ||| ≤ K || y || y, and hence y ≤ K || y ||.



The formulation of the next lemma involves the notation and conclusion in Lemma 8.1.114. Lemma 8.1.115 For i = 1, 2, let Hi be a Hilbert space over K, let ||| · ||| be a continuous norm on Hi , let (Yi , || · ||) be the dual Banach space over K consisting of those y ∈ Hi such that the linear form φi (y) := ( · |y) on Hi is ||| · |||-continuous, and let F : H2 → H1 be a ||| · |||-continuous linear mapping. Then F is  · -continuous, the adjoint F • maps Y1 into Y2 , and F • , regarded as a mapping form (Y1 , || ·|| ) to (Y2 , || ·|| ), is w∗ -continuous (relative to the dualities with the canonical complete preduals) with || F • || = ||| F |||.

526

Selected topics in the theory of non-associative normed algebras

Proof

By the closed graph theorem, F is  · -continuous. Let F  : (H1 , ||| · |||) → (H2 , ||| · |||)

denote the transpose of F : (H2 , ||| · |||) → (H1 , ||| · |||). Then, for x ∈ H2 and y ∈ Y1 , we have (x|φ2−1 [F  (φ1 (y))]) = [F  (φ1 (y))](x) = φ1 (y)(F(x)) = (F(x)|y) = (x|F • (y)), and hence, up to the inclusion of Y2 ⊆ H2 , we have φ2−1 F  φ1 = (F • )|Y1 . Thus F • (Y1 ) ⊆ Y2 and, up the natural identifications, F • , regarded as a mapping from Y1 to Y2 , is nothing other than F  . Since F  is w∗ -continuous relative to the corresponding dualities of (Hi , ||| · |||) with the completion of (Hi , ||| · |||), so is F • (regarded as a mapping from Y1 to Y2 ). Finally, since φ2−1 F  φ1 = (F • )|Y1 , and the mappings φi are isometries between the corresponding Banach spaces, we have that || F • || = ||| F  ||| = ||| F |||.  Let A be a semi-H ∗ -algebra over K with zero annihilator. In the remaining part of this subsection, we will consider the norm ||| · ||| on (the vector space of) A given by ||| a ||| := max{La , Ra } for every a ∈ A. τ c(A) will stand for the subspace of A consisting of those elements b ∈ A such that the linear form φ(b) := ( · |b) on A is ||| · |||-continuous, and  · τ will denote the norm on τ c(A) defined by bτ := ||| φ(b) ||| for every b ∈ τ c(A). Elements of τ c(A) are called trace-class elements of A, and the norm  · τ is called the trace-norm on τ c(A). Theorem 8.1.116 Let A be a semi-H ∗ -algebra over K with zero annihilator. Then (τ c(A),  · τ ) becomes a dual Banach space in a natural way. Moreover, τ c(A) contains AA, hence it is a dense ideal of A, and the following assertions hold: (i) a ≤ ρaτ for every a ∈ τ c(A), where ρ stands for the norm of the product of A. (ii) abτ ≤ ab∗ , and hence abτ ≤ μab for all a, b ∈ A, where μ denotes the norm of the involution of A (cf. Corollary 8.1.12(i)). (iii) abτ ≤ μρabτ and baτ ≤ μρbτ a for all a ∈ A and b ∈ τ c(A). (iv) abτ ≤ μρ 2 aτ bτ for all a, b ∈ τ c(A). Furthermore, for each a ∈ A, the mappings b → ab and b → ba from τ c(A) to τ c(A) are w∗ -continuous. In particular the product of τ c(A) is separately w∗ -continuous (i.e. (τ c(A), w∗ ) is a topological algebra in the sense of Definition 6.1.93). Proof The norm ||| · ||| satisfies ||| · ||| ≤ ρ ·  on A. It follows from Lemma 8.1.114 that (τ c(A),  · τ ) becomes a dual Banach space in a natural way, and that assertion (i) holds.

8.1 H ∗ -algebras

527

Let a, b be in A. Then for every c ∈ A we have that |(c|ab)| = |(cb∗ |a)| ≤ cb∗ a ≤ Lc ab∗  ≤ ||| c |||ab∗ . Therefore ab lies in τ c(A) and abτ ≤ ab∗ . In this way we have shown that τ c(A) contains AA and that assertion (ii) holds. As any subspace of A containing AA, τ c(A) is a dense ideal of A, the denseness being a consequence of Corollary 8.1.12(ii). Assertions (iii) and (iv) are straightforward consequences of (i) and (ii). Let a be in A. Then, by Fact 8.1.1(i) and Lemma 8.1.113, La∗ : A → A and Ra∗ : A → A are ||| · |||-continuous. Therefore, by Lemma 8.1.115, La = (La∗ )• and Ra = (Ra∗ )• (cf. (8.1.4)) leave τ c(A) invariant and, regarded as operators from τ c(A) to itself, they are w∗ -continuous. Now the proof of the theorem is complete.  Remark 8.1.117 (a) Let A be a semi-H ∗ -algebra over K with zero annihilator. Then assertions (i) to (iv) in the above theorem have their respective topological consequences, namely: (i) The inclusion (τ c(A),  · τ ) → (A,  · ) is continuous. (ii) The mapping (a, b) → ab from (A, ·)×(A, ·) to (τ c(A), ·τ ) is continuous. (iii) The mappings (a, b) → ab and (a, b) → ba from (A,  · ) × (τ c(A),  · τ ) to (τ c(A),  · τ ) are continuous, i.e. up the multiplication of the norm  ·  by a suitable positive number, (τ c(A),  · τ ) becomes a complete normed (A,  · )bimodule (cf. §4.6.3). (iv) The mapping (a, b) → ab from (τ c(A),  · τ ) × (τ c(A),  · τ ) to (τ c(A),  · τ ) is continuous, i.e. up the multiplication of the norm  · τ by a suitable positive number, (τ c(A),  · τ ) becomes a complete normed algebra (cf. §1.1.3). (b) Again let A be a semi-H ∗ -algebra over K with zero annihilator, and let  denote the adjoint of ∗, i.e. the unique conjugate-linear mapping  : A → A satisfying (a|b∗ ) = (b|a ) for all a, b ∈ A. Then, clearly,  is an involutive mapping. Moreover, τ c(A) is -invariant, and we have: (v) a τ = aτ for every a ∈ τ c(A). Indeed, for a ∈ τ c(A) and b ∈ A, we have |(b|a )| = |(a|b∗ )| = |(b∗ |a)| ≤ aτ ||| b∗ |||. On the other hand, for b ∈ A, we have that Lb∗  = (Lb )•  = Lb , and analogously Rb∗  = Rb , hence ||| b∗ ||| = ||| b ||| for every b ∈ A.

(8.1.16)

Therefore, if a ∈ τ c(A), then a ∈ τ c(A), and a τ ≤ aτ . By the way, the reader can easily check that, for a, b ∈ A, the equality (ab) = b∗ a∗ holds. It follows from (v) that for all a1 , . . . , an , b1 , . . . , bn ∈ A we have   n n n n      ai bi = ai bi = (ai bi ) = b∗i a∗i . (8.1.17) i=1

τ

i=1

τ

i=1

τ

i=1

τ

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Selected topics in the theory of non-associative normed algebras

(c) Now let A be an H ∗ -algebra over K with zero annihilator. Then, by Corollary 8.1.12(iii), ∗ is isometric. This implies that μ = 1 in assertions (ii) to (iv) of Theorem 8.1.116 and that  = ∗, hence, by part (b) of the present remark, τ c(A) is a ∗-subalgebra of A, and we have (v’) a∗ τ = aτ for every a ∈ τ c(A). Corollary 8.1.118 Let A be a complete normed algebra over K, let B be an H ∗ algebra over K with zero annihilator, and let F : A → (τ c(B),  ·τ ) be a dense-range algebra homomorphism. Then F is continuous. Proof Let i stand for the inclusion mapping (τ c(B),  · τ ) → (B,  · ). Then, by Theorem 8.1.116, i is a continuous dense-range algebra homomorphism. Therefore i ◦ F : A → B is a dense-range algebra homomorphism, which is continuous thanks to Corollary 8.1.83. The continuity of F now follows from the closed graph theorem.  Proposition 8.1.119 Let A and B be semi-H ∗ -algebras over K with zero annihilator, and let F : A → B be a continuous algebra homomorphism. Then F(τ c(A)) ⊆ τ c(B), and F, regarded as a mapping from (τ c(A),  · τ ) to (τ c(B),  · τ ), is w∗ -continuous with Fτ ≤ FF ∗ . Proof

Let b be in B. Then, by Lemma 8.1.61, we have LF• (b) = F • Lb F ∗ , hence LF• (b)  ≤ F • Lb F ∗  ≤ F||| b |||F ∗ .

Analogously, we have RF• (b)  ≤ F||| b |||F ∗ , hence ||| F • (b) ||| ≤ F||| b |||F ∗ . Therefore F • is ||| · |||-continuous with ||| F • ||| ≤ FF ∗ . Now the result follows from Lemma 8.1.115.  The following corollary shows how, for a semi-H ∗ -algebra A with zero annihilator, τ c(A) is determined in a purely algebraic way. Corollary 8.1.120 Let A and B be semi-H ∗ -algebras over K with zero annihilator, and let F : A → B be a bijective algebra homomorphism. Then F(τ c(A)) = τ c(B). Proof By Corollary 8.1.85, F is continuous. Therefore the result follows by applying Proposition 8.1.119 to both F and F −1 .  In some particular cases, for example if A = HS (H) for some complex Hilbert space H (or more generally if A is any alternative complex H ∗ -algebra with zero annihilator), the above corollary becomes trivial. Indeed, by Theorem 8.1.111 (or Corollary 8.1.153), we have that τ c(A) = AA in this case. However, the equality τ c(A) = AA above is far from being true in general. Indeed, according to Theorem 8.1.125, there exists a topologically simple Lie H ∗ -algebra such that lin(AA) is not  · τ -dense in τ c(A). We recall that derivations of a semi-H ∗ -algebra with zero annihilator are automatically continuous (cf. Corollary 8.1.82).

8.1 H ∗ -algebras

529

Proposition 8.1.121 Let A be a semi-H ∗ -algebra over K with zero annihilator, and let D be a derivation of A. Then D leaves τ c(A) invariant, and D, regarded as a mapping from τ c(A) to itself, is w∗ -continuous with Dτ ≤ D + D∗ . Proof Let a be in A. Then, by the second equality in Lemma 8.1.55(iv), we have that D• La = LD• (a) − La D∗ , and hence LD• (a)  = D• La + La D∗  ≤ (D•  + D∗ )La  ≤ (D + D∗ )||| a |||. Analogously, we have RD• (a)  ≤ (D + D∗ )||| a |||, and therefore ||| D• (a) ||| ≤ (D + D∗ )||| a |||. Thus D• is ||| · |||-continuous with ||| D• ||| ≤ D + D∗ , and the proof is concluded by applying Lemma 8.1.115 again.  We note that the 1 -sum of any family of (complete) normed algebras over K is a (complete) normed algebra over K in a natural way, and recall that closed ideals of a semi-H ∗ -algebra with zero annihilator are semi-H ∗ -algebras in a natural way (cf. Proposition 8.1.13(v)). Theorem 8.1.122 Let A be a semi-H ∗ -algebra over K with zero annihilator, and let {Ai }i∈I be the family of all minimal closed ideals of A. Then, in a canonical 1 way, (τ c(A),  · τ ) is isometrically isomorphic to the 1 -sum ⊕ i∈I (τ c(Ai ),  · τ ). ∗ Moreover, the canonical bijective algebra homomorphism is w -bicontinuous. 2 Ai , and that, for Proof Note that, according to Corollary 8.1.18, we have A = ⊕ i∈I i ∈ I and ai ∈ Ai , ||| ai ||| has the same meaning in both A and Ai . Then observe that, if Fi is a bounded linear operator on Ai for each i ∈ I, and if the family {Fi } is bounded, then the mapping {ai } → {Fi (ai )} defines a bounded linear operator F on A with F = sup{Fi  : i ∈ I}. On the other hand, if {ai } is in A, then it follows  from the inequalities i∈I ai 2 < +∞ and ||| ai ||| ≤ ρai  that {ai } is an element c0 c0 of the c0 -sum ⊕i∈I (Ai , ||| · |||). Therefore the mapping  : (A, ||| · |||) → ⊕i∈I (Ai , ||| · |||) defined by ({ai }) := {ai } is a linear isometry. (Indeed, choose Fi equal to Lai or Rai in the observation above, and note that then F is equal to L{ai } or R{ai } , c00 c0 (Ai , ||| · |||) is dense in ⊕i∈I (Ai , ||| · |||) respectively.) Moreover, since the c00 -sum ⊕i∈I and is contained in the range of , it follows that  has dense range. Hence

 : (  A, ||| · |||) →

c0 = i , ||| · |||), (A i∈I

the unique continuous linear extension of  to the corresponding completions, is an isometric linear bijection. Therefore, up to natural identifications, the transpose mapping  : 

1 = i∈I

(τ c(Ai ),  · τ ) ≡

1 = i∈I

i , ||| · |||) → ( (A A, ||| · |||) ≡ (τ c(A),  · τ )

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Selected topics in the theory of non-associative normed algebras

is also an isometric linear bijection, which is w∗ -bicontinuous because 6 −1 ) . Finally, it follows easily from the definition of the adjoint mapping  )−1 = ( ( 1   that  is nothing other than the mapping {ai } → {ai } from ⊕ i∈I (τ c(Ai ),  · τ ) to

2 (τ c(A),  · τ ) ⊆ A = ⊕i∈I Ai , hence it is an algebra homomorphism, whose inverse mapping becomes the wanted isometric bijective algebra homomorphism.  and let P Lemma 8.1.123 Let A ⎫be a semi-H ∗ -algebra over K with zero ⎧ ⎧ annihilator, ⎫ be a closed



left ⎬ right ⎩ two-sided ⎭

ideal of (τ c(A),  · τ ). Then P is a



left ⎬ right ⎩ two-sided ⎭

ideal of A.

Proof Suppose that P is a closed left ideal of (τ c(A),  · τ ). Let x and a be in P and A, respectively. Then, by the denseness of τ c(A) in A assured by Theorem 8.1.116, there exists a sequence an in τ c(A) such that a =  · -limn an . Then, by assertion (ii) in Theorem 8.1.116, we have that ax =  · τ -limn an x. Therefore, since an x belongs to P for every n, and P is  · τ -closed in τ c(A), we see that ax lies in P. Therefore P is a left ideal of A. Similarly, if P is a closed right ideal of (τ c(A),  · τ ), then P is a right ideal of A.  Proposition 8.1.124 Let A be a semi-H ∗ -algebra over K with zero annihilator. Then the following conditions are equivalent: (i) τ c(A) is prime. (ii) τ c(A) has no direct summand other than zero and τ c(A). (iii) A is topologically simple. Moreover, if K = C, and if the above conditions are fulfilled, then τ c(A) is centrally closed. Proof The implication (i)⇒(ii) is clear, whereas the one (ii)⇒(iii) follows straightforwardly from Theorem 8.1.122 and Proposition 8.1.13(iv). To prove that (iii)⇒(i) we argue by contradiction. Suppose that τ c(A) is not prime. Then there are nonzero closed ideals P, Q of (τ c(A),  · τ ) with PQ = 0. Now suppose in addition that A is topologically simple. Then, by Lemma 8.1.123, P and Q are  · -dense in A, and hence AA = 0, the desired contradiction. Finally, suppose that K = C, and that A is topologically simple. Then, since τ c(A) is dense in A (cf. Theorem 8.1.116), it follows from Corollary 8.1.34 and Lemma 8.1.46 that τ c(A) is centrally closed.  Let A be a semi-H ∗ -algebra over K with zero annihilator. If (τ c(A),  · τ ) is topologically simple, then, by Proposition 8.1.124, A is topologically simple. As the next theorem shows, the converse need not be true. Theorem 8.1.125 Let H be any infinite-dimensional complex Hilbert space, and let A denote the closed real ∗-subalgebra of the Lie complex H ∗ -algebra [HS (H)]ant consisting of all skew-adjoint elements of HS (H). Then we have:

8.1 H ∗ -algebras

531

(i) A is a topologically simple Lie real H ∗ -algebra. (ii) τ c(A) = T C (H) ∩ A, and the topology of the trace-norm of τ c(A) coincides with the restriction to τ c(A) of the topology of the trace-norm of T C (H) (cf. Theorem 8.1.111). (iii) The linear hull of [A, A] is not dense in τ c(A) for the topology of the trace-norm of τ c(A). (iv) τ c(A) is not topologically simple for the topology of the trace-norm of τ c(A). Proof Assertion (i) is well-known. Indeed, it follows from Example 8.1.3, Fact 8.1.154, and Propositions 8.1.157 and 8.1.87 (the later with  = −∗). Let · denote the Hilbert–Schmidt norm of HS (H) as well as the corresponding operator norm on BL(HS (H)), and let ||| · ||| denote the norm of H as well as the corresponding operator norm on BL(H). Then, according to Fact 8.1.110, we have ||| F ||| = LF  = RF  for every F ∈ HS (H).

(8.1.18)

To determine the trace-class algebra (τ c(A),  · τ c(A) ) of A, for F ∈ A let us consider the bounded linear operator adF on (A,  · ) defined by adF (G) := [F, G] for every G ∈ A, and the norm ||| · |||A on A defined by ||| F |||A := adF . Since [F, G] = FG − GF ≤ FG + GF ≤ ||| F |||G + G||| F ||| = 2||| F |||G, we see that ||| F |||A ≤ 2||| F |||. On the other hand, if F ∈ A is a finite-rank operator on H, and if x is any norm-one element of H, then we can choose a norm-one element y ∈ H such that (y|F(H)) = 0 = (y|x), and then we see that x y − y x ∈ A and that [F, x

y−y

x] = F(x = F(x)

y−y

x) − (x

y+y

y−y



F (x) = F(x)

x)F y−y

F(x).

Moreover, keeping in mind that (u1 u2 |v1 v2 ) = (u1 |v1 )(v2 |u2 ) for all ui , vi , ∈ H (i = 1, 2) [809, Lemma II.1.5(vii)], we realize that x

y−y

x2 = 2 and [F, x

y−y

x]2 = 2F(x)2 .

Therefore F(x) ≤ ||| F |||A , hence ||| F ||| ≤ ||| F |||A because of the arbitrariness of x ∈ SH . Now, if F is an arbitrary element in A, then taking a sequence Fn in A of finiterank operators on H  · -converging to F, and keeping in mind the continuity of the norm and of the product of A, as well as the fact that ||| · ||| ≤  ·  on HS (H), we get that ||| Fn |||A → ||| F |||A and ||| Fn ||| → ||| F |||, and we conclude that ||| F ||| ≤ ||| F |||A . Summarizing, we have proved that ||| F ||| ≤ ||| F |||A ≤ 2||| F ||| for every F ∈ A.

(8.1.19)

Let T be in T C (H) ∩ A. Then, denoting by  · τ the trace-norm on T C (H), and keeping in mind §8.1.112 and (8.1.18), for each S ∈ A we have that |(S|T)| ≤ ||| S |||Tτ ≤ ||| S |||A Tτ ,

532

Selected topics in the theory of non-associative normed algebras

and hence T ∈ τ c(A) and Tτ c(A) ≤ Tτ . Therefore T C (H) ∩ A ⊆ τ c(A). Conversely, if T ∈ τ c(A) and S ∈ HS (H), then, writing S = S1 + iS2 with S1 , S2 ∈ A, we clearly have that ||| Si ||| ≤ ||| S ||| (i = 1, 2), and hence |(S|T)| ≤ |(S1 |T)| + |(S2 |T)| ≤ ||| S1 |||A Tτ c(A) + ||| S2 |||A Tτ c(A) ≤ 2(||| S1 ||| + ||| S2 |||)Tτ c(A) ≤ 4||| S |||Tτ c(A) . Therefore T ∈ T C (H) and Tτ ≤ 4Tτ c(A) . Thus τ c(A) ⊆ T C (H) ∩ A. Summarizing again, we have proved that τ c(A) = T C (H) ∩ A, and that Tτ c(A) ≤ Tτ ≤ 4Tτ c(A) for every T ∈ τ c(A), which proves assertion (ii). Since the trace-form τ on T C (H) is a  · τ -continuous commutative linear form (cf. again Theorem 8.1.111) which does not vanish on τ c(A), it follows from (ii) that lin([A, A]) is contained in the proper  · τ c(A) -closed subspace ker(τ ) ∩ τ c(A) of τ c(A), and assertion (iii) follows. Finally, since lin([A, A]) is a nonzero ideal of τ c(A), assertion (iv) follows from assertion (iii).  Remark 8.1.126 (a) Let H be a complex Hilbert space, let (A,  · ) stand for the topologically simple complex H ∗ -algebra HS (H), and suppose that there exists a nonzero H ∗ -ideal I of (τ c(A), ∗,  · τ ) (cf. the paragraph immediately before Lemma 8.1.99). Take a complex H ∗ -algebra B, together with a continuous algebra ∗-homomorphism φ : B → (τ c(A), ∗,  · τ ) such that φ(B) = I, and let i denote the inclusion (τ c(A),  · τ ) → (A,  · ). Since (τ c(A),  · τ ) is topologically simple, and i is a continuous dense-range algebra homomorphism (by §8.1.112 and Theorem 8.1.116), it follows that i ◦ φ : B → (A,  · ) is a dense-range algebra homomorphism. Therefore, by Corollary 8.1.98, i ◦ φ is surjective, which implies that τ c(A) = A, and then that  ·  and  · τ are equivalent norms on τ c(A) (by the Banach isomorphism theorem). Now suppose that H is infinite-dimensional. We claim that (τ c(A), ∗,  · τ ) has no nonzero H ∗ -ideal. Indeed, otherwise, by the preceding paragraph, the norms  ·  and  · τ would be equivalent on τ c(A). But this is certainly impossible. For, given an arbitrary natural number n, we can find pairwise orthogonal norm-one elements x1 , . . . , xn ∈ H, and hence, by [809, Theorems II.1.2 and III.1.2], we have n  k=1

xk

xk =



n and

n  k=1

xk

= n.

xk τ

(b) Let H be any infinite-dimensional complex Hilbert space. We claim that (T C (H),  · τ ) is a complete norm ideal on H, but is not an ultraprime normed algebra. Indeed, we already know that (T C (H),  · τ ) is a complete norm ideal on H (see Theorem 8.1.111 and [809, Lemma III.1.9(iii)]). Now, if the normed algebra (T C (H),  · τ ) were ultraprime, then by Proposition 6.1.142(a), the trace-norm  · τ and the operator norm ||| · ||| would be equivalent on T C (H). But this is not possible since, given an arbitrary natural number n, we can find pairwise orthogonal

8.1 H ∗ -algebras

533

norm-one elements x1 , . . . , xn ∈ H, and hence, by [809, Theorems I.1.1 and III.1.2],  n we have ||| nk=1 xk xk ||| = 1 and xk τ = n. k=1 xk We will show later that, given an associative complex H ∗ -algebra A with zero annihilator, there exists a unique  · τ -continuous linear form f on τ c(A) such that f (ab) = f (ba) = (a|b∗ ) for all a, b ∈ A (see Theorem 8.1.139(iii)–(iv)). In particular, this means that the bilinear form (a, b) → (a|b∗ ) on A can be ‘linearized’ by means of a continuous linear form f defined on some subspace X of A containing AA and endowed with a suitable topology. In what follows, we are going to discuss this result in the setting of ( possibly non-associative) semi-H ∗ -algebras. We begin by considering the case that X = A, ignoring any continuity condition on the linear form f . Let X be a normed space over K. The weak operator topology of BL(X) is defined as the topology of pointwise convergence in BL(X) when X is endowed with the weak topology, and will be denoted by WOT. Both the strong and weak operator topologies are Hausdorff locally convex topologies, and the following result (whose proof will not be discussed here) holds. Proposition 8.1.127 [726, Theorem VI.1.4] Let X be a Banach space over K, and let  be a linear form on BL(X). Then the following conditions are equivalent:  (i) There exist x1 , . . . , xn ∈ X and f1 , . . . , fn ∈ X  such that (T) = ni=1 fi (T(xi )) for every T ∈ BL(X). (ii)  is WOT-continuous. (iii)  is SOT-continuous. As a consequence, the WOT- and SOT-closed convex subsets of BL(X) are the same. Let A be a normed algebra over K. By a left approximate unit in A we mean a net uλ in A such that limλ uλ a = a for every a ∈ A. The notion of a right approximate unit explains by itself. We note that if A has a one-sided approximate unit, then A has zero annihilator. Lemma 8.1.128 Let A be a semi-H ∗ -algebra over K, and let C be a ∗-invariant convex subset of A. Suppose that A has a left approximate unit consisting of elements of C. Then A has a left approximate unit consisting of self-adjoint elements of C. Proof Let uλ be the left approximate unit whose existence has been supposed, and let a be in A. Then for every b ∈ A we have that lim(u∗λ a|b) = lim(a|uλ b) = (a|b). λ

Therefore a = w-lim u∗λ a. Since hλ := 12 (uλ + u∗λ ) ∈ H(A, ∗) ∩ C

λ

clearly a = w-lim uλ a, it follows that, writing for each λ, we have a = w-lim hλ a. Since a is arbitrary in A, the last equality means that IA = WOT-lim Lhλ , and hence IA belongs to the WOT-closure of the convex set LH(A,∗)∩C . By Proposition 8.1.127, IA lies in the SOT-closure of LH(A,∗)∩C and therefore there exists a net kμ in H(A, ∗) ∩ C such that a =  · -limμ kμ a for every a ∈ A, i.e. kμ is a left approximate unit in A consisting of self-adjoint elements of C. 

534

Selected topics in the theory of non-associative normed algebras

Proposition 8.1.129 For a semi-H ∗ -algebra A over K, the following conditions are equivalent: (i) A has a left approximate unit. (ii) There exists a linear form f on A such that f (ab) = (a|b∗ ) for all a, b ∈ A. Moreover, if the above conditions are fulfilled, then: (iii) A has a left approximate unit consisting of self-adjoint elements. (iv) The linear form f given by (ii) can be chosen in such a way that f  = f (cf. Remark 8.1.117(b)). Proof Suppose that A has a left approximate unit uλ . Then for all a1 , . . . , an , b1 , . . . , bn ∈ A we have    n n n      ∗ ∗ (ai |bi ) = lim (ai |uλ bi ) = lim ai bi  uλ ,  λ λ i=1

i=1

i=1

hence limλ (x|uλ ) exists for every x ∈ lin(AA), and the mapping g : x → limλ (x|uλ ) is a linear form on lin(AA) satisfying g(ab) = (a|b∗ ) for all a, b ∈ A. By extending g to a linear form on A, condition (ii) holds. Conversely, suppose that (i) does not hold. Then IA does not belong to the SOTclosure of the subspace LA . Therefore, by the Hahn–Banach theorem, there exists an SOT-continuous linear form  on BL(A) such that (La ) = 0 for every a ∈ A and (IA ) = 0. But, by Proposition 8.1.127 and the Riesz–Fr´echet representation  theorem, there are a1 , . . . , an , b1 , . . . , bn ∈ A such that (T) = ni=1 (T(ai )|b∗i ) for every T ∈ BL(A). It follows that 0 = (La∗ ) =

n  i=1

for every a ∈ A, hence

(La∗ (ai )|b∗i ) = n

i=1 ai bi

n n n    (a∗ ai |b∗i ) = (ai |ab∗i ) = (ai bi |a) i=1

i=1

i=1

= 0; and

n n   ∗ 0 = (IA ) = (IA (ai )|bi ) = (ai |b∗i ). i=1

i=1

This shows ostensibly that condition (ii) cannot hold. Now to conclude the proof it is enough to show that (i)⇒(iii) and that (ii)⇒(iv). But the first implication follows from Lemma 8.1.128 taking there C = A. Let f be any of the linear forms given by (ii). Then for a, b ∈ A we have f  (ab) = f ((ab) ) = f (b∗ a∗ ) = (b∗ |a) = (a|b∗ ). Therefore 12 ( f + f  ) is -invariant and satisfies the same requirement as that of f .  Corollary 8.1.130 For a semi-H ∗ -algebra A over K, the following conditions are equivalent: (i) A has an approximate unit. (ii) A is an H ∗ -algebra and there exists a linear form f on A such that f (ab) = (a|b∗ ) for all a, b ∈ A.

8.1 H ∗ -algebras

535

(iii) A is an H ∗ -algebra and has a left approximate unit. (iv) A has an approximate unit consisting of self-adjoint elements. Moreover, if the above conditions are fulfilled, then: (v) Each linear form f on A given by condition (ii) is commutative and associative, i.e. it satisfies f (ab) = f (ba) and f [(ab)c] = f [a(bc)] for all a, b, c ∈ A. (vi) The linear form f given by (ii) can be chosen in such a way that f ∗ = f . Proof (i)⇒(ii) By a part of the assumption (i) and the implication (i)⇒(ii) in Proposition 8.1.129, there exists a linear form f on A satisfying f (ab) = (a|b∗ ) for all a, b ∈ A. Moreover, by the whole assumption (i) (that there is an approximate unit uλ in A), for every a ∈ A we have a2 = lim(uλ a|a) = lim(uλ |aa∗ ) = lim(a∗ uλ |a∗ ) = a∗ 2 , λ

λ

λ

hence ∗ is an isometric mapping. Therefore, by Fact 8.1.1(ii), A is an H ∗ -algebra. (ii)⇒(iii) By the implication (ii)⇒(i) in Proposition 8.1.129. (iii)⇒(iv) Suppose that (iii) holds. Then, by the implication (i)⇒(iii) in Proposition 8.1.129, A has a left approximate unit consisting of self-adjoint elements. But, since ∗ is a continuous (cf. Corollary 8.1.12(i)) algebra involution, such a left approximate unit becomes also a right approximate unit. (iv)⇒(i) This is clear. Now that we have proved that conditions (i) to (iv) are equivalent, let us conclude the proof by showing that (ii) implies (v) and (vi). Suppose that condition (ii) is fulfilled, and note that (ii) implies that A has zero annihilator. If f is any linear form on A satisfying f (ab) = (a|b∗ ) for all a, b ∈ A, then it is enough to keep in mind that ∗ is an isometric (cf. Corollary 8.1.12(iii)) conjugate-linear algebra involution on A to obtain that f (ab) = (a|b∗ ) = (b|a∗ ) = f (ba) and f [(ab)c] = (ab|c∗ ) = (a|c∗ b∗ ) = (a|(bc)∗ ) = f [a(bc)] for all a, b, c ∈ A, and (v) is proved. On the other hand, by Remark 8.1.117(c), we have that  = ∗ on A. Therefore (vi) follows from the implication (ii)⇒(iv) in Proposition 8.1.129.  Now we are going to explore the possibility of ‘linearizing’ the bilinear form (a, b) → (a|b∗ ) on a semi-H ∗ -algebra A by means of a ‘very’ continuous linear form f defined on some subspace X of A containing AA and endowed with a suitable topology. Among the choices of X, we consider two cases, namely that of X = A endowed with the norm topology, and that of X equal to the dual Banach space (τ c(A),  · τ ) endowed with the w∗ -topology (cf. Theorem 8.1.116). Fact 8.1.131 For a semi-H ∗ -algebra A over K, the following assertions hold: (i) A has at most one left unit. (ii) If A has a left unit u, then A has zero annihilator, and the equality u∗ = u holds. (iii) If A has a left unit (or more generally, if there is some b ∈ A such that Lb is bijective), then:

536

Selected topics in the theory of non-associative normed algebras (a) τ c(A) = A. (b) The norms  · τ and  ·  are equivalent on A. (c) (τ c(A),  · τ ) is a reflexive Banach space, hence  · -,  · τ -, w-, and w∗ -continuous linear forms on (τ c(A),  · τ ) are the same.

Proof As we already noted in the general case of left approximate units, the existence of a left unit in A implies that A has zero annihilator (the first conclusion in (ii)). Let u, v be left units in A. Then (u − v)A = 0, hence, by Theorem 8.1.10(v), u − v ∈ Ann(A) = 0, and therefore u = v. This proves (i). Let u be a left unit in A. Then Lu = IA , hence, taking adjoints according to (8.1.4), we have that Lu∗ = (Lu )• = (IA )• = IA , and therefore u∗ is a left unit in A. The equality u∗ = u now follows from (i), and proves the second conclusion in (ii). Suppose that there is some b ∈ A such that Lb is bijective, and note that this assumption implies that A has zero annihilator. Then, by Theorem 8.1.116, for a ∈ A we have a = b[Lb−1 (a)] ∈ AA ⊆ τ c(A), hence A ⊆ τ c(A), and (iii)(a) follows. Now, by Remark 8.1.117(i), the identity mapping (τ c(A),  · τ ) → (A,  · ) is continuous, hence bicontinuous by the Banach isomorphism theorem. This proves (iii)(b). Finally, since (A,  · ) is a Hilbert space, (iii)(c) follows straightforwardly from (iii)(b).  We remember that, given a semi-H ∗ -algebra A over K with zero annihilator, (τ c(A),  · τ ) is a dual Banach space containing AA (cf. Theorem 8.1.116). Proposition 8.1.132 For a semi-H ∗ -algebra A over K, the following conditions are equivalent: (i) There exists a  · -continuous linear form g on A such that g(ab) = (a|b∗ ) for all a, b ∈ A. (ii) A has a left unit. (iii) A has zero annihilator, and there exists a w∗ -continuous linear form f on the dual Banach space (τ c(A),  · τ ) such that f (ab) = (a|b∗ ) for all a, b ∈ A. Moreover, if the above conditions are fulfilled, then τ c(A) = A, the linear forms g and f given by (i) and (iii) are unique, and the equalities g = f = f  hold. Proof (i)⇒(ii) By the assumption (i) and the Riesz–Fr´echet representation theorem, there exists u ∈ A such that (a|b∗ ) = (ab|u) = (a|ub∗ ) for all a, b ∈ A. Therefore u is a left unit in A. (ii)⇒(iii) Let u be the left unit in A whose existence is assured by the assumption (ii). Then A has zero annihilator, and the mapping f : a → (a|u) from (τ c(A),  · τ ) to K satisfies the requirements in condition (iii). (iii)⇒(i) Suppose that (iii) holds. Then, since w∗ -continuous linear forms on a dual Banach space identify with elements in its (complete) predual, it is enough to see f as an element of the completion of (A, ||| · |||) to realize that there exists a sequence un in A such that: (a) un is a Cauchy sequence in (A, ||| · |||). (b) f (a) = limn (a|un ) for every a ∈ A.

8.1 H ∗ -algebras

537

Since ||| a ||| = max{La , Ra } for every a ∈ A, if follows from (a) that Lun is a  · -Cauchy sequence in BL(A), hence, denoting by T its  · -limit, for all a, b ∈ A we have that (a|T(b∗ )) = limn (a|un b∗ ) = limn (ab|un ). But, by (b), we have that f (ab) = limn (ab|un ). Therefore (a|T(b∗ )) = f (ab) = (a|b∗ ) for all a.b ∈ A, hence T = IA . In this way we have shown that IA lies in the  · -closure of LA . Therefore, taking b ∈ A with IA − Lb  < 1, we know that Lb is a bijective operator on A, and hence, by Fact 8.1.131(iii), τ c(A) = A and f is  · -continuous. Thus condition (i) is fulfilled with g = f . The last conclusion in the proposition follows from Fact 8.1.131(iii), Corollary 8.1.12(ii), and the implication (ii)⇒(iv) in Proposition 8.1.129.  For the structure of semi-H ∗ -algebras having a left unit, the reader is referred to Corollary 8.2.52. We note that an algebra A has a unit if (and only if) it has a left unit and a right unit. Indeed if u and v are left and right units in A, respectively, then we have u = uv = v, and hence A has a unit. Corollary 8.1.133 For a semi-H ∗ -algebra A over K, the following conditions are equivalent: (i) A has a unit. (ii) A is an H ∗ -algebra with zero annihilator, and there exists a w∗ -continuous linear form f on (τ c(A),  · τ ) such that f (ab) = (a|b∗ ) for all a, b ∈ A. (iii) A is an H ∗ -algebra and has a left unit. Moreover, if the above conditions are fulfilled, then: (iv) The linear form f given by condition (ii) is unique and satisfies f ∗ = f , f (ab) = f (ba) and f [(ab)c] = f [a(bc)] for all a, b, c ∈ A. Proof (i)⇒(ii) By the implication (i)⇒(iii) in Corollary 8.1.130 and the implication (ii)⇒(iii) in Proposition 8.1.132. (ii)⇒(iii) By the implication (iii)⇒(ii) in Proposition 8.1.132. (iii)⇒(i) By the assumption (iii), the involution of A is an algebra involution, so that the adjoint of the left unit whose existence is assumed becomes a right unit. Finally, suppose that condition (i) holds. Then (iv) follows from the last conclusion in Proposition 8.1.132 and the implications (i)⇒(v) and (i)⇒(vi) in Corollary 8.1.130.  Remark 8.1.134 (a) Let B be a normed algebra over K which is also a dual Banach space in such a way that, for each b ∈ B, the operator Rb on B is w∗ continuous. Let uλ be a bounded left approximate unit in B. Then there are w∗ cluster points of the net uλ in B (thanks to the w∗ -compactness of closed balls in B), and one can straightforwardly verify that each of these points becomes a left unit in B. Examples of normed algebras B as above are the extremal algebra Ea(K) of a compact convex subset K of C (cf. p. 647 of Volume 1 and Theorem 4.6.74), all JBWalgebras, all non-commutative JBW ∗ -algebras (cf. Corollary 5.1.30(iii)), all normed

538

Selected topics in the theory of non-associative normed algebras

algebras whose normed space is reflexive (so in particular all semi-H ∗ -algebras), and the algebras (τ c(A),  · τ ) where A is any semi-H ∗ -algebra with zero annihilator (cf. Theorem 8.1.116). We note that, when B is anyone of the normed algebras in the examples just quoted, it satisfies the additional requirement that, for each b ∈ B, the operator Lb on B is w∗ -continuous. (b) Now let A be a semi-H ∗ -algebra. Then it follows from the above part of the present remark that the equivalent conditions (i) to (iii) in Proposition 8.1.132 are also equivalent to the following: (iv) (A,  · ) has a bounded left approximate unit. (v) A has zero annihilator and (τ c(A),  · τ ) has a bounded left approximate unit. (vi) A has zero annihilator and τ c(A) has a left unit. Analogously we realize that the equivalent conditions (i) to (iii) in Corollary 8.1.133 are also equivalent to the following: (vii) (A,  · ) has a bounded approximate unit. (viii) A has zero annihilator and (τ c(A),  · τ ) has a bounded approximate unit. (ix) A has zero annihilator and τ c(A) has a unit. In relation to conditions (vi) and (ix) above, we note that, by the denseness of τ c(A) in (A,  · ), a (left) unit of τ c(A) is a (left) unit of A. To conclude our discussion about the possibility of ‘linearizing’ the bilinear form (a, b) → (a|b∗ ) on a semi-H ∗ -algebra A, we are going to explore the possibility of such a ‘linearization’ by means of a  · τ -continuous linear form on τ c(A). A subset S of a normed algebra A over K is said to be operator-bounded if there is a positive number M such that ||| s ||| := max{Ls , Rs } ≤ M for every s ∈ S. When the positive constant M above merits emphasis, we say that S is operator-bounded by M. This notion applies in particular to (ranges of) nets in A. Proposition 8.1.135 For a semi-H ∗ -algebra A over K, the following conditions are equivalent: (i) A has an operator-bounded left approximate unit. (ii) A has zero annihilator, and there exists a ·τ -continuous linear form f on τ c(A) such that f (ab) = (a|b∗ ) for all a, b ∈ A. Moreover, if the above conditions are fulfilled, then: (iii) A has an operator-bounded left approximate unit consisting of self-adjoint elements. (iv) The linear form f given by condition (ii) can be chosen in such a way that f = f. Proof (i)⇒(ii) By the assumption (i), there is M > 0 and a left approximate unit uλ in A such that ||| uλ ||| ≤ M for every λ. Thus for every λ and all a1 , . . . , an , b1 , . . . , bn ∈ A we have

8.1 H ∗ -algebras

  n n      ai bi  (ai bi |uλ ) ≤   i=1

i=1

539

||| uλ ||| ≤ M

n 

ai bi

i=1

τ

, τ

and therefore     n  n  n n            ∗  ∗  ai bi  (ai |bi ) = lim  (ai |uλ bi ) = lim  (ai bi |uλ ) ≤ M     λ  λ  i=1

i=1

n

i=1

i=1

. τ

n

∗ i=1 (ai |bi )

In this way i=1 ai bi → becomes a (well-defined)  · τ -continuous linear form on lin(AA). By extending it via the normed version of the Hahn–Banach theorem, we obtain the desired linear form f in condition (ii). (ii)⇒(i) We argue by contradiction, hence we assume that (ii) holds but that (i) does not hold. Let f be the linear form given by (ii), and write M := f τ . Then, since (i) is not fulfilled, IA does not belong to the SOT-closure of the absolutely convex set S := {La : a ∈ A and ||| a ||| ≤ M}. It follows from the Hahn–Banach separation theorem (see for example [689, Corollary 34.4]) that there exists an SOT-continuous linear form  on BL(A) such that sup{|(S)| : S ∈ S } < |(IA )|. But, by Proposition  8.1.127, there are a1 , . . . , an , b1 , . . . , bn ∈ A such that (T) = ni=1 (T(ai )|b∗i ) for every T ∈ BL(A). It follows that  $ n F n     ∗ ∗ ∗ ∗  bi ai = sup  (a|bi ai ) : ||| a ||| ≤ M M   i=1 i=1 τ  F $ n    ∗  = sup  (La (ai )|bi ) : ||| a ||| ≤ M = sup{|(S)| : S ∈ S } < |(IA )|   i=1   n   n n          f (ai bi ) ≤ M ai bi , =  (ai |b∗i ) =      i=1

hence

n

∗ ∗ i=1 bi ai τ

i=1

i=1

τ

n

i=1 ai bi τ . But, by the equality (8.1.17) in Remark n ∗ ∗ i=1 bi ai τ = i=1 ai bi τ , the desired contradiction.

< n

8.1.117(b), we have Now to conclude the proof it is enough to show that (i)⇒(iii) and that (ii)⇒(iv). Suppose that (i) is fulfilled. Then there is M > 0 and a left approximate unit uλ in A such that ||| uλ ||| ≤ M for every λ. Then (iii) follows from Lemma 8.1.128 by taking there C = {a ∈ A : ||| a ||| ≤ M} (cf. assertion (8.1.16) in Remark 8.1.117(b)). Now suppose that (ii) holds. Then, keeping in mind Remark 8.1.117(v), and arguing as in the proof of the implication (ii)⇒(iv) in Proposition 8.1.129, assertion (iv) follows.  Remark 8.1.136 (a) Let A be a semi-H ∗ -algebra over K, and let M be a positive number. Then, looking at the proof of Proposition 8.1.135, we realize that A has a left approximate unit operator-bounded by M if and only if A has zero annihilator, and there exists a  · τ -continuous linear form f on τ c(A) such that f (ab) = (a|b∗ ) for all a, b ∈ A and f τ ≤ M.

540

Selected topics in the theory of non-associative normed algebras

(b) Again let A be a semi-H ∗ -algebra over K, and suppose that there exists a left approximate unit in A operator-bounded by 1. Then, for all a1 , . . . , an ∈ A we have n 

ai a∗i

i=1

= τ

n 

ai 2 .

(8.1.20)

i=1

Indeed, by Theorem 8.1.116(ii), the implication (i)⇒(ii) in Proposition 8.1.135, and part (a) of the present remark, we have  n  n n n     ∗ ∗ 2 ∗ ai ai ≤ ai ai τ ≤ ai  = f ai ai i=1

τ

i=1

≤ f τ

i=1 n  i=1

ai a∗i

≤ τ

i=1 n  i=1

ai a∗i

. τ

(c) Finally, let A be a semi-H ∗ -algebra over K having a left approximate unit and a right approximate unit, both operator-bounded by 1. Then we are in a position to apply part (b) of the present remark to both A and to the opposite semi-H ∗ -algebra A(0) of A, obtaining in this way that, for every a ∈ A, the equalities aa∗ τ = a2 = a∗ aτ hold. By replacing a with a∗ we get a∗  = a, and hence, by Fact 8.1.1(ii), A is an H ∗ -algebra. Corollary 8.1.137 For a semi-H ∗ -algebra A over K, the following conditions are equivalent: (i) A has an operator-bounded approximate unit. (ii) A is an H ∗ -algebra with zero annihilator, and there exists a  · τ -continuous linear form f on τ c(A) such that f (ab) = (a|b∗ ) for all a, b ∈ A. (iii) A is an H ∗ -algebra and has an operator-bounded left approximate unit. (iv) A has an operator-bounded approximate unit consisting of self-adjoint elements. Moreover, if the above conditions are fulfilled, then: (v) The linear form f given by condition (ii) satisfies f (ab) = f (ba) and f [(ab)c] = f [a(bc)] for all a, b, c ∈ A, and can be chosen in such a way that f ∗ = f . Proof (i)⇒(ii) By the implication (i)⇒(iii) in Corollary 8.1.130 and the implication (i)⇒(ii) in Proposition 8.1.135. (ii)⇒(iii) By the implication (ii)⇒(i) in Proposition 8.1.135. (iii)⇒(iv) By the second requirement in the assumption (iii) and the implication (i)⇒(iii) in Proposition 8.1.135, A has an operator-bounded left approximate unit hλ consisting of self-adjoint elements. Then, by the first requirement in the assumption (iii), hλ is an approximate unit. (iv)⇒(i) This is clear.

8.1 H ∗ -algebras

541

Finally, suppose that condition (ii) holds. Then (v) follows from the implication (ii)⇒(iv) in Proposition 8.1.135, Remark 8.1.117(c), and the argument in the proof of the implication (ii)⇒(v) in Corollary 8.1.130.  Now that our discussion about the possibility of ‘linearizing’ the bilinear form (a, b) → (a|b∗ ) on a semi-H ∗ -algebra A has been concluded, we are going to apply it, together with relevant results proved in Volume 1 of our work, to derive outstanding facts concerning alternative H ∗ -algebras. (We recall that, according to Proposition 8.1.23(ii), alternative semi-H ∗ -algebras with zero annihilator are H ∗ -algebras.) Lemma 8.1.138 Let X be a normed algebra over K, and let C be a convex subset of X. If there exists a left approximate unit in X consisting of elements of C (the closure of C in X), then there exists also a left approximate unit in X consisting of elements of C . Proof Assume that there is no left approximate unit in X consisting of elements of C . Then IX does not belong to the SOT-closure of LC in BL(X). Therefore there exists an SOT-continuous linear form  on BL(X) such that M := sup ({(Lc ) : c ∈ C }) < ((IX )). Since  is  · -continuous (because the strong operator topology of BL(X) is weaker than the norm topology), and the mapping a → La from (X,  · ) to (BL(X),  · ) is continuous, it follows that sup ({(Ld ) : d ∈ C }) = M < ((IX )). Thus IX cannot belong to the SOT-closure of LC in BL(X), i.e. there is no left approximate unit in X consisting of elements of C .  To be applied in the proof of Theorem 8.1.139, we note that, if A is an H ∗ -algebra, then, by (8.1.4) and (8.1.5), for a ∈ A we have Ra = [(La )∗ ]• , and hence, if in addition A has zero annihilator, then, by Corollary 8.1.12(iii), we have ||| a ||| := max{La , Ra } = La  = Ra . Another result which will be applied in the proof of Theorem 8.1.139 is the wellknown fact that, for a reflexive Banach space X, the closed unit ball of BL(X) is WOTcompact (see for example [726, Exercise VI.9.6]). Actually this follows from the more general result, already commented in Example 2.9.68, that, for a dual Banach space X, BL(X) identifies with the dual of a (possibly incomplete) normed space, namely the projective tensor product X∗ ⊗π X, in such a way that the corresponding weak∗ topology of BL(X) converts into the topology of pointwise convergence when X is endowed with its weak∗ topology. Concerning assertion (vi) in Theorem 8.1.139, we should say that by a minimal w∗ -closed L-summand of a dual Banach space we mean a nonzero w∗ -closed L-summand L which contains no nonzero w∗ -closed L-summand other than L. Concerning the formulations of Theorems 8.1.139 and 8.1.140, some remarks should be done. Let A be a normed algebra over K with zero annihilator. Then, according to Lemma 8.1.113, ||| · ||| is a vector space norm on A making the product

542

Selected topics in the theory of non-associative normed algebras

of A separately continuous. In the particular case that A is associative, the fact that Lab = La Lb and Rab = Rb Ra for all a, b ∈ A, allows us to realize that ||| · ||| is an algebra norm on A, and consequently to conclude that the completion of (A, ||| · |||) is a normed algebra ‘in a natural way’. Analogously, in the case that A is a Lie algebra, the fact that Ra = −La and Lab = [La , Lb ] for all a, b ∈ A implies that ||| ab ||| ≤ 2||| a |||||| b ||| for all a, b ∈ A, so that the product of A is ( jointly) ||| · |||-continuous and, up the multiplication of ||| · ||| by 2, the completion of (A, ||| · |||) is a normed algebra ‘in a natural way’. Nevertheless, in general, even if A is a complex H ∗ -algebra, we do not know whether the product of A is ||| · |||-continuous, so there is no way to extend it by continuity to the completion of (A, ||| · |||) (By assertion (8.1.16) in Remark 8.1.117(b), we only know that the involution of A is ||| · |||-isometric, so that it can be uniquely extended to an isometric conjugate-linear algebra involution on the completion of (A, ||| · |||).) Therefore, when in Theorem 8.1.139 (respectively, Theorem 8.1.140) we assert (respectively, suppose) that the complete predual of (τ c(A),  · τ ) is an alternative (respectively, a non-associative) C∗ -algebra ‘in a natural way’, we are asserting (respectively, assuming) that (A, ||| · ||| , ∗) is an alternative (respectively, a non-associative) ‘pre’-C∗ -algebra, i.e. a normed alternative (respectively, possibly non-associative) ∗-algebra such that ||| a∗ a ||| = ||| a |||2 for every a ∈ A. In particular we are asserting (respectively, assuming) that ||| · ||| is an algebra norm on A. Theorem 8.1.139 For an alternative complex H ∗ -algebra A with zero annihilator, the following assertions hold: (i) The (complete) predual of (τ c(A),  · τ ) becomes an alternative C∗ -algebra in a natural way. (ii) (A,  · ) has an approximate unit operator-bounded by 1 and consisting of selfadjoint elements. (iii) There exists a unique  · τ -continuous linear form f on τ c(A) such that f (ab) = (a|b∗ ) for all a, b ∈ A. (iv) The linear form f in assertion (iii) above satisfies f ∗ = f , f τ ≤ 1, f (ab) = f (ba), and f [(ab)c] = f [a(bc)] for all a, b, c ∈ A. n n ∗ 2 (v) For all a1 , . . . , an ∈ A we have that i=1 ai ai τ = i=1 ai  . (vi) A is associative if and only if the dual Banach space (τ c(A),  · τ ) has no eightdimensional minimal w∗ -closed L-summand. Proof Let C denote the  · -closure of LA in BL(A), then, since La∗ = (La )• and La•b = La • Lb for all a, b ∈ A, C becomes a closed self-adjoint subalgebra of BL(A)sym , and hence, by Facts 3.3.2 and 3.3.4, it is a JB∗ -algebra. Moreover, the mapping  : a → La is a dense-range algebra ∗-homomorphism from Asym to C. Therefore, by Proposition 4.5.36(ii), there exists an alternative C∗ -algebra B such that Bsym = C as JB∗ -algebras, and , regarded as a mapping from A to B, becomes an algebra ∗-homomorphism. Note that B = C as Banach spaces, and that the involution of B is the same as that of C (which, by the way, is being denoted by • since shortly before Proposition 8.1.23). Let denote the alternative product of B. Then, for a ∈ A we have ||| a ||| = La  = (a), and hence ||| a∗ a ||| = (a∗ a) = (a∗ )

(a) = (a)•

(a) = (a)2 = ||| a |||2 .

8.1 H ∗ -algebras

543

Analogously, for a, b ∈ A we have ||| ab ||| ≤ ||| a |||||| b |||. It follows that the completion of (A, ||| · |||) (equal to the predual of τ c(A)) becomes an alternative C∗ -algebra in the most natural way, which proves (i). Let (X, ||| · |||) denote the alternative C∗ -algebra completion of (A, ||| · |||). By Proposition 3.5.23, (X, ||| · |||) has an approximate unit bounded by 1 and consisting of self-adjoint elements. By Lemma 8.1.138 (with C = BH[(A,||| · |||),∗] ), (X, ||| · |||) has an approximate unit uλ bounded by 1 and consisting of self-adjoint elements of A. We claim that Luλ converges to IA in the weak operator topology of BL(A). Since BBL(A) is WOT-compact, and Luλ is a net in BBL(A) , it is enough to show that IA is the unique WOT-cluster point of Luλ in BL(A). Let T be a WOT-cluster point of Luλ , and let a, b be in A. Then (T(a)|b) is a cluster point of (uλ a|b). On the other hand, when b lies in τ c(A), we have that |(uλ a − a|b)| ≤ ||| uλ a − a |||bτ for every λ, hence (a|b) = limλ (uλ a|b). Keeping in mind that τ c(A) is dense in A, it follows that (T(a)|b) = (a|b) for all a, b ∈ A, and therefore T = IA , as desired. As a consequence of the claim just proved, IA belongs to the WOT-closure of the convex set {Lh : h ∈ H(A, ∗) and ||| h ||| ≤ 1}. Since WOT- and SOT-closed subsets of BL(A) are the same (cf. Proposition 8.1.127), there exists a left approximate unit in (A,  · ) operator-bounded by 1 and consisting of self-adjoint elements. Such a left approximate unit is an approximate unit because the involution of A is a continuous algebra involution. In this way, (ii) has been proved. The existence of a linear form on τ c(A) with the properties asserted in assertion (iii) and (iv) follows from assertion (ii), Corollary 8.1.137, and Remark 8.1.136(a), whereas the uniqueness of such a linear form follows from Corollary 8.1.153. Assertion (v) follows from (ii) and Remark 8.1.136(b). By the definitions of an M-ideal and of an L-summand (cf. Definition 5.1.12) and the bipolar theorem, w∗ -closed L-summands of a dual Banach space are in a natural order-reversing bijective correspondence with M-ideals of its predual. Therefore assertion (vi) follows from (i) and Theorem 6.2.2.  The above theorem has the following converse. Theorem 8.1.140 Let A be a complex H ∗ -algebra with zero annihilator. Suppose that (A,  · ) has an approximate unit operator-bounded by 1, and that the predual of (τ c(A),  · τ ) is a non-associative C∗ -algebra in the natural way. Then A is alternative. Proof Let uλ be the approximate unit in (A,  · ) whose existence has been supposed. Since the topology of the norm ||| · ||| on A is weaker than that of the natural norm  · , for every a ∈ A we have a = ||| · |||-limλ uλ a. Let (X, ||| · |||) denote the nonassociative C∗ -algebra completion of (A, ||| · |||), let x be in X, and let ε > 0. Take a ∈ A with ||| x − a ||| ≤ ε. Then, since the net uλ is operator-bounded by 1, for every λ we have ||| x − uλ x ||| ≤ ||| x − a ||| + ||| a − uλ a ||| + ||| uλ (a − x) ||| ≤ ε + ||| a − uλ a ||| + ||| uλ |||||| a − x ||| ≤ 2ε + ||| a − uλ a |||,

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and hence lim supλ ||| x − uλ x ||| ≤ 2ε. By letting ε → 0, we obtain that x = ||| · |||limλ uλ x. Analogously, x = ||| · |||-limλ xuλ . Since x is arbitrary in X, and the net uλ is operator-bounded by 1, it follows that uλ is an approximate unit in (X, ||| · |||) bounded by 1. Therefore, by Theorem 3.5.53, X is an alternative algebra. Since A ⊆ X as algebras, A is alternative.  Combining Theorems 8.1.139 and 8.1.140, we get the following non-associative characterization of alternative complex H ∗ -algebras with zero annihilator. Corollary 8.1.141 A complex H ∗ -algebra A with zero annihilator is alternative if and only if (A,  · ) has an approximate unit operator-bounded by 1, and the predual of (τ c(A),  · τ ) is a non-associative C∗ -algebra in the natural way. The above corollary, together with assertion (vi) in Theorem 8.1.139, provides us with a non-associative characterization of associative complex H ∗ -algebras with zero annihilator. Remark 8.1.142 Let A be an alternative complex H ∗ -algebra with zero annihilator, and let (X, ||| · |||) stand for the predual of (τ c(A),  · τ ), namely the completion of (A, ||| · |||). Then, according to assertion (i) in Theorem 8.1.139, (X, ||| · |||) becomes an alternative C∗ -algebra in a natural way. Therefore, by Corollary 3.5.35, the bidual (X  , ||| · |||) of (X, ||| · |||) becomes an alternative W ∗ -algebra also in a natural way. Moreover, we are provided with the following chain of continuous injective algebra ∗-homomorphisms: (τ c(A),  · τ ) → (A,  · ) → (X, ||| · |||) → (X  , ||| · |||). Since (τ c(A),  · τ ) ≡ (X, ||| · |||) ,

(8.1.21)

(X  , ||| · |||) ≡ (τ c(A),  · τ ) ,

(8.1.22)

we have

and hence, keeping in mind the separate w∗ -continuity of the product of (X  , ||| · |||) (cf. Corollary 5.1.30(iii)), it is easily seen that τ c(A) is an ideal of (X  , ||| · |||). Then, since X is equal to the ||| · |||-closure of τ c(A) in (X  , ||| · |||), it follows that (X, ||| · |||) is an ideal of (X  , ||| · |||). Now recall that the mapping φ : (τ c(A),  · τ ) → (A, ||| · |||) , defined by φ(b)(a) := (a|b) for all a ∈ A and b ∈ τ c(A), is the germ of the isometric conjugate-linear identification (τ c(A),  · τ ) ≡ (X, ||| · |||) , and that τ c(A) is an ideal of (X  , ||| · |||). It follows from assertion (iii) in Theorem 8.1.139 that the mapping (b)(x) := f (xb∗ ) for all x ∈ X and  : (τ c(A),  · τ ) → (A, ||| · |||) , defined by φ φ b ∈ τ c(A), describes the natural isometric conjugate-linear identification (8.1.21), and that the mapping ψ : (X  , ||| · |||) → (τ c(A),  · τ ) , defined by ψ(y)(b) := f (y∗ b) for all y ∈ X  and b ∈ τ c(A), describes the natural isometric conjugate-linear identification (8.1.22).

8.1 H ∗ -algebras

545

According to §8.1.112, in the particular case that A is equal to the associative HS (H) of all Hilbert–Schmidt operators on a complex Hilbert space H, (τ c(A),  · τ ) is the algebra T C (H) of all trace-class operators on H, (X, ||| · |||) is the C∗ -algebra K(H) of all compact operators on H, and (X  , ||| · |||) is the W ∗ -algebra BL(H) of all bounded linear operators on H.

H ∗ -algebra

8.1.9 Historical notes and comments Example 8.1.5 is due to Cuenca [714, 1.2.20]. Proposition 8.1.6 is nothing other than a formalization of the fact (originally due to Ambrose [20]) that, as commented in [714, pp. vii–viii], the involution of a semi-H ∗ -algebra with zero annihilator is uniquely determined by the inner product. Proposition 8.1.7, Fact 8.1.8, and Theorem 8.1.9 are also due to Cuenca [714, Section 2.1]. As acknowledged by him, the proof of Proposition 8.1.7 is inspired by that of Albert’s finite-dimensional forerunner [827] (see also [808, Theorem 5.4]). Most results in Subsection 8.1.2 are due to Cuenca and Rodr´ıguez [199]. Some additional information, taken from [696, Section 34] and [714, Chapter 1], has been included. Although easily deducible from known results, Corollaries 8.1.20–8.1.22 are new. There are forerunners of Theorems 8.1.10 and 8.1.16 for associative (and even alternative), Jordan, or Lie H ∗ -algebras. These forerunners will be mentioned later. Without any doubt, Theorem 8.1.16 is the star result in Subsection 8.1.2, since all previously known arguments to prove the different forerunners of this theorem were unable to work in the general non-associative case. The following result is taken also from [714, Chapter 1]. Proposition 8.1.143 Let A be a Lie semi-H ∗ -algebra over K with zero annihilator. Then A is an H ∗ -algebra. Proof The Jacobi identity (cf. p. 581 of Volume 1) reads as Lab = [La , Lb ]. Therefore, taking adjoints according to (8.1.4), and applying again the Jacobi identity, we  have L(ab)∗ = [Lb∗ , La∗ ] = Lb∗ a∗ . Thus (ab)∗ − b∗ a∗ ∈ Ann(A) = 0. Algebras with a semiprime multiplication algebra were first studied by Jacobson [981] in a finite-dimensional context, obtaining the following description theorem. Theorem 8.1.144 For a finite-dimensional algebra A over a field F, the following conditions are equivalent: (i) M (A) is semiprime. (ii) A = ⊕ni=0 Bi is a direct sum of ideals, one of them, say B0 , is an algebra with zero product, and the others are simple algebras. If the above conditions are fulfilled, then M (A) ∼ = ⊕ni=0 M (Bi ). It follows from Proposition 3.6.39(i) and Theorem 3.6.57 that, for a simple finitedimensional algebra B over F (as are the algebras Bi , 1 ≤ i ≤ n, in the above theorem), M (B) is isomorphic to a full matrix algebra over a division algebra over F.

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Selected topics in the theory of non-associative normed algebras

A new proof of Theorem 8.1.144 has been given in [879, Theorem 5.13], and an extension to the setting of finitely generated algebras with artinian multiplication algebra has been established by Finston [950, Theorem 2.7]. As a consequence of Theorem 8.1.144, a finite-dimensional algebra is a direct sum of simple ideals if and only if it is multiplicatively semiprime (see the beginning of Subsection 8.2.2 for the definition). Motivated by this fact, Albert [8] (see also [350, pp. 1090–1]) proposed a notion of radical of a finite-dimensional algebra (which is known today as the Albert radical) and proved that, if A is a finite-dimensional algebra, then the Albert radical of A is the smallest ideal R of A such that A/R is multiplicatively semiprime (cf. §4.4.82). Moreover, he gave the first example of a prime algebra which is not multiplicatively semiprime (cf. Example 8.1.24). Without any finiteness condition, a systematic study of the relationship between the lattice of ideals in an algebra and that of ideals in the multiplication algebra was initiated by Pritchard [1054]. Again without any finiteness condition, multiplicatively prime algebras were considered first by Cabrera and Mohammed [886], where Proposition 8.1.25 was proved, and the behaviour of these algebras concerning extended centroid and central closure was studied. (For the meaning of the central closure QA of a semiprime algebra A, the reader is referred to [148].) Indeed, if A is a multiplicatively prime algebra, then (in a natural way) CA = CM (A) , and QM (A) = MCA (QA ). As a consequence, QA is also a multiplicatively prime algebra. These properties remain true in the more large class of multiplicatively semiprime algebras [888, 882]. As a matter of fact, prime associative algebras are multiplicatively prime [886, Proposition 4], and a similar result holds for most nearly associative algebras (see [876, 878, 881, 889, 896]). Corollary 8.1.26 had been pointed out earlier in [526] with the following argument: if A is a topologically simple algebra, and if F, G are in M (A) with FM (A)G = 0 and G = 0, then F = 0 because the linear hull of M (A)G(A) is a nonzero ideal of A and hence it is dense in A. Before Example 8.1.24, we pointed out that simple algebras are multiplicatively prime. In fact, the multiplication algebra of a simple algebra is primitive (cf. Definition 3.6.12), hence prime in view of Proposition 3.6.16(i). Indeed, for a simple algebra A, the inclusion M (A) → L(A) becomes an irreducible representation of M (A) (cf. Definitions 3.6.33 and 3.6.35) with zero kernel, hence M (A) is a primitive algebra (cf. Theorem 3.6.38(i)). In [889] Cabrera and Mohammed introduced totally multiplicatively prime normed algebras and proved Propositions 8.1.27 and 8.1.37, as well as Theorem 8.1.32 (that complex semi-H ∗ -algebras are totally multiplicatively prime in a nice way). Corollaries 8.1.33 (that topologically simple complex semi-H ∗ -algebras are totally prime in a nice way) and 8.1.34 (that topologically simple complex semi-H ∗ algebras are centrally closed) had been shown earlier by the authors in [149] and [148], respectively. Lemma 8.1.29 is due to Cuenca and Rodr´ıguez [198], although the proof we have given is new. In [889] it is also proved that, given a non-empty set X and 1 ≤ p < ∞, the absolute-valued algebra Fp (X, K) over K is totally multiplicatively prime if and

8.1 H ∗ -algebras

547

only if p = 1. (Here, as in §2.8.18, Fp (X, K) stands for the free non-associative algebra F (X, K) over K generated by X, endowed with the absolute value  · p .) As a consequence, F (X, K) is multiplicatively prime. This shows the abundance of normed multiplicatively prime algebras that are not totally multiplicatively prime. In [889] it is proved also that M (F (X, K)) can be seen as the free associative algebra over K generated by X ∪ Xr , where X and Xr are disjoint copies of X. Moreover, the operator norm on M (F1 (X, K)) coincides with the natural 1 -norm, and we have that WF,a 1 = F1 a1 for all F ∈ M (F1 (X, K)) and a ∈ F1 (X, K). Therefore, according to pp. 258–9 of Volume 1, the ‘free normed non-associative algebra’ over K generated by the set X is totally multiplicatively prime in a nice way. Conversely, it is noteworthy that, in [428, 1013], Mathieu introduced several ‘bounded algebras of quotients’ for ultraprime normed associative algebras. For example, the bounded symmetric algebra of quotients Qb (A) of an ultraprime normed associative algebra A can be characterized as the maximal normed associative algebra extension Q of A satisfying the conditions in Proposition 6.1.127 and converting the inclusion A → Q into a topological embedding. As one of the most relevant results in [1013], Mathieu showed that, if A is an ultraprime normed associative complex algebra, then Qb (A) is ultraprime. Following this approach, in [890, 891] the socalled normed algebras of quotients with bounded evaluation were introduced in the setting of totally prime normed associative algebras, and it is proved that, if A is a totally prime (respectively, totally multiplicatively prime) normed associative complex algebra, then its normed symmetric algebra of quotients with bounded evaluation is totally prime (respectively, totally multiplicatively prime). To conclude our comments on Subsection 8.1.3, we describe in the following diagram the relationship between the different normed-refinements of primeness. The diagram summarizes Proposition 8.1.27, and Theorems 6.1.63, 6.1.78, and 8.1.32 (cf. also Remarks 8.1.28 and 8.1.35, and Proposition 8.1.36).

Totally prime algebras

Ultraprime algebras

Totally multiplicatively prime algebras

Prime non-commutative JB ∗ -algebras

Topologically simple H ∗ -algebras

548

Selected topics in the theory of non-associative normed algebras

Results from Proposition 8.1.38 to Theorem 8.1.41 are due to Villena [624]. Forerunners of Theorem 8.1.41 for alternative, Jordan, and Mal’cev H ∗ -algebras had been shown earlier by Zalar [1125, 1126]. Since associative complex H ∗ -algebras with zero annihilator are semisimple (see [795, Theorem 4.10.29] or Corollary 8.1.146 below), the associative forerunner of Theorem 8.1.41 follows from the celebrated Johnson–Sinclair theorem [355] already reviewed in p. 130 of Volume 1. The next proposition, taken from [714], is useful to discuss other particular cases of Theorem 8.1.41. Proposition 8.1.145 Let A be a non-commutative Jordan complex H ∗ -algebra. Then J-Rad(A) = Ann(Asym ). Therefore A is J-semisimple if (and only if ) Ann(Asym ) = 0. Proof Since J-Rad(A) = J-Rad(Asym ) (cf. Proposition 4.4.17(iii)), and Asym is an H ∗ -algebra in a natural way, we may suppose that A is a Jordan algebra. The inclusion Ann(A) ⊆ J-Rad(A) is clear. Let a be in J-Rad(A). Then, by Lemma 4.4.26, we have r(a) = 0, and hence, by Theorem 4.1.17 and Propositions 1.1.107 and 4.1.30(ii), also r(La ) = 0. Therefore, if in addition a is self-adjoint, then La = 0, so a lies in Ann(A). Since J-Rad(A) is ∗-invariant, the inclusion J-Rad(A) ⊆ Ann(A) follows.  As a first consequence of the above proposition, Jordan complex H ∗ -algebras with zero annihilator are J-semisimple. Therefore the particularization of Theorem 8.1.41 to Jordan H ∗ -algebras (which was unknown before the publication of [624]) is contained in the later and more involved result by Villena [625] (cf. Remark 7.2.9(a)) asserting that derivations of J-semisimple complete normed Jordan algebras are continuous. The next result follows straightforwardly from Corollary 2.4.10, Propositions 8.1.23(i) and 8.1.145, and the comments in Definition 4.4.12. Corollary 8.1.146 Let A be an alternative complex H ∗ -algebra. Then Rad(A) = Ann(A). Therefore A is semisimple if (and only if ) Ann(A) = 0. Keeping in mind the above corollary, the particularization of Theorem 8.1.41 to alternative H ∗ -algebras is contained in Villena’s earlier result [1117] asserting that derivations of semisimple complete normed alternative algebras are continuous. §8.1.147 With the exception of Lemmas 8.1.44 and 8.1.47, results from Lemma 8.1.42 to Theorem 8.1.52 are due to Rodr´ıguez [526]. Lemma 8.1.44 is folklore whereas Lemma 8.1.47 is originally due to Sinclair [1090]. Theorem 8.1.53 and the first assertion in Corollary 8.1.54 are new. The second assertion in this corollary is well-known: actually, by Lemma 8.2.18 and Theorem 4.4.43, which were proved in [259] and [516], respectively, each real or complex semi-H ∗ -algebra with zero annihilator has a unique complete algebra norm topology. Before the publication of [526], results from Proposition 8.1.49 to Proposition 8.1.51 had no associative

8.1 H ∗ -algebras

549

forerunners. The associative forerunner of Theorem 8.1.52 can be considered as folklore (see for example Theorem 3.2 of the Albrecht–Dales paper [829]). §8.1.148 Subsection 8.1.5 is due to Cuenca and Rodr´ıguez [198], although Corollaries 8.1.66 and 8.1.68 were not noticed there, and the reorganization of the proof of Theorem 8.1.56 (through Lemma 8.1.55) is new. Before the publication of [198], the results in this subsection had no associative forerunner. In [1042] it is proved that the algebra C(C) of complex octonions can be structured as an H ∗ -algebra, and, to showing the essential uniqueness of the H ∗ -algebra structure on C(C) (as we have done in §8.1.69 for any topologically simple complex H ∗ algebra), forerunners of results in Subsection 8.1.5 for C(C) are roughly established. If one looks at [1179, Section III.6], then one finds these forerunners in their correct formulation. The forerunner of Corollary 8.1.67 for Lie H ∗ -algebras is due to Balachandran [850]. Proposition 8.1.71 is due to Villena [624]. The forerunner for alternative algebras had been established earlier by Rodr´ıguez [518], who also proved Theorem 8.1.76. The proof we have given of this theorem (through Lemmas 6.1.76 and 8.1.72 to 8.1.75) is new. §8.1.149 Subsection 8.1.7 is structured from the paper of Mart´ınez and the authors [142], where Proposition 8.1.77 and (with the exception of Lemma 8.1.90) results from Proposition 8.1.87 to Corollary 8.1.97 are proved. Lemma 8.1.90 becomes a small refinement of Theorem X.5 of [752]. Corollaries 8.1.79, 8.1.82, 8.1.86, and 8.1.91 were not noticed in the papers containing their complex forerunners, namely [85], [624], [624]–[518], and [198], respectively. As we commented in relation to their complex forerunners, Corollaries 8.1.80, 8.1.81, 8.1.84, and the first assertion in Corollary 8.1.85 are new. According to §8.1.147, the second assertion in Corollary 8.1.85 is well-known. Corollary 8.1.83 and its complex forerunner (namely Theorem 8.1.52 already reviewed in §8.1.147) are proved in [526]. The notion of an H ∗ -ideal, Lemma 8.1.99, and Theorem 8.1.101 are due to Beltita [687, Proposition 7.35 and Theorem 7.38]. The real versions of Lemmas 8.1.103 and 8.1.104 and of the first conclusions in Theorem 8.1.107 and Corollary 8.1.108 are proved in [889]. The real versions of Corollary 8.1.98 and of the second conclusions in Theorem 8.1.107 and Corollary 8.1.108 are new. The real versions of von Neumann’s bicommutant and Kaplansky’s density theorems, given by Proposition 8.1.106, are known. Indeed, they can be found as Theorems 4.3.8 and 4.4.1 of [768]. Theorem 8.1.109 is proved for the first time in [525]. The proof we have given follows the indications in [903, Section 4] for a severe simplification of the original one. Invoking Theorem 8.1.92 instead of Theorem 8.1.64, minor changes to the proof of Theorem 8.1.109 allow us to obtain the following. Proposition 8.1.150 Let A be an H ∗ -algebra over K with zero annihilator, and let τ be a (linear) algebra involution of A. Then A is bicontinuously isomorphic to a suitable H ∗ -algebra over K in such a way that τ converts into a ∗-involution.

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As commented in [1075], invoking the structure results contained in Corollary 8.1.18 and Theorem 8.1.151, and keeping in mind the Schatten–von Neumann Theorem 8.1.111, a theory of trace-class elements, in any associative complex H ∗ -algebra with zero annihilator, can be established in an almost straightforward way. Nevertheless, Saworotnow [1074] was able to develop such a theory in an intrinsic way, avoiding any reference to structure results. (Actually Fact 8.1.110, which has been the motivating result in our development, underlies Saworotnow’s paper.) Almost simultaneously, Saworotnow and Friedell, in the paper [1075] quoted above, attained the same objective through an equivalent but different approach. Motivated by the Saworotnow–Friedell papers quoted above, Mart´ınez [424] developed a satisfactory theory of trace-class elements in any (possibly nonassociative) complex H ∗ -algebra with zero annihilator, which has become the foundation of Subsection 8.1.8. Actually, unless Theorem 8.1.125 (which has been suggested to us by D. Beltita in a recent private communication), results from Lemma 8.1.113 to Corollary 8.1.137 are just a revisiting of Mart´ınez’ paper, trying to extend its results to real or complex semi-H ∗ -algebras with zero annihilator. Towards this goal, we have been obliged to reelaborate much material, so that it turns out to be rather difficult to say whether or not our results are known in [424]. The inequalities (8.1.19) in the proof of Beltita’s Theorem 8.1.125 recall the spirit of Stampfli’s paper [1099] (see also [678, Section 4.1]). Theorems 8.1.139 and 8.1.140, as well as Corollary 8.1.141, are new. Indeed, they could not be shown in [424] because their proofs (the only ones we know) depend on results published much later. According to Fern´andez-Polo, Garc´es, Peralta, and Villanueva [948], if H1 and H2 are complex Hilbert spaces, then every surjective isometry from ST C (H1 ) to ST C (H2 ) admits a unique extension to a surjective linear or conjugate-linear isometry from T C (H1 ) to T C (H2 ). It would be interesting to explore the question whether the above result remains true if we replace T C (Hi ) with τ c(Ai ), being the Ai semi-H ∗ algebras with zero annihilator. For a detailed history of Hilbert–Schmidt and trace-class operators on a complex Hilbert space the reader is referred to Pietsch [790, pp. 137–9]. According to him, the theory of these operators is originally due to Schatten and von Neumann [1078, 1079, 1080]. Now that we have widely discussed the content of the previous subsections of this section, we are going to complement this information by giving a more ordered history of H ∗ -algebras, and by reviewing the classification theorems of topologically simple H ∗ -algebras in the most familiar classes of algebras. (To this end, we found an updated version of [525, Section E] useful.) We note that, in view of Propositions 8.1.23 and 8.1.143 and Corollaries 8.1.79 and 8.1.81, topologically simple semi-H ∗ algebras in the most familiar classes of algebras are in fact H ∗ -algebras. H ∗ -algebras appeared first in the literature in the hands of Steen [598], but its theory began with the work of Ambrose [20] on associative complex H ∗ -algebras. Ambrose’s paper inspired the subsequent study of H ∗ -algebras in the most familiar

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classes of non-associative algebras and even the treatment of general (possibly nonassociative) H ∗ -algebras, as we have done in this section. In fact, Ambrose proved the associative complex version of Theorem 8.1.16 and determined all topologically simple associative complex H ∗ -algebras. In an equivalent formulation, more suitable for our development, this determination reads as follows. Theorem 8.1.151 Up to a positive multiple of the inner product, every topologically simple associative complex H ∗ -algebra is totally isomorphic to the H ∗ -algebra HS (H) of all Hilbert–Schmidt operators on a suitable complex Hilbert space H (cf. Example 8.1.3). The corresponding description of topologically simple associative real H ∗ algebras was obtained by Kaplansky [374] in analogous terms as those in Theorem 8.1.151, but involving real, complex, or quaternionic Hilbert spaces. This result has been rediscovered several times (see [55, 142, 163]), and in fact it follows from Proposition 8.1.87 and Theorem 8.1.151, once the ∗-involutions on HS (H) (for H a complex Hilbert space) are determined. (Note that, an operator τ on an H ∗ -algebra A is a ∗-involution if and only if τ ◦ ∗ is an involutive conjugate-linear algebra ∗-automorphism of A.) Because such a determination will be useful for a simplified statement of structure theorems for Jordan and Lie complex H ∗ -algebras, we formulate it here. Fact 8.1.152 [142] Given a complex Hilbert space H, the ∗-involutions on the topologically simple associative H ∗ -algebra HS (H) are the mappings of the form F → JF ∗ J −1 for a suitable conjugation or anticonjugation J on H. We recall that a conjugation (respectively, anticonjugation) on the complex Hilbert space H is a conjugate-linear isometry J : H → H with J 2 = IH (respectively, J 2 = −IH ). Conjugations always exist, while anticonjugations exist if and only if the dimension of H is either an even number or infinite. In any case, when they exist, they are essentially unique [738, Lemma 7.5.6]. The theory of associative complex H ∗ -algebras is included in several books such as [696, Section 34], [783, Section 25.5], [795, Section IV.10 and A.1.3], and [1162, Section 14.3]. Topologically simple alternative H ∗ -algebras which are not associative are easily determined by applying the following results: (i) The algebra C(C) of complex octonions can be regarded, in an essentially unique way, as a complex H ∗ -algebra (cf. §8.1.148). (ii) Slater’s theorem on prime alternative algebras (cf. Theorem 6.1.33). (iii) The centroid of a topologically simple H ∗ -algebra over K is C if K = C (cf. Lemma 8.1.29), and is R or C if K = R (cf. Corollary 8.1.89). As it can be expected, the topologically simple alternative non-associative H ∗ algebras are: the algebra of complex octonions, in the complex case; and the same algebra regarded as a real algebra, together with the two real octonions algebras, in

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the real case. (The reader is referred to [1144, Theorems 8.2 and 8.4] for a different approach to the proof of this result.) As a consequence, we have the following. Corollary 8.1.153 Let A be an alternative complex H ∗ -algebra with zero annihilator. Then τ c(A) = AA. Proof By Proposition 8.1.13(iv) and Theorem 8.1.122, we may suppose that A is topologically simple. If A is associative, then, keeping in mind §8.1.112, the result follows from Theorems 8.1.111 and 8.1.151. Otherwise A = C(C), hence it has a unit u, and therefore, by Theorem 8.1.116, A = uA ⊆ AA ⊆ τ c(A) ⊆ A.  Alternative algebras satisfy the so-called Moufang identities (cf. Lemma 2.3.60). In the remarkable paper [912], Cuenca proves that, if a real or complex H ∗ -algebra A satisfies some of the Moufang identities, then A is alternative. Cuenca’s proof involves the main results in Subsection 8.1.2, together with Proposition 8.1.7, Fact 8.1.8, and Theorem 8.1.88. Jordan complex H ∗ -algebras were studied first by Viola Devapakkiam and Rema in [1118] and [1119]. They proved that every simple finite-dimensional Jordan complex algebra (hence in particular the simple exceptional Jordan algebra H3 (C(C))) can be structured as an H ∗ -algebra. Moreover, they showed that Jordan complex H ∗ -algebras with zero annihilator have dense socle, and gave the first steps for the classification of separable topologically simple Jordan complex H ∗ -algebras. The results in these papers laid the foundations for the definitive classification of topologically simple Jordan complex H ∗ -algebras, obtained later by Cuenca and Rodr´ıguez in [199]. To formulate this classification theorem in a nice way, let us introduce some natural concepts and facts that will be also useful in other contexts. If τ is a linear algebra involution on an algebra A, we will denote by H(A, τ ) and S(A, τ ) the sets of all τ -hermitian and τ -skew elements of A, respectively. That is: H(A, τ ) = {h ∈ A : τ (h) = h} , S(A, τ ) = {s ∈ A : τ (s) = −s}. H(A, τ ) and S(A, τ ) are subalgebras of Asym and Aant , respectively. In the case that A is a (semi-)H ∗ -algebra, it is clear that, for a continuous linear algebra ∗-involution τ on A, H(A, τ ) and S(A, τ ) are ∗-invariant closed subalgebras of the (semi-)H ∗ algebras Asym and Aant , respectively (cf. §8.1.4), and in this way they will be considered as new (semi-)H ∗ -algebras. Note that, if A has zero annihilator, then the assumption of continuity of τ is superfluous (cf. Corollary 8.1.85). Every isometric linear algebra involution on an H ∗ -algebra with zero annihilator is a ∗-involution (a consequence of Proposition 8.1.6 explicitly formulated in [144, Proposition 1.7]), and the converse is true if the H ∗ -algebra is topologically simple (cf. Corollary 8.1.97). By a topologically τ -simple H ∗ -algebra we mean an H ∗ -algebra with linear algebra involution τ , nonzero product, and no nonzero proper τ -invariant closed ideals. Of course, topologically τ -simple H ∗ -algebras have zero annihilator, and

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every topologically simple H ∗ -algebra is topologically τ -simple for every linear algebra involution τ . Moreover, we have the following easy result (see for example [144, Theorem 1.5]). Fact 8.1.154 Let (A, τ ) be a topologically τ -simple H ∗ -algebra with isometric linear involution, and suppose that A is not topologically simple. Then A = B ⊕ 2 B0 , where B is a suitable topologically simple H ∗ -algebra, and τ is the exchange involution. In this case we have natural isomorphisms H(A, τ ) ∼ = Bsym and S(A, τ ) ∼ = Bant . Given a complex Hilbert space H with a conjugation , the Hilbert space C ⊕ 2 H, with Jordan product (λ + x)(μ + y) := (λμ + (x|y )) + (λy + μx) and H ∗ -algebra involution (λ + x)∗ := λ + x , becomes a Jordan complex H ∗ -algebra, called the Jordan H ∗ -algebra of the involutive Hilbert space (H, ), and denoted by J(H, ). Now we can formulate the classification theorem for topologically simple Jordan complex H ∗ -algebras [199] as follows. Theorem 8.1.155 Up to a positive multiple of the inner product, the topologically simple Jordan complex H ∗ -algebras are H3 (C(C)), the Jordan H ∗ -algebras J(H, ) of an involutive complex Hilbert space (H, ) with dim(H) ≥ 2, and the Jordan H ∗ algebras H(A, τ ) of all τ -hermitian elements in a topologically τ -simple associative complex H ∗ -algebra A with isometric linear involution τ (cf. Theorem 8.1.151, and Facts 8.1.152 and 8.1.154). The paper [199] contains also a classification theorem for topologically simple non-commutative Jordan complex H ∗ -algebras, asserting that these are either anticommutative, commutative (cf. Theorem 8.1.155), simple quadratic, or real mutations of topologically simple associative complex H ∗ -algebras (cf. Theorem 8.1.151). The quadratic complex H ∗ -algebras are relatively well-described (see [911] and [141]), they are automatically non-commutative Jordan algebras, and, except for the non simple two-dimensional case, they are very similar to simple quadratic non-commutative JB∗ -algebras constructed in Theorem 3.5.5. Indeed, it is enough to relax there the assumption x ∧ y ≤ x y in the definition of an H-algebra (cf. Definition 2.6.4) to the simple continuity of ∧, and take as concluding norm in the construction the natural Hilbert norm on the complexification of a real Hilbert space. Cuenca and S´anchez [917] classified topologically simple Jordan (and even noncommutative Jordan) real H ∗ -algebras. The reader is referred to the original paper for the detailed formulations of their results. Summarized forms of these results can be found in [525, pp. 156–7]. The papers of Schue [557] and [558] on Lie H ∗ -algebras became the first incursions in the study of H ∗ -algebras from a non-associative point of view. It was proved in [557] the particular Lie version of Theorem 8.1.16, and the following was also shown there. Proposition 8.1.156 Every simple finite-dimensional Lie complex algebra can be structured as an H ∗ -algebra.

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Schue’s papers contain also the classification of separable topologically simple Lie complex H ∗ -algebras, as well as fine Cartan decompositions for an arbitrary Lie complex H ∗ -algebra with zero annihilator. Since Schue’s paper [557] until 1972, there was a great activity in the field of Lie H ∗ -algebras (see [740] and references therein). We only cite here the almost simultaneous papers of Balachandran [851], de la Harpe [971], and Unsain [1111] (where, in the spirit of Theorem 8.1.88, the classification of topologically simple Lie real H ∗ -algebras is reduced to that of complex ones), and the following reformulated version of results in Sections 3 and 4 of Balachandran’s paper [53]. Proposition 8.1.157 If (A, τ ) is an infinite-dimensional topologically τ -simple associative complex H ∗ -algebra with isometric linear algebra involution, then the Lie complex H ∗ -algebra S(A, τ ) is topologically simple (cf. Theorem 8.1.151, and Facts 8.1.152 and 8.1.154). Schue’s work has become the key tool for the following classification theorem of topologically simple Lie complex H ∗ -algebras, obtained by Cuenca, Garc´ıa, and Mart´ın [197], and reproved later by Neher [460] using different methods. Theorem 8.1.158 Every infinite-dimensional topologically simple Lie complex H ∗ -algebra is of the form S(A, τ ) for some (infinite-dimensional) topologically τ -simple associative complex H ∗ -algebra A with isometric linear algebra involution τ (cf. Theorem 8.1.151, Facts 8.1.152 and 8.1.154, and Proposition 8.1.157). For additional information about Lie H ∗ -algebras the reader is referred to Chapters 7 and 8 of Beltita’s book [687] (already quoted in §8.1.149) where, for the first time in a book, general ( possibly non-associative) real or complex H ∗ -algebras are considered in a systematic way, and, among other interesting results, Theorems 8.1.10, 8.1.16, and 8.1.158, Corollary 8.1.91, and assertion (8.1.11) in the proof of Theorem 8.1.62 are stablished, most times referring the reader to the original papers for a proof. Note also that Lemma 7.16 of [687] is a consequence of Corollaries 8.1.97 and 8.1.98, and that Proposition 7.17 of [687] is nothing other than the specialization of Theorem 8.1.88 to the case of Lie H ∗ -algebras. According to Beltita’s comments in [687, p. xi], ‘Roughly speaking, there is a tight connection between equivariant monotone operators and the H ∗ -ideals of complete normed Lie ∗-algebras. . . . The basic observation is that one can have such ideals, and they play an important role, even in the case of topologically simple complete normed Lie ∗-algebras.’ By a Malcev algebra we mean an anticommutative algebra A such that J(x, y, xz) = J(x, y, z)x for all x, y, z ∈ A, where J(x, y, z) := (xy)z + (yz)x + (zx)y. All Lie algebras become examples of Malcev algebras. Other examples are provided by the algebras of the form Bant for every alternative algebra B. Schue’s Cartan decomposition for Lie H ∗ -algebras has also become useful in the treatment of Malcev H ∗ -algebras [141], where the following theorem is proved.

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Theorem 8.1.159 Every topologically simple Malcev complex H ∗ -algebra is either a Lie algebra or the Malcev H ∗ -algebra of all trace-zero elements (cf. §2.5.11) in the alternative H ∗ -algebra C(C) of complex octonions. (Here, the set of all trace-zero elements in C(C) is regarded as a ∗-subalgebra of C(C)ant .) A very simple new proof of the above theorem was provided later by the authors [148] by combining Corollary 8.1.34 with the main result in Filippov’s paper [949]. Topologically simple Malcev non-Lie real H ∗ -algebras are easily obtained from Theorem 8.1.159 by applying Theorem 8.1.88 (see [1144, Theorem 8.3] for details). Another interesting work on H ∗ -algebras satisfying more or less familiar identities is that of Mart´ınez and the authors [144] on structurable complex H ∗ -algebras. A structurable algebra is an algebra A with a linear algebra involution τ satisfying (i) [s, x, y] = −[x, s, y] = [x, y, s] (ii) [a, b, c] − [c, a, b] = [b, a, c] − [c, b, a] (iii) 23 [[a2 , a], b] = [b, a2 , a] − [b, a, a2 ] for all x, y ∈ A, s ∈ S(A, τ ), and a, b, c ∈ H(A, τ ). Structurable algebras were introduced by Allison in [831], and, in the finite-dimensional setting, their structure theory was completed by the works of Smirnov [1092] and Schafer [1076]. Examples of structurable algebras are alternative algebras with any linear algebra involution and Jordan algebras with the identity operator as involution. The main interest of structurable algebras relies on the fact that, in the finite-dimensional case, they give, by means of an extended Kantor–Koecher–Tits construction, all isotropic simple Lie algebras [832]. The paper [144] begins with a systematic study of complex H ∗ -algebras with a linear algebra involution τ . After showing the appropriate version of Theorem 8.1.16 in their context (thus reducing the general case of algebras with zero annihilator to the particular topologically τ -simple one), they prove the complex version of Proposition 8.1.150. Moreover, the essential uniqueness of the H ∗ -structure on topologically τ -simple complex H ∗ -algebras with isometric linear involution τ is obtained. As a consequence, the theory of structurable complex H ∗ algebras reduces to that of topologically τ -simple ones with τ an isometric linear algebra involution. Then they prove the following theorem (see also Remark 1.10(b) in the Allison–Faulkner paper [833], where the part of the theorem concerning the so-called ‘Smirnov’s structurable algebra’ [144, Proposition 4.7] is reproved in a simpler way). Theorem 8.1.160 Every finite-dimensional τ -simple structurable complex algebra (A, τ ) can be structured as an H ∗ -algebra in such a way that τ becomes isometric. Now, with the Allison–Smirnov classification of finite-dimensional τ -simple structurable algebras, the interest must be focussed in the infinite-dimensional case. Certainly we already know the examples provided by the cases (A, τ ), where, either A is a topologically τ -simple associative complex H ∗ -algebra with isometric linear algebra involution τ (cf. Theorem 8.1.151, and Facts 8.1.152 and 8.1.154, again), or

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A is a topologically simple Jordan complex H ∗ -algebra (cf. Theorem 8.1.155) and τ is the identity operator on A. In order to build further infinite-dimensional examples, Saworotnow’s theory of Hilbert modules over associative complex H ∗ -algebras with zero annihilator (see [1073, 953, 1093]) is recaptured in the paper we are reviewing. If we wanted to state with some precision the construction made in [144] of certain structurable H ∗ -algebras from some Hilbert modules, then we would give a quite long list of concepts and results, a fact that seems unsuitable for the philosophy of the review we are doing. It is therefore better to appeal to the imagination and, consequently, to think that: (i) Hilbert modules are the natural generalizations of complex Hilbert spaces when the base field C is replaced by an associative complex H ∗ -algebra A with zero annihilator. It is noteworthy that the A-valued inner product of a Hilbert A-module is subjected to the condition of taking its values into the dense ∗-invariant ideal τ c(A) (cf. Theorem 8.1.116 and Remark 8.1.117(c)). (ii) ‘Involutive Hilbert modules’ over associative complex H ∗ -algebras with zero annihilator and an isometric linear algebra involution, introduced in [144], are then reasonable non-commutative variants of involutive complex Hilbert spaces (note that the only linear algebra involution on C is the identity). (iii) Structurable complex H ∗ -algebras constructed in [144] from involutive Hilbert modules correspond in this setting to Jordan complex H ∗ -algebras of an involutive complex Hilbert space (see the definition before Theorem 8.1.155). Now, with the lack of precision accepted above, the main result in [144] can be summarized as follows (see [144, Theorems 2.8 and 5.1] together with [140, 1.9.4]). Theorem 8.1.161 If (A, τ ) is a topologically τ -simple structurable complex H ∗ -algebra, and if τ is isometric on A, then one of the following assertions holds: (i) (ii) (iii) (iv)

A is finite dimensional (cf. Theorem 8.1.160). A is associative (cf. Theorem 8.1.151, and Facts 8.1.152 and 8.1.154). A is a Jordan algebra and τ is the identity on A (cf. Theorem 8.1.155). (A, τ ) is a structurable H ∗ -algebra constructed from an involutive Hilbert module over a topologically τ -simple associative complex H ∗ -algebra E with isometric linear involution τ .

Among other tools, the proof of the above theorem involves Theorem 8.1.159. Involutive Hilbert modules arising in case (iv) of Theorem 8.1.161 are also precisely described in [144, Theorem 3.10 ]. Theorem 8.1.161 has been applied in [935] to show that every structurable complex H ∗ -algebra (A, τ ) with zero annihilator has an approximate unit operator-bounded by 1, which consists of ∗- and τ -invariant elements. For the actual relevance of this result, see Remark 8.1.136 and Corollary 8.1.137. Theorem 8.1.161, together with nice ideas in Schafer’s paper [1077], has become the key tool for the proof in [140] of the following infinite-dimensional version of the Alison–Kantor–Koecher–Tits construction.

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Theorem 8.1.162 There is a canonical correspondence (A, τ ) → K (A, τ ) from the category S of structurable complex H ∗ -algebras with zero annihilator and isometric linear involution onto the category L of Lie complex H ∗ -algebras with zero annihilator. Moreover, for (A, τ ) ∈ S , this correspondence induces an order-isomorphism from the complete lattice of all τ -invariant closed ideals of A onto that of closed ideals of K (A, τ ). As a consequence, for (A, τ ) ∈ S , A is topologically τ -simple if and only if K (A, τ ) is topologically simple. The proof of the above theorem has needed (and hence encouraged) further developments of the theory of Hilbert modules (over associative complex H ∗ -algebras with zero annihilator). These have been provided in [885] (see also [1023]), where the notion of an orthonormal basis for a Hilbert module has been introduced, orthonormal bases have been characterized among ‘orthonormal systems’ by means of ‘Parseval’s equality’ or ‘Fourier’s expansion’, and the existence of orthonormal bases, as well as the coincidence of the cardinal numbers of all these bases, has been proved. Then, as needed for the proof of Theorem 8.1.162, a satisfactory theory of operators of Hilbert–Schmidt type on Hilbert modules has been developed. For additional information about Hilbert modules, the reader is referred to [849], [1162, Section 12.6], and references therein. By a (non-associative) ∗-triple over K we mean a vector space X over K endowed with a triple product {· · · } : X × X × X → X which is linear in the outer variables and conjugate-linear in the middle variable. In this way, Jordan ∗-triples, introduced in Definition 4.1.32, become examples of ∗-triples. By an H ∗ -triple over K we mean a ∗-triple X over K which is also a Hilbert space in such a way that the equalities ({xyz}|w) = (x|{wzy}) = (z|{yxw}) hold for all x, y, z, w ∈ X. The first H ∗ -triples to appear in the literature are the complex Jordan H ∗ -triples, which were introduced (under the name of Hilbert triple systems) and studied in detail by Kaup [994] in relation to the classification of bounded symmetric domains in complex Hilbert spaces. In the same way as C∗ -algebras are examples of JB∗ -triples (cf. Fact 4.1.41), associative H ∗ -algebras give rise to Jordan H ∗ -triples. Indeed, every associative H ∗ -algebra becomes a Jordan H ∗ -triple under the triple product (x, y, z) → 12 (xy∗ z + zy∗ x). Thus HS (H), for a complex Hilbert space H (cf. Example 8.1.3), and its closed ∗-subtriples become examples of Jordan H ∗ -triples. Among them, we emphasize the spaces HS (H, K), of all Hilbert– Schmidt operators between ( possibly different) complex Hilbert spaces H and K, which can be introduced as the set of those operators F ∈ HS (H ⊕ 2 K) such that F(K) = 0 and F(H) ⊆ K. Kaup proved the peculiar variants of Theorems 8.1.10 and 8.1.16 in his context, and gave the following description of topologically simple Jordan H ∗ -triples (see also Neher’s book [1177], where concepts and results are also suitably extended to the real case).

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Theorem 8.1.163 Up to a positive multiple of the inner product and up to a multiplication of the triple product by ±1, topologically simple Jordan H ∗ -triples are the ‘Hilbert variants’ of Cartan factors (cf. §7.1.4), namely, the Jordan H ∗ -triples of the form HS (H, K), {x ∈ HS (H) : σ x∗ σ = −x}, and {x ∈ HS (H) : σ x∗ σ = x}, where H and K are complex Hilbert spaces, and σ is a conjugation on H; and the Cartan (JBW ∗ -triple) factors of type IV, V, and VI (endowed with their equivalent natural Hilbert norms). We note that, as consequence of Proposition 4.1.34, Jordan complex H ∗ -algebras become Jordan H ∗ -triples under the triple product {xyz} := x(y∗ z) − y∗ (zx) + z(xy∗ ), and that Kaup’s theorem above shows ostensibly that every topologically simple ‘positive’ Jordan H ∗ -triple can be seen as a closed ∗-subtriple of a suitable Jordan complex H ∗ -algebra. With a little care in using Kaup’s theorem and the ternary versions of Theorems 8.1.10 and 8.1.16, it can be shown that actually every positive Jordan H ∗ -triple can be seen as a closed ∗-subtriple of a Jordan complex H ∗ -algebra. Talking about H ∗ -triples, we encourage the reader to look at the paper of Zalar [1129], where, through many nice concepts, and providing elegant proofs, most of the theory of general non-associative H ∗ -algebras is translated to the setting of nonassociative H ∗ -triples. Let us also cite the characterization up to equivalent renorming of Jordan H ∗ -triples having ‘finite capacity’ (or ‘finite rank’, in a more familiar terminology in our context) in the sense of Loos [1004]. This characterization has been provided in [943], and contains an earlier result in [925], which in its turn extends one in [974, Theorem 6.2]. Theorem 8.1.164 A Jordan-Banach ∗-triple is bicontinuously isomorphic to a Jordan H ∗ -triple of finite rank if and only if it is strongly regular and the set of its division generalized tripotents is bounded. Let us say that a Jordan ∗-triple X is said to be strongly regular if for every x ∈ X we have that x ∈ Q2x (X) (cf. p. 506 of Volume 1 for notation), and that division generalized tripotents are those elements e ∈ X such that {eee} = ±e and the e-homotope algebra X (e) is a J-division Jordan algebra (cf. Proposition 4.1.35, and Definitions 4.1.13 and 4.1.36). In the line of developing further the theory of Hilbert modules (over a complex H ∗ -algebra with zero annihilator), the paper [884] provides a non-structural proof of the existence of a natural categorical one-to-one correspondence between Hilbert modules and ‘associative’ H ∗ -triples (‘of the second kind’) with zero annihilator, obtaining as a consequence the structure of topologically simple associative H ∗ triples, becoming these in essence of the form HS (H, K), for H and K complex Hilbert spaces, now endowed with the triple product (x, y, z) → xy∗ z (see also [1129] for a more direct proof of this last result). By Theorem 8.1.161, every non-trivial problem in structurable complex H ∗ -algebras reduces in essence to the consideration of the problem in structurable H ∗ -algebras constructed from involutive Hilbert modules. In this line, derivations of

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such structurable H ∗ -algebras commuting with the linear involution have been studied in [883], showing the existence of a natural bicontinuous Lie-algebra isomorphism from the Banach–Lie algebra of these derivations onto some (perfectly determined) Banach–Lie algebra of bounded ‘differential operators’ on the parameterizing Hilbert module. The notion of a differential operator arising in this result is the natural version for modules of that appearing for vector spaces over division algebras in the study of derivations on prime associative algebras with nonzero socle (see [753]). The proof of the result above turns again over the correspondence between Hilbert modules and associative H ∗ -triples with zero annihilator, through which bounded differential operators convert into (automatically continuous) ‘generalized derivations’, which are fully described. We remark that, thanks to [883, Proposition 2.2], generalized derivations of associative H ∗ -triples with zero annihilator can be seen as ‘derivation pairs’ in the sense of Zalar [1128], and then we refer to this last paper for further information about the automatic continuity of derivation pairs on alternative and Jordan H ∗ -triples. The reader is also referred to [1010, Section 3] for a more detailed review of the results of [883] commented above. The transition from H ∗ -algebras to certain ternary H ∗ -structures, different from the H ∗ -triples considered above, can be done through the study of two-graded H ∗ -algebras provided by Cuenca and Mart´ın in [915]. Two-graded H ∗ -algebras are H ∗ -algebras A which splits into an orthogonal direct sum A = A0 ⊕ A1 of self-adjoint (closed) subspaces Ai (i = 0, 1) such that Ai Aj ⊆ Ai+j (sum in module two). The study of two-graded H ∗ -algebras may be reduced to the study of the topologically simple ones (in a graded sense) in a similar way as it is done for ungraded H ∗ -algebras. It is easy to prove that the only topologically simple two-graded associative H ∗ -algebras over K are of one (and only one) of the following two types: (i) B ⊕ 2 B with B a topologically simple associative H ∗ -algebra over K, the product given by (x, y)(u, v) := (xu + yv, xv + yu), H ∗ -algebra involution (x, y)∗ := (x∗ , y∗ ), even part B ⊕ {0}, and odd part {0} ⊕ B. (ii) An (a priori ungraded) topologically simple associative H ∗ -algebra A over K, with even part the Hermitian elements relative to an involutive ∗-automorphism σ of A, and odd part the skew-hermitian elements. Furthermore the grading involutive ∗-automorphisms σ can be precisely determined, arising a number of non-isomorphic H ∗ -algebras which are not worth to describe here. The interesting matter here is that the odd part of any two-graded topologically simple associative H ∗ -algebra is a topologically simple associative ‘ternary H ∗ -algebra’ (‘of the first kind’), and moreover, any topologically simple associative ‘ternary H ∗ -algebra’ arises in this way (up to the sign of the ‘ternary H ∗ -algebra involution’) (see [902] for the precise classification of topologically simple associative ternary H ∗ -algebras). In this way binary methods can be used to deal with ternary structures in an associative context, and this idea can be also exploited in a Jordan setting. In a purely algebraic context, the same link appears between prime associative two-graded algebras with nonzero socle and

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prime associative ternary algebras with nonzero socle (see [913, 914]). Now let us formulate in a summarized form the classification of topologically simple twograded Jordan H ∗ -algebras obtained in [915]. Theorem 8.1.165 Topologically simple two-graded Jordan H ∗ -algebras over K are of one of the following types: (i) J ⊕ 2 J, with J a topologically simple Jordan H ∗ -algebra over K, the product given by (x, y)(u, v) :=(xu+yv, xv+yu), H ∗ -algebra involution (x, y)∗ := (x∗ , y∗ ), even part J ⊕ {0}, and odd part {0} ⊕ J. (ii) The two-graded Jordan H ∗ -algebra obtained by symmetrization of the product of a two-graded associative H ∗ -algebra A over K that is topologically simple without reference to the grading of A. (iii) The two-graded Jordan H ∗ -algebra obtained by symmetrization of the product of a topologically simple associative H ∗ -algebra A over K with a ∗-involution τ , with even part H(A, τ ), and odd part S(A, τ ). (iv) The two-graded Jordan H ∗ -algebra H(A, τ ), where A is as in (ii), and τ is a ∗-involution on A preserving the grading of A. (v) A two-graded Jordan H ∗ -algebra of quadratic type. (vi) A two-graded Jordan H ∗ -algebra of exceptional type. Since associative ternary H ∗ -algebras (of the first kind) have already arisen in our recent comments, and we have even outlined their structure theory, it seems suitable to formulate the concept of a ternary H ∗ -algebra with some precision, allowing it to work also outside the associative environment. Let us therefore say that a (nonassociative) ternary H ∗ -algebra is a real or complex Hilbert space A together with a trilinear triple product {· · · } : A × A × A → A, and an involutive mapping ∗ : A → A, which is linear in the real case, conjugate-linear in the complex case, and satisfies {xyz}∗ = {y∗ z∗ x∗ } and ({xyz}|w) = (x|{wz∗ y∗ }) = (z|{y∗ x∗ w}) = (y|{x∗ wz∗ }) for all x, y, z, w ∈ A. Jordan ternary H ∗ -algebras (i.e. Jordan triples that are ternary H ∗ -algebras) have been first considered and studied by Castell´on and Cuenca in [900]. Later, a complete structure theory of Jordan ternary H ∗ -algebras has been achieved by Castell´on, Cuenca, and Mart´ın [903, 904]. We formulate here this theory in a very condensed form, leaving to the imagination of the reader the technical notion, involved in the statement, of a ‘polarized’ topologically (∗-)simple Jordan ternary H ∗ -algebra. Theorem 8.1.166 Every real unpolarized topologically simple Jordan ternary H ∗ -algebra, that is neither finite-dimensional nor of quadratic type, is, up to a positive factor of the inner product, and up the sign of the involution and the triple product, isometrically ∗-isomorphic to the odd part of a real topologically simple two-graded Jordan H ∗ -algebra (cf. Theorem 8.1.165). Moreover, the simple finitedimensional Jordan ternary H ∗ -algebras, as well as those of quadratic type, are fully described.

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Cuenca and his coauthors have considered another type of a ternary H ∗ -structure, by imposing on a Hilbert space A with a trilinear triple product {· · · } and an involutive conjugate-linear mapping ∗, the axioms {xyz}∗ = {x∗ y∗ z∗ } and ({xyz}|w) = (x|{wz∗ y∗ }) = (z|{y∗ x∗ w}) = (y|{z∗ wx∗ }) for every x, y, z, w ∈ A (see also [1127], and note that, if the triple product is symmetric in the outer variables — as it fortunately happens in the Jordan case — then nothing but particular ternary H ∗ -algebras arise). In this setting they have proved the essential uniqueness (up to the sign of the involution) of the ‘H ∗ -structure’ in the topologically simple case [162], and have shown fine classification theorems in the associative (of the second kind) and alternative context. The interested reader is referred to the Castell´on–Cuenca papers [163, 901] for the precise versions of these classification theorems, as well as for further information about these topics. 8.2 Extending the theory of H∗ -algebras: generalized annihilator normed algebras Introduction Generalized annihilator normed algebras need not be pre-Hilbert spaces, nor to have a conjugate-linear involution, but share with semi-H ∗ -algebras with zero annihilator the properties of being semiprime and that the annihilator of any closed proper ideal is different from zero (see Definition 8.2.3 and Fact 8.2.4). As one of the key tools in the theory, we prove that the annihilator of each maximal closed ideal of a generalized annihilator normed algebra contains a minimal closed ideal (see Lemma 8.2.6). Then we recapture the notion of weak radical of an algebra, which allows us to establish the general non-associative uniquenessof-norm theorem (cf. Definition 4.4.39 and Theorem 4.4.43), and prove that any generalized annihilator complete normed real or complex algebra with zero weak radical is the closure of the direct sum of its minimal closed ideals, which are indeed topologically simple normed algebras (see Theorem 8.2.17). In Lemma 8.2.18 we show that the weak radical of any real or complex semi-H ∗ -algebra coincides with its annihilator. Therefore, Theorem 8.2.17 provides us with a new proof of Theorem 8.1.16 with completely different arguments (see §8.2.19 for details). To continue our review of this section, let us note that, as any self-adjoint algebra of bounded linear operators on a Hilbert space, the multiplication algebra of a semiH ∗ -algebra is semiprime. Therefore every semi-H ∗ -algebra with zero annihilator is multiplicatively semiprime, i.e. both it and its multiplication algebra are semiprime. Rather surprisingly, multiplicative semiprime algebras can be characterized in terms musically related to generalized annihilator normed algebras (see §8.2.26 and the equivalence (i)⇔(iv) in Theorem 8.2.31). This characterization is applied to show that generalized annihilator normed algebras are multiplicatively semiprime. Actually we characterize generalized annihilator normed algebras among those normed algebras which are multiplicatively semiprime (see Theorem 8.2.33). Generalized complemented normed algebras are defined as those normed algebras having zero annihilator and satisfying that every closed ideal is a direct summand.

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As the main result in the theory, we prove that, if A is a generalized complemented complete normed algebra with zero weak radical, and if {Ai }i∈I stands for the family of its minimal closed ideals, then for each a ∈ A there exists a unique summable  family {ai }i∈I in A such that ai ∈ Ai for every i ∈ I, and a = i∈I ai (see Theorem 8.2.44). Since generalized complemented normed algebras are generalized annihilator, this refines Theorem 8.2.17 (reviewed at the beginning of this introduction) in its particular setting. 8.2.1 The main result Let A be an algebra over K, and let I be an ideal of A. Then the set {a ∈ A : aI = 0 = Ia} is a subspace of A, and hence contains a largest ideal of A. This last ideal will be called the annihilator of I relative to A, and will be denoted by Ann(I). (Note that, as we shall see in Remark 8.2.2, this definition is consistent with the one given in §6.1.66 for ideals of associative algebras.) The following fact is straightforward. Fact 8.2.1 Let A be an algebra over K. We have: (i) If I, J are ideals of A with I ⊆ J, then Ann(J) ⊆ Ann(I). (ii) For any ideal I of A we have I ⊆ Ann(Ann(I)).  4 (iii) For any family {Iα } of ideals of A we have Ann( α Iα ) = α Ann(Iα ). Moreover, if A is normed, then (iv) For any ideal I of A we have: (a) Ann(I) is closed in A. (b) Ann(I) = Ann(I). We note that, if A is a semiprime algebra over K, and if I is an ideal of A, then, by Fact 6.1.75, Ann(I) is the largest ideal J of A which satisfies any of the conditions IJ = 0, I ∩ J = 0, or JI = 0, hence in particular I ∩ Ann(I) = 0. These facts will be applied without notice in what follows. Remark 8.2.2 Let A be an algebra over K, and let I be an ideal of A. If A is associative, then Ann(I) = {a ∈ A : aI = 0 = Ia}, since the right-hand side of the above equality is itself an ideal of A. This remains true if A is alternative, by using that the associator alternates in this case. On the other hand, Zel’manov [1130, Lemma 3] has proved that, if A is a Jordan algebra, then Ann(I) = {a ∈ A : aI = 0 = [a, A, I]}. Definition 8.2.3 A normed algebra A over K is said to be generalized annihilator if it is semiprime and if Ann(I) = 0 for every closed proper ideal I of A.

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As a motivating example, we have the following fact, which follows from Proposition 8.1.13(i) and Remark 8.1.14. Fact 8.2.4 Up to multiplication of the inner product by a positive number, semi-H ∗ -algebras over K with zero annihilator are generalized annihilator complete normed algebras. Topologically simple normed algebras become trivial examples of generalized annihilator normed algebras. In fact the aim of this subsection is to show that, under the additional assumption of some kind of semisimplicity, the theory of generalized annihilator complete normed algebras can be reduced in some way to the case of topologically simple complete normed algebras. As usual in our work, given a non-empty subset S of a normed space X over K, we denote by lin(S) the closed linear hull of S in X. Proposition 8.2.5 Let A be a generalized annihilator normed algebra over K. Then: (i) For every ideal I of A we have A = I ⊕ Ann(I). (ii) If I is a closed ideal of A, then closed ideals of I are precisely the closed ideals of A contained in I. (iii) Minimal closed ideals of A are topologically simple normed algebras. (iv) If I, J are closed ideals of A, then lin(IJ) is a closed ideal of A. (v) For every ideal I of A we have Ann(I) = {a ∈ A : aI = 0 = Ia}. Proof

(8.2.1)

Let I be an ideal of A. Then, by Fact 8.2.1 and semiprimeness of A, we have Ann(I ⊕ Ann(I)) = Ann(I ⊕ Ann(I)) = Ann(I) ∩ Ann(Ann(I)) = 0.

Since I ⊕ Ann(I) is a closed ideal of A, and A is actually a generalized annihilator, assertion (i) follows. Assertion (ii) follows from (i), whereas (iii) follows from (ii). On the other hand, assertion (iv) is proved by realizing that lin(IJ) is a closed ideal of I ∩ J, and then by applying (ii). To prove (v) it is enough to show that the righthand side of the equality (8.2.1), say Z, is an ideal of A. Let z and a be in Z and A, respectively, and according to assertion (i), write a = limn (yn + zn ) with yn ∈ I and zn ∈ Ann(I). Then az = lim(yn + zn )z = lim zn z ∈ Ann(I) ⊆ Z. n

n

Therefore AZ ⊆ Z. Analogously, ZA ⊆ Z, so that, as desired, Z is an ideal of A.



Assertion (v) in the above proposition will not be applied in what follows, but has its own interest (cf. Remark 8.2.2). Let A be a normed algebra over K. By a maximal closed ideal of A we mean a closed proper ideal M of A which is not contained in any closed proper ideal of A other than M.

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The following lemma will be crucial in the proof of the main results of this subsection. Lemma 8.2.6 Let A be a generalized annihilator normed algebra, and let M be a maximal closed ideal of A. Then lin(Ann(M) Ann(M)) is a minimal closed ideal of A. Moreover, we have lin(Ann(M) Ann(M)) = lin(A Ann(M)) = lin(Ann(M)A). Proof Write I := lin(Ann(M)A) which, by Proposition 8.2.5(iv), is a closed ideal of A. If I were zero, then, since M is a closed proper ideal of A, we would have 0 = Ann(M) ⊆ Ann(A) = 0, a contradiction. Therefore I = 0. Now, if J is a nonzero closed ideal of A contained in I, then J ⊆ Ann(M), and hence, by maximality of M, we have A = J + M. Therefore Ann(M)A = Ann(M)(J + M) ⊆ Ann(M)J ⊆ J. Hence I = J. Thus we have proved that I is a minimal closed ideal of A. On the other hand, since 0 = lin(Ann(M) Ann(M)) ⊆ I, and lin(Ann(M) Ann(M)) is a closed ideal of A (again by Proposition 8.2.5(iv)), we derive from the minimality of I that lin(Ann(M) Ann(M)) = I. By the symmetry of these arguments, we have that lin(Ann(M) Ann(M)) = lin(A Ann(M)), and the proof is complete.  Assertion (i) in the next fact generalizes Corollary 8.1.12(ii). Fact 8.2.7 Let A be a generalized annihilator normed algebra over K. We have: (i) A = lin(AA). (ii) If M is any maximal closed ideal of A, then the normed algebra A/M is topologically simple. Proof Write I := lin(AA). Then for x, y ∈ Ann(I) we have xy ∈ I ∩ Ann(I) = 0. Therefore Ann(I) Ann(I) = 0, and hence Ann(I) = 0. To now we have applied only that A is semiprime. But, since A is in fact generalized annihilator, we deduce that A = I, and (i) is proved. Let M be any maximal closed ideal of A. Since closed ideals of A/M are in a one-to-one correspondence with the closed ideals of A containing M, A/M has no nonzero closed proper ideals. Moreover, by (i), A/M has nonzero product. Thus A/M is topologically simple, and the proof is complete.  The next result is an appropriate variant of Theorem 8.1.88. Theorem 8.2.8 Let A be a topologically simple (complete) normed real algebra. Then one of the following conditions holds for A: (i) There exists a topologically simple (complete) normed complex algebra B, and an isometric involutive conjugate-linear algebra automorphism  of B, such that A = H(B, ).

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(ii) There exist topologically simple (complete) normed complex algebras B and C, and continuous copies  A and  C of A and C in BR and B, respectively, such that  C is an ideal of B contained in  A. Proof Let AC be the normed complexification of A (cf. Proposition 1.1.98), and let  denote the canonical involution of AC (cf. Lemma 1.1.97). If AC is topologically simple, then, by taking (B, ) := (AC , ), condition (i) holds for A. Suppose that AC is not topologically simple. Note that, for every -invariant subspace Z of AC we have Z = (Z ∩ A) ⊕ i(Z ∩ A). Let I be any closed proper ideal of AC . Then I ∩ I  ∩ A is a closed proper ideal of A, so I ∩ I  ∩ A = 0, hence I ∩ I  = 0, and II  = 0 = I  I. Therefore we have that I  ⊆ Ann(I) for every closed proper ideal of AC .

(8.2.2)

Now let I be a closed ideal of AC with II = 0. Then, by (8.2.2), J := (I + I  ) ∩ A is an ideal of A with JJ = 0, so J = 0, hence I = 0. Therefore AC is semiprime, which, together with (8.2.2) again, gives us that AC is a generalized annihilator normed algebra.

(8.2.3)

Let F denote the family of all closed proper ideals of AC ordered by inclusion. We claim that F is inductive. Let C be a chain of F . To prove that C has a bound in F we may suppose that there exists a nonzero member J ∈ C and that J ⊆ I for every I ∈ C . Then, by (8.2.2) once more, for every I ∈ C we have I ⊆ Ann(I  ) ⊆ Ann(J  ). But Ann(J  ) is a proper ideal of AC , since J = 0 and (8.2.3) applies. Therefore Ann(J  ) becomes a bound of C in F . Now that we have proved the claim, it follows from Zorn’s lemma that there exists a maximal closed ideal M of AC . The assumption that AC is not topologically simple is waiting for an application, and this is the appropriate moment of applying it to be sure that M = 0 and, consequently, that Ann(M) is a proper ideal, so that (8.2.2) can be invoked with I := Ann(M). By Lemma 8.2.6, there exists a minimal closed ideal C of AC with C ⊆ Ann(M), which implies C ⊆ [Ann(M)] ⊆ Ann[Ann(M)] = M

(8.2.4)

(the equality being true by maximality of M). Write B := AC /M, and let π : AC → B stand for the natural quotient algebra homomorphism. Then, by Fact 8.2.7(ii), B is a topologically simple (complete) normed complex algebra. Moreover, in view of (8.2.4), for x ∈ C we have π(x) = π(x + x ) ∈ π(A), hence π(C) ⊆ π(A). On the other hand, by Proposition 8.2.5(iii), C is a topologically simple (complete) normed complex algebra. Since π is injective on both A and C, it is enough to put  A := π(A) and  C := π(C) to realize that condition (ii) holds for A. 

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Selected topics in the theory of non-associative normed algebras

Remark 8.2.9 It is easy to realize that continuous copies of a topologically simple normed algebra into any normed algebra are topologically simple, and that nonzero ideals of a topologically simple normed algebra are topologically simple. Anyone of these facts implies that the ideal  C of B in condition (ii) of Theorem 8.2.8 is a topologically simple normed algebra. Therefore, if one is not interested in conditions involving completeness (for instance, if the topologically simple normed real algebra A is not complete), then the appropriate formulation of condition (ii) of Theorem 8.2.8 is the following: (iii) There exists a topologically simple normed complex algebra B, a nonzero ideal D of B, and a continuous copy  A of A in BR such that  A contains D. On the other hand, since continuous copies of a normed algebra with minimality of norm topology (cf. Definition 4.4.22) into any normed algebra are bicontinuous copies, and the algebra  A in condition (iii) above is dense in B, we obtain that, if A is a topologically simple complete normed real algebra with minimality of norm topology, and if condition (i) of Theorem 8.2.8 does not hold for A, then there exists a topologically simple complete normed complex algebra B such that A is bicontinuously isomorphic to BR . Let C be an associative algebra over K. We recall that a subalgebra D of C is said to be a quasi-full subalgebra of C if D contains the quasi-inverses of the elements of D which are quasi-invertible in C (cf. §3.6.41). Now let B be an arbitrary algebra over K. According to Definition 4.4.39(b), the smallest quasi-full subalgebra of L(B) containing LB ∪ RB is called the quasi-full multiplication algebra of B, and is denoted by QFM (B). As pointed out in Remark 4.4.41, for a complete normed algebra A over K, the Banach isomorphism theorem implies that QFM (A) is contained in BL(A). Proposition 8.2.10 Let A be a generalized annihilator complete normed algebra over K. Then every closed ideal of A is invariant under QFM (A). Two preparatory lemmas are needed in the proof of this proposition. Lemma 8.2.11 Let C and D be quasi-full subalgebras of an associative algebra B over K such that CD = 0 = DC. Then C + D is a quasi-full subalgebra of B. Proof For x, y ∈ B write x◦y := x+y−xy. Then ◦ becomes an associative operation on B satisfying: (i) x ◦ 0 = x = 0 ◦ x for every x ∈ B; (ii) x ◦ y = 0 = y ◦ x if and only if x is quasi-invertible in B with quasi-inverse y. As a consequence, by elementary computation in (associative) semigroups, we have (iii) if x ◦ y = y ◦ x, and if x ◦ y is quasi-invertible in B, then both x and y are quasiinvertible in B.

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Now let c and d be in C and D, respectively, and suppose that c + d is quasiinvertible in B. Since c ◦ d = c + d = d ◦ c, it follows that both c and d are quasiinvertible in B. But C and D are quasi-full subalgebras of B, so the quasi-inverses c% of c and d% of d lie in C and D, respectively. Then (c + d)% = (c ◦ d)% = d% ◦ c% = d% + c% belongs to C + D. Therefore C + D is a quasi-full subalgebra of B.



Lemma 8.2.12 Let B be a quasi-full subalgebra of a complete normed associative algebra over K, and let C be a quasi-full subalgebra of B. Then the closure C of C in B is a quasi-full subalgebra of B. Proof Since B is a quasi-full subalgebra of a complete normed associative algebra, it follows from Proposition 1.1.15 and Example 3.6.42 that q-Inv(B) (the set of quasiinvertible elements of B) is open in B, and that the mapping b → b% is continuous. Let x be in C ∩ q-Inv(B), say x = limn xn with xn ∈ C for every n ∈ N. Since q-Inv(B) is open, we may suppose that xn ∈ q-Inv(B) for every n. Therefore, by the continuity of the mapping b → b% and the quasi-fullness of C in B, we have that x% = limn xn% belongs to C, so proving that C is a quasi-full subalgebra of B.  The above lemma is a variant of Proposition 4.1.116. Proof of Proposition 8.2.10 Let P be a closed ideal of A, and let us write Q := Ann(P). We now consider the sets P := {F ∈ QFM (A) : F(A) ⊆ P, F(Q) = 0}, Q := {G ∈ QFM (A) : G(A) ⊆ Q, G(P) = 0}. Both P and Q are quasi-full subalgebras of QFM (A) (in fact each of them is the intersection of two one-sided ideals). Moreover, PQ = QP = 0, clearly. Hence P + Q, the closure of P +Q in QFM (A), is a quasi-full subalgebra of QFM (A), by Lemmas 8.2.11 and 8.2.12, and therefore it is also a quasi-full subalgebra of L(A), because QFM (A) is a quasi-full subalgebra of L(A). Now let a be in A. By Proposition 8.2.5(i), a = limn ( pn + qn ) with pn ∈ P and qn ∈ Q for every n. Then La = limn (Lpn + Lqn ) and Ra = limn (Rpn + Rqn ), hence La and Ra belong to P + Q. Then P + Q = QFM (A), by definition of QFM (A). Since P is closed and clearly invariant under P +Q, it follows that P is also invariant  under P + Q = QFM (A), as required. Let B be an algebra over K, and write W (B) to denote the set of those elements b ∈ B such that Lb and Rb belong to the Jacobson radical of QFM (B). Clearly, W (B) is a subspace of B, so it contains a largest subspace invariant under QFM (B). According to Definition 4.4.39(c), this latter subspace is called the weak radical of B, and is denoted by w-Rad(B). For a primitive ideal P of QFM (B), denote by P the largest QFM (B)-invariant subspace of B contained in the set {x ∈ B : Lx , Rx ∈ P}. Subspaces like the one P above, which are clearly ideals of B, will be called weak primitive ideals of B.

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Lemma 8.2.13 Let B be an algebra over K. Then every weak primitive ideal of B is a proper ideal, and w-Rad(B) is equal to the intersection of all weak primitive ideals of B. Proof Let P be a primitive ideal of QFM (B), and let P denote the weak primitive ideal of B associated with P. Then, since P is a quasi-full subalgebra of QFM (B) and P = QFM (B), P cannot contain LB ∪ RB . Hence P = B, clearly. The remainder follows straightforwardly from the definitions of the weak radical (recalled above), of a weak primitive ideal (introduced above), and of the Jacobson radical of an associative algebra (cf. Definition 3.6.12).  Lemma 8.2.14 Let A be a complete normed algebra over K. Then every weak primitive ideal of A is closed in A. As a consequence, w-Rad(A) is closed in A. Proof Since QFM (A) is a quasi-full subalgebra of the complete normed associative algebra BL(A), it follows from the definition of a primitive ideal (cf. Definition 3.6.12) and the implication (vii)⇒(iv) in Proposition 3.6.43 that every primitive ideal P of QFM (A) is closed in QFM (A). Hence it is easy to conclude that the weak primitive ideal P of A associated with P is closed. The consequence follows from Lemma 8.2.13.  Proposition 8.2.15 Let A be a generalized annihilator complete normed algebra over K. Then every weak primitive ideal P of A is a maximal closed ideal of A. Proof By Lemma 8.2.14, P is closed. Suppose that Q is a closed ideal of A containing P. Write S := Ann(Q), and consider the sets Q := {F ∈ QFM (A) : F(A) ⊆ Q, F(S) = 0}, S := {G ∈ QFM (A) : G(A) ⊆ S, G(Q) = 0}. Actually, the condition that F(A) ⊆ Q can be dropped in the definition of Q because, if F ∈ QFM (A), and if F(S) = 0 then, by Propositions 8.2.5(i) and 8.2.10, we have F(A) = F(Q + S) ⊆ F(Q) ⊆ Q. An analogous remark holds for S . It follows from Proposition 8.2.10 that Q and S are ideals of QFM (A). Furthermore, LQ ∪ RQ ⊆ Q and LS ∪ RS ⊆ S . Now let P be the primitive ideal of QFM (A) defining P. Then, since QS = 0 and P is prime (cf. Proposition 3.6.16(i)), we have that either Q ⊆ P or S ⊆ P. If the first alternative occurs, then Q ⊆ {x ∈ A : Lx , Rx ∈ P}. But Q is invariant under QFM (A) by Proposition 8.2.10, so that Q ⊆ P by definition of P and thus Q = P. If the second alternative occurs, then S ⊆ {x ∈ A : Lx , Rx ∈ P} and S ⊆ P as above, so Ann(Q) = S ⊆ P ⊆ Q, hence Ann(Q) = 0 because A is semiprime, and Q = A since A is in fact generalized annihilator. Therefore P is a maximal closed ideal.  The main result of this subsection will be an easy consequence of the following.

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Lemma 8.2.16 Let A be a generalized annihilator complete normed algebra over K. Then every closed ideal of A not contained in w-Rad(A) contains a minimal closed ideal of A. Proof Let B be a closed ideal of A not contained in w-Rad(A). Then, by Lemma 8.2.13, there exists a weak primitive ideal P of A such that B is not contained in P. But, by Proposition 8.2.15, P is a maximal closed ideal of A, and hence A = B + P. Then lin(Ann(P)A) ⊆ B, and lin(Ann(P)A) is a minimal closed ideal of A thanks to Lemma 8.2.6.  Now we can conclude the proof of the main result in this subsection, namely the following. Theorem 8.2.17 Let A be a generalized annihilator complete normed algebra over K with zero weak radical. Then A is the closure of the direct sum of its minimal closed ideals, and these are topologically simple (complete) normed algebras. Proof Let {Mα } be the family of all minimal closed ideals of A, and write  B := α Mα . If B were not equal to A, then we would have Ann(B) = 0 and so, by Lemma 8.2.16, Ann(B) would contain a minimal closed ideal Mβ . Thus, by assertions (i), (iii), and (iv)(b) in Fact 8.2.1, we would obtain Mβ ⊆ Ann(B) ⊆ Ann(Mβ),  hence Mβ = 0, which is a contradiction. Therefore A = B = α Mα . Let Mβ be one of the minimal closed ideals of A. Then for α = β we have Mβ ∩ Mα = 0 by minimality,   so Mβ Mα = 0. Therefore Mβ α=β Mα = 0, and hence Mβ ∩ ( α=β Mα ) = 0 by  semiprimeness of A. Thus the sum α Mα is direct. Finally, that each Mα is a topologically simple normed algebra follows from Proposition 8.2.5(iii).  Now we are going to realize how the star result of Subsection 8.1.2 can be derived from the above theorem. To this end, we only need the following. Lemma 8.2.18 Let A be a semi-H ∗ -algebra over K. Then w-Rad(A) = Ann(A). Proof Let a be in w-Rad(A). Then La∗ La lies in the Jacobson radical of QFM (A), and hence, by Corollary 3.6.23, we have that r(La∗ La ) = 0. On the other hand, since La∗ La = (La )• La (cf. (8.1.4)), we have r(La∗ La ) = La 2 . It follows that La = 0. Similarly Ra = 0. Therefore a lies in Ann(A). This proves the inclusion w-Rad(A) ⊆ Ann(A). The reverse inclusion follows from Proposition 4.4.59(i).  §8.2.19 New proof of Theorem 8.1.16 Let A be a semi-H ∗ -algebra over K with zero annihilator. Then, by Fact 8.2.4, A is a generalized annihilator (complete) normed algebra. On the other hand, by Lemma 8.2.18, A has zero weak radical. Therefore the crucial fact that A is the closure of the direct sum of its minimal closed ideals follows from Theorem 8.2.17. The remaining information in the statement of Theorem 8.1.16 is verified as in the first paragraph of its original proof.  Other interesting consequences of Theorem 8.2.17 will be derived in what follows.

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Selected topics in the theory of non-associative normed algebras

For the notions of J-radical and of J-semisimplicity of a Jordan-admissible algebra over K, the reader is referred to Definition 4.4.12. Let A be a Jordan-admissible algebra over K. Since w-Rad(A) ⊆ J-Rad(A) (cf. Proposition 4.4.59(i)-(ii)), it follows from Theorem 8.2.17 that, if A is J-semisimple, complete normed, and generalized annihilator, then the conclusion in Theorem 8.2.17 holds for A, and of course minimal closed ideals of A are Jordan-admissible algebras. For later discussion, we emphasize here the particularization to non-commutative Jordan algebras of the result just established. Corollary 8.2.20 Let A be a generalized annihilator complete normed J-semisimple non-commutative Jordan algebra over K. Then A is the closure of the direct sum of its minimal closed ideals, and these are topologically simple (complete) normed J-semisimple (non-commutative Jordan) algebras. Proof In view of the comments immediately above, it is enough to show that minimal closed ideals of A are J-semisimple. But this follows from the hereditary property of the J-radical of a (commutative) Jordan algebra (see for example [822, Corollary 14.4.2]), together with Proposition 4.4.17(iii).  Let us say that a Jordan-admissible algebra A over K is a J-radical algebra if J-Rad(A) = A. It is noteworthy that, as we already noticed in Remark 4.4.68(b), anticommutative algebras are J-radical non-commutative Jordan algebras. Therefore Corollary 8.2.20 becomes useless in the setting of anticommutative algebras. This is not the case of Theorem 8.2.17 which, for example, works successfully for Lie H ∗ -algebras with zero annihilator (cf. Lemma 8.2.18 and its immediately later application). Now let A be a non-commutative Jordan algebra over K. Then, since J-Rad(A) ⊆ Rad(A) (cf. Proposition 4.4.59(iii)), it follows from Corollary 8.2.20 that if A is semisimple, complete normed, and generalized annihilator, then the conclusion in Corollary 8.2.20 holds for A, but, of course, this has no special interest. In the particular case that A is alternative, we have in fact J-Rad(A) = Rad(A) (cf. the comments in Definition 4.4.12), and actually all people prefer to speak about semisimplicity instead of J-semisimplicity. Thus we have the following. Corollary 8.2.21 Let A be a generalized annihilator complete normed semisimple alternative algebra over K. Then A is the closure of the direct sum of its minimal closed ideals, and these are either topologically simple (complete) normed associative algebras or isomorphic to C(C) if K = C, and to C(C)R , C(R), or O if K = R. The disjunctive above on minimal closed ideals follows from a theorem of Kleinfeld [390] on primitive alternative algebras (see also [822, Theorem 1 in p. 201]), the Gelfand–Mazur theorem (cf. Corollary 1.1.43 and Proposition 2.5.40), and the following. Fact 8.2.22 Topologically simple complete normed semisimple algebras over K are primitive.

8.2 Generalized annihilator normed algebras

571

Proof Let A be a topologically simple complete normed semisimple algebra over K. Since primitive ideals of any complete normed algebra are closed proper ideals (cf. Corollary 3.6.13), and there are primitive ideals of A (by semisimplicity), it follows from the topological simplicity of A that zero is a primitive ideal of A.  We note that, keeping in mind that primitive algebras are prime (by Proposition 3.6.16(i)), Kleinfeld’s theorem quoted above follows easily from Slater’s Theorem 6.1.33 (see [822, Section 10.1] for details). As a straightforward consequence of Corollary 8.2.21, we obtain the following. Corollary 8.2.23 Let A be a generalized annihilator complete normed semisimple associative algebra over K. Then A is the closure of the direct sum of its minimal closed ideals, and these are topologically simple (complete) normed semisimple associative algebras. Let us prove two more corollaries. Corollary 8.2.24 Let A be a generalized annihilator non-commutative JB∗ -algebra. Then A is the c0 -sum of the family of its minimal closed ideals, and these are topologically simple non-commutative JB∗ -algebras. Proof Since every non-commutative JB∗ -algebra is J-semisimple (cf. Lemma 4.4.28(iii)), it follows from Corollary 8.2.20 and Proposition 3.4.13 that A is the closure of the direct sum of its minimal closed ideals, and that these are topologically simple non-commutative JB∗ -algebras. If M1 , M2 , . . . , Mn are minimal closed ideals of A, then the sum M1 + M2 + · · · + Mn is direct, and therefore, as a consequence of Proposition 3.4.4 and §3.4.11, for xi ∈ Mi (i = 1, 2, . . . , n) we have x1 + x2 + · · · + xn  = max{x1 , x2 , . . . , xn }. We now apply an ‘extension-by-continuity’ argument.



The above corollary, together with Fact 8.2.22 and the comments immediately before it, allows us to derive the following. Corollary 8.2.25 Let A be a generalized annihilator alternative C∗ -algebra. Then A is the c0 -sum of the family of its minimal closed ideals, and these are either topologically simple (associative) C∗ -algebras or equal to the alternative C∗ -algebra C(C) of complex octonions (cf. Proposition 2.6.8 and Corollary 3.4.76).

8.2.2 Generalized annihilator algebras are multiplicatively semiprime An algebra A over K is said to be multiplicatively semiprime whenever both A and M (A) are semiprime algebras. All semi-H ∗ -algebras over K with zero annihilator become examples of multiplicatively semiprime algebras. Indeed, if A is a semiH ∗ -algebra with zero annihilator, then A is semiprime (cf. Remark 8.1.14). Moreover, since M (A) is a self-adjoint subset of the C∗ -algebra BL(A), we see that, for F ∈ M (A), the condition FM (A)F = 0 implies FF • F = 0, hence

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Selected topics in the theory of non-associative normed algebras 0 = F • FF • F = F • F2 = F4 ,

and so F = 0. Thus M (A) is semiprime. In what follows we are going to show that all generalized annihilator normed algebras are multiplicatively semiprime. In fact we will obtain a characterization of generalized annihilator normed algebras among those normed algebras which are multiplicatively semiprime. §8.2.26 Let A be an algebra over K. The annihilator operation on the lattice of all ideals of A induces a ‘closure operation’ in that lattice. Indeed, for any ideal I of A, the π-closure of I is defined by π

I := Ann(Ann(I)) and, as a consequence of Fact 8.2.1(i)–(ii), for all I, J ideals of A we have that π

π

(i) I ⊆ J implies I ⊆ J . π (ii) I ⊆ I . π π π π (iii) Ann(I) = Ann(I ), and hence I = (I ) . For subspaces S of A and N of M (A), we define Sann := {F ∈ M (A) : F(S) = 0} and Nann := {a ∈ A : N (a) = 0}. (In the case that S is an ideal of A, the meaning of Sann was already introduced in the proof of Proposition 8.1.25.) It is clear that Sann is a left ideal of M (A) and that Nann is a subspace of A. Moreover, if S1 , S2 are subspaces of A with S1 ⊆ S2 , then S2ann ⊆ S1ann , and, if N1 , N2 are subspaces of M (A) with N1 ⊆ N2 , then (N2 )ann ⊆ (N1 )ann . The ε-closure of a subspace S of A and the ε -closure of a subspace N of M (A) are defined by ε

S := (Sann )ann and N

ε

:= (Nann )ann .

For any subspaces S, S1 , S2 of A and subspaces N , N1 , N2 of M (A) it is immediate to verify that ε

ε

ε

ε

(i) S1 ⊆ S2 implies S1 ⊆ S2 , and N1 ⊆ N2 implies N1 ⊆ N2 . ε

ε

(ii) S ⊆ S , and N ⊆ N . ε ε (iii) Sann = (S )ann = (Sann ) , ε

Nann = (Nann ) = (N

ε

and

hence

)ann , and hence N

ε

S

ε

= (N

ε ε

= (S ) . ε

ε

Analogously,

) .

The next result describes the behaviour of these closures in relation to the products of A and of M (A). Proposition 8.2.27 Let A be an algebra over K. We have: ε

ε

(i) If F ∈ M (A), and if S is a subspace of A, then F(S ) ⊆ F(S) . As a consequence, ε ε ε S1 S2 ⊆ lin(S1 S2 ) for all subspaces S1 , S2 of A. ε

ε

(ii) N1 N2 ⊆ lin(N1 N2 ) for all subspaces N1 , N2 of M (A).

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573

Proof Let F be an element of M (A) and let S be a subspace of A. It is clear that F(S)ann (F(x)) = 0 for every x ∈ S, and therefore F(S)ann F ⊆ Sann . It follows from ε ε ε this that F(S)ann F(S ) = 0, and so F(S ) ⊆ F(S) . Now, let S1 , S2 be subspaces of A. Then for each x ∈ S1 we have ε

ε

ε

ε

xS2 = Lx (S2 ) ⊆ Lx (S2 ) ⊆ lin(S1 S2 ) , ε

ε

ε

ε

and so S1 S2 ⊆ lin(S1 S2 ) . Analogously, we obtain that S1 S2 ⊆ lin(S1 S2 ) . Finally, we have ε

ε ε

ε ε

ε

ε ε

ε

S1 S2 ⊆ lin(S1 S2 ) ⊆ lin( lin(S1 S2 ) ) = ( lin(S1 S2 ) ) = lin(S1 S2 ) . If N1 , N2 are subspaces of M (A), then clearly we have N1 N2 (( lin(N1 N2 ))ann ) = 0, ε

therefore N2 (( lin(N1 N2 ))ann ) ⊆ (N1 )ann , hence N1 N2 (( lin(N1 N2 ))ann ) = 0, ε

ε

and finally N1 N2 ⊆ lin(N1 N2 ) .



Let A be an algebra over K. It is clear that, if I is an ideal of A and if P is an ideal of M (A), then I ann and Pann are ideals of M (A) and A, respectively. As a ε

consequence, I and P

ε

are ideals of A and M (A), respectively.

Proposition 8.2.28 Let A be an algebra over K. Then: (i) For any ideal I of A, we have: ε π (a) I ⊆ I . ε ε (b) Ann(I) = Ann(I ) = Ann(I). ε π π ε π (c) (I ) = (I ) = I . (ii) When in addition M (A) is semiprime, for any ideal P of M (A) we have: (a) P

ε

π

⊆P . ε

ε

(b) Ann(P) = Ann(P ) = Ann(P). ε

π

 π ε

π

(c) (P ) = (P ) = P . Proof Let I be an ideal of A. It follows from Proposition 8.2.27(i) and the equalities ε ε ε ε I Ann(I) = 0 = Ann(I)I that I Ann(I) = 0 = Ann(I) I , and as a consequence ε ε Ann(I) ⊆ Ann(I ). This fact, together with the chain of inclusions ε

ε

Ann(I ) ⊆ Ann(I) ⊆ Ann(I) , allows us to conclude the proof of (i)(b). Moreover, by taking annihilator to both ε π

ε

π

sides of the equality Ann(I ) = Ann(I), we obtain (I ) = I , and also, as a consequence, assertion (i)(a) holds. The remaining part of (i)(c) follows from the chain of inclusions π

π ε

π π

π

I ⊆ (I ) ⊆ (I ) = I . Now, suppose that M (A) is semiprime, and fix an ideal P of M (A). It follows from Proposition 8.2.27(ii) and the equalities P Ann(P) = 0 = Ann(P)P

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Selected topics in the theory of non-associative normed algebras ε

ε

that P Ann(P) = 0 = Ann(P) P. Therefore, keeping in mind that M (A) is ε

ε

semiprime and Fact 6.1.75, we deduce that Ann(P) ⊆ Ann(P) ⊆ Ann(P ). But the reverse inclusions are obviously true, and so we have proved (ii)(b). Finally, arguing as in the proof of (i)(a) and (i)(c), we deduce (ii)(a) and (ii)(c) from (ii)(b).  Let A be an algebra over K. As in the proof of Proposition 8.1.25, given an ideal I of A, we will consider the ideal [I : A] of M (A) defined by [I : A] := {F ∈ M (A) : F(A) ⊆ I}. Note that LI , RI ⊆ [I : A], and hence we have that I[I : A]ann = [I : A]ann I = 0. Therefore [I : A]ann ⊆ Ann(I), and so ε

Ann(I)ann ⊆ [I : A] .

(8.2.5)

Proposition 8.2.29 Let A be an algebra over K. Then the following assertions are equivalent: (i) M (A) is semiprime. (ii) I ann ∩ [I : A] = 0 for every ideal I of A. Moreover, in the case that the above equivalent conditions are fulfilled, we have Ann(I)ann ⊆ Ann(I ann ) for every ideal I of A. Proof The implication (i)⇒(ii) follows from Fact 6.1.75 and the obvious equality I ann [I : A] = 0 for every ideal I of A. Now, suppose that (ii) holds, and suppose that P is an ideal of M (A) such that PP = 0. Then P ⊆ ( lin(P(A)))ann ∩ [ lin(P(A)) : A] = 0, and hence P = 0. Thus M (A) is semiprime, and we have proved that (ii) implies (i). Finally, suppose that the equivalent conditions in the statement are fulfilled, and fix an ideal I of A. Then, by (ii), [I : A] ⊆ Ann(I ann ), and hence, by Proposition 8.2.28(ii)(b), we derive that ε

[I : A] ⊆ Ann(I ann ).

(8.2.6) 

Now the proof concludes by combining the inclusions (8.2.5) and (8.2.6). Proposition 8.2.30 Let A be a multiplicatively semiprime algebra. Then P for every ideal P of M (A). ε

ε

=P

π

π

Proof Let P be an ideal of M (A). Since we know that P ⊆ P (cf. Proposition 8.2.28(ii)(a)), we only need to prove the reverse inclusion. Note that π

π

P (P + Ann(P))LPann (A) = P PLPann (A) ⊆ P(Pann A) ⊆ P(Pann ) = 0, π

hence P (P + Ann(P))LPann = 0, and so π

π

LPann P (P + Ann(P))LPann P = 0.

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575

Since P + Ann(P) is an essential ideal of M (A) (cf. Lemma 6.1.67(iii)), it follows π π from Proposition 6.1.134 that LPann P = 0. Therefore Pann lin(P (Pann )) = 0, π

π

and hence lin(P (Pann )) lin(P (Pann )) = 0. Now, the semiprimeness of A gives π

ε

π

that P (Pann ) = 0, that is to say P ⊆ P , and the proof is complete.



The notions of -closed ideal, -dense ideal, minimal -closed ideal, and maximal -closed ideal for  equal to some of the π, ε or ε closures (which will be used from now on) are explained by themselves. It follows from Proposition 8.2.5(i) that a normed algebra A over K is generalized annihilator if and only if, for every ideal I of A, we have A = I ⊕ Ann(I). Therefore the equivalence (i)⇔(v) in the next theorem shows how multiplicatively semiprime algebras become purely algebraic substitutes of generalized annihilator normed algebras. Theorem 8.2.31 Let A be an algebra over K. Then the following conditions are equivalent: (i) (ii) (iii) (iv) (v)

A is multiplicatively semiprime. A is semiprime and Ann(I ann ) = Ann(I)ann for every ideal I of A. ε π A is semiprime and I = I for every ideal I of A. A is semiprime and for each ε-closed proper ideal I of A, we have Ann(I) = 0. ε A = I ⊕ Ann(I) for every ideal I of A.

Proof Suppose that A is multiplicatively semiprime, and let I be an ideal of A. Keeping in mind Proposition 8.2.28(i)(b), we see that ε

ε

ε

ε

0 = I ∩ Ann(I ) = I ∩ Ann(I) = (I ann + Ann(I)ann )ann , ε

hence M (A) = I ann + Ann(I)ann , and so, by Proposition 8.2.28(ii)(b), 0 = Ann(I ann + Ann(I)ann ) = Ann(I ann ) ∩ Ann(Ann(I)ann ), π

and hence Ann(I ann ) ⊆ Ann(I)ann . Now, by Proposition 8.2.29, we deduce that π Ann(I ann ) = Ann(I)ann . Therefore, using Proposition 8.2.30, we see that ε

Ann(I ann ) = Ann(I)ann = Ann(I)ann , and the implication (i)⇒(ii) is proved. Suppose that (ii) holds, and let I be an ideal of A. Applying (ii) to both I and Ann(I), it follows that π

π

I ann = Ann(Ann(I ann )) = Ann(Ann(I)ann ) = Ann(Ann(I))ann = (I )ann . π

π

π

Since clearly (I )ann ⊆ I ann ⊆ I ann , we deduce that I ann = (I )ann . It follows from ε

π ε

ε

π

this that I = (I ) , and finally that I = I . Thus we have proved the implication (ii)⇒(iii). The implication (iii)⇒(iv) is trivial.

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Selected topics in the theory of non-associative normed algebras

Suppose that (iv) holds, and let I be an ideal of A. Keeping in mind the semiprimeness of A and Proposition 8.2.28(i)(b), we see that ε

0 = Ann(I) ∩ Ann(Ann(I)) = Ann(I ⊕ Ann(I) ), ε

and hence I ⊕ Ann(I) = A. Thus the proof of the implication (iv)⇒(v) is finished. Suppose that (v) holds. Then, clearly, A is semiprime. Let I be an ideal of A. The ε equality A = I ⊕ Ann(I) gives that 0 = (I ⊕ Ann(I))ann = I ann ∩ Ann(I)ann . Since [I : A](Ann(I)) ⊆ I ∩ Ann(I) = 0, we have that [I : A] ⊆ Ann(I)ann . Therefore 0 = I ann ∩ [I : A], and the semiprimeness of M (A) follows from Proposition 8.2.29.  For any semiprime algebra A over K, it is clear that the mapping I → Ann(I) becomes an order-reversing bijection on the lattice of all π-closed ideals of A, and hence it interchanges the set of all minimal π-closed ideals of A with the set of all maximal π-closed ideals of A. In view of the the implication (i)⇒(iii) in the above proposition, whenever in addition A is multiplicatively semiprime, we can replace ‘π-closed’ by ‘ε-closed’ in the above comment. Let A be a normed algebra over K. It follows from the continuity of the operators in ε M (A) that, for each ideal I of A, we have (I)ann = I ann , and hence I ⊆ I . Therefore, ε-closed ideals of A are closed. The converse is not true. For example, the complete normed associative algebra BL(H) of all bounded linear operators in a nonzero real or complex Hilbert space H is a multiplicatively prime algebra (since every prime associative algebra is multiplicatively prime [886, Proposition 4]), and hence every one of its nonzero ideals is ε-dense (cf. Proposition 8.1.25), but the ideal K(H) of all compact operators in H is a closed proper ideal whenever H is infinite-dimensional. Concerning minimal closed ideals, we have the following. Lemma 8.2.32 Let A be a normed semiprime algebra over K, and let I be a minimal ε closed ideal of A. Then I is a minimal ε-closed ideal of A. ε

Proof Let J be a nonzero ε-closed ideal of A contained in I . Then, by Proposition 8.2.27(i), we have ε

ε

ε

0 = JJ ⊆ JI ⊆ lin(JI) ⊆ J ∩ I , and hence J ∩ I is a nonzero closed ideal of A. Since I is a minimal closed ideal, it ε ε follows that J ∩ I = I. Hence I ⊆ J, and so J = I . Thus I is a minimal ε-closed ideal of A.  Next we will determine the relation between the generalized annihilator normed algebras and the multiplicatively semiprime algebras. Theorem 8.2.33 Let A be a normed algebra over K. Then, the following conditions are equivalent:

8.2 Generalized annihilator normed algebras

577

(i) A is a generalized annihilator normed algebra. (ii) A is multiplicatively semiprime and every ε-dense ideal of A is dense. Moreover, in the case that A satisfies the above equivalent conditions, we have: (1) The set of all maximal ε-closed ideals of A coincides with the set of all maximal closed ideals of A. ε (2) The set {I : I is a minimal closed ideal of A} coincides with the set of all minimal ε-closed ideals of A. Proof Suppose that A is generalized annihilator. Then, Proposition 8.2.5(i), we have A = I + Ann(I) for every ideal I of A. Now, the relationship between the · and ε closures, together with the implication (v)⇒(i) in Theorem 8.2.31, give that A is multiplicatively semiprime. Moreover, if I is an ε-dense ideal of A, then Ann(I) = 0, and so I is dense in A. Thus the proof of the implication (i)⇒(ii) is complete. The implication (ii)⇒(i) is a direct consequence of the implication (v)⇒(i) in Theorem 8.2.31. Now, suppose that A satisfies the equivalent conditions in the statement and let us show (1) and (2). If M is a maximal ε-closed ideal of A, then M is closed and, for each closed proper ε ideal I of A containing M, we derive from (ii) that I is an ε-closed proper ideal of ε A containing M, hence I = M, and so I = M. Thus, M is a maximal closed ideal of A. To prove the converse, let us fix a maximal closed ideal I of A. If J is an ε-closed proper ideal of A containing I, then, since J is closed, we see that J = I. In particular, ε ε since by (ii), I is also an ε-closed proper ideal of A, it follows that I = I, that is, I is ε-closed. Summarizing, I is a maximal ε-closed ideal of A. Thus we have proved (1). ε If I is a minimal closed ideal of A, then, by Lemma 8.2.32, I is a minimal ε-closed ideal of A. In order to prove the converse, let us fix a minimal ε-closed ideal B of A. Since B is a closed ideal of A, it follows from Proposition 8.2.5(iv) that lin(BB) is a nonzero closed ideal of A contained in B. Moreover, if I is a nonzero closed ideal of A contained in lin(BB), then, keeping in mind (1) and the fact that Ann(B) is a maximal ε-closed ideal of A, we deduce that A = I + Ann(B), and so BB ⊆ BA ⊆ B(I + Ann(B)) ⊆ I. Therefore, I = lin(BB). Thus lin(BB) is a minimal closed ideal of A. The proof concludes by noting that ε

B = lin(BB) .



8.2.3 Generalized complemented normed algebras A normed algebra A over K is said to be generalized complemented if it has zero annihilator and if every closed ideal of A is a direct summand of A. Clearly, topologically simple normed algebras become examples of generalized complemented normed algebras. It is also clear that generalized complemented normed algebras

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Selected topics in the theory of non-associative normed algebras

are generalized annihilator. Moreover, by Proposition 8.1.13(i), semi-H ∗ -algebras with zero annihilator are generalized complemented. As a consequence of Lemma 5.1.1, we are provided with the following. Fact 8.2.34 A normed algebra A over K is generalized complemented if and only if for every closed ideal M of A we have A = M ⊕ Ann(M). The following proposition provides us with two different manners of computing algebraically the closure of an ideal in a generalized complemented normed algebra. Proposition 8.2.35 Let A be a generalized complemented normed algebra over K (for example, a semi-H ∗ -algebra over K with zero annihilator), and let M be any π ε ideal of A. Then M = M = M . Proof

By Fact 8.2.34, we have A = M ⊕ Ann(M), and analogously A = Ann(Ann(M)) ⊕ Ann(M). π

Therefore, since Ann(Ann(M)) ⊇ M, we conclude that M = Ann(Ann(M)) = M . Finally, by the implication (i)⇒(ii) in Theorem 8.2.33 and the implication (i)⇒(iii) ε  in Theorem 8.2.31, we have M = M . §8.2.36 In the following lemmata X will denote a Banach space over K, and {Xi }i∈I will stand for a family of closed subspaces of X satisfying the following axiom:   X= Xi ⊕ Xi for every countable substet J of I. (8.2.7) i∈J

i∈I\J

As a consequence we have X = ⊕i∈I Xi . Another clear consequence is the following. Lemma 8.2.37 For each countable subset J of I, there exists a positive number K (J) such that max{x, y} ≤ K (J)x + y for all x ∈ ⊕i∈J Xi and y ∈ ⊕i∈I\J Xi . Lemma 8.2.38 Suppose that there exists a finite subset F of I, and a positive number M > 0 such that u ≤ M u + v for all disjoint subsets K and L of I \ F with K ∪ L = I \ F, and all u ∈ ⊕i∈K Xi and v ∈ ⊕i∈L Xi . Then there exists a positive number K such that x ≤ K x + y for every subset J of I, and all x ∈ ⊕i∈J Xi and y ∈ ⊕i∈I\J Xi . Proof

According to Lemma 8.2.37, write N := max{K (E) : E a subset of F}.

Now let J be any subset of I, and let x and y be in ⊕i∈J Xi and ⊕i∈I\J Xi , respectively. Write K := (I \ F) ∩ J and L := (I \ F) ∩ (I \ J), so that K and L are disjoint subsets of I \ F with K ∪ L = I \ F. Write also x = x1 + x2 with x1 ∈ ⊕i∈F∩J Xi and x2 ∈ ⊕i∈K Xi , and y = y1 + y2 with y1 ∈ ⊕i∈F∩(I\J) Xi and y2 ∈ ⊕i∈L Xi . Then we have x ≤ x1  + x2  ≤ N x1 + y1  + M x2 + y2 . But x1 + y1 ∈ ⊕i∈F Xi and x2 + y2 ∈ ⊕i∈I\F Xi , hence max{x1 + y1 , x2 + y2 } ≤ N x1 + y1 + x2 + y2  = N x + y.

(8.2.8)

8.2 Generalized annihilator normed algebras

579

Therefore, invoking (8.2.8), we obtain x ≤ (N + M )N x + y. Thus the conclusion in the lemma holds with K := (N + M )N .  Lemma 8.2.39 There exists a positive number K such that x ≤ K x + y for every subset J of I, and all x ∈ ⊕i∈J Xi and y ∈ ⊕i∈I\J Xi . Proof We argue by contradiction, hence we assume that the lemma is not true. As a first consequence, there exists a subset J of I, and elements x1 ∈ ⊕i∈J Xi and y1 ∈ ⊕i∈I\J Xi such that x1  > x1 + y1 . Note that x1 ∈ ⊕i∈F1 Xi and y1 ∈ ⊕i∈G1 Xi for suitable finite subsets F1 and G1 of J and I \ J, respectively, and that F1 ∩ G1 = ∅. Starting from F1 , G1 , x1 , y1 just obtained, we are going to construct four sequences Fn , Gn , xn , yn , with the following conditions: (i) For each n ∈ N, Fn and Gn are finite subsets of I, and moreover, Fn ∩ Gm = ∅ for all n, m ∈ N . (ii) For each n ∈ N we have xn ∈ ⊕i∈Fn Xi , yn ∈ ⊕i∈Gn Xi , and xn  > nxn + yn . Assume inductively that, for some n ∈ N, we have found finite subsets F1 , . . . , Fn , G1 , . . . , Gn of I such that Fk ∩ G = ∅ for all k, = 1, . . . , n, and elements x1 , . . . , xn , y1 , . . . , yn such that xk ∈ ⊕i∈Fk Xi , yk ∈ ⊕i∈Gk Xi , and xk  > kxk + yk  for every 1 ≤ k ≤ n. Write F := (∪1≤k≤n Fk ) ∪ (∪1≤k≤n Gk ), and note that, since we assume that the present lemma is not true, and this assumption means that the conclusion of Lemma 8.2.38 fails, we realize that the assumption in Lemma 8.2.38 must fail. Therefore there exist disjoint subsets K and L of I \F with K ∪L = I \F, and elements xn+1 ∈ ⊕i∈K Xi and yn+1 ∈ ⊕i∈L Xi such that xn+1  > (n + 1)xn+1 + yn+1 . But clearly there are finite subsets Fn+1 and Gn+1 of K and L, respectively, such that xn+1 ∈ ⊕i∈Fn+1 Xi and yn+1 ∈ ⊕i∈Gn+1 Xi . Thus, keeping in mind the induction hypothesis and the definition of F, the finite subsets F1 , . . . , Fn , Fn+1 , G1 , . . . , Gn , Gn+1 of I satisfy that Fk ∩ G = ∅ for all k, = 1, . . . , n, n + 1, and the elements x1 , . . . , xn , xn+1 , y1 , . . . , yn , yn+1 satisfy that xk ∈ ⊕i∈Fk Xi , yk ∈ ⊕i∈Gk Xi , and xk  > kxk + yk  for every 1 ≤ k ≤ n + 1. Now that we have constructed the sequences Fn , Gn , xn , yn fulfilling conditions (i) and (ii) above, we write H := ∪n∈N Fn , and apply Lemma 8.2.37 to obtain that xn  ≤ K (H)xn + yn  for every n ∈ N. Therefore nxn + yn  < xn  ≤ K (H)xn + yn , hence n < K (H) for every n ∈ N, the desired contradiction.



Lemma 8.2.40 For each x ∈ X there exists a unique summable family {xi }i∈I in X  such that xi ∈ Xi for every i ∈ I, and x = i∈I xi . Proof Let K be the positive number given by Lemma 8.2.39, and let J be any subset of I. Then for x ∈ ⊕i∈J Xi and y ∈ ⊕i∈I\J Xi we have max{x, y} ≤ K x + y ≤ 2K max{x, y}, and clearly these inequalities remain true for x ∈ ⊕i∈J Xi and y ∈ ⊕i∈I\J Xi . Therefore (⊕i∈J Xi ) ∩ (⊕i∈I\J Xi ) = 0, and (⊕i∈J Xi ) ⊕ (⊕i∈I\J Xi ) is a complete (hence closed)

580

Selected topics in the theory of non-associative normed algebras

subspace of X. Since ⊕i∈I Xi ⊆ (⊕i∈J Xi ) ⊕ (⊕i∈I\J Xi ) and ⊕i∈I Xi = X, it follows that X = (⊕i∈J Xi ) ⊕ (⊕i∈I\J Xi ). Thus, denoting by πJ the unique linear projection on X with πJ (X) = ⊕i∈J Xi and ker(πJ ) = ⊕i∈I\J Xi , we realize that πJ is continuous with πJ  ≤ K . Now, if J1 , . . . , Jn are pairwise disjoint subsets of I, then for every   x ∈ ⊕i∈I Xi we have clearly π∪nk=1 Jk (x) = nk=1 πJk (x), and hence π∪nk=1 Jk = nk=1 πJk by the denseness of ⊕i∈I Xi in X and the continuity of the projections πsomething . Now let x be in X, and for each i ∈ I write xi := π{i} (x) ∈ Xi . We claim that the  family {xi }i∈I is summable in X with x = i∈I xi . Let ε > 0. Since X = ⊕i∈I Xi , there  exists a finite subset F0 of I, and elements yi (i ∈ F0 ) such that x − i∈F0 yi  ≤ Kε . Now, for every finite subset F of I containing F0 we have  =  xi − yi ∈ Xi and, since πF (x −



i∈F

i∈F0

i∈F xi ) = 0,

also =   x− xi = xi ∈ Xi , i∈F

hence x−

 i∈F

 xi ≤ K

i∈F

x−

 i∈F

 xi

i∈I\F

i∈I\F

⎛ ⎞    xi − yi ⎠ = K x − yi ≤ ε. +⎝ i∈F

i∈F0

i∈F0

Thus the claim has been proved.  Suppose that x = i∈I zi for some summable family {zi }i∈I in X such that zi ∈ Xi for every i ∈ I. Then for every j ∈ I we have π{ j} (x) = zj . Therefore xj = zj for every j ∈ I because, by definition, xj = π{ j} (x). Now the proof is complete.  Now we are going to apply the above discussion on Banach spaces to generalized complemented normed algebras. Fact 8.2.41 Let A be a generalized complemented complete normed algebra over K, and let M, N be closed ideals of A such that M ∩ N = 0. Then M + N is closed in A. Proof By Fact 8.2.34, we have A = M ⊕ Ann(M), and hence, since A is a Banach space, and M and Ann(M) are closed subspaces of A, the above direct sum is topological direct. On the other hand, since MN = NM = 0, we see that N ⊆ Ann(M). Therefore, since N is closed in Ann(M), it follows that M ⊕ N is a closed subspace of A, as required.  Proposition 8.2.42 Let A be a generalized complemented complete normed algebra  over K, and let {Ai }i∈I be a family of closed ideals of A such that A = i∈I Ai and Ai Aj = 0 whenever i, j ∈ I with i = j. Then for each a ∈ A there exists a unique  summable family {ai }i∈I in A such that ai ∈ Ai for every i ∈ I, and a = i∈I ai .   Proof Let J be any subset of I. Write M := i∈J Ai and N := i∈I\J Ai . Then M and N are closed ideals of A with MN = 0 and A = M + N. Therefore, since M ∩N = 0

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581

(cf. Fact 6.1.75), it follows from Fact 8.2.41 that A = M ⊕ N. Now, keeping in mind the arbitrariness of J, we are under the protection of Axiom (8.2.7) in §8.2.36, so that the result follows from Lemma 8.2.40.  Now let us do a purely algebraic interlude. Proposition 8.2.43 Let A be an algebra over K, and let P be a direct summand of A. Then we have: (i) P is invariant under QFM (A). (ii) w-Rad(P) = w-Rad(A) ∩ P. Proof Take any ideal Q of A such that A = P ⊕ Q. We begin by proving the following Claim Let C and D quasi-full subalgebras of QFM (A) such that (a) C D = DC = 0 = C ∩ D (b) C contains LP ∪ RP , and D contains LQ ∪ RQ . Then we have: (1) C and D are ideals of QFM (A) satisfying QFM (A) = C ⊕ D. (2) Ideals of C are ideals of QFM (A). Proof By assumption (a) and Lemma 8.2.11, C ⊕ D, is a quasi-full subalgebra of QFM (A), and therefore it is also a quasi-full subalgebra of L(A), because QFM (A) is a quasi-full subalgebra of L(A). Let a be in A. On the other hand, by assumption (b), we have LA ∪ RA ⊆ C + D. It follows from the definition of QFM (A) that C ⊕ D = QFM (A). Keeping in mind assumption (a) again, this completes the proof of assertion (1). Assertion (2) is a straightforward consequence of (1).  Now consider the sets P := {F ∈ QFM (A) : F(A) ⊆ P, F(Q) = 0}, Q := {G ∈ QFM (A) : G(A) ⊆ Q, G(P) = 0}. Both P and Q are quasi-full subalgebras of QFM (A) (in fact each of them is the intersection of two one-sided ideals). Moreover, it becomes clear that PQ = QP = 0 = P ∩ Q.

(8.2.9)

Therefore, by assertion (1) in the claim, P is an ideal of QFM (A) and QFM (A) = P ⊕ Q.

(8.2.10)

Since P is clearly invariant under P and Q, it follows that P is also invariant under QFM (A). Thus assertion (i) in the proposition has been proved. Let R denote the quasi-full subalgebra of P generated by LP ∪ RP . Then, keeping in mind (8.2.9), we have RQ = QR = 0 = R ∩ Q, and therefore, by assertion (1)

582

Selected topics in the theory of non-associative normed algebras

in the claim, we have QFM (A) = R ⊕ Q. Since R ⊆ P, it follows from (8.2.10) that R = P. Now note that P(A) ⊆ P, and consequently, for F ∈ P, let (F) stand for the restriction of F to P, regarded as a mapping from P to P. Then  : F → (F) becomes an injective algebra homomorphism from P to L(P) satisfying (F)( p) = F( p) for all F ∈ P and p ∈ P,

(8.2.11)

(Lp ) = LpP and (Rp ) = RPp for every p ∈ P.

(8.2.12)

so in particular

On the other hand, since −1 (B) is a quasi-full subalgebra of P whenever B is any quasi-full subalgebra of L(P), we have in particular that −1 (QFM (P)) is a quasifull subalgebra of P. Therefore, since −1 (QFM (P)) contains LP ∪ RP , it also contains R, and hence, since P = R, it follows that (P) = QFM (P). Thus  can and will be seen as a bijective algebra homomorphism from P to QFM (P). To continue our argument, it should be kept in mind that the Jacobson radical of any associative algebra is the largest quasi-invertible ideal of the algebra (cf. Theorem 3.6.38). Thus −1 [Rad(QFM (P))] is a quasi invertible ideal of P, so a quasi invertible ideal of QFM (A) (by assertion (2) in the claim), and hence −1 [Rad(QFM (P))] ⊆ Rad(QFM (A)). Therefore, recalling the definition of the weak radical of any algebra, and keeping in mind (8.2.12), for p ∈ w-Rad(P) we have that Lp , Rp ∈ Rad(QFM (A)). On the other hand, (regarded as a subspace of P) w-Rad(P) is invariant under QFM (P), so (regarded as a subspace of A) it it is invariant under P (by (8.2.11)), so it is invariant under QFM (A) (because Q(w-Rad(P)) ⊆ Q(P) = 0 and (8.2.10) applies). It follows that w-Rad(P) ⊆ w-Rad(A) ∩ P. The proof of the reverse inclusion is left to the reader. (Actually, this last inclusion will not be applied in what follows.)  Now, combining Theorem 8.2.17 with Propositions 8.2.42 and 8.2.43(ii), we obtain the main result in this subsection, namely the following. Theorem 8.2.44 Let A be a generalized complemented complete normed algebra over K with zero weak radical, and let {Ai }i∈I be the family of all minimal closed ideals of A. Then: (i) For each i ∈ I, Ai becomes a topologically simple complete normed algebra over K with zero weak radical. (ii) For each a ∈ A there exists a unique summable family {ai }i∈I in A such that  ai ∈ Ai for every i ∈ I, and a = i∈I ai . Now, arguing as in the proofs of Corollaries 8.2.20 and 8.2.21, with Theorem 8.2.44 instead of Theorem 8.2.17, we obtain Corollaries 8.2.45 and 8.2.46 which follow. Corollary 8.2.45 Let A be a generalized complemented complete normed J-semisimple non-commutative Jordan algebra over K, and let {Ai }i∈I be the family of all minimal closed ideals of A. Then:

8.2 Generalized annihilator normed algebras

583

(i) For each i ∈ I, Ai becomes a topologically simple complete normed J-semisimple non-commutative Jordan algebra over K. (ii) For each a ∈ A there exists a unique summable family {ai }i∈I in A such that  ai ∈ Ai for every i ∈ I, and a = i∈I ai . Corollary 8.2.46 Let A be a generalized complemented complete normed semisimple alternative algebra over K, and let {Ai }i∈I be the family of all minimal closed ideals of A. Then: (i) For each i ∈ I, Ai is either a topologically simple complete normed semisimple associative algebra over K, or isomorphic to C(C) if K = C, and to C(C)R , C(R), or O if K = R. (ii) For each a ∈ A there exists a unique summable family {ai }i∈I in A such that  ai ∈ Ai for every i ∈ I, and a = i∈I ai . 8.2.4 Historical notes and comments Save for Theorem 8.2.8, which is new and has no associative forerunner, the content of Subsection 8.2.1 is due to Fern´andez and Rodr´ıguez [259] (1986). Of course, some results which were known before the publication of [259] have been included. This is the case of Corollary 8.2.21, which is originally due to Fern´andez [1157] (1983), and of Corollary 8.2.23, which is originally due to Civin and Yood [175] (1965). In fact, Theorem 3.2 of [175] contains the following generalization of Corollary 8.2.23. Theorem 8.2.47 Let A be a quasi-full subalgebra of a complete normed associative algebra over K, and suppose that A is semisimple and generalized annihilator. Then A is the closure of the direct sum of its minimal closed ideals. An interesting variant of the above theorem can be found in [1123, Corollary 2.7] (1972). Assertion (iv) in Proposition 8.2.5 is trivial in the associative setting, even if we remove the assumption that the normed algebra A be generalized annihilator. Once this assertion has been happily established in the ( possibly non-associative) generalized annihilator setting, the proof of Lemma 8.2.6 is the same as that of its associative forerunner, which is due to Yood [1123, Lemma 2.3]. It is proved in [259] that primitive ideals of a generalized annihilator complete normed algebra over K are weak primitive ideals. Therefore, by Lemma 8.2.14, primitive ideals of a generalized annihilator complete normed algebra over K are maximal closed ideals. This follows also from the facts that primitive ideals of any algebra are prime ideals and are closed whenever the algebra is complete normed, and that, as proved also in [259], the maximal closed ideals of a generalized annihilator normed algebra over K are the closed prime ideals. The associative forerunner of this last result is due to Yood [1123, Lemma 2.1]. §8.2.48 Generalized annihilator normed algebras owe their name to the fact that, in the associative setting, particular cases of them, called annihilator normed (associative) algebras, had been introduced and studied earlier by Bonsall and

584

Selected topics in the theory of non-associative normed algebras

Goldie [864] (1954) (see also [696, Section 32]). Annihilator condition is inherited by minimal closed ideals, which are then describable (see [696, Theorem 32.20] for details). Annihilator normed alternative algebras were considered first in [1043] (1983), and their theory attains its climax in [1157], where the appropriate variant of Corollary 8.2.21 for annihilator normed alternative algebras is proved. In their turn, dual normed (associative) algebras, introduced and studied by Kaplansky [374, 375] (1948–9), become particular cases of annihilator algebras. The appropriate Jordan variant of the duality condition has been introduced and studied by Bunce [870] (1982) in the setting of JB-algebras. Dual JB-algebras are generalized annihilator and their closed ideals are dual JB-algebras. By the way, with arguments similar to those in the proof of Corollary 8.2.24, one can prove the following generalization of [870, Corollary 2.2(iii)]. Corollary 8.2.49 [259] Let A be a generalized annihilator JB-algebra. Then A is the c0 -sum of the family of its minimal closed ideals, and these are topologically simple JB-algebras. It follows from the above comments and corollary that the theory of dual JB-algebras reduces to the topologically simple case. The topologically simple dual JB-algebras are classified in [870] (see also [520, Theorem 5.4]). All results in Subsection 8.2.2 were proved by Cabello and Cabrera in [876] (2004). The proof of Proposition 8.2.30 is new. The original proof in [876] involved non-trivial results on the extended centroid and the central closure of a semiprime associative algebra. In [1123, Theorem 2.6], Yood proved that a normed associative algebra A over K is the closure of the direct sum of its minimal closed ideals if and only if A is generalized annihilator and the intersection of its closed prime ideals reduces to zero. An interesting purely algebraic non-associative variant of Yood’s theorem just quoted can be seen in [876, Theorem 3.7]. In this variant, multiplicatively semiprime algebras play the role of generalized annihilator normed algebras, and the ε-closure plays the role of the norm-closure (cf. §8.2.26 and the equivalence (i)⇔(iv) in Theorem 8.2.31). As acknowledged by the authors of [876], their Theorem 3.7 just reviewed was motivated by the lattice version of Yood’s theorem shown in [946, Theorem 6.3]. It is also proved in [876] that an algebra A with zero annihilator is the ε-closure of the direct sum of its minimal ε-closed ideals if and only if A is an essential subdirect product of a family of multiplicatively prime algebras. The proof of this result strongly relies on Theorem 4.1 in the Fern´andez–Garc´ıa paper [941]. Actually, [941, Theorem 4.1] becomes a related result working in very general settings. The fact that the norm-closure of an ideal of a generalized complemented normed algebra coincides with the π-closure is folklore, whereas the fact that the normclosure of such an ideal coincides with the ε-closure is new (cf. Proposition 8.2.35). Results from §8.2.36 to Lemma 8.2.40, as well as the associative forerunner of Corollary 8.2.46, are due to Bachelis [846]. Bachelis’ original argument involved previous results of Ruckle [1067] which, in our development, have been either

8.2 Generalized annihilator normed algebras

585

avoided or incorporated with a proof. Proposition 8.2.43 and Theorem 8.2.44 are due to Fern´andez and Rodr´ıguez [259], although no proof of Proposition 8.2.43 was given there. Our proof of Proposition 8.2.43 is close to that of [520, Proposition 5.1]. Different purely algebraic substitutes of generalized complemented normed algebras can be found in [877, 879, 880]. To conclude our comments on this section, let us explore the structure of generalized annihilator complete normed algebras having a left unit. As a matter of fact, in a purely algebraic meaning, they are ‘generalized complemented’, i.e. all ideals of such an algebra are direct summands. Exercise 8.2.50 Let A be an algebra over K with a left unit. Prove that 

(i) Every proper

right two-sided





ideal of A is contained in a maximal

right two-sided



ideal

of A. Prove also that, if A is complete normed, then: 

(ii) The closure of a proper Hint

right two-sided





ideal of A is a proper

right two-sided

Please mimic the proofs of Fact 1.1.52 and Proposition 1.1.49.



ideal of A. 

Theorem 8.2.51 Let A be a generalized annihilator complete normed algebra over K having a left unit. Then the family of all minimal ideals of A is finite, A is the direct sum of its minimal ideals, and these are complete normed simple algebras having a left unit. Proof Since every dense ideal of A is the whole algebra (by Exercise 8.2.50(ii)), it follows from Proposition 8.2.5(i) that A = I ⊕ Ann(I) for every ideal I of A.

(8.2.13)

Replacing I with Ann(I) in (8.2.13), we obtain A = Ann(Ann(I)) ⊕ Ann(I), and hence, since Ann(Ann(I)) ⊇ I, a new application of (8.2.13) yields Ann(Ann(I)) = I. Therefore every ideal of A is closed in A, and the mapping I → Ann(I) becomes an involutive order antiautomorphism of the set of all ideals of A ordered by inclusion. As a consequence, the annihilator of any maximal ideal of A is a minimal ideal of A. On the other hand, it follows from (8.2.13) that ideals of A are algebras with a left unit and that ideals of an ideal of A are ideals of A, and hence that minimal ideals of A are complete normed simple algebras with a left unit. Now let P := ⊕λ∈ Iλ denote the (automatically direct) sum of all minimal ideals of A, and assume that P = A. Then, by Exercise 8.2.50(i), there exists a maximal ideal M of A containing P. But, since Ann(M) is a minimal ideal of A, we have Ann(M) ⊆ P. Therefore Ann(M) ⊆ M, and hence Ann(M) = 0, and finally M = A, a contradiction. Therefore A is the direct sum of its minimal ideals. As a consequence, denoting by u the left unit of A whose existence has been assumed, there exists a  finite subset  of  such that u = λ∈ xλ with xλ ∈ Iλ for every λ ∈ .

586

Selected topics in the theory of non-associative normed algebras

To conclude the proof it only remains to show that the family of all minimal ideals of A is finite. But, if this were not true, then there μ ∈  \ , and  would exist  taking xμ ∈ Iμ \ {0}, we would have xμ = uxμ = λ∈ xλ xμ = λ∈ xλ xμ = 0, a contradiction.  The arguments in the above proof are close to those in the proof of [520, Corollary 5.3], where the unital particular case of Theorem 8.2.51 is shown. Since algebras with a left unit have zero annihilator, and semi-H ∗ -algebras with zero annihilator are generalized annihilator complete normed algebras, the next result follows from Theorem 8.2.51. Corollary 8.2.52 Let A be a semi-H ∗ -algebra over K with a left unit. Then the family of all minimal ideals of A is finite, A is the orthogonal sum of its minimal ideals, and these are simple semi-H ∗ -algebras with a left unit. The specialization of the above corollary to H ∗ -algebras follows from the fact that with a left unit are the same as semi-H ∗ -algebras with a unit (cf. the equivalence (i)⇔(iii) in Corollary 8.1.133).

H ∗ -algebras

8.3 Continuing the theory of non-associative normed algebras Introduction As the main result in Subsection 8.3.1 we prove that algebra homomorphisms from complete normed complex algebras to complete normed complex algebras with no nonzero two-sided topological divisor of zero are continuous (see Theorem 8.3.9). In Subsection 8.3.2 we prove that complete normed J-semisimple non-commutative Jordan complex algebras each element of which has a finite J-spectrum are a finite direct sum of closed simple ideals which are either finite-dimensional or quadratic (see Corollary 8.3.22), and derive that complete normed semisimple alternative complex algebras each element of which has a finite spectrum are finite-dimensional (see Corollary 8.3.24). After the usual subsection devoted to historical notes and comments, we include a comprehensive survey on normed Jordan algebras. 8.3.1 Continuity of homomorphisms into normed algebras without topological divisors of zero Let F be a bounded linear operator on a normed space X. As in the paragraph preceding Lemma 2.8.1, we denote by k(F) the largest non-negative number k satisfying kx ≤ F(x) for every x in X. In this way F is bounded below if and only if k(F) > 0. The following lemma refines Lemma 2.8.2. Lemma 8.3.1 Let X and Y be Banach spaces over K, let  : X → Y be a dense-range (possibly discontinuous) linear mapping, and let F and G be in BL(X) and BL(Y), respectively, such that G is non bijective and the equality F = G holds. Then k(G) ≤ F.

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Proof We may suppose that G is bounded below (since otherwise we have k(G) = 0 and nothing is to prove). Let 0 < δ < k(G). We have  (G − δIY ) − G = δ < k(G), so that, since G is bounded below and non bijective, it follows from Lemma 2.8.1 that G − δIY is bounded below and non bijective. This implies that the range of G − δIY is a proper closed subspace of Y. Then, since the equality (F − δIX ) = (G − δIY ) holds, and  has dense range, we deduce that F − δIX cannot be bijective. Therefore we have δ ≤ F , and the proof is concluded by letting δ → k(G).  Notation 8.3.2 Let X and Y be Banach spaces over K, and let  : X → Y be a denserange linear mapping. Then the set  X := {F ∈ BL(X) : there is G ∈ BL(Y) satisfying F = G} is a (possibly non closed) subalgebra of BL(X), for F ∈  X there exists a unique G ∈ BL(Y) satisfying F = G, and the mapping F → G from  X to BL(Y) becomes , and the an algebra homomorphism. Such a homomorphism will be denoted by     symbol Y will stand for the closure in BL(Y) of the range of . X (respectively,  Y) will be considered as a normed (respectively, complete normed) algebra under the restriction of the natural norm of BL(X) (respectively, BL(Y)). Proposition 8.3.3 Let X and Y be Banach spaces over K, and let  : X → Y be a dense-range linear mapping. Suppose that there is some operator in  Y which is  consists only of bounded below and non bijective. Then the separating set for  operators which are not bounded below. Proof We argue by contradiction, so that we assume the existence of some operator ) which is bounded below. G ∈ S( In a first step we assume additionally that the operator G above is non bijective. (Fn ) for some sequence Fn in  X with lim Fn = 0. Since the set of Write G := lim  those elements in BL(Y) which are bounded below and non bijective is open (by (Fn ) is non bijective for all Lemma 2.8.1), there is no restriction in assuming that   n. Then, by Lemma 8.3.1, for n ∈ N we have k((Fn )) ≤ Fn , and therefore, since the function k(.) is continuous on BL(Y) (by Lemma 2.8.1), we obtain k(G) = 0, contradicting that G is bounded below. To conclude the proof, remove now the assumption in the preceding paragraph that G is non-bijective. By the assumption in the theorem, there exists an element (say T) ), and the in  Y which is bounded below and non-bijective. Since G belongs to S( separating set for an algebra homomorphism between normed algebras is an ideal of ). Then TG is an the closure of the range (cf. Lemma 1.1.58), also TG lies in S(  element of S() which is bounded below and non-bijective. But this is not possible in view of the first step of the proof.  §8.3.4 Now let A and B be complete normed algebras over K, and let  : A → B be B  a dense-range algebra homomorphism. Since for a ∈ A the equalities LaA = L(a) B A and Ra = R(a)  hold, with the convention of symbols in Notation 8.3.2 we have

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(LaA ) = LB , and  (RAa ) = RB . It follows from the denseness that LaA , RAa ∈  A,  (a) (a) of the range of  and the continuity of the mappings b → LbB and b → RBb from B to BL(B) that LbB , RBb ∈  B for every b ∈ B. Therefore the multiplication algebra M (B) of B is contained in  B. Then the corollary below follows straightforwardly from Proposition 8.3.3. Corollary 8.3.5 Let B be a complete normed algebra over K such that there is some operator in M (B) which is bounded below and non-bijective. Then, for every complete normed algebra A over K and every dense-range algebra homomorphism  consists only of operators which are not bounded  : A → B, the separating set for  below. Let A, B and  be as in §8.3.4. As pointed out in the proof of Proposition 8.1.49, ). Therefore, invoking Corollary for b ∈ S(), the operators LbB and RBb lie in S( 8.3.5, we obtain the following. Corollary 8.3.6 Let B be a complete normed algebra over K such that there is some operator in M (B) which is bounded below and non bijective. Then, for every complete normed algebra A over K and every dense-range algebra homomorphism  : A → B, the separating set for  consists only of two-sided topological divisors of zero in B. Therefore, if in addition B has no nonzero two-sided topological divisor of zero, then dense-range algebra homomorphisms from complete normed algebras over K into B are continuous. The next corollary follows straightforwardly from the above one. Corollary 8.3.7 Let B be a complete normed algebra over K. Suppose that B is not a quasi-division algebra (cf. Definition 2.5.35) and has no nonzero two-sided topological divisor of zero. Then dense-range algebra homomorphisms from complete normed algebras over K into B are continuous. Proposition 8.3.8 Let A and B be complete normed algebras over K, let  : A → B be a surjective algebra homomorphism, and suppose that B has no nonzero two-sided topological divisor of zero. Then  is continuous. Proof If B is not a quasi-division algebra, then the continuity of  follows from Corollary 8.3.7. Otherwise, since quasi-division algebras are simple, the continuity of  follows from Corollary 4.4.51.  Now we can conclude the proof of the main result in this subsection. Namely the following. Theorem 8.3.9 Let A and B be complete normed complex algebras, let  : A → B be an algebra homomorphism, and suppose that B has no nonzero two-sided topological divisor of zero. Then  is continuous. Proof Since the absence of nonzero two-sided topological divisors of zero is inherited by every subalgebra of B, we may replace B with the closure of the range of ,

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and suppose without loss of generality that  has dense range. If B is not a quasidivision algebra, then the continuity of  follows from Corollary 8.3.7. Otherwise, by Theorem 2.7.7, B is finite-dimensional, so  is surjective, and so the continuity of  follows from Proposition 8.3.8.  We recall that, according to Proposition 2.7.65, complete normed infinitedimensional unital complex algebras with no nonzero two-sided topological divisor of zero do exist. The question whether Theorem 8.3.9 is true with ‘real’ instead of ‘complex’ remains unanswered to date (even if ‘two-sided topological divisor of zero’ is replaced with ‘one-sided topological divisor of zero’, cf. §2.7.53). The answer to this question is closely related to Problem 2.7.45. 8.3.2 Complete normed Jordan algebras with finite J-spectrum Let A be a Jordan algebra over K. For an element a ∈ A, the a-homotope A(a) of A is the algebra whose underlying vector space is the same as that A and whose product is given by x · y := Ux,y (a). a

The a-homotope of A is again a Jordan algebra, as can easily be checked using similar arguments to those in the proofs of Propositions 4.1.34 and 4.1.35. Following [777, Definition III.1.4.1], a pair (z, y) of elements of A is called a quasi-invertible pair if z is quasi-J-invertible in the homotope A(y) , and an element z of A is called properly quasi-invertible if it remains quasi-J-invertible in all homotopes A(y) , i.e. all pairs (z, y) are quasi-invertible. Following [777, Definition III.1.2.1], a structural transformation T on A is a linear operator for which there exists a linear operator T ∗ on A such that both T and T ∗ extend to A1 in such a way that UT(x) = TUx T ∗ on A1 for all x ∈ A1 . If T ∗ is also structural with T ∗∗ = T, then the pair (T, T ∗ ) is called a structural pair on A. The fundamental identity in Proposition 3.4.15 says that all operators Ux determine a structural pair (Ux , Ux ). Theorem 8.3.10 [777, Theorem III.1.4.3] Let A be a Jordan algebra over K. We have: (i) (z, y) is a quasi-invertible pair in A if and only if (y, z) is. (ii) If (T, T ∗ ) is a structural pair on A, then (T(z), y) is quasi-invertible if and only if (z, T ∗ (y)) is quasi-invertible. Theorem 8.3.11 [777, Theorem III.1.5.1] The Jacobson radical of a Jordan algebra A over K coincides with the set of all properly quasi-invertible elements of A. Lemma 8.3.12 Let A be a J-semisimple unital Jordan complex algebra. If J-sp(A, a) is a singleton for every a ∈ A, then A = C.

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Proof

Suppose that J-sp(A, a) is a singleton for every a ∈ A, and consider the set N := {x ∈ A : J-sp(A, x) = {0}}.

Note that N = A\J-Inv(A) ⊆ {x ∈ A : 1 ∈ / J-sp(A, x)} = {x ∈ A : x is quasi-J-invertible in A}. (8.3.1) Let x be in N. By Theorem 4.1.3(vi), for every a ∈ A, Ux (a) ∈ / J-Inv(A), and hence Ux (a) is quasi-J-invertible in A, that is (Ux (a), 1) is a quasi-invertible pair. For each a ∈ A, by Theorem 8.3.10(ii) considering the structural transformation Ux on A, we see that (a, x2 ) is a quasi-invertible pair, i.e. a is quasi-J-invertible in the Jordan 2 homotope A(x ) . Therefore, by Theorem 8.3.10(i), x2 is quasi-J-invertible in the Jordan homotope A(a) for every a ∈ A, so that x2 is properly quasi-invertible, hence, by Theorem 8.3.11, x2 ∈ J-Rad(A), and so x2 = 0. Now note that for each a ∈ A there are unique elements λ ∈ C and x ∈ N such that a = λ1 + x. (8.3.2) More precisely, J-sp(A, a) = {λ} if and only if a = λ1 + x for x ∈ N. In order to prove that N is an ideal of A, note that CN ⊆ N, and consequently x + y ∈ N whenever x and y are linearly dependent elements in N. Suppose that x, y ∈ N are linearly independent, and write x + y = α1 + u and x − y = β1 + v for α, β ∈ C and u, v ∈ N. It follows from the first paragraph of the proof that 0 = 2(x2 + y2 ) = (x + y)2 + (x − y)2 = (α1 + u)2 + (β1 + v)2 = (α 2 + β 2 )1 + 2(αu + βv), and hence, by (8.3.2), α 2 + β 2 = 0 and αu + βv = 0. Therefore (α + β)x + (α − β)y = α(x + y) + β(x − y) = (α 2 + β 2 )1 + (αu + βv) = 0, hence α + β = α − β = 0, so α = β = 0, and so x + y ∈ N. Thus N is a subspace of A. Moreover, by (8.3.2), we have A = C1 ⊕ N. Note also that, for all x, y ∈ N we have 0 = (x + y)2 = 2xy, and consequently NN = 0. Now, for a ∈ A, writing a = α1 + u with α ∈ C and u ∈ N, we see that ax = (α1 + u)x = αx ∈ N for every x ∈ N. Thus N is an ideal of A. Since N is quasi-J-invertible (by (8.3.1)), we have N ⊆ J-Rad(A) = 0, hence N = 0, and as a result A = C.  Lemma 8.3.13 Let A be a complete normed unital J-semisimple Jordan complex algebra all elements of which have finite J-spectra. If A contains no idempotent different from 0 and 1, then A = C. Proof Suppose that A contains no idempotent different from 0 and 1. Then, by Proposition 4.1.89, the J-spectrum of each element of A reduces to a singleton. Therefore, by Lemma 8.3.12, A = C. 

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Lemma 8.3.14 Let A be a complete normed unital Jordan complex algebra all elements of which have finite J-spectra. Then A contains no infinite sequence of nonzero pairwise orthogonal idempotents. Proof Assume by the contrary that A contains an infinite sequence en of nonzero pairwise orthogonal idempotents. Take a sequence λn of pairwise different complex   numbers such that n∈N |λn |en  < ∞, and set x := ∞ n=1 λn en . Noticing that for / J-Inv(A) (by Theorem each n ∈ N we have Uλn 1−x (en ) = 0, and hence λn 1 − x ∈ 4.1.3(ii)), we realize that λn ∈ J-sp(A, x). This contradicts that J-sp(A, x) is finite.  In the next statement we collect some hereditary properties of the J-radical of a Jordan algebra. Theorem 8.3.15 [777, Theorem III.1.6.1] Let A be a Jordan algebra over K. We have: (i) If an element of a strict inner ideal B of A is quasi-J-invertible in A, then its quasi-J-inverse lies back in B. (ii) J-Rad(Ai (e)) = Ai (e) ∩ J-Rad(A) for i = 0, 1 and every idempotent e in A. (iii) J-Rad(I) = I ∩ J-Rad(A) for every ideal I of A. (iv) If A is J-semisimple, then A is nondegenerate (cf. Definition 6.1.49(a)). Let A be a Jordan algebra over K. An idempotent e in A is called primitive if e = 0 and e cannot be written as e = e1 + e2 where e1 and e2 are nonzero orthogonal idempotent elements. If e = e1 + e2 where the ei are orthogonal idempotents then eei = ei so ei ∈ A1 (e) (cf. Lemma 2.5.3). On the other hand, if e1 is an idempotent element contained in A1 (e), then e2 = e − e1 is an idempotent orthogonal to e1 and e = e1 + e2 . It follows that a nonzero idempotent e is primitive if and only if the subalgebra A1 (e) contains no idempotent = 0, e. Clearly the unit element is the unique nonzero idempotent in a J-division Jordan algebra. Hence e is primitive if A1 (e) is a J-division algebra. We shall call an idempotent e completely primitive if A1 (e) is a J-division algebra.  A unital Jordan algebra A over K is said to have finite capacity if 1 = ni=1 ei where the ei are completely primitive pairwise orthogonal idempotents in A. The minimum n for which this holds will be called the capacity of A. Lemma 8.3.16 Let A be a complete normed unital Jordan complex algebra all elements of which have finite J-spectra. We have: (i) For each nonzero idempotent e ∈ A, A1 (e) contains a primitive idempotent of A. (ii) 1 is a finite sum of pairwise orthogonal primitive idempotents of A. (iii) If in addition A is J-semisimple, then A has finite capacity; more precisely,  1 = ni=1 ei where the ei are nonzero pairwise orthogonal idempotents in A such that A1 (ei ) = Cei . Proof First we show that there are primitive idempotents in A. Assume that this is not true. Then each nonzero idempotent e ∈ A can be written as e = α(e) + β(e) with

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α(e), β(e) nonzero orthogonal idempotents of A. This allows us to define inductively two sequences en and fn of nonzero idempotents in A by e1 := α(1), f1 := β(1), en+1 := α( fn ), and fn+1 := β( fn ). It is easily realized that en becomes a sequence of nonzero pairwise orthogonal idempotents in A, which contradicts Lemma 8.3.14. Now let e ∈ A be a nonzero idempotent. Then, since A1 (e) = Ue (A) is a complete normed unital Jordan complex algebra all elements of which have finite J-spectra (by Theorem 8.3.15(i)), we can repeat the argument in the preceding paragraph (with A1 (e) instead of A) to obtain that A1 (e) contains a primitive idempotent of A1 (e). But, clearly, such an idempotent is primitive in A. Thus assertion (i) is proved. Now that the existence of primitive idempotents in A is not in doubt, we can apply Zorn’s lemma to find a maximal family (relative to the inclusion) of pairwise orthogonal primitive idempotents in A. In view of Lemma 8.3.14, this family is  finite, say {e1 , . . . , en }. Since e := 1 − ni=1 ei is an idempotent orthogonal to all ei , it follows from assertion (i) and the maximality of the family {e1 , . . . , en } that e = 0,  i.e. 1 = ni=1 ei (which proves assertion (ii)). Suppose that A is J-semisimple, and let i = 1, . . . , n. Then it follows from Theorem 8.3.15(ii) that the complete normed unital Jordan complex algebra A1 (ei ) is J-semisimple. Therefore, since all elements of A1 (ei ) have finite spectra, and the idempotent ei is primitive, it follows from Lemma 8.3.13 that A1 (ei ) = Cei .  Lemma 8.3.17 Let A be a complete normed unital J-semisimple Jordan complex algebra all elements of which have finite J-spectra, let B be a closed ideal of A, and let e ∈ A be an idempotent such that A1 (e) ∩ B = 0. Then A1 (e) ∩ B contains a primitive idempotent of A. Proof (a) Let C be any proper inner ideal of A. If there is c ∈ C such that 0∈ / J-sp(A, c), then c ∈ J-Inv(A), so 1 = Uc (a) for some a ∈ A (cf. Theorem 4.1.3(ii)), hence 1 ∈ C, and then A = U1 (A) ⊆ C, contrary to the assumption that C is proper. Therefore 0 ∈ J-sp(A, c) for every c ∈ C. (b) Write C := A1 (e) ∩ B, which is a nonzero strict inner ideal of A. If c ∈ C satisfies J-sp(A, c) = 0, then, by Theorem 4.1.17, we have that r(c) = 0, hence, by Lemma 4.1.15, c is quasi-J-invertible in A, and finally, by Theorem 8.3.15(i), c is quasi-J-invertible in C. Thus, if J-sp(A, c) = 0 for every c ∈ C, then C = J-Rad(C) = C ∩ J-Rad(A1 (e)) = C ∩ J-Rad(A) = 0,

(8.3.3)

which is a contradiction. (Note that the second equality in (8.3.3) follows from the fact that C is an ideal of A1 (e) and Theorem 8.3.15(iii), whereas the the third equality in (8.3.3) follows from Theorem 8.3.15(ii).) Therefore there exists c ∈ C such that J-sp(A, c) = 0. (c) Suppose additionally that C = A. Then, by parts (a) and (b) above, there exists c ∈ C such that J-sp(A, c) ⊇ {0, λ} for some 0 = λ ∈ C. Since J-sp(A, c) is finite, it follows from Proposition 4.1.89 that there exists an idempotent p different from 0 and 1 in the closed J-full subalgebra of A generated by c. Since C1 + C is a closed J-full subalgebra of A (by Theorem 8.3.15(i)), it follows that p = μ1 + d

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for suitable μ ∈ C and d ∈ C. Now, since J-sp(A, p) = {0, 1}, we realize that J-sp(A, d) = {−μ, 1 − μ}, which, in view of (a) again, is equivalent to either μ = 0 or μ = 1. Therefore either p ∈ C or 1 − p ∈ C. In any case C contains a nonzero idempotent (say q). Then, by Lemma 8.3.16(i), A1 (q) contains a primitive idempotent of A. Since A1 (q) = Uq (A) ⊆ C, the lemma is proved under the additional assumption that C = A. But, if C = A, then the lemma is not in doubt (cf. Lemma 8.3.16(ii)).  Lemma 8.3.18 Let B be a nondegenerate Jordan algebra, and let f ∈ B be an idempotent such that B0 ( f ) = 0. Then f is a unit for B. Proof Let x = 2Uf ,1−f (x) be in B 1 ( f ) (cf. (6.1.11) in Lemma 6.1.80). Since the 2 subalgebra of B generated by f and x is special (cf. Theorem 3.1.55), and B0 ( f ) = 0, computing in an associative enveloping algebra we have x2 = [ fx(1 − f ) + (1 − f )xf ]2 = [ fx(1 − f )]2 + [(1 − f )xf ]2 + (1 − f )xf 2 x(1 − f ) + fx(1 − f )2 xf = fx(1 − f )2 xf = Uf Ux (1 − f ). Therefore, applying twice Proposition 3.4.15, for every b ∈ B we have Ux2 (b) = Uf Ux U1−f Ux Uf (b) ∈ Uf Ux (B0 ( f )) = 0, and hence Ux2 = 0 on B. Therefore x2 = 0 because B is nondegenerate. Since x is arbitrary in B 1 ( f ), we can linearize the last equality to obtain that B 1 ( f )B 1 ( f ) = 0. 2 2 2 Therefore, by Lemma 6.1.86(ii), B 1 ( f ) is an ideal of B. Since B 1 ( f )B 1 ( f ) = 0 and B 2 2 2 is a semiprime algebra (because nondegeneracy implies semiprimeness), we realize that B 1 ( f ) = 0. Finally, since B0 ( f ) = 0, it follows that B = B1 ( f ), i.e., as desired, f 2 is a unit for B.  Fact 8.3.19 Let A be a complete normed unital J-semisimple Jordan complex algebra all elements of which have finite J-spectra, and let B be a closed ideal of A. Then B has a unit. Proof We can suppose that B = 0. Then, since B = A ∩ B = A1 (1) ∩ B, it follows from Lemma 8.3.17 that B contains primitive idempotents of A. Therefore, by Zorn’s lemma, there exists a maximal family (relative to the inclusion) of pairwise orthogonal idempotents of B which are primitive in A. In view of Lemma 8.3.14, this  family is finite, say {e1 , . . . , en }. Write f := ni=1 ei ∈ B and e := 1 − f . If A1 (e) ∩ B were not zero, then, by Lemma 8.3.17 again, A1 (e) ∩ B would contain a primitive idempotent of A (say en+1 ), and {e1 , . . . , en , en+1 } would be a family of pairwise orthogonal idempotents of B which are primitive in A, contrary to the maximality of {e1 , . . . , en }. Therefore A1 (e) ∩ B = 0, i.e. B0 ( f ) = A0 ( f ) ∩ B = 0. Since B is a nondegenerate Jordan algebra (by Theorem 8.3.15(iii)–(iv)), it follows from Lemma 8.3.18 that f is a unit for B. 

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Let A be a Jordan algebra over K. An idempotent e ∈ A is called absolutely primitive if e = 0 and every element in A1 (e) has the form αe + z where α ∈ K and  z is nilpotent. The Jordan algebra A is called reduced if A is unital and 1 = ni=1 ei where the ei are absolutely primitive idempotents of A. Theorem 8.3.20 [754, Corollary 1 in p. 203] Any reduced simple Jordan algebra over K is either finite-dimensional or is a Jordan algebra of a nondegenerate symmetric bilinear form on an infinite-dimensional vector space over K. Now we can conclude the proof of the main result in this subsection, namely the following. Theorem 8.3.21 Let A be a complete normed unital J-semisimple non-commutative Jordan complex algebra all elements of which have finite J-spectra. Then: (i) The family of all minimal ideals of A is finite. (ii) A is the direct sum of its minimal ideals. (iii) Each minimal ideal of A is a unital simple algebra, and is either finitedimensional or a complete normed infinite-dimensional quadratic complex algebra (cf. Proposition 3.5.4). Proof Suppose at first that A is commutative. Clearly A is semiprime. Let B be any closed proper ideal of A. Then, by Fact 8.3.19, B has a unit e = 1, which becomes a central idempotent of A (cf. Lemma 5.1.8). Therefore 0 = (1−e)A ⊆ Ann(B). Thus A is generalized annihilator (cf. Definition 8.2.3). It follows from Theorem 8.2.51 that the family of all minimal ideals of A is finite, that A is the direct sum of its minimal ideals, and that each minimal ideal of A is a complete normed unital simple algebra. Now let I be a minimal ideal of A. Then, by assertions (i) and (iii) in Theorem 8.3.15, I is a complete normed unital J-semisimple Jordan complex algebra all elements of which have finite J-spectra. Therefore, as a consequence of Lemma 8.3.16(ii), I is reduced. Since I is simple, it follows from Theorem 8.3.20 that, as desired, I is either finite-dimensional or is a complete normed infinite-dimensional quadratic complex algebra. Now remove the assumption that A is commutative. Then, keeping in mind Proposition 4.4.17(iii), we realize that Asym is a complete normed unital J-semisimple Jordan complex algebra all elements of which have finite J-spectra. Therefore, by the preceding paragraph and Lemma 8.1.75, minimal ideals of Asym are ideals of A, and then, clearly, they are minimal ideals of A. Now the proof is concluded by invoking again the preceding paragraph.  An apparently more general version of Theorem 8.3.21 is the following. Corollary 8.3.22 Let A be a nonzero complete normed J-semisimple non-commutative Jordan complex algebra such that J-sp(A1 , a) is finite for every a ∈ A. Then the conclusion in Theorem 8.3.21 holds. Proof Clearly A1 is a complete normed unital non-commutative Jordan complex algebra such that J-sp(A1 , x) is finite for every x ∈ A1 . Moreover, since A is a closed

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ideal of A1 , it follows from Proposition 4.4.17(iii) and Theorem 8.3.15(iii) that A1 is J-semisimple. Therefore, by Lemma 3.6.10 and Fact 8.3.19, A is unital. Then, since J-sp(A1 , a) = {0} ∪ J-sp(A, a) for every a ∈ A, the result follows from Theorem 8.3.21  Corollary 8.3.23 Let A be a complete normed unital semisimple alternative complex algebra all elements of which have finite spectra. Then A is finite-dimensional. Proof In view of Theorem 8.3.21, it is enough to show that every simple quadratic alternative complex algebra is finite-dimensional. Let A be a quadratic alternative complex algebra. By Corollary 2.5.19(i), the algebraic norm function n on A admits composition, Therefore, if A is simple, then n is nondegenerate, and A is finitedimensional thanks to Theorem 6.1.31.  Corollary 8.3.24 Let A be a nonzero complete normed semisimple alternative complex algebra such that J-sp(A1 , a) is finite for every a ∈ A, Then A is finitedimensional. Proof Argue as in the proof of Corollary 8.3.22, with Corollary 8.3.23 instead of Theorem 8.3.21.  We conclude this subsection by proving the following. Proposition 8.3.25 Let A be a complete normed J-semisimple non-commutative Jordan algebraic complex algebra. Then the conclusion in Theorem 8.3.21 holds. Proof Since A is algebraic, it follows from Propositions 1.3.4(ii) and 4.1.86 that J-sp(A1 , a) is finite for every a ∈ A. Therefore, since A is J-semisimple, the result follows from Corollary 8.3.22. 

8.3.3 Historical notes and comments Subsection 8.3.1 has been taken from [529]. Other results on general non-associative automatic continuity are reviewed in what follows. As we commented in pp. 494 and 597 of Volume 1, in the Cedilnik–Rodr´ıguez paper [165] it is shown that Problem 4.1.110 has an affirmative answer if the algebra B in that problem is algebraic, and a proof of Corollary 4.4.55 is given. The following variant of Corollary 4.4.55 is also proved in [165]. Proposition 8.3.26 Let A be a complete normed power-associative algebraic algebra over K. Then the following conditions are equivalent: (i) Algebra homomorphisms from complete normed algebras over K to A are continuous. (ii) Algebra homomorphisms from complete normed, associative, and commutative algebras over K to A are continuous. (iii) A has no isotropic element.

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Then the authors of [165] raise the question whether power-associativity can be removed in the above proposition. Certainly an affirmative answer to this question would unify Corollary 4.4.55 and Proposition 8.3.26. Our next comments follow almost verbatim Runde’s review [1068]. The following are typical questions investigated in the field of automatic continuity: Given a complete normed associative algebra A, is every algebra homomorphism into A continuous? Is every algebra homomorphism onto A continuous? or, less generally, does A have a unique complete normed or Fr´echet algebra topology? Is every derivation of A automatically continuous? It is known, for instance, that if A is semisimple then every algebra homomorphism onto A is continuous [353] (see also Theorem 4.4.45 and Corollary 4.4.61), and so is every derivation of A [355] (see also Remark 7.2.9(a)). In [462], Newmann, Rodr´ıguez, and Velasco treat these questions for algebras of vector-valued functions such as C0 (, A) and L1 (G, A) (see [767, Example 4.1.6 and p. 530]), where  is a locally compact Hausdorff topological space, G is a locally compact topological abelian group, and A is an arbitrary complete normed associative algebra. These algebras were previously investigated by Kantrowitz and Neumann [991], and by Neumann and Velasco [1019] in the case where A was assumed to be unital. In [462], not only the demand that A be unital is dropped, but it also need no longer be associative. The existence of a unit is replaced by the existence of what the authors call an ‘admissible set of non-divisors of zero’ (if A is unital, then {1} is admissible). Not only are the results from the previous papers extended to this more general setting, but also results are obtained which are new even in the case where A is unital and associative: For the algebra C0 (, A), the authors obtain a very nice dichotomy for the automatic continuity questions mentioned above in terms of whether  has an isolated point or not. If A is an integral domain, then the set A \ {0} is admissible. Hence, the methods developed in [462] apply, in particular, to the situation considered in the paper by Garimella [952]. Let us finally note that an associative version of [462] is contained in Section 5.6 of the Laursen–Neumann book [767] Kaplansky [378] proved that a complete normed semisimple associative complex algebra whose elements have finite spectra is finite-dimensional (the associative forerunner of Corollary 8.3.24). Later this result was rediscovered by Hirschfeld and Johnson [327], and was refined by Aupetit [682, Theorem 3.2.1] who proved Theorem 2.8.73 already reviewed in Volume 1. The Jordanization of Kaplansky’s theorem begins with the paper of Aupetit and Zra¨ıbi [845], where it is shown that a complete normed Jordan complex algebra A, with the property that the J-spectra of all elements in some non-empty open subset of A reduce to singletons, satisfies either A = C or A = J-Rad(A). The definitive Jordanization of Kaplansky’s theorem is due to Benslimane and Kaidi [91], who determine the structure of complete normed J-semisimple non-commutative Jordan complex algebras whose elements have finite J-spectra, as stated in Theorem 8.3.21 and Corollary 8.3.22, and prove Corollaries 8.3.23 and 8.3.24. The Benslimane–Kaidi argument begins by proving Lemma 8.3.12, which becomes a clever purely algebraic version of the Aupetit–Zra¨ıbi result, and continues

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with the proofs of Lemmas 8.3.13, 8.3.14, and 8.3.16(iii). In our opinion, the paper [91] contains minor ambiguities concerning an occasional argument and a reference to Jacobson’s book [754]. To clarify them, we have adapted a part of Kaplansky’s proof of the associative forerunner of Theorem 8.3.21 (see [378, Lemma 7]) to prove assertions (i) and (ii) of Lemma 8.3.16. Then, after proving Lemmas 8.3.17 and 8.3.18 and Fact 8.3.19, and after invoking Theorem 8.2.51, our reference to [754] is limited to the formulation of Theorem 8.3.20. Let us say that Lemma 8.3.18, just quoted, picks up an argument in the proof of [822, Theorem 15.3.6]. Theorems 8.3.10, 8.3.11, and 8.3.15, whose proofs have not been discussed, are originally due to McCrimmon [435, 1017]. Proposition 8.3.25 is due to Benslimane, Fern´andez, and Kaidi [89]. 8.3.4 Normed Jordan algebras after Aupetit’s paper [40]: a survey As we commented in p. 592 of Volume 1, Aupetit paper [40], proving the uniquenessof-norm theorem for J-semisimple complete normed Jordan algebras (cf. Corollary 4.4.14), became an inflection point in the theory of normed Jordan algebras. Most results on this topic preceding Aupetit’s paper [40] just quoted (like those in [54, 82, 422, 495, 628, 653, 759, 775]) have been already reviewed in our development, in many cases with a complete proof, and several subsequent results in the same line (like some in [48, 85, 86, 91, 361, 370, 403, 423, 488, 516, 540, 586, 710, 714, 750, 814]) have also been included. The present subsection is devoted to surveying most subsequent relevant progress in the theory of general normed Jordan algebras, which have not been previously included in our work. In this goal, we found the survey papers [41] and [525] useful, and we will try to update them. The subharmonicity of the spectral radius on normed Jordan algebras, already applied in [40], was further exploited in the paper by Aupetit and Zra¨ıbi [845], where the following Jordan version of [683, Theorem 7.1.13] was shown. Theorem 8.3.27 If f is a holomorphic mapping from an open subset  of C into a complete normed unital Jordan complex algebra A, then the mapping λ → J-sp(A, f (λ)) from  to the compact subsets of C is an analytic multivalued function (cf. [683, Chapter VII]). This theorem was applied in [845] to prove that, if A is a complete normed Jordan complex algebra, then the spectral radius is subadditive and submultiplicative on A if and only if A/J-Rad(A) is associative (see [682, Theorem 2.1.2] for the associative forerunner). As a consequence of Corollary 8.3.24, complete normed semisimple associative complex algebras, whose elements have finite spectra, are finite-dimensional. It is well-known that the same conclusion holds if the requirement of semisimplicity with finite spectrum is replaced with von Neumann regularity (Kaplansky [992]) or

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semiprimeness and coincidence with the socle (Tullo [1105]). The Jordanization of these results can be seen in the papers of Benslimane–Kaidi [91] and Benslimane– Fern´andez–Kaidi [89], respectively, where it is proved that under the above requirements (in the appropriate Jordan versions), complete normed non-commutative Jordan algebras are finite direct sums of closed simple ideals which are either finite-dimensional or quadratic. One of the main tools in the proof of these facts is the concept and basic theory of the socle of a nondegenerate non-commutative Jordan algebra. In the particular case of alternative algebras, nondegeneracy is equivalent to semiprimeness, so that, in the general case, nondegeneracy can be seen as a ‘J-semiprimeness’. The concept of socle for semiprime associative algebras was introduced by J. Dieudonn´e [922]. The socle of such an algebra is defined as the sum of all minimal left ideals (equal also to the sum of all minimal right ideals), and is therefore an ideal. The reader is referred to Jacobson’s book [753] for full information about the socle in the associative setting. The theory of the socle in the Jordan setting begins with the work by Osborn and Racine [1033], where the socle of a nondegenerate Jordan algebra is defined as the sum of all minimal inner ideals of the algebra, and it is proved that the socle is an ideal (see also [777, p. 329]). The non-commutative Jordan version of the Osborn–Racine theory is done in [257]. Among the several papers in which the theory of the socle for nondegenerate non-commutative Jordan algebras was further developed (without leaving the purely algebraic setting), we cite the one of Fern´andez–Rodr´ıguez [258], where an analytic incursion was made, showing that the socle of a complete normed nondegenerate non-commutative Jordan algebra coincides with the largest von Neumann regular ideal. The structure theory for complete normed prime nondegenerate non-commutative Jordan complex algebras with nonzero socle was developed in [489]. Moreover, in this paper, a more precise description is obtained in the case that additionally the algebra has minimality of norm topology and, as a consequence, prime non-commutative JB∗ -algebras with nonzero socle are determined. An appropriate version of [489] in the setting of Lie algebras has been recently provided by Fern´andez [253, 254]. In [250], Fern´andez introduced modular annihilator Jordan algebras as those nondegenerate Jordan algebras A such that the Jordan algebra A/soc(A) is J-radical (where soc(A) denotes the socle of A). After developing the basic algebraic theory for such algebras, complete normed modular annihilator Jordan algebras were considered, showing as main results that compact complete normed nondegenerate Jordan complex algebras are modular annihilator, and that JB-algebras are modular annihilator if and only if they are dual in the sense of Bunce [870] (cf. §8.2.48). (We recall that a normed Jordan algebra A is said to be compact if, for every x ∈ A, the operator Ux is compact.) In a subsequent paper [251], following the standard terminology in the associative case, Fern´andez introduced Jordan–Riesz algebras as those complete normed Jordan complex algebras A such that, for every x ∈ A, zero is the unique possible accumulation point of J-sp(A1 , x). It was proved that J-semisimple Jordan–Riesz algebras are modular annihilator, and that the converse

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is true in several interesting cases. In the same paper, it was shown that the socle of any normed nondegenerate Jordan algebra is an algebraic ideal. Also the question of the coincidence of the socle and the largest algebraic ideal in the complete normed J-semisimple case was considered, giving the first steps for an eventual future solution. A great part of the production on nongeometric aspects of normed Jordan algebras has focused on a deeper understanding of the behaviour of the socle of complete normed Jordan algebras. Thus Aupetit’s techniques have been tested, for the first time in this field, by Aupetit himself and Baribeau [45], where, as the main result, the following Jordan version of [683, Theorem 5.7.8] is proved. Theorem 8.3.28 If A is a J-semisimple complete normed Jordan complex algebra, and if the J-spectrum of every element in A is at most countable, then the socle of A is nonzero. The Aupetit–Baribeau paper also contains a rather involved structure theorem for separable complete normed Jordan complex algebras with the property that the spectrum of every element is at most countable. Roughly speaking, the knowledge of such algebras is reduced to that of separable complete normed modular annihilator Jordan algebras [45, Theorem 19]. The proof of this structure theorem relies on Theorem 8.3.28 and on the spectral characterization of complete normed modular annihilator Jordan algebras, which was provided almost simultaneously by Benslimane and Rodr´ıguez [95], and concludes Fern´andez’ abovementioned work [251]. The proof of this characterization consists of a Jordanization of Aupetit’s associative methods in [841], and reads as follows. Theorem 8.3.29 A J-semisimple complete normed non-commutative Jordan complex algebra is modular annihilator (if and) only if it is a Jordan–Riesz algebra. Then, as pointed out in [89], an appropriate version of Theorem 8.3.29 for real algebras can be easily derived. A significant advance regarding the question of the coincidence of the socle and the largest algebraic ideal in complete normed Jordan algebras was provided in the paper of Benslimane, Jaa, and Kaidi [90]. For a good understanding of some results reviewed in what follows, we point out that the appropriate variants of Propositions 4.1.34 and 4.1.35 hold when ‘Jordan ∗-triple’ is replaced with ‘Jordan triple’ (cf. Definition 7.1.19). The papers of Fern´andez, Garc´ıa, and S´anchez [256] and Loos [1003], introducing socle in Jordan triples and Jordan pairs (cf. §7.1.31) respectively, contain interesting new methods and concepts that even clarify the classical theory of the socle for associative and Jordan algebras. The paper [256] also contains a very fine structure theorem for the so-called reduced simple Jordan triples, and, extending the corresponding result for complete normed Jordan algebras, it is proved that the socle of a complete normed nondegenerate Jordan triple coincides with the largest von Neumann regular ideal. With the above-mentioned structure theorem, the main result in [256] is proved. This result can be summarized as follows.

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Theorem 8.3.30 Every complete normed von Neumann regular complex Jordan triple is a direct sum of a finite number of closed simple ideals, each of which is either (i) finite-dimensional, (ii) a Jordan triple coming from an infinite-dimensional complete normed simple quadratic Jordan complex algebra, or (iii) a Jordan triple associated to an infinite-dimensional complete normed simple von Neumann regular associative triple of the second kind. Complete normed simple von Neumann regular associative complex triples of the first or second kind are also described [256, Theorem 6.5]. For a complete overview of the theory of the socle in Jordan pairs, the reader is referred to [939, 947]. Richer information will surely be obtained from the forthcoming book of Fern´andez [1158] where a careful reorganization of papers like [661, 856, 931, 940, 942, 944, 1135] is achieved. According to him, the aim of the book is to show how Jordan systems can be used to solve questions in Lie theory. To this end, given a Lie algebra L (over a ring of scalars  in which 6 is invertible), he associates to any ‘Jordan element’ a ∈ L (ad3a L = 0) a Jordan algebra Ja , and more generally, he associates to any ‘abelian inner ideal’ B of L ([B, [B, L]] ⊂ B and [B, B] = 0) a Jordan pair SubL B. Most properties of the Lie algebra are inherited by these associated Jordan systems, and in addition, the natures of the Jordan element and of the abelian inner ideal are reflected in the structure of the attached Jordan systems. These facts turn out to be crucial for applications of Jordan theory to Lie algebras. Actually the notion of an abelian inner ideal plays a fundamental role in the structure theory of Lie algebras. Indeed, in the classical setting of finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero, minimal abelian inner ideals are precisely the one-dimensional subspaces generated by an ‘extremal element’. In the general setting of a nondegenerate Lie algebra L over a ring of scalars, the sum of the minimal abelian inner ideals of L is an ideal, called the socle of L, which is a direct sum of simple nondegenerate Lie algebras. The notion of the socle of a nondegenerate Jordan pair is deeply studied in Fern´andez’ book, and is applied in the Lie-Jordan connection outlined above. For example, a way of proving the eventual existence of a nonzero socle in a given nondegenerate Lie algebra consists of showing that it contains an abelian inner ideal whose associated Jordan pair satisfies the descending chain condition for their principal inner ideals. In [402], Loos introduces the concept of a properly algebraic (respectively, properly spectrum-finite) element of a Jordan triple as an element that is algebraic (respectively, has finite J-spectrum) in every Jordan algebra homotope, extends the notion of J-semisimplicity to Jordan triples, and proves the following. Theorem 8.3.31 For a complete normed J-semisimple complex Jordan triple, the socle, the set of properly algebraic elements, the largest properly spectrum-finite ideal, and the largest von Neumann regular ideal all coincide. In fact Theorem 8.3.31 is a straightforward consequence of an analogous result also proved in [402] for Jordan-Banach pairs. The method of proof makes essential

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use of Jordan pairs which are only ‘half-Banach’, only one of the spaces is complete but both normed. The main tools are structural transformations [1003] and subquotients [1005]. Once Theorem 8.3.31 is established, applying the main result of [90] and following some indications in [525, Section G] Loos proves the following. Corollary 8.3.32 The socle of a complete normed J-semisimple Jordan complex algebra coincides with the largest algebraic (equivalently, spectrum-finite) ideal. A non-commutative version of the above corollary, as well as a more intrinsic proof (avoiding Jordan pairs), has been provided later by Wilkins [1121]. In [43], Aupetit introduced the concept of rank for an element in a complete normed unital J-semisimple Jordan complex algebra, and proved that the socle of such an algebra coincides with the set of all finite rank elements. In [844] it was shown that the trace and determinant on the socle of a complete normed associative algebra can be developed in a purely spectral and analytic way, that is to say internally, without using operators on the algebra. Using an appropriate definition of multiplicity of a spectral value, and extending former results obtained in the associative case [844], the concepts of trace and determinant of elements with finite J-spectrum in complete normed Jordan algebras were introduced in [46], and the trace and J-spectrum preserving linear mappings in complete normed Jordan algebras were studied in [44]. In [404], Loos yields purely algebraic proofs of many results which were established in [44, 46]. In [405], Loos introduces nuclear elements and defines a notion of metric and bounded approximation property in complete normed Jordan triples analogous to that for complete normed associative algebras, showing that the trace form introduced in [404] may be extended to the nuclear elements, assuming the bounded approximation property. Combining associative techniques from Grabiner [963] and Sinclair–Tullo [1091], Benslimane and Boudi [87] proved that complete normed Noetherian alternative complex algebras are finite-dimensional. Related results can be found in [88, 119, 120]. Now consider the following. Fact 8.3.33 Let A be an associative algebra, let M be a maximal modular left ideal of A, and let f be a partially defined centralizer on A with dom( f )  M. Then, denoting by π : A → A/M the quotient mapping, there is an element f in the centralizer set for the irreducible left A-module A/M such that π( f (x)) = f (π(x)) for every x ∈ dom( f ). Proof [523] First of all we show that f (M ∩ dom( f )) ⊆ M. Indeed, otherwise we have A = M + f (M ∩ dom( f )), so there exists a right modular unit u for M which lies in f (M ∩ dom( f )), and writing u = f (m) with m in M ∩ dom( f ), for arbitrary x ∈ dom( f ) we have that x − f (x)m = x − xf (m) = x − xu ∈ M, so x lies in M, contradicting the assumption dom( f )  M. On the other hand, π(dom( f )) is a nonzero submodule of the irreducible left A-module A/M, so A/M = π(dom( f )).

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Now π(x) → π( f (x)) (x ∈ dom( f )) is a well-defined mapping (say f ) from A/M into A/M, and clearly f belongs to the centralizer set for the left A-module A/M.  Now recall that, in an associative setting, the concept of a normed Q-algebra was introduced in §3.6.41 as a normed associative complex algebra for which the set of quasi-invertible elements is open. It is easily derived from Corollary 3.6.44(ii) and Fact 8.3.33 that primitive associative complex normed Q-algebras are centrally closed (see [523] for details). As a consequence complete normed primitive associative complex algebras are centrally closed. In order to obtain the appropriate Jordan version of this last assertion, general non-associative methods have been developed in [523], providing as desired the following. Theorem 8.3.34 Every complete normed J-primitive Jordan complex algebra is prime and centrally closed (cf. §7.2.1 for the notion of a J-primitive Jordan algebra). By a Lie triple homomorphism between Jordan algebras A and B over K we mean any linear mapping  : A → B satisfying ([x, y, z]) = [φ(x), (y), (z)] for all x, y, z ∈ A. Given a Jordan algebra A over K, we define the centre modulo the radical, Z (A), of A by Z (A) := {a ∈ A : [a, A, A] ⊆ J-Rad(A)}. Since for any commutative algebra A the equality Z(A) = {a ∈ A : [a, A, A] = 0} holds (a consequence of the equality (3.4.11) before Remak 3.4.29), it follows that the centre modulo the radical of a Jordan algebra A is equal to zero whenever A is J-semisimple and has zero centre. As the main result in [130], Breˇsar, Cabrera, Fosner, and Villena prove the following. Theorem 8.3.35 Let A and B be complete normed Jordan algebras over K, and let  : A → B be a surjective Lie triple homomorphism. Then the separating space S() is contained in Z (B), the centre modulo the radical of B. Therefore  is continuous whenever B is J-semisimple and has zero centre. The proof of the above theorem relies on Zel’manov’s prime Theorem 6.1.132 for Jordan algebras, and on the recently developed theory of functional identities [686, 702, 868]. Other tools involved in the proof are Proposition 7.2.15 and Theorem 8.3.34. Now recall that the centre modulo the radical of an associative algebra was defined in the paragraph immediately before Proposition 4.4.78, and that Lie homomorphisms between associative algebras are defined as those linear mappings preserving commutators. Corollary 8.3.36 Let A and B be complete normed associative algebras over K, and let  : A → B be a surjective Lie homomorphism. Then the separating space S() is contained in Z (B), the centre modulo the radical of B. Therefore  is continuous whenever B is semisimple and has zero centre. Proof Since Asym and Bsym are complete normed Jordan algebras, and  becomes a surjective Lie triple homomorphism from Asym to Bsym (thanks to the equality

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(3.1.9) in the proof of Lemma 3.1.53), it follows from Theorem 8.3.35 that S() ⊆ Z (Bsym ). On the other hand, since Rad(Bsym ) = Rad(B) (by Proposition 4.4.17), and Z(Bsym /Rad(Bsym )) = Z(B/Rad(B)) (by Proposition 4.3.45), we realize that Z (Bsym ) = Z (B). Therefore S() ⊆ Z (B), as desired.  Corollary 8.3.36 is the main result in the Aupetit–Mathieu paper [47] fully discussed in pp. 597-598 of Volume 1. Before closing, let us list a few more references which are related to the subject matter of this section: Aupetit [839, 840, 842, 843], Benslimane–Boudi [857, 858], Boudi–Marhnine–Zarhouti [865], Bouhya–Fern´andez [121], Cao–Turovskii [897], Fern´andez–Marhnine–Zarhouti [945], Hessenberger [322, 323, 324, 325, 978], Hessenberger–Maouche [326], Maouche [412, 413, 1007, 1008], and Wilkins [636]. 8.4 The joint spectral radius of a bounded set Introduction For any bounded subset S of a normed space, we put S := sup{s : s ∈ S}. In Proposition 4.5.2(i), we proved the following. Theorem 8.4.1 Let A be a normed associative algebra over K. Then for each bounded and multiplicatively closed subset S of A there exists an equivalent algebra norm ||| . ||| on A such that ||| S ||| ≤ 1. Then we applied Theorem 8.4.1 to show that, for every element a in a normed associative algebra A over K, we have r(a) = inf{||| a ||| : ||| . ||| ∈ En(A)},

(8.4.1)

where En(A) denotes the set of all equivalent algebra norms on A (cf. Corollary 4.5.3). In this section we are going to present a full non-associative discussion of Theorem 8.4.1 and of the equality (8.4.1). Subsection 8.4.1 begins with an aperitif where we show how the nilpotency of a normed algebra A can be determined in terms of the equivalent algebra norms on A. Indeed, we prove that A is nilpotent if and only if there exists a natural number n in such a way that, for each ε ∈]0, 1], we can find an equivalent algebra norm ||| · ||| on A such that εn  ·  ≤ ||| · ||| ≤ ε ·  (Theorem 8.4.7). The subsection continues by introducing the notion of ( joint) spectral radius r(S) of a bounded subset S of any normed algebra A (Definition 8.4.10). Then we prove one of the key results in the section, namely that, if A is a normed algebra, and if S is a bounded subset of A with r(S) < 1, then the multiplicatively closed subset of A generated by S is bounded, and has the same spectral radius as S (Proposition 8.4.17). We also prove that, if A is a normed algebra, and if S is a bounded subset of A which has non-empty interior in A or is contained in the nucleus of A, then r(S) = inf{||| S ||| : ||| . ||| ∈ En(A)}

(8.4.2)

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(Proposition 8.4.19). In particular the equality (8.4.2) holds whenever A is any normed associative algebra and S is an arbitrary bounded subset of A. Then we involve in our development one of the key ideas in the section. Indeed, for a normed algebra A, the failure or success of A in relation to the conclusion of Theorem 8.4.1 can be quantified by means of a non-negative extended real number β(A) (Definition 8.4.28). The situation β(A) = +∞ means that A becomes a complete disaster concerning the conclusion of Theorem 8.4.1, whereas the inequality β(A) ≤ 1 can be interpreted as that Theorem 8.4.1 remains ‘approximately’ true for A. Actually, the mere requirement β(A) < +∞ (called the ‘multiplicative boundedness property’ for A) is enough to develop the most part of the theory of the joint spectral radius in parallel with the case where A is associative. We prove that β(A) lies in {0} ∪[1, +∞] for every normed algebra A, and that, for each λ ∈ {0} ∪ [1, +∞], there is a suitable normed algebra A such that β(A) = λ (Theorem 8.4.34). We also show that the equality (8.4.2) is true for every bounded subset S of a given normed algebra A if and only if β(A) ≤ 1 (Corollary 8.4.37). To conclude our review of Subsection 8.4.1, let us emphasize the existence of a (necessarily non-associative) normed algebra A satisfying β(A) ≤ 1, but failing to the conclusion of Theorem 8.4.1 (see Example 8.4.30). In Subsection 8.4.2, we introduce topologically nilpotent normed algebras as those normed algebras whose closed unit balls have zero spectral radius. Among the results obtained, we emphasize the following ones: (i) A normed algebra A is topologically nilpotent if and only if there are ‘arbitrarily small’ equivalent algebra norms on A, if and only if β(A) = 0 (Theorem 8.4.44). (ii) A normed associative algebra A is topologically nilpotent if and only if so is the normed Jordan algebra Asym obtained by symmetrization of its product (Theorem 8.4.49). (iii) Every non-topologically-nilpotent normed algebra can be equivalently algebrarenormed in such a way that the spectral radius of the corresponding closed unit ball is arbitrarily close to 1 (Theorem 8.4.51). (iv) Every topologically nilpotent complete normed algebra is equal to its weak radical (see Definition 4.4.39 and Corollary 8.4.66). As another relevant result in the subsection, we emphasize the one asserting that, if S is a bounded, closed, and absolutely convex subset of a complete normed complex algebra A, then there exists a sequence {sn } in S such that 1

lim sup fn (s1 , . . . , sn ) n = r(S), n→∞

where, for each n, fn is a suitable ‘way of multiplying’ s1 , . . . , sn in the given order (Theorem 8.4.70). We involve also in our development the notion of a finitely quasinilpotent normed algebra (meaning that all finite subsets of the algebra have zero spectral radius). Clearly, topological nilpotency implies finite quasi-nilpotency, but, even in the associative and commutative case, the converse is not true (Example 8.4.75). We prove that, if a normed algebra is finitely quasi-nilpotent and has the

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multiplicative boundedness property, then its multiplication ideal is finitely quasinilpotent (see §3.6.53 and Corollary 8.4.80). Moreover, the converse assertion is true in the associative case (Corollary 8.4.82). This allows us to put in relation two outstanding problems in the theory of complete normed radical associative complex algebras (see Problems 8.4.83 and 8.4.84). As another remarkable result, we prove that, in the finite-dimensional case, topological nilpotency, finite quasi-nilpotency, and nilpotency are equivalent notions (Corollary 8.4.90). In Subsection 8.4.3, we show that, for every member A in a large class of normed algebras (which contains all commutative C∗ -algebras, all JB-algebras, and all absolute-valued algebras), the conclusion in Theorem 8.4.1 has the following stronger form: for each bounded and multiplicatively closed subset S of A we have S ≤ 1 (Corollary 8.4.94). Actually, the fact just reviewed is a consequence of a more general result, proved in Proposition 8.4.92, which allows us to show that, if A is a normed algebra such that for each ε > 0 there is an equivalent algebra norm ||| · ||| on A satisfying ||| ab ||| ≥ (1 − ε)||| a |||||| b ||| for all a, b ∈ A, then there exists a unique equivalent norm | · | on A such that (A, | · |) becomes an absolute-valued algebra (Corollary 8.4.98). In Subsection 8.4.4, we involve in our development tensor products of algebras. Thus, in Proposition 8.4.104 (respectively, Proposition 8.4.106), we prove that the projective tensor product of two normed algebras is topologically nilpotent (respectively, finitely quasi-nilpotent) if some of them is so, and that the converse is true if some of them is associative. Moreover, associativity in the above converse cannot be removed (Example 8.4.107). We also prove that a normed algebra A is topologically nilpotent if and only if so is the normed algebra C0 (E, A) for some (equivalently, every) Hausdorff locally compact topological space E (Corollary 8.4.110). The results obtained about tensor products of normed algebras are then applied to show that most notions introduced in the section can be non-trivially exemplified into a class of algebras ‘almost arbitrarily’ prefixed. As a sample, we can find non-nilpotent topologically nilpotent complete normed algebras, as well as non-topologically-nilpotent finitely quasi-nilpotent complete normed algebras, in the class of non-associative alternative algebras, and in the class of Lie algebras (Corollary 8.4.115).

8.4.1 Basic notions and results As a matter of fact, we are going to show, by means of an easy example, how Theorem 8.4.1 does not remain true if the assumption that A is associative is altogether removed, nor even if this assumption is relaxed to the one that A is a non-commutative Jordan algebra, and if S is one of the simplest multiplicatively closed subsets of A, namely a singleton reduced to an idempotent. Let A be an algebra over K, and let λ be in K. According to §5.10.109, the λ-mutation of A, denoted by A(λ) , is defined as the algebra whose vector space is that of A, and whose product (say ) is defined by a b := λab + (1 − λ)ba. If A

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is a normed algebra, then the algebra A(λ) will be considered without notice as a normed algebra under the norm σλ  · , where σλ := |λ| + |1 − λ|. We note that, if A is a non-commutative Jordan algebra, then so is A(λ) . Example 8.4.2 Let B be the two-dimensional normed associative algebra over K with basis {u, v}, multiplication table given by u2 = u, uv = v, and vu = v2 = 0, and norm defined by αu + βv := |α| + |β|. Now, let μ be in [1, +∞[, put A := B(μ) , and note that A is a non-commutative Jordan algebra and that u is an idempotent in A. Since uv = μv in A, we have ||| u ||| ≥ μ for every algebra norm ||| · ||| on A. To continue our non-associative discussion of Theorem 8.4.1, we prove the following easy result. Lemma 8.4.3 Let A be an algebra over K, and let S be an absorbent and multiplicatively closed subset of A. Then the Minkowski functional of S, + , a pS (a) := inf λ ∈ R+ : ∈ S , λ is submultiplicative. ab Proof Let a, b be in A, and let λ, μ be in R+ such that λa ∈ S and μb ∈ S. Then λμ ∈S since S is multiplicatively closed. Therefore pS (ab) ≤ λμ, and, by letting λ → pS (a) and μ → pS (b), we get pS (ab) ≤ pS (a)pS (b), as desired. 

Given a normed space X and a bounded subset S of X, we put S := sup{s : s ∈ S}. Let A be an algebra, and let S be a subset of A. Since the intersection of any family of multiplicatively closed subsets of A is again a multiplicatively closed subset of A, it follows that the intersection of all multiplicatively closed subsets of A containing S is the smallest multiplicatively closed subset of A containing S. This subset is called the multiplicatively closed subset of A generated by S, and is denoted by MC(S). Proposition 8.4.4 Let A be a nonzero normed algebra over K, and let S be a subset of A. Then the following conditions are equivalent: (i) There exists δ > 0 such that MC [(δBA ) ∪ S] is bounded. (ii) There exists an equivalent algebra norm ||| · ||| on A such that |||S||| ≤ 1. Moreover, if the above conditions are fulfilled, then the norm ||| · ||| in (ii) can be chosen in such a way that 1 1  ·  ≤ ||| · ||| ≤  · , M δ where M := MC [(δBA ) ∪ S]. Proof

(i)⇒(ii) Suppose that (i) holds, and put M := MC [(δBA ) ∪ S].

(8.4.3)

8.4 The joint spectral radius of a bounded set

607

Since the absolutely convex hull of MC [(δBA )∪S] (say U) is multiplicatively closed, bounded by M, and absorbent (because in fact it contains δBA ), it follows from Lemma 8.4.3 that the Minkowski functional of U (say ||| · |||) is an algebra norm on A satisfying (8.4.3) and |||S||| ≤ 1. In particular, condition (ii) is fulfilled. (ii)⇒(i) Suppose that (ii) holds. Let δ > 0 be such that δ ||| · ||| ≤  · . Then (δBA ) ∪ S ⊆ B(A,||| · |||) , hence MC [(δBA ) ∪ S] ⊆ B(A,||| · |||) because B(A,||| · |||) is multiplicatively closed. Therefore MC [(δBA ) ∪ S] is bounded, and condition (i) is fulfilled.  Corollary 8.4.5 Let A be a normed algebra over K, and let S be a bounded and multiplicatively closed subset of A with non-empty interior. Then there exists an equivalent algebra norm ||| · ||| on A such that |||S||| ≤ 1. Proof The assumptions on S imply that the absolutely convex hull of S is bounded and multiplicatively closed, and contains 0 as an interior point. Now apply Proposition 8.4.4.  According to Example 8.4.2, the assumption in the above corollary that S has nonempty interior cannot be altogether removed. Let A be an algebra over K. According to an old convention in our work, given subsets B and C of A, we put BC := {xy : (x, y) ∈ B × C}. Now let S be a subset of A, and let n be in N. We define Sn inductively by S1 := S and Sn :=

n−1 

Sk Sn−k for n > 1.

k=1

Thus elements of Sn are precisely the products of n elements of S, ‘no matter how associated’. Fact 8.4.6 Let A be an algebra over K, let S be a multiplicatively closed subset of A, and let n be in N. Then Sn+1 ⊆ Sn . Proof We argue by induction on n. The fact is true for n = 1 because S is multiplicatively closed. Let n > 1, and assume inductively that Sm+1 ⊆ Sm for every m ∈ N with m < n. Let x be in Sn+1 . Then we have x = yz with y ∈ Sp , z ∈ Sq , p, q ∈ N, and p + q = n + 1. Since both p and q are < n + 1, and some of them (say p) is ≥ 2, we can apply the induction hypothesis, with m := p − 1, to get x ∈ Sp Sq ⊆ Sp−1 Sq ⊆ Sp−1+q = Sn . Since x is arbitrary in Sn+1 , we realize that Sn+1 ⊆ Sn , as desired.  Let A be an algebra over K, and let S be a subset of A. In agreement with §2.8.36, we say that S is nilpotent if there exists a natural number n such that Sm = 0 for every m ≥ n. We recall that, if S is nilpotent, then the smallest such an n is called the index of nilpotency of S. We note that, if S is multiplicatively closed ( for example, if S = A), and if Sn = 0 for some n, then, by Fact 8.4.6, we have Sm = 0 for every m ≥ n. Theorem 8.4.7 Let A be a normed algebra over K, and let n be a natural number. Then the following assertions are equivalent:

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Selected topics in the theory of non-associative normed algebras

(i) A is nilpotent of index ≤ n + 1. (ii) For every ε ∈]0, 1], there exists an algebra norm ||| · ||| on A such that εn  ·  ≤ ||| · ||| ≤ ε · . Proof (i)⇒(ii) Let ε be in ]0, 1]. By the assumption (i), MC ( 1ε BA ) is bounded by ( 1ε )n . By applying Proposition 8.4.4, with S := 1ε BA and δ := 1ε , we find the desired algebra norm ||| · ||| on A such that εn  ·  ≤ ||| · ||| ≤ ε · . (ii)⇒(i) Let ε be in ]0, 1]. By the assumption (ii), there exists an algebra norm ||| · ||| on A such that εn  ·  ≤ ||| · ||| ≤ ε · . Now, let a be a product of n + 1 elements of A (say a1 , . . . , an+1 ), no matter how associated. We have that εn a ≤ |||a||| ≤ |||a1 ||| · · · |||an+1 ||| ≤ εn+1 a1  · · · an+1 , and hence that a ≤ εa1  · · · an+1 . By letting ε → 0, we get a = 0.



Let A be an algebra. According to the notation introduced at the beginning of Subsection 2.3.4, for a, b, c ∈ A, we put [a, b, c] := (ab)c − a(bc). According to Definition 6.2.7, the nucleus of A is defined as the set of those elements a ∈ A such that [a, A, A] = [A, a, A] = [A, A, a] = 0. Keeping in mind that, for x, y, z, w ∈ A, the equality [x, y, z]w + x[y, z, w] = [xy, z, w] − [x, yz, w] + [x, y, zw] holds (a fact already noticed in (2.5.18)), we realize that the nucleus of A is a subalgebra of A. Now, a reasonable non-associative generalization of Theorem 8.4.1 is the following. Theorem 8.4.8 Let A be a normed algebra over K, and let S be a bounded and multiplicatively closed subset of A contained in the nucleus of A. Then there exists an equivalent algebra norm ||| · ||| on A such that |||S||| ≤ 1. Proof Let M ≥ 1 be a bound for S. We put ε := subsets of A given by

1 M,

and claim that the family F of

F := {S, εBA , εBA S, εSBA , εSBA S} (where SBA S := (SBA )S = S(BA S) by nuclearity of S) has the property that, whenever X and Y are in F , we have XY ⊆ Z for some Z ∈ F . Indeed, F fulfils such a property according to the following table: XY | S εBA εBA S εSBA εSBA S

| | | | |

S

εBA

εBA S

εSBA

εSBA S

S εBA S εBA S εSBA S εSBA S

εSBA εBA εBA εBA εSBA

εSBA S εBA εBA S εSBA εSBA S

εSBA εBA εBA εSBA εSBA

εSBA S εBA S εBA S εSBA S εSBA S

8.4 The joint spectral radius of a bounded set

609

The above table can be easily verified by applying that S is nuclear and multiplicatively closed, and that ε = M1 ≤ 1 (which implies εS ⊆ BA ). As a sample, we show that the inclusion XY ⊆ Z holds in the case that X = Y = Z = εSBA S. Indeed, we have (εSBA S)(εSBA S) = [(εSBA S)(εS)](BA S) = [(εSBA )(εSS)](BA S) ⊆ [(εSBA )(εS)](BA S) ⊆ [(εSBA )BA ](BA S) = [εS(BA BA )](BA S) ⊆ (εSBA )(BA S) = εS(BA BA )S ⊆ εSBA S. It follows from the definition of F and the claim just proved that T := S ∪ (εBA ) ∪ (εBA S) ∪ (εSBA ) ∪ (εSBA S) coincides with the multiplicatively closed subset of A generated by (εBA ) ∪ S. Since T is bounded (by M), Proposition 8.4.4 applies.  §8.4.9 Let A be a normed unital algebra over K. Then, taking S := {1} in the above theorem, we rediscover the fact, already shown in Proposition 1.1.111, that there exists an equivalent algebra norm on A converting A into a norm-unital normed algebra over K. Definition 8.4.10 Let A be a normed algebra over K, and let S be any bounded subset of A. Note that, for n in N, we have Sn  ≤ Sn . We define the spectral radius, r(S), of S by 1

r(S) := lim sup Sn  n ≤ S. n→∞

The following fact is easily verified. Fact 8.4.11 Let A and B be normed algebras over K, let S be a bounded subset of A, and let  : A → B be a continuous algebra homomorphism. Then r((S)) ≤ r(S). As a consequence, given a normed algebra A over K and a bounded subset S of A, equivalent algebra norms on A give the same spectral radius for S, and hence r(S) ≤ inf{|||S||| : ||| · ||| ∈ En(A)},

(8.4.4)

where En(A) denotes the set of all equivalent algebra norms on A. The obvious properties r(T) ≤ r(S) whenever T is a subset of S, and r(λS) = |λ|r(S) whenever λ belongs to K, will be applied in what follows without notice. If the normed algebra A is associative, then we easily realize that Sn+m  ≤ Sn Sm  for all n, m ∈ N, and consequently, it is enough to invoke Lemma 1.1.17 to get the following generalization of Corollary 1.1.18(i). Lemma 8.4.12 Let A be a normed associative algebra over K, and let S be a bounded subset of A. Then 1

1

r(S) = lim Sn  n = inf{Sn  n : n ∈ N}. n→∞

In the absence of associativity, the conclusion in Lemma 8.4.12 need not remain true. Indeed, we have the following.

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Selected topics in the theory of non-associative normed algebras

Example 8.4.13 Let A be the two-dimensional normed algebra over K with basis {u, v}, multiplication table given by u2 = v2 = v and uv = vu = 0, and norm defined by ? λu + μv := |λ|2 + |μ|2 . Put S := BK u. Then S is a bounded, complete, and absolutely convex subset of A. However, it is easily realized that, for n ∈ N, we have S2n  = 1 and S2n+1  = 0. 1 Therefore, the sequence {Sn  n } has no limit, and moreover, 1

r(S) = 1 = 0 = inf{[Sn ] n : n ∈ N}. §8.4.14 Let A be a normed algebra over K, and let a be in A. To this moment, in our work the symbol r(a) had a meaning only if a generates an associative subalgebra of A (cf. §3.4.61), and, as a consequence, whenever A is power-associative and a is arbitrary in A. Now we can extend the sense of that symbol for arbitrary A and arbitrary a ∈ A, by simply considering the singleton {a}, and putting r(a) := r({a}), where the right-hand side of this equality should be understood in the sense of Definition 8.4.10. We recall that, in this general setting, a substitute of the spectral radius of a, denoted by s(a), was already introduced in §4.4.2, by means of the equality 1

s(a) := inf{a[n]  2n : n ∈ N ∪ {0}}, where a[n] is defined inductively by a[0] := a and a[n+1] := (a[n] )2 . As we remarked there, we have also s(a) = r(a) whenever a generates an associative subalgebra of A. As a matter of fact, this last equality need not be true in general. Indeed, we have the following. Proposition 8.4.15 Let A be a normed algebra over K, and let a be in A. Then: (i) s(a) ≤ r(a). (ii) There are choices of A and a such that A is two-dimensional and Jordanadmissible, s(a) = 0, and a = r(a) = 1. Proof

n

It is easily realized that a[n] lies in {a}2 for every n ∈ N ∪ {0}. Therefore 1

n

1

s(a) := inf{a[n]  2n : n ∈ N ∪ {0}} ≤ lim sup {a}2  2n n→∞

1

≤ lim sup {a}n  n = r(a). n→∞

Now let A be the two-dimensional normed algebra over K with basis {a, b}, multiplication table given by a2 = b, b2 = 0, and ab = −ba = a, and norm defined by λa + μb := |λ| + |μ|.

8.4 The joint spectral radius of a bounded set

611

Then we have a[2] = 0 and Lan (a2 ) = a (which implies a ∈ {a}n+2 ) for every n ∈ N. Therefore s(a) = 0 and 1

1 = a ≤ lim sup {a}n+2  n+2 = r(a) ≤ a = 1. n→∞

On the other hand, we have (x • y) • z = 0 whenever x, y, z ∈ {a, b}, which implies that  Asym is associative. Therefore A is Jordan-admissible. Fact 8.4.16 Let A be a normed algebra over K, and let S be a bounded and multiplicatively closed subset of A. Then r(S) ≤ 1. Proof

We have Sn ⊆ S for every n ∈ N, since S is multiplicatively closed. Therefore 1

1

r(S) = lim sup Sn  n ≤ lim S n ≤ 1. n→∞

n→∞



Proposition 8.4.17 Let A be a normed algebra over K, and let S be a bounded subset of A such that r(S) < 1. Then MC(S) is bounded, and r(MC(S)) = r(S). Proof The assumption that r(S) < 1 implies that Sn  < 1 for n big enough, and hence that the set ∪n∈N Sn (which is nothing other than MC(S)) is bounded. Let 0 < ε ≤ 1 − r(S). Then there exists nε ∈ N such that Sn  ≤ (r(S) + ε)n whenever n ≥ nε . Let n be in N with n ≥ nε , and let w be in (MC(S))n . Then w lies in Sm for some m ≥ n ≥ nε . Therefore, we have w ≤ (r(S) + ε)m ≤ (r(S) + ε)n , the last inequality being true because m ≥ n and r(S) + ε ≤ 1. Since w is arbitrary in (MC(S))n , the above implies (MC(S))n  ≤ (r(S) + ε)n . Now, from the arbitrariness of n under the condition n ≥ nε we deduce r(MC(S)) ≤ r(S) + ε. Finally, let ε → 0.  Corollary 8.4.18 Let A be a normed algebra over K, and let S be a bounded subset of A. We have:   (i) r(S) = inf{k ∈ R+ : MC Sk is bounded }. (ii) If MC(S) is bounded, then r(MC(S)) = r(S).   Proof Let k > 0 be such that MC Sk is bounded. Then, by Fact 8.4.16, we have      S S ≤ k r MC ≤ k. r(S) = k r k k S Therefore r(S) ≤ m := inf{k ∈ R+ : MC k is bounded }. If r(S)    < m, then we can take l ∈ R with r(S) < l < m, so that MC Sl is unbounded and r Sl < 1, which implies in view of Proposition 8.4.17 that MC Sl is bounded, a contradiction. Thus r(S) = m. Suppose that MC(S) is bounded. Then, by Fact 8.4.16, we have r(S) ≤ r(MC(S)) ≤ 1, hence r(MC(S)) = r(S) if r(S) = 1. Otherwise, by Proposition 8.4.17, we have also r(MC(S)) = r(S). 

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Selected topics in the theory of non-associative normed algebras

Proposition 8.4.19 Let A be a normed algebra over K, and let S be a bounded subset of A. If S has non-empty interior in A, or is contained in the nucleus of A, then r(S) = inf{|||S||| : ||| · ||| ∈ En(A)}. Proof Suppose that S has non-empty  1 interior  in A (respectively, is contained in the nucleus of A). Let ε > 0. Since r r(S)+ε S < 1, Proposition 8.4.17 applies, giving  1  us that MC r(S)+ε S is bounded. On the other hand, since S has non-empty interior  1  (respectively, is contained in the nucleus of A), the same happens for MC r(S)+ε S . It follows from Corollary 8.4.5 (respectively, from Theorem 8.4.8) that there exists an equivalent algebra norm ||| · ||| on A such that |||S||| ≤ r(S) + ε. Keeping in mind the inequality (8.4.4), the proof is complete.  Now we are provided with the following generalization of Corollary 4.5.3 (see also the equality (8.4.1)). Corollary 8.4.20 Let A be a normed associative algebra over K, and let S be any bounded subset of A. Then r(S) = inf{||| S ||| : ||| . ||| ∈ En(A)}. Applying Proposition 8.4.19 with S := BA , we derive the following. Corollary 8.4.21 Let A be a normed algebra, and let ε > 0. Then there exists an equivalent algebra norm ||| · ||| on A such that ||| · ||| ≤ (r(BA ) + ε) · . For the proof of the next results, and even for the formulation of some results below, like Theorem 8.4.70 and Corollaries 8.4.72 and 8.4.73, we need some formalizations concerning the ‘ways of multiplying’ n elements of a non-associative algebra in a given order. To illustrate the situation, consider the case n = 3, and, consequently, take arbitrary elements a1 , a2 , a3 in an algebra A. If A is associative, then a1 a2 a3 stands unambiguously for their product in the given order. However, if we do not know that the algebra A is associative, then, in principle, we have two different ways of multiplying them in the given order, namely a1 (a2 a3 ) and (a1 a2 )a3 . These ways of multiplying do not depend on the algebra A, nor on the elements a1 , a2 , a3 , and therefore become members of an abstract set, which is denoted by W3 , and is rigorously defined as a consequence of the following. Notation 8.4.22 Let n be in N. We denote by Fn the free non-associative algebra on n indeterminates (say x1 , . . . , xn ) over K (cf. §2.8.17). As in §2.8.26, given any algebra A over K, a1 , . . . , an ∈ A, and f ∈ Fn , we denote by f (a1 , . . . , an ) the valuation of f at (a1 , . . . , an ), i.e. the image of f under the unique algebra homomorphism  : Fn → A satisfying (xi ) = ai for i = 1, . . . , n. Now, let Wn denote the subset of Fn consisting of those non-associative words f = f (x1 , . . . , xn ) such that f (y1 , . . . , yn ) = y1 · · · yn , where y1 , . . . , yn stand for the generators of the free associative algebra on n indeterminates over K. Then, by passing to valuations of its members at n-tuples of elements of an arbitrary algebra, Wn becomes the set of

8.4 The joint spectral radius of a bounded set

613

all ‘ways of multiplying’ n elements a1 , . . . , an of the algebra in the given order. We note that Wn does not contain all non-associative words in Fn which are of degree 1 in each of the indeterminates. For example, x2 x1 does not lie in W2 . Anyway, if A is any algebra over K, and if S is a subset of A, then every element of Sn can be written as f (s1 , . . . , sn ) for suitable f ∈ Wn and s1 , . . . , sn ∈ S. This will be applied without notice in what follows. Given any subset S of a normed space X, we denote by |co|(S) the closed absolutely convex hull of S in X. Lemma 8.4.23 Let A be a normed algebra, and let S be a bounded subset of A. Then we have Sn  = (|co|(S))n  for every n ∈ N, and hence r(S) = r(|co|(S)). Proof Let n be in N, and let f be in Wn . Then, for all s1 , . . . , sn ∈ S, we have f (s1 , . . . , sn ) ≤ Sn  because f (s1 , . . . , sn ) lies in Sn . Since the mapping (a1 , . . . , an ) → f (a1 , . . . , an ) from A× . n. . ×A to A is n-linear and continuous, we derive that f (|co|(S), . n. ., |co|(S)) is bounded by Sn . Since f is an arbitrary element of Wn , this means that (|co|(S))n is bounded by Sn  or, equivalently, that (|co|(S))n  ≤ Sn .  Proposition 8.4.24 Let A be a normed algebra over K, and let S be a bounded subset of A. Then there exists a countable subset T of S such that r(T) = r(S). Let n be in N. Then there exist s1,n , s2,n , . . . , sn,n ∈ S and fn ∈ Wn such that  n n fn (s1,n , s2,n , . . . , sn,n ) ≥ Sn . (8.4.5) n+1  Now, put T := m∈N {s1,m , s2,m , . . . , sm,m }.Then T is a countable subset of S. Moren n n over, it follows from (8.4.5) that T n  ≥ n+1 S . By passing to n-th roots, and taking sup limit as n → ∞, we get r(T) ≥ r(S).  Proof

§8.4.25 Let A be an algebra over K, and let S be a subset of A. We define inductively S[n] by S[0] := S and S[n+1] := (S[n] )2 . Suppose in what follows that A is normed and that S is bounded. Then we have 1 S[n+1]  ≤ S[n] 2 , and hence the sequence S[n]  2n is decreasing. Therefore we can define the number 1

1

s(S) := inf{S[n]  2n : n ∈ N ∪ {0}} = lim S[n]  2n . n→∞

We note that, if A is associative, then we have Lemma 8.4.12, we have n

S[n]

n

= S2 for every n, and hence, by

1

s(S) = lim S2  2n = r(S). n→∞

Arguing as in the proof of Proposition 8.4.15(i), we get the following.

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Selected topics in the theory of non-associative normed algebras

Fact 8.4.26 Let A be a normed algebra over K, and let S be a bounded subset of A. Then s(S) ≤ r(S). A subset S of a normed algebra A over K is said to be idempotent if S ⊆ |co|(S2 ). Proposition 8.4.27 Let A be a normed algebra over K, and let S be a nonzero bounded idempotent subset of A. Then 1 ≤ s(S). Proof It is straightforward that the square of any idempotent subset of A is an idempotent subset of A. Therefore, by induction we obtain that S ⊆ |co|(S[n] ) for 1 1 every n ∈ N ∪ {0}, and hence S 2n ≤ S[n]  2n . The result now follows by letting n → ∞.  As we said in the introduction, our main goal in this section is to discuss the validity of Theorem 8.4.1 in the non-associative setting. For a more precise nonassociative discussion of that theorem, we introduce here the following. Definition 8.4.28 Let A be a normed algebra over K. Given a positive number k, we say that A satisfies the norm-k boundedness property if, for each bounded and multiplicatively closed subset S of A, there exists an equivalent algebra norm ||| · ||| on A such that |||S||| ≤ k. We say that A satisfies the multiplicative boundedness property if it satisfies the norm-k boundedness property for some k > 0. When A satisfies the multiplicative boundedness property, we put β(A) := inf{k ∈ R+ : A satisfies the norm-k boundedness property}. Otherwise, we put β(A) := +∞. The norm-1 boundedness property (in short, NBP) becomes specially relevant because, in view of Theorem 8.4.1, it is satisfied by all normed associative algebras over K. Let us say that a normed algebra A over K satisfies the approximate norm-1 boundedness property (in short, ANBP) if β(A) ≤ 1. We note that both norm-k boundedness property for fixed k > 0, the multiplicative boundedness property, the extended number β(·), the NBP, and the ANBP are algebraic-topological notions, i.e. they do not change if we replace the norm of the algebra with any equivalent algebra norm. The following fact is straightforward. Fact 8.4.29 Let A be a normed algebra over K, and let B be any subalgebra of A. Then β(B) ≤ β(A). Obviously, the NBP implies the ANBP. Nevertheless, the converse is not true. Indeed, we have the following. Example 8.4.30 Let A be the free vector space over K on a countably infinite set of generators, say {p} ∪ {an : n ∈ N}. Thus every element x ∈ A can be written in a unique way as  αn an , (8.4.6) x = βp + n∈N

8.4 The joint spectral radius of a bounded set

615

where β ∈ K and {αn }n∈N is a quasi-null sequence in K. Define the norm on A by x := sup {2|β| , sup{n|αn | : n ∈ N}} and the multiplication by an am = 0 = pan for all n, m ∈ N, p2 = p, and an p = an+1 for every n ∈ N. It is a matter of routine to verify that A is a normed algebra. Let us now show that A does not satisfy the NBP. Indeed, clearly the singleton {p} is a bounded and multiplicatively closed subset of A. Suppose to derive a contradiction that there exists an equivalent algebra norm ||| · ||| on A such that ||| p ||| ≤ 1. Then, for every n ∈ N we have ||| an+1 ||| = ||| an p ||| ≤ ||| an ||| ||| p ||| ≤ ||| an |||, so by induction ||| an ||| ≤ ||| a1 |||. However, since an  = n for every n ∈ N, the norms  ·  and ||| · ||| are not equivalent, the desired contradiction. Finally, let us show that A satisfies the ANBP. Indeed, let ε > 0 and let S be a bounded and multiplicatively closed subset of A. Let K ≥ 1, K ≥ sup{s : s ∈ S}. Keeping in mind (8.4.6), we define a new norm on A by ||| x ||| := sup {(1 + ε)|β| , sup{cn |αn | : n ∈ N}} ,  i(1+ε)n−i  where cn := inf : 1 ≤ i ≤ n . Noticing that cn+1 ≤ (1 + ε)cn , we easily K realize that ||| · ||| is an algebra norm on A. Moreover, since cn ≤ Kn , we see that the topology of ||| · ||| is weaker than that of  · . On the other hand, computing the intervals of decreasing and increasing of the function t → t(1 + ε)n−t from R+ to R,   n−1 , Kn for every n ∈ N, which implies we get that cn = min (1+ε) K $ F , +c (1 + ε)n−1 n : n ∈ N = min : n ∈ N > 0, inf n nK so the topology of  ·  is weaker than that of ||| · |||. It follows that  ·  and ||| · ||| are equivalent norms on A. To conclude that A satisfies the ANBP, let x be in S, and write x as in (8.4.6). Then we have that |αn | ≤ Kn for every n ∈ N (by the definition of K), and |β| ≤ 1 (since otherwise, for n ∈ N we would have (Lx )n (x) ∈ S, and hence K ≥ (Lx )n (x) ≥ 2|β|n+1 → ∞, a contradiction). Keeping in mind the above inequalities and that cn ≤ Kn for every n ∈ N, it is easy to check that ||| x ||| ≤ 1 + ε. Remark 8.4.31 Let  A stand for the completion of the normed algebra A in the above example. Since A does not satisfy the NBP, and the NBP is inherited by subalgebras, it becomes clear that  A cannot satisfy the NBP. It is not so clear but true that  A satisfies the ANBP. To realize this, note that A is linearly isometric in a natural way to the space c00 of all quasi-null sequences in K endowed with the sup norm, and that, consequently  A is linearly isometric to c0 in a natural way. Therefore every element x ∈  A can be written in a unique way as x = βp +

∞  n=1

αn an ,

(8.4.7)

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Selected topics in the theory of non-associative normed algebras

where β ∈ K and {αn }n∈N is a sequence in K with limn→∞ nαn = 0. Keeping in mind this fact, minor changes to the argument in the last paragraph of Example 8.4.30 show that  A satisfies indeed the ANBP. Lemma 8.4.32 Let A be a normed algebra over K, and let λ be in K \ { 12 }. Then |2λ − 1| (|λ| + |1 − λ|)2 (λ) β(A) ≤ β(A ) ≤ β(A). |2λ − 1| (|λ| + |1 − λ|)2 Proof To prove the second inequality, it is enough to show that, if A satisfies the norm-k boundedness property for some k > 0, then A(λ) satisfies the norm (|λ|+|1−λ|)2  |2λ−1| k boundedness property. Let k > 0 be such that A satisfies the norm-k boundedness property, and let S be a bounded and multiplicatively closed subset of A(λ) . Then the absolutely convex hull of S (say T) is bounded   and multiplicatively λ closed in A(λ) . Since A = [A(λ) ](μ) with μ := 2λ−1 ∈ K \ 12 , it follows xy = μx

y + (1 − μ)y

x ∈ σμ T

whenever x and y are in T. In this way, σ1μ T becomes a bounded and multiplicatively closed subset of A. By the assumption on k, there exists ||| · ||| ∈ En(A) such that ||| S ||| ≤ ||| T ||| ≤ σμ k. This implies that σλ ||| · ||| ∈ En(A(λ) ) and σλ ||| S ||| ≤ σλ σμ k. Since S is an arbitrary bounded and multiplicatively closed subset of A(λ) , and   2 (λ) satisfies the norm- (|λ|+|1−λ|)2 k σλ σμ = (|λ|+|1−λ|) |2λ−1| , the above shows that A |2λ−1| boundedness property, as desired. λ Writing the inequality just shown with (A(λ) , μ) instead of (A, λ), where μ := 2λ−1 as above, we get the first inequality in the statement.  Proposition 8.4.33 Let A be a normed algebra over K satisfying the multiplicative boundedness property. Then A(λ) fulfils the multiplicative boundedness property for 1   every λ in K \ 2 , and the mapping λ → β(A(λ) ), from K \ 12 to R, is continuous. Proof The firstconclusion follows straightforwardly from Lemma 8.4.32. Let λ  (μ) = [A(λ) ](η) . Therefore, and μ be in K \ 12 . Put η := μ+λ−1 2λ−1 . Then we have A noticing that η = 12 , we can apply Lemma 8.4.32, with (A(λ) , η) instead of (A, λ), to get (|μ + λ − 1| + |λ − μ|)2 |2λ − 1||2μ − 1| (λ) (μ) β(A ) ≤ β(A ) ≤ β(A(λ) ). |2λ − 1||2μ − 1| (|μ + λ − 1| + |λ − μ|)2 By fixing λ, the above implies limμ→λ β(A(μ) ) = β(A(λ) ).



We recall that the class of non-commutative Jordan algebras is closed under mutations of its members, and hence contains all mutations of associative algebras. Theorem 8.4.34 For every normed algebra A over K, we have β(A) ∈ {0} ∪ [1, +∞]. Moreover, for every λ ∈ {0} ∪ [1, +∞], there exists a normed two-dimensional noncommutative Jordan algebra A such that β(A) = λ .

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Proof Let A be a normed algebra over K such that β(A) < 1. Take k < 1 such that A satisfies the norm-k boundedness property. Then, since BA is a bounded and multiplicatively closed subset of A, there exists  · 1 ∈ En(A) satisfying  · 1 ≤ k · . Assume inductively that for some n ∈ N there exists  · n ∈ En(A) satisfying  · n ≤ kn  · . Then, since B(A,·n ) is a bounded and multiplicatively closed subset of A, there exists ·n+1 ∈ En(A) satisfying ·n+1 ≤ k·n ≤ kn+1 ·. Now, let ε > 0. Then, given any bounded subset S of A, it is enough to choose a bound M > 0 for S, and n ∈ N with kn ≤ Mε , to have  · n ∈ En(A) and Sn ≤ ε. As a consequence, A satisfies the norm-ε boundedness property. Since ε is an arbitrary positive number, we derive β(A) = 0. Taking A equal to the algebra over K whose vector space is K2 , whose product is identically zero, and whose norm is any norm on K2 , A becomes a two-dimensional non-commutative Jordan normed algebra such that β(A) = 0. Now we are going to show that all numbers in [1, +∞[ are of the form β(A) for a suitable normed two-dimensional non-commutative Jordan algebra A over K. Let B be the two-dimensional normed associative algebra over K, and let u be the element of B, both given by Example 8.4.2. As we showed there, we have ||| u ||| ≥ μ for every ||| · ||| ∈ En(B(μ) ). Since u2 = u in B(μ) (i.e. the singleton {u} is a multiplicatively closed subset of B(μ) ), it follows that k ≥ μ whenever k is any positive number such that B(μ) satisfies the norm-k boundedness property, and hence that β(B(μ) ) ≥ μ. In this way we have shown that the mapping f : μ → β(B(μ) ), from [1, +∞[ to [0, +∞], takes its values into [1, +∞], and is unbounded. On the other hand, we have f (1) = 1 because B(1) = B and B is associative. By keeping in mind Proposition 8.4.33, it follows that f ([1, +∞[) = [1, +∞[. Therefore, given 1 ≤ λ < +∞, there exists 1 ≤ μ < +∞ such that f (μ) = λ, and hence we are provided with a normed two-dimensional non-commutative Jordan algebra A := B(μ) such that β(A) = λ. Now let A stand for the two-dimensional anticommutative normed algebra over K with basis {u, v}, multiplication table given by u2 = v2 = 0 and uv = −vu = v, and norm defined by λu + μv := |λ| + |μ|. Then, for every ||| · ||| ∈ En(A), we have ||| v ||| = ||| uv ||| ≤ ||| u |||||| v |||, and hence 1 ≤ ||| u |||. Assume that there exists k > 0 such that A satisfies the norm-k boundedness property. Then, since {2ku, 0} becomes a multiplicatively closed subset of A, there is ||| · ||| ∈ En(A) such that ||| 2ku ||| ≤ k,  which implies ||| u ||| ≤ 12 , a contradiction. Therefore we have β(A) = +∞. Remark 8.4.35 Of course, the easiest choice of a normed algebra A over K such that β(A) = 0 (respectively, β(A) = 1) is A = K endowed with the zero product (respectively, A = K endowed with the usual product). Since, up to a bijective algebra homomorphism, there are no one-dimensional algebras over K others than the two ones in the above sentence, it follows that, for λ ∈]1, +∞], the two-dimensional algebra A given by Theorem 8.4.34 such that β(A) = λ is of the smallest possible dimension. Such an algebra A cannot be associative because normed associative algebras have the NBP.

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Proposition 8.4.36 Let A be a normed algebra over K. Then the following conditions are equivalent: (i) A satisfies the multiplicative boundedness property. (ii) There exists a non-negative real number k such that, for every bounded subset S of A, we have inf{|||S||| : ||| · ||| ∈ En(A)} ≤ k r(S). Moreover, when these conditions hold, the minimum non-negative real number k in condition (ii) is equal to β(A). Proof (i)⇒(ii)  Let S be any bounded subset of A, and let ε > 0. Keeping in mind S that r r(S)+ε < 1, we can apply Proposition 8.4.17 and the assumption (i) to find ||| · ||| ∈ En(A) such that |||S||| ≤ (β(A) + ε)(r(S) + ε). Consequently, inf{|||S||| : |||· ||| ∈ En(A)} ≤ (β(A) +ε)(r(S) +ε). Since ε is an arbitrary positive number, we deduce inf{|||S||| : ||| · ||| ∈ En(A)} ≤ β(A)r(S). This shows that condition (ii) holds with k := β(A), and hence that the minimum possible k is less than or equal to β(A). (ii)⇒(i) Keeping in mind Fact 8.4.16, the assumption (ii) gives almost straightforwardly that (i) holds, and that β(A) is less than or equal to the minimum possible k in condition (ii).  By putting together Proposition 8.4.36 and the inequality (8.4.4), we derive the following. Corollary 8.4.37 Let A be a normed algebra over K. Then the following assertions are equivalent: (i) A satisfies the ANBP. (ii) For every bounded subset S of A, we have r(S) = inf{|||S||| : ||| · ||| ∈ En(A)}. Since normed associative algebras satisfy the NBP (cf. Theorem 8.4.1), we can apply Corollary 8.4.37 to get a new proof of Corollary 8.4.20. As usual in our work, for any normed space X over K, we denote by BL(X) the normed associative algebra over K of all bounded linear operators on X. We note that each equivalent norm ||| · ||| on X gives rise to an equivalent algebra norm on BL(X) (namely, the operator norm on BL(X) corresponding to ||| · |||), which will be denoted also by ||| · |||. §8.4.38 Let A be an algebra over K. As usual in our work, for a in A, we denote by La = LaA (respectively, Ra = RAa ) the operator of left (respectively, right) multiplication by a on A, and, given a subset S of A, we put

8.4 The joint spectral radius of a bounded set

619

LS = LSA := {La : a ∈ S} and RS = RAS := {Ra : a ∈ S}. It is clear that, if A is in fact a normed algebra, then both LA and RA are contained in BL(A). Proposition 8.4.39 Let A be a normed algebra over K satisfying the multiplicative boundedness property, and let S be a bounded subset of A. Then we have r(LS ∪ RS ) ≤ β(A)r(S)

(8.4.8)

r(LS RS ) ≤ β(A)2 r(S)2 .

(8.4.9)

and

Proof have

Let ||| · ||| be any equivalent algebra norm on A. By the inequality (8.4.4), we r(LS ∪ RS ) ≤ |||LS ∪ RS |||

(respectively, r(LS RS ) ≤ |||LS RS |||). On the other hand, the inequality |||LS ∪ RS ||| ≤ ||| S ||| (respectively, |||LS RS ||| ≤ ||| S |||2 ) is clear. Therefore we derive r(LS ∪ RS ) ≤ ||| S ||| (respectively, r(LS RS ) ≤ ||| S |||2 ). Now, the assumption that A satisfies the multiplicative boundedness property allows us to apply Proposition 8.4.36 to obtain (8.4.8) (respectively, (8.4.9)).  Corollary 8.4.40 Let A be a normed associative algebra over K, and let S be a bounded subset of A. Then we have r(LS ) = r(RS ) = r(LS ∪ RS ) = r(S) and r(LS RS ) = r(S)2 . Proof Since A is associative, we have β(A) ≤ 1, and hence, by Proposition 8.4.39, we get r(LS ∪ RS ) ≤ r(S) and r(LS RS ) ≤ r(S)2 . Now, since the inequalities r(LS ) ≤ r(LS ∪ RS ) and r(RS ) ≤ r(LS ∪ RS ) are clear, it only remains to show that r(S) ≤ r(LS ), r(S) ≤ r(RS ), and r(S)2 ≤ r(LS RS ). Let n be in N, and let x1 , . . . , xn+1 be in S. Then we have x1 · · · xn+1 = Lx1 · · · Lxn (xn+1 ). From the arbitrariness of x1 , . . . , xn+1 in S, we deduce Sn+1  ≤ (LS )n S, and, passing to n-th roots and taking sup limit as n → ∞, we get r(S) ≤ r(LS ). The inequality r(S) ≤ r(RS ) is verified in an analogous way. Let n be in N, and let x1 , . . . , x2n+1 be in S. Then we have x1 · · · x2n+1 = Mx1 ,x2n+1 Mx2 ,x2n · · · Mxn ,xn+2 (xn+1 ),

620

Selected topics in the theory of non-associative normed algebras

where, for a, b ∈ A, Ma,b := La Rb . From the arbitrariness of x1 , . . . , x2n+1 in S, we deduce S2n+1  ≤ (LS RS )n S. Passing to n-th roots, and applying Lemma 8.4.12, we can take limit as n → ∞ to get r(S)2 ≤ r(LS RS ).  8.4.2 Topologically nilpotent normed algebras Let A be a normed algebra over K, and let S be a subset of A. We say that S is quasi-nilpotent if S is bounded and r(S) = 0. Clearly, nilpotent bounded subsets of A are quasi-nilpotent. The normed algebra A is said to be topologically nilpotent if its closed unit ball is quasi-nilpotent. Clearly, normed nilpotent algebras over K are topologically nilpotent, but the converse is not true. Indeed, we have the following. Example 8.4.41 Let A stand for the complete normed commutative and associative algebra over K of all continuous K-valued functions on [0, 1], with the sup norm and convolution multiplication ∗ defined by  t ( f ∗ g)(t) := f (s)g(t − s)ds. 0

For f ∈ BA and t ∈ [0, 1] we have |f (t)| ≤ 1 = |( f1 ∗ · · · ∗ fn )(t)| ≤

tn−1 (n−1)!

t0 0! .

Assume inductively that

for all f1 , . . . , fn ∈ BA and t ∈ [0, 1].

Then for f1 , . . . , fn+1 ∈ BA and t ∈ [0, 1] we have  t      |( f1 ∗ · · · ∗ fn+1 )(t)| =  ( f1 ∗ · · · ∗ fn )(s)fn+1 (t − s)ds ≤ 0

t 0

(8.4.10)

sn−1 tn ds = . (n − 1)! n!

Therefore (8.4.10) holds for every n ∈ N, so (BA ≤ for every n ∈ N, and so r(BA ) = 0. On the other hand, an easy induction argument shows that the function tn−1 for e ∈ A defined by e(t) = 1 for every t ∈ [0, 1] satisfies that (e ∗ . n. . ∗ e)(t) = (n−1)! all t ∈ [0, 1] and n ∈ N. It follows that A becomes a non-nil (hence non-nilpotent) topologically nilpotent normed algebra. )n 

1 (n−1)!

A first easy characterization of topological nilpotency is given by the following. Proposition 8.4.42 Let A be a normed algebra over K. Then the following conditions are equivalent: (i) A is topologically nilpotent. (ii) All bounded subsets of A are quasi-nilpotent. (iii) All bounded countable subsets of A are quasi-nilpotent. Proof (i)⇒(ii) Since each bounded subset of A is contained in a positive multiple of BA . (ii)⇒(iii) This is clear. (iii)⇒(i) By Proposition 8.4.24. 

8.4 The joint spectral radius of a bounded set

621

A straightforward consequence of the above characterization is the following. Corollary 8.4.43 Topological nilpotency is preserved by passing to algebra equivalent renormings. Theorem 8.4.44 Let A be normed algebra over K. Then the following assertions are equivalent: (i) A is topologically nilpotent. (ii) For every ε > 0, there exists an equivalent algebra norm ||| · ||| on A such that ||| · ||| ≤ ε · . (iii) β(A) = 0. Proof (i)⇒(ii) By Corollary 8.4.21. (ii)⇒(iii) Noticing that the assumption (ii) implies that A satisfies (a very strong form of) the norm-k boundedness property for every positive number k, the desired conclusion (iii) follows. (iii)⇒(i) Let ε > 0. Then, by the assumption (iii), A satisfies the norm-ε boundedness property, and hence, since BA is a bounded and multiplicatively closed subset of A, there exists ||| · ||| ∈ En(A) such that BA ⊆ εB(A,||| · |||) . This implies r(BA ) ≤ ε. Now, let ε → 0.  The following straightforward consequence of the implication (i)⇒(ii) in Theorem 8.4.44 is worth emphasizing. Corollary 8.4.45 Let A be a topologically nilpotent normed algebra over K. Then A satisfies a refined form of the NBP. Indeed, for each bounded subset S of A there exists an equivalent algebra norm ||| · ||| on A such that ||| S ||| ≤ 1. Proposition 8.4.46 Let A be a topologically nilpotent normed algebra over K. We have: (i) A contains no nonzero bounded idempotent subset. (ii) If A = 0, then A has no bounded approximate unit (cf. the paragraph immediately before Lemma 3.5.21). Proof Assertion (i) follows from Fact 8.4.26 and Proposition 8.4.27. Suppose that A = 0 and that A has an approximate unit bounded by M > 0. Then we have BA ⊆ M(BA )2 , and hence MBA is a nonzero bounded idempotent subset of A, contradicting (i). This proves assertion (ii).  Let A be an algebra over K. According to the paragraph immediately before 1 Definition 2.2.7, we write Asym instead of A( 2 ) . Now let S be a subset of A. Given n ∈ N, the set Sn depends on the product of A. Therefore we write (S• )n instead of Sn when S is regarded as a subset of Asym . Analogously, when A is normed and S is bounded, we write r(S• ) to denote the spectral radius of S considered as a subset of Asym .

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Selected topics in the theory of non-associative normed algebras

Corollary 8.4.47 Let A be a normed algebra over K. Then we have:   (i) For λ in K \ 12 , A is topologically nilpotent if and only if so is A(λ) . (ii) If A is topologically nilpotent, then so is Asym . (iii) There are choices of A such that Asym is topologically nilpotent, but A is not topologically nilpotent. Proof Assertion (i) follows straightforwardly from Lemma 8.4.32 and the equivalence (i)⇔(iii) in Theorem 8.4.44. Let S be any subset of A, and let n be in N. An easy induction argument shows that every element of (S• )n becomes a convex combination of elements of Sn . Therefore, if S is bounded, then we get (S• )n  ≤ Sn , which, in view of the arbitrariness of n, implies r(S• ) ≤ r(S). When the last inequality applies with S := BA , assertion (ii) follows. To prove (iii), take A equal to the two-dimensional anticommutative normed algebra in the last paragraph of the proof of Theorem 8.4.34. We know that β(A) = +∞, so that, by Theorem 8.4.44, A is not topologically nilpotent. However, since A is  anticommutative, Asym is nilpotent of index 2. Lemma 8.4.48 Let A be an associative algebra over K, let S be a multiplicatively closed subset of A, and let n be in N. Then every element in S2n−1 can be written as a sum of 3n−1 terms, each of which is the product of ±1 by an element of (S• )n . Proof The assertion in the lemma is obviously true in the case that n = 1. Assume inductively that the assertion in the lemma is true for a given n ∈ N. Let w be in S2n+1 , say w = x1 · · · x2n+1 with x1 , . . . , x2n+1 ∈ S. Then, denoting by • the product of Asym , applying the magic (and, on the other hand, obvious) formula x1 · · · x2n+1 = x1 • (x2 · · · x2n+1 ) − x2 • (x3 · · · x2n+1 x1 ) + (x1 x2 ) • (x3 · · · x2n+1 ), and keeping in mind that S is multiplicatively closed in A, we deduce that w = y1 • z1 − y2 • z2 + y3 • z3 , where y1 := x1 , y2 := x2 , y3 := x1 x2 are elements of S, and z1 := (x2 x3 )x4 · · · x2n+1 , z2 := x3 · · · x2n (x2n+1 x1 ), z3 := x3 · · · x2n+1 lie in S2n−1 . By the induction hypothesis, z1 (and analogously z2 and z3 ) can be written as a sum of 3n−1 terms, each of which is the product of ±1 by an element of (S• )n . Therefore w can be written as a sum of 3n terms, each of which is the product of ±1 by an element of (S• )n+1 . In this way, we have proved that the assertion in the lemma is true with n + 1 instead of the given n.  Theorem 8.4.49 Let A be a normed associative algebra over K, and let S be a multiplicatively closed subset of A. Then S is quasi-nilpotent in A if and only if it is quasi-nilpotent in Asym . Proof The ‘only if’ part follows from the inequality r(S• ) ≤ r(S), which we showed in the proof of Corollary 8.4.47, even without the assumptions that A is

8.4 The joint spectral radius of a bounded set

623

associative and that S is multiplicatively closed. Let n be in N. Then, by Lemma 8.4.48, we have S2n−1  ≤ 3n−1 (S• )n . Passing to n-th roots, keeping in mind that 1 r(S) = limn→∞ Sn  n (by Lemma 8.4.12), and taking sup limit as n → ∞, we derive r(S)2 ≤ 3 r(S• ). Now, the ‘if’ part follows from the last inequality.  By taking S := BA in the above theorem, we get the following. Corollary 8.4.50 Let A be a normed associative algebra over K. Then A is topologically nilpotent if and only if so is the normed Jordan algebra Asym . Theorem 8.4.51 Let A be a normed algebra over K, and put R(A) := {r(B(A,|||·|||) ) : ||| · ||| ∈ En(A)}. Then either R(A) = {0} or ]0, 1[⊆ R(A) ⊆]0, 1], depending on whether or not A is topologically nilpotent. Proof It is clear that R(A) = {0} if and only if A is topologically nilpotent. Now observe that ]0, 1]r ⊆ R(A) whenever r is in R(A). Indeed, if r is equal to r(B(A,|||·|||) ) for some ||| · ||| ∈ En(A), and if 0 < ε ≤ 1, then 1ε ||| · ||| lies in En(A) and εr = r(B(A, 1 ||| · |||) ) ∈ R(A). It follows from this observation that, to conclude ε the proof, it is enough to show that, if A is not topologically nilpotent, then ρ := sup R(A) = 1. Let ε > 0. By Corollary 8.4.21, there exists an algebra norm ||| · ||| on A satisfying 1k  ·  ≤ ||| · ||| ≤ (r(BA ) + ε) ·  for some positive number k. Now, let n be in N. For any product (say a) of n elements of BA (say a1 , . . . , an ), no matter how associated, we have a ≤ k|||a||| ≤ kPn |||a1 ||| · · · |||an ||| ≤ kPn (r(BA ) + ε)n , where Pn := |||(B(A,|||·|||) )n |||. If follows that Mn ≤ kPn (r(BA ) + ε)n , where Mn := (BA )n . By passing to n-th roots, and taking sup limit as n → ∞, we derive that r(BA ) ≤ r(B(A,|||·|||) )(r(BA ) + ε), and hence that r(BA ) ≤ ρ(r(BA ) + ε). By letting ε → 0, we obtain r(BA ) ≤ ρ r(BA ). Now, if A is not topologically nilpotent, then the above implies ρ ≥ 1. But the inequality ρ ≤ 1 is obvious.  Remark 8.4.52 Let A be a normed algebra. Of course, the situation R(A) = {0} occurs if and only if A is topologically nilpotent. On the other hand, the situation R(A) =]0, 1] occurs for example in the case that A has a unit 1 because then there exists ||| · ||| ∈ En(A) such that ||| 1 ||| = 1 (cf. §8.4.9). The problem if R(A) =]0, 1] whenever A is not topologically nilpotent [928, p. 275] seems to remain open to date, even when A is associative, complex, and complete. By looking at the proof of Theorem 8.4.51, we realize that R(A) =]0, 1] whenever A is not topologically nilpotent, and there exists ||| · ||| ∈ En(A) such that ||| · ||| ≤ r(BA ) · . §8.4.53 Let A be a normed algebra over K, and let I be a closed ideal of A. Then the inequality r(BA/I ) ≤ r(BA ) holds (indeed, consider the natural quotient mapping  : A → A/I, note that (BA ) is contained in BA/I and contains the open unit ball of A/I, and apply Fact 8.4.11 and Lemma 8.4.23). Therefore, if A is topologically nilpotent, then so is A/I for every closed ideal I of A.

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Selected topics in the theory of non-associative normed algebras

Let A be an algebra over K. We recall that the annihilator of A was defined by the equality Ann(A) := {a ∈ A : aA = Aa = 0} (cf. Definition 1.1.10). Clearly, Ann(A) is an ideal of A, and, in the case that A is normed, Ann(A) is closed in A. Proposition 8.4.54 Let A be a normed algebra over K. Then: (i) β(A) ≤ β(A/Ann(A)). (ii) A is topologically nilpotent if and only if so is A/Ann(A). Proof Denoting by π : A → A/Ann(A) the natural surjection, for a, b ∈ A and x, y ∈ Ann(A), we have ab = (a + x)(b + y), and hence ab ≤ a + xb + y, which implies ab ≤ π(a)π(b).

(8.4.11)

To prove (i) it is enough to show that, if A/Ann(A) satisfies the norm-k boundedness property for some k > 0, then A satisfies the norm-k boundedness property. Let k > 0 be such that A/Ann(A) satisfies the norm-k boundedness property, and let S be a bounded and multiplicatively closed subset of A. Then π(S) is a bounded and multiplicatively closed subset of A/Ann(A), so that there exists | · | ∈ En(A/Ann(A)) such that |π(S)| ≤ k. Taking δ > 0 such that δ ·  ≤ | · | on A/Ann(A) and δ 2  ·  ≤ k on S, putting |||a||| := max{δ 2 a, |π(a)|} for every a ∈ A, and keeping in mind (8.4.11), we easily realize that ||| · ||| ∈ En(A) and that |||S||| ≤ k. Therefore A satisfies the norm-k boundedness property, as desired. The ‘if’ part of (ii) follows from (i) and Theorem 8.4.44, whereas the ‘only if’ part of (ii) follows from §8.4.53.  Now we are going to involve completeness in our development. Theorem 8.4.55 Let A be a complete normed algebra over K. Then A is topologically nilpotent if (and only if) it is generated as an algebra by some of its quasi-nilpotent subsets. Proof Suppose that there exists a quasi-nilpotent subset S of A which generates A as an algebra. Keeping in mind Proposition 8.4.17 and Lemma 8.4.23, the quasinilpotency of S implies that |co|(MC(S)) is a quasi-nilpotent subset of A. On the other hand, the fact that A is generated by S implies that |co|(MC(S)) is absorbent in A. It follows that |co|(MC(S)) is a quasi-nilpotent ‘barrel’ in A. Here by a barrel we mean an absorbent, closed, and absolutely convex subset. Since barrels in a Banach space are neighbourhoods of zero, we conclude that A is topologically nilpotent, as desired. 

8.4 The joint spectral radius of a bounded set

625

We will realize later (see Example 8.4.75) that the above theorem does not remain true if we replace ‘generated as an algebra’ with ‘generated as a normed algebra’. Proposition 8.4.56 Let A be a nonzero topologically nilpotent complete normed algebra over K. Then the linear hull of A2 cannot be equal to A. Proof Assume to the contrary that the linear hull of A2 is equal to A. Then the set |co|((BA )2 ) is absorbent in A, and hence |co|((BA )2 ) is a barrel in A. Therefore there exists δ > 0 such that δBA ⊆ |co|((BA )2 ). Now 1δ BA is a nonzero bounded idempotent subset of A, which is impossible in view of Proposition 8.4.46(i).  The following lemma follows straightforwardly from Proposition 8.4.36. Lemma 8.4.57 Let A be a normed algebra over K satisfying the multiplicative boundedness property, let S be a quasi-nilpotent subset of A, and let ε > 0. Then there exists an equivalent algebra norm ||| · ||| on A such that ||| S ||| ≤ ε. Let A be an algebra over K. A subalgebra B of A is called a quasi-full subalgebra of A if, whenever b is in B, a is in A, and a + b − ab = b + a − ba = 0, we have a ∈ B. Since the intersection of quasi-full subalgebras of A is another quasi-full subalgebra of A, it follows that, for any non-empty subset S of A, there is a smallest quasifull subalgebra of A which contains S. This subalgebra will be called the quasi-full subalgebra of A generated by S. Suppose that A is associative. Then, for x ∈ A there exists at most one y ∈ A such that x + y − xy = y + x − yx = 0 and, if such a y does exist, then we say that x is quasi-invertible in A, and that y is the quasi-inverse of x (cf. Definition 3.6.19). Thus quasi-full subalgebras of A in the meaning defined above coincide with quasi-full subalgebras of A as introduced in §3.6.41. A normed algebra over K is said to be quasi-nil if all its elements are quasinilpotent. We recall that normed radical associative algebras over K are quasi-nil (cf. Corollary 3.6.23) and that, more generally, normed J-radical non-commutative Jordan algebras over K are quasi-nil (cf. Lemma 4.4.26). Lemma 8.4.58 Let A be a normed algebra over K satisfying the multiplicative boundedness property, and let S be a quasi-nilpotent subset of A. Then the quasifull subalgebra of A generated by S is quasi-nil. Proof By Lemma 8.4.57, for each n ∈ N there exists  · n ∈ En(A) such that Sn ≤ 1n . Now, consider the set B of those elements x ∈ A such that limn→∞ xn = 0. Clearly, B becomes a subalgebra of A containing S. Moreover, if b is in B, if a is in A, and if b + a − ba = a + b − ab = 0, then we have an ≤

bn 1 − bn

for n big enough, which implies that a lies in B. Therefore B is a quasi-full subalgebra of A. On the other hand, the inequality r(a) ≤ an holds for all a ∈ A and n ∈ N, and implies r(b) = 0 whenever b is in B (i.e. B is quasi-nil). It follows that the quasi-full subalgebra of A generated by S is quasi-nil. 

626

Selected topics in the theory of non-associative normed algebras

Proposition 8.4.59 Let A be a normed algebra over K satisfying the multiplicative boundedness property, and let S be a quasi-nilpotent subset of A. Then the quasi-full subalgebra of BL(A) generated by LS ∪ RS is quasi-nil. Proof Keeping in mind Lemma 8.4.58 and the fact that normed associative algebras fulfil the NBP, it is enough to show that LS ∪ RS is a quasi-nilpotent subset of BL(A). But this follows from the assumptions that A satisfies the multiplicative boundedness property and that S is quasi-nilpotent, by applying Proposition 8.4.39.  Given an algebra A and a subalgebra B of A, we denote by LBA (respectively, RAB ) the set of all left (respectively, right) multiplication operators on A by elements of B (cf. §8.4.38). Corollary 8.4.60 Let A be a normed algebra over K satisfying the multiplicative boundedness property, and let B be a topologically nilpotent subalgebra of A. Then the quasi-full subalgebra of BL(A) generated by LBA ∪ RAB is quasi-nil. Proof Clearly, the quasi-full subalgebra of BL(A) generated by LBA ∪ RAB coincides A ∪ RA . Now, since B with the quasi-full subalgebra of BL(A) generated by LB B BB B is quasi-nilpotent (because B is supposed to be topologically nilpotent) and A is supposed to fulfil the multiplicative boundedness property, the result follows by  applying Proposition 8.4.59 with BB instead S. Invoking Corollary 8.4.45, we derive the following. Corollary 8.4.61 Let A be a topologically nilpotent normed algebra over K. Then the quasi-full subalgebra of BL(A) generated by LA ∪ RA is quasi-nil. From now on, it is convenient to recall that the (Jacobson) radical of an associative algebra A can be characterized as the largest quasi-invertible ideal of A (cf. Theorem 3.6.21), so that an associative algebra is a radical algebra if and only if all its elements are quasi-invertible. Lemma 8.4.62 Let A be a complete normed associative algebra over K, let S be a quasi-nilpotent subset of A. Then the quasi-full subalgebra of A generated by S is a radical algebra. Proof Let C stand for the quasi-full subalgebra of A generated by S. Since normed associative algebras satisfy the NBP, we can apply Lemma 8.4.58 to get that every element of C is quasi-nilpotent. Then, by Lemma 1.1.20, every element of C has a quasi-inverse in A. Finally, since C is a quasi-full subalgebra of A, every element of C has a quasi-inverse in C, i.e. C is a radical algebra, as desired.  Let A be an algebra over K, let B be a quasi-full subalgebra of A, and let S be any subset of B. Keeping in mind that ‘to be a quasi-full subalgebra of’ is a transitive relation, and that B remains quasi-full in any subalgebra of A containing B, it is easily realized that the quasi-full subalgebra of A generated by S is equal to the quasi-full subalgebra of B generated by S.

8.4 The joint spectral radius of a bounded set

627

As usual in our work, for every vector space X over K, we denote by L(X) the associative algebra over K of all linear operators on X. Given an algebra A over K, and a subset S of A, we denote by QFM A (S) the quasi-full subalgebra of L(A) generated by LS ∪ RS . Now notice that, if X is a Banach space over K, then, by the Banach isomorphism theorem, BL(X) becomes a quasi-full subalgebra of L(X). It follows that, if A is a complete normed algebra over K, and if S is any subset of A, then QFM A (S) is contained in BL(A). More precisely, by the paragraph immediately above, QFM A (S) is equal to the quasi-full subalgebra of BL(A) generated by LS ∪ RS . In the case that S = A, all these facts were already noticed in Remark 4.4.41. Theorem 8.4.63 Let A be a complete normed algebra satisfying the multiplicative boundedness property, and let S be a quasi-nilpotent subset of A. Then QFM A (S) is a radical algebra. Proof Keeping in mind Lemma 8.4.62 and the comments immediately above, it is enough to show that LS ∪ RS is a quasi-nilpotent subset of BL(A). But this follows from the assumptions that A satisfies the multiplicative boundedness property and that S is quasi-nilpotent, by applying Proposition 8.4.39.  By applying the above Theorem with BB instead of S, we derive the following. Corollary 8.4.64 Let A be a complete normed algebra over K satisfying the multiplicative boundedness property, and let B be a topologically nilpotent subalgebra of A. Then QFM A (B) is a radical algebra. Let A be an algebra over K. In Definition 4.4.39, we introduced the so-called quasifull multiplication algebra QFM (A) := QFM A (A) of A, and defined the weak radical w-Rad(A) of A as the largest QFM (A)-invariant subspace of A contained in the subspace {a ∈ A : {La , Ra } ⊆ Rad(QFM (A))}, where Rad(·) means Jacobson radical. There, we were interested in the case that w-Rad(A) = 0 because then A has at most one complete algebra norm topology (cf. Theorem 4.4.43). Here, we focus in the completely opposite situation, namely that A = w-Rad(A). To this end, the following straightforward fact will be useful. Fact 8.4.65 Let A be an algebra over K. Then A = w-Rad(A) if and only if QFM (A) is a radical algebra. Now, invoking Corollaries 8.4.45 and 8.4.64, and Fact 8.4.65, we derive the following. Corollary 8.4.66 Let A be a topologically nilpotent complete normed algebra over K. Then A = w-Rad(A). Remark 8.4.67 Let A be a topologically nilpotent complete normed algebra over K. It follows from Corollary 8.4.66 and Proposition 4.4.59 that A = Rad(A), meaning

628

Selected topics in the theory of non-associative normed algebras

that A has no proper modular left ideal (cf. Definition 3.6.12), and that A = s-Rad(A), meaning that A has no proper modular ideal (cf. Definition 3.6.6). Moreover, if in addition A is Jordan-admissible, then A = J-Rad(A), meaning that every element of A is quasi-J-invertible in A (cf. Definition 4.4.12). This last result can be ostensibly refined. Indeed, as we will point out in Fact 8.4.68, the assumption that A is complete normed can be relaxed to the one that A is a (Jordan-admissible) normed Q-algebra, and the assumption that A is topologically nilpotent can be relaxed to the one that A is quasi-nil. Fact 8.4.68 Let A be a quasi-nil (Jordan-admissible) normed Q-algebra over K. Then A = J-Rad(A). Proof Let a be in A. By Proposition 8.4.15(i), we have s(a) = 0. Then, by Lemma 4.4.21(iii), a is quasi-J-invertible in A. The result now follows from the arbitrariness of a ∈ A.  As a consequence, we re-encounter the fact that quasi-nil associative normed Q-algebras are radical (cf. Corollary 3.6.46). The following corollary to Theorem 8.4.63 has its own interest. Indeed, it shows that the multiplicative boundedness property has relevant consequences of a purely algebraic nature. Corollary 8.4.69 Let A be a complete normed algebra over K satisfying the multiplicative boundedness property. Then QFM A (S) is a radical algebra whenever S is any nilpotent subset of A. Proof Let S be a nilpotent subset of A. If S is bounded, then, by Theorem 8.4.63, QFM A (S) is a radical algebra. Otherwise,   s T := : s ∈ S \ {0} s is a bounded nilpotent subset of A with QFM A (T) = QFM A (S).



Theorem 8.4.70 Let A be a normed complex algebra, and let S be a bounded, complete, and absolutely convex subset of A. Then there exists a sequence (sn , fn ), with sn ∈ S and fn ∈ Wn , such that 1

lim sup fn (s1 , . . . , sn ) n = r(S). n→∞

Proof

It is enough to show that the inequality 1

lim sup fn (s1 , . . . , sn ) n ≥ r(S)

(8.4.12)

n→∞

holds for some sequence {(sn , fn )}, with sn ∈ S and fn ∈ Wn . In its turn, since (8.4.12) holds whenever r(S) = 0, we may suppose that r(S) > 0.

8.4 The joint spectral radius of a bounded set

629

We consider the topological space X := S N , which turns out to be completely metrizable (because S is complete) under the distance d given by ∞

d((si ), (ti )) =

1  1 si − ti , M 2i+1 i=1

where M := max{S, 1}. For each δ ∈]0, 1[ and k ∈ N, let Xk,δ stand for the set of those sequences (si ) ∈ X such that there exists n ≥ k such that 1

max{f (s1 , . . . , sn ) n : f ∈ Wn } > (1 − δ)r(S). It is easily realized that Xk,δ is an open subset of X. Suppose that there exists (si ) ∈ ∩∞ k=2 Xk, 1k . For each i ∈ N, choose fi ∈ Wi such that fi (s1 , . . . , si ) = max{f (s1 , . . . , si ) : f ∈ Wi }. Then, for every k ∈ N\{1}, there exists n ≥ k satisfying   1 1 n fn (s1 , . . . , sn ) > 1 − r(S), k 1

which implies that lim supn→∞ fn (s1 , . . . , sn ) n ≥ r(S), and hence that the sequence {(sn , fn )} satisfies the requirements in the conclusion of the theorem. Therefore, our desired conclusion will follow from the statement ∩∞ k=2 Xk, 1k = ∅, which, in its turn, follows from Baire’s category theorem if we show that each of the sets Xk,δ is dense in X.  Now, since X = 0≤ρ 1 − δ r(S). Keeping in mind the definition of 2 > Sn , the last inequality implies the existence of u1 , . . . , un ∈ S and fn ∈ Fn with 2πir n fn (u1 , . . . , un ) ≥ (1 − δ) 2 r(S)n . Let εr = e m ε (1 ≤ r ≤ m) and ε0 = 0. Set zr := fn ((x1 + εr u1 ), . . . , (xm + εr um ), um+1 , . . . , un ).  m m q m Then, keeping in mind that m r=1 εr = 0 for q = 1, . . . , m − 1, and r=1 εr = mε , m m we obtain r=1 (zr − z0 ) = mε fn (u1 , . . . , un ), and hence 2m max{z0 , z1 , . . . , zm } ≥ 

m 

(zr − z0 )

r=1 n

= mεm fn (u1 , . . . , un ) ≥ mεm (1 − δ) 2 r(S)n , so that there exist p ∈ {0, 1, . . . , m} such that zp  ≥

n εm (1 − δ) 2 r(S)n . 2

630

Selected topics in the theory of non-associative normed algebras

For i ∈ N, set

⎧ xi + εp ui ⎪ ⎪ ⎨ yi = ui ⎪ ⎪ ⎩ xi

(1 ≤ i ≤ m) (m < i ≤ n) (i > n).

Note that, if 1 ≤ i ≤ m, then xi + εp ui ∈ S because xi ∈ ρS, ui ∈ S, |ρ| + |εp | ≤ ρ + ε ≤ 1, and S is absolutely convex. Therefore y := (yi ) lies in X. On the other hand, we have m ∞ 1  1 1  1 xi − yi  + xi − yi  d(x, y) = d((xi ), (yi )) = M M 2i+1 2i+1 i=1

i=m+1

m ∞ 1  1 1 ε 1 1  1 ε + 2M < ≤ ε. + ≤ M M M 2 2m 2i+1 2i+1 i=1

i=m+1

Finally, 1

1

max{f (y1 , . . . , yn ) n : f ∈ Wn } ≥ fn (y1 , . . . , yn ) n @ m 1 1 n ε = zp  n ≥ (1 − δ) 2 r(S) > (1 − δ)r(S), 2 

that is, y lies in Xk,δ . As an immediate consequence of the above theorem, we obtain the following.

Corollary 8.4.71 Let A be a normed associative complex algebra, and let S be a bounded, complete, and absolutely convex subset of A. Then there exists a sequence sn in S, such that 1

lim sup s1 · · · sn  n = r(S). n→∞

The main consequence of Theorem 8.4.70 is the following. Corollary 8.4.72 Let A be a complete normed complex algebra. Then the following statements are equivalent: (i) A is topologically nilpotent. (ii) For any sequence (an , fn ), with an ∈ BA and fn ∈ Wn , we have 1

lim fn (a1 , · · · an ) n = 0.

n→∞

By putting together Lemma 8.4.23 and Theorem 8.4.70 we derive the following. Corollary 8.4.73 Let A be a complete normed complex algebra, and let S be a bounded subset of A. Then there exists a sequence (sn , fn ), with sn ∈ |co|(S) and fn ∈ Wn , such that 1

lim sup fn (s1 , · · · sn ) n = r(S). n→∞

8.4 The joint spectral radius of a bounded set

631

Let A be an algebra over K, and let S be a subset of A. Following [822, p. 67], we denote by M A (S) the subalgebra of L(A) generated by LS ∪ RS . Now suppose that A is normed. It follows from Proposition 8.4.59 that, if A fulfils the multiplicative boundedness property, and if S is quasi-nilpotent, then M A (S) is quasi-nil. In what follows, we are going to realize that the result just formulated can be largely improved. Indeed, we are obtaining in Theorem 8.4.79 a strong conclusion under a weaker assumption. To this end, we say that S is finitely quasi-nilpotent if every finite subset of S is quasi-nilpotent. Clearly, topologically nilpotent normed algebras are finitely quasi-nilpotent, and finitely quasi-nilpotent normed algebras are quasi-nil. First of all, we discuss the new notion in the associative setting. Proposition 8.4.74 Let A be a normed associative and commutative algebra over K. Then A is finitely quasi-nilpotent if (and only if) it is quasi-nil. Proof Let S = {s1 , . . . , sm } be a finite subset of A. To prove that S is quasi-nilpotent we may suppose that S ≤ 1. Let n be in N, and let x be in Snm . Then there are k1 , . . . , km ∈ N ∪ {0} such that x = s1 k1 s2 k2 · · · sm km and k1 + · · · + km = nm. Now, clearly, there is j ∈ {1, . . . , m} such that kj ≥ n, hence x ≤ sj kj  = sj kj −n sj n  ≤ sj n  ≤ max{s1 n , . . . , sm n }. Since x is arbitrary in Snm , we derive that Snm  ≤ max{s1 n , . . . , sm n }, hence, invoking Lemma 8.4.12, we have ) + ,* 1 1 1 1 m r(S) = lim Snm  nm ≤ max lim s1 n  n , . . . , lim sm n  n n→∞

n→∞

1 m

= [max{r(s1 ), . . . , r(sm )}] = 0.

n→∞



As the next example shows, finite quasi-nilpotency does not imply topological nilpotency, even in the complete normed associative and commutative case. Example 8.4.75 Let A stand for the complete normed associative and commutative algebra over K of all Lebesgue integrable K-valued functions on [0, 1], with the L1 norm  1 f  := |f (t)|dt, 0

and convolution multiplication defined by  t f (s)g(t − s)ds. ( f ∗ g)(t) := 0

Let B stand for the set of all continuous K-valued functions on [0, 1]. Then, clearly, B is a subalgebra of A. Moreover, if we endow B with the sup norm  · ∞ , then we re-encounter (with a different name) the normed algebra over K in Example 8.4.41.

632

Selected topics in the theory of non-associative normed algebras

Denote by S the closed unit ball of (B,  · ∞ ). Then, since (B,  · ∞ ) is topologically nilpotent, we have r(S) = 0 in (B,  · ∞ ). Therefore, since the inclusion (B,  · ∞ ) → A = (A,  · ) is a continuous algebra homomorphism, it follows from Fact 8.4.11 that r(S) = 0 in A. As a first consequence, since B equals the linear hull of S, and B is dense in A, we realize that A is generated as a normed algebra by a quasi-nilpotent subset, namely S. As another consequence, for every f ∈ B we have r( f ) = 0 in A. Since B is dense in A, and the spectral radius is continuous on A (a fact easily derivable from Corollary 1.1.115), it follows that A is quasi-nil. On the other hand, the sequence un of functions in A defined by $ n if 0 ≤ t < 1n un (t) := 0 if 1n ≤ t ≤ 1 is clearly bounded by 1, and, as we show in what follows, becomes an approximate unit for A. Let f be in B, and let ε > 0. Then the uniform continuity of f provides us with a natural number m such that |f (t) − f (t − s)| < ε for all n ≥ m, t ∈ [0, 1], and s ∈ [0, 1n ].

(8.4.13)

Now let n be arbitrary in N. Then for t ∈ [0, 1n ] we have  t  t (un ∗ f )(t) = un (s)f (t − s)ds = n f (t − s)ds, 0

whereas for t

∈ [ 1n , 1] 

we have 1 n

(un ∗ f )(t) =

0

 un (s)f (t − s)ds +

0

t 1 n



1 n

un (s)f (t − s)ds = n

f (t − s)ds.

0

Therefore  f − un ∗ f  =

1 n

0

 |f (t) − (un ∗ f )(t)|dt +

1 1 n

|f (t) − (un ∗ f )(t)|dt

    1   t  1    n     = f (t − s)ds dt + f (t − s)ds f (t) − n f (t) − n  dt    1   0 0 0 n    1  t  1   1  1  n n n   |f (t)|dt + n |f (t − s)|ds dt + f (t−s)ds dt. ≤ f (t)−n 1   0 0 0 0 

1 n

n

Now we are showing natural bounds for each term in the above sum. Indeed,  1 n 1 |f (t)|dt ≤ f ∞ , n 0   1  t  1 n n 1 n |f (t − s)|ds dt ≤ n f ∞ tdt = f ∞ , 2n 0 0 0

8.4 The joint spectral radius of a bounded set

633

and, for n ≥ m,      1   1  1  1  1    n n n    f (t − s)ds dt = f (t)ds − n f (t − s)ds dt n f (t) − n  1  1  0 0 0  n n      1   1  1  1 n   n ( f (t) − f (t − s))ds dt ≤ n |f (t) − f (t − s)|ds dt =n  1  1  0 0 n n   1 1 ε < ε, ≤ n 1− n n 3 where we have applied (8.4.13). Therefore f − un ∗ f  < 2n f ∞ + ε whenever n ≥ m, and therefore lim supn→∞ f − un ∗ f  ≤ ε. By letting ε → 0, we get limn→∞ un ∗ f = f . Since f is arbitrary in B, and B is dense in A, and the sequence un is bounded, it follows that un is an approximate unit for A. Summarizing, A becomes a quasi-nil complete normed associative and commutative algebra, is generated as a normed algebra by a quasi-nilpotent subset, and has a bounded approximate unit. Consequently, by Propositions 8.4.74 and 8.4.46(ii), A is finitely quasi-nilpotent but not topologically nilpotent. Moreover, since A contains B as a subalgebra, and B is non-nil, we realize that A is non-nil.

Let A be a normed algebra over K, and let n be a natural number. We say that A is n-quasi-nilpotent if all subsets of A with cardinality ≤ n are quasi-nilpotent. Commutativity cannot be removed in Proposition 8.4.74. Indeed, we have the following. Theorem 8.4.76 For each n ∈ N there exists an n-quasi-nilpotent normed associative algebra over K which is not (n + 1)-quasi-nilpotent. Proof Let us fix n ∈ N. By a refinement of the famous Golod result [295] (see [677, pp. 60–6] or [763, Example 23.2.5]), there is a non-nilpotent associative algebra B over K generated by n + 1 elements and such that every subalgebra of B generated by n elements is nilpotent. Moreover, this algebra can be materialized as the quotient B = A/I, where A is the free associative algebra over K on n + 1 indeterminates x0 , . . . , xn , and I is the ideal of A generated by a suitable set of homogeneous polynomials (= linear combinations of associative words of the same degree). Let Am stand for the subspace of A consisting of all homogeneous polynomials of degree m and put Im := I ∩ Am , for every m ∈ N. Then we have A = ⊕m∈N Am . Moreover, denoting by Pm the projection from A onto Am corresponding to the above decomposition, it follows from the homogeneity of the generators of I that Pm (I) ⊆ I. Now let us convert A into a normed algebra over K by supplying it with   the 1 -norm, namely i λi wi := i |λi | for each finite linear combination of pairwise different associative words wi in the indeterminates. We note that, for each m ∈ N, the projection Pm is contractive. Let (zk )k∈N be a sequence in I converging to some a ∈ A, and let m be in N. Then Pm (zk ) ∈ Im for every k ∈ N and limk→∞ Pm (zk ) = Pm (a), hence Pm (a) ∈ Im in virtue of the finite-dimensionality

634

Selected topics in the theory of non-associative normed algebras

of Im . Therefore, by the arbitrariness of m ∈ N, we realize that a lies in I. Thus we have proved that I is a closed ideal of A. So B is a normed algebra, and r (T) = 0 for every T = {b1 , . . . , bn } ⊆ B (since T generates a nilpotent subalgebra). Therefore B is an n-quasi-nilpotent algebra. Set X := {x0 , . . . , xn } ⊆ A and S := {b0 , . . . , bn } ⊆ B, where bi := xi + I ∈ B for each i ∈ {0, . . . , n}, and let m be in N. The contractiveness of Pm and the fact that Pm (I) ⊆ I imply that x + I = x + Im  for every x ∈ Am . Therefore one may calculate Sm  as max{x + Im  : x ∈ X m }. We claim that Sm  = 1. Assume, to the contrary, that x + Im  < 1 for every x ∈ X m . Then for each x ∈ X m there is f (x) ∈ Im such that  x + f (x)  < 1. Let x∈X m μ(x) f (x) = 0 for some numbers μ(x) ∈ K. Then  x∈X m

|μ(x) | =

 x∈X m

μ(x) x +

 x∈X m

μ(x) f (x) ≤



|μ(x) |x + f (x) .

x∈X m

If any of the numbers μ(x) are different from zero then we receive a contradiction. Therefore the system of all f (x) is linearly independent. Since Am equals the linear hull of X m , it follows that the system of all f (x) is a basis of Am . But then Am ⊆ I, so Am ⊆ I, hence B = A/I is nilpotent, again a contradiction. Thus Sm  = 1 for every m ∈ N whence r (S) = 1. Therefore B is not (n + 1)quasi-nilpotent.  Now, retaking an old convention in our work, given an algebra A over K and a subset S of A, we denote by A(S) the subalgebra of A generated by S. Fact 8.4.77 Let A be an algebra over K, let S be a subset of A, and let F (S) stand for the family of all finite subsets of S. Then:  (i) A(S) = T∈F (S) A(T). (ii) Every finite subset of A(S) is contained in A(R) for a suitable R ∈ F (S). Proof Keeping in mind that F (S) (ordered by inclusion) is a directed set, and that the mapping T → A(T) from F (S) to the family of all subsets of A preserves  inclusion, we see that B := T∈F (S) A(T) is a subalgebra of A. Since B contains S and is contained in A(S), assertion (i) follows. Let {x1 , . . . , xm } be any finite subset of A(S). By assertion (i), for each i ∈ {1, . . . , m}  there exists Ti ∈ F (S) with xi ∈ A(Ti ). By putting R := m i=1 Ti ∈ F (S), we have {x1 , . . . , xm } ⊆ A(R).  Lemma 8.4.78 Let A be a normed algebra over K, and let S be a finitely quasinilpotent subset of A. Then A(S) is finitely quasi-nilpotent. Proof Let T be any finite subset of A(S). We must show that T is quasi-nilpotent. By Fact 8.4.77(ii), T is contained in A(R) for a suitable R ∈ F (S). Now, since S is finitely quasi-nilpotent, R is quasi-nilpotent. Therefore, by Proposition 8.4.17 and Lemma 8.4.23, |co| [MC(R)] is also quasi-nilpotent. Since

8.4 The joint spectral radius of a bounded set

635



A(R) = t>0 t |co| [MC(R)], and T is a finite subset of A(R), we have T ⊆ t |co| [MC(R)] for some t > 0, hence T is quasi-nilpotent.  Theorem 8.4.79 Let A be a normed algebra over K satisfying the multiplicative boundedness property, and let S be a finitely quasi-nilpotent subset of A. Then M A (S) is finitely quasi-nilpotent. Proof Keeping in mind Lemma 8.4.78, it is enough to show that LS ∪ RS is a finitely quasi-nilpotent subset of BL(A). But this follows from the assumptions that A satisfies the multiplicative boundedness property and that S is finitely quasi-nilpotent, by applying Proposition 8.4.39.  For any algebra A over K, write M  (A) := M A (A), which is nothing other than the classical multiplication ideal of A, already introduced in §3.6.53. It follows from Corollary 8.4.61 that, if A is a topologically nilpotent normed algebra, then M  (A) is quasi-nil. A better result, which follows straightforwardly from Theorem 8.4.79, is the following. Corollary 8.4.80 Let A be a finitely quasi-nilpotent normed algebra over K satisfying the multiplicative boundedness property. Then M  (A) is finitely quasi-nilpotent. Remark 8.4.81 For a normed algebra A over K, consider the following conditions: (i) A is topologically nilpotent. (ii) A is finitely quasi-nilpotent and satisfies the multiplicative boundedness property. Then (i)⇒(ii). Indeed, that topological nilpotency implies finite quasi-nilpotency is clear, whereas that topological nilpotency implies the multiplicative boundedness property follows from Corollary 8.4.45. Actually, keeping in mind Theorem 8.4.1 and Example 8.4.75, we realize that (ii) is strictly weaker than (i). Concerning Condition (ii), we will see later that finite quasi-nilpotency does not imply the multiplicative boundedness property (see Corollary 8.4.116). Let A be an associative algebra. Then M  (A) contains three distinguished subalgebras, namely LA , RA , and the algebra E (A) of the so-called elementary operators on A (i.e. the operators of the form a→

n 

bi aci

i=1

for suitable n-tuples (b1 , . . . , bn ) and (c1 , . . . , cn ) of elements of A) [919]. It is easily realized that M  (A) = LA + RA + E (A). Corollary 8.4.82 Let A be a normed associative algebra over K. Then the following conditions are equivalent: (i) A is finitely quasi-nilpotent. (ii) LA is finitely quasi-nilpotent.

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Selected topics in the theory of non-associative normed algebras

(iii) RA is finitely quasi-nilpotent. (iv) E (A) is finitely quasi-nilpotent. (v) M  (A) is finitely quasi-nilpotent. Proof Since normed associative algebras satisfy the NBP, the implication (i)⇒(v) follows from Corollary 8.4.80. On the other hand, the implications (v)⇒(ii), (v)⇒(iii), and (v)⇒(iv) are clear. (ii)⇒(i) (respectively, (iii)⇒(i) or (iv)⇒(i)) Let S be any finite subset of A. Then LS (respectively, RS or LS RS ) is a finite subset of LA (respectively, RA or E (A)). It follows from the assumption (ii) (respectively, (iii) or (iv)) and Corollary 8.4.40 that S is quasi-nilpotent.  In relation to the above corollary, the following problem seems to remain open. Problem 8.4.83 Is every complete normed radical associative complex algebra finitely quasi-nilpotent? Of course, the above problem has interest only in the non-commutative case (compare Proposition 8.4.74). If Problem 8.4.83 had an affirmative answer, then, as a consequence of Corollary 8.4.82, we would be provided also with an affirmative answer to the following. Problem 8.4.84 Let A be a complete normed radical associative complex algebra. Is M  (A) a quasi-nil algebra? Lemma 8.4.85 Let A be a normed non-commutative Jordan algebra over K, and let a be an algebraic quasi-nilpotent element of A. Then a is nilpotent. Proof In view of Proposition 4.1.86, we may suppose that A is associative, commutative, and finite-dimensional. Certainly, there exist n ∈ N and p ∈ K[x] such that an [1 − ap(a)] = 0. By Corollary 1.1.115, we have r(ap(a)) ≤ r(a)r( p(a)) = 0 < 1. Therefore, by Lemma 1.1.20, 1 − ap(a) is invertible in A1 , hence an = 0.  Corollary 8.4.86 Let A be a normed algebra over K satisfying the multiplicative boundedness property, and let S be a finitely quasi-nilpotent subset of A such that M A (S) is algebraic of bounded degree. Then M A (S) is nilpotent. Proof Since A is supposed to fulfil the multiplicative boundedness property, and S is supposed to be finitely quasi-nilpotent, Theorem 8.4.79 applies, giving as a consequence that M A (S) is quasi-nil. Since we also suppose that M A (S) is algebraic, it follows from Lemma 8.4.85 that M A (S) is a nil algebra. Now, the assumption that M A (S) is algebraic of bounded degree reads as that M A (S) is a nil algebra of bounded index. By the Nagata–Higman theorem (see for example [822, Corol lary 6.1]), M A (S) is nilpotent. Now we need to invoke the following. Fact 8.4.87 Let A be an algebra over K. Then A is nilpotent if and only if M  (A) is nilpotent.

8.4 The joint spectral radius of a bounded set

637

Proof Suppose that A is nilpotent, say An = 0 for some n ≥ 2. Then Am = 0 for every m ≥ n. Let T be in [M  (A)]n−1 . Then T is a sum of terms each of which is a product of at least n − 1 linear operators Si , each Si being either Lai or Rai (ai ∈ A). Hence, for x ∈ A, T(x) is a sum of terms each of which lies in Am for a suitable m ≥ n. Hence T(x) = 0 for every x ∈ A. Since T is arbitrary in [M  (A)]n−1 , we see that [M  (A)]n−1 = 0, and M  (A) is indeed nilpotent. n To prove the converse, we claim that for every n ∈ N we have A2 ⊆ [M  (A)]n (A). Since the above inclusion is clearly true for n = 1, assume inductively that it is true n+1 for some n, and let x be in A2 . Then x = yz with y ∈ Ap , z ∈ Aq , p, q ∈ N, and p + q = 2n+1 . If p ≥ 2n , then n

x = Rz (y) ∈ M  (A)(Ap ) ⊆ M  (A)(A2 ) ⊆ [M  (A)]n+1 (A), where we have applied Fact 8.4.6 for the first inclusion. Otherwise, we have q ≥ 2n n+1 and, arguing similarly, we get x ∈ [M  (A)]n+1 (A). Since x is arbitrary in A2 , the claim has been proved. Now suppose that M  (A) is nilpotent, say [M  (A)]n = 0 for n some n. It follows from the claim that A2 = 0, and A is indeed nilpotent.  Corollary 8.4.88 Let A be a normed algebra over K. Then the following conditions are equivalent: (i) A is nilpotent. (ii) A is topologically nilpotent, and M  (A) is algebraic of bounded degree. (iii) A is finitely quasi-nilpotent and has the multiplicative boundedness property, and M  (A) is algebraic of bounded degree. Proof (i)⇒(ii) Keeping in mind the ‘only if’ part of Fact 8.4.87, the assumption (i) implies that M  (A) is nilpotent. But this largely implies that M  (A) is algebraic of bounded degree. (ii)⇒(iii) By Remark 8.4.81. (iii)⇒(i) The assumption (iii) allows us to apply Corollary 8.4.86 to obtain that  M  (A) is nilpotent. Finally, by the ‘if’ part of Fact 8.4.87, A is nilpotent. Let A be a finite-dimensional algebra over K. We know that there are always algebra norms on A, and that all these norms are equivalent. Therefore we will not emphasize that A is normed when we are discussing properties (like the topologically nilpotency, the multiplicative boundedness property, or the quasi-nilpotency of a given subset) which are only of algebraic and topological kind. This convention will be applied without notice in what follows. The following theorem becomes a large non-associative generalization of Lemma 8.4.85. Theorem 8.4.89 Let A be a finite-dimensional algebra over K. Then finitely quasinilpotent subsets of A are nilpotent. Proof Let S be a finitely quasi-nilpotent subset of A. Let B stand for the subalgebra of A generated by S, and let T be a basis of B. The finite dimensionality of A implies that T is a finite subset of B, and hence, by Lemma 8.4.78, that T is quasi-nilpotent.

638

Selected topics in the theory of non-associative normed algebras

By Proposition 8.4.17 and Lemma 8.4.23, |co|(MC(T)) is a quasi-nilpotent bounded subset of B. On the other hand, |co|(MC(T)) is absorbent in B, which, keeping in mind that B is finite-dimensional, implies that |co|(MC(T)) is a neighbourhood of zero in B. It follows that B is topologically nilpotent. By keeping in mind again that B is finite-dimensional, and applying the implication (ii)⇒(i) in Corollary 8.4.88, we derive that B is nilpotent, and hence that S is nilpotent.  After Theorem 8.4.89, we realize that, in the setting of finite-dimensional algebras, all notions related to that of quasi-nilpotent subsets, become in fact purely algebraic notions. In particular we have the following. Corollary 8.4.90 Let A be a finite-dimensional algebra over K. Then the following conditions are equivalent: (i) (ii) (iii) (iv) (v)

A is nilpotent. A is topologically nilpotent. A is finitely quasi-nilpotent. A is n-quasi-nilpotent, where n := dim(A). A is m-quasi-nilpotent, where m stands for the minimum natural number such that A is generated (as an algebra) by a subset of cardinality m.

Proof The implications (i)⇒(ii)⇒(iii)⇒(iv)⇒(v) are clear, whereas the one (v)⇒(i) follows from Theorem 8.4.89 by keeping in mind that nilpotent subsets generate nilpotent algebras. 

8.4.3 Involving nearly absolute-valued algebras Definition 8.4.91 Given a nonzero normed algebra A over K, we denote by η(A) = η(A,  · ) the largest non-negative real number η such that the inequality a2  ≥ ηa2 holds for every a ∈ A. We note that η(A) ≤ 1. A normed algebra A over K is said to satisfy a norm square inequality (in short, NSI) if A = 0 and η(A) > 0. It is straightforward that, if A satisfies a NSI, then so does every equivalent algebra renorming of A. Proposition 8.4.92 Let A be a normed algebra over K satisfying a NSI. Then: (i) For every bounded subset S of A we have S ≤

1 η(A) s(S).

1 , (ii) For every bounded and multiplicatively closed subset S of A we have S ≤ η(A) 1 boundedness property. and hence A satisfies (a strong form of) the norm- η(A) 1 . (iii) 1 ≤ β(A) ≤ η(A) (iv) For every continuous algebra homomorphism  from any normed algebra 1 . over K to A we have  ≤ η(A) 1 ||| · ||| for every algebra norm ||| · ||| on A generating a topology (v)  ·  ≤ η(A) stronger than the natural one.

8.4 The joint spectral radius of a bounded set

639

Proof An easy induction argument shows that, for x ∈ A and n ∈ N ∪ {0}, we have n n x[n]  ≥ [η(A)]2 −1 x2 . Now, let S be a bounded subset of A, let s be in S, and let n n n be in N ∪ {0}. Since s[n] ∈ S[n] , it follows that S[n]  ≥ [η(A)]2 −1 s2 . Keeping in mind the arbitrariness of s ∈ S and of n ∈ N ∪ {0}, assertion (i) follows. Assertion (ii) follows from (i) and Facts 8.4.16 and 8.4.26. 1 in (iii) is a straightforward consequence of (ii). On the The inequality β(A) ≤ η(A) other hand, it follows from (i) that r(BA ) ≥ η(A) > 0, hence β(A) ≥ 1 by Theorems 8.4.34 and 8.4.44, which completes the proof of (iii). Assertion (iv) follows from (i), Fact 8.4.11, the inequality (8.4.4), and Proposition 8.4.15(i). Assertion (v) follows straightforwardly from (iv).  An interesting consequence of assertion (v) in Proposition 8.4.92 is that every normed algebra A over K satisfying a NSI can be equivalently renormed in such a way that it satisfies ‘the best’ possible NSI. Indeed, we have the following. Proposition 8.4.93 Let A be a nonzero normed algebra over K, and put η∗ (A) := sup{η(A, ||| · |||) : ||| · ||| ∈ En(A)}. Then there exists || · || ∈ En(A) such that η(A, || · ||) = η∗ (A). Proof We may suppose that A satisfies a NSI. Take a sequence  · n in En(A) such that lim η(A,  · n ) = η∗ (A),

n→∞

and put k := min{η(A,  · n ) : n ∈ N} > 0. By Proposition 8.4.92(v), for every n ∈ N we have · ≤

1 1 1  · n and  · n ≤  ·  ≤  · . η(A) η(A,  · n ) k

(8.4.14)

Now, take an ultrafilter U on N refining the Fr´echet filter (of all cofinite subsets of N). Since for each a ∈ A the sequence an is bounded (by 1k a), we can define a real-valued function || · || on A by || a || := limU an . It is straightforward that || · || becomes a sub-multiplicative seminorm on A satisfying || a2 || ≥ η∗ (A) || a ||2 for every 1 || · || and || · || ≤ η∗1(A)  · . It a ∈ A. On the other hand, by (8.4.14), we have  ·  ≤ η(A) follows that || · || is an equivalent algebra norm on A such that η(A, || · ||) = η∗ (A).  Let A be a normed algebra over K. We say that A satisfies the norm square equality (in short, NSE) if A = 0 and the equality a2  = a2 holds for every a ∈ A. Relevant examples of normed algebras satisfying the norm square equality are all commutative C∗ -algebras and all JB-algebras ( provided they are nonzero), all absolute-valued algebras over K, and all smooth-normed algebras over K (cf. Subsection 2.6.1). Clearly, the normed algebra A satisfies the NSE if and only if A satisfies a NSI with η(A) = 1. Therefore it is enough to invoke Propositions 8.4.92 and 8.4.93 to get Corollaries 8.4.94 and 8.4.95.

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Selected topics in the theory of non-associative normed algebras

Corollary 8.4.94 Let A be a normed algebra over K satisfying the NSE. Then: (i) For every bounded subset S of A we have S = s(S) = r(S). (ii) For every bounded and multiplicatively closed subset S of A we have S ≤ 1, and hence A satisfies (a strong form of) the NBP. (iii) β(A) = 1. (iv) Continuous algebra homomorphisms from normed algebras over K to A are contractive. (v)  ·  ≤ ||| · ||| for every algebra norm ||| · ||| on A generating a topology stronger than the natural one. Corollary 8.4.95 Let A be a normed algebra over K such that η∗ (A) = 1. Then there exists a unique equivalent algebra norm || · || on A such that (A, || · ||) satisfies the NSE. Remark 8.4.96 It follows from Fact 8.4.29 and the implication (iii)⇒(i) in Theorem 8.4.44 that both the multiplicative boundedness property and the topological nilpotency pass from a normed algebra to any of its subalgebras. Moreover, in view of §8.4.53, the topological nilpotency passes from a normed algebra to any of its quotients. It is noteworthy that the multiplicative boundedness property need not pass from a normed algebra to its quotients. Indeed, if B is any normed algebra over K which does not satisfy the multiplicative boundedness property, then, by Corollary 2.8.20, there exist an absolute-valued algebra A over K and a closed ideal I of A such that B = A/I. Thus A/I does not satisfy the multiplicative boundedness property, whereas, by Corollary 8.4.94(iii), A does. As we did in the third paragraph after Theorem 2.8.88, given a nonzero normed algebra A over K, we denote by ρ(A) = ρ(A,  · ) the largest non-negative real number ρ such that the inequality ab ≥ ρab holds for all a, b ∈ A. We note that ρ(A) ≤ η(A) ≤ 1.

(8.4.15)

We recall that, according to the paragraph immediately before Corollary 2.5.56 (respectively, §2.5.1), a normed algebra A over K is said to be nearly absolute-valued (respectively, absolute-valued) if A = 0 and ρ(A) > 0 (respectively ρ(A) = 1). It is straightforward that, if A is nearly absolute-valued, then so is every equivalent algebra renorming of A. Moreover, if A is nearly absolute-valued (respectively, absolute-valued), then A satisfies a NSI (respectively, the NSE), and therefore the conclusions in Proposition 8.4.92 (respectively, Corollary 8.4.94) hold verbatim for A. In the case that A is nearly absolute-valued, it could be convenient to lose strength in these conclusions. Thus, invoking (8.4.15), we obtain the following. Corollary 8.4.97 Let A be a nearly absolute-valued algebra over K. Then: (i) For every bounded subset S of A we have S ≤

1 ρ(A) s(S).

1 (ii) For every bounded and multiplicatively closed subset S of A we have S ≤ ρ(A) , 1 and hence A satisfies (a strong form of) the norm- ρ(A) boundedness property.

8.4 The joint spectral radius of a bounded set

641

1 (iii) 1 ≤ β(A) ≤ ρ(A) . (iv) For every continuous algebra homomorphism  from any normed algebra 1 . over K to A we have  ≤ ρ(A) 1 ||| · ||| for every algebra norm ||| · ||| on A generating a topology (v)  ·  ≤ ρ(A) stronger than the natural one.

Now, reducing the proof in an obvious way to the case that A is nearly absolutevalued, and arguing as in the proof of Proposition 8.4.93 (with ρ(·) instead of η(·), and Corollary 8.4.97 instead of Proposition 8.4.92), we obtain Corollary 8.4.98 Let A be a normed algebra over K, and put ρ ∗ (A) := sup{ρ(A, ||| · |||) : ||| · ||| ∈ En(A)}. Then there exists || · || ∈ En(A) such that ρ(A, || · ||) = ρ ∗ (A). As a consequence, if ρ ∗ (A) = 1, then there exists a unique equivalent norm || · || on A such that (A, || · ||) becomes an absolute-valued algebra. Fact 8.4.99 Let A stand for either the real algebra H of Hamilton’s quaternions or the complex algebra M2 (C) of all 2 × 2 complex matrices, and let λ be in [1, +∞]. λ+1 Then β(A( 2 ) ) = λ. Proof We know that, denoting by 1 the unit of A, there is a basis {1, u, v, w} of A satisfying u2 = v2 = w2 = −1, uv = −vu = w, vw = −wv = u, and wu = −uw = v (cf. λ+1 §2.5.1). Now, denote by T the operator of left multiplication by u on A( 2 ) . Then we have T 2 (v) = −λ2 v, which implies λ ≤ r(T). This inequality, together with the one λ+1 r(T) ≤ β(A( 2 ) ), obtained by keeping in mind that the spectral radius of u relative to λ+1 λ+1 A( 2 ) equals 1 and applying Corollary 8.4.39, leads to λ ≤ β(A( 2 ) ). On the other hand, since β(A) ≤ 1 (because A is associative), we can apply Lemma 8.4.32 to get λ+1 β(A( 2 ) ) ≤ λ.  Lemma 8.4.100 Let A be an absolute-valued algebra over K, and let λ be in K. Then ρ(A(λ) ) ≥

| |λ| − |1 − λ| | . σλ

Therefore, if (λ) = 12 , then A(λ) is nearly absolute-valued. Proof

Denote by a and a

the product of A(λ) . Then, for all a, b ∈ A we have

b ≥ |λ|ab − |1 − λ|ba = (|λ| − |1 − λ|)ab b ≥ |1 − λ|ba − |λ|ab = (|1 − λ| − |λ|)ab, hence a

b ≥ | |λ| − |1 − λ| | ab.



We recall that every nearly absolute-valued algebra satisfies a NSI. Now, keeping in mind that H is an absolute-valued algebra, we can apply Proposition 8.4.92(iii), Fact 8.4.99, and Lemma 8.4.100, to derive the following variant of Theorem 8.4.34.

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Selected topics in the theory of non-associative normed algebras

Corollary 8.4.101 For every normed algebra A over K satisfying a NSI, we have β(A) ∈ [1, +∞]. Moreover, for every λ ∈ [1, +∞], there exists a four-dimensional nearly absolute-valued non-commutative Jordan real algebra A such that β(A) = λ. In p. 282 of Volume 1 of our work, we found examples of nearly absolute-valued real algebras which are not absolute-valued algebras in any equivalent algebra norm. Now, invoking Corollary 8.4.94(iii), Fact 8.4.99, and Lemma 8.4.100, we are even provided with examples of nearly absolute-valued real algebras which do not satisfy the NSE in any equivalent algebra norm. 8.4.4 Involving tensor products Given normed spaces X and Y over K, a norm  ·  on X ⊗ Y is said to be a cross norm if the equality x ⊗ y = xy holds for every (x, y) ∈ X × Y. Proposition 8.4.102 Let A and B be normed algebras over K, let  ·  be a cross algebra norm on A ⊗ B, and consider A ⊗ B as a normed algebra over K under the norm  ·  . We have: (i) If S and T are bounded subsets of A and B, respectively, then (a) r(S ⊗ T) ≤ r(S)r(T); (b) if in addition A is associative, then r(S ⊗ T) = r(S)r(T). (ii) If A is associative, and if A ⊗ B is topologically nilpotent, then A or B is topologically nilpotent. Proof Let S and T be bounded subsets of A and B, respectively. To prove (i)(a), it is enough to show that, for every n ∈ N, the inequality (S ⊗ T)n  ≤ Sn T n  holds. Let n be in N, and let f be in Wn . Then, for all x1 , . . . , xn ∈ S and y1 , . . . , yn ∈ T, we have f (x1 ⊗ y1 , . . . , xn ⊗ yn ) = f (x1 , . . . , xn ) ⊗ f (y1 , . . . , yn ), which implies f (x1 ⊗ y1 , . . . , xn ⊗ yn ) ≤ Sn T n  because f (x1 , . . . , xn ) and f (y1 , . . . , yn ) lie in Sn and T n , respectively. By the arbitrariness of x1 , . . . , xn ∈ S, y1 , . . . , yn ∈ T, and f ∈ Wn , we have (S ⊗ T)n  ≤ Sn T n , as desired. Suppose that A is associative. Let n be a natural number such that min{Sn , T n } > 0,

8.4 The joint spectral radius of a bounded set

643

and let 0 < ε < min{Sn , T n }. Then there are x1 , . . . , xn ∈ S, y1 , . . . , yn ∈ T, and f ∈ Wn such that x1 · · · xn  ≥ Sn  − ε and f (y1 , . . . , yn ) ≥ T n  − ε. Since f (x1 ⊗ y1 , . . . , xn ⊗ yn ) = (x1 · · · xn ) ⊗ f (y1 , . . . , yn ), we obtain (S ⊗ T)n  ≥ f (x1 ⊗ y1 , . . . , xn ⊗ yn ) = x1 · · · xn f (y1 , . . . , yn ) ≥ (Sn  − ε)(T n  − ε). By letting ε → 0, we derive (S ⊗ T)n  ≥ Sn T n . Now note that the inequality just proved is in fact true for every n ∈ N because it is trivially true if min{Sn , T n } = 0. By keeping in mind that A is associative, and applying consequently Lemma 8.4.12, it follows (S ⊗ T)n  ≥ r(S)n T n  for every n ∈ N, which implies (by passing to n-th roots and taking sup limit as n → ∞) r(S ⊗ T) ≥ r(S)r(T). Thus (i)(b) has been proved. Now suppose that A is associative and that A ⊗ B is topologically nilpotent. Then, by (i)(b), we have r(BA )r(BB ) = r(BA ⊗ BB ) ≤ r(BA⊗B ) = 0. Therefore r(BA ) = 0 or r(BB ) = 0. This proves (ii).  Corollary 8.4.103 Let A be a normed algebra over K, let S a bounded subset of A, let  ·  be a cross algebra norm on A ⊗ A, and consider A ⊗ A as a normed algebra over K under the norm  ·  . Then r(S ⊗ S) = r(S)2 . Proof In view of Proposition 8.4.102(i)(a), it is enough to show that (S ⊗ S)n  ≥ Sn 2 for every natural number n such that Sn  > 0. Let n be such a natural number, and let 0 < ε < Sn . Then there are y1 , . . . , yn ∈ S and f ∈ Wn such that f (y1 , . . . , yn ) ≥ Sn  − ε. Since f (y1 ⊗ y1 , . . . , yn ⊗ yn ) = f (y1 , . . . , yn ) ⊗ f (y1 , . . . , yn ), we obtain (S ⊗ S)n  ≥ f (y1 ⊗ y1 , . . . , yn ⊗ yn ) = f (y1 , . . . , yn )2 ≥ (Sn  − ε)2 . By letting ε → 0, we derive (S ⊗ S)n  ≥ Sn 2 , as desired.



As we noticed before Proposition 1.1.98, given normed algebras A and B over K, the algebra A ⊗ B becomes a normed algebra under the projective tensor norm  · π . Such a normed algebra will be denoted by A ⊗π B.

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Selected topics in the theory of non-associative normed algebras

Proposition 8.4.104 Let A and B be normed algebras over K. We have: r(BA⊗π B ) ≤ r(BA )r(BB ). If A is associative, then r(BA⊗π B ) = r(BA )r(BB ). r(BA⊗π A ) = r(BA )2 . If A is topologically nilpotent, then so is A ⊗π B. If A is associative, and if A ⊗π B is topologically nilpotent, then A or B is topologically nilpotent. (vi) A is topologically nilpotent if and only if so is A ⊗π A.

(i) (ii) (iii) (iv) (v)

Proof Since  · π is a cross algebra norm on A ⊗ B, and BA⊗π B is the closed convex hull of BA ⊗ BB in A ⊗π B, assertions (i) and (ii) (respectively, assertion (iii)) follow (respectively, follows) from Lemma 8.4.23 by invoking Proposition 8.4.102 (respectively, Corollary 8.4.103). Finally, assertions (iv), (v), and (vi) follow straightforwardly from assertions (i), (ii), and (iii), respectively.  Remark 8.4.105 Let A and B be normed algebras over K. Mainly in the case that A and B are complete, we could be interested in the complete projective tensor product π B, namely the completion of (A ⊗ B,  · π ). It is noteworthy that, since BA⊗π B A⊗ is dense in BA⊗ π B , it follows from Lemma 8.4.23 that Proposition 8.4.104 remains π A) instead of A ⊗π B (occasionally, A ⊗π A).  true with A⊗π B (occasionally, A⊗ Proposition 8.4.106 Let A and B be normed algebras over K. We have: (i) If A is finitely quasi-nilpotent, then so is A ⊗π B. (ii) If A is associative, and if A ⊗π B is finitely quasi-nilpotent, then A or B is finitely quasi-nilpotent. (iii) A is finitely quasi-nilpotent if and only if so is A ⊗π A. Proof Suppose that A is finitely quasi-nilpotent, and let {z1 , . . . , zn } be any finite subset of A ⊗π B. To prove (i) we must show that {z1 , . . . , zn } is quasi-nilpotent. To this end, we may suppose that {z1 , . . . , zn }π < 1. Then, for each i = 1, . . . , n, there i are xi1 , . . . , xini ∈ BA , yi1 , . . . , yini ∈ BB , and αi1 , . . . , αini ∈ [0, 1] such that nj=1 αij = 1 ni n and zi = j=1 αij xij ⊗ yij . Now put S := ∪i=1 {xi1 , . . . , xini }. Since S is a finite subset of A, and A is suppose to be finitely quasi-nilpotent, S is quasi-nilpotent. Therefore, by Proposition 8.4.102(i)(a), S ⊗ BB is a quasi-nilpotent subset of A ⊗π B. Finally, since {z1 , . . . , zn } ⊆ co(S ⊗ BB ), Lemma 8.4.23 applies giving us that {z1 , . . . , zn } is quasi-nilpotent, as desired. Now assume that A is associative but not finitely quasi-nilpotent, and that A ⊗π B is finitely quasi-nilpotent. Let T be any finite subset of B. To prove (ii) we must show that T is quasi-nilpotent. Since A is not finitely quasi-nilpotent, there exists a finite subset S of A with r(S) = 0. Then, since S ⊗ T is a finite subset of A ⊗ B, and A ⊗π B is finitely quasi-nilpotent, we have r(S ⊗ T) = 0 in A ⊗π B. Moreover, since A is associative, it follows from Proposition 8.4.102(i)(b) that r(S ⊗ T) = r(S)r(T). Therefore r(T) = 0, as desired.

8.4 The joint spectral radius of a bounded set

645

The ‘only if’ part of assertion (iii) follows from assertion (i). Suppose that A ⊗π A is finitely quasi-nilpotent, and let S be any finite subset of A. Then S ⊗ S is a finite subset of A ⊗ B, and hence, by Corollary 8.4.103, S is quasi-nilpotent. Thus the ‘if’ part of assertion (iii) has been proved.  Associativity cannot be removed in those assertions in Propositions 8.4.104 and 8.4.106 which involve it. Indeed, we have the following. Example 8.4.107 Take a vector space over K with basis {u, v}, convert it into an algebra with multiplication table u2 = uv = v and vu = v2 = 0, and define a norm on it by αu + βv := |α| + |β|. It is easily realized that  ·  becomes an algebra norm, giving rise in this way to a two-dimensional normed algebra A such that (AA)A = 0. Moreover, we have Lu n (u) = v for every n ∈ N, which implies r(BA ) = 1 because u = v = 1. Now, let B stand for the opposite algebra of A (cf. §1.1.36), which satisfies B(BB) = 0, and becomes a normed algebra under the norm of A. The equalities r(BB ) = r(BA ) = 1 are now clear, and hence neither A nor B is topologically nilpotent. Nevertheless, the facts (AA)A = 0 and B(BB) = 0 imply that A ⊗ B is nilpotent (of index 3), so topologically nilpotent, and so finitely quasi-nilpotent. Finally, since A and B are finite-dimensional non-topologically-nilpotent algebras, it follows from Corollary 8.4.90 that they are not finitely quasi-nilpotent. Proposition 8.4.108 Let A be a normed algebra over K such that there exist ρ > 0 and a nonzero bounded idempotent subset S of A in such a way that ρS is multiplicatively closed, let B be any normed algebra over K, let  ·  be a cross algebra norm on A ⊗ B, and put A ⊗ B := (A ⊗ B,  ·  ). Then: (i) ρβ(B) ≤ β(A ⊗ B). (ii) B is topologically nilpotent whenever so is A ⊗ B. Proof To prove assertion (i), it is enough to show that, if A ⊗ B satisfies the norm-k boundedness property for some k > 0, then B satisfies the norm-(ρ −1 k) boundedness property. Let k > 0 be such that A⊗ B satisfies the norm-k boundedness property, and let T be any bounded and multiplicatively closed subset of B. Then, since ρS ⊗ T is a bounded and multiplicatively closed subset of A ⊗ B, there exists ||| · ||| ∈ En(A ⊗ B) such that ||| ρS ⊗ T ||| ≤ k. Now, define a (vector space) norm || · || on B by || y || := sup{||| x ⊗ y ||| : x ∈ S}, and note that, since S is an idempotent subset of A, we have || y || ≤ sup{||| x ⊗ y ||| : x ∈ S2 }

(8.4.16)

for every y ∈ B. The fact that ||| · ||| and  ·  are equivalent norms on A ⊗ B, and the cross property of  ·  , imply that || · || is equivalent to the given norm on B. On the other hand, the inequality ||| ρS ⊗ T ||| ≤ k implies that || T || ≤ ρ −1 k. Therefore, to conclude the proof of the first conclusion, it is enough to show that || · || is an algebra norm on B. Let x1 , x2 be in S, and let y1 , y2 be in B. Then we have

646

Selected topics in the theory of non-associative normed algebras ||| (x1 x2 ) ⊗ (y1 y2 ) ||| = ||| (x1 ⊗ y1 )(x2 ⊗ y2 ) ||| ≤ ||| x1 ⊗ y1 |||||| x2 ⊗ y2 ||| ≤ || y1 || || y2 || .

By the arbitrariness of x1 , x2 in S, we derive sup{||| x ⊗ (y1 y2 ) ||| : x ∈ S2 } ≤ || y1 || || y2 || . Finally, invoking the inequality (8.4.16), we conclude || y1 y2 || ≤ || y1 || || y2 ||. Assertion (ii) follows from assertion (i) and the equivalence (i)⇔ (iii) in Theorem 8.4.44.  Remark 8.4.109 The assumption on the normed algebra A done in the above proposition is automatically fulfilled if A = 0 and either A has a bounded approximate unit or A is complete and equals the linear hull of A2 . Indeed, in both cases, there exists δ > 0 such that δBA is an idempotent subset of A (cf. the proofs of Propositions 8.4.46(ii) and 8.4.56), and hence the assumption on A is fulfilled by taking ρ := 1δ and S := δBA . Let A be a normed algebra and let E be a Hausdorff locally compact topological space. Then the space C0 (E, A) (of all A-valued continuous functions on E vanishing at infinity) becomes a normed algebra under the operations defined pointwise, and the sup norm. Corollary 8.4.110 Let E be a Hausdorff locally compact topological space, and let A be a normed algebra over K. Then we have: (i) β(A) ≤ β(C0 (E, A)). (ii) A is topologically nilpotent if and only if so is C0 (E, A). Proof It is well-known that the injective tensor product C0 (E, K) ⊗ε A can be seen isometrically as a subalgebra of C0 (E, A) (see for instance [717, Example 4.2(2)]). In view of Fact 8.4.29, this implies β(C0 (E, K) ⊗ε A) ≤ β(C0 (E, A)). On the other hand, it is also well-known that C0 (E, K) has an approximate unit bounded by one. Now assertion (i) follows by applying Remark 8.4.109 and Proposition 8.4.108(i). Let x1 , . . . , xn be in BC0 (E,A) , let f be in Wn , and let t be in E. Then x1 (t), . . . , xn (t) lie in BA , and hence we have f (x1 , . . . , xn )(t) = f (x1 (t), . . . , xn (t)) ≤ (BA )n . From the arbitrariness of t in E we deduce f (x1 , . . . , xn ) ≤ (BA )n , and then from the arbitrariness of x1 , . . . , xn in BC0 (E,A) and f in Wn we derive BC0 (E,A)  ≤ (BA )n .

8.4 The joint spectral radius of a bounded set

647

By passing to n-th roots and taking sup limit as n → ∞, we get r(BC0 (E,A)) ) ≤ r(BA ). Therefore, if A is topologically nilpotent, then so is C0 (E, A). Conversely, if C0 (E, A) is topologically nilpotent, then so is C0 (E, K) ⊗ε A, and hence, by Proposition 8.4.108(ii), A is topologically nilpotent.  Corollary 8.4.111 Let A be a non-commutative C∗ -algebra, and let λ be in [1, +∞]. λ+1 Then β(A( 2 ) ) = λ. Proof Since β(A) ≤ 1 (because A is associative), it follows from Lemma 8.4.32 λ+1 that β(A( 2 ) ) ≤ λ. On the other hand, since A is a non-commutative C∗ -algebra, it follows from Corollary 6.1.21 that it contains (as a closed ∗-subalgebra) a copy of either M2 (C) or C0 (]0, 1], M2 (C)). Therefore, by Facts 8.4.29 and 8.4.99, and λ+1 Corollary 8.4.110(i), we have β(A( 2 ) ) ≥ λ.  In §3.5.32, we introduced the notion of an identity over K, and later (in Subsection 3.5.7), given a set of identities I (which may be empty), we defined the variety of algebras over K determined by I as the class of all algebras over K satisfying all identities in I. An identity is said to be homogeneous if it is a linear combination of non-associative words which are all of the same degree in each of the indeterminates (cf. §3.4.41). Thus all non-associative words involved in a homogeneous identity f have a common global degree, which is called the degree of f . We note that, since K is infinite, every variety of K-algebras can be determined by a suitable set of homogeneous identities [822, Corollary 1.2]. Lemma 8.4.112 Let M be a variety of K-algebras, and let A and B be algebras over K. We have: (i) If A is associative and commutative, and if B is a member of M , then A ⊗ B lies in M . (ii) If A is power-associative and non-nil, and if A ⊗ B is a member of M (respectively, if A ⊗ B is nilpotent), then B lies in M (respectively, B is nilpotent). Proof Assertion (i) follows from Corollary 1.2 of [822] just quoted and [822, Theorem 1.6]. Suppose that A is power-associative and non-nil, and that A ⊗ B lies in M . Take a non-nilpotent element u in A, and let f = f (x1 , . . . , xm ) be any of the homogeneous identities determining the variety M . Then, for all y1 , . . . , ym ∈ B, we have 0 = f (u ⊗ y1 , . . . , u ⊗ ym ) = up ⊗ f (y1 , . . . , ym ), where p stands for the degree of f . Since up = 0, we derive f (y1 , . . . , ym ) = 0, so B satisfies the identity f , and so B is a member of M . This proves the bracketfree version of assertion (ii). The bracketed version of (ii) follows from the above by noticing that, for n ≥ 2 the class of all nilpotent K-algebras of index ≤ n is a variety. 

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Selected topics in the theory of non-associative normed algebras

Lemma 8.4.113 Let M and N be varieties of K-algebras, and let B a normed member of M \ N . We have: (i) If A stands for the normed algebra over K in Example 8.4.41, and if B is not π B is a non-nilpotent topologically nilpotent member of nilpotent, then A⊗ M \N . (ii) If A stands for the normed algebra over K in Example 8.4.75, and if B is not topologically nilpotent (respectively, has not the multiplicative boundedness property), then A ⊗π B is a finitely quasi-nilpotent member of M \ N which is not topologically nilpotent (respectively, which has not the multiplicative boundedness property). Proof Let A stand for the normed algebra over K in Example 8.4.41, and suppose π B is topologically nilpotent because so is A, and that B is not nilpotent. Then A⊗ Proposition 8.4.104.(iv) and Remark 8.4.105 apply. On the other hand, since A is associative, commutative, and non-nil, and B is a non-nilpotent member of M \ N ,  π B) is a nonwe can apply Lemma 8.4.112 to derive that A ⊗ B (and hence A⊗ nilpotent member of M \ N . Thus assertion (i) has been proved. Now, let A stand for the normed algebra over K in Example 8.4.75, and suppose that B is not topologically nilpotent (respectively, has not the multiplicative boundedness property). Then A ⊗π B is finitely quasi-nilpotent because so is A, and Proposition 8.4.106(i) applies. Moreover, since A is associative, commutative, and non-nil, and B is a member of M \ N , we can apply Lemma 8.4.112 to derive that A ⊗ B lies in M \ N . Finally, since A has a bounded approximate unit, and B is not topologically nilpotent (respectively, has not the multiplicative boundedness property), we can apply Proposition 8.4.108(ii) (respectively, Proposition 8.4.108(i)) and Remark 8.4.109 to get that A⊗π B is not topologically nilpotent (respectively, has not the multiplicative boundedness property). Thus assertion (ii) has been proved.  Theorem 8.4.114 Let M and N be varieties of K-algebras. We have: (i) If there is a normed non-nilpotent member in M \ N , then there is also a nonnilpotent topologically nilpotent complete normed member in M \ N . (ii) If there is a normed member in M \ N which is not topologically nilpotent (respectively, which has not the multiplicative boundedness property), then there is also a finitely quasi-nilpotent normed member in M \ N which is not topologically nilpotent (respectively, which has not the multiplicative boundedness property). (iii) If there is a finite-dimensional member in M \N which is not nilpotent (respectively, which has not the multiplicative boundedness property), then there is also a finitely quasi-nilpotent complete normed member in M \ N which is not topologically nilpotent (respectively, which has not the multiplicative boundedness property). Proof The present assertions (i) and (ii) follow straightforwardly from assertions (i) and (ii), respectively, in Lemma 8.4.113. Keeping in mind that, for finite-dimensional algebras, nilpotency is equivalent to topological nilpotency (by

8.4 The joint spectral radius of a bounded set

649

Corollary 8.4.90), that the projective tensor product of a Banach space and a finitedimensional space is a Banach space, and that the normed algebra A over K in Example 8.4.75 is complete, the present assertion (iii) follows from assertion (ii) in Lemma 8.4.113.  Corollary 8.4.115 There are non-nilpotent topologically nilpotent complete normed algebras over K, as well as non-topologically-nilpotent finitely quasi-nilpotent complete normed algebras over K, in each of the following classes: (i) (ii) (iii) (iv) (v)

The class of associative algebras which are not commutative. The class of alternative algebras which are not associative. The class of Jordan algebras which are not associative. The class of Lie algebras. The class of commutative power-associative algebras which are not Jordan algebras.

Proof Let M (respectively, N ) stand for the variety of associative, alternative, Jordan, Lie, or commutative power-associative (respectively, commutative, associative, associative, zero, or Jordan) algebras over K. Then it is well known that there are non-nilpotent finite-dimensional members in M \ N . Now apply Theorem 8.4.114(i), as well as the bracket-free version of Theorem 8.4.114(iii).  Corollary 8.4.116 There are finitely quasi-nilpotent complete normed algebras over K failing to enjoy the multiplicative boundedness property in each of the following classes: (i) The class of Lie algebras. (ii) The class of non-commutative Jordan algebras which are not anticommutative. Proof Let M (respectively, N ) stand for the variety of Lie or non-commutative Jordan (respectively, zero or anticommutative) algebras over K. Now note that the last paragraph of the proof of Theorem 8.4.34 shows the existence of a twodimensional Lie algebra A over K failing to enjoy the multiplicative boundedness property. Consequently, the three-dimensional algebra A ⊕ K is non-commutative Jordan, is not anticommutative, and fails to enjoy the multiplicative boundedness property (cf. Fact 8.4.29). It follows that, in any case, there is a finite-dimensional member in M \ N which has not the multiplicative boundedness property, so that the bracketed version of Theorem 8.4.114(iii) applies.  According to §2.8.18, the free non-associative algebra over K on a countably infinite set of indeterminates can be endowed with an absolute value. By passing to the completion, we are then provided with an example of a complete normed algebra over K, which has the multiplicative boundedness property, is not topologically nilpotent (cf. Theorem 8.4.44 and Corollary 8.4.94(iii)), and does not satisfy any identity. Now we can complete the picture by proving the following. Corollary 8.4.117 There are topologically nilpotent complete normed algebras over K, as well as finitely quasi-nilpotent normed algebras over K without the multiplicative boundedness property, which do not satisfy any identity.

650

Selected topics in the theory of non-associative normed algebras

Proof In the whole proof, M stands for the variety of all algebras over K, f is any identity over K, Nf stands for the variety of those algebras over K which satisfy the identity f , and B denotes the free non-associative algebra over K on a countably infinite set of indeterminates, endowed with an absolute value. We note that B is not a member of Nf . In the present paragraph, A stands for the normed algebra over K in example π B is a topologically nilpotent complete normed 8.4.41. By Lemma 8.4.113(i), A⊗ π B does not satisfy any algebra, and lies in M \ Nf . By the arbitrariness of f , A⊗ identity. To conclude the proof, take any normed algebra C without the multiplicative boundedness property, and put D := C ⊕∞ B, so that D has not the multiplicative boundedness property (cf. Fact 8.4.29), and lies in M \ Nf . Now, let A stand for the normed algebra over K in Example 8.4.75. By the bracketed version of Lemma 8.4.113(ii), A ⊗π D becomes a finitely quasi-nilpotent normed algebra without the multiplicative boundedness property, and lies in M \ Nf . By the arbitrariness of f , A ⊗π D does not satisfy any identity.  It follows from Corollaries 8.4.45 and 8.4.117 that there are non-topologicallynilpotent finitely quasi-nilpotent normed algebras over K, which do not satisfy any identity. However, we do not know any answer to the following. Problem 8.4.118 Is there a finitely quasi-nilpotent complete normed algebra over K which fails to enjoy the multiplicative boundedness property (or at least fails to be topologically nilpotent), and does not satisfy any identity? 8.4.5 Historical notes and comments In an early paper [544], Rota and Strang prove Theorem 8.4.1, introduce the notion of ( joint) spectral radius of a bounded subset of a normed associative algebra, point out Lemma 8.4.12, and apply Theorem 8.4.1 to derive Corollary 8.4.20 in the way commented immediately after Corollary 8.4.37. The Rota–Strang paper remained forgotten for many years. Nevertheless, the idea of the spectral radius of a bounded subset of a normed associative algebra underlines the definition of topologically nilpotent normed associative algebras. Indeed, as we have done even in the non-associative setting, such algebras could have been introduced as those normed associative algebras such that the spectral radius of their closed unit balls is equal to zero. Following Palmer’s review in [786, 4.8.8], ‘These [algebras] were introduced by J. K. Miziolek, T. M¨uldner and A. Rek [1020] as a class of topological algebras. Recently their study was revived by Peter G. Dixon [927] and continued by Dixon and Vladimir M¨uller [928], Dixon and George A. Willis [929].’ We refer the reader to Palmer’s whole review of the papers just quoted [786, pp. 515–7] for a comprehensive view of the theory of topologically nilpotent normed associative algebras, noticing however that, as we will see in §8.4.121, an error enters the formulation of the encyclopedic Theorem 4.8.8 of [786].

8.4 The joint spectral radius of a bounded set

651

Actually, some of the results in [927, 928, 929, 1020] have been included in our development. Indeed, Examples 8.4.41 and 8.4.75 are sketched in [1020, Examples 2.3 and Example 2.5] (see also [929, Example 2.2 and p. 47]), and the associative forerunners of Propositions 8.4.46(ii) and 8.4.104(iv) are proved also in [1020]. On the other hand, the associative forerunner of Proposition 8.4.56 is proved in [927, Theorem 4.1] and is collected in [786, Theorem 4.8.9]. In fact, it is shown there that, if A is a nonzero topologically nilpotent complete normed algebra, and if X is any nonzero Banach left A-module (cf. the paragraph immediately before Definition 3.6.33), then the linear hull of AX cannot be equal to X. The particularizations of Lemma 8.4.12 and of Corollary 8.4.71 to the case that S = BA , as well as the associative forerunner of Theorem 8.4.51, can be found in [928, Proposition 2, and Theorems 3 and 5]. To conclude our review of topologically nilpotent normed associative algebras, let us refer the reader to the book of Doran–Wichmann [1156, Section 11] and to §8.4.121. Let us also say that topologically nilpotent normed associative algebras have proved useful in providing significant positive answers to the question of splitting radical extensions of certain complete normed associative algebras (see [962, 847] for details). The Rota–Strang spectral radius is rediscovered in the papers of Shulman [1083] and Turovskii [615], where a special attention is paid to the spectral radius of finite subsets of a normed associative algebra. Actually, in [615], the notions of n-quasinilpotent and of finitely quasi-nilpotent subset of a normed associative algebra are introduced, and Theorem 8.4.76, as well as the associative forerunners of Proposition 8.4.24 and of Lemma 8.4.78, are proved. The paper [615] contains also a finite-subset generalization of the associative forerunner of Theorem 4.5.14. For more information about the Rota–Strang spectral radius of finite subsets of normed associative algebras, the reader is referred to M¨uller’s survey paper [1030], and to the papers of Soltysiak [1095] and Rosenthal–Soltysiak [1066]. According to an oral communication from Turovskii, Problem 8.4.83 belongs to Shulman, and was published first in [1106, p. 153] (1984). It appears again in [615, p. 174] (1985), and is emphasized in both [1030, p. 262] and [1084, Question 15] (1994). In [861], Berger and Wang prove that for each m ∈ N and each bounded subset S of Mm (K) we have   r(S) = lim sup sup r(a)1/n : a ∈ Sn . (8.4.17) n→∞

Since every finite-dimensional associative algebra over K can be seen as a subalgebra of Mm (K) for some m ∈ N, the Berger–Wang theorem just reviewed actually shows that the equality (8.4.17) holds for every bounded subset S of any finite-dimensional associative algebra over K. The Berger–Wang proof uses results from the theory of rings. A proof using tools from analysis and geometry (convexity), as well as some elementary results from [861], can be found in Elsner’s paper [936]. The Berger– Wang theorem plays a very important role in the modern theory of finite-dimensional dynamical systems (see [1165] and references therein).

652

Selected topics in the theory of non-associative normed algebras

In [569, 1108], Shulman and Turovskii prove that the Berger–Wang formula (8.4.17) holds for every precompact subset S of a normed associative algebra A, whenever A satisfies various compactness conditions. A simplest condition of this kind – that A consists of compact operators, the most general one – that A is hypocompact, which means that each quotient A/I of A by a proper closed ideal I has a nonzero compact element (an element a is compact if the map x '→ axa is compact). As a consequence, if a normed associative algebra is hypocompact and quasi-nil, then it is finitely quasi-nilpotent. Thus Problem 8.4.83 (and hence Problem 8.4.84) has a positive answer if the algebra A in question is hypocompact. The variant for compact operators of the Berger–Wang theorem, commented above, allows one to study invariant subspaces for a wide class of associative algebras, Lie algebras and semigroups of compact operators. In [569] a peculiar notion of spectral radius of a bounded subset of any normed Lie algebra is also introduced (see §8.4.124). Concerning our development, one of the most significant results in [569] is the associative forerunner of Proposition 8.4.17 [569, Proposition 2.7]. The Shulman–Turovskii paper contains also the associative forerunner of Corollary 8.4.18(i) [569, Corollary 2.5]. More information about the paper [569] can be found in §§8.4.120 and 8.4.119. The work of Shulman and Turovskii on the joint spectral radius increases in later years with relevant applications to the theory of topological radicals (see [1107, 1085, 1086, 1108, 1087, 1088] and their joint papers with P. Cao [897] and E. Kissin [998]). In [452], Moreno and Rodr´ıguez discuss the validity of Theorem 8.4.1 in the non-associative setting. After observing by means of easy examples (cf. Example 8.4.2) that Theorem 8.4.1 does not remain true in general if the associativity of the algebra A is removed, they prove Theorem 8.4.8 and realize that Theorem 8.4.1 is far from being characteristic of the associativity. Consequently, they introduce the NBP (cf. Definition 8.4.28) for a normed ( possibly non-associative) algebra, and show that both normed nilpotent algebras and normed algebras fulfilling the NSE satisfy the NBP. For normed nilpotent algebras, this follows from either Theorem 8.4.7 or Corollary 8.4.45, whereas the result concering normed algebras fulfilling the NSE is collected in Corollary 8.4.94. They also introduce in [452] the notion of spectral radius of an element of any normed algebra A (cf. §8.4.14), and prove that, if A satisfies the NBP, then we have r(a) = inf{||| a ||| : ||| · ||| ∈ En(A)} for every a ∈ A (compare Corollary 8.4.37). In a later paper [453], Moreno and Rodr´ıguez introduce the notion of ( joint) spectral radius of a bounded subset of a normed ( possibly non-associative) algebra (cf. Definition 8.4.10), as well as the multiplicative boundedness property for a normed algebra (cf. Definition 8.4.28), and prove most of the results in this section which involve normed non-associative algebras. In fact, our development follows essentially the approach in [452, 453], incorporating those results of other authors which are invoked there. Regarding the latter, we have helped ourselves with the books of Palmer [786, p. 516 and Example 4.8.3] for Example 8.4.41, Doran–Wichmann [1156, Example 11.3] for Example 8.4.75, and Schafer [808, Theorem 2.4] for Fact

8.4 The joint spectral radius of a bounded set

653

8.4.87. Concerning Theorem 8.4.76 and other results in the section involving normed associative algebras, we have helped ourselves with priceless private communications of Shulman and Turovskii. Let us also say that Proposition 8.4.74 seems to be folklore. As in similar occasions, some refinements of the results in [452, 453] have been made. Some of them, mainly in Subsection 8.4.3, are due to Moreno and Rodr´ıguez (unpublished), whereas the others are new. According to the authors of [453], the question whether the ANBP implies the NBP remained an open problem for them when they submitted the paper by the first time. The negative answer to this question, first published in [453] and collected in Example 8.4.30, is due to the referee of the paper. Some generalizations of associative results to the non-associative setting, like Proposition 8.4.24, have no special merit, but in general, this is not the case. Indeed, in the associative setting, most proofs use Lemma 8.4.12 and/or Corollary 8.4.20, which do not remain true when associativity is removed. In particular, this obstacle occurs (and is successfully overcome) in the proofs of Proposition 8.4.17 and Theorem 8.4.70. Nevertheless, as acknowledged in [453], most clever ideas in the proof of Theorem 8.4.70 are taken almost verbatim from that of its associative forerunner, namely Theorem 3 in the Dixon–M¨uller paper [928]. In the following notes, we are going to complement the material developed in the section. §8.4.119 Let A be normed associative algebra over K. It is proved in [569, Proposition 3.1] that the mapping S → r(S), from the family of all bounded subsets of A (endowed with the topology of the Hausdorff metric) to R, is upper semicontinuous. Now suppose that K = C, let  be a domain in C, and let F be a family of holomorphic mappings from  to A such that (i) F(z) := {f (z) : f ∈ F} is bounded in A for each z ∈ , (ii) limz→z0 supf ∈F f (z) − f (z0 ) = 0 for every z0 ∈ . Then, as proved in [569, Theorem 3.5], the functions z → log r(F(z)) and z → r(F(z)) are subharmonic on . It would be interesting to know whether in these results the assumption of associativity for A can be removed. Concerning the former, an affirmative answer could be expected by simply replacing r(·) with s(·) (cf. §8.4.25). Indeed, according to Theorem 4.5.14, the non-associative generalization of the result is true when F reduces to a singleton. §8.4.120 Let A be a normed algebra over K, and let S be a subset of A. Following [569], we write  t |co|(MC(S)), Abs(S) := t>0

and note that Abs(S) is a subalgebra of A, which contains the subalgebra of A generated by S, and is contained in the closed subalgebra of A generated by S. As pointed out in [453], it is enough to look at the proof of Theorem 8.4.55 to realize that this theorem remains true if the condition that A is generated as an algebra by some of its

654

Selected topics in the theory of non-associative normed algebras

quasi-nilpotent subsets is replaced with the weaker (but less natural) one that there exists a quasi-nilpotent subset S of A such that Abs(S) = A. As a consequence, if A is complete, if S is quasi-nilpotent, and if Abs(S) is closed in A, then Abs(S) is topologically nilpotent. In the associative case, this result becomes Proposition 2.12 of [569]. §8.4.121 For a complete normed associative complex algebra A, consider the following conditions: (i) A is topologically nilpotent. (ii) There is some finite constant C satisfying −3.2n n

1

sup{an  n : a ∈ BA } ≤ Cn

for all n ∈ N.

(iii) For each element a ∈ A, there is some finite constant C = C(a) satisfying 1

an  n ≤ Cn

−3.2n n

for all n ∈ N.

Then conditions (ii) and (iii) are equivalent. Indeed, the implication (ii)⇒(iii) is clear, whereas the converse implication depends on a Baire category argument (see the proof of [927, Theorem 2.1] for details). On the other hand, as a consequence of a quantitative version of the Nagata–Higman theorem proved in [927, Theorem 3.1], condition (ii) implies condition (i) [927, Theorem 3.2]. However, contrary to what asserted in [786, Theorem 4.8.8], (i) does not imply (ii). The following counterexample was communicated to the authors of [453] by V. M¨uller to be included in their paper. Consider the complete normed associative complex algebra A of those  j formal power series ∞ j=1 αj x (with one generator x and complex coefficients αj ) such that ∞ 

αj xj :=

∞ 

j=1

|αj |

j=1

1 < ∞. jj

Let n be in N. Then we have x = 1 and 1

xn  n =

1 , n

(8.4.18)

so that A does not satisfy condition (ii) above. For j ∈ N, set aj := j j x j . Then we have aj1 · · · ajn  ≤

1 nn

for all j1 , . . . , jn ∈ N. Indeed, since for p, q ∈ N the inequality qq 1 ≤ p+q ( p + q) (1 + p)1+p

(8.4.19)

8.4 The joint spectral radius of a bounded set

655

holds, we have j

aj1 · · · ajn  =

j

j11 · · · jnn ( j1 + · · · + jn ) j1 +···+jn j



j

(1 + j1 + · · · + jn−1 )1+j1 +···+jn−1 j



j

n−1 j11 · · · jn−1



(2 + j1 + · · · + jn−2 )2+j1 +···+jn−2

j

n−3 j11 · · · jn−3

(3 + j1 + · · · + jn−3

)3+j1 +···+jn−3

j

n−2 j11 · · · jn−2

≤ ··· ≤

1 , nn

as desired. Since BA = |co|({aj : j ∈ N}), and x = a1 , we derive from (8.4.18) and 1

(8.4.19) that (BA )n  n = 1n . By letting n → ∞, we realize that r(BA ) = 0, i.e. A satisfies condition (i). §8.4.122 Let A be an associative algebra over K. A straightforward consequence (of the proof) of the Nagata–Higman theorem is that, if the Jordan algebra Asym is nilpotent of index n, then A is nilpotent of index ≤ 2n − 1. As pointed out in [453], a better result follows from Lemma 8.4.48. Indeed, if Asym is nilpotent of index n, then A is nilpotent of index ≤ 2n − 1. We do not know whether or not, for every n ≥ 2, there is a choice of A such that A is nilpotent of index 2n − 1, and Asym is nilpotent of index n. §8.4.123 Let A be a normed algebra over K. According to §8.4.53, if A is topologically nilpotent, then so are both I and A/I for every closed ideal I of A. It is proved by Dixon [927, Theorem 5.1] that, if A is associative, and if A/I is topologically nilpotent, for some topologically nilpotent closed ideal I of A, then A is topologically nilpotent. It is noteworthy that, as shown in [453], the result just reviewed need not remain true if associativity is removed. Indeed, take A equal to the two-dimensional anticommutative algebra over K with basis {u, v}, and product determined by u2 = v2 = 0 and uv = −vu = v. We know from the last paragraph of the proof of Theorem 8.4.34 that β(A) = +∞, which implies, by Theorem 8.4.44, that A is not topologically nilpotent. However, by putting I := Kv, I becomes a closed ideal of A such that both I and A/I are nilpotent of index 2. After this counterexample, it would be interesting to know if Dixon’s result reviewed above remains true whenever the requirement that A is associative is relaxed to the one that A satisfies the multiplicative boundedness property. §8.4.124 Thinking about non-associative generalizations of the Rota–Strang spectral radius of a bounded subset S of a normed algebra A, we are already provided with two possibilities, namely that we have called spectral radius, r(S), of S (cf. Definition 8.4.10), and the unnamed s(S) (cf. §8.4.25). There is a third possibility. Indeed, for any subset S of an algebra A, define inductively Sn by S1 := S and Sn+1 := SSn , and, when A is normed and S is bounded, put 1

t(S) := lim sup Sn  n . n→∞

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Selected topics in the theory of non-associative normed algebras

(To avoid laterality, we could have defined Sn by S1 := S and Sn+1 := (SSn ) ∪ (Sn S), but this is irrelevant for the meaning of t(S) in the case we are interested in, namely that A be anticommutative.) Anyway, it becomes clear that t(S) ≤ r(S), and that this inequality becomes an equality if A is associative. In the Shulman–Turovskii paper [569, pp. 436–7], the symbol t(S) is taken as the definition of spectral radius of the bounded set S, when A is a normed Lie algebra. Nevertheless, even in the case of normed Lie algebras, the notion underlying the symbol t(·) seems to us rather untractable. With the aim of amending this feeling, the authors of [453] tried to prove the existence of a universal constant C satisfying r(S) ≤ Ct(S) for every bounded subset S of any normed Lie algebra. However, they had no success in this goal, nor even in that of proving that normed Lie algebras A satisfying t(BA ) = 0 are topologically nilpotent. In what follows, a by-product of their effort, in relation to the questions just suggested, is presented. As usual, the product of a Lie algebra will be denoted by [·, ·]. Lemma 8.4.125 Let A be a Lie algebra over K, let S be a subset of A, and let p, q be in N. Then we have [Sp , Sq ] ⊆ 2min{p,q}−1 |co|(Sp+q ). Proof

It is enough to show that [Sp , Sq ] ⊆ 2p−1 |co|(Sp+q ).

(8.4.20)

To this end, we argue by induction on p. The case p = 1 is obvious. Assume inductively that the inclusion (8.4.20) holds for every q and some fixed p. Then, by the Jacobi identity, we have [Sp+1 , Sq ] = [[S, Sp ], Sq ] ⊆ [S, [Sp , Sq ]] − [Sp , [S, Sq ]] = [S, [Sp , Sq ]] − [Sp , Sq+1 ] ⊆ [S, 2p−1 |co|(Sp+q )] − 2p−1 |co|(Sp+q+1 ) |co|(Sp+q+1 ) − |co|(Sp+q+1 ) ⊆ 2p ⊆ 2p |co|(Sp+q+1 ). 2 Therefore the inclusion (8.4.20) holds for every q and p + 1 instead of p.



Keeping in mind the above lemma, an easy induction argument shows the following. Corollary 8.4.126 Let A be a Lie algebra over K, let S be a subset of A, let n be in N. Then we have Sn ⊆ 2φ(n) |co|(Sn ), where φ : N → N ∪ {0} is defined inductively by φ(1) := 0 and φ(n) := max{φ( p) + φ(q) + min{p, q} − 1 : p, q ∈ N with p + q = n} for n > 1.

8.4 The joint spectral radius of a bounded set

657

With the notation in the above corollary, suppose that A is normed, and that S is bounded. Then we have 1

Sn  n ≤ 2

φ(n) n

1

Sn  n .

(8.4.21)

φ(n) n

However, as a matter of fact, the sequence has no finite sup limit because φ(2n) ≥ 2φ(n) + n − 1 for every n ∈ N. Anyway, we have the following. Proposition 8.4.127 Let A be a normed Lie algebra over K, and let S be a bounded subset of A such that   1 1 Sn  n = o n (8.4.22) 2 as n → ∞. Then S is quasi-nilpotent. Proof An easy induction argument shows that, for n ∈ N, we have that φ(n) ≤ n2 , 1 1 and hence, by (8.4.21), that Sn  n ≤ 2n Sn  n . On the other hand, our assumption 1 (8.4.22) reads as that limn→∞ 2n Sn  n = 0. It follows that 1

r(S) = lim Sn  n = 0. n→∞



Proposition 8.4.127 has a flavour similar to that of Theorem 3.2 in Dixon’s paper [927], already reviewed in §8.4.121. §8.4.128 In §4.4.77, we applied Theorem 8.4.1 to show that normed associative algebras having minimality of norm (cf. Definition 4.4.33) have also minimality of norm topology (cf. Definition 4.4.22). Now we can prove a generalization of this fact. Note at first that, by Theorem 8.4.1 and Corollary 8.4.45, both associative algebras and nilpotent algebras enjoy the NBP in any algebra norm. Now, let us say that a normed algebra satisfies the continuous NBP if (A, ||| · |||) satisfies the NBP whenever ||| · ||| is any continuous algebra norm on A. Then, looking at the argument in §4.4.77, we realize that, if a normed algebra satisfies the continuous NBP and has minimality of norm, then it has minimality of norm topology. Note however that, by the implication (i)⇒(ii) in Theorem 8.4.44, nonzero topologically nilpotent normed algebras cannot have minimality of norm. §8.4.129 As we commented in §2.8.80, generalized standard algebras are defined by a suitable finite set of identities [553], and, roughly speaking, they compose the minimum variety of algebras containing all alternative algebras and all Jordan algebras. As we noted there, generalized standard algebras are non-commutative Jordan algebras [553, Theorem 2], and hence they are power-associative. Let A be a finite-dimensional generalized standard algebra over K. Then assertions (i) to (v) in Corollary 8.4.90 are equivalent to the following: (vi) A is quasi-nil. Indeed, suppose that A is quasi-nil. Since A is finite-dimensional and powerassociative, we derive from Lemma 8.4.85 that A is a nil algebra. Since finitedimensional generalized standard nil algebras are nilpotent [553, Theorem 4], we conclude that A is nilpotent.

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Selected topics in the theory of non-associative normed algebras

§8.4.130 Let A be a finite-dimensional Lie algebra over K. Then, as a by-product of [453, Theorems 8.5 and 8.10], assertions (i) to (v) in Corollary 8.4.90 are equivalent to the following ones: (vi) (vii) (viii) (ix) (x) (xi) (xii)

A satisfies the NBP. A satisfies the ANBP. A satisfies the multiplicative boundedness property. A is 2-quasi-nilpotent. t(S) = 0 for every bounded subset S of A (cf. §8.4.124). t(S) = 0 for every finite subset S of A. t({x, y}) = 0 for all x, y ∈ A.

It is noteworthy that the proofs of the results in [453] invoked above involve the main theorems in [188], as well as celebrated theorems of Zel’manov in [1132, 1136]. §8.4.131 Recently, the undergraduate students J. D. Poyato and A. Rueda raised with us the question whether there are nonzero complete normed associative and commutative algebras with no maximal ideal. We passed the question to G. Dales, who, in a private communication, answered it affirmatively by showing that, actually, algebras such as that in Example 8.4.75 have no maximal ideal. A part of the argument is of a purely algebraic nature, and is contained in Lemma 8.4.132 and Proposition 8.4.133. Lemma 8.4.132 Let A be an algebra over K. We have: (i) If there is a proper ideal I of A such that A/I has maximal left ideals, then A has maximal left ideals. (ii) If lin(A2 ) = A, then A has maximal left ideals. (iii) If A is associative and radical, then every maximal left ideal of A contains A2 . Proof Suppose that there is a proper ideal I of A such that A/I has a maximal left ideal J. Then, denoting by π : A → A/I the quotient algebra homomorphism, π −1 (J) becomes a maximal left ideal of A. This proves assertion (i). Now suppose that lin(A2 ) = A. Then I := lin(A2 ) is a proper ideal of A, and A/I is a nonzero algebra with zero product. Therefore all subspaces of A/I are left ideals of A/I, and maximal subspaces of A/I are maximal left ideals of A/I. Since maximal subspaces do exist, it follows from assertion (i) that A has maximal left ideals. Thus assertion (ii) has been proved. Finally, suppose that A is associative and that there is a maximal left ideal M of A which does not contain A2 . Then the left A-module A/M (cf. Example 3.6.34) has no nonzero proper submodule and A(A/M) = 0, i.e. A/M is an irreducible left A-module (cf. Definition 3.6.35). But the existence of an irreducible left A-module implies that A is not radical. Indeed, by Theorem 3.6.38(i), the kernel of the associated irreducible representation is a primitive ideal of A, and hence, by Definition 3.6.12, A is not radical. Now the proof of the lemma is complete. 

8.4 The joint spectral radius of a bounded set

659

Proposition 8.4.133 Let A be an associative algebra over K. Then the following conditions are equivalent: (i) A has no maximal left ideal. (ii) A is radical and lin(A2 ) = A. (iii) A has no maximal right ideal. Proof (i)⇒(ii) Suppose that A has no maximal left ideal. Then, a fortiori, A has no maximal modular left ideal (cf. the paragraph immediately before Fact 3.6.3). Therefore, by Definition 3.6.12, A is a radical algebra. On the other hand, by Lemma 8.4.132(ii), lin(A2 ) = A. (ii)⇒(i) Now suppose that A is a radical algebra and that lin(A2 ) = A. Then, by Lemma 8.4.132(iii), A has no maximal left ideal. Keeping in mind that condition (ii) means the same for both A and the opposite algebra of A (cf. Corollary 3.6.22), the equivalence (ii)⇔(iii) follows from that (i)⇔(ii) proved above.  Now recall that, by Cohen’s factorization theorem [909] (see also [696, Corollary 11.11]), every complete normed associative algebra A over K having a bounded approximate unit is equal to A2 . Therefore it is enough to invoke Proposition 8.4.133 to get the following. Corollary 8.4.134 Let A be a complete normed associative radical algebra over K having a bounded approximate unit. Then A has neither maximal left ideals nor maximal right ideals. In particular, the algebra A over K in Example 8.4.75 has no maximal ideal. Remark 8.4.135 (a) Associative algebras over K such as that in Sasiada’s celebrated example [550] have neither maximal left ideals nor maximal right ideals. Indeed, every simple radical algebra over K fulfils condition (ii) in Proposition 8.4.133. (b) Let A be an algebra over K. Keeping in mind that assertions (i) and (ii) in Lemma 8.4.132 remain true with ‘(two-sided) ideal’ instead of ‘left ideal’, we realize that, if A has no maximal ideal, then A equals its strong radical s-Rad(A), and lin(A2 ) = A. Nevertheless, even if A is associative, the converse is not true. Indeed, if A denotes the algebra in Sasiada’s example, then A = s-Rad(A) and lin(A2 ) = A, but zero is a maximal ideal of A. (c) For rings, a commutative version of Proposition 8.4.133 is known. Indeed, as proved by Henriksen [976], a commutative ring R has no maximal ideal if and only if R is radical and R2 + pR = R for every prime p ∈ Z, where $ n F  R2 := xi yi : n ∈ N; xi , yi ∈ R . i=1

§8.4.136 We do not know whether normed generalized standard algebras satisfy the NBP, nor even whether alternative or Jordan normed algebras satisfy the

660

Selected topics in the theory of non-associative normed algebras

multiplicative boundedness property. Certainly, the algebra O of Cayley numbers (cf. Subsection 2.5.1) satisfies the NBP, since it is an absolute-valued algebra and Corollary 8.4.94(ii) applies. Nevertheless, we do not know whether or not the alternative C∗ -algebra of complex octonions C(C) = C ⊗π O (cf. Proposition 2.6.8) satisfies the multiplicative boundedness property. The same can be said of the real alternative C∗ -algebra C(R), which can be introduced as the real subalgebra of C(C) consisting of all fixed points for the conjugate-linear automorphism of C(C) obtained by composing the C∗ -algebra involution with the standard involution. Despite the comments in the preceding paragraph, as we show immediately below, real or complex non-commutative JB∗ -algebras (and hence real or complex alternative C∗ -algebras) satisfy a peculiar variant of the multiplicative boudedness property. The proof applies the straightforward fact that, if S is a ∗-invariant subset of any ∗-algebra, then Sn is ∗-invariant for every n ∈ N. Proposition 8.4.137 Let A be a normed ∗-algebra over K such that Ua (a∗ ) ≥ a3 for every a ∈ A. Then:

√ (i) For every bounded ∗-invariant subset S of A we have S ≤ 3 r(S). (ii) For every bounded, ∗-invariant, and multiplicatively closed subset S of A we √ have S ≤ 3. (iii) If B is any normed ∗-algebra over √ K, and if  : B → A is a continuous algebra ∗-homomorphism, then (b) ≤ 3 max{b, b∗ } for every b ∈ B. (iv) If ||| · ||| is an algebra √ norm on A generating a topology stronger than the natural one, then a ≤ 3 max{||| a |||, ||| a∗ |||}. Proof

Let S be a bounded ∗-invariant subset of A. For s ∈ S we have clearly s3 ≤ Us (s∗ ) = s(ss∗ + s∗ s) − s2 s∗  ≤ 3S3 ,

and hence S3 ≤ 3S3 . Applying the above with Sn instead of S, and keeping in mind that (Sn )3 ⊆ S3n , we realize that Sn 3 ≤ 3S3n  for every n ∈ N.

(8.4.23)

We claim that for every m ∈ N we have m

S3 ≤ 3

3m −1 2

m

S3 .

(8.4.24)

Indeed, certainly the inequality (8.4.24) holds for m = 1. Assume inductively that it holds for some m ∈ N. Then, applying (8.4.23), we obtain ! " 3m+1 −3 3m+1 −1 m+1 m 3 m m+1 = S3 ≤ 3 2 S3 3 ≤ 3 2 S3 . S3 Therefore, as claimed, (8.4.24) holds for every m ∈ N, and hence, passing to 3m -roots and taking sup limit at m → ∞, assertion (i) follows.

8.4 The joint spectral radius of a bounded set

661

Assertion (ii) follows from assertion (i) and Fact 8.4.16. Let B be a normed ∗-algebra over K, let  : B → A be a continuous algebra ∗-homomorphism, and let b be in B. Then, by assertion (i) and Fact 8.4.11, we have √ √ (b) ≤ {(b), (b)∗ } ≤ 3 r({(b), (b)∗ }) = 3 r(({b, b∗ })) √ √ √ ≤ 3 r({b, b∗ }) ≤ 3 {b, b∗ } = 3 max{b, b∗ }. Thus assertion (iii) has been proved. Assertion (iv) follows straightforwardly from (iii).



Of course, the assumption on A in the above proposition is fulfilled if A is a real or complex non-commutative JB∗ -algebra. In the particular case that A is a real or complex alternative C∗ -algebra, a better conclusion can be obtained. Indeed, as we show immediately below, such an algebra satisfies a peculiar variant of the NBP. The proof applies the fact, easily realized by induction, that, if S is any subset of an algebra, then we have (S2 )[n] ⊆ (S[n] )2 for every n ∈ N ∪ {0}.

(8.4.25)

Proposition 8.4.138 Let A be a normed ∗-algebra over K such that a∗ a = a2 for every a ∈ A. Then: (i) For every bounded ∗-invariant subset S of A we have S = s(S) = r(S). (ii) For every bounded, ∗-invariant, and multiplicatively closed subset S of A we have S ≤ 1. (iii) If B is any normed ∗-algebra over K, and if  : B → A is a continuous algebra ∗-homomorphism, then (b) ≤ max{b, b∗ } for every b ∈ B. (iv) If ||| · ||| is an algebra norm on A generating a topology stronger than the natural one, then a ≤ max{||| a |||, ||| a∗ |||}. Proof An easy induction argument shows that, for x ∈ A and n ∈ N ∪ {0}, we n+1 have x2 = (x∗ x)[n] . Now, let S be a bounded ∗-invariant subset of A, let s be in S, and let n be in N ∪ {0}. Since (s∗ s)[n] lies in (S2 )[n] , it follows from (8.4.25) 1 n+1 that s2 ≤ (S2 )[n]  ≤ (S[n] )2  ≤ S[n] 2 , and hence S ≤ S[n]  2n . Therefore 1 S ≤ inf{S[n]  2n : n ∈ N ∪ {0}} = s(S). Since s(S) ≤ r(S) ≤ S (cf. Fact 8.4.26), assertion (i) follows. Assertion (ii) follows from assertion (i) and Fact 8.4.16. Assertion (iii) follows from (i) by arguing as in the third paragraph of the proof of Proposition 8.4.137. Finally, assertion (iv) follows straightforwardly from (iii).  Remark 8.4.139 (a) Since (real or complex) alternative C∗ -algebras are noncommutative JB∗ -algebras, and the conclusion in Proposition 8.4.138 is true when A is an alternative C∗ -algebra, one can wonder whether this conclusion remains true

662

Selected topics in the theory of non-associative normed algebras

when A is a non-commutative JB∗ -algebra. As a matter of fact, the answer to the question just raised is ! negative. Indeed, let A stand for the unital JB∗ -algebra √ " sym 0 2 [M2 (C)] , and set x := 0 0 ∈ A. Then we have x2 = 0 and xx∗ = 1 in A. Therefore the ∗-invariant set S := {0, x, x∗ , 1} is multiplicatively closed in A, and hence, by Fact 8.4.16, r(S) ≤ 1. But, since 1 ∈ √ Sn for every n ∈ N, we must have in fact r(S) = 1. Nevertheless, the equality S = 2 holds. √ This example suggests√the question whether in Proposition 8.4.137 the constant 3 can be replaced with 2, at least when A is a non-commutative JB∗ -algebra. (b) Let A be a (real or complex) non-commutative JB∗ -algebra. Then every algebra norm on A generates a topology stronger than the natural one. Indeed, in the complex case this was proved in Theorem 4.4.29, and the real case follows from the complex one by invoking Proposition √ 1.1.98 and Fact 4.2.55. Therefore, by Proposition 8.4.137(iv), we have a ≤ 3 max{||| a |||, ||| a∗ |||} for every algebra norm ||| · ||| on A and every a ∈ A. Actually, √ 4.4.28(i)(a) that, in the above √ it follows from Lemma with 2. Thus, as above, we suspect inequality, the constant 3 can be replaced √ √ that in Proposition 8.4.137 the constant 3 can be replaced with 2, at least when A is a non-commutative JB∗ -algebra. In the particular case that A is an alternative C∗ -algebra, Proposition 8.4.138(iv) gives a ≤ max{||| a |||, ||| a∗ |||} for every algebra norm ||| · ||| on A and every a ∈ A. But this is already known for us (cf. the paragraph immediately before Theorem 4.4.29). (c) Let A be a normed ∗-algebra over K such that there is k > 0 satisfying Ua (a∗ ) ≥ ka3 for every a ∈ A.

(8.4.26)

Then, arguing as in the proof of Proposition 8.4.137, we realize that ? the conclusions √ in that proposition remain true if we replace the constant 3 with 3k . Now suppose that A = 0. Then it is easy to see that k ≤ 3, so we can define κ(A) as the maximum possible k in (8.4.26), and we can set κ ∗ (A) := sup{κ(A, ||| · |||) : ||| · ||| ∈ En(A)}. Now, arguing as in the proof of Proposition 8.4.93 (with κ(·) instead of η(·), and the variant of Proposition 8.4.137 just suggested instead of Proposition 8.4.92), we realize that there exists || · || ∈ En(A) such that κ(A, || · ||) = κ ∗ (A). (d) Let A be a normed ∗-algebra over K such that there is k > 0 satisfying a∗ a ≥ ka2 for every a ∈ A.

(8.4.27)

(Note that, by Lemma 3.4.65, condition (8.4.27) holds with k = 12 whenever A is any JB∗ -algebra.) Arguing as in the proof of Proposition 8.4.138, we realize that for every bounded ∗-invariant subset S of A we have S ≤ 1k s(S), and that assertions (ii) to (iv) in that proposition remain true if the right-hand sides of the inequalities appearing there are multiplied by 1k . Now suppose that A = 0. Then k ≤ 1, so we can define λ(A) as the maximum possible k in (8.4.27), and we can set

8.4 The joint spectral radius of a bounded set

663

λ∗ (A) := sup{λ(A, ||| · |||) : ||| · ||| ∈ En(A)}. Now, arguing as in the proof of Proposition 8.4.93 (with λ(·) instead of η(·), and the variant of Proposition 8.4.138 just suggested instead of Proposition 8.4.92), we realize that there exists || · || ∈ En(A) such that λ(A, || · ||) = λ∗ (A). As a consequence, if λ∗ (A) = 1, then there exists a unique equivalent algebra norm | · | on A such that |a∗ a| = |a|2 for every a ∈ A. Note that λ∗ (A) = 1 means that for each ε > 0 there is ||| · ||| ∈ En(A) such that ||| a∗ a ||| ≥ (1 − ε)||| a |||2 for every a ∈ A.

References

Bibliography of Volume 1 (updated) Papers [1] M. D. Acosta, J. Becerra, and A. Rodr´ıguez, Weakly open sets in the unit ball of the projective tensor product of Banach spaces. J. Math. Anal. Appl. 383 (2011), 461–73. [2] J. F. Adams, On the structure and applications of the Steenrod algebra. Comment. Math. Helv. 32 (1958), 180–214. [3] C. A. Akemann and G. K. Pedersen, Complications of semicontinuity in C∗ -algebra theory. Duke Math. J. 40 (1973), 785–95. 346 [4] C. A. Akemann and G. K. Pedersen, Facial structure in operator algebra theory. Proc. London Math. Soc. 64 (1992), 418–48. [5] C. A. Akemann and N. Weaver, Geometric characterizations of some classes of operators in C∗ -algebras and von Neumann algebras. Proc. Amer. Math. Soc. 130 (2002), 3033–7. [6] A. A. Albert, On a certain algebra of quantum mechanics. Ann. of Math. 35 (1934), 65–73. Reprinted in [692], pp. 85–93. [7] A. A. Albert, Quadratic forms permitting composition. Ann. of Math. 43 (1942), 161–77. Reprinted in [692], pp. 219–35. [8] A. A. Albert, The radical of a non-associative algebra. Bull. Amer. Math. Soc. 48 (1942), 891–7. Reprinted in [692], pp. 277–83. 546 [9] A. A. Albert, Absolute valued real algebras. Ann. of Math. 48 (1947), 495–501. Reprinted in [691], pp. 643–9. [10] A. A. Albert, A structure theory for Jordan algebras. Ann. of Math. 48 (1947), 546–67. Reprinted in [692], pp. 401–22. [11] A. A. Albert, On the power-associativity of rings. Summa Brasil. Math. 2 (1948), 21–32. Reprinted in [692], pp. 423–35. [12] A. A. Albert, Power-associative rings. Trans. Amer. Math. Soc. 608 (1948), 552–93. Reprinted in [692], pp. 437–78. 273 [13] A. A. Albert, Absolute valued algebraic algebras. Bull. Amer. Math. Soc. 55 (1949), 763–8. Reprinted in [691], pp. 651–6. [14] A. A. Albert, A note of correction. Bull. Amer. Math. Soc. 55 (1949), 1191. [15] E. M. Alfsen, F. W. Shultz, and E. Størmer, A Gelfand–Neumark theorem for Jordan algebras. Adv. Math. 28 (1978), 11–56. 398, 401 [16] G. R. Allan, Some simple proofs in holomorphic spectral theory. In Perspectives in operator theory, pp. 9–15, Banach Center Publ. 75, Polish Acad. Sci. Inst. Math., Warsaw, 2007.

664

References

665

[17] K. Alvermann, The multiplicative triangle inequality in noncommutative JB- and JB∗ -algebras. Abh. Math. Sem. Univ. Hamburg 55 (1985), 91–6. [18] K. Alvermann, Real normed Jordan Banach algebras with an involution. Arch. Math. (Basel) 47 (1986), 135–50. [19] K. Alvermann and G. Janssen, Real and complex non-commutative Jordan Banach algebras. Math. Z. 185 (1984), 105–13. xix, xx, 19, 267, 268, 334, 335, 398, 411, 421 [20] W. Ambrose, Structure theorems for a special class of Banach algebras. Trans. Amer. Math. Soc. 57 (1945), 364–886. xxiii, xxv, 477, 545, 550 [21] J. A. Anquela, F. Montaner, and T. Cort´es, On primitive Jordan algebras. J. Algebra 163 (1994), 663–74. xxii, 462, 466 [22] C. Aparicio, F. G. Oca˜na, R. Pay´a, and A. Rodr´ıguez, A non-smooth extension of Fr´echet differentiability of the norm with applications to numerical ranges. Glasgow Math. J. 28 (1986), 121–37. [23] C. Aparicio and A. Rodr´ıguez, Sobre el espectro de derivaciones y automorfismos de las a´ lgebras de Banach. Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp.) 79 (1985), 113–18. [24] H. Araki and G. A. Elliott, On the definition of C∗ -algebras. Publ. Res. Inst. Math. Sci. 9 (1973/74), 93–112. [25] J. Arazy, Isometries of Banach algebras satisfying the von Neumann inequality. Math. Scand. 74 (1994), 137–51. 136, 207 [26] J. Arazy and B. Solel, Isometries of nonselfadjoint operator algebras. J. Funct. Anal. 90 (1990), 284–305. 207 [27] W. Arendt, P. R. Chernoff, and T. Kato, A generalization of dissipativity and positive semigroups. J. Operator Theory 8 (1982), 167–80. [28] R. Arens, Operations induced in function classes. Monatsh. Math. 55 (1951), 1–19. [29] R. Arens, The adjoint of a bilinear operation. Proc. Amer. Math. Soc. 2 (1951), 839–48. [30] R. Arens and I. Kaplansky, Topological representation of algebras. Trans. Amer. Math. Soc. 63 (1948), 457–81. [31] S. A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces. J. Amer. Math. Soc. 13 (2000), 243–94. [32] S. A. Argyros and R. G. Haydon, A hereditarily indecomposable L∞ -space that solves the scalar-plus-compact problem. Acta Math. 206 (2011), 1–54. [33] S. A. Argyros, J. L´opez-Abad, and S. Todorcevic, A class of Banach spaces with few non-strictly singular operators. J. Funct. Anal. 222 (2005), 306–84. [34] R. M. Aron and R. H. Lohman, A geometric function determined by extreme points of the unit ball of a normed space. Pacific J. Math. 127 (1987), 209–31. [35] A. Arosio, Locally convex inductive limits of normed algebras. Rend. Sem. Mat. Univ. Padova 51 (1974), 333–59. [36] H. Auerbach, Sur les groupes lin´eaires I, II, III. Studia Math. 4 (1934), 113–27; Ibid. 4 (1934), 158–66; Ibid. 5 (1935), 43–9. [37] B. Aupetit, Symmetric almost commutative Banach algebras. Notices Amer. Math. Soc. 18 (1971), 559–60. [38] B. Aupetit, Caract´erisation spectrale des alg`ebres de Banach commutatives. Pacific J. Math. 63 (1976), 23–35. [39] B. Aupetit, Caract´erisation spectrale des alg`ebres de Banach de dimension finie. J. Funct. Anal. 26 (1977), 232–50. [40] B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras. J. Funct. Anal. 47 (1982), 1–6. x, 597 [41] B. Aupetit, Recent trends in the field of Jordan–Banach algebras. In [1189], pp. 9–19. 597

666

References

[42] B. Aupetit, Analytic Multifunctions and their applications. In Complex potential theory (ed. P. M. Gauthier), pp. 1–74, Kluwer Academic Publishers, Dordrecht– Boston–London 1994. [43] B. Aupetit, Spectral characterization of the socle in Jordan–Banach algebras. Math. Proc. Cambridge Philos. Soc. 117 (1995), 479–89. 601 [44] B. Aupetit, Trace and spectrum preserving linear mappings in Jordan–Banach algebras. Monatsh. Math. 125 (1998), 179–87. 601 [45] B. Aupetit and L. Baribeau, Sur le socle dans les alg`ebres de Jordan–Banach. Canad. J. Math. 41 (1989), 1090–100. 599 [46] B. Aupetit and A. Maouche, Trace and determinant in Jordan–Banach algebras. Publ. Mat. 46 (2002), 3–16. 601 [47] B. Aupetit and M. Mathieu, The continuity of Lie homomorphisms. Studia Math. 138 (2000), 193–99. 603 [48] B. Aupetit and M. A. Youngson, On symmetry of Banach Jordan algebras. Proc. Amer. Math. Soc. 91 (1984), 364–6. 597 [49] A. Avil´es and P. Koszmider, A Banach space in which every injective operator is surjective. Bull. London Math. Soc. 45 (2013), 1065–74. [50] R. Baer, Radical ideals. Amer. J. Math. 65 (1943), 537–68. [51] J. C. Baez, The octonions. Bull. Amer. Math. Soc. 39 (2002), 145–205. Errata. Ibid. 42 (2005), 213. [52] M. Baillet, Analyse spectrale des op´erateurs hermitiens d’une espace de Banach. J. London Math. Soc. 19 (1979), 497–508. [53] V. K. Balachandran, Simple L∗ -algebras of classical type. Math. Ann. 180 (1969), 205–19. 554 [54] V. K. Balachandran and P. S. Rema, Uniqueness of the norm topology in certain Banach Jordan algebras. Publ. Ramanujan Inst. 1 (1969), 283–9. 597 [55] V. K. Balachandran and N. Swaminatan, Real H ∗ -algebras. J. Funct. Anal. 65 (1986), 64–75. 551 [56] P. Bandyopadhyay, K. Jarosz, and T. S. S. R. K. Rao, Unitaries in Banach spaces. Illinois J. Math. 48 (2004), 339–51. [57] B. A. Barnes, Locally B∗ -equivalent algebras. Trans. Amer. Math. Soc. 167 (1972), 435–42. [58] B. A. Barnes, Locally B∗ -equivalent algebras II. Trans. Amer. Math. Soc. 176 (1973), 297–303. [59] B. A. Barnes, A note on invariance of spectrum for symmetric Banach ∗-algebras. Proc. Amer. Math. Soc. 126 (1998), 3545–7. [60] T. J. Barton and Y. Friedman, Bounded derivations of JB∗ -triples. Quart. J. Math. Oxford 41 (1990), 255–68. xx, 274, 337, 338 [61] F. L. Bauer, On the field of values subordinate to a norm. Numer. Math. 4 (1962), 103–11. [62] J. Becerra, M. Burgos, A. Kaidi, and A. Rodr´ıguez, Banach algebras with large groups of unitary elements. Quart. J. Math. Oxford 58 (2007), 203–20. [63] J. Becerra, M. Burgos, A. Kaidi, and A. Rodr´ıguez, Banach spaces whose algebras of operators have a large group of unitary elements. Math. Proc. Camb. Phil. Soc. 144 (2008), 97–108. [64] J. Becerra, M. Burgos, A. Kaidi, and A. Rodr´ıguez, Nonassociative unitary Banach algebras. J. Algebra 320 (2008), 3383–97. [65] J. Becerra, S. Cowell, A. Rodr´ıguez, and G. V. Wood, Unitary Banach algebras. Studia Math. 162 (2004), 25–51.

References

667

[66] J. Becerra, G. L´opez, A. M. Peralta, and A. Rodr´ıguez, Relatively weakly open sets in closed balls of Banach spaces, and real JB∗ -triples of finite rank. Math. Ann. 330 (2004), 45–58. 243, 274, 336 [67] J. Becerra, G. L´opez, and A. Rodr´ıguez, Relatively weakly open sets in closed balls of C∗ -algebras. J. London Math. Soc. 68 (2003), 753–61. 274, 336, 421 [68] J. Becerra, A. Moreno, and A. Rodr´ıguez, Absolute-valuable Banach spaces. Illinois J. Math. 49 (2005), 121–38. [69] J. Becerra and A. M. Peralta, Subdifferentiability of the norm and the Banach–Stone theorem for real and complex JB∗ -triples. Manuscripta Math. 114 (2004), 503–16. [70] J. Becerra and A. Rodr´ıguez, Isometric reflections on Banach spaces after a paper of A. Skorik and M. Zaidenberg. Rocky Mountain J. Math. 30 (2000), 63–83. [71] J. Becerra and A. Rodr´ıguez, Transitivity of the norm on Banach spaces having a Jordan structure, Manuscripta Math. 102 (2000), 111–27. 210 [72] J. Becerra and A. Rodr´ıguez, Characterizations of almost transitive superrreflexive Banach spaces. Comment. Math. Univ. Carolinae 42 (2001), 629–36. [73] J. Becerra and A. Rodr´ıguez, Transitivity of the norm on Banach spaces. Extracta Math. 17 (2002), 1–58. [74] J. Becerra and A. Rodr´ıguez, Strong subdifferentiability of the norm on JB∗ -triples. Quart. J. Math. Oxford 54 (2003), 381–90. [75] J. Becerra and A. Rodr´ıguez, Big points in C∗ -algebras and JB∗ -triples. Quart. J. Math. Oxford 56 (2005), 141–64. [76] J. Becerra and A. Rodr´ıguez, Absolute-valued algebras with involution, and infinitedimensional Terekhin’s trigonometric algebras. J. Algebra 293 (2005), 448–56. [77] J. Becerra and A. Rodr´ıguez, Non self-adjoint idempotents in C∗ - and JB∗ -algebras. Manuscripta Math. 124 (2007), 183–93. [78] J. Becerra and A. Rodr´ıguez, C∗ - and JB∗ -algebras generated by a non-self-adjoint idempotent. J. Funct. Anal. 248 (2007), 107–27. [79] J. Becerra and A. Rodr´ıguez, Locally uniformly rotund points in convex-transitive Banach spaces. J. Math. Anal. Appl. 360 (2009), 108–18. [80] J. Becerra, A. Rodr´ıguez, and G. V. Wood, Banach spaces whose algebras of operators are unitary: a holomorphic approach. Bull. London Math. Soc. 35 (2003), 218–24. 210 [81] A. Beddaa and M. Oudadess, On a question of A. Wilansky in normed algebras. Studia Math. 95 (1989), 175–7. [82] H. Behncke, Hermitian Jordan Banach algebras. J. London Math. Soc. 20 (1979), 327–33. 597 [83] H. Behncke and F. O. Nyamwala, Two projections and one idempotent. Int. J. Pure Appl. Math. 52 (2009), 501–10. [84] J. P. Bell and L. W. Small, A question of Kaplansky. J. Algebra 258 (2002), 386–8. [85] A. Bensebah, JV-alg`ebres et JH ∗ -alg`ebres. Canad. Math. Bull. 34 (1991), 447–55. 549, 597 [86] A. Bensebah, Weakness of the topology of a JB∗ -algebra. Canad. Math. Bull. 35 (1992), 449–54. 597 [87] M. Benslimane and N. Boudi, Alternative Noetherian Banach Algebras. Extracta Math. 12 (1997), 41–6. 601 [88] M. Benslimane and N. Boudi, Noetherian Jordan Banach algebras are finitedimensional. J. Algebra 213 (1999), 340–50. 601 [89] M. Benslimane, A. Fern´andez, and A. Kaidi, Caract´erisation des alg`ebres de Jordan– Banach de capacit´e finie. Bull. Sci. Math. 112 (1988), 473–80. 597, 598, 599

668

References

[90] M. Benslimane, O. Jaa, and A. Kaidi, The socle and the largest spectrum finite ideal. Quart. J. Math. Oxford 42 (1991), 1–7. 599, 601 [91] M. Benslimane and A. Kaidi, Structure des alg`ebres de Jordan–Banach non commutatives complexes r´eguli`eres ou semi-simples a` spectre fini. J. Algebra 113 (1988), 201–6. xxv, 596, 597, 598 [92] M. Benslimane and N. Merrachi, Alg`ebres de Jordan Banach v´erifiant xx−1  = 1. J. Algebra 206 (1998), 129–34. [93] M. Benslimane and N. Merrachi, Alg`ebres a` puissances associatives norm´ees sans J-diviseurs de z´ero. Algebras Groups Geom. 16 (1999), 355–61. [94] M. Benslimane and A. Moutassim, Some new class of absolute-valued algebras with left-unit. Adv. Appl. Clifford Algebras 21 (2011), 31–40. [95] M. Benslimane and A. Rodr´ıguez, Caract´erisation spectrale des alg`ebres de Jordan Banach non commutatives complexes modulaires annihilatrices. J. Algebra 140 (1991), 344–54. 599 [96] M. I. Berenguer and A. R. Villena, Continuity of Lie derivations on Banach algebras. Proc. Edinburgh Math. Soc. 41 (1998), 625–30. [97] M. I. Berenguer and A. R. Villena, Continuity of Lie mappings of the skew elements of Banach algebras with involution. Proc. Amer. Math. Soc. 126 (1998), 2717–20. [98] M. I. Berenguer and A. R. Villena, Continuity of Lie isomorphisms of Banach algebras. Bull. London Math. Soc. 31 (1999), 6–10. [99] M. I. Berenguer and A. R. Villena, On the range of a Lie derivation on a Banach algebra. Comm. Algebra 28 (2000), 1045–50. [100] E. Berkson, Some characterizations of C∗ -algebras. Illinois J. Math. 10 (1966), 1–8. [101] A. Beurling, Sur les int´egrales de Fourier absolument convergentes et leur application a` fonctionelle. In Neuvi`eme congr`es des math´ematiciens scandinaves. Helsingfors, 1938, pp. 345–66. [102] D. K. Biss, J. D. Christensen, D. Dugger, and D. C. Isaksen, Large annihilators in Cayley–Dickson algebras II. Bol. Soc. Mat. Mexicana 13 (2007), 269–92. [103] D. K. Biss, J. D. Christensen, D. Dugger, and D. C. Isaksen, Eigentheory of Cayley– Dickson algebras. Forum Math. 21 (2009), 833–51. [104] D. K. Biss, D. Dugger, and D. C. Isaksen, Large annihilators in Cayley–Dickson algebras. Comm. Algebra 36 (2008), 632–64. [105] D. Blecher and B. Magajna, Dual operator systems. Bull. London Math. Soc. 43 (2011), 311–20. [106] D. P. Blecher, Z.-J. Ruan, and A. M. Sinclair, A characterization of operator algebras. J. Funct. Anal. 89 (1990), 188–201. [107] R. P. Boas, Entire functions of exponential type. Bull. Amer. Math. Soc. 48 (1942), 839–49. [108] H. F. Bohnenblust and S. Karlin, Geometrical properties of the unit sphere of a Banach algebra. Ann. of Math. 62 (1955), 217–29. [109] B. Bollob´as, An extension to the theorem of Bishop and Phelps. Bull. London Math. Soc. 2 (1970), 181–2. [110] B. Bollob´as, The numerical range in Banach algebras and complex functions of exponential type. Bull. London Math. Soc. 3 (1971), 27–33. [111] F. F. Bonsall, A minimal property of the norm in some Banach algebras. J. London Math. Soc. 29 (1954), 156–64. [112] F. F. Bonsall, Locally multiplicative wedges in Banach algebras. Proc. London Math. Soc. 30 (1975), 239–56. [113] F. F. Bonsall, Jordan algebras spanned by Hermitian elements of a Banach algebra. Math. Proc. Cambridge Philos. Soc. 81 (1977), no. 1, 3–13.

References

669

[114] F. F. Bonsall, Jordan subalgebras of Banach algebras. Proc. Edinburgh Math. Soc. 21 (1978), 103–10. [115] F. F. Bonsall and M. J. Crabb, The spectral radius of a Hermitian element of a Banach algebra. Bull. London Math. Soc. 2 (1970), 178–80. [116] F. F. Bonsall and J. Duncan, Dually irreducible representations of Banach algebras, Quart. J. Math. Oxford 19 (1968), 97–111. [117] R. Bott and J. Milnor, On the parallelizability of the spheres. Bull. Amer. Math. Soc. 64 (1958), 87–9. [118] A. B¨ottcher and I. M. Spitkovsky, A gentle guide to the basics of two projections theory. Linear Algebra Appl. 432 (2010), 1412–59. [119] N. Boudi, On Jordan Banach algebras with countably generated inner ideals. Algebra Colloq. 17 (2010), 211–22. 601 [120] N. Boudi, H. Marhnine, C. Zarhouti, A. Fern´andez, and E. Garc´ıa, Noetherian Banach Jordan pairs. Math. Proc. Cambridge Philos. Soc. 130 (2001), 25–36. 601 [121] K. Bouhya and A. Fern´andez, Jordan-∗-triples with minimal inner ideals and compact JB∗ -triples. Proc. London Math. Soc. 68 (1994), 380–98. 241, 461, 603 [122] K. Boyko, V. Kadets, M. Mart´ın, and D. Werner, Numerical index of Banach spaces and duality. Math. Proc. Cambridge Philos. Soc. 142 (2007), 93–102. [123] O. Bratteli and D. W. Robinson, Unbounded derivations of C∗ -algebras. II. Comm. Math. Phys. 46 (1976), 11–30. [124] R. B. Braun, Structure and representations of non-commutative C∗ -Jordan algebras. Manuscripta Math. 41 (1983), 139–71. xx, xxi, 398, 411, 421 [125] R. B. Braun, A Gelfand–Neumark theorem for C∗ -alternative algebras. Math. Z. 185 (1984), 225–42. xx, xxi, 411 [126] R. B. Braun, W. Kaup, and H. Upmeier, A holomorphic characterization of Jordan C∗ -algebras. Math. Z. 161 (1978), 277–90. 210 [127] M. Brelot, Points irr´eguliers et transformations continues en th´eorie du potentiel. J. Math. Pures Appl. 19 (1940), 319–37. [128] M. Brelot, Sur les ensembles effil´es. Bull. Sci. Math. 68 (1944), 12–36. [129] M. Bremner and I. Hentzel, Identities for algebras obtained from the Cayley–Dickson process. Comm. Algebra 29 (2001), 3523–34. [130] M. Breˇsar, M. Cabrera, M. Fosner, and A. R. Villena, Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan–Banach algebras. Studia Math. 169 (2005), 207–28. 602 ˇ ˇ [131] M. Breˇsar, P. Semrl, and S. Spenko, On locally complex algebras and low-dimensional Cayley–Dickson algebras. J. Algebra 327 (2011), 107–25. [132] A. Browder, States, numerical ranges, etc.. In Proceedings of the Brown informal analysis seminar. Summer, 1969 (revised version, Summer of 2012). [133] A. Browder, On Bernstein’s inequality and the norm of Hermitian operators. Amer. Math. Monthly 78 (1971), 871–3. [134] R. Brown, On generalized Cayley–Dickson algebras. Pacific J. Math. 20 (1967), 415–22. [135] L. J. Bunce, F. J. Fern´andez-Polo, J. Mart´ınez, and A. M. Peralta, A Saitˆo–Tomita– Lusin theorem for JB∗ -triples and applications. Quart. J. Math. Oxford 57 (2006), 37–48. [136] L. J. Bunce and A. M. Peralta, Images of contractive projections on operator algebras. J. Math. Anal. Appl. 272 (2002), 55–66. 210 [137] M. J. Burgos, A. M. Peralta, M. I. Ram´ırez, and E. E. Ruiz, Von Neumann regularity in Jordan–Banach triples. In [704], pp. 67–88.

670

References

[138] R. C. Busby, Double centralizers and extensions of C∗ -algebras. Trans. Amer. Math. Soc. 132 (1968), 79–99. [139] F. Cabello, Regards sur le probl`eme des rotations de Mazur. Extracta Math. 12 (1997), 97–116. [140] M. Cabrera, A. El Marrakchi, J. Mart´ınez, and A. Rodr´ıguez, An Allison– Kantor–Koecher–Tits construction for Lie H ∗ -algebras. J. Algebra 164 (1994), 361–408. xxv, 556 [141] M. Cabrera, J. Mart´ınez, and A. Rodr´ıguez, Malcev H ∗ -algebras. Math. Proc. Cambridge Phil. Soc. 103 (1988), 463–71. xxv, 553, 554 [142] M. Cabrera, J. Mart´ınez, and A. Rodr´ıguez, Nonassociative real H ∗ -algebras. Publ. Mat. 32 (1988), 267–74. xxiv, 549, 551 [143] M. Cabrera, J. Mart´ınez, and A. Rodr´ıguez, A note on real H ∗ -algebras. Math. Proc. Cambridge Phil. Soc. 105 (1989), 131–2. [144] M. Cabrera, J. Mart´ınez, and A. Rodr´ıguez, Structurable H ∗ -algebras. J. Algebra 147 (1992), 19–62. xxv, 552, 553, 555, 556 [145] M. Cabrera, A. Moreno, and A. Rodr´ıguez, Normed versions of the Zel’manov prime theorem: positive results and limits. In: Operator theory, operator algebras and related topics (Timisoara, 1996) (eds. A. Gheondea, R. N. Gologan and D. Timotin), Theta Found., Bucharest, 1997, pp. 65–77. 462, 463, 465 [146] M. Cabrera, A. Moreno, and A. Rodr´ıguez, Zel’manov’s theorem for primitive Jordan–Banach algebras. J. London Math. Soc. 57 (1998), 231–44. xxii, 463, 464, 465 [147] M. Cabrera, A. Moreno, A. Rodr´ıguez, and E. I. Zel’manov, Jordan polynomials can be analytically recognized. Studia Math. 117 (1996), 137–47. xxii, 437, 470 [148] M. Cabrera and A. Rodr´ıguez, Extended centroid and central closure of semiprime normed algebras: a first approach. Comm. Algebra 18 (1990), 2293–326. xxiii, 408, 546, 555 [149] M. Cabrera and A. Rodr´ıguez, Nonassociative ultraprime normed algebras. Quart. J. Math. Oxford 43 (1992), 1–7. xxi, xxiii, 409, 546 [150] M. Cabrera and A. Rodr´ıguez, New associative and nonassociative Gelfand–Naimark theorems. Manuscripta Math. 79 (1993), 197–208. [151] M. Cabrera and A. Rodr´ıguez, Zel’manov theorem for normed simple Jordan algebras with a unit. Bull. London Math. Soc. 25 (1993), 59–63. xxii, 467, 468 [152] M. Cabrera and A. Rodr´ıguez, Non-degenerately ultraprime Jordan–Banach algebras: a Zel’manovian treatment. Proc. London Math. Soc. 69 (1994), 576–604. xxii, 409, 468, 469 [153] M. Cabrera and A. Rodr´ıguez, A new simple proof of the Gelfand–Mazur–Kaplansky theorem. Proc. Amer. Math. Soc. 123 (1995), 2663–6. 408 [154] M. Cabrera and A. Rodr´ıguez, On the Gelfand–Naimark axiom a∗ a = a∗ a. Quart. J. Math. Oxford 63 (2012), 855–60. [155] A. Calder´on, A. Kaidi, C. Mart´ın, A. Morales, M. Ram´ırez, and A. Rochdi, Finite-dimensional absolute-valued algebras. Israel J. Math. 184 (2011), 193–220. [156] A. J. Calder´on and C. Mart´ın, Two-graded absolute valued algebras. J. Algebra 292 (2005), 492–515. [157] J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. of Math. 42 (1941), 839–73. [158] S. R. Caradus, Operators of Riesz type. Pacific J. Math. 18 (1966), 61–71. [159] E. Cartan, Les groupes bilin´eaires et les syst`emes de nombres complexes. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 12 (1898), 1–99. Reprinted in Oeuvres Compl`etes, Vol II, Paris, Gauthier-Villars, 1952.

References

671

[160] H. Cartan, Th´eorie g´en´erale du balayage en potentiel newtonien. Ann. Univ. Grenoble. Sect. Sci. Math. Phys. 22 (1946), 221–80. [161] P. G. Casazza, Approximation properties. In Handbook of the geometry of Banach spaces, Vol. I (eds. W. B. Johnson and J. Lindenstrauss), pp. 271–316, North-Holland, Amsterdam, ´ 2001. Addenda and corrigenda: In [757], pp. 1817–18. [162] A. Castellon and J. A. Cuenca, Isomorphisms of H ∗ -triple systems. Ann. Scuola Norm. Sup.´Pisa Sci. Fis. Mat. 19 (1992), 507–14. 561 [163] A. Castellon and J. A. Cuenca, Associative H ∗ -triple systems. In [734], pp. 45–67. 551, 561 [164] A. Cayley, On Jacobi’s elliptic functions, in reply to the Rev. B. Bronwin; and on quaternions. Philos. Mag. 26 (1845), 208–11. [165] A. Cedilnik and A. Rodr´ıguez, Continuity of homomorphisms into complete normed algebraic algebras. J. Algebra 264 (2003), 6–14. 595, 596 [166] A. Cedilnik and B. Zalar, Nonassociative algebras with submultiplicative bilinear form. Acta Math. Univ. Comenian. 63 (1994), 285–301. [167] K.-C. Chan and D. Z. Dokovic, Conjugacy classes of subalgebras of the real sedenions. Canad. Math. Bull. 49 (2006), 492–507. [168] A. Chandid and A. Rochdi, Mutations of absolute valued algebras. Int. J. Algebra 2 (2008), 357–68. [169] M. D. Choi and E. Effros, Injectivity and operator spaces. J. Funct. Anal. 24 (1974), 156–209. on it. λ [170] W. Chonghu, Some geometric properties of Banach spaces and x+λy−x J. Nanjing Univ. 6 (1989), 37–45.  [171] C.-H. Chu, T. Dang, B. Russo, and B. Ventura, Surjective isometries of real C∗ -algebras. J. London Math. Soc. 47 (1993), 97–118. [172] C.-H. Chu, B. Iochum, and G. Loupias, Grothendieck’s theorem and factorization of operators in Jordan triples. Math. Ann. 284 (1989), 41–53. xix, 2, 245, 267, 343, 344, 346 [173] C.-H. Chu and P. Mellon, Jordan structures in Banach spaces and symmetric manifolds. Expos. Math. 16 (1998), 157–80. 211 [174] C.-H. Chu and M. V. Velasco, Automatic continuity of homomorphisms in nonassociative Banach algebras. Canadian J. Math. 65 (2013), 989–1004. [175] P. Civin and B. Yood, Lie and Jordan structures in Banach algebras. Pacific J. Math. 15 (1965), 775–97. 583 [176] S. B. Cleveland, Homomorphisms of non-commutative ∗-algebras. Pacific J. Math. 13 (1963), 1097–109. [177] M. A. Cobalea and A. Fern´andez, Prime noncommutative Jordan algebras and central closure. Algebras Groups Geom. 5 (1988), 129–36. 408 [178] P. M. Cohn, On homomorphic images of special Jordan algebras. Canadian J. Math. 6 (1954), 253–64. [179] M. D. Contreras and R. Pay´a, On upper semicontinuity of duality mappings. Proc. Amer. Math. Soc. 121 (1994), 451–9. [180] M. D. Contreras, R. Pay´a, and W. Werner, C∗ -algebras that are I-rings. J. Math. Anal. Appl. 198 (1996), 227–36. [181] E. R. Cowie, An analytic characterization of groups with no finite conjugacy classes. Proc. Amer. Math. Soc. 87 (1983), 7–10. [182] M. J. Crabb, J. Duncan, and C. M. McGregor, Some extremal problems in the theory of numerical ranges. Acta Math. 128 (1972), 123–42. [183] M. J. Crabb, J. Duncan, and C. M. McGregor, Characterizations of commutativity for C∗ -algebras. Glasgow Math. J. 15 (1974), 172–5. 400

672

References

[184] M. J. Crabb and C. M. McGregor, Numerical ranges of powers of Hermitian elements. Glasgow Math. J. 28 (1986), 37–45. [185] M. J. Crabb and C. M. McGregor, Polynomials in a Hermitian element. Glasgow Math. J. 30 (1988), 171–6. [186] B. Cuartero and J. E. Gal´e, Locally PI-algebras over valued fields. In Aportaciones Matem´aticas en Memoria del Profesor V. M. Onieva, pp. 137–45, Santander, Universidad de Cantabria, 1991. [187] B. Cuartero and J. E. Gal´e, Bounded degree of algebraic topological algebras. Comm. Algebra 422 (1994), 329–37. [188] B. Cuartero, J. E. Gal´e, A. Rodr´ıguez, and A. M. Slinko, Bounded degree of weakly algebraic topological Lie algebras. Manuscripta Math. 81 (1993), 129–39. 658 [189] D. F. Cudia, The geometry of Banach spaces. Smoothness. Trans. Amer. Math. Soc. 110 (1964), 284–314. [190] J. A. Cuenca, On one-sided division infinite-dimensional normed real algebras. Publ. Mat. 36 (1992), 485–8. [191] J. A. Cuenca, On composition and absolute-valued algebras. Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 717–31. [192] J. A. Cuenca, On an Ingelstam’s theorem. Comm. Algebra 35 (2007), 4057–67. [193] J. A. Cuenca, On structure theory of pre-Hilbert algebras. Proc. R. Soc. Edinburgh 139A (2009), 303–19. [194] J. A. Cuenca, Some classes of pre-Hilbert algebras with norm-one central idempotent. Israel J. Math. 193 (2013), 343–58. [195] J. A. Cuenca, A new class of division algebras and Wedderburn’s and Frobenius’ Theorems. Preprint 2012. [196] J. A. Cuenca, Third-power associative absolute vaued algebras with a nonzero idempotent commuting with all idempotents. Publ. Mat. 58 (2014), 469–84. [197] J. A. Cuenca, A. Garc´ıa, and C. Mart´ın. Structure theory for L∗ -algebras, Math. Proc. Cambridge Phil. Soc. 107 (1990), 361–5. xxv, 554 [198] J. A. Cuenca and A. Rodr´ıguez, Isomorphisms of H ∗ -algebras. Math. Proc. Cambridge Phil. Soc. 97 (1985), 93–9. xxiv, 546, 549 [199] J. A. Cuenca and A. Rodr´ıguez, Structure theory for noncommutative Jordan H ∗ -algebras. J. Algebra 106 (1987), 1–14. xxiii, xxv, 545, 552, 553 [200] J. A. Cuenca and A. Rodr´ıguez, Absolute values on H ∗ -algebras. Comm. Algebra 23 (1995), 1709–40. 481 [201] J. Cuntz, Locally C∗ -equivalent algebras. J. Funct. Anal. 23 (1976), 95–106. [202] J. M. Cusack, Jordan derivations on rings. Proc. Amer. Math. Soc. 53 (1975), 321–4. [203] H. G. Dales, Norming nil algebras. Proc. Amer. Math. Soc. 83 (1981), 71–4. [204] H. G. Dales, On norms on algebras. In [773], pp. 61–96. [205] T. Dang, Real isometries between JB∗ -triples. Proc. Amer. Math. Soc. 114 (1992), 971–80. [206] T. Dang and B. Russo, Real Banach Jordan triples. Proc. Amer. Math. Soc. 122 (1994), 135–45. [207] E. Darp¨o and A. Rochdi, Classification of the four-dimensional power-commutative real division algebras. Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), 1207–23. [208] W. J. Davis, Separable Banach spaces with only trivial isometries. Rev. Roumaine Math. Pures Appl. 16 (1971), 1051–4. [209] W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczy´nski, Factoring weakly compact operators. J. Funct. Anal. 17 (1974), 311–27. [210] O. Diankha, A. Diouf, and A. Rochdi, A brief statement on the absolute-valued algebras with one-sided unit. Int. J. Algebra 7 (2013), 833–8.

References

673

[211] L. E. Dickson, On quaternions and their generalization and the history of the eight square theorem. Ann. of Math. 20 (1919), 155–71. [212] L. E. Dickson, Linear algebras with associativity not assumed. Duke Math. J. 1 (1935), 113–25. [213] S. Dineen, The second dual of a JB∗ -triple system. In Complex Analysis, Functional Analysis and Approximation Theory (ed. J. M´ugica), pp. 67–9, North-Holland Math. Stud. 125, North-Holland, Amsterdam–New York, 1986. xix, 2, 211, 236, 241 [214] R. S. Doran, A generalization of a theorem of Civin and Yood on Banach ∗-algebras. Bull. London Math. Soc. 4 (1972), 25–6. [215] G. V. Dorofeev, An example of a solvable but not nilpotent alternative ring. Uspekhi Mat. Nauk 15 (1960), 147–50. [216] J. Duncan, Review of [425]. Math. Rev. 87e:46067. [217] J. Duncan, C. M. McGregor, J. D. Pryce, and A. J. White, The numerical index of a normed space. J. London Math. Soc. 2 (1970), 481–8. [218] J. Duncan and P. J. Taylor, Norm inequalities for C∗ -algebras. Proc. Roy. Soc. Edinburgh Sect. A 75 (1975/76), 119–29. 400 [219] N. Dunford, Spectral theory I. Convergence to projections. Trans. Amer. Math. Soc. 54 (1943), 185–217. [220] S. Dutta and T. S. S. R. K. Rao, Norm-to-weak upper semi-continuity of the pre-duality map. J. Anal. 12 (2005), 1–10. [221] P. Eakin and A. Sathaye, On automorphisms and derivations of Cayley–Dickson algebras. J. Algebra 129 (1990), 263–78. [222] C. M. Edwards, On Jordan W ∗ -algebras. Bull. Sci. Math. 104 (1980), 393–403. xix, xx, 2, 19 [223] C. M. Edwards, Multipliers of JB-algebras. Math. Ann. 249 (1980), 265–72. [224] C. M. Edwards, F. J. Fern´andez-Polo, C. S. Hoskin, and A. M. Peralta, On the facial structure of the unit ball in a JB∗ -triple. J. Reine Angew. Math. 641 (2010), 123–44. [225] C. M. Edwards and G. T. R¨uttimann, On the facial structure of the unit balls in a JBW ∗ -triple and its predual. J. London Math. Soc. 38 (1988), 317–32. [226] C. M. Edwards and G. T. R¨uttimann, The facial and inner ideal structure of a real JBW ∗ -triple. Math. Nachr. 222 (2001), 159–84. [227] R. E. Edwards, Multiplicative norms on Banach algebras. Math. Proc. Cambridge Phil. Soc. 47 (1951), 473–4. [228] E. G. Effros and E. Størmer, Jordan algebras of self-adjoint operators. Trans. Amer. Math. Soc. 127 (1967), 313–16. [229] E. G. Effros and E. Størmer, Positive projections and Jordan structure in operator algebras. Math. Scand. 45 (1979), 127–38. 210 [230] M. Eidelheit, On isomorphisms of rings of linear operators. Studia Math. 9 (1940), 97–105. [231] S. Eilenberg, Extensions of general algebras. Ann. Soc. Polon. Math. 21 (1948), 125–34. [232] A. Elduque and J. M. P´erez, Infinite-dimensional quadratic forms admitting composition. Proc. Amer. Math. Soc. 125 (1997), 2207–16. [233] A. J. Ellis, The duality of partially ordered normed linear spaces. J. London Math. Soc. 39 (1964), 730–44. [234] A. J. Ellis, Minimal decompositions in base normed spaces. In Foundations of quantum mechanics and ordered linear spaces (Advanced Study Inst., Marburg, 1973), pp. 30–2. Lecture Notes in Phys. 29, Springer, Berlin, 1974. [235] M. L. El-Mallah, Quelques r´esultats sur les alg`ebres absolument valu´ees. Arch. Math. 38 (1982), 432–7.

674

References

[236] M. L. El-Mallah, Sur les alg`ebres absolument valu´ees qui v´erifient l’identit´e (x, x, x) = 0. J. Algebra 80 (1983), 314–22. [237] M. L. El-Mallah, On finite dimensional absolute valued algebras satisfying (x, x, x) = 0. Arch. Math. 49 (1987), 16–22. [238] M. L. El-Mallah, Absolute valued algebras with an involution. Arch. Math. 51 (1988), 39–49. [239] M. L. El-Mallah, Absolute valued algebras containing a central idempotent. J. Algebra 128 (1990), 180–7. [240] M. L. El-Mallah, Absolute valued algebraic algebra satisfying (x, x, x) = 0. Pure Math. Appl. 8 (1997), 39–52. [241] M. L. El-Mallah, Semi-algebraic absolute valued algebras with an involution. Comm. Algebra 31 (2003), 3135–41. [242] M. L. El-Mallah, H. Elgendy, A. Rochdi, and A. Rodr´ıguez, On absolute valued algebras with involution. Linear Algebra Appl. 414 (2006), 295–303. [243] M. L. El-Mallah and A. Micali, Sur les alg`ebres norm´ees sans diviseurs topologiques de z´ero. Bol. Soc. Mat. Mexicana 25 (1980), 23–8. [244] M. L. El-Mallah and A. Micali, Sur les dimensions des alg`ebres absolument valu´ees. J. Algebra 68 (1981), 237–46. [245] P. Enflo, A counterexample to the approximation problem in Banach spaces. Acta Math. 130 (1973), 309–17. [246] T. S. Erickson, W. S. Martindale III, and J. M. Osborn, Prime nonassociative algebras. Pacific J. Math. 60 (1975), 49–63. 495 [247] J. Esterle, Normes d’alg`ebres minimales, topologie d’alg`ebre norm´ee minimum sur certaines alg`ebres d’endomorphismes continus d’un espace norm´e. C. R. Acad. Sci. Paris S´er. A-B 277 (1973), A425–7. [248] K. Fan and I. Glicksberg, Some geometrical properties of the spheres in a normed linear space. Duke Math. J. 25 (1958), 553–68. [249] K. E. Feldman, New proof of Frobenius hypothesis on the dimensions of real algebras without divisors of zero. Moscow Univ. Math. Bull. 55 (2000), 48–50. [250] A. Fern´andez, Modular annihilator Jordan algebras. Comm. Algebra 13 (1985), 2597–613. 598 [251] A. Fern´andez, Noncommutative Jordan Riesz algebras. Quart. J. Math. Oxford 39 (1988), 67–80. 598, 599 [252] A. Fern´andez, Noncommutative Jordan algebras containing minimal inner ideals. In [749], pp. 153–76. [253] A. Fern´andez, Banach–Lie algebras spanned by extremal elements. In [704], pp. 125–32. 598 [254] A. Fern´andez, Banach–Lie algebras with extremal elements. Quart. J. Math. Oxford 62 (2011), 115–29. 598 [255] A. Fern´andez, E. Garc´ıa, and A. Rodr´ıguez, A Zel’manov prime theorem for JB∗ -algebras. J. London Math. Soc. 46 (1992), 319–35. xxi, xxii, 405, 408, 437, 466, 467 [256] A. Fern´andez, E. Garc´ıa, and E. S´anchez, von Neumann regular Jordan Banach triple systems. J. London Math. Soc. 42 (1990), 32–48. 599, 600 [257] A. Fern´andez and A. Rodr´ıguez, Primitive noncommutative Jordan algebras with nonzero socle. Proc. Amer. Math. Soc. 96 (1986), 199–206. 598 [258] A. Fern´andez and A. Rodr´ıguez, On the socle of a noncommutative Jordan algebra. Manuscripta Math. 56 (1986), 269–78. 598 [259] A. Fern´andez and A. Rodr´ıguez, A Wedderburn theorem for nonassociative complete normed algebras. J. London Math. Soc. 33 (1986), 328–38. xxv, 548, 583, 584, 585

References

675

[260] F. J. Fern´andez-Polo, J. Mart´ınez, and A. M. Peralta, Surjective isometries between real JB∗ -triples. Math. Proc. Cambridge Phil. Soc. 137 (2004), 709–23. [261] F. J. Fern´andez-Polo, J. Mart´ınez, and A. M. Peralta, Geometric characterization of tripotents in real and complex JB∗ -triples. J. Math. Anal. Appl. 295 (2004), 435–43. [262] F. J. Fern´andez-Polo, J. J. Garc´es, and A. M. Peralta, A Kaplansky theorem for JB∗ -triples. Proc. Amer. Math. Soc. 140 (2012), 3179–91. [263] C. Finet, Uniform convexity properties of norms on superreflexive Banach spaces. Israel J. Math. 53 (1986), 81–92. [264] C. Foias, Sur certains theoremes de J. von Neumann concernant les ensembles spectraux. Acta Sci. Math. 18 (1957), 15–20. [265] J. W. M. Ford, A square root lemma for Banach ∗-algebras. J. London Math. Soc. 42 (1967), 521–2. [266] E. Formanek, The Nagata–Higman Theorem. Acta Appl. Math. 21 (1990), 185–92. [267] C. Franchetti, Lipschitz maps and the geometry of the unit ball in normed spaces. Arch. Math. 46 (1986), 76–84. [268] C. Franchetti and R. Pay´a, Banach spaces with strongly subdifferentiable norm. Boll. Un. Mat. Ital. 7 (1993), 45–70. [269] Y. Friedman and B. Russo, Structure of the predual of a JBW ∗ -triple. J. Reine Angew. Math. 356 (1985), 67–89. 236, 274, 332, 337 [270] Y. Friedman and B. Russo, The Gelfand–Naimark theorem for JB∗ -triples. Duke Math. J. 53 (1986), 139–48. xxii, 459 ¨ [271] F. G. Frobenius, Uber lineare Substitutionen und bilineare Formen. J. Reine Angew. Math. 84 (1878), 1–63. Reprinted in Gesammelte Abhandlungen Band I, pp. 343–405. [272] J. Froelich, Unital multiplications on a Hilbert space. Proc. Amer. Math. Soc. 117 (1993), 757–9. [273] M. Fukamiya, On a theorem of Gelfand and Neumark and the B∗ -algebra. Kumamoto J. Sci. Ser. A 1 (1952), 17–22. 686 [274] R. Fuster and A. Marquina, Geometric series in incomplete normed algebras. Amer. Math. Monthly 91 (1984), 49–51. [275] J. E. Gal´e, Weakly compact homomorphisms in nonassociative algebras. In [734], pp. 167–71. [276] J. E. Gal´e, T. J. Ransford, and M. C. White, Weakly compact homomorphisms. Trans. Amer. Math. Soc. 331 (1992), 815–24. 336 [277] E. Galina, A. Kaplan, and L. Saal, Charged representations of the infinite Fermi and Clifford algebras. Lett. Math. Phys. 72 (2005), 65–77. [278] E. Galina, A. Kaplan, and L. Saal, Spinor types in infinite dimensions. J. Lie Theory 15 (2005), 457–95. ¨ [279] V. Gantmacher, Uber schwache totalstetige Operatoren. Rec. Math. [Mat. Sbornik] 7 (1940), 301–8. [280] L. T. Gardner, On isomorphisms of C∗ -algebras. Amer. J. Math. 87 (1965), 384–96. [281] L. T. Gardner, An elementary proof of the Russo–Dye theorem. Proc. Amer. Math. Soc. 90 (1984), 181. [282] S. Garibaldi and H. P. Petersson, Wild Pfister forms over Henselian fields, K-theory, and conic division algebras. J. Algebra 327 (2011), 386–465. [283] I. M. Gelfand, Abstrakte Funktionen und lineare Operatoren, Mat. Sb. 4 (1938), 235–86. [284] I. M. Gelfand, Normierte Ringe. Rec. Math. [Mat. Sbornik] 9 (1941), 3–24. [285] I. M. Gelfand and M. A. Naimark, On the embedding of normed rings into the ring of operators in Hilbert space. Mat. Sbornik 12 (1943), 197–213.

676

References

[286] I. M. Gelfand and A. N. Kolmogorov, Rings of continuous functions on topological spaces. Doklady Akad. Nauk SSSR 22 (1939), 11–15. [287] J. R. Giles, D. A. Gregory, and B. Sims, Geometrical implications of upper semicontinuity of the duality mapping of a Banach space. Pacific J. Math. 79 (1978), 99–109. [288] B. Gleichgewicht, A remark on absolute-valued algebras. Colloq. Math. 11 (1963), 29–30. [289] B. W. Glickfeld, A metric characterization of C(X) and its generalization to C∗ -algebras. Illinois J. Math. 10 (1966), 547–56. [290] J. G. Glimm and R. V. Kadison, Unitary operators in C∗ -algebras. Pacific J. Math. 10 (1960), 547–56. [291] G. Godefroy and V. Indumathi, Norm-to-weak upper semi-continuity of the duality and pre-duality mappings. Set-Valued Anal. 10 (2002), 317–30. [292] G. Godefroy, V. Montesinos, and V. Zizler, Strong subdifferentiability of norms and geometry of Banach spaces. Comment. Math. Univ. Carolinae 36 (1995), 493–502. [293] G. Godefroy and T. S. S. R. K. Rao, Renorming and extremal structures. Illinois J. Math. 48 (2004), 1021–9. [294] I. C. Gohberg, A. S. Markus, and I. A. Fel’dman, Normally solvable operators and ideals associated with them. Bul. Akad. Stiince RSS Moldoven 10 (1960), 51–70. Amer. Math. Soc. Transl. 61 (1967), 63–4. [295] E. S. Golod, On nil-algebras and finitely approximable p-groups. Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 273–6. 633 [296] E. A. Gorin, Bernstein inequalities from the perspective of operator theory. (In Russian.) Vestnik Khar’kov. Gos. Univ. 205 (1980), 77–105, 140. [297] W. T. Gowers and B. Maurey, The unconditional basic sequence problem. J. Amer. Math. Soc. 6 (1993), 851–74. [298] S. Grabiner, The nilpotence of Banach nil algebras. Proc. Amer. Math. Soc. 21 (1969), 510. [299] D. A. Gregory, Upper semi-continuity of subdifferential mappings. Canad. Math. Bull. 23 (1980), 11–19. [300] P. Greim and M. Rajalopagan, Almost transitivity in C0 (L). Math. Proc. Cambridge Phil. Soc. 121 (1997), 75–80. [301] R. Grza´slewicz and P. Scherwentke, On strongly extreme and denting contractions in L (C(X), C(Y)). Bull. Acad. Sinica 25 (1997), 155–60. [302] U. Haagerup, R. V. Kadison, and G. K. Pedersen, Means of unitary operators, revisited. Math. Scand. 100 (2007), 193–7. [303] A. Ha¨ıly, A non-semiprime associative algebra with zero weak radical. Extracta Math. 12 (1997), 53–60. [304] A. Ha¨ıly, A. Kaidi, and A. Rodr´ıguez, Algebra descent spectrum of operators. Israel J. Math. 177 (2010), 349–68. [305] M. Hamana, Injective envelopes of C∗ -algebras. J. Math. Soc. Japan 31 (1979), 181–97. [306] W. R. Hamilton, Note by Professor Sir W. R. Hamilton, respecting the researches of John T. Graves, esq. Trans. R. Irish Acad., 1848, Science 338–41. [307] L. Hammoudi, Nil-algebras and infinite groups. J. Math. Sci. (NY) 144 (2007), 4004–12. [308] H. Hanche-Olsen, A note on the bidual of a JB-algebra. Math. Z. 175 (1980), 29–31. [309] M. L. Hansen and R. V. Kadison, Banach algebras with unitary norms. Pacific J. Math. 175 (1996), 535–52.

References

677

[310] F. Hansen and G. K. Pedersen, Jensen’s inequality for operators and L¨owner’s theorem. Math. Ann. 258 (1981/82), 229–41. [311] P. Harmand and T. S. S. R. K. Rao, An intersection property of balls and relations with M-ideals. Math. Z. 197 (1988), 277–90. [312] L. A. Harris, The numerical range of holomorphic functions in Banach spaces. Amer. J. Math. 93 (1971), 1005–19. [313] L. A. Harris, Banach algebras with involution and M¨obius transformations. J. Funct. Anal. 11 (1972), 1–16. [314] L. A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces. In Proceedings on Infinite Dimensional Holomorphy (Internat. Conf., Univ. Kentucky, Lexington, Ky., 1973), pp. 13–40. Lecture Notes in Math. 364, Springer, Berlin, 1974. 210, 211 [315] L. A. Harris, The numerical range of functions and best approximation. Math. Proc. Cambridge Philos. Soc. 76 (1974), 133–41. [316] L. A. Harris and R. V. Kadison, Schurian algebras and spectral additivity. J. Algebra 180 (1996), 175–86. [317] R. Harte and M. Mbekhta, On generalized inverses in C∗ -algebras. Studia Math. 103 (1992), 71–7. [318] S. Heinrich, Ultraproducts in Banach space theory. J. Reine Angew. Math. 313 (1980), 72–104. 211, 236 [319] S. Hejazian and A. Niknam, A Kaplansky theorem for JB∗ -algebras. Rocky Mountain J. Math. 28 (1998), 977–82. [320] I. N. Herstein, Jordan homomorphism. Trans. Amer. Math. Soc. 81 (1956), 331–41. [321] I. N. Herstein, Jordan derivations of prime rings. Proc. Amer. Math. Soc. 8 (1957), 1104–10. [322] G. Hessenberger, A spectral characterization of the socle of Banach Jordan systems. Math. Z. 223 (1996), 561–8. 603 [323] G. Hessenberger, Inessential and Riesz elements in Banach Jordan systems. Quart. J. Math. Oxford 47 (1996), 337–47. 603 [324] G. Hessenberger, The trace in Banach Jordan pairs. Bull. London Math. Soc. 30 (1998), 561–8. 603 [325] G. Hessenberger, An improved characterization of inessential elements in Banach Jordan systems and the generalized Ruston characterization. Arch. Math. (Basel) 74 (2000), 438–40. 603 [326] G. Hessenberger and A. Maouche, On Banach Jordan systems with at most two spectral values. Far East J. Math. Sci. (FJMS) 7 (2002), 115–20. 603 [327] R. A. Hirschfeld and B. E. Johnson, Spectral characterization of finite-dimensional algebras. Indag. Math. 34 (1972), 19–23. 596 [328] L. Hogben and K. McCrimmon, Maximal modular inner ideals and the Jacobson radical of a Jordan algebra. J. Algebra 68 (1981), 155–69. [329] W. Holsztynski, Continuous mappings induced by isometries of spaces of continuous functions. Studia Math. 26 (1966), 133–6. [330] G. Horn, Classification of JBW ∗ -triples of Type I. Math. Z. 196 (1987), 271–91. xxii, 340, 344, 438, 459 [331] G. Horn, Coordinatization theorem for JBW ∗ -triples. Quart. J. Math. Oxford 38 (1987), 321–35. xxi, 411, 416 [332] X.-J. Huang and C.-K. Ng, An abstract characterization of unital operator spaces. J. Operator Theory 67 (2012), 289–98. [333] T. Huruya, The normed space numerical index of C∗ -algebras. Proc. Amer. Math. Soc. 63 (1977), 289–90.

678

References

¨ [334] A. Hurwitz, Uber die Composition der quadratischen Formen von beliebig vielen Variabelm. Nach. Ges. Wiss. G¨ottingen (1898), 309–16. [335] K. Imaeda and M. Imaeda, Sedenions: algebra and analysis. Appl. Math. Comp. 115 (2000), 77–88. [336] L. Ingelstam, A vertex property for Banach algebras with identity, Math. Scand. 11 (1962), 22–32. [337] L. Ingelstam, Hilbert algebras with identity. Bull. Amer. Math. Soc. 69 (1963), 794–6. [338] L. Ingelstam, Non-associative normed algebras and Hurwitz’ problem. Ark. Mat. 5 (1964), 231–8. [339] L. Ingelstam, Real Banach algebras. Ark. Mat. 5 (1964), 239–79. [340] B. Iochum, G, Loupias, and A. Rodr´ıguez, Commutativity of C∗ -algebras and associativity of JB∗ -algebras. Math. Proc. Cambridge Phil. Soc. 106 (1989), 281–91. xxi, 398, 399, 400, 401 [341] J. M. Isidro, W. Kaup, and A. Rodr´ıguez, On real forms of JB∗ -triples. Manuscripta Math. 86 (1995), 311–35. 242, 243, 459 [342] J. M. Isidro and A. Rodr´ıguez, Isometries of JB-algebras. Manuscripta Math. 86 (1995), 337–48. [343] J. M. Isidro and A. Rodr´ıguez, On the definition of real W*-algebras. Proc. Amer. Math. Soc. 124 (1996), 3407–10. 244 [344] N. Jacobson, Structure theory of simple rings without finiteness assumptions. Trans. Amer. Math. Soc. 57 (1945), 228–45. [345] N. Jacobson, The radical and semisimplicity for arbitrary rings. Amer. J. Math. 67 (1945), 300–20. [346] N. Jacobson, A topology for the set of primitive ideals in an arbitrary ring. Proc. Nat. Acad. Sci. USA 31 (1945), 333–8. [347] N. Jacobson, Structure theory for algebraic algebras of bounded degree. Ann. of Math. 46 (1945), 695–707. [348] N. Jacobson, A theorem on the structure of Jordan algebras. Proc. Nat. Acad. Sci. USA 42 (1956), 140–7. [349] N. Jacobson, Composition algebras and their automorphisms. Rend. Circ. Mat. Palermo 7 (1958), 55–80. [350] N. Jacobson, Abraham Adrian Albert 1905–1972. Bull. Amer. Math. Soc. 80 (1974), 1075–100. Reprinted in [691], pp. xlv–lxiii. 546 [351] N. Jacobson and C. E. Rickart, Jordan homomorphisms of rings. Trans. Amer. Math. Soc. 69 (1950), 479–502. [352] B. E. Johnson, Centralisers on certain topological algebras. J. London Math. Soc. 39 (1964), 603–14. [353] B. E. Johnson, The uniqueness of the (complete) norm topology. Bull. Amer. Math. Soc. 73 (1967), 537–9. 596 [354] B. E. Johnson, Continuity of derivations on commutative Banach algebras. Amer. J. Math. 91 (1969), 1–10. [355] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90 (1968), 1067–73. 466, 548, 596 [356] P. Jordan, J. von Neumann, and E. Wigner, On an algebraic generalization of the quantum mechanical formalism. Ann. of Math. 35 (1934), 29–64. [357] V. Kadets, O. Katkova, M. Mart´ın, and A. Vishnyakova, Convexity around the unit of a Banach algebra. Serdica Math. J. 34 (2008), 619–28. [358] R. V. Kadison, Isometries of operator algebras. Ann. of Math. 54 (1951), 325–38.

References

679

[359] R. V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. of Math. 56 (1952), 494–503. [360] R. V. Kadison and G. K. Pedersen, Means and convex combinations of unitary operators. Math. Scand. 57 (1985), 249–66. [361] A. Kaidi, Structure des alg`ebres de Jordan–Banach non commutatives reelles de division. In [749], pp. 119–24. 597 [362] A. Kaidi, J. Mart´ınez, and A. Rodr´ıguez, On a non-associative Vidav–Palmer theorem. Quart. J. Math. Oxford 32 (1981), 435–42. [363] A. Kaidi, A Morales, and A. Rodr´ıguez, Prime non-commutative JB∗ -algebras. Bull. London Math. Soc. 32 (2000), 703–8. xxii, 421 [364] A. Kaidi, A. Morales, and A. Rodr´ıguez, Geometrical properties of the product of a C∗ -algebra. Rocky Mountain J. Math. 31 (2001), 197–213. [365] A. Kaidi, A Morales, and A. Rodr´ıguez, A holomorphic characterization of C∗ - and JB∗ -algebras. Manuscripta Math. 104 (2001), 467–78. xviii, 272, 273 [366] A. Kaidi, A. Morales, and A. Rodr´ıguez, Non-associative C∗ -algebras revisited. In [1141], pp. 379–408. xx, 275, 330, 346, 421, 422 [367] A. Kaidi, M. I. Ram´ırez, and A. Rodr´ıguez, Absolute-valued algebraic algebras are finite-dimensional. J. Algebra 195 (1997), 295–307. [368] A. Kaidi, M. I. Ram´ırez, and A. Rodr´ıguez, Absolute-valued algebraic algebras. In [705], pp. 103–9. [369] A. Kaidi, M. I. Ram´ırez, and A. Rodr´ıguez, Nearly absolute-valued algebras. Comm. Algebra 30 (2002), 3267–84. [370] A. Kaidi and A. S´anchez, J-diviseurs topologiques de z´ero dans une alg`ebre de Jordan n.c. norm´ee. In [733], pp. 193–7. 597 [371] H. Kamowitz and S. Scheinberg, The spectrum of automorphisms of Banach algebras. J. Funct. Anal. 4 (1969), 268–76. [372] I. Kaplansky, Topological rings. Amer. J. Math. 69 (1947), 153–83. [373] I. Kaplansky, Topological methods in valuation theory. Duke Math. J. 14 (1947), 527–41. [374] I. Kaplansky, Dual rings. Ann. of Math. 49 (1948), 689–701. xxv, 551, 584 [375] I. Kaplansky, Normed algebras. Duke Math. J. 16 (1949), 399–418. 584 [376] I. Kaplansky, Topological representation of algebras II. Trans. Amer. Math. Soc. 68 (1950), 62–75. [377] I. Kaplansky, Infinite-dimensional quadratic forms permitting composition. Proc. Amer. Math. Soc. 4 (1953), 956–60. 360 [378] I. Kaplansky, Ring isomorphisms of Banach algebras. Canad. J. Math. 6 (1954), 374–81. 596, 597 [379] I. Kaplansky, ‘Problems in the theory of rings’ revisited. Amer. Math. Monthly 77 (1970), 445–54. [380] W. Kaup, Algebraic characterization of symmetric complex Banach manifolds. Math. Ann. 228 (1977), 39–64. 2, 209, 210, 211, 241 [381] W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183 (1983), 503–29. xix, 2, 209, 210, 211, 241 [382] W. Kaup, Contractive projections on Jordan C∗ -algebras and generalizations. Math. Scand. 54 (1984), 95–100. xix, 207, 210 [383] W. Kaup, On real Cartan factors. Manuscripta Math. 92 (1997), 191–222. 460 [384] W. Kaup, JB∗ -triple. In [741], pp. 218–19. 462 [385] W. Kaup and H. Upmeier, Jordan algebras and symmetric Siegel domains in Banach spaces. Math. Z. 157 (1977), 179–200. 211 [386] K. Kawamura, On a conjecture of Wood. Glasgow Math. J. 47 (2005), 1–5.

680

References

[387] J. L. Kelley and R. L. Vaught, The positive cone in Banach algebras. Trans. Amer. Math. Soc. 74 (1953), 44–55. [388] M. Kervaire, Non-parallelizability of the n sphere for n > 7. Proc. Nat. Acad. Sci. USA 44 (1958), 280–3. [389] D. C. Kleinecke, On operator commutators. Proc. Amer. Math. Soc. 8 (1957), 535–6. [390] E. Kleinfeld, Primitive alternative rings and semi-simplicity. Amer. J. Math. 77 (1955), 725–30. 570 [391] J. J. Koliha, Isolated spectral points. Proc. Amer. Math. Soc. 124 (1996), 3417–24. [392] P. Koszmider, M. Mart´ın, and J. Mer´ı, Isometries on extremely non-complex Banach spaces. J. Inst. Math. Jussieu 10 (2011), 325–48. [393] N. Krupnik, S. Roch, and B. Silbermann, On C∗ -algebras generated by idempotents. J. Funct. Anal. 137 (1996), 303–19. [394] S. H. Kulkarni, A very simple and elementary proof of a theorem of Ingelstam. Amer. Math. Monthly 111 (2004), 54–8. [395] K. Kunen and H. Rosenthal, Martingale proofs of some geometrical results in Banach space theory. Pacific J. Math. 100 (1982), 153–75. [396] A. Kurosch, Ringtheoretische Probleme, die mit dem Burnsideschen Problem u¨ ber periodische Gruppen in Zusammenhang stehen. Bull. Acad. Sci. URSS. S´er. Math. [Izvestia Akad. Nauk SSSR] 5 (1941), 233–40. [397] S. Kuwata, Born–Infeld Lagrangian using Cayley–Dickson algebras. Internat. J. Modern Physics A 19 (2004), 1525–48. [398] N. J. Laustsen, On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces. Glasgow Math. J. 45 (2003), 11–19. [399] C.-W. Leung, C.-K. Ng, and N.-C. Wong, Geometric unitaries in JB-algebras. J. Math. Anal. Appl. 360 (2009), 491–4. ˚ Lima, The metric approximation property, norm-one projections and intersection [400] A. properties of balls. Israel J. Math. 84 (1993), 451–75. [401] O. Loos, Fitting decomposition in Jordan systems. J. Algebra 136 (1991), 92–102. [402] O. Loos, Properly algebraic and spectrum-finite ideals in Jordan systems. Math. Proc. Cambridge Philos. Soc. 114 (1993), 149–61. 600 [403] O. Loos, On the set of invertible elements in Banach Jordan algebras. Results Math. 29 (1996), 111–14. 597 [404] O. Loos, Trace and norm for reduced elements of Jordan pairs. Comm. Algebra 25 (1997), 3011–42. 601 [405] O. Loos, Nuclear elements in Banach Jordan pairs. In [705], pp. 111–17. 601 [406] R. J. Loy, Maximal ideal spaces of Banach algebras of derivable elements. J. Austral. Math. Soc. 11 (1970), 310–12. [407] G. Lumer, Semi-inner-product spaces. Trans. Amer. Math. Soc. 100 (1961), 29–43. [408] G. Lumer, Complex methods, and the estimation of operator norms and spectra from real numerical ranges. J. Funct. Anal. 10 (1972), 482–95. [409] G. Lumer and R. S. Phillips, Dissipative operators in a Banach space. Pacific J. Math. 11 (1961), 679–98. 208 [410] I. G. Macdonald, Jordan algebras with three generators. Proc. London Math. Soc. 10 (1960), 395–408. [411] B. Magajna, Weak* continuous states on Banach algebras. J. Math. Anal. Appl. 350 (2009), 252–5. [412] A. Maouche, Caract´erisations spectrales du radical et du socle d’une paire de Jordan–Banach. Canad. Math. Bull. 40 (1997), 488–97. 603 [413] A. Maouche, Spectrum preserving linear mappings for scattered Jordan–Banach algebras. Proc. Amer. Math. Soc. 127 (1999), 3187–90. 603

References

681

[414] J. C. Marcos, A. Rodr´ıguez, and M. V. Velasco, A note on topological divisors of zero and division algebras. Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. 109 (2015), 93–100. [415] J. C. Marcos and M. V. Velasco, The Jacobson radical of a non-associative algebra and the uniqueness of the complete norm topology. Bull. London Math. Soc. 42 (2010), 1010–20. [416] J. C. Marcos and M. V. Velasco, Continuity of homomorphisms into power-associative complete normed algebras. Forum Math. 25 (2013), 1109–25. [417] M. Mart´ın, The group of isometries of a Banach space and duality. J. Funct. Anal. 255 (2008), 2966–76. [418] M. Mart´ın and J. Mer´ı, Numerical index of some polyhedral norms on the plane. Linear Multilinear Algebra 55 (2007), 175–90. [419] M. Mart´ın, J. Mer´ı, and R. Pay´a, On the intrinsic and the spatial numerical range. J. Math. Anal. Appl. 318 (2006), 175–89. [420] M. Mart´ın, J. Mer´ı, and A. Rodr´ıguez, Finite-dimensional Banach spaces with numerical index zero. Indiana Univ. Math. J. 53 (2004), 1279–89. [421] M. Mart´ın and R. Pay´a, Numerical index of vector-valued function spaces. Studia Math. 142 (2000), 269–80. [422] J. Mart´ınez, JV-algebras. Math. Proc. Cambridge Philos. Soc. 87 (1980), 47–50. 597 [423] J. Mart´ınez, Holomorphic functional calculus in Jordan–Banach algebras. In [749], pp. 125–34. 597 [424] J. Mart´ınez, Tracial elements for nonassociative H ∗ -algebras. In [733], pp. 257–68. xxv, 550 [425] J. Mart´ınez, J. F. Mena, R. Pay´a, and A. Rodr´ıguez, An approach to numerical ranges without Banach algebra theory. Illinois J. Math. 29 (1985), 609–25. 673 [426] J. Mart´ınez and A. M. Peralta, Separate weak∗ -continuity of the triple product in dual real JB∗ -triples. Math. Z. 234 (2000), 635–46. 243 [427] J. Mart´ınez and A. Rodr´ıguez, Imbedding elements whose numerical range has a vertex at zero in holomorphic semigroups. Proc. Edinburgh Math. Soc. 28 (1985), 91–5. [428] M. Mathieu, Rings of quotients of ultraprime Banach algebras, with applications to elementary operators. In [773], pp. 297–317. xxii, 408, 468, 547 [429] B. Maurey, Banach spaces with few operators. In [757], pp. 1247–97. [430] P. Mazet, La preuve originale de S. Mazur pour son th´eor`eme sur les alg`ebres norm´ees. Gaz. Math., Soc Math. Fr. 111 (2007), 5–11. ¨ [431] S. Mazur, Uber konvexe Mengen in linearen normierten R¨aumen. Studia Math. 4 (1933), 70–84. [432] S. Mazur, Sur les anneaux lin´eaires. C. R. Acad. Sci. Paris 207 (1938), 1025–7. [433] K. McCrimmon, Norms and noncommutative Jordan algebras. Pacific J. Math. 15 (1965), 925–56. [434] K. McCrimmon, Macdonald’s theorem with inverses. Pacific J. Math. 21 (1967), 315–25. [435] K. McCrimmon, The radical of a Jordan algebra. Proc. Nat. Acad. Sci. USA 62 (1969), 671–8. 597 [436] K. McCrimmon, Noncommutative Jordan rings. Trans. Amer. Math. Soc. 158 (1971), 1–33. xxi, 273, 348, 411 [437] K. McCrimmon and E. Zel’manov, The structure of strongly prime quadratic Jordan algebras. Adv. Math. 69 (1988), 133–222. xxi, 348, 365, 464 [438] R. McGuigan, Strongly extreme points in Banach spaces. Manuscripta Math. 5 (1971), 113–22.

682

References

[439] A. McIntosh, Functions and derivations of C∗ -algebras. J. Funct. Anal. 30 (1978), 264–75. [440] A. Medbouhi and A. Tajmouati, Calcul fonctionnel harmonique dans les alg`ebres de Banach alternatives involutives et applications. Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 141–52. [441] J. F. Mena and A. Rodr´ıguez, Weakly compact operators on non-complete normed spaces. Expo. Math. 27 (2009), 143–51. [442] J. F. Mena and A. Rodr´ıguez, Compact and weakly compact operators on noncomplete normed spaces. In [704], pp. 205–11. [443] A. Meyberg, Identit¨aten und das Radikal in Jordan-Tripelsystemen. Math. Ann. 197 (1972), 203–20. [444] V. D. Milman, The infinite-dimensional geometry of the unit sphere of a Banach space. Soviet Math. Dokl. 8 (1967), 1440–4. [445] J. Milnor, Some consequences of a theorem of Bott. Ann. of Math. 68 (1958), 444–9. [446] R. T. Moore, Hermitian functionals on B-algebras and duality characterizations of C∗ -algebras. Trans. Amer. Math. Soc. 162 (1971), 253–65. [447] A. Moreno, Distinguishing Jordan polynomials by means of a single Jordan-algebra norm. Studia Math. 122 (1997), 67–73. xxii, 470 [448] A. Moreno and A. Rodr´ıguez, On the Zel’manovian classification of prime JB∗ -triples. J. Algebra 226 (2000), 577–613. xxii, 422, 458, 460, 461 [449] A. Moreno and A. Rodr´ıguez, On the Zel’manovian classification of prime JB∗ - and JBW ∗ -triples. Comm. Algebra 31 (2003), 1301–28. xxii, 458, 459, 460, 461 [450] A. Moreno and A. Rodr´ıguez, Introducing Analysis in Zel’manov’s theorems for Jordan systems. In Comptes Rendus de la Premi`er Rencontre Maroco-Andalouse sur les Alg`ebres et leurs Applications, T´etouan, Septembre 2001, pp. 8–36, Universit´e Abdelmalek Essaadi, Facult´e des Sciences, T´etouan, 2003. 462 [451] A. Moreno and A. Rodr´ıguez, A bilinear version of Holsztynski’s theorem on isometries of C(X)-spaces. Studia Math. 166 (2005), 83–91. [452] A. Moreno and A. Rodr´ıguez, On multiplicatively closed subsets of normed algebras. J. Algebra 323 (2010), 1530–52. xxvi, 652, 653 [453] A. Moreno and A. Rodr´ıguez, Topologically nilpotent normed algebras. J. Algebra 368 (2012), 126–68. xxvi, 652, 653, 654, 655, 656, 658 [454] G. Moreno, The zero divisors of the Cayley–Dickson algebras over the real numbers. Bol. Soc. Mat. Mexicana 4 (1998), 13–28. [455] G. Moreno, Alternative elements in the Cayley–Dickson algebras. In Topics in mathematical physics, general relativity and cosmology in honor of Jerzy Pleba˜nski, pp. 333–46, World Sci. Publ., Hackensack, NJ, 2006. [456] A. Moutassim and A. Rochdi, Sur les alg`ebres pr´ehilbertiennes v´erifiant a2  ≤ a2 . Adv. appl. Clifford alg. 18 (2008), 269–78. [457] M. Nagumo, Einige analytishe Untersuchunger in linearen metrishen Ringen. Jap. J. Math. 13 (1936), 61–80. [458] M. Nakamura, Complete continuities of linear operators. Proc. Japan Acad. 27 (1951), 544–7. [459] J. C. Navarro-Pascual and M. A. Navarro, Unitary operators in real von Neumann algebras. J. Math. Anal. Appl. 386 (2012), 933–8. [460] E. Neher, Generators and relations for 3-graded Lie algebras, J. Algebra 155 (1993), 1–35. xxv, 554 [461] J. von Neumann, On an algebraic generalization of the quantum mechanical formalism I. Mat. Sb. 1 (1936), 415–84.

References

683

[462] M. M. Neumann, A. Rodr´ıguez, and M. V. Velasco, Continuity of homomorphisms and derivations on algebras of vector-valued functions. Quart. J. Math. Oxford 50 (1999), 279–300. 596, 709 [463] J. I. Nieto, Normed right alternative algebras over the reals. Canad. J. Math. 24 (1972), 1183–6. [464] J. I. Nieto Gateaux differentials in Banach algebras. Math. Z. 139 (1974), 23–34. [465] E. Oja, On bounded approximation properties of Banach spaces. In Banach algebras 2009, pp. 219–31, Banach Center Publ. 91, Polish Acad. Sci. Inst. Math., Warsaw, 2010. [466] T. Okayasu, A structure theorem of automorphisms of von Neumann algebras. Tˆohoku Math. J. 20 (1968), 199–206. [467] S. Okubo, Pseudo-quaternion and pseudo-octonion algebras. Hadronic J. 1 (1978), 1250–78. [468] S. Okubo and J. M. Osborn, Algebras with nondegenerate associative symmetric forms permitting composition. Comm. Algebra 9 (1981), 1233–61. [469] C. L. Olsen and G. K. Pedersen, Convex combinations of unitary operators in von Neumann algebras. J. Funct. Anal. 66 (1986), 365–80. [470] J. M. Osborn, Quadratic division algebras. Trans. Amer. Math. Soc. 105 (1962), 202–21. [471] J. M. Osborn, Varieties of algebras. Adv. Math. 8 (1972), 163–369. ¨ [472] A. Ostrowski, Uber einige L¨osungen der Funktionalgleichung (x) (y) = (xy). Acta Math. 41 (1918), 271–84. [473] R. S. Palais, The classification of real division algebras. Amer. Math. Monthly 75 (1968), 366–8. [474] T. W. Palmer, Characterizations of C∗ -algebras. Bull. Amer. Math. Soc. 74 (1968), 538–40. [475] T. W. Palmer, Unbounded normal operators on Banach spaces. Trans. Amer. Math. Soc. 133 (1968), 385–414. [476] T. W. Palmer, Characterizations of C∗ -algebras II. Trans. Amer. Math. Soc. 148 (1970), 577–88. [477] T. W. Palmer, Real C∗ -algebras. Pacific J. Math. 35 (1970), 195–204. [478] T. W. Palmer, Spectral algebras. Rocky Mountain J. Math. 22 (1992), 293–328. [479] A. L. T. Paterson, Isometries between B∗ -algebras. Proc. Amer. Math. Soc. 22 (1969), 570–2. [480] A. L. T. Paterson and A. M. Sinclair, Characterization of isometries between C∗ -algebras. J. London Math. Soc. 5 (1972), 755–61. [481] R. Pay´a, J. P´erez, and A. Rodr´ıguez, Non-commutative Jordan C∗ -algebras. Manuscripta Math. 37 (1982), 87–120. xx, xxi, 3, 19, 398, 411, 421 [482] R. Pay´a, J. P´erez, and A. Rodr´ıguez, Type I factor representations of non-commutative JB∗ -algebras. Proc. London Math. Soc. 48 (1984), 428–44. xx, xxi, 398, 411, 421 [483] G. K. Pedersen, Measure theory in C∗ -algebras II. Math. Scand. 22 (1968), 63–74. [484] G. K. Pedersen, The λ-function in operator algebras. J. Operator Theory 26 (1991), 345–81. [485] A. M. Peralta, On the axiomatic definition of real JB∗ -triples. Math. Nachr. 256 (2003), 100–6. [486] A. M. Peralta, Positive definite hermitian mappings associated to tripotent elements. Expo. Math. 33 (2015), 252–8. [487] L. A. Peresi, Other nonassociative algebras. In The concise handbook of algebra (eds. A. V. Mikhalev and G. F. Pilz), pp. 343–7, Dordrecht, Kluwer Academic, 2002.

684

References

[488] J. P´erez, L. Rico, and A. Rodr´ıguez, Full subalgebras of Jordan–Banach algebras and algebra norms on JB∗ -algebras. Proc. Amer. Math. Soc. 121 (1994), 1133–43. 336, 597 [489] J. P´erez, L. Rico, A. Rodr´ıguez, and A. R. Villena, Prime Jordan–Banach algebras with nonzero socle. Comm. Algebra 20 (1992), 17–53. 598 [490] S. Perlis, A characterization of the radical of an algebra. Bull. Amer. Math. Soc. 48 (1942), 128–32. [491] R. R. Phelps, Extreme points in function algebras. Duke Math. J. 32 (1965), 267–77. [492] S. Popa, On the Russo–Dye theorem. Michigan Math. J. 28 (1981), 311–15. [493] V. Pt´ak, On the spectral radius in Banach algebras with involution. Bull. London Math. Soc. 2 (1970), 327–34. [494] V. Pt´ak, Banach algebras with involution. Manuscripta Math. 6 (1972), 245–90. [495] P. S. Putter and B. Yood, Banach Jordan ∗-algebras. Proc. London Math. Soc. 41 (1980), 21–44. 597 [496] I. Raeburn and A. M. Sinclair, The C∗ -algebra generated by two projections. Math. Scand. 65 (1989), 278–90. [497] R. Raffin, Anneaux a puissances commutatives and anneaux flexibles. C. R. Acad. Sci. Paris 230 (1950), 804–6. [498] F. Rambla, A counterexample to Wood’s conjecture. J. Math. Anal. Appl. 317 (2006), 659–67. [499] T. J. Ransford, A short proof of Johnson’s uniqueness-of-norm theorem. Bull. London Math. Soc. 21 (1989), 487–8. [500] C. E. Rickart, The singular elements of a Banach algebra. Duke Math. J. 14 (1947), 1063–77. [501] C. E. Rickart, The uniqueness of norm problem in Banach algebras. Ann. of Math. 51 (1950), 615–28. [502] C. E. Rickart, On spectral permanence for certain Banach algebras. Proc. Amer. Math. Soc. 4 (1953), 191–6. [503] C. E. Rickart, An elementary proof of a fundamental theorem in the theory of Banach algebras. Michigan Math. J. 5 (1958), 75–8. ´ [504] F. Riesz, Sur certains syst`emes singuliers d’´equations int´egrales. Ann. Sci. Ecole Norm. Sup. 28 (1911), 33–62. ¨ [505] F. Riesz, Uber lineare Funktionalgleichungen. Acta Math. 41 (1916), 71–98. [506] A. G. Robertson, A note on the unit ball in C∗ -algebras. Bull. London Math. Soc. 6 (1974), 333–5. [507] A. G. Robertson and M. A. Youngson, Positive projections with contractive complements on Jordan algebras. J. London Math. Soc. 25 (1982), 365–74. 210 [508] A. Rochdi, Eight-dimensional real absolute valued algebras with left unit whose automorphism group is trivial. Int. J. Math. Math. Sci. 70 (2003), 4447–54. [509] A. Rochdi and A. Rodr´ıguez, Absolute valued algebras with involution. Comm. Algebra 37 (2009), 1151–9. [510] A. Rodr´ıguez, Algebras estelares de Banach con elemento unidad. In Actas de las I Jornadas Matem´aticas Hispano-Lusitanas, pp. 189–210, Madrid, 1973. [511] A. Rodr´ıguez, Derivaciones en a´ lgebras normadas y commutatividad. Cuadernos del Dpto. de Estad´ıstica Matem´atica (Granada) 2 (1975), 51–68. [512] A. Rodr´ıguez, Teorema de estructura de los Jordan-isomorfismos de las C∗ -´algebras. Rev. Mat. Hispano-Americana 37 (1977), 114–28. [513] A. Rodr´ıguez, Rango num´erico y derivaciones cerradas de las a´ lgebras de Banach. In V Jornadas luso-espanholas de matem´aticas, pp. 179–200, Aveiro, 1978.

References

685

[514] A. Rodr´ıguez, A Vidav–Palmer theorem for Jordan C∗ -algebras and related topics. J. London Math. Soc. 22 (1980), 318–32. [515] A. Rodr´ıguez, Non-associative normed algebras spanned by hermitian elements. Proc. London Math. Soc. 47 (1983), 258–74. 210 [516] A. Rodr´ıguez, The uniqueness of the complete algebra norm topology in complete normed nonassociative algebras. J. Funct. Anal. 60 (1985), 1–15. 548, 597 [517] A. Rodr´ıguez, Mutations of C∗ -algebras and quasiassociative JB∗ -algebras. Collectanea Math. 38 (1987), 131–5. [518] A. Rodr´ıguez, Jordan axioms for C∗ -algebras. Manuscripta Math. 61 (1988), 279–314. xxiv, 474, 549 [519] A. Rodr´ıguez, Automatic continuity with application to C∗ -algebras. Math. Proc. Cambridge Philos. Soc. 107 (1990), 345–7. [520] A. Rodr´ıguez, An approach to Jordan–Banach algebras from the theory of nonassociative complete normed algebras. In [749], pp. 1–57. 584, 585, 586 [521] A. Rodr´ıguez, One-sided division absolute valued algebras. Publ. Mat. 36 (1992), 925–54. [522] A. Rodr´ıguez, Closed derivations of Banach algebras. In Homenaje a Pablo Bobillo Guerrero, pp. 21–8, Univ. Granada, Granada, 1992. [523] A. Rodr´ıguez, Primitive nonassociative normed algebras and extended centroid. In [734], pp. 233–43. 408, 601, 602 [524] A. Rodr´ıguez, Absolute valued algebras of degree two. In [733], pp. 350–6. [525] A. Rodr´ıguez, Jordan structures in Analysis. In [764], pp. 97–186. xxii, 338, 340, 408, 462, 467, 549, 550, 553, 597, 601 [526] A. Rodr´ıguez, Continuity of densely valued homomorphisms into H ∗ -algebras. Quart. J. Math. Oxford 46 (1995), 107–18. xxiv, 546, 548, 549 [527] A. Rodr´ıguez, Multiplicative characterization of Hilbert spaces and other interesting classes of Banach spaces. In Encuentro de An´alisis Matem´atico, Homenaje al profesor B. Rodr´ıguez-Salinas (eds. F. Bombal, F. L. Hern´andez, P. Jim´enez y J. L. de Mar´ıa), Rev. Mat. Univ. Complutense Madrid 9 (1996), 149–89. [528] A. Rodr´ıguez, Isometries and Jordan isomorphisms onto C∗ -algebras. J. Operator Theory 40 (1998), 71–85. 274, 346 [529] A. Rodr´ıguez, Continuity of homomorphisms into normed algebras without topological divisors of zero. Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp.) 94 (2000), 505–14. xxv, 595 [530] A. Rodr´ıguez, Sobre el tama˜no de los conjuntos de n´umeros. In Actas del Encuentro de Matem´aticos Andaluces, Sevilla 2000, Volumen I, pp. 235–48, Secretariado de Publicaciones de la Universidad de Sevilla, Sevilla, 2001. [531] A. Rodr´ıguez, A numerical range characterization of uniformly smooth Banach spaces. Proc. Amer. Math. Soc. 129 (2001), 815–21. [532] A. Rodr´ıguez, Banach–Jordan algebra. In [741], pp. 55–7. 462 [533] A. Rodr´ıguez, Absolute-valued algebras, and absolute-valuable Banach spaces. In Advanced Courses of Mathematical Analysis I, Proceedings of the First International School C´adiz, Spain 22–27 September 2002 (eds. A. Aizpuru-Tom´as and F. Le´on-Saavedra), pp. 99–155, World Scientific, 2004. [534] A. Rodr´ıguez, Numerical ranges of uniformly continuous functions on the unit sphere of a Banach space. J. Math. Anal. Appl. 297 (2004), 472–6. 208 [535] A. Rodr´ıguez, On Urbanik’s axioms for absolute valued algebras with involution. Comm. Algebra 36 (2008), 2588–92. [536] A. Rodr´ıguez, Banach space characterizations of unitaries: a survey. J. Math. Anal. Appl. 369 (2010), 168–78.

686

References

[537] A. Rodr´ıguez, Approximately norm-unital products on C∗ -algebras, and a nonassociative Gelfand–Naimark theorem. J. Algebra 347 (2011), 224–46. 421 [538] A. Rodr´ıguez, A. M. Slinko, and E. I. Zel’manov, Extending the norm from Jordan– Banach algebras of hermitian elements to their associative envelopes. Comm. Algebra 22 (1994), 1435–55. xxii, xxiii, 437, 470, 471, 472 [539] A. Rodr´ıguez and M. V. Velasco, Continuity of homomorphisms and derivations on Banach algebras with an involution. In Function Spaces; Proceedings of the Third Conference on Function Spaces, May 19–23, 1998, Southern Illinois University at Edwardsville (ed. K. Jarosz), pp. 289–98, Contemp. Math. 232, Amer. Math. Soc., Providence, RI, 1999. xxii, 474, 475 [540] A. Rodr´ıguez and M. V. Velasco, A non-associative Rickart’s dense-rangehomomorphism theorem. Quart. J. Math. Oxford 54 (2003), 367–76. 597 [541] M. Rordam, Advances in the theory of unitary rank and regular approximation. Ann. of Math. 128 (1988), 153–72. [542] H. Rosenthal, The Lie algebra of a Banach space. In Banach Spaces, pp. 129–57, Lecture Notes in Math. 1166, 1985. [543] H. Rosenthal, Functional Hilbertian sums. Pacific J. Math. 124 (1986), 417–67. [544] G.-C. Rota and W. G. Strang, A note on the joint spectral radius. Indag. Math. 22 (1960), 379–81. xxvi, 650 [545] L. H. Rowen, A short proof of the Chevalley–Jacobson density theorem, and a generalization. Amer. Math. Monthly 85 (1978), 185–6. [546] Z.-J. Ruan, Subspaces of C*-algebras. J. Funct. Anal. 76 (1988), 217–30. [547] B. Russo, Structure of JB∗ -triples. In [764], pp. 209–80. [548] B. Russo and H. A. Dye, A note on unitary operators in C∗ -algebras. Duke Math. J. 33 (1966), 413–16. [549] S. Sakai, On a conjecture of Kaplansky. Tˆohoku Math. J. 12 (1960), 31–3. [550] E. Sasiada and P. M. Cohn, An example of a simple radical ring. J. Algebra 5 (1967), 373–7. 659 [551] R. D. Schafer, On the algebras formed by the Cayley–Dickson process. Amer. J. Math. 76 (1954), 435–46. [552] R. D. Schafer, Noncommutative Jordan algebras of characteristic 0. Proc. Amer. Math. Soc. 6 (1955), 472–5. [553] R. D. Schafer, Generalized standard algebras. J. Algebra 12 (1969), 386–417. 657 [554] J. A. Schatz, Review of [273]. Math. Rev. 14 (1953), 884. ¨ [555] J. Schauder, Uber lineare, vollstetige Funktionaloperationen. Studia Math. 2 (1930), 183–96. [556] I. J. Schoenberg, A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal. Proc. Amer. Math. Soc. 3 (1952), 961–4. [557] J. R. Schue, Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc. 95 (1960), 69–80. 553, 554 [558] J. R. Schue, Cartan decompositions for L∗ -algebras. Trans. Amer. Math. Soc. 98 (1961), 334–49. 553 [559] 1. I. Schur, Neue Begr¨undung der Theorie der Gruppencharaktere. Sitzungsberichte der Preuβischen Akademie der Wissenschaften 1905 Physik-Math. Klasse 406–32. Reprinted in Gesammelte Abhandlungen Vol. I pp. 143–69, Spriger, Berlin, 1973. [560] I. E. Segal, Irreducible representations of operator algebras. Bull. Amer. Math. Soc. 53 (1947), 73–88. [561] B. Segre, La teoria delle algebre ed alcune questione di realt`a. Univ. Roma, Ist. Naz. Alta. Mat., Rend. Mat. e Appl., serie 5 13 (1954), 157–88. [562] S. Shelah and J. Steprans, A Banach space on which there are few operators. Proc. Amer. Math. Soc. 104 (1988), 101–5.

References

687

[563] D. Sherman, A new proof of the noncommutative Banach–Stone theorem. Quantum probability, 363–75, Banach Center Publ. 73, Polish Acad. Sci. Inst. Math., Warsaw, 2006. [564] G. Shilov, On the extension of maximal ideals. C. R. (Doklady) Acad. Sci. URSS 29 (1940), 83–4. [565] S. Shirali and J. W. M. Ford, Symmetry in complex involutory Banach algebras. II. Duke Math. J. 37 (1970), 275–80. [566] F. V. Shirokov, Proof of a conjecture by Kaplansky. (In Russian.) Uspekhi Mat. Nauk 11 (1956) 167–8. [567] A. I. Shirshov, On special J-rings. (In Russian.) Mat. Sb. 38 (80) (1956), 149–66. [568] A. I. Shirshov, On some non-associative nil-rings and algebraic algebras. Mat. Sb. 41 (83) (1957), 381–94. [569] V. S. Shulman and Yu. V. Turovskii, Joint spectral radius, operator semigroups, and a problem of W. Wojtynski. J. Funct. Anal. 177 (2000), 383–441. xxvi, 652, 653, 654, 656 [570] F. W. Shultz, On normed Jordan algebras which are Banach dual spaces. J. Funct. Anal. 31 (1979), 360–76. 19, 398, 401 [571] A. A. Siddiqui, Self-adjointness in unitary isotopes of JB∗ -algebras. Arch. Math. 87 (2006), 350–8. [572] A. A. Siddiqui, Asymmetric decompositions of vectors in JB∗ -algebras. Arch. Math. (Brno) 42 (2006), 159–66. [573] A. A. Siddiqui, JB∗ -algebras of topological stable rank 1. Int. J. Math. Math. Sci. 2007, Art. ID 37186, 24 pp. [574] A. A. Siddiqui, A proof of the Russo–Dye theorem for JB∗ -algebras. New York J. Math. 16 (2010), 53–60. [575] A. A. Siddiqui, Convex combinations of unitaries in JB∗ -algebras. New York J. Math. 17 (2011), 127–37. [576] A. A. Siddiqui, The λu -function in JB∗ -algebras. New York J. Math. 17 (2011), 139–47. [577] A. M. Sinclair, Jordan automorphisms on a semisimple Banach algebra. Proc. Amer. Math. Soc. 25 (1970), 526–8. [578] A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras. Proc. Amer. Math. Soc. 24 (1970), 209–14. [579] A. M. Sinclair, The states of a Banach algebra generate the dual. Proc. Edinburgh Math. Soc. 17 (1970/71), 341–4. [580] A. M. Sinclair, The norm of a hermitian element in a Banach algebra. Proc. Amer. Math. Soc. 28 (1971), 446–50. [581] A. M. Sinclair, The Banach algebra generated by a derivation. In Spectral theory of linear operators and related topics (Timisoara/Herculane, 1983), pp. 241–50, Oper. Theory Adv. Appl. 14, Birkh¨auser, Basel, 1984. [582] I. M. Singer and J. Wermer, Derivations on commutative normed algebras. Math. Ann. 129 (1955), 260–4. [583] A. Skorik and M. Zaidenberg, On isometric reflexions in Banach spaces. Math. Physics, Analysis, Geometry 4 (1997), 212–47. [584] V. G. Skosyrskii, Strongly prime noncommutative Jordan algebras. (In Russian.) Trudy Inst. Mat. (Novosibirk) 16 (1989), 131–64, Issled. po Teor. Kolets i Algebr, 198–9. 411, 422 [585] V. G. Skosyrskii, Primitive Jordan algebras. Algebra Logic 31 (1993), 110–20. xxii, 462 [586] A. M. Slinko, Complete metric Jordan algebras. Mat. Zametki 447 (1990), 100–5. English translation: Math. Notes 47 (1990), 491–4. 597

688 [587] [588] [589] [590] [591] [592] [593] [594] [595] [596] [597] [598] [599] [600] [601] [602] [603] [604] [605] [606] [607] [608] [609] [610] [611] [612] [613] [614]

References A. M. Slinko, 1, 2, 4, 8, . . . What comes next? Extracta Math. 19 (2004), 155–61. M. F. Smiley, The radical of an alternative ring. Ann. of Math. 49 (1948), 702–9. M. F. Smiley, Right H ∗ -algebras. Proc. Amer. Math. Soc. 4 (1953), 1–4. M. F. Smiley, Real Hilbert algebras with identity. Proc. Amer. Math. Soc. 16 (1965) 440–1. R. R. Smith, On Banach algebra elements of thin numerical range. Math. Proc. Cambridge Philos. Soc. 86 (1979), 71–83. R. R. Smith, The numerical range in the second dual of a Banach algebra. Math. Proc. Cambridge Philos. Soc. 89 (1981), 301–7. ˇ V. Smulyan, Sur la structure de la sph`ere unitaire dans l’espace de Banach. Math. Sb. 9 (1941), 545–61. I. N. Spatz, Smooth Banach algebras. Proc. Amer. Math. Soc. 22 (1969), 328–9. I. Spitkovsky, Once more on algebras generated by two projections. Linear Algebra Appl. 208/209 (1994), 377–95. J. Spurn´y, A note on compact operators on normed linear spaces. Expo. Math. 25 (2007), 261–3. L. L. Stach´o, A projection principle concerning biholomorphic automorphisms. Acta Sci. Math. (Szeged) 44 (1982), 99–124. xix, 207, 210 S. W. P. Steen, Introduction to the theory of operators. V. Metric rings, Proc. Cambridge Philos. Soc. 36 (1940), 139–49. 550 W. F. Stinespring, Positive functions on C∗ -algebras. Proc. Amer. Math. Soc. 6, (1955), 211–16. M. H. Stone, Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41 (1937), 375–481. E. Størmer, On the Jordan structure of C∗ -algebras. Trans. Amer. Math. Soc. 120 (1965), 438–47. E. Størmer, Jordan algebras of type I. Acta Math. 115 (1966), 165–84. 339 E. Størmer, Irreducible Jordan algebras of self-adjoint operators. Trans. Amer. Math. Soc. 130 (1968), 153–66. 339 R. Strasek and B. Zalar, Uniform primeness of classical Banach Lie algebras of compact operators. Publ. Math. Debrecen 61 (2002), 325–40. E. Strzelecki, Metric properties of normed algebras. Studia Math. 23 (1963), 41–51. E. Strzelecki, Power-associative regular real normed algebras. J. Austral. Math. Soc. 6 (1966), 193–209. D. Suttles, A counterexample to a conjecture of Albert. Notices Amer. Math. Soc., 19 (1972), A-566. A. Szankowski, B(H ) does not have the approximation property. Acta Math. 147 (1981), 89–108. J. Talponen, A note on the class of superreflexive almost transitive Banach spaces. Extracta Math. 23 (2008), 1–6. P. A. Terekhin, Trigonometric algebras. J. Math. Sci. (New York) 95 (1999), 2156–60. M. P. Thomas, The image of a derivation is contained in the radical. Ann. of Math. 128 (1988), 435–60. M. P. Thomas, Primitive ideals and derivations on noncommutative Banach algebras. Pacific J. Math. 159 (1993), 139–52. O. Toeplitz, Das algebraische Analogon zu einem Satze von Fejer. Math. Z. 2 (1918), 187–97. D. M. Topping, An isomorphism invariant for spin factors. J. Math. Mech. 15 (1966), 1055–63.

References

689

[615] Yu. V. Turovskii, On spectral properties of Lie subalgebras and on the spectral radius of subsets of a Banach algebra. (In Russian.) Spectral theory of operators and its applications, 6 (1985), 144–81. xxvi, 651 [616] M. Uchiyama, Operator monotone functions which are defined implicitly and operator inequalities. J. Funct. Anal. 175 (2000), 330–47. [617] K. Urbanik, Absolute valued algebras with an involution. Fundamenta Math. 49 (1961), 247–58. [618] K. Urbanik, Remarks on ordered absolute valued algebras. Colloq. Math. 11 (1963), 31–9. [619] K. Urbanik and F. B. Wright, Absolute valued algebras. Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 8 (1960), 285–6. [620] K. Urbanik and F. B. Wright, Absolute valued algebras. Proc. Amer. Math. Soc. 11 (1960), 861–6. [621] M. V. Velasco, Spectral theory for non-associative complete normed algebras and automatic continuity. J. Math. Anal. Appl. 351 (2009), 97–106. [622] E. Vesentini, On the subharmonicity of the spectral radius. Boll. Un. Mat. Ital. 1 (1968), 427–9. [623] I. Vidav, Eine metrische Kennzeichnung der selbstadjungierten Operatoren. Math. Z. 66 (1956), 121–8. [624] A. R. Villena, Continuity of derivations on H ∗ -algebras. Proc. Amer. Math. Soc. 122 (1994), 821–6. xxiv, 548, 549 [625] A. R. Villena, Derivations on Jordan–Banach algebras. Studia Math. 118 (1996), 205–29. 466, 548 [626] A. R. Villena, Automatic continuity in associative and nonassociative context. Irish Math. Soc. Bulletin 46 (2001), 43–76. [627] A. R. Villena, Epimorphisms onto derivation algebras. Proc. Edinburgh Math. Soc. 46 (2003), 379–82. [628] C. Viola Devapakkiam, Jordan algebras with continuous inverse. Math. Japon. 16 (1971), 115–25. 597 [629] B. J. Vowden, On the Gelfand–Neumark theorem. J. London Math. Soc. 42 (1967), 725–31. [630] B. Walsh, Classroom notes: The scarcity of cross products on Euclidean spaces. Amer. Math. Monthly 74 (1967), 188–94. [631] H. M. Wark, A non-separable reflexive Banach space on which there are few operators. J. London Math. Soc. 64 (2001), 675–89. [632] J. Wichmann, Hermitian ∗-algebras which are not symmetric. J. London Math. Soc. 8 (1974), 109–12. [633] J. Wichmann, On commutative B∗ -equivalent algebras. Notices Amer. Math. Soc. (1975), Abstract 720-46-19. [634] J. Wichmann, The symmetric radical of an algebra with involution. Arch. Math. (Basel) 30 (1978), 83–8. [635] A. Wilansky, Letter to the Editor. Amer. Math. Monthly 91 (1984), 531. [636] T. J. D. Wilkins, Inessential ideals in Jordan–Banach algebras. Bull. London Math. Soc. 29 (1997), 73–81. 603 [637] P. Wojtaszczyk, Some remarks on the Daugavet equation. Proc. Amer. Math. Soc. 115 (1992), 1047–52. [638] G. V. Wood, Maximal symmetry in Banach spaces. Proc. Royal Irish Acad. 82A (1982), 177–86. [639] G. V. Wood, Maximal algebra norms. In Trends in Banach spaces and operator theory (Memphis, TN, 2001), pp. 335–45, Contemp. Math. 321, Amer. Math. Soc., Providence, RI, 2003.

690

References

[640] F. B. Wright, Absolute valued algebras. Proc. Nat. Acad. Sci. USA 39 (1953), 330–2. [641] J. D. M. Wright, Jordan C∗ -algebras. Michigan Math. J. 24 (1977), 291–302. xx, 347, 398, 401 [642] J. D. M. Wright and M. A. Youngson, A Russo Dye theorem for Jordan C∗ -algebras. In Functional analysis: surveys and recent results (Proc. Conf., Paderborn, 1976) (eds. K. D. Bierstedt and F. Fuchssteiner), pp. 279–82, NorthHolland Math. Stud. 27, North-Holland, Amsterdam, 1977. [643] J. D. M. Wright and M. A. Youngson, On isometries of Jordan algebras. J. London Math. Soc. 17 (1978), 339–44. [644] W. Wu, Locally pre-C∗ -equivalent algebras. Proc. Amer. Math. Soc. 131 (2003), 555–62. [645] B. Yood, Topological properties of homomorphisms between Banach algebras. Amer. J. Math. 76 (1954), 155–67. [646] B. Yood, Homomorphisms on normed algebras. Pacific J. Math. 8 (1958), 373–81. [647] B. Yood, Faithful ∗-representations of normed algebras. Pacific J. Math. 10 (1960), 345–63. [648] B. Yood, Ideals in topological rings. Canad. J. Math. 16 (1964), 28–45. [649] B. Yood, On axioms for B∗ -algebras. Bull. Amer. Math. Soc. 76 (1970), 80–2. [650] D. Yost, A base norm space whose cone is not 1-generating. Glasgow Math. J. 25 (1984), 35–6. 690 [651] D. Yost, Review of [650]. Zbl. Mat. 0544.46014. [652] M. A. Youngson, A Vidav theorem for Banach Jordan algebras. Math. Proc. Cambridge Philos. Soc. 84 (1978), 263–72. [653] M. A. Youngson, Equivalent norms on Banach Jordan algebras. Math. Proc. Cambridge Philos. Soc. 86 (1979), 261–9. 597 [654] M. A. Youngson, Hermitian operators on Banach Jordan algebras, Proc. Edinburgh Math. Soc. 22 (1979), 93–104. [655] M. A. Youngson, Nonunital Banach Jordan algebras and C∗ -triple systems. Proc. Edinburgh Math. Soc. 24 (1981), 19–29. 210 [656] B. Zalar, Inner product characterizations of classical Cayley–Dickson algebras. In [733], pp. 405–9. [657] B. Zalar, On Hilbert spaces with unital multiplication. Proc. Amer. Math. Soc. 123 (1995), 1497–501. [658] I. Zalduendo, A simple example of a noncommutative Arens product. Publ. Mat. 35 (1991), 475–7. [659] W. Zelazko, On generalized topological divisors of zero in real m-convex algebras. Studia Math. 28 (1966/1967), 241–4. [660] G. Zeller-Meier, Sur les automorphismes des alg`ebres de Banach. C. R. Acad. Sci. Paris S´er. A-B 264 (1967), A1131–2. [661] E. I. Zel’manov, Absolute zero-divisors and algebraic Jordan algebras. Siberian Math. J. 23 (1982), 841–54. 600 [662] E. I. Zel’manov, On prime Jordan algebras II. Siberian Math. J. 24 (1983), 89–104. xxi, 348, 364, 365, 366, 367, 401, 437 [663] E. I. Zel’manov, On prime Jordan triple systems III. Siberian Math. J. 26 (1985), 71–82. xxii, 421, 437, 438, 459, 461 [664] J. Zem´anek, A note on the radical of a Banach algebra. Manuscripta Math. 20 (1977), 191–6. [665] H. Zettl, A characterization of ternary rings of operators. Adv. Math. 48 (1983), 117–43. 459 [666] K. A. Zhevlakov, Solvability of alternative nil-rings. Sibirsk. Mat. Z. 3 (1962), 368–77.

References

691

[667] K. A. Zhevlakov, Coincidence of Smiley and Kleinfeld radicals in alternative rings. Algebra Logic 8 (1969), 175–81. [668] M. Zorn, Theorie der alternativen Ringe. Abh. Math. Sem. Univ. Hamburg 8 (1930), 123–47. [669] M. Zorn, Alternativk¨orper und quadratische Systeme. Abh. Math. Sem. Univ. Hamburg 9 (1933), 395–402.

Books [670] A. A. Albert, Structure of algebras. Revised printing, Amer. Math. Soc. Colloq. Publ. 24, American Mathematical Society, Providence, RI, 1961. [671] F. Albiac and N. J. Kalton, Topics in Banach space theory. Grad. Texts in Math. 233, Springer, New York, 2006. 213, 236 [672] E. M. Alfsen and F. W. Shultz, Non-commutative spectral theory for affine function spaces on convex sets. Mem. Amer. Math. Soc. 6 (1976), no. 172. [673] E. M. Alfsen and F. W. Shultz, Geometry of state spaces of operator algebras. Mathematics: Theory and Applications. Birkh¨auser Boston, Inc., Boston, MA, 2003. 331, 411 [674] C. D. Aliprantis and K. C. Border, Infinite dimensional analysis. A hitchhiker’s guide. Third edition. Springer, Berlin, 2006. [675] G. R. Allan, Introduction to Banach spaces and algebras. Prepared for publication and with a preface by H. Garth Dales. Oxf. Grad. Texts Math. 20, Oxford University Press, Oxford, 2011. [676] D. Amir, Characterizations of inner product spaces. Oper. Theory Adv. Appl. 20, Birkh¨auser Verlag, Basel–Boston–Stuttgart, 1986. [677] V. A. Andrunakievich and Yu. M. Rjabuhin, Radicals of algebras and structural theory. (In Russian.) Contemporary Algebra ‘Nauka’, Moscow, 1979. 633 [678] P. Ara and M. Mathieu, Local multipliers of C∗ -algebras. Springer Monogr. Math., Springer, London, 2003. 401, 408, 550 [679] S. A. Argyros and A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces. Mem. Amer. Math. Soc. 170 (2004), no. 806. [680] R. B. Ash and W. P. Novinger, Complex variables. Dover Publications, 2007. [681] L. Asimow and A. J. Ellis, Convexity theory and its applications in functional analysis. London Math. Soc. Monogr. 16, Academic Press, London, 1980. [682] B. Aupetit, Propri´et´es spectrales des alg`ebres de Banach. Lecture Notes in Math. 735, Springer, Berlin, 1979. 401, 436, 498, 596, 597 [683] B. Aupetit, A primer on spectral theory. Universitext, Springer, New York, 1991. 597, 599 [684] S. Ayupov, A. Rakhimov, and S. Usmanov, Jordan, real and Lie structures in operator algebras. Math. Appl. 418, Kluwer Academic Publishers Group, Dordrecht, 1997. [685] S. Banach, Theory of linear operations. Translated from the French by F. Jellett. With comments by A. Pełczy´nski and Cz. Bessaga. North-Holland Math. Library 38, North-Holland Publishing Co., Amsterdam, 1987. [686] K. I. Beidar, W. S. Martindale III, and A. V. Mikhalev, Rings with generalized identities. Monographs Textbooks Pure Appl. Math. 196, Marcel Dekker, New York, 1996. 401, 602 [687] D. Beltita, Smooth homogeneous structures in operator theory. Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 137, Chapman & Hall/CRC, Boca Raton, FL, 2006. xxiv, xxv, 154, 155, 174, 549, 554 [688] D. Beltita and M. Sabac, Lie algebras of bounded operators. Oper. Theory Adv. Appl. 120, Birkh¨auser Verlag, Basel, 2001.

692

References

[689] S. K. Berberian, Lectures in functional analysis and operator theory. Grad. Texts in Math. 15, Springer, New York–Heidelberg, 1974. 157, 183, 539 [690] D. P. Blecher and Ch. Le Merdy, Operator algebras and their modules – an operator space approach. London Math. Soc. Monogr. New Series 30, Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford, 2004. [691] R. E. Block, N. Jacobson, J. M. Osborn, D. J. Saltman, and D. Zelinsky (eds.), A. Adrian Albert, Collected mathematical papers: Associative algebras and Riemann matrices, Part 1. Amer. Math. Soc., Providence, RI, 1993. 664, 678 [692] R. E. Block, N. Jacobson, J. M. Osborn, D. J. Saltman, and D. Zelinsky (eds.), A. Adrian Albert, Collected mathematical papers: Nonassociative algebras and miscellany, Part 2. Amer. Math. Soc., Providence, RI, 1993. 664, 699 [693] R. P. Boas, Entire functions. Academic Press Inc., New York, 1954. [694] F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras. London Math. Soc. Lecture Note Ser. 2, Cambridge University Press, Cambridge, 1971. [695] F. F. Bonsall and J. Duncan, Numerical ranges II. London Math. Soc. Lecture Note Ser. 10, Cambridge University Press, Cambridge, 1973. [696] F. F. Bonsall and J. Duncan, Complete normed algebras. Ergeb. Math. Grenzgeb. 80, Springer, Berlin, 1973. xvii, 436, 545, 551, 584, 659 ´ ements de math´ematique. Fasc. XXXII. Th´eories spectrales. Chapitre [697] N. Bourbaki, El´ I: Alg`ebres norm´ees. Chapitre II: Groupes localement compacts commutatifs. Actualit´es Scientifiques et Industrielles 1332, Hermann, Paris, 1967. [698] O. Bratteli, Derivations, dissipations and group actions on C∗ -algebras. Lecture Notes in Math. 1229, Springer, Berlin, 1986. [699] O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics II. Texts Monographs Phys., Springer, New York, 1981. [700] H. Braun and M. Koecher, Jordan-Algebren. Grundlehren Math. Wiss. 128, Springer, Berlin–New York, 1966. [701] M. Brelot, On topologies and boundaries in potential theory. (Enlarged edition of a course of lectures delivered in 1966). Lecture Notes in Math. 175, Springer, Berlin–New York, 1971. [702] M. Breˇsar, M. A. Chebotar, and W. S. Martindale III, Functional identities. Front. Math., Birkh¨auser Verlag, Basel, 2007. 602 [703] S. R. Caradus, W. E. Pfaffenberger, and B. Yood, Calkin algebras and algebras of operators on Banach spaces. Lecture Notes in Pure and Appl. Math. 9, Marcel Dekker Inc., New York, 1974. [704] J. Carmona, A. Morales, A. M. Peralta, and M. I. Ram´ırez (eds.), Proceedings of Jordan structures in algebra and analysis meeting. Tribute to El Amin Kaidi for his 60th birthday. Almer´ıa 2009. Editorial C´ırculo Rojo, Almer´ıa 2010. 669, 674, 682 [705] A. Castell´on, J. A. Cuenca, A. Fern´andez, and C. Mart´ın (eds.), Proceedings of the International Conference on Jordan Structures (M´alaga, 1997). Univ. M´alaga, M´alaga, 1999. 679, 680, 700, 707 [706] J. Cerd`a, Linear functional analysis. Grad. Stud. Math. 116, American Mathematical Society, Providence, RI; Real Sociedad Matem´atica Espa˜nola, Madrid, 2010. [707] S. B. Chae, Holomorphy and calculus in normed spaces. With an appendix by Angus E. Taylor. Monographs Textbooks Pure Appl. Math. 92. Marcel Dekker Inc., New York, 1985. 23, 53 [708] A. Chandid, Alg`ebres absolument valu´ees qui satisfont a` (xp , xq , xr ) = 0. PhD thesis, Casablanca 2009. [709] G. Choquet, Cours d’analyse. Tome II: Topologie. Masson et Cie, Editeurs, Paris, 1964.

References

693

[710] C.-H. Chu, Jordan structures in geometry and analysis. Cambridge Tracts in Math. 190, Cambridge University Press, New York, 2012. xix, 174, 207, 209, 210, 597, 711 [711] J. B. Conway, A course in functional analysis. Second edition. Grad. Texts in Math. 96, Springer, New York, 1990. [712] J. H. Conway and D. A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry. A K Peters, Ltd., Natick, MA, 2003. [713] E. R. Cowie, Isometries in Banach algebras. PhD thesis, Swansea 1981. [714] J. A. Cuenca, Sobre H ∗ -´algebras no asociativas. Teor´ıa de estructura de las H ∗ -´algebras de Jordan no conmutativas semisimples. PhD thesis, Granada 1982, Secretariado de Publicaciones de la Universidad de M´alaga, 1984. xxiii, 545, 548, 597 [715] H. G. Dales, Banach algebras and automatic continuity. London Math. Soc. Monogr. New Series 24, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2000. 267, 414 [716] M. M. Day, Normed linear spaces. Ergeb. Math. Grenzgeb. 21, Springer, Berlin, 1973. [717] A. Defant and K. Floret, Tensor norms and operator ideals. North-Holland Math. Stud. 176, North-Holland Publishing Co., Amsterdam, 1993. 267, 356, 416, 646 [718] J. Diestel, Geometry of Banach spaces – Selected topics, Lecture Notes in Math. 485, Springer, Berlin–Heidelberg–New York, 1975. [719] J. Diestel, J. H. Fourie, and J. Swart, The metric theory of tensor products. Grothendieck’s r´esum´e revisited. American Mathematical Society, Providence, RI, 2008. [720] J. Diestel and J. J. Uhl Jr., Vector measures. Mathematical Surveys 15, American Mathematical Society, Providence, RI, 1977. [721] S. Dineen, The Schwarz lemma. Oxford Math. Monogr., Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1989. 79, 87, 88, 207, 211 [722] R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs Surveys Pure Appl. Math. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. [723] J. Dixmier, Les alg`ebres d’op´erateurs dans l’espace hilbertien (alg`ebres de von Neumann). Deuxi`eme e´ dition, revue et augment´ee. Cahiers Scientifiques 25, ´ Gauthier-Villars Editeur, Paris, 1969. 492, 506 [724] J. Dixmier, Les C∗ -alg`ebres et leurs repr´esentations. Deuxi`eme e´ dition. Cahiers ´ Scientifiques 29, Gauthier-Villars Editeur, Paris, 1969. [725] R. S. Doran and V. A. Belfi, Characterizations of C∗ -algebras: The Gelfand– Naimark theorems. Monographs Textbooks Pure Appl. Math. 101, Marcel Dekker, New York–Basel, 1986. xvii, 401 [726] N. Dunford and J. T. Schwartz, Linear operators. Part I. General theory. With the assistance of W. G. Bade and R. G. Bartle. Reprint of the 1958 original. Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988. 12, 533, 541 [727] H.-D. Ebbinghaus, H. Hermes, F. Hirzbruch, H. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert, Numbers. Grad. Texts in Math. 123, Springer, New York, 1991. 409 [728] A. Elduque and H. Ch. Myung, Mutations of alternative algebras. Math. Appl. 278, Kluwer Academic Publishers Group, Dordrecht, 1994. [729] M. Fabian, P. Habala, P. H´ajek, V. Montesinos, and V. Zizler, Banach space theory. The basis for linear and nonlinear analysis. CMS Books Math., Springer, New York, 2011. 247, 249, 288, 338

694

References

[730] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces. Vol. 1. Function spaces. Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 129, Chapman & Hall/CRC, Boca Raton, FL, 2003. [731] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces. Vol. 2. Vector-valued function spaces. Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 138, Chapman & Hall/CRC, Boca Raton, FL, 2008. [732] Y. Friedmann and T. Scarr, Physical applications of homogeneous balls. Prog. Math. Phys. 40, Birh¨auser, Boston, 2005. 207 [733] S. Gonz´alez (ed.), Nonassociative algebra and its applications (Oviedo, 1993). Math. Appl. 303, Kluwer Academic Publishers, Dordrecht–Boston–London 1994. 679, 681, 685, 690, 702, 706 [734] S. Gonz´alez and H. Ch. Myung (eds.), Nonassociative algebraic models. Proceedings of the workshop held at the Universidad de Zaragoza, Zaragoza, April 1989. Nova Science Publishers, Inc., Commack, NY, 1992. 671, 675, 685, 701, 702, 709 [735] K. R. Goodearl, Notes on real and complex C∗ -algebras. Shiva Mathematics Series 5. Shiva Publishing Ltd., Nantwich, 1982. [736] A. Grothendieck, Produits tensoriels topologiques et espaces nucl´eaires. Mem. Amer. Math. Soc. (1955), no. 16. [737] W. R. Hamilton, On quaternions, or on a new system of imaginaries in Algebra. Philosophical Magazine, (1844–1850), ed. D. R. Wilkins, 2000. http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/OnQuat.pdf [738] H. Hanche-Olsen and E. Størmer, Jordan operator algebras. Monographs and Studies in Mathematics 21, Pitman, Boston, 1984. xviii, xx, xxi, 3, 8, 9, 11, 12, 13, 14, 286, 330, 332, 339, 347, 348, 362, 398, 399, 401, 416, 551 [739] P. Harmand, D. Werner, and W. Werner, M-ideals in Banach spaces and Banach algebras. Lecture Notes in Math. 1547, Springer, Berlin, 1993. 19, 266, 267 [740] P. de la Harpe, Classical Banach–Lie algebras and Banach–Lie groups of operators in Hilbert space. Lecture Notes in Math. 285, Springer, Berlin–New York, 1972. 554 [741] M. Hazewinkel (ed.), Encyclopaedia of mathematics, suplement III. Kluwer Academic Publishers, 2002. 679, 685, 703 [742] A. Ya. Helemskii, Banach and locally convex algebras. Translated from Russian by A. West. Oxford Sci. Publ., The Clarendon Press, Oxford University Press, New York, 1993. [743] I. N. Herstein, Topics in ring theory. The University of Chicago Press, Chicago, 1969. 367, 510 [744] I. N. Herstein, Noncommutative rings. Reprint of the 1968 original. With an afterword by Lance W. Small. Carus Mathematical Monographs 15, Mathematical Association of America, Washington, DC, 1994. [745] E. Hille, Functional analysis and semi-groups. Amer. Math. Soc. Colloq. Publ. 31, American Mathematical Society, New York, 1948. [746] E. Hille and R. S. Phillips, Functional analysis and semi-groups. Third printing of the revised edition of 1957. Amer. Math. Soc. Colloq. Publ. 31, American Mathematical Society, Providence, RI, 1974. [747] K. Hoffman, Banach spaces of analytic functions. Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962. [748] B. Iochum, Cˆones autopolaires et alg`ebres de Jordan. Lecture Notes in Math. 1049, Springer, Berlin, 1984. [749] B. Iochum and G. Loupias (eds.), Colloque sur les alg`ebres de Jordan, Montpellier, Octobre 1985. Ann. Sci. Univ. ‘Blaise Pascal’, Clermont II, S´er. Math. 27 (1991). 674, 679, 681, 685, 702

References

695

[750] R. Iordanescu, Jordan structures in analysis, geometry and physics. Editura Academiei Romˆane, Bucharest, 2009. 597 [751] J. M. Isidro and L. L. Stach´o, Holomorphic automorphism groups in Banach spaces: an elementary introduction. North-Holland Math. Stud. 105, North-Holland Publishing Co., Amsterdam, 1985. xix, 53, 87, 88, 135, 174, 207, 211 [752] N. Jacobson, Lie algebras. Interscience Tracts in Pure and Applied Mathematics 10, Interscience Publishers, New York–London, 1962. 549 [753] N. Jacobson, Structure of rings. Amer. Math. Soc. Colloq. Publ. 37, Providence, RI, 1964. 500, 559, 598 [754] N. Jacobson, Structure and representations of Jordan algebras. Amer. Math. Soc. Colloq. Publ. 39, Providence, RI, 1968. 472, 594, 597 [755] N. Jacobson, Structure theory of Jordan algebras. University of Arkansas Lecture Notes in Mathematics 5, University of Arkansas, Fayetteville, AR, 1981. 361 [756] G. J. O. Jameson, Topology and normed spaces. Chapman and Hall, London; Halsted Press [John Wiley & Sons], New York, 1974. [757] W. B. Johnson and J. Lindenstrauss (eds.), Handbook of the geometry of Banach spaces. Vol. 2. North-Holland, Amsterdam, 2003. 671, 681 [758] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I. Elementary theory. Grad. Stud. Math. 15, American Mathematical Society, Providence, RI, 1997. 330, 366, 459 [759] A. Kaidi, Bases para una teor´ıa de las a´ lgebras no asociativas normadas. PhD thesis, Granada 1978. 597 [760] I. L. Kantor and A. S. Solodovnikov, Hypercomplex numbers. An elementary introduction to algebras. Translated from Russian by A. Shenitzer. Springer, New York, 1989. [761] I. Kaplansky, Rings of operators. Math. Lecture Note Ser., W. A. Benjamin, Inc., New York–Amsterdam, 1968. xxi, 398 [762] I. Kaplansky, Algebraic and analytic aspects of operator algebras. CBMS Regional Conf. Ser. in Math. 1, American Mathematical Society, Providence, RI, 1970. [763] M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the theory of groups. Translated from the second Russian edition by Robert G. Burns. Grad. Texts in Math. 62, Springer-Verlag, New York–Berlin, 1979. 633 [764] W. Kaup, K. McCrimmon, and H. P. Petersson (eds.), Jordan algebras, Proceedings of the conference held in Oberwolfach, Germany, August 9–15, 1992. Walter de Gruyter, Berlin–New York, 1994. 685, 686 [765] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions. Amer. Math. Soc. Colloq. Publ. 44, Providence, RI, 1998. [766] T. Y. Lam, A first course in noncommutative rings. Second edition. Grad. Texts in Math. 131, Springer, New York, 2001. [767] K. B. Laursen and M. M. Neumann, An introduction to local spectral theory. London Math. Soc. Monogr. New Series 20, Clarendon Press, Oxford, 2000. 596 [768] B. Li, Real operator algebras. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. 549 [769] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I. Sequence spaces. Ergeb. Math. Grenzgeb. 92, Springer, Berlin–New York, 1977. 261 [770] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II. Function spaces. Ergeb. Math. Grenzgeb. 97, Springer, Berlin–New York, 1979. [771] O. Loos, Jordan pairs. Lecture Notes in Math. 460, Springer, Berlin–New York, 1975. 445 [772] O. Loos, Bounded symmetric domains and Jordan pairs. Lecture Notes, University of California, Irvine, 1977. 211, 337, 441, 444, 454, 460

696

References

[773] R. J. Loy (ed.), Conference on Automatic Continuity and Banach Algebras (Canberra, 1989). Proc. Centre Math. Anal. Austral. Nat. Univ. 21, Austral. Nat. Univ., Canberra, 1989. 672, 681 ´ [774] M. Mart´ın, Indice num´erico de un espacio de Banach. PhD thesis, Granada 2000. [775] J. Mart´ınez, Sobre a´ lgebras de Jordan normadas completas. PhD thesis, Granada 1977. Tesis Doctorales de la Universidad de Granada 149, Secretariado de Publicaciones de la Universidad de Granada, 1977. 597 [776] R. D. Mauldin (ed.), The Scottish book. Mathematics from the Scottish caf´e. Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979. Birkh¨auser, Boston, Mass., 1981. [777] K. McCrimmon, A taste of Jordan algebras. Universitext, Springer, New York, 2004 361, 364, 401, 589, 591, 598 [778] R. E. Megginson, An introduction to Banach space theory. Grad. Texts in Math. 183, Springer, New York, 1998. 7, 220, 340 [779] A. Meyberg, Lectures on algebras and triple systems. Lecture notes, The University of Virginia, Charlottesville, 1972. [780] V. M¨uller, Spectral theory of linear operators and spectral systems in Banach algebras. Second edition. Oper. Theory Adv. Appl. 139, Birkh¨auser Verlag, Basel, 2007. [781] G. J. Murphy, C∗ -algebras and operator theory. Academic Press, Inc., Boston, MA, 1990. 330, 408 [782] H. Ch. Myung, Malcev-admissible algebras. Progr. Math. 64, Birkh¨auser Boston, Inc., Boston, MA, 1986. [783] M. A. Naimark, Normed algebras. Translated from the second Russian edition by Leo F. Boron. Third edition. Wolters–Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics, Wolters–Noordhoff Publishing, Groningen, 1972. 551 [784] F. G. Oca˜na, El axioma de Sakai en JV-´algebras. PhD thesis, Granada 1979. Tesis Doctorales de la Universidad de Granada 224, Secretariado de Publicaciones de la Universidad de Granada, 1979. [785] S. Okubo, Introduction to octonion and other non-associative algebras in physics. Montroll Memorial Lecture Ser. Math. Phys. 2, Cambridge University Press, Cambridge, 1995. [786] T. W. Palmer, Banach algebras and the general theory of ∗-algebras, Volume I, algebras and Banach algebras. Encyclopedia Math. Appl. 49, Cambridge University Press, Cambridge, 1994. xxvi, 650, 651, 652, 654 [787] T. W. Palmer, Banach algebras and the general theory of ∗-algebras, Volume II, ∗-algebras. Encyclopedia Math. Appl. 79, Cambridge University Press, Cambridge, 2001. xvii [788] V. Paulsen, Completely bounded maps and operator algebras. Cambridge Stud. Adv. Math. 78, Cambridge University Press, Cambridge, 2002. 273 [789] G. Pedersen, Analysis now. Grad. Texts in Math. 118, Springer, New York, 1989. 330 [790] A. Pietsch, History of Banach spaces and linear operators. Birkh¨auser Boston, Inc., Boston, MA, 2007. 236, 550 [791] G. Pisier, Introduction to operator space theory. London Math. Soc. Lecture Note Ser. 294, Cambridge University Press, Cambridge, 2003. [792] D. Przeworska-Rolewicz, Linear spaces and linear operators. Institute of Mathematics of the Polish Academy of Sciences, Warsaw, 2007. [793] T. Ransford, Potential theory in the complex plane. London Math. Soc. Stud. Texts 28, Cambridge University Press, Cambridge, 1995. [794] Yu. P. Razmyslov, Identities of algebras and their representations. Transl. Math. Monogr. 138, American Mathematical Society, Providence, RI, 1994.

References

697

[795] C. E. Rickart, General theory of Banach algebras. The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, NJ–Toronto–London–New York, 1960. 335, 481, 524, 548, 551 [796] F. Riesz and B. Sz-Nagy, Lec¸ons d’analyse fonctionelle. Cinqui`eme e´ dition. Gauthier-Villars, Paris; Akad´emiai Kiad´o, Budapest, 1968. 330 [797] S. Roch, P. A. Santos, and B. Silbermann, Non-commutative Gelfand theories. A tool-kit for operator theorists and numerical analysts. Universitext, Springer London, Ltd., London, 2011. [798] A. Rodr´ıguez, Contribuci´on a la teor´ıa de las C∗ -´algebras con unidad. PhD thesis, Granada 1974. Tesis Doctorales de la Universidad de Granada 57, Secretariado de Publicaciones de la Universidad de Granada, 1974. 336 [799] A. Rodr´ıguez, N´umeros hipercomplejos en dimensi´on infinita. Discurso de ingreso en la Academia de Ciencias Matem´aticas, F´ısico-Qu´ımicas y Naturales de Granada, Granada, 1993. 409 [800] S. Rolewicz, Metric linear spaces. Math. Appl. 20, D. Reidel Publishing Co., Dordrecht; PWN-Polish Scientific Publishers, Warsaw, 1985. 196 [801] H. E. Rose, Linear algebra. A pure mathematical approach. Birkh¨auser Verlag, Basel, 2002. [802] L. H. Rowen, Ring theory, Volume I. Pure Appl. Math. 127. Academic Press, Inc., Boston, 1988. [803] W. Rudin, Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. 425, 433 [804] W. Rudin, Functional analysis. Second edition. Internat. Ser. Pure Appl. Math., McGraw-Hill, Inc., New York, 1991. [805] B. P. Rynne and M. A. Youngson, Linear functional analysis. Second edition. Springer Undergrad. Math. Ser., Springer London, Ltd., London, 2008. [806] S. Sakai, C∗ -algebras and W ∗ -algebras. Ergeb. Math. Grenzgeb. 60, Springer, New York–Heidelberg, 1971. 19, 329, 331, 339, 399, 421 [807] S. Sakai, Operator algebras in dynamical systems. The theory of unbounded derivations in C∗ -algebras. Encyclopedia Math. Appl. 41, Cambridge University Press, Cambridge, 1991. [808] R. D. Schafer, An introduction to nonassociative algebras. Corrected reprint of the 1966 original. Dover Publications, Inc., New York, 1995. 208, 360, 361, 401, 545, 652 [809] R. Schatten, Norm ideals of completely continuous operators. Second printing. Ergeb. Math. Grenzgeb. 27, Springer, Berlin–New York, 1970. 409, 481, 494, 524, 531, 532, 533 [810] A. M. Sinclair, Automatic continuity of linear operators. London Math. Soc. Lecture Note Ser. 21, Cambridge University Press, Cambridge–New York–Melbourne, 1976. 499 [811] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis. Second edition. John Wiley & Sons, New York–Chichester–Brisbane, 1980. [812] J. P. Tian, Evolution algebras and their applications. Lecture Notes in Math. 1921, Springer, Berlin, 2008. [813] D. M. Topping, Jordan algebras of self-adjoint operators. Mem. Amer. Math. Soc. (1965), no. 53. 400 [814] H. Upmeier, Symmetric Banach Manifolds and Jordan C∗ -algebras. North-Holland Math. Stud. 104, North-Holland, Amsterdam, 1985. xix, 52, 53, 87, 135, 174, 207, 211, 597 [815] H. Upmeier Jordan algebras in analysis, operator theory, and quantum mechanics. CBMS Regional Conf. Ser. in Math. 67, American Mathematical Society, Providence, RI, 1987. 1987. 53, 87, 135, 174, 207, 211

698

References

[816] A. R. Villena, Algunas cuestiones sobre las JB∗ -´algebras: centroide, centroide extendido y z´ocalo. PhD thesis, Granada 1990. [817] A. Wilansky, Modern methods in topological vector spaces. McGraw-Hill International Book Co., New York, 1978. [818] A. Wilansky, Topology for analysis. Reprint of the 1970 edition. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1983. [819] R. Wisbauer, Modules and algebras: bimodule structure and group actions on algebras. Pitman Monographs Surveys Pure Appl. Math. 81, Longman, Harlow, 1996. [820] W. Zelazko, Banach algebras. Translated from Polish by Marcin E. Kuczma. Elsevier Publishing Co., Amsterdam–London–New York; PWN-Polish Scientific Publishers, Warsaw, 1973. [821] J. Zem´anek, Properties of the spectral radius in Banach algebras. PhD thesis, Warszawa 1977. [822] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative. Translated from Russian by Harry F. Smith. Pure Appl. Math. 104, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York–London, 1982. 360, 361, 401, 413, 414, 415, 570, 571, 597, 631, 636, 647

References

699

Additional Bibliography to Volume 2 Papers [823] J. F. Aarnes, On the Mackey-topology for a von Neumann algebra. Math. Scand. 22 (1968), 87–107. 332 [824] M. D. Acosta, An inequality for norm of operators. Extracta Math. 10 (1995), 115–8. 469 [825] M. D. Acosta, Denseness of norm attaining mappings. RACSAM. Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. 100 (2006), 9–30. 338 [826] C. A. Akemann, The dual space of an operator algebra. Trans. Amer. Math. Soc. 126 (1967), 286–302. xx, 268, 274, 332, 334, 335 [827] A. A. Albert, A theory of trace-admissible algebras. Proc. Nat. Acad. Sci. USA 35 (1949), 317–22. Reprinted in [692], pp. 491–6. 545 [828] A. A. Albert and N. Jacobson, On reduced exceptional simple Jordan algebras. Ann. of Math. 66 (1957), 400–17. Reprinted in [692], pp. 677–94. 361 [829] E. Albrecht and H. G. Dales, Continuity of homomorphisms from C∗ -algebras and other Banach algebras. In Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), pp. 375–96, Lecture Notes in Math. 975, Springer, Berlin-New York, 1983. 549 [830] E. M. Alfsen and E. G. Effros, Structure in real Banach spaces. I, II. Ann. of Math. 96 (1972), 98–128; ibid. 96 (1972), 129–73. 19, 398 [831] B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras. Math. Ann. 237 (1978), 133–56. 555 [832] B. N. Allison, Models of isotropic simple Lie algebras. Comm. Algebra 7 (1979), 1835–75. 555 [833] B. N. Allison and J. R. Faulkner, The algebra of symmetric octonion tensors. Nova J. Algebra Geom. 2 (1993), 47–57. 555 [834] J. A. Anquela, M. Cabrera, and A. Moreno, Eater ideals in Jordan algebras. J. Pure Appl. Algebra 125 (1998), 1–17. 464 [835] P. Ara, The extended centroid of C∗ -algebras. Archiv. Math. 54 (1990), 358–64. 408 [836] P. Ara, On the symmetric algebra of quotients of a C∗ -algebra. Glasgow Math. J. 32 (1990), 377–9. 408 [837] J. Arazy, An application of infinite-dimensional holomorphy to the geometry of Banach spaces. In Geometrical aspects of functional analysis (1985/86), pp. 122–50, Lecture Notes in Math. 1267, Springer, Berlin, 1987. xix, 87, 135, 174, 207 [838] R. Arens and G. Goldberg, Quadrative seminorms and Jordan structures on algebras. Linear Algebra Appl. 181 (1993), 269–78. 470 [839] B. Aupetit, Analytic multivalued functions in Banach algebras and uniform algebras. Adv. Math. 44 (1982), 18–60. 603 [840] B. Aupetit, Some uses of subharmonicity in functional analysis. In Spectral theory (Warsaw, 1977), pp. 31–7, Banach Center Publ. 8, Polish Acad. Sci. Inst. Math., Warsaw, 1982. 603 [841] B. Aupetit, Inessential elements in Banach algebras. Bull. London Math. Soc. 18 (1986), 493–7. 599 [842] B. Aupetit, Spectral characterization of the radical in Banach and Jordan-Banach algebras. Math. Proc. Cambridge Philos. Soc. 114 (1993), 31–5. 603 [843] B. Aupetit, Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras. J. London Math. Soc. 62 (2000), 917–24. 603 [844] B. Aupetit and H. du T. Mouton, Trace and determinant in Banach algebras. Studia Math. 121 (1996), 115–36. 601

700

References

[845] B. Aupetit and A. Zra¨ıbi, Propri´et´es analytiques du spectre dans les alg`ebres de Jordan-Banach. Manuscripta Math. 38 (1982), 381–6. 596, 597 [846] G. F. Bachelis, Homomorphisms of annihilator Banach algebras. Pacific J. Math. 25 (1968), 229–47. xxv, 584 [847] W. G. Bade, P. C. JR. Curtis, and A. M. Sinclair, Raising bounded groups and splitting of radical extensions of conmmutative Banach algebras. Studia Math. 141 (1) (2000), 85–98. 651 [848] J. W. Baker and J. S. Pym, A remark on continuous bilinear mappings. Proc. Edinburgh Math. Soc. 17 (1970/71), 245–8. 436 [849] D. Baki´c and B. Guljaˇs, Operators on Hilbert H ∗ -modules. J. Operator Theory 46 (2001), 123–37. 557 [850] V. K. Balachandran, Automorphisms of L∗ -algebras. Tˆohoku Math. J. 22 (1970), 163–73. 549 [851] V. K. Balachandran, Real L∗ -algebras. Indian J. Pure Appl. Math. 3 (1972), 1224–46. 554 [852] T. J. Barton, T. Dang, and G. Horn, Normal representations of Banach Jordan triple systems. Proc. Amer. Math. Soc. 102 (1988), 551–5. 211, 241 [853] T. J. Barton and Y. Friedman, Grothendieck’s inequality for JB∗ -triples and applications. J. London Math. Soc. 36 (1987), 513–23. xx, 268, 274, 275, 337, 338, 345, 346 [854] T. J. Barton and R. M. Timoney, Weak∗ -continuity of Jordan triple products and its applications. Math. Scand. 59 (1986), 177–91. xix, 2, 211, 236, 237 [855] K. I. Beidar, A. V. Mikhalev, and A. M. Slinko, A criterion for primeness of nondegenerate alternative and Jordan algebras. Trudy Moskov. Mat. Obshch. 50 (1987), 130–7; translation in Trans. Moscow Math. Soc. 50 (1988), 129–37. xxii, 408, 468 [856] G. Benkart, On inner ideals and ad-nilpotent elements of Lie algebras. Trans. Amer. Math. Soc. 232 (1977), 61–81. 600 [857] M. Benslimane and N. Boudi, Isomorphisms of Jordan-Banach algebras. Extracta Math. 11 (1996), 414–6. 603 [858] M. Benslimane and N. Boudi, Ring isomorphisms of Jordan-Banach algebras. In Commutative ring theory (F`es, 1995), pp. 105–11, Lect. Notes Pure Appl. Math. 185, Dekker, New York, 1997. 603 [859] M. Benslimane and L. Mesmoudi, Characterization of commutative normed algebras. In [705], pp. 53–7. 436 [860] M. Benslimane, L. Mesmoudi, and A. Rodr´ıguez, A characterization of commutativity for non-associative normed algebras. Extracta Math. 15 (2000), 121–43. xxii, 436 [861] M. A. Berger and Y. Wang, Bounded semigroups of matrices. Linear Algebra Appl. 166 (1992), 21–7. 651 [862] D. P. Blecher and V. I. Paulsen, Multipliers of operator spaces, and the injective envelope. Pacific J. Math. 200 (2001), 1–17. 273 [863] F. Bombal, On (V∗ ) sets and Pełczy´nski’s property (V∗ ). Glasgow Math. J. 32 (1990), 109–20. 266 [864] F. F. Bonsall and A. W. Goldie, Annihilator algebras. Proc. London Math. Soc. 4 (1954), 154–67. 584 [865] N. Boudi, H. Marhnine, and C. Zarhouti, Additive derivations on Jordan Banach pairs. Comm. Algebra 32 (2004), 3609–25. 466, 603 [866] H. N. Boyadjiev and M. A. Youngson, Alternators on Banach Jordan algebras. C. R. Acad. Bulgare Sci. 33 (1980), 1589–90. 400 [867] R. Braun, W. Kaup, and H. Upmeier, On the automorphisms of circular and Reinhardt domains in complex Banach spaces. Manuscripta Math. 25 (1978), 97–133. 207 [868] M. Breˇsar, Functional identities: A survey. Contemp. Math. 259 (2000), 93–109. 602

References

701

[869] M. Breˇsar and A. R. Villena, The range of a derivation on a Jordan-Banach algebra. Studia Math. 146 (2001), 177–200. 466, 707 [870] L. J. Bunce, The theory and structure of dual JB-algebras. Math. Z. 180 (1982), 525–34. 584, 598 [871] L. J. Bunce, On compact action in JB-algebras. Proc. Edinburgh Math. Soc. 26 (1983), 353–60. 400 [872] L. J. Bunce, Norm preserving extensions in JBW ∗ -triple preduals. Quart. J. Math. Oxford 52 (2001), 133–6. 338, 339, 346 [873] L. J. Bunce and C.-H. Chu, Compact operations, multipliers and Radon-Nikodym property in JB∗ -triples. Pacific J. Math. 153 (1992), 249–65. xx, 275, 346, 450, 459, 461 [874] L. J. Bunce and C.-H. Chu, Real contractive projections on commutative C∗ -algebras. Math. Z. 226 (1997), 85–101. 210 [875] L. J. Bunce, C.-H. Chu, L. L. Stach´o, and B. Zalar, On prime JB∗ -triples. Quart. J. Math. Oxford 49 (1998), 279–90. 409 [876] J. C. Cabello and M. Cabrera, Structure theory for multiplicatively semiprime algebras. J. Algebra 282 (2004), 386–421. xxv, 546, 584 [877] J. C. Cabello, M. Cabrera, and A. Fern´andez, π -complemented algebras through pseudocomplemented lattices. Order 29 (2012), 463–79. 585 [878] J. C. Cabello, M. Cabrera, G. L´opez, and W. S. Martindale III, Multiplicative semiprimeness of skew Lie algebras. Comm. Algebra 32 (2004), 3487–501. 546 [879] J. C. Cabello, M. Cabrera, and E. Nieto, ε-complemented algebras. J. Algebra 349 (2012), 234–67. 546, 585 [880] J. C. Cabello, M. Cabrera, A. Rodr´ıguez, and R. Roura, A characterization of π -complemented algebras. Comm. Algebra 41 (2013), 3067–79. 585 [881] J. C. Cabello, M. Cabrera, and R. Roura, A note on the multiplicative primeness of degenerate Jordan algebras. Sib. Math. J. 51 (2010), 818–23. 546 [882] J. C. Cabello, M. Cabrera, and R. Roura, π -complementation in the unitisation and multiplication algebras of a semiprime algebra. Comm. Algebra 40 (2012), 3507–31. 546 [883] M. Cabrera, A. El Marrakchi, J. Mart´ınez, and A. Rodr´ıguez. Bounded differential operators on Hilbert modules and derivations of structurable H ∗ -algebras. Comm. Algebra 21 (1993), 2905–45. 559 [884] M. Cabrera, J. Mart´ınez, and A. Rodr´ıguez, Hilbert modules over H ∗ -algebras in relation with Hilbert ternary rings. In [734], pp. 33–44. 558 [885] M. Cabrera, J. Mart´ınez, and A. Rodr´ıguez, Hilbert modules revisited: orthonormal bases and Hilbert–Schmidt operators. Glasgow Math. J. 37 (1995), 45–54. 557 [886] M. Cabrera and A. A. Mohammed, Extended centroid and central closure of the multiplication algebra. Comm. Algebra 27 (1999), 5723–36. 546, 576 [887] M. Cabrera and A. A. Mohammed, A representation theorem for algebras with commuting involutions. Linear Algebra Appl. 306 (2000), 25–31. 476 [888] M. Cabrera and A. A. Mohammed, Extended centroid and central closure of multiplicatively semiprime algebras. Comm. Algebra 29 (2001), 1215–33. 546 [889] M. Cabrera and A. A. Mohammed, Totally multiplicatively prime algebras. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 1145–62. xxiii, 408, 546, 547, 549 [890] M. Cabrera and A. A. Mohammed, Algebra of quotients with bounded evaluation of a normed prime algebra. In: Operator theory and Banach algebras. Proceedings of the International Conference in Analysis, Rabat (Moroco), April 12–14, 1999 (eds. M. Chidami, R. Curto, M. Mbekhta, F.-H. Vasilescu and J. Zem´anek). Theta Found., Bucharest, 2003, pp. 39–50. 547

702

References

[891] M. Cabrera and A. A. Mohammed, Algebras of quotients with bounded evaluation of a normed semiprime algebra. Studia Math. 154 (2003), 113–35. 547 [892] M. Cabrera, A. Moreno, and A. Rodr´ıguez, On primitive Jordan-Banach algebras. In [733], pp. 54–9. 463 [893] M. Cabrera, A. Moreno, and A. Rodr´ıguez, On the behaviour of Jordan-algebra norms on associative algebras. Studia Math. 113 (1995), 81–100. xxii, 469, 470, 471, 474 [894] M. Cabrera and A. Rodr´ıguez, A characterization of the centre of a nondegenerate Jordan algebra. Comm. Algebra 21 (1993), 359–69. 469 [895] M. Cabrera and A. Rodr´ıguez, Zel’manov’s theorem for nondegenerately ultraprime Jordan-Banach algebras. In [733], pp. 60–5. 462 [896] M. Cabrera and A. R. Villena, Multiplicative-semiprimeness of nondegenerate Jordan algebras. Comm. Algebra 32 (2004), 3995–4003. 546 [897] P. Cao and Yu. V. Turovskii, Topological radicals, VI. Scattered elements in Banach Jordan and associative algebras. Studia Math. 235 (2016), 171–208. 603, 652 ´ Cartan, Sur les domaines bornes homogenes de l’espace de n variables complexes. [898] E. Abh. Math. Sem. Univ. Hamburg 11 (1935), 116–62. 210 [899] H. Cartan, Les fonctions de deux variables complexes et le probl`eme de la repr´esentation analytique. J. Math. Pures Appl., 9e s´erie, 10 (1931), 1–114. 87 [900] A. Castell´on and J. A. Cuenca, Compatibility in Jordan H ∗ -triple systems. Boll. Un. Mat. Ital. B 4 (1990), 433–47. 560 [901] A. Castell´on and J. A. Cuenca, Alternative H ∗ -triple systems. Comm. Algebra 20 (1992), 3191–206. 561 [902] A. Castell´on, J. A. Cuenca, and C. Mart´ın, Ternary H ∗ -algebras. Boll. Un. Mat. Ital. B 6 (1992), 217–28. 559 [903] A. Castell´on, J. A. Cuenca, and C. Mart´ın, Jordan H ∗ -triples. In [733], pp. 66–72. 549, 560 [904] A. Castell´on, J. A. Cuenca, and C. Mart´ın, Special Jordan H ∗ -triple system. Comm. Algebra 28 (2000), 4699–706. 560 [905] M. Chaperon, On the Cauchy–Kowalevski theorem. Enseign. Math. 55 (2009), 359–71. 137 [906] C.-H. Chu and B. Iochum, Weakly compact operators on Jordan triples. Math. Ann. 281 (1988), 451–8. 335, 340, 341 [907] C.-H. Chu and B. Iochum, Complementation of Jordan triples in von Neumann algebras. Proc. Amer. Math. Soc. 108 (1990), 19–24. 336 [908] C.-H. Chu, A. Moreno, and A. Rodr´ıguez, On prime real JB∗ -triples. In Function spaces (Edwardsville, IL, 1998), pp. 105–9, Contemp. Math. 232, Amer. Math. Soc., Providence, RI, 1999. 346 [909] P. J. Cohen, Factorization in group algebras. Duke Math. J. 26 (1959), 199–205. 659 [910] M. J. Crabb, A new prime C∗ -algebra that is not primitive. J. Funct. Anal. 236 (2006), 630–3. 467 [911] J. A. Cuenca, Sur la th´eorie de structure des H ∗ -alg`ebres de Jordan non commutatives. In [749], pp. 143–52. 553 [912] J. A. Cuenca, Moufang H ∗ -algebras. Extracta Math. 17 (2002), 239–46. 552 [913] J. A. Cuenca, A. Garc´ıa, and C. Mart´ın, Jacobson density for associative pairs and its applications. Comm. Algebra 17 (1989), 2595–610. 560 [914] J. A. Cuenca, A. Garc´ıa, and C. Mart´ın, Prime associative superalgebras with nonzero socle. Algebras Groups Geom. 11 (1994), 359–69. 560 [915] J. A. Cuenca and C. Mart´ın, Jordan two-graded H ∗ -algebras. In [734], pp. 91–100. 559, 560

References

703

[916] J. A. Cuenca and F. Miguel, On a Hopf’s theorem. Preprint 2016. 409 [917] J. A. Cuenca and A. S´anchez, Structure theory for real noncommutative Jordan H ∗ -algebras. J. Algebra 164 (1994), 481–99. 553 [918] J. Cuntz, On the continuity of semi-norms on operator algebras. Math. Ann. 220 (1976), 171–83. 399 [919] R. E. Curto, Spectral theory of elementary operators. In Elementary operators and applications (Blaubeuren, 1991) (Ed. by M. Mathieu), pp. 3–52, World Sci. Publ., River Edge, NJ, 1992. 635 [920] A. D’Amour, Quadratic Jordan systems of Hermitian type. J. Algebra 149 (1992), 197–233. xxii, 438, 448, 459, 461 [921] A. D’Amour and K. McCrimmon, The structure of quadratic Jordan systems of Clifford type. J. Algebra 234 (2000), 31–89. xxii, 438, 452, 453, 454, 459 [922] J. Dieudonn´e, Sur le socle d’un anneau et les anneaux simples infinis. Bull. Soc. Math. France 70 (1942), 46–75. 598 [923] S. Dineen, Complete holomorphic vector fields on the second dual of a Banach space. Math. Scand. 59 (1986), 131–42. 236 [924] S. Dineen and R. M. Timoney, The centroid of a JB∗ -triple system. Math. Scand. 62 (1988), 327–42. 346 [925] S. Dineen and R. M. Timoney, The algebraic metric of Harris on JB∗ -triple systems. Proc. Roy. Irish Acad. Sect. A 90 (1990), 83–97. 558 [926] J. Dixmier, Sur les C∗ -alg`ebres. Bull. Soc. Math. France 88 (1960), 95–112. 467 [927] P. G. Dixon, Topologically nilpotent Banach algebras and factorization. Proc. Roy. Soc. Edinburgh Sect A 119 (1991), 329–41. xxvi, 650, 651, 654, 655, 657 [928] P. G. Dixon and V. M¨uller, A note on topologically nilpotent Banach algebras. Studia Math. 102 (1992), 269–75. xxvi, 623, 650, 651, 653 [929] P. G. Dixon and G. A. Willis, Approximate identities in extensions of topologically nilpotent Banach algebras. Proc. Roy. Soc. Edinburgh 122A (1992), 45–52. xxvi, 650, 651 [930] R. S. Doran and J. Wichmann, The Gelfand–Naimark theorems for C∗ -algebras. Enseignement Math. 23 (1977), 153–80. xvii [931] C. Draper, A. Fern´andez, E. Garc´ıa, and M. G´omez, The socle of a nondegenerate Lie algebra. J. Algebra 319 (2008), 2372–94. 600 [932] G. A. Edgar, An ordering for the Banach spaces. Pacific J. Math. 108 (1983), 83–98. 266 [933] C. M. Edwards and G. T. R¨uttimann, A characterization of inner ideals in JB∗ -triples. Proc. Amer. Math. Soc. 116 (1992), 1049–57. 339 [934] A. El Kinani and M. Oudadess, Un crit`ere g´en´eralis´e de Le Page et commutativit´e. C. R. Math. Rep. Acad. Sci. Canada 18 (1996), 71–4. 436 [935] A. El Marrakchi, Sur l’unit´e approch´ee des H ∗ -alg`ebres structurables avec annulateur z´ero. Int. J. Math. Game Theory Algebra 13 (2003), 267–79. 556 [936] L. Elsner, The generalized spectral-radius theorem: an analytic-geometric proof. Linear Algebra Appl. 220 (1995), 151–9. 651 [937] G. Emmanuele, On the Banach spaces with the property (V ∗ ) of Pełczy´nski. Ann. Mat. Pura Appl. 152 (1988), 171–81. 266 [938] A. B. A. Essaleh, A. M. Peralta, and M. I. Ram´ırez, Pointwise-generalized-inverses of linear maps between C∗ -algebras and JB∗ -triples. Operators and Matrices (to appear). 273, 274 [939] A. Fern´andez, Banach-Jordan pair. In [741], pp. 57–8. 462, 600 [940] A. Fern´andez, A Jordan approach to finitary Lie algebras. Quart. J. Math. 67 (2016), 565–71. 600

704

References

[941] A. Fern´andez and E. Garc´ıa, Algebraic lattices and nonassociative structure. Proc. Amer. Math. Soc. 126 (1998), 3211–21. 584 [942] A. Fern´andez, E. Garc´ıa, and M. G´omez, The Jordan algebras of a Lie algebra. J. Algebra 308 (2007), 164–77. 600 [943] A. Fern´andez, E. Garc´ıa, E. S´anchez, and M. Siles, Strong regularity and Hermitian Hilbert triples. Quart. J. Math. Oxford 45 (1994), 43–55. 558 [944] A. Fern´andez and A. Yu. Golubkov, Lie algebras with an algebraic adjoint representation revisited. Manuscripta Math. 140 (2013), 363–76. 600 [945] A. Fern´andez, H. Marhnine, and C. Zarhouti, Derivations on Banach-Jordan pairs. Quart. J. Math. Oxford 52 (2001), 269–83. 466, 603 [946] A. Fern´andez and M. I. Toc´on, Pseudocomplemented semilatices, boolean algebras, and compatible products. J. Algebra 242 (2001), 60–91. 584 [947] A. Fern´andez and M. I. Toc´on, Strongly prime Jordan pairs with nonzero socle. Manuscripta Math. 111 (2003), 321–40. 600 [948] F. J. Fern´andez-Polo, J. J. Garc´es, A. M. Peralta, and I. Villanueva, Tingley’s problem for spaces of trace class operators. Linear Algebra Appl. 529 (2017), 294–323. 550 [949] V. T. Filippov, On the theory of Mal’cev algebras. Algebra i Logika 16 (1977), 101–8. 555 [950] D. Finston, On multiplication algebras. Trans. Amer. Math. Soc. 293 (1986), 807–18. 546 [951] Y. Friedman and B. Russo, Contractive projections on operator triple systems. Math. Scand. 52 (1983), 279–311. 210, 331 [952] R. V. Garimella, On continuity of derivations and epimorphisms on some vectorvalued group algebras. Bull. Austral. Math. Soc. 56 (1997), 209–15. 596 [953] G. R. Giellis, A characterization of Hilbert modules. Proc. Amer. Math. Soc. 36 (1972), 440–2. 556 ´ [954] G. Godefroy, Epluchabilit´ e et unicit´e du pr´edual. S´eminaire Choquet, 17e ann´ee (1977/78), Initiation a´ l’analyse, Fasc. 2, Comm. No. 11, 3 pp., Secr´etariat Math., Paris, 1978. 8 [955] G. Godefroy, Parties admissibles d’un espace de Banach. Applications. Ann. Sci. ´ Ecole Norm. Sup. (4) 16 (1983), 109–22. 237, 267 [956] G. Godefroy, Existence and uniqueness of isometric preduals: a survey. In Banach space theory (Iowa City, IA, 1987) (Ed. by B.-L. Lin), pp. 131–93, Contemp. Math., 85, Amer. Math. Soc., Providence, 1989. 238, 241 [957] G. Godefroy and B. Iochum, Arens-regularity of Banach algebras and the geometry of Banach spaces. J. Funct. Anal. 80 (1988), 47–59. xix, 245, 267, 421 [958] G. Godefroy and P. Saab, Quelques espaces de Banach ayant les propri´et´es (V) ou (V ∗ ) de A. Pełczy´nski. C. R. Acad. Sci. Paris S´er. I Math. 303 (1986), 503–6. 245, 266 [959] G. Godefroy and P. Saab, Weakly unconditionally convergent series in M-ideals. Math. Scand. 64 (1989), 307–18. 266 [960] G. Godefroy and M. Talagrand, Classes d’espaces de Banach a` pr´edual unique. C. R. Acad. Sci. Paris S´er. I Math. 292 (1981), 323–5. 238 [961] G. Godefroy and M. Talagrand, Nouvelles classes d’espaces de Banach a` pr´edual ´ unique. S´eminaire d’Analyse Fonctionnelle de l’Ecole Polytechnique, 1980/81. 238, 239, 240, 241 [962] E. A. Gorin and V. Ja. Lin, A condition on the radical of a Banach algebra guaranteeing strong decomposability. (Russian) Mat. Zametki 2 (1967), 589-592; translation in Mathematical Notes 2 (1967), 851–2. 651

References

705

[963] S. Grabiner, Finitely generated, Noetherian, and Artinian Banach modules. Indiana Univ. Math. J. 26 (1977), 413–25. 601 [964] A. Grothendieck, R´esum´e de la th´eorie m´etrique des produits tensoriels topologiques. Bol. Soc. Mat. S˜ao Paulo 8 (1953), 1–79. Reprinted in Resenhas 2 (1996), 401–80. xx, 275, 342, 343 [965] A. Grothendieck, Sur les applications lin´eaires faiblement compactes d’espaces du type C(K). Canadian J. Math. 5 (1953), 129–73. 334 [966] S. L. Gulick, Commutativity and ideals in the biduals of topological algebras. Pacific J. Math. 18 (1966), 121–37. 268 [967] U. Haagerup, Solution of the similarity problem for cyclic representations of C∗ -algebras. Ann. Math. 118 (1983), 215–40. 342 [968] U. Haagerup, The Grothendieck inequality for bilinear forms on C∗ -algebras. Adv. in Math. 56 (1985), 93–116. 343, 344 [969] U. Haagerup and H. Hanche-Olsen, Tomita–Takesaki theory for Jordan algebras. J. Operator Theory 11 (1984), 343–64. 339 [970] U. Haagerup and T. Itoh, Grothendieck type norms for bilinear forms on C∗ -algebras. J. Operator Theory 34 (1995), 263–83. 342 [971] P. de la Harpe, Classification des L∗ -alg`ebres semi-simples r´eelles s´eparables. C. R. Acad. Sci. Paris S´er. A-B 272 (1971), A1559–61. 554 [972] L. A. Harris, Schwarz’s lemma in normed linear spaces. Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1014–7. 207 [973] L. A. Harris, A continuous form of Schwarz’s lemma in normed linear spaces. Pacific J. Math. 38 (1971), 635–9. 207 [974] L. A. Harris, Analytic invariants and the Schwarz–Pick inequality. Israel J. Math. 34 (1979), 177–97. 558 [975] L. A. Harris, A generalization of C∗ -algebras. Proc. London Math. Soc. 42 (1981), 331–61. 459 [976] M. Henriksen, A simple characterization of commutative rings without maximal ideals. Amer. Math. Monthly 82 (1975), 502–5. 659 [977] C. W. Henson and L. C. Moore, Jr., Subspaces of the nonstandard hull of a normed space. Trans. Amer. Math. Soc. 197 (1974), 131–43. 236 [978] G. Hessenberger, Analytic properties of the spectrum in Banach Jordan systems. Collect. Math. 47 (1996), 277–84. 603 [979] G. Horn, Characterization of the predual and ideal structure of a JBW ∗ -triple. Math. Scand. 61 (1987), 117–33. xix, 2, 211, 237, 238, 239, 241 [980] G. Horn and E. Neher, Classification of continuous JBW ∗ -triples. Trans. Amer. Math. Soc. 306 (1988), 553–78. 340, 344, 459 [981] N. Jacobson, A note on non-associative algebras. Duke Math. J. 3 (1937), 544–8. 545 [982] R. C. James, Uniformly non-square Banach spaces. Ann. of Math. 80 (1964), 542–50. 261 [983] F. B. Jamjoom, A. M. Peralta, and A. A. Siddiqui, Jordan weak amenability and orthogonal forms on JB∗ -algebras. Banach J. Math. Anal. 9 (2015), 126–45. 346 [984] F. B. Jamjoom, A. M. Peralta, A. A. Siddiqui, and H. M. Tahlawi, Approximation and convex decomposition by extremals and the λ-function in JBW ∗ -triples. Quart. J. Math. Oxford 66 (2015), 583–603. 346 [985] H. Jarchow, Weakly compact operators on C(K) and C∗ -algebras. In Functional analysis and its applications (Nice, 1986) (Ed. by H. Hogbe-Nlend), pp. 263–99, ICPAM Lecture Notes, World Sci. Publishing, Singapore, 1988. 274, 335 [986] J.-S. Jeang and C.-C. Ko, On the commutativity of C∗ -algebras. Manuscripta Math. 115 (2004), 195–8. 401

706

References

[987] G. Ji and J. Tomiyama, On characterizations of commutativity of C∗ -algebras. Proc. Amer. Math. Soc. 131 (2003), 3845–9. 401 [988] W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces. Israel J. Math. 9 (1971), 488–506. 236 [989] R. V. Kadison, Operator algebras with a faithful weakly-closed representation. Ann. of Math. 64 (1956), 175–81. 19 [990] N. J. Kalton, Rademacher series and decoupling. New York J. Math. 11 (2005), 563–95. 343 [991] R. Kantrowitz and M. M. Neumann, Automatic continuity of homomorphisms and derivations on algebras of continuous vector-valued functions. Czechoslovak Math. J. 45(120) (1995), 747–56. 596 [992] I. Kaplansky, Regular Banach algebras. J. Indian Math. Soc. 12, (1948), 57–62. 597 [993] W. Kaup, Jordan algebras and holomorphy. In Functional analysis, holomorphy, and approximation theory (Proc. Sem., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1978) (Ed. by S. Machado), pp. 341–65, Lecture Notes in Math., 843, Springer, Berlin, 1981. 211 ¨ [994] W. Kaup, Uber die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension. I. Math. Ann. 257 (1981), 463–86. II Math. Ann. 262 (1983), 57–75. 211, 557 [995] W. Kaup, Hermitian Jordan triple systems and the automorphisms of bounded symmetric domains. In [733], pp. 204–14. 210 [996] W. Kaup, On spectral and singular values in JB∗ -triples. Proc. Roy. Irish Acad. Sect. A 96 (1996), 95–103. 210 [997] W. Kaup and H. Upmeier, Banach spaces with biholomorphically equivalent unit balls are isomorphic. Proc. Amer. Math. Soc. 58 (1976), 129–33. 207 [998] E. Kissin, V. S. Shulman, and Yu. V. Turovskii, Topological radicals and Frattini theory of Banach Lie algebras. Integral Equations Operator Theory 74 (2012), 51–121. 652 [999] C. Le Page, Sur quelques conditions entraˆınant la commutativit´e dans les alg`ebres de Banach. C. R. Acad. Sci. Paris S´er. A–B 265 (1967), A235–A237. xxii, 347, 423, 436 [1000] C. S. Lin, On commutativity of C∗ -algebras. Glasgow Math. J. 29 (1987), 93–7. 400 [1001] J. Lindenstrauss, On operators which attain their norm. Israel J. Math. 1 (1963), 139–148. xx, 274, 337, 338 [1002] J. Lindenstrauss and H. P. Rosenthal, The Lp -spaces. Israel J. Math. 7 (1969), 325–49. 236 [1003] O. Loos, On the socle of a Jordan pair. Collect. Math. 40 (1989), 109–25. 599, 601 [1004] O. Loos, Finiteness conditions in Jordan pairs. Math. Z. 206 (1991), 577–87. 558 [1005] O. Loos and E. Neher, Complementation of inner ideals in Jordan pairs. J. Algebra 166 (1994), 255–95. 601 [1006] M. Mackey and P. Mellon, A Schwarz lemma and composition operators. Integral Equations Operator Theory 48 (2004), 511–24. 210 [1007] A. Maouche, Extension d’un th´eor`eme de Harte et applications aux alg`ebres de Jordan-Banach. Monatsh. Math. 122 (1996), 205–13. 603 [1008] A. Maouche, Diagonalization in Banach algebras and Jordan-Banach algebras. Extracta Math. 19 (2004), 257–60. 603 [1009] M. Mart´ın, Norm-attaining compact operators. J. Funct. Anal. 267 (2014), 1585–92. 338 [1010] J. Mart´ınez, Derivations of structurable H ∗ -algebras. In: Nonassociative algebra and its applications (S˜ao Paulo, 1998), pp. 219–27, Lecture Notes in Pure and Appl. Math. 211, Dekker, New York, 2000. 559

References

707

[1011] M. Mathieu, Weakly compact homomorphisms from C∗ -algebras are of finite rank. Proc. Amer. Math. Soc. 107 (1989), 761–2. 336 [1012] M. Mathieu, Elementary operators on prime C∗ -algebras, I. Math. Ann. 284 (1989), 223–44. xxi, 408 [1013] M. Mathieu, The symmetric algebra of quotients of an ultraprime Banach algebra. J. Austral. Math. Soc. Ser. A 50 (1991), 75–87. 468, 469, 547 [1014] M. Mathieu, Review of [869]. MR1853520 (2002k:46123). 466 [1015] M. Mathieu and V. Runde, Derivations mapping into the radical. II. Bull. London Math. Soc. 24 (1992), 485–7. 466 [1016] K. McCrimmon, On Herstein’s theorems relating Jordan and associative algebras. J. Algebra 13 (1969), 382–92. 367 [1017] K. McCrimmon, A characterization of the radical of a Jordan algebra. J. Algebra 18 (1971), 103–11. 597 [1018] K. McCrimmon, The Zel’manov approach to Jordan homomorphisms of associative algebras. J. Algebra 123 (1989), 457–77. 465 [1019] M. M. Neumann and M. V. Velasco, Continuity of epimorphisms and derivations on vector-valued group algebras. Arch. Math. (Basel) 68 (1997), 151–8. 596 [1020] J. K. Miziolek, T. M¨uldner, and A. Rek, On topologically nilpotent algebras. Studia Math. 43 (1972), 41–50. xxvi, 650, 651 [1021] Gh. Mocanu, Sur certains types de commutativit´e dans les alg`ebres de Banach. An. Univ. Bucuresti Mat. 37 (1988), 36–41. 436 [1022] S. Mohammadzadeh and H. R. E. Vishki, Arens regularity of module actions and the second adjoint of a derivation. Bull. Aust. Math. Soc. 77 (2008), 465–76. 268 [1023] L. Moln´ar, Modular bases in a Hilbert A-module. Czechoslovak Math. J. 42 (117) (1992), 649–56. 557 [1024] L. Moln´ar, A few conditions for a C∗ -algebra to be commutative. Abstr. Appl. Anal. 2014, Art. ID 705836, 4 pp. 401 [1025] A. Moreno, Extending the norm from special Jordan triple systems to their associative envelopes. In Banach algebras ’97 (Blaubeuren), pp. 363–75, de Gruyter, Berlin, 1998. xxiii, 475, 476 [1026] A. Moreno, The triple-norm extension problem: the nondegenerate complete case. Studia Math. 136 (1999), 91–7. xxiii, 475, 476 [1027] A. Moreno, Some recent results about the norm extension problem. In [705], pp. 125–31. xxiii, 462, 474, 475 [1028] A. Moreno and A. Rodr´ıguez, The norm extension problem: positive results and limits. II Congress on Examples and Counterexamples in Banach Spaces (Badajoz, 1996). Extracta Math. 12 (1997), 165–71. 462, 473, 474 [1029] A. Moreno and A. Rodr´ıguez, Algebra norms on tensor products of algebras, and the norm extension problem. Linear Algebra Appl. 269 (1998), 257–305. xxiii, 472, 473, 474, 475 [1030] V. M¨uller, Nil, nilpotent and PI-algebras. In [1189], pp. 259–65. xxvi, 651 [1031] M. Neufang, On Mazur’s property and property (X). J. Operator Theory 60 (2008), 301–16. 238, 239 [1032] T. Ogasawara, A theorem on operator algebras. J. Sci. Hiroshima Univ. Ser. A. 18 (1955), 307–9. 400 [1033] J. M. Osborn and M. L. Racine, Jordan rings with nonzero socle. Trans. Amer. Math. Soc. 251 (1979), 375–87. 598 [1034] M. Oudadess, Commutativit´e de certaines alg`ebres de Banach. Bol. Soc. Mat. Mexicana 28 (1983), 9–14. 436 [1035] G. K. Pedersen and E. Størmer, Traces on Jordan algebras. Canad. J. Math. 34 (1982), 370–3. 339

708

References

[1036] A. Pełczy´nski, Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Polon. Sci. 10 (1962), 641–8. 266 [1037] A. M. Peralta, Little Grothendieck’s theorem for real JB∗ -triples. Math. Z. 237 (2001), 531–45. 337, 346 [1038] A. M. Peralta, New advances on the Grothendieck’s inequality problem for bilinear forms on JB∗ -triples. Math. Inequal. Appl. 8 (2005), 7–21. 346 [1039] A. M. Peralta, Some remarks on weak compactness in the dual space of a JB∗ -triple. Tˆohoku Math. J. 58 (2006), 149–59. 335, 340 [1040] A. M. Peralta and A. Rodr´ıguez, Grothendieck’s inequalities for real and complex JBW ∗ -triples. Proc. London Math. Soc. 83 (2001), 605–25. xx, 268, 274, 338, 340, 341, 342, 344, 345, 346 [1041] A. M. Peralta and A. Rodr´ıguez, Grothendieck’s inequalities revisited. In [1141], pp. 409–23. 346 [1042] I. P´erez de Guzm´an, Structure theorems for alternative H ∗ -algebras. Math. Proc. Cambridge Philos. Soc. 94 (1983), 437–46. xxv, 549 [1043] I. P´erez de Guzm´an, Annihilator alternative algebras. Pacific J. Math. 107 (1983), 89–94. 584 [1044] H. Pfitzner, L-summands in their biduals have Pełczy´nski’s property (V∗ ). Studia Math. 104 (1993), 91–8. xix, 245, 267 [1045] H. Pfitzner, Separable L-embedded Banach spaces are unique preduals. Bull. Lond. Math. Soc. 39 (2007), 1039–44. 241 [1046] H. Pfitzner, Phillips’ lemma for L-embedded Banach spaces. Arch. Math. (Basel) 94 (2010), 547–53. 267 [1047] H. Pfitzner, A well-known criterion of Pełczy´nski’s Property (V∗ ). Private communication, September 2014. xix, 2, 245, 266, 267 [1048] H. Pfitzner, Predual uniqueness coming from the generalized property (X) a` la Godefroy–Talagrand. Private communication, April 2017. 238, 241 [1049] R. S. Phillips, On symplectic mappings of contraction operators. Studia Math. 31 (1968) 15–27. 207 [1050] J. Pinter-Lucke, Commutativity conditions for rings: 1950–2005. Expo. Math. 25 (2007), 165–74. 401 [1051] G. Pisier, Grothendieck’s theorem for noncommutative C∗ -algebras, with an appendix on Grothendieck’s constants. J. Funct. Anal. 29 (1978), 397–415. 342, 343 [1052] G. Pisier, Grothendieck’s theorem, past and present. Bull. Amer. Math. Soc. 49 (2012), 237–323. xx, 342, 343 [1053] R. A. Poliquin and V. E. Zizler, Optimization of convex functions on w∗ -compact sets. Manuscripta Math. 68 (1990), 249–270. 338 [1054] F. L. Pritchard, Ideals in the multiplication algebra of a nonassociative K-algebra. Comm. Algebra 21 (1993), 4541–59. 546 [1055] I. I. Pyatetskii-Shapiro, On a problem proposed by E. Cartan. Dokl. Akad. Nauk SSSR 124 (1959), 272–3. 210 [1056] J. S. Pym, The convolution of functionals on spaces of bounded functions. Proc. London Math. Soc. 15 (1965), 84–104. 267 [1057] N. Randrianantoanina, Pełczy´nski’s property (V∗ ) for symmetric operator spaces. Proc. Amer. Math. Soc. 125 (1997), 801–6. 267 [1058] J. W. Robbin, On the existence theorem for differential equations. Proc. Amer. Math. Soc. 19 (1968), 1005–6. 137 [1059] A. Rodr´ıguez, La continuidad del producto de Jordan implica la del ordinario en el caso completo semiprimo. In: Contribuciones en probabilidad, estad´ıstica matem´atica, ense˜nanza de la matem´atica y an´alisis, pp. 280–8, Secretariado de Publicaciones de la Universidad de Granada, Granada, 1979. xxiii, 474

References

709

[1060] A. Rodr´ıguez, A note on Arens regularity. Quart. J. Math. Oxford 38 (1987), 91–3. 267, 268 [1061] A. Rodr´ıguez, On the strong∗ topology of a JBW ∗ -triple. Quart. J. Math. Oxford 42 (1991), 99–103. xx, 274, 335, 337, 338, 339 [1062] A. Rodr´ıguez, Non-associative normed algebras: geometric aspects. In [1189], pp. 299–311. 421, 462 [1063] A. Rodr´ıguez, Estructuras de Jordan en An´alisis. Rev. Real Acad. Cienc. Exact. F´ıs. Natur. Madrid 88 (1994), 309–17. 462 [1064] A. Rodr´ıguez and M. V. Velasco, Continuity of homomorphisms and derivations on normed algebras which are tensor products of algebras with involution. Rocky Mountain J. Math. 32 (2002), 1045–55. xxiii, 474, 475 [1065] A. Rodr´ıguez and A. R. Villena, Centroid and extended centroid of JB∗ -algebras, In [734], pp. 223–32. 408 [1066] P. Rosentahl and A. Soltysiak, Formulas for the joint spectral radius of noncommuting Banach algebra elements. Proc. Amer. Math. Soc. 123 (9) (1995), 2705–8. 651 [1067] W. H. Ruckle, The infinite sum of closed subspaces of an F-space. Duke Math. J. 31 (1964), 543–54. 584 [1068] V. Runde, Review of [462]. MR1706325 (2000f:46067). 596 [1069] B. Russo, Derivations and projections on Jordan triples: an introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis. In Advanced courses of mathematical analysis V, pp. 118–227, World Sci. Publ., Hackensack, NJ, 2016. 346 [1070] S. Sakai, A characterization of W ∗ -algebras. Pacific J. Math. 6 (1956), 763–73. 19 [1071] S. Sakai, Weakly compact operators on operator algebras. Pacific J. Math. 14 (1964), 659–64. 274, 336 [1072] S. Sakai, On topologies of finite W ∗ -algebras. Illinois J. Math. 9 (1965), 236–41. 334 [1073] P. P. Saworotnow, A generalized Hilbert space. Duke Math. J. 35 (1968), 191–7. 556 [1074] P. P. Saworotnow, Trace-class and centralizers of an H ∗ -algebra. Proc. Amer. Math. Soc. 26 (1970), 101–4. 550 [1075] P. P. Saworotnow and J. C. Friedell, Trace-class for an arbitrary H ∗ -algebra. Proc. Amer. Math. Soc. 26 (1970), 95–100. 550 [1076] R. D. Schafer, On structurable algebras. J. Algebra 92 (1985), 400–12. 555 [1077] R. D. Schafer, Invariant forms on central simple structurable algebras. J. Algebra 122 (1989), 112–7. 556 [1078] R. Schatten, The cross-space of linear transformations. Ann. of Math. 47 (1946), 73–84. 550 [1079] R. Schatten and J. von Neumann, The cross-space of linear transformations. II. Ann. of Math. 47 (1946), 608–30. 550 [1080] R. Schatten and J. von Neumann, The cross-space of linear transformations. III. Ann. of Math. 49 (1948), 557–82. 550 [1081] S. Sherman, Order in operator algebras. Amer. J. Math. 73 (1951), 227–32. 400 [1082] I. P. Shestakov, Speciality and deformations of algebras. In: Algebra: Proceedings of the International Algebraic Conference on the Occasion of the 90th Birthday of A. G. Kurosh (Moscow, 1998), pp. 345–56, de Gruyter, Berlin, 2000. xxii, 470 [1083] V. S. Shulman, On invariant subspaces of Volterra operators. (Russian) Funk. Anal. i Prilozen 18 (1984), 84–5. xxvi, 651 [1084] V. S. Shulman, Invariant subspaces and spectral mapping theorems. In [1189], pp. 313–25. 651 [1085] V. S. Shulman and Yu. V. Turovskii, Topological radicals. I. Basic properties, tensor products and joint quasinilpotence. In Topological algebras, their applications, and

710

[1086]

[1087]

[1088]

[1089] [1090] [1091] [1092] [1093] [1094] [1095] [1096] [1097] [1098] [1099] [1100] [1101] [1102] [1103] [1104] [1105] [1106]

[1107]

References related topics (Ed. by K. Jarosz and A. Soltysiak), pp. 293–333, Banach Center Publ. 67, Polish Acad. Sci., Warsaw, 2005. 652 V. S. Shulman and Yu. V. Turovskii, Topological radicals, II. Applications to spectral theory of multiplication operators. In Elementary operators and their applications (Belfast, 2009) (Ed. by R. E. Curto and M. Mathieu), pp. 45–114, Oper. Theory Adv. Appl., 212, Birkh¨auser/Springer Basel AG, Basel, 2011. 652 V. S. Shulman and Yu. V. Turovskii, Topological radicals and the joint spectral radius. (Russian) Funktsional. Anal. i Prilozhen. 46 (2012), 61–82; translation in Funct. Anal. Appl. 46 (2012), 287–304. 652 V. S. Shulman and Yu. V. Turovskii, Topological radicals, V. From algebra to spectral theory. In Algebraic methods in functional analysis. The Victor Shulman anniversary volume (Ed. by I. G. Todorov and L. Turowska), pp. 171–280, Oper. Theory Adv. Appl., 233, Birkh¨auser/Springer Basel AG, Basel, 2013. 652 S. Simons, On the Dunford–Pettis property and Banach spaces that contain c0 . Math. Ann. 216 (1975), 225–31. 266 A. M. Sinclair, Homomorphisms from C∗ -algebras. Proc. London Math. Soc. 29 (1974), 435–52. Corrigendum: Proc. London Math. Soc. 32 (1976), 322. 548 A. M. Sinclair and A. W. Tullo, Noetherian Banach algebras are finite dimensional. Math. Ann. 211 (1974), 151–3. 601 O. N. Smirnov, Simple and semisimple structurable algebras. (Russian) Algebra i Logika 29 (1990), 571–96; translation in Algebra and Logic 29 (1990), 377–94. 555 J. F. Smith, The structure of Hilbert modules. J. London Math. Soc. 8 (1974), 741–9. 556 R. R. Smith and J. D. Ward, M-ideal structure in Banach algebras. J. Funct. Anal. 27 (1978), 337–49. 19 A. Soltysiak, On the joint spectral radii of commuting Banach algebra elements. Studia Math. 105 (1993), 93–9. 651 P. J. Stacey, Type I2 JBW-algebras. Quart. J. Math. Oxford 33 (1982), 115–27. 416 L. L. Stach´o, A short proof of the fact that biholomorphic automorphisms of the unit ball in certain Lp spaces are linear. Acta Sci. Math. (Szeged) 41 (1979), 381–3. 207 L. L. Stach´o, On nonlinear projections of vector fields. In Convex analysis and chaos (Sakado, 1998), pp. 47–54, Josai Math. Monogr. 1, Josai Univ., Sakado, 1999. 207 J. G. Stampfli, The norm of a derivation. Pacific J. Math. 33 (1970), 737–47. 550 A. C. P. Steptoe, JC∗ -triples and inner ideals in universally reversible JC∗ -algebras. Quart. J. Math. 56 (2005), 397–402. 460 J. Stern, Some applications of model theory in Banach space theory. Ann. Math. Logic 9 (1976), 49–121. 236 M. H. Stone, Boundedness properties in function-lattices. Canadian J. Math. 1 (1949), 176–86. 421 M. Takesaki, On the conjugate space of operator algebra. Tˆohoku Math. J. 10 (1958), 194–203. 334 D. C. Taylor, The strict topology for double centralizer algebras. Trans. Amer. Math. Soc. 150 (1970) 633–43. 400 A. W. Tullo, Conditions on Banach algebras which imply finite dimensionality. Proc. Edinburgh Math. Soc. 20 (1976), 1–5. 598 Yu. V. Turovskii, Spectral properties of elements of normed algebras, and invariant subspaces. (Russian) Funktsional. Anal. i Prilozhen. 18 (1984), 77–8; translation in Funct. Anal. Appl. 18 (1984), 152–4. 651 Yu. V. Turovskii and V. S. Shulman, Radicals in Banach algebras, and some problems in the theory of radical Banach algebras. (Russian) Funktsional. Anal. i Prilozhen. 34 (2001), no. 4, 88–91; translation in Funct. Anal. Appl. 35 (2001), 312–4. 652

References

711

[1108] Yu. V. Turovskii and V. S. Shulman, Topological radicals and the joint spectral radius. (Russian) Funktsional. Anal. i Prilozhen. 46 (2012), 61–82; translation in Funct. Anal. Appl. 46 (2012), 287–304. 652 ¨ [1109] A. Ulger, Weakly compact bilinear forms and Arens regularity. Proc. Amer. Math. Soc. 101 (1987), 697–704. 268 [1110] H. Umegaki, Weak compactness in an operator space. K¯odai Math. Sem. Rep. 8 (1956), 145–51. 334 [1111] I. Unsain, Classification of the simple separable real L∗ -algebras. J. Differential Geometry 7 (1972), 423–51. 554 ¨ [1112] H. Upmeier, Uber die Automorphismengruppen von Banach-Mannigfaltigkeiten mit invarianter Metrik. Math. Ann. 223 (1976), 279–88. 174 [1113] H. Upmeier, Remarks on the Campbell–Hausdorff formula for holomorphic vector fields. Private communication, June 2017. xix, 174 [1114] J.-P. Vigu´e, Le groupe des automorphismes analytiques d’un domaine born´e d’un espace de Banach complexe. Application aux domaines born´es sym´etriques. Ann. Sci. ´ Ecole Norm. Sup. (4) 9 (1976), 203–81. xix, 87, 159, 174, 211 [1115] J.-P. Vigu´e, Les domaines born´es sym´etriques d’un espace de Banach complexe et les syst´emes triples de Jordan. Math. Ann. 229 (1977), 223–31. 211 [1116] J.-P. Vigu´e, Sur la d´ecomposition d’un domaine born´e sym´etrique en produit continu ´ de domaines born´es sym´etriques irr´eductibles. Ann. Sci. Ecole Norm. Sup. (4) 14 (1981), 453–63. 241 [1117] A. R. Villena, Continuity of derivations on a complete normed alternative algebra. J. Inst. Math. Comput. Sci. Math. Ser. 3 (1990), 99–106. 548 [1118] C. Viola Devapakkiam, Hilbert space methods in the theory of Jordan algebras. I. Math. Proc. Cambridge Philos. Soc. 78 (1975), 293–300. xxv, 552 [1119] C. Viola Devapakkiam and P. S. Rema, Hilbert space methods in the theory of Jordan algebras. II. Math. Proc. Cambridge Philos. Soc. 79 (1976), 307–19. xxv, 552 [1120] N. Weaver, A prime C∗ -algebra that is not primitive. J. Funct. Anal. 203 (2003), 356–61. 467 [1121] T. J. D. Wilkins, Spectrum-finite ideals of Jordan-Banach algebras. J. London Math. Soc. 54 (1996), 359–68. 601 [1122] F. J. Yeadon, A note on the Mackey topology of a von Neumann algebra. J. Math. Anal. Appl. 45 (1974), 721–2. 336 [1123] B. Yood, Closed prime ideals in topological rings. Proc. London Math. Soc. 24 (1972), 307–23. 583, 584 [1124] M. A. Youngson, Review of [710]. Bull. Lond. Math. Soc. 44 (2012), 1304–7. xix [1125] B. Zalar, Continuity of derivations on Mal’cev H ∗ -algebras. Math. Proc. Cambridge Philos. Soc. 110 (1991), 455–9. 548 [1126] B. Zalar, On derivations of H ∗ -algebras. University of Ljubljana, Preprint Series of the Department of Mathematics, Vol. 29 (1991), 356. 548 [1127] B. Zalar, A note on the definition of a nonassociative H ∗ -triple. University of Ljubljana, Preprint Series of the Department of Mathematics, Vol. 30 (1992), 361. 561 [1128] B. Zalar, Continuity of derivation pairs on alternative and Jordan H ∗ -triples. Boll. Un. Mat. Ital. B 7 (1993), 215–30. 559 [1129] B. Zalar, Theory of Hilbert triple systems. Yokohama Math. J. 41 (1994), 95–126. 558 [1130] E. I. Zel’manov, Jordan algebras with finiteness conditions. Algebra Logic 17 (1978), 450–7. 562 [1131] E. I. Zel’manov, On prime Jordan algebras. Algebra Logic 18 (1979), 162–75. 364, 437

712

References

[1132] E. I. Zel’manov, Lie algebras with algebraic associated representation. (Russian) Mat. Sb. (N.S.) 121(163) (1983), 545–61; translation in Math. USSR-Sb. 49 (1984), 537–52. 658 [1133] E. I. Zel’manov, Primary Jordan triple systems. Sibirsk. Mat. Zh. 24 (1983), 23–37. xxii, 437, 438, 459 [1134] E. I. Zel’manov, Primary Jordan triple systems. II. Sibirsk. Mat. Zh. 25 (1984), 50–61. xxii, 437, 438, 447, 459 [1135] E. I. Zel’manov, Lie algebras with a finite grading. Math. USSR Sb. 52 (1985), 347–85. 600 [1136] E. I. Zel’manov, On Engel Lie algebras. Soviet Math. Dokl. 35 (1987), 44–7. 658 [1137] V. Zizler, On some extremal problems in Banach spaces. Math. Scand. 32 (1973), 214–24. xx, 274, 337

Books [1138] R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications. Second edition. Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988. 135 [1139] K. Alvermann, Normierte nichtkommutative Jordan-Algebren vom Typ I. PhD thesis, Braunschweig 1982. 421 [1140] E. Behrends, M-structure and the Banach–Stone theorem. Lecture Notes in Math. 736, Springer, Berlin, 1979. 350, 351, 398 [1141] K. D. Bierstedt, J. Bonet, M. Maestre, and J. Schmets (eds.), Recent progress in functional analysis. Proceedings of the International Functional Analysis Meeting on the Occasion of the 70th Birthday of Professor Manuel Valdivia, Valencia, Spain, July 3-7, 2000. North Holland Math. Studies 189, Elsevier, Amsterdam 2001. 679, 708 [1142] A. Bonfiglioli and R. Fulci, Topics in noncommutative algebra. The theorem of Campbell, Baker, Hausdorff and Dynkin. Lecture Notes in Math. 2034, Springer, Heidelberg, 2012. 154, 155, 174 ´ ements de math´ematique: groupes et alg`ebres de Lie. Chapitres 2 [1143] N. Bourbaki, El´ et 3. Springer-Verlag, Berlin, 2006. 154, 155, 174 [1144] M. Cabrera, H ∗ -´algebras no asociativas reales. H ∗ -´algebras de Malcev complejas y reales. PhD thesis, Granada 1987. Tesis Doctorales de la Universidad de Granada, Secretariado de Publicaciones de la Universidad de Granada, 1988. 552, 555 [1145] H. Cartan, Sur les groupes de transformations analytiques. Act. Sci. Ind. 198, Hermann, Paris, 1935. 87 [1146] H. Cartan, Cours de calcul diff´erentiel. Second edition. Hermann, Paris, 1977. 53 [1147] M. Chaperon, Calcul diff´erentiel et calcul int´egral. Second edition. Dunod, Paris, 2008. 53 [1148] R. Coleman, Calculus on normed vector spaces. Universitext. Springer, New York, 2012. 53, 135, 136 [1149] J. B. Conway, Functions of one complex variable. Second edition. Grad. Texts in Math. 11. Springer-Verlag, New York-Berlin, 1978. 44, 55, 58, 101, 185 [1150] H. G. Dales, F. K. Dashiell, Jr., A. T.-M. Lau, and D. Strauss, Banach Spaces of Continuous Functions as Dual Spaces. CMS Books Math., Springer, New York, 2016. 421 [1151] J. Diestel, Sequences and series in Banach spaces. Grad. Texts in Math. 92. Springer-Verlag, New York, 1984. 239, 245, 253, 258, 266, 267, 279 ´ ements d’analyse. Tome I. Fondements de l’analyse moderne. Third [1152] J. Dieudonn´e, El´ edition. Translated from the English by D. Huet. With a foreword by Gaston Julia. Cahiers Scientifiques XXVIII. Gauthier-Villars, Paris, 1981. 53, 135, 136

References

713

´ ements d’analyse. Tome II: Chapitres XII a` XV. Cahiers Scientifiques [1153] J. Dieudonn´e, El´ XXXI Gauthier-Villars, Paris, 1968. 335 [1154] S. Dineen, Complex analysis in locally convex spaces. North-Holland Math. Stud. 57. North-Holland Publishing Co., Amsterdam, 1981. 53 [1155] S. Dineen, Complex analysis on infinite-dimensional spaces. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1999. xxvi, 53 [1156] R. S. Doran and J. Wichmann, Approximate identities and factorization in Banach modules. Lecture Notes in Math. 768, Springer, Berlin, 1979. xxvi, 651, 652 [1157] A. Fern´andez, Teoremas de Wedderburn-Zorn en a´ lgebras alternativas normadas completas, PhD thesis, Universidad de Granada 1983. 583, 584 [1158] A. Fern´andez, Jordan techniques in Lie algebras. Book in progress. 600 [1159] T. Franzoni and E. Vesentini, Holomorphic maps and invariant distances. NorthHolland Math. Stud. 69, North-Holland Publishing Co., Amsterdam, 1980. 53, 87 [1160] M. Herv´e, Analyticity in infinite-dimensional spaces. de Gruyter Studies in Mathematics, 10. Walter de Gruyter & Co., Berlin, 1989. 53 [1161] J. Horv´ath, Topological vector spaces and distributions. Volume I. AddisonWesley Publishing Co., Reading, Massachusetts-London-Don Mills, Ontario, 1966. 7, 290 [1162] V. I. Istrˇatescu, Inner product structures. Theory and applications. Mathematics and its Applications, 25. D. Reidel Publishing Co., Dordrecht, 1987. 551, 557 [1163] H. Jarchow, Locally convex spaces. Mathematische Leitf¨aden. B. G. Teubner, Stuttgart, 1981. 335 [1164] P. T. Johnstone, Stone spaces. Reprint of the 1982 edition. Cambridge Studies in Advanced Mathematics, 3. Cambridge University Press, Cambridge, 1986. 421 [1165] R. Jungers, The joint spectral radius. Theory and applications. Lecture Notes in Control and Information Sciences, 385. Springer-Verlag, Berlin, 2009. 651 [1166] V. Kadets, M. Mart´ın, J. Mer´ı, and A. P´erez, Spear operators between Banach spaces. Lecture Notes in Math., Springer. 209 [1167] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II. Advanced theory. Grad. Stud. Math. 16, American Mathematical Society, Providence, RI, 1997. 19 [1168] J. L. Kelley and I. Namioka, Linear topological spaces. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, NJ, 1963. 288 [1169] M. Koecher, An elementary approach to bounded symmetric domains, Lecture Notes (Rice University, Houston, TX, 1969). 211 [1170] S. Lang, Differential and Riemannian manifolds. Third edition. Grad. Texts in Math. 160. Springer-Verlag, New York, 1995. 135 [1171] J. M. Lee, Introduction to smooth manifolds. Second Edition. Grad. Texts in Math. 218. Springer, New York, 2013. 135 [1172] J. Lelong-Ferrand and J. M. Arnaudi`es, Cours de math´ematiques. Tome 2: Analyse. Fourth edition. Dunod, Paris, 1977. 53 [1173] P. Mellon, An introduction to bounded symmetric domains. Textos de Matem´atica. S´erie B. Universidade de Coimbra, Departamento de Matem´atica, Coimbra, 2000. 210, 211 ´ [1174] A. Moreno, Algebras de Jordan-Banach primitivas, PhD thesis, Universidad de Granada 1995. 470 [1175] J. Mujica, Complex analysis in Banach spaces. Dover Publications, Inc. Mineola, New York, 2010. 53, 87

714

References

[1176] R. Narasimhan, Several complex variables. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago and London, 1971. 87, 207 [1177] E. Neher, Jordan triple systems by the grid approach. Lecture Notes in Math. 1280, Springer-Verlag, Berlin, 1987. 557 [1178] G. K. Pedersen, C∗ -algebras and their automorphism groups. London Math. Soc. Monogr. 14, Academic Press, London-New York, 1979. 408 ´ [1179] I. P´erez de Guzm´an, Algebras alternativas normadas. Teorema de estructura para H ∗ -´algebras alternativas. PhD thesis, Granada 1977. Tesis Doctorales de la Universidad de Granada 266, Secretariado de Publicaciones de la Universidad de Granada, 1980. 549 [1180] S. Reich and D. Shoikhet, Nonlinear semigroups, fixed points, and geometry of domains in Banach spaces. Imperial College Press, London, 2005. 79, 87, 88, 135 [1181] C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. 154, 174 [1182] J. R. Ringrose, Compact non-self-adjoint operators. Van Nostrand Reinhold Co., London, 1971. 481, 524 [1183] K. Saitˆo and J. D. M. Wright, Monotone complete C∗ -algebras and generic dynamics. Springer Monographs in Mathematics. Springer, London, 2015. 19 [1184] I. Singer, Bases in Banach spaces. I. Die Grundlehren der mathematischen Wissenschaften, Band 154. Springer-Verlag, New York-Berlin, 1970. 239 [1185] M. Takesaki, Theory of operator algebras. I. Reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences, 124. Operator Algebras and Noncommutative Geometry, 5. Springer-Verlag, Berlin, 2002. 332, 334, 335 ¨ [1186] H. Upmeier, Uber die Automorphismengruppen beschr¨ankter Gebiete in Banachr¨aumen. PhD thesis, T¨ubingen 1975. 174 [1187] J.-P. Vigu´e, Le groupe des automorphismes analytiques d’un domaine born´e d’un espace de Banach complexe. Application aux domaines born´es sym´etriques. PhD thesis, Publication Math´ematique d’Orsay, 1976. 87, 135, 174, 211 [1188] S. Willard, General topology. Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581]. Dover Publications, Inc., Mineola, NY, 2004. 87, 280 [1189] J. Zem´anek (Editor), Functional analysis and operator theory (Warsaw, 1992). Banach Center Publ., 30, Polish Acad. Sci., Warsaw, 1994. 665, 707, 709

Symbol index for Volume 1

1 (the unit of a unital algebra), 2 1 (the unit of the unital extension), 33  · π (projective tensor norm), 31 | · |n (natural C∗ -norm on Mn (C)), 166  · n (n ∈ N), 167–175 {· · ·}, 127, 130, 324, 463 [a, b] = ab − ba, 126 [a, b, c] = (ab)c − a(bc), 151 π (natural product on the range of the projection π), 154

1 Xλ ( 1 -sum of the family {Xλ }), 109 ⊕λ∈ ∞ X ( -sum of the family {X }), 271 ⊕ i∈I i ∞ i [ij] (for i, j = 1, 2), 538

a • b = 12 (ab + ba), 122 a−1 (the [J-]inverse of a), 5, 188, 453, 473 a[n] (plenary powers of a), 566 a% (the quasi-[J-]inverse of a), 431, 585 Ann(A) (annihilator of A), 4 Aut(A) = Aut(A, A), 384 Aut(A, B) (isomorphisms from A onto B), 384 Aut+ (A) = {F ∈ Aut(A) : F • = F, sp(F) ⊆ R+ 0 }, 384 Aut∗ (A, B) (∗-isomorphisms from A onto B), 387 A(a) (subalgebra of A generated by a), 262 A(S) (subalgebra of A generated by S), 9 ¯ A(S) (closed subalgebra of A generated by S), 9 A1 (= A or A1 depending on whether or not A is unital), 407 A1 (unital extension of A), 33 AC (complexification of A), 32 AR (real algebra underlying A), 97 Ak (e) k = 1, 12 , 0 (Peirce subspaces of A relative to the idempotent e), 178 A = {x ∈ A : [J−]sp(A, x) ⊆ }, 65, 486 AU (ultrapower of A), 272 A(0) (opposite algebra of A), 13 A(u) (u-isotope of A), 519

A+ (positive part of A), 47, 613 Aant (antisymmetrized algebra of A), 560 AK = {x ∈ A : V(A, 1, x) ⊆ K}, 649 Asym (symmetrized algebra of A), 122 (Ai )U (ultraproduct of the family {Ai }), 271 A (E) (flexible quadratic algebra of the pre-H-algebra E), 204 A (K) (associative algebra of the compact set K ⊆ [1, ∞[), 537 A (U, ϑ, K), 257 Ap (U, ϑ, K) (1 ≤ p < ∞), 257 A (V, ×, (·, ·)) (quadratic algebra of (V, ×, (·, ·))), 182 ∗A (for A = C, H, or O), 278 A∗ (for A = C, H, or O), 278 ∗ A (for A = H, or O), 220 An n ∈ N ∪ {0} (Cayley–Dickson algebras), 199 A-Rad(A) (A-radical of A), 580 BL(X, Y) (bounded linear operators from X to Y), 3 BL(X) = BL(X, X), 3 B(I, X) (bounded functions from I to X), 117, 307 B(x, y) (Bergmann operator of (x, y)), 509 BC = {xy : (x, y) ∈ B × C}, 2 βu,K (r) = inf{1 − u + rx : x ∈ KBX , τ (u, x) ≤ −1}, 299 B(A) (Baer radical of A), 601 B(K) (Jordan algebra of the compact set K ⊆ [1, ∞[), 553 BX (closed unit ball of X), 2 co(S) (convex hull of S), 28 |co|(S) (absolutely convex hull of S), 99 co(S) (closed convex hull of S), 99 |co|(S) (closed absolutely convex hull of S), 99 c0 (null sequences in K), 3

715

716

Symbol index for Volume 1

CA (extended centroid of A), 195 Cb (E, A) (bounded continuous functions from E to A), 3 CbC (E) = Cb (E, C), 150 Cp ([1, 2], S3 ) = {α ∈ C([1, 2], S3 ) : α(1) ∈ Rp}, 560 Cp (K, C3 ) = {α ∈ C(K, C3 ) : α(1) ∈ Cp}, 555 Cp (K, M2 (C)) = {α ∈ C(K, M2 (C)) : α(1) ∈ Cp}, 545 CK (E) = C0K (E) when E is compact, 3

C0K (E) (K-valued continuous functions on E vanishing at infinity), 3 C0T (E) = {x ∈ C0C (E) : x(zt) = zx(t) ∀(z, t) ∈ T × E}, 498 C(C) (complex octonions), 205 C(R) (Cayley–Dickson doubling of M2 (R)), 218 C(E, A) (continuous functions from E to A), 3 C0 (E, X) (X-valued continuous functions on E vanishing at infinity), 330 C D(A) (Cayley–Dickson doubling of A), 176 CN (X, K) (free complete normed non-associative K-algebra on X), 261 C3 (three-dimensional spin factor), 553 ∗

C (McClay algebra), 216 deg(A) (degree of A), 212 dens(E) (density character of E), 257 dom(·) (domain of a partially defined operator), 194, 640 Der∗ (A), 384 Dis(X, u) (dissipative elements of X relative to u), 291 D(X, u) (states of X relative to u), 94 DY (X, u) = D(X, u) ∩ Y, 99 ∗ Dw (X, x) = D(X, x) ∩ X∗ , 285 D(K) (Banach space of K), 645 dfˆ (a) : X → X (formal differential of f at a), 652 δX (u, ·) : R+ → R (modulus of midpoint local convexity of X at u), 111 A (characters on A), 21  = A , 21 exp(a) (exponential of a), 10, 342  an (exp −1)(a) = ∞ n=1 n! , 609 ext(S) (extreme points of S), 107 Ea(K) (extremal algebra of K), 647 Ea1 (K) (derivable elements of Ea(K)), 651 Ean (K) n ∈ N (n-times derivable elements of Ea(K)), 669 Ea∞ (K) = ∩n∈N Ean (K), 669 η : [1, ∞[→ M2 (C), 537 ηK = η|K , 537 ηij : [1, ∞[→ M2 (C), 537 ηijK = (ηij )|K , 537

f (a), 46, 57–59, 479, 484, 648 f˜ : A → A, 66, 486 f ∗ (x) = f (x∗ ) for (x, f ) ∈ X × X  , 146 f [ij] (for f ∈ CC (K) and i, j = 1, 2), 538 f  (for f ∈ Ea1 (K)), 651 F  : Y  → X  (transpose of the operator F : X → Y), 29 F ⊗ G (operator tensor product of F and G), 30 F ∗ : K → H (adjoint of the operator F : H → K), 38 F K (E) (K-valued functions on E), 2 Fi (f )(x1 , . . . , xn ) (0 ≤ i ≤ n), 370 F(X, Y) (finite-rank operators from X to Y), 73 F(X) = F(X, X), 75 F (X, K) (free non-associative K-algebra on X), 258 Fp (X, K) (1 ≤ p < ∞), 258 F : A (K) → C(K, M2 (C)), 538 G : A → CC () (Gelfand representation for complete normed unital associative and commutative complex algebras), 22 G : J → C0T () (Gelfand representation for complex Banach Jordan ∗-triples), 500  (a contour in C), 58 A (centroid of A), 4  (A) (left centralizers on A), 254 G (X) (surjective linear isometries on X), 332 G : B(K) → C(K, C3 ), 553 H(X, ∗) (∗-invariant elements of (X, ∗)), 39 ˆ 2 (Hilbert tensor product of H1 and H2 ), H1 ⊗H 417 H3 (O) (Albert exceptional Jordan algebra), 337 H () (C-valued holomorphic functions on ), 59 H (algebra of Hamilton quaternions), 176 id(x0 ) = {e ∈ A : ex0 = x0 }, 437 Ind (z0 ) (index of z0 with respect to ), 58 Inv(A) (invertible elements of A), 5 IX (identity mapping on X), 2 (I : A) = {x ∈ A : xA + Ax ⊆ I}, 602 (z) (imaginary part of z), 132 J-Inv(A) (J-invertible elements of A), 453, 475 J-Rad(A) (Jacobson radical of A), 569 J-sp(A, a) (J-spectrum of a relative to A), 456, 476 J (e) (e-homotope algebra of J), 465 Jk (e) k = 1, 12 , 0 (Peirce subspaces of J relative to the tripotent e), 505 ker(x0 ) = {a ∈ A : ax0 = 0}, 437 k(F) = max{k ≥ 0 : kx ≤ F(x) ∀x ∈ X}, 250 K(X, u) = ∩f ∈D(X,u) ker(f ), 351

Symbol index for Volume 1 K(X, Y) (compact operators from X to Y), 70 K(X) = K(X, X), 75 K = R or C, 1 K[x] (polynomials over K in the indeterminate x), 9 K(x) (fractions over K in the indeterminate x), 57 lin(S) (linear hull of S), 351 L(X, Y) (linear mappings from X to Y), 1 L(X) = L(X, X), 1 La (left multiplication by a), 13 LS := {Lx : x ∈ S}, 433 LaX (left multiplication by a on the bimodule X), 637 LxB = (Lx )|B , 348 L(x, y)(z) = {xyz}, 465 L(J, J) = {L(x, y) : x, y ∈ J}, 468 J (nonzero triple homomorphisms from J to C), 499  = J , 499 m : Z  × X → Y  , 124 m : Y  × Z  → X  , 124 m : X  × Y  → Z  , 124 mt = m , 124 mr (y, x) = m(x, y), 126 m∗ (x, y) = (m(x∗ , y∗ ))∗ , 146 M(A) (algebra of multipliers of A), 126, 325 Mn (X) (n × n matrices with entries in the vector space X), 166 Mn (A) (n × n matrices with entries in the algebra A), 167 M∞ (K) (infinite matrices over K with a finite number of nonzero entries), 267 Ma,b (x) = axb, 601 M (X) (free monad generated by X), 258 M (B)A , 357 M  (A) (multiplication ideal of A), 443

717

pA (product of A), 408 Pk (e) k = 1, 12 , 0 (Peirce projections relative to e), 178, 505 + , ϕ(X, u, r) = sup u+rx−1 − τ (u, x) : x ∈ BX , r 299 π1 () = {x : (x, f ) ∈  for some f }, 106 (X) = {(x, f ) : x ∈ SX , f ∈ D(X, x)}, 106 (Y, X) = {(y, x ) ∈ SY × SX  : x ∈ D(X, y)}, 116 P (algebra of pseudo-octonions), 220 q-Inv(A) (quasi-invertible elements of A), 440 Qx (y) = {xyx}, 506 Qx,z (y) = {xyz}, 507 p(x) ∈ K(x) : q(a) ∈ Inv(A)}, 57 Qa = { q(x) QFM (A) (quasi-full multiplication algebra of A), 578

r(a) (spectral radius of a), 6, 381 Rad(A) (radical of A), 429 Ra (right multiplication by a), 13 RS := {Rx : x ∈ S}, 433 RX a (right multiplication by a on the bimodule X), 637 (z) (real part of z), 95 s(a) (substitute of the spectral radius of a), 566 sp(A, a) (spectrum of a relative to A), 12 sp(a) = sp(A, a), 12 s-Rad(A) (strong radical of A), 20, 427 Sc (commutant of S), 24 Scc = (Sc )c , 24 σ (x) (triple spectrum of x), 504 S3 (three-dimensional real spin factor), 560 § (algebra of sedenions), 199 SX (unit sphere of X), 2 S() (separating space of ), 18

n(a) (algebraic norm of a), 181 n(X, u) (numerical index of (X, u)), 98 nY (X, u), 99 ∗ nw (X, u) = nX∗ (X, u), 295 nR (X, u) (real numerical index of (X, u)), 353 N(X) (spatial numerical index of X), 105 N (X, K) (free normed non-associative K-algebra on X), 258

t(a) (trace of a), 181 τ (u, x) = max (V(X, u, x)), 291 τ t : CC (F) → CC (E) for τ : E → F, 45 ϑ : M2 (C) → M2 (C), 553–554 * : A (K) → A (K), 552–554 T = §C , 10

ωK (z) = max{|ewz | : w ∈ K}, 645 O (algebra of Cayley numbers), 176

uw-Rad(A) (ultra-weak radical of A), 580 U(X, u) = {f ∈ BP(X) : f (x, u) = f (u, x) = x ∀x ∈ X}, 405 Ua = La (La + Ra ) − La2 , 121

  p(a) = nk=0 αk ak for p(x) = nk=0 αk xk ∈ K[x], 9 p(a1 , . . . , an ) (valuation of p at (a1 , . . . , an )), 262 P(X) (continuous products on X), 405

Ua,b = 12 [La (Lb + Rb ) + Lb (La + Ra )] − La•b , 364, 453 UaX (x) = a(ax + xa) − a2 x for x ∈ X, 637 UxB = (Ux )|B , 348

718

Symbol index for Volume 1

v(X, u, x) (numerical radius of x relative to (X, u)), 98 v(x) = v(X, u, x), 98 V(X, u, x) (numerical range of x relative to (X, u)), 94 V(x) = V(X, u, x), 94 w-Rad(A) (weak radical of A), 578 W(f ) (spatial numerical range of f : SY → X), 116, 308 W(T) (spatial numerical range of T : X → X), 107 W(T) (spatial numerical range of T : Y → X), 116 W (A) = {a ∈ A : La , Ra ∈ Rad(QFM (A))}, 578 W(X, Y) (weakly compact operators from X to Y), 70 W(X) = W(X, X), 75 x(2n+1) = {xx(2n−1) x} (triple powers of x), 468 X  ((topological) dual of X), 2

X  (bidual of X), 2 (X, u) (numerical-range space), 94 X ⊗π Y (projective tensor product of X and Y), 31 X ⊕1 Y ( 1 -sum of X and Y), 109 XR (real vector space underlying X), 95 XC (complexification of X), 31 Xn (continuous n-linear mappings from X n to X), 370 XU (ultrapower of X), 271 (Xi )U (ultraproduct of the family {Xi }), 271 X (U, K) (free vector space over K generated by U), 257 y ⊗ f : x → f (x)y, 73 Z(A) (centre of A), 192 Z (B) (centre modulo the radical of B), 597

Subject index for Volume 1

abelian Jordan ∗-triple, 468 A-bimodule, 636 A-bimodule relative to V , 643 absolute value, 176 absolute-valued algebra, 176 absolute-valued C∗ -algebra, 416 absolute-valued left semi-H ∗ -algebra, 253 adjoint operation (bilinear), 124 adjoint operator, 39 Albert isotopic (absolute-valued algebras), 211 Albert radical, 599 algebra, 1 algebra admitting power-associativity, 493 algebra antihomomorphism, 13 algebra homomorphism, 2 algebra involution, 39 algebra isomorphism, 2 algebra norm, 2 algebra with hermitian multiplication, 581 algebraic algebra, 180 algebraic algebra of bounded degree, 212 algebraic element, 180 algebraic norm function, 181 algebraically J-unitary element, 513 algebraically unitary element, 102, 367 almost norming subspace, 99 almost transitive normed space, 302 α-property, 135–137 alternative algebra, 152 alternative bimodule, 643 alternative C∗ -algebra, 153 alternative C∗ -complexification, 524 alternative C∗ -representation, 610 alternative W ∗ -algebra, 409 annihilator of an algebra, 4 approximate unit, 404 approximation problem, 90 approximation property, 90 A-radical, 580

Arens regular bilinear mapping, 126 Arens regular normed algebra, 126 Artin theorem, 153 associative algebra, 1 associative and commutative bimodule, 644 associative bimodule, 643 associator, 151 A-submodule (of a left A-module), 436 automorphism of an n-algebra, 371 Baer chain, 601 Baer radical, 599 Banach Jordan ∗-triple, 465 Banach–Steinhaus closure theorem, 74 Banach–Stone theorem, 151 Bergmann operator, 509 bicommutant, 24 big point, 333 Birkhoff–Witt theorem, 581 Bishop–Phelps–Bollob´as theorem, 287 bounded below (operator), 27 bounded index, 265 Brown–McCoy radical, 20 Calkin algebra, 93 canonical derivation of Ea(K), 651 canonical involution of the complexification, 31 of a matrix algebra, 167 carrier space, 22 Cayley algebra, 176 Cayley numbers, 176 Cayley–Dickson algebra, 199 Cayley–Dickson doubling (of a Cayley algebra), 176 Cayley–Dickson doubling process, 176 central algebra over K, 4 central element, 192

719

720

Subject index for Volume 1

centralizer (on an algebra), 4 centralizer set for a left A-module, 439 centre, 192 centre modulo the radical, 597 centroid, 4 character, 20 closeable operator, 651 closed curve, 58 closed J-full subalgebra generated by a subset, 483 closed operator, 641 closed ∗-subalgebra generated by a subset, 419 closed subalgebra generated by a subset, 9 closed subtriple generated by a subset, 466 closure of a closable operator, 651 commutant, 24 commutative algebra, 1 commutative subset, 24 commutator, 126 compact operator, 70 complete holomorphic vector field, 174 complete normed algebra, 2 complete tripotent, 517 complex extreme point, 321 complexification, 31 composition algebra, 186 cone, 49 continuous functional calculus, 46, 479 contour, 58 contour surrounds K in , 58 convex cone, 49 convex-transitive normed space, 333 core of a subspace (of an algebra), 429 cross-product algebra, 187 CS-closed set, 294 C∗ -algebra, 39 C∗ -algebra of multipliers, 126 C∗ -complexification, 524 C∗ -equivalent algebra, 632 C∗ -isotope algebra, 415 C∗ -norm, 141 C∗ -representation, 610 C∗ -seminorm, 141 C∗ -unital extension, 609 curve, 58 cyclic vector, 437 degree of a non-associative word global, 258 in each indeterminate, 373 degree of an algebra, 212 densely defined operator, 641 density character, 257 denting point, 118

derivation of an algebra, 122 of an n-algebra, 371 descending chain condition, 583 direct product of algebras, 33 disc algebra, 315 dissipative element, 97 distinguished element (of a numerical range space), 94 division algebra, 192 division alternative algebra, 188 division associative algebra, 15 divisor of zero (joint, left, one-sided, right, two-sided), 27 duality mapping, 284 e-homotope algebra, 465 eigenvalue, 80 eigenvector, 80 element acting weakly as a unit, 316 equivalent non-commutative JB∗ -representations, 618 essential ideal, 149 exponential, 10, 342 extended centroid, 195 extremal algebra of K, 647 finite-rank operator, 73 (first) Arens extension, 125 (first) Arens product, 125 flexible algebra, 149 flexible quadratic algebra of a pre-H-algebra, 204 (Fr´echet) derivative of a function at a point, 8 (Fr´echet) differentiable function at a point, 8 free complete normed non-associative algebra, 261 free non-associative algebra, 258 free (non-associative) monad, 258 free normed non-associative algebra, 259 Frobenius–Zorn theorem, 191 full subalgebra, 22, 480 fundamental formula for Jordan algebras, 364 for Jordan ∗-triples, 508 Gˆateaux derivative of the norm, 204 Gelfand homomorphism theorem non-unital version, 428 unital version, 23 Gelfand representation of a complete normed unital associative and commutative complex algebra, 22 of a complex Banach Jordan ∗-triple, 500 Gelfand space, 22 Gelfand theory, 22

Subject index for Volume 1 Gelfand topology, 22 Gelfand transform of an element, 22 Gelfand–Beurling formula associative, 15 Jordan, 458 Gelfand–Mazur theorem complex, 15 real, 194 Gelfand–Mazur–Kaplansky theorem, 197 Gelfand–Naimark theorem commutative, 40 non-commutative, 40 non-unital non-associative, 415 unital non-associative, 343 generalized standard algebra, 278 generated as a normed algebra by a subset, 25 generated as a normed ∗-algebra by a subset, 538 generator of Ea(K), 647 geometric functional calculus, 648 geometrically unitary element, 100 H-algebra, 208 hereditarily indecomposable Banach space, 247 hermitian Banach Jordan ∗-triple, 465 hermitian element, 97 hermitian Jordan-admissible complex ∗-algebra, 613 Hilbert tensor product, 417 hole, 29 holomorphic functional calculus, 64, 485 holomorphic vector field, 174 H ∗ -algebra, 222 Hurwitz theorem, 217 ideal (left, right, two-sided), 16 ideal generated by a subset, 583 idempotent, 3 identity, 406 index of a point with respect to a contour, 58 index of nilpotency, 265 inner ideal, 594 intrinsic numerical range, 308 inverse element, 5, 187 invertible element, 5, 187 involution on a set, 39 irreducible left A-module, 437 irreducible representation, 437 isomorphic left A-modules, 439 isotropic element, 179 i-special Jordan algebra, 425 Jacobson density theorem, 445 Jacobson radical of an associative algebra, 429 of a Jordan-admissible algebra, 569

721

JB-algebra, 319 JB-algebra of multipliers, 325 JB∗ -admissible algebra, 406 JB∗ -algebra, 345 JB∗ -complexification, 524 JB∗ -representation, 610 JB∗ -triple, 130 JB∗ -triple complexification, 524 JBW-algebra, 323 JBW ∗ -triple, 528 JC-algebra, 320 JC∗ -algebra, 345 J-division Jordan algebra, 457 J-division Jordan-admissible algebra, 475 J-divisor of zero, 460, 478, 496 J-full subalgebra, 476 J-full subalgebra generated by a subset, 483 J-inverse element, 451, 473 J-invertible element, 451, 473 Johnson uniqueness-of-norm theorem, 565 Johnson–Aupetit–Ransford theorem, 570 Jordan A-bimodule, 637 Jordan-admissible algebra, 163 Jordan algebra, 162 Jordan derivation, 122 Jordan homomorphism, 122 Jordan identity, 162 Jordan ∗-triple, 463 Jordan triple identity, 463 J-primitive ideal, 594 J-primitive Jordan algebra, 594 J-semisimple Jordan-admissible algebra, 569 J-spectrum, 456 J-unitary element, 512 K-extreme point, 321 Kadison isometry theorem, 131 Kadison–Paterson–Sinclair theorem, 127 Kernel of a numerical-range space, 351 Kleinecke–Shirokov theorem, 442 Kurosh’s problem, 276 left A-module, 436 left A-module corresponding to a representation, 436 left centralizer, 254 left-division algebra, 192 left multiplication operator, 13 left powers, 665 left semi-H ∗ -algebra, 237 left standard representation, 436 left unit, 219 Lie algebra, 581 locally C∗ -equivalent algebra, 632 locally finite algebra, 276

722

Subject index for Volume 1

locally nilpotent algebra, 277 logarithm (of an element of an algebra), 68 L-summand, 314 Macdonald’s theorem, 389 matricial L∞ -property, 170 2 -property, 170 matricial L∞ maximal ideal (left, right, two-sided), 17 maximal modular ideal (left, right, two-sided), 427 maximal modular inner ideal, 594 M-ideal, 315 minimal ideal (left, right, two-sided), 179 minimality of norm, 576 minimality of norm topology, 572 minimum norm, 596 minimum norm topology, 596 minimum polynomial, 180 modular ideal (left, right, two-sided), 426 modular unit (left, right), 426 module homomorphism, 439 module multiplication, 436 modulus of midpoint local convexity, 111 monad, 258 multilinear identity, 406 multiplication, 1 multiplication ideal, 443 multiplicatively nil ideal, 601 Nagata–Higman theorem, 267 n-algebra, 371 natural involution of a V-algebra, 134 n-contractive operator, 169 nearly absolute-valued algebra, 198 nice algebra, 122 nil algebra, 265 nil algebra of bounded index, 265 nilpotent subset, 265 n-linear non-associative word, 373 non-associative C∗ -algebra, 170 non-associative polynomial, 262 non-associative word, 258 non-commutative JB∗ -algebra, 345 non-commutative JB∗ -complexification, 524 non-commutative JB∗ -representation, 610 that factors through another, 618 non-commutative JB∗ -unital extension, 609 non-commutative JBW ∗ -algebra, 531 non-commutative Jordan A-bimodule, 637 non-commutative Jordan algebra, 163 non-thin set at a point, 612 norm-unital normed algebra, 34 normal element, 42, 365 normal subset, 418 normed A-bimodule, 638 normed algebra, 2

(normed) algebra completion, 35 normed complexification, 31 normed n-algebra, 371 normed Q-algebra associative, 440 Jordan-admissible, 572 normed unital extension, 609 norming subspace, 99 nowhere dense subset, 302 numerical index, 98 numerical radius, 98 numerical range, 94 numerical-range order, 143 numerical-range space, 94 octonions, 176 complex, 205 one-parameter semigroup, 10 one-sided division algebra, 192 one-sided semi-H ∗ -algebra, 252 operator algebra, 173 operator space, 175 operator system, 175 operator that factors through a space, 87 opposite algebra, 13 order defined by a proper convex cone, 49 orthogonal idempotents, 54 orthogonal subtriples, 514 partial isometry, 552 partially defined centralizer, 194 partially defined derivation, 642 partially defined linear operator, 640 Peirce decomposition of a Jordan ∗-triple, 505 of a power-associative algebra, 179 plenary powers, 566 polynomial function, 263 polynomial functional calculus, 54 positive element of a C∗ -algebra, 47 of a JB-algebra, 328 of a non-commutative JB∗ -algebra, 383 positive hermitian Banach Jordan ∗-triple, 465 positive linear functional, 141 power-associative A-bimodule, 651 power-associative algebra, 164 power-commutative algebra, 165 pre-duality mapping, 285 pre-H-algebra, 204 prime algebra, 194 prime ideal, 430 primitive algebra, 429 primitive ideal, 429 product, 1

Subject index for Volume 1 product of an n-algebra, 371 projective tensor norm, 31 projective tensor product, 31 proper cone, 49 proper ideal, 16 pseudo-octonions, 220 quadratic algebra, 180 quadratic commutative algebra of a real pre-Hilbert space, 232 quadratic form admitting composition, 183 quadratic operator, 506 quasi-division algebra, 192 quasi-full multiplication algebra, 578 quasi-full subalgebra, 440 quasi-full subalgebra generated by a subset, 578 quasi-inverse, 431 quasi-invertible element, 431 quasi-invertible subset, 431 quasi-J-full subalgebra, 594 quasi-J-inverse element, 585 quasi-J-invertible element, 568 quasi-J-invertible subset, 568 quaternions, 176 quotient algebra, 18 quotient involution, 145 radical, 429 radical algebra, 429 rational functional calculus, 57 real alternative C∗ -algebra, 521 real C∗ -algebra, 521 real JB∗ -algebra, 521 real JB∗ -triple, 522 real non-commutative JB∗ -algebra, 521 real numerical index, 353 regular A-bimodule, 639 regular left A-module, 436 representation (of an associative algebra), 436 representation corresponding to a left A-module, 436 Rickart’s dense-range-homomorphism theorem non-associative to Jordan-admissible, 458 non-associative to non-unital associative, 427 non-associative to unital associative, 20 Riesz–Schauder theory, 86 right-division algebra, 192 right multiplication operator, 13 right semi-H ∗ -algebra, 252 Russo–Dye theorem, 140 Russo–Dye–Palmer theorem, 141 scalar-plus-compact property, 248 scalar-plus-strictly-singular property, 248 Schoenberg theorem, 216

723

Schur lemma, 445 second Arens extension, 126 second Arens product, 126 second commutant, 24 sedenions, 199 self-adjoint element, 42 semi-H ∗ -algebra, 254 semi-L-summand, 314 semi-M-ideal, 315 semiprime algebra, 128 semiprime ideal, 430 semisimple algebra, 429 separating points (family of mappings), 22 separating space (of an operator), 18 Shirshov–Cohn theorem, 337 with inverses, 491 simple algebra, 18 Singer–Wermer theorem, 391, 443 smooth normed space at a norm-one element, 203 smooth-normed algebra, 204 solvable algebra, 269 spatial numerical index, 105 spatial numerical range, 107, 116, 308 special Jordan algebra, 337 spectral mapping theorem for the continuous functional calculus, 47, 479 for the holomorphic functional calculus, 64, 484 spectral radius, 6, 381 spectrum of an element, 12 split null A-extension, 639 split null X-extension, 636 standard involution of a Cayley algebra, 176 of a free non-associative algebra, 258 standard left A-module, 436 standard normed unital extension, 609 ∗-algebra, 39 ∗-mapping, 39 ∗-subalgebra, 39 state of X relative to u, 94 Stone–Weierstrass theorem unital version, 41 unit-free version, 53 strict inner ideal, 594 strictly singular operator, 248 strong radical, 20, 427 strong subdifferentiability of the norm, 299 strongly associative subalgebra of a Jordan algebra, 356 strongly exposed point, 118 strongly exposed subset, 299 strongly extreme point, 111 strongly semisimple algebra, 20, 427 subalgebra, 2 subalgebra generated by a subset, 9

724

Subject index for Volume 1

subharmonic function, 611 submean inequality, 611 subtriple, 465 subtriple generated by a subset, 466 super-trigonometric algebra, 201 symmetry (of a unital JB-algebra), 321 τ -point, 299 three-dimensional real spin factor, 560 three-dimensional spin factor, 553 topological divisor of zero (joint, left, one-sided, right, two-sided), 27 topological group, 6 topological J-divisor of zero, 460, 478, 496 topologically nilpotent algebra, 604 topologically simple algebra, 82 totally disconnected, 399 trace function, 181 transitive normed space, 217 transpose mapping of a continuous mapping, 45 transpose of an involution, 146 transpose of an operator, 29 trigonometric algebra, 200 triple homomorphism, 471 triple powers, 468 triple product, 127, 130, 324, 463 triple spectrum, 504 tripotent, 505 u-isotope JB∗ -algebra, 519 ultra-weak radical, 580 ultrapower, 271 ultraproduct, 271 uniform Fr´echet differentiability of the norm, 304 uniform strongly subdifferentiability of the norm, 301 uniformly non-square normed space, 230 uniformly smooth normed space, 304 unit, 2

unital A-bimodule, 637 unital algebra, 2 unital extension, 33 unital ∗-representation, 233 unitary element, 43, 368, 471 unitary normed algebra, 119 upper semicontinuity (of a set-valued function), 284 Urbanik–Wright theorem commutative, 216 non-commutative, 216 V-algebra, 134 variety of algebras, 424 vertex, 99 Vidav algebra, 134 Vidav–Palmer theorem associative, 142 alternative, 153 non-associative, 348 V -normal element, 424 V -normal subset, 424 von Neumann inequality, 174 von Neumann lemma, 7, 457 Vowden theorem, 421 weak radical, 578 weakly compact operator, 70 Weil algebra, 588 w∗ -superbig point, 334 w∗ -unitary element, 295 w∗ -vertex, 295 x-modular strict inner ideal, 594 X-valued partially defined derivation, 640 zero-annihilator ideal (z-ideal), 602 zero-annihilator radical (z-radical), 602 Zorn’s vector matrices, 177

Symbol index for Volume 2

S = sup{s : s ∈ S}, 603 f E := supx∈E f (x), 54

C(E, A) (continuous functions from E to A), 53 c(X, A) (core of X in A), 413 C D(A) (Cayley–Dickson doubling of A), 359

ANBP (approximate norm-1 boundedness property), 614 A(k, n) = {α = (α1 , . . . , αk ) ∈ Nk :| α |= n}, 20 A0 (k, n) = {α = (α1 , . . . , αk ) ∈ (N∪{0})k :| α |= n}, 20 A(a) (a-homotope of A), 589 Ann(I) (annihilator of the ideal I), 562 Ann(X) (annihilator of the Jordan ∗-triple X), 225 ada (x) = [a, x], 176 Ak (e) k = 1, 12 , 0 (Peirce subspaces of A relative to the idempotent e), 375 ∗ A+ ∗ (cone of w -continuous positive linear functionals on A), 286 A(λ) (λ-mutation of A), 325 Aut() (biholomorphic mappings from  onto ), 61 aut() (complete holomorphic vector fields on ), 138 Aut0 () = Aut() ∩ P 1 (X, X), 187 Aut0 () (connected component of I in (Aut(), Ta )), 183 aut0 () = aut() ∩ P 1 (X, X), 179 aut0 () = aut() ∩ (P 0 (X, X) ⊕ P 2 (X, X)), 179 BLn (X, Y) (bounded n-linear operators from X× . n. . ×X to Y), 20 BLsn (X, Y) (bounded symmetric n-linear operators from X× . n. . ×X to Y), 20 β(A), 614 Cb (E, A) (bounded continuous functions from E to A), 54 C(F) (split octonion algebra over F), 360

Df (x0 ) (derivative of f at x0 ), 22 Df (x0 ; x) (derivative of f at x0 in the direction x), 42 X (open unit ball of X), 46 exp : aut() → Aut() (exponential mapping), 143 En(A) (set of all equivalent algebra norms on A), 603, 609 F ∗ (a) := (F(a∗ ))∗ (definition of the operator F ∗ : A → B for an operator F : A → B with A and B semi-H ∗ -algebras), 488 F • : B → A (adjoint of the operator F : A → B for A and B semi-H ∗ -algebras), 488 g : H (U, X ) → H (g(U), Y ) (g : X → Y biholomorphic mapping), 112 g♦ : H (U, X) → H (g(U), Y) (g : X → Y biholomorphic mapping), 112 H(X, u) (hermitian elements of X relative to u), 1 H3 (C, ), 361 H (, Y) (holomorphic mappings from  to Y), 54 Hb (, Y) (bounded holomorphic mappings from  to Y), 54 Hb (B, Y) (holomorphic mappings from  to Y which are bounded on B), 55 H0 (, Y) (holomorphic mappings from  to Y which are bounded on every B  ), 56 H S (H) (Hilbert–Schmidt operators on H), 481

725

726

Symbol index for Volume 2

[I : A] = {F ∈ M (A) : F(A) ⊆ I} , 574 π I = Ann(Ann(I)), 572 k(F) = max{k ≥ 0 : kx ≤ F(x) ∀x ∈ X}, 586 Kδ = K + δX , 55 K   (K lies strictly inside ), 55  : H (U, Y) → H (U, Y) (differential operator on H (U, Y) associated to ), 106 n (nth power of  as a differential operator), 116 lin(S) (closed linear hull of S), 563 m(X, X∗ ) (Mackey topology of a dual Banach space X), 290 MC(S) (multiplicatively closed subset generated by S), 606 M(A) (algebra of multipliers of A), 315 Mult(X), 319 M (A) (multiplication algebra), 367 M(A), 406 NBP (norm-1 boundedness property), 614 Nann = {a ∈ A : N (a) = 0}, 572 NEP (norm extension problem), 470 N (A) (nucleus of A), 413 ε

N = (Nann )ann , 572 NSE (norm square equality), 639 NSI (norm square inequality), 638 Na,b (F, G) = F(a)G(b), 368 PA,f ,x0 (derivative of f at x0 with respect the tree A), 31 Pk (e) k = 1, 12 , 0 (Peirce projections relative to e), 375

(X) = {(x, f ) : x ∈ SX , f ∈ D(X, x)}, 193 P(X, Y) (polynomials from X to Y), 24 P n (X, Y) (homogeneous polynomials of degree n from X to Y), 22 Q(A) (symmetric Martindale algebra of quotients), 403 r(S) (spectral radius of S), 609 rb ( f , x0 ) (radius of boundedness of f at x0 ), 46 S(A, τ ) = {a ∈ A : τ (a) = −a}, 445 Sann = {F ∈ M (A) : F(S) = 0}, 572 s∗ (X, X∗ ) (strong∗ topology), 302 σλ , 606 ε S = (Sann )ann , 572 SOT (strong operator topology), 492 τ c(A) (trace class elements of A), 526 Tf ,x0 ,n (nth degree Taylor polynomial of f at x0 ), 24 t+ (x) (positive lifetime of x), 94 t− (x) (negative lifetime of x), 94 Tp (M) (tangent space of M at p), 160  T(M) := p∈M Tp (M) (tangent bundle of M), 162 T C (H) (trace-class operators on H), 524 Trees(m, n) (trees of degree m and height n), 30 WF,a (T) = FT(a), 489 WOT (weak operator topology), 533 X0 = aut()(0), 179 X∗ (predual of the dual Banach space X), 4 Xs (symmetric part of a complex Banach space X), 187

Subject index for Volume 2

absolutely primitive idempotent in a Jordan algebra, 594 Albert ring, 364 algebraic norm function, 359 alternative C∗ -algebra of multipliers, 321 alternative W ∗ -algebra, 4 alternative W ∗ -factor, 348 analytic Banach manifold, 159 analytic mapping, 36, 160 analytic structure, 159 analytic subset, 184 analytic vector field, 163 annihilator of a Jordan ∗-triple, 225 annihilator of an ideal, 562 approximate norm-1 boundedness property, 614 approximate unit, 533 asymptotic development, 26 atlas, 159 Banach Lie algebra of a Banach Lie group, 168 Banach Lie group, 165 binarion algebra over F, 359 capacity of a unital Jordan algebra, 591 Carath´eodory semidistance, 80 Cartan factor, 438 Cauchy inequalities, 45, 46 Cauchy integral formula, 45, 49 Cayley algebra, 359 Cayley–Dickson doubling (of a Cayley algebra), 359 Cayley–Dickson doubling process, 359 Cayley–Dickson ring, 360 central localization, 360 central order in an algebra, 360 centrally closed algebra over K, 368 chart, 159

circular subgroup, 175 circular vector field, 176 compact open topology, 53 compatible tripotents, 227 complete vector field, 97 completely primitive idempotent of a Jordan algebra, 591 cross norm, 642 direct summand of an algebra, 3 of a Jordan ∗-triple, 225 directional derivative, 41 domain homogeneous, 81 symmetric, 210 dual Banach space, 4 ε-closure of a subspace, 572 ε -closure of a subspace, 572 equivalent JBW ∗ -representations, 235 non-commutative JBW ∗ -representations, 13 essential triple ideal, 450 essentially defined double centralizer, 402 even-swapping ∗-involution, 442 exponential of a vector field, 98 factor representation of a JB∗ -triple, 439 factor representation of a non-commutative JB∗ -algebra, 350 finite capacity of a unital Jordan algebra, 591 finitely quasi-nilpotent subset, 631 flow, 97 (Fr´echet) differentiable function at a point, 22 (Fr´echet) derivative of a function at a point, 22

727

728

Subject index for Volume 2

G-analytic mapping, 43 generalized annihilator normed algebra, 562 generalized complemented normed algebra, 577 generalized real Cartan factor, 460 generalized real spin triple factor, 460 Hilbert–Schmidt operator, 481 homogeneous polynomial, 21 homotope of a Jordan algebra, 589 H ∗ -algebra, 480 H ∗ -ideal of a normed ∗-algebra, 518 hyper-Stonean topological space, 414 idempotent subset of a normed algebra, 614 indicator, 391 inner ideal of a Jordan algebra, 462 of a Jordan ∗-triple, 227 of a non-commutative Jordan algebra, 283 intrinsic triple product, 198 involutive algebra, 464 i-special Jordan triple, 445 JB∗ -triple of multipliers, 319 JBW-factor, 348 JBW-representation, 13 JBW ∗ -algebra, 4 JBW ∗ -factor, 348 JBW ∗ -representation, 235 JBW ∗ -triple factor, 438 JC∗ -triple, 440 Jordan algebra of a symmetric bilinear form, 365 Jordan pair, 452 Jordan triple, 444 Jordan triple of Clifford type, 452 Jordan triple of hermitian type, 447 J-primitive ideal, 462 J-primitive Jordan algebra, 462 L-embedded Banach space, 12 linear vector field, 98 local flow of a vector field, 88 L-projection, 5

1 -sequence, 248 L-summand, 5 λ-mutation of an algebra, 325 Mackey topology, 290 matricial decomposition of a C∗ -algebra, 441 matricially decomposed C∗ -algebra, 441 maximal M-ideal, 412 maximal closed ideal of a normed algebra, 563 maximal modular inner ideal, 462 M-embedded Banach space, 16 M-ideal, 5

minimal closed ideal, 485 minimal self-adjoint idempotent, 293 monotone complete JB-algebra, 9 non-commutative JB∗ -algebra, 14 M-projection, 3 M-summand, 3 multi-index, 20 multiplication algebra, 367 multiplicative boundedness property, 614 multiplicatively prime algebra, 489 multiplicatively semiprime algebra, 571 non-commutative JB∗ -algebra of multipliers, 317 non-commutative JBW ∗ -algebra, 4 non-commutative JBW ∗ -factor, 348 type I, 351 non-commutative JBW ∗ -representation, 13 nondegenerate non-commutative Jordan algebra, 364 norm extension problem, 470 norm ideal (of operators), 409 norm square equality, 639 norm square inequality, 638 norm-k boundedness property, 614 normal linear functional on a JB-algebra, 9 on a non-commutative JB∗ -algebra, 14 normed Jordan algebra of a continuous symmetric bilinear form, 463 nucleus of an algebra, 413 octonion algebra over F, 359 one-parameter group of automorphisms, 126 orthogonal tripotents, 227 π-closure of an ideal, 572 partial Banach Jordan ∗-triple structure, 181 Peirce decomposition of a non-commutative Jordan algebra, 375, 380 Poincar´e distance, 79 polydisc, 48 polynomial, 24 positive linear functional on a JB-algebra, 9 on a non-commutative JB∗ -algebra, 14 power series, 32 predual, 4 prime Jordan ∗-triple, 438 primitive M-ideal, 351 primitive idempotent of a Jordan algebra, 591 property (V ∗ ), 248 purely alternative algebra, 413 purely non-associative algebra, 413

Subject index for Volume 2 quasi-centralizer on an algebra, 502 quasi-nil normed algebra, 625 quasi-nilpotent subset, 620 quaternion algebra over F, 359 radius of convergence of a power series, 33 of restricted convergence of a power series, 33 radius of boundedness, 46 real alternative W ∗ -algebra, 17 real Cartan factor, 460 real JBW ∗ -algebra, 17 real JBW ∗ -triple, 242 real JBW ∗ -triple factor, 460 real JC∗ -triple, 445 real non-commutative JBW ∗ -algebra, 17 real spin triple factor, 460 real W ∗ -algebra, 17 reduced Jordan algebra, 594 (semi-)H ∗ -algebra complexification, 511 semi-H ∗ -algebra realification, 510 semi-H ∗ -algebra, 480 special Jordan triple, 445 spectral radius of a bounded subset, 609 spin triple factor, 439 split composition algebra, 360 standard involution of a Cayley algebra, 359 ∗-involution on a ∗-algebra, 362 ∗-tight envelope, 471 Stonean topological space, 414 strict inner ideal, 462 strong operator topology, 492 strong∗ topology of a dual Banach space, 311 of a JBW ∗ -triple, 302 of a non-commutative JBW ∗ -algebra, 276 support idempotent, 284 support tripotent, 300 symmetric Martindale algebra of quotients, 403 symmetric part of a complex Banach space, 187

729

tangent bundle, 162 tangent space, 160 τ -prime, 365 Taylor polynomial, 24 Taylor series, 35 topological algebra, 384 topologically interconnected idempotents, 385 topologically nilpotent normed algebra, 620 totally multiplicatively prime normed algebra, 491 totally prime normed algebra, 368 trace function, 359 trace-class elements of a semi-H ∗ -algebra, 526 trace-class operator, 524 tree, 30 tree-derivative, 31 T-topology (topology of local uniform convergence), 56 Ta -topology (analytic topology), 159 ultra-τ -prime normed associative algebra, 468 ultraprime normed algebra, 369 vector field, 88 weak operator topology, 533 weak primitive ideal, 567 weakly analytic mapping, 44 weakly G-analytic mapping, 44 weakly unconditionally Cauchy series, 245 w∗ -closed subtriple generated by, 226 W ∗ -algebra, 4 W ∗ -factor, 348 x-modular strict inner ideal, 462 Zel’manov’s prime theorem for Jordan algebras, 405 Zorn’s vector matrices, 360