Lectures on von Neumann algebras [Second ed.] 9781108496841, 1108496849

Table of contents :
Cover
Front Matter
CAMBRIDGE–IISc SERIES
Lectures on von Neumann
Algebras
Copyright
Dedication
CONTENTS
PREFACE TO 2nd
EDITION
PREFACE TO 1st
EDITION
INTRODUCTION
1 TOPOLOGIES ON
SPACES OF
OPERATORS
2 BOUNDED LINEAR
OPERATORS IN
HILBERT SPACES
3 VON NEUMANN
ALGEBRAS
4 THE GEOMETRY OF
PROJECTIONS AND
THE CLASSIFICATION
OF VON NEUMANN
ALGEBRAS
5 LINEAR FORMS ON
ALGEBRAS OF
OPERATORS
6 RELATIONS BETWEEN
A VON NEUMANN
ALGEBRA AND ITS
COMMUTANT
7 FINITE VON NEUMANN
ALGEBRAS
8 SPATIAL
ISOMORPHISMS AND
RELATIONS BETWEEN
TOPOLOGIES
9 UNBOUNDED LINEAR
OPERATORS IN
HILBERT SPACES
10 THE THEORY OF
STANDARD VON
NEUMANN ALGEBRAS
APPENDIX
BIBLIOGRAPHY
SUBJECT INDEX
NOTATION INDEX

Citation preview

Lectures on von Neumann Algebras The first edition of this book was first published in Romanian in 1975, and then translated into English in 1979. The authors have revised the second edition with the addition of several topics. This valuable text discusses the fundamental concepts of von Neumann algebras including bounded and unbounded linear operators in Hilbert spaces, the von Neumann bicommutant theorem, the Kaplansky density theorem, reduced and induced von Neumann algebras, the geometry of projections and the classification of von Neumann algebras, the type relation with the commutant, a detailed study of the predual of a von Neumann algebra, the finite von Neumann algebras, the von Neumann BT-theorem and the Tomita theory of standard forms of von Neumann Algebras. Each chapter of the book has a section of exercises and a section of comments. The revised text covers new material including factors associated to discrete groups, the first two examples of non-isomorphic type II1 factors, the original Murray–von Neumann proof of the existence of the trace on factors of type II1 , the Dixmier averaging theorem in case of factors, the proof of the BT-theorem of von Neumann, a detailed theory of standard forms for abelian, finite and semi-finite von Neumann algebras (these being the classical cases), as well as Tomita’s theorem for von Neumann algebras having a cyclic and separating vector, as a prelude to the general theory of standard forms. Also the Jones index is briefly described in a section of comments. The book is written in an easy to understand manner, with complete proofs, the only prerequisites being some elementary understanding of functional analysis. Serban ¸ Valentin Stratil ̆ ă is a professor in the Department of Mathematics at the University of Bucharest and senior researcher at the Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy. His areas of research include functional analysis, harmonic analysis, operator algebras and representation theory. László Zsidó, former researcher at the Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, is a professor in the Department of Mathematics at the University of Rome ‘Tor Vergata’. His areas of research include functional analysis, harmonic analysis, operator algebras, generalized functions and ergodic theory. Both authors received the ‘Simion Stoilow’ Prize of the Romanian Academy in 1975 for this book. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/EDEB93BAEFB2CFF1F4806E1EBC28E161

CAMBRIDGE–IISc SERIES Cambridge–IISc Series aims to publish the best research and scholarly work on different areas of science and technology with emphasis on cutting-edge research. The books will be aimed at a wide audience including students, researchers, academicians and professionals and will be published under three categories: research monographs, centenary lectures and lecture notes. The editorial board has been constituted with experts from a range of disciplines in diverse fields of engineering, science and technology from the Indian Institute of Science, Bangalore. IISc Press Editorial Board: Amaresh Chakrabarti, Professor and Chair, Centre for Product Design and Manufacturing Diptiman Sen, Professor, Centre for High Energy Physics Prabal Kumar Maiti, Professor, Department of Physics S. P. Arun, Associate Professor, Centre for Neuroscience Titles in print in this series: • Continuum Mechanics: Foundations and Applications of Mechanics by C. S. Jog • Fluid Mechanics: Foundations and Applications of Mechanics by C. S. Jog • Noncommutative Mathematics for Quantum Systems by Uwe Franz and Adam Skalski • Mechanics, Waves and Thermodynamics by Sudhir Ranjan Jain • Finite Elements: Theory and Algorithms by Sashikumaar Ganesan and Lutz Tobiska • Ordinary Differential Equations: Principles and Applications by A. K. Nandakumaran, P. S. Datti and Raju K. George • Biomaterials Science and Tissue Engineering: Principles and Methods by Bikramjit Basu • Knowledge Driven Development: Bridging Waterfall and Agile Methodologies by Manoj Kumar Lal

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Cambridge IISc Series

Lectures on von Neumann Algebras Second Edition

̆ ă Serban ¸ Valentin Stratil László Zsidó

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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 to 321, 3rd Floor, Plot No.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/9781108496841 © Cambridge University Press 2019 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 Romanian edition published by Editura Academiei, 1975 First English translation published by Editura Academiei and Abacus Press, 1979 Second edition published by Cambridge University Press, 2019 Printed in India A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Stratil ̆ a,̆ Serban, ¸ 1943- author. | Zsidó, László, author. Title: Lectures on von Neumann algebras/Serban ¸ Valentin Stratil ̆ a,̆ László Zsidó. Other titles: Lectii de algebre von Neumann. English Description: 2nd edition. | Cambridge; New York, NY: Cambridge University Press, 2019. | Series: Cambridge–IISc series | Originally published in Romanian in 1975. First English translation, 1979. | Includes bibliographical references and indexes. Identifiers: LCCN 2019001647 | ISBN 9781108496841 (hardback: alk. paper) Subjects: LCSH: Von Neumann algebras. | Hilbert space. Classification: LCC QA326 .S7613 2019 | DDC 512/.556–dc 3 LC record available at https://lccn.loc.gov/2019001647 ISBN 978-1-108-49684-1 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.

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To the memory of Sanda Stratil ̆ ă (1944–2011)

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CONTENTS

Preface to 2nd edition Preface to 1st edition

ix xi

Introduction

1

1 Topologies on spaces of operators

4

2 Bounded linear operators in Hilbert spaces

15

3 von Neumann algebras

60

4 The geometry of projections and the classification of von Neumann algebras

87

5 Linear forms on algebras of operators

115

6 Relations between a von Neumann algebra and its commutant

145

7 Finite von Neumann algebras

162

8 Spatial isomorphisms and relations between topologies

197

9 Unbounded linear operators in Hilbert spaces

215

10 The theory of standard von Neumann algebras

276

Appendix

399

Bibliography

403

Subject index

422

Notation index

426

vii

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PREFACE TO 2nd EDITION

This book is an elementary self-contained exposition, with complete proofs, of the fundamentals of the theory of von Neumann algebras. Although this theory has considerably evolved in the last 40 years since the first edition of this book was printed, the fundamentals of the theory remain the same. Besides many achievements in the operator algebras themselves, new connected powerful mathematical theories have appeared, such as the Noncommutative Geometry of Alain Connes and the Free Probability Theory of Dan-Virgil Voiculescu, and this makes it even more necessary to have an elementary presentation of the fundamentals of the theory. The first edition was intensively used in the main mathematical centres in the world and was very soon out of print, and we received many demands for the book that we could not fulfil. In this new edition we have added, among other things, some material concerning factors associated to ICC groups, the BT-Theorem of J. von Neumann, the Murray–von Neumann proof of the existence of the trace on type II1 factors, the Averaging Theorem of J. Dixmier in case of factors, and a preliminary presentation of Tomita’s Theorem first in the classical case of a cyclic trace vector and then in an introduction to Chapter X, in the case of a cyclic and separating vector. In the comments sections, we have quoted some new results, among which are the Jones index of subfactors and the Connes classification of hyperfinite factors. The extensive bibliography of the first edition has been considerably reduced, mainly to quoted books or articles. The main consequences of Tomita’s theory, presented in Chapter 10, are the subject of a direct continuation of the present book, namely Modular Theory in Operator Algebras by S. ¸ Strătilă (1981). We are grateful to V. Sunder (Chennai Institute of Mathematical Sciences, India) and Gadadhar Misra (Indian Institute of Science, Bangalore, India) for proposing a revised edition of this book, especially to Gadadhar Misra for the help in the LaTeX conversion of the book, and to the Cambridge University Press for publishing this second edition of the book.

ix

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Lectures on von Neumann algebras

We are also grateful to Alexandru Negrescu who typed in LaTeX all the supplementary material for the Second Edition (c. 60 p). The Authors Bucharest, September 2018

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PREFACE TO 1st EDITION

This book is based on lectures delivered in July–August 1972 at the Suceava Summer School organized by the Institute of Mathematics of the Academy of the Socialist Republic of Romania in cooperation with the Society of Mathematical Sciences. The study of the algebras of operators in Hilbert spaces was initiated by F. J. Murray and J. von Neumann in connection with some problems of theoretical physics. The wealth of the mathematical facts contained in their fundamental papers interested many mathematicians. This soon led to the crystallization of a new branch of mathematics: the theory of algebras of operators. The first systematic exposition of this theory appeared in the well-known monograph by J. Dixmier 1957/1969, which was subtitled ‘Algèbres de von Neumann’. It expounded almost all the significant results achieved until its appearance. Afterwards, the theory continued to develop, for it had important applications in the theory of group representations, in mathematical physics and in other branches of mathematics. Of great importance were the results obtained by M. Tomita, who exhibited canonical forms for arbitrary von Neumann algebras. In recent times, fine classifications and structure theorems have been obtained for von Neumann algebras especially by A. Connes. The present book contains what we consider to be the fundamental part of the theory of von Neumann algebras. The book also contains the essential elements of the spectral theory in Hilbert spaces. The material is divided into 10 chapters; besides the basic text, each chapter has two complementary sections: exercises and comments (including bibliographical comments). The reader is expected to know only some elementary facts from functional analysis. In writing this book, we made use of existing books and courses (J. Dixmier 1957/1969, I. Kaplansky 1955/1968, J. R. Ringrose 1966/1967, 1972a, b, S. Sakai 1962, 1971, M. Takesaki 1969/1970, 1970 and D. M. Topping 1971), as well as many articles, some of them available only as preprints. Some of the exercises are borrowed from J. Dixmier’s book 1957/1969.

xi

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xii

Lectures on von Neumann algebras

Thanks to Grigore Arsene and Dan Voiculescu for their help in the writing of this book, for the useful discussions and for the bibliographical information they gave us. We thank Sanda Stratil ̆ ă for compiling the bibliography and for carefully typing the manuscript. The Authors

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INTRODUCTION

In the study of operator algebras, there are two main methods, the first is of an algebraic character whereas the second is more analytic. The algebraic method proceeds by a successive reduction of problems concerning the arbitrary operators to problems about positive operators and from these to problems about projections, where one can avail oneself of the lattice-theoretical geometry of projections. In this geometry, the main notion is that of equivalence, and the main result is the comparison theorem, an important technical device being the polar decomposition of operators. These methods are elementary, but they afford a clear classification of the von Neumann algebras into general types. The results obtained by these methods are shown in Chapter 4 and in the first sections of Chapter 7. The analytic method, which is more complex and profound, consists of a systematic manipulation with linear forms defined on operator algebras; they may be bounded or unbounded. Here the important facts are concentrated around certain results which extend the classical Lebesgue–Radon–Nikodym theorem, the main technical tool here being the polar decomposition of linear forms. The analytic methods permit the analysis of relations existing between the given algebra and its commutant, as well as of those that relate the predual of the given algebra to the Hilbert space in which this algebra is operating. Chapter 6 shows the relations existing between the type of the given algebra and its commutant, whereas Chapters 7 and 8 exhibit the quantitative relations that measure the relative wealth of the given algebra and its commutant. For finite von Neumann algebras, the existence of a trace, which measures the relative dimension of projections, allows the evaluation of the quantitative relations between the given algebra and its commutant by a coupling function of a metric nature. In other, more general, cases, the coupling between the given algebra and its commutant can be measured only by projective objects, namely cardinals associated with central projections, but the information thus obtained is not always satisfactory.

1

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2

Lectures on von Neumann algebras

The von Neumann algebras that are well equilibrated with their commutants are called standard von Neumann algebras, and the main result of Chapter 10 is that any von Neumann algebra is isomorphic to a standard von Neumann algebra in a canonical form. This has been known for a long time in the case of the semifinite von Neumann algebras; to be extended to the general case, it required a new technique, namely a ‘polar decomposition’ for the involution of the algebra. Chapter 10 is dedicated to the study of the canonical forms of the von Neumann algebras as well as to some applications to the theory of arbitrary von Neumann algebras. The theory of operator algebras is based on two fundamental results: the density theorem of J. von Neumann and the density theorem of I. Kaplansky, both shown in Chapter 3. This book covers results of M. Takesaki’s work (1970) and those of J. Dixmier’s book (1957/1969), with the exception of the reduction theory and some examples of factors included there. The reduction theory aims at decomposing an arbitrary von Neumann algebra into the family of von Neumann algebras with trivial centres (the so-called factors), in such a manner that the algebra be obtainable from this family, whereas its properties will be derivable from those of the factors. In this manner, the reduction theory transfers to the factors the purely non-commutative part of the algebra, whereas the commutative part is reflected in the space of the indices of the family of factors; the main problem of the structure and classification of the von Neumann algebras is thus reduced to the corresponding problems for factors. For the reduction theory, one can read J. Dixmier’s book (1957/1969), as well as the expository article by L. Zsidó (1973b), based on the ideas of S. Sakai (1964). Both develop the classical reduction theory of J. von Neumann but from seemingly different points of view, which can easily be shown to be similar. For factor theory, we recommend the works of J. Dixmier (1957/1969, 1969), S. Sakai (1971), D. McDuff (1970b), H. Araki and E. J. Woods (1968) and A. Connes (1974c, 1975b, h, 1976b, c, d). Important results concerning the structure of von Neumann algebras are contained in the works of A. Connes (1973c, 1974b) and M. Takesaki (1973c, g). Our exposition refers to the spatial theory of von Neumann algebras, which considers them as being subalgebras of the algebra of all bounded linear operators on a Hilbert space. S. Sakai obtained in (1956b) the abstract characterization of von Neumann algebras and developed the theory of von Neumann algebras by non-spatial methods. Thus, in S. Sakai’s book (1971) the reader can find some of the results we present here with different proofs. Also, S. Sakai’s book (1971) contains some other results that are not included in this book. ‘Algebras of operators’ usually designate something more general than von Neumann algebras, the so-called 𝐶 ∗ -algebras. In our exposition, we have only incidentally referred to the 𝐶 ∗ -algebras, but this theory makes full use of the theory of von Neumann algebras. For this theory, as well as for its applications to the theory of group representations, we refer the reader to J. Dixmier’s monograph (1964/1968). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:37, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.003

Introduction

3

Other topics connected with the theory of operator algebras, but not treated in this book, are the following: the problem of the generation of von Neumann algebras (see Saitô 1972), non-commutative harmonic analysis and duality theory for locally compact groups (see Eymard 1964; Takesaki 1971/1972; Walter 1972, 1974), non-commutative ergodic theory (see A. Guichardet 1974), applications to the theory of operators (see Douglas, 1972) and connections with some problems of theoretical physics (see Kastler 1964, 1967; Emch 1972; Ruelle 1969). Although rather a long time has elapsed since the publication of the works by F. J. Murray and J. von Neumann and their results are included in the books mentioned above, we consider that their works are still worth reading for those interested in the theory of operator algebras. This book is self-contained with complete proofs. The exercises contain results that enrich the text and that can be proved with the methods described in it; the more difficult exercises are marked by an asterisk, whereas some of the exercises that offer no difficulty are used in the main text and are marked by the symbol ‘!’. The final sections of each chapter include complements that contain additional results, bibliographical references, as well as the names of the mathematicians to whom the results contained in each chapter are to be ascribed.

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1

TOPOLOGIES ON SPACES OF OPERATORS

In this chapter, we introduce the main topologies in the space B (H ) of all bounded linear operators on a Hilbert space. Lemma 1.1. Let E be a vector space, 𝜑 a linear form on E and 𝑝1 , 𝑝2 , … , 𝑝𝑛 semi-norms on E , such that 𝑛

𝑥 ∈ E.

|𝜑(𝑥)| ≤ ∑ 𝑝𝑘 (𝑥), 𝑘=1

Then there exist linear forms 𝜑1 , … , 𝜑𝑛 on E , such that 𝑛

𝜑 = ∑𝑘=1 𝜑𝑘 , |𝜑𝑘 (𝑥)| ≤ 𝑝𝑘 (𝑥),

𝑥 ∈ E,

𝑘 = 1, … , 𝑛.

Proof. Let E 𝑛 be the Cartesian product of 𝑛 copies of E , D ⊂ E 𝑛 the diagonal of E 𝑛 and 𝑝 the semi-norm on E 𝑛 defined by 𝑛

𝑝(𝑥1 , … , 𝑥𝑛 ) = ∑ 𝑝𝑘 (𝑥𝑘 ),

(𝑥1 , … , 𝑥𝑛 ) ∈ E 𝑛 .

𝑘=1

and 𝜑0̃ the linear form on D defined by 𝑥 ∈ E.

𝜑0̃ (𝑥, … , 𝑥) = 𝜑(𝑥),

From the hypothesis, we immediately infer that the linear form 𝜑0̃ on D is majorized on D by the semi-norm 𝑝. With the Hahn–Banach theorem, we infer that there exists a linear form 𝜑̃ on E 𝑛 , having the following properties: ̃ … , 𝑥) = 𝜑0̃ (𝑥, … , 𝑥), 𝜑(𝑥, ̃ 1 , … , 𝑥𝑛 )| ≤ 𝑝(𝑥1 , … , 𝑥𝑛 ), |𝜑(𝑥

𝑥 ∈ E,

(𝑥1 , … , 𝑥𝑛 ) ∈ E 𝑛 .

We then define the forms 𝜑𝑘 by the relations: ̃ … , 0, 𝑥, 0, … , 0), 𝜑𝑘 (𝑥) = 𝜑(0,

𝑥 ∈ E , 𝑘 = 1, … , 𝑛,

where, in the right-hand member, 𝑥 stands on the 𝑘th place. 4

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5

Topologies on spaces of operators The linear forms thus defined satisfy the conditions of the statement.

Q.E.D.

1.2. Let E be a Banach space, E ∗ the dual of E and F a vector subspace of E ∗ . We denote by 𝜎(E ; F ) the weak topology defined in E by the family F of linear forms; then the 𝜎(E ; F )-topology is defined by the family of semi-norms {𝑝𝜑 ; 𝜑 ∈ F }, where 𝑥 ∈ E.

𝑝𝜑 (𝑥) = |𝜑(𝑥)|,

We consider the norm topology in E ∗ , and we denote by F̄ the closure of F in this topology. We denote by E 1 the closed unit ball in E . Lemma. Let E be a Banach space, F ⊂ E ∗ a vector subspace and 𝜑 a linear form on E : (i) 𝜑 is 𝜎(E ; F )-continuous iff 𝜑 ∈ F . (ii) 𝜑 is 𝜎(E ; F )-continuous on E 1 iff 𝜑 ∈ F̄ . (iii) The topologies 𝜎(E ; F ) and 𝜎(E ; F̄ ) coincide on E 1 . (iv) If F is closed in the norm topology and 𝜑 is 𝜎(E ; F )-continuous on E 1 , then 𝜑 is 𝜎(E ; F )-continuous on E . Proof. (i) Obviously, if 𝜑 ∈ F , then 𝜑 is 𝜎(E ; F )-continuous. Conversely, if 𝜑 is 𝜎(E ; F )-continuous, then there exist 𝜓1 , … , 𝜓𝑛 ∈ F , such that 𝑛

|𝜑(𝑥)| ≤ ∑ 𝑝𝜓𝑘 (𝑥),

𝑥 ∈ E.

𝑘=1

By virtue of Lemma 1.1, there exist linear forms 𝜑1 , … , 𝜑𝑛 on E , such that 𝑛

𝜑 = ∑𝑘=1 𝜑𝑘 , |𝜑𝑘 (𝑥)| ≤ 𝑝𝜓𝑘 (𝑥) = |𝜓𝑘 (𝑥)|,

𝑥 ∈ E,

𝑘 = 1, … , 𝑛.

If 𝜓𝑘 = 0, then 𝜑𝑘 = 0. If 𝜓𝑘 ≠ 0, then there exists an 𝑥𝑘 ∈ E , such that 𝜓𝑘 (𝑥𝑘 ) = 1, and for any 𝑥 ∈ E , we have |𝜑𝑘 (𝑥 − 𝜓𝑘 (𝑥)𝑥𝑘 )| ≤ |𝜓𝑘 (𝑥 − 𝜓𝑘 (𝑥)𝑥𝑘 )| = 0. Consequently, we have 𝑛

𝜑𝑘 = 𝜑𝑘 (𝑥𝑘 )𝜓𝑘 ∈ F and 𝜑 = ∑ 𝜑𝑘 ∈ F . 𝑘=1

(ii) It is easily seen that if 𝜑 ∈ F̄ , then the restriction of 𝜑 to E 1 is 𝜎(E ; F )-continuous. Conversely, let 𝜑 be a linear form on E , whose restriction to E 1 is 𝜎(E ; F ) continuous. Then 𝜑 is norm continuous and therefore 𝜑 ∈ E ∗ . Let 𝜀 > 0 be an arbitrary positive real Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:50:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.004

6

Lectures on von Neumann algebras

number. Since the restriction of 𝜑 to E 1 is 𝜎(E ; F )-continuous at 0, we infer that there exist linear forms 𝜓1 , … , 𝜓𝑛 ∈ F , such that 𝑛

‖𝑥‖ ≤ 1, ∑ 𝑝𝜓𝑘 (𝑥) < 1 ⇒ |𝜑(𝑥)| < 𝜀. 𝑘=1

Hence, we immediately infer that for any 𝑥 ∈ E , we have 𝑛

|𝜑(𝑥)| ≤ 𝜖‖𝑥‖ + ‖𝜑‖ ∑ 𝑝𝜓𝑘 (𝑥). 𝑘=1

By virtue of Lemma 1.1, it follows that there exist linear forms 𝜑1 , 𝜑2 on E , such that 𝜑 = 𝜑1 + 𝜑2 , |𝜑1 (𝑥)| ≤ 𝜀‖𝑥‖, 𝑥 ∈ E , 𝑛

|𝜑2 (𝑥)| ≤ ‖𝜑‖ ∑ 𝑝𝜓𝑘 (𝑥), 𝑥 ∈ E . 𝑘=1

Consequently, 𝜑2 ∈ F and ‖𝜑−𝜑2 ‖ = ‖𝜑1 ‖ ≤ 𝜀. Since 𝜀 > 0 was arbitrary, we get 𝜑 ∈ F̄ . Statements (iii) and (iv) immediately follow from (i) and (ii), respectively. Q.E.D. 1.3. Let H be a Hilbert space and B (H ) the space of all bounded linear operators on H . We consider B (H ) as a Banach space only with respect to the usual operator norm: ‖𝑥‖ = sup{‖𝑥𝜉‖; 𝜉 ∈ H , ‖𝜉‖ = 1}. For 𝜉, 𝜂 ∈ H , we define a linear form 𝜔𝜉,𝜂 on B (H ) by 𝜔𝜉,𝜂 (𝑥) = (𝑥𝜉|𝜂), 𝑥 ∈ B (H ). Obviously, 𝜔𝜉,𝜂 ∈ B (H )∗ , and it is easily checked that ‖𝜔𝜉,𝜂 ‖ = ‖𝜉‖ ⋅ ‖𝜂‖. The form 𝜔𝜉,𝜉 will be simply denoted by 𝜔𝜉 . Let B (H )∼ be the vector space generated in B (H )∗ by the forms 𝜔𝜉,𝜂 , 𝜉, 𝜂 ∈ H , whereas B (H )∗ denotes the norm closure of B (H )∼ in B (H )∗ . Besides the norm topology, we shall also consider the following topologies in B (H ): the weak operator topology or the wo-topology, which is the topology defined by the family of semi-norms B (H ) ∋ 𝑥 ↦ |(𝑥𝜉|𝜂)|, 𝜉, 𝜂 ∈ H ; in other words, it is just the 𝜎(B (H ); B (H )∼ )-topology; the strong operator topology or the so-topology, which is the topology defined by the family of semi-norms B (H ) ∋ 𝑥 ↦ ||𝑥𝜉||, 𝜉 ∈ H ; the ultraweak operator topology or the w-topology, which is, by definition, the 𝜎(B (H ); B (H )∗ )-topology. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:50:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.004

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We now apply Lemma 1.2, where we make E = B (H ), F = B (H )∼ and F̄ = B (H )∗ , and by taking into account the terminology just introduced, we get the following: Lemma. Let H be a Hilbert space. Then, (i) B (H )∼ is the set of all wo-continuous linear forms on B (H ). (ii) B (H )∗ is the set of all w-continuous linear forms on B (H ). (iii) A linear form 𝜑 on B (H ) is w-continuous iff its restriction to B (H )1 is wo-continuous. (iv) In B (H )1 , the wo-topology and the w-topology coincide. Theorem 1.4. A linear form 𝜑 on B (H ) is wo-continuous iff it is so-continuous. Proof. It is easy to see that the so-topology is finer (stronger) than the wo-topology; therefore, any wo-continuous linear form is so-continuous. Conversely, if 𝜑 is so-continuous, then there exist non-zero vectors 𝜉1 , … , 𝜉𝑛 ∈ H , such that 𝑛

|𝜑(𝑥)| ≤ ∑ ‖𝑥𝜉𝑘 ‖, 𝑥 ∈ B (H ). 𝑘=1

From Lemma 1.1, there exist linear forms 𝜑1 , … , 𝜑𝑛 on B (H ), such that 𝑛

𝜑 = ∑ 𝜑𝑘 𝑘=1

𝑥 ∈ B (H ),

|𝜑𝑘 (𝑥)| ≤ ‖𝑥𝜉𝑘 ‖,

𝑘 = 1, … , 𝑛.

Let 𝑘 ∈ {1, … , 𝑛} be any fixed index. We obviously have H = {𝑥𝜉𝑘 ; 𝑥 ∈ B (H )}. As a consequence of what we have already proved, the mapping 𝑥𝜉𝑘 ↦ 𝜑𝑘 (𝑥) is a bounded linear form on H . With Riesz’ theorem, we infer that there exists an 𝜂𝑘 ∈ H , such that 𝜑𝑘 (𝑥) = (𝑥𝜉𝑘 |𝜂𝑘 ), 𝑥 ∈ B (H ). Consequently, for any 𝑘, there exists an 𝜂𝑘 ∈ H , such that 𝜑𝑘 = 𝜔𝜉𝑘 ,𝜂𝑘 . Therefore, 𝑛

𝜑 = ∑ 𝜑𝑘 ∈ B (H )∼ , 𝑘=1

i.e., 𝜑 is wo-continuous.

Q.E.D.

1.5. From Theorem 1.4 and from Mackey’s theorem, we infer the following: Corollary. A convex subset of B (H ) is wo-closed iff it is so-closed. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:50:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.004

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1.6. From the Hahn–Banach theorem and from Theorem 1.4, we infer the following. Corollary. Let M ⊂ B (H ) be a vector subspace. A linear form on M is wo-continuous iff it is so-continuous. Lemma 1.7. The Banach space B (H ) is isomorphic to the dual of the Banach space B (H )∗ by the mapping given by the canonical bilinear form: B (H ) × B (H )∗ ∋ (𝑥, 𝜑) ↦ 𝜑(𝑥). Proof. Let 𝑥 ∈ B (H ). By the formula Φ𝑥 (𝜑) = 𝜑(𝑥), 𝜑 ∈ B (H )∗ , one defines a bounded linear form on B (H )∗ , such that ‖Φ𝑥 ‖ ≤ ‖𝑥‖. In fact, we have the equality ‖Φ𝑥 ‖ = ‖𝑥‖, as can be inferred from the following computation: ‖𝑥‖ = sup{‖𝑥𝜉‖; 𝜉 ∈ H , ‖𝜉‖ = 1} = sup{|(𝑥𝜉|𝜂)|; 𝜉, 𝜂 ∈ H , ‖𝜉‖ = ‖𝜂‖ = 1} = sup{|𝜔𝜉,𝜂 (𝑥)|; 𝜉, 𝜂 ∈ H , ‖𝜉‖ = ‖𝜂‖ = 1} ≤ sup{|Φ𝑥 (𝜔𝜉,𝜂 ); ‖𝜔𝜉,𝜂 ‖ = 1} ≤ ‖Φ𝑥 ‖. Conversely, let Φ ∈ (B (H )∗ )∗ . By the formula ̃ 𝜂) = Φ(𝜔𝜉,𝜂 ), 𝜉, 𝜂 ∈ H , 𝜑(𝜉, we define a bounded sesquilinear form on H . With the help of Riesz’ theorem, we get a uniquely determined operator 𝑥 ∈ B (H ), such that ̃ 𝜂) = (𝑥𝜉|𝜂), 𝜉, 𝜂 ∈ H . 𝜑(𝜉, It follows that Φ(𝜔𝜉,𝜂 ) = Φ𝑥 (𝜔𝜉,𝜂 ), for any 𝜉, 𝜂 ∈ H . Consequently, Φ and Φ𝑥 coincide on B (H )∼ . This shows that Φ = Φ𝑥 . Q.E.D. Theorem 1.8. For any Hilbert space H , the closed unit ball B (H )1 of B (H ) is wo-compact. Proof. According to Lemma 1.7, B (H ) = (B (H )∗ )∗ ; from the Alaoglu theorem, it follows that B (H )1 is w-compact. By taking into account statement (iv) from Lemma 1.3, we infer that B (H )1 is wo-compact. Q.E.D. Lemma 1.9. Let B be a Banach space and B ∗ ⊂ B ∗ a norm-closed vector subspace, such that B = (B ∗ )∗ through the canonical bilinear form on B × B ∗ . Let M ⊂ B be a 𝜎(B ; B ∗ )-closed vector subspace. We denote M ∗ = {𝜑|M ; 𝜑 ∈ B ∗ } ⊂ M ∗ . Then, Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:50:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.004

9

Topologies on spaces of operators (i) M ∗ is a norm-closed vector subspace of M ∗ . (ii) For any 𝜓 ∈ M ∗ and any 𝜀 > 0, there exists a 𝜑 ∈ B ∗ , such that 𝜓 = 𝜑|M ,

‖𝜑‖ ≤ ‖𝜓‖ + 𝜀.

(iii) M = (M ∗ )∗ , through the canonical bilinear form on M × M ∗ . Proof. (i) Let M ∘ = {𝜑 ∈ B ∗ ; 𝜑|M = 0} be the polar of M in B ∗ . Since M is 𝜎(B ; B ∗ )-closed, from the bipolar theorem we infer that M = M ∘∘ = {𝑥 ∈ B ; 𝜑(𝑥) = 0 for any 𝜑 ∈ M ∘ }. The mapping B ∗ ∋ 𝜑 ↦ 𝜑|M ∈ M ∗ is linear, of norm ≤ 1, and its kernel is equal to M ∘ . Consequently, it induces a linear mapping B ∗ |M ∘ ∋ 𝜑|M ∘ ↦ 𝜑|M ∈ M ∗ . We shall show that this mapping is isometric. Let 𝜑0 ∈ B ∗ , such that ‖𝜑0 |M ∘ ‖ = 1, i.e., dist (𝜑0 , M ∘ ) = 1. From the Hahn–Banach theorem, we infer that there exists a bounded linear form Φ on B ∗ , such that ‖Φ‖ = Φ(𝜑0 ) = 1 and Φ(𝜑) = 0 for any 𝜑 ∈ M ∘ . Since B = (B ∗ )∗ , we infer that there exists an 𝑥 ∈ B , such that ‖𝑥‖ = 𝜑0 (𝑥) = 1 and 𝜑(𝑥) = 0 for any 𝑥 ∈ M ∘ . It follows that 𝑥 ∈ M ∘∘ = M , and therefore, ‖𝜑0 |M ‖ ⩾ 𝜑0 (𝑥) = 1. Thus, the mapping B ∗ |M ∘ → M ∗ we have just defined is an isometric isomorphism. Since B ∗ |M ∘ is complete, it follows that M ∗ is a complete subspace of M ∗ , and therefore, it is norm closed. (ii) For any 𝜓 ∈ M ∗ there exists, by virtue of what we have just proved, a 𝜑0 ∈ B ∗ , such that 𝜓 = 𝜑0 |M and ‖𝜓‖ = ‖𝜑0 |M ∘ ‖ = dist(𝜑0 , M ∘ ). Then, for any 𝜖 > 0, there exists a 𝜑1 ∈ M ∘ such that ‖𝜑0 + 𝜑1 ‖ ≤ ‖𝜓‖ + 𝜖. Let us define 𝜑 = 𝜑0 + 𝜑1 ∈ B ∗ . Then 𝜑|M = 𝜓 and ‖𝜑‖ ≤ ‖𝜓‖ + 𝜖. (iii) For any 𝑥 ∈ M , the mapping Ψ𝑥 ∶ M ∗ ∋ 𝜓 ↦ 𝜓(𝑥) is a bounded linear form on M ∗ . By taking into account (ii) and the canonical identification B = (B ∗ )∗ , we get ‖Ψ𝑥 ‖ = sup{|𝜓(𝑥)|; 𝜓 ∈ M ∗ , ‖𝜓‖ < 1} = sup{|𝜑(𝑥)|; 𝜑 ∈ B ∗ , ‖𝜑‖ < 1} = ‖𝑥‖. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:50:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.004

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Conversely, let 𝜓 ∈ (M ∗ )∗ . We define a linear form on B ∗ by the formula Φ(𝜑) = Ψ(𝜑|M ), 𝜑 ∈ B ∗ . Then Φ ∈ (B ∗ )∗ and Φ|M ∘ = 0. Since (B ∗ )∗ = B , there exists an 𝑥 ∈ B , such that Φ(𝜑) = 𝜑(𝑥) for any 𝜑 ∈ B ∗ and 𝜑(𝑥) = 0 for any 𝜑 ∈ M ∘ . Consequently, 𝑥 ∈ M ∘∘ = M and Ψ(𝜑|M ) = 𝜑|M (𝑥) = Ψ𝑥 (𝜑|M ), for any 𝜑 ∈ B ∗ , i.e., Ψ = Ψ𝑥 . Q.E.D. 1.10. Let H be a Hilbert space and B (H ) the Banach space of all bounded linear operators on H . Let M ⊂ B (H ) be a 𝑤-closed vector subspace. We introduce the following notations: M ∼ = the wo-dual of M , i.e., the set of all the wo-continuous linear forms on M . Obviously, M ∼ is a vector subspace of M ∗ . M ∗ = the 𝑤-dual of M , i.e., the set of all 𝑤-continuous linear forms on M . Obviously, M ∗ is a vector subspace of M ∗ . Theorem. Let M ⊂ B (H ) be a w-closed vector subspace. Then the following statements are true: (i) M ∼ = {𝜑|M ; 𝜑 ∈ B (H )∼ }. (ii) M ∗ = {𝜑|M ; 𝜑 ∈ B (H )∗ }. (iii) M ∗ = M ∼ , i.e., M ∗ is equal to the norm closure of M ∼ in M ∗ . (iv) M = (M ∗ )∗ , i.e., M is identified, as a normed space, to the dual of M ∗ through the canonical bilinear form on M × M ∗ . (v) For any 𝜓 ∈ M ∗ and any 𝜀 > 0, there exists a 𝜑 ∈ B (H )∗ , such that 𝜑|M = 𝜓, ‖𝜑‖ ≤ ‖𝜓‖ + 𝜀. (vi) If 𝜓 is a linear form on M , then 𝜓 ∈ M ∗ iff the restriction of 𝜓 to the closed unit ball M 1 of M is wo-continuous. Proof. The statements (i) and (ii) follow from Lemma 1.3 (i) and (ii), with the help of the Hahn–Banach theorem. The statements (iv) and (v), as well as the fact that M ∗ is a closed subset of M ∗ , follow from Lemma 1.9, by taking into account the statement (ii) from above and Lemma 1.7. By virtue of statements (i) and (ii) from above, the bounded linear mapping 𝜑 ↦ 𝜑|M maps B (H )∼ on M ∼ and B (H )∗ on M ∗ . Since by Definition 1.3, B (H )∼ is uniformly dense in B (H )∗ , it follows that M ∼ is uniformly dense in M ∗ . Thus, statement (iii) is proved. From statements (i) and (ii) from above, it follows that the topology induced on M by the wo-topology (resp., the w-topology) of B (H ) is the 𝜎(M ; M ∼ )-topology (resp., the 𝜎(M ; M ∗ )-topology). By virtue of statement (iii), M ∗ = M ∼ . Consequently, statement (vi) follows from Lemma 1.2 (ii). Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:50:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.004

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1.11. Let M ⊂ B (H ) be a w-closed vector subspace. The norm-closed vector subspace M ∗ of the dual M ∗ of M is called the predual of M . This term is justified by statement (iv) from Theorem 1.10. EXERCISES In the exercises that follow some elementary notions about bounded linear operators on Hilbert spaces are assumed, although these are expounded in Chapter 2. !E.1.1 Let {𝑥𝑖 } ⊂ B (H ) be a directed set. Then, 𝑠𝑜

wo

𝑥𝑖 → 0 ⇔ 𝑥𝑖∗ 𝑥𝑖 → 0. *E.1.2 Let 𝜑 be a wo-continuous linear form on B (H ). Then there exist two families of mutually orthogonal vectors {𝜉1 , … , 𝜉𝑛 }, {𝜂1 , … , 𝜂𝑛 } ⊂ H , such that 𝑛

𝜑 = ∑ 𝜔𝜉𝑘 ,𝜂𝑘 , 𝑘=1 𝑛

‖𝜑‖ = ∑ ‖𝜉𝑘 ‖ ‖𝜂𝑘 ‖. 𝑘=1

E.1.3 With the help of E.1.2, show that for any w-continuous linear form 𝜑 on ∞ B (H ), there exist two sequences {𝜉𝑘 }, {𝜂𝑘 ‖ ⊂ H , such that ∑𝑘=1 ‖𝜉𝑘 ‖2 < ∞ +∞, ∑𝑘=1 ‖𝜂𝑘 ‖2 < +∞ and ∞

𝜑 = ∑ 𝜔𝜉𝑘 ,𝜂𝑘 . 𝑘=1

In particular, the w-topology in B (H ) is defined by the system of semi-norms ∞

𝑥 ↦ | ∑ (𝑥𝜉𝑘 |𝜂𝑘 )|, 𝑘=1 ∞

∞

where {𝜉𝑘 }𝑘 , {𝜂𝑘 }𝑘 ⊂ H , ∑𝑘=1 ‖𝜉𝑘 ‖2 < +∞, ∑𝑘=1 ‖𝜂𝑘 ‖2 < +∞. E.1.4 The ultrastrong topology on B (H ) is defined by the system of semi-norms ∞

𝑥 → ( ∑ ‖𝑥𝜉𝑘 ‖2 )1/2 , 𝑘=1 ∞

where {𝜉𝑘 }𝑘 ⊂ H , ∑𝑘=1 ‖𝜉𝑘 ‖2 < +∞. ▶

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Show that in the closed unit ball of B (H ), the ultrastrong topology coincides with the strong topology. Prove the following relations between the indicated topologies: ≤

weak topology ∧⧵

strong topology ∧⧵

≤

ultraweak topology

ultrastrong topology ≤ norm topology

!E.1.5 Show that the *-mapping B (H ) ∋ 𝑥 ↦ 𝑥 ∗ ∈ B (H ) is weakly, ultraweakly and norm continuous. E.1.6 Let H be an infinitely dimensional Hilbert space and {𝜉𝑛 } an orthonormal sequence in H . One defines the following operators: 𝑣𝑛 ∶ 𝜉 ↦ (𝜉|𝜉𝑛 )𝜉1 , 𝑛 = 1, 2, … Show that 𝑣𝑛 → 0 in the ultrastrong topology, but, for any 𝑛, one has = 1. Infer that

‖𝑣𝑛∗ 𝜉1 ‖

(i) the *-operator is not strongly (resp., ultrastrongly) continuous on B (H ). (ii) the strong (resp., ultrastrong) topology is strictly finer (i.e., stronger) than the weak (resp., ultraweak) topology in B (H ). (iii) the norm topology is strictly finer than the ultrastrong topology in B (H ). !E.1.7 Show that, for any 𝑎 ∈ B (H ), the mappings B (H ) ∋ 𝑥 ↦ 𝑎𝑥 ∈ B (H ) B (H ) ∋ 𝑥 ↦ 𝑥𝑎 ∈ B (H ) are weakly, strongly, ultraweakly and ultrastrongly continuous. !E.1.8 Which of the following mappings B (H )1 × B (H ) ∋ (𝑥, 𝑦) ↦ 𝑥𝑦 ∈ B (H ) B (H ) × B (H )1 ∋ (𝑥, 𝑦) ↦ 𝑥𝑦 ∈ B (H ) is strongly and ultrastrongly continuous? ▶

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Topologies on spaces of operators

13

E.1.9 Let V be an ultrastrong neighbourhood of 0 ∈ B (H ). With the help of E.1.6 (iii), show that there exists an 𝜉 ∈ H , such that sup{‖𝑥𝜉‖; 𝑥 ∈ V } = +∞. Infer from this that, by endowing the space B (H ) × B (H ) with the product of the ultrastrong topologies and B (H ) with the weak topology, the mapping B (H ) × B (H ) ∋ (𝑥, 𝑦) ↦ 𝑥𝑦 ∈ B (H ) is not continuous. E.1.10 Let H be an infinitely dimensional Hilbert space and {𝜉𝑛 } an orthonormal sequence in H . We define the operators 𝑒𝑛 ∶ 𝜉 → (𝜉|𝜉𝑛 )𝜉𝑛 , 𝑛 = 1, 2, … 𝑥𝑛,𝑚 = 𝑒𝑛 + 𝑛𝑒𝑚 , 𝑚, 𝑛 = 1, 2, … Show that 0 is ultrastrongly adherent to the set {𝑥𝑛,𝑚 }𝑛,𝑚=1,2,…, but no sequence in this set converges weakly to 0. Infer from this fact that B (H ) is not metrizable with respect to any of the topologies – weak, strong, ultraweak and ultrastrong.

COMMENTS C.1.1 If X is a Banach space and X ∗ is its dual, then for any 𝜆 > 0, we write X ∗𝜆 = {𝜑 ∈ X ∗ ; ‖𝜑‖ ≤ 𝜆}. ̆ Theorem (Krein–Smulian). Let X be a Banach space and let K ⊂ X ∗ be a convex subset. Then K is 𝜍(X ∗ ; X )-closed iff for any 𝜆 > 0, the set K ∩ X ∗𝜆 is 𝜍(X ∗ ; X )-closed. In X ∗ , one defines the 𝑏𝜍(X ∗ ; X )-topology as being the finest topology in X ∗ , which coincides with the 𝜍(X ∗ ; X )-topology in X ∗𝜆 , 𝜆 > 0. One shows that the 𝑏𝜍(X ∗ ; X )-topology is a locally convex topology (namely, the topology of uniform convergence on the sequences from X that converge ̆ to 0). This is the main fact needed in the proof of the Krein–Smulian theorem. Indeed, this fact once established, the theorem easily follows by taking into account Lemma 1.2 (iv) (one takes F = F = X and E = X ∗ ) and Mackey’s theorem.

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14

Lectures on von Neumann algebras ̆ For the full proof of the Krein–Smulian theorem, we refer the reader to N. Dunford and J. Schwartz (1958/1963/1970), Section V.5.7. ̆ By taking into account Theorem 1.10, from the Krein–Smulian theorem, the following results can be obtained: Corollary. Let M ⊂ B (H ) be a w-closed vector subspace. A convex subset K ⊂ M is w-closed iff, for any 𝜆 > 0, the set K ∩ M 𝜆 is w-closed. Corollary. Let M ⊂ B (H ) be a vector subspace. Then M is w-closed iff M 1 is wo-compact. If M ⊂ B (H ) is a *-algebra, the result stated in this corollary will be proved by another method in 3.11.

C.1.2 With the help of the Tikhonov theorem, one can easily prove the following general result of ‘weak’ compacity, proved for the first time by R. V. Kadison: Let X , X ′ be vector spaces, F a set of linear forms on X ′ , which separate the elements of X ′ and L (X , X ′ ) the vector space of all linear mappings from X into X ′ , endowed with the topology 𝜍, generated by the sets D (𝑥, V ′ ) = {𝐴 ∈ L (X , X ′ ); 𝐴𝑥 ∈ V ′ }, where 𝑥 runs over X , whereas V ′ runs over the set of all 𝜍(X ′ ; F )-open subsets of X ′ . Theorem. If the subset S ⊂ X linearly generates X , whereas S ′ ⊂ X ′ is a 𝜍(X ′ ; F )-compact subset, then the set 𝔊 = {𝐴 ∈ L (X , X ′ ); 𝐴 S ⊂ S ′ } is 𝜍-compact. Corollary. If (S 𝑖 )𝑖∈𝐼 , is a family of subsets of S , whereas (S ′𝑖 )𝑖∈𝐼 is a family of 𝜍(X ′ , F )-compact subsets of S ′ , then the set 𝔊0 = {𝐴 ∈ 𝔊; 𝐴S 𝑖 ⊂ S ′𝑖 , 𝑖 ∈ 𝐼} is 𝜍-compact. Alaoglu’s theorem and Theorem 1.8 are particular cases of this result. For the proof of the theorem and for other applications, we refer the reader to R. V. Kadison (1961) (see also Arsene [1973]). C.1.3 Bibliographical comments. The reader can easily find the few general facts from functional analysis needed in the treatise by N. Dunford and J. Schwartz (1958/1963/1970, Chaps. II and V ). The preceding exposition follows that of J. R. Ringrose (1972b).

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2

BOUNDED LINEAR OPERATORS IN HILBERT SPACES

This chapter contains the basic facts about bounded linear operators in Hilbert spaces, which are necessary in order to develop the elementary part of the theory of von Neumann algebras. 2.1. In the first chapter, we considered only the Banach space structure of B (H ). By taking into consideration the multiplication of the operators, B (H ) becomes a Banach algebra, i.e., for any 𝑥, 𝑦 ∈ B (H ), we have ‖𝑥𝑦‖ ≤ ‖𝑥‖‖𝑦‖. For any 𝑥 ∈ B (H ), the relation (𝑥𝜉|𝜂) = (𝜉|𝑥 ∗ 𝜂), 𝜉, 𝜂 ∈ H determines an operator 𝑥 ∗ ∈ B (H ), called the adjoint of 𝑥. The mapping B (H ) ∋ 𝑥 ↦ 𝑥 ∗ ∈ B (H ) is called the canonical involution on B (H ) or the *-operation on B (H ). Thus, B (H ) becomes, in a canonical manner, an involutive algebra or a *-algebra, i.e., for any 𝑥, 𝑦 ∈ B (H ), 𝜆 ∈ ℂ, we have (𝑥 + 𝑦)∗ = 𝑥 ∗ + 𝑦 ∗ , ̄ ∗, (𝜆𝑥)∗ = 𝜆𝑥 (𝑥𝑦)∗ = 𝑦 ∗ 𝑥 ∗ , 𝑥∗∗ = 𝑥. We shall call a *-algebra of operators any *-subalgebra of B (H ). The notions of *-homomorphism and *-isomorphism between *-algebras of operators are now obvious. The connection existing among the norm, the multiplication, and the *-operation in B (H ) is expressed by the following:

15

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Lemma. For any 𝑥 ∈ B (H ), we have the equality: ‖𝑥 ∗ 𝑥‖ = ‖𝑥‖2 . Proof. We have ‖𝑥 ∗ ‖ = sup ‖𝑥 ∗ 𝜉‖ = sup sup |(𝑥∗ 𝜉|𝜂)| = sup sup |(𝜉|𝑥𝜂)| ≤ ‖𝑥‖, ‖𝜉‖=1

‖𝜉‖=1 ‖𝜂‖=1

‖𝜉‖=1 ‖𝜂‖=1

which shows that ‖𝑥‖2 = sup ‖𝑥𝜉‖2 = sup (𝑥𝜉|𝑥𝜉) = sup (𝑥 ∗ 𝑥𝜉|𝜉) ≤ ‖𝑥 ∗ 𝑥‖ ≤ ‖𝑥‖2 . ‖𝜉‖=1

‖𝜉‖=1

‖𝜉‖=1

Q.E.D. In the preceding proof, we have also obtained the equality ‖𝑥 ∗ ‖ = ‖𝑥‖, 𝑥 ∈ B (H ). 2.2. Any Banach algebra A , which is a *-algebra and in which the equality ‖𝑥 ∗ 𝑥‖ = ‖𝑥‖2 holds for any 𝑥 ∈ A , is called a 𝐶 ∗ -algebra. Lemma 2.2 shows that B (H ) is a 𝐶 ∗ -algebra in a canonical manner. Moreover, any *-algebra of operators, which is also closed for the norm topology, is a 𝐶 ∗ -algebra; these algebras will be called 𝐶 ∗ -algebras of operators (or concrete 𝐶 ∗ -algebras). For any subset I ⊂ B (H ), there exists a smallest 𝐶 ∗ -algebra of operators, which contains I . This one will be called the 𝐶 ∗ -algebra generated by I and it will be denoted by C ∗ (I ). A special class of 𝐶 ∗ -algebras of operators, which is the subject of this book, is the class of von Neumann algebras. Any *-algebra of operators M ⊂ B (H ), which contains the identity operator 1H and which is so-closed (or, equivalently, wo-closed [see 1.5.]) is called a von Neumann algebra. For any subset I ⊂ B (H ), there exists a smallest von Neumann algebra that contains I ; it will be called the von Neumann algebra generated by I , and it will bc denoted by R (I ). In what follows, we shall be concerned only with 𝐶 ∗ -algebras of operators, which will be called, simply, 𝐶 ∗ -algebras. 2.3. For any 𝑥 ∈ B (H ), its resolvent set is defined by 𝜌(𝑥) = {𝜆 ∈ ℂ; 𝜆 − 𝑥 is invertible}; its complement 𝜎(𝑥) = ℂ ⧵ 𝜌(𝑥) is the spectrum of 𝑥. It is easy to check that if 𝜆0 ∈ 𝜌(𝑥), then {𝜆 ∈ ℂ; |𝜆 − 𝜆0 | < ‖(𝜆0 − 𝑥)−1 ‖−1 } ⊂ 𝜌(𝑥), Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

17

Bounded linear operators in Hilbert spaces and, for any 𝜆 such that |𝜆 − 𝜆0 | < ‖(𝜆0 − 𝑥)−1 ‖−1 , we have ∞

(𝜆 − 𝑥)−1 = ∑ (𝜆0 − 𝜆)𝑛 (𝜆0 − 𝑥)−𝑛−1 . 𝑛=0

In particular, 𝜌(𝑥) is an open subset of ℂ, and the function 𝜌(𝑥) ∋ 𝜆 ↦ (𝜆 − 𝑥)−1 ∈ B (H ) is analytic for the norm topology of B (H ). On the other hand, we have {𝜆 ∈ ℂ; |𝜆| > ‖𝑥‖} ⊂ 𝜌(𝑥), and for any 𝜆, such that |𝜆| > ‖𝑥‖, we have ∞

(𝜆 − 𝑥)−1 = ∑ 𝜆−𝑛−1 𝑥 𝑛 . 𝑛=1

In particular, we have 𝜎(𝑥) ⊂ {𝜆 ∈ ℂ; |𝜆| ⩽ ‖𝑥‖}, which shows that 𝜎(𝑥) is a compact subset of ℂ. Lemma. For any 𝑥 ∈ B (H ), the spectrum 𝜎(𝑥) is non-empty, the sequence (‖𝑥𝑛 ‖1/𝑛 )𝑛≥1 converges and lim ‖𝑥 𝑛 ‖1/𝑛 = sup{|𝜆|; 𝜆 ∈ 𝜎(𝑥)}. 𝑛→∞

Proof. Let 𝛼(𝑥) = inf𝑛>0 ‖𝑥 𝑛 ‖1/𝑛 . It is easy to check that if 𝑥, 𝑦 ∈ B (H ) and 𝑥𝑦 = 𝑦𝑥, then 𝛼(𝑥𝑦) ≤ 𝛼(𝑥)𝛼(𝑦). Let 𝜀 > 0. There exists a natural number 𝑛𝜀 , such that ‖𝑥 𝑛𝜀 ‖1/𝑛𝜀 ≤ 𝛼(𝑥) + 𝜀. For any 𝑛 > 0, there exist natural numbers 𝑞, 𝑟, which are uniquely determined by the condition: 𝑛 = 𝑛𝜀 𝑞 + 𝑟, 0 ≤ 𝑟 ≤ 𝑛𝜀 − 1. We have ‖𝑥 𝑛 ‖ = ‖𝑥 𝑛𝜀 𝑞 𝑥 𝑟 ‖ ≤ ‖𝑥 𝑛𝜀 ‖𝑞 ‖𝑥‖𝑟 ≤ (𝛼(𝑥) + 𝜀)𝑛𝜀 𝑞 ‖𝑥‖𝑟 = (𝛼(𝑥) + 𝜀)𝑛−𝑟 ‖𝑥‖𝑟 , and therefore, we have ‖𝑥 𝑛 ‖1/𝑛 ≤ (𝛼(𝑥) + 𝜀)

1−

𝑟 𝑛

𝑟

‖𝑥‖ 𝑛 ;

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Lectures on von Neumann algebras

consequently, we can write 𝛼(𝑥) ≤ lim inf ‖𝑥 𝑛 ‖1/𝑛 ≤ lim sup ‖𝑥 𝑛 ‖1/𝑛 ≤ 𝛼(𝑥) + 𝜀. 𝑛→∞

𝑛→∞ 1

Since 𝜀 > 0 was arbitrary, it follows that the limit lim𝑛→∞ ‖𝑥 𝑛 ‖ 𝑛 exists, and it is equal to 𝛼(𝑥). ∞ ∞ If |𝜆| > 𝛼(𝑥), then ∑𝑛=0 ‖𝑥𝑛 ‖/𝜆𝑛 < +∞, which implies that the series ∑𝑛=1 𝑥 𝑛 /𝜆𝑛 converges with respect to the norm. It follows that 1 − (𝑥/𝜆) is invertible, which implies that 𝜆 − 𝑥 is invertible. Consequently, we have 𝛼(𝑥) ≥ sup{|𝜆|; 𝜆 ∈ 𝜎(𝑥)}. In order to prove the reversed inequality, as well as the fact that the spectrum 𝜎(𝑥) is non-empty, we distinguish two cases. If 𝛼(𝑥) = 0, then 0 ∈ 𝜎(𝑥), i.e., 𝑥 is not invertible. Indeed, if 𝑥 is invertible, then 1 = 𝛼(1) = 𝛼(𝑥𝑥 −1 ) ≤ 𝛼(𝑥)𝛼(𝑥 −1 ) = 0, which is a contradiction. If 𝛼(𝑥) > 0, let us assume that 𝛼(𝑥) > sup{|𝜆|; 𝜆 ∈ 𝜎(𝑥)}, in order to get a contradiction. Since 𝜎(𝑥) is a compact set, there exists an 𝑟 ∈ (0, 𝛼(𝑥)), such that 𝜎(𝑥) ⊂ {𝜆 ∈ ℂ; |𝜆| ≤ 𝑟}. Consequently, we have 𝐷 = {𝜆 ∈ ℂ; |𝜆| > 𝑟} ⊂ 𝜌(𝑥). It is easily checked that for any bounded linear form 𝜑 on B (H ), the function 𝜆 ↦ 𝜑((𝜆 − 𝑥)−1 ) is analytic in 𝐷. Moreover, for |𝜆| > 𝛼(𝑥), we have ∞

𝜑((𝜆 − 𝑥)−1 ) = ∑ 𝜆−𝑛−1 𝜑(𝑥 𝑛 ). 𝑛=0

It follows that by the formula 𝑓(𝜇) = {

0

for 𝜇 = 0 1

−1

𝜑(( − 𝑥) ) 𝜇

for 0 < |𝜇| < 𝑟−1

,

one defines an analytic function in the set 𝐷 −1 = {𝜇 ∈ ℂ; |𝜇| < 𝑟−1 }. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

19

Bounded linear operators in Hilbert spaces Since the Taylor expansion of 𝑓 at 0 is ∞

𝑓(𝜇) = ∑ 𝜇𝑛+1 𝜑(𝑥 𝑛 ), 𝑛=0

the same formula holds for any 𝜇 ∈ 𝐷 −1 . −1 Now let 𝜆0 ∈ ℂ be such that 𝑟 < |𝜆0 | < 𝛼(𝑥). Then 𝜆−1 0 ∈ 𝐷 , and therefore, for any bounded linear form 𝜑 on B (H ), we have lim 𝜆−𝑛−1 𝜑(𝑥 𝑛 ) = 0. 0

𝑛→∞

From the Banach–Steinhaus theorem, we infer that sup |𝜆0 |−𝑛−1 ‖𝑥 𝑛 ‖ = 𝑐 < +∞. 𝑛>0

Consequently, for any 𝑛 > 0, we have ‖𝑥 𝑛 ‖ ≤ 𝑐|𝜆0 |𝑛+1 , whence 1+

𝛼(𝑥) = lim ‖𝑥 𝑛 ‖1/𝑛 ≤ lim 𝑐1/𝑛 |𝜆0 | 𝑛→∞

1 𝑛

𝑛→∞

= |𝜆0 | < 𝛼(𝑥), Q.E.D.

which is a contradiction. One usually denotes 1

|𝜎(𝑥)| = sup{|𝜆|; 𝜆 ∈ 𝜎(𝑥)} = lim ‖𝑥 𝑛 ‖ 𝑛 , 𝑛→∞

and the number |𝜎(𝑥)| is called the spectral radius of 𝑥. Obviously, we have |𝜎(𝑥)| ≤ ‖𝑥‖, and for any 𝑥, 𝑦 ∈ B (H ), such that 𝑥𝑦 = 𝑦𝑥, we have |𝜎(𝑥𝑦)| ≤ |𝜎(𝑥)| ⋅ |𝜎(𝑦)|. It is easily seen that 𝜆 ∈ 𝜎(𝑥 ∗ ) ⇔ 𝜆 ̄ ∈ 𝜎(𝑥); consequently, we have |𝜎(𝑥 ∗ )| = |𝜎(𝑥)|. 2.4. Now let 𝑥 ∈ B (H ) and 𝑝(𝜆) = 𝛼0 + 𝛼1 𝜆 + … + 𝛼𝑛 𝜆𝑛 be a polynomial. One defines 𝑝(𝑥) = 𝛼0 + 𝛼1 𝑥 + … + 𝛼𝑛 𝑥 𝑛 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

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Lectures on von Neumann algebras

Lemma. With the above notations, we have 𝜎(𝑝(𝑥)) = {𝑝(𝜆); 𝜆 ∈ 𝜎(𝑥)}. Proof. If 𝑝 is a constant, the assertion of the lemma is obvious. Therefore, let 𝑛 ≥ 1, 𝛼𝑛 ≠ 0. If 𝜆0 ∈ 𝜎(𝑥), then 𝜆0 − 𝑥 is not invertible, whereas from 𝑛

𝑛

𝑘−1

𝑗 𝑝(𝜆0 ) − 𝑝(𝑥) = ∑ 𝛼𝑘 (𝜆𝑘0 − 𝑥 𝑘 ) = (𝜆0 − 𝑥) ∑ 𝛼𝑘 ∑ 𝜆𝑘−1 0 𝑥 , 𝑘=0

𝑘=1

𝑗=1

we infer that 𝑝(𝜆0 )−𝑝(𝑥) is not invertible too. It follows that 𝑝(𝜆0 ) ∈ 𝜎(𝑝(𝑥)). Conversely, if 𝜇 ∉ {𝑝(𝜆); 𝜆 ∈ 𝜎(𝑥)}, and if 𝜆1 , … , 𝜆𝑛 are the zeros of the polynomial 𝑝(𝜆) − 𝜇, then 𝜆1 , … , 𝜆𝑛 ∉ 𝜎(𝑥). Since 𝑝(𝑥) − 𝜇 = 𝛼𝑛 (𝑥 − 𝜆1 ) … (𝑥 − 𝜆𝑛 ), it follows that 𝑝(𝑥) − 𝜇 is invertible, which shows that 𝜇 ∉ 𝜎(𝑝(𝑥)).

Q.E.D.

2.5. An operator 𝑥 ∈ B (H ) is said to be normal if 𝑥𝑥 ∗ = 𝑥 ∗ 𝑥. It is easy to see that if 𝑥 ∈ B (H ) is normal, then, for any 𝜉 ∈ H , we have ‖𝑥 ∗ 𝜉‖ = ‖𝑥𝜉‖, and conversely. The operator 𝑥 ∈ B (H ) is said to be self-adjoint or hermitian if 𝑥 ∗ = 𝑥. Obviously, any hermitian operator is normal. Lemma. Let 𝑥 ∈ B (H ). Then (i) if 𝑥 is normal, |𝜎(𝑥)| = ‖𝑥‖. (ii) if 𝑥 is self-adjoint, 𝜎(𝑥) ⊂ ℝ. Proof. If 𝑥 ∈ B (H ) is self-adjoint, then 𝑛

−𝑛

|𝜎(𝑥)| = lim ‖𝑥 2 ‖2 𝑛→∞

= ‖𝑥‖,

from Lemma 2.1. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

Bounded linear operators in Hilbert spaces

21

If 𝑥 ∈ B (H ) is normal, then 𝑥 ∗ 𝑥 is self-adjoint, 𝑥 and 𝑥 ∗ commute, and therefore, by taking into account what we have proved in Sections 2.1 and 2.3, we have ‖𝑥‖2 = ‖𝑥 ∗ 𝑥‖ = |𝜎(𝑥 ∗ 𝑥)| ≤ |𝜎(𝑥 ∗ | |𝜎(𝑥)| ≤ ‖𝑥 ∗ ‖ ‖𝑥‖ = ‖𝑥‖2 , whence |𝜎(𝑥)| = ‖𝑥‖. Now, if 𝑥 ∈ B (H ) is self-adjoint and 𝜆 = 𝛼 + i𝛽 ∈ 𝜎(𝑥), 𝛼, 𝛽 ∈ ℝ, then for any 𝑛 ≥ 1, the operator 𝑥𝑛 = 𝑥 − 𝛼 + i𝑛𝛽 is normal and i(𝑛+1)𝛽 ∈ 𝛼(𝑥𝑛 ). By taking into account what we have proved in Section 2.3, we have (𝑛 + 1)2 𝛽 2 ≤ |𝜎(𝑥𝑛 )|2 ≤ ‖𝑥𝑛 ‖2 = ‖𝑥𝑛∗ 𝑥𝑛 ‖ = ‖(𝑥 − 𝛼 − i𝑛𝛽)(𝑥 − 𝛼 + i𝑛𝛽)‖ = ‖(𝑥 − 𝛼)2 + 𝑛2 𝛽 2 ‖ = sup sup (((𝑥 − 𝛼)2 𝜉|𝜂) + (𝑛2 𝛽 2 𝜉|𝜂)) ‖𝜂‖=1 ‖𝜉‖=1

≤ ‖𝑥 − 𝛼‖2 + 𝑛2 𝛽 2 . Since 𝑛 ≥ 1 is arbitrary, we have 𝛽 = 0, which shows that 𝜆 ∈ ℝ.

Q.E.D.

2.6. The following theorem will enable us to construct ‘convenient’ elements from B (H ). For any compact space Ω, we shall denote by C (Ω) the set of all continuous complex functions, which are defined on Ω. With the pointwise defined algebraic operations and with the *-operation defined by complex conjugation, the set C (Ω) becomes canonically a commutative 𝐶 ∗ -algebra, if we endow it with the uniform norm. Any element from C (Ω) is ‘normal’, whereas the ‘self-adjoint’ elements are the real functions. The ‘spectrum’ of an element 𝑓 ∈ C (Ω) coincides with the range 𝑓(Ω) of the function 𝑓. Theorem (of operational calculus with continuous functions). Let 𝑥 ∈ B (H ) be a self–adjoint operator. Then there exists a unique mapping C (𝜎(𝑥)) ∋ 𝑓 ↦ 𝑓(𝑥) ∈ B (H ), such that (i) if 𝑓 is a polynomial, 𝑓(𝜆) = 𝛼0 + 𝛼1 𝑥 + … + 𝛼𝑛 𝜆𝑛 , then 𝑓(𝑥) = 𝛼0 + 𝛼1 𝑥 + … + 𝛼𝑛 𝑥𝑛 . (ii) ‖𝑓(𝑥)‖ = ‖𝑓‖, for any 𝑓 ∈ C (𝜎(𝑥)). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

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Moreover, this mapping is a *-isomorphism of the 𝐶 ∗ -algebra C (𝜎(𝑥)) onto the 𝐶 ∗ - algebra C ∗ ({𝑥, 1}). Proof. The set of all polynomials can be mapped by restriction, which is a *-homomorphism, onto a *-subalgebra P (𝜎(𝑥)) in C (𝜎(𝑥)). By taking into account the Stone–Weierstrass theorem, we infer that P (𝜎(𝑥)) is dense in C (𝜎(𝑥)) for the norm topology. For any ‘polynomial’ 𝑝 ∈ C (𝜎(𝑥)), we define 𝑝(𝑥) as we have done in Section 2.4. [By using the Lemmas from 2.4 and 2.5, it is easy to show that if 𝑝 is a polynomial and 𝑝|𝜎(𝑥) = 0, then 𝑝(𝑥) = 0; this shows that 𝑝(𝑥) is correctly defined for any 𝑝 ∈ P (𝜎(𝑥)) (translator’s note).] The set {𝑝(𝑥); 𝑝 ∈ P (𝜎(𝑥))} is a *-subalgebra, and it is dense in C ∗ ({𝑥, 1}) with respect to the norm topology. For any 𝑝 ∈ P (𝜎(𝑥)), we have ‖𝑝(𝑥)‖ = |𝜎(𝑝(𝑥))| = sup{|𝜇|; 𝜇 ∈ 𝜎(𝑝(𝑥))} = sup{|𝑝(𝜆)|; 𝜆 ∈ 𝜎(𝑥)} = ‖𝑝‖. Consequently, the mapping P (𝜎(𝑥)) ∋ 𝑝 ↦ 𝑝(𝑥) ∈ B (H ) is correctly defined and isometric. Thus, there exists a unique isometric extension of this mapping to C (𝜎(𝑥)), which proves the existence and the uniqueness of the mapping having properties (i) and (ii). The relations (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥), (𝑓𝑔)(𝑥) = 𝑓(𝑥)𝑔(𝑥), (𝜆𝑓)(𝑥) = 𝜆𝑓(𝑥), ̄ 𝑓(𝑥) = (𝑓(𝑥))∗ ,

𝑓, 𝑔 ∈ C (𝜎(𝑥)), 𝑓, 𝑔 ∈ C (𝜎(𝑥)),

𝜆 ∈ ℂ, 𝑓 ∈ C (𝜎(𝑥)), 𝑓 ∈ C (𝜎(𝑥)),

are easy to prove, first for polynomials, and then by tending to the limit, for arbitrary continuous functions. This shows that the mapping just defined is indeed a *-isomorphism of the 𝐶 ∗ -algebra C (𝜎(𝑥)) onto the 𝐶 ∗ -algebra C ∗ ({𝑥, 1}). Q.E.D. If 𝑥, 𝑦 ∈ B (H ) are commuting self-adjoint operators, 𝑥𝑦 = 𝑦𝑥, and 𝑓 ∈ C (𝜎(𝑥)), 𝑔 ∈ C (𝜎(𝑦)), then 𝑓(𝑥) and 𝑔(𝑦) are commuting normal operators: 𝑓(𝑥)𝑔(𝑦) = 𝑔(𝑦)𝑓(𝑥); indeed, this fact is obvious if 𝑓 and 𝑔 are polynomials; for the general case, one tends to the limit. If 𝑓(0) = 0, then 𝑓(𝑥) ∈ C ∗ ({𝑥}), because in this case, 𝑓 can be approximated by polynomials without constant terms. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

Bounded linear operators in Hilbert spaces

23

2.7. By taking into account the isomorphism we have just obtained, as well as its uniqueness, one can immediately get the following: Corollary. Let 𝑥 ∈ B (H ) be a self-adjoint operator. For any 𝑓 ∈ C (𝜎(𝑥)), we have 𝜎(𝑓(𝑥)) = {𝑓(𝜆); 𝜆 ∈ 𝜎(𝑥)}, whereas if 𝑓 is real and 𝑔 ∈ C (𝜎(𝑓(𝑥))), then 𝑔(𝑓(𝑥)) = (𝑔 ∘ 𝑓)(𝑥). Corollary 2.8. If 𝑥 ∈ B (H ) is an invertible operator, then 𝑥−1 ∈ C ∗ ({𝑥, 1}). Proof. If 𝑥 is self-adjoint, the statement immediately follows from Theorem 2.6. Now let 𝑥 ∈ B (H ) be any invertible operator, and 𝑦 = 𝑥−1 . Then 𝑦 ∗ = (𝑥 ∗ )−1 , 𝑦𝑦 ∗ = (𝑥 ∗ 𝑥)−1 , and therefore, 𝑦𝑦 ∗ ∈ C ∗ ({𝑥 ∗ 𝑥, 1}) ⊂ C ∗ ({𝑥, 1}). Since 𝑥 ∗ ∈ C ∗ ({𝑥, 1}), we have 𝑥 −1 = 𝑦 = 𝑦(𝑦 ∗ 𝑥 ∗ ) = (𝑦𝑦 ∗ )𝑥∗ ∈ C ∗ ({𝑥, 1}). Q.E.D. 2.9. An operator 𝑥 ∈ B (H ) is said to be positive if it is self-adjoint and 𝜎(𝑥) ⊂ ℝ+ = {𝜆 ∈ ℝ; 𝜆 ≥ 0}. Corollary. If 𝑥 ∈ B (H ) is a positive operator, then there exists a unique positive operator 𝑎 ∈ B (H ), such that 𝑎2 = 𝑥. Proof. Since we have the inclusion 𝜎(𝑥) ⊂ ℝ+ , the function defined by 𝑓(𝜆) = 𝜆1/2 , 𝜆 ∈ 𝜎(𝑥), belongs to C (𝜎(𝑥)). From Corollary 2.7, by denoting 𝑎 = 𝑓(𝑥), we have 𝑎2 = 𝑥 and 𝑎 is a positive operator. Let 𝑏 ∈ B (H ) be an arbitrary positive operator, such that 𝑏2 = 𝑥. Let us consider a sequence {𝑝𝑛 } of polynomials that converges uniformly on 𝜎(𝑥) to the function 𝑓 we have just defined. Since 𝜎(𝑏) ⊂ ℝ+ , from 𝜎(𝑥) = {𝜆2 ; 𝜆 ∈ 𝜎(𝑏)}, we infer that the polynomials 𝑝𝑛 (𝜆2 ) converge uniformly on 𝜎(𝑏) to the identical function. By taking into account Theorem 2.6, we have lim ‖𝑏 − 𝑝𝑛 (𝑥)‖ = lim ‖𝑏 − 𝑝𝑛 (𝑏2 )‖ = 0,

𝑛→∞

𝑛→∞

which shows that 𝑏 = lim𝑛→∞ 𝑝𝑛 (𝑥) = 𝑓(𝑥) = 𝑎.

Q.E.D.

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The unique positive operator 𝑎 ∈ B (H ), such that 𝑎2 = 𝑥 is denoted by 𝑥 1/2 . We observe that 𝑥1/2 ∈ C ∗ ({𝑥}). Corollary 2.10. Let 𝑥 ∈ B (H ) be a self-adjoint operator. Then there exist positive operators 𝑎, 𝑏 ∈ B (H ), such that 𝑥 = 𝑎 − 𝑏, 𝑎𝑏 = 0, and they are uniquely determined by these conditions. Proof. Let us consider on 𝜎(𝑥) the continuous functions defined by the equalities 𝑓(𝜆) = {

𝑔(𝜆) = {

𝜆

for

𝜆≥0

0

for

𝜆≤0

0

for

𝜆≥0

−𝜆

for

𝜆 ≤ 0.

Then the operators 𝑎 = 𝑓(𝑥), 𝑏 = 𝑔(𝑥) satisfy the two conditions required in the corollary, which proves the existence part of the corollary. In order to prove the uniqueness, we observe that (𝑎 + 𝑏)2 = 𝑥 2 . By taking into account a remark we made in Section 2.6, we get 𝑎 + 𝑏 = (𝑎1/2 + 𝑏1/2 )2 . This shows that 𝑎+𝑏 and 𝑥 2 are positive operators (see 2.5 and 2.7), and from Corollary 2.9, we infer that 𝑎 + 𝑏 = (𝑥 2 )1/2 . This implies that

1 1 𝑎 = ((𝑥2 )1/2 + 𝑥), 𝑏 = ((𝑥 2 )1/2 − 𝑥). 2 2 Q.E.D.

The operators 𝑎 and 𝑏, given by this corollary, are denoted: 𝑎 = 𝑥 + , 𝑏 = 𝑥 − . We observe that 𝑥 + , 𝑥 − ∈ C ∗ ({𝑥}). Thus, we have 𝑥 = 𝑥∗ − 𝑥− 𝑥 + 𝑥 − = 0.

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Bounded linear operators in Hilbert spaces

25

Lemma 2.11. Let 𝑥 ∈ B (H ). Then, (i) 𝑥 = 0 ⇔ (𝑥𝜉|𝜉) = 0 for any 𝜉 ∈ H . (ii) 𝑥 is self-adjoint ⇔ (𝑥𝜉|𝜉) ∈ ℝ for any 𝜉 ∈ H . Proof. It is easy to prove the following ‘polarization formula’: 4(𝑥𝜉|𝜂) = (𝑥(𝜉 + 𝜂)|𝜉 + 𝜂) − (𝑥(𝜉 − 𝜂)|𝜉 − 𝜂) +i(𝑥(𝜉 + i𝜂)|𝜉 + i𝜂) − i(𝑥(𝜉 − i𝜂)|𝜉 − i𝜂), which holds for any 𝜉, 𝜂 ∈ H . If (𝑥𝜉|𝜉) = 0 for any 𝜉 ∈ H , by taking into account the polarization formula, we infer that (𝑥𝜉|𝜂) = 0, for any 𝜉, 𝜂 ∈ H ; in particular, we have ‖𝑥𝜉‖2 = (𝑥𝜉|𝑥𝜉) = 0, for any 𝜉 ∈ H , which shows that 𝑥 = 0. If 𝑥 is self-adjoint, then for any 𝜉 ∈ H , (𝑥𝜉|𝜉) = (𝜉|𝑥𝜉) = (𝑥𝜉|𝜉), i.e., (𝑥𝜉|𝜉) is real. Conversely, if (𝑥𝜉|𝜉) is real, we have ((𝑥 − 𝑥 ∗ )𝜉|𝜉) = (𝑥𝜉|𝜉) − (𝑥∗ 𝜉|𝜉) = (𝑥𝜉|𝜉) − (𝜉|𝑥𝜉) = (𝑥𝜉|𝜉) − (𝑥𝜉|𝜉) = 0. With the first part of the lemma, we infer that 𝑥 ∗ = 𝑥.

Q.E.D.

2.12. The following proposition characterizes the positive operators: Proposition. For any operator 𝑥 ∈ B (H ), the following statements are equivalent: (i) 𝑥 is positive. (ii) There exists a positive operator 𝑎 ∈ B (H ), such that 𝑥 = 𝑎2 . (iii) There exists an operator 𝑦 ∈ B (H ), such that 𝑥 = 𝑦 ∗ 𝑦. (iv) (𝑥𝜉|𝜉) ≥ 0, for any 𝜉 ∈ H . Proof. The implication (i) ⇔ (ii) follows from Corollary 2.9, whereas the implications (ii) ⇒ (iii) ⇒ (iv) are trivial. Let us now assume that (𝑥𝜉|𝜉) ≥ 0 for any 𝜉 ∈ H . From Lemma 2.11, 𝑥 is self-adjoint. From Corollary 2.10, we infer that there exist positive operators 𝑥 + , 𝑥 − ∈ C ∗ ({𝑥}), such that 𝑥 = 𝑥 + − 𝑥 − , 𝑥 + 𝑥 − = 0. Then, for any 𝜉 ∈ H , we have 0 ≤ (𝑥(𝑥 − 𝜉)|𝑥 − 𝜉) = (𝑥 − 𝑥𝑥 − 𝜉|𝜉) = −((𝑥 − )3 𝜉|𝜉) = −(𝑥 − (𝑥− 𝜉)|𝑥 − 𝜉) ≤ 0,

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which shows that ((𝑥− )3 𝜉|𝜉) = 0, 𝜉 ∈ H . From Lemma 2.11, it follows that (𝑥− )3 = 0. By taking into account Theorem 2.6, we hence infer that 𝑥 − = 0. Consequently, 𝑥 = 𝑥 + is a positive operator. Q.E.D. 2.13. An operator 𝑒 ∈ B (H ) is said to be a projection if 𝑒∗ = 𝑒 and 𝑒2 = 𝑒. Any projection is a positive operator. If 𝑒 is a projection, then 1−𝑒 is a projection too; it is sometimes denoted by 𝑒⊥ (= 1 − 𝑒). If 𝑒 ∈ B (H ) is a projection, then 𝑒H is a closed subspace of H , called the projection subspace of 𝑒, whereas (1−𝑒)H is the orthogonal complement (𝑒H )⟂ of 𝑒H . Conversely, to any closed subspace S ⊂ H , there corresponds a unique projection 𝑒 ∈ B (H ), such that 𝑒H = S ; this projection will be called the projection on the subspace S , and it will also sometimes be denoted by S ; e.g., we have S ⟂ = 1 − S . The set of all projection in B (H ) will be denoted by PB (H ) . It is easily checked that if 𝑒 ∈ PB (H ) , 𝑒 ≠ 0, then ‖𝑒‖ = 1. Two projections 𝑒1 , 𝑒2 ∈ PB (H ) are said to be orthogonal if 𝑒1 𝑒2 = 0. The following statements are easily checked: 𝑒1 𝑒2 ∈ PB (H ) ⇔ 𝑒1 𝑒2 = 𝑒2 𝑒1 , 𝑒1 + 𝑒2 ∈ PB (H ) ⇔ 𝑒1 𝑒2 = 0. Let 𝑥 ∈ B (H ). We introduce the following notations: n(𝑥) = the projection on the kernel of 𝑥 ∶ {𝜉 ∈ H ; 𝑥𝜉 = 0}, l(𝑥) = the projection on the closure of the range of 𝑥 ∶ 𝑥H , r(𝑥) = 1 − n(𝑥). It is easily checked that r(𝑥) = l(𝑥 ∗ ), and l(𝑥) (resp., r(𝑥)) is the smallest projection 𝑒 ∈ B (H ), such that 𝑒𝑥 = 𝑥 (resp., 𝑥𝑒 = 𝑥). One says that l(𝑥) (resp., r(𝑥)) is the left support (resp., the right support) of 𝑥. If 𝑥 is self-adjoint, then the following notation is used: s(𝑥) = l(𝑥) = r(𝑥), and one says that s(𝑥) is the support of 𝑥. For any 𝑥 ∈ B (H ), the following relations hold: l(𝑥) = s(𝑥𝑥 ∗ ), r(𝑥) = s(𝑥 ∗ 𝑥). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

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An operator 𝑣 ∈ B (H ) is said to be a partial isometry if there exists a closed subspace S ⊂ H , such that ‖𝑣𝜉‖ = ‖𝜉‖, for any 𝜉 ∈ S , 𝑣𝜉 = 0, for any 𝜉 ∈ S ⟂ . The closed subspace S (resp., 𝑣S ) is called the initial subspace (resp., the final subspace) of the partial isometry 𝑣. It is easy to check that r(𝑣) = 𝑣 ∗ 𝑣, l(𝑣) = 𝑣𝑣 ∗ . The projection 𝑣∗ 𝑣 (resp., 𝑣𝑣∗ ) is called the initial projection (resp., the final projection) of the partial isometry 𝑣. The projection subspace of the initial (resp., of the final) projection of 𝑣 is just the initial (resp., the final) subspace of 𝑣. Conversely, if 𝑣 ∈ B (H ) and if 𝑣∗ 𝑣 is a projection, then 𝑣𝑣 ∗ is also a projection, whereas 𝑣 is a partial isometry whose initial subspace is S = (𝑣 ∗ 𝑣)H . Indeed, by taking into account the relation (𝑣 ∗ 𝑣)2 = 𝑣 ∗ 𝑣, it follows that ‖𝑣 − 𝑣𝑣 ∗ 𝑣‖2 = ‖(𝑣 ∗ − 𝑣 ∗ 𝑣𝑣∗ )(𝑣 − 𝑣𝑣 ∗ 𝑣)‖ = … = 0, whence 𝑣 = 𝑣𝑣 ∗ 𝑣 and (𝑣𝑣 ∗ )2 = 𝑣𝑣 ∗ . We then have 𝜉 ∈ S ⇒ 𝜉 = 𝑣 ∗ 𝑣𝜉 ⇒ ‖𝜉‖2 = (𝜉|𝜉) = (𝑣 ∗ 𝑣𝜉|𝜉) = ‖𝑣𝜉‖2 , 𝜉 ∈ S ⟂ ⇒ 0 = 𝑣 ∗ 𝑣𝜉 ⇒ 0 = 𝑣𝑣 ∗ 𝑣𝜉 = 𝑣𝜉. Theorem 2.14 (the polar decomposition). For any operator 𝑥 ∈ B (H ), there exists a unique positive operator 𝑎 ∈ B (H ) and a unique partial isometry 𝑣 ∈ B (H ), such that 𝑥 = 𝑣𝑎, 𝑣∗ 𝑣 = s(𝑎). Proof. We define 𝑎 = (𝑥 ∗ 𝑥)1/2 and the operator 𝑣0 on 𝑎H by the relation 𝑣0 (𝑎𝜉) = 𝑥𝜉,

𝜉∈H.

Since, for any 𝜉 ∈ H , we have ‖𝑣0 (𝑎𝜉)‖2 = ‖𝑥𝜉‖2 = (𝑥 ∗ 𝑥𝜉|𝜉) = (𝑎2 𝜉|𝜉) = ‖𝑎𝜉‖2 , it follows that the operator 𝑣0 can be extended, in a unique manner, to an isometric operator (i.e., one which conserves the norm), for which we shall keep the same notation, Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

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defined on the space 𝑎H = s(𝑎)H . We then define a partial isometry 𝑣 ∈ B (H ) by the relations: 𝑣𝜉 = {

𝑣0 𝜉

for

𝜉 ∈ s(𝑎)H ,

0

for

𝜉 ∈ (s(𝑎)H )⟂ .

It is now easy to check that 𝑥 = 𝑣𝑎, ∗

𝑣 𝑣 = s(𝑎), which establishes the existence part of the statement. In order to prove the uniqueness, let us remark that from the conditions of the statement, it follows that 𝑥 ∗ 𝑥 = 𝑎𝑣 ∗ 𝑣𝑎 = 𝑎s(𝑎)𝑎 = 𝑎2 , which implies that 𝑎 = (𝑥 ∗ 𝑥)1/2 . Then one can easily see that the partial isometry 𝑣 necessarily maps according to the above definition. Q.E.D. The operator 𝑎 = (𝑥 ∗ 𝑥)1/2 is called the absolute value (or the modulus) of 𝑥 and is denoted by |𝑥|. We remark that |𝑥| ∈ C ∗ ({𝑥}). The relations 𝑥 = 𝑣|𝑥|, 𝑣 ∗ 𝑣 = s(|𝑥|) are called the polar decomposition of 𝑥. 2.15. Let 𝑥 ∈ B (H ) and let 𝑥 = 𝑣|𝑥|, 𝑣 ∗ 𝑣 = s(|𝑥|) be the polar decomposition of 𝑥. It is easy to check the relations 𝑥∗ = 𝑣 ∗ (𝑣|𝑥|𝑣∗ ),

(𝑣 ∗ )∗ 𝑣∗ = s(𝑣|𝑥|𝑣 ∗ ),

and therefore, according to Theorem 2.14, they yield the polar decomposition of the operator 𝑥 ∗ . In particular, we have |𝑥 ∗ | = 𝑣|𝑥|𝑣 ∗ and 𝑥 = |𝑥 ∗ |𝑣,

𝑣𝑣 ∗ = s(|𝑥∗ |).

For this reason, one sometimes says that 2.14 is the left polar decomposition, whereas the preceding formulas yield the right polar decomposition of the operator 𝑥. It is easy to check the following relations: r(𝑥) = s(|𝑥|) = 𝑣 ∗ 𝑣, l(𝑥) = s(|𝑥∗ |) = 𝑣𝑣 ∗ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

29

Bounded linear operators in Hilbert spaces If 𝑥 ∈ B (H ) is a self-adjoint operator, and if 𝑥 = 𝑣|𝑥|, 𝑣 ∗ 𝑣 = s(|𝑥|) is its polar decomposition, then the following relations are immediately obtained: |𝑥| = 𝑥 + + 𝑥 − , 𝑣 = s(𝑥+ ) − s(𝑥− ); in particular, we have 𝑣 = 𝑣 ∗ .

2.16. We shall sometimes denote by B (H )ℎ the set of all self-adjoint operators in B (H ). The notion of positive operator allows the introduction of an order relation in B (H )ℎ . Namely, for 𝑥, 𝑦 ∈ B (H )ℎ , we shall write 𝑥 ≤ 𝑦 if the operator 𝑦 − 𝑥 is positive; by taking into account Proposition 2.12, it is easy to check that the relation ‘≤’ is indeed an order relation in B (H )ℎ . From now on, we shall use the notation 𝑥 ≥ 0 in order to express the fact that the operator 𝑥 is positive and we shall sometimes denote B (H )+ = {𝑥 ∈ B (H ); 𝑥 ≥ 0}. The set B (H )ℎ is a real wo-closed vector subspace of B (H ), whereas B (H )+ is a wo-closed convex cone. If 𝑎, 𝑏 ∈ B (H )+ and 𝑎𝑏 = 𝑏𝑎, then 𝑎𝑏 ∈ B (H )+ . If 𝑎, 𝑏 ∈ B (H )ℎ and 𝑎 ≤ 𝑏, then 𝑥 ∗ 𝑎𝑥 ≤ 𝑥 ∗ 𝑏𝑥, for any 𝑥 ∈ B (H ). Proposition. Let {𝑥𝑖 }𝑖∈𝐼 ⊂ B (H )ℎ be an increasing net, such that there exists a 𝑦 ∈ B (H )ℎ for which 𝑥𝑖 ≤ 𝑦, 𝑖 ∈ 𝐼. Then there exists an 𝑥 ∈ B (H )ℎ , such that 𝑥 = sup 𝑥𝑖 . 𝑖∈𝐼

Moreover, 𝑥 is the limit of the net (𝑥𝑖 )𝑖∈𝐼 for the so-topology in B (H ). Proof. We can assume, without any loss of generality, that 0 ≤ 𝑥𝑖 ≤ 1, 𝑖 ∈ 𝐼. For any 𝜉 ∈ H , we define 𝐹(𝜉, 𝜉) = sup(𝑥𝑖 𝜉|𝜉) = lim(𝑥𝑖 𝜉|𝜉), 𝑖∈𝐼

𝑖∈𝐼

and for any 𝜉, 𝜂 ∈ H , 1 𝐹(𝜉, 𝜂) = [𝐹(𝜉 + 𝜂, 𝜉 + 𝜂) − 𝐹(𝜉 − 𝜂, 𝜉 − 𝜂) + i𝐹(𝜉 + i𝜂, 𝜉 + i𝜂) − i𝐹(𝜉 − i𝜂, 𝜉 − i𝜂)]. 4 Then 𝐹(., .) is a bounded positive sesquilinear form, whose norm is ≤1. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

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According to the Riesz theorem, there exists a unique operator 𝑥 ∈ B (H ), ‖𝑥‖ ≤ 1, such that 𝐹(𝜉, 𝜂) = (𝑥𝜉|𝜂),

𝜉, 𝜂 ∈ H ,

and according to Proposition 2.12, we have 𝑥 ≥ 0. Since, for any 𝑖 ∈ 𝐼 and any 𝜉 ∈ H , we have (𝑥𝜉|𝜉) = 𝐹(𝜉, 𝜉) ≥ (𝑥𝑖 𝜉|𝜉), it follows that 𝑥 is an upper bound of the net (𝑥𝑖 )𝑖∈𝐼 . On the other hand, if 𝑥0 ∈ B (H )ℎ is an upper bound of the family (𝑥𝑖 )𝑖∈𝐼 , then for any 𝜉 ∈ H , we have (𝑥𝜉|𝜉) = 𝐹(𝜉, 𝜉) = lim(𝑥𝑖 𝜉|𝜉) ≤ (𝑥0 𝜉|𝜉), 𝑖∈𝐼

which implies that 𝑥 ≤ 𝑥0 . Hence 𝑥 = sup 𝑥𝑖 in B (H )ℎ . 𝑖∈𝐼

Finally, for any 𝜉 ∈ H , we have ‖(𝑥 − 𝑥𝑖 )𝜉‖2 ≤ ‖(𝑥 − 𝑥𝑖 )1/2 ‖2 ‖(𝑥 − 𝑥𝑖 )1/2 𝜉‖2 ≤ ((𝑥 − 𝑥𝑖 )𝜉|𝜉) = 𝐹(𝜉, 𝜉) − (𝑥𝑖 𝜉|𝜉) → 0, which implies that 𝑥 is the limit of the net (𝑥𝑖 )𝑖∈𝐼 for the so-topology in B (H ).

Q.E.D.

If (𝑥𝑖 )𝑖∈𝐼 ⊂ B (H )ℎ is an increasing net and if 𝑥 ∈ B (H )ℎ belongs to the wo-closure of this family, then 𝑥 = sup𝑖∈𝐼 𝑥𝑖 , and therefore, 𝑥 belongs to the so-closure of the same family. In this case, we shall write 𝑥𝑖 ↑ 𝑥. This notation means that the net (𝑥𝑖 )𝑖∈𝐼 is increasing, 𝑥 = sup 𝑥𝑖 in B (H )ℎ and that 𝑥 belongs to the so-closure of the family (𝑥𝑖 )𝑖∈𝐼 . The same notation will be used for the ‘increasing convergence’ of real numbers. 2.17. Let 𝑥 ∈ B (H ), 0 ≤ 𝑥 ≤ 1 and 𝑒 ∈ PB (H ) . Then 𝑥 ≤ 𝑒 ⇔ 𝑥 = 𝑥𝑒.

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Indeed, from the relation 𝑥 ≤ 1, we infer that 𝑒𝑥𝑒 ≤ 𝑒, whence if 𝑥 = 𝑥𝑒, we deduce that 𝑥 ≤ 𝑒. Conversely, if 0 ≤ 𝑥 ≤ 𝑒, then we successively get: 0 = (1 − 𝑒)0(1 − 𝑒) ≤ (1 − 𝑒)𝑥(1 − 𝑒) ≤ (1 − 𝑒)𝑒(1 − 𝑒) = 0, (1 − 𝑒)𝑥(1 − 𝑒) = 0, ((1 − 𝑒)𝑥 1/2 )((1 − 𝑒)𝑥 1/2 )∗ = 0, (1 − 𝑒)𝑥 1/2 = 0, (1 − 𝑒)𝑥 = 0, which implies that 𝑥 = 𝑒𝑥 = 𝑥𝑒. In particular, if 𝑒1 , 𝑒2 ∈ PB (H ) , then 𝑒1 ⩽ 𝑒2 iff 𝑒1 = 𝑒1 𝑒2 . It is easy to check that we have 𝑒1 ≤ 𝑒2 iff 𝑒1 H ⊂ 𝑒2 H . Let (𝑒𝑖 )𝑖∈𝐼 ⊂ PB (H ) be any family of projections. One can define the following projections: ⋁

𝑒𝑖 = the projection on the subspace ∑ 𝑒𝑖 H .

⋀

𝑒𝑖 = the projection on the subspace

𝑖∈𝐼

𝑖∈𝐼

𝑖∈𝐼

⋂

𝑒𝑖 H .

𝑖∈𝐼

One can then immediately check that ⋁𝑖∈𝐼 𝑒𝑖 , (resp., ⋀𝑖∈𝐼 𝑒𝑖 ) is the least upper bound (resp., the greatest lower bound) of the family (𝑒𝑖 )𝑖∈𝐼 with respect to the order relation induced on PB (H ) by the order relation just defined in B (H )ℎ . If the set of indices is finite, 𝐼 = (1, … , 𝑛), then one also uses the following notations: 𝑛

𝑒1 ∨ … ∨ 𝑒𝑛 =

⋁

𝑒𝑖 , for

⋀

𝑒𝑖 , for

𝑖=1 𝑛

𝑒1 ∧ … ∧ 𝑒𝑛 =

⋁

𝑒𝑖 ,

⋀

𝑒𝑖 .

𝑖∈𝐼

𝑖=1

𝑖∈𝐼

From the preceding results and from Proposition 2.16, we get the following: Corollary. (i) P B (H ) is a complete lattice. (ii) If (𝑒𝑖 )𝑖∈𝐼 is an increasing net, then 𝑒𝑖 ↑

⋁

𝑒𝑖 .

𝑖∈𝐼

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32

Lectures on von Neumann algebras

(iii) If (𝑒𝑖 )𝑖∈𝐼 is a family of mutually orthogonal projections, then the family (𝑒𝑖 )𝑖∈𝐼 is summable for the so-topology, and ∑ 𝑒𝑖 = 𝑖∈𝐼

⋁

𝑒𝑖 .

𝑖∈𝐼

We observe that for any 𝑒1 , … , 𝑒𝑛 ∈ PB (H ) , we have s(𝑒1 + … + 𝑒𝑛 ) = 𝑒1 ∨ … ∨ 𝑒𝑛 . 2.18. We shall now extend the operational calculus given by Theorem 2.6 to a larger class than that of the continuous functions. In order to do this, we shall need the following: Lemma. Let 𝑥 ∈ B (H ) be a self-adjoint operator and let {𝑓𝑛 } and {𝑔𝑛 } be two bounded increasing sequences of positive functions from C (𝜎(𝑥)), such that sup 𝑓𝑛 (𝜆) ≤ sup 𝑔𝑛 (𝜆), 𝑛

𝜆 ∈ 𝜎(𝑥).

𝑛

Then sup 𝑓𝑛 (𝑥) ≤ sup 𝑔𝑛 (𝑥). 𝑛

𝑛

Proof. By taking into account Theorem 2.6 and Proposition 2.16, we infer the existence of the elements sup𝑛 𝑓𝑛 (𝑥), sup𝑛 𝑔𝑛 (𝑥) and the relations 𝑓𝑛 (𝑥) ↑ sup 𝑓𝑛 (𝑥),

𝑔𝑛 (𝑥) ↑ sup 𝑔𝑛 (𝑥).

𝑛

𝑛

Let 𝑛 be a natural number and 𝜀 > 0. For any 𝜆 ∈ 𝜎(𝑥), we have 𝑓𝑛 (𝜆) − 𝜀 < 𝑓𝑛 (𝜆) ≤ sup 𝑓𝑚 (𝜆) ≤ sup 𝑔𝑚 (𝜆); 𝑚

𝑚

consequently, there exists a neighbourhood 𝑉𝜆 of 𝜆 and a natural number 𝑚𝜆 , such that 𝑓𝑛 (𝜇) − 𝜀 < 𝑔𝑚𝜆 (𝜇), 𝜇 ∈ 𝑉𝜆 . Since 𝜎(𝑥) is compact, it follows that there exists a natural number 𝑚𝑛 , such that 𝑓𝑛 − 𝜀 ≤ 𝑔𝑚𝑛 in C (𝜎(𝑥)). Consequently, by taking into account Theorem 2.6, we have 𝑓𝑛 (𝑥) − 𝜀 ≤ 𝑔𝑚𝑛 (𝑥) ≤ sup 𝑔𝑚 (𝑥), 𝑚

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33

Bounded linear operators in Hilbert spaces whence, by taking into account the fact that 𝜀 > 0 is arbitrary, we get 𝑓𝑛 (𝑥) ≤ sup 𝑔𝑚 (𝑥); 𝑚

since 𝑛 is arbitrary, we have sup 𝑓𝑛 (𝑥) ≤ sup 𝑔𝑚 (𝑥). 𝑛

𝑚

Q.E.D. 2.19. Let 𝑥 ∈ B (H ) be a self-adjoint operator. We shall write 𝑚(𝑥) = inf {𝜆; 𝜆 ∈ 𝜎(𝑥)},

𝑀(𝑥) = sup{𝜆; 𝜆 ∈ 𝜎(𝑥)}.

Since 𝜎(𝑥) is compact, we have 𝑚(𝑥), 𝑀(𝑥) ∈ 𝜎(𝑥). For any 𝜆 ∈ ℝ, we shall consider the continuous functions: 1

1, ⎧ ⎪ 𝜆 𝑓𝑛 (𝑡) = 𝑛(𝜆 − 𝑡), ⎨ ⎪ ⎩ 0,

for 𝑡 ∈ (−∞, 𝜆 − ], 1

𝑛

for 𝑡 ∈ [𝜆 − , 𝜆], 𝑛

for 𝑡 ∈ [𝜆, +∞).

Then we have 𝑓𝑛𝜆 (𝑡) ↑ 𝜒(−∞,𝜆) (𝑡), (𝑓𝑛𝜆 )2 (𝑡) ↑ 𝜒(−∞,𝜆) (𝑡),

𝑡 ∈ ℝ, 𝑡 ∈ ℝ,

where by 𝜒𝐷 we denote the characteristic function of the set 𝐷 ⊂ ℝ. According to Proposition 2.16 and Lemma 2.18, there exists a projection 𝑒𝜆 ∈ B (H ), such that 𝑓𝑛𝜆 (𝑥) ↑ 𝑒𝜆 . In what follows we shall prove some properties of the projections 𝑒𝜆 : (i) 𝑒𝜆 ∈ R ({𝑥}); in particular, 𝑒𝜆 commutes with any operator commuting with 𝑥. This fact follows from the definition of the projections 𝑒𝜆 , from Theorem 2.6 and from the obvious equality R ({𝑥}) = so-closure of C ∗ ({𝑥, 1}). 𝜆1 ≤ 𝜆2 ⇒ 𝑒𝜆1 ≤ 𝑒𝜆2 .

(ii) Indeed, for any 𝑛, we have

𝜆

𝜆

𝑓𝑛 1 ≤ 𝑓𝑛 2 in C (𝜎(𝑥)), Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

34

Lectures on von Neumann algebras which implies that 𝜆

𝜆

𝑓𝑛 1 (𝑥) ≤ 𝑓𝑛 2 (𝑥); the required property now follows by tending to the limit. (iii)

𝜆𝑛 ↑ 𝜆 ⇒ 𝑒𝜆𝑛 ↑ 𝑒𝜆 . Indeed, we then have 𝑓𝑛𝜆 ↑ 𝜒(−∞,𝜆) , pointwise; by taking into account Lemma 2.18, the definition of the projection 𝑒𝜆 and (ii), we get 𝜆

𝑒𝜆 ≥ 𝑒𝜆𝑛 ≥ 𝑓𝑛 𝑛 (𝑥) ↑ 𝑒𝜆 , and therefore, 𝑒𝜆𝑛 ↑ 𝑒𝜆 . (iv)

𝜆 ≤ 𝑚(𝑥) ⇒ 𝑒𝜆 = 0; 𝜆 > 𝑀(𝑥) ⇒ 𝑒𝜆 = 1. Indeed, if 𝜆 ≤ 𝑚(𝑥), then 𝑓𝑛𝜆 = 0 in C (𝜎(𝑥)), for any 𝑛, whereas for 𝜆 > 𝑀(𝑥), we have 𝑓𝑛𝜆 = 1 in C (𝜎(𝑥)), if 𝑛 is sufficiently great.

(v)

𝑥𝑒𝜆 ≤ 𝜆𝑒𝜆 , 𝑥(1 − 𝑒𝜆 ) ≥ 𝜆(1 − 𝑒𝜆 ). Indeed, we have the following relations: 𝑡𝑓𝑛𝜆 (𝑡) ≤ 𝜆𝑓𝑛𝜆 (𝑡), 𝑡 ∈ ℝ, 𝑡(1 − 𝑓𝑛𝜆 (𝑡)) ≥ (𝜆 −

1 )(1 − 𝑓𝑛𝜆 (𝑡)), 𝑡 ∈ ℝ, 𝑛

whence 𝑥𝑓𝑛𝜆 (𝑥) ≤ 𝜆𝑓𝑛𝜆 (𝑥), 𝑥(1 − 𝑓𝑛𝜆 (𝑥)) ≥ (𝜆 −

1 )(1 − 𝑓𝑛𝜆 (𝑥)), 𝑛

and the stated inequalities can be obtained by tending to the limit. From property (v), we infer that if 𝜇 ≤ 𝜆, then 𝜇(𝑒𝜆 − 𝑒𝜇 ) ≤ 𝑥(𝑒𝜆 − 𝑒𝜇 ) ≤ 𝜆(𝑒𝜆 − 𝑒𝜇 ). Let now 𝛿 > 0, 𝜀 > 0, and let Δ = {𝑚(𝑥) = 𝜆0 < 𝜆1 < … < 𝜆𝑛 = 𝑀(𝑥) + 𝛿}

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35

Bounded linear operators in Hilbert spaces

be a partition of the interval [𝑚(𝑥), 𝑀(𝑥) + 𝛿], whose norm is ‖Δ‖ = sup{𝜆𝑖 − 𝜆𝑖−1 ; 𝑖 = 1, 2, … , 𝑛} < 𝜀. We shall now consider the ‘Darboux sums’: 𝑛

𝑠(Δ) = ∑ 𝜆𝑖−1 (𝑒𝜆𝑖 − 𝑒𝜆𝑖−1 ), 𝑖=1 𝑛

𝑆(Δ) = ∑ 𝜆𝑖 (𝑒𝜆𝑖 − 𝑒𝜆𝑖−1 ). 𝑖=1

By taking into account the preceding results, we can easily prove the following relations: 𝑠(Δ) ≤ 𝑥 ≤ 𝑆(Δ), ‖𝑆(Δ) − 𝑠(Δ)‖ < 𝜀, and these enable us to write +∞

(vi)

𝑥=∫

𝑀(𝑥)+0

𝜆 𝑑𝑒𝜆 = ∫

−∞

𝜆 𝑑𝑒𝜆 ,

𝑚(𝑥)

where the integral is to be considered as a vector Stieltjes integral, which converges with respect to the norm. Assertions (i)–(vi) make up what is usually called the spectral theorem for the self-adjoint operator 𝑥, whereas the family of projections (𝑒𝜆 )𝜆 is called the spectral scale of the self-adjoint operator 𝑥. For any 𝜉, 𝜂 ∈ H , we shall consider the function 𝑒𝜉,𝜂 defined by the relation 𝑒𝜉,𝜂 (𝜆) = (𝑒𝜆 𝜉|𝜂),

𝜆 ∈ ℝ.

Then the functions 𝑒𝜉,𝜉 are positive, increasing and 𝑒𝜉,𝜉 ≤ ‖𝜉‖2 , whereas the functions 𝑒𝜉,𝜂 are of bounded variation, and their total variation can be majorized with the help of the Cauchy–Buniakovsky inequality 𝑉(𝑒𝜉,𝜂 ) ≤ ‖𝜉‖‖𝜂‖. We recall that any function of bounded variation on ℝ determines a bounded, Borel measure, whose norm is equal to the total variation of the function; the integral corresponding to such a measure is usually called the Lebesgue–Stieltjes integral. In particular, the functions 𝑒𝜉,𝜂 determine bounded Borel measures; by the same method as in the proof of statement (iv), one can show that the support of the measures defined in this manner is contained in the spectrum 𝜎(𝑥) of 𝑥. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

36

Lectures on von Neumann algebras From property (vi), or by direct verification, it follows that

(vii)

+∞

(𝑥𝜉|𝜂) = ∫

𝜆 𝑑𝑒𝜉,𝜂 (𝜆), 𝜉, 𝜂 ∈ H .

−∞

2.20. For any topological space Ω, we shall denote by B (Ω) the set of all bounded complex Borel functions, defined on Ω. The set B (Ω) can be endowed canonically with a structure of a 𝐶 ∗ -algebra, with the usual algebraic operations and with the uniform norm (i.e., the sup-norm). If the space Ω is metrizable, then, by virtue of a theorem of Baire, B (Ω) is the smallest class of functions, closed with respect to the pointwise convergence of the bounded sequences, which contains the bounded continuous functions. Theorem (of operational calculus with Borel functions). Let 𝑥 ∈ B (H ) be a selfadjoint operator. There exists a unique mapping B (𝜎(𝑥)) ∋ 𝑓 ↦ 𝑓(𝑥) ∈ B (H ), such that (i) if 𝑓 is a polynomial, 𝑓(𝜆) = 𝛼0 + 𝛼1 𝜆 + … + 𝛼𝑛 𝜆𝑛 , then 𝑓(𝑥) = 𝛼0 + 𝛼1 𝑥 + … + 𝛼𝑛 𝑥 𝑛 . (ii) if 𝑓, 𝑓𝑛 ∈ B (𝜎(𝑥)), sup𝑛 ‖𝑓𝑛 ‖ < +∞ and 𝑓𝑛 ↦ 𝑓 pointwise, then 𝑓𝑛 (𝑥) → 𝑓(𝑥) for the so-topology in B (H ). Moreover, this mapping is a *-homomorphism of the 𝐶 ∗ -algebra B (𝜎(𝑥)) into the von Neumann algebra R ({𝑥}), and it is an extension of the *-isomorphism given by Theorem 2.6. Proof. Any mapping that satisfies conditions (i) and (ii) obviously coincides with the mapping given by Theorem 2.6, when restricted to C (𝜎(𝑥)). In this way, the uniqueness is an immediate consequence of the theorem of Baire, already mentioned above. In order to prove the existence, as well as the other properties of the mapping, described in the statement, we shall define, for any 𝑓 ∈ B (𝜎(𝑥)): +∞

𝐹𝑓 (𝜉, 𝜂) = ∫

𝑓(𝜆)𝑑𝑒𝜉,𝜂 (𝜆), 𝜉, 𝜂 ∈ H ,

−∞

where we used the Lebesgue–Stieltjes integral; more precisely, the function 𝑓 can be extended to a Borel function on ℝ, whereas the integral does not depend on this extension, since the support of the measure is included in 𝜎(𝑥) (see Section 2.19). Then 𝐹𝑓 (., .) is a bounded sesquilinear form, defined on H × H : |𝐹𝑓 (𝜉, 𝜂)| ≤ ‖𝑓‖𝑉(𝑒𝜉,𝜂 ) ≤ ‖𝑓‖‖𝜉‖‖𝜂‖,

𝜉, 𝜂 ∈ H .

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Bounded linear operators in Hilbert spaces

From the theorem of Riesz, it follows that there exists a unique operator 𝑓(𝑥) ∈ B (H ), such that +∞

(𝑓(𝑥)𝜉|𝜂) = ∫

𝑓(𝜆)𝑑𝑒𝜉,𝜂 (𝜆),

𝜉, 𝜂 ∈ H ,

−∞

In this manner, we have defined the mapping: B (𝜎(𝑥)) ∋ 𝑓 ↦ 𝑓(𝑥) ∈ B (H ). It is easy to show that this mapping is linear. By taking into account the relation 𝑒𝜉,𝜂 ̄ = ∗ ̄ 𝑒𝜂,𝜉 , 𝜉, 𝜂 ∈ H , it is easy to show that 𝑓(𝑥) = (𝑓(𝑥)) . Let 𝑓, 𝑔 ∈ B (𝜎(𝑥)). For any 𝜉, 𝜂 ∈ H , we have +∞ ∗

(𝑓(𝑥)𝑔(𝑥)𝜉|𝜂) = (𝑔(𝑥)𝜉|(𝑓(𝑥)) 𝜂) = ∫

𝑔(𝜆)𝑑𝑒𝜉,(𝑓(𝑥))∗ 𝜂 (𝜆).

−∞

But we have 𝑑𝑒𝜉,(𝑓(𝑥))∗ 𝜂 (𝜆) = (𝑒𝜆 𝜉|(𝑓(𝑥))∗ 𝜂) = (𝑓(𝑥)𝑒𝜆 𝜉|𝜂) +∞

=∫

𝜆

𝑓(𝜇)𝑑𝑒𝑒𝜆 𝜉,𝜂 (𝜇) = ∫ 𝑓(𝜇)𝑑𝑒𝜉,𝜂 (𝜇),

−∞

−∞

where the last equality follows from 2.19 (ii). Consequently, we have +∞

(𝑓(𝑥)𝑔(𝑥)𝜉|𝜂) = ∫

𝜆

𝑔(𝜆)𝑑 (∫ 𝑓(𝜇)𝑑𝑒𝜉,𝜂 (𝜇))

−∞ +∞

=∫

−∞

𝑔(𝜆)𝑓(𝜆)𝑑𝑒𝜉,𝜂 (𝜆)

−∞ +∞

=∫

(𝑓𝑔)(𝜆)𝑑𝑒𝜉,𝜂 (𝜆) = ((𝑓𝑔)(𝑥)𝜉|𝜂);

−∞

the second equality above is obvious if 𝑔 is the characteristic function of an interval; this 𝜆 fact already implies that the measures 𝑑 (∫−∞ 𝑓(𝜇)𝑑𝑒𝜉,𝜂 (𝜇)) and 𝑓(𝜆)𝑑𝑒𝜉,𝜂 (𝜆) are equal, and therefore, the same equality is true for any 𝑔 ∈ B (𝜎(𝑥)). We have thus shown that (𝑓𝑔)(𝑥) = 𝑓(𝑥)𝑔(𝑥). Consequently, the mapping 𝑓 ↦ 𝑓(𝑥) is a *-homomorphism of the 𝐶 ∗ -algebra B (𝜎(𝑥)) into B (H ). If 𝑓0 (𝜆) = 1, 𝜆 ∈ 𝜎(𝑥), then, obviously, 𝑓0 (𝑥) = 1. If 𝑓1 (𝜆) = 𝜆, 𝜆 ∈ 𝜎(𝑥), then, by taking into account 2.19 (vii), we get 𝑓1 (𝑥) = 𝑥. Since the mapping 𝑓 ↦ 𝑓(𝑥) is multiplicative, we now immediately get property (i). For any 𝑓 ∈ B (𝜎(𝑥)) and any 𝜉 ∈ H , we have the relation +∞

(*)

2

‖𝑓(𝑥)𝜉‖ = ∫

|𝑓(𝜆)|2 𝑑𝑒𝜉,𝜉 (𝜆).

−∞

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38

Lectures on von Neumann algebras

Indeed, we have ‖𝑓(𝑥)𝜉‖2 = (𝑓(𝑥)𝜉|𝑓(𝑥)𝜉) = (𝑓(𝑥)∗ 𝑓(𝑥)𝜉|𝜉) ̄ = (𝑓(𝑥)𝑓(𝑥)𝜉|𝜉) = (|𝑓|2 (𝑥)𝜉|𝜉) +∞

=∫

|𝑓(𝜆)|2 𝑑𝑒𝜉,𝜉 (𝜆).

−∞

Property (ii) now easily follows since +∞ 2

‖(𝑓𝑛 (𝑥) − 𝑓(𝑥)𝜉‖ = ∫

|𝑓𝑛 (𝜆) − 𝑓(𝜆)|2 𝑑𝑒𝜉,𝜉 (𝜆),

−∞

whereas the integral converges to 0, by virtue of the dominated convergence theorem of Lebesgue. Finally, the set {𝑓 ∈ B (𝜎(𝑥)); 𝑓(𝑥) ∈ R ({𝑥})} contains the polynomials and it is closed with respect to the pointwise convergence of bounded sequences; therefore, from the above-mentioned theorem of Baire, it equals B (𝜎(𝑥)). Q.E.D. If 𝑥, 𝑦 ∈ B (H ) are commuting self-adjoint operators (i.e., 𝑥𝑦 = 𝑦𝑥), and if 𝑓 ∈ B (𝜎(𝑥)), 𝑔 ∈ B (𝜎(𝑦)), then 𝑓(𝑥) and 𝑔(𝑦) are commuting normal operators. If 𝑥 ∈ B (H ) is a self-adjoint operator, and if 𝑒 ∈ B (H ) is a projection that commutes with 𝑥, then for any function 𝑓 ∈ B (𝜎(𝑥)), 𝑓(0) = 0, we have 𝑓(𝑒𝑥) = 𝑒𝑓(𝑥). Indeed, this equality is easily checked for 𝑓, a polynomial without the constant term, and then, by tending to the limit, it obtains for any 𝑓 ∈ B (𝜎(𝑥)), 𝑓(0) = 0. Relation (*) from the proof of the theorem is useful in other situations too. For example, with its help, one can easily prove that if 𝑓𝑛 , 𝑓 ∈ B (𝜎(𝑥)), and 𝑓𝑛 → 𝑓 uniformly, then 𝑓𝑛 (𝑥) → 𝑓(𝑥) for the norm topology. We also observe that, since it is a *-homomorphism, the mapping 𝑓 ↦ 𝑓(𝑥) is positive. 2.21. The following fact has already been established in Section 2.19, but we mention it again due to its special usefulness: Corollary. Let 𝑥 ∈ B (H ) be a positive operator and 𝛼 > 0 a positive number. Then there exists a projection 𝑒 ∈ R ({𝑥}), such that 𝑥𝑒 ≥ 𝛼𝑒, 𝑥(1 − 𝑒) ≤ 𝛼(1 − 𝑒). We observe that one can take 𝑒 = 𝜒(𝛼,+∞) (𝑥) or 𝑒 = 𝜒[𝛼,+∞) (𝑥), which shows that the projection 𝑒 is not uniquely determined by the preceding conditions. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

39

Bounded linear operators in Hilbert spaces 2.22. If 𝑥 ∈ B (H ) is a self-adjoint operator, then s(𝑥) = 𝜒ℝ\{0} (𝑥).

Indeed, from the obvious equality 𝜆 ⋅ 𝜒ℝ\{0} (𝜆) = 𝜆, 𝜆 ∈ ℝ, it follows that 𝑥𝜒ℝ\{0} (𝑥) = 𝑥, which implies that s(𝑥) ≤ 𝜒ℝ\{0} (𝑥). In contrast, from the relation 𝑥s(𝑥) = 𝑥, it follows that 𝑓(𝑥)s(𝑥) = 𝑓(𝑥), first for 𝑓 a polynomial without constant term, and then, by tending to the limit, for any 𝑓 ∈ B (𝜎(𝑥)), 𝑓(0) = 0. In particular, we have 𝜒ℝ\{0} (𝑥)s(𝑥) = 𝜒ℝ\{0} (𝑥), and therefore, 𝜒ℝ\{0} (𝑥) ⩽ s(𝑥). From the proof, we also inferred that for any 𝑓 ∈ B (𝜎(𝑥)), such that 𝑓(0) = 0, we have s(𝑓(𝑥)) ≤ s(𝑥). If 𝑓𝑛 ∈ B (𝜎(𝑥)), sup𝑛 ‖𝑓𝑛 ‖ < +∞, and 𝑓𝑛 → 𝜒ℝ\{0} , then 𝑠𝑜

𝑓𝑛 (𝑥) → s(𝑥). For example, if 𝑥 ≥ 0, we have 𝑠𝑜

𝑛𝑥(1 + 𝑛𝑥)−1 → s(𝑥), 𝑠𝑜

𝑥 1/𝑛 → s(𝑥). Also, it is easy to show that there exists a sequence of polynomials without constant terms, in 𝑥, which is so-convergent to s(𝑥). Corollary. Let 𝑥 ∈ B (H ) be a positive operator . Then there exists a sequence of projections {𝑒𝑛 } ⊂ R ({𝑥}), such that 1 𝑥𝑒𝑛 ≥ 𝑒𝑛 , 𝑛 𝑒𝑛 ↑ s(𝑥). One can take 𝑒𝑛 = 𝜒( 1 ,+∞) (𝑥). 𝑛

2.23. Already the spectral theorem (2.19 (vi)) implied that any self-adjoint operator 𝑥 ∈ B (H ) is the limit, for the norm topology, of linear combinations of projections from R ({𝑥}). In particular, any von Neumann algebra coincides with the norm-closed linear span of its projections. These results can be further strengthened by the following: Corollary. Let 𝑥 ∈ B (H ), 0 ≤ 𝑥 ≤ 1. Then there exists a sequence of projections {𝑒𝑛 } ⊂ R ({𝑥}), such that ∞ 1 𝑥 = ∑ 𝑛 𝑒𝑛 ; 2 𝑛=1 the series converges in the norm topology. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

40

Lectures on von Neumann algebras

Proof. The sequence {𝑒𝑛 } can be defined inductively in the following manner: According to Corollary 2.21, there exists a projection 𝑒1 ∈ R ({𝑥}), such that 1 𝑥𝑒1 ≥ 𝑒1 , 2 1 𝑥(1 − 𝑒1 ) ≤ (1 − 𝑒1 ). 2 From Corollary 2.21, then exists a projection 𝑒𝑛 ∈ R ({𝑥}), such that 𝑛−1

1 1 𝑒𝑘 ) 𝑒𝑛 ≥ 𝑛 𝑒𝑛 , 𝑘 2 2 𝑘=1

(𝑥 − ∑

𝑛−1

1 1 𝑒 ) (1 − 𝑒𝑛 ) ≤ 𝑛 (1 − 𝑒𝑛 ). 𝑘 𝑘 2 2 𝑘=1

(𝑥 − ∑

One can easily prove by induction, and by using the hypothesis that 0 ≤ 𝑥 ≤ 1, that we have 𝑛 1 1 0 ≤ 𝑥 − ∑ 𝑘 𝑒𝑘 ≤ 𝑛 , 2 2 𝑘=1 whence the desired assertion immediately follows.

Q.E.D.

The preceding corollary corresponds to the dyadic decomposition of the real numbers between 0 and 1. 2.24. If 𝑥 ∈ B (H ) is an arbitrary operator, then the operators i 1 𝑥1 = (𝑥 + 𝑥 ∗ ), 𝑥2 = (𝑥∗ − 𝑥) ∈ C ∗ ({𝑥}) 2 2 are self-adjoint, and 𝑥 = 𝑥1 + i𝑥2 . An operator 𝑢 ∈ B (H ) is said to be unitary if it maps isometrically H onto H . It is easily checked that 𝑢 ∈ B (H ) is unitary iff 𝑢∗ 𝑢 = 𝑢𝑢∗ = 1. Proposition. Let 𝑥 ∈ B (H ) be an arbitrary operator. Then 𝑥 is a linear combination of unitary operators from C ∗ ({𝑥, 1}). Proof. Because of the preceding remark, we can assume, without any loss of generality, that 𝑥 ∗ = 𝑥, ‖𝑥‖ ≤ 1. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

Bounded linear operators in Hilbert spaces

41

In this case, we define 𝑢 = 𝑥 + i(1 − 𝑥 2 )1/2 , and it is easy to check that 𝑢 is unitary and 1 𝑥 = (𝑢 + 𝑢∗ ). 2 Q.E.D. In fact, we proved that any operator (resp., any self-adjoint, positive operator) is a linear combination of 4 (resp., a linear combination with positive coefficients of 2) unitary operators from the 𝐶 ∗ -algebra with identity, generated by it. We observe, therefore, that any 𝐶 ∗ -algebra of operators (resp., any 𝐶 ∗ -algebra of operators with identity) is the vector space generated by the self-adjoint (resp., the unitary) operators it contains. 2.25. For arbitrary-bounded linear operators on H , one can define an operational calculus with functions analytic on a neighbourhood of the spectrum. Let 𝑥 ∈ B (H ). We shall denote by A (𝜎(𝑥)) the set of all analytic functions defined on a neighbourhood of the spectrum 𝜎(𝑥) of 𝑥 (neighbourhood that can depend on the considered function). By identifying two functions from A (𝜎(𝑥)), if they coincide on a neighbourhood of the spectrum 𝜎(𝑥), we can define canonically, in A (𝜎(𝑥)), an algebra structure. For any 𝑓 ∈ A (𝜎(𝑥)), we shall consider closed rectifiable Jordan curves, with the positive orientation, Γ1 , Γ2 , … , Γ𝑘 , such that the interiors of these curves be mutually disjoint, the union of the interiors of these curves contain 𝜎(𝑥), whereas the closure of this region be included in the domain of 𝑓. We denote Γ = {Γ1 , … , Γ𝑘 } and define 𝑓A (𝑥) = (2𝜋i)−1 ∫ 𝑓(𝜆)(𝜆 − 𝑥)−1 𝑑𝜆, Γ

as a Cauchy integral, which converges in norm. From the well-known theorem of Cauchy, 𝑓A (𝑥) does not depend on the choice of Γ. Theorem (of operational calculus with analytic functions). Let 𝑥 ∈ B (H ) be an arbitrary operator: (i) If 𝑓 is a polynomial, 𝑓(𝜆) = 𝛼0 + 𝛼1 𝜆 + … + 𝛼𝑛 𝜆𝑛 , then 𝑓 ∈ A (𝜎(𝑥)) and 𝑓A (𝑥) = 𝛼0 + 𝛼 1 𝑥 + … + 𝛼 𝑛 𝑥 𝑛 . (ii) The mapping A (𝜎(𝑥) ∋ 𝑓 ↦ 𝑓A (𝑠) ∈ B (H ) is an algebra homomorphism. Moreover, 𝑓A (𝑥) ∈ C ∗ ({𝑥, 1}), for any 𝑓 ∈ A (𝜎(𝑥)). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

42

Lectures on von Neumann algebras

Proof. The mapping 𝑓 ↦ 𝑓A (𝑥) is obviously linear. Consequently, in order to prove (i), it suffices to consider the case 𝑓(𝜆) = 𝜆𝑛 , 𝑛 ≥ 0. Let Γ be a circle centered at 0 and of radius > ‖𝑥‖, positively oriented. For any 𝜆 ∈ Γ, we have ∞

(𝜆 − 𝑥)−1 = ∑ 𝜆−𝑘−1 𝑥 𝑘 , 𝑘=0

the series being convergent in norm. Hence ∞

𝑓A (𝑥) = (2𝜋i)

−1

𝑛

∫ 𝜆 (𝜆 − 𝑥)

−1

𝑑𝜆 = ∑ ((2𝜋i)−1 ∫(𝜆𝑛−𝑘−1 𝑑𝜆)𝑥 𝑘 = 𝑥 𝑛 .

Γ

Γ

𝑘=0

We still have to prove the multiplicativity of the mapping 𝑓 ↦ 𝑓A (𝑥). Let 𝑓, 𝑔 ∈ A (𝜎(𝑥)) and Γ1 , … , Γ𝑘 , Γ1′ , … , Γ𝑗′ be closed rectifiable Jordan curves, positively oriented, such that the interiors of the curves Γ1 , … , Γ𝑘 be mutually disjoint and their union include 𝜎(𝑥), the closure of this union be included in the union of the mutually disjoint interiors of the curves Γ1′ , … , Γ𝑗′ , whereas the closure of this last union be included in the intersection of the domains of 𝑓 and 𝑔. We denote Γ = {Γ1 , … , Γ𝑘 } and Γ′ = {Γ1′ , … , Γ𝑗′ }. From the identity (𝜆 − 𝑥)−1 − (𝜇 − 𝑥)−1 = (𝜇 − 𝜆)(𝜆 − 𝑥)−1 (𝜇 − 𝑥)−1 , we infer that 𝑓A (𝑥)𝑔A (𝑥) = −(4𝜋 2 )−1 (∫ 𝑓(𝜆)(𝜆 − 𝑥)−1 𝑑𝜆) (∫ 𝑔(𝜇)(𝜇 − 𝑥)−1 𝑑𝜇) Γ′

Γ

= −(4𝜋 2 )−1 ∫ ∫ 𝑓(𝜆)𝑔(𝜇)(𝜆 − 𝑥)−1 (𝜇 − 𝑥)−1 𝑑𝜆𝑑𝜇 Γ

Γ′

= −(4𝜋 2 )−1 ∫ ∫ Γ

Γ′

𝑓(𝜆)𝑔(𝜇) ((𝜆 − 𝑥)−1 − (𝜇 − 𝑥)−1 )𝑑𝜆𝑑𝜇 𝜇−𝜆

= −(4𝜋 2 )−1 ∫ 𝑓(𝜆)(𝜆 − 𝑥)−1 (∫ Γ

Γ′

+(4𝜋 2 )−1 ∫ 𝑔(𝜇)(𝜇 − 𝑥)−1 (∫ Γ′

Γ

𝑔(𝜇) 𝑑𝜇) 𝑑𝜆 𝜇−𝜆 𝑓(𝜆) 𝑑𝜆) 𝑑𝜇 𝜇−𝜆

= (2𝜋i)−1 ∫ 𝑓(𝜆)𝑔(𝜆)(𝜆 − 𝑥)−1 𝑑𝜆 = (𝑓𝑔)A (𝑥). Γ

Q.E.D. The mapping A (𝜎(𝑥)) ∈ 𝑓 ↦ 𝑓A (𝑥) ∈ B (H ) is also called the Dunford operational calculus for the operator 𝑥. Corollary 2.26. Let 𝑥 ∈ B (H ) and 𝑓 ∈ A (𝜎(𝑥)). Then 𝜎(𝑓A (𝑥)) = {𝑓(𝜆); 𝜆 ∈ 𝜎(𝑥)}. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

43

Bounded linear operators in Hilbert spaces

Proof. Let 𝜆0 ∈ 𝜎(𝑥). We shall consider, on the domain of 𝑓, the analytic function 𝑔 defined by the formula: 𝑓(𝜆0 )−𝑓(𝜆)

𝑔(𝜆) = {

𝜆0 −𝜆

𝑓 ′ (𝜆0 )

for 𝜆 ≠ 𝜆0 for 𝜆 = 𝜆0 .

Then, from Theorem 2.25 (ii), we have 𝑓(𝜆0 ) − 𝑓A (𝑥) = (𝜆0 − 𝑥)𝑔A (𝑥); hence, the invertibility of 𝑓(𝜆0 ) − 𝑓A (𝑥) implies the invertibility of 𝜆0 − 𝑥, a contradiction. Consequently, 𝑓(𝜆0 ) − 𝑓A (𝑥) is not invertible and 𝑓(𝜆0 ) ∈ 𝜎(𝑓A (𝑥)). Conversely, let 𝜇0 ∈ 𝜎(𝑓A (𝑥)). If 𝜇0 ∉ {𝑓(𝜆); 𝜆 ∈ 𝜎(𝑥)}, then the formula ℎ(𝜆) =

1 𝜇0 − 𝑓(𝜆)

defines a function from A (𝜎(𝑥)) and from Theorem 2.25 (ii), we have ℎA (𝑥)(𝜇0 − 𝑓A (𝑥)) = 1, contrary to the assumption that 𝜇0 ∈ 𝜎(𝑓A (𝑥)). Therefore, we have 𝜇0 ∈ {𝑓(𝜆); 𝜆 ∈ 𝜎(𝑥)}. Q.E.D. Corollary 2.27. Let 𝑥 ∈ B (H ), 𝑓 ∈ A (𝜎(𝑥)) and 𝑔 ∈ A (𝜎(𝑓A (𝑥))). Then we have 𝑔A (𝑓A (𝑥)) = (𝑔 ∘ 𝑓)A (𝑥). Proof. Let Γ1 , Γ2 , … , Γ𝑘 , Γ1′ , Γ2′ , … , Γ𝑗′ be positively oriented, closed, rectifiable, Jordan curves, such that the interiors of the curves Γ1 , Γ2 , … , Γ𝑘 be mutually disjoint and their union include 𝜎(𝑥), the closure of this union be included in the domain of 𝑓 and its image by 𝑓 be included in the mutually disjoint union of the interiors of the curves Γ1′ , Γ2′ , … , Γ𝑗′ , whereas the closure of this union be included in the domain of 𝑔. We denote Γ = {Γ1 , … , Γ𝑘 } and Γ′ = {Γ1′ , … , Γ𝑗′ }. For any 𝜇 ∈ Γ′ , the formula ℎ(𝜆) =

1 𝜇 − 𝑓(𝜆)

determines a function ℎ ∈ A (𝜎(𝑥)). From Theorem 2.25 (ii), we have (𝜇 − 𝑓A (𝑥))ℎA (𝑥) = 1, and therefore, (𝜇 − 𝑓A (𝑥))−1 = ℎA (𝑥). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

44

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Therefore, we have 𝑔A (𝑓A (𝑥)) = (2𝜋i)−1 ∫ 𝑔(𝜇)(𝜇 − 𝑓A (𝑥))−1 𝑑𝜇 Γ′

= −(4𝜋 2 )−1 ∫ 𝑔(𝜇) (∫ Γ′

Γ

1 (𝜆 − 𝑥)−1 𝑑𝜆) 𝜇 − 𝑓(𝜆)

= (2𝜋i)−1 ∫(𝑔 ∘ 𝑓)(𝜆)(𝜆 − 𝑥)−1 𝑑𝜆 = (𝑔 ∘ 𝑓)A (𝑥). Γ

Q.E.D. 2.28. On the set ℂ\{𝜆; Re 𝜆 ≤ 0, Im 𝜆 = 0}, we define the function ln by the formula: ln 𝜆 = ln |𝜆| + arg 𝜆,

−𝜋 < arg 𝜆 < 𝜋.

On the same set, and for any 𝛼 ∈ ℂ, we define the function 𝜆 ↦ 𝜆𝛼 = exp(𝛼 ln 𝜆). The functions 𝜆 ↦ ln 𝜆 and 𝜆 ↦ 𝜆𝛼 are analytic in their domain of definition. Therefore, for any operator 𝑥 ∈ B (H ), such that 𝜎(𝑥) ⊂ ℂ\{𝜆; Re 𝜆 ≤ 0, Im 𝜆 = 0}, the operators ln 𝑥 and 𝑥 𝛼 , 𝛼 ∈ ℂ, is well defined. Corollary. Let 𝑥 ∈ B (H ) be such that 𝜎(𝑥) ⊂ ℂ\{𝜆; Re 𝜆 ≤ 0, Im 𝜆 = 0}. Then the mapping ℂ ∋ 𝛼 ↦ 𝑥 𝛼 ∈ B (H ) is an entire function (with respect to the norm topology in B (H )). Proof. Let 𝑦 = ln 𝑥. From Corollary 2.27, for any 𝛼 ∈ ℂ, we have 𝑥𝛼 = exp(𝛼𝑦). By taking into account Theorem 2.25, it is easy to verify the relation ∞

1 𝑛 𝑛 𝑦 𝛼 , 𝑛 𝑛=0

exp(𝛼𝑦) = ∑

whence it immediately follows that the mapping 𝛼 ↦ exp(𝛼𝑦) is an entire function. Q.E.D. 2.29. The following proposition yields a natural connection between the operational calculus with continuous functions (2.6) and the operational calculus with analytic functions (2.25). We obviously have A (𝜎(𝑥)) ⊂ C (𝜎(𝑥)), 𝑥 ∈ B (H ). Proposition. Let 𝑥 ∈ B (H ) be a self-adjoint operator and 𝑓 ∈ A (𝜎(𝑥)). Then we have 𝑓A (𝑥) = 𝑓(𝑥). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

Bounded linear operators in Hilbert spaces

45

Proof. For any 𝑔 ∈ A (𝜎(𝑥)), the operator 𝑔A (𝑥) is normal. By virtue of Corollary 2.26, we have 𝜎(𝑔A (𝑥)) = {𝑔(𝜆); 𝜆 ∈ 𝜎(𝑥)}, and therefore, from Lemmas 2.3 and 2.5, we have ‖𝑔A (𝑥)‖ = |𝜎(𝑔A (𝑥))| = sup{|𝑔(𝜆)|; 𝜆 ∈ 𝜎(𝑥)}. Therefore, the mapping

𝑔|𝜍(𝑥) ↦ 𝑔A (𝑥), 𝑔 ∈ A (𝜎(𝑥))

is isometric. Since {𝑔|𝜍(𝑥) ; 𝑔 ∈ A (𝜎(𝑥))} is a dense subset of C (𝜎(𝑥)), the preceding mapping can be uniquely extended to an isometric mapping of C (𝜎(𝑥)) into B (H ). By taking into account Theorem 2.25 (i), and also the uniqueness part of Theorem 2.6, we infer that this extension coincides with the mapping C (𝜎(𝑥)) ∋ 𝑓 ↦ 𝑓(𝑥) ∈ B (H ), which was defined in Theorem 2.6.

Q.E.D.

We observe that from the preceding proposition it follows, in particular, that for any self-adjoint operator 𝑥 ∈ B (H ) and any 𝑓 ∈ A (𝜎(𝑥)), the operator 𝑓A (𝑥) depends only on 𝑓|𝜍(𝑥) . 2.30. Let 𝛼 ∈ ℂ, Re 𝛼 > 0. We consider the mapping [0, +∞) ∋ 𝜆 ↦ 𝜆𝛼 ∈ ℂ, defined on (0, +∞) as in 2.28 and equal to zero at 0. This mapping is Borel measurable and bounded on compact sets. Thus, for any positive operator 𝑥 ∈ B (H ), the (normal) operator 𝑥𝛼 makes sense. From Proposition 2.29, this definition is compatible with that given in 2.28. Corollary. Let 𝑥 ∈ B (H ) be a positive operator and 𝜉 ∈ H ; then the mapping 𝛼 ↦ 𝑥𝛼 𝜉 ∈ H is continuous on {𝛼; Re 𝛼 > 0} and analytic in {𝛼; Re 𝛼 > 0}, with respect to the norm topology in H . Proof. From Corollary 2.22, there exists a sequence of operators {𝑒𝑛 } ⊂ R ({𝑥}), such that 𝑥𝑒𝑛 ≥ We write 𝑥𝑛 =

1 𝑒 , 𝑒 ↑ s(𝑥). 𝑛 𝑛 𝑛

1 (1 − 𝑒𝑛 ) + 𝑥𝑒𝑛 . 𝑛

Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

46 Since 𝜎(𝑥𝑛 ) ⊂ {𝜆; Re 𝜆 ≥ mapping

Lectures on von Neumann algebras 1 𝑛

}, from Corollary 2.28, it follows that, for any 𝜉 ∈ 𝑒𝑛 H , the 𝛼 ↦ 𝑥 𝛼 𝜉 = (𝑥𝑛 )𝛼 𝜉

is continuous on {𝛼; Re 𝛼 ⩾ 0} and analytic in {𝛼; Re 𝛼 > 0}. Obviously, if 𝜉 ∈ n(𝑥)H (with the notation from 2.13), the mapping 𝛼 ↦ 𝑥𝛼 𝜉 = 0 is continuous on {𝛼; Re 𝛼 > 0} and analytic in {𝛼; Re 𝛼 > 0}). Let now 𝜉 ∈ H be arbitrary. Since the set ∞

n(𝑥)H ∪

⋃

𝑒𝑛 H

𝑛=1

is total in H , there exists a sequence {𝜉𝑘 }, which converges to 𝜉, and is such that the mappings 𝛼 ↦ 𝑥 𝛼 𝜉𝑘 ; 𝑘 = 1, 2, … be continuous on {𝛼; Re 𝛼 ⩾ 0} and analytic in {𝛼; Re 𝛼 > 0}. Since the mappings 𝛼 ↦ 𝑥 𝛼 𝜉𝑘 converge uniformly, on any compact subset of {𝛼; Re 𝛼 > 0}, to the mapping 𝛼 ↦ 𝑥 𝛼 𝜉, it follows that the mapping 𝛼 ↦ 𝑥 𝛼 𝜉 is continuous on {𝛼; Re 𝛼 ⩾ 0} and analytic in {𝛼; Re 𝛼 > 0}. Q.E.D. 2.31. In the preceding sections, we already used the fact that, since the function exp is an entire function, for any operator 𝑥 ∈ B (H ), the operator exp(𝑥) makes sense and the relation ∞ 1 exp(𝑥) = ∑ 𝑥 𝑛 𝑛! 𝑛=0 holds. We hence infer that, for any 𝑥 ∈ B (H ), we have (exp(𝑥))∗ = 𝑒𝑥𝑝(𝑥 ∗ ); also, if 𝑥, 𝑦 ∈ B (H ), 𝑥𝑦 = 𝑦𝑥, we have exp(𝑥) exp(𝑦) = exp(𝑥 + 𝑦). In particular, if 𝑥 ∈ B (H ) is self-adjoint, then the operator exp(𝑖𝑥) is unitary. Proposition. Let 𝑥1 , 𝑥2 , 𝑦 ∈ B (H ). If 𝑥1 , 𝑥2 are normal and if 𝑥1 𝑦 = 𝑦𝑥2 , then 𝑥1∗ 𝑦 = 𝑦𝑥2∗ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

Bounded linear operators in Hilbert spaces

47

Proof. The function 𝑓 ∶ ℂ ∋ 𝜆 ↦ exp(−𝜆𝑥1∗ )𝑦 exp(𝜆𝑥2∗ ) ∈ B (H ) is an entire analytic function with respect to the norm topology of B (H ). From the relation 𝑦𝑥2 = 𝑥1 𝑦, we infer that, for any 𝜆 ∈ ℂ, we have ̄ 1 )𝑦 exp(−𝜆𝑥 ̄ 2 ), 𝑦 = exp(𝜆𝑥 and therefore, we have ̄ 1 )𝑦 exp(−𝜆𝑥 ̄ 2 ) exp(𝜆𝑥2∗ ) 𝑓(𝜆) = exp(−𝜆𝑥1∗ ) exp(𝜆𝑥 ̄ 1 ))𝑦 exp(i(i𝜆𝑥 ̄ 2 − i𝜆𝑥2∗ )). = exp(i(i𝜆𝑥1∗ − i𝜆𝑥 ̄ 1 and i𝜆𝑥 ̄ 2 − i𝜆𝑥2∗ are self-adjoint, their exponentials are Since the operators i𝜆𝑥1∗ − i𝜆𝑥 unitary operators; therefore, the function 𝑓 is also bounded. From the Liouville theorem, it follows that 𝑓 is constant. Consequently, its derivative is equal to zero: 0 = 𝑓 ′ (𝜆) = −𝑥1∗ exp(−𝜆𝑥1∗ )𝑦 exp(𝜆𝑥2∗ ) + exp(−𝜆𝑥1∗ )𝑦 exp(𝜆𝑥2∗ )𝑥2∗ . In particular, we have 𝑓 ′ (0) = 0, which implies that 𝑥1∗ 𝑦 = 𝑦𝑥2∗ . Q.E.D. From the preceding proposition, we infer, in particular, that if an operator 𝑦 commutes with a normal operator 𝑥, then it commutes with its adjoint 𝑥 ∗ too. 2.32. In this section, we recall the structure of operators in Hilbert direct sums of Hilbert spaces and we introduce some notations. Let H be a Hilbert space, 𝛾 an arbitrary cardinal number, 𝐼 a set of indices, whose cardinal is 𝛾, and (H 𝑖 )𝑖∈𝐼 a family of Hilbert spaces, such that H 𝑖 = H for any 𝑖 ∈ 𝐼. We consider the Hilbert direct sum ˜ 𝛾 = ⊕𝑖∈𝐼 H 𝑖 . H ˜ 𝛾 are the families 𝜉 ̃ = (𝜉𝑖 )𝑖∈𝐼 ⊂ H , such that The elements of the Hilbert space H ˜ 𝛾 , we ∑𝑖∈𝐼 ‖𝜉𝑖 ‖2 < +∞, whereas for any two elements 𝜉 ̃ = (𝜉𝑖 )𝑖∈𝐼 , 𝜂 ̃ = (𝜂𝑖 )𝑖∈𝐼 ∈ H have, by definition (𝜉|̃ 𝜂)̃ = ∑(𝜉𝑖 |𝜂𝑖 ). 𝑖∈𝐼

For any 𝑖0 ∈ 𝐼, we consider the operator ˜𝛾, 𝑢𝑖0 ∶ H ∋ 𝜉 ↦ 𝑢𝑖0 (𝜉) ∈ H Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

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where, for any 𝑢 ∈ H , we define 𝑢𝑖0 (𝜉) = (𝜉𝑖 )𝑖∈𝐼 ,

𝜉𝑖 = {

0 for 𝑖 ≠ 𝑖0 𝜉 for 𝑖 = 𝑖0 .

The adjoint of this operator is ˜ 𝛾 ∋ 𝜉 ̃ ↦ 𝑢𝑖∗ (𝜉) ∈ H , 𝑢𝑖∗0 ∶ H 0 ˜ 𝛾 , we have where, for any 𝜉 ̃ = (𝜉𝑖 )𝑖∈𝐼 ∈ H 𝑢𝑖∗0 (𝜉)̃ = 𝜉𝑖0 . It is easily checked that for any 𝑖 ∈ 𝐼, the operator 𝑢𝑖 is a linear isometric operator, and 𝑢𝑖∗ 𝑢𝑖 = the identity on H , ˜ 𝛾 onto H 𝑖 . 𝑢𝑖 𝑢𝑖∗ = the projection of H ˜ 𝛾 ), one can associate a ‘matrix’ (𝑥𝑖𝑘 ) of operators from To any operator 𝑥 ∈ B (H B (H ), by the relation 𝑥𝑖𝑘 = 𝑢𝑖∗ ∘ 𝑥 ∘ 𝑢𝑘 , 𝑖, 𝑘 ∈ 𝐼, with its help the operator 𝑥 can be recovered by the formula: (*)

𝑥 = ∑ 𝑢𝑖 ∘ 𝑥𝑖𝑘 ∘ 𝑢𝑘∗ , 𝑖,𝑘∈𝐼

where the series is so-convergent. Conversely, if 𝛾 is a finite cardinal number, then to any ‘matrix’ of elements from B (H ), ˜ 𝛾 ) by the formula (*). there corresponds an operator from B (H If 𝛾 is an infinite cardinal, then, of course, only those ‘matrices’ that satisfy the ˜ 𝛾 ). For example, for any convergence condition from (*) can yield operators from B (H family (𝑥𝑖 )𝑖∈𝐼 ⊂ B (H ), such that sup𝑖∈𝐼 ‖𝑥𝑖 ‖ < +∞, the ‘matrix’ (𝛿𝑖𝑘 𝑥𝑖 ) yields ˜ 𝛾 ), which leaves invariant all the subspaces H 𝑖 ; obviously, any an operator from B (H ˜ 𝛾 ), having this property, is of this form. operator from B (H In particular, for any 𝑥 ∈ B (H ), we can consider the operator ˜ 𝛾 ), 𝑥̃ = (𝛿𝑖𝑘 𝑥) ∈ B (H ˜ 𝛾 ). It is easy to see that if an operator which commutes with all operators 𝑢𝑖 𝑢𝑘∗ ∈ B (H ˜ 𝛾 ) commutes with all the operators 𝑢𝑖 𝑢∗ ∈ B (H ˜ 𝛾 ), then 𝑥𝑖𝑘 = 0 for (𝑥𝑖𝑘 ) ∈ B (H 𝑘 𝑖 ≠ 𝑘 and 𝑥𝑖𝑖 = 𝑥𝑘𝑘 for any 𝑖, 𝑘 ∈ 𝐼; consequently, there exists an 𝑥 ∈ B (H ), such that (𝑥𝑖𝑘 ) = 𝑥.̃ Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.005

49

Bounded linear operators in Hilbert spaces Let X ⊂ B (H ). We shall use the notations: ˜ 𝛾 ); 𝑥𝑖𝑘 ∈ X for any 𝑖, 𝑘 ∈ 𝐼} Mat𝛾 (X ) = {(𝑥𝑖𝑘 ) ∈ B (H ˜ 𝛾 ); 𝑥 ∈ X }. X 𝛾̃ = {𝑥̃ ∈ B (H ˜ 𝛾 ) = Mat𝛾 (B (H )). Therefore, we have B (H ˜ 𝛾 ), it is easy to check that we have If 𝜆 ∈ ℂ and (𝑥𝑖𝑘 ), (𝑦𝑖𝑘 ) ∈ B (H (𝑥𝑖𝑘 ) + (𝑦𝑖𝑘 ) = (𝑥𝑖𝑘 + 𝑦𝑖𝑘 ), 𝜆(𝑥𝑖𝑘 ) = (𝜆𝑥𝑖𝑘 ), ∗ (𝑥𝑖𝑘 )∗ = (𝑥𝑘𝑖 ),

(𝑥𝑖𝑘 )(𝑦𝑖𝑘 ) = (∑ 𝑥𝑖𝑗 𝑦𝑗𝑘 ), 𝑗∈𝐼

the series in the right-hand member of the last equality being so-convergent. In particular, for 𝜆 ∈ ℂ, 𝑥, 𝑦 ∈ B (H ), we have 𝑥̃ + 𝑦 ̃ = (𝑥 + 𝑦), ̃ 𝜆𝑥̃ = (𝜆𝑥), ̃ (𝑥)∗ = (𝑥 ∗ ), ̃ 𝑥̃𝑦 ̃ = (𝑥𝑦). ̃ In what follows, if 𝛾 is a finite cardinal number, i.e., a natural number 𝑛, we shall write 𝑛 instead of 𝛾, whereas if 𝛾 is an infinite cardinal number, known from the context, then we shall omit it. 2.33. Let H , K be two Hilbert spaces and H ⊗ K their (algebraic) tensor product as vector spaces. In H ⊗ K , one can define a unique pre-Hilbert structure, such that (𝜉1 ⊗ 𝜂1 |𝜉2 ⊗ 𝜂2 ) = (𝜉1 |𝜉2 )(𝜂1 |𝜂2 ), for any 𝜉1 , 𝜉2 ∈ H , 𝜂1 , 𝜂2 ∈ H . This pre-Hilbert structure is separated. The Hilbert space obtained by the completion of the pre-Hilbert space H ⊗ K is called the Hilbert tensor ̄ . product of the spaces H and K , and it is denoted by H ⊗K Let 𝑥 ∈ B (H ), 𝑦 ∈ B (H ). The tensor product 𝑥 ⊗ 𝑦 of the linear operators 𝑥, 𝑦 is a continuous linear operator on H ⊗ K . Indeed, since 𝑥 ⊗ 𝑦 = (𝑥 ⊗ 1)(1 ⊗ 𝑦), we can assume, for example, that 𝑦 = 1. Let 𝑛

∑ 𝜉𝑘 ⊗ 𝜂𝑘 ∈ H ⊗ K ; 𝑘=1

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we can assume that the vectors 𝜂𝑘 are mutually orthogonal. We then have 𝑛

𝑛

‖(𝑥 ⊗ 1) ∑ 𝜉𝑘 ⊗ 𝜂𝑘 ‖2 = ‖ ∑ 𝑥𝜉𝑘 ⊗ 𝜂𝑘 ‖2 𝑘=1 𝑛

𝑘=1 𝑛

𝑛

= ∑ ‖𝑥𝜉𝑘 ‖2 ‖𝜂𝑘 ‖2 ≤ ‖𝑥‖2 ∑ ‖𝜉𝑘 ‖2 ‖𝜂𝑘 ‖2 = ‖𝑥‖2 ‖ ∑ 𝜉𝑘 ⊗ 𝜂𝑘 ‖2 , 𝑘=1

𝑘=1

𝑘=1

and the assertion is proved. Consequently, 𝑥 ⊗ 𝑦 can be extended in a unique manner to a ̄ ∈ B (H ⊗K ̄ ). continuous operator 𝑥 ⊗𝑦 It is easily verified that the mapping ̄ ∈ B (H ⊗K ̄ ) B (H ) × B (K ) ∋ (𝑥, 𝑦) ↦ (𝑥 ⊗𝑦) is bilinear (one can show that the corresponding linear mapping from the algebraic tensor product is injective); also, for any 𝑥1 , 𝑥2 ∈ B (H ), 𝑦1 , 𝑦2 ∈ B (K ), we have ̄ 1 )(𝑥2 ⊗𝑦 ̄ 2 ) = 𝑥1 𝑥2 ⊗𝑦 ̄ 1 𝑦2 , (𝑥1 ⊗𝑦 and for any 𝑥 ∈ B (H ), 𝑦 ∈ B (H ), we have ̄ ∗ = 𝑥 ∗ ⊗𝑦 ̄ ∗. (𝑥⊗𝑦) In particular, if 𝑥 ∈ B (H ) and 𝑦 ∈ B (K ) are self-adjoint (resp., normal, unitary or ̄ ∈ B (H ⊗K ̄ ) is a self-adjoint (resp., normal, unitary, projection) operators, then 𝑥⊗𝑦 projection) operator. Also if 𝑥 ∈ B (H ) and 𝑦 ∈ B (K ) are positive operators, then ̄ ∈ B (H ⊗K ̄ ) is a positive operator. If 𝑥 ∈ B (H ), 𝑦 ∈ B (K ) and 𝑥 = 𝑢|𝑥|, 𝑦 = 𝑥 ⊗𝑦 ̄ = (𝑢⊗𝑣)(|𝑥| ̄ ̄ 𝑣|𝑦| are the polar decompositions of these operators, then 𝑥 ⊗𝑦 ⊗|𝑦|) is the ̄ ∈ B (H ⊗K ̄ ). polar decomposition of 𝑥 ⊗𝑦 2.34. Let H , K be Hilbert spaces, (𝜂𝑖 )𝑖∈𝐼 an orthonormal basis in K and 𝛾 = dim K = card 𝐼. In what follows we shall show that the orthonormal basis we have chosen allows a ˜𝛾. ̄ canonical identification of the Hilbert spaces H ⊗K and H Indeed, for any 𝑖 ∈ 𝐼, the linear, isometric mapping ̄ H ∋ 𝜉 ↦ 𝜉 ⊗ 𝜂𝑖 ∈ H ⊗K determines a canonical identification of the Hilbert space H with a closed subspace H 𝑖 of ̄ . The spaces H 𝑖 are mutually orthogonal, whereas their union ⋃ H 𝑖 is total in H ⊗K 𝑖∈𝐼 ̄ . Consequently, H ⊗K ̄ H ⊗K is the Hilbert direct sum of the spaces H 𝑖 . Therefore, the mapping (𝜉𝑖 )𝑖∈𝐼 ↦ ∑ 𝜉𝑖 ⊗ 𝜂𝑖 𝑖∈𝐼

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Bounded linear operators in Hilbert spaces establishes a canonical identification ˜ 𝛾 = H ⊗K ̄ . H

̄ ) can be represented Once this identification has been done, the operators from B (H ⊗K by ‘matrices’ of operators from B (H ). For example, it is easily checked that, for any 𝑥 ∈ B (H ), we have ̄ 𝑥̃ = 𝑥 ⊗1. 2.35. As a particular case of 2.32, consider a Hilbert space H with an orthonormal basis {𝜀𝑖 }𝑖∈𝐼 and 𝑥, 𝑦 ∈ B (H ) and define the matrix coefficients of 𝑥 and 𝑦 with respect to this basis: 𝑥𝑖𝑗 = (𝑥𝜀𝑗 ∣ 𝜀𝑖 ) ∈ ℂ, 𝑦𝑖𝑗 = (𝑦𝜀𝑗 ∣ 𝜀𝑖 ) , (𝑖, 𝑗 ∈ 𝐼). Clearly (𝑥 + 𝑦)𝑖𝑗 = 𝑥𝑖𝑗 + 𝑦𝑖𝑗 , (𝜆𝑥)𝑖𝑗 = 𝜆𝑥𝑖𝑗 ,

(𝜆 ∈ ℂ),

∗

(𝑥 )𝑖𝑗 = 𝑥 𝑗𝑖 , and also (𝑥𝑦)𝑖𝑗 = ∑ 𝑥𝑖𝑘 𝑦𝑘𝑗 , 𝑘∈𝐼

since (𝑥𝑦)𝑖𝑗 = (𝑥𝑦𝜀𝑗 ∣ 𝜀𝑖 ) = (𝑦𝜀𝑗 ∣ 𝑥 ∗ 𝜀𝑖 ) = (∑ (𝑦𝜀𝑗 ∣ 𝜀𝑘 ) 𝜀𝑘 ∣ 𝑥 ∗ 𝜀𝑖 ) 𝑘

= ∑ (𝑦𝜀𝑗 ∣ 𝜀𝑘 ) (𝑥𝜀𝑘 ∣ 𝜀𝑖 ) = ∑ 𝑥𝑖𝑘 𝑦𝑘𝑗 . 𝑘

𝑘

This computation is not just formal; it ensures the summability of the families {𝑥𝑖𝑘 𝑦𝑘𝑗 }𝑘 , for all 𝑖, 𝑗 ∈ 𝐼. Notice also that ∑ |𝑥𝑖𝑗 |2 = ‖𝑥𝜀𝑗 ‖2 ≤ ‖𝑥‖2 ,

for all 𝑗 ∈ 𝐼,

𝑖

∑ |𝑥𝑖𝑗 |2 = ‖𝑥 ∗ 𝜀𝑖 ‖2 ≤ ‖𝑥‖2 ,

for all 𝑖 ∈ 𝐼.

𝑗

It is obvious that 𝑥 = 0 ⇔ 𝑥𝑖𝑗 = 0,

(∀)𝑖, 𝑗 ∈ 𝐼.

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Of course, if card(𝐼) = 𝑛 < ∞, this is just the usual identification B (ℂ𝑛 ) ∋ 𝑥 ≡ [𝑥𝑖𝑗 ] ∈ Mat𝑛 (ℂ). When H is infinite dimensional, the same matrix formalism can be used to check egalities in B (H ) or to perform computations such as (𝑥𝜉 ∣ 𝑒𝑖 ) = ∑ 𝑥𝑖𝑗 (𝜉 ∣ 𝑒𝑗 ) ,

(𝑥 ∈ B (H ), 𝜉 ∈ H ).

𝑗

EXERCISES E.2.1 Let 𝑥, 𝑦 ∈ B (H ). If 1 − 𝑥𝑦 is invertible, then 1 − 𝑦𝑥 is also invertible. Infer that 𝜎(𝑥𝑦)\{0} = 𝜎(𝑦𝑥)\{0}. E.2.2 Let 𝑥 ∈ B (H ) be self-adjoint. Show that ‖𝑥‖ = sup{|(𝑥𝜉|𝜉)|; 𝜉 ∈ H , ‖𝜉‖ = 1}. !E.2.3 Let 𝑒, 𝑓 ∈ B (H ) be projections. Show that 1(𝑒𝑓) = 𝑒 − 𝑒 ∧ (1 − 𝑓),

r(𝑒𝑓) = 𝑓 − (1 − 𝑒) ∧ 𝑓;

1(𝑒(1 − 𝑓)) = 𝑒 − 𝑒 ∧ 𝑓,

r(𝑒(1 − 𝑓)) = 𝑒 ∨ 𝑓 − 𝑓.

E.2.4 Let 𝑒, 𝑓 ∈ B (H ) be projections. Then the sequence {(𝑒𝑓)𝑛 } so-converges to 𝑒 ∧ 𝑓. E.2.5 For any projection 𝑒 ∈ B (H ), the operator 𝑠 = 1 − 2𝑒 is a symmetry (i.e., self-adjoint and unitary). Conversely, any symmetry is of this form. !E.2.6 Let 𝑎, 𝑏 ∈ B (H ), 0 ≤ 𝑎 ≤ 𝑏. Show that there exists an 𝑥 ∈ B (H ), ‖𝑥‖ ≤ 1, such that 𝑎1/2 = 𝑥𝑏1/2 . !E.2.7 Let 𝑎, 𝑏 ∈ B (H ), 0 ≤ 𝑎 ≤ 𝑏, 𝑎𝑏 = 𝑏𝑎. Show that 0 < 𝑎2 < 𝑏2 . Infer that 𝑎H ⊂ 𝑏H . E.2.8 Let 𝑎, 𝑏 ∈ B (H ), 0 ≤ 𝑎 ≤ 𝑏, a invertible. Then 𝑏 is invertible and 0 ≤ 𝑏−1 ≤ 𝑎−1 . E.2.9 Let {𝑥𝑖 }𝑖∈𝐼 ⊂ B (H ) be a net of normal operators, which so-converges to the normal operator 𝑥 ∈ B (H ). Then the net {𝑥𝑖∗ }𝑖∈𝐼 is so-convergent to 𝑥 ∗ . In other words, the restriction of the *-operation to the set of normal operators is so-continuous. E.2.10 Show that if 𝑥 ∈ B (H ) is normal, then ‖𝑥‖2 = ‖𝑥 2 ‖. ▶

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E.2.11 Two operators 𝑥, 𝑦 ∈ B (H ) are said to be similar (resp., unitarily equivalent) if there exists an invertible (resp., unitary) operator 𝑠 ∈ B (H ), such that 𝑦 = 𝑠𝑥𝑠−1 . Show that two normal similar operators are unitarily equivalent. E.2.12 Let A be a commutative Banach algebra with unit 1 ∈ A . Show that any element 𝑥 ∈ A , such that ‖1 − 𝑥‖ < 1 is invertible. Infer that any maximal ideal 𝔐 of A is closed and any non-zero element from the Banach algebra A /𝔐 is invertible. Then, with the help of the Liouville theorem, infer that A /𝔐 consists of the scalar multiples of the unit element. E.2.13 Let 𝑥 ∈ B (H ) be normal and 𝜆 ∈ ℂ. Then 𝜆 ∈ 𝜎(𝑥) iff 𝜆 − 𝑥 belongs to a maximal ideal of C ∗ ({𝑥, 1}). E.2.14 Let 𝑥 ∈ B (H ) be normal and 𝑝(., .) be a complex polynomial in two variables, Then, 𝜎(𝑝(𝑥, 𝑥 ∗ )) = {𝑝(𝜆, 𝜆); 𝜆 ∈ 𝜎(𝑥)}. E.2.15 Extend Theorem 2.6 and Corollary 2.7 to the case of normal operators 𝑥 ∈ B (H ). E.2.16 Let 𝑥 ∈ B (H ) be a self-adjoint operator, {𝑒𝜆 } its spectral scale and 𝑓 ∈ C (𝜎(𝑥)). Then +∞

𝑓(𝑥) = ∫

𝑓(𝜆)𝑑𝑒𝜆 ,

−∞

where the integral is a norm-convergent vector Stieltjes integral. E.2.17 Let 𝑥 ∈ B (H ) be a self-adjoint operator. For any Borel subset 𝐷 of the spectrum 𝜎(𝑥) of 𝑥, we define the spectral projection of 𝑥, which corresponds to the 𝐷 by the formula 𝑒(𝐷) = 𝜒𝐷 (𝑥). Then, for any 𝑓 ∈ B (𝜎(𝑥)), we have ‖𝑓(𝑥)‖ = inf sup |𝑓(𝜆)| 𝑒(𝐷)=1 𝜆∈𝐷

and 𝜎(𝑓(𝑥)) =

⋂

{𝑓(𝜆); 𝜆 ∈ 𝐷} ⊂ {𝑓(𝜆); 𝜆 ∈ 𝜎(𝑥)}.

𝑒(𝐷)=1

E.2.18 Let 𝑥 ∈ B (H ) be a self-adjoint operator and 𝑓 ∈ B (𝜎(𝑥)) real. Then, for any 𝑔 ∈ B ({𝑓(𝜆); 𝜆 ∈ 𝜎(𝑥)}), we have 𝑔(𝑓(𝑥)) = (𝑔 ∘ 𝑓)(𝑥). ▶

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*E.2.19 An operator 𝑥 ∈ B (H ) is said to be compact if for any bounded subset S ⊂ H , the set 𝑥(S ) is relatively compact. One usually denotes by K (H ) (the set of all compact operators from B (H ). Show that K (H ) is the smallest non-zero, norm-closed, two-sided ideal of B (H ). If H is infinitely dimensional and separable, then K (H ) is the only proper, norm-closed, two-sided ideal of B (H ). *E.2.20 Let 𝑥 ∈ B (H ) and 𝜆 ∈ ℂ. One says that 𝜆 is an eigenvalue of 𝑥 if E 𝜆 = ker(𝑥 − 𝜆) ≠ 0; in this case, the non-zero vectors from E 𝜆 are called eigenvectors, and the dimension of E 𝜆 is called the multiplicity of the eigenvalue 𝜆. If 𝜆 is an eigenvalue of 𝑥, then 𝜆 ∈ 𝜎(𝑥). The eigenvectors that correspond to different eigenvalues of a self-adjoint operator are orthogonal. Show that if 𝑥 ∈ K (H ) and 0 ≠ 𝜆 ∈ 𝜎(𝑥), then 𝜆 is an eigenvalue of finite multiplicity of 𝑥. Thus, the spectrum of a compact operator either is a finite set or forms a sequence converging to zero. E.2.21 Let 𝑥 ∈ K (H ), 𝑥 ≥ 0, 𝜎(𝑥)\{0} = {𝜆1 , 𝜆2 , …} and let 𝑒𝑘 be the orthogonal projection onto E 𝜆𝑘 . Then the projections 𝑒𝑘 are mutually orthogonal and 𝑠 = ∑ 𝜆𝑘 𝑒𝑘 , 𝑘

the series being norm convergent. *E.2.22 Let 𝑥 ∈ B (H ), ‖𝑥‖ ≤ 1. Show that there exists an isometry 𝑣 of H into a Hilbert space K and a unitary operator 𝑢 ∈ B (K ), such that (i) 𝑥 𝑛 = 𝑣 ∗ 𝑢𝑛 𝑣; 𝑛 = 1, 2, … (ii) the set ∑𝑛∈ℤ 𝑢𝑛 𝑣(H ) is total in K . The pair (𝑣 ∶ H → K , 𝑢) is called the minimal unitary dilation of the ‘contraction’ 𝑥, and it is unique in an obvious sense. E.2.23 Let 𝑥 ∈ B (H ), ‖𝑥‖ ≤ 1 and 𝑝 be a complex polynomial. With the help of the unitary dilation of 𝑥, prove the following von Neumann inequality ‖𝑝(𝑥)‖ ≤ sup{|𝑝(𝜆)|; 𝜆 ∈ ℂ, |𝜆| = 1}. E.2.24 Let 𝑥 ∈ B (H ), ‖𝑥‖ ≤ 1. Show that for any 𝜉 ∈ H , we have 𝑥𝜉 = 𝜉 ⇔ 𝑥 ∗ 𝜉 = 𝜉, whence infer that n(1 − 𝑥) = n(1 − 𝑥 ∗ ) ≤ l(𝑥) ∧ r(𝑥). ▶

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By denoting 𝑦 = 𝑥 − n(1 − 𝑥), show that n(1 − 𝑦) = n(1 − 𝑦 ∗ ) = 0. E.2.25 Let 𝑥 ∈ B (H ), ‖𝑥‖ ≤ 1. Show that for any 𝜉 ∈ H , we have the following convergence: 1 (𝜉 + 𝑥𝜉 + ⋯ + 𝑥 𝑛−1 𝜉) → 𝑝𝜉, 𝑛 where 𝑝 is the orthogonal projection onto {𝜁 ∈ H ; 𝑥𝜁 = 𝜁}. This result is the mean ergodic theorem of von Neumann. (Hint: Let H 1 = {𝜁 ∈ K ; 𝑥𝜁 = 𝜁} and H 2 = {𝜂 − 𝑥𝜂; 𝜂 ∈ H }; in accordance with E.2.24, we have K 1 = K ⊥2 , and therefore, K = H 1 ⊗ H 2 ; the convergence can be checked, separately, for 𝜉 ∈ H 1 and 𝜉 ∈ H 2 . The proof we have sketched here is by B. Sz.-Nagy. For other information, we refer the reader to N. Dunford and J. Schwartz (1958/1963/1970, Chap. VIII).

COMMENTS C.2.1 In this section, we will state briefly some results concerning the theory of abstract 𝐶 ∗ -algebras. In doing so, we will repeat the operational calculus for normal operators. If A is a 𝐶 ∗ -algebra, in the algebra A ̃ obtained by the adjunction of the unit element to A , one can introduce canonically a structure of a 𝐶 ∗ -algebra. For simplicity’s sake, we shall assume in what follows that all 𝐶 ∗ -algebras encountered have a unit. The ideas and results from Sections 2.3 to 2.5 also extend to the case of the abstract 𝐶 ∗ -algebras and with the same proofs. Let A be a commutative 𝐶 ∗ -algebra. A character of A is any non-zero homomorphism 𝜔 ∶ A → ℂ. For any 𝑥 ∈ A , the element 𝜔(𝑥) − 𝑥 belongs to the kernel ker 𝜔, which is a two-sided ideal of A , and therefore, 𝜔(𝑥) ∈ 𝜍(𝑥). It follows that |𝜔(𝑥)| ≤ |𝜍(𝑥)| ≤ ‖𝑥‖, and if 𝑥 is self-adjoint, then 𝜔(𝑥) is real. Consequently, ‖𝜔‖ = 1 and 𝜔(𝑥 ∗ ) = 𝜔(𝑥), 𝑥 ∈ A . The set ΩA of all characters of A , endowed with the topology induced by the 𝜍(A ∗ , A )-topology, is a compact space, called the spectrum of A . Any element 𝑥 ∈ A determines a function 𝑥̂ ∈ C (ΩA ), given by ̂ 𝑥(𝜔) = 𝜔(𝑥),

𝜔 ∈ ΩA .

The mapping 𝜔 ↦ ker 𝜔 establishes a bijection between the set of all characters of A and the maximal (two-sided) ideals of A (see E.2.12). If

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Lectures on von Neumann algebras 𝑥 ∈ A and 𝜆 ∈ 𝜍(𝑥), then 𝜆 − 𝑥 belongs to a maximal ideal of A (E.2.13); therefore, there exists an 𝜔 ∈ ΩA , such that 𝜔(𝑥) = 𝜆. Consequently, for any 𝑥 ∈ A , we have ̂ ‖𝑥‖ = |𝜍(𝑥)| = sup{|𝜔(𝑥)|; 𝜔 ∈ ΩA } = ‖𝑥‖. From the preceding results and by taking into account the Stone–Weierstrass theorem, we infer the following. Theorem (the Gelfand representation). Let A be a comnutative 𝐶 ∗ -algebra. The mapping 𝑥 ↦ 𝑥̂ establishes an isometric *-isomorphism of the 𝐶 ∗ -algebra A onto the 𝐶 ∗ -algebra C (ΩA ) . Let 𝑥 be a normal element of an arbitrary 𝐶 ∗ -algebra, e.g., a normal operator from B (H ). If A = C ∗ ({𝑥, 1}), then the mapping 𝜔 ↦ 𝜔(𝑥) establishes a homeomorphism of ΩA onto 𝜍(𝑥). Consequently, there exists an isometric *-isomorphism C (𝜍(𝑥)) ∋ 𝑓 ↦ 𝑓(𝑥) ∈ C ∗ ({𝑥, 1}), which is called the operational calculus for the normal element 𝑥. In particular, Theorem 2.6 also extends to normal operators (see E.2.15). The notion of positive operator extends, with the same definition, to the notion of a positive element, whereas the results from 2.7 to 2.10 extend to arbitrary 𝐶 ∗ -algebras, with the same proof. Also, the equivalence of statements (i), (ii) and (iii) from Proposition 2.12 remains true, but, in order to prove this, the following remarks are necessary. Let A be a 𝐶 ∗ -algebra and A + = {𝑥 ∈ A ; 𝑥 ≥ 0}; a self-adjoint element 𝑥 ∈ A , ‖𝑥‖ ≤ 1, is positive iff ‖1−𝑥‖ ≤ 1; this can easily be proved using the Gelfand representation; with the help of this result, one can easily prove that A + is a closed convex cone and A + ∩ (−A + ) = {0}. If 𝑥 ∈ A and 𝑥∗ 𝑥 ∈ (−A + ), then 𝑥 = 0. Indeed, let 𝑥 = ℎ+𝑖𝑘, where ℎ, 𝑘 ∈ A are self-adjoint. From the hypothesis −𝑥∗ 𝑥 ∈ A + and from E.2.1, it follows that −𝑥𝑥 ∗ ∈ A + . Then one can immediately check that 𝑥∗ 𝑥 = 2ℎ2 + 2𝑘 2 + (−𝑥𝑥 ∗ ) ∈ A + , hence 𝑥∗ 𝑥 ∈ A + ∩ (−A + ) = {0}, which implies that 𝑥 = 0. With the help of these hints, the implication (iii) ⇒ (i) from 2.12 follows with a slight modification of the argument just used in the implication (iv) ⇒ (i) from 2.12.

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Using the Gelfand representation and the first remark from Section 3.12, one can show that any injective *-homomorphism of 𝐶 ∗ -algebras is isometric. Let A be an abstract 𝐶 ∗ -algebra. For any element 𝑥 ∈ A , 𝑥 ≠ 0, we have −𝑥∗ 𝑥 ∉ A + . From the Krein–Rutman theorem, there exists a positive form (see 5.1) 𝜑𝑥 on A , such that 𝜑𝑥 (−𝑥 ∗ 𝑥) < 0. With the help of the form 𝜑𝑥 , one gets, as in Section 5.18, a *-homomorphism 𝜋𝜑𝑥 ∶ A → B (H 𝜑𝑥 ), 𝜋𝜑𝑥 (𝑥) ≠ 0. By denoting 𝜋 ∶ A → B (H ) the direct sum of the mappings 𝜋𝜑𝑥 , 𝑥 ∈ A , it follows that 𝜋 is an injective *-homomorphism of the 𝐶 ∗ -algebra A into the 𝐶 ∗ -algebra B (H ). We can thus obtain the following: Theorem. Any 𝐶 ∗ -algebra is isometrically *-isomorphic with a 𝐶 ∗ -algebra of operators on a Hilbert space. I. M. Gelfand and M. A. Naimark defined the notion of a 𝐶 ∗ -algebra by the following axioms: (I) A is *-algebra. (II) A is a Banach space, with the vector structure of (I). (III) ‖𝑥𝑦‖ ≤ ‖𝑥‖‖𝑦‖, for any 𝑥, 𝑦 ∈ A . (IV) ‖𝑥 ∗ 𝑥‖ = ‖𝑥 ∗ ‖‖𝑥‖, for any 𝑥 ∈ A . (V) ‖𝑥 ∗ ‖ = ‖𝑥‖, for any 𝑥 ∈ A . (VI) 1 + 𝑥 ∗ 𝑥 is invertible in A , for any 𝑥 ∈ A . With this definition, they proved the preceding theorem, and they made the conjecture that axiom (VI) and axiom (V) follow from the other axioms. M. Fukamiya (1952; Fukamiya considered only the commutative case, but his arguments were general, as noticed by I. Kaplansky and recorded by J. A. Schatz in his review of the paper of M. Fulkamiya 1952.) and J. L. Kelley and R. L. Vaught (1953) proved that, indeed axiom (VI) follows from the other axioms, that is, they proved the equivalence of statements (i), (ii) and (iii) from 2.12 for the abstract case, using the arguments we have briefly mentioned above. J. Glimm and R. V. Kadison (1960) proved that axiom (V) also follows from axioms (I) to (IV), if A has the unit element; J. Vowden (1967) proved the same for the general case, thus solving positively and completely the Gelfand–Naimark conjecture. The conjunction of axioms (IV) and (V) is obviously equivalent to ′

(IV ) ‖𝑥 ∗ 𝑥‖ = ‖𝑥‖2 ; for any 𝑥 ∈ A ,

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Lectures on von Neumann algebras this being the axiom we have adopted here (2.2). H. Araki and G. A. Elliott (1973) have shown that from axioms (I), (II) and (IV′ ), axiom (III) follows; they have also shown that axiom (III) follows from axioms (I), (II) and (IV), by assuming that the *-operation is continuous (see also Sebestyén [1973, 1974]). For the theory of 𝐶 ∗ -algebras, the reader is also referred to the books of J. Dixmier (1957/1969) and M. A. Naimark (1956/1968), where they can find a detailed exposition of the arguments presented in this section (see also Doran and Wichmann [1977]).

C.2.2 Many results from the theory of 𝐶 ∗ -algebras extend in a natural, but not trivial manner, to more general Banach algebras with involution. The older results in this direction can be found in the classical books of M. A. Naimark (1956/1968) and C. E. Rickart (1960). An elegant exposition of the new results can be found in V. Pták (1972) and F. Bonsall and J. Duncan (1973). C.2.3 To any normal operator 𝑥 in a separable Hilbert space, one can associate canonically a class of absolute continuity of finite Borel measures on 𝜍(𝑥), called the spectral type of 𝑥, and a function, defined on 𝜍(𝑥) and taking values in ℕ ∪ {+∞}, measurable with respect to the spectral type, and called the spectral multiplicity function of 𝑥. One can prove that two normal operators in separable Hilbert spaces are unitarily equivalent iff they have the same spectral type and the same spectral multiplicity function. The spectral type and the spectral multiplicity function allow the construction of a canonical form of the normal operator, called the spectral representation. This theory is presented, in detail, in the books of P. R. Halmos (1951) and N. Dunford and J. Schwartz (1958/1963/1970, Chap. X). C.2.4 Bibliographical comments. There exist many treatises and monographs containing the spectral theory of bounded linear operators in Hilbert spaces. We mention especially the books by F. Riesz and B. Sz.-Nagy (1952/1953/1954/1965), N. Dunford and J. Schwartz (1958/1963/1970, Chaps. VI, VII, IX and X), P. R. Halmos (1967) and C. T. Ionescu-Tulcea (1956). In our exposition, we used these sources, as well as a course by J. R. Ringrose 1966/1967 (see also Foia s¸ [1966]). For the analytic operational calculus, we refer the reader to N. Dunford and J. Schwartz (1958/1963/1970, Chap. VII). The case of the finite-dimensional Hilbert spaces is masterfully treated in the book by P. R. Halmos (1963). One was able to develop an analytic operational calculus of several commuting operators; this culminates with a deep theorem due to G. Shilov, R. Arens and A. Calderón (see Bourbaki [1967]). New investigations in this direction have been initiated by J. L. Taylor (1970a, b, 1973).

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Proposition 2.31 is known as the Fuglede–Putnam theorem, whereas the proof given here is due to M. Rosenblum (1958; see also Putnam [1967]). A systematic approach to the theory of unitary dilation (see E.2.22) can be found in the book by B. Sz.-Nagy and C. Foia s¸ (1968). The unitary dilation theorems are due to M. A. Naimark and B. Sz.-Nagy, whereas extensions of such theorems to 𝐶 ∗ -algebras were made by W. F. Stinespring (1955) and W. Arveson (1972; see also Suciu [1973]).

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3

VON NEUMANN ALGEBRAS

In this chapter, we present the density theorems of J. von Neumann and I. Kaplansky, and we introduce the elementary operations on von Neumann algebras. 3.1. Let H be a Hilbert space. For any subsets X ⊂ B (H ) and S ⊂ H , we write X S = {𝑥𝜉; 𝑥 ∈ X , 𝜉 ∈ S }, [X S ] = the closed vector subspace generated by X S . The projection in B (H ), which corresponds to the closed vector subspace [X S ], will be denoted by [X S ] too. If S = {𝜉}, 𝜉 ∈ H , we shall simply write X S = X 𝜉, [X S ] = [X 𝜉]. For any subset X ⊂ B (H ), we shall denote by X ′ the commutant of X : X ′ = {𝑥 ′ ∈ B (H ); 𝑥 ′ 𝑥 = 𝑥𝑥 ′ , for any 𝑥 ∈ X } and by X

″

the bicommutant of X : X ″ = (X ′ )′ ;

by induction we can define the (𝑛 + 1)-th commutant of X to be the commutant of the 𝑛-th commutant of X . It is now easy to see that, whereas the obvious inclusion X ⊂ X″ may be strict, for any 𝑘 ≥ 1 we have the (2𝑘 − 1)-th commutant of X = X ′ , the (2𝑘)-th commutant of X = X ″ . For any subset X ⊂ B (H ), X ′ is an algebra that contains the identity operator 1 ∈ B (H ); moreover, it is easy to check that X ′ is so-closed (equivalently, it is wo-closed (1.4)). If X = X ∗ , then X ′ is a von Neumann algebra (see 2.2). In particular, if M is a von Neumann algebra, then the commutant M ′ is a von Neumann algebra. The passage to the commutant is the first elementary operation on von Neumann algebras. 60

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If X ⊂ B (H ), X = X ∗ and 𝑒 ∈ B (H ) are a projection, then 𝑒 ∈ X ′ iff 𝑥𝑒 = 𝑒𝑥𝑒 for any 𝑥 ∈ X . In particular, if X = X ∗ , then [X 𝜉] ∈ X ′ for any 𝜉 ∈ H . 3.2. The fundamental result of the theory of von Neumann algebras is the following: Theorem (von Neumann’s density theorem). Let A ⊂ B (H ), 1 ∈ A , be a *-algebra operators. Then the so-closure of A coincides with the bicommutant of A . Proof. It is sufficient to show that A is so-dense in A ″ . To this end, let us choose an element 𝑥″ ∈ A ″ . Let 𝜉 ∈ H . Then [A 𝜉] ∈ A ′ , and this implies, in particular, that the subspace [A 𝜉] is invariant for any operator from A ″ . Since 1 ∈ A , it follows that 𝜉 ∈ [A 𝜉], and therefore, 𝑥 ″ 𝜉 ∈ [A 𝜉]. Let now 𝜉1 , … , 𝜉𝑛 ∈ H . We introduce the notations (see 2.32): 𝜉 ̃ = (𝜉1 , … , 𝜉𝑛 ) ∈ H 𝑛̃ , A 𝑛̃ = {𝑥;̃ 𝑥 ∈ A } ⊂ B (H 𝑛̃ ). Then A 𝑛̃ ⊂ B (H 𝑛̃ ) is a *-algebra, 1 ∈ A 𝑛̃ and ′ ′ (A 𝑛̃ )′ = {(𝑥𝑖𝑗 ); 𝑥𝑖𝑗 ∈ A ′ , 𝑖, 𝑗 = 1, … , 𝑛} = Mat𝑛 (A ′ ).

Indeed, the relation ′ ′ ′ ′ 0 = 𝑥(𝑥 ̃ 𝑖𝑗 ) − (𝑥𝑖𝑗 )𝑥̃ = (𝑥𝑥𝑖𝑗 − 𝑥𝑖𝑗 𝑥), in B (H 𝑛̃ )

is satisfied for any 𝑥̃ ∈ A 𝑛̃ , iff the relations ′ ′ 𝑥𝑥𝑖𝑗 = 𝑥𝑖𝑗 𝑥, in B (H ),

𝑖, 𝑗 = 1, … , 𝑛

are satisfied for any 𝑥 ∈ A . ′ Consequently, for any (𝑥𝑖𝑗 ) ∈ (A 𝑛̃ )′ , we have ′ ′ ′ ′ ″ 𝑥̃″ (𝑥𝑖𝑗 ) − (𝑥𝑖𝑗 )𝑥̃″ = (𝑥 ″ 𝑥𝑖𝑗 − 𝑥𝑖𝑗 𝑥 ) = 0,

i.e., 𝑥̃″ ∈ (A 𝑛̃ )″ . According to the fist part of the proof, it follows that ̃ 𝑥̃″ 𝜉 ̃ ∈ [A 𝑛̃ 𝜉]. Hence, there exists a sequence {𝑥𝑚 } ∈ A , such that lim ‖(𝑥 ″ − 𝑥𝑚 )𝜉𝑘 ‖ = 0,

𝑚→∞

It follows that 𝑥 ″ is so-adherent to A .

𝑘 = 1, … , 𝑛. Q.E.D.

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Corollary 3.3. Let M ⊂ B (H ), 1 ∈ M be a *-algebra of operators. Then the following statements are equivalent: (i) M is a von Neumann algebra. (ii) M = M ″ . Corollary 3.4. Let M ⊂ B (H ) be a von Neumann algebra and 𝑥 ∈ B (H ). Then the following properties are equivalent: (i) 𝑥 ∈ M . (ii) 𝑥𝑒′ = 𝑒′ 𝑥, for any projection 𝑒′ ∈ M ′ . (iii) 𝑢′∗ 𝑥𝑢′ = 𝑥, for any unitary operator 𝑢′ ∈ M ′ . Proof. By taking into account 2.23 (resp., 2.24), from property (ii) (resp., (iii)) we infer that 𝑥 ∈ M ″ , and therefore, with Corollary 3.3, we have 𝑥 ∈ M . Q.E.D. Corollary 3.5. Let M ⊂ B (H ) be a von Neumann algebra and 𝑥 ∈ M . Then l(𝑥), r(𝑥) ∈ M . Proof. We recall (see 2.13) that l(𝑥) is the smallest projection 𝑒 ∈ B (H ), such that 𝑒𝑥 = 𝑥. Let 𝑢′ ∈ M be a unitary operator and 𝑒 ∈ B (H ) a projection. Then 𝑢′∗ 𝑥𝑢′ = 𝑥, and hence, 𝑒𝑥 = 𝑥 iff (𝑢′∗ 𝑒𝑢′ )𝑥 = 𝑥. Consequently, we have 𝑢′∗ l(𝑥)𝑢′ = l(𝑥). From Corollary 3.4, it follows that l(𝑥) ∈ M . One can similarly prove that r(𝑥) ∈ M . Q.E.D. Corollary 3.6. Let M ⊂ B (H ) be a von Neunmann algebra and 𝑥 ∈ M . If 𝑥 = 𝑣|𝑥| is the polar decomposition of 𝑥, then |𝑥|, 𝑣 ∈ M . Proof. Since M is closed in the norm topology, we have |𝑥| = (𝑥 ∗ 𝑥)1/2 ∈ M . Let 𝑢′ ∈ M ′ be a unitary operator. Then 𝑢′∗ 𝑥𝑢′ = 𝑥, 𝑢′∗ |𝑥|𝑢′ = |𝑥|, hence 𝑥 = (𝑢′∗ 𝑣𝑢′ )|𝑥|, and 𝑢′∗ 𝑣𝑢′ is a partial isometry whose initial projection equals the support of |𝑥|. From the uniqueness of the polar decomposition (see Theorem 2.14) it follows that 𝑢′∗ 𝑣𝑢′ = 𝑣. With Corollary 3.4, it follows that 𝑣 ∈ M . Q.E.D. 3.7. We recall (see 2.17) that the set of all projections in B (H ) is a complete lattice in a canonical manner. Let M ⊂ B (H ) be a von Neumann algebra. We shall denote by P M the set of all projections in M . We shall consider on P M the order relation, which is induced by the order relation already defined in the set of all projections in B (H ). Corollary. Let M ⊂ B (H ) be a von Neumann algebra. Then P M is a complete lattice. Proof. Let {(𝑒𝑖 )𝑖∈𝐼 ⊂ P M and 𝑒 = ⋁𝑖∈𝐼 𝑒𝑖 ∈ B (H ). For any unitary operator 𝑢′ ∈ M ′ , we have 𝑢′∗ 𝑒𝑢′ =

(𝑢′∗ 𝑒𝑖 𝑢′ ) = 𝑒 = 𝑒, ⋁ ⋁ 𝑖 𝑖∈𝐼

𝑖∈𝐼

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and hence, with Corollary 3.4, we have 𝑒 ∈ M . It follows that 𝑒 ∈ P M is the least upper bound of the family {𝑒𝑖 }𝑖∈𝐼 in P M . Q.E.D. 3.8. Let M ⊂ B (H ) be a von Neumann algebra and 𝜉 ∈ H . We shall write 𝑝𝜉 = [M ′ 𝜉],

𝑝𝜉′ = [M 𝜉].

By taking into account the last remark from Section 3.1 and also Corollary 3.3, we get the following: Corollary. Let M ⊂ B (H ) be a von Neumann algebra and 𝜉 ∈ H . Then 𝑝𝜉 ∈ M ,

𝑝𝜉′ ∈ M ′ .

The projections of the form 𝑝𝜉 (resp., 𝑝𝜉′ ), 𝜉 ∈ H , are called the cyclic projections in M (resp., M ′ ). If {𝜉𝑛 } is a sequence of vectors in H , and if the projections 𝑝𝜉′ 𝑛 are mutually orthogonal, ∞

then ⋁𝑛=1 𝑝𝜉𝑛 is a cyclic projection in M . Indeed, we can assume that ‖𝜉𝑛 ‖ ≤ 2−𝑛 , and ∞ we then define 𝜉 = ∑𝑛=1 𝜉𝑛 ∈ H . Since the projections 𝑝𝜉′ 𝑛 are mutually orthogonal and 𝑝𝜉′ 𝑛 𝜉𝑛 = 𝜉𝑛 , we have 𝑝𝜉′ 𝑛 𝜉 = 𝜉𝑛 . Then we have 𝑝𝜉 = [M ′ 𝜉] ≥ [M ′ 𝑝𝜉′ 𝑛 𝜉] = [M ′ 𝜉𝑛 ] = 𝑝𝜉𝑛 , ∞

which implies that 𝑝𝜉 ≥ ⋁𝑛=1 𝑝𝜉𝑛 ; since the reversed inequality is obvious, the proof is complete. Q.E.D. A set of vectors S ⊂ H are said to be totalizing for M if [M S ] = H ; it is said to be separating for M , if 𝑥 ∈ M , 𝑥𝜉 = 0 for any 𝜉 ∈ S ⇒ 𝑥 = 0. It is easy to prove that S is totalizing (resp., separating) for M iff S is separating (resp., totalizing) for M ′ . A vector 𝜉 ∈ H is said to be cyclic (or totalizing) for M (resp., separating for M ) iff the set {𝜉} is totalizing (resp., separating) for M . The vector 𝜉 ∈ M is cyclic (resp., separating) for M iff 𝑝𝜉′ = 1 (resp., 𝑝𝜉 = 1). 3.9. Let M ⊂ B (H ) be a von Neumann algebra and M ′ ⊂ B (H ) its commutant. Then Z = M ∩ M′ is the common centre of the algebras M and M ′ . It is obvious that Z ⊂ B (H ) is a (commutative) von Neumann algebra. We shall denote by R (M , M ′ ) ⊂ B (H ) the von Neumann algebra generated by M ∪ M ′ . It is easy to check that R (M , M ′ ) = Z ′ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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A von Neumann algebra is said to be a factor if its centre is equal to the set of all scalar multiples of the unit operator. A projection in M will be called a central projection if it belongs to the centre of M . A factor is characterized by the fact that its only central projections are 0 and 1. Corollary. Let M ⊂ B (H ) be a von Neumann algebra and 𝑥 ∈ M . The set of all central projections 𝑝, such that 𝑝𝑥 = 𝑥, has a smallest element, denoted by z(𝑥), which can be calculated by the formula: z(𝑥) = [(M 𝑥)H ]. Proof. By taking into account Corollary 3.7 (applied to Z ), we define z(𝑥) by the formula: z(𝑥) = ∧{𝑝 ∈ P Z ; 𝑝𝑥 = 𝑥} ∈ P Z . If 𝑝 ∈ P Z and 𝑝𝑥 = 𝑥, then 𝑝H ⊃ 𝑥H . It follows that z(𝑥)H ⊃ 𝑥H , and therefore, z(𝑥)𝑥 = 𝑥. Let 𝑝 = [(M 𝑥)H ]. Since [(M 𝑥)H ] is a subspace invariant with respect to the operators in M and M ′ , it follows that 𝑝 ∈ M ′ ∩ M = Z . Since [(M 𝑥)H ] ⊃ 𝑥H , it follows that 𝑝𝑥 = 𝑥. Hence 𝑝 ≥ z(𝑥). On the other hand, it is obvious that z(𝑥)H is invariant with respect to the operators in M and z(𝑥)H ⊃ 𝑥H ; hence, we have z(𝑥)H ⊃ [(M 𝑥)H ], i.e., z(𝑥) ≥ 𝑝. Consequently, z(𝑥) = [(M 𝑥)H ]. Q.E.D. The projection z(𝑥) will be called the central support of 𝑥. We shall consider, in particular, the central support of the projections in M . Obviously, z(𝑥) = z(l(𝑥)) = z(r(𝑥)). It is easy to check that for any 𝑒 ∈ P M , we have z(𝑒) = ∨{𝑢∗ 𝑒𝑢; 𝑢 ∈ M , unitary}. 3.10. A very important result in the theory of von Neumann algebras is contained in the following: Theorem (I. Kaplansky’s density theorem). Let M ∈ B (H ) be a von Neumann algebra and A ⊂ M a so-dense *-subalgebra of M . Then the unit ball (resp., the self-adjoint part of the unit ball, the positive part of the unit ball) of A is so-dense in the unit ball (resp., in the self-joint part of the unit ball, in the positive part of the unit ball) of M . Proof. For any subset X ∈ B (H ), we shall denote, for this proof, by X 1 (resp., X ℎ [X + ]) the unit ball (resp., the self-adjoint [the positive part]) of X . In accordance with Corollary 1.5, the statement is equivalent to the following assertions: + ℎ A 1 (resp. A ℎ1 [A + 1 ]) is wo-dense in M 1 (resp. M 1 [M 1 ]).

In order to carry out the proof, we shall assume, without any loss of generality, that A is closed for the norm topology. Without any further comment, we shall use the theorem (2.6) on the operational calculus. (I) The mapping

1 M ∋ 𝑥 ↦ (𝑥 + 𝑥 ∗ ) ∈ M ℎ 2

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is wo-continuous, and the image of A by this mapping is A ℎ . Since A is wo-dense in M , it follows that A ℎ is wo-dense in M ℎ . (II) Let now 𝑥 ∈ M ℎ1 . The function [−1, 1] ∋ 𝑡 ↦ 2𝑡(1 + 𝑡 2 )−1 ∈ [−1, 1] is continuous, strictly increasing and surjective; therefore, it has a continuous inverse. It follows that there exists an element 𝑦 ∈ M ℎ , such that 𝑥 = 2𝑦(1 + 𝑦 2 )−1 . In accordance with (I), there exists a net {𝑏𝑖 } ∈ A ℎ , which is so-convergent to 𝑦. For any 𝑖, we define 𝑎𝑖 = 2𝑏𝑖 (1 + 𝑏𝑖2 )−1 . Then {𝑎𝑖 } ⊂ A ℎ1 and for any 𝑖, we have 𝑎𝑖 − 𝑥 = (1 + 𝑏𝑖2 )−1 (2𝑏𝑖 (1 + 𝑦 2 ) − (1 + 𝑏𝑖2 )2𝑦)(1 + 𝑦 2 )−1 = 2(1 + 𝑏𝑖2 )−1 (𝑏𝑖 − 𝑦)(1 + 𝑦 2 )−1 + 2(1 + 𝑏𝑖2 )−1 𝑏𝑖 (𝑦 − 𝑏𝑖 )𝑦(1 + 𝑦 2 )−1 , and this equality obviously shows that the net {𝑎𝑖 } is so-convergent to 𝑥. Consequently, A ℎ1 is so-dense in M + 1. + 1/2 (III) Let 𝑥 ∈ M 1 and 𝑦 = 𝑥 . In accordance with (II), there exists a net {𝑏𝑖 } ⊂ A ℎ1 , which is so-convergent to 𝑦. We denote 𝑎𝑖 = 𝑏𝑖∗ 𝑏𝑖 ∈ A + 1 . Then the net {𝑎𝑖 } is wo-convergent to 𝑥, as one can see from the following formula: |(𝑥 − 𝑎𝑖 )𝜉|𝜂)| = = = ≤

|((𝑦 ∗ 𝑦 − 𝑏𝑖∗ 𝑏𝑖 )𝜉|𝜂)| |(𝑦 ∗ (𝑦 − 𝑏𝑖 )𝜉|𝜂)| + |((𝑦 ∗ − 𝑏𝑖∗ )𝑏𝑖 𝜉|𝜂)| |((𝑦 − 𝑏𝑖 )𝜉|𝑦𝜂)| + (𝑏𝑖 𝜉|(𝑦 − 𝑏𝑖 )𝜂)| ‖(𝑦 − 𝑏𝑖 )𝜉‖‖𝜂‖ + ‖(𝑦 − 𝑏𝑖 )𝜂‖‖𝜉‖, 𝜉, 𝜂 ∈ H .

+ Consequently, A + 1 is wo-dense in M 1 . (IV) Let us now consider the Hilbert space H 2̃ and the *-algebra Mat2 (A ) ⊂ B (H 2̃ ), as well as the von Neumann algebra Mat2 (M ) ∈ B (H 2̃ ). It is easily verified that Mat2 (A ) is so-dense in Mat2 (M ). Let 𝑥 ∈ M 1 . Then the element

(

0 𝑥

𝑥∗ ) ∈ Mat2 (M ) 0

is self-adjoint, and its norm is less than 1. From (II), we infer that there exists a net {(

′ 𝑥11 ′ 𝑥21

′ 𝑥12 ⊂ (Mat2 (A ))ℎ1 , ′ )} 𝑥22

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′ which is so-convergent to the element given by the preceding matrix. It follows that ‖𝑥21 ‖≤ ′ 1, and the net {𝑥21 } is so-convergent to 𝑥. Consequently, A 1 is so-dense in M 1 . Q.E.D.

Corollary 3.11. Let M ⊂ B (H ), M ∋ 1, be a *-algebra. Then the following statements are equivalent: (i) M is a von Neumann algebra. (ii) M is w-closed. (iii) M 1 is w-compact where by M 1 we here denoted the closed unit ball of M . Proof. If M is a von Neumann algebra, then M is wo-closed and therefore 𝑤-closed. If M is w-closed, then from Theorem 1.10, we have M = (M ∗ )∗ , which implies that M 1 is 𝑤-compact, in accordance with Alaoglu’s theorem. We still have to prove the implication (iii) ⇒ (i); in other words, we must prove that M is wo-closed if we know that M 1 is wo-closed (see 1.2 (iii) and 1.10). If 𝑥 ∈ B (H ) is a wo-adherent point to M 1 , then in accordance with Kaplansky’s density theorem, there exists a net {𝑥𝑖 } ⊂ M , ‖𝑥𝑖 ‖ ≤ ‖𝑥‖, which is wo-convergent to 𝑥. Since M 1 is wo-closed, it follows that 𝑥 ∈ M . Q.E.D. 3.12. Let A 𝑗 ⊂ B (H ), A 𝑗 ∋ 1, 𝑗 = 1, 2, be two 𝐶 ∗ -algebras of operators and let 𝜋 ∶ A 1 → A 2 be a *-homomorphism, such that 𝜋(1) = 1. Then ‖𝜋‖ = 1. Indeed, it is easily verified that for any element 𝑥1 ∈ A 1 , we have 𝜎(𝜋(𝑥1∗ 𝑥1 )) ⊂ 𝜎(𝑥1∗ 𝑥1 ), and therefore, by taking into account Lemma 2.5, we get ‖𝜋(𝑥1 )‖2 = ‖𝜋(𝑥1 )∗ 𝜋(𝑥1 )‖ = ‖𝜋(𝑥1∗ 𝑥1 )‖ ≤ ‖𝑥1∗ 𝑥1 ‖ = ‖𝑥1 ‖2 . Let M 𝑗 ⊂ B (H 𝑗 ), 𝑗 = 1, 2, be two von Neumann algebras and let 𝜋 ∶ M 1 → M 2 be a wo-continuous *-homomorphism, such that 𝜋(1) = 1. Then 𝜋 is 𝑤-continuous. Indeed, it is sufficient to show that, for any 𝑤-continuous linear form 𝜑2 on M 2 , the restriction of the linear form 𝜑2 ∘ 𝜋 to the unit ball of M 1 is 𝑤-continuous. However, this fact is obvious since the 𝑤-topology coincides with the wo-topology on the unit ball of a von Neumann algebra, whereas 𝜋 is wo-continuous and ‖𝜋‖ = 1. Corollary. Let 𝜋 ∶ M 1 → M 2 ⊂ B (H 2 ) be a 𝑤-continuous *-homomorphism between two von Neumann algebras, such that 𝜋(1) = 1. Then 𝜋(M 1 ) ⊂ B (H 2 ) is a von Neumann algebra. Proof. In accordance with Corollary 3.11, it is sufficient to show that the closed unit ball of the *-algebra 𝜋(M 1 ) is w-compact. Let 𝑥2 ∈ 𝜋(M 1 ), such that ‖𝑥2 ‖ < 𝛼 < 1. Then there exists an 𝑥1 ∈ M 1 , such that 𝑥2 = 𝜋(𝑥1 ). Let 𝑥1 = 𝑣|𝑥1 |, 𝑣, |𝑥1 | ∈ M 1 , be the polar decomposition of 𝑥1 (see Corollary 3.6), and let 𝑒 ∈ P M , in accordance with Corollary 2.21, be such that |𝑥1 |𝑒 ≥ 𝛼𝑒, |𝑥1 |(1 − 𝑒) ≤ 𝛼(1 − 𝑒). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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von Neumann algebras We have 𝜋(|𝑥1 |)𝜋(𝑒) ≥ 𝛼𝜋(𝑒), whence

𝛼‖𝜋(𝑒)‖ ≤ ‖𝜋(|𝑥1 |)‖ = ‖𝜋(𝑣 ∗ 𝑣|𝑥1 |)‖ ≤ ‖𝜋(𝑣|𝑥1 |)‖ = ‖𝑥2 ‖,

and therefore, ‖𝜋(𝑒)‖ ≤ (‖𝑥2 ‖/𝛼) < 1, since 𝜋(𝑒) is a projection, we have 𝜋(𝑒) = 0. It follows that 𝑣|𝑥1 |(1 − 𝑒) ∈ M 1 , ‖𝑣|𝑥1 |(1 − 𝑒)‖ < 1 and 𝑥2 = 𝜋(𝑣|𝑥1 |(1 − 𝑒)). Consequently, we have {𝑥2 ∈ 𝜋(M 1 ); ‖𝑥2 ‖ < 1} = 𝜋({𝑥1 ∈ M 1 ; ‖𝑥1 ‖ < 1}). Since the closed unit ball of M 1 is w-compact, and since 𝜋 is w-continuous, it follows that the closed unit ball of 𝜋(M 1 ), i.e., {𝑥2 ∈ 𝜋(M 1 ); ‖𝑥2 ‖ ≤ 1} = 𝜋({𝑥1 ∈ M 1 ; ‖𝑥1 ‖ ⩽ 1}) is w-compact.

Q.E.D.

3.13. Let 𝑥 ∈ X ∈ B (H ) and 𝑒 ∈ P B (H ) . We shall write 𝑥𝑒 = 𝑒𝑥|𝑒H ∈ B (𝑒H ), X 𝑒 = {𝑥𝑒 ; 𝑥 ∈ X } ⊂ B (𝑒H ). As another consequence of Kaplansky’s density theorem, we shall prove a theorem that will enable us to introduce other elementary operations on von Neumann algebras. Theorem. Let M ∈ B (H ) be a von Neummn algebra and 𝑒 ∈ P M . Then we have that (i) M 𝑒 ⊂ B (𝑒H ) and (M ′ )𝑒 ⊂ B (𝑒H ) are von Neumann algebras. (ii) (M 𝑒 )′ = (M ′ )𝑒 . Proof. The mapping

M ′ ∋ 𝑥 ′ ↦ 𝑥𝑒′ ∈ (M ′ )𝑒 ⊂ B (𝑒H )

is a wo-continuous *-homomorphism, which is surjective; from Section 3.12, we infer that (M ′ )𝑒 is a von Neumann algebra. It is obvious that M 𝑒 ⊂ ((M ′ )𝑒 )′ . Conversely, any element in ((M ′ )𝑒 )′ is of the form 𝑥𝑒 , where 𝑥 ∈ B (H ), 𝑥 = 𝑒𝑥𝑒. For any 𝑥 ′ ∈ M ′ , we have 𝑥𝑒 𝑥𝑒′ = 𝑥𝑒′ 𝑥𝑒

in B (H ),

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𝑥𝑥 ′ = 𝑥 ′ 𝑥

in B (H ),

which implies that 𝑥 ∈ M ″ = M (see 3.3). Thus, we proved that M 𝑒 = ((M ′ )𝑒 )′ ; in particular, M 𝑒 is a von Neumann algebra. By passing to the commutant in this equality, we get (M 𝑒 )′ = (M ′ )𝑒 , because (M ′ )𝑒 , is a von Neumann algebra (see Corollary 3.3).

Q.E.D.

Henceforth we shall write M ′𝑒 = (M 𝑒 )′ = (M ′ )𝑒 . 3.14. We now introduce other elementary operations on von Neumann algebras: the reduction and the induction. Let M ⊂ B (H ) be a von Neumann algebra and 𝑒 ∈ P M . The von Neumann algebra M 𝑒 is called the reduced von Neumann algebra of M with respect to 𝑒. It is easy to check that the mapping 𝑒M 𝑒 ∋ 𝑥 ↦ 𝑥𝑒 ∈ M 𝑒 is a *-isomorphism of *-algebras. Let M ⊂ B (H ) be a von Neumann algebra and 𝑒′ ∈ P M ′ . The von Neumann algebra M 𝑒′ is called the induced von Neumann algebra of M with respect to 𝑒′ . In the proof of Theorem 3.13, we observed and used the fact that the mapping M ∋ 𝑥 ↦ 𝑥𝑒 ′ ∈ M 𝑒 ′ is a wo-continuous *-homomorphism of von Neumann algebras. This *-homomorphism is called the canonical induction determined by 𝑒′ ∈ P M ′ . Proposition. Let M ⊂ B (H ) be a von Neumann algebra and 𝑒′ ∈ P M ′ . Then the canonical induction M → M 𝑒′ is a *-isomorphism iff z(𝑒′ ) = 1. Proof. We have (1 − z(𝑒′ ))𝑒′ = 0, and therefore, if the canonical induction is a *-isomorphism, we have z(𝑒′ ) = 1. Conversely, if z(𝑒′ ) = 1, then in accordance with Corollary 3.9, we have [(M ′ 𝑒′ )H ] = H . If 𝑥 ∈ M and 𝑥𝑒′ = 0, then 𝑥𝑒′ H = 0, and therefore, we have 𝑥[(M ′ 𝑒′ )H ] = 0; hence 𝑥 = 0. Consequently, the canonical induction is a *-isomorphism. Q.E.D. 3.15. If 𝑝 is a central projection in the von Neumann algebra M ⊂ B (H ), then the mappings M 𝑝 ∋ 𝑥 ↦ 𝑥𝑝 ∈ M 𝑝 , M ′ 𝑝 ∋ 𝑥 ′ ↦ 𝑥𝑝′ ∈ M ′𝑝 Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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von Neumann algebras establish the canonical identifications M 𝑝 = M 𝑝 ⊂ B (𝑝H ), M ′ 𝑝 = M ′𝑝 ⊂ B (𝑝H ).

Corollary. Let M ⊂ B (H ) be a von Neumann algebra with the centre Z , and let 𝑒 ∈ P M . Then the common centre of the algebras M 𝑒 , M ′𝑒 is equal to Z 𝑒 . Proof. If 𝑒 is a central projection, the assertion is obvious. Consequently, we can assume, without any loss of generality, that z(𝑒) = 1. Then, in accordance with Proposition 3.14, the canonical induction M ′ → M ′𝑒 is a *-isomorphism, and therefore, the centre of M ′𝑒 is the image of the centre Z of M ′ by this *-isomorphism. Q.E.D. In particular, if M is a factor, then M 𝑒 and M ′𝑒 are also factors. 3.16. Before introducing a new elementary operation on von Neumann algebra, namely the tensor product, we prove some commutation relations for ‘matrices’ of operators; such relations have already been considered in the proofs of Theorems 3.2 and 3.10. Let H be a Hilbert space, 𝛾 any cardinal number and 𝐼 a set of indices, such that card 𝐼 = 𝛾. We use the notations already introduced in Section 2.32. ˜ 𝛾 , Mat𝛾 (X ) ⊂ B (H ˜ 𝛾 ), Lemma. Let X ⊂ B (H ) be any subset. Then for the sets X the following relations ˜ 𝛾 )′ = Mat𝛾 (X ′ ), (X ˜ 𝛾 )″ = ( X ˜ ″ )𝛾 , (X hold; if X ∋ 0, 1, then we also have the following relations: ′

˜ )𝛾 , (Mat𝛾 (X ))′ = (X

(Mat𝛾 (X ))″ = Mat𝛾 (X ″ ).

˜ 𝛾 )′ = Mat𝛾 (X ′ ) can be proved in the same manner as in the proof Proof. The relation (X of Theorem 3.2. We observe that, for any 𝑖0 , 𝑘0 ∈ 𝐼, we have (notation as in Section 2.32) 𝑢𝑖0 𝑢𝑘∗ 0 = (𝛿𝑖0 𝑖 , 𝛿𝑘0 𝑘 ). Consequently, if X ∋ 0, 1, then all the operators 𝑢𝑖 𝑢𝑘∗ belong to Mat𝛾 (X ), and therefore, in accordance with a remark from Section 2.32, it follows that (Mat𝛾 (X ))′ ⊂ B˜ (H )𝛾 . ˜ 𝛾 ⊂ Mat𝛾 (X ), and therefore, On the other hand, we have X ˜ 𝛾 )′ = Mat𝛾 (X ′ ). (Mat𝛾 (X ))′ ⊂ (X Consequently, we have ′

˜ )𝛾 . (Mat𝛾 (X ))′ ⊂ B˜ (H )𝛾 ∩ Mat𝛾 (X ′ ) = (X ′

˜ )𝛾 is proved. Since the reversed inclusion is obvious, the relation (Mat𝛾 (X ))′ = (X The other relations are immediate consequences of the already proved ones. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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3.17. Let M ⊂ B (K ), N ⊂ B (K ) be von Neumann algebras. Then, 𝑛

M ⊗ N = { ∑ 𝑥𝑘 ⊗𝑦𝑘 ; 𝑥𝑘 ∈ M , 𝑦𝑘 ∈ N , 𝑛 = 1, 2, …} 𝑘=1

is a *-algebra of operators on H ⊗H . The von Neumann algebra generated in B (H ⊗H ) by M ⊗ N is denoted by M ⊗N , and it is called the tensor product of the von Neumann algebras M and N . We recall that for any Hilbert space H , one denotes by C (H ) the set of all the scalar multiples of the identity operator on H ; obviously, C (H ) ⊂ B (H ) is a von Neumann algebra. It is easy to check that B (H )′ = C (H ),

C (H )′ = B (H ).

In the following proposition, we use the identifications we have agreed upon in Section 2.34. Proposition. Let M ⊂ B (H ) be a von Neumann algebra and K a Hilbert space, 𝛾 = dim K . Then we have ˜ 𝛾. (i) M ⊗C (K ) = M ⊗C (K ) = M (ii) M ⊗B (K ) = Mat𝛾 (M ). (iii) (M ⊗C (K ))′ = M ′ ⊗B (K ), (M ⊗B (K ))′ = M ′ ⊗C (K ). ˜ 𝛾 is an immediate consequence of Section 2.34, Proof. The relation M ⊗C (K ) = M ″ ˜ 𝛾 )″ = ( M ˜ )𝛾 = M ˜ 𝛾 , and assertion (i) is whereas from Lemma 3.16, it follows that (M proved. For any von Neumann algebra N ⊂ B (H ), it is easy to prove that M ⊗N = R (M ⊗C (K ) ∪ C (H )⊗N ), whence we get (M ⊗N )′ = (M ⊗C (K ))′ ∩ (C (H )⊗N )′ . In particular, we have (M ⊗B (K ))′ = (M ⊗C (K ))′ ∩ (C (H )⊗B (K ))′ = (M ⊗C (K ))′ ∩ (B (H )⊗C (K )) = M ′ ⊗C (K ), where the two last equalities are easily checked by direct verification. We then have (M ⊗C (K ))′ = (M ″ ⊗C (K ))′ = (M ′ ⊗B (K ))″ = M ′ ⊗B (K ), which proves assertion (iii). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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von Neumann algebras Finally, by taking into account properties (i), (iii) and Lemma 3.16, we get ˜′ 𝛾 )′ M ⊗B (K ) = (M ′ ⊗C (K ))′ = (M = (Mat𝛾 (M ))″ = Mat𝛾 (M ″ ) = Mat𝛾 (M ), and assertion (ii) is proved.

Q.E.D.

3.18. If M ⊂ B (H ) is a von Neumann algebra and K a Hilbert space, then the mapping M ∋ 𝑥 ↦ 𝑥⊗1 ∈ M ⊗C (K ) is a *-isomorphism, called amplification. With the usual identifications, this isomorphism can also be written in the following form: ˜ 𝛾, M ∋ 𝑥 ↦ 𝑥̃ ∈ M

𝛾 = dim K .

With the notations from Section 2.32, let 𝑒𝑖 = 𝑢𝑖 𝑢𝑖∗ be the projection of H 𝛾̃ onto H 𝑖 . Then 𝑒𝑖 ∈ Mat𝛾 (C (K )) = C (K )⊗B (K ), for any 𝑖 ∈ 𝐼; it follows that 𝑒𝑖 ∈ M ⊗B (K ),

𝑖 ∈ 𝐼,

𝑒𝑖 ∈ M ′ ⊗B (K ) = (M ⊗C (K ))′ ,

𝑖 ∈ 𝐼.

We can, therefore, consider the reduced algebra (M ⊗B (K ))𝑒𝑖 ⊂ B (H 𝑖 ) and the induced algebra (M ⊗C (K ))𝑒𝑖 ⊂ B (H 𝑖 ). It is easily seen that the mapping 𝑥 ↦ 𝑢𝑖 𝑥𝑢𝑖∗ |K 𝑖 is a *-isomorphism of M onto (M ⊗B (K ))𝑒𝑖 = (M ⊗C (K ))𝑒𝑖 . Thus, the passage from M to M ⊗B (K ) (resp., to M ⊗C (K )) and the passage from a von Neumann algebra to a reduced algebra (resp., to an induced algebra) are reciprocal operations (see also Section 4.22), whereas the *-isomorphism reciprocal to an amplification is an induction. 3.19. Since the closed unit ball of a von Neumann algebra is 𝑤-compact (see 3.11), it has extreme points, by virtue of the Krein-Milman theorem. The following lemma, which describes the nature of these extreme points, will be used in the proof of Theorem 5.16. Lemma. Let M be a von Neumann algebra and 𝑣 an extreme point of the closed unit ball of M . Then 𝑣 is a partial isometry. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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Proof. We must show that 𝑣∗ 𝑣 is a projection, and in order to achieve this, we must show that 𝜎(𝑣 ∗ 𝑣) ⊂ {0, 1}. If this be not true, let 𝜆 ∈ 𝜎(𝑣 ∗ 𝑣), 0 < 𝜆 < 𝛼 < 1, and 𝜀 > 0 be such that 𝜀(1 + 𝜀𝛼)2 < 1. Let us define 𝛼 = 𝜀𝜒[0,𝛼] (𝑣 ∗ 𝑣). It is easily verified, by taking into account the operational calculus, that the following relations hold ‖(1 + 𝑎)𝑣 ∗ 𝑣(1 + 𝑎)‖ ≤ 1, ‖(1 − 𝑎)𝑣 ∗ 𝑣(1 − 𝑎)‖ ≤ 1. Consequently, the elements 𝑣1 = 𝑣(1 + 𝑎),

𝑣2 = 𝑣(1 − 𝑎)

belong to the closed unit ball of M and 1 1 𝑣 = 𝑣1 + 𝑣2 . 2 2 Since 𝑣 is an extreme point, it follows that 𝑣1 = 𝑣2 = 𝑣, whence 𝑣𝑎 = 0. Consequently, we have 𝑣∗ 𝑣𝑎 = 0, and therefore, 𝑣∗ 𝑣𝜒[0,𝛼] (𝑣 ∗ 𝑣) = 0, a contradiction. Q.E.D. 3.20. In the last sections of this chapter, we consider some general properties of the ideals of the algebras of operators. Proposition. Let A be a 𝐶 ∗ -algebra and 𝔑 ⊂ A a left ideal of A . Then there exists a net {𝑢𝛼 )𝛼∈Γ ⊂ 𝔑, such that (i) 0 ≤ 𝑢𝛼 ≤ 1, 𝛼 ∈ Γ. (ii) 𝛼 ≤ 𝛽 ⇒ 𝑢𝛼 ≤ 𝑢𝛽 . (iii) ‖𝑥 − 𝑥𝑢𝛼 ‖ →𝛼 0, for any 𝑥 ∈ 𝔑. Proof. We denote by Γ the set of all pairs (𝑛, 𝐹), where 𝑛 is a natural integer and 𝐹 is a finite subset of 𝔑. Endowed with the order relation (𝑛, 𝐹) ≤ (𝑚, 𝐺) ⇔ 𝑛 ≤ 𝑚 and 𝐹 ⊂ 𝐺, Γ becomes a directed set. For any 𝛼 = (𝑛, 𝐹) ∈ Γ, we define 𝑣𝛼 = ∑ 𝑥 ∗ 𝑥 ∈ 𝔑,

𝑢𝛼 = (𝑛−1 + 𝑣𝛼 )−1 𝑣𝛼 .

𝑥∈𝐹

We observe that 𝑢𝛼 = 𝑓𝑛 (𝑣𝛼 ), where 𝑓𝑛 (𝑡) = (𝑛−1 + 𝑡)−1 𝑡. Since 0 ≤ 𝑓𝑛 (𝑡) ≤ 1, for any 𝑡 > 0, from Theorem 2.6, it follows that 0 ≤ 𝑢𝛼 ≤ 1. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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von Neumann algebras If 𝛼 = (𝑛, 𝐹) ≤ (𝑚, 𝐺) = 𝛽, then 𝑣𝛼 ≤ 𝑣𝛽 , and therefore, with E.2.8, we have (𝑛−1 + 𝑣𝛼 )−1 ≥ (𝑛−1 + 𝑣𝛽 )−1 .

Since 𝑛−1 (𝑛−1 + 𝑡)−1 ≥ 𝑚−1 (𝑚−1 + 𝑡)−1 , for any 𝑡 ≥ 0, with Theorem 2.6, we infer that 𝑛−1 (𝑛−1 + 𝑣𝛽 )−1 ≥ 𝑚−1 (𝑚−1 + 𝑣𝛽 )−1 . Consequently, we have 1 − 𝑛−1 (𝑛−1 + 𝑣𝛼 )−1 ≤ 1 − 𝑛−1 (𝑛−1 + 𝑣𝛽 )−1 ≤ 1 − 𝑚−1 (𝑚−1 + 𝑣𝛽 )−1 , i.e., 𝑢𝛼 ≤ 𝑢 𝛽 . Finally, for any 𝛼 = (𝑛, 𝐹) ∈ Γ, we have ∑ [𝑥(1 − 𝑢𝛼 )]∗ [𝑥(1 − 𝑢𝛼 )] = (1 − 𝑢𝛼 )𝑣𝛼 (1 − 𝑢𝛼 ) 𝑥∈𝐹

= 𝑛−2 (𝑛−1 + 𝑣𝛼 )−2 𝑣𝛼 ≤ 4−1 𝑛−1 , because we have (𝑛−1 + 𝑡)−2 𝑡 ≤ 4−1 𝑛, for any 𝑡 ∈ ℝ. We therefore have [𝑥(1 − 𝑢𝛼 )]∗ [𝑥(1 − 𝑢𝛼 )] ≤ 4−1 𝑛−1 , whence

‖𝑥(1 − 𝑢𝛼 )‖2 ≤ 4−1 𝑛−1 ,

𝑥 ∈ 𝐹,

𝑥 ∈ 𝐹.

It follows that ‖𝑥 − 𝑥𝑢𝛼 ‖ →𝛼 0, for any 𝑥 ∈ ℝ. Q.E.D. The net {𝑢𝛼 }𝛼∈Γ is called an approximate unit for the left ideal 𝔑. We observe that property (iii) remains in force for any 𝑥 belonging to the norm closure 𝔑 of 𝔑. In the preceding proposition, we did not assume the existence of a unit element in the 𝐶 ∗ -algebra A ∈ B (H ), all inverses being considered in B (H ). In particular, it follows that any two-sided ideal of A , which is also dense in A for the norm topology, contains an approximate unit {𝑢𝛼 }𝛼∈Γ for A ; in this case, by replacing, in property (iii), the element 𝑥 by 𝑥∗ , it follows that we also have ‖𝑥 − 𝑢𝛼 𝑥‖ → 0, for any 𝑥 ∈ A . Corollary. Let M be a von Neumann algebra and 𝔑 ⊂ M a left ideal. Then there exists a unique projection 𝑒 ∈ M , such that 𝑤

𝔑 = M 𝑒. Any approximate unit of 𝔑 is so-convergent to 𝑒. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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Proof. Let {𝑢𝛼 } be an approximate unit of 𝔑, in accordance with the preceding proposition, 𝑤

and let 𝑒 = sup𝛼 𝑢𝛼 ∈ 𝔑 (see 2.16). For any 𝑥 ∈ 𝔑, we have a 𝑥 − 𝑥𝑢𝛼 → 0; hence 𝑥 = 𝑥𝑒. Consequently, we have 𝑥 = 𝑥𝑒, 𝑤

𝑤

for any 𝑥 ∈ 𝔑 . In particular, we have 𝑒2 = 𝑒, i.e., 𝑒 is a projection. We have M 𝑒 ⊂ 𝔑 , 𝑤

𝑤

since 𝔑 is a left ideal, and also 𝔑 ⊂ M 𝑒, as we have already proved; it follows that 𝑤

𝔑 = M 𝑒. The uniqueness of the projection 𝑒 is immediate.

Q.E.D.

We observe that the adjoint of a left ideal of M is a right ideal. Consequently, the 𝑤-closed left (resp., right) ideals of M are of the form M 𝑒 (resp., 𝑒M ), where 𝑒 is a projection in M . If 𝔑 = M 𝑒 is a 𝑤-closed two-sided ideal of M , then for any unitary 𝑢 ∈ M , we have M 𝑒 = 𝔑 = 𝑢𝔑𝑢∗ = M (𝑢𝑒𝑢∗ ), which implies that 𝑒 = 𝑢𝑒𝑢∗ . From Corollary 3.4, we infer that 𝑒 ∈ M ′ , and therefore, 𝑒 is central. Consequently, the 𝑤-closed two-sided ideals of M are of the form M 𝑝, where 𝑝 is a central projection, and conversely. 3.21. Let A be a 𝐶 ∗ -algebra. A subset 𝔉 ⊂ A + is said to be a face if it has the property 𝑎, 𝑏 ∈ 𝔉,

𝑐 ∈ A +,

𝑐 ≤ 𝑎 + 𝑏 ⇒ 𝑐 ∈ 𝔉.

It is easy to prove that if 𝔉 ⊂ A + is a face, then 𝔉 + 𝔉 ⊂ 𝔉, ℝ+ 𝔉 ⊂ 𝔉. For any face 𝔉 ⊂ A + , one defines 𝔑 = {𝑥 ∈ A ; 𝑥 ∗ 𝑥 ∈ 𝔉}, 𝑛

𝔐 = 𝔑∗ 𝔑 = { ∑ 𝑦𝑗∗ 𝑥𝑗 ; 𝑥𝑗 , 𝑦𝑗 ∈ 𝔑, 𝑛 ∈ ℕ} . 𝑗=1

Proposition. Let 𝔉 ⊂ A + be a face. Then (i) 𝔑 is a left ideal and 𝔐 is a *-subalgebra of A . (ii) 𝔐+ = 𝔉 and 𝔐 is the linear hull of 𝔐+ . (iii) there exists an approximate unit of 𝔑, which is contained in 𝔐+ . Proof. (i) Let 𝑥, 𝑦 ∈ 𝔑 and 𝑎 ∈ A . We have (𝑥 + 𝑦)∗ (𝑥 + 𝑦) ≤ 2(𝑥 ∗ 𝑥 + 𝑦 ∗ 𝑦) ∈ 𝔉, and therefore, we have 𝑥 + 𝑦 ∈ 𝔑. We then have (𝑎𝑥)∗ (𝑎𝑥) = 𝑥 ∗ 𝑎∗ 𝑎𝑥 ≤ ‖𝑎‖2 𝑥∗ 𝑥 ∈ 𝔉, which implies that 𝑎𝑥 ∈ 𝔑. Consequently, 𝔑 is a left ideal of A . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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(ii) Let 𝑎 ∈ 𝔉. From Proposition 2.12, we infer that there exists an 𝑥 ∈ A , such that 𝑎 = 𝑥 ∗ 𝑥. Then we have 𝑥 ∈ 𝔑, and therefore, 𝑥∗ 𝑥 ∈ 𝔐 ∩ A + = 𝔐+ . It is easy to prove the following polarization relation: 3

𝑦 ∗ 𝑥 = 4−1 ∑ i𝑘 (𝑥 + i𝑘 𝑦)∗ (𝑥 + i𝑘 𝑦),

𝑥, 𝑦 ∈ A .

𝑘=0 𝑛

If 𝑎 = ∑𝑗=1 𝑦𝑗∗ 𝑥𝑗 ∈ 𝔐+ , 𝑦𝑗 , 𝑥𝑗 ∈ 𝔑, then by taking into account the polarization relation, we get 𝑛

𝑎 = 4−1 ∑ (𝑥𝑗 + 𝑦𝑗 )∗ (𝑥𝑗 + 𝑦𝑗 ) − (𝑥𝑗 − 𝑦𝑗 )∗ (𝑥𝑗 − 𝑦𝑗 ) 𝑗=1 𝑛

≤ 4−1 ∑ (𝑥𝑗 + 𝑦𝑗 )∗ (𝑥𝑗 + 𝑦𝑗 ) ∈ 𝔉, 𝑗=1

and therefore, we have 𝑎 ∈ 𝔉. We have thus shown that 𝔐+ = 𝔉. It is obvious that 𝔐 contains the linear hull of 𝔐+ , whereas the reversed inclusion easily follows from the polarization relation. (iii) We shall use the notations introduced in the proof of Proposition 3.20. For any 𝛼 = (𝑎, 𝐹) ∈ Γ, we have 𝑣𝛼 ∈ 𝔉 = 𝔐+ . Since (𝑛−1 + 𝑡)−1 𝑡 ≤ 𝑛𝑡, for any 𝑡 ≥ 0, by taking into account Theorem 2.6, we get 𝑢𝛼 = (𝑛−1 + 𝑣𝛼 )−1 𝑣𝛼 ≤ 𝑛𝑣𝛼 ∈ 𝔉, and since 𝔐+ is a face, we get

𝑢𝛼 ∈ 𝔐 + . Q.E.D.

A subalgebra 𝔐 of the 𝐶 ∗ -algebra A , such that 𝔐+ is a face, and 𝔐, itself, is the linear hull of 𝔐+ , is called a facial subalgebra. Any facial subalgebra is self-adjoint. The following corollary characterizes the reduced algebras of a von Neumann algebra. Corollary. Let M be a von Neumann algebra and 𝔐 ⊂ M a 𝑤-closed subset. The following statements are equivalent: (i) 𝔐 is a facial subalgebra of M . (ii) There exists a projection 𝑒 ∈ M , such that 𝔐 = 𝑒M 𝑒. Proof. We assume that 𝔐 is a facial subalgebra. Since 𝔐 is 𝑤-closed, the face 𝔉 = 𝔐+ is 𝑤-closed, and therefore, the left ideal 𝔑 = {𝑥 ∈ M ; 𝑥 ∗ 𝑥 ∈ 𝔉} is 𝑤-closed. In accordance with Corollary 3.20, there exists a projection 𝑒 ∈ M , such that 𝔑 = M 𝑒. In accordance with the preceding proposition, we have 𝔐 = 𝔑∗ 𝔑 = (M 𝑒)∗ (M 𝑒) = 𝑒M 𝑒. We have thus proved that (i) ⇒ (ii). The implication (ii) ⇒ (i) offers no difficulties. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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Let 𝔐 be a two-sided ideal of the von Neumann algebra M . If 𝑥 ∈ M and 𝑥 = 𝑣|𝑥| is the polar decomposition of 𝑥 (see 3.6), then |𝑥| = 𝑣 ∗ 𝑥 ∈ 𝔐, and 𝑥 ∗ = |𝑥|𝑣 ∗ ∈ 𝔐 (see 2.15). Consequently, any two-sided ideal of a von Neumann algebra is self-adjoint. A face 𝔉 ⊂ M + is said to be invariant if 𝑎 ∈ 𝔉, 𝑢 ∈ M , unitary ⇒ 𝑢𝑎𝑢∗ ∈ 𝔉. Using the polar decomposition, it is easy to show that a face 𝔉 ⊂ M + is invariant iff 𝑥 ∈ M , 𝑥 ∗ 𝑥 ∈ 𝔉 ⇒ 𝑥𝑥 ∗ ∈ 𝔉. The following corollary characterizes the positive parts of the two-sided ideals of von Neumann algebras. Corollary. Let M be a von Neumann algebra. Then mappings 𝔐 ↦ 𝔐+ , 𝔉 ↦ the linear hull of 𝔉, are reciprocal bijections between the set of all two-sided ideals 𝔐 ⊂ M and the set of all invariant faces 𝔉 ⊂ M + . Proof. Let 𝔐 ⊂ M be a two-sided ideal and 𝑏 ∈ 𝔐+ . If 𝑎 ∈ M + and 𝑎 ≤ 𝑏, then in accordance with exercise E.2.6, there exists an 𝑥 ∈ M , such that 𝑎1/2 = 𝑥𝑏1/2 . It follows that 𝑎 = 𝑥𝑏𝑥 ∗ ∈ 𝔐+ . We infer that 𝔐+ is a face, obviously invariant, since 𝔐 is a two-sided ideal. The remaining assertions in the corollary are easily verified by taking into account the preceding proposition. Q.E.D. EXERCISES E.3.1 Let M ⊂ B (H ) be a von Neumann algebra, {𝑥𝑛 } ⊂ M and 𝑥 ∈ B (H ). If we have 𝑥𝑛 𝜉 → 𝑥𝜉 weakly, and 𝑥𝑛∗ 𝜉 → 𝑥 ∗ 𝜉 weakly, for any 𝜉 belonging to a total subset of H , then 𝑥 ∈ M . Let H be a separable Hilbert space and {𝜉𝑘 } an orthonormal basis of H . We define the operators 𝑥𝑛 , 𝑥, 𝑦 ∈ B (H ) by the formulas 2𝜉2 − 3𝜉3 ⎧ ⎪ 3 𝑥𝑛 𝜉𝑘 = −𝑛𝜉2 + 𝑛𝜉3 2 ⎨ ⎪ ⎩0

for 𝑘 = 2, for 𝑘 = 𝑛, for 𝑘 ≠ 2, 𝑛. ▶

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𝑥𝜉𝑘 = {

2𝜉2 − 3𝜉3

for 𝑘 = 2,

0

for 𝑘 ≠ 2. ∞

𝑦𝜉𝑘 = {

1

∑𝑗=1 𝜉𝑗 1 𝑘

𝑗

𝜉1

for 𝑘 = 1, for 𝑘 ≠ 1.

Show that 𝑥𝑛 𝜉𝑘 → 𝑥𝜉𝑘 for any 𝑘, but although we have 𝑥𝑛 𝑦 = 𝑦𝑥𝑛 , for any 𝑛, we have (𝑥𝑦𝜉1 |𝜉2 ) ≠ (𝑦𝑥𝜉1 |𝜉2 ); it follows that 𝑥 ∉ R ({𝑥𝑛 }). E.3.2 A von Neumann algebra M ⊂ B (H ) is a factor iff for any 𝑥, 𝑦 ∈ M , there exists an 𝑎 ∈ M , such that 𝑥𝑎𝑦 ≠ 0. E.3.3 Let M ⊂ B (H ) be a von Neumann algebra and {𝑒𝑖 }𝑖∈𝐼 ⊂ P M . Then we have z( 𝑒𝑖 ) = z(𝑒 ). ⋁ ⋁ 𝑖 𝑖∈𝐼

𝑖∈𝐼

E.3.4 Let A ⊂ B (H ) be a *-algebra and 𝑥 ∈ B (H ) an invertible operator, such that the mapping 𝑎 ↦ 𝑥 −1 𝑎𝑥 be a *-automorphism of A . Then there exists a unitary 𝑢 ∈ B (H ), such that 𝑥−1 𝑎𝑥 = 𝑢∗ 𝑎𝑢, for any 𝑎 ∈ A . If, moreover, A is a von Neumann algebra and 𝑥 ∈ A , then one can find such a 𝑢, having the above properties and, moreover, belonging to A . E.3.5 One says that a von Neumann algebra M ⊂ B (H ) is of countable type (or countably decomposable) if any family of mutually orthogonal non-zero projections in M is at most countable. Any von Neumann algebra in a separable Hilbert space is of countable type. Show that the closed unit ball of a von Neumann algebra of countable type is so-metrizable, whereas the closed unit ball of a von Neumann algebra in a separable Hilbert space is wo-metrizable. E.3.6 Let A ⊂ M ⊂ N ⊂ B (H ) be von Neumann algebras, such that A be included in the centre of N . For any 𝑥 ∈ N , 𝑥 ∉ M , there exists a 𝑝 ∈ P A , 𝑝 ≠ 0, such that 𝑞 ∈ P A , 0 ≠ 𝑞 ≤ 𝑝 ⇒ 𝑥𝑞 ∉ M . If M 𝑞 ≠ N 𝑞, for any 𝑞 ∈ P A , 𝑞 ≠ 0, then there exists a projection 𝑒 ∈ N , such that 𝑒𝑞 ∉ M , for any 𝑞 ∈ P A , 𝑞 ≠ 0. !E.3.7 A von Neumann algebra M ⊂ B (H ) is of countable type iff there exists an M -separating orthonormal sequence in H . !E.3.8 A commutative von Neumann algebra is of countable type iff it has a separating vector. ▶

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!E.3.9 A vector 𝜉 ∈ H is called a trace vector for the von Neumann algebra M ⊂ B (H ) if (𝑥𝑦𝜉|𝜉) = (𝑦𝑥𝜉|𝜉), 𝑥, 𝑦 ∈ M . Obviously, if M is commutative, then any 𝜉 ∈ H is a trace vector for M . Let M ⊂ B (H ) be a von Neumann algebra, which has a cyclic trace vector 𝜉 ∈ H . Then 𝜉 is separating for M and the mapping M 𝜉 ∋ 𝑥𝜉 ↦ 𝑥 ∗ 𝜉 ∈ M 𝜉 is isometric; hence, it extends in a unique manner to a conjugation 𝐽 on H ; i.e., an antilinear, involutive (𝐽 2 = 1) and isometric mapping 𝐽 ∶ H → H . Show that 𝐽(𝑥 ′ 𝜉) = (𝑥 ′ )∗ 𝜉, 𝑥 ′ ∈ M ′ . (Hint: 𝑥 ′ 𝜉 = lim𝑛→∞ 𝑥𝑛 𝜉, 𝑥𝑛 ∈ M .) Infer from the preceding properties that the mapping 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 is a *-antiisomorphism of M onto M ′ , which acts identically on the centre of M . !E.3.10 A von Neumann algebra M ⊂ B (H ) is said to be maximal abelian if it is commutative and maximal (with respect to the inclusion) with this property, in B (H ). The von Neumann algebra M is maximal Abelian iff M = M ′ . Any commutative von Neumann algebra in B (H ) is included in a maximal Abelian von Neumann algebra in B (H ). Show that a commutative von Neumann algebra, of countable type, is maximal Abelian iff it has a cyclic vector. E.3.11 Let M ⊂ B (H ) be a von Neumann algebra with the centre Z = M ∩ M ′ . ′ Show that for subsets {𝑥𝑖𝑗 ; 1 ≤ 𝑖, 𝑗 ≤ 𝑛} ⊂ M and {𝑥𝑖𝑗 ; 1 ≤ 𝑖, 𝑗 ≤ 𝑛} ⊂ M ′ , the following assertions are equivalent: 𝑛

′ (i) ∑𝑘=1 𝑥𝑖𝑘 𝑥𝑘𝑗 = 0, 1 ≤ 𝑖, 𝑗 ≤ 𝑛. (ii) there exists a subset {𝑧𝑖𝑗 ; 1 ≤ 𝑖, 𝑗 ≤ 𝑛} ⊂ Z , such that 𝑛

𝑛

∑ 𝑥𝑖𝑘 𝑧𝑘𝑗 = 0,

′ ′ ∑ 𝑧𝑖𝑘 𝑥𝑘𝑗 = 𝑥𝑖𝑗 ,

𝑘=1

𝑘=1

1 ≤ 𝑖, 𝑗 ≤ 𝑛.

′ ′ (Hint: (𝑥𝑖𝑗 ) ∈ M ⊗B (H 𝑛 ), (𝑥𝑖𝑗 ) ∈ M ′ ⊗B (H 𝑛 ); (i) ⇒ (𝑥𝑖𝑗 )(𝑥𝑖𝑗 ) = 0.)

▶

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E.3.12 Let M ⊂ B (H ) be a factor, {𝑥1 , … , 𝑥𝑛 } ⊂ M a linearly independent subset of M , and {𝑥1′ , … , 𝑥𝑛′ } ⊂ M ′ . Then 𝑛

∑ 𝑥𝑘 𝑥𝑘′ = 0 ⇔ 𝑥𝑘′ = 0,

1 ≤ 𝑘 ≤ 𝑛.

𝑘=1

Infer that the mapping 𝑛

𝑛

∑ 𝑥𝑘 ⊗𝑥𝑘′ ↦ ∑ 𝑥𝑘 𝑥𝑘′ 𝑘=1

𝑘=1

is a *-isomorphism of the *-algebra M ⊗ M ′ onto the *-algebra generated in B (H ) by M and M ′ . E.3.13 Let M ⊂ B (H ) be a von Neumann algebra, {𝑥1 , … , 𝑥𝑛 } ⊂ M and 𝑛 {𝜉1 , … , 𝜉𝑛 } ⊂ H be such that ∑𝑘=1 𝑥𝑘 𝜉𝑘 = 0. Then there exists a subset {𝑥𝑖𝑗 ; 1 ≤ 𝑖, 𝑗 ≤ 𝑛} ⊂ M , such that 𝑛

∑ 𝑥𝑘 𝑥𝑘𝑗 = 0,

1 ≤ 𝑗 ≤ 𝑛,

𝑘=1 𝑛

∑ 𝑥𝑖𝑘 𝜉𝑘 = 𝜉𝑖 ,

1 ≤ 𝑖 ≤ 𝑛.

𝑘=1

In other words, H is a flat M -module. E.3.14 Let M ⊂ B (H ) be a von Neumann algebra and 𝑒 ∈ P M . Show that (i) if X ⊂ M is a *-subalgebra and M = R (X ), then M 𝑒 = R (X 𝑒 ). (ii) if X ′ ⊂ M ′ and M ′ = R (X ′ ), then M ′𝑒 = R (X ′𝑒 ). !E.3.15 Let M 1 ⊂ B (H 1 ), M 2 ⊂ B (H 2 ) be von Neumann algebras and 𝑒1 ∈ P M 1 , 𝑒2 ∈ P M 2 . Then 𝑒1 ⊗𝑒2 ∈ P M 1 ⊗M 2 , and the following relations hold: (M 1 ⊗M 2 )𝑒1 ⊗𝑒2 = (M 1 )𝑒1 ⊗(M 2 )𝑒2 , (M ′1 ⊗M ′2 )𝑒1 ⊗𝑒2 = (M ′1 )𝑒1 ⊗(M ′2 )𝑒2 . E.3.16 Let M 1 ⊂ B (H 1 ), M 2 ⊂ B (H 2 ) be von Neumann algebras. Show that if M 1 = R (X 1 ), M 2 = R (X 2 ), then M 1 ⊗M 2 = R (X ), where we have denoted X = {𝑥1 ⊗1; 𝑥1 ∈ X 1 } ∪ {1⊗𝑥2 ; 𝑥2 ∈ X 2 }. ▶

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!E.3.17 State and prove an associative property for the tensor product of von Neumann algebras. E.3.18 Let A be a 𝐶 ∗ -algebra, 𝔑 a left ideal of A and {𝑢𝛼 }𝛼∈Γ an approximate unit for 𝔑. Show that 𝔑 = {𝑥 ∈ A ; ‖𝑥 − 𝑥𝑢𝛼 ‖ → 0}. E.3.19 Let A be a 𝐶 ∗ -algebra and 𝔑 ⊂ A a closed left ideal of A . With the help of an approximate unit of 𝔑, show that 𝑎 ∈ 𝔑+ , 𝜆 > 0 ⇒ 𝑎 𝜆 ∈ 𝔑 + . (Hint: it is sufficient to prove that 𝑎1/2 ∈ 𝔑+ ). E.3.20 Let A be a 𝐶 ∗ -algebra, 𝑎 ∈ A + and 𝔑𝑎 , the closed left ideal of A generated by 𝑎 in A . Show that (i) 𝑢𝑛 = (𝑛−1 + 𝑎)−1 𝑎 is an approximate unit of 𝔑𝑎 . (ii) if 𝑥 ∈ A and 𝑥 ∗ 𝑥 ≤ 𝑎, then 𝑥 ∈ 𝔑𝑎 .

COMMENTS C.3.1 A wo-closed subalgebra A of B (H ), which contains the identity operator (but which is not assumed to be self-adjoint), is said to be transitive if 0 and H are the only closed linear subspaces K ⊂ H , which are A -invariant, i.e., such that 𝑥 ∈ A ⇒ 𝑥(K ) ⊂ K . Obviously, B (H ) is transitive. From the von Neumann density theorem (3.2), it easily follows that any self-adjoint transitive subalgebra A ⊂ B (H ) coincides with B (H ). If H is finitely dimensional, then a classical theorem of Burnside states that any transitive subalgebra A ⊂ B (H ) coincides with B (H ). The problem of establishing whether this statement is, or is not, true in the general case, is still unresolved, nor is the weaker problem of the existence of an invariant closed non-trivial vector subspace, for any operator in B (H ). The first problem is known as the ‘problem of the transitive algebras’, whereas the second is known as the ‘problem of the invariant subspaces’. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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An important contribution towards the solution of this problem, which has a surprisingly simple proof, was obtained by V. I. Lomonosov (1973). We state it in the form given by H. Radjavi and P. Rosenthal (1973): Theorem. Any transitive subalgebra A ⊂ B (H ), which contains a non-zero compact operator, coincides with B (H ). Corollary. Any operator 𝑥 ∈ B (H ), which commutes with a non-zero compact operator, has a non-trivial closed invariant vector subspace. The result contained in the preceding corollary was obtained independently by D. Voiculescu (1976), who used methods completely different from those of V. I. Lomonosov. A vector subspace K ⊂ H is said to be para-closed if it is the range of an operator in B (H ). C. Foiaṣ (1972) has shown that any subalgebra A ⊂ B (H ), whose only invariant para-closed vector subspaces are 0 and H , coincides with B (H ); this is another extension of the Burnside theorem. Extensions to von Neumann algebras have been obtained by D. Voiculescu (1972) and C. Peligrad (1975). For other information, we refer the reader to H. Radjavi and P. Rosenthal (1973). C.3.2 The essential fact in the proof of the density theorem of I. Kaplansky (3.10) is the so-continuity on B (H )ℎ of the mapping 𝑥 ↦ 𝑓(𝑥), where 𝑓(𝑡) = 2𝑡(1+ 𝑡 2 )−1 . The continuity in the so-topology of the functions of normal operators has been studied by I. Kaplansky (1951) and more recently by R. V. Kadison (1968). We mention the following results of R. V. Kadison: Theorem. Let Ω ⊂ ℂ be a subset, such that (Ω\Ω) ∩ Ω = ∅ and let 𝑓 ∶ Ω → ℂ be a complex function. Then the following properties are equivalent: (i) The mapping 𝑥 ↦ 𝑓(𝑥) is so-continuous on the set of all normal operators, whose spectrum is contained in Ω; (ii) The function 𝑓 is continuous, bounded on bounded sets and such that 𝑓(𝑧)/𝑧 is bounded at infinity. Therefore, the theorem is valid for open or closed set Ω. In particular, by taking Ω = ℝ in the preceding theorem, it follows that any continuous and bounded function 𝑓 ∶ ℝ ↦ ℂ is so-continuous on the set of all self-adjoint operators, a result that extends the fact that was used in the proof of the density theorem of I. Kaplansky. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.006

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C.3.3 Using the density theorem of I. Kaplansky, R. V. Kadison (1957) proved the following: Theorem. Let A ⊂ B (H ) be a wo-dense 𝐶 ∗ -algebra. Then for any 𝑥 ∈ B (H ) and any finite-dimensional vector subspace K ⊂ H , there exists an 𝑎 ∈ A , such that 𝑥𝜉 = 𝑎𝜉,

𝜉 ∈ H.

Moreover, if 𝑥 is self-adjoint, then a can be chosen so as to be self-adjoint too, whereas if 𝑥 is unitary and A contains the identity operator, then 𝑎 can be chosen to be unitary. A 𝐶 ∗ -algebra A ⊂ B (H ) is said to be irreducible if the only closed vector subspaces of K , which are invariant for all operators in A , are 0 and H ; the 𝐶 ∗ -algebra A is said to be strictly irreducible if the only invariant (not necessarily closed) vector subspace H is 0 and H . From the preceding theorem, one can infer the following: Corollary. A 𝐶 ∗ -algebra A ⊂ B (H ) is irreducible iff it is strictly irreducible. For the proof of these results, with more precise formulations, we refer to the books of J. Dixmier (1964) and S. Sakai (1971). M. Tomita (1960) (see also Saitô [1967]) has obtained extensions of these results. For an exposition of these extensions, we refer to L. Zsidó (1973c). A study of the strict irreducibility for the representations of the Banach algebras with involution has been carried out by B. A. Barnes (1972). C.3.4 Let M ⊂ B (H ) be a von Neumann algebra. For any set X ⊂ M + , we denote by X 𝜍 (resp., X 𝛿 ) the set of all elements of M + , which are suprema (resp., infima) of increasing sequences (resp., decreasing sequences) of elements in X ; we also denote by X 𝑚 (resp., X 𝑚 ) the set of all elements of M + , which are suprema (resp., infima) of bounded increasingly (resp., decreasingly) directed subsets of X . G. K. Pedersen (1971, 1972), proved the following remarkable result: Theorem. If M ⊂ B (H ) is a von Neumann algebra of countable type, and if A ⊂ M is a 𝐶 ∗ -algebra, which is wo-dense in M , then + 𝜍 ((A + 1 ) )𝛿 = M 1 .

If M is not assumed to be of countable type, then this theorem fails to be true even in the commutative case. Nevertheless, we have the following result, due to G. K. Pedersen (loc. cit.):

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Theorem. If M ⊂ B (H ) is a von Neumann algebra and if A ⊂ M is a wo-dense 𝐶 ∗ -subalgebra, then 𝑚 𝑚 (((A + = M+ 1 ) )𝑚 ) 1.

From these theorems, one can get the older results of R. V. Kadison (1956, 1957), which we shall now state: 𝑚 Corollary. Let M ⊂ B (H ) be a 𝐶 ∗ -algebra. If (M + ⊂ M , or if H is 1) + 𝜍 separable and (M 1 ) ⊂ M , then M is a von Neumann algebra. A proof of the results of G. K. Pedersen can be found in L. Zsidó (1973b). ̆ ă (1973a). The results of R. V. Kadison are also presented in S. ¸ Stratil

C.3.5 J. Dixmier and O. Maréchal (1971) proved the following: Theorem. The set of all cyclic vectors of a von Neumann algebra M ⊂ B (H ) is a 𝐺𝛿 -set, which is either empty, or dense in H . We observe that for the proof of this result, they also showed that any element 𝑥 in a von Neumann algebra M is the so-limit of a sequence {𝑥𝑛 } of invertible elements in M , such that ‖𝑥𝑛 ‖ ≤ ‖𝑥‖. We also mention the following result of M. Broise (1967): Theorem. Let M , N ⊂ B (H ) be von Neumann algebras, such that N ′ is abelian and of countable type. If M and N have the same cyclic vectors, then M =N. C.3.6 Let (Ω, B , 𝜇) be a finite measure space. We shall now consider the Hilbert space H 𝜇 = L 2 (𝜇) and for any 𝑓 ∈ L ∞ (𝜇), we shall denote by 𝑥𝑓 ∈ B (H 𝜇 ) the operator given by the multiplication by 𝑓. Then M 𝜇 = {𝑥𝑓 ; 𝑓 ∈ L ∞ (𝜇)} ⊂ B (H ) is a maximal abelian von Neumann algebra of countable type. On the other hand, if M ⊂ B (H ) is a maximal abelian von Neumann algebra of countable type, then M has a cyclic and separating vector (see E.3.10), which determines in a natural manner a measure 𝜇 on the spectrum of M (see C.2.1). It is easily shown that M is *-isomorphic with M 𝜇 , and therefore, M and M 𝜇 are spatially isomorphic (E.5.22). Consequently, the maximal abelian von Neumann algebras of countable type can be described in a simple manner, modulo the spatial isomorphism. This description can be easily extended to arbitrary maximal abelian von Neumann algebras.

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C.3.7 In this chapter, we introduced the following elementary operations on von Neumann algebras: the passing to the commutant, the reduction, the induction and the tensor product; the amplification, which is a particular case of the latter. Another important operation, which did not explicitly appear in our presentation, is of course the direct sum: if M ⊂ B (H ) and N ⊂ B (H ) are von Neumann algebras, then M ⊕ N = {𝑥 ⊕ 𝑦 ∈ B (H ⊕ H ); 𝑥 ∈ M , 𝑦 ∈ 𝑁 } is a von Neumann algebra. Another operation is the cross-product, which we shall describe in what follows. Let M ⊂ B (H ) be a von Neumann algebra, Aut(M ) the group of all *-automorphisms of M , 𝐺 a locally compact group, d𝑔 its left Haar measure, and 𝐺 ∋ 𝑔 ↦ 𝜋𝑔 ∈ Aut(M ) a homomorphism of groups, such that for any 𝑥 ∈ M , the mapping 𝐺 ∋ 𝑔 ↦ 𝜋𝑔 (𝑥) ∈ M is continuous for the wo-topology in M . One considers the space K (𝐺, H ) of all functions defined on 𝐺 and taking values in H , which have compact supports and are continuous for the norm topology; we endow it with the scalar product (𝑓1 |𝑓2 ) = ∫ (𝑓1 (𝑔)|𝑓2 (𝑔)) d𝑔, 𝐺

and we denote by L (𝐺; H ) the Hilbert space obtained by completion. For any 𝑥 ∈ M , the operator 𝑡𝑥 ∈ B (L 2 (𝐺, H )) is defined by the relations 2

′ ′ (𝑡𝑥 (𝑓))(𝑔′ ) = 𝜋𝑔−1 ′ (𝑥)(𝑓(𝑔 )), 𝑓 ∈ K (𝐺; H ), 𝑔 ∈ 𝐺,

whereas for any 𝑔 ∈ 𝐺, one defines the (unitary) operator ᵆ𝑔 ∈ B (L 2 (𝐺; H )) by the relations (ᵆ𝑔 (𝑓))(𝑔′ ) = 𝑓(𝑔−1 𝑔′ ),

𝑓 ∈ K (𝐺; H ), 𝑔′ ∈ 𝐺.

The von Neumann algebra generated in B (L 2 (𝐺, H )) by the operators 𝑡𝑥 , 𝑥 ∈ M , and ᵆ𝑔 , 𝑔 ∈ 𝐺, is called the cross-product of M by the action 𝜋 of 𝐺, and it is denoted by R (M , 𝜋) or, simply, by M × 𝐺.

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If 𝐺 is discrete, the preceding construction appears in the work of F. J. Murray and J. von Neumann (1932) in connection with the construction of different types of factors (see C.4.3), whereas systematic expositions of this construction appear in J. Dixmier (1957/1969, Chap. I, §9.2), T. Turumaru (1958), N. Suzuki (1959), and so on. In the general case, this construction first appears in the paper of S. Doplicher, D. Kastler and D. W. Robinson (1966); it is systematically studied by M. Takesaki (1973). If 𝐺 is a separable abelian locally compact group, which acts by *-automorphisms of the von Neumann algebra M ⊂ B (H ), where H is assumed to be separable, then the group 𝐺̂ of the characters of 𝐺 acts in a natural manner by *-automorphisms of M × 𝐺; M. Takesaki (1973) has shown that the von Neumann algebra (M × 𝐺) × 𝐺̂ is *-isomorphic to the von Neumann algebra M ⊗B (L 2 (𝐺)). In particular, if M is properly infinite, then the following isomorphism (M × 𝐺) × 𝐺̂ ≅ M holds, thus yielding a duality theorem. For extensions to the non-abelian case, ̆ a,̆ D. V. Voiculescu, L. Zsidó (1976a,b) and M. Landstad we refer to S. ¸ Stratil (1977, 1979). Another definition of a cross-product, which is better adapted to the construction of factors, has been given by W. Krieger (1970). Finally, let us mention the fact that one can define a notion of infinite tensor product for von Neumann algebras, for which we refer to J. von Neumann (1938), D. Bures (1963), A. Guichardet (1969) and H. Araki and E. J. Woods (1968). C.3.8 The extreme points of the closed unit ball of a 𝐶 ∗ -algebra have been thoroughly studied, the fundamental results being the following two theorems of R. V. Kadison (1951), which we state in the improved version given by S. Sakai (1971): Theorem. The closed unit ball of a 𝐶 ∗ -algebra A has extreme points iff A has the unit element. In this case, the unit element is an extreme point. Theorem. An element 𝑥 of the closed unit ball of a 𝐶 ∗ -algebra A is extreme iff (1 − 𝑥𝑥 ∗ )A (1 − 𝑥 ∗ 𝑥) = {0}.

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Lectures on von Neumann algebras Lemma 3.19 is an easy consequence of the last theorem. R. V. Kadison used these results in order to study the isometries between algebras of operators, in generalizing the classical theorem of S. Banach and M. H. Stone (see Dunford and Schwartz [1958/1963/1970], V.8.8). We also mention the following strong results due to B. Russo and H. A. Dye (1966): Theorem. In any 𝐶 ∗ -algebra A with a unit element, the uniformly closed convex hull of the set {ᵆ ∈ A ; ᵆ unitary} coincides with the closed unit ball of A . For other related results, we refer to P. E. Miles (1964), B. Yood (1969), J. B. Conway and J. Szücs (1973), F. F. Bonsall and J. Duncan (1973).

C.3.9 Bibliographical comments. Theorem 3.2 is due to J. von Neumann (1940), whereas Theorem 3.10 to I. Kaplansky (1950). Elementary operations on von Neumann algebras have been considered by many authors, among whom we mention F. J. Murray and J. von Neumann (1936), J. von Neumann (1938), J. Dixmier (1952), I. E. Segal (1951), Y. Misonou (1954) and M. Tomita (1953, 1954). For a detailed exposition of the properties of ideals of algebras of ̆ ă operators, as well as for the corresponding references, we refer to S. ¸ Stratil (1973a). The term ‘von Neumann algebra’ was introduced by J. Dixmier (1957/1969), F. J. Murray and J. von Neumann called these algebras ‘rings of operators’. I. M. Gelfand and M. A. Naimark called the Banach algebras ‘normed rings’; the term ‘Banach algebra’ was introduced by E. Hille. The terms ‘𝐶 ∗ algebra’ and ‘𝑊 ∗ -algebra’ (see C.5.3) have been introduced by I. E. Segal. Sometimes, for 𝐶 ∗ -algebras, one uses the equivalent term ‘𝐵 ∗ -algebras’; this double terminology is related to the problems discussed at the end of Section C.2.1. In writing this chapter, we used the books by J. Dixmier (1957/1969, 1964/1968), and also the course by D. M. Topping (1971).

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4

THE GEOMETRY OF PROJECTIONS AND THE CLASSIFICATION OF VON NEUMANN ALGEBRAS

In this chapter, we study the relations existing between the lattice operations and the equivalence of projections, and we also introduce the classification of von Neumann algebras according to types. 4.1. Let M ⊂ B (H ) be a von Neumann algebra. We shall denote by P M the set of the projections in M . Then 𝒫M is a complete lattice (see Corollary 3.7). Two projections 𝑒, 𝑓 ∈ P M are said to be equivalent, and this relation is denoted by 𝑒 ∼ 𝑓, if there exists a partial isometry 𝑢 ∈ M , such that 𝑒 = 𝑢∗ 𝑢 and 𝑓 = 𝑢𝑢∗ ; then 𝑢𝑒 = 𝑢 = 𝑓𝑢. We say that 𝑒 is dominated by 𝑓, and we denote this relation by 𝑒 ≺ 𝑓, if 𝑒 is equivalent to a subprojection of 𝑓. The relation ‘∼’ is an equivalence relation in P M , whereas the relation ‘≺’ is a preorder relation in 𝒫M . If 𝑒 ≺ 𝑝𝜉 , 𝜉 ∈ H , then there exists an 𝜂 ∈ H , such that 𝑒 = 𝑝𝜂 . If 𝑒 ∼ 𝑓, then z(𝑒) = z(𝑓); in particular, if 𝑒 ∼ 0, then 𝑒 = 0. If 𝑒 ∼ 𝑓 and 𝑝 ∈ P X , then 𝑒𝑝 ∼ 𝑓𝑝. If 𝑒 ∼ 𝑓 via the partial isometry 𝑢, then the mapping 𝑒M 𝑒 ∋ 𝑥 ↦ 𝑢𝑥𝑢∗ ∈ 𝑓M 𝑓 is an isomorphism of *-algebras. The image 𝑢𝑔𝑢∗ of the projection 𝑔 ∈ 𝑒M 𝑒 is a projection equivalent to 𝑔. If 𝑒 = ∨𝑖∈𝐼 𝑒𝑖 , where 𝑒𝑖 is a mutually orthogonal projection, and if 𝑓 ∼ 𝑒, then there exists a family {𝑓𝑖 )𝑖∈𝐼 ⊂ P M , where 𝑓𝑖 is a mutually orthogonal projection, such that 𝑓 = ∨𝑖∈𝐼 𝑓𝑖 and 𝑓𝑖 ∼ 𝑒𝑖 , for any 𝑖 ∈ 𝐼. Proposition 4.2. Let {𝑒𝑖 }𝑖∈𝐼 , {𝑓𝑖 }𝑖∈𝐼 ⊂ P M , where the 𝑒𝑖 are mutually orthogonal projections, and the 𝑓𝑖 are also mutually orthogonal projections. If 𝑒𝑖 ∼ 𝑓𝑖 , for any 𝑖 ∈ 𝐼, then ∨𝑖∈𝐼 𝑒𝑖 ∼ ∨𝑖∈𝐼 𝑓𝑖 . Proof. Let 𝑢𝑖 ∈ M be such that 𝑒𝑖 = 𝑢𝑖∗ 𝑢𝑖 ,

𝑓𝑖 = 𝑢𝑖 𝑢𝑖∗ ,

𝑖 ∈ 𝐼, 87

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and for any finite subset 𝐽 ⊂ 𝐼, let 𝑢𝐽 = ∑ 𝑢𝑖 . 𝑖∈𝐽

We then have a net {𝑢𝐽 }𝐽 of operators, about which we shall prove that it is so-convergent. In order to do this, we have only to show that for any 𝜉 ∈ H , the net {𝑢𝐽 𝜉}𝐽 is convergent (in accordance with the Banach–Steinhaus theorem). Since the vectors 𝑢𝑖 𝜉 are mutually orthogonal, this is equivalent to the following condition: ∑ ‖𝑢𝑖 𝜉‖2 = lim ∑ ‖𝑢𝑖 𝜉‖2 < +∞; 𝐽

𝑖∈𝐼

𝑖∈𝐽

indeed, we have ∑ ‖𝑢𝑖 𝜉‖2 = ∑(𝑢𝑖∗ 𝑢𝑖 𝜉|𝜉) = ∑(𝑒𝑖 𝜉|𝜉) = (( 𝑖∈𝐼

𝑖∈𝐼

𝑖∈𝐼

⋁

𝑒𝑖 )𝜉|𝜉) < +∞.

𝑖∈𝐼

Consequently, the operators 𝑢 = ∑ 𝑢𝑖 , 𝑖∈𝐼

𝑣 = ∑ 𝑢𝑖∗ 𝑖∈𝐼

exist by so-convergence, and they belong to M . From 𝑢 = 𝑠𝑜-∑𝑖∈𝐼 𝑢𝑖 , it follows that 𝑢∗ = 𝑤𝑜-∑𝑖∈𝐼 𝑢𝑖∗ , and therefore, 𝑣 = 𝑢∗ . From the equalities 𝑢𝑖∗ 𝑢𝑘 = 𝛿𝑖𝑘 𝑒𝑖 , it follows that 𝑢∗ 𝑢 = ∨𝑖∈𝐼 𝑒𝑖 , 𝑢𝑢∗ = ∨𝑖∈𝐼 𝑓𝑖 . Q.E.D. Theorem 4.3. For any 𝑥 ∈ M , one has l(𝑥) ∼ r(𝑥). Proof. Let 𝑥 = 𝑢|𝑥| be the polar decomposition of 𝑥 in M . Then the partial isometry 𝑢 implements the equivalence l(𝑥) ∼ s(|𝑥|). From n(|𝑥|) = n(𝑥), we also infer that s(|𝑥|) = l(|𝑥|) = l(𝑥 ∗ ) = r(𝑥). Q.E.D. Corollary 4.4 (the parallelogram rule). For any 𝑒, 𝑓 ∈ P M , one has the following relations: (i) 𝑒 ∨ 𝑓 − 𝑓 ∼ 𝑒 − 𝑒 ∧ 𝑓. (ii) 𝑒 − 𝑒 ∧ (1 − 𝑓) ∼ 𝑓 − (1 − 𝑒) ∧ 𝑓. Proof. It follows from Theorem 4.3 and exercise E.2.3.

Q.E.D.

Corollary 4.5. Let 𝑒, 𝑓 ∼ P M . The following assertions are equivalent: (i) 𝑒M 𝑓 ≠ {0}. (ii) There exist 𝑒1 , 𝑓1 ∈ P M , 0 ≠ 𝑒1 ≤ 𝑒, 0 ≠ 𝑓1 ≤ 𝑓, such that 𝑒1 ∼ 𝑓1 . (iii) z(𝑒)z(𝑓) ≠ 0. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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Proof. (i) ⇒ (ii): if 𝑥 ∈ M and 𝑒𝑥𝑓 ≠ 0, then 𝑒1 = l(𝑒𝑥𝑓) and 𝑓1 = r(𝑒𝑥𝑓) satisfy condition (ii). (ii) ⇒ (i): if 𝑒 ≥ 𝑒1 = 𝑢∗ 𝑢, 𝑓 ≥ 𝑓1 = 𝑢𝑢∗ ≠ 0, then we have 𝑒𝑢∗ 𝑓 = 𝑢∗ ≠ 0. (i) ⇒ (iii): if z(𝑒)z(𝑓) = 0, then 𝑒𝑥𝑓 = z(𝑒)𝑒𝑥𝑓z(𝑓) = 0, for any 𝑥 ∈ M . (iii) ⇒ (i): by taking into account Corollary 3.9, from 𝑒M 𝑓 = {0}, it follows that 𝑒z(𝑓) = 0, and therefore z(𝑒)z(𝑓) = 0. Q.E.D. Theorem 4.6 (the comparison theorem). For any 𝑒, 𝑓 ∈ P M , there exists a 𝑝 ∈ P X , such that 𝑒𝑝 ≺ 𝑓𝑝, 𝑒(1 − 𝑝) ≻ 𝑓(1 − 𝑝). Proof. Let ({𝑒𝑖 }𝑖∈𝐼 , {𝑓𝑖 }𝑖∈𝐼 ) be a maximal pair of families of mutually orthogonal projections, such that 𝑒𝑖 ≤ 𝑒, 𝑓𝑖 ≤ 𝑓, 𝑒𝑖 ∼ 𝑓𝑖 , 𝑖 ∈ 𝐼. In accordance with Proposition 4.2, it follows that 𝑒1 =

⋁

𝑒𝑖 ∼

𝑖∈𝐼

⋁

𝑓𝑖 = 𝑓1 .

𝑖∈𝐼

If 𝑒2 = 𝑒−𝑒1 and 𝑓2 = 𝑓−𝑓1 , then, due to the maximality of the chosen pair and to Corollary 4.5, it follows that z(𝑒2 )z(𝑓2 ) = 0. Let us define 𝑝 = z(𝑓2 ). Then we have 𝑒𝑝 = 𝑒1 𝑝 + 𝑒2 𝑝 = 𝑒1 𝑝 + 𝑒2 z(𝑒2 )z(𝑓2 ) = 𝑒1 𝑝 ∼ 𝑓1 𝑝 ≤ 𝑓𝑝, 𝑓(1 − 𝑝) = 𝑓1 (1 − 𝑝) + 𝑓2 (1 − 𝑝) = 𝑓1 (1 − 𝑝) ∼ 𝑒1 (1 − 𝑝) ≤ 𝑒(1 − 𝑝). Q.E.D. Theorem 4.7 (von Neumann’s Schröder–Bernstein-type theorem). Let 𝑒, 𝑓 ∈ P M . If 𝑒 ≺ 𝑓 and 𝑓 ≺ 𝑒, then 𝑒 ∼ 𝑓. Proof. Let 𝑤, 𝑣 ∈ M be such that 𝑤𝑤 ∗ = 𝑒,

𝑤 ∗ 𝑤 ≤ 𝑓,

𝑣𝑣 ∗ = 𝑓,

𝑣 ∗ 𝑣 ≤ 𝑒.

For 𝑔 ∈ P M , 𝑔 ≤ 𝑓, we define 𝜑(𝑔) = 𝑓 − 𝑤 ∗ (𝑒 − 𝑣 ∗ 𝑔𝑣)𝑤. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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It is easy to see that 𝜑 is an increasing function on the complete lattice {𝑔 ∈ P M ; 𝑔 ≤ 𝑓}. Let X = {𝑔; 𝑔 ≤ 𝜑(𝑔)} and ℎ = ⋁𝑔∈X 𝑔. If 𝑔 ∈ X , then 𝑔 ≤ ℎ, and therefore, 𝑔 ≤ 𝜑(𝑔) ≤ 𝜑(ℎ); hence ℎ ≤ 𝜑(ℎ). Consequently, we have 𝜑(ℎ) ≤ 𝜑(𝜑(ℎ)), and therefore, 𝜑(ℎ) ∈ X ; it follows that 𝜑(ℎ) ≤ ℎ. Consequently, we have ℎ = 𝑓 − 𝑤 ∗ (𝑒 − 𝑣 ∗ ℎ𝑣)𝑤. The partial isometries ℎ𝑣 and (𝑒 − 𝑣 ∗ ℎ𝑣)𝑤 yield the equivalences: ℎ ∼ 𝑣 ∗ ℎ𝑣 and 𝑓 − ℎ ∼ 𝑒 − 𝑣 ∗ ℎ𝑣. Consequently, we have 𝑒 ∼ 𝑓.

Q.E.D.

4.8. A projection 𝑒 ∈ P M is said to be abelian if the reduced algebra 𝑒M 𝑒 is commutative. A projection 𝑒 ∈ P M is said to be finite if 𝑓 ∈ P M , 𝑓 ≤ 𝑒, 𝑓 ∼ 𝑒 ⇒ 𝑓 = 𝑒. One says that a projection 𝑒 ∈ P M is properly infinite if 𝑝 ∈ P Z , 𝑝𝑒 finite ⇒ 𝑝𝑒 = 0. Any abelian projection is finite. If 𝑒 is finite and 𝑓 ≺ 𝑒, then 𝑓 is finite. If 𝑒 is abelian and 𝑓 ≺ 𝑒, then 𝑓 is abelian. Proposition 4.9. Let 𝑒, 𝑓 ∈ P M . If 𝑒 is abelian and 𝑓 ≤ 𝑒, then 𝑓 = 𝑒z(𝑓). Proof. Since 𝑒M 𝑒 is commutative, for any 𝑥 ∈ M , we have 𝑓𝑥(𝑒 − 𝑓) = 𝑓(𝑒𝑥𝑒)(𝑒 − 𝑓) = 𝑓(𝑒 − 𝑓)(𝑒𝑥𝑒) = 0; hence 𝑓M (𝑒 − 𝑓) = {0}. In accordance with Corollary 4.5, it follows that z(𝑓)z(𝑒 − 𝑓) = 0. In particular, we have 𝑓 = 𝑒z(𝑓). Q.E.D. Proposition 4.10. Let 𝑒, 𝑓 ∈ P M be abelian projections. If z(𝑒) ≤ z(𝑓), then 𝑒 ≺ 𝑓. If z(𝑒) = z(𝑓), then 𝑒 ∼ 𝑓. Proof. In accordance with Theorem 4.7, it is sufficient to prove only the first assertion. For this, in accordance with the comparison theorem (4.6), we can suppose that 𝑓 ≤ 𝑒. But then 𝑓 = 𝑒z(𝑓), in accordance with Proposition 4.9. Therefore, we have 𝑒 = 𝑒z(𝑒) = 𝑒z(𝑓) = 𝑓. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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Proposition 4.11. Let 𝑒 ∈ P M be such that it does not contain any non-zero abelian subprojection. Then there exist 𝑒1 , 𝑒2 ∈ P M , such that 𝑒 = 𝑒1 + 𝑒2 ,

𝑒1 𝑒2 = 0,

𝑒1 ∼ 𝑒2 .

Proof. Let ({𝑒1,𝑖 }𝑖∈𝐼 , {𝑒2,𝑖 }𝑖∈𝐼 ) be a maximal pair of families of mutually orthogonal projections, such that 𝑒1,𝑖 ≤ 𝑒,

𝑒2,𝑖 ≤ 𝑒,

𝑒1,𝑖 𝑒2,𝑖 = 0,

𝑒1,𝑖 ∼ 𝑒2,𝑖 ,

𝑖 ∈ 𝐼.

In accordance with Proposition 4.2, it follows that 𝑒1 =

⋁

𝑒1,𝑖 ∼

𝑖∈𝐼

⋁

𝑒2,𝑖 = 𝑒2 ,

𝑒1 𝑒2 = 0.

𝑖∈𝐼

If ℎ = 𝑒 − 𝑒1 − 𝑒2 ≠ 0, then ℎ is not abelian; consequently, ℎ contains a non-zero subprojection 𝑔, which is not central in ℎ M ℎ. It follows that 𝑔M (ℎ − 𝑔) ≠ {0}. In accordance with Corollary 4.5, this result contradicts the maximality of the chosen pair; consequently, ℎ = 0, i.e., 𝑒 = 𝑒1 + 𝑒2 . Q.E.D. Proposition 4.12. Let 𝑒 ∈ P M . Then 𝑒 is properly infinite iff there exists a countable family {𝑒𝑛 } ⊂ P M of mutually orthogonal projections, which are bounded from above by 𝑒, and such that 𝑒=

⋁

𝑒𝑛

𝑛

𝑒𝑛 ∼ 𝑒,

for any 𝑛.

Corollary. A projection 𝑒 ∈ P M is properly infinite, iff there exist 𝑒1 , 𝑒2 ∈ P M , such that 𝑒 = 𝑒1 + 𝑒2 , 𝑒1 𝑒2 = 0, 𝑒1 ∼ 𝑒2 ∼ 𝑒. Proof. We proceed by steps: (I) If 𝑒 is not finite, then 𝑒 contains an infinite set of mutually orthogonal subprojections, which are equivalent and different from zero. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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Lectures on von Neumann algebras Indeed, let 𝑒1 ≤ 𝑒, 𝑒1 ≠ 𝑒, 𝑒1 ∼ 𝑒 and 𝑓1 = 𝑒 − 𝑒1 . There exist 𝑒2 , 𝑓2 ∈ P M , such that 𝑒1 = 𝑒2 + 𝑓2 , 𝑒2 𝑓2 = 0, 𝑒1 ∼ 𝑒2 , 𝑓1 ∼ 𝑓2 . Then there exist 𝑒3 , 𝑓3 ∈ P M , such that 𝑒2 = 𝑒3 + 𝑓3 ,

𝑒3 𝑓3 = 0,

𝑒2 ∼ 𝑒3 ,

𝑓2 ∼ 𝑓3 .

Now we proceed by induction. The required set is {𝑓1 , 𝑓2 , 𝑓3 , …}. (II) If {𝑒𝑖 }𝑖∈𝐼 is an infinite family of mutually orthogonal, equivalent projections, 𝑒 = ⋁𝑖∈𝐼 𝑒𝑖 and if 𝑓 ∈ P M , 𝑓𝑒 = 0, 𝑓 ≺ 𝑒𝑖 , then there exist {ℎ𝑖 }𝑖∈𝐼 ⊂ P M , such that the ℎ𝑖 be mutually orthogonal and 𝑒 + 𝑓 = ⋁𝑖∈𝐼 ℎ𝑖 , 𝑒𝑖 ∼ ℎ𝑖 for any 𝑖 ∈ 𝐼. Indeed, let 𝑖0 ∈ 𝐼 and let 𝑓𝑖0 , 𝑔𝑖0 ∈ P M be such that 𝑒𝑖0 = 𝑓𝑖0 + 𝑔𝑖0 ,

𝑓𝑖0 𝑔𝑖0 = 0,

𝑓𝑖0 ∼ 𝑓.

For any 𝑖 ∈ 𝐼, there exist 𝑓𝑖 , 𝑔𝑖 ∈ P M , such that 𝑒𝑖 = 𝑓𝑖 + 𝑔𝑖 ,

𝑓𝑖 𝑔𝑖 = 0,

𝑓𝑖 ∼ 𝑓𝑖0 ,

𝑔𝑖 ∼ 𝑔𝑖0 .

Let 𝜑 be a bijection between the sets {𝑔𝑖 }𝑖∈𝐼 and {𝑓, 𝑓𝑖 }𝑖∈𝐼 , and let ℎ𝑖 = 𝑔𝑖 + 𝜑(𝑔𝑖 ),

𝑖 ∈ 𝐼.

Now the assertion is obvious. (III) If {𝑒𝑖 }𝑖∈𝐼 is a maximal infinite family of mutually orthogonal, equivalent projections, then there exists a 𝑝 ∈ P Z , 𝑝 ≠ 0, and {𝑔𝑖 }𝑖∈𝐼 ⊂ P M , where the 𝑔𝑖 ’s are mutually orthogonal, such that 𝑝 = ⋁𝑖∈𝐼 𝑔𝑖 and 𝑔𝑖 ∼ 𝑝𝑒𝑖 , for any 𝑖 ∈ 𝐼. Indeed, let 𝑒 = ⋁𝑖∈𝐼 𝑒𝑖 and 𝑖0 ∈ 𝐼. Since the relation 𝑒𝑖0 ≺ 1 − 𝑒 is not possible (due to the maximality of the family), by applying the comparison theorem to 𝑒𝑖0 and 1 − 𝑒, it follows that there exists a 𝑝 ∈ P M , 𝑝 ≠ 0, such that 𝑝(1 − 𝑒) < 𝑝𝑒𝑖0 . But 𝑝 = 𝑝(1 − 𝑒) ∨ (⋁𝑖∈𝐼 𝑝𝑒𝑖 ), and the assertion follows from (II). (IV) If 1 ∈ M is properly infinite, then 1 is the least upper bound of a countable family of equivalent, mutually orthogonal projections. Indeed, with (I), there exists a maximal infinite family of equivalent, mutually orthogonal projections, whereas from (III), we infer that there exists a 𝑝 ∈ P Z , 𝑝 ≠ 0, which is the least upper bound of a countable family of equivalent, mutually orthogonal projections. Since 1 − 𝑝 is also properly infinite, the proof proceeds by transfinite induction. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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(V) If 𝑒 ∈ P M is properly infinite, then, in accordance with (IV), there exists a countable family {𝑓𝑛 } of equivalent, mutually orthogonal projections, such that ⋁𝑛 𝑓𝑛 = 𝑒. Let us define 𝑒𝑛 = ∨{𝑓𝑚 ; 𝑛 divides 𝑚 and 𝑓𝑚 𝑒𝑘 = 0 for 𝑘 < 𝑛}. Then 𝑒 = ⋁𝑛 𝑒𝑛 and 𝑒𝑛 ∼ 𝑒, for any 𝑛; the first part of the proposition is proved. The other assertions follow immediately. Q.E.D. 4.13. One says that a projection 𝑒 ∈ P M is of countable type if any family of mutually orthogonal non-zero projections, majorized by 𝑒, is at most countable. Proposition. Let 𝑒, 𝑓 ∈ P M with 𝑒 of countable type and 𝑓 properly infinite. If z(𝑒) ≤ z(𝑓), then 𝑒 ≺ 𝑓. Proof. Let {𝑒𝑖 }𝑖∈𝐼 be a maximal family of mutually orthogonal non-zero projections, such that 𝑒𝑖 ≤ 𝑒, 𝑒𝑖 ≺ 𝑓, 𝑖 ∈ 𝐼. By virtue of Corollary 4.5 and of the maximality of the chosen family, we infer that 𝑒=

⋁

𝑒𝑖 .

𝑖∈𝐼

Since 𝑒 is of countable type, the set 𝐼 is at most countable. With Proposition 4.12, it follows that there exists a family {𝑓𝑖 }𝑖∈𝐼 of mutually orthogonal projections, such that 𝑓=

⋁

𝑓𝑖 , 𝑓𝑖 ∼ 𝑓,

𝑖 ∈ 𝐼.

𝑖∈𝐼

It follows that 𝑒𝑖 ≺ 𝑓𝑖 , 𝑖 ∈ 𝐼, hence 𝑒 < 𝑓.

Q.E.D.

Lemma 4.14. Let {𝑒𝑖 }𝑖∈𝐼 be a family of finite (resp., abelian) projections, whose central supports are mutually orthogonal. Then 𝑒 = ⋁𝑖∈𝐼 𝑒𝑖 is a finite (resp., abelian) projection. Proof. Let 𝑓 ∈ P M , 𝑓 ≤ 𝑒, 𝑓 ∼ 𝑒. Since 𝑒 − 𝑒𝑖 ≤ ⋁𝑛≠𝑖 z(𝑒𝑛 ), it follows that 𝑒𝑖 = 𝑒z(𝑒𝑖 ). Since 𝑒𝑖 is finite, it follows that 𝑒𝑖 = 𝑓z(𝑒𝑖 ) ≤ 𝑓,

𝑖 ∈ 𝐼.

Consequently, 𝑒 = 𝑓. For the case of abelian projections, the proof is similar.

Q.E.D.

Proposition 4.15. If 𝑒, 𝑓 ∈ P M are finite projections, then 𝑒 ∨ 𝑓 is a finite projection. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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Proof. We can assume that 𝑒 ∨ 𝑓 = 1. We shall also assume that 1 is not finite and we shall get a contradiction. From Lemma 4.14, there exists a finite central projection 𝑝 ∈ P Z , such that 1 − 𝑝 be properly infinite. Therefore, we can assume that 1 is properly infinite. From Corollary 4.12, there exists a 𝑔 ∈ P M , such that 𝑔 ∼ 1 ∼ 1 − 𝑔. From the parallelogram rule (4.4), we get 1 − 𝑒 = 𝑒 ∨ 𝑓 − 𝑒 ∼ 𝑓 − 𝑒 ∧ 𝑓 ≤ 𝑓, hence 1 − 𝑒 is a finite projection. We now apply the comparison theorem (4.6) to the projections 𝑔 ∧ (1 − 𝑒) and (1 − 𝑔) ∧ 𝑒. It follows that we can consider that one of the following relations holds, without any loss of generality: (1)

𝑔 ∧ (1 − 𝑒) ≺ (1 − 𝑔) ∧ 𝑒,

(2)

(1 − 𝑔) ∧ 𝑒 ≺ 𝑔 ∧ (1 − 𝑒).

In the first case, by taking into account the parallelogram rule, we get 𝑔 = 𝑔 ∧ (1 − 𝑒) + (𝑔 − 𝑔 ∧ (1 − 𝑒)) ≺ (1 − 𝑔) ∧ 𝑒 + (𝑔 ∨ (1 − 𝑒) − (1 − 𝑒)) ≤ 𝑒, and this contradicts the finiteness of 𝑒. In the second case, a similar argument leads to 1−𝑔 ≺ 1 − 𝑒, thus contradicting the finiteness of 1 − 𝑒. The proposition is proved. Q.E.D. 4.16. Let M ⊂ B (H ) be a von Neumann algebra. M is said to be finite if 1 is a finite projection. M is said to be semifinite if any non-zero central projection contains a non-zero finite projection. M is said to be of type I if any non-zero central projection contains a non-zero abelian projection. M is said to be of type II if it is semifinite, and it does not contain any non-zero abelian projection. M is said to be of type III if it does not contain any non-zero finite projection. M is said to be of type 𝐼fin if it is finite and of type I. M is said to be of type 𝐼∞ if it is not finite and it is of type I. M is said to be of type 𝐼𝐼1 if it is finite and of type II. M is said to be of type 𝐼𝐼∞ if it is not finite, but it is of type II.

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Other terms used in this connection are introduced by the following definitions: M is said to be discrete if it is of type I. M is said to be continuous if it is of type II or III. M is said to be properly infinite if 1 is a properly infinite projection. M is said to be purely infinite if it is of type III. Theorem 4.17 (of classification). Let M ⊂ B (H ) be a von Neumann algebra. Then 5 there exist unique projections 𝑝𝑖 ∈ P Z , 𝑖 = 1, 2, 3, 4, 5, such that ∑𝑖=1 𝑝𝑖 = 1 and M 𝑝1 is of type 𝐼𝑓𝑖𝑛 , M 𝑝2 is of type 𝐼∞ , M 𝑝3 is of type 𝐼𝐼1 , M 𝑝4 is of type 𝐼𝐼∞ , M 𝑝5 is of type 𝐼𝐼𝐼. Proof. The theorem follows from the superposition of the following three decompositions: (i) There exist unique projections 𝑝0 , 𝑞0 ∈ P Z , such that 𝑝0 𝑞0 = 0, 𝑝0 + 𝑞0 = 1, M 𝑝0 is semifinite and M 𝑞0 is purely infinite. Indeed, let us define 𝑝0 = ∨{𝑝 ∈ P Z ; M 𝑝 is semifinite} and 𝑞0 = 1 − 𝑝0 . (ii) There exist unique projections 𝑝0 , 𝑞0 ∈ P Z , such that 𝑝0 𝑞0 = 0, 𝑝0 + 𝑞0 = 1, M 𝑝0 is finite and M 𝑞0 is properly infinite. Indeed, let us define 𝑝0 = ∨{𝑝 ∈ P Z ; 𝑝 is finite} and 𝑞0 = 1 − 𝑝0 . The fact that 𝑝0 is finite follows from Lemma 4.14. (iii) There exist unique projections 𝑝0 , 𝑞0 ∈ P Z , such that 𝑝0 𝑞0 = 0, 𝑝0 + 𝑞0 = 1, M 𝑝0 is discrete and M 𝑞0 is continuous. Indeed, let us define 𝑝0 = ∨{𝑝 ∈ P Z ; M 𝑝 is discrete} and 𝑞0 = 1 − 𝑝0 . Q.E.D. Corollary 4.18. A factor M 𝐼fin , 𝐼∞ , 𝐼𝐼1 , 𝐼𝐼∞ , 𝐼𝐼𝐼.

⊂

B (H ) is of one and only one of the types

Proposition 4.19. Let M ⊂ B (H ) be a von Neumann algebra. Then M is discrete (resp., semifinite) iff M contains an abelian (resp., finite) projection, whose central support is equal to 1. Proof. Let {𝑒𝑖 }𝑖∈𝐼 be a maximal family of abelian (resp., finite) non-zero projections, whose central supports be mutually orthogonal. With Lemma 4.14, we infer that the projection 𝑒 = ⋁𝑖∈𝐼 𝑒𝑖 is abelian (resp., finite), whereas from the maximality of the chosen family, we infer that z(𝑒) = 1. Q.E.D.

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Corollary 4.20. M is discrete (resp., semifinite) iff any non-zero projection in M contains an abelian (resp., finite) non-zero projection. Proof. Assuming that M is discrete (resp., semifinite), let 𝑒 ∈ P M be an abelian (resp., finite) projection, such that z(𝑒) = 1 and let 𝑓 ∈ P M , 𝑓 ≠ 0. With the comparison theorem, we infer that there exists a 𝑝 ∈ P Z , such that 𝑒𝑝 ≺ 𝑓𝑝 and 𝑒(1 − 𝑝) ≻ 𝑓(1 − 𝑝). It follows that 𝑓(1 − 𝑝) is abelian (resp., finite), whereas 𝑓𝑝 contains an abelian (resp., finite) projection, which is equivalent to 𝑒𝑝. If 𝑝 ≠ 0, then 𝑒𝑝 ≠ 0, since z(𝑒) = 1. Q.E.D. 4.21. The following table shows the classification of von Neumann algebras. Type I Type Ifin Type I∞ Discrete Discrete Semifinite Semifinite Finite Properly infinite

Type II Type II1 Type II∞ Continuous Continuous Semifinite Semifinite Finite Properly infinite

Type III Continuous Properly infinite Properly infinite

4.22. Two von Neumann algebras M 1 ⊂ B (H 1 ) and M 2 ⊂ B (H 2 ) are said to be spatially isomorphic if there exists a unitary operator 𝑢 ∶ H 1 → H 2 (i.e. 𝑢 is isometric and surjective), such that 𝑢M 1 𝑢−1 = M 2 . In this case, M 1 and M 2 are *-isomorphic. The following theorem provides a link between the geometry of projections and the tensor product (see also Section 3.18). Theorem. Let M ⊂ B (H ) be a von Neumann algebra and {𝑒𝑖 }𝑖∈𝐼 ⊂ M a family of equivalent, mutually orthogonal projections, such that ∑𝑖∈𝐼 𝑒𝑖 = 1. We denote by H 𝐼 a Hilbert space whose dimension is equal to card (I). Then M is spatially isomorphic to ̄ (H 𝐼 ), for any 𝑖 ∈ 𝐼. M 𝑒𝑖 ⊗B Proof. Let us fix an index 𝑖0 ∈ 𝐼. For any 𝑖 ∈ 𝐼, let 𝑣𝑖 ∈ M be a partial isometry, such that 𝑣𝑖∗ 𝑣𝑖 = 𝑒𝑖0 ,

𝑣𝑖 𝑣𝑖∗ = 𝑒𝑖 .

Let then {𝜂𝑖 }𝑖∈𝐼 be an orthonormal basis in H 𝐼 . We define the linear operator ̄ 𝐼. 𝑢 ∶ H ∋ 𝜉 ↦ ∑ 𝑣𝑖∗ (𝜉) ⊗ 𝜂𝑖 ∈ 𝑒𝑖0 (H )⊗H 𝑖∈𝐼

It is easy to see that it is a unitary operator. Let 𝑥 ∈ B (H ), 𝜁 ∈ 𝑒𝑖0 (H ) and 𝑖 ∈ 𝐼. We then have 𝑢𝑥𝑢−1 (𝜁 ⊗ 𝜂𝑖 ) = 𝑢𝑥𝑣𝑖 (𝜁) = ∑ 𝑣𝑘∗ 𝑥𝑣𝑖 (𝜁) ⊗ 𝜂𝑘 . 𝑘∈𝐼

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∗ If we denote by 𝑤𝑘,𝑖 the partial isometry in B (H 𝐼 ), such that 𝑤𝑘,𝑖 𝑤𝑘,𝑖 = [ℂ𝜂𝑖 ], whereas ∗ 𝑤𝑘,𝑖 𝑤𝑘,𝑖 = [ℂ𝜂𝑘 ], we have

̄ 𝑘,𝑖 . 𝑢𝑥𝑢−1 = ∑ (𝑣𝑘∗ 𝑥𝑣𝑖 )⊗𝑤 𝑖,𝑘∈𝐼

Hence we immediately infer that ̄ (H 𝐼 ), 𝑢M 𝑢−1 ⊂ M 𝑒𝑖 ⊗B 0

and it is easy to see that ̄ (H 𝐼 ) ⊂ (M 𝑒 ⊗B ̄ (H 𝐼 ))′ . 𝑢M ′ 𝑢−1 ⊂ (M 𝑒𝑖 )′ ⊗C 𝑖 0

0

By taking into account Corollary 3.3, we deduce that ̄ (H 𝐼 ). 𝑢M 𝑢−1 = M 𝑒𝑖 ⊗B 0

Q.E.D. 4.23. In this section, we exhibit the relations existing between the two-sided ideals of a von Neumnnn algebra and the ideals of its centre. Let M be a von Neumann algebra, whose centre is Z . Lemma 1. For any ideal ℑ of Z , we have (M ℑ) ∩ Z = ℑ. Proof. Let 𝑎 ∈ (M ℑ) ∩ Z , 𝑎 ≥ 0. With the help of the polar decomposition (see 3.6), we can write 𝑛

𝑎 = ∑ 𝑥𝑗 𝑦𝑗 , 𝑥𝑗 = 𝑥𝑗∗ ∈ M , 0 ≤ 𝑦𝑗 ∈ ℑ. 𝑗=1 𝑛

Then 0 ≤ 𝑎 ≤ ∑𝑗=1 ‖𝑥𝑗 ‖𝑦𝑗 . With the help of exercise E.2.6, we find an element 𝑧 ∈ Z , 𝑛

such that 𝑎 = 𝑧 ∑𝑗=1 ‖𝑥𝑗 ‖𝑦𝑗 . Hence 𝑎 ∈ ℑ. From the foregoing result and with the help of the polar decomposition, we get the inclusion (M ℑ) ∩ Z ⊂ ℑ. Since the reversed inclusion is obvious, the lemma is proved. Q.E.D. Lemma 2. Let 𝔐 and 𝔑 be two-sided ideals in M . Then we have (𝔐 + 𝔑) ∩ Z = 𝔐 ∩ Z + 𝔑 ∩ Z . Proof. Let 𝑎 ∈ (𝔐 + 𝔑) ∩ Z , 𝑎 ≥ 0 and let 𝑥 ∈ 𝔐, 𝑥 = 𝑥 ∗ , 𝑦 ∈ 𝔑, 𝑦 = 𝑦 ∗ be such that 𝑎 = 𝑥 + 𝑦. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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Since 𝑎 ∈ Z , it follows that 𝑥 and 𝑦 commute. Let 𝑒 = s((2𝑥 − 𝑎)+ ) (see 2.10 and 2.15); it follows that 𝑒 ∈ M and 𝑎 ≤ 𝑎𝑒 ≤ 2𝑥𝑒,

0 ≤ 𝑎(1 − 𝑒) ≤ 2𝑦(1 − 𝑒).

From exercise E.2.6, we infer that there exist 𝑠, 𝑡 ∈ M , such that 𝑎𝑒 = 𝑠𝑥𝑒,

𝑎(1 − 𝑒) = 𝑡𝑦(1 − 𝑒),

hence 𝑎𝑒 ∈ 𝔐 and 𝑎(1 − 𝑒) ∈ 𝔑. With the comparison theorem (4.6), we infer that there exists a projection 𝑝 ∈ Z , such that 𝑒𝑝 ≺ (1 − 𝑒)𝑝, (1 − 𝑒)(1 − 𝑝) ≺ 𝑒(1 − 𝑝). Consequently, there exist two partial isometries 𝑢, 𝑣 ∈ M , such that 𝑢∗ 𝑢 = 𝑒𝑝, 𝑢𝑢∗ ≤ (1 − 𝑒)𝑝 and 𝑣 ∗ 𝑣 = (1 − 𝑒)(1 − 𝑝), 𝑣𝑣 ∗ ≤ 𝑒(1 − 𝑝). Hence we infer that 𝑢∗ (1 − 𝑒)𝑝𝑢 = 𝑒𝑝,

𝑣 ∗ 𝑒(1 − 𝑝)𝑣 = (1 − 𝑒)(1 − 𝑝).

Thus, by taking into account the fact that 𝑎 ∈ Z , from what we have already proved we infer that 𝑢∗ 𝑎(1 − 𝑒)𝑝𝑢 = 𝑎𝑒𝑝 ∈ 𝔐 ∩ 𝔑, 𝑣∗ 𝑎𝑒(1 − 𝑝)𝑣 = 𝑎(1 − 𝑒)(1 − 𝑝) ∈ 𝔐 ∩ 𝔑. Consequently, we have 𝑎𝑝 = 𝑎(1 − 𝑒)𝑝 + 𝑎𝑒𝑝 ∈ 𝔑 𝑎(1 − 𝑝) = 𝑎(1 − 𝑒)(1 − 𝑝) + 𝑎𝑒(1 − 𝑝) ∈ 𝔐 and 𝑎 = 𝑎(1 − 𝑝) + 𝑎𝑝 ∈ 𝔐 ∩ Z + 𝔑 ∩ Z . We have thus proved the inclusion (𝔐+𝔑)∩Z ⊂ 𝔐∩Z +𝔑∩Z . The reversed inclusion is obvious. Q.E.D. Theorem. Let M be a von Neumann algebra and Z its centre. For any ideal ℑ of Z , the set of all two-sided ideals 𝔐 of M , such that 𝔐 ∩ Z = ℑ has a smallest element, denoted by 𝔐0 (ℑ), and a greatest element, denoted by 𝔐∞ (ℑ). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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Proof. With Lemma 1, the ideal 𝔐0 (ℑ) = M ℑ obviously is the smallest element of the considered set. This set is inductively ordered by inclusion, and therefore, it has a maximal element 𝔐∞ (ℑ). From Lemma 2, it follows that the considered set is increasingly directed, and therefore, 𝔐∞ (ℑ) is its greatest element. Q.E.D. From the preceding theorem, one can easily infer the following: Corollary. Let M be a von Neumann algebra with the centre Z . Then the mapping 𝔐↦𝔐∩Z =ℑ is a bijection between the set of all maximal two-sided ideals 𝔐 of M and the set of all maximal ideals ℑ of Z . 4.24. We record here the main properties of the cyclic projections 𝑝𝜉 ∈ M (and 𝑝𝜉′ ∈ M ′ , by symmetry). Proposition. Let 𝜉 ∈ H , M ⊂ B (H ) a von Neumann algebra, and 𝑒 ∈ M a projection. Then, (1) 𝜉 ∈ 𝑒H ⇒ 𝑝𝜉 ≤ 𝑒. (2) 𝑒 ≤ 𝑝𝜉 ⇒ 𝑒 = 𝑝𝑒𝜉 . (3) 𝑒 ∼ 𝑝𝜉 (via 𝑣 ∈ M ) ⇒ 𝑒 = 𝑝𝑣∗ 𝜉 , 𝑝𝑣′ ∗ 𝜉 = 𝑝𝜉′ . (4) 𝑒 ≺ 𝑝𝜉 ⇒ 𝑒 = 𝑝𝜂 , for some 𝜂 ∈ H . (5) 𝑝𝑥𝜉 ≺ 𝑝𝜉 , (∀)𝑥 ∈ M ; if 𝜉 ∈ r(𝑥)H , then 𝑝𝑥𝜉 ∼ 𝑝𝜉 . (6) 𝑝𝑥′ 𝜉 ≤ 𝑝𝜉 , (∀)𝑥 ′ ∈ M ′ ; if 𝑥 ′ is invertible, then 𝑝𝑥′ 𝜉 = 𝑝𝜉 . (7) 𝑞𝑝𝜉 = 𝑝𝑞𝜉 , for any central projection 𝑞 ∈ Z M . Proof. (1) We have 𝑒𝑥 ′ 𝜉 = 𝑥 ′ 𝑒𝜉 = 𝑥 ′ 𝜉, (∀)𝑥 ′ ∈ M ′ , hence 𝑒𝜂 = 𝜂, for all 𝜂 ∈ M ′ 𝜉. (2) We have 𝑝𝑒𝜉 ≤ 𝑒 by (1) and ′ 𝑒 ≤ 𝑝𝜉 ⇒ 𝑒H = 𝑒𝑝𝜉 H = 𝑒M ′ 𝜉 = 𝑒M ′ 𝜉 = M ′ 𝑒𝜉 = 𝑝𝑒𝜉 H.

(3) If 𝑣∗ 𝑣 = 𝑒, 𝑣𝑣 ∗ = 𝑝𝜉 , then 𝑝𝑣∗ 𝜉 = M ′ 𝑣∗ 𝜉 = 𝑣∗ M ′ 𝜉 = 𝑣 ∗ M ′ 𝜉 = 𝑒H = 𝑒, and 𝑝𝑣′ ∗ 𝜉 = M 𝑣∗ 𝜉 ≥ M 𝑣𝑣∗ 𝜉 = M 𝑝𝜉 𝜉 = M 𝜉 = 𝑝𝜉′ ≥ M 𝑣∗ 𝜉 = 𝑝𝑣′ ∗ 𝜉 . (4) Follows from (2) and (3). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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(5) We have 𝑝𝑥𝜉 = M ′ 𝑥𝜉 = 𝑥M ′ 𝜉 = 𝑥M ′ 𝜉 = l(𝑥𝑝𝜉 ) ∼ r(𝑥𝑝𝜉 ) ≤ 𝑝𝜉 . If 𝜉 ∈ r(𝑥)H , then 𝑝𝜉 ≤ r(𝑥) by (1); hence r(𝑥𝑝𝜉 ) = 𝑝𝜉 . (6) and (7) are obvious. Q.E.D. 4.25. We will now provide a large class of II1 -factors. Let 𝐺 be a discrete group with neutral element 𝑒 ∈ 𝐺. Then 𝑙2 (𝐺) is a Hilbert space with orthonormal basis {𝛿𝑡 ; 𝑡 ∈ 𝐺} where 𝛿𝑡 (𝑠) = 𝛿𝑡𝑠 = {

1, if 𝑠 = 𝑡, 0, if 𝑠 ≠ 𝑡.

Any operator 𝑋 ∈ B (𝑙 2 (𝐺)) defines and is defined by a matrix with entries 𝑋𝑠,𝑡 = (𝑋𝛿𝑡 ∣ 𝛿𝑠 ) = (𝑋𝛿𝑡 ) (𝛿𝑠 ) ∈ ℂ

(𝑠, 𝑡 ∈ 𝐺).

For any 𝑔 ∈ 𝐺, we define the left translation 𝜆(𝑔) ∈ B (𝑙 2 (𝐺)) and the right translation 𝜌(𝑔) ∈ B (𝑙 2 (𝐺)) by 𝑔−1 𝑠

[𝜆(𝑔)𝜉] (𝑠) = 𝜉(𝑔−1 𝑠), for 𝜉 ∈ 𝑙 2 (𝐺) with matrix entries 𝜆(𝑔)𝑠,𝑡 = 𝛿𝑡

𝑠 = 𝛿𝑔𝑡 ,

𝑠𝑔

[𝜌(𝑔)𝜉] (𝑠) = 𝜉(𝑠𝑔), for 𝜉 ∈ 𝑙 2 (𝐺) with matrix entries 𝜌(𝑔)𝑠,𝑡 = 𝛿𝑡 . It is clear that ‖𝜆(𝑔)𝜉‖ = ‖𝜉‖ = ‖𝜌(𝑔)𝜉‖ and 𝜆(𝑔1 𝑔2 ) = 𝜆(𝑔1 )𝜆(𝑔2 ),

𝜌(𝑔1 𝑔2 ) = 𝜌(𝑔1 )𝜌(𝑔2 ),

𝜆(𝑒) = 𝐼 = 𝜌(𝑒).

In particular, 𝜆 (𝑔−1 ) = 𝜆(𝑔)−1 , 𝜌 (𝑔−1 ) = 𝜌(𝑔)−1 and also (exercise): 𝜆 (𝑔−1 ) = 𝜆(𝑔)∗ ,

𝜌 (𝑔−1 ) = 𝜌(𝑔)∗ .

Denote L (𝐺) = {𝜆(𝑔); 𝑔 ∈ 𝐺}″ ⊂ B (𝑙2 (𝐺)),

R (𝐺) = {𝜌(𝑔); 𝑔 ∈ 𝐺}″ ⊂ B (𝑙 2 (𝐺)).

Obviously, L (𝐺) ⊂ R (𝐺)′ . Now, for any 𝑋 ∈ B (𝑙 2 (𝐺)), we compute by 2.35: 𝑟𝑔

[𝑋𝜌(𝑔)]𝑠,𝑡 = ∑ 𝑋𝑠,𝑟 𝜌(𝑔)𝑟,𝑡 = ∑ 𝑋𝑠,𝑟 𝛿𝑡 = 𝑋𝑠,𝑡𝑔−1 , 𝑟

𝑟 𝑠𝑔

[𝜌(𝑔)𝑋]𝑠,𝑡 = ∑ 𝜌(𝑔)𝑠,𝑟 𝑋𝑟,𝑡 = ∑ 𝛿𝑟 𝑋𝑟,𝑡 = 𝑋𝑠𝑔,𝑡 . 𝑟

𝑟

It follows that 𝑋 ∈ R (𝐺)′ ⇔ 𝑋𝑠,𝑡𝑔−1 = 𝑋𝑠𝑔,𝑡 ⇔ 𝑋𝑠𝑔,𝑡𝑔 = 𝑋𝑠,𝑡 ,

(∀)𝑠, 𝑡, 𝑔 ∈ 𝐺.

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101

Classification of von Neumann algebras Let 𝑋 ∈ R (𝐺)′ and consider 𝛼 = 𝑋𝛿𝑒 ∈ 𝑙 2 (𝐺),

𝛼(𝑔) = (𝑋𝛿𝑒 )(𝑔) = 𝑋𝑔,𝑒 .

For any 𝑙 2 (𝐺) ∋ 𝜉 = ∑(𝜉 ∣ 𝛿𝑡 )𝛿𝑡 = ∑ 𝜉(𝑡)𝛿𝑡 , we have 𝑡

𝑡

(𝑋𝜉)(𝑠) = (𝑋𝜉 ∣ 𝛿𝑠 ) = ∑ 𝜉(𝑡) (𝑋𝛿𝑡 ∣ 𝛿𝑠 ) = ∑ 𝜉(𝑡)𝑋𝑠,𝑡 𝑡

𝑡

= ∑ 𝜉(𝑡)𝑋𝑠𝑡−1 ,𝑒 = ∑ 𝛼 (𝑠𝑡 𝑡

−1

) 𝜉(𝑡) = ∑ 𝛼(𝑔)𝜉 (𝑔−1 𝑠) = (𝛼 ∗ 𝜉) (𝑠),

𝑡

𝑔

the last equation being the definition of the convolution 𝛼 ∗ 𝜉. Thus, 𝛼 ∈ 𝑙 2 (𝐺), 𝛼 ∗ 𝜉 ∈ 𝑙 2 (𝐺) (∀)𝜉 ∈ 𝑙 2 (𝐺), and the left convolution operator 𝑇𝛼 ∶ 𝑙 2 (𝐺) ∋ 𝜉 ↦ 𝛼 ∗ 𝜉 ∈ 𝑙 2 (𝐺) is bounded. Call such a vector 𝛼 ∈ 𝑙 2 (𝐺) left bounded and notice that (𝑇𝛼 )𝑠,𝑡 = 𝛼(𝑠𝑡 −1 ). Also, notice that any basic vector is left bounded and 𝑇𝛿𝑔 = 𝜆(𝑔). We have proved that R (𝐺)′ = {𝑇𝛼 ∈ B (𝑙2 (𝐺)); 𝛼 ∈ 𝑙 2 (𝐺) left bounded} ⊃ L (𝐺). The last inclusion is actually an equality. Indeed, we have seen that R (𝐺)′ = {left convolutors}. Similarly, with similar definitions, L (𝐺)′ = {right convolutors}. It is straightforward to check that any left convolutor commutes with any right convolutor. Consequently, R (𝐺)′ ⊂ (L (𝐺)′ )′ = L (𝐺)″ = L (𝐺). Thus (1)

L (𝐺) = R (𝐺)′ = {𝑇𝛼 ∈ B (𝑙 2 (𝐺)); 𝛼 ∈ 𝑙 2 (𝐺) left bounded}.

It is immediate to check that, formally, 𝑇𝛼 𝑇𝛽 = 𝑇𝛼∗𝛽 and (𝑇𝛼 )∗ = 𝑇𝛼∗ , where 𝛼 ∗ (𝑔) = 𝛼(𝑔−1 )

(𝑔 ∈ 𝐺).

This proves that 𝛼, 𝛽 ∈ 𝑙 2 (𝐺) left bounded ⇒ 𝛼 ∗ 𝛽, 𝛼 ∗ ∈ 𝑙 2 (𝐺) left bounded. It is convenient to use the notation 𝑇𝛼 =∶ ∑ 𝛼(𝑔)𝜆(𝑔), 𝑔∈𝐺

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where the symbol ∑ has no summability meaning. The convenience is that the equality 𝑇𝛼 𝑇𝛽 = 𝑇𝛼∗𝛽 is translated into the formal distributivity for the sums ∑, the equality (𝑇𝛼 )∗ = 𝑇𝛼∗ means simply to commute ∑ and ∗, and 𝑇𝛼 + 𝑇𝛽 = 𝑇𝛼+𝛽 means ∑ 𝛼(𝑔)𝜆(𝑔) + ∑ 𝛽(𝑔)𝜆(𝑔) = ∑(𝛼(𝑔) + 𝛽(𝑔))𝜆(𝑔). 𝑔

𝑔

𝑔

In order to practice this notation, we show that (2)

𝑋 ∈ L (𝐺) ∩ L (𝐺)′ ⇔ 𝑋 = 𝑇𝛼 where 𝛼(𝑔) = 𝛼(𝑠𝑔𝑠−1 ),

(∀)𝑠, 𝑔 ∈ 𝐺.

Indeed, if 𝑋 ∈ L (𝐺) ∩ L (𝐺)′ , then 𝑋 = 𝑇𝛼 = ∑ 𝛼(𝑔)𝜆(𝑔) and 𝑋𝜆(𝑠) = 𝜆(𝑠)𝑋, for every 𝑔

𝑠 ∈ 𝐺, where 𝑋𝜆(𝑠) = ∑ 𝛼(𝑔)𝜆(𝑔)𝜆(𝑠) = ∑ 𝛼(𝑔)𝜆(𝑔𝑠) = ∑ 𝛼(𝑔𝑠−1 )𝜆(𝑔), 𝑔

𝑔

𝑔

𝜆(𝑠)𝑋 = ∑ 𝛼(𝑔)𝜆(𝑠)𝜆(𝑔) = ∑ 𝛼(𝑔)𝜆(𝑠𝑔) = ∑ 𝛼(𝑠−1 𝑔)𝜆(𝑔), 𝑔

𝑔

𝑔

hence 𝛼(𝑔𝑠−1 ) = 𝛼 (𝑠−1 𝑔), that is 𝛼(𝑔) = 𝛼 (𝑠𝑔𝑠−1 ) ,

(∀)𝑠, 𝑔 ∈ 𝐺.

This condition says that 𝛼 is constant on the inner conjugacy classes of 𝐺. Let us call 𝐺 an ICC group if 𝐺 has only infinite conjugacy classes, except, of course, the conjugacy class of 𝑒 ∈ 𝐺. If 𝐺 is an ICC group, then for the above 𝛼, we necessarily have 𝛼(𝑔) = 0, for all 𝑔 ≠ 𝑒, since 𝛼 ∈ 𝑙 2 (𝐺), and therefore 𝑋 = 𝑇𝛼 = ∑ 𝛼(𝑔)𝜆(𝑔) = 𝛼(𝑒) ⋅ 𝐼. 𝑔

We have proved that (3)

L (𝐺) is a factor ⇔ 𝐺 is an 𝐼𝐶𝐶 group.

The vector 𝛿𝑒 ∈ 𝑙 2 (𝐺) has several remarkable properties with respect to L (𝐺). First, 𝛿𝑒 ∈ 𝑙2 (𝐺) is a cyclic vector for L (𝐺), that is L (𝐺)𝛿𝑒 = 𝑙 2 (𝐺), since 𝜆(𝑔) ∈ L (𝐺) and 𝜆(𝑔)𝛿𝑒 = 𝛿𝑔 (𝑔 ∈ 𝐺). It is also a separating vector since if 𝑇𝛼 𝛿𝑒 = 0, then 𝛼(𝑒) = (𝑇𝛼 𝛿𝑒 ∣ 𝛿𝑒 ) = 0; hence 𝛼 = 0, 𝑇𝛼 = 0, when 𝐺 is an ICC group. Moreover, (𝑇𝛼 𝑇𝛽 𝛿𝑒 ∣ 𝛿𝑒 ) = (𝛼 ∗ 𝛽)(𝑒) = (𝛽 ∗ 𝛼)(𝑒) = (𝑇𝛽 𝑇𝛼 𝛿𝑒 ∣ 𝛿𝑒 ). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

103

Classification of von Neumann algebras This means that 𝛿𝑒 ∈ 𝑙 2 (𝐺) is a trace vector for L (𝐺), that is (𝑋𝑌𝛿𝑒 ∣ 𝛿𝑒 ) = (𝑌𝑋𝛿𝑒 ∣ 𝛿𝑒 ) (∀)𝑋, 𝑌 ∈ L (𝐺).

It follows that the linear form L (𝐺) ∋ 𝑋 ↦ 𝜏(𝑋) = (𝑋𝛿𝑒 ∣ 𝛿𝑒 ) is a wo-continuous faithful trace on L (𝐺): 𝜏(𝑋𝑌) = 𝜏(𝑌𝑋) and 𝑋 ≥ 0, 𝜏(𝑋) = 0 ⇒ 𝑋 = 0

(𝑋, 𝑌 ∈ L (𝐺)).

This obviously implies that, for an ICC group, L (𝐺) is a finite factor. For 𝐺 infinite, L (𝐺) is infinite dimensional; hence it cannot be of type 𝐼𝑛 (i.e. a full matrix algebra). Conclusion: (4)

If 𝐺 is an 𝐼𝐶𝐶 group, then L (𝐺) is a type II1 -factor.

4.26. There are many examples of ICC groups. The free group with two generators 𝐺 = 𝔽2 is an obvious example. Another example is the group 𝑆(∞) of finite permutations of ℕ, that is 𝑆(∞) = {𝜎 ∶ ℕ → ℕ bijection; (∃)𝑛0 ∈ ℕ such that 𝜎(𝑛) = 𝑛, (∀)𝑛 ≥ 𝑛0 }. That 𝑆(∞) is an ICC group is clear from the following example: (

3𝑛 𝑛3

)(

1 2 3 ... 9 ∗ ∗ 7 ... ∗

)(

3𝑛

... ... ... 𝑛 )=( ), 𝑛 > 7 𝑛3 ... ... ... 7

The factors L (𝑆(∞)) and L (𝔽2 ) were the first two examples of type II1 -factors given by Murray and von Neumann, and they showed that Theorem. L (𝑆(∞)) and L (𝔽2 ) are not isomorphic. Proof. We first mention a fact which will be proved later, namely that the non-zero trace 𝜏 on a type II1 -factor M (exists and) is unique and hence is an isomorphism invariant for M , as well as the ‖ ⋅ ‖2 -norm: ‖𝑥‖2 = 𝜏(𝑥 ∗ 𝑥)1/2

(𝑥 ∈ M ).

The proof is based on an isomorphism invariant property of M , called property Γ: (∀)𝑥1 , ..., 𝑥𝑛 ∈ M , (∀)𝜀 > 0, (∃)𝑢 ∈ M unitary, 𝜏(𝑢) = 0 such that ‖𝑢∗ 𝑥𝑘 𝑢 − 𝑥𝑘 ‖2 < 𝜀, (1) L (𝑆(∞)) has property Γ. Indeed, the finite sums of the form

∑

(∀)𝑘 = 1, 2, ..., 𝑛. 𝛼(𝑔)𝜆(𝑔) where

𝑔∈𝑆(𝑚)

𝑆(𝑚) = {𝜎 ∈ 𝑆(∞); 𝜎(𝑘) = 𝑘,

(∀)𝑘 > 𝑚}

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are so-dense, and hence ‖ ⋅ ‖2 -dense in L (𝑆(∞)). Consequently, we may assume that 𝑥1 = 𝜆(𝑔1 ), ..., 𝑥𝑛 = 𝜆(𝑔𝑛 ) with 𝑔1 , ..., 𝑔𝑛 ∈ 𝑆(𝑚). If 𝑔 is the transposition 𝑔 = (2𝑚, 2𝑚 + 1), then 𝑢 = 𝜆(𝑔) satisfies 𝜏(𝑢) = 0 and 𝑢∗ 𝑥𝑘 𝑢 − 𝑥𝑘 = 0,

(∀)𝑘 = 1, 2, ..., 𝑛.

(2) L (𝔽2 ) has not the property Γ. Let 𝐺 = 𝔽2 and 𝑎, 𝑏 ∈ 𝔽2 , the two free generators of 𝔽2 . Let 𝐺 ⊃ 𝐹 = {the set of all 𝑔 ∈ 𝔽2 ending with some 𝑎𝑛 , 𝑛 ∈ ℤ\{0}}. Denote 𝑔1 = 𝑎, 𝑔2 = 𝑏, 𝑔3 = 𝑏−1 . Then, (i) 𝐺 = {𝑒} ∪ 𝐹 ∪ 𝑔1 𝐹𝑔1−1 and

(ii) 𝐹, 𝑔2 𝐹𝑔2−1 , 𝑔3 𝐹𝑔3−1 are mutually disjoint.

Suppose L (𝐺) has property (Γ). Then, given 𝜀 > 0 and 𝑥𝑘 = 𝜆(𝑔𝑘 ) (𝑘 = 1, 2, 3), there is an unitary 𝑢 = ∑ 𝛼(𝑔)𝜆(𝑔) ∈ L (𝐺) with zero trace 𝛼(𝑒) = 𝜏(𝑢) = 0 and (∗) 𝜀2 > ‖𝑢∗ 𝑥𝑘 𝑢 − 𝑥𝑘 ‖22 = ‖𝑢 − 𝑥𝑘∗ 𝑢𝑥𝑘 ‖22 = ∑ |𝛼(𝑔) − 𝛼(𝑔𝑘 𝑔𝑔𝑘−1 )|2

(𝑘 = 1, 2, 3).

For any set 𝑆 ⊂ 𝐺, put 𝜈(𝑆) = ∑ |𝛼(𝑔)|2 . Note that 𝑔∈𝑆

𝜈(𝐺) = ‖𝑢‖22 = 1, so 0 ≤ 𝜈(𝑆) ≤ 1. Below we use (*), a triangle inequality and |𝑥 − 𝑦| ≤ 2|√𝑥 − √𝑦|, for 0 ≤ 𝑥, 𝑦 ≤ 1: |𝜈(𝑆) − 𝜈(𝑔𝑘 𝑆𝑔𝑘−1 )| ≤ 2 |𝜈(𝑆)1/2 − 𝜈(𝑔𝑘 𝑆𝑔𝑘−1 )1/2 | ≤ 1/2

≤ 2 ( ∑ |𝛼(𝑔) −

𝛼(𝑔𝑘 𝑔𝑔𝑘−1 )|2 )

≤ 2𝜀.

𝑔∈𝑆

Now, from (i) and 𝛼(𝑒) = 0, we get 1 = 𝜈(𝐺) ≤ 𝜈(𝐹) + 𝜈(𝑔1 𝐹𝑔1−1 ) + 𝜈({𝑒}) ≤ 2𝜈(𝐹) + 2𝜀, i.e., 𝜈(𝐹) ≥

1 2

− 𝜀, while, from (ii), we get 1 = 𝜈(𝐺) ≤ 𝜈(𝐹) + 𝜈(𝑔2 𝐹𝑔2−1 ) + 𝜈(𝑔3 𝐹𝑔3−1 ) ≤ 3𝜈(𝐹) − 4𝜀,

i.e., 𝜈(𝐹) ≤ contradictory.

1 3

+

4 3

𝜀. Thus

1 2

− 𝜀

≤

1 3

+

4 3

𝜀, which, for 𝜀 small enough, is Q.E.D.

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105

Classification of von Neumann algebras 4.27. The following question is still open: Are the factors L (𝔽2 ) and L (𝔽3 ) isomorphic or not? If instead of factors, we consider ‘the reduced 𝐶 ∗ -algebras’: C ∗𝑟 (𝐺) = 𝐶 ∗ -subalgebra of B (𝑙 2 (𝐺)) generated by {𝜆(𝑔); 𝑔 ∈ 𝐺},

M. Pimsner and D. V. Voiculescu (1982) have shown, by computing the ‘K-theory’ that C ∗𝑟 (𝔽𝑛 ) are pairwise non-isomorphic. Working on the above question D. V. Voiculescu created the free probability theory. Two of his students, Ken Dykema (1994) and Florin Rădulescu (1994), proved, independently of each other, using Voiculescu free probability theory, that either all L (𝔽𝑛 ) are isomorphic or are pairwise non-isomorphic, but the above question is still open (2018). 4.28. Another comment concerning L (𝑆(∞)) is in order. Since 𝑆(∞) =

∞

⋃ 𝑆(𝑚), 𝑚=1

it follows that L (𝑆(∞)) is the w-closure of the union of an increasing sequence of finite-dimensional * subalgebras. Such factors are called approximately finite dimensional (AFD) or hyperfinite. The later terminology is improper (but in use) since L (𝑆(∞)) ⊗ B (𝑙 2 (ℕ)) is hyperfinite but infinite, namely of type II∞ . Murray and von Neumann proved that all hyperfinite type II1 factors are *-isomorphic – for the proof, we refer to Dixmier (1957/1969). The problem whether all hyperfinite factors of type II∞ are *-isomorphic remained long time unsolved. It was Alain Connes (1976) who proved that indeed all hyperfinite type II∞ factors are *-isomorphic. Furthermore, due to the works of Alain Connes (1975a, 1976, 1985), Wolfgang Krieger (1976) and Uffe Haagerup (1987), the hyperfinite factors of type III are now completely classified. Another difficult problem was whether all factors of type II1 are or not of the form L (𝐺) for 𝐺 an ICC discrete group. It was again Alain Connes (1975a, b) who showed that there are factors of type II1 not anti-isomorphic to themselves and hence not of the form L (𝐺) (where the anti-isomorphism is implied by the inversion 𝑔 ↦ 𝑔−1 ).

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In the exercises in which the symbols M and Z are not otherwise explained, they will denote a von Neumann algebra and its centre. E.4.1 Let 𝑒, 𝑓 ∈ P M . Then 𝑒 ≺ 𝑓 ⇔ then exists a 𝑔 ∈ P M , such that 𝑒 ≤ 𝑔 ∼ 𝑓. E.4.2 Let 𝑒, 𝑓 ∈ P M . Then there exists a 𝑝 ∈ P X , such that 𝑒𝑝 ≺ 𝑓𝑝, (1 − 𝑒)(1 − 𝑝) ≺ (1 − 𝑓)(1 − 𝑝). E.4.3 Let 𝑒1 , 𝑒2 , 𝑓1 , 𝑓2 ∈ P M , 𝑒1 𝑒2 = 𝑓1 𝑓2 = 0. If 𝑒1 + 𝑒2 = 𝑓1 + 𝑓2 , 𝑒1 ∼ 𝑒2 , 𝑓1 ∼ 𝑓2 , then 𝑒1 ∼ 𝑓1 . E.4.4 Let 𝑒, 𝑓 ∈ P M , 𝑎 = 𝑒 + 𝑓 − 1 and 𝑠 = 1 − 2𝑠(𝑎− ). Then 𝑠 is a symmetry and 𝑠𝑒𝑓𝑠 = 𝑓𝑒. E.4.5 Let 𝑒, 𝑓 ∈ P M , 𝑒 ∧ (1 − 𝑓) = (1 − 𝑒) ∧ 𝑓 = 0. Then there exists a symmetry 𝑠 ∈ M , such that 𝑠𝑒𝑠 = 𝑓. E.4.6 For any 𝑒, 𝑓 ∈ P M , there exists a symmetry 𝑠 ∈ M , such that 𝑠(𝑒 ∨𝑓 −𝑒)𝑠 = 𝑓 − 𝑒 ∧ 𝑓. In particular, the parallelogram rule now easily follows (see 4.4). E.4.7 For any pair of equivalent projections 𝑒, 𝑓 ∈ M , there exist projections 𝑒1 , 𝑒2 , 𝑓1 , 𝑓2 ∈ P M , 𝑒1 𝑒2 = 𝑓1 𝑓2 = 0 and unitary elements 𝑢1 , 𝑢2 ∈ M , such that 𝑒 = 𝑒1 + 𝑒2 , 𝑢1∗ 𝑒1 𝑢1

= 𝑓1 ,

𝑓 = 𝑓1 + 𝑓2 , 𝑢2∗ 𝑒2 𝑢2 = 𝑓2 .

!E.4.8 Let 𝑒, 𝑓 ∈ P M , with 𝑒 finite, 𝑓 properly infinite. Then 𝑒 ≺ 𝑓 ⇔ z(𝑒) ≤ z(𝑓). !E.4.9 Let 𝑒, 𝑓 ∈ P M be equivalent finite projections. Then 1 − 𝑒 ∼ 1 − 𝑓. Consequently, there exists a unitary element 𝑢 ∈ M , such that 𝑢∗ 𝑒𝑢 = 𝑓. ▶

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E.4.10 Let 𝑒 ∈ P M be a continuous projection (i.e. the reduced algebra M 𝑒 is continuous). For any natural number 𝑛, there exists a finite set {𝑒1 , … , 𝑒𝑛 } ⊂ 𝑛 P M of equivalent, mutually orthogonal projections, such that 𝑒 = ∑𝑖=1 𝑒𝑖 . E.4.11 Let M be a (properly) infinite factor and 𝑒, 𝑓 ∈ P M . Then 𝑒 ∨ 𝑓 ∼ 1 ⇔ 𝑒 ∼ 1 or 𝑓 ∼ 1. E.4.12 A projection 𝑒 ∈ P M is said to be minimal relatively to the central support if 𝑒 ≠ 0 and 𝑓 ∈ P M , 𝑓 ≤ 𝑒, z(𝑓) = z(𝑒) ⇒ 𝑓 = 𝑒.

E.4.13 !E.4.14

E.4.15

E.4.16

E.4.17

Show that a non-zero projection is minimal relatively to the central support iff it is abelian. A factor is of type I iff it contains a minimal projection, and in this case, it is *-isomorphic to a B (H ), the dimension of H being uniquely determined. Any finitely dimensional von Neumann algebra is of type Ifin . Infer from this result that if M is continuous and 𝑒 ∈ P M , 𝑒 ≠ 0, then dim(𝑒H ) = +∞. One says that M is homogeneous of type I𝛾 (resp., uniform of type S𝛾 ), where 𝛾 is any cardinal, if there exists a family {𝑒𝑖 }𝑖∈𝐼 of abelian (resp., finite) equivalent, mutually orthogonal projections in M , such that ∑𝑖∈𝐼 𝑒𝑖 = 1, and card 𝐼 = 𝛾. In this case, M is of type I (resp., semifinite). Show that for any von Neumann algebra M of type I (resp., semifinite and properly infinite), there exists a family Γ of distinct cardinals (resp., distinct infinite cardinals) and a family {𝑝𝛾 }𝛾∈Γ of mutually orthogonal central projections, such that ∑𝛾∈Γ 𝑝𝛾 = 1, and moreover, M 𝑝𝛾 be homogeneous of type I𝛾 (resp., uniform of type S𝛾 ). If 𝑛 is a natural number, and M is homogeneous of type I𝛾 , then any family of non-zero, equivalent, mutually orthogonal projections in M has at most 𝑛 elements. In particular, if 𝑚, 𝑛 are natural numbers, whereas M is homogeneous of type I𝑚 and of type I𝑛 , then 𝑚 = 𝑛 (see also Section 8.4). Let M be a von Neumann algebra, which is homogeneous, of type I𝛾 . Show that there exists a commutative von Neumann algebra Z , which is *-isomorphic to the centre of M , and a Hilbert space H , such that M be ̄ (H ). spatially isomorphic to Z ⊗B In particular, any factor of type I is *-isomorphic to B (H ), where H is a suitable Hilbert space. 𝑛 Let 𝑒, 𝑓1 , … , 𝑓𝑛 ∈ P M , with abelian 𝑓1 , … , 𝑓𝑛 . If 𝑒 ≤ ∑𝑘=1 𝑓𝑘 , then there 𝑚 exist mutually orthogonal, abelian 𝑒1 , … , 𝑒𝑚 ∈ P M , such that 𝑒 = ∑𝑗=1 𝑒𝑗 . ▶

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E.4.18 Let M be a von Neumann algebra of type I (resp., of type II; resp., of type III), and 𝑒 ∈ P M . Then M 𝑒 is of type I (resp., of type II; resp., of type III). E.4.19 Show that any factor of type II is spatially isomorphic to the tensor product of a factor of type II1 by a B (H ). !E.4.20 Let M be a finite von Neumann algebra and H a Hilbert space. Show that the ̄ (H ) is finite if dim (H ) < +∞, and properly von Neumann algebra M ⊗B infinite, if dim (H ) = +∞. !E.4.21 Let 𝑒 ∈ P M be such that 𝑝 ∈ P Z , 𝑒𝑝 abelian ⇒ 𝑒𝑝 = 0. Show that there exist 𝑒1 , 𝑒2 ∈ P M , 𝑒1 𝑒2 = 0, such that 𝑒 = 𝑒1 + 𝑒2 z(𝑒1 ) = z(𝑒2 ) = z(𝑒). E.4.22 Show that any abelian von Neumann algebra is *-isomorphic to a maximal abelian von Neumann algebra. E.4.23 Let M be a von Neumann algebra and ‘≈’ an equivalence relation in P M , such that (i) If {𝑒𝑖 }𝑖∈𝐼 and {𝑓𝑖 }𝑖∈𝐼 are families of mutually orthogonal projections in M , such that 𝑒𝑖 ≈ 𝑓𝑖 , 𝑖 ∈ 𝐼, then ∑ 𝑒𝑖 ≈ ∑ 𝑓𝑖 ; 𝑖∈𝐼

𝑖∈𝐼

(ii) If 𝑒, 𝑓 ∈ P M and if there exists a unitary element 𝑢 ∈ M , such that 𝑒 = 𝑢∗ 𝑓𝑢, then 𝑒 ≈ 𝑓. With the help of Proposition 4.12 and of E.4.9, show that if 𝑒, 𝑓 ∈ P M , and 𝑒∼𝑓 then 𝑒 ≈ 𝑓. ▶

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Consequently, the relation ‘∼’ is the minimal completely additive extension of the relation of ‘unitary equivalence’. E.4.24 Let M be a factor. Show that (1) any non-zero two-sided ideal of M is w-dense in M . (2) if M is finite, or of type III and of countable type, then M has no non-trivial two-sided ideals. (3) if M is semifinite and properly infinite, then the linear hull of the set of all finite projections in M is equal to the smallest non-trivial two-sided ideal of M . (4) if M is of type III, but it is not of countable type, then the linear hull of the set of all projections in M , which are of countable type, is the smallest non-trivial two-sided ideal of M . E.4.25 Prove the Second Comparison Theorem: For 𝑒, 𝑓 ∈ P M , there exists a 𝑝 ∈ P Z such that 𝑒𝑝 ≺ 𝑓𝑝 and (1 − 𝑒)(1 − 𝑝) ≺ (1 − 𝑓)(1 − 𝑝). Hint: Look at the proof in 4.15.

COMMENTS C.4.1 The geometry of projections developed in an algebraic frame, started by C. E. Rickart (1950) and by I. Kaplansky (1951, 1952, 1968), for the so-called *-Baer rings. A complete exposition of the results obtained in this direction can be found in the book by S. K. Berberian (1972). A 𝐶 ∗ -algebra, which is, at the same time, a *-Baer ring, is called an 𝐴𝑊 ∗ -algebra (algebraic 𝑊 ∗ -algebra; for the notion of a 𝑊 ∗ -algebra, which is, essentially, the same as that of von Neumann algebra, see C.5.3). For 𝐴𝑊 ∗ -algebras, almost all results in this chapter are true, with essentially the same proofs. Any commutative 𝐴𝑊 ∗ -algebra is *-isomorphic to C (Ω), where Ω is a stonean space (a Hausdorff compact space, in which the closure of any open set is open), whereas any commutative von Neumann algebra (commutative 𝑊 ∗ -algebra) is *-isomorphic to C (Ω), where Ω is a hyperstonem space (cf. Dixmier 1951a). For expositions of these results, see Bade (1971) and Zsidó (1973). C.4.2 R. V. Kadison and G. K. Pedersen (1970) defined an equivalence relation for the positive elements of a von Neumann algebra M , namely two elements

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Lectures on von Neumann algebras 𝑎, 𝑏 ∈ M , 𝑎, 𝑏 ≥ 0, are said to be equivalent if there exists a family {𝑥𝑖 }𝑖∈𝐼 ⊂ M , such that 𝑎 = ∑ 𝑥𝑖∗ 𝑥𝑖 , 𝑖∈𝐼

𝑏 = ∑ 𝑥𝑖 𝑥𝑖∗ . 𝑖∈𝐼

They developed a theory for this equivalence relation, which is similar to the geometry of projections, and they showed that, for projections, the equivalence they introduced coincides with the usual equivalence of projections (4.1). C.4.3 From exercises E.4.14 and E.4.16, there follows a complete description of the structure of von Neumann algebras of type I. The structure of the continuous von Neumann algebras is still far from being understood, even for the case of separable Hilbert spaces. Since in this case, a ‘reduction theory’ exists, which reduces the study of general von Neumann algebras to that of the factors – theory that was developed by J. von Neumann (1949) (see also Dixmier [1957/1969]; and Zsidò [1973]), the difficulty remains the classification of the continuous factors. F. J. Murray and J. von Neumann (1943) constructed two non-isomorphic factors of type II1 . J. Schwartz (1963a) constructed a third factor of type II1 . S. Sakai (1968d) and W. M. Ching (1969) added two more examples of non-isomorphic factors of type II1 . J. Dixmier and E. C. Lance (1969) constructed two other factors of type II1 , whereas G. Zeller-Meier (1969) succeeded in constructing another two new factors. Thus, in 1969, only nine non-isomorphic factors of type II1 were known. D. McDuff (1969a) constructed a countable family of mutually non-isomorphic factors of type II1 ; afterwards, D. McDuff (1969b) and S. Sakai (1970b) have shown that there exists a family of mutually non-isomorphic factors of type II1 , having the power of the continuum. In the same article, S. Sakai has shown the existence of a family, having the power of the continuum, of mutually non-isomorphic factors of type II∞ . As far as the factors of type III are concerned, F. J. Murray and J. von Neumann have shown the existence of a factor of type III. L. Pukánszky (1956) found a second example of a factor of type III, and afterwards, J. Schwartz (1963b) found a third example. Thus, in 1967, only three mutually non-isomorphic factors of type III, were known. R. T. Powers (1967) proved the existence of a family of ‘hyperfinite’ factors of type III, having the power of the continuum; in 1968, H. Araki and E. J. Woods (1968) found another family of mutually non-isomorphic hyperfinite factors of type III, having the power of the continuum. S. Sakai (1970a) showed that there also exists a family of mutually non-isomorphic non-hyperfinite factors of type III, having the power Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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of the continuum. Expositions on this state of the theory of factors of type III can be found in J. Schwartz (1970) and S. Sakai (1971). Afterwards, the theory of factors of type III greatly expanded, overcoming the stage of the ‘fight with cardinals’. These investigations started with the paper of A. Connes (1973) and were developed by A. Connes (1973b, 1974b, c, 1975a, b, 1976c, d) and M. Takesaki (1973c, g). The main technical instruments used in these investigations are the cross-products, infinite tensor products and, especially, the theory of Tomita, which we shall present in Chapter 10. C.4.4 Let M ⊂ B (H ) a von Neumann algebra and Z be the centre of M . For any 𝑥 ∈ M , we denote by U (𝑥) the convex hull of the set {ᵆ∗ 𝑥ᵆ; ᵆ ∈ M , unitary} and by U (𝑥)𝑛 (resp., U (𝑥)w ) the closure of U (𝑥) in the uniform topology (resp., the 𝑤-topology). Following an idea of J. von Neumann, J. Dixmier (1969) introduced the sets K (𝑥) = Z ∩ U (𝑥)𝑛 , C (𝑥) = Z ∩ U (𝑥)w . The fundamental result obtained by J. Dixmier is the following: Theorem 1. Let M ⊂ B (H ) be a von Neumann algebra. For any 𝑥 ∈ M , one has that K (𝑥) ≠ ∅. With the help of the spectral theorem, the proof reduces to the case in which 𝑥 is a projection, whereas, in this case, the proof uses the geometry of projections (see Dixmier [1957/1969], Chap. III, §5). The significance of this theorem is discussed in C.7.1. As far as the inclusion K (𝑥) ⊂ C (𝑥) is concerned, H. Halpern (1972) and ̆ ă and L. Zsidó (1972/1973) have proved the following: S. ¸ Stratil Theorem 2. Let M ⊂ B (H ) be a properly infinite von Neumann algebra. Then the following properties are equivalent: (i) For any 𝑥 ∈ M , the equality K (𝑥) = C (𝑥) holds. (ii) M is of countable type. In the case of finite von Neumann algebras, the equality K (𝑥) = C (𝑥) always holds; more precisely, these sets reduce to a single element (see C.7.1). By studying the derivations of von Neumann algebras, S. Sakai (1966) proved the following: Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.007

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Lectures on von Neumann algebras Theorem 3. If M ⊂ B (H ) is a von Neumann algebra of type III and of countable type, then K (𝑥) ⧵ {0} ≠ ∅, for any 𝑥 ∈ M , 𝑥 ≠ 0. In contrast, L. Zsidó (1974) obtained the following result: Theorem 4. Let M ⊂ B (H ) be a properly infinite von Neumann algebra. Then, for any 𝑥 ∈ 𝑀, the set C (𝑥) is the w-closed convex hull of the set w

Z ∩ {ᵆ∗ 𝑥ᵆ; ᵆ ∈ M , ᵆ unitary} . C.4.5 Here we give the proof of ‘The Averaging Theorem of Dixmier’, which is Theorem 1 from C.4.4 for factors. Thus, let M ⊂ B (H ) be a factor. It suffices to prove that K (𝑥) ∩ ℂ ⋅ 1 ≠ ∅,

(1)

for 𝑥 = 𝑥 ∗ ∈ M . Indeed, for an arbitrary 𝑥 ∈ M , we can then find a convex combination ‖ ‖ 1 ‖‖∑ 𝛼𝑘 ᵆ𝑘 (Re 𝑥)ᵆ𝑘∗ − 𝜆𝑛 ‖‖ < , ‖𝑘 ‖ 𝑛

(𝜆𝑛 ∈ ℂ, 𝛼𝑘 ≥ 0, ∑ 𝛼𝑘 = 1, ᵆ𝑘 ∈ 𝑈(M )) 𝑘

and then a convex combination ‖ ‖ ‖∑ 𝛽𝑗 𝑣𝑗 (∑ 𝛼𝑘 ᵆ𝑘 (Im 𝑥)ᵆ∗ ) 𝑣 ∗ − 𝜇𝑛 ‖ ≤ 1 , 𝑗 𝑘 ‖ ‖ 𝑛 ‖𝑗 ‖ 𝑘 (𝜇𝑛 ∈ ℂ, 𝛽𝑗 ≥ 0, ∑ 𝛽𝑗 = 1, 𝑣𝑗 ∈ 𝑈(M )) 𝑗

concluding that ‖ ‖ ‖∑ 𝛽𝑗 𝑣𝑘 (𝑣𝑗 ᵆ𝑘 ) 𝑥(𝑣𝑗 ᵆ𝑘 )∗ − (𝜆𝑛 + i𝜇𝑛 )‖ ≤ 2 , ‖ ‖ 𝑛 ‖𝑗,𝑘 ‖ and 𝜆 + i𝜇 ∈ K (𝑥) ∩ ℂ ⋅ 1 for any limit point of the bounded sequence {𝜆𝑛 + i𝜇𝑛 }. Thus, let 𝑥 = 𝑥 ∗ ∈ M , 𝑥 ∉ ℂ ⋅ 1, let 𝑚 = inf{𝜆; 𝜆 ∈ 𝜍(𝑥)} < 𝑀 = sup{𝜆; 𝜆 ∈ 𝜍(𝑥)}, and for each unitary, ᵆ ∈ 𝑈(M ) denote 𝑇𝑢 𝑥 =

1 (𝑥 + ᵆ∗ 𝑥ᵆ) ∈ K (𝑥). 2

It is enough to show that there is ᵆ ∈ 𝑈(M ) such that (2)

diam 𝜍(𝑇𝑢 𝑥) ≤

3 diam 𝜍(𝑥) 4

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Classification of von Neumann algebras since then, by iteration, we can find ᵆ1 , ᵆ2 , ..., ᵆ𝑛 ∈ 𝑈(M ) with 𝑛

3 diam 𝜍 (𝑇𝑢𝑛 ...𝑇𝑢2 𝑇𝑢1 𝑥) ≤ ( ) diam 𝜍(𝑥) = 𝜀𝑛 → 0 4 and 𝑦𝑛 = 𝑇𝑢𝑛 ...𝑇𝑢2 𝑇𝑢1 𝑥 ∈ K (𝑥), ‖𝑦𝑛 − 𝜆𝑛 ‖ ≤ 𝜀𝑛 → 0 with some 𝜆𝑛 ∈ 𝜍(𝑦𝑛 ), hence a subsequence of {𝑦𝑛 } converges in norm to some scalar 𝜆 ∈ K (𝑥) ∩ ℂ ⋅ 1. 1 To prove (2), consider 𝑒 = 𝜒(−∞;𝑡) (𝑥) ∈ P R (𝑥) ⊂ P M , with 𝑡 = (𝑚 + 2 𝑀). Notice that 𝑒 ≠ 0 ≠ 1 − 𝑒 (since 𝑚 < 𝑀), 𝑒 commutes with 𝑥, and 𝑥𝑒 ≤ 𝑡𝑒,

𝑥(1 − 𝑒) ≥ 𝑡(1 − 𝑒).

By comparison (Theorem 4.6), we have, for instance, 𝑒 ≺ 1 − 𝑒: 𝑣∗ 𝑣 = 𝑒, 𝑣𝑣 ∗ = 𝑒1 ≤ 1 − 𝑒,

e

e1

υ

υ*

e

e1

for some 𝑣 ∈ M .

1– e– ≤ e1 1– e– ≤ e1 1– e– ≤ e1

Then ᵆ = 𝑣 + 𝑣 + (1 − 𝑒 − 𝑒1 ) ∈ M is a selfadjoint unitary, 𝑒1 = ᵆ𝑒ᵆ∗ = 𝜒(−∞;𝑡) (ᵆ𝑥ᵆ∗ ) and ∗

𝑥 = 𝑥𝑒 + 𝑥(1 − 𝑒) ≥ 𝑚𝑒 + 𝑡(1 − 𝑒), ᵆ𝑥ᵆ∗ = (ᵆ𝑥ᵆ∗ )𝑒1 + (ᵆ𝑥ᵆ∗ )(1 − 𝑒1 ) ≥ 𝑚𝑒1 + 𝑡(1 − 𝑒1 ) = 𝑚𝑒1 + 𝑡𝑒 + 𝑡(1 − 𝑒 − 𝑒1 ) ≥ 𝑚𝑒1 + 𝑡𝑒 + 𝑚(1 − 𝑒 − 𝑒1 ) = 𝑚(1 − 𝑒) + 𝑡𝑒, 1

hence (𝑚 + 𝑡) ≤ 𝑇𝑢 𝑥 ≤ 𝑀 and therefore 2

1 3 3 diam 𝜍(𝑇𝑢 𝑥) ≤ 𝑀 − (𝑚 + 𝑡) = (𝑀 − 𝑚) = diam 𝜍(𝑥). 2 4 4 Q.E.D. C.4.6 Bibliographical comments. The results in this chapter were obtained by F. J. Murray and J. von Neumann for the case of factors. The reduction theory of J. von Neumann (1949) provided the possibility of extending these results to von Neumann algebras with a separable predual. In general, these results have been obtained by global methods by J. Dixmier (1951b) and I. Kaplansky (1951a, 1952, 1955b). In the book by J. Dixmier (1957/1969), the classification of von Neumann algebras is performed on the basis of apparently different

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Lectures on von Neumann algebras criteria from those we have given here (loc. cit., Chap. III, §§6, 8), but one can prove that it is equivalent to what we have given here (loc. cit., Chap. II, §§1, 2, 8). The proof of Theorem 4.7 is due to A. Lebow. The result in Theorem 4.23, for closed ideals, as well as in Corollary 4.23, is due to Y. Misonou (1952). In the general form given here, the result belongs to D. V. Voiculescu (1972b). We note that if the ideal 𝔍 of Z is closed, then 𝔐∞ (ℑ) is closed too (see Dixmier [1957/1969], Chap. III, §5). In our exposition, we used J. Dixmier (1957/1969) and I. Kaplansky (1955b). The results E.4.4, E.4.5 and E.4.6 appeared in the course by D. M. Topping (1971).

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LINEAR FORMS ON ALGEBRAS OF OPERATORS

5

This chapter is dedicated to the study of the predual of a von Neumann algebra, i.e., to the study of the w-topology in the algebra. In this manner, the algebra appears to be, and it is studied as, the dual Banach space of its predual. 5.1. Let A ⊂ B (H ) be a *-algebra of operators. By a form on A , we shall mean any linear functional on A . To any form 𝜑 on A , one can associate its adjoint form 𝜑∗ , defined by 𝜑∗ (𝑎) = 𝜑(𝑎∗ ), 𝑎 ∈ A . A form 𝜑 is said to be self-adjoint (or hermitian) if 𝜑 = 𝜑∗ . The form 𝜑 is self-adjoint iff it takes real values at the self-adjoint elements of the *-algebra A . Any form 𝜑 has a unique decomposition 𝜑 = 𝜑1 + i𝜑2 , where 𝜑1 and 𝜑2 are self-adjoint forms. The form 𝜑 is bounded iff 𝜑1 and 𝜑2 are bounded. A form 𝜑 on A is said to be positive if it takes positive values at positive elements of A . If A is a 𝐶 ∗ -algebra of operators, then any positive form on A is self-adjoint. Proposition 5.2. Any positive form 𝜑 on a 𝐶 ∗ -algebra A is bounded. Proof. Let

𝛼 = sup{𝜑(𝑎); 𝑎 ∈ A , 𝑎 ≥ 0, ‖𝑎‖ ≤ 1}.

If 𝛼 = +∞, then there exists a sequence {𝑎𝑛 } ⊂ A , 𝑎𝑛 ≥ 0, ‖𝑎𝑛 ‖ ≤ 1, such that 𝜑(𝑎𝑛 ) ≥ 𝑛. ∞ In contrast, for any sequence (𝜆𝑛 )𝑛 of positive numbers, such that ∑𝑛=1 𝜆𝑛 < +∞, the ∞ series ∑𝑛=1 𝜆𝑛 𝑎𝑛 converges to an element 𝑎 ∈ A , and since 𝜑 is positive, it follows that 𝑚

𝑚

∑ 𝜆𝑛 𝜑(𝑎𝑛 ) = 𝜑 ( ∑ 𝜆𝑛 𝑎𝑛 ) ≤ 𝜑(𝑎), 𝑛=1

𝑚 = 1, 2, … .

𝑛=1 ∞

It follows that the series ∑𝑛=1 𝜆𝑛 𝜑(𝑎𝑛 ) converges. Since the sequence {𝜆𝑛 }, 𝜆𝑛 ≥ 0, such ∞ that ∑𝑛=1 𝜆𝑛 < +∞ was arbitrary, this fact contradicts the relation 𝜑(𝑎𝑛 ) ≥ 𝑛, 𝑛 ≥ 1. Consequently, we have 𝛼 < +∞. 115

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Now, for any 𝑥 = 𝑥 ∗ ∈ A , ‖𝑥‖ ≤ 1, we have |𝜑(𝑥)| ≤ 𝜑(𝑥 + ) + 𝜑(𝑥 − ) ≤ 2𝛼, whence ‖𝜑‖ ≤ 4𝛼.

Q.E.D.

Proposition 5.3 (the Schwarz inequality). Let 𝜑 be a positive form on the *-algebra A . Then, for any 𝑎, 𝑏 ∈ A , we have |𝜑(𝑎𝑏)|2 ≤ 𝜑(𝑎𝑎∗ )𝜑(𝑏∗ 𝑏). Proof. If 𝜑(𝑎𝑏) = 0, the inequality is obvious and so we can assume that 𝜑(𝑎𝑏) ≠ 0. For any 𝜆 ∈ ℂ, we have 𝜑((𝜆𝑎 + 𝑏∗ )(𝜆𝑎 + 𝑏∗ )∗ ) ≥ 0, i.e., |𝜆|2 𝜑(𝑎𝑎∗ ) + 𝜆𝜑(𝑎𝑏) + 𝜆𝜑(𝑎𝑏) + 𝜑(𝑏∗ 𝑏) ≥ 0. If we take in this inequality 𝜆 = 𝑡(|𝜑(𝑎𝑏)|/𝜑(𝑎𝑏)), 𝑡 ∈ ℝ, it follows that 𝑡 2 𝜑(𝑎𝑎∗ ) + 2𝑡|𝜑(𝑎𝑏)| + 𝜑(𝑏∗ 𝑏) ≥ 0. If we now write that the discriminant of this real polynomial is negative, we get |𝜑(𝑎𝑏)|2 − 𝜑(𝑎𝑎∗ )𝜑(𝑏∗ 𝑏) ≤ 0. Q.E.D. Proposition 5.4. Let 𝜑 be a bounded for on the 𝐶 ∗ -algebra A , assumed to have the unit element. Then 𝜑 is positive iff 𝜑(1) = ‖𝜑‖. Proof. If 𝜑 is positive, with the help of the Schwarz inequality, we easily get 𝜑(1) = ‖𝜑‖. Conversely, assume that 𝜑(1) = ‖𝜑‖ = 1 and that there exists an 𝑎 ∈ A , 𝑎 ≥ 0, such that 𝜑(𝑎) is not positive. Then there exists a disk {𝜆; |𝜆 − 𝜆0 | ≤ 𝑟} in the complex plane, which contains the spectrum of 𝑎, but does not contain 𝜑(𝑎). Since the spectrum of the normal operator 𝑎 − 𝜆0 is included in the disk {𝜆; |𝜆| ≤ 𝑟}, we have ‖𝑎 − 𝜆0 ‖ ≤ 𝑟. It follows that |𝜑(𝑎) − 𝜆0 | = |𝜑(𝑎) − 𝜆0 𝜑(1)| = |𝜑(𝑎 − 𝜆0 )| ≤ ‖𝑎 − 𝜆0 ‖ ≤ 𝑟 a contradiction.

Q.E.D.

Proposition 5.5. Let M ⊂ B (H ) be a von Neumann algebra and 𝜑 a bounded form on M . Then 𝜑 is positive iff it takes positive values at all the projections in M . Proof. The proposition is an immediate consequence of Corollary 2.23.

Q.E.D.

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5.6. Let 𝜑 be a bounded form on a von Neumann algebra M ⊂ B (H ). One says that 𝜑 is completely additive if, for any family {𝑒𝑖 }𝑖∈𝐼 of mutually orthogonal projections in M , one has 𝜑 (∑ 𝑒𝑖 ) = ∑ 𝜑(𝑒𝑖 ). 𝑖∈𝐼

𝑖∈𝐼

Obviously, any w-continuous form on M is completely additive. As we shall see later (Theorem 5.11), the converse is also true. In order to prove this, we need some preparation. Let 𝜑 be a bounded self-adjoint form on M and 𝛼 a real number. By taking into account Corollary 2.23, it easily follows that, if there exists an 𝑎 ∈ M , 𝑎 ≥ 0, ‖𝑎‖ ≤ 1, such that 𝜑(𝑎) > 𝛼, then there exists an 𝑒 ∈ P M , such that 𝜑(𝑒) > 𝛼. Consequently, if 𝜑(𝑒) ≤ 𝛼 for any 𝑒 ∈ P M , then ‖𝜑‖ ≤ 4𝛼. Lemma 5.7. Let 𝜑 be a bounded, self-adjoint, completely additive form on M . For any 𝑒 ∈ P M , there exists an 𝑓 ∈ P M , 𝑓 ≤ 𝑒, such that 𝜑(𝑓) ≥ 𝜑(𝑒), whereas the restriction of 𝜑 to 𝑓M 𝑓 be positive. Proof. Let {𝑒𝑖 }𝑖∈𝐼 be a maximal family of mutually orthogonal projections, majorized by 𝑒 and such that 𝜑(𝑒𝑖 ) < 0, 𝑖 ∈ 𝐼; let 𝑓 = 𝑒 − ∑𝑖∈𝐼 𝑒𝑖 . Since the considered family is maximal, 𝜑 is positive at any projection which is majorized by 𝑓, and therefore, according to Proposition 5.5, it is positive on 𝑓M 𝑓. On the other hand, since 𝜑(𝑒𝑖 ) < 0, 𝑖 ∈ 𝐼, and since 𝜑 is completely additive, we have 𝜑(𝑓) ≥ 𝜑(𝑒). Q.E.D. Lemma 5.8. Let 𝜑 be a completely additive positive form on M and let 𝑒 ∈ P M , 𝑒 ≠ 0. Then there exists an 𝑓 ∈ P M , 𝑓 ≠ 0, 𝑓 ≤ 𝑒, and a 𝜉 ∈ H , such that |𝜑(𝑥𝑓)| ≤ ‖𝑥𝑓𝜉‖,

𝑥 ∈ M.

Proof. There exists an 𝜂 ∈ H , such that (𝜔𝜂 − 𝜑)(𝑒) = ‖𝑒𝜂‖2 − 𝜑(𝑒) > 0. From Lemma 5.7, we infer that there exists an 𝑓 ∈ P M , 𝑓 ≤ 𝑒, such that the restriction of 𝜔𝜂 − 𝜑 to 𝑓M 𝑓 be positive and (𝜔𝜂 − 𝜑)(𝑓) ≥ (𝜔𝜂 − 𝜑)(𝑒). It follows that 𝑓 ≠ 0 and for any 𝑥 ∈ M , ‖𝑥𝑓𝜂‖2 − 𝜑(𝑓𝑥 ∗ 𝑥𝑓) = (𝜔𝜂 − 𝜑)(𝑓𝑥 ∗ 𝑥𝑓) ≥ 0. Using the Schwarz inequality (5.3), we get |𝜑(𝑥𝑓)|2 ≤ 𝜑(1)𝜑(𝑓𝑥 ∗ 𝑥𝑓) ≤ 𝜑(1)‖𝑥𝑓𝜂‖2 , and therefore, we can choose 𝜉 to be equal to a suitable scalar multiple of 𝜂.

Q.E.D.

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Lemma 5.9. Any positive form 𝜑, which is completely additive on M , is w-continuous. Proof. Let {𝑒𝑖 }𝑖∈𝐼 be a maximal family of mutually orthogonal non-zero projections in M , such that for any 𝑖 ∈ 𝐼, there exists a 𝜉𝑖 ∈ H with the property that 𝑥 ∈ M,

|𝜑(𝑥𝑒𝑖 )| ≤ ‖𝑥𝑒𝑖 𝜉𝑖 ‖, From Lemma 5.8, we infer that ∑ 𝑒𝑖 = 1. 𝑖∈𝐼

Since 𝜑 is completely additive, we get ∑ 𝜑(𝑒𝑖 ) = 𝜑(1). 𝑖∈𝐼

Let 𝜀 > 0. Then there exists a finite subset 𝐽 ⊂ 𝐼 such that, if we denote 𝑒 = ∑ 𝑒𝑖 ,

𝑓 = ∑ 𝑒𝑖 ,

𝑖∈ 𝐽

𝑖∈𝐼⧵𝐽

we have 𝜑(𝑓) ≤ 𝜀2 ‖𝜑‖−1 . We now define the bounded forms 𝜑1 , 𝜑2 on M by 𝜑1 (𝑥) = 𝜑(𝑥𝑒),

𝑥 ∈ M,

𝜑2 (𝑥) = 𝜑(𝑥𝑓),

𝑥 ∈ M.

Then 𝜑 = 𝜑1 + 𝜑2 , 𝜑1 is w-continuous, because |𝜑1 (𝑥)| ≤ ∑ ‖𝑥𝑒𝑖 𝜉𝑖 ‖,

𝑥 ∈ M,

𝑖∈𝐽

whereas ‖𝜑2 ‖ ≤ 𝜀, as a consequence of the following computations: |𝜑2 (𝑥)|2 = |𝜑(𝑥𝑓)|2 ≤ 𝜑(1)𝜑(𝑓𝑥 ∗ 𝑥𝑓) ≤ ‖𝜑‖‖𝑥‖2 𝜑(𝑓) ≤ 𝜀2 ‖𝑥‖2 ,

𝑥 ∈ M.

Hence we obtained a 𝜑1 ∈ M ∗ , such that ‖𝜑 − 𝜑1 ‖ ≤ 𝜀. Since M ∗ is uniformly closed in M ∗ (cf. Theorem 1.10), we infer that 𝜑 ∈ M ∗ , and this shows that 𝜑 is w-continuous. Q.E.D. Lemma 5.10. Let 𝑒 ∈ B (H ) be a projection, 𝑥 ∈ B (H ), ‖𝑥‖ ≤ 1 and 𝛼, 𝛽, 𝛾 be real numbers, such that 𝛼, 𝛽, 𝛼𝛽 − 𝛾2 ≥ 0. Then 𝛼𝑒 + 𝛽(1 − 𝑒) + 𝛾(𝑒𝑥(1 − 𝑒) + (1 − 𝑒)𝑥 ∗ 𝑒) ≥ 0. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

119

Linear forms on algebras of operators Proof. For any real numbers 𝑠, 𝑡, we have 𝛼𝑠2 + 𝛽𝑡 2 − 2|𝛾|𝑠𝑡 ≥ 0. Hence, for any 𝜉 ∈ H , we have 𝛼(𝑒𝜉|𝜉)) + 𝛽((1 − 𝑒)𝜉|𝜉) + 𝛾(𝑒𝑥(1 − 𝑒)𝜉|𝜉) + 𝛾((1 − 𝑒)𝑥 ∗ 𝑒𝜉|𝜉) = 𝛼‖𝑒𝜉‖2 + 𝛽‖(1 − 𝑒)𝜉‖2 + 2𝛾Re (𝑥(1 − 𝑒)𝜉|𝑒𝜉) ≥ 𝛼‖𝑒𝜉‖2 + 𝛽‖(1 − 𝑒)𝜉‖2 − 2|𝛾|‖(1 − 𝑒)𝜉‖‖𝑒𝜉‖ ≥ 0.

Q.E.D. Theorem 5.11. A bounded form on a von Neumann algebra is w-continuous iff it is completely additive. Proof. Let 𝜑 be a completely additive bounded form on the von Neumann algebra M . We can assume that 𝜑 is self-adjoint and ‖𝜑‖ ≤ 1. We write 𝜇 = sup{𝜑(𝑎);

𝑎 ∈ M,

0 ≤ 𝑎 ≤ 1}.

Then 0 ≤ 𝜇 ≤ 1. Let 𝜀 > 0, 𝜀 ≤ 3/4. There exists an 𝑎 ∈ M , 0 ≤ 𝑎 ≤ 1, such that 𝜑(𝑎) > 𝜇 − 𝜀. From Corollary 5.6 and Lemma 5.7, we infer that there exists a projection 𝑒1 ∈ M , such that 𝜑(𝑒1 ) > 𝜇 − 𝜀, whereas the restriction of 𝜑 to 𝑒1 M 𝑒1 be positive. From Lemma 5.9, we infer that the restriction of 𝜑 to 𝑒1 M 𝑒1 is w-continuous. Let 𝑒2 = 1 − 𝑒1 . We define the forms 𝜑𝑖𝑗 ∈ M ∗ by 𝜑𝑖𝑗 (𝑥) = 𝜑(𝑒𝑖 𝑥𝑒𝑗 ),

𝑥 ∈ M ; 𝑖, 𝑗 = 1, 2.

Then 𝜑11 is w-continuous and 𝜑 = 𝜑11 + 𝜑12 + 𝜑21 + 𝜑22 . If 𝑓 ∈ P M , 𝑓 ≤ 𝑒2 , then 𝜇 ≥ 𝜑(𝑒1 + 𝑓) = 𝜑(𝑒1 ) + 𝜑(𝑓) > 𝜇 − 𝜀 + 𝜑(𝑓), and therefore, 𝜑(𝑓) < 𝜀. We now investigate the norms ‖𝜑12 ‖ and ‖𝜑21 ‖. Let 𝑥 ∈ M , ‖𝑥‖ ≤ 1. We denote 𝑦 = (1 − 𝜀)𝑒1 + 𝜀𝑒2 + 𝜀1/2 (1 − 𝜀)1/2 (𝑒1 𝑥𝑒2 + 𝑒2 𝑥 ∗ 𝑒1 ). Then

1 − 𝑦 = 𝜀𝑒1 + (1 − 𝜀)𝑒2 − 𝜀1/2 (1 − 𝜀)1/2 (𝑒1 𝑥𝑒2 + 𝑒2 𝑥 ∗ 𝑒1 ).

With Lemma 5.10, we infer that 0 ≤ 𝑦 ≤ 1; hence 𝜇 ≥ 𝜑(𝑦) = (1 − 𝜀)𝜑(𝑒1 ) + 𝜀𝜙(𝑒2 ) + 𝜀1/2 (1 − 𝑒)1/2 𝜑(𝑒1 𝑥𝑒2 + 𝑒2 𝑥 ∗ 𝑒1 ) > (1 − 𝜀)(𝜇 − 𝜀) − 𝜀 + 2𝜀1/2 (1 − 𝜀)1/2 Re 𝜑(𝑒1 𝑥𝑒2 ). We hence obtain 2+𝜇−𝜀 1 1 3 Re 𝜑12 (𝑥) = Re 𝜑(𝑒1 𝑥𝑒2 ) ≤ 𝜀1/2 ≤ 𝜀1/2 = 3𝜀1/2 . 1/2 2 2 (1 − 𝜀) (1/4)1/2 Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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Therefore, we have ‖𝜑12 ‖ ≤ 3𝜀1/2 , and analogously, ‖𝜑21 ‖ ≤ 3𝜀1/2 . We denote by 𝜓 the restriction of (−𝜑) to 𝑒2 M 𝑒2 . Then ‖𝜓‖ ≤ 1, and for any projection 𝑓 ≤ 𝑒2 , we have 𝜓(𝑓) > −𝜀. By repeating for 𝜓 the preceding argument, we get the projections 𝑓1 , 𝑓2 , such that 𝑓1 + 𝑓2 = 𝑒2 and by denoting 𝜓𝑖𝑗 (𝑥) = 𝜓(𝑓𝑖 𝑥𝑓𝑗 ),

𝑥 ∈ 𝑒2 M 𝑒2 ;

𝑖, 𝑗 = 1, 2,

the following properties should hold: 𝜓11 is w-continuous, ‖𝜓12 ‖ ≤ 3𝜀1/2 , ‖𝜓21 ‖ ≤ 3𝜀1/2 and for any projection 𝑓 ≤ 𝑓2 , 𝜓22 (𝑓) < 𝜀. It follows that |𝜓22 (𝑓)| < 𝜀, for any projection 𝑓 ≤ 𝑓2 ; hence ‖𝜓22 ‖ ≤ 4𝜀 (see 5.6). Consequently, we have ‖𝜓 − 𝜓11 ‖ ≤ 4𝜀 + 6𝜀1/2 . We define the form 𝜓0 on M by 𝜓0 (𝑥) = 𝜓11 (𝑒2 𝑥𝑒2 ),

𝑥 ∈ M.

Then 𝜓0 is w-continuous and ‖𝜑 − 𝜑11 − 𝜓0 ‖ ≤ 4𝜀 + 12𝜀1/2 . Since M ∗ is closed in M ∗ and 0 < 𝜀 < 3/4 is arbitrary, it follows that 𝜑 ∈ M ∗ , i.e., 𝜑 is w-continuous. Q.E.D. Corollary 5.12. A bounded form on a von Neumann algebra is w-continuous iff its restrictions to maximal commutative von Neumann subalgebras of M are w-continuous. Corollary 5.13. Any *-isomorphism between two von Neumann algebras is w-continuous. Proof. Let M ⊂ B (H ), N ⊂ B (H ) be two von Neumann algebras and let 𝜋∶M → N be a *-isomorphism between them. Then for any element 𝑥 ∈ M , we have 𝜎(𝑥 ∗ 𝑥) = 𝜎(𝜋(𝑥 ∗ 𝑥)) (see Corollary 2.8), and hence, by taking into account Lemma 2.5, we get ‖𝑥‖2 = ‖𝑥 ∗ 𝑥‖ = ‖𝜋(𝑥 ∗ 𝑥)‖ = ‖𝜋(𝑥)∗ 𝜋(𝑥)‖ = ‖𝜋(𝑥)‖2 , i.e., 𝜋 is an isometry. Let 𝜓 be a w-continuous form on N . Then 𝜑 = 𝜓 ∘ 𝜋 is a bounded form on M . If {𝑒𝑖 }𝑖∈𝐼 is 𝑛 family of mutually orthogonal projections in M , then {𝜋(𝑒𝑖 )}𝑖∈𝐼 is a family of mutually orthogonal projections in N and 𝜋(∑ 𝑒𝑖 ) = 𝜋( 𝑖∈𝐼

⋁

𝑒𝑖 ) =

𝑖∈𝐼

⋁

𝜋(𝑒𝑖 ) = ∑ 𝜋(𝑒𝑖 ). 𝑖∈𝐼

𝑖∈𝐼

Consequently, we have 𝜑(∑ 𝑒𝑖 ) = 𝜓(𝜋(∑ 𝑒𝑖 )) = 𝜓(∑ 𝜋(𝑒𝑖 )) = ∑ 𝜓(𝜋(𝑒𝑖 )) = ∑ 𝜑(𝑒𝑖 ). 𝑖∈𝐼

𝑖∈𝐼

𝑖∈𝐼

𝑖∈𝐼

𝑖∈𝐼

Thus, 𝜑 is completely additive, and by virtue of Theorem 5.11, 𝜑 is w-continuous. It follows that 𝜋 is w-continuous. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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5.14. Another consequence of Theorem 5.11 is the characterization of the weakly relatively compact subsets of the predual of a von Neumann algebra. Theorem. Let M ⊂ B (H ) be a von Neumann algebra, M ∗ the predual of M and F ⊂ M ∗ a norm-bounded subset. The following assertions are equivalent: (i) F is 𝜎(M ∗ , M )-relatively compact. (ii) For any countable family {𝑒𝑛 } of mutually orthogonal projections in M , one has 𝜑(𝑒𝑛 ) → 0, uniformly for 𝜑 ∈ F . Proof. (ii)⇒(i). Since F is a bounded subset of M ∗ ⊂ M ∗ , it follows that its 𝜎(M ∗ ; M )-closure F in M ∗ is 𝜎(M ∗ ; M )-compact. It is, therefore, sufficient to show that F ⊂ M ∗ ; indeed, we then have F ⊂ F ⊂ M ∗ and F is 𝜎(M ∗ ; M )-compact, since the 𝜎(M ∗ ; M )-topology is just the restriction of the 𝜎(M ∗ ; M )-topology to M ∗ . Let 𝜑 ∈ F . There then exists a net {𝜑𝑘 }𝑘∈𝐾 ⊂ F , which is 𝜎(M ∗ ; M )-convergent to 𝜑. Let {𝑒𝑖 }𝑖∈𝐼 be a family of mutually orthogonal projections in M and 𝑒 = ∑𝑖∈𝐼 𝑒𝑖 . We then have 𝜑(𝑒) = lim 𝜑𝑘 (𝑒), 𝑘∈𝐾

𝜑(𝑒𝑖 ) = lim 𝜑𝑘 (𝑒𝑖 ), for any 𝑖 ∈ 𝐼, 𝑘∈𝐾

𝜑𝑘 (𝑒) = ∑ 𝜑𝑘 (𝑒𝑖 ), uniformly for 𝑘 ∈ 𝐾. 𝑖∈𝐾

In fact, we have 𝜓(𝑒) = ∑𝑖∈𝐼 𝜓(𝑒𝑖 ), uniformly for 𝜓 ∈ F . Indeed, if this be not true, then there exists a sequence {𝜓𝑛 }𝑛 ⊂ F , a sequence (𝐽𝑛 )𝑛 of finite mutually disjoint subsets 𝐽𝑛 ⊂ 𝐼, and a 𝛿 > 0, such that for any 𝑛, we have | ∑𝑖∈𝐽 𝜓𝑛 (𝑒𝑖 )| ≥ 𝛿. We define 𝑓𝑛 = ∑𝑖∈𝐽 𝑒𝑖 . 𝑛 𝑛 Then {𝑓𝑛 }𝑛 is a countable family of mutually orthogonal projections in M , and for any 𝑛, we have |𝜓𝑛 (𝑓𝑛 )| ≥ 𝛿, a contradiction if (ii) is taken into account. It follows that 𝜑(𝑒) = ∑ 𝜑(𝑒𝑖 ), 𝑖∈𝐼

hence 𝜑 is completely additive, and therefore, by virtue of Theorem 5.11, 𝜑 is w-continuous, i.e., 𝜑 ∈ M ∗ . (i) ⇒ (ii). We shall proceed by contradiction. Hence, there exists a sequence {𝑒𝑛 }𝑛 of mutually orthogonal projections in M , a sequence {𝜑𝑛 } ⊂ F and a 𝛿 > 0, such that, for any 𝑛, we have |𝜑𝑛 (𝑒𝑛 )| ≥ 4𝛿. Since F is 𝜎(M ∗ ; M )-relatively compact, we can assume that the sequence {𝜑𝑛 } is 𝜎(M ∗ ; M )-convergent to a form 𝜑 ∈ M ∗ . Since the sequence {𝑒𝑛 } is w-convergent to 0, we have lim𝑛→∞ 𝜑(𝑒𝑛 ) = 0, and therefore, we can assume that, for any 𝑛, we have |𝜑(𝑒𝑛 )| ≤ 𝛿. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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The sequence of forms 𝜓𝑛 = 𝜑𝑛 − 𝜑 ∈ M ∗ is 𝜎(M ∗ ; M )-convergent to 0, and for any 𝑛, we have (1) |𝜓𝑛 (𝑒𝑛 )| ≥ 3𝛿. We shall now construct an increasing sequence {𝑛(1), 𝑛(2), …} of natural numbers, with the following properties: |𝑘−1 | |∑ 𝜓 | (𝑒 ) 𝑛(𝑘) 𝑛(𝑗) | < 𝛿, | | 𝑗=1 |

(2)

for any 𝑘 = 2, 3, …

∞

∑

(3)

|𝜓𝑛(𝑘) (𝑒𝑗 )| < 𝛿,

for any 𝑘 = 1, 2, …

𝑗=𝑛(𝑘+1)

In order to do this, let us first observe that for any 𝜓 ∈ M ∗ , we have ∞

∑ |𝜓(𝑒𝑛 )| < +∞, 𝑛=1 ∞

because for any bounded sequence {𝜆𝑛 } of scalars, the series ∑𝑛=1 𝜆𝑛 𝑒𝑛 is w-convergent; ∞ hence the series ∑𝑛=1 𝜆𝑛 𝜓(𝑒𝑛 ) is convergent. We begin the construction by taking 𝑛(1) = 1, and we assume that 𝑛(1), … , 𝑛(𝑝−1) have already been constructed, such that condition (2) be satisfied for 𝑘 = 2, … , 𝑝 − 1, whereas condition (3) be satisfied for 𝑘 = 1, … , 𝑝 − 2. Since {𝜓𝑛 } is 𝜎(M ∗ ; M )-convergent to 0 and ∞ since ∑𝑗=1 |𝜓𝑛(𝑝−1) (𝑒𝑗 )| < +∞, for a sufficiently great 𝑛, the following inequalities are satisfied: |𝑝−1 | | ∑ 𝜓 (𝑒 )| < 𝛿, 𝑛 𝑛(𝑗) | | | 𝑗=1 | ∞

∑ |𝜓𝑛(𝑝−1) (𝑒𝑗 )| < 𝛿. 𝑗=1

Consequently, relation (2) is satisfied for 𝑘 = 𝑝, whereas relation (3) is satisfied for 𝑘 = 𝑝 − 1, if we choose 𝑛(𝑝) > 𝑛(𝑝 − 1) to be sufficiently great. The required construction is thus possible by induction. From relation (3), it follows that ∞

(4)

∑ |𝜓𝑛(𝑘) (𝑒𝑛(𝑗) )| < 𝛿,

𝑘 = 1, 2, …

𝑗=𝑘+1 ∞

We now consider the projection 𝑓 = ∑𝑗=1 𝑒𝑛(𝑗) ∈ M . We then have ∞

𝜓𝑛(𝑘) (𝑓) = ∑ 𝜓𝑛(𝑘) (𝑒𝑛(𝑗) ),

𝑘 = 1, 2, …

𝑗=1

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𝑘 = 1, 2, …

thus contradicting the fact that the sequence {𝜓𝑛 }𝑛 is 𝜎(M ∗ ; M )-convergent to 0. Q.E.D. 5.15. The w-continuous positive forms on a von Neumann algebra are also called normal forms. Let 𝜑 be a normal form on M and let 𝑎 ∈ M , 𝑎 ≥ 0, be such that 𝜑(𝑎) = 0. Then 𝜑(s(𝑎)) = 0. Indeed, in accordance with Corollary 2.22, there exists an increasing sequence {𝑒𝑛 } ⊂ P M , such that ⋁𝑛 𝑒𝑛 = s(𝑎), and 𝑎𝑒𝑛 ≥ (1/𝑛)𝑒𝑛 , for any 𝑛; it follows that 𝜑(𝑒𝑛 ) = 0, for any 𝑛, and therefore, 𝜑(s(𝑎)) = 0. If 𝑒, 𝑓 ⊂ P M , 𝜑(𝑒) = 0 and 𝜑(𝑓) = 0, then 𝜑(𝑒 ∨ 𝑓) = 0, because 𝑒 ∨ 𝑓 = s(𝑒 + 𝑓). Consequently, if 𝜑 is a normal form on M , then the family {𝑒 ∈ P M ; 𝜑(𝑒) = 0} is increasingly directed, and therefore, by denoting by 1 − s(𝜑) the least upper bound of this family, we infer that 𝜑(1 − s(𝜑)) = 0. The projection s(𝜑) is called the support of 𝜑. One says that 𝜑 is faithful if s(𝜑) = 1. With the help of the Schwarz inequality, one can easily prove that 𝜑(𝑥) = 𝜑(𝑥s(𝜑)) = 𝜑(s(𝜑)𝑥),

𝑥 ∈ M.

From the definition of the support, it follows that 𝑥 ∈ M , 𝑥 ≥ 0,

𝜑(𝑥) = 0 ⇒ s(𝜑)𝑥s(𝜑) = 0;

in particular, the form 𝜑 is faithful iff the implication 𝑥 ∈ M , 𝑥 ≥ 0,

𝜑(𝑥) = 0 ⇒ 𝑥 = 0.

holds. Let 𝜑 be a form on M and 𝑎 ∈ M . We then define the forms (𝐿𝑎 𝜑)(𝑥) = 𝜑(𝑎𝑥), 𝑥 ∈ M , (𝑅𝑎 𝜑)(𝑥) = 𝜑(𝑥𝑎), 𝑥 ∈ M , (𝑇𝑎 𝜑)(𝑥) = 𝜑(𝑎∗ 𝑥𝑎), 𝑥 ∈ M . If 𝜑 is bounded (resp., w-continuous), then 𝐿𝑎 𝜑, 𝑅𝑎 𝜑, 𝑇𝑎 𝜑 are bounded (resp., w-continuous). 5.16. The following result often allows the reduction of problems on w-continuous forms to problems on normal forms. Theorem (of polar decomposition for forms). Let 𝜑 be a w-continuous form on the von Neumann algebra M . Then there exists a normal form |𝜑| and a partial isometry 𝑣 ∈ M , uniquely determined by the following conditions: 𝜑 = 𝑅𝑣 |𝜑|, 𝑣 𝑣 = s(|𝜑|). ∗

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Proof. The set {𝑥 ∈ M ; ‖𝑥‖ ≤ 1, 𝜑(𝑥) = ‖𝜑‖} is a non-empty, w-compact, convex part of M . Let 𝑢 be an extreme point of this set. Then 𝑢 is an extreme point of the unit ball of M ; consequently, by virtue of Proposition 3.19, it is a partial isometry. We define 𝜓 = 𝑅ᵆ 𝜑. Since 𝜓(1) = 𝜑(𝑢) = ‖𝜑‖ ≥ ‖𝜓‖ ≥ 𝜓(1). It follows that 𝜓(1) = ‖𝜓‖. From Proposition 5.4, we infer that 𝜓 is positive, and therefore, it is normal. We define 𝑣 = 𝑢∗ s(𝜓). Since 𝑢 is a partial isometry, we have 𝑢 = 𝑢𝑢∗ 𝑢, whence 𝜓(1 − 𝑢𝑢∗ ) = 𝜑(𝑢 − 𝑢𝑢∗ 𝑢) = 0, and therefore, 𝑢𝑢∗ ≥ s(𝜓). It follows that 𝑣∗ 𝑣 = s(𝜓), and for any 𝑥 ∈ M , we have 𝜓(𝑥) = 𝜓(𝑥s(𝜓)) = 𝜑(𝑥s(𝜓)𝑢) = 𝜑(𝑥𝑣 ∗ ). We shall now prove that 𝜑 = 𝑅𝑣 𝜓, i.e., 𝜑(𝑥) = 𝜓(𝑥𝑣), for any 𝑥 ∈ M . Indeed, if this is not true, then there exists an 𝑥 ∈ M , ‖𝑥‖ ≤ 1, such that 𝜑(𝑥(1 − 𝑣𝑣 ∗ )) = 𝛼 > 0. For any natural number 𝑛, we have ‖𝑛𝑣∗ + 𝑥(1 − 𝑣𝑣 ∗ )‖2 = ‖(𝑛𝑣 ∗ + 𝑥(1 − 𝑣𝑣 ∗ ))(𝑛𝑣 + (1 − 𝑣𝑣 ∗ )𝑥∗ )‖ = ‖𝑛2 𝑣∗ 𝑣 + 𝑥(1 − 𝑣𝑣 ∗ )𝑥 ∗ ‖ ≤ 𝑛2 + 1; hence

|𝜑(𝑛𝑣 ∗ + 𝑥(1 − 𝑣𝑣 ∗ ))| ≤ ‖𝜑‖(𝑛2 + 1)1/2 .

In contrast, we have 𝜑(𝑛𝑣 ∗ + 𝑥(1 − 𝑣𝑣 ∗ )) = 𝜓(𝑛) + 𝜑(𝑥(1 − 𝑣𝑣 ∗ )) = ‖𝜑‖𝑛 + 𝛼. Thus we have

‖𝜑‖𝑛 + 𝛼 ≤ ‖𝜑‖(𝑛2 + 1)1/2 ,

an impossible inequality if 𝑛 is sufficiently great. It follows that, if we denote |𝜑| = 𝜓, we have 𝜑 = 𝑅𝑣 |𝜑| and 𝑣∗ 𝑣 = s(|𝜑|), the existence part of the theorem being thus established. Let 𝜓 and 𝜓′ be two normal forms on M , and 𝑣, 𝑣 ′ partial isometrics in M , such that 𝜑 = 𝑅𝑣 𝜓 = 𝑅𝑣′ 𝜓′ and 𝑣∗ 𝑣 = s(𝜓), 𝑣 ′∗ 𝑣′ = s(𝜓′ ). We have 𝜓(1) = 𝜓(𝑣 ∗ 𝑣) = 𝜑(𝑣 ∗ ) = 𝜓 ′ (𝑣 ∗ 𝑣′ ) = 𝜓 ′ (𝑣 ′∗ 𝑣′ 𝑣∗ 𝑣 ′ ) = 𝜑(𝑣 ′∗ 𝑣′ 𝑣∗ ) = 𝜓(𝑣 ′∗ 𝑣′ 𝑣∗ 𝑣) = 𝜓(𝑣 ′∗ 𝑣′ ); hence 𝑣′∗ 𝑣′ ≥ s(𝜓) = 𝑣 ∗ 𝑣. Analogously, we have 𝑣∗ 𝑣 ≥ 𝑣 ′∗ 𝑣′ ; hence 𝑣∗ 𝑣 = 𝑣 ′∗ 𝑣′ = 𝑒. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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Since 𝑣′∗ 𝑣′ = 𝑣 ′∗ 𝑣′ 𝑣′∗ 𝑣𝑣∗ 𝑣 = 𝑒𝑣 ′∗ 𝑣𝑒 ∈ 𝑒M 𝑒, we can write 𝑣′∗ 𝑣 = 𝑎 + i𝑏, where 𝑎, 𝑏 ∈ 𝑒M 𝑒 are self-adjoint. We have 𝜓(𝑎) + i𝜓(𝑏) = 𝜓(𝑣 ′∗ 𝑣) = 𝜑(𝑣 ′∗ ) = 𝜓 ′ (𝑣′∗ 𝑣′ ) = ‖𝜓 ′ ‖ = ‖𝜓‖; hence 𝜓(𝑎) = ‖𝜓‖. Hence, 𝜓(𝑒 − 𝑎) = 0. Since 𝑒 − 𝑎 ≥ 0, we find that 𝑎 = 𝑒. It follows that ‖𝑒 + i𝑏‖ ≤ 1, whence 𝑏 = 0. Consequently, we have 𝑣′∗ 𝑣 = 𝑒. Since 𝑣′∗ 𝑣 = 𝑒, we also have 𝑣∗ 𝑣′ = 𝑒, and we can write 𝑣′ 𝑣′∗ = 𝑣 ′ 𝑣′∗ 𝑣′ 𝑣′∗ = 𝑣 ′ 𝑒𝑣 ′∗ = 𝑣 ′ 𝑣′∗ 𝑣𝑣∗ 𝑣′ 𝑣′∗ , 𝑣′ 𝑣′∗ (1 − 𝑣𝑣 ∗ )𝑣′ 𝑣′∗ = 0, (1 − 𝑣𝑣 ∗ )𝑣′ 𝑣′∗ = 0, 𝑣′ 𝑣′∗ = 𝑣𝑣 ∗ 𝑣′ 𝑣 ′∗ ≤ 𝑣𝑣 ∗ . Analogously, we find that 𝑣𝑣 ∗ ≤ 𝑣 ′ 𝑣′∗ , and therefore, 𝑣𝑣 ∗ = 𝑣 ′ 𝑣′∗ = 𝑓. We finally have 𝑣 = 𝑣𝑣 ∗ 𝑣 = 𝑓𝑣 = 𝑣 ′ 𝑣′∗ 𝑣 = 𝑣 ′ 𝑒 = 𝑣 ′ 𝑣 ′∗ 𝑣′ = 𝑣 ′ , and for any 𝑥 ∈ M , 𝜓(𝑥) = 𝜓(𝑥𝑣 ∗ 𝑣) = 𝜑(𝑥𝑣 ∗ ) = 𝜓 ′ (𝑥𝑣 ∗ 𝑣′ ) = 𝜓 ′ (𝑥𝑒) = 𝜓 ′ (𝑥). The uniqueness part of the theorem is thus proved.

Q.E.D.

5.17. Theorem 5.16 then gives us Theorem (the Jordan decomposition). Let 𝜑 be a w-continuous self-adjoint form on the von Neumann algebra M . Then there exist normal forms 𝜑1 and 𝜑2 , uniquely determined by the conditions 𝜑 = 𝜑1 − 𝜑2 , s(𝜑1 )s(𝜑2 ) = 0. Proof. Let 𝜓 be the normal form on M and 𝑣 ∈ M the partial isometry, which, in accordance with Theorem 5.16, satisfiy the properties: 𝜑 = 𝑅𝑣 𝜓, 𝑣 𝑣 = s(𝜓). ∗

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We now define the normal form 𝜓0 = 𝑇𝑣 𝜓. Then we have s(𝜓0 ) = 𝑣𝑣 ∗ . Since 𝜑 is self-adjoint, for any 𝑥 ∈ M , we have 𝜑(𝑥) = 𝜑(𝑥 ∗ ) = 𝜓(𝑥 ∗ 𝑣) = 𝜓(𝑣 ∗ 𝑥) = 𝜓(𝑣 ∗ 𝑥𝑣 ∗ 𝑣) = 𝜓0 (𝑥𝑣 ∗ ). From the uniqueness part of Theorem 5.16, we get 𝑣 = 𝑣 ∗ and 𝜓 = 𝜓0 . It follows that 𝑣 = 𝑒1 −𝑒2 , where 𝑒1 and 𝑒2 are orthogonal projections. We define 𝜑1 = 𝑅𝑒1 𝜑 and 𝜑2 = −𝑅𝑒2 𝜑. Since 𝜓 = 𝜓0 and since s(𝜓) = 𝑒1 + 𝑒2 , we infer that, for any 𝑥 ∈ M , we have 𝜓(𝑥) = 𝜓(𝑒1 𝑥𝑒1 + 𝑒2 𝑥𝑒2 ). We hence infer that 𝜑1 and 𝜑2 are positive. The other conditions in the statement of the theorem are now easily verified, and thus, the existence part of the theorem is proved. Let now 𝜑1′ and 𝜑2′ be normal forms whose supports 𝑒1′ , 𝑒2′ be orthogonal, and such that 𝜑 = 𝜑1′ − 𝜑2′ . We denote 𝜓′ = 𝜑1′ + 𝜑2′ and 𝑣′ = 𝑒1′ − 𝑒2′ . It is easily seen that 𝜑 = 𝑅𝑣′ 𝜓′ and 𝑣′∗ 𝑣′ = s(𝜓 ′ ); hence, from the uniqueness part of Theorem 5.16, we have 𝑣′ = 𝑣 and 𝜓′ = 𝜓. It follows that 𝑒1′ − 𝑒2′ = 𝑒1 − 𝑒2 , 𝑒1′ + 𝑒2′ = s(𝜓 ′ ) = s(𝜓) = 𝑒1 + 𝑒2 ; hence 𝑒1′ = 𝑒1 , 𝑒2′ = 𝑒2 . We now immediately get the equalities 𝜑1′ = 𝜑1 ,

𝜑2′ = 𝜑2 . Q.E.D.

5.18. Let A be a 𝐶 ∗ -algebra with the unit element, H a Hilbert space and 𝜋 ∶ A → B (H ), 𝜋(1) = 1, a *-homomorphism. Then any vector 𝜉 ∈ H determines a positive form 𝜑 = 𝜔𝜉 ∘ 𝜋 on A . Conversely, let 𝜑 be a positive form on a 𝐶 ∗ -algebra A , assumed to have the unit element. With the help of the Schwarz inequality, it is easy to show that the set 𝔑𝜑 = {𝑎 ∈ A ; 𝜑(𝑎∗ 𝑎) = 0} is a left ideal of A . For any 𝑎 ∈ A , we shall denote by 𝑎𝜑 the canonical image of 𝑎 in A 𝜑 = A /𝔑𝜑 . We define on A 𝜑 a scalar product by the relation: (𝑎𝜑 |𝑏𝜑 )𝜑 = 𝜑(𝑏∗ 𝑎),

𝑎𝜑 , 𝑏𝜑 ∈ A 𝜑 .

Then A 𝜑 becomes a separated pre-Hilbert space. We denote by H 𝜑 the Hilbert space obtained by the completion of A 𝜑 . For any 𝑥 ∈ A , we define 𝜋𝜑0 (𝑥)𝑎𝜑 = (𝑥𝑎)𝜑 ,

𝑎𝜑 ∈ A 𝜑 .

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Since ‖𝜋𝜑0 (𝑥)𝑎𝜑 ‖𝜑 ≤ ‖𝑥‖‖𝑎𝜑 ‖𝜑 , it follows that 𝜋𝜑0 (𝑥) can be uniquely extended by continuity to an operator 𝜋𝜑 (𝑥) ∈ B (H 𝜑 ). It is easily seen that the mapping 𝜋𝜑 ∶ A ∋ 𝑥 ↦ 𝜋𝜑 (𝑥) ∈ B (H 𝜑 ) is a *-homomorphism, 𝜋𝜑 (1) = 1 and 𝜑 = 𝜔1𝜑 ∘ 𝜋𝜑 . We observe that the vector 1𝜑 ∈ H 𝜑 is cyclic for 𝜋𝜑 (A ), i.e., [𝜋𝜑 (A )1𝜑 ] = H 𝜑 . Proposition. Let M be a von Neumann algebra and 𝜑 a normal form on M . Then 𝜋𝜑 is a w-continuous *-homomorphism, 𝜋𝜑 (1) = 1, 𝜋𝜑 (M ) ⊂ B (H 𝜑 ) is a von Neumann algebra and 𝜑 = 𝜔1𝜑 ∘ 𝜋𝜑 , 1𝜑 ∈ H 𝜑 is a cyclic vector for 𝜋𝜑 (M ). Moreover, if s(𝜑) = 1, then 𝜋𝜑 is a *-isomorphism of M onto 𝜋𝜑 (M ) and 1𝜑 is a separating vector for 𝜋𝜑 (M ). Proof. Let {𝑥𝑖 }𝑖∈𝐼 ⊂ M , 𝑥 ∈ M , 𝑥𝑖 ↑ 𝑥. Then 𝜋𝜑 (𝑥) is an upper bound for the increasing net {𝜋(𝑥𝑖 )}𝑖∈𝐼 , and since 𝜑 is normal, for any 𝑎 ∈ M , we have lim(𝜋𝜑 (𝑥𝑖 )𝑎𝜑 |𝑎𝜑 ) = lim 𝜑(𝑎∗ 𝑥𝑖 𝑎) = 𝜑(𝑎∗ 𝑥𝑎) = (𝜋𝜑 (𝑥)𝑎𝜑 |𝑎𝜑 ). 𝑖∈𝐼

𝑖∈𝐼

Consequently, we have 𝜋𝜑 (𝑥𝑖 ) ↑ 𝜋𝜑 (𝑥). In particular, 𝜋𝜑 is a completely additive mapping. Since ‖𝜋𝜑 ‖ ≤ 1, by virtue of Theorem 5.11 (see also E.5.17), it follows that 𝜋𝜑 is a w-continuous *-homomorphism. The fact that 𝜋𝜑 (M ) ⊂ B (H 𝜑 ) is a von Neumann algebra is a consequence of Corollary 3.12. Let us now assume that s(𝜑) = 1. If 𝑥 ∈ M and 𝜋𝜑 (𝑥)1𝜑 = 0, then 𝑥𝜑 = 0, i.e., 𝜑(𝑥 ∗ 𝑥) = 0, whence 𝑥 = 0. Q.E.D. By taking into account E.5.6, from the preceding proposition, it follows that any von Neumann algebra of countable type is *-isomorphic to a von Neumann algebra, which has a separating cyclic vector. 5.19. If 𝜑 and 𝜓 are linear forms on a *-algebra of operators, we shall write 𝜑 ≤ 𝜓 if 𝜓 − 𝜑 is a positive form.

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Lemma. Let A ⊂ B (H ), A ∋ 1, be a *-algebra, 𝜉 ∈ H and 𝜑 a positive form on A , such that 𝜑 ≤ 𝜔𝜉 . Then there exists an 𝑎′ ∈ A ′ , 0 ≤ 𝑎′ ≤ 1, such that 𝜑 = 𝜔𝑎′ 𝜉 . Proof. For any 𝑥, 𝑦 ∈ A , we have |𝜑(𝑦 ∗ 𝑥)| ≤ 𝜑(𝑦 ∗ 𝑦)1/2 𝜑(𝑥 ∗ 𝑥)1/2 ≤ ‖𝑦𝜉‖‖𝑥𝜉‖. If we write (𝑥𝜉|𝑦𝜉)0 = 𝜑(𝑦 ∗ 𝑥), we define a positive sesquilinear form of norm ≤ 1 on [A 𝜉]. Hence there exists a linear operator 𝑎0 on [A 𝜉], 0 ≤ 𝑎0 ≤ 1, such that 𝜑(𝑦 ∗ 𝑥) = (𝑥𝜉|𝑦𝜉)0 = (𝑎0 𝑥𝜉|𝑦𝜉). For any 𝑥, 𝑦, 𝑧 ∈ A , we have (𝑎0 𝑥𝑦𝜉|𝑧𝜉) = 𝜑(𝑧∗ 𝑥𝑦) = (𝑎0 𝑦𝜉|𝑥 ∗ 𝑧𝜉) = (𝑥𝑎0 𝑦𝜉|𝑧𝜉); hence 𝑎0 (𝑥|[A 𝜉]) = (𝑥|[A 𝜉])𝑎0 . Thus, if we denote 𝑒0 = [A 𝜉] ∈ A ′ , it follows that 𝑒0 ∘ 𝑎0 ∘ 𝑒0 ∈ A ′ . Let 𝑎′ = (𝑒0 ∘ 𝑎0 ∘ 𝑒0 )1/2 . Then 𝑎′ ∈ A ′ , 0 ≤ 𝑎′ ≤ 1, and for any 𝑥 ∈ A , we have 𝜑(𝑥) = (𝑎0 𝑥𝜉|𝜉) = (𝑎′2 𝑥𝜉|𝜉) = 𝜔𝑎′ 𝜉 (𝑥). Q.E.D. Lemma 5.20. Let 𝜑 be a positive form on the 𝐶 ∗ -algebra A and 𝑎 ∈ A . If 𝐿𝑎 𝜑 ≥ 0, then 𝐿𝑎 𝜑 ≤ ‖𝑎‖𝜑. Proof. Let 𝑥 ∈ A , 𝑥 ≥ 0. Then (𝐿𝑎 𝜑)(𝑥) = 𝜑(𝑎𝑥) = 𝜑(𝑎𝑥 1/2 𝑥 1/2 ) ≤ 𝜑(𝑎𝑥𝑎∗ )1/2 𝜑(𝑥)1/2 . Since 𝐿𝑎 𝜑 ≥ 0, it follows that 𝜑(𝑎𝑥𝑎∗ ) = (𝐿𝑎 𝜑)(𝑥𝑎∗ ) = (𝐿𝑎 𝜑)(𝑎𝑥 ∗ ) = (𝐿𝑎 𝜑)(𝑎𝑥 ∗ ) = 𝜑(𝑎2 𝑥),

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Linear forms on algebras of operators hence 𝐿𝑎2 𝜑 ≥ 0, (𝐿𝑎 𝜑)(𝑥) ≤ (𝐿𝑎2 𝜑)(𝑥)1/2 𝜑(𝑥)1/2 . We proceed analogously with the forms 𝐿𝑎 𝜑, … , 𝐿𝑎2 𝑛−1 𝜑, and by induction, we get 𝑛

𝑛

(𝐿𝑎 𝜑)(𝑥) ≤ (𝐿𝑎2𝑛 𝜑)(𝑥)1/2 𝜑(𝑥)1/2+…+1/2 𝑛

𝑛

𝑛

≤ ‖𝜑‖1/2 ‖𝑎‖‖𝑥‖1/2 𝜑(𝑥)1/2+…+1/2 . By tending to the limit for 𝑛 → ∞, it follows that (𝐿𝑎 𝜑)(𝑥) ≤ ‖𝑎‖𝜑(𝑥). Q.E.D. 5.21. The following result, due to S. Sakai, is an extension of the Radon–Nikodym theorem from measure theory; it is very important in itself and also because of its applications. Theorem (of Radon–Nikodym type). Let 𝜑 and 𝜓 be two normal forms on the von Neumann algebra M ⊂ B (H ), such that 𝜑 ≤ 𝜓. Then there exists an 𝑎 ∈ M uniquely determined by the properties 0 ≤ 𝑎 ≤ 1, s(𝑎) ≤ s(𝜓), 𝜑 = 𝐿𝑎 𝑅𝑎 𝜓. Proof. Without any loss of generality, we can assume that s(𝜓) = 1. Then, because of Proposition 5.18, we can assume that 𝜓 = 𝜔𝜉 , with a suitable vector 𝜉 ∈ H . Since s(𝜓) = 1, we have 𝑥 ∈ M , 𝑥𝜉 = 0 ⇒ 𝑥 = 0, [M ′ 𝜉] = H . From Lemma 5.19, there exists an 𝑎′ ∈ M ′ , 0 ≤ 𝑎′ ≤ 1, such that 𝜑 = 𝜔𝑎′ 𝜉 . We now consider on M ′ the forms 𝜑′ and 𝑓 ′ , determined by the relations: 𝜑′ (𝑥′ ) = (𝑥 ′ 𝜉|𝜉), 𝑓 ′ (𝑥′ ) = (𝑥 ′ 𝜉|𝑎′ 𝜉). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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From the polar decomposition theorem (5.16; see also exercise E.5.10), it follows that there exists a normal form 𝑔′ on M ′ and a partial isometry 𝑣′ ∈ M ′ , such that 𝑓 ′ = 𝐿𝑣 ′ 𝑔 ′ , 𝑔′ = 𝐿𝑣′∗ 𝑓 ′ . Thus, we have 𝑔′ = 𝐿𝑣′∗ 𝐿𝑎′ 𝜑′ = 𝐿𝑎′ 𝑣′∗ 𝜑′ , and with Lemma 5.20, we infer that 𝑔′ ≤ 𝜑′ = 𝜔𝜉 . If we apply again Lemma 5.19, we get a 𝑏 ∈ M , 0 ≤ 𝑏 ≤ 1, such that 𝑔′ (𝑥 ′ ) = 𝜔𝑏𝜉 (𝑥 ′ ) = (𝑥 ′ 𝜉|𝑏2 𝜉),

𝑥′ ∈ M ′ .

Let us denote 𝑎 = 𝑏2 . For any 𝑥′ ∈ M ′ , we have (𝑥′ 𝜉|𝑎𝜉) = (𝐿𝑎′ 𝑣′∗ 𝜑′ )(𝑥 ′ ) = (𝑎′ 𝑣′∗ 𝑥 ′ 𝜉|𝜉) = (𝑥 ′ 𝜉|𝑣 ′ 𝑎′ 𝜉), (𝑥′ 𝜉|𝑎′ 𝜉) = 𝑓 ′ (𝑥′ ) = 𝑔′ (𝑣 ′ 𝑥′ ) = 𝜑′ (𝑎′ 𝑣′∗ 𝑣′ 𝑥 ′ ) = (𝑥 ′ 𝜉|𝑣 ′∗ 𝑣′ 𝑎′ 𝜉). Since [M 𝜉] = H , it follows that 𝑎𝜉 = 𝑣 ′ 𝑎′ 𝜉, 𝑎′ 𝜉 = 𝑣 ′∗ 𝑣′ 𝑎′ 𝜉. Hence, for any 𝑥 ∈ M , we have 𝜑(𝑥) = (𝑥𝑎′ 𝜉|𝑎′ 𝜉) = (𝑥𝑣 ′∗ 𝑣′ 𝑎′ 𝜉|𝑎′ 𝜉) = (𝑥𝑣 ′ 𝑎′ 𝜉|𝑣 ′ 𝑎′ 𝜉) = (𝑥𝑎𝜉|𝑎𝜉) = 𝜔𝜉 (𝑎𝑥𝑎) = 𝜓(𝑎𝑥𝑎), and the existence part of the theorem is proved. In order to prove the uniqueness, let 𝑎, 𝑏 ∈ M , 0 ≤ 𝑎, 𝑏 ≤ 1, be such that for any 𝑥 ∈ M , we have 𝜔𝜉 (𝑎𝑥𝑎) = 𝜔𝜉 (𝑏𝑥𝑏). Since ‖𝑥𝑎𝜉‖2 = 𝜔𝜉 (𝑎𝑥 ∗ 𝑥𝑎) = 𝜔𝜉 (𝑏𝑥 ∗ 𝑥𝑏) = ‖𝑥𝑏𝜉‖2 , we can define a partial isometry 𝑢′ ∈ M ′ by the relation 𝑢′ (𝑥𝑎𝜉) = 𝑥𝑏𝜉. We now consider the following normal forms on M ′ 𝑔𝑎′ (𝑥′ ) = (𝑥 ′ 𝑎𝜉|𝜉), 𝑔𝑏′ (𝑥′ ) = (𝑥 ′ 𝑏𝜉|𝜉),

𝑥′ ∈ M ′ , 𝑥′ ∈ M ′ .

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Then we have 𝑔𝑏′ (𝑥′ ) = (𝑥 ′ 𝑢′ 𝑎𝜉|𝜉) = (𝑅ᵆ′ 𝑔𝑎′ )(𝑥 ′ ), 𝑢′∗ 𝑢′ = s(𝑔𝑎′ ). From the uniqueness of the polar decomposition of 𝑔𝑎′ , it follows that the partial isometry 𝑢′ maps identically [M 𝑎𝜉] onto [M 𝑎𝜉]. In particular, we have 𝑎𝜉 = 𝑢′ 𝑎𝜉 = 𝑏𝜉, whence 𝑎 = 𝑏. Q.E.D. 5.22. As applications of the Radon–Nikodym-type theorem, in the following sections, we present some fundamental results that are essentially due to J. von Neumann. These results are at the basis of the subsequent theory and are themselves Radon–Nikodym-type theorems. Let M ⊂ B (H ) be a fixed von Neumann algebra and 𝜉 ∈ H . The restriction of the form 𝜔𝜉 (1.3) to M will be denoted also by 𝜔𝜉 , whereas the restriction of the form 𝜔𝜉 to M ′ will be denoted by 𝜔𝜉′ . With the notations already introduced in Section 3.8, we have the following relations, which can be easily verified: z(𝑝𝜉 ) = z(𝑝𝜉′ ) = [M M ′ 𝜉], s(𝜔𝜉 ) = s(𝜔𝜉 |M ) = 𝑝𝜉 , s(𝜔𝜉′ ) = s(𝜔𝜉 |M ′ ) = 𝑝𝜉′ . Lemma. Let 𝜑 be a normal form on the von Neumann algebra M ⊂ B (H ), and 𝜉 ∈ M . If 𝜑 ≥ 𝜔𝜉 and s(𝜑) = 𝑝𝜉 , then there exists an 𝜂 ∈ H , such that 𝜑 = 𝜔𝜂 and 𝑝𝜂′ = 𝑝𝜉′ . Proof. Since 𝜑 ≥ 𝜔𝜉 , and from the Radon–Nikodym-type Theorem 5.21, there exists an 𝑎 ∈ M , 0 ≤ 𝑎 ≤ 1, such that 𝜔𝜉 = 𝐿𝑎 𝑅𝑎 𝜑, s(𝑎) = s(𝜑). Since s(𝜔𝜉 ) = s(𝜑), it follows that s(𝑎) = s(𝜑) = 𝑝𝜉 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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From Corollary 2.22, there exists a sequence {𝑒𝑛 } ⊂ P M , such that 𝑒𝑛 and 𝑎 commute and 1 𝑎𝑒𝑛 ≥ 𝑒𝑛 , 𝑛 𝑒𝑛 ↑ s(𝑎). Then 𝑎𝑒𝑛 is invertible in 𝑒𝑛 M 𝑒𝑛 , hence there exists an 𝑥𝑛 ∈ 𝑒𝑛 M 𝑒𝑛 , 𝑥𝑛 ≥ 0, such that 𝑎𝑥𝑛 = 𝑒𝑛 . We define 𝜂𝑛 = 𝑥𝑛 𝜉. Then, for any 𝑛 ≥ 𝑚, we have ‖𝜂𝑛 − 𝜂𝑚 ‖2 = ((𝑥𝑛 − 𝑥𝑚 )𝜉|(𝑥𝑛 − 𝑥𝑚 )𝜉) = 𝜔𝜉 ((𝑥𝑛 − 𝑥𝑚 )2 ) = 𝜑(𝑎(𝑥𝑛 − 𝑥𝑚 )2 𝑎) = 𝜑(𝑒𝑛 − 𝑒𝑚 ), and since 𝜑(𝑒𝑛 ) converges to 𝜑(s(𝑎)), it follows that {𝜂𝑛 } is a Cauchy sequence. Let 𝜂 = lim𝑛→∞ 𝜂𝑛 . For any 𝑥 ∈ M , we have 𝜑(𝑥) = 𝜑(s(𝑎)𝑥s(𝑎)) = lim 𝜑(𝑒𝑛 𝑥𝑒𝑛 ) = lim 𝜑(𝑎𝑥𝑛 𝑥𝑥𝑛 𝑎) 𝑛→∞

𝑛→∞

= lim (𝑥𝑥𝑛 𝜉|𝑥𝑛 𝜉) = (𝑥𝜂|𝜂) = 𝜔𝜂 (𝑥), 𝑛→∞

hence 𝜑 = 𝜔𝜂 . On the other hand, we have 𝜂𝑛 ∈ M 𝜉 for any 𝑛, hence 𝜂 ∈ [M 𝜉], and therefore, 𝑝𝜂′ ≤ 𝑝𝜉′ . Conversely, we have 𝑎𝜂 = lim 𝑎𝑥𝑛 𝜉 = lim 𝑒𝑛 𝜉 = s(𝑎)𝜉 = 𝑝𝜉 (𝜉) = 𝜉; 𝑛→∞

𝑛→∞

i.e., 𝜉 ∈ M 𝜂, whence 𝑝𝜉′ ≤ 𝑝𝜂′ .

Q.E.D.

Theorem 5.23. Let 𝜓 be a normal form on the von Neumann algebra M ⊂ B (H ) and 𝜉 ∈ H . If s(𝜓) ≤ 𝑝𝜉 , then there exists an 𝜂 ∈ [M 𝜉] ∩ [M ′ 𝜉], such that 𝜓 = 𝜔𝜂 . Moreover, if s(𝜓) = 𝑝𝜉 , then there exists an 𝜂 ∈ H , such that 𝜓 = 𝜔𝜂 and 𝑝𝜂′ = 𝑝𝜉′ . Proof. Let 𝜑 = 𝜓 + 𝜔𝜉 . By virtue of Lemma 5.22, there exists an 𝜂0 ∈ H , such that 𝜑 = 𝜔𝜂0 and 𝑝𝜂′ 0 = 𝑝𝜉′ . Since 𝜓 ≤ 𝜑, from the Radon–Nikodym-type theorem, there exists an 𝑎 ∈ M , 0 ≤ 𝑎 ≤ 1, such that 𝜓 = 𝐿𝑎 𝑅𝑎 𝜑, s(𝑎) ≤ s(𝜑) = 𝑝𝜉 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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Linear forms on algebras of operators Let us define 𝜂 = 𝑎𝜂0 . Then, for any 𝑥 ∈ M , we have 𝜓(𝑥) = 𝜔𝜂0 (𝑎𝑥𝑎) = 𝜔𝜂 (𝑥),

i.e., 𝜓 = 𝜔𝜂 . Since 𝜂 = 𝑎𝜂0 ∈ M 𝜂0 , it follows that 𝜂 ∈ [M 𝜉]. On the other hand, we have 𝜂 = 𝑎𝜂0 ∈ s(𝑎)H ⊂ 𝑝𝜉 H = [M ′ 𝜉]. Let us now assume that s(𝜓) = 𝑝𝜉 . Then, with the preceding notations, it follows that s(𝑎) = 𝑝𝜂0 . By taking into account Corollary 2.22, we find two sequences {𝑒𝑛 } ⊂ P M and {𝑥𝑛 } ⊂ 𝑒𝑛 M 𝑒𝑛 , 𝑥𝑛 ≥ 0, such that 𝑎𝑥𝑛 = 𝑥𝑛 𝑎 = 𝑒𝑛 , 𝑒𝑛 ↑ s(𝑎). We denote 𝜂𝑛 = 𝑥𝑛 𝜂 = 𝑥𝑛 𝑎𝜂0 = 𝑒𝑛 𝜂0 . Since 𝜂𝑛 = 𝑥𝑛 𝜂 ∈ M 𝜂, we infer that 𝜂0 = 𝑝𝜂0 (𝜂0 ) = s(𝑎)(𝜂0 ) = lim 𝑒𝑛 𝜂0 ∈ [M 𝜂]. 𝑛→∞

Thus, we have 𝑝𝜂′ 0 ≤ 𝑝𝜂′ . But 𝑝𝜂′ 0 = 𝑝𝜉′ , hence 𝑝𝜉′ ≤ 𝑝𝜂′ . On the other hand, it is obvious that 𝑝𝜂′ ≤ 𝑝𝜉′ . Consequently, we have 𝑝𝜂′ = 𝑝𝜉′ . Q.E.D. Corollary 5.24. Let M ⊂ B (H ) be a von Neumann algebra with the property that there exists a vector 𝜉 ∈ H , which is separating for M . Then, for any normal (resp., normal and faithful) form 𝜑 on M , there exists a vector 𝜂 ∈ H , such that 𝜑 = 𝜔𝜂 (resp., 𝜑 = 𝜔𝜂 and 𝑝𝜂′ = 𝑝𝜉′ ). Corollary 5.25. Let M 1 ⊂ B (H 1 ), M 2 ⊂ B (H 2 ) be von Neumann algebras, such that there exist vectors 𝜉1 ∈ H 1 , 𝜉2 ∈ H 2 , which are cyclic and separating for M 1 (resp., M 2 ). For any *-isomorphism 𝜋 ∶ M 1 → M 2 , there exists a unitary operator 𝑢 ∶ H 1 → H 2 such that, for any 𝑥1 ∈ M 1 , we have 𝜋(𝑥1 ) = 𝑢 ∘ 𝑥1 ∘ 𝑢∗ . In particular, 𝜋 is wo-continuous. Proof. We define a normal form 𝜑 on M 1 by 𝜑(𝑥1 ) = 𝜔𝜉2 (𝜋(𝑥1 )),

𝑥1 ∈ M 1 .

Then 𝜑 is faithful, because 𝜉2 is separating for M 2 , whereas 𝜋 is a *-isomorphism. From Corollary 5.24, it follows that there exists a vector 𝜂1 , which is cyclic for M 1 , such that 𝜑 = 𝜔𝜂1 . But in this case, 𝜂1 is also separating for M 1 because 𝜑 is faithful. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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We now define a linear mapping 𝑢0 ∶ M 1 𝜂1 → M 2 𝜂2 by 𝑢0 (𝑥1 𝜂1 ) = 𝜋(𝑥1 )𝜂2 . This mapping is isometric: ‖𝑥1 𝜂1 ‖2 = 𝜔𝜂1 (𝑥1∗ 𝑥1 ) = 𝜑(𝑥1∗ 𝑥1 ) = 𝜔𝜉2 (𝜋(𝑥1∗ 𝑥1 )) = ‖𝜋(𝑥1 )𝜉2 ‖2 . Since the vectors 𝜂1 , 𝜉2 are cyclic, respectively, for M 1 , M 2 , it follows that the mapping 𝑢0 can be uniquely extended, by continuity, to a unitary operator 𝑢 ∶ H 1 → H 2 . For any 𝑥1 , 𝑦1 ∈ M 1 , we have 𝜋(𝑥1 )𝜋(𝑦1 )𝜉2 = 𝜋(𝑥1 𝑦1 )𝜉2 = 𝑢0 𝑥1 𝑦1 𝜂1 = (𝑢0 𝑥1 )(𝑦1 𝜂1 ) = 𝑢0 𝑥1 𝑢0−1 𝜋(𝑦1 )𝜉2 ; hence 𝜋(𝑥1 ) = 𝑢 ∘ 𝑥1 ∘ 𝑢−1 .

Q.E.D. EXERCISES

In the exercises in which the symbols M ⊂ B (H ) are not explained, they will denote a von Neumann algebra M that acts on the Hilbert space H . E.5.1 Let A be a 𝐶 ∗ -algebra and 𝜑 a bounded form on A . If there exists an 𝑎 ∈ A , 0 ≠ 𝑎 ≥ 0, such that 𝜑(𝑎) = ‖𝜑‖‖𝑎‖, then 𝜑 is positive. !E.5.2 Let A ⊂ B (H ), A ∋ 1, be a *-algebra. If 𝜉, 𝜂 ∈ H and if 𝜔𝜉,𝜂 |A is positive, then there exists a 𝜁 ∈ H , such that 𝜔𝜉,𝜂 |A = 𝜔𝜁 |A . Infer that if 𝜑 is a positive wo-continuous form, then there exist 𝜉1 , … , 𝜉𝑛 ∈ H , such that 𝑛 𝜑 = ∑𝑘=1 𝜔𝜉𝑘 |A . E.5.3 Let 𝜑 be a normal form on M and {𝑥𝑖 } a net in the closed unit ball of M . Then we have 𝑠𝑜 𝜑(𝑥𝑖∗ 𝑥𝑖 ) → 0 ⇔ 𝑥𝑖 s(𝜑) → 0. E.5.4 Let 𝜑, 𝜓 be normal on M . Then s(𝜓) ≤ s(𝜑) iff on the closed unit ball of M the topology determined by the seminorm 𝑥 ↦ 𝜑(𝑥 ∗ 𝑥)1/2 is stronger than the topology determined by the seminorm 𝑥 ↦ 𝜓(𝑥 ∗ 𝑥)1/2 . !E.5.5 On a von Neumann algebra, one considers the 𝑠-topology given by the semi-norms 𝑠𝜑 (𝑥) = 𝜑(𝑥 ∗ 𝑥)1/2 , 𝑥 ∈ M , where 𝜑 runs over all normal forms on M . Show that if 𝜑0 is a normal form on M , such that s(𝜑0 ) = 1, then, on the closed unit ball of M , the 𝑠-topology is determined by the norm 𝑠𝜑0 . ▶

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!E.5.6 A projection 𝑒 ∈ M is of countable type iff it is the support of a normal form on M . In particular, M is of countable type if there exists a normal form 𝜑0 on M , such that s(𝜑0 ) = 1. E.5.7 Prove the following implications: H is separable ⇒ the predual M ∗ of M is separable ⇒ M is of countable type ⇒ the closed unit ball of M is 𝑠-metrizable. E.5.8 Show that, for any net {𝑥i } ⊂ M , one has 𝑠

w

𝑥i → 0 ⇒ 𝑥i∗ 𝑥i → 0. Infer that on the closed unit ball of M the 𝑠-topology coincides with so-topology. Show that for any form 𝜑 on M , 𝜑 is s-continuous ⇔ 𝜑 is w-continuous. E.5.9 Let 𝜑 be a normal form on M . For any 𝑒 ∈ P M , 𝑒 ≠ 0, there exist 𝑓 ∈ P M , 0 ≠ 𝑓 ≤ 𝑒 and 𝜉 ∈ M , such that 𝜑(𝑓𝑥𝑓) = 𝜔𝜉 (𝑥),

𝑥 ∈ M.

!E.5.10 Let 𝜑 be a w-continuous form on M and 𝜑 = 𝑅𝑣 |𝜑| its polar decomposition. Then the polar decomposition of the form 𝜑∗ is 𝜑∗ = 𝑅𝑣∗ |𝜑∗ |,

|𝜑∗ | = 𝐿𝑣∗ 𝑅𝑣 |𝜑|.

In particular, 𝜑 = 𝐿𝑣 |𝜑∗ |,

𝑣𝑣 ∗ = s(|𝜑∗ |).

E.5.11 Let 𝜑 be a w-continuous form on M . Then |𝜑| is the unique normal form 𝜓 on M with the properties ‖𝜓‖ = ‖𝜑‖, |𝜑(𝑥)|2 ≤ ‖𝜑‖𝜓(𝑥𝑥 ∗ ), 𝑥 ∈ M . E.5.12 Let 𝜑, 𝜓 be w-continuous forms on M . Then, for any 𝑥 ∈ M , we have ||𝜑 + 𝜓|(𝑥)|2 ≤ (‖𝜑‖ + ‖𝜓‖)(|𝜑|(𝑥𝑥 ∗ ) + |𝜓|(𝑥𝑥 ∗ )). E.5.13 Let 𝜑 be a normal form on M and 𝑎 ∈ M . Then |𝐿𝑎 𝜑| ≤ ‖𝑎‖𝜑. ▶

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E.5.14 Let 𝜑 be a w-continuous form on M and 𝑒 ∈ P M . Show that 𝐿𝑒 𝜑 = 𝜑 ⇔ ‖𝐿𝑒 𝜑‖ = ‖𝜑‖. !E5.15 Let 𝜑, 𝜓 be normal forms on M . Show that s(𝜑)s(𝜓) = 0 ⇒ ‖𝜑 − 𝜓‖ = ‖𝜑‖ + ‖𝜓‖. E.5.16 Any normal form on the von Neumann algebra M ⊂ B (H ) extends to a normal form on B (H ). Infer that any w-continuous form 𝜑 on M extends to a w-continuous form 𝜑̃ on B (H ), such that ‖𝜑‖̃ = ‖𝜑‖. E.5.17 Let M 1 , M 2 be von Neumann algebras and Φ ∶ M 1 → M 2 a bounded linear mapping. Then Φ is w-continuous iff Φ is completely additive. *E.5.18 Show that the assertions (i) and (ii) from Theorem 5.14 are equivalent to the following assertion: (iii) There exists a normal form 𝜑0 on M , such that for any 𝜀 > 0, there exists a 𝛿 > 0, with the property: 𝑥 ∈ M,

‖𝑥‖ ≤ 1,

𝜑0 (𝑥∗ 𝑥 + 𝑥𝑥 ∗ ) ≤ 𝛿 ⇒ |𝜑(𝑥)| ≤ 𝜀, for any 𝜑 ∈ F .

E.5.19 Let M ∗ be the predual of the von Neumann algebra M . If F is a 𝜎(M ∗ , M )-relatively compact part of M + ∗ = {𝜑 ∈ M ∗ ; 𝜑 ≥ 0}, then the set {𝐿𝑥 𝜑; 𝜑 ∈ F , 𝑥 ∈ M , ‖𝑥‖ ≤ 1} is also 𝜎(M ∗ , M )-relatively compact. E.5.20 Produce an example in order to show that there exist 𝜎(M ∗ , M )-relatively compact parts F ⊂ M ∗ , such that the set {|𝜑|; 𝜑 ∈ F } is not 𝜎(M ∗ , M )-relatively compact. E.5.21 Show that the following assertions are equivalent: (i) Any normal form on M is of the type 𝜔𝜉 , 𝜉 ∈ H . (ii) For any 𝑒 ∈ P M of countable type, M 𝑒 ⊂ B (H ) has a separating vector. E.5.22 Show that two *-isomorphic maximal abelian von Neumann algebras are spatially isomorphic. E.5.23 Let 𝜑, 𝜓 be two completely additive positive forms on the von Neumann algebra M , 𝑝 ∈ M a non-zero projection and assume that 𝜑(𝑝) ≤ 𝜓(𝑝). Show that there is a non-zero projection 𝑒 ∈ M , 𝑒 ≤ 𝑝 such that 𝜑(𝑥) ≤ 𝜓(𝑥),

(∀)𝑥 ∈ M , 0 ≤ 𝑥 ≤ 𝑒. ▶

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E.5.24 Let 𝜑, 𝜓 be two completely additive positive forms on the von Neumann algebra M , 𝜑 ≠ 0, 𝜓 ≠ 0. For any non-zero projection 𝑝 ∈ M and any 𝜀 > 0, there exist a non-zero projection 𝑒 ∈ M , 𝑒 ≤ 𝑝, and 𝜃 > 0, such that (∀)𝑓 ∈ P M , 𝑓 ≤ 𝑒.

𝜃𝜑(𝑓) ≤ 𝜓(𝑓) ≤ 𝜃(1 + 𝜀)𝜑(𝑓),

[Hint: Assume 𝜑(1) = 𝜓(1) and (using E.5.23) assume that 𝜑(𝑓) ≤ 𝜓(𝑓),

(∀)𝑓 ∈ P M , 𝑓 ≤ 𝑝.

Then define 𝜃 = sup{𝜆 > 0; 𝜆𝜑(𝑓) ≤ 𝜓(𝑓), (∀)𝑓 ∈ P M , 𝑓 ≤ 𝑝}. Apply E.5.23 again to find 0 ≠ 𝑒 ∈ P M , 𝑒 ≤ 𝑝, such that (∀)𝑓 ∈ P M , 𝑓 ≤ 𝑒.]

𝜓(𝑓) ≤ 𝜃(1 + 𝜀)𝜑(𝑓),

COMMENTS C.5.1. Besides the uniform topology (i.e., the norm topology) and the topologies wo, w, so, on a von Neumann algebras M , one also considers the following topologies: the s-topology, given by the seminorms 𝑠𝜑 (𝑥) = 𝜑(𝑥 ∗ 𝑥)1/2 ,

𝑥 ∈ M,

where 𝜑 runs over the set of all normal forms on M (see E.5.5); the 𝑠∗ -topology, given by the seminorms 𝑠𝜑 (𝑥) = 𝜑(𝑥 ∗ 𝑥)1/2 ,

𝑥 ∈ M,

∗ 𝑠𝜑 (𝑥)

𝑥 ∈ M,

∗

= 𝜑(𝑥 𝑥)

1/2

,

where 𝜑 runs over the set of all normal forms on M ; the 𝜏-topology = 𝜏(M ; M ∗ ), i.e., the Mackey topology associated with the topology w = 𝜍(M ; M ∗ ); this topology is given by the seminorms 𝑃M (𝑥) = sup |𝜑(𝑥)|, 𝜑∈K

𝑥 ∈ M,

where K runs over the set of all 𝜍(M ∗ ; M )-compact, convex, equilibrated subsets of M ∗ , and it is the finest locally convex topology on M , which determines the same set of linear, continuous forms on M , as the w-topology (see N. Bourbaki, Espaces vectoriels topologiques).

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Lectures on von Neumann algebras The general relations existing between these topologies are represented in the following diagram: (1) w𝑜 ≤ 𝑠𝑜 (2)∧

∧(3)

w ≤ 𝑠

≤

(4)

(5)

𝑠∗

≤

𝜏.

(6)

Relations (1) and (2) are obvious from the definitions of the corresponding topologies (see 1.3 and 1.10); relations (3) and (4) easily follow from Proposition 5.3 and E.5.8; relation (5) is trivial, whereas relation (6) follows from E.5.8, if we observe that the *-operation is 𝜏-continuous. As far as the restrictions of these topologies to the closed unit ball of M are concerned (denoted below by the subscript 1 to the corresponding symbol of the topology), we have the following relations: w𝑜1 ≤ 𝑠𝑜1 (𝑎)‖

‖(𝑏)

w1 ≤ 𝑠1

≤

𝑠1∗

=

𝜏1 .

(𝑐) Equality (a) has already been established (1.3 and 1.10), equality (b) follows from (a) with the help of E.5.8, whereas equality (c) is proved by C. A. Akemann (1967). The cases in which the equalities w𝑜 = w, 𝑠𝑜 = 𝑠 hold are discussed in Chapter 8. For other results concerning topologies on von Neumann algebras, we refer to J. F. Aarnes (1968), C. A. Akemann (1967), S. Sakai (1957, 1965), P. C. Shields (1959). C.5.2 One calls a derivation of an algebra A any linear mapping 𝜗 ∶ A → A , such that 𝜗(𝑥𝑦) = 𝑥𝜗(𝑦) + 𝜗(𝑥)𝑦, 𝑥, 𝑦 ∈ A . Any element 𝑎 ∈ A determines an inner derivation 𝜗𝑎 ∶ A ∋ 𝑥 ↦ 𝑎𝑥 − 𝑥𝑎 ∈ A . The study of the derivations of algebras of operators has been started by I. Kaplansky (1958), who proved that any derivation of a von Neumann algebra of type I is inner (aided by the fact that any derivation of a commutative Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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139

𝐶 ∗ -algebra is identically zero, a result due to I. M. Singer) and made the conjecture that any derivation of a 𝐶 ∗ -algebra is uniformly continuous. This conjecture has been positively solved by S. Sakai (1960), and afterwards, B. E. Johnson and A. M. Sinclair (1968) showed that any derivation of a semi-simple Banach algebra is continuous. With the help of the results of I. Kaplansky and S. Sakai, already mentioned, R. V. Kadison (1966) showed that any derivation of a 𝐶 ∗ -algebra A ⊂ B (H ) is wo-continuous and extends to an inner derivation of B (H ); S. Sakai (1966), with the help of Kadison’s result, proved the following theorem: Theorem 1. Any derivation of a von Neumann algebra is inner. Other proofs of this theorem have been given by B. E. Johnson and J. R. Ringrose (1969), and by W. B. Arveson (1974); D. Olesen (1974) has shown that any derivation of an 𝐴𝑊 ∗ -algebra (see C.3.1) is inner by extending the arguments of W. B. Arveson. On the other hand, S. Sakai (1971) has shown that any derivation of a simple 𝐶 ∗ -algebra with the unit element is inner, whereas other results in this direction have been obtained by S. Sakai (1971, 1973, 1976), D. Olesen and G. K. Pedersen and C. A. Akemann, G. A. Elliott, G. K. Pedersen and J. Tomiyama (1976), and others. As an extension of the study of the derivations, the theory of the cohomology of algebras of operators and of general Banach algebras was also developed: R. V. Kadison and J. R. Ringrose (1967), B. E. Johnson, R. V. Kadison and J. R. Ringrose (1972), B. E. Johnson (1972) and I. C. Craw (1972, 1973). Along the study of the derivations, significant results were obtained in the theory of automorphisms of algebras of operators, for which we refer the reader to the works of R. V. Kadison and J. R. Ringrose (1966, 1967, 1974) and H. J. Borchers (1966). We recall that any *-isomorphism between two von Neumann algebras is w-continuous, hence s-continuous (5.13). In connection with the continuity of the algebraic isomorphisms, we mention the following result of T. Okayasu (1968), which we state in the form given by S. Sakai (1971): Theorem 2. Let Φ ∶ A → B be an algebraic isomorphism of 𝐶 ∗ -algebras. Then there exists a derivation 𝜗 of A and a *-isomorphism Ψ ∶ A → B , such that Φ = Ψ exp(𝜗). In particular, one infers from this result that any pair of algebraically isomorphic 𝐶 ∗ -algebras is *-isomorphic (Gardner 1965) and that any Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:57:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.008

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Lectures on von Neumann algebras algebraic isomorphism between two 𝐶 ∗ -algebras is uniformly (i.e. norm) continuous (Rickart 1960). On the other hand, from Theorems 1 and 2, one obtains the following result: Corollary. Let Φ ∶ M → N be an algebraic isomorphism between von Neumann algebras. Then there exists an invertible positive element 𝑎 ∈ M and a *-isomorphism Ψ ∶ M → N , such that Φ(𝑥) = Ψ(𝑎𝑥𝑎−1 ),

𝑥 ∈ M.

Thus, any algebraic isomorphism between von Neumann algebras is continuous for the topologies w and 𝑠. Let 𝜗 be a derivation of a von Neumann algebra M . By virtue of Theorem 1, there exists an 𝑎 ∈ M , such that 𝜗 = 𝜗𝑎 . In connection with the selection of the element 𝑎, one knows that there exists a unique 𝑎𝜗 ∈ M , 𝜗 = 𝜗𝑎𝜗 , such that, for any central projection 𝑝 ∈ M , one has ‖𝑝𝜗‖ = 2‖𝑝𝑎𝜗 ‖. In particular, we have ‖𝜗‖ = 2 inf{‖𝑎‖;

𝜗 = 𝜗𝑎 }.

This result has been obtained by J. G. Stampfli (1970), for the case in which M = B (H ), by P. Gajendragadkar (1972), for the case M = von Neumann algebra with a separable predual, and by L. Zsidó (1973), for the general case. For other information concerning the norm of the derivations, we refer the reader to C. Apostol and L. Zsidó (1973). The study of isometries between von Neumann algebras has been carried out by R. V. Kadison (1951, 1956), who obtained the following result: Theorem 3. Let Φ ∶ A → B be a linear isomorphism between two 𝐶 ∗ algebras with the unit element, such that Φ(1) = 1. The following assertions are equivalent: (i) Φ is an isometry; (ii) For any 𝑎 ∈ A ℎ , one has Φ(𝑎) ∈ B ℎ and Φ(𝑎𝑛 ) = Φ(𝑎)𝑛 , 𝑛 ∈ ℕ; (iii) For any 𝑥 ∈ M , one has: 𝑥 ≥ 0 ⇔ Φ(𝑥) ≥ 0. If A and B are von Neumann algebras, then the preceding assertions are equivalent to the following one: (iv) There exist central projections 𝑝 ∈ A , 𝑞 ∈ B , such that Φ induces a *-isomorphism of A 𝑝 onto B 𝑞, and a *-anti-isomorphism of A (1 − 𝑝) onto B (1 − 𝑞).

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Condition (ii) is equivalent to the fact that Φ is a Jordan *-isomorphism, i.e., Φ commutes with the *-operation and conserves the ‘Jordan product’, 1 𝑎 ∘ 𝑏 = (𝑎𝑏 + 𝑏𝑎). Condition (iii) is equivalent to the fact that Φ is a 2 bipositive linear isomorphism, such that Φ(1) = 1. Thus, the linear isometries that map 1 to 1, the bipositive linear isomorphisms that map 1 to 1 and the Jordan *-isomorphisms are equivalent notions, whereas in the case of von Neumann algebras, they are characterized by condition (iv) in terms of *-isomorphisms and *-anti-isomorphisms. Consequently, Theorem 3 is a general, non-commutative, extension of the Banach–Stone theorem (see Dunford and Schwarz [1958/1963/1970], Chap. V, 8.8). R. V. Kadison’s proof is based on the study of the extreme points of the closed unit ball and on some older results of N. Jacobson and C. E. Rickart (1950). Another proof of the equivalence (i) ⇔ (ii), based on the notion of numerical range, was obtained by A. L. T. Paterson (1969), whereas extensions of the theorem to 𝐶 ∗ -algebras without the unit element were given by L. A. Harris (1969) and A. L. T. Paterson and A. M. Sinclair (1972). Other results concerning the Jordan structure of 𝐶 ∗ -algebras are contained in some papers by E. Størmer, D. M. Topping et al. With the help of the above theorem of Kadison and of the Tomita theory, A. Connes (1974b) obtained the characterization of von Neumann algebras as ordered linear spaces. A linear mapping Φ ∶ A → B between the 𝐶 ∗ -algebras A and B is said to be 𝑛-positive, 𝑛 ∈ ℕ, if the natural extension Φ𝑛 ∶ Mat𝑛 (A ) → Mat𝑛 (B ) is a positive mapping. If Φ is 𝑛-positive, for any 𝑛 ∈ ℕ, then Φ is called completely positive. The fundamental result concerning the completely positive mappings is the following theorem of W. F. Stinespring (1955): Theorem 4. For any linear mapping Φ ∶ A → B (H ), the following assertions are equivalent: (i) Φ is completely positive; (ii) There exists a *-representation 𝜋 ∶ A → B (K ) and a bounded operator 𝑣 ∶ H → K , such that Φ(𝑎) = 𝑣 ∗ 𝜋(𝑎)𝑣,

𝑎 ∈ A.

Corollary. If Φ ∶ A → B (H ) is completely positive, then |Φ(𝑎)|2 = Φ(|𝑎|)2 ,

𝑎 ∈ A.

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142

Lectures on von Neumann algebras If either A or B is commutative, then any positive linear mapping Φ ∶ A → B is completely positive. Thus, from the preceding corollary it trivially follows the ‘Schwarz-type inequality’ of R. V. Kadison (1956). For the study of positive and completely positive mappings, we refer to W. B. Arveson (1972), R. V. Kadison (1951, 1956), E. Størmer (1963). A detailed analysis of the 𝑛-positive mappings can be found in Man-Duen Choi (1972). Some applications of the completely positive mappings to the theory of operators and algebras of operators can be found in W. B. Arveson (1972), I. Suciu (1973) and L. Zsidó (1973c). Linear mappings between 𝐶 ∗ -algebras, which map unitary elements to unitary elements, have been studied by B. Russo and H. A. Dye (1966) and B. Russo (1966).

C.5.3 We recall (C.2.1) that the 𝐶 ∗ -algebras of operators possess an axiomatic description. The following theorem of S. Sakai (1956) (see also [1962, 1971]) allows an axiomatic description of von Neumann algebras: Theorem. A 𝐶 ∗ -algebra M is *-isomorphic to a von Neumann algebra iff it is the dual of a Banach space. A proof of this theorem has also been obtained by J. Tomiyama (1959). One calls a 𝑊 ∗ -algebra any 𝐶 ∗ -algebra, which is the dual of a Banach space. On account of the preceding theorem, von Neumann algebras are also called concrete 𝑊 ∗ -algebras. The proof of the preceding theorem required that some results, already known for von Neumann algebras, be obtained by non-spatial arguments. These are essentially developed methods, used in the theory of abstract 𝐶 ∗ -algebras, the compactness of the closed unit balls in 𝑊 ∗ -algebras and the ̆ Krein–Smulian theorem (C.1.1). Among the abstract 𝐶 ∗ -algebra techniques, we mention the ‘Arens trick’, presented in the proof of Lemma 2.5. A characteristic sample is the proof of Theorem 5.16. The advantage of this technique lies in the invariance with respect to *-isomorphisms of the results obtained with its help. Not incidentally, all the results, which are invariant with respect to *-isomorphisms, have proofs of this nature. The book of S. Sakai (1971) is an excellent exposition of the theory based on these ideas. If A is a 𝐶 ∗ -algebra and 𝜑 ∈ A ∗ , 𝜑 ≥ 0, we have already defined the representation 𝜋𝜑 ∶ A → B (H 𝜑 ), of A into B (H 𝜑 ) (5.18). If we denote by 𝜋 ∶ A → B (H ) the direct sum of all representations 𝜋𝜑 , one defines the enveloping von Neumann algebra of the 𝐶 ∗ -algebras A as being the von Neumann algebra R (𝜋(A )) ⊂ B (H ) (see Dixmier [1964/1968], §12).

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The bidual A ∗∗ of the 𝐶 ∗ -algebra A can be organized, in a natural manner, as a 𝑊 ∗ -algebra, and it is *-isomorphic to the enveloping von Neumann algebra of A (see, e.g. Tomita [1967]). The multiplication that one introduces in A ∗∗ is a natural one (first defined by Arens (1951)) and called the Arens multiplication. Any continuous linear form on the 𝐶 ∗ -algebra A can be extended in a unique manner, by continuity, as a w-continuous linear form on the bidual 𝑊 ∗ -algebra A ∗∗ ; hence, (A ∗∗ )∗ = A ∗ . Any *-representation 𝜋 ∶ A → B (H ) can be extended in a unique manner to a normal *-representation ̃ ∗∗ ) ∶= R (𝜋(A )). These facts allow the 𝜋̃ ∶ A ∗∗ → B (H ) and 𝜋(A extension of some results, known for w-continuous linear forms on von Neumann algebras, to the bounded linear forms on 𝐶 ∗ -algebras. ̆ ă and L. Zsidó (1975a) there is a unified exposition of the topics In S. ¸ Stratil recorded in this section and in Section C.2.1. C.5.4 The theory of operator algebras allowed the extension of the integration theory to a non-commutative framework. The first results in this direction were obtained by J. Dixmier (1953) and I. E. Segal (1953). For further developments, we recommend the works of S. K. Berberian, T. Ogasawara and K. Yoshinaga, H. Umegaki, L. Pukánszky, E. Nelson, A. R. Padmanabhan, K. Saitô, E. Christensen, F. J. Yeadon, E. C. Lance, U. Haagerup, A. Connes and I. Cuculescu. On the other hand, extensions of a different kind of the measure theory to the 𝐶 ∗ -algebras have been obtained by G. K. Pedersen and F. Combes. An almost complete exposition of the results obtained in this direction can be ̆ ă (1973). found in S. ¸ Stratil C.5.5 Bibliographical comments. Theorem 5.11 and Corollary 5.12 are due to M. Takesaki (1958), but in our exposition, we followed the proof of B. E. Johnson, R. V. Kadison and J. R. Ringrose (1972). Theorem 5.14 is due to C. A. Akemann (1967). Theorems 5.16 and 5.21 were obtained by S. Sakai (1958, 1965), whereas Theorem 5.17 by A. Grothendieck (1957). The construction from Section 5.18, with the help of which, to any positive form 𝜑 on a 𝐶 ∗ -algebra A , one associates a ‘cyclic representation’ 𝜋𝜑 of A , is sometimes called the Gelfand–Naimark–Segal construction or, briefly, the GNS construction. Theorem 5.23 is essentially due to F. J. Murray and J. von Neumann (1936), whereas our proof follows that in the paper of J. Vowden (1967) (see also the talk by Kadison [1958]). Corollary 5.25 is due to F. J. Murray and J. von Neumann (1937) and sometimes appears in the literature under the name The Spatial Theorem of J. von Neumann.

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Lectures on von Neumann algebras Crucial results in the theory of operator algebras are Radon–Nikodym-type theorems. Under the same hypotheses as those in Theorem 5.21, S. Sakai (1962, Remark, p. 1.46; 1971, Proposition 1.24.4) has also shown that there exists an ℎ ∈ M , 0 ≤ ℎ ≤ 1, such that 𝜑=

1 (𝐿 𝜓 + 𝑅ℎ 𝜓) 2 ℎ

(see Lemma 2 in the Introduction of Chapter 10) We shall come to the Radon–Nikodym-type theorems again in C.6.1, C.6.2 and in Chapter 10. In our exposition, we also followed J. Dixmier (1957/1969), J. R. Ringrose (1972) and S. Sakai (1971).

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6

RELATIONS BETWEEN A VON NEUMANN ALGEBRA AND ITS COMMUTANT

In this chapter, we shall show that the passage to the commutant of a von Neumann algebra is an operation that conserves the type, and, afterward, we shall give two applications of this result. The following lemmas can be looked upon as being corollaries to Theorem 5.23. We shall use the notations already introduced in Sections 3.8 and 5.22. Lemma 6.1. Let M ⊂ B (H ) be a von Neumann algebra and 𝜉, 𝜂 ∈ H . If 𝑝𝜉 ≺ 𝑝𝜂 (resp., 𝑝𝜉 ∼ 𝑝𝜂 ), in M , then 𝑝𝜉′ ≺ 𝑝𝜂′ (resp., 𝑝𝜉′ ∼ 𝑝𝜂′ ) in M ′ . Proof. By virtue of the Schröder–Bernstein-type theorem (4.7), it is sufficient to prove the first assertion of the theorem. Let 𝑣 ∈ M be a partial isometry, such that 𝑣∗ 𝑣 = 𝑝𝜉 ,

𝑣𝑣 ∗ ≤ 𝑝𝜂 .

We denote 𝜂0 = 𝑣𝜉. Then 𝜉 = 𝑣 ∗ 𝜂0 ∈ M 𝜂0 . On the other hand, we have 𝜂0 ∈ [M ′ 𝜂]; hence s(𝜔𝜂0 ) ≤ 𝑝𝜂 . From Theorem 5.23, there exists a 𝜉0 ∈ [M 𝜂], such that 𝜔𝜂0 = 𝜔𝜉0 . If we define

𝑣′ (𝑥𝜉0 ) = 𝑥𝜂0 ,

𝑥 ∈ M,

and if we observe that ‖𝑥𝜉0 ‖2 = 𝜔𝜉0 (𝑥∗ 𝑥) = 𝜔𝜂0 (𝑥∗ 𝑥) = ‖𝑥𝜂0 ‖2 , it follows that we thus define a partial isometry 𝑣′ ∈ M ′ , such that 𝑣′∗ 𝑣′ = 𝑝𝜉′ 0 , Consequently, we have

𝑣 ′ 𝑣′∗ = 𝑝𝜂′ 0 .

𝑝𝜉′ ≤ 𝑝𝜂′ 0 ∼ 𝑝𝜉′ 0 ≤ 𝑝𝜂′ . Q.E.D. 145

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Corollary. Let M ⊂ B (H ) be a von Neumann algebra having a separating vector 𝜉 ∈ H and a cyclic vector 𝜂 ∈ H . Then M has a vector 𝜁 ∈ H , which is simultaneously cyclic and separating. Proof. By assumption 𝑝𝜉 = 1, 𝑝𝜂′ = 1, so that 𝑝𝜂 ≤ 1 = 𝑝𝜉 and the Lemma implies 1 = 𝑝𝜂′ ≺ 𝑝𝜉′ ≤ 1; hence 1 ∼ 𝑝𝜉′ . Now use 4.24.(3) for the commutant M ′ , the projection 1 ∈ M ′ and the cyclic projection 𝑝𝜉′ to obtain a vector 𝜁 ∈ H such that 1 = 𝑝𝜁′

and

𝑝𝜁 = 𝑝𝜉 = 1. Q.E.D.

Lemma 6.2. Let M ⊂ B (H ) be a von Neumann algebra and let 𝜉 ∈ H . Then 𝑝𝜉 is an abelian projection in M iff 𝑝𝜉′ is an abelian projection in M ′ . Proof. Let us assume, e.g., that 𝑝𝜉 is an abelian projection in M . We now consider the von Neumann algebra N = M 𝑝′ ⊂ B (𝑝𝜉′ H ), 𝜉

whose commutant is

N ′ = M ′𝑝′ ⊂ B (𝑝𝜉′ H ). 𝜉

We must thus show that N ′ is abelian. For any 𝜂 ∈ 𝑝𝜉′ H , we denote 𝑞𝜂 = [N ′ 𝜂] ∈ N ,

𝑞𝜂′ = [N 𝜂] ∈ N ′ .

Since 𝑝𝜉 is abelian in M , the projection 𝑞𝜉 = (𝑝𝜉 )𝑝′ is abelian in N . Moreover, the vector 𝜉

𝜉 is cyclic for the abelian von Neumann algebra N 𝑞𝜉 . With exercise E.3.10, we infer that N

′ 𝑞𝜉

= N 𝑞𝜉 ;

hence N ′𝑞𝜉 is abelian. Since z(𝑝𝜉 ) = z(𝑝𝜉′ ), it is easy to see that, in the von Neumann algebra N , the central support of the projection 𝑞𝜉 is equal to the unit element. With Proposition 3.14, we infer that N ′ is *-isomorphic to N ′𝑞𝜉 ; hence N ′ is abelian. Q.E.D. Lemma 6.3. Let M ⊂ B (H ) be a von Neumann algebra and let 𝜉 ∈ H . Then 𝑝𝜉 is a finite projection in M iff 𝑝𝜉′ is a finite projection in M ′ . Proof. We shall get at a contradiction, if we suppose that 𝑝𝜉 is finite, whereas 𝑝𝜉′ is not. Without any loss of generality, we can assume that 𝑝𝜉′ is properly infinite.

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Let 𝑞 be a central projection, such that 𝑞𝑝𝜉 be abelian. Then 𝑝𝑞𝜉 = 𝑞𝑝𝜉 is abelian; ′ With Lemma 6.2, we infer that 𝑝𝑞𝜉 = 𝑞𝑝𝜉′ is abelian, hence it is finite. Since 𝑝𝜉′ is properly infinite, it follows that 𝑞𝑝𝜉 = 0, and since z(𝑝𝜉 ) = z(𝑝𝜉′ ), we have 𝑞𝑝𝜉 = 0. By taking into account exercise E.4.21, it follows that there exist 𝑒, 𝑓 ∈ P M , such that 𝑝𝜉 = 𝑒 + 𝑓, 𝑒𝑓 = 0, z(𝑒) = z(𝑓) = z(𝑝𝜉 ). If we denote 𝛼 = 𝑒𝜉, 𝛽 = 𝑓𝜉, we have 𝑒 = 𝑝𝛼 , 𝑓 = 𝑝𝛽 . We now consider the projections 𝑒′ = 𝑝𝛼′ , 𝑓 ′ = 𝑝𝛽′ . We shall show that 𝑒′ and 𝑓 ′ are finite projections in M ′ . Indeed, if 𝑒′ is not finite, then there exists a central projection 𝑞 ≠ 0, such that the projection 𝑞𝑒′ be properly infinite and z(𝑞𝑒′ ) = 𝑞 ≤ z(𝑝𝜉′ ). Hence the projections 𝑞𝑒′ and 𝑞𝑝𝜉′ are properly infinite, of countable type (see exercise E.5.6) and their central supports are equal. From Proposition 4.13, it follows that 𝑞𝑒′ ∼ 𝑞𝑝𝜉′ ; hence, with Lemma 6.1, we have 𝑞𝑒 ∼ 𝑞𝑝𝜉 . But 𝑞𝑝𝜉 = 𝑞𝑒 + 𝑞𝑓 and 𝑞𝑓 ≠ 0, because 0 ≠ 𝑞 ≤ z(𝑝𝜉′ ) = z(𝑝𝜉 ) = z(𝑓). Hence 𝑞𝑝𝜉 is not finite, but this contradicts the fact that 𝑝𝜉 is finite. Analogously, one shows that 𝑓 ′ is finite. Since 𝜉 = 𝛼 + 𝛽, it follows that ′ 𝑝𝜉′ = 𝑝𝛼+𝛽 ≤ 𝑝𝛼′ ∨ 𝑝𝛽′ = 𝑒′ ∨ 𝑓 ′ , which is finite (4.15);

hence 𝑝𝜉′ is finite, in contradiction with our assumption that 𝑝𝜉′ is properly infinite. This proves the Lemma. Q.E.D. Theorem 6.4. Let M ⊂ B (H ) be a von Neumann algebra. Then M is of type I (resp., of type II; resp., of type III) iff M ′ is of type I (resp., of type II; resp., of type III). Proof. It is sufficient to prove that if M ′ is discrete (resp., semifinite), then M is discrete (resp., semifinite; see Table 4.21). Let 𝑞 be a non-zero central projection. Since M is discrete (resp., semifinite), there exists an abelian (resp., finite) non-zero projection 𝑒′ ∈ M , such that 𝑒′ ≤ 𝑞. Let 𝜉 ∈ 𝑒′ (H ). Then 𝑝𝜉′ ≤ 𝑒′ is an abelian (resp., finite) projection in M ′ , such that z(𝑝𝜉′ ) ≤ 𝑞. From Lemma 6.2 (resp., 6.3), it follows that 𝑝𝜉 is an abelian (resp., finite) projection in M , such that z(𝑝𝜉 ) ≤ 𝑞. Thus, any non-zero central projection contains an abelian (resp., finite) non-zero projection in M ; hence M is, indeed, discrete (resp., semifinite). Q.E.D. Corollary 6.5. Let M ⊂ B (H ) be a von Neumann algebra. Then M is discrete (resp., semifinite) iff M is *-isomorphic to a von Neumann algebra whose commutant is abelian (resp., finite).

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Proof. If M is discrete (resp., semifinite), then according to Theorem 6.4, M ′ is discrete (resp., semifinite). With Proposition 4.19, we infer that there exists an abelian (resp., finite) projection 𝑒′ ∈ M ′ , such that z(𝑒′ ) = 1. Then (M 𝑒′ )′ = (M ′ )𝑒′ is an abelian (resp., finite) von Neumann algebra and M is *-isomorphic to M 𝑒′ (see Proposition 3.14 and Theorem 3.13). The converse is an immediate consequence of Theorem 6.4 and of the evident fact that types of von Neumann algebras are conserved by *-isomorphisms. Q.E.D. 6.6. Let M ⊂ B (H ) be a von Neumann algebra, M ′ its commutant and Z = M ∩M ′ its centre. With the help of Corollary 3.3, it is easy to see that Z ′ is the smallest von Neumann algebra included in B (H ), which contains M and M ′ , i.e., it is the von Neumann algebra R (M , M ′ ), generated by M and M ′ . From Theorem 6.4, the following corollary obviously obtains: Corollary. Let M ⊂ B (H ) be a von Neumann algebra. Then R (M , M ′ ) is a von Neumann algebra of type I. Theorem 6.7. Let M ⊂ B (H ) be a commutative von Neumann algebra with 𝜉 ∈ H a cyclic vector, M 𝜉 = H . Then M is maximal commutative M′ = M. Proof. Let 𝑥 ′ ∈ M ′ . As M 𝜉 = H , there is a sequence 𝑎𝑛 ∈ M such that 𝑎𝑛 𝜉 ↦ 𝑥 ′ 𝜉. Notice that the sequence ‖𝑎𝑛∗ 𝜉‖ = ‖𝑎𝑛 𝜉‖ is bounded and for any 𝑏 ∈ M , (𝑎𝑛∗ 𝜉 ∣ 𝑏𝜉) = (𝑏∗ 𝜉 ∣ 𝑎𝑛 𝜉) ↦ (𝑏∗ 𝜉 ∣ 𝑥 ′ 𝜉) = (𝑥 ′∗ 𝜉 ∣ 𝑏𝜉); hence

𝑎𝑛∗ 𝜉 ↦ 𝑥 ′∗ 𝜉 weakly,

since M 𝜉 = H . Consider also 𝑦 ′ ∈ M ′ . For any 𝑏, 𝑐 ∈ M , we have (𝑥 ′ 𝑦 ′ 𝑏𝜉 ∣ 𝑐𝜉) = (𝑦 ′ 𝑏𝜉 ∣ 𝑐𝑥 ′∗ 𝜉) = lim(𝑦 ′ 𝑏𝜉 ∣ 𝑐𝑎𝑛∗ 𝜉) 𝑛

′

= lim(𝑦 𝑏𝑎𝑛 𝜉 ∣ 𝑐𝜉) = (𝑦 ′ 𝑏𝑥 ′ 𝜉 ∣ 𝑐𝜉) = (𝑦 ′ 𝑥 ′ 𝑏𝜉 ∣ 𝑐𝜉), 𝑛

and as M 𝜉 = H , 𝑥 ′ 𝑦 ′ = 𝑦 ′ 𝑥 ′ . Hence 𝑥 ′ ∈ M ″ = M .

Q.E.D.

6.8. The proof of Theorem 6.7 does not really require the commutativity of M (except for the last chain of equalities), but just (𝑎𝑏𝜉 ∣ 𝜉) = (𝑏𝑎𝜉 ∣ 𝜉), for 𝑎, 𝑏 ∈ M . Recall that, given a von Neumann algebra M ⊂ B (H ), a vector 𝜉 ∈ 𝐻 is called an M -trace vector if 𝜔𝜉 is a trace on M , that is (𝑎𝑏𝜉 ∣ 𝜉) = (𝑏𝑎𝜉 ∣ 𝜉),

(∀)𝑎, 𝑏 ∈ M .

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149

Now, if the above proof works, what should be the conclusion for a non-commutative M ? Notice that, unexplicitly but essentially, the isometric conjugate-linear map 𝑎𝜉 ↦ 𝑎∗ 𝜉

(𝑎 ∈ M ),

carried the whole proof. It is this map again that will enable us to formulate and prove a more general version of Theorem 6.7. Theorem. Let M ⊂ B (H ) be a von Neumann algebra having a cyclic trace vector 𝜉 ∈ H . Then the vector 𝜉 is also separating, there is a unique conjugation 𝐽 ∶ H → H such that 𝐽𝑎𝜉 = 𝑎∗ 𝜉 (𝑎 ∈ M ), and the map

𝑗 ∶ M ∋ 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 ∈ M ′

is a *-anti-isomorphism acting identically on Z M = M ∩ M ′ : 𝐽𝑧∗ 𝐽 = 𝑧,

(∀)𝑧 ∈ Z M .

Proof. If 𝑥 ∈ M , 𝑥𝜉 = 0, then, for any 𝑎, 𝑏 ∈ M : (𝑥𝑎𝜉 ∣ 𝑏𝜉) = (𝑏∗ 𝑥𝑎𝜉 ∣ 𝜉) = (𝑎𝑏∗ 𝑥𝜉 ∣ 𝜉) = 0, hence 𝑥 = 0 since M 𝜉 = H . The trace property of 𝜉 also shows that the conjugate linear map M 𝜉 ∋ 𝑎𝜉 ↦ 𝑎∗ 𝜉 ∈ M 𝜉 is correctly defined and isometric, thus defining a unique conjugation 𝐽 ∶ H ↦ H such that (1)

𝐽𝑎𝜉 = 𝑎∗ 𝜉,

(∀)𝑎 ∈ M .

It is obvious that 𝐽𝑥 ∗ 𝐽 ∈ M ′ for 𝑥 ∈ M . Indeed, for 𝑦, 𝑎 ∈ M , (2)

𝐽𝑥 ∗ 𝐽𝑦𝑎𝜉 = 𝐽𝑥 ∗ 𝑎∗ 𝑦 ∗ 𝜉 = 𝑦𝑎𝑥𝜉 = 𝑦𝐽𝑥 ∗ 𝑎∗ 𝜉 = 𝑦𝐽𝑥 ∗ 𝐽𝑎𝜉;

hence (𝐽𝑥 ∗ 𝐽)𝑦 = 𝑦(𝐽𝑥 ∗ 𝐽) as M 𝜉 = H . As in the proof of Theorem 6.7, the essential step is that (3)

𝐽𝑎′ 𝜉 = 𝑎′∗ 𝜉,

(∀)𝑎′ ∈ M ′ ,

and the proof is the same: let 𝑎𝑛 ∈ M , 𝑎𝑛 𝜉 → 𝑎′ 𝜉. Then 𝑎𝑛∗ 𝜉 = 𝐽𝑎𝑛 𝜉 → 𝐽𝑎′ 𝜉 by the definition of 𝐽, and for any 𝑏 ∈ M , (𝑎𝑛∗ 𝜉 ∣ 𝑏𝜉) = (𝜉 ∣ 𝑎𝑛 𝑏𝜉) = (𝜉 ∣ 𝑏𝑎𝑛 𝜉) → (𝜉 ∣ 𝑏𝑎′ 𝜉) = (𝑎′∗ 𝜉 ∣ 𝑏𝜉); Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:15, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.009

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hence 𝑎𝑛∗ 𝜉 → 𝑎′∗ 𝜉 weakly, which proves that indeed 𝐽𝑎′ 𝜉 = 𝑎′∗ 𝜉. Using (3) instead of (1) and arguing as in (2) for M ′ instead of M , we see that 𝐽𝑥 ′∗ 𝐽 ∈ M ″ = M ,

𝑥′ ∈ M ′ .

Thus 𝐽M 𝐽 ⊂ M ′ , 𝐽M ′ 𝐽 ⊂ M ; hence 𝐽𝐽M ′ 𝐽𝐽 ⊂ 𝐽M 𝐽, that is M ′ ⊂ 𝐽M 𝐽 since 𝐽 2 = 𝐼, and 𝐽M 𝐽 = M ′ . As 𝐽 2 = 𝐼 and 𝐽 ∗ = 𝐽, for any 𝑥, 𝑦 ∈ M , we have 𝑗(𝑥𝑦) = 𝐽(𝑥𝑦)∗ 𝐽 = 𝐽𝑦 ∗ 𝑥 ∗ 𝐽 = 𝐽𝑦 ∗ 𝐽𝐽𝑥 ∗ 𝐽 = 𝑗(𝑦)𝑗(𝑥) 𝑗(𝑥∗ ) = 𝐽𝑥𝐽 = (𝐽𝑥 ∗ 𝐽)∗ = 𝑗(𝑥)∗ . Finally, for 𝑧 ∈ Z M = M ∩ M ′ and any 𝑎 ∈ M , we have 𝐽𝑧∗ 𝐽𝑎𝜉 = 𝐽𝑧∗ 𝑎∗ 𝜉 = 𝑎𝑧𝜉 = 𝑧𝑎𝜉, hence 𝑗(𝑧) = 𝐽𝑧∗ 𝐽 = 𝑧.

Q.E.D.

Notice that if M is commutative, then M = Z M , and the Theorem 6.8 says that the inclusion map 𝑗 ∶ M → M ′ is surjective, so that Theorem 6.7 follows as a trivial consequence and 𝐽 disappears from the statement. 6.9. A von Neumann algebra M ⊂ B (H ) is said to be in standard form if there is a conjugation 𝐽 ∶ H → H , such that the map 𝑗 ∶ M ∋ 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 ∈ M ′ is a *-anti-isomorphism of M onto M ′ acting identically on Z M . Theorem 6.8 states that any von Neumann algebra M ⊂ B (H ) with a cyclic (and separating) trace vector is in standard form. Such a statement makes sense even if we omit the assumption on 𝜉 ∈ H of being a trace vector. And is true! Theorem (Minoru Tomita). Every von Neumann algebra M ⊂ B (H ) with a cyclic and separating vector 𝜉 ∈ H is in standard form. This Theorem will be proved in the Introduction of Chapter 10, which is devoted to even more general cases. The converse of the Tomita’s theorem is rather easy: any countably decomposable von Neumann algebra M ⊂ B (H ), which is in standard form has a cyclic and separating vector. (E.7.15) 6.10. To get an idea of what ‘more general cases’ should mean, let us consider first a 𝜎-finite positive measure 𝜇 on a set X . Every 𝑓 ∈ L ∞ (𝜇) defines a bounded linear operator: 𝜋𝜇 (𝑓) ∶ L 2 (𝜇) ∋ 𝜉 ↦ 𝜋𝜇 (𝑓)𝜉 = 𝑓𝜉 ∈ L 2 (𝜇),

‖𝜋𝜇 (𝑓)‖ = ‖𝑓‖∞ ,

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and M ∶= {𝜋𝜇 (𝑓); 𝑓 ∈ L ∞ (𝜇)} ⊂ B (L 2 (𝜇)) is a maximal commutative von Neumann algebra. (If 𝜇 is finite, this is an obvious consequence of Theorem 6.7.) ′

Proof. We show that the commutant 𝜋𝜇 (L ∞ (𝜇)) ⊂ B (L 2 (𝜇)) is commutative. Let 𝔄 be the set of all functions 𝑓 ∈ L 2 (𝜇), such that the linear mapping 𝑇𝑓𝑜 ∶ L 2 (𝜇) ∩ L ∞ (𝜇) ∋ 𝜉 ↦ 𝑓𝜉 ∈ L 2 (𝜇) is bounded with respect to the L 2 -norm, and let 𝑇𝑓 ∶ L 2 (𝜇) ↦ L 2 (𝜇) the extension of 𝑇𝑓𝑜 to the whole L 2 (𝜇). It is clear that 𝑓 ∈ 𝔄 ⇒ 𝑓 ∈ 𝔄 and 𝑇𝑓 = 𝑇𝑓∗ and 𝑓, 𝑔 ∈ 𝔄 ⇒ 𝑇𝑓 𝑇𝑔 = 𝑇𝑔 𝑇𝑓 since, for any 𝜉, 𝜂 ∈ L 2 (𝜇) ∩ L ∞ (𝜇), we have (𝑇𝑓 𝑇𝑔 𝜉 ∣ 𝜂) = (𝑇𝑔 𝜉 ∣ 𝑇𝑓 𝜂) = ∫ 𝑓𝑔𝜉𝜂 d𝜇 = (𝑇𝑓 𝜉 ∣ 𝑇𝑔 𝜂) = (𝑇𝑔 𝑇𝑓 𝜉 ∣ 𝜂). ′

Let 𝑥 ′ , 𝑦 ′ ∈ 𝜋𝜇 (L ∞ (𝜇)) , 𝜂 ∈ L 2 (𝜇) ∩ L ∞ (𝜇) and consider 𝑓 = 𝑥 ′ 𝜂 ∈ L 2 (𝜇),

𝑔 = 𝑦 ′ 𝜂 ∈ L 2 (𝜇).

Then, as easily verified, 𝑓, 𝑔 ∈ 𝔄 and 𝑇𝑓 = 𝑥 ′ 𝜋𝜇 (𝜂), 𝑇𝑔 = 𝑦 ′ 𝜋𝜇 (𝜂), and moreover 𝜋𝜇 (𝜂)𝑥 ′ 𝑦 ′ 𝜋𝜇 (𝜂) = 𝑇𝑓 𝑇𝑔 = 𝑇𝑔 𝑇𝑓 = 𝜋𝜇 (𝜂)𝑦 ′ 𝑥 ′ 𝜋𝜇 (𝜂). Since 𝜋𝜇 (L 2 (𝜇) ∩ L ∞ (𝜇)) is a w-dense ideal in 𝜋𝜇 (L ∞ (𝜇)), it follows that 𝑥 ′ 𝑦 ′ = 𝑦′ 𝑥′ . Q.E.D. 6.11. Consider now a semifinite von Neumann algebra M with a normal semifinite faithful trace 𝜑 ∶ M + → [0, ∞], and let us briefly recall the ‘GNS-construction’ in this case (compare with section 5.18). Since 𝜑 is a semifinite trace, the set 𝔑𝜑 = {𝑥 ∈ M ; 𝜑(𝑥 ∗ 𝑥) < ∞} is a w-dense two-sided ideal in M , and since 𝜑 is faithful, 𝔑𝜑 is a separated pre-Hilbert space with the scalar product (𝑎 ∣ 𝑏)𝜑 = 𝜑(𝑏∗ 𝑎);

𝑎, 𝑏 ∈ 𝔑𝜑 .

The completion of 𝔑𝜑 is a Hilbert space H 𝜑 and the image of 𝑎 ∈ 𝔑𝜑 in H 𝜑 will be denoted by 𝑎𝜑 ∈ H 𝜑 . The equation 𝜋𝜑 (𝑥)𝑎𝜑 = (𝑥𝑎)𝜑

(𝑥 ∈ M , 𝑎 ∈ 𝔑𝜑 )

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defines a ∗-representation 𝜋𝜑 ∶ M → B (H 𝜑 ), which is injective (since 𝜑 is faithful and semifinite) and w-continuous (since 𝜑 is normal). Hence 𝜋𝜑 (M ) ⊂ B (H 𝜑 ) is a von Neumann algebra ∗ -isomorphic to M . We show that 𝜋𝜑 (M ) ⊂ B (H 𝜑 ) is a standard von Neumann algebra with the conjugation 𝐽 = 𝐽𝜑 ∶ H 𝜑 → H 𝜑 , defined by 𝐽 ∶ 𝔑𝜑 ∋ 𝑎𝜑 ↦ 𝑎𝜑∗ ∈ 𝔑𝜑 . Proof. This map is isometric since 𝜑 is a trace. In what follows we shall identify 𝜋𝜑 (𝑥) ≡ 𝑥, (∀)𝑥 ∈ M , so we consider M ⊂ B (H 𝜑 ). We have 𝐽M 𝐽 ⊂ M ′ since, for 𝑥, 𝑦 ∈ M and 𝑎 ∈ 𝔑𝜑 , we have 𝐽𝑥𝐽𝑦𝑎𝜑 = 𝐽𝑥𝐽(𝑦𝑎)𝜑 = 𝐽(𝑥𝑎∗ 𝑦 ∗ )𝜑 = (𝑦𝑎𝑥 ∗ )𝜑 = 𝑦(𝑎𝑥 ∗ )𝜑 = 𝑦𝐽(𝑥𝑎∗ )𝜑 = 𝑦𝐽𝑥𝐽𝑎𝜑 . In order to prove 𝐽M ′ 𝐽 ⊂ M , we need some preparation. (a) For every 𝜂 ∈ H , consider the linear operator 𝑅𝑜𝜂 ∶ 𝔑𝜑 ∋ 𝑎𝜑 ↦ 𝑎𝜂 ∈ H , let 𝔄′ = {𝜂 ∈ H ; 𝑅𝑜𝜂 is bounded}, and for 𝜂 ∈ 𝔄′ , denote by 𝑅𝜂 ∈ B (H ) the extension of 𝑅𝑜𝜂 . Notice that 𝔑𝜑 ⊂ 𝔄′ and 𝑅𝑏𝜑 = 𝐽𝑏∗ 𝐽, for all 𝑏𝜑 ∈ 𝔑𝜑 , since, for all 𝑎 ∈ 𝔑𝜑 , we have 𝑅𝑜𝑏𝜑 𝑎𝜑 = 𝑎𝑏𝜑 = (𝑎𝑏)𝜑 = 𝐽(𝑏∗ 𝑎∗ )𝜑 = 𝐽𝑏∗ 𝐽𝑎𝜑 . (b) We have 𝜂 ∈ 𝔄′ ⇒ 𝐽𝜂 ∈ 𝔄′ and 𝑅𝐽𝜂 = 𝑅∗𝜂 . It is enough to show that 𝑅𝐽𝜂 ⊂ 𝑅∗𝜂 , and this follows from the following equalities with arbitrary 𝑎, 𝑏 ∈ 𝔑: (𝑅𝑜𝐽𝜂 𝑎𝜑 ∣ 𝑏𝜑 )𝜑 = (𝑎𝐽𝜂 ∣ 𝑏𝜑 )𝜑 = (𝐽𝜂 ∣ (𝑎∗ 𝑏)𝜑 )𝜑 = (𝐽(𝑎∗ 𝑏)𝜑 ∣ 𝜂)𝜑 = (𝑏∗ 𝑎𝜑 ∣ 𝜂)𝜑 = (𝑎𝜑 ∣ 𝑏𝜂)𝜑 = (𝑎𝜑 ∣ 𝑅𝜂 𝑏𝜑 )𝜑 . (c) We have 𝑅𝜂 ∈ M ′ for any 𝜂 ∈ 𝔄′ since, for 𝑥 ∈ M , 𝑎 ∈ 𝔑𝜑 , 𝑅𝜂 𝑥𝑎𝜑 = 𝑅𝜂 (𝑥𝑎)𝜑 = 𝑥𝑎𝜂 = 𝑥𝑅𝜂 𝑎𝜑 . (d) If 𝜂 ∈ 𝔄′ and 𝑥′ ∈ M ′ , then, as easily checked, 𝑥 ′ 𝜂 ∈ 𝔄′ and 𝑅𝑥′ 𝜂 = 𝑥 ′ 𝑅𝜂 . 𝑠

(e) There is a net 𝔑𝜑 ∋ 𝑎𝑘 → 1, since 𝜑 is semifinite. Let 𝜂𝑘 = (𝑎𝑘∗ )𝜑 ∈ 𝔑𝜑 . Then, for every 𝑥 ′ ∈ M ′ we have 𝑠

𝑅𝑥′ 𝜂𝑘 = 𝑥 ′ 𝑅𝜂𝑘 = 𝑥 ′ 𝐽𝑎𝑘 𝐽 → 𝑥 ′ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:58:15, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.009

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Thus, in order to prove that 𝐽M ′ 𝐽 ⊂ M , it is sufficient to show that 𝐽𝑅𝜂 𝐽 ∈ M = (M ′ )′ , for all 𝜂 ∈ 𝔄′ . Indeed, for every 𝑥′ ∈ M ′ and every 𝑎, 𝑏 ∈ 𝔑𝜑 ⊂ M , we have (𝐽𝑅𝜂 𝐽𝑥 ′ 𝑎𝜑 ∣ 𝑏𝜑 )𝜑 = (𝑎𝜑 ∣ 𝑥 ′∗ 𝐽𝑅𝐽𝜂 𝐽𝑏𝜑 )𝜑 = (𝑎𝜑 ∣ 𝑥 ′∗ 𝐽𝑏∗ 𝐽𝜂)𝜑 = (𝑎𝜑 ∣ 𝑥 ′∗ 𝑅𝑏𝜑 𝜂)𝜑 = (𝑎𝜑 ∣ 𝑅𝑥′∗ 𝑏𝜑 𝜂)𝜑 = (𝑅𝐽𝑥′∗ 𝑏𝜑 𝑎𝜑 ∣ 𝜂)𝜑 = (𝑎𝐽𝑥 ′∗ 𝑏𝜑 ∣ 𝜂)𝜑 = (𝐽𝑥 ′∗ 𝑏𝜑 ∣ 𝑎∗ 𝜂)𝜑 = (𝐽𝑥 ′∗ 𝑏𝜑 ∣ 𝑅𝜂 𝐽𝑎𝜑 )𝜑 = (𝐽𝑅𝜂 𝐽𝑎𝜑 ∣ 𝑥 ′∗ 𝑏𝜑 )𝜑 = (𝑥 ′ 𝐽𝑅𝜂 𝐽𝑎𝜑 ∣ 𝑏𝜑 )𝜑 Therefore 𝑗 ∶ M ∋ 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 ∈ M ′ is a ∗-antiautomorphism and, as easily checked, 𝑗(𝑧) = 𝑧 for all 𝑧 ∈ Z M . Q.E.D. It is instructive to notice that

𝔄′ = 𝔑 𝜑 .

Indeed, let 𝜂 ∈ 𝔄′ . Then 𝐽𝑅𝜂 𝐽 ∶= 𝑥 ∗ ∈ M , so 𝑅𝜂 = 𝐽𝑥 ∗ 𝐽, and hence, for 𝑎 ∈ 𝔑𝜑 , we have 𝑎𝜂 = 𝑅𝜂 𝑎𝜑 = 𝐽𝑥 ∗ 𝐽𝑎𝜑 = (𝑎𝑥)𝜑 . 𝑠

Now, when 𝔑𝜑 ∋ 𝑎i → 1, we get (by the normality of 𝜑): 𝜑(𝑥 ∗ 𝑥) ≤ lim inf 𝜑(𝑥 ∗ 𝑎i∗ 𝑎i 𝑥) = lim inf ‖(𝑎i 𝑥)𝜑 ‖2𝜑 i

=

i

lim inf ‖𝑎i 𝜂‖2𝜑 i

= ‖𝜂‖ < ∞;

hence 𝑥 ∈ 𝔑𝜑 , and we see that 𝑅𝜂 𝑎𝜑 = (𝑎𝑥)𝜑 = 𝑎𝑥𝜑 = 𝑅𝑥𝜑 𝑎𝜑 , and therefore, 𝑅𝜂 = 𝑅𝑥𝜑 and 𝜂 = 𝑥𝜑 ∈ 𝔑𝜑 by a general remark: 𝜉, 𝜂 ∈ 𝔄′ and 𝑅𝜉 = 𝑅𝜂 ⇒ 𝜉 = 𝜂. Indeed, with 𝔑𝜑 ∋ 𝑎i → 1 ∈ M , we have 𝜉 = lim 𝑎i 𝜉 = lim 𝑅𝜉 (𝑎i )𝜑 = lim 𝑅𝜂 (𝑎i )𝜑 = lim 𝑎i 𝜂 = 𝜂. i

i

i

i

In the examples 6.10 and 6.11, the role of the ‘cyclic and separating trace vector’ from 6.8 was played by a whole ∗-algebra 𝔄′ = 𝔑𝜑 contained in the Hilbert space. EXERCISES !E.6.1 Let M ⊂ B (H ) be a von Neumann algebra, 𝑒′ ∈ P M ′ and 𝜉 ∈ H , such that 𝑒′ ∼ 𝑝𝜉′ . Then there exists an 𝜂 ∈ H , such that 𝑒′ = 𝑝𝜂′ ,

𝑝𝜉 = 𝑝𝜂 . ▶

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E.6.2 Let M ⊂ B (H ) be a von Neumann algebra and 𝑒, 𝑓 ∈ P M . If M 𝑒 ⊂ B (𝑒H ) has a separating vector, M 𝑓 ⊂ B (𝑓H ) has a cyclic vector and z(𝑒) ≤ z(𝑓), then 𝑒 ≺ 𝑓. !E.6.3 If a von Neumann algebra has a cyclic vector and a separating vector, then it has a vector that is both cyclic and separating (compare with C.3.5). E.6.4 If M is a finite von Neumann algebra and if it has a finite totalizing family, then M ′ is also finite. E.6.5 Let H be a separable Hilbert space and M ⊂ B (H ) a finite von Neumann algebra with a properly infinite commutant. Then there exists a sequence {𝜉𝑛 } ⊂ H , such that the projections 𝑝𝜉′ 𝑛 be mutually orthogonal, ∞

E.6.6

E.6.7

E.6.8

!E.6.9

∑𝑛=1 𝑝𝜉′ 𝑛 = 1 and for any 𝑛, 𝑝𝜉𝑛 = 1. (Hint: see 7.18.) Let H be a separable Hilbert space and M ⊂ B (H ) a properly infinite von Neumann algebra with a properly infinite commutant. Then M has a separating cyclic vector. Let M be a von Neumann algebra, with the centre Z . A projection 𝑝 ∈ Z is the central support of a cyclic projection iff it is of countable type in Z . Infer from this result the statement in E.3.8. Let M ⊂ B (H ) be a von Neumann algebra. For any 𝑥 ∈ M and any 𝜉 ∈ H , we have 𝑝𝑥𝜉 ≺ 𝑝𝜉 . If, moreover, 𝜉 ∈ [𝑥 ∗ H ], then 𝑝𝑥𝜉 ∼ 𝑝𝜉 . Let M ⊂ B (H ) be a von Neumann algebra, and 𝐽 ∶ H → H a conjugation, such that 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 is a *-anti-isomorphism of M onto M ′ . Then, for any 𝜉 ∈ H , we have 𝑝𝜉′ = 𝐽𝑝𝐽𝜉 𝐽.

E.6.10 Let M be a von Neumann algebra of type I (resp., II; resp., III) and 𝑒′ ∈ P M ′ . Then M 𝑒′ is of type I (resp., II; resp., III). E.6.11 Let M be a von Neumann algebra and 𝑒 ∈ M a minimal projection in M . Then z(𝑒) is a minimal projection in Z , whereas M z(𝑒) is a factor of type I. Infer from this result that the least upper bound of the set of all minimal projections in M is a central projection.

COMMENTS C.6.1 Let M ⊂ B (H ) be a von Neumann algebra and 𝜉 ∈ H . By taking into account Theorem 5.23, which is the basis of all results in this chapter, it is

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only natural to enquire about the structure of the vectors in the space [M 𝜉]. Since any such vector is the limit of a sequence of vectors of the form 𝑥𝑛 𝜉, where 𝑥𝑛 ∈ M , it is natural to search for the conditions that make true the following assertion: for any 𝜂 ∈ [M 𝜉], there exists a closed (see Section 9.1.) ⎧ ⎪ (𝑇) operator 𝑇 in H , affiliated (see Section 9.7.) to M , such that ⎨ ⎪ ⎩ 𝜂 = 𝑇𝜉. This deep problem was first considered by F. J. Murray and J. von Neumann (1936) in connection with the results presented in this chapter. They gave a partially positive answer, by showing that the following statement is always true: for any 𝜂 ∈ [M 𝜉], there exists a closed operator 𝑇 ⎧ ⎪ (𝐵𝑇) in H , affiliated to M , and an operator 𝐵 ∈ M , such that ⎨ ⎪ ⎩ 𝜂 = 𝐵𝑇𝜉. More precisely, Theorem (BT theorem). Let M ⊂ B (H ) be a von Neumann algebra, 𝜉 ∈ H and 𝜁 ∈ M 𝜉. Then there are 𝑎 ∈ M + , 𝑥 ∈ M and 𝜂 ∈ M 𝜉 such that 𝑎𝜂 = 𝜉, 𝑥𝜂 = 𝜁 and s(𝑎)𝜂 = 𝜂. Proof. Assume ‖𝜉‖ = 1, ‖𝑦‖ = 1. Since 𝑦 ∈ M 𝜉, we can write ∞

𝜁 = ∑ 𝑦𝑛 𝜉 with 𝑦𝑛 ∈ M , ‖𝑦𝑛 𝜉‖ < 4−𝑛 . 𝑛=0

Let 𝑏𝑛 ∈ M + such that 𝑛

𝑏𝑛2 = 1 + ∑ 4𝑘 𝑦𝑘∗ 𝑦𝑘

(𝑛 ∈ ℕ).

𝑘=0

Then {𝑏𝑛 }𝑛 ⊂ M + is an increasing sequence; hence {𝑏𝑛−1 }𝑛 ⊂ M + is a decreasing sequence, and therefore, there is 𝑎 ∈ M + such that 𝑏𝑛−1 ↓ 𝑎. On the other hand, 𝑛

𝑛

‖𝑏𝑛 𝜉‖2 = ‖𝜉‖2 + ∑ 4𝑘 ‖𝑦𝑘 𝜉‖2 ≤ 1 + ∑ 4−𝑘 < 3 𝑘=0

𝑘=0

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Lectures on von Neumann algebras and H 1 is weakly compact; hence {𝑏𝑛 𝜉}𝑛 has a weak-limit point 𝜂 ′ ∈ M 𝜉. Let 𝜂 = s(𝑎)𝜂 ′ ∈ M ′ . Finally, for 𝑛 ≤ 𝑚, 𝑚 −1 ∗ −1 −1 −1 4𝑛 𝑏 𝑚 𝑦𝑛 𝑦𝑛 𝑏 𝑚 ≤ 𝑏𝑚 = (1 + Σ)−1/2 Σ(1 + Σ)−1/2 ≤ 1. ( ∑ 4𝑘 𝑦𝑘∗ 𝑦𝑘 ) 𝑏𝑚 𝑘=0 so

−1 Since 𝑏𝑚 ⟶ 𝑎, it follows that 4𝑛 𝑎𝑦𝑛∗ 𝑦𝑛 𝑎 ≤ 1 and ‖𝑦𝑛 𝑎‖ ≤ 2−𝑛 , which insures the norm convergence of the series ∞

𝑥 ∶= ∑ 𝑦𝑛 𝑎 ∈ M . 𝑛=0

We have 𝑎𝜂 ′ = 𝜉 since for any 𝜃 ∈ H and any 𝜀 > 0, we can find 𝑛 ∈ ℕ such that |(𝑎𝜂 ′ − 𝜉 ∣ 𝜃)| = |(𝜂 ′ ∣ 𝑎𝜃) − (𝑏𝑛 𝜉 ∣ 𝑎𝜃) + (𝑏𝑛 𝜉 ∣ 𝑎𝜃) − (𝜉 ∣ 𝜃)| ≤ |(𝜂 ′ − 𝑏𝑛 𝜉 ∣ 𝑎𝜃)| + |(𝑏𝑛 𝜉 ∣ 𝑎𝜃 − 𝑏𝑛−1 𝜃)| ≤ |(𝜂 ′ − 𝑏𝑛 𝜉 ∣ 𝑎𝜃)| + ‖𝑏𝑛 𝜉‖‖𝑎𝜃 − 𝑏𝑛−1 𝜃‖ 𝜀 𝜀 ≤ + √3 ⋅ = 𝜀. 2 √ 2 3 ∞

∞

Thus, 𝑎𝜂 = 𝑎s(𝑎)𝜂 ′ = 𝑎𝜂 ′ = 𝜉 and 𝑥𝜂 = ∑ 𝑦𝑛 𝑎𝜂 = ∑ 𝑦𝑛 𝜉 = 𝜁. Q.E.D. 𝑛=0

𝑛=0

This proof is an ultimate refinement, due to Christian Skau, of the original proof. The Theorem is called the BT-theorem since it allows to express any 𝜁 ∈ M 𝜉 under the form 𝜁 = 𝐵𝑇𝜉, with 𝐵 = 𝑥 ∈ M a bounded operator and 𝑇 = ‘𝑎−1 ’ an ‘unbounded’ operator affiliated to M . Notice that the BT-theorem allows another proof of Lemma 6.1, which we restate for convenience of notation: Let M ⊂ B (H ) be a von Neumann algebra and 𝜉, 𝜂 ∈ H . Then 𝑝𝜂 ≺ 𝑝𝜉 in M ⇔ 𝑝𝜂′ ≺ 𝑝𝜉′ in M ′ .

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Proof. Indeed, assume 𝑝𝜂′ ≺ 𝑝𝜉′ via 𝑣′ ∈ M ′ , 𝑣 ′∗ 𝑣′ = 𝑝𝜂′ , 𝑣 ′ 𝑣′∗ ≤ 𝑝𝜉′ . Then 𝜂 = 𝑣 ′∗ 𝜁 for 𝜁 = 𝑣 ′ 𝜂 ∈ M 𝜉 = 𝑝𝜉′ . By the BT-theorem, there are 𝑎 ∈ M + , 𝑥 ∈ M and 𝜂0 ∈ M 𝜉 such that 𝑎𝜂0 = 𝜉, 𝑥𝜂0 = 𝜁 and s(𝑎)𝜂0 = 𝜂0 and using 4.24.(6) and 4.24.(5), we conclude 𝑝𝜂 = 𝑝𝑣′∗ 𝜁 ≤ 𝑝𝜁 = 𝑝𝑥𝜂0 ≺ 𝑝𝜂0 ∼ 𝑝𝑎𝜂0 = 𝑝𝜉 . Q.E.D. C.6.1 bis. As far as statement (𝑇) is concerned, H. A. Dye (1952) has shown that the projection 𝑝𝜁 , 𝜁 ∈ H , is finite iff the following implication holds: 𝜉 ∈ [M 𝜁] ⇒ for 𝜉 the statement (𝑇) is true. H. A. Dye called a von Neumann algebra M ⊂ B (H ) essentially finite if any cyclic projection 𝑝𝜉 ∈ M , 𝜉 ∈ H , is finite. Thus, the statement (𝑇) is true for any vector in H iff M is essentially finite. In particular, if M is finite, then statement (𝑇) is true for any vector in H , a result already known to F. J. Murray and J. von Neumann. The proof of this fact immediately follows from the (𝐵𝑇)-theorem, with the help of exercise E.9.26. On the other hand, if M ⊂ B (H ) is a von Neumann algebra, 𝜉 ∈ H , and 𝜓 is a normal form on M , such that s(𝜓) ≤ s(𝜔𝜉 ), then Theorem 5.23 shows that there exists an 𝜂 ∈ [M 𝜉], such that 𝜓 = 𝜔𝜂 . Having in mind the problem (𝑇), there naturally arises the question whether the vector 𝜂 ∈ [M 𝜉], such that 𝜓 = 𝜔𝜂 , can be chosen so that 𝜂 = 𝑇𝜉, where 𝑇 is a closed operator in H , affiliated to M . Therefore, a new problem arises, namely to establish the conditions under which the following statement is true: for any normal form 𝜓 on M , such that s(𝜓) ≤ s(𝜔𝜉 ), there exists a ⎧ ⎪ (𝑅𝑁) closed operator 𝑇 in H , affiliated to M , such that ⎨ ⎪ 𝜓 = 𝜔𝑇𝜉 ⎩ Since the condition s(𝜓) ≤ s(𝜔𝜉 ) is a condition of ‘absolute continuity’, and since the operator 𝑇 plays the role of a ‘density’, the statement (𝑅𝑁) is obviously analogous to the classical Radon–Nikodym theorem.

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Lectures on von Neumann algebras By taking into account Theorem 5.23, it is obvious that (𝑇) ⇒ (𝑅𝑁). In particular, if M is essentially finite, then the statement (𝑅𝑁) is true for any vector in H . In fact, the statement (𝑅𝑁) is true for any vector in H , without any restriction on M , as we shall see in Chapter 10, where we shall make more precise considerations concerning the density 𝑇.

C.6.2 Let us consider again a von Neumann algebra M ⊂ B (H ) and a vector 𝜉 ∈ H . Concerning the commutant M ′ ⊂ B (H ), the analogous statement to statement (𝑇) is the following: for any 𝜂 ∈ [M ′ 𝜉], then exists a closed operator ⎧ ⎪ (𝑇 ′ ) 𝑇 ′ in H , affiliated to M ′ , such that ⎨ ⎪ 𝜂 = 𝑇 ′ 𝜉. ⎩ If 𝜓 is a normal form on M , such that s(𝜓) ≤ s(𝜔𝜉 ), Theorem 5.23 shows that there exists an 𝜂 ∈ [M ′ 𝜉], such that 𝜓 = 𝜔𝜂 . By taking into account statement (𝑇 ′ ), there naturally arises the question whether the vector 𝜂 ∈ [M ′ 𝜉], such that 𝜓 = 𝜔𝜂 , can be chosen so that 𝜂 = 𝑇 ′ 𝜉, where 𝑇 ′ is a closed operator in H , affiliated to M ′ . Consequently, a new problem arises, namely to establish the conditions under which the following statement is true: for any normal form 𝜓 on M , such that s(𝜓) ≤ s(𝜔𝜉 ), there exists a ⎧ ⎪ (𝐷) positive self-adjoint operator 𝐴′ in H , affiliated to M ′ , such that ⎨ ⎪ 𝜓 = 𝜔𝐴′ 𝜉 . ⎩ Together with this problem, one poses the analogous problem, concerning the statement:

(𝐷 ′ )

′ ′ ′ ′ ⎧ for any normal form 𝜓 on M , such that s(𝜓 ) ≤ s(𝜔𝜉 ), ⎪ ⎪ there exists a positive self-adjoint operator 𝐴 in H , affiliated to M ,

⎨ such that ⎪ ⎪ ⎩

′ 𝜓′ = 𝜔𝐴𝜉 .

The solution to problems (𝐷) and (𝐷 ′ ) was given by H. A. Dye (1952), along with the solution to problem (𝑇) (C.6.1). We remark that statements (𝐷) and (𝐷 ′ ) are also of the Radon–Nikodym type, but the density now belongs to the commutant. A similar, but trivial, situation was considered in Lemma 5.19 (see, also, exercise E.9.33).

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159

We now state the theorem of H. A. Dye, and we also give a sketch of the proof using some of the results in Chapter 9. Theorem. Let M ⊂ B (H ) be a von Neumann algebra and 𝜁 ∈ H . The following statements are equivalent: (1) 𝑝𝜁 is a finite projection in M . (2) 𝜉 ∈ [M 𝜁] ⇒ for 𝜉 the statement (𝑇) is true. (3) 𝜉 ∈ [M ′ 𝜁] ⇒ for 𝜉 the statement (𝐷) is true. (1′ ) 𝑝𝜁′ is a finite projection in M ′ . (2′ ) 𝜉 ∈ [M ′ 𝜁] ⇒ for 𝜉 the statement (𝑇 ′ ) is true. (3′ ) 𝜉 ∈ [M 𝜁] ⇒ for 𝜉 the statement (𝐷 ′ ) is true. Proof. The proof proceeds according to the following diagram:

(1)

(2)

(3)

(1')

(2')

(3')

The equivalence (1) ⇔ (1′ ) coincides with Lemma 6.3. The implication (1) ⇒ (2) immediately follows from the (𝐵𝑇)-theorem, with the help of exercise E.9.26. The equivalence (2′ ) ⇔ (3) easily obtains from Theorem 5.23 and exercise E.9.32. It is obvious that the implication (1′ ) ⇒ (2′ ) and the equivalence (2) ⇔ (3′ ) obtain in a similar manner. We have still to prove the implication (2′ ) ⇒ (1), since the implication (2) ⇒ (1′ ) is obviously similar to this one. Let us assume that the projection 𝑝𝜁 is not finite in M . Then we can assume that 𝑝𝜁 is properly infinite. It follows that there exist a projection 𝑒 ∈ M , 𝑒 ≤ 𝑝𝜁 , 𝑒 ≠ 𝑝𝜁 , and a partial isometry 𝑣 ∈ M , such that 𝑣 ∗ 𝑣 = 𝑝𝜁 ,

𝑣𝑣 ∗ = 𝑒,

𝑣𝑝𝜁 = 𝑣 = 𝑒𝑣 = 𝑝𝜁 𝑣;

n(1 − 𝑣) = n(1 − 𝑣 ∗ ) = 0. Therefore, the operator 1 − 𝑣 is injective, and (1 − 𝑣)H is a dense subspace of H . From exercise E.9.8, we infer that 𝐴 = 𝑖(1 + 𝑣)(1 − 𝑣)−1

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Lectures on von Neumann algebras is a closed (symmetric) linear operator in H , such that D 𝐴 = (1 − 𝑣)H . Of course, 𝐴 is affiliated to M . We have 𝜉 = (1 − 𝑣)𝜁 ∈ D 𝐴 . It is easily verified that 𝜉 ∈ [M ′ 𝜁], and since n(1 − 𝑣 ∗ ) = 0, it follows that we even have the equality 𝑝𝜉 = 𝑝𝜁 . Hence, if we again use the relations n(1 − 𝑣) = n(1 − 𝑣 ∗ ) = 0, we get (*)

D 𝐴 ∩ [M ′ (𝑝𝜉 − 𝑒)𝜉] = {0}.

Let us now assume that hypothesis (2′ ) holds. We shall first show that for any projection 𝑞 ∈ M , 0 ≠ 𝑞 ≤ 𝑝𝜉 , there exists a projection 𝑟′ ∈ M ′ , such that 𝑟′ 𝜉 ≠ 0

and

𝑟′ 𝜉 ∈ [M ′ 𝑞𝜉].

Indeed, we have 𝑞𝜉 ∈ [M ′ 𝜉], and therefore, from hypothesis (2′ ), there exists a closed operator 𝑇 ′ in H , affiliated to M ′ , such that 𝑞𝜉 = 𝑇 ′ 𝜉. Then, for any 𝑥′ ∈ M ′ , we have 𝑥′ 𝑇 ′ 𝜉 ∈ [M ′ 𝑞𝜉]. By taking into account the polar decomposition Theorem (9.28) and the operational calculus with positive self-adjoint operators (9.11 and 9.13), it is easily seen that there exist an 𝑥′ ∈ M ′ and a projection 𝑟′ ∈ M ′ , such that 𝑟′ 𝜉 = 𝑥 ′ 𝑇 ′ 𝜉 ≠ 0. A familiar argument, based on the Zorn Lemma, shows that for any projection 𝑞 ∈ M , 0 ≠ 𝑞 ≤ 𝑝𝜉 , there exists a projection 𝑞 ′ ∈ M ′ , such that [M ′ 𝑞𝜉] = [M ′ 𝑞 ′ 𝜉]. In particular, let 𝑞 = 𝑝𝜉 − 𝑒. Since 𝜉 ∈ D 𝐴 , and since 𝐴 is affiliated to M , it follows that (see E.9.25): M ′ 𝑞 ′ 𝜉 ⊂ D 𝐴 ∩ [M ′ 𝑞𝜉]; hence (**)

D 𝐴 ∩ [M ′ (𝑝𝜉 − 𝑒)𝜉] is dense in [M ′ (𝑝𝜉 − 𝑒)𝜉].

The contradiction between relations (*) and (**) proves the implication (2′ ) ⇒ (1), and thus, the theorem is also proved. Q.E.D.

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Corollary. Let M ⊂ B (H ) be a von Neumann algebra. The following statements are equivalent: (1) M is essentially finite. (2) The statement (𝑇) is true for any vector 𝜉 ∈ H . (3) The statement (𝐷) is true for any vector 𝜉 ∈ H . We stress the fact that, while the statement (𝑅𝑁) is always true, the statement (𝐷), of Radon–Nikodym type, of H. A. Dye, depends on finiteness conditions. C.6.3 From Lemma 6.3, it follows that the von Neumann algebra M ⊂ B (H ) is essentially finite iff M ′ ⊂ B (H ) is essentially finite. It is easily seen that any essentially finite von Neumann algebra is semifinite. Conversely, according to Corollary 6.5, any semifinite von Neumann algebra is *-isomorphic to an essentially finite von Neumann algebra. In contrast, B (H ) is essentially finite, but it is properly infinite if H is infinitely dimensional. C.6.4 Bibliographical comments. Lemma 6.1 was proved, for the case of the factors, by F. J. Murray and J. von Neumann (1936). Lemmas 6.2, 6.3 and Theorem 6.4. are stated by I. Kaplansky (1950), whereas proofs have been given by H. A. Dye (1952), E. L. Griffin (1955), J. Dixmier (1954), R. Pallu de la Barrière (1954) and R. V. Kadison (1957b). In the proofs of Lemmas 6.2 and 6.3, we followed D. M. Topping (1971) and R. V. Kadison (1957b), respectively. For another proof of Lemma 6.2, see L. Zsidó (1973b), I.7.7, whereas another proof of Lemma 6.3 is proposed in exercise E.7.20. A proof of Lemma 6.1 based directly on the polar decomposition theorem was given by R. Herman and M. Takesaki (1970).

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FINITE VON NEUMANN ALGEBRAS

7

In this chapter, we study the lattice of the finite projections in a von Neumann algebra, and we characterize the finite von Neumann algebras with the help of the traces. Theorem 7.1. Let M be a von Neumann algebra and 𝑒, 𝑓, 𝑔 ∈ PM . If 𝑒 ≤ 𝑔 and if the projection (𝑒 ∨ 𝑓) ∧ 𝑔 is finite, then (𝑒 ∨ 𝑓) ∧ 𝑔 = 𝑒 ∨ (𝑓 ∧ 𝑔). Proof. Let ℎ = (𝑒 ∨ 𝑓) ∧ 𝑔, 𝑘 = 𝑒 ∨ (𝑓 ∧ 𝑔). The relation 𝑘 ≤ ℎ is obvious in view of the hypothesis that 𝑒 ≤ 𝑔. On the other hand, we have 𝑒 ∨ 𝑓 = (𝑒 ∨ (𝑓 ∧ 𝑔)) ∨ 𝑓 ≤ ((𝑒 ∨ 𝑓) ∧ 𝑔) ∨ 𝑓 ≤ 𝑒 ∨ 𝑓; hence ℎ ∨ 𝑓 = 𝑘 ∨ 𝑓 = 𝑒 ∨ 𝑓, and 𝑔 ∧ 𝑓 ≤ (𝑒 ∨ (𝑓 ∧ 𝑔)) ∧ 𝑓 ≤ ((𝑒 ∨ 𝑓 ∧ 𝑔) ∧ 𝑓 ≤ 𝑔 ∧ 𝑓; therefore, we have ℎ ∧ 𝑓 = 𝑘 ∧ 𝑓 = 𝑔 ∧ 𝑓. By taking into account the parallelogram rule (4.4), we get ℎ−𝑓∧𝑔 =ℎ−ℎ∧𝑓 ∼ℎ∨𝑓−𝑓 =𝑒∨𝑓−𝑓 = 𝑘 ∨ 𝑓 − 𝑓 ∼ 𝑘 − 𝑘 ∧ 𝑓 = 𝑘 − 𝑓 ∧ 𝑔. It follows that ℎ ∼ 𝑘. But 𝑘 ≤ ℎ, and by hypothesis, ℎ is finite. Consequently, we have ℎ = 𝑘. Q.E.D. 7.2. One says that a projection 𝑒 ∈ M is piecewise of countable type if there exists a family {𝑞𝑘 }𝑘∈𝐾 of mutually orthogonal central projections, such that ∑𝑘∈𝐾 𝑞𝑘 = 1, and 𝑒𝑞𝑘 is of countable type, for any 𝑘 ∈ 𝐾. 162

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Lemma. Any finite projection 𝑒 in a von Neumann algebra M is piecewise of countable type. Proof. Let 𝜑 be a normal form on M , with 0 ≠ 𝑒0 = s(𝜑) ≤ 𝑒, and let E be a maximal family of mutually orthogonal subprojections of 𝑒, which are all equivalent to 𝑒0 . Since 𝑒 is finite, this family is finite. Let E = {𝑒1 , … , 𝑒𝑛 }. By applying the comparison theorem (4.6) 𝑛 to the projections 𝑒0 , 𝑒 − ∑𝑖=1 𝑒𝑖 , and by taking into account the maximality of the family E , it follows that there exists a central projection 𝑞 ≠ 0, such that 𝑛

𝑞 (𝑒 − ∑ 𝑒𝑖 ) ≺ 𝑞𝑒0 . 𝑖=1

In accordance with exercise E.5.6, 𝑒0 is of countable type. From the preceding results, it follows that 𝑞𝑒 is of countable type, 𝑞 ≠ 0. Let now {𝑞𝑘 }𝑘∈𝐾 be a maximal family of mutually orthogonal central non-zero projections, such that 𝑞𝑘 𝑒 is of countable type, for any 𝑘 ∈ 𝐾. From the maximality of the family and from the first part of the proof, it follows that ∑𝑘∈𝐾 𝑞𝑘 = 1. Consequently, 𝑒 is piecewise of countable type. Q.E.D. Lemma 7.3. Let M be a von Neumann algebra, 𝑓 ∈ PM , and {𝑒𝑛 } an increasing sequence of finite projections in M . If 𝑒𝑛 ≺ 𝑓 for any 𝑛, then ⋁𝑛 𝑒𝑛 ≺ 𝑓. Proof. We shall construct an increasing sequence {𝑓𝑛 } ⊂ PM , such that 𝑓𝑛 ≤ 𝑓, 𝑓𝑛 ∼ 𝑒𝑛 ;

𝑛 = 1, 2, …

Then, by taking into account Proposition 4.2 and exercise E.4.9, we shall obtain ⋁ 𝑛

𝑒𝑛 ∼

⋁

𝑓𝑛 ≤ 𝑓.

𝑛

For the construction, we shall proceed by induction. Let us assume that 𝑓1 , … , 𝑓𝑛−1 have been already constructed. By hypothesis, there exists an equivalence between 𝑒𝑛 and a subprojection of 𝑓. We deduce that there exists a 𝑔 ∈ PM , 𝑔 ≤ 𝑓, such that 𝑒𝑛−1 ∼ 𝑔 and 𝑒𝑛 − 𝑒𝑛−𝑙 ≺ 𝑓 − 𝑔; But 𝑒𝑛−1 ∼ 𝑓𝑛−1 , whence, in view of exercise E.4.9, 𝑓 − 𝑓𝑛−1 ∼ 𝑓 − 𝑔. Consequently, there exists an ℎ ∈ PM , such that 𝑒𝑛 − 𝑒𝑛−1 ∼ ℎ ≤ 𝑓 − 𝑓𝑛−1 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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We then define 𝑓𝑛 = 𝑓𝑛−1 + ℎ.

Q.E.D.

Theorem 7.4. Let M be a von Neumann algebra, 𝑓 ∈ PM and {𝑒𝑖 }𝑖∈𝐼 an increasingly directed family of finite projections in M . If ⋁𝑖∈𝐼 𝑒𝑖 is piecewise of countable type, and 𝑒𝑖 ≺ 𝑓, for any 𝑖 ∈ 𝐼, then ⋁𝑖∈𝐼 𝑒𝑖 ≺ 𝑓. Proof. We shall first assume that 𝑒 = ⋁𝑖∈𝐼 𝑒𝑖 is of countable type. In accordance with exercise E.5.6, there exists a normal form 𝜑 on M , such that 𝑒 = s(𝜑). It follows that 𝜑(𝑒) = sup 𝜑(𝑒𝑖 ); 𝑖∈𝐼

hence there exists a sequence {𝑖𝑛 } ⊂ 𝐼, such that 𝜑(𝑒) = sup 𝜑(𝑒𝑖𝑛 ). 𝑛

We now define by induction the sequence {𝑒𝑛 }, in the following manner: 𝑒1 = 𝑒𝑖1 , 𝑒𝑛 = 𝑒𝑖 , where 𝑖 ∈ 𝐼, 𝑒𝑖 ≥ 𝑒𝑖𝑛 and 𝑒𝑖 ≥ 𝑒𝑘 for 𝑘 < 𝑛. The definition is possible, because the family {𝑒𝑖 }𝑖∈𝐼 is increasingly directed. Then {𝑒𝑛 } is an increasing sequence of finite projections in M , 𝑒𝑛 ≺ 𝑓, and 𝑒 = ⋁𝑛 𝑒𝑛 , because 𝜑(𝑒) = sup𝑛 𝜑(𝑒𝑛 ) and 𝑒 = s(𝜑). With Lemma 7.3, we infer that 𝑒 ≺ 𝑓. Let us now assume that 𝑒 is piecewise of countable type and let {𝑞𝑘 }𝑘∈𝐾 be a family of mutually orthogonal central projections, such that ∑𝑘∈𝐾 𝑞𝑘 = 1, such that 𝑒𝑞𝑘 is of countable type, for any 𝑘 ∈ 𝐾. By virtue of the first part of the proof, it follows that 𝑒𝑞𝑘 ≺ 𝑓𝑞𝑘 , for any 𝑘 ∈ 𝐾. By taking into account Proposition 4.2, we infer that 𝑒 ≺ 𝑓. Q.E.D. Corollary 7.5. Let M be a von Neumann algebra, 𝑓 ∈ PM and {𝑒𝑖 }𝑖∈𝐼 an increasingly directed family of projections in M . If ⋁𝑖∈𝐼 𝑒𝑖 is finite and if 𝑒𝑖 ≺ 𝑓, for any 𝑖 ∈ 𝐼, then ⋁𝑖∈𝐼 𝑒𝑖 ≺ 𝑓. Corollary 7.6. Let M be a von Neumann algebra, 𝑓 ∈ PM and {𝑒𝑖 }𝑖∈𝐼 an increasingly directed family of projections in M . If ⋁𝑖∈𝐼 𝑒𝑖 is finite, then (𝑒 ∧ 𝑓) = ( 𝑒𝑖 ) ∧ 𝑓. ⋁ 𝑖 ⋁ 𝑖∈𝐼

𝑖∈𝐼

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165

Finite von Neumann algebras Proof. Let 𝑒=

⋁

𝑒𝑖 ,

𝑔=

𝑖∈𝐼

⋁

(𝑒𝑖 ∧ 𝑓).

𝑖∈𝐼

We must show that 𝑒 ∧ 𝑓 = 𝑔. The relation 𝑞 ≤ 𝑒 ∧ 𝑓 is obvious. We now consider the projection ℎ = 𝑒 ∧ 𝑓 − 𝑔. The following computation, based on the parallelogram rule (4.4), shows that ℎ ≺ 𝑒 − 𝑒𝑖 : ℎ ≤ 𝑒 ∧ 𝑓 − 𝑒𝑖 ∧ 𝑓 = 𝑒 ∧ 𝑓 − (𝑒 ∧ 𝑓) ∧ 𝑒𝑖 ∼ (𝑒 ∧ 𝑓) ∨ 𝑒𝑖 − 𝑒𝑖 ≤ 𝑒 − 𝑒𝑖 . We now show that 𝑒𝑖 ≺ 𝑒 − ℎ, for any 𝑖. If this is not true, by taking into account the comparison theorem (4.6), we would find a central projection 𝑞 ≠ 0 and a projection 𝑔𝑖 ≤ 𝑞𝑒𝑖 , 𝑔𝑖 ≠ 𝑞𝑒𝑖 , such that 𝑞(𝑒 − ℎ) ∼ 𝑔𝑖 . On the other hand, from what we already proved, it follows that there exists a projection ℎ𝑖 ≤ 𝑞(𝑒 − 𝑒𝑖 ), such that 𝑞ℎ ∼ ℎ𝑖 . Consequently, we have 𝑞𝑒 ∼ 𝑔𝑖 + ℎ𝑖 ≤ 𝑞𝑒,

𝑔𝑖 + ℎ𝑖 ≠ 𝑞𝑒,

and this result contradicts the finiteness of 𝑞𝑒. According to Corollary 7.5, it follows that 𝑒=

⋁

𝑒𝑖 ≺ 𝑒 − ℎ.

𝑖∈𝐼

Since 𝑒 is finite, we infer that ℎ = 0.

Q.E.D.

7.7. Let L be a lattice. One says that L is a modular lattice if 𝑒, 𝑓, 𝑔 ∈ L ,

𝑒 ≤ 𝑔 ⇒ (𝑒 ∨ 𝑓) ∧ 𝑔 = 𝑒 ∨ (𝑓 ∧ 𝑔).

One says that L is a complemented lattice if it has a smallest element 0, a greatest element 1 and if, for any 𝑒 ∈ L , there exists an 𝑒′ ∈ L , such that 𝑒 ∧ 𝑒′ = 0, 𝑒 ∨ 𝑒′ = 1. One says that L is upper (resp., lower) continuous if 𝑓 ∈ L , {𝑒𝑖 }𝑖∈𝐼 ⊂ L increasingly (resp., decreasingly) directed and ⋁𝑖∈𝐼 𝑒𝑖 ∈ L (resp., ⋀𝑖∈𝐼 𝑒𝑖 ∈ L ) ⇒ ⋁𝑖∈𝐼 (𝑒𝑖 ∧ 𝑓) = (⋁𝑖∈𝐼 𝑒𝑖 ) ∧ 𝑓 (resp., ⋀𝑖∈𝐼 (𝑒𝑖 ∨ 𝑓) = (⋀𝑖∈𝐼 𝑒𝑖 ) ∨ 𝑓). One says that L is a continuous geometry if L is an upper and lower continuous, complemented, modular lattice. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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From Corollary 3.7, Theorem 7.1 and Corollary 7.6, the following theorem obtains: Theorem. Let M be a finite von Neumann algebra. Then the lattice PM is a continuous geometry. 7.8. We shall now construct the central trace on a finite von Neumann algebra. Let M be a von Neumann algebra and 𝜑 a w-continuous form on M . We shall consider the sets Q 𝜑 = {𝑇ᵆ 𝜑 ∶ 𝑢 ∈ M , unitary} ⊂ M ∗ , K 𝜑 = the norm closed convex hull of Q 𝜑 in M ∗ . Since M is the dual of the Banach space M ∗ , from the Mackey theorem, we now infer that the convex set K 𝜑 is 𝜎(M ∗ ; M )-closed. Lemma. Let M be a finite von Neumann algebra and 𝜑 a w-continuous form. Then the set K 𝜑 is 𝜎(M ∗ ; M )-compact. Proof. We must show that the set K 𝜑 is 𝜎(M ∗ ; M )-relatively compact. From the Akemann theorem (5.14), it is sufficient to show that, for any sequence {𝑒𝑛 } of orthogonal projections in M , we have lim 𝜓(𝑒𝑛 ) = 0, uniformly for 𝜓 ∈ K 𝜑 . 𝑛→∞

It is sufficient to prove the uniformity of this convergence with respect to 𝜓 ∈ Q 𝜑 . In order to prove this property, we shall assume that it is not true. Then there exists a 𝛿 > 0, a subsequence {𝑓𝑛 } of the sequence {𝑒𝑛 } and a sequence {𝑢𝑛 } of unitary operators in M , such that, by denoting 𝜓𝑛 = 𝑇ᵆ𝑛 𝜑, we should have |𝜓𝑛 (𝑓𝑛 )| ≥ 𝛿, for any 𝑛 = 1, 2, … We denote 𝑔𝑛 = 𝑢𝑛∗ 𝑓𝑛 𝑢𝑛 ∈ PM . Then we have (1)

𝑔𝑛 ∼ 𝑓𝑛 and |𝜑(𝑔𝑛 )| ≥ 𝛿, for any 𝑛 = 1, 2, …

We shall define 𝑚

ℎ𝑚,𝑛 =

⋁

𝑔𝑘 ;

𝑘=𝑛 ∞

ℎ𝑛 =

∞

⋁

𝑔𝑘 =

⋀

ℎ𝑛 .

𝑘=𝑛 ∞

ℎ=

1 ≤ 𝑛 ≤ 𝑚,

⋁

ℎ𝑚,𝑛 ;

𝑛 = 1, 2, …

𝑚=𝑛

𝑛=1

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167

Finite von Neumann algebras We shall show that 𝑘=𝑚

(2)

ℎ𝑚,𝑛 ≺ ∑ 𝑓𝑘 , for any 𝑚 ≥ 𝑛 ≥ 1. 𝑘=𝑛

We proceed by induction on 𝑚 ≥ 𝑛. For 𝑚 = 𝑛, we have ℎ𝑛,𝑛 = 𝑔𝑛 ∼ 𝑓𝑛 . If relation (2) is true for 𝑚 = 𝑟, then, since 𝑔𝑟+1 ∨ ℎ𝑟,𝑛 − ℎ𝑟,𝑛 ∼ 𝑔𝑟+1 − 𝑔𝑟+1 ∧ ℎ𝑟,𝑛 ≤ 𝑔𝑟+1 ∼ 𝑓𝑟+1 , it follows that

𝑘=𝑟

𝑘=𝑟+1

ℎ𝑟+1,𝑛 = 𝑔𝑟+1 ∨ ℎ𝑟,𝑛 ≺ ∑ 𝑓𝑘 + 𝑓𝑟+1 = ∑ 𝑓𝑘 . 𝑘=𝑛

𝑘=𝑛

For any 𝑛, the sequence {ℎ𝑚,𝑛 }𝑚 is increasing, and from relation (2), we infer that ∞

ℎ𝑚,𝑛 ≺ ∑ 𝑓𝑘 . 𝑘=𝑛

According to Lemma 7.3, we now infer that ∞

ℎ𝑛 ≺ ∑ 𝑓𝑘 . 𝑘=𝑛

By taking into account exercise E.4.9, we get ∞

1 − ∑ 𝑓𝑘 ≺ 1 − ℎ𝑛 ≤ 1 − ℎ, 𝑘=𝑛

and by applying again Lemma 7.3, it follows that ∞

1=

⋁

𝑛=1

∞

(1 − ∑ 𝑓𝑘 ) ≺ 1 − ℎ. 𝑘=𝑛

Consequently, ℎ = 0, since M is finite. ∞ Since the sequence {ℎ𝑛 } is decreasing, ⋀𝑛=1 ℎ𝑛 = ℎ = 0 and 𝑔𝑛 ≤ ℎ𝑛 , it follows that the sequence {𝑔𝑛 } is wo-convergent to 0. On the other hand, on the closed unit ball of M the wo-topology coincides with the w-topology, and therefore, the sequence {𝑔𝑛 } is w-convergent to 0. Consequently, we have that lim𝑛 𝜑(𝑔𝑛 ) = 0, which contradicts relation (1). Q.E.D. 7.9. One calls a central form on an algebra A any form 𝜑, such that 𝜑(𝑥𝑦) = 𝜑(𝑦𝑥), for any 𝑥, 𝑦 ∈ A . A form 𝜑 on a 𝐶 ∗ -algebra A ∋ 1 is central iff it is unitarily invariant, i.e., 𝑇ᵆ 𝜑 = 𝜑, for any unitary 𝑢 ∈ A . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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Lemma. Let 𝜑 be a w-continuous central form on the von Neumann algebra M , whose center is Z . Then ‖𝜑‖ = ‖𝜑|Z ‖. In particular, 𝜑 ≥ 0 iff 𝜑|Z ≥ 0. Proof. Let 𝜑 = 𝑅𝑣 |𝜑| be the polar decomposition (5.16) of 𝜑. From the equality 𝜑 = 𝑅ᵆ𝑣ᵆ∗ 𝑇ᵆ |𝜑|, which is true for any unitary 𝑢 ∈ M , and from the uniqueness of the polar decomposition, it follows that 𝑣 and |𝜑| are unitarily invariant. It follows that 𝑣 ∈ Z and |𝜑| is central. Consequently, we have ‖𝜑‖ = ‖ |𝜑| ‖ = |𝜑|(1) = 𝜑(𝑣 ∗ ) ≤ ‖𝜑|Z ‖ ⋅ ‖𝑣 ∗ ‖ ≤ ‖𝜑‖; hence ‖𝜑‖ = ‖𝜑|Z ‖. The second assertion follows from the first, by taking into account Proposition 5.4. Q.E.D. Lemma 7.10. Let M be a finite von Neumann algebra, Z its centre. Then any w-continuous form 𝜔 on Z uniquely extends to a bounded central form 𝜑𝜔 on M . Moreover, 𝜑𝜔 is w-continuous, ‖𝜑𝜔 ‖ = ‖𝜔‖ and 𝜔 ≥ 0 implies 𝜑𝜔 ≥ 0. Proof. The uniqueness part of the lemma, as well as the two last assertions of the Lemma, follow from Lemma 7.9. In order to prove the existence and the w-continuity of the form 𝜑𝜔 , we first consider a w-continuous form 𝜑 on M , such that 𝜑|Z = 𝜔 (see Theorem 1.10). We now apply the Ryll–Nardzewski fixed point theorem (see Theorem A.3 in the Appendix) for the following particular case. X = M ∗ in the uniform (norm) topology is a separated locally convex vector space, whose dual is M . K = K 𝜑 is a weakly compact, convex, non-empty subset of X (in accordance with Lemma 7.8). E = {𝑇ᵆ |K ; 𝑢 ∈ M , unitary} is a non-contracting semi-group of weakly continuous affine mappings of K into K , since any 𝑇ᵆ is a linear isometry of M ∗ onto M ∗ . It follows that there exists a 𝜑𝜔 ∈ K 𝜑 ⊂ M ∗ , such that 𝑇ᵆ 𝜑𝜔 = 𝜑𝜔 , for any unitary 𝑢 ∈ M . Consequently, 𝜑𝜔 is a w-continuous central form on M . On the other hand, since 𝜑|Z = 𝜔, it follows that 𝜓|Z = 𝜔, for any 𝜓 ∈ K 𝜑 ; in particular, we have 𝜑𝜔 |Z = 𝜔. Q.E.D.

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Theorem 7.11. Let M be a von Neumann algebra, Z its centre. Then M is finite iff there exists a mapping ♮ ∶ M ∋ 𝑥 ↦ 𝑥♮ ∈ Z , having the following properties: (i) ♮ is linear and bounded. (ii) (𝑥𝑦)♮ = (𝑦𝑥)♮ , for any 𝑥, 𝑦 ∈ M . (iii) 𝑧♮ = 𝑧, for any 𝑧 ∈ Z . The mapping ♮ having properties (i)–(iii) is unique. Moreover, the mapping ♮ also has the following properties. (iv) ‖♮‖ = 1. (v) ♮ is w-continuous. (vi) (𝑧𝑥)♮ = 𝑧𝑥 ♮ , for any 𝑥 ∈ M , 𝑧 ∈ Z . (vii) 𝑥 ∈ M , 𝑥 ≥ 0 ⇒ 𝑥 ♮ ≥ 0. (viii) 𝑥 ∈ M , 𝑥 ≥ 0, 𝑥 ♮ = 0 ⇒ 𝑥 = 0. (ix) 𝑥 ♮ ∈ 𝑐𝑜 n {𝑢𝑥𝑢∗ ; 𝑢 ∈ M , unitary} for every 𝑥 ∈ M . Proof. It is easy to see that the existence of the mapping ♮ implies the finiteness of the von Neumann algebra M . Let us now assume that the von Neumann algebra M is finite. We shall first prove the uniqueness of the mapping ♮. In order to do this, it is sufficient to prove that for any w-continuous form 𝜔 on Z , we have 𝜔(𝑥 ♮ ) = 𝜑𝜔 (𝑥), where 𝜑𝜔 is the unique bounded central form on M , such that 𝜑𝜔 |Z = 𝜔 (7.10). But this fact is obvious, since from conditions (i)–(iii), we infer that the mapping 𝑥 ↦ 𝜔(𝑥 ♮ ),

𝑥 ∈ M,

is a bounded central form on M , which extends 𝜔. We now prove the existence of a mapping ♮, having properties (i)–(ix). In accordance with Lemma 7.10, the mapping 𝐸 ∶ Z ∗ ↦ M ∗ , defined by 𝐸𝜔 = 𝜑𝜔 ,

𝜔 ∈ Z ∗,

is linear and isometric. By taking into account the canonical identifications M = (M ∗ )∗ , Z = (Z ∗ )∗ , we now define the mapping ♮ as being the transpose of the mapping 𝐸, ♮ = 𝑡 𝐸 ∶ M → Z . In other words, the mapping ♮ is determined by the relations: 𝜔(𝑥 ♮ ) = 𝜑𝜔 (𝑥),

𝜔 ∈ Z ∗, 𝑥 ∈ M .

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Properties (i)–(iv) and (vii) are easily verified. Since 𝑡𝐸 is weakly continuous, ♮ is w-continuous; hence property (v) is established. It is now sufficient to prove property (vi) only for the unitary elements 𝑧 ∈ Z . Let 𝑤 ∈ Z be unitary and let us define the mapping 𝑊 ∶ M → Z by 𝑊(𝑥) = 𝑤 ∗ (𝑤𝑥)♮ , 𝑥 ∈ M . Then 𝑊 satisfies conditions (i)–(iii); hence, in view of the uniqueness, we have 𝑊 = ♮. Hence 𝑤 ∗ (𝑤𝑥)♮ = 𝑥 ♮ , i.e., (𝑤𝑥)♮ = 𝑤𝑥 ♮ , for any 𝑥 ∈ M . Let us now prove property (viii). Let 𝑥 ∈ M , 𝑥 ≥ 0, 𝑥 ≠ 0. There then exists a non-zero positive normal form 𝜔 on Z , such that 𝑝 = s(𝜔) ≤ z(𝑥). Since 𝜑𝜔 is a normal positive central form (7.10), it is easy to see that s(𝜑𝜔 ) is unitarily invariant; hence it is a central projection, whence s(𝜑𝜔 ) = 𝑝. The relation 𝜑𝜔 (𝑥) = 0 implies 𝑥𝑝 = 0, and this is not possible, because 0 ≠ 𝑝 ≤ z(𝑥). Consequently, we have 𝜑𝜔 (𝑥) ≠ 0, whence 𝑥 ♮ ≠ 0. The last assertion (ix) follows using the fact that 𝜑(𝑥) = 𝜑(𝑥 ♮ ) for every bounded central form 𝜑 on M , the proof of Lemma 7.10 and the Hahn–Banach theorem. Q.E.D. 7.12. If M is a finite von Neumann algebra, the mapping ♮, introduced by Theorem 7.11, is also called the canonical central trace on M . Let 𝑒, 𝑓 ∈ PM . Then 𝑒 ≺ 𝑓 (resp., 𝑒 ∼ 𝑓) iff 𝑒♮ ≤ 𝑓 ♮ (resp., 𝑒♮ = 𝑓 ♮ ). Indeed, let us assume that 𝑒♮ ≤ 𝑓 ♮ . From the comparison theorem (4.6), there exists a projection 𝑝 ∈ Z , such that 𝑒𝑝 ≺ 𝑓𝑝, 𝑒(1−𝑝) ≻ 𝑓(1−𝑝). By taking into account the properties of the mapping ♮ and, especially, property (viii), it follows that 𝑒(1 − 𝑝) ∼ 𝑓(1 − 𝑝) and hence 𝑒 ≺ 𝑓. In particular, if M is a finite factor, the mapping ♮ has scalar values. The restriction of the mapping ♮ to PM is also denoted by 𝑑, and it is called the normalized dimension function on PM . Two projections 𝑒, 𝑓 ∈ M are equivalent iff they have the same dimension: 𝑑(𝑒) = 𝑑(𝑓)! 7.13. Let M be a von Neumann algebra and M + = {𝑥 ∈ M ; 𝑥 ≥ 0}. One calls a trace on M + any function 𝜇 ∶ M + → [0, +∞], having the properties 𝜇(𝑥 + 𝑦) = 𝜇(𝑥) + 𝜇(𝑦),

𝑥, 𝑦 ∈ M + , 𝑥 ∈ M + , 𝜆 ≥ 0,

𝜇(𝜆𝑥) = 𝜆𝜇(𝑥),

𝑥 ∈ M.

𝜇(𝑥 ∗ 𝑥) = 𝜇(𝑥𝑥 ∗ ), Then 𝜇 obviously is unitarily invariant. One says that a trace 𝜇 on M + is faithful if

𝑥 ∈ M + , 𝜇(𝑥) = 0 ⇒ 𝑥 = 0. One says that a trace 𝜇 on M + is normal if for any family {𝑥𝑖 }𝑖∈𝐼 ⊂ M + , which is increasingly directed and bounded, one has that 𝜇(sup 𝑥𝑖 ) = sup 𝜇(𝑥𝑖 ). 𝑖∈𝐼

𝑖∈𝐼

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One says that a trace 𝜇 on M + is finite if 𝜇(𝑥) ≤ +∞, for any 𝑥 ∈ M . The restriction to M + of any positive central form on M is a finite trace on M + . Conversely, any finite trace on M + uniquely extends to a positive central form on M . One says that a trace 𝜇 on M + is semifinite if for any 0 ≠ 𝑥 ∈ M + , there exists a 𝑦 ∈ M + , 𝑦 ≠ 0, 𝑦 ≤ 𝑥, such that 𝜇(𝑦) < +∞. If 𝜇 is a normal semifinite trace on M + , then for any 𝑥 ∈ M + , we have 𝜇(𝑥) = sup{𝜇(𝑦); 𝑦 ≤ 𝑥, 𝜇(𝑦) < +∞}. Indeed, if 𝜇(𝑥) < +∞, the assertion if obvious. If 𝜇(𝑥) = +∞, one considers a maximal totally ordered family {𝑦i }i∈𝐼 ⊂ M + , such that 0 ≠ 𝑦i ≤ 𝑥 and 𝜇(𝑦i ) < +∞, and then one easily proves that 𝜇(supi∈𝐼 𝑦i ) = +∞. One says that a family of traces {𝜇𝑘 }𝑘∈𝐾 on M + is sufficient if for any 𝑥 ∈ M + , 𝑥 ≠ 0, there exists a 𝑘 ∈ 𝐾, such that 𝜇𝑘 (𝑥) ≠ 0. One defines the support s(𝜇) of a normal trace 𝜇 on M + as being the projection complementary to the greatest projection in M , which is annihilated by 𝜇. Since 𝜇 is unitarily invariant, s(𝜇) is a central projection. The normal trace 𝜇 is faithful iff s(𝜇) = 1. A family {𝜇𝑘 }𝑘∈𝐾 of normal traces is sufficient iff ⋁𝑘∈𝐾 s(𝜇𝑘 ) = 1. If a von Neumann algebra possesses a sufficient family of semifinite normal traces, then it possesses a faithful semifinite normal trace. Corollary 7.14. A von Neumann algebra is finite iff it possesses a sufficient family of finite normal traces. Corollary 7.15. A von Neumann algebra is semifinite iff it possesses a faithful semifinite normal trace. Proof. Let M be a von Neumann algebra and let 𝜇 be a faithful semifinite normal trace on M + . For any central projection 0 ≠ 𝑝 ∈ M , there exists an element 𝑥 ∈ M + , 𝑥 ≠ 0, 𝑥 ≤ 𝑝, such that 𝜇(𝑥) < +∞. There then exists a projection 𝑒 ∈ M , 𝑒 ≠ 0, and an 𝜀 > 0, such that 𝑒 commutes with 𝑥 and 𝑒𝑥 ≥ 𝜀𝑒 (see Corollary 2.22). Then 1 1 𝜇(𝑒) ≤ 𝜇(𝑥𝑒) ≤ 𝜇(𝑥) < +∞. Thus 𝜇(𝑒) < +∞, and this implies that the projection 𝜀 𝜀 𝑒 is finite. Hence, M is semifinite. Conversely, let M be a semifinite von Neumann algebra. In order to show that M possesses a faithful semifinite normal trace, it is sufficient to show that M possesses a sufficient family of semifinite normal traces. We can assume that M is a uniform von Neumann algebra, i.e., there exists a family {𝑒𝑖 }𝑖∈𝐼 of equivalent finite mutually orthogonal projections, such that ∑𝑖∈𝐼 𝑒𝑖 = 1 (see exercise E.4.14). Let 𝑒0 be one of these projections and, for any 𝑖 ∈ 𝐼, let 𝑣𝑖 ∈ M be a partial isometry such that 𝑣𝑖∗ 𝑣𝑖 = 𝑒0 , 𝑣𝑖 𝑣𝑖∗ = 𝑒𝑖 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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We now define, for any 𝑥 ∈ M , 𝑥𝑖𝑘 = 𝑣𝑖∗ 𝑥𝑣𝑘 ∈ 𝑒0 M 𝑒0 ,

𝑖, 𝑘 ∈ 𝐼.

Then we have 𝑥 = ∑ 𝑒𝑖 𝑥𝑒𝑘 = ∑ 𝑣𝑖 𝑥𝑖𝑘 𝑣𝑘∗ = (𝑥𝑖𝑘 ), 𝑖,𝑘∈𝐼

𝑖,𝑘∈𝐼

where the last equality is a notation. One says that 𝑥𝑖𝑘 is the (𝑖, 𝑘)-component in the matrix representation with respect to the basis {𝑣𝑖 𝑣𝑘∗ }𝑖,𝑘∈𝐼 . It is easy to see that ∗ (𝑥∗ )𝑖𝑘 = 𝑥𝑘𝑖 and (𝑥𝑦)𝑖𝑘 = ∑ 𝑥𝑖𝑗 𝑥𝑗𝑘 . 𝑗∈𝐼

The von Neumann algebra 𝑒0 M 𝑒0 is finite. For any finite normal trace 𝜇0 on (𝑒0 M 𝑒0 )+ , we define a function on M + by 𝜇(𝑥) = ∑ 𝜇0 (𝑥𝑖𝑖 ),

𝑥 ∈ M +.

𝑖∈𝐼

It is easily verified that 𝜇 is a semifinite normal trace on M + and that the set of all semifinite normal traces on M + , obtained in this manner, is sufficient. Q.E.D. 7.16. A von Neumann algebra M is said to be homogeneous of type 𝐼𝑛 if there exists a family {𝑒𝑖 }𝑖∈𝐼 of equivalent abelian mutually orthogonal projections, such that ∑𝑖∈𝐼 𝑒𝑖 = 1, and card 𝐼 = 𝑛. In this case, M is of type I and M is finite iff 𝑛 is finite. Conversely, for any finite von Neumann algebra, of type 𝐼, there exists a family {𝑝𝑛 }𝑛=1,2,… of mutually ∞ orthogonal central projections, such that ∑𝑛=1 𝑝𝑛 = 1, uniquely determined by the condition that 𝑀𝑝𝑛 be of type 𝐼𝑛 , 𝑛 = 1, 2, … (see exercise E.4.14). Proposition. Let M be a von Neumann algebra of type 𝐼𝑛 , 𝑛 finite. Then (PM )♮ coincides with the set of all elements of the form 𝑛

𝑘 𝑞 , 𝑛 𝑘 𝑘=1 ∑

where 𝑞1 , … , 𝑞𝑛 are mutually orthogonal central projections. Proof. Let 𝑒0 ∈ M be an abelian projection, such that z(𝑒0 ) = 1 (see Proposition 4.19). Since M is homogeneous, of type 𝐼𝑛 , there exists a family of 𝑛 abelian, mutually orthogonal projections, equivalent to 𝑒0 , whose sum equals 1. Consequently, we have (𝑒0 𝑞)♮ =

1 𝑞. 𝑛

for any central projection 𝑞. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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Therefore, it is obvious that any operator of the form given in the statement of the theorem belongs to (PM )♮ . Let now 𝑒 ∈ PM and let {𝑒1 , … , 𝑒𝑘 } be a maximal family of subprojections of 𝑒, which are mutually orthogonal and equivalent to 𝑒0 z(𝑒). There then exists a non-zero central projection 𝑞 ≤ z(𝑒), such that 𝑘

𝑞𝑒 = ∑ 𝑞𝑒𝑖 𝑖=1

(see Proposition 4.10). It follows that (𝑞𝑒)♮ =

𝑘 𝑞. 𝑛

Consequently, there exists a family {𝑞𝑖 }𝑖∈𝐼 , of mutually orthogonal central projections, such that ∑𝑖∈𝐼 𝑞𝑖 = z(𝑒), and for any 𝑖 ∈ 𝐼, there exists a natural 1 ≤ 𝑘𝑖 ≤ 𝑛, such that (𝑞𝑖 𝑒)♮ =

𝑘𝑖 𝑞, 𝑛 𝑖

𝑖 ∈ 𝐼.

We define 𝑞𝑘 =

∑

𝑞𝑖 ,

𝑘 = 1, 2, … , 𝑛.

𝑖∈𝐼,𝑘𝑖 =𝑘

Then we have

𝑛

♮

𝑘 𝑞𝑘 . 𝑛 𝑘=1

𝑒 = ∑

Q.E.D. 7.17. In the case of a von Neumann algebra of type II1 , we have the following ‘Darboux property’ for the restriction of the mapping ♮ to PM . Proposition. Let M be a von Neumann algebra of type II1 . For any 𝑒, 𝑓 ∈ PM , and any 𝑧 ∈ Z , such that 𝑒♮ ≤ 𝑧 ≤ 𝑓 ♮ , there exists a 𝑔 ∈ PM , such that 𝑒 ≤ 𝑔 ≤ 𝑓 and 𝑔♮ = 𝑧. Corollary. Let M be a von Neumann algebra of type II1 . Then (PM )♮ coincides with the set of all elements: 𝑧 ∈ Z , 0 ≤ 𝑧 ≤ 1. Proof. We shall first show that for any 𝑒 ∈ PM , 𝑒 ≠ 0, and any 𝜀 > 0, there exists an 𝑒𝜀 ∈ ♮ PM , 0 ≠ 𝑒𝜀 ≤ 𝑒, such that 𝑒𝜀 ≤ 𝜀z(𝑒𝜀 ). Indeed, by taking into account Proposition 4.11, for any 𝑛 = 1, 2, …, we can find a family {𝑒1𝑛 , … , 𝑒2𝑛𝑛 } of equivalent, mutually orthogonal, 1 non-zero subprojections of 𝑒, whose sum is 𝑒. Then (𝑒1𝑛 )♮ = 𝑛 𝑒♮ and so it is sufficient to 2 choose 𝑛, such that 1/2𝑛 ≤ 𝜀. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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Let now E be a maximal totally ordered family of projections ℎ in M , such that 𝑒 ≤ ℎ ≤ 𝑓,

ℎ♮ ≤ 𝑧.

We denote by 𝑔 the least upper bound of the family E . It is obvious that 𝑒 ≤ 𝑔 ≤ 𝑓 and 𝑔♮ ≤ 𝑧. If 𝑧 − 𝑔♮ ≠ 0, then there exists an 𝜀 > 0 and a central projection 𝑝 ≠ 0, such that (𝑧 − 𝑔♮ )𝑝 ≥ 𝜀𝑝. It follows that (𝑓 −𝑔)𝑝 ≠ 0, since, if this is not true, then 𝑔♮ 𝑝 = 𝑓 ♮ 𝑝 ≥ 𝑧𝑝, and this relation contradicts the preceding one. In accordance with the first part of the proof, there exists an ♮ 𝑒𝜀 ∈ PM , 0 ≠ 𝑒𝜀 ≤ (𝑓 − 𝑔)𝑝, such that 𝑒𝜀 ≤ 𝜀𝑝. But the existence of the element 𝑔 + 𝑒𝜀 contradicts the maximality of the family E . Consequently, we have 𝑧 = 𝑔♮ . Q.E.D. 7.18. If M is a finite von Neumann algebra whose commutant M ′ is finite, then there exists a remarkable connection between the canonical central traces on M and M ′ . In order to establish the connection, the following lemma is necessary: Lemma. Let M ⊂ B (H ) be a von Neumann algebra of countable type. Then there exist 𝑝, 𝑞 ∈ P Z , 𝑝𝑞 = 0, 𝑝 + 𝑞 = 1 and 𝜉, 𝜂 ∈ H , such that 𝑝 = 𝑝𝜉 , 𝑞 = 𝑝𝜂′ . Proof. Let {𝜉𝑛 )𝑛∈𝐼 ⊂ H , ‖𝜉𝑛 ‖ = 1, be a maximal family, such that the projections 𝑝𝜉𝑛 , 𝑛 ∈ 𝐼, are mutually orthogonal and the projections 𝑝𝜉′ 𝑛 , 𝑛 ∈ 𝐼, are also mutually orthogonal. Since M is of countable type, it follows that 𝐼 is at most countable. We denote 1 𝑒 = ∑𝑛 𝑝𝜉𝑛 , 𝑒′ = ∑𝑛 𝑝𝜉′ 𝑛 . If we define 𝜉0 = ∑𝑛 𝑛 𝜉𝑛 , it follows that 𝑒 = 𝑝𝜉0 , and 2 𝑒′ = 𝑝𝜉′ 0 . On the other hand, from the maximality of the family {𝜉𝑛 }, we infer that (1 − 𝑒)(1 − 𝑒′ ) = 0. By taking into account Corollary 3.9, we obtain z(1 − 𝑒)z(1 − 𝑒′ ) = 0. Let us denote 𝑝 = 1 − z(1 − 𝑒), 𝑞 = z(1 − 𝑒). Then, from the relations 𝑝 = 1 − z(1 − 𝑒) ≤ 𝑒 𝑞 = z(1 − 𝑒) ≤ 1 − z(1 − 𝑒′ ) ≤ 𝑒′ , Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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𝑝 = 𝑝𝜉 ,

Q.E.D. Theorem 7.19 (of coupling). Let M ⊂ B (H ) be a finite von Neumann algebra, whose commutant M ′ ⊂ B (H ) is also finite. Let ♮, ♮′ be the canonical central traces on M , M ′ , respectively. Then, for any 𝜉, 𝜂 ∈ H , we have ′

′

(𝑝𝜉 )♮ (𝑝𝜂′ )♮ = (𝑝𝜂 )♮ (𝑝𝜉′ )♮ . Proof. If {𝑞𝑖 }𝑖∈𝐼 is a family of mutually orthogonal central projections, such that ∑𝑖∈𝐼 𝑞𝑖 = 1, then it is sufficient to prove the result in the statement of the theorem for each of the von Neumann algebras M 𝑞𝑖 ⊂ B (𝑞𝑖 H ). In accordance with Theorem 4.17, and with exercise E.4.14, we can assume that M and ′ M are either of type II1 or of homogeneous type 𝐼fin . By Lemma 7.2, we may assume that M is of countable type. Since M is of countable type, Lemma 7.18 can be applied. Consequently, without any loss of generality, we can assume that there exists a vector 𝜉0 ∈ H , such that 𝑝𝜉0 = 1. In this case, we shall show that, for any 𝜉 ∈ H , we have ′

′

(𝑝𝜉′ )♮ = (𝑝𝜉′ 0 )♮ (𝑝𝜉 )♮ , and the theorem will be proved. ′ Since s((𝑝𝜉′ 0 )♮ ) = z((𝑝𝜉′ 0 )) = z(𝑝𝜉0 ) = 1, from Corollary 2.22, we infer that there 1

′

exists a sequence {𝑞𝑛 } of central projections, 𝑞𝑛 ↑ 1, such that (𝑝𝜉′ 0 )♮ 𝑞𝑛 ≥ 𝑞𝑛 , for any 𝑛. 𝑛 Without any loss of generality, we can assume that there exists an 𝜀 > 0, such that ′

(𝑝𝜉′ 0 )♮ ≥ 𝜀. We denote by C (resp., by C ′ ) the set of all elements 𝑧 ∈ Z , such that there exists a ′ 𝜉 ∈ H , for which 𝑧 = (𝑝𝜉 )♮ (resp., 𝑧 = (𝑝𝜉′ )♮ ). Since any projection that is dominated by a cyclic projection is cyclic and since any cyclic projection in M ′ is dominated by 𝑝𝜉′ 0 (since 𝑝𝜉0 = 1 and as a result of Lemma 6.1), from 7.12, we infer that ′

′

C = (PM )♮ and C ′ = {𝑧 ∈ (PM ′ )♮ ; 𝑧 ≤ (𝑝𝜉′ 0 )♮ }. By taking into account Propositions 7.16 and 7.17, the structure of the sets C and C ′ becomes evident in the case of the type II1 , as well as in the case of the homogeneous type Ifin . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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We now define a mapping Ψ0 ∶ C → C ′ by the relations ′

Ψ0 ((𝑝𝜉 )♮ ) = (𝑝𝜉′ )♮ ,

𝜉∈H.

By virtue of Lemma 6.1 and of Section 7.12, this mapping is correctly defined, injective, surjective and both it, as well as its inverse, preserve the order relation. Since z(𝑝𝜉′ ) = z(𝑝𝜉 ), it follows that s(Ψ0 (𝑧)) = s(𝑧), 𝑧 ∈ C . ′ ′ In what follows we shall need to note: if 𝑒 ∈ PM , 𝑒′ ∈ PM and Ψ0 (𝑒♮ ) = (𝑒′ )♮ , then there exists a 𝜉 ∈ H , such that 𝑒 = 𝑝𝜉 , 𝑒′ = 𝑝𝜉′ . Indeed, there exists an 𝜂 ∈ H , such ′

′

that 𝑒 = 𝑝𝜂 ; hence, (𝑝𝜂′ )♮ = Ψ0 ((𝑝𝜂 )♮ ) = Ψ0 (𝑒♮ ) = (𝑒′ )♮ . According to 7.12, we infer that 𝑒′ ∼ 𝑝𝜂′ , and now the desired result can be obtained by applying exercise E.6.1. If 𝑧1 , 𝑧2 , 𝑧1 + 𝑧2 ∈ C , and Ψ0 (𝑧1 ) + Ψ0 (𝑧2 ) ∈ C ′ , then Ψ0 (𝑧1 + 𝑧2 ) = Ψ0 (𝑧1 ) + Ψ0 (𝑧2 ). Indeed, there exist 𝑒1 , 𝑒2 ∈ PM , 𝑒1′ , 𝑒2′ ∈ PM ′ , such that 𝑒1′ 𝑒2′ = 0,

𝑒1 𝑒2 = 0,

′

𝑧1 = (𝑒1 )♮ ,

Ψ0 (𝑧1 ) = (𝑒1′ )♮ ,

𝑧2 = (𝑒2 )♮ ,

Ψ0 (𝑧2 ) = (𝑒2′ )♮ .

′

In view of the preceding remark, there exist 𝜉1 , 𝜉2 ∈ H , such that 𝑒1 = 𝑝𝜉1 ,

𝑒1′ = 𝑝𝜉′ 1 ,

𝑒2 = 𝑝𝜉2 ,

𝑒2′ = 𝑝𝜉′ 2 ,

It is now easy to see that, if we denote 𝜉 = 𝜉1 + 𝜉2 , we have 𝑒1 + 𝑒2 = 𝑝𝜉 ,

𝑒1′ + 𝑒2′ = 𝑝𝜉′ .

Consequently, we have Ψ0 (𝑧1 + 𝑧2 ) = Ψ0 ((𝑒1 )♮ + (𝑒2 )♮ ) = Ψ0 ((𝑒1 + 𝑒2 )♮ ) ′

= Ψ0 ((𝑝𝜉 )♮ ) = (𝑝𝜉′ )♮ = (𝑒1′ + 𝑒2′ )♮ ′

′

′

= (𝑒1′ )♮ + (𝑒2′ )♮ = Ψ0 (𝑧1 ) + Ψ0 (𝑧2 ). If 𝜆 ≥ 0, 𝑧 ∈ C , 𝜆𝑧 ∈ C and 𝜆Ψ0 (𝑧) ∈ C ′ , then Ψ0 (𝜆𝑧) = 𝜆Ψ0 (𝑧). Indeed, for rational 𝜆, the result can be obtained from the additivity property we have just proved. Then, for an arbitrary 𝜆 > 0, the desired result can be obtained by taking into account the monotony property of the mapping Ψ0 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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Let L (resp., L ′ ) be the vector space generated in Z by C (resp., by C ′ ). In view of the additivity and homogeneity properties of the mapping Ψ0 , it follows that there exists a unique linear mapping Ψ ∶ L → L ′, which extends Ψ0 . We note that s(Ψ(𝑧)) ≤ s(𝑧), 𝑧 ∈ L . ′ Since Ψ(1) = (𝑝𝜉′ 0 )♮ ≥ 𝜀, it follows that Ψ(1) is invertible. We now define the mapping Φ∶L →Z by the relations

1 Ψ(𝑧), 𝑧 ∈ L . Ψ(1) In the case of the homogeneous type 𝐼fin , the set L consists of all linear combinations with rational coefficients, which have the same denominator, of central projections, and Φ is a linear mapping such that Φ(1) = 1. In the case of the type II1 , we have L = Z , and Φ is a positive linear mapping of Z into Z , such that Φ(1) = 1; it is easily verified that Φ is bounded (0 ≤ 𝑧 ≤ 1 ⇒ 0 ≤ Φ(𝑧) ≤ 1, whence ‖Φ‖ ≤ 4). In each case, we have s(Φ(𝑧)) ≤ s(𝑧). Let 𝑞 be a central projection. Since 1 = 𝑞 + (1 − 𝑞), it follows that Φ(𝑧) =

1 = Φ(𝑞) + Φ(1 − 𝑞). If we multiply this relation by 𝑞, we get Φ(𝑞) = 𝑞. Consequently, Φ is the identity mapping. Then, for any vector 𝜉 ∈ H , we have ′

′

(𝑝𝜉′ )♮ = Ψ((𝑝𝜉 )♮ ) = Ψ(1)Φ((𝑝𝜉 )♮ ) = (𝑝𝜉′ 0 )♮ (𝑝𝜉 )♮ . Q.E.D. 7.20. Let Q be the set of all families {(𝑧𝑖 , 𝑞𝑖 )}𝑖∈𝐼 , where 𝑞𝑖 are mutually orthogonal central projections, such that ∑𝑖∈𝐼 𝑞𝑖 = 1, 𝑧𝑖 ∈ Z , and s(|𝑧𝑖 |) ≤ 𝑞𝑖 , 𝑖 ∈ 𝐼. We shall say that the elements {(𝑧𝑖 , 𝑞𝑖 )}𝑖∈𝐼 and {(𝑧𝑘′ , 𝑞𝑘′ )}𝑘∈𝐾 are equivalent if for any 𝑖 ∈ 𝐼 and 𝑘 ∈ 𝐾, we have 𝑧𝑖 𝑞𝑘′ = 𝑧𝑘′ 𝑞𝑖 (they ‘coincide on intersections’). We shall denote Z ̃ by the quotient set of Q by the preceding equivalence relation and by {(𝑧𝑖 , 𝑞𝑖 )}∼𝑖∈𝐼 the equivalence class of {(𝑧𝑖 , 𝑞𝑖 )}𝑖∈𝐼 . We shall define the following operations on Z :̃ {(𝑧𝑖 , 𝑞𝑖 )}∼𝑖∈𝐼 + {(𝑧𝑘′ , 𝑞𝑘′ )}∼𝑘∈𝐾 = {(𝑧𝑖 𝑞𝑘′ + 𝑧𝑘′ 𝑞𝑖 , 𝑞𝑖 𝑞𝑘′ )}∼(𝑖,𝑘)∈𝐼×𝐾 𝜆{(𝑧𝑖 , 𝑞𝑖 )}∼𝑖∈𝐼 = {(𝜆𝑧𝑖 , 𝑞𝑖 )}∼𝑖∈𝐼 . {(𝑧𝑖 , 𝑞𝑖 )}∼𝑖∈𝐼 {(𝑧𝑘′ , 𝑞𝑘′ )}∼𝑘∈𝐾 = {(𝑧𝑖 𝑧𝑘′ , 𝑞𝑖 𝑞𝑘′ )}∼(𝑖,𝑘)∈𝐼×𝐾 ({(𝑧𝑖 , 𝑞𝑖 )}∼𝑖∈𝐼 )∗ = {(𝑧𝑖∗ , 𝑞𝑖 )}∼𝑖∈𝐼 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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It is easily verified that Z ,̃ endowed with the operations already defined, is an involutive algebra with the unit element. Thus, the notions of a positive element and of an invertible element make sense. The mapping 𝑧 ↦ {(𝑧, 1)}∼ is an injective *-homomorphism of Z into Z ;̃ hence Z ̃ is an extension of Z . From the coupling theorem (7.19), the following corollary easily obtains: Corollary. Let M ⊂ B (H ) be a finite von Neumann algebra, whose commutant M ′ ⊂ B (H ) is also finite. Then there exists an element 𝔠M ,M ′ ∈ Z ,̃ which is positive and invertible, such that for any 𝜉 ∈ H we have ′

(𝑝𝜉′ )♮ = 𝔠M ,M ′ (𝑝𝜉 )♮ . The element 𝔠M ,M ′ is uniquely determined by this condition. If M ⊂ B (H ) is a finite factor, whose commutant M ′ ⊂ B (H ) is also finite, then Z and Z ̃ coincide with the field ℂ of the scalars; hence 𝔠M ,M ′ is a non-zero positive number. The element 𝔠M ,M ′ is called the coupling element or the coupling function; in the case of factors, it is called the coupling constant. Obviously, we have 𝔠M ′ ,M = (𝔠M ,M ′ )−1 . If 𝑝 is a central projection, then the centre of M 𝑝 obviously identifies with Z 𝑝 . If we assume that this identification is already performed, we have 𝔠M 𝑝 ,M ′𝑝 = (𝔠M ,M ′ )𝑝. 7.21. Let M be a finite von Neumann algebra, whose commutant M ′ is also finite, and let Z be their centre. Let 𝑒′ ∈ M ′ be a projection whose central support z(𝑒′ ) = 1. In accordance with Proposition 3.14, the canonical induction M → M 𝑒′ is a *-isomorphism, and this fact allows for the canonical identification of the common centre of the algebras M 𝑒′ , and M ′𝑒 , with Z . This identification induces a canonical identification of the extension of the common centre of the algebras M 𝑒 , M ′𝑒′ , with Z ̃ (see 7.20). We shall assume that these identifications have been performed. Let now ♮, ♮′ .♮𝑒′ , ♮′𝑒′ be the canonical central traces on M , M ′ , M 𝑒′ , M ′𝑒′ . Then for any 𝑥 ∈ M , we have (𝑥𝑒′ )♮𝑒′ = 𝑥 ♮ , and for any, 𝑥′ ∈ M ′ , ′

′

′

(𝑥𝑒′ ′ )♮𝑒′ = ((𝑒′ )♮ )−1 (𝑒′ 𝑥 ′ 𝑒′ )♮ .

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′

The first relation is obvious. For the second, we first remark that, since z(𝑒′ ) = 1, (𝑒′ )♮ is an invertible element in Z ̃ (see 2.22). Then, one sees that, for any 𝑥′ ∈ M ′ , we have ′

′

((𝑒′ )♮ )−1 (𝑒′ 𝑥 ′ 𝑒′ )♮ ∈ Z . Finally, the mapping ′

′

M ′𝑒′ ∋ 𝑥𝑒′ ′ ↦ ((𝑒′ )♮ )−1 (𝑒′ 𝑥′ 𝑒′ )♮ ∈ Z satisfies the conditions (i), (ii) and (iii) from Theorem 7.11, and therefore, it coincides with the canonical central trace on M ′𝑒′ . With these preparations, we can now state the following: Proposition. Let M ⊂ B (H ) be a finite von Neumann algebra, whose commutant M ′ ⊂ B (H ) is finite. If 𝑒′ ∈ M ′ is a projection whose central support z(𝑒′ ) = 1, then ′

𝔠M 𝑒′ ,M ′ ′ = ((𝑒′ )♮ )−1 𝔠M ,M ′ . 𝑒

Proof. Let 𝜉 ∈ 𝑒 H ⊂ H and let us consider the projections ′

𝑝𝜉 = [M ′ 𝜉] ∈ M ,

𝑝𝜉′ = [M 𝜉] ∈ M ′ ,

𝑞𝜉 = [M ′𝑒′ 𝜉] ∈ M 𝑒′ ,

𝑞𝜉′ = [M 𝑒′ 𝜉] ∈ M 𝑒′ ′ .

Then 𝑞𝜉 = (𝑝𝜉 )𝑒′ and 𝑞𝜉′ = (𝑝𝜉′ )𝑒′ . Consequently, we have ′

′

′

′

′

(𝑞𝜉′ )♮𝑒′ = ((𝑝𝜉′ )𝑒′ )♮𝑒′ = ((𝑒′ )♮ )−1 (𝑝𝜉′ )♮ = ((𝑒′ )♮ )−1 𝔠M,M′ (𝑝𝜉 )♮ , ′

′

′

′

= ((𝑒′ )♮ )−1 𝔠M,M′ ((𝑝𝜉 )𝑒′ )♮𝑒′ = ((𝑒′ )♮ )−1 𝔠M,M′ (𝑞𝜉 )♮𝑒′ . The uniqueness of the coupling element now implies the formula in the statement of the proposition. Q.E.D. 7.22. Let M ⊂ B (H ) be a finite von Neumann algebra whose commutant M ′ ⊂ B (H ) is also finite, and let 𝑛 be a natural number. Let H 𝑛 be a Hilbert space of dimension 𝑛. We now consider the von Neumann algebras M ̃ 𝑛 = M ⊗C (H 𝑛 ) ⊂ B (H ⊗H 𝑛 ),

M̃

′ 𝑛

= M ′ ⊗B (H 𝑛 ) ⊂ B (H ⊗H 𝑛 ).

′ ′ Then M ̃ 𝑛 and M ̃ 𝑛 are finite, and there exists a projection 𝑒𝑛′ ∈ M ̃ 𝑛 , z(𝑒𝑛′ ) = 1, such that the canonical induction M ̃ 𝑛 → (M ̃ 𝑛 )𝑒𝑛′ be the inverse of the canonical amplification M → M ̃ 𝑛 (see 3.18 and E.4.20).

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From Proposition 7.12, we deduce the following: Corollary. 𝔠M̃

̃′ 𝑛 ,M 𝑛

= 𝑛𝔠M ,M ′ .

7.23. Finally, we present a topological criterion of finiteness: Proposition. A von Neumann algebra M is finite iff the *-operation is s-continuous on the closed unit ball of M . Proof. Let us first assume that M is finite. We shall show that the set {𝐿𝑎 𝜑; 𝜑 a finite normal trace on M , 𝑎 ∈ M } is total in M ∗ . Indeed, if 𝑥 ∈ M , and 𝜑(𝑎𝑥) = 0 for any finite normal trace 𝜑 on M , and any 𝑎 ∈ M , then 𝜑(𝑥 ∗ 𝑥) = 0, for any finite normal trace 𝜑 on M ; hence, in view of Corollary 7.14, we have 𝑥 = 0. Since (M ∗ )∗ = M , the assertion is now obvious. Let {𝑥𝑖 } be a net in the closed unit ball of M , which is s-convergent to 0. Then, for any normal trace 𝜑 on M and any 𝑎 ∈ M , we have |𝜑(𝑎𝑥𝑖 𝑥𝑖∗ )| = |𝜑(𝑥𝑖∗ 𝑎𝑥𝑖 )| ≤ 𝜑(𝑥𝑖∗ 𝑎𝑎∗ 𝑥𝑖 )1/2 𝜑(𝑥𝑖∗ 𝑥𝑖 )1/2 ≤ ‖𝑎‖𝜑(𝑥𝑖∗ 𝑥𝑖 ) → 0. By taking into account the first part of the proof, it follows that the net {𝑥𝑖 } is s-convergent to 0. Consequently, the *-operation is s-continuous on the closed unit ball of M . Conversely, let us assume that M is not finite. According to Proposition 4.12, there exists a sequence {𝑒𝑛 } of equivalent mutually orthogonal, non-zero projections in M . Let {𝑣𝑛 } be a sequence of partial isometries in M , such that 𝑣𝑛∗ 𝑣𝑛 = 𝑒𝑛 ,

𝑣𝑛 𝑣𝑛∗ = 𝑒1 , 𝑛 = 1, 2, … .

Then it is easy to see that 𝑠

𝑠

𝑣𝑛 → 0, but 𝑣𝑛∗ ↛ 0, and this shows that the *-operation is not s-continuous on the closed unit ball of M . Q.E.D. EXERCISES E.7.1 A von Neumann algebra is finite iff 𝑥, 𝑦 ∈ M ,

𝑥𝑦 = 1 ⇒ 𝑦𝑥 = 1. ▶

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E.7.2 Let M be a von Neumann algebra and 𝑥, 𝑦 ∈ M . If n(𝑥𝑦) is a finite projection, then n(𝑥𝑦) − n(𝑦) ∼ n(𝑥) ∧ l(𝑦). E.7.3 Let M be a von Neumann algebra, Z its centre and let 𝑑 ∶ PM → Z + be a mapping having the properties: (i) (ii) (iii) (iv) (v)

𝑑(𝑒 + 𝑓) = 𝑑(𝑒) + 𝑑(𝑓), for any 𝑒, 𝑓 ∈ PM , 𝑒𝑓 = 0. 𝑑(𝑒) = 𝑑(𝑓), for any 𝑒, 𝑓 ∈ PM , 𝑒 ∼ 𝑓. 𝑑(1) = 1. 𝑑(𝑝𝑒) = 𝑝𝑑(𝑒), for any 𝑒 ∈ PM , 𝑝 ∈ PZ . 𝑑(𝑒) = 0 ⇒ 𝑒 = 0, for any 𝑒 ∈ PM .

Show that M is finite, and for any 𝑒 ∈ PM , we have 𝑑(𝑒) = 𝑒♮ . E.7.4 Let M be a finite von Neumann algebra and Z its centre. The following assertions are equivalent: (i) M is of countable type. (ii) Z is of countable type. (iii) There exists a faithful finite normal trace on M + . E.7.5 Let {𝜉𝑖 }𝑖∈𝐼 be an orthonormal basis in H . For any 𝑥 ∈ B (H ), 𝑥 ≥ 0, the number 𝑡𝑟(𝑥) = ∑(𝑥𝜉𝑖 |𝜉𝑖 ) = ∑ ‖𝑥 1/2 𝜉𝑖 ‖2 𝑖∈𝐼

𝑖∈𝐼

does not depend on the chosen basis in H . and the mapping 𝑥 ↦ 𝑡𝑟(𝑥) is a faithful semifinite normal trace on B (H )+ ; it is called the canonical trace on B (H )+ . E.7.6 Let us denote T 𝑟(H ) = {𝑎 ∈ B (H ); ‖𝑎‖𝑡𝑟 = 𝑡𝑟(|𝑎|),

𝑡𝑟(|𝑎|) < +∞},

𝑎 ∈ T (H ).

Then T 𝑟(H ) is a Banach space for the norm ‖⋅‖𝑡𝑟 , and 𝑡𝑟 extends by linearity to a bounded linear form on the Banach space T 𝑟(H ), which is also denoted by 𝑡𝑟. ▶

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*E.7.7 The set T 𝑟(H ) is a two-sided ideal in B (H ), contained in the closed two-sided ideal K (H ) of all the compact operators on H . Thus, any element 𝑎 ∈ T 𝑟(H ) determines a form 𝜑𝑎 on B (H ), given by the formula: 𝑥 ∈ B (H ).

𝜑𝑎 (𝑥) = 𝑡𝑟(𝑎𝑥) = 𝑡𝑟(𝑥𝑎), Show that the mapping 𝑎 ↦ 𝜑𝑎 |K (H )

is an isometric isomorphism of the Banach space T 𝑟(H ) onto the Banach space K (H )∗ , whereas the mapping 𝑎 ↦ 𝜑𝑎 is an isometric isomorphism of the Banach space T 𝑟(H ) onto the Banach space B (H )∗ . In particular, B (H ) identifies canonically with K (H )∗∗ . E.7.8 With the help of E.7.7, show that for any normal form 𝜑 on B (H ), there ∞ exists an orthogonal sequence {𝜉𝑛 } ⊂ H , ∑𝑛=1 ‖𝜉𝑛 ‖2 < +∞, such that ∞

𝜑(𝑥) = ∑ (𝑥𝜉𝑛 |𝜉𝑛 ),

𝑥 ∈ B (H ).

𝑛=1

With the help of a polar decomposition theorem, infer then that, for any w-continuous form 𝜑 on B (H ), there exist orthogonal sequences {𝜉𝑛 }, ∞ ∞ {𝜂𝑛 } ⊂ H , ∑𝑛=1 ‖𝜉𝑛 ‖2 < +∞, ∑𝑛=1 ‖𝜂𝑛 ‖2 < +∞, such that ∞

𝜑(𝑥) = ∑ (𝑥𝜉𝑛 |𝜂𝑛 ),

𝑥 ∈ B (H ).

𝑛=1 ∞

‖𝜑‖ = ∑ ‖𝜉𝑛 ‖‖𝜂𝑛 ‖. 𝑛=1

E.7.9 Let M be a semifinite von Neumann algebra. Then M is continuous iff there exists a decreasing sequence {𝑒𝑛 } of finite projections in M , such that, for any 𝑛, z(𝑒𝑛 ) = 1, 𝑒𝑛 − 𝑒𝑛+1 ∼ 𝑒𝑛+1 . ▶

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E.7.10 Show that a von Neumann algebra M is properly infinite (resp., of type III) iff any finite (resp., semifinite) normal trace on M + is identically zero. Infer from this result that if M is a von Neumann algebra of type III, and if 𝜇 is a non-zero normal trace on M + , then 𝜇(𝑥) = +∞, for any 𝑥 ∈ M + , 𝑥 ≠ 0. E.7.11 Let M be a von Neumann algebra and 𝜇 a normal trace on M + . For any family {𝑒𝑖 }𝑖∈𝐼 ⊂ M of mutually orthogonal projections, such that ∑𝑖∈𝐼 𝑒𝑖 = 1, we have 𝜇(𝑥) = ∑ 𝜇(𝑒𝑖 𝑥𝑒𝑖 ), 𝑥 ∈ M + . 𝑖∈𝐼

Infer from this result that for any semifinite normal trace 𝜇 on M + , there exists a family {𝜑𝑖 }𝑖∈𝐼 of normal forms on M , whose supports s(𝜑𝑖 ) are mutually orthogonal, and such that 𝜇(𝑥) = ∑ 𝜑𝑖 (𝑥),

𝑥 ∈ M +.

𝑖∈𝐼

E.7.12 Let M be a finite von Neumann algebra with centre Z and 𝜇 a normal semifinite trace on M + . Then 𝜇(𝑥 ♮ ) ≤ 𝜇(𝑥),

𝑥 ∈ M +.

Infer that 𝜇|Z + is a normal semifinite trace on Z + , there exists a family {𝜇𝑖 }𝑖∈𝐼 of finite normal traces on M with mutually orthogonal supports, such that 𝜇(𝑥) = ∑ 𝜇𝑖 (𝑥),

𝑥 ∈ M +,

𝑖∈𝐼

and hence 𝜇(𝑥 ♮ ) = 𝜇(𝑥),

𝑥 ∈ M +.

E.7.13 Let M be a semifinite von Neumann algebra of a countable type with center Z , 𝜇 a normal semifinite faithful trace on M + and 𝑒, 𝑓 ∈ PM . Show that (1) 𝑒 ≺ 𝑓 ⇔ 𝜇(𝑒𝑝) ≤ 𝜇(𝑓𝑝) for all 𝑝 ∈ PZ . (2) 𝑒 is finite ⇔ for every 0 ≠ 𝑝 ∈ PZ , there exists 0 ≠ 𝑞 ∈ PZ , 𝑞 ≤ 𝑝, such that 𝜇(𝑞𝑒) < +∞ ⇔ there exists a sufficient family {𝜇𝑖 }𝑖∈𝐼 of normal semifinite traces on M + such that 𝜇𝑖 (𝑒) < +∞, 𝑖 ∈ 𝐼. E.7.14 Show that any two normal traces 𝜇, 𝑣 on a factor M are proportional. Extend this result for two normal semifinite faithful traces on a von Neumann algebra. (Hint: if M is a finite factor, then 𝜇(𝑥) = 𝛼𝑥 ♮ , 𝑥 ∈ M , for some 𝛼 ∈ [0, +∞). The required statement for the general case is given as a consequence of Corollary 1 in C.10.4; for the proof here use 5.21 and 7.11.) ▶

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!E.7.15 One says that a von Neumann algebra M ⊂ B (H ) is standard if there exists a conjugation 𝐽 ∶ H → H , such that the mapping 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 be a *-anti-isomorphism of M onto M ′ , which acts identically on the centre. Show that any standard von Neumann algebra of countable type has a separating cyclic vector. !E.7.16 Any maximal abelian von Neumann algebra is standard. Consequently, any abelian von Neumann algebra is *-isomorphic to a standard von Neumann algebra. !E.7.17 Let M ⊂ B (H ) be a finite von Neumann algebra of countable type. The following assertions are equivalent: (i) M is standard. (ii) M has a separating cyclic trace vector. (iii) M ′ is finite and 𝔠M ,M ′ = 1. !E.7.18 Any finite von Neumann algebra is *-isomorphic to a standard von Neumann algebra. (Hint: if M is of countable type, apply the construction given in Section 5.18, to a faithful w-continuous central positive form.) E.7.19 Any semifinite von Neumann algebra is *-isomorphic to a standard von Neumann algebra. E.7.20 With the help of E.7.17, give a simpler proof to Lemma 6.3. E.7.21 Let M ⊂ B (H ) be a factor of type 𝐼𝑚 , whose commutant is of type 𝐼𝑛 , where 𝑚 and 𝑛 are natural numbers. Prove directly Theorem 7.19 in this case and show that dim H = 𝑚𝑛 and 𝑐M ,M ′ = 𝑚/𝑛. E.7.22 Let M be a von Neumann algebra and 𝑒 ∈ PM . The following assertions are equivalent: (i) 𝑒 is the least upper bound of a finite family of minimal projections in M . (ii) The mapping M ∋ 𝑥 ↦ 𝑒𝑥𝑒 ∈ 𝑒M 𝑒 is continuous for the w-topology in M and the uniform (norm) topology in 𝑒M 𝑒. *E.7.23 Let M be a finite von Neumann algebra and G ⊂ M a multiplicative group of invertible elements such that sup{‖𝑔‖; 𝑔 ∈ G } < +∞. With the help of the Ryll–Nardzewski fixed point theorem, show that there exists an invertible element 𝑎 ∈ M + , such that for any 𝑔 ∈ G the element 𝑎𝑔𝑎−1 be unitary. ▶

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E.7.24 Let M ⊂ B (H ) be a finite factor with finite commutant M ′ ⊂ B (H ). Adopt the notation (C.7.5): dim M (H ) = 𝑐M ,M ′ . Show that dim M (H ) ≤ 1 ⇔ (∃)𝜉 ∈ H cyclic for M ∶ 𝑝𝜉′ = 1. dim M (H ) ≥ 1 ⇔ (∃)𝜉 ∈ H separating for M ∶ 𝑝𝜉 = 1. dim M (H ) = 1 ⇔ (∃)𝜉 ∈ H cyclic and separating for M . (Hint: Lemma 7.18.) E.7.25 Let M ⊂ B (H ) be a finite factor. Show that, on M , the w-topology coincides with the w𝑜-topology. More precisely, for 𝑛 ∈ ℕ, we have 𝑛

dimM (H ) ≥

1 ⇔ (∃)𝜉1 , … , 𝜉𝑛 ∈ H with 𝑝 =1 ⋁ 𝜉𝑘 𝑛 𝑘=1

𝑛

⇔ (∀)𝜑 ∈

M+ ∗ , (∃)𝜉1 , … , 𝜉𝑛

∈ H with 𝜑 = ∑ 𝜔𝜉𝑘 |M 𝑘=1 𝑛

⇔ (∀)𝜓 ∈ M ∗ , (∃)𝜉1 , … , 𝜉𝑛 , 𝜂1 , … , 𝜂𝑛 ∈ H with 𝜓 = ∑ 𝜔𝜉𝑘 ,𝜂𝑘 |M . 𝑘=1

In particular, show that the normalized trace 𝜏 on M can be written as 𝑛

𝜏 = ∑ 𝜔𝜉𝑘 , with 𝑝𝜉1 + ... + 𝑝𝜉𝑛 = 1

(𝑛 ∈ ℕ).

𝑘=1 1

1

(Hint: Assume dim M (H ) ≥ and consider 𝑒 ∈ M , 𝜏(𝑒) = . Then 𝑛 𝑛 dim 𝑒M 𝑒 (𝑒H ) ≥ 1.) E.7.26 Let M be a factor of type II∞ and 𝑒 ∈ M a finite non-zero projection. Show that 𝑒 divides 1, i.e., there exists an infinite family of mutually orthogonal projections {𝑒𝑖 }𝑖∈𝐼 ⊂ M such that 𝑒𝑖 ∼ 𝑒, (∀)𝑖 ∈ 𝐼 and ∑ 𝑒𝑖 = 1. 𝑖∈𝐼

Let 𝑣𝑖 ∈ M , 𝑣𝑖∗ 𝑣𝑖 = 𝑒, 𝑣𝑖 𝑣𝑖∗ = 𝑒𝑖 (𝑖 ∈ 𝐼). For any 𝑥 ∈ M + define 𝜇(𝑥) = ∑ 𝜏 (𝑣𝑖∗ 𝑥𝑣𝑖 ) , 𝑖∈𝐼

▶

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where 𝜏 is the normalized trace on 𝑒M 𝑒. Show that 𝜇 is a normal semifinite faithful trace on M . (Actually M = 𝑒M 𝑒⊗B (𝑙2 (𝐼)) and 𝜇 = 𝜏⊗ Tr, where Tr is the usual trace on 𝐵 (𝑙2 (𝐼)) constructed as above starting with 𝑒 ∈ 𝐵 (𝑙 2 (𝐼)) a minimal projection.)

COMMENTS C.7.1 Let M be a finite von Neumann algebra and 𝑥 ∈ M . With the notations from C.4.4, it is easy to see that 𝑧 ∈ C (𝑥) ⇒ 𝑧 = 𝑥 ♮ . According to Theorem 1 from C.4.4 (or by Theorem 7.11. (ix)), it follows that (*)

K (𝑥) = C (𝑥) = {𝑥 ♮ }.

The first proof of Theorem 7.11 was given by J. Dixmier (1949) who extended the arguments of F. J. Murray and J. von Neumann, used by them for the case of factors. The culminating point of J. Dixmier’s proof consists in showing that the set K (𝑥) reduces to a single element. Although this proof is much longer than that given above, we consider it to be very illuminating (see Dixmier [1957/1969], Chap. III, §8). The shortening of the classical proof of Theorem 7.11 has been an open problem for a long time (see, e.g., Kadison [1955, 1963]). The proof given above has been obtained by F. J. Yeadon (1971). A description of the sets K (𝑥) and C (𝑥), analogous to that given by the relation (*), for properly infinite von Neumann algebras, has been obtained by ̆ ă and L. Zsidó (1972/1973) (see also Stratil ̆ ă H . Halpern (1972) and S. ¸ Stratil [1973a]). C.7.2 Some phenomena that take place in the lattice of all the projections in a von Neumann algebra also appear in the abstract frame of lattice theory (see, e.g., von Neumann [1936/1937/1960]; S. Maeda [1958]; F. Maeda and S. Maeda [1971]; Loomis [1955]; Skornyakov [1961]). We also mention the remarkable result contained in the title of I. Kaplansky’s paper (1955). C.7.3 Let M be a von Neumann algebra and 𝜇 a trace on M + . Then the set {𝑥 ∈ M + ; 𝜇(𝑥) < +∞}

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is the positive part of a two-sided ideal 𝔐𝜇 of M , and there exists a unique linear form on 𝔐𝜇 , which coincides with 𝜇 on 𝔐+ 𝜇 . This linear form will also be denoted by 𝜇. One can show that for any 𝑎 ∈ 𝔐𝜇 , one has 𝜇(𝑎𝑥) = 𝜇(𝑥𝑎),

𝑥 ∈ M,

and if 𝜇 is normal, then the linear form M ∋ 𝑥 ↦ 𝜇(𝑎𝑥) is w-continuous. Obviously, 𝜇 is finite iff 𝔐𝜇 = M . If 𝜇 is normal, then 𝜇 is semifinite iff 𝔐𝜇 is w-dense in M . The proofs of these results can easily be obtained using the results in Section 3.21 (see also 10.14 and Dixmier [1957/1969], Ch. I, §6). C.7.4 Let M and N be von Neumann algebras. According to the types of the algebras M and N , the type of the tensor product M ⊗N is completely described by the following table: M

N

M ⊗N

Type I𝑚

Type I𝑛

Type I𝑚𝑛

Type I

Type I

Type I

Finite

Finite

Finite

Semifinite

Semifinite

Semifinite

Continuous

Arbitrary

Continuous

Properly infinite

Arbitrary

Properly infinite

Type III

Arbitrary

Type III

For the proofs, we refer the reader to J. Dixmier (1957/1969, Chap. III, §8.7), and S. Sakai (1971), 2.6. We mention the fact that only the last implication in this table offers some difficulties. This implication has been proved by S. Sakai (1957), who, to this end, also obtained the topological criterion of finiteness (7.23). C.7.5 Here we describe the Murray–von Neumann proof of the existence of the trace on a type II1 -factor M . I. The first goal is to construct a ‘dimension function,’ which is a completely additive function ∆ ∶ PM → [0, 1], Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

188

Lectures on von Neumann algebras such that, for 𝑒, 𝑓 ∈ PM , 𝑒 ≺ 𝑓 ⇔ ∆(𝑒) ≤ ∆(𝑓). This is accomplished by imitating the dyadic decomposition for numbers 𝑟 ∈ [0, 1]: 𝑟 = ∑ 2−𝑛𝑘 , 𝑘

where, by recurrence, 𝑛𝑘 = 𝑛𝑘 (𝑟) is the smallest 𝑛 ∈ ℕ such that 𝑘−1 −𝑛

2

≤ 𝑟 − ∑ 2−𝑛𝑗

(stop if this is zero).

𝑗=1

Thus, let 𝑒𝑛 ∈ PM (of trace 2−𝑛 ) be such that (by recurrence) 𝑛−1

𝑛

𝑒𝑛 ≤ 1 − ∑ 𝑒𝑘 and 𝑒𝑛 ∼ 1 − ∑ 𝑒𝑘 𝑘=1

(𝑛 ∈ ℕ).

𝑘=1

Call any 𝑓 ∈ PM , which is equivalent to some 𝑒𝑛 a fundamental projection. Notice that (2−𝑛 → 0): (1)

𝑓 ∈ PM , 𝑓 ≺ 𝑒𝑛 (∀)𝑛 ∈ ℕ ⇒ 𝑓 = 0.

(2)

0 ≠ 𝑓 ∈ PM ⇒ (∃)𝑛 ∈ ℕ, 𝑒𝑛 ≺ 𝑓.

(3)

∑ 𝑒𝑛 = 1.

∞ 𝑛=1

(4) 𝑓, 𝑓𝑛 ∈ PM , 𝑓𝑛 ≤ 𝑓, 𝑓𝑛 increasing, 𝑓 − 𝑓𝑛 ≺ 𝑒𝑛 , (∀)𝑛 ∈ ℕ ⇒ 𝑓𝑛 ↑ 𝑓. Indeed, if 0 ≠ 𝑓 ∼ 𝑓𝑛 ≤ 𝑒𝑛 , (∀)𝑛 ∈ ℕ, then {𝑓𝑛 }𝑛∈ℕ would be an infinite sequence of mutually orthogonal and equivalent non-zero projections in contradiction with the finiteness of 𝑀. This proves (1) and (2) follow by comparison. Since ∞

𝑛

1 − ∑ 𝑒𝑘 ≤ 1 − ∑ 𝑒𝑘 ∼ 𝑒𝑛 , 𝑘=1

𝑘=1

also (3) follows from (1). Finally, for (4), notice that 𝑓𝑛 ↑ ℎ ≤ 𝑓 and 𝑓 − ℎ ≤ 𝑓 − 𝑓𝑛 ≺ 𝑒𝑛 , (∀)𝑛 ∈ ℕ; hence ℎ = 𝑓 by (1). Let 𝑒 ∈ PM . Define a strictly increasing sequence 𝑛𝑘 = 𝑛𝑘 (𝑒) ∈ ℕ and a sequence of projections 𝑓𝑘 ∈ PM by recurrence: 𝑛𝑘 is the smallest 𝑛 ∈ ℕ such that 𝑘−1

𝑒𝑛 ≺ 𝑒 − ∑ 𝑓𝑗

(stop if this is zero)

𝑗=1

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Finite von Neumann algebras and then 𝑓𝑘 is defined by 𝑘−1

𝑒𝑛𝑘 ∼ 𝑓𝑘 ≤ 𝑒 − ∑ 𝑓𝑗 . 𝑗=1

Then define ‘the dimension of 𝑒’ by ∞

∆(𝑒) = ∑ 2−𝑛𝑘 ,

∆(0) = 0.

𝑘=1

The function ∆ ∶ PM ∋ 𝑒 ↦ ∆(𝑒) ∈ [0, 1] is well defined, i.e. 𝑛𝑘 = 𝑛𝑘 (𝑒) is uniquely determined by 𝑒. This is obvious by induction since 𝑓′ , 𝑓 ″ ∈ PM , 𝑓 ′ ≤ 𝑒, 𝑓 ″ ≤ 𝑒, 𝑓 ′ ∼ 𝑓 ″ ⇒ 𝑒 − 𝑓 ′ ∼ 𝑒 − 𝑓 ″ ⇒ 𝑛1 (𝑒 − 𝑓 ′ ) = 𝑛1 (𝑒 − 𝑓 ″ ); hence the (several possible) choices of the 𝑓𝑘 ’s do not affect the 𝑛𝑘 ’s. Actually, for any strictly increasing sequence {𝑛𝑘 }𝑘 ⊂ ℕ, we clearly have ∆(𝑒) = ∑ 2−𝑛𝑘 ⇔ 𝑒 ∼ ∑ 𝑒𝑛𝑘 .

(5)

𝑘

𝑘

Thus, the map ∆ ∶ PM → [0, 1] is surjective. Moreover, (6)

𝑒 ≺ 𝑓 ⇔ ∆(𝑒) ≤ ∆(𝑓),

𝑒 ∼ 𝑓 ⇔ ∆(𝑒) = ∆(𝑓).

Indeed 𝑒 ≺ 𝑓 if and only if, for the first 𝑘 ∈ ℕ with 𝑛𝑘 (𝑒) ≠ 𝑛𝑘 (𝑓), we have 𝑛𝑘 (𝑒) > 𝑛𝑘 (𝑓) and the same is true for the dyadic expansion of real numbers 𝑟, 𝑠 ∈ [0, 1]. This already proves that PM / ∼ = [0, 1].

(7)

It is an easy exercise to show that if 𝑒, 𝑓 ∈ PM , 𝑒 ≤ 𝑓 and ∆(𝑒) ≤ 𝑟 ≤ ∆(𝑓), then there is 𝑔 ∈ PM , 𝑒 ≤ 𝑔 ≤ 𝑓, such that ∆(𝑔) = 𝑟. The dimension function ∆ ∶ PM → [0, 1] is additive: (8)

𝑒, 𝑓 ∈ PM , 𝑒𝑓 = 0 ⇒ ∆(𝑒 + 𝑓) = ∆(𝑒) + ∆(𝑓).

Indeed, it is clear that ∆(𝑒) and ∆(𝑓) are small ⇒ ∆(𝑒 + 𝑓) is small; hence we may assume that both 𝑒 and 𝑓 are finite sums of fundamental projections, and further we may assume that 𝑓 ∼ 𝑒𝑛 in which case (8) becomes obvious.

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190

Lectures on von Neumann algebras Moreover, the dimension function ∆ ∶ PM → [0, 1] is completely additive: 𝑓 = ∑ 𝑓𝑛 in PM ⇒ ∆(𝑓) = ∑ ∆(𝑓𝑛 ).

(9)

𝑛∈𝐼

𝑛∈𝐼

Indeed, for each finite subset 𝐹 ⊂ 𝐼, we have, by the additivity of ∆, ∑ ∆(𝑓𝑛 ) = ∆ ( ∑ 𝑓𝑛 ) ≤ ∆(𝑓); hence ∑ ∆(𝑓𝑛 ) ≤ ∆(𝑓). 𝑛∈𝐹

𝑛∈𝐹

𝑛∈𝐼

In particular, this shows that the set {𝑛 ∈ 𝐼; ∆(𝑓𝑛 ) ≠ 0} and hence the set {𝑛 ∈ 𝐼; 𝑓𝑛 ≠ 0} is countable, so we may assume 𝐼 = ℕ. Suppose that ∞

∑ ∆(𝑓𝑛 ) < ∆(𝑓). 𝑛=1 ∞

Then, for some 𝑚 ∈ ℕ, we have 2−𝑚 + ∑𝑛=1 ∆(𝑓𝑛 ) < ∆(𝑓); hence there is a projection 𝑓0 ∈ 𝑀 such that 𝑒𝑚 ∼ 𝑓0 ≤ 𝑓. ∞

Since ∆(𝑓1 ) ≤ ∑ ∆(𝑓𝑛 ) ≤ ∆(𝑓) − 2−𝑚 = ∆(𝑓 − 𝑓0 ), there is a projection 𝑛=1

𝑔1 ∈ 𝑀 with 𝑓1 ∼ 𝑔1 ≤ 𝑓 − 𝑓0 . ∞

Since ∆(𝑓2 ) ≤ ∑ ∆(𝑓𝑛 ) ≤ ∆(𝑓) − 2−𝑚 − ∆(𝑓1 ) = ∆(𝑓 − 𝑓0 − 𝑔1 ), there is 𝑛=2

a projection 𝑔2 ∈ 𝑀 with 𝑓2 ∼ 𝑔 2 ≤ 𝑓 − 𝑓 0 − 𝑔 1 , ∞

𝑁−1

and by induction,…, since ∆(𝑓𝑁 ) ≤ ∑ ∆(𝑓𝑛 ) ≤ ∆ (𝑓 − 𝑓0 − ∑ 𝑔𝑛 ), there is a projection 𝑔𝑁 ∈ PM with

𝑛=𝑁

𝑛=1

𝑁−1

𝑓𝑁 ∼ 𝑔𝑁 ≤ 𝑓 − 𝑓0 − ∑ 𝑔𝑛 . 𝑛=1

We thus get a sequence {𝑔𝑛 } ⊂ PM of mutually orthogonal projections: 𝑓𝑛 ∼ 𝑔 𝑛 ≤ 𝑓 − 𝑓 0 It follows that

∞

(𝑛 ∈ ℕ).

∞

𝑓 = ∑ 𝑓𝑛 ∼ ∑ 𝑔𝑛 ≤ 𝑓 − 𝑓0 ≠ 𝑓, 𝑛=1

𝑛=1

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191

Finite von Neumann algebras in contradiction with the finiteness of (any) 𝑓 ∈ PM . Hence ∆(𝑓) = ∑ ∆(𝑓𝑛 ). 𝑛

II. The next goal is to extend the dimension function ∆ ∶ PM → [0, 1] to a trace 𝜏 on the whole M . Such a trace will be automatically w-continuous since it is completely additive on projections. By applying E.5.24 to the dimension function ∆ and to the restriction to PM of some vector form 𝜔𝜉 (𝜉 ∈ 𝐻, ‖𝜉‖ = 1) and 𝜀 > 0, we get a non-zero projection 0 ≠ 𝑒 ∈ M and 𝜃 > 0, such that 𝜃∆(𝑓) ≤ 𝜔𝜉 (𝑓) ≤ 𝜃(1 + 𝜀)∆(𝑓),

(∀)𝑓 ∈ PM , 𝑓 ≤ 𝑒.

Of course, we can choose 𝑒 ∈ M a fundamental projection, with the future benefit that a fundamental projection divides the identity 1 ∈ M . Let 𝜑𝜀 = 1 𝜔𝜉 |𝑒M 𝑒 . Then 𝜃

(1)

∆(𝑓) ≤ 𝜑𝜀 (𝑓) ≤ (1 + 𝜀)∆(𝑓),

(∀)𝑓 ∈ PM , 𝑓 ≤ 𝑒.

Since ∆ takes equal values on equivalent projections, we may expect 𝜑𝜀 to be ‘almost a trace’. More precisely, a positive linear form 𝜑 on M is called an 𝜀-trace if 𝜑(ᵆ𝑥ᵆ∗ ) ≤ (1 + 𝜀)𝜑(𝑥),

(∀)𝑥, ᵆ ∈ M , 𝑥 ≥ 0, ᵆ unitary.

It is an easy exercise to check that 𝜑 ∈ 𝑀+∗ is an 𝜀-trace if and only if (2)

𝜑(𝑥𝑥 ∗ ) ≤ (1 + 𝜀)𝜑(𝑥 ∗ 𝑥),

(∀)𝑥 ∈ M .

With this definition, we see now that (3)

𝜑𝜀 is an 𝜀-trace on 𝑒M 𝑒.

Indeed, let ᵆ ∈ 𝑈(𝑒M 𝑒), that is ᵆ ∈ M , ᵆ∗ ᵆ = ᵆᵆ∗ = 𝑒. If 𝑥 = 𝑓 ∈ P 𝑒M 𝑒 , then, by (1): 𝜑𝜀 (ᵆ𝑓ᵆ∗ ) ≤ (1 + 𝜀)∆(ᵆ𝑓ᵆ∗ ) = (1 + 𝜀)∆(𝑓) ≤ (1 + 𝜀)𝜑𝜀 (𝑓), and if 𝑥 ∈ M , 0 ≤ 𝑥 ≤ 𝑒, then (2) follows for 𝜑𝜀 using the dyadic decomposition of 𝑥 (2.23). Since 𝑒 is a fundamental projection, say 𝑒 ∼ 𝑒𝑁 , 𝑒 divides the identity and we have M = 𝑀𝑎𝑡2𝑁 (𝑒M 𝑒) = 𝑒M 𝑒 ⊗ 𝑀𝑎𝑡2𝑁 (ℂ),

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192

Lectures on von Neumann algebras and then 𝜏𝜀 = 𝜑𝜀 ⊗ Tr is an 𝜀-trace on M ,

(4)

where Tr is the usual trace (not normalized) on 𝑀𝑎𝑡2𝑁 (ℂ). 2𝑁

More precisely, let 1 = ∑ 𝑓𝑛 with 𝑒 ∼ 𝑓𝑛 ∈ PM and 𝑣𝑛 ∈ M , 𝑛=1

𝑣𝑛∗ 𝑣𝑛 = 𝑒, 𝑣𝑛 𝑣𝑛∗ = 𝑓𝑛 , (∀)𝑛 = 1, ..., 2𝑁 . Then 2𝑁

(∀)𝑥 ∈ M ,

𝜏𝜀 (𝑥) = ∑ 𝜑𝜀 (𝑣𝑛∗ 𝑥𝑣𝑛 ), 𝑛=1

and it is immediate to check that 𝜏𝜀 is an 𝜀-trace. Moreover, for any projection 𝑝 ∈ M , we have (1 + 𝜀)−1 ∆(𝑝) ≤ 𝜏𝜀 (𝑝) ≤ (1 + 𝜀)2 ∆(𝑝).

(5)

Indeed, given 𝑝 ∈ PM , there is 𝑞 ∈ PM such that 𝑞 ∼ 𝑝 and 𝑞𝑓𝑛 = 𝑓𝑛 𝑞,

(∀)𝑛 = 1, ..., 2𝑁 .

This follows obviously from the dyadic decomposition of 𝑝 into fundamental projections since all 𝑓𝑛 ∼ 𝑒𝑁 . Then also 1 − 𝑞 ∼ 1 − 𝑝 (since M is finite), and therefore, there is a unitary ᵆ ∈ M such that 𝑝 = ᵆ∗ 𝑞ᵆ and 𝑞𝑓𝑛 = 𝑓𝑛 𝑞,

(∀)𝑛 = 1, ..., 2𝑁 .

It follows that 𝑣𝑛∗ 𝑞𝑣𝑛 ∈ P𝑒M 𝑒 , 𝑣𝑛∗ 𝑞𝑣𝑛 ∼ 𝑓𝑛 𝑞𝑓𝑛 ,

(∀)𝑛 = 1, ..., 2𝑁 .

These allow us to check the relations (5): 𝜏𝜀 (𝑝) ≤ (1 + 𝜀)𝜏𝜀 (𝑞) = (1 + 𝜀) ∑ 𝜑𝜀 (𝑣𝑛∗ 𝑞𝑣𝑛 ) ≤ (1 + 𝜀)2 ∑ ∆(𝑣𝑛∗ 𝑞𝑣𝑛 ) 𝑛

𝑛

= (1 + 𝜀)2 ∑ ∆(𝑓𝑛 𝑞𝑓𝑛 ) = (1 + 𝜀)2 ∑ ∆(𝑞) = (1 + 𝜀)2 ∑ ∆(𝑝) 𝑛

𝑛

−1

𝑛

𝜏𝜀 (𝑝) ≥ (1 + 𝜀) 𝜏𝜀 (𝑞) = (1 + 𝜀)

∑ 𝜑𝜀 (𝑣𝑛∗ 𝑞𝑣𝑛 ) 𝑛

−1

−1

= (1 + 𝜀)

−1

∑ ∆(𝑓𝑛 𝑞𝑓𝑛 ) = (1 + 𝜀) 𝑛

≥ (1 + 𝜀)

−1

∑ ∆(𝑣𝑛∗ 𝑞𝑣𝑛 )

∑ ∆(𝑞) = (1 + 𝜀) 𝑛

𝑛 −1

∑ ∆(𝑝). 𝑛

Finally, we now see that the following limit exists: (6)

𝜏 ∶= lim 𝜏𝜀 ∈ M ∗ 𝜀→0

(norm convergence)

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Finite von Neumann algebras and 𝜏 is a faithful w-continuous normalized trace on M 𝜏|PM = ∆.

(7)

Indeed, for any 𝑝 ∈ PM , we have 𝜏(𝑝) = lim 𝜏𝜀 (𝑝) = ∆(𝑝) by (5). 𝜀→0

∞

For any 𝑥 ∈ M , 0 ≤ 𝑥 ≤ 1, with dyadic decomposition 𝑥 = ∑ 2−𝑛 𝑝𝑛 , 𝑛=1

we have, again by (5): |𝜏𝜀 (𝑥)−𝜏𝛿 (𝑥)| ≤ ∑ 2−𝑛 ∆(𝑝𝑛 ) [(1 + 𝜀)2 − (1 + 𝛿)−1 ] ≤ (1+𝜀)2 −(1+𝛿)−1 → 0, 𝑛

as 𝜀 → 0, 𝛿 → 0. It follows that |𝜏𝜀 (𝑥) − 𝜏𝛿 (𝑥)| ≤ 4 [(1 + 𝜀)2 − (1 + 𝛿)−1 ] ‖𝑥‖, which proves (6). From 𝜏𝜀 (𝑥𝑥 ∗ ) ≤ (1 + 𝜀)𝜏𝜀 (𝑥∗ 𝑥), with 𝜀 → 0, we get 𝜏(𝑥𝑥 ∗ ) = 𝜏(𝑥 ∗ 𝑥), (∀)𝑥 ∈ M ; hence 𝜏 is a trace on M . It is w-continuous since all 𝜏𝜀 were w-continuous (or since 𝜏|PM = ∆ is completely additive). It is faithful and 𝜏(1) = 1 since 𝜏|PM = ∆ implies s(𝜏) = 1 and ∆(1) = 1. The wo-continuity of the trace on a finite factor is an obvious consequence of the fact that on a finite factor, the w-topology coincides with the wo-topology (E.7.25). C.7.6 The fundamental group of a type II1 factor. Let M ⊂ B (H ) be a type II1 factor with normalized trace 𝜏. Consider also the type II∞ factor N = M ⊗B (𝑙2 (ℕ)) with the normal semifinite faithful trace 𝜇 = 𝜏⊗ Tr . For any 𝑡 ∈ [0; 1], denote by M 𝑡 , the ∗-isomorphism class of a reduced algebra 𝑒M 𝑒 with 𝑒 ∈ PM , 𝜏(𝑒) = 𝑡. Lemma. M 𝑡 is ∗-isomorphic to M if and only if there is a ∗-automorphism 𝜍 ∶ N → N such that 𝜇 ∘ 𝜍 = 𝑡𝜇. Proof. Let 𝜋 ∶ M → 𝑒M 𝑒 be a ∗-isomorphism. Since the projections 𝑒⊗1 and 1⊗1 are both infinite in N , they are equivalent; hence 𝑤 ∗ 𝑤 = 𝑒⊗1, 𝑤𝑤 ∗ = 1⊗1,

for some 𝑤 ∈ N .

Then the mapping 𝜌 ∶ 𝑒M 𝑒⊗B (𝑙 2 (ℕ)) ∋ 𝑥 ↦ 𝑤𝑥𝑤 ∗ ∈ M ⊗B (𝑙2 (ℕ)) = N is a ∗-isomorphism, and 𝜍 = 𝜌 ∘ (𝜋 ⊗ id) ∶ N → N Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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Lectures on von Neumann algebras is a ∗-automorphism. If 𝑝 ∈ B (𝑙 2 (ℕ)) is any minimal projection, then 𝜇(1 ⊗ 𝑝) = 1, while (𝜇 ∘ 𝜍)(1 ⊗ 𝑝) = 𝜇(𝑤(𝑒 ⊗ 𝑝)𝑤 ∗ ) = 𝜇(𝑒 ⊗ 𝑝) = 𝜏(𝑒) = 𝑡; hence 𝜇 ∘ 𝜍 = 𝑡𝜇. Conversely, let 𝜍 ∶ N → N be a ∗-automorphism, 𝜇 ∘ 𝜍 = 𝑡𝜇 and let 𝑝 ∈ B (𝑙 2 (ℕ)) be any minimal projection. Then 𝜇(𝜍(1 ⊗ 𝑝)) = 𝑡𝜇(1 ⊗ 𝑝) = 𝜇(𝑒 ⊗ 𝑝); hence 𝑒 ⊗ 𝑝 ∼ 𝜍(1 ⊗ 𝑝) in N . It follows that 𝑒M 𝑒 ≡ (𝑒 ⊗ 𝑝) (M ⊗B (𝑙2 (ℕ))) (𝑒 ⊗ 𝑝) ≡ 𝜍(1 ⊗ 𝑝) (M ⊗B (𝑙 2 (ℕ))) 𝜍(1 ⊗ 𝑝) ≈ (1 ⊗ 𝑝) (M ⊗B (𝑙 2 (ℕ))) (1 ⊗ 𝑝) ≡ 𝑀, where ≈ means ∗-isomorphism and ≡ means spatial isomorphism.

Q.E.D.

If 𝑡 > 1, write M 𝑡 ≈ M if M 1 ≈ M . By the above Lemma, the set 𝑡

𝐺(M ) = {𝑡 ∈ (0, ∞); M 𝑡 ≈ M } is a multiplicative subgroup of 𝑅∗+ = (0, ∞), called the fundamental group of the II1 -factor M . Since 1936, when Murray and von Neumann introduced this isomorphy invariant, it is not yet known what is 𝐺(𝐿(𝐹𝑛 )),

𝑛 ∈ ℕ, 𝑛 ≥ 2.

However, the pioneering work of D. V. Voiculescu on free random variables leads to the conclusion that 𝐺(𝐿(𝐹∞ )) = 𝑅∗+ and that either all 𝐺(𝐿(𝐹𝑛 )) = {1} or all 𝐺(𝐿(𝔽𝑛 )) = 𝑅∗+ . On the other hand, there was a breakthrough discovery of Alain Connes (1980a) that for ‘rigid’ type II1 factors the fundamental group is countable. This, together with an example of a ‘rigid’ type II1 -factor, proves at once the existence of an uncountable family of different isomorphy classes of type II1 factors. C.7.7 Let M ⊂ B (H ) be a finite factor with normalized trace 𝜏 and M ′ ⊂ B (H ) its commutant also assumed finite, with normalized trace 𝜏 ′ . Then 𝑐M ,M ′ ∈ (0, ∞) and is also denote by dim M (H ) = 𝑐M ,M ′ =

𝜏(𝑝𝜉 ) 𝜏 ′ (𝑝𝜉′ )

(𝜉 ∈ H )

and called the dimension of H with respect to M ⊂ B (H ). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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Finite von Neumann algebras

Vaughan Jones (1983) considered N ⊂ M a finite subfactor with finite commutant N ′ ⊃ M ′ and defined the index of the subfactor N of M by [M ∶ N ] =

dimN (H ) . dimM (H )

Taking into account Proposition 8.2 and Proposition 7.21, we see that the definition of [M ∶ N ] is actually independent of the realization M ⊂ B (H ), i.e., is independent of H , being an intrinsic property of the inclusion N ⊂ M . If M ⊂ B (H ) has a cyclic and separating vector 𝜉0 ∈ H , then the projection 𝑒 = N 𝜉0 ∈ N

′

plays an important role, and with 𝜏 ′ the normalized trace on N ′ , we have 𝜆 ∶= 𝜏 ′ (𝑒) = [M ∶ N ]−1 . The basic construction of V. Jones is the von Neumann factor M 1 = {M ∪ {𝑒}}″ ⊂ B (H ) and [M 1 ∶ M ] = [M ∶ N ]. It is clear that [M ∶ N ] ≥ 1. By a repeated use of the basic construction, V. Jones proved the following striking result: If [M ∶ N ] < 4, then [M ∶ N ] ∈ {4 cos2

𝜋 ; 𝑛 = 3, 4, ...} . 𝑛

Moreover, V. Jones proved that, for each 𝑛 = 3, 4, ..., there is a subfactor N of the type II1 hyperfinite factor M such that [M ∶ N ] = 4 cos2

𝜋 . 𝑛

On the other hand, with a rather simple argument, V. Jones proved that if M is the hyperfinite factor of type II1 , then for any 𝑟 ∈ [4; ∞), there is a subfactor N ⊂ M such that [M ∶ N ] = 𝑟. However, an essentially still open problem is for what 𝑟 ∈ (4; ∞), there is an irreducible inclusion N ⊂ M , that is N ′ ∩ M = ℂ ⋅ 1, such that [M ∶ N ] = 𝑟? (M = the hyperfinite II1 factor.) An important contribution to the theory of index is due to M. Pimsner and S. Popa (1986). The book of F. Goodman, P. de la Harpe and V. Jones (1989) is devoted to a detailed study of the index of subfactors. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.010

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Lectures on von Neumann algebras The Jones index had important applications to the theory of knots. Namely, V. Jones (1985) found a polynomial invariant for knots that distinguishes a knot from its mirror image, and A. Ocneanu (see Freyd, P., … Ocneanu, A. [1985]) found an even more refined polynomial invariant in two variables. We quote also a nice technical remark by H. Wenzl (1987).

C.7.8 Bibliographical comments. The results in this section are essentially due to F. J. Murray and J. von Neumann, who proved them in the case of factors. The globalization of these results to arbitrary von Neumann algebras was begun by J. Dixmier (1949) and I. Kaplansky (1950), and continued by H. A. Dye, E. L. Griffin, R. V. Kadison, R. Pallu de la Barrière, and others. We mention the fact that Lemma 7.18 is due to E. L. Griffin (1953/1955). In writing this chapter, we referred to I. Kaplansky (1955/1968), J. R. Ringrose (1972b), R. V. Kadison (1952) and J. Dixmier (1957/1969).

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8

SPATIAL ISOMORPHISMS AND RELATIONS BETWEEN TOPOLOGIES

In this chapter, we consider the cases in which a *-isomorphism between von Neumann algebras is implemented by a unitary operator, as well as the cases in which the w-topology coincides with the wo-topology. 8.1. Let M 1 ⊂ B (H 1 ) and M 2 ⊂ B (H 2 ) be von Neumann algebras. One says that a *-isomorphism 𝜋 ∶ M 1 → M 2 is a spatial isomorphism (or that it is unitarily implemented) if there exists a unitary operator 𝑢 ∶ H 1 → H 2 , such that 𝜋(𝑥1 ) = 𝑢 ∘ 𝑥1 ∘ 𝑢∗ ,

𝑥1 ∈ M 1 .

While any *-isomorphism is w-continuous (5.13), but it is not necessarily wo-continuous, it is obvious that any spatial isomorphism is wo-continuous. This is one of the main properties that distinguish the two kinds of isomorphisms. The problem that we consider in this chapter is to find some sufficiently simple and general conditions under which a given *-isomorphism is spatial. For example, according to Corollary 5.25, any *-isomorphism between two von Neumann algebras with vectors, which are both cyclic and separating, is spatial. 8.2. In what follows we shall essentially use the following simple proposition, which establishes a ‘canonical form’ for the *-homomorphisms of a von Neumann algebra onto another one and makes more precise the problem of the unitary implementation of a *-isomorphism. Proposition. Let M 1 ⊂ B (H 1 ) and M 2 ⊂ B (H 2 ) be von Neumann algebras and 𝜋 ∶ M 1 → M 2 a w-continuous *-homomorphism, such that 𝜋(M 1 ) = M 2 . Then there exist a von Neumann algebra M ⊂ B (H ), two projections 𝑒1′ , 𝑒2′ ∈ M ′ ⊂ B (H ) and two spatial isomorphisms

197

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Lectures on von Neumann algebras 𝜋1 ∶ M 1 → M 𝑒1′ ⊂ B (𝑒1′ H ) 𝜋2 ∶ M 2 → M 𝑒2′ ⊂ B (𝑒2′ H ),

such that z(𝑒1′ ) = 1 and (𝜋2 ∘ 𝜋 ∘ 𝜋1−1 )(𝑥𝑒1′ ) = 𝑥𝑒2′ ,

𝑥 ∈ M.

Moreover, 𝜋 is a *-isomorphism ⇔ z(𝑒2′ ) = 1, 𝜋 is a spatial isomorphism ⇔ 𝑒1′ ∼ 𝑒2′ in M ′ . Proof. Let H = H 1 ⊕ H 2 . The mapping 𝜋1̂ ∶ M 1 ∋ 𝑥1 → 𝑥1 ⊕ 𝜋(𝑥1 ) ∈ B (H ) is an injective w-continuous *-homomorphism; hence, in accordance with Corollary 3.12, we have M = 𝜋1̂ (M 1 ) = {𝑥1 ⊕ 𝜋(𝑥1 ); 𝑥1 ∈ M 1 } ⊂ B (H ) is a von Neumann algebra, whereas 𝜋1̂ ∶ M 1 → M is a *-isomorphism. We now consider the canonical isometries 𝑢1 ∶ H 1 → H ,

𝑢1 (𝜉1 ) = 𝜉1 ⊕ 0,

𝜉1 ∈ H 1 ,

𝑢2 ∶ H 2 → H ,

𝑢2 (𝜉2 ) = 0 ⊕ 𝜉2 ,

𝜉2 ∈ H 2 ;

the projections 𝑒1′ = 𝑢1 ∘ 𝑢1∗ , 𝑒2′ = 𝑢2 ∘ 𝑢2∗ ∈ M ′ ⊂ B (H ) and the spatial isomorphisms 𝜋1 ∶ M 1 ∋ 𝑥1 ↦ 𝑢1 ∘ 𝑥1 ∘ 𝑢1∗ ∈ M 𝑒1′ ⊂ B (𝑒1′ H ), 𝜋2 ∶ M 2 ∋ 𝑥2 ↦ 𝑢2 ∘ 𝑥2 ∘ 𝑢2∗ ∈ M 𝑒2′ ⊂ B (𝑒2′ H ). Since the canonical induction M → M 𝑒1′ coincides with the *-isomorphism 𝜋1 ∘ 𝜋1̂ −1 from Proposition 3.14, we infer that z(𝑒1′ ) = 1. Consequently, the mapping 𝜋̂ ∶ M 𝑒1′ ∋ 𝑥𝑒1′ ↦ 𝑥𝑒2′ ∈ M 𝑒2′ is correctly defined. It is immediately verified that 𝜋̂ = 𝜋2 ∘ 𝜋 ∘ 𝜋1−1 . The other assertions in the statement of the proposition easily follow. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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The proof can be sketched by the following commutative diagram: M π^ 1 π1

Me'1

M1

π^

Me'2

π

π2 M2

Q.E.D. 8.3. Let now M 1 ⊂ B (H 1 ) and M 2 ⊂ B (H 2 ) be finite von Neumann algebras, whose commutants M ′1 ⊂ B (H 1 ) and M ′2 ⊂ B (H 2 ) are also finite. According to Section 7.20, let Z 1̃ and Z 2̃ be the extensions of the centres Z 1 and Z 2 of the algebras M 1 , M 2 , and let 𝔠M 1 ,M ′1 ∈ Z 1̃ , 𝔠M 2 ,M ′2 ∈ Z 2̃ be the corresponding coupling elements. Any *-isomorphism 𝜋 ∶ M 1 → M 2 induces a *-isomorphism 𝜋̃ ∶ Z 1̃ → Z 2̃ , uniquely determined by the condition that the restriction of 𝜋̃ to Z 1 coincides with the restriction of 𝜋 to Z 1 . Theorem. Let M 1 and M 2 be finite von Neumann algebras, whose commutants are also finite. A *-isomorphism 𝜋 ∶ M 1 → M 2 is spatial iff 𝜋(𝔠 ̃ M 1 ,M ′1 ) = 𝔠M 2 ,M ′2 . Proof. If 𝜋 is spatial, then obviously, 𝜋(𝔠 ̃ M 1 ,M ′1 ) = 𝔠M 2 ,M ′2 . Conversely, let us assume that this condition is satisfied. According to Proposition 8.2, we can assume that there exists a von Neumann algebra M ⊂ B (H ) and two projections 𝑒1′ , 𝑒2′ ∈ M ′ , such that z(𝑒1′ ) = z(𝑒2′ ) = 1, 𝜋 ∶ M 1 = M 𝑒1′ ∋ 𝑥𝑒1′ ↦ 𝑥𝑒2′ ∈ M 𝑒2′ = M 2 . On one hand, M is finite since it is *-isomorphic to M 1 and M 2 . On the other hand, since M ′𝑒′ = M ′1 and M ′𝑒′ = M ′2 are finite, it follows that 𝑒1′ , 𝑒2′ ∈ M ′ are finite 1 2 projections. In accordance with Proposition 4.15, 𝑒1′ ∨ 𝑒2′ ∈ M ′ is a finite projection. Therefore, we can assume that M ′ is finite. The centres of the algebras M 𝑒1′ and M 𝑒2′ canonically identify with the centre of the algebra M , and these identifications induce canonical identifications of the corresponding extensions. Assuming that these identifications have been performed, the condition 𝜋(𝔠 ̃ M 1 ,M ′1 ) = 𝔠M 2 ,M ′2 becomes 𝔠M 1 ,M ′1 = 𝔠M 2 ,M ′2 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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Let now ♮′ be the canonical central trace on M ′ . According to Proposition 7.21, we have ′

𝔠M , M ′ = (𝑒1′ )♮ 𝔠M 1 ,M ′1 , ′

𝔠M ,M ′ = (𝑒2′ )♮ 𝔠M 2 ,M ′2 . It follows that

′

′

(𝑒1′ )♮ = (𝑒2′ )♮ ; hence, according to 7.12, 𝑒1′ ∼ 𝑒2′ , and this implies that the *-isomorphism 𝜋 is spatial.

Q.E.D.

8.4. In order to define the spatial invariants, which will be used in the other theorems of unitary implementation, we need the following: Lemma. Let M be a properly infinite von Neumann algebra, and let, for any 𝑗 = 1, 2, {𝑒𝑗,𝑖𝑗 }𝑖𝑗 ∈𝐼𝑗 be a family of equivalent, mutually orthogonal projections in M , which are piecewise of countable type, such that ∑𝑖 ∈𝐼 𝑒𝑗,𝑖𝑗 = 1. Then card 𝐼1 = card 𝐼2 . 𝑗

𝑗

Proof. Without any loss of generality, we can assume that 𝑒𝑗,𝑖𝑗 are of countable type, 𝑖𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, 2. For each 𝑖1 ∈ 𝐼1 , we denote 𝐼2,𝑖1 = {𝑖2 ∈ 𝐼2 ; 𝑒1,𝑖1 𝑒2,𝑖2 𝑒1,𝑖1 ≠ 0}. Obviously, we have (*)

𝐼2 =

⋃

𝐼2,𝑖1 .

𝑖1 ∈𝐼1

Since 𝑒1,𝑖1 is of countable type, according to exercise E.5.6, we infer that there exists a normal form 𝜑𝑖1 on M , such that s(𝜑𝑖1 ) = 𝑒1,𝑖1 . Then, we have +∞ > 𝜑1 (𝑒1,𝑖1 ) = 𝜑𝑖1 (𝑒1,𝑖1 ( ∑ 𝑒2,𝑖2 ) 𝑒1,𝑖1 ) 𝑖2 ∈𝐼2

= ∑ 𝜑𝑖1 (𝑒1,𝑖1 𝑒2,𝑖2 𝑒1,𝑖1 ) 𝑖2 ∈𝐼2

= ∑ 𝜑𝑖1 (𝑒1,𝑖1 𝑒2,𝑖2 𝑒1,𝑖1 ); 𝑖2 ∈𝐼2,𝑖1

hence 𝐼2,𝑖2 is at most countable. Since 𝐼2 is an infinite set, from relation (*), it follows that card 𝐼1 ≤ card 𝐼2 . The reversed inequality can be obtained analogously.

Q.E.D.

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8.5. Let M be a properly infinite von Neumann algebra and let 𝛾 be an infinite cardinal. One says that M is uniform of type 𝛾 if there exists a family {𝑒𝑖 }𝑖∈𝐼 of equivalent, mutually orthogonal projections in M , piecewise of countable type, such that ∑𝑖∈𝐼 𝑒𝑖 = 1 and card 𝐼 = 𝛾. According to Lemma 7.2, any finite projection is piecewise of countable type. Consequently, a semifinite, properly infinite von Neumann algebra, which is uniform of type 𝑆𝛾 (see exercise E.4.14), is also uniform of type 𝛾. By taking into account exercise E.4.14, Lemma 8.4 shows that for any semifinite von Neumann algebra M , there exist a family Γ of distinct infinite cardinals and a family {𝑝1 , 𝑝𝛾 }𝛾∈Γ of mutually orthogonal central projections, uniquely determined by the conditions: 𝑝1 + ∑ 𝑝𝛾 = 1, 𝛾∈Γ

M 𝑝1 is finite, M 𝑝𝛾 is uniform of type 𝑆𝛾 . On the other hand, since any abelian projection is finite (4.8), any properly infinite von Neumann algebra, which is homogeneous of type 𝐼𝛾 (see exercise E.4.14) is also uniform of type 𝛾. By taking into account exercise E.4.14, Lemma 8.4 and exercise E.4.15 show that for any discrete von Neumann algebra M , there exists a family Γ of distinct cardinals and a family {𝑝𝛾 }𝛾∈Γ of mutually orthogonal, central, non-zero projections, uniquely determined by the conditions: ∑ 𝑝𝛾 = 1, 𝛾∈Γ

M 𝑝𝛾 is homogeneous of type 𝐼𝛾 . In the general case of a properly infinite von Neumann algebra, we have the following result: Proposition. Let M be a properly infinite von Neumann algebra. Then there exist a family Γ of distinct cardinals and a family {𝑝𝛾 }𝛾∈Γ of non-zero, mutually orthogonal, central projections, uniquely determined by the conditions: ∑ 𝑝𝛾 = 1, 𝛾∈Γ

M 𝑝𝛾 𝑖𝑠 𝑢𝑛𝑖𝑓𝑜𝑟𝑚, 𝑜𝑓 𝑡𝑦𝑝𝑒 𝐼𝛾 . Proof. The existence part of the proposition easily follows with the help of a usual argument based on the Zorn lemma and on the comparison theorem (4.6), whereas the uniqueness part of the proposition follows from Lemma 8.4. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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8.6. Let M ⊂ B (H ) be a von Neumann algebra whose commutant M ′ ⊂ B (H ) is properly infinite. Let Γ′ and {𝑝𝛾′ ′ }𝛾′ ∈Γ′ ⊂ M ′ ∩ M be the families canonically associated to M ′ in accordance with Proposition 8.5. One can then define the symbol 𝔲M ′ = (Γ′ , {𝑝𝛾′ ′ }𝛾′ ∈Γ′ ), which can be called the uniformity of M . If 𝜋 ∶ M 𝑙 → M 2 is a *-isomorphism between two von Neumann algebras, whose commutants are properly infinite, and if ′ 𝔲M ′ = (Γ𝑗′ , {𝑝𝑗,𝛾 ′ }𝛾 ′ ∈Γ′ ),

𝑗 = 1, 2,

𝑗

𝑗

then we shall write that 𝜋(𝔲 ̃ M ′ ) = 𝔲M ′ , 1

Γ1′

Γ2′

′ 𝜋(𝑝1,𝛾 ′)

′ 𝑝2,𝛾 ′,

′

2

Γ1′

Γ2′ .

= and = for any 𝛾 ∈ = The following theorem of spatial isomorphism contains the case of the semifinite algebras whose commutants are properly infinite, as well as the case of the algebras of type III (see Theorem 6.4). if

Theorem. Let M 1 and M 2 be von Neumann algebras whose commutants are properly infinite. A *-isomorphism 𝜋 ∶ M 1 → M 2 is spatial iff 𝜋(𝔲 ̃ M ′ ) = 𝔲M ′ . 1

2

Proof. If 𝜋 is spatial, then obviously 𝜋(𝔲 ̃ M ′ ) = 𝔲M ′ . Conversely, let us assume that this 1 2 condition is satisfied. Then we can assume that M ′1 as well as M ′2 are uniform of the same type 𝛾. According to Proposition 8.2, we can assume that there exist a von Neumann algebra M ⊂ B (H ) and two projections 𝑒1′ , 𝑒2′ ∈ M ′ , such that z(𝑒1′ ) = z(𝑒2′ ) = 1, 𝜋 ∶ M 1 = M 𝑒1′ ∋ 𝑥𝑒1′ ↦ 𝑥𝑒2′ ∈ M 𝑒2′ = M 2 . Since, by assumption, M ′1 and M ′2 are uniform, of the same type, it follows that, for each 𝑗 = 1, 2, there exists an infinite family ′ {𝑒𝑗,𝑖 }𝑖∈𝐼 ⊂ M ′𝑗 = M ′𝑒′ , 𝑗

of equivalent, mutually orthogonal projections, piecewise of countable type, such that ′ ∑ 𝑒𝑗,𝑖 = 𝑒𝑗′ ,

𝑗 = 1, 2

𝑖∈𝐼

(i.e., the unit element of the corresponding algebra). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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By performing a partition by countable subsets of the set 𝐼 and by considering the ′ corresponding sums of those projections 𝑒𝑗,𝑖 , whose indices 𝑖 belong to the same subset ′ of the partition, it is easy to see that we can assume that the projections 𝑒𝑗,𝑖 are properly infinite. ′ ′ For any 𝑖 ∈ 𝐼, 𝑒1,𝑖 and 𝑒2,𝑖 are then properly infinite projections, piecewise of countable ′ ′ ′ type, in M , and z(𝑒1,𝑖 ) = z(𝑒2,𝑖 ) = 1. According to Proposition 4.13, it follows that ′ ′ 𝑒1,𝑖 ∼ 𝑒2,𝑖 , 𝑖 ∈ 𝐼.

Therefore, we have 𝑒1′ ∼ 𝑒2′ , and hence, 𝜋 is a spatial isomorphism.

Q.E.D.

8.7. Let M ⊂ B (H ) be a discrete von Neumann algebra. Its commutant M ′ ⊂ B (H ) is again a discrete von Neumann algebra (according to Theorem 6.4). By taking into account Section 8.5, we obtain a family Γ′ of distinct cardinals and a family {𝑝𝛾′ ′ }𝛾′ ∈Γ′ of mutually orthogonal central projections, uniquely determined by the conditions ∑ 𝑝𝛾′ ′ = 1, 𝛾 ′ ∈Γ′

M ′ 𝑝𝛾′ ′ is homogeneous, of type 𝐼𝛾′ . We can then define the symbol 𝔳M ′ = (Γ′ , {𝑝𝛾′ ′ }𝛾′ ∈Γ′ ), which can be called the homogeneity of M ′ . If 𝜋 ∶ M 𝑙 → M 2 is a *-isomorphism between two discrete von Neumann algebras and if ′ 𝔳M ′ = (Γ𝑗′ , {𝑝𝑗,𝛾 ′ }𝛾 ′ ∈Γ′ ),

𝑗 = 1, 2,

𝑗

𝑗

then we shall write that 𝜋(𝔳 ̃ M ′ ) = 𝔳M ′ 1

if

Γ1′

=

Γ2′

and

′ 𝜋(𝑝1,𝛾 ′)

=

′ 𝑝2,𝛾 ′,

′

for any 𝛾 ∈

2

Γ1′

=

Γ2′ .

Theorem. Let M 1 and M 2 be discrete von Neumann algebras. Then a *-isomorphism 𝜋 ∶ M 1 → M 2 is spatial iff 𝜋(𝔳 ̃ M ′ ) = 𝔳M ′ . 1

2

Proof. The proof is similar to that of Theorem 8.6. Instead of Proposition 4.13, one uses proposition 4.10. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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8.8. Let 𝜋 ∶ M → M be a *-automorphism of the von Neumann algebra M , whose centre is Z . Obviously, we have 𝜋(Z ) = Z . We shall say that 𝜋 acts identically on the centre if 𝜋(𝑧) = 𝑧, for any 𝑧 ∈ Z . It is obvious that if M is a factor, then any *-automorphism of M acts identically on the centre. If 𝜋 acts identically on the centre, then 𝜋 conserves the invariants 𝔠M , M ′ , 𝔲M ′ , 𝔳M ′ already introduced, and according to the case. In the following sections, we state some obvious consequences of Theorems 8.3, 8.6 and 8.7. Corollary 8.9. Any *-automorphism of a von Neumann algebra, whose commutant is properly infinite, which acts identically on the centre, is spatial. Corollary 8.10. Any *-automorphism of a finite von Neumann algebra, which acts identically on the centre, is spatial. Corollary 8.11. Any *-automorphism of a discrete von Neumann algebra, which acts identically on the centre, is inner. Proof. It follows from 8.11 and 6.5. Corollary 8.12. Any *-isomorphism between von Neuman algebras, whose commutants are properly infinite and of countable type, is spatial. Corollary 8.13. Any *-isomorphism between von Neumann algebras of type III, which operate in separable Hilbert spaces, is spatial. 8.14. In what follows we shall study the relations existing between the various topologies already defined in a von Neumann algebra. We recall that, besides the norm (uniform) topology and the topologies wo, so and w, which were defined in Section 1.3, we have also considered the topology s, which has been defined in exercise E.5.5. Between these topologies, we have the following relations of strength: wo ≤ 𝑠𝑜 w ≤ 𝑠 ≤ 𝑛, where by 𝑛 we have denoted the norm (uniform) topology. We are now concerned with the precise relations existing between the topologies w and wo and between the topologies s and so. On the closed unit ball of M , the restrictions of the topologies w and wo (resp., s and so) coincide (see 1.2, 1.10 and E.5.8). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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The s-continuous (resp., the so-continuous) linear forms coincide with the w-continuous (resp., the wo-continuous) linear forms (see 1.4 and E.5.8). It follows that the w-topology (resp., the wo-topology) is the weakened topology associated with the s-topology (resp., the so-topology). On the other hand, a net {𝑥𝑖 } in M is s-convergent to 0 (resp., so-convergent to 0) iff the net {𝑥𝑖∗ 𝑥𝑖 } is w-convergent to 0 (resp., wo-convergent to 0; see E.5.8). Consequently, the s-topology coincides with the so-topology iff the w-topology coincides with the wo-topology. Obviously, these conditions are equivalent to the condition that any w-continuous linear form be wo-continuous; hence (see 5.16), to the condition that any normal form be wo-continuous. The fundamental result of the problem we are concerned with is contained in Corollary 5.24: if the von Neumann algebra M ⊂ B (H ) has a separating vector, then any normal form on M is an 𝜔𝜉 , 𝜉 ∈ H ; in particular, in this case, the w-topology coincides with the wo-topology. It is therefore only natural to begin our investigations with the study of the conditions under which a von Neumann algebra has separating vectors. Lemma 8.15. Let M ⊂ B (H ) be a von Neumann algebra whose commutant M ′ ⊂ B (H ) is properly infinite. Then M has a separating vector iff M is of countable type. Proof. If M ⊂ B (H ) has a separating vector 𝜉 ∈ H , then s(𝜔𝜉 ) = 𝑝𝜉 = 1, and according to exercise E.5.6, M is of countable type. Conversely, let M be of countable type and M ′ properly infinite. In order to show that M has a separating vector, we can assume that, in accordance with Lemma 7.18, there exists an 𝜂 ∈ H , such that 𝑝𝜂′ = 1. Then the projections 𝑝𝜂 and 1 in M are of countable type, properly infinite (see Lemma 6.3), and they have the same central support. Proposition 4.13 now implies that 𝑝𝜂 ∼ 1. Let 𝑣 ∈ M , such that 𝑣∗ 𝑣 = 𝑝𝜂 , 𝑣𝑣 ∗ = 1 and 𝜉 = 𝑣𝜂. Then we get 𝑝𝜉 = 1; hence 𝜉 is a separating vector for M . Q.E.D. Theorem 8.16. Let M ⊂ B (H ) be a von Neumann algebra whose commutant M ′ ⊂ B (H ) is properly infinite. For any w-continuous linear form 𝜑 on M , there exist 𝜉, 𝜂 ∈ H , such that 𝜑 = 𝜔𝜉,𝜂 . In particular, the w-topology coincides with the wo-topology, and the s-topology coincides with the so-topology. Proof. Let first 𝜓 be a normal form on M and 𝑒 = s(𝜓). Then M 𝑒 is a von Neumann algebra of countable type, whose commutant is properly infinite. According to Lemma 8.15, there exists a 𝜁 ∈ H , such that 𝑒 = 𝑝𝜁 . Since s(𝜓) = 𝑝𝜁 , from Theorem 5.23, we infer that there exists an 𝜂 ∈ H , such that 𝜓 = 𝜔𝜂 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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Let now 𝜑 be a w-continuous linear form on M , and let 𝜑 = 𝑅𝑣 𝜓 be its polar decomposition (5.16). Since 𝜓 is a normal form, there exists an 𝜂 ∈ H , such that 𝜓 = 𝜔𝜂 . Then 𝜑 = 𝜔𝑣𝜂,𝜂 . Q.E.D. Corollary 8.17. Let M ⊂ B (H ) be an arbitrary von Neumann algebra. For any w-continuous linear form 𝜑 on M , there exist two sequences {𝜉𝑛 }, {𝜂𝑛 } ⊂ H , such that ∞ 𝑛=1

∞

∞

𝜑 = ∑ 𝜔𝜉𝑛 ,𝜂𝑛 ,

∑ ‖𝜉‖2 < +∞,

∑ ‖𝜂‖2 < +∞.

𝑛=1

𝑛=1

Proof. Let us consider the separable Hilbert space 𝑙 2 and the von Neumann algebra M ̃ = M ⊗C (𝑙 2 ) ⊂ B (H ⊗𝑙 2 ). Then H ⊗𝑙 2 identifies with the Hilbert space ∞

H ̃ = {{𝜉𝑛 } ⊂ H ; ∑ ‖𝜉𝑛 ‖2 < +∞} , 𝑛=1

and for any 𝑥̃ = 𝑥⊗1 ∈ M ̃ and any 𝜉 ̃ = {𝜉𝑛 } ∈ H ,̃ we have 𝑥̃𝜉 ̃ = {𝑥𝜉𝑛 }. According to Section 3.18, the amplification M ∋ 𝑥 ↦ 𝑥̃ ∈ M ̃ is a *-isomorphism, whereas, according to Proposition 3.17, and to exercise E.4.20, (M ̃ )′ = M ′ ⊗B (𝑙 2 ) ⊂ B (H ⊗𝑙 2 ) is a properly infinite von Neumann algebra. Let 𝜑 be a w-continuous linear form on M and let us define the w-continuous linear form 𝜑̃ on M ̃ by the relation 𝑥̃ = 𝑥⊗1 ∈ M ̃ .

𝜑(̃ 𝑥)̃ = 𝜑(𝑥),

According to Theorem 8.16, there exist 𝜉 ̃ = {𝜉𝑛 }, 𝜂 ̃ = {𝜂𝑛 } ∈ H ,̃ such that 𝜑̃ = 𝜔𝜉,̃ 𝜂̃ . It follows that

∞

𝜑 = ∑ 𝜔𝜉𝑛 ,𝜂𝑛 . 𝑛=1

Q.E.D.

Lemma 8.18. Let M ⊂ B (H ) be a finite von Neumann algebra of countable type, whose commutant M ′ ⊂ B (H ) is finite. Then M has a separating vector iff 𝔠M , M ′ ≤ 1. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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Proof. Let 𝜉 ∈ H be a separating vector for M . Then we have 𝑝𝜉 = 1, 𝑝𝜉′ ≤ 1; hence ′

1 ≥ (𝑝𝜉′ )♮ = 𝔠M , M ′ (𝑝𝜉 )♮ = 𝔠M , M ′ . Conversely, let us assume that 𝔠M , M ′ ≤ 1 and then we shall prove that M has a separating vector. According to Lemma 7.18, we can assume that there exists a 𝜉 ∈ H , such that 𝑝𝜉′ = 1. Then ′

1 = (𝑝𝜉′ )♮ = 𝔠M , M ′ (𝑝𝜉 )♮ ≤ (𝑝𝜉 )♮ ≤ 1, hence 𝑝𝜉 = 1. Thus, 𝜉 is a separating vector for M .

Q.E.D.

Theorem 8.19. Let M ⊂ B (H ) be a finite von Neumann algebra, whose commutant M ′ ⊂ B (H ) is finite. Then the w-topology (resp., the s-topology) coincides with the wo-topology (resp., the so-topology) iff 𝔠M , M ′ ∈ Z . Moreover, if 𝑛 is a natural number, then the following conditions are equivalent: (i) for any normal form 𝜓 on M , there exist 𝑛 vectors 𝜂1 , … , 𝜂𝑛 , such that 𝑛

𝜓 = ∑ 𝜔𝜂𝑘 . 𝑘=1

(ii) For any w-continuous linear form 𝜑 on M , there exist 𝑛 pairs of vectors (𝜉1 , 𝜂1 ), … , (𝜉𝑛 , 𝜂𝑛 ), such that 𝑛

𝜑 = ∑ 𝜔𝜉𝑘 ,𝜂𝑘 𝑘=1

(iii) 𝔠M , M ′ ≤ 𝑛. Proof. It is easy to see that assertion (i) (resp., (ii); resp., (iii)) for the algebra M and 𝑛 = 𝑛0 is equivalent to assertion (i) (resp., (ii); resp., (iii)) for the algebra M ̃ 𝑛0 = M ⊗C (H 𝑛0 ) and 𝑛 = 1, where H 𝑛0 is a Hilbert space of dimension 𝑛0 (see Corollary 7.22 and the proof of Corollary 8.17). Consequently, in order to prove the equivalence of the assertions (i)–(iii), we can assume that 𝑛 = 1. Then (i) ⇒ (ii), according to the polar decomposition theorem (5.16); (ii) ⇒ (i), according to exercise E.5.2, whereas the equivalence (i) ⇔ (iii) follows by taking into account Lemma 8.18 and Proposition 7.21. We remark that the equivalence (i) ⇔ (ii) holds in any von Neumann algebra. If 𝔠M ,M ′ ∈ Z , then there exists a natural number 𝑛, such that 𝔠M , M ′ ≤ 𝑛; from what we have already proved it clearly follows that the w-topology coincides with the wo-topology. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.011

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If 𝑧 ̃ ∈ Z ,̃ 𝑧 ̃ ≥ 0 (see 7.20) and if there exists a 𝑧 ∈ Z , such that 𝑧 ̃ ≤ 𝑧, then 𝑧 ̃ ∈ Z . Consequently, if 𝔠M , M ′ ∉ Z , then there exists a sequence {𝑞𝑛 } of non-zero central projections of countable type, which are mutually orthogonal such that, for any 𝑛, we have ∞ 𝔠M , M ′ 𝑞𝑛 ≥ 𝑛𝑞𝑛 . Then 𝑞 = ∑𝑛=1 𝑞𝑛 is of countable type and 𝔠M , M ′ 𝑞 ∉ Z 𝑞. Thus, we can assume that M is of countable type. Let 𝜑 be a faithful normal form on M . If the w-topology coincides with the wo-topology, then there exist a natural number 𝑛 and vectors 𝜉1 , … , 𝜉𝑛 ∈ H , such that 𝑛

𝜑 = ∑ 𝜔𝜉𝑘 . 𝑘=1

˜ 𝑛 is a separating vector for the von Neumann algebra M ̃ 𝑛 . Then 𝜉 ̃ = {𝜉1 , … , 𝜉𝑛 } ∈ H ˜ 𝑛 . In particular (see According to Corollary 5.24, any normal form on M ̃ 𝑛 is an 𝜔𝜉 ̃ , 𝜉 ̃ ∈ H 𝑛

the proof of Corollary 8.17), any normal form on M is equal to a sum ∑𝑘=1 𝜔𝜉𝑘 . From the first part of the proof, we infer that 𝔠M , M ′ ≤ 𝑛; hence 𝔠M , M ′ ∈ Z . Q.E.D. Corollary 8.20. Let M ⊂ B (H ) be a finite factor. Then the w-topology (resp., the s-topology) coincides with the wo-topology (resp., the so-topology). 8.21. Our study of the relation existing between the topologies w and wo (resp., s and so) is completed by the following: Theorem. Let M ⊂ B (H ) be a properly infinite von Neumann algebra, whose commutant M ′ ⊂ B (H ) is finite. Then, for any wo-continuous positive form 𝜑 on M , its support s(𝜑) is a finite projection in M . In particular, the w-topology (resp., the s-topology) is strictly stronger than the wo-topology (resp., the so-topology). Proof. Since M ′ is finite, for any 𝜉 ∈ H , 𝑝𝜉′ is a finite projection in M ′ , hence (see 6.3), 𝑝𝜉 is a finite projection in M . If 𝜑 is a wo-continuous positive form on M , then there exist 𝜉1 , … , 𝜉𝑛 ∈ H , such that 𝑛

𝜑 = ∑ 𝜔𝜉𝑘 , 𝑘=1

whence

𝑛

s(𝜑) =

⋁

𝑝𝜉𝑘 .

𝑘=1

By taking into account Proposition 4.15, it follows that s(𝜑) is finite.

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The final part of the theorem follows from the obvious remark that, on a properly infinite von Neumann algebra, there exist normal forms whose supports are infinite. Q.E.D. Corollary 8.22. Let M ⊂ B (H ) be an infinite factor. Then the w-topology (resp., the s-topology) coincides with the wo-topology (resp., the so-topology) iff the commutant M ′ ⊂ B (H ) is infinite. EXERCISES *E.8.1 Let M be a von Neumann algebra, whose commutant is M ′ and whose centre is Z . One says that 𝑒 ∈ PM is a maximally cyclic projection in M if it is cyclic and if 𝑓 ∈ PM cyclic, 𝑒 ≺ 𝑓 ⇒ 𝑒 ∼ 𝑓. If 𝑒 ∈ PM is a maximally cyclic projection in M and if 𝑝 ∈ PZ , then 𝑒𝑝 ∈ PM 𝑝 is maximally cyclic in M𝑝 . With the help of the comparison theorem, one infers that the maximally cyclic projections in M are mutually equivalent. Show that if Z is of countable type, then any cyclic projection in M is contained in a maximally cyclic projection. If, moreover, M and M ′ are properly infinite, then the set of all maximally cyclic projections in M coincides with the set of all properly infinite projections of countable type, whose central support is equal to 1. E.8.2 Let M ⊂ B (H ) be a von Neumann algebra with the cyclic vector 𝜉 ∈ H and 𝑒 ∈ PM . The following assertions are equivalent: (i) 𝑒 is maximally cyclic. (ii) 𝑒 ∼ 𝑝𝜉 . (iii) 𝑒 ∼ 𝑝𝜂 ⇒ 𝑝𝜂′ = 1, 𝜂 ∈ H . E.8.3 Let M 1 ⊂ B (H 1 ) and M 2 ⊂ B (H 2 ) be von Neumann algebras with the cyclic vectors 𝜉1 ∈ H 1 and 𝜉2 ∈ H 2 , respectively. A *-isomorphism 𝜋 ∶ M 1 → M 2 is spatial iff it conserves the maximal cyclicity of the projections (i.e., 𝑒1 ∈ P M 1 maximally cyclic ⇒ 𝜋(𝑒1 ) ∈ P M 1 maximally cyclic). E.8.4 Let M 1 and M 2 be properly infinite von Neumann algebras, whose commutants are finite and whose centres are of countable type. A *-isomorphism 𝜋 ∶ M 1 → M 2 is spatial iff it conserves the maximal cyclicity of the projections. ▶

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!E.8.5 Any *-isomorphism between two standard von Neumann algebras is spatial. E.8.6 Prove the assertion from E.7.18 with the help of Theorem 8.16. E.8.7 Let M ⊂ B (H ) be a von Neumann algebra. Then the w-topology on M is determined by the family of seminorms: |∞ | 𝑥 ↦ || ∑ (𝑥𝜉𝑛 |𝜂𝑛 )|| , |𝑛=1 | ∞

where {𝜉𝑛 }, {𝜂𝑛 } ⊂ H , ∑𝑛=1 (‖𝜉𝑛 ‖2 + ‖𝜂𝑛 ‖2 ) < +∞, whereas the s-topology on M is determined by the family of seminorms 1/2

∞ 2

𝑥 ↦ ( ∑ ‖𝑥𝜉𝑛 ‖ )

,

𝑛=1 ∞

where {𝜉𝑛 } ⊂ H , ∑𝑛=1 ‖𝜉𝑛 ‖2 < +∞. In particular, the s-topology coincides with the ultrastrong topology. !E.8.8 Let M 1 and M 2 be von Neumann algebras and 𝜋 ∶ M 1 → M 2 a w-continuous *-homomorphism, such that 𝜋(M 1 ) = M 2 . Then there exist ˜ 1, an amplification 𝜋1 ∶ M 1 → M ˜ ˜ ˜ 1 )′ and an induction 𝜋2 ∶ M 1 → (M 1 )𝑒′ , 𝑒′ ∈ (M ˜ 1 )𝑒 ′ → M 2 , a spatial isomorphism 𝜋2 ∶ (M such that 𝜋 = 𝜋3 ∘ 𝜋2 ∘ 𝜋1 . (Hint: if M 2 has a cyclic vector 𝜉2 , then, with a suitable amplification, we ˜ 1 𝜉1̃ ].) have (𝜔𝜉2 ∘ 𝜋)∼ = 𝜔𝜉1̃ and one defines 𝑒′ = [M E.8.9 Let M ⊂ B (H ) be a finite factor and 𝜇 a finite normal trace on M + . From Corollary 8.20, one obviously infers that there exists a finite family {𝜉1 , … , 𝜉𝑛 } ⊂ H , such that 𝑛

𝜇 = ( ∑ 𝜔𝜉𝑘 ) |M + . 𝑘=1

With the help of E.5.9, show that the preceding representation can be chosen in such a manner, that the projections 𝑝𝜉1 … , 𝑝𝜉𝑛 be mutually orthogonal. E.8.10 Let M ⊂ B (H ) be a von Neumann algebra and let 𝜇 be a normal trace on M + . Then there exists a family {𝜉𝑖 }𝑖∈𝐼 ⊂ H , such that 𝜇 = (∑ 𝜔𝜉𝑖 ) |M + . 𝑖∈𝐼

▶

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whereas if 𝜇 is, moreover, semifinite, then this representation can be obtained in such a manner, that the projections 𝑝𝜉𝑖 , 𝑖 ∈ 𝐼, be mutually orthogonal. In particular, if M is a von Neumann algebra of countable type, then for any semifinite normal trace 𝜇 on M + , there exists a sequence {𝜉𝑘 } ⊂ H , such that {𝑝𝜉𝑘 } be mutually orthogonal and such that ∞

𝜇 = ( ∑ 𝜔𝜉𝑘 ) |M + . 𝑘=1

E.8.11 Show that, on a von Neumann algebra M , the s-topology coincides with the so-topology iff any projection, which is the least upper bound of a countable family of cyclic projections in M , is equal to the least upper bound of a finite family of cyclic projections in M . E.8.12 Let M be a von Neumann algebra. Show that if any w-continuous form on M is a finite sum of forms 𝜔𝜉,𝜂 , 𝜉, 𝜂 ∈ H , then there exists a natural number 𝑛, such that any w-continuous form on M is the sum of at most 𝑛 forms 𝜔𝜉,𝜂 . Infer from this result that if any projection of countable type in M is the least upper bound of a finite family of cyclic projections in M , then there exists a natural number 𝑛 such that any projection of countable type in M , is the least upper bound of at most 𝑛 cyclic projections in M . E.8.13 Prove the assertion from E.3.8, with the help of Lemma 7.18. E.8.14 Let 𝜋1 (resp., 𝜋2 ) be a w-continuous *-homomorphism of the von Neumann algebra M 1 (resp., M 2 ) onto the von Neumann algebra N 1 (resp., N 2 ). Show that there exists a unique w-continuous *-homomorphism 𝜋 of the von Neumann algebra M 1 ⊗M 2 onto the von Neumann algebra N 1 ⊗N 2 , such that 𝜋(𝑥1 ⊗𝑥2 ) = 𝜋1 (𝑥1 )⊗𝜋2 (𝑥2 ), 𝑥1 ∈ M 1 , 𝑥2 ∈ M 2 . If 𝜋1 and 𝜋2 are *-isomorphisms, then 𝜋 is a *-isomorphism (Hint: use exercise E.8.8). E.8.15 Let 𝜑1 (resp., 𝜑2 ) be a w-continuous linear form on the von Neumann algebra M 1 (resp., M 2 ). Show that there exists a unique w-continuous linear form 𝜑 on the von Neumann algebra M 1 ⊗M 2 , such that 𝜑(𝑥1 ⊗𝑥2 ) = 𝜑1 (𝑥1 )𝜑2 (𝑥2 ),

𝑥1 ∈ M 1 , 𝑥2 ∈ M 2 ,

If 𝜑1 and 𝜑2 are positive, then 𝜑 is positive. If 𝜑1 and 𝜑2 are faithful, then 𝜑 is faithful. (Hint: use Corollary 8.17.)

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COMMENTS C.8.1 We have not yet discussed the unitary implementation of the *-isomorphisms between von Neumann algebras of type II∞ , whose commutants are of type II1 (see Table 4.21). In the cases dealt with by Theorems 8.3, 8.6 and 8.7, the invariants 𝔠M , M′ , 𝔲M′ , 𝔳M′ , which decide on the unitary implementability of the *-isomorphisms, are expressed in terms of cardinal numbers and central elements. In particular, in these cases, any *-automorphism of a factor is spatial. In contrast to these cases, R. V. Kadison (1963) showed that there exist factors of type II∞ , whose commutants are of type II1 , which possess *-automorphisms, which are not spatial. It follows that the conceivable invariants that would decide on the unitary implementability of the *-isomorphisms between von Neumann algebras of type II∞ , whose commutants are of type II1 , are not of the same type as those for the cases already studied. R. V. Kadison (1957b) indicated such a system of invariants. Independently of the type of the von Neumann algebras, we have the fundamental result given by Corollary 5.25 and recalled in Section 8.1. Extensions of this result are contained in exercises E.8.3. (R. V. Kadison) and E.8.5 (J. Dixmier and I. E. Segal; see also E.8.4). C.8.2 Let M ⊂ B (H ) be a von Neumann algebra, 𝐺 a locally compact group and 𝑔 ↦ 𝜋𝑔 a wo-measurable representation (with respect to the Haar measure) of 𝐺, by *-automorphisms of M . The problem arises whether there exists a so-continuous unitary representation 𝑔 ↦ ᵆ𝑔 of 𝐺, in H , such that we have 𝜋𝑔 (𝑥) = ᵆ𝑔 𝑥ᵆ𝑔∗ ,

𝑥 ∈ M , 𝑔 ∈ 𝐺.

In general, the answer to this problem is negative, even if each *-automorphism 𝜋𝑔 is spatial. The following theorem gives a positive result in this direction: Theorem 1. Let M ⊂ B (H ) be a von Neumann algebra, 𝐺 a locally compact group and 𝑔 ↦ 𝜋𝑔 a wo-measurable representation of 𝐺 by *-automorphisms of M . If (i) H is separable and 𝐺 is separable and (ii) The commutant M ′ ⊂ B (H ) is properly infinite,

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213

then there exists a so-continuous unitary representation 𝑔 ↦ ᵆ𝑔 of 𝐺 in H , such that 𝜋𝑔 (𝑥) = ᵆ𝑔 𝑥ᵆ𝑔∗ ,

𝑥 ∈ M , 𝑔 ∈ 𝐺.

R. R. Kallman (1971a) proved this theorem by additionally assuming that M is semifinite and that the representation 𝑔 ↦ 𝜋𝑔 is wo-continuous. Under the above stated, more general, conditions, the theorem has been formulated and proved by M. Henle (1970). The simple and elegant proof of M. Henle reduces the problem to the result contained in Corollary 8.12. A *-automorphism of M is said to be inner if it is implemented by a unitary operator in M . Another positive result in connection with the above-mentioned problem has been obtained by R. R. Kallman (1971a) and C. C. Moore (1950, III, IV): Theorem 2. Let {𝜋𝑡 }𝑡∈ℝ be a wo-continuous one-parameter group of inner *-automorphisms of the von Neumann algebra M ⊂ B (H ). If H is separable, then there exists a so-continuous one-parameter group {ᵆ𝑡 }𝑡∈ℝ of unitary operators in M , such that 𝜋𝑡 (𝑥) = ᵆ𝑡 𝑥ᵆ𝑡∗ ,

𝑥 ∈ M , 𝑡 ∈ ℝ.

A particular case of this theorem was previously proved by R. V. Kadison (1965). A simple proof, in the case of factors, has been given by F. Hansen (1977). An assertion, equivalent to the fact that any derivation of a von Neumann algebra is inner, is that for any one-parameter group {𝜋𝑡 }𝑡∈ℝ of *-automorphisms of the von Neumann algebra M , which, moreover, is norm-continuous, there exists an invertible operator 𝑎 ∈ M , 0 ≤ 𝑎 ≤ 1, such that 𝜋𝑡 (𝑥) = 𝑎i𝑡 𝑥𝑎−i𝑡 , 𝑥 ∈ M , 𝑡 ∈ ℝ. H. J. Borchers (1966) gave a condition for the inner implementability of wo-continuous groups, with several real parameters, of spatial automorphisms. We state his result only in the case of a single parameter. Theorem 3. Let {𝜋𝑡 }𝑡∈ℝ be a wo-continuous one-parameter group of *-automorphisms of the von Neumann algebra M . Then the following conditions are equivalent: (i) There exists a 𝑏 ∈ B (H ), 𝑏 ≥ 0, s(𝑏) = 1, such that 𝜋𝑡 (𝑥) = 𝑏i𝑡 𝑥𝑏−i𝑡 ,

𝑥 ∈ M , 𝑡 ∈ ℝ.

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Lectures on von Neumann algebras (ii) There exists an 𝑎 ∈ M , 𝑎 ≥ 0, 𝑎 ≤ 1, s(𝑎) = 1, such that 𝜋𝑡 (𝑥) = 𝑎i𝑡 𝑥𝑎−i𝑡 ,

𝑥 ∈ M , 𝑡 ∈ ℝ.

Variants of the proof of H. J. Borchers appear in G. Dell’Antonio (1966), R. V. Kadison (1969), and S. Sakai (1971). Another proof has been given by W. B. Arveson (1974) (see also L. Zsidó [1977]). In connection with the conditions under which a *-automorphism of a von Neumann algebra is inner, we mention the following remarkable result due to R. V. Kadison and J. R. Ringrose (1967): Theorem 4. Any *-automorphism 𝜋 of a von Neumann algebra, such that ‖𝜋 − 1‖ < 2, is inner. A simple proof of this theorem, in which the bound 2 is replaced by √3 can be found in J. Dixmier’s book (1957/1969). For other criteria, we refer the reader to S. Sakai (1971: 167–68). C.8.3 Bibliographical comments. Theorem 8.3 is due to F. J. Murray and J. von Neumann (1937, 1943), for the case of factors, and to H. A. Dye (1953), E. L. Griffin (1953/1955), and R. Pallu de la Barrière (1952, 1954), for the general case. Theorem 8.6 is due to E. L. Griffin (1953, 1955), whereas Theorem 8.7 to I. Kaplansky (1952). The results on the comparison of the topologies wo and w are due to J. Dixmier (1953), J. A. Dye (1952), I. Kaplansky (1950), R. Pallu de la Barrière (1954), and others. The decomposition given in exercise E.8.8 is due to J. Dixmier (1954). The result given in exercise E.8.14 is due to F. J. Murray and J. von Neumann (1943), Y. Misonou (1954) and T. Turumaru (1952, 1953, 1954, 1956). In our exposition of these results, we used J. Dixmier (1957/1969), R. V. Kadison (1957b), J. R. Ringrose (1966/1967) and S. Sakai (1971).

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9

UNBOUNDED LINEAR OPERATORS IN HILBERT SPACES

This chapter contains the fundamental results from the theory of (unbounded) linear operators in Hilbert spaces that will be used in the next chapter. 9.1. Let H and K be Hilbert spaces. One says that 𝑇 is a linear operator from H into K if 𝑇 is a linear mapping of a vector subspace D 𝑇 of H into the vector space K ; in this case, D 𝑇 is called the domain of definition of 𝑇. If H = K , one also says that 𝑇 is a linear operator in H . When no danger of confusion could arise, we shall not indicate the spaces between which the linear operators act. Let 𝑆 and 𝑇 be linear operators. One says that they are equal, and one denotes by 𝑇 = 𝑆 this relation, if D 𝑇 = D 𝑆 and 𝑇𝜉 = 𝑆𝜉, for any 𝜉 ∈ D 𝑇 = D 𝑆 . One says that 𝑇 is an extension of 𝑆 (or that 𝑆 is a restriction of 𝑇), and one denotes by 𝑇 ⊃ 𝑆 (or 𝑆 ⊂ 𝑇) this relation, if D 𝑇 ⊃ D 𝑆 and 𝑇𝜉 = 𝑆𝜉, for any 𝜉 ∈ D 𝑆 . For the linear operators 𝑇 and 𝑆, one defines the multiplication by scalars 𝜆 ∈ ℂ, 𝜆𝑇: D 𝜆𝑇 = D 𝑇 , (𝜆𝑇)𝜉 = 𝜆(𝑇𝜉),

𝜉 ∈ D 𝜆𝑇 ;

the addition 𝑇 + 𝑆 D 𝑇+𝑆 = D 𝑇 ∩ D 𝑆 , (𝑇 + 𝑆)𝜉 = 𝑇𝜉 + 𝑆𝜉,

𝜉 ∈ D 𝑇+𝑆 ;

the composition 𝑆 ∘ 𝑇 = 𝑆𝑇: D 𝑆𝑇 = {𝜉 ∈ D 𝑇 ; 𝑇𝜉 ∈ D 𝑆 }, (𝑆𝑇)𝜉 = 𝑆(𝑇𝜉),

𝜉 ∈ D 𝑆𝑇 ;

the inverse 𝑇 −1 (if the mapping 𝑇 ∶ D 𝑇 → K is injective): D 𝑇 −1 = 𝑇D 𝑇 , 𝑇 −1 𝜂 = 𝜉 ⇔ 𝑇𝜉 = 𝜂,

𝜂 ∈ D 𝑇 −1 . 215

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It is easy to verify the associativity of the addition and of the composition, as well as the following distributivity relations: (𝑆1 + 𝑆2 )𝑇 = 𝑆1 𝑇 + 𝑆2 𝑇, 𝑆(𝑇1 + 𝑇2 ) ⊃ 𝑆𝑇1 + 𝑆𝑇2 . We recall that by the Hilbert sum of the Hilbert spaces H and K , the following vector space is meant H ⊕ K = {(𝜉, 𝜂); 𝜉 ∈ H , 𝜂 ∈ K }, in which the scalar product is given by ((𝜉1 , 𝜂1 ) (𝜉2 , 𝜂2 )) = (𝜉1 |𝜉2 ) + (𝜂1 |𝜂2 ),

𝜉1 , 𝜉2 ∈ H , 𝜂1 , 𝜂2 ∈ K .

Let 𝑇 be a linear operator from H into K . The set G 𝑇 = {(𝜉, 𝑇𝜉); 𝜉 ∈ D 𝑇 } ⊂ H ⊕ K is a vector subspace, called the graph of 𝑇. A vector subspace G of H ⊕ K is the graph of a linear operator from H into K iff (0, 𝜂) ∈ G ⇒ 𝜂 = 0. Obviously, we have 𝑇 = 𝑆 (resp., 𝑇 ⊃ 𝑆) iff G 𝑇 = G 𝑆 (resp., G 𝑇 ⊃ G 𝑆 ). One says that 𝑇 is densely defined if D 𝑇 is dense in H . One says that 𝑇 is preclosed if it is densely defined and if the closure of G 𝑇 in H ⊕K is the graph of a linear operator, denoted by 𝑇 and called the closure of 𝑇. Thus, 𝑇 is preclosed iff it is densely defined and {𝜉𝑛 } ⊂ D 𝑇 , 𝜉𝑛 → 0,

{𝑇𝜉𝑛 } converges ⇒ 𝑇𝜉𝑛 → 0.

One says that 𝑇 is closed if it is preclosed and 𝑇 = 𝑇, i.e. if 𝑇 is densely defined and G 𝑇 is closed in H ⊕ K . Thus, 𝑇 is closed iff it is densely defined and {𝜉𝑛 } ⊂ D 𝑇 ,

𝜉𝑛 → 𝜉0 ,

𝑇𝜉𝑛 → 𝜂0 ⇒ 𝜉0 ∈ D 𝑇 ,

𝑇𝜉0 = 𝜂0 .

One says that 𝑇 is bounded if sup{‖𝑇𝜉‖; 𝜉 ∈ D 𝑇 , ‖𝜉‖ ≤ 1} < +∞. If this condition is not satisfied, one says that 𝑇 is unbounded and then 𝑇 is continuous at no point of its domain of definition. If 𝑇 is densely defined and bounded, then 𝑇 is preclosed and 𝑇 is everywhere defined (D 𝑇 = H ). Conversely, if 𝑇 is closed and everywhere defined, then 𝑇 is bounded, in accordance with the closed graph theorem. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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If 𝑇 is closed and 𝑇 −1 exists, then 𝑇 −1 is closed. For any closed operator 𝑇, from H into K , the kernel of 𝑇 is a closed vector subspace of H . We denote the projection of H onto this subspace by n(𝑇) and r(𝑇) = 1 − n(𝑇). We shall also denote by l(𝑇) the projection of K onto the closure of the vector subspace 𝑇D 𝑇 . One says that r(𝑇) (resp., l(𝑇)) is the right (resp., left) support of 𝑇. If 𝑇 is bounded, these notations are in accordance with those already introduced in Section 2.13. 9.2. Let 𝑇 be a densely defined linear operator from H into K . The set D = {𝜂 ∈ H ; the form D 𝑇 ∋ 𝜉 ↦ (𝑇𝜉|𝜂) is bounded} is a vector subspace of K . Since D 𝑇 is dense in H , from the Riesz theorem, one infers that, for any 𝜂 ∈ D , there exists a unique element 𝜂 ∗ ∈ H , such that 𝜉 ∈ D 𝑇.

(𝑇𝜉|𝜂) = (𝜉|𝜂 ∗ ),

We now define a linear operator 𝑇 ∗ from K into H , called the adjoint of 𝑇, by the relations D ∗𝑇 = D , 𝑇 ∗ 𝜂 = 𝜂∗ ,

𝜂 ∈ D 𝑇∗ .

Thus, 𝑇 ∗ is determined by the relations (𝑇𝜉|𝜂) = (𝜉|𝑇 ∗ 𝜂),

𝜉 ∈ D 𝑇 , 𝜂 ∈ D 𝑇∗ .

It is easily verified that if the operators 𝑇, 𝑆, 𝑇 + 𝑆, 𝑆𝑇 are densely defined and 𝜆 ∈ ℂ, then (𝜆𝑇)∗ = 𝜆𝑇 ∗ , 𝑇 ⊃ 𝑆 ⇒ 𝑇 ∗ ⊂ 𝑆∗, (𝑇 + 𝑆)∗ ⊃ 𝑇 ∗ + 𝑆 ∗ , (𝑆𝑇)∗ ⊃ 𝑇 ∗ 𝑆 ∗ , and if 𝑇 −1 exists and is densely defined, then (𝑇 −1 )∗ = (𝑇 ∗ )−1 . Proposition. If 𝑇 is a densely defined linear operator and if 𝑥 is a bounded, everywhere defined linear operator, then (𝑇 + 𝑥)∗ = 𝑇 ∗ + 𝑥 ∗ , (𝑥𝑇)∗ = 𝑇 ∗ 𝑥 ∗ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Proof. Since 𝑥 is everywhere defined, D 𝑇+𝑥 = D 𝑇 . Hence, from the fact that 𝑥 is bounded and from the relation ((𝑇 + 𝑥)𝜉|𝜂) = (𝑇𝜉|𝜂) + (𝑥𝜉|𝜂), it follows that D 𝑇 ∗ = D (𝑇+𝑥)∗ . Thus, the first relation from the statement of the proposition follows by observing that, for any 𝜉 ∈ D 𝑇 = D 𝑇+𝑥 , 𝜂 ∈ D 𝑇 ∗ = D (𝑇+𝑥)∗ , we have (𝜉|(𝑇 + 𝑥)∗ 𝜂) = ((𝑇 + 𝑥)𝜉|𝜂) = (𝑇𝜉|𝜂) + (𝑥𝜉|𝜂) = (𝜉|𝑇 ∗ 𝜂) + (𝜉|𝑥 ∗ 𝜂) = (𝜉|(𝑇 ∗ + 𝑥 ∗ )𝜂). Since 𝑥 is everywhere defined, we have D 𝑥𝑇 = D 𝑇 , and as a result of a remark made just above the proposition, we have (𝑥𝑇)∗ ⊃ 𝑇 ∗ 𝑥∗ . Let 𝜂 ∈ D (𝑥𝑇)∗ and 𝜉 ∈ D 𝑇 . From the relation (𝜉|(𝑥𝑇)∗ 𝜂) = (𝑥𝑇𝜉|𝜂) = (𝑇𝜉|𝑥 ∗ 𝜂), it follows that 𝑥∗ 𝜂 ∈ D 𝑇 ∗ ; hence 𝜂 ∈ D 𝑇 ∗ 𝑥∗ , and 𝑇 ∗ 𝑥 ∗ 𝜂 = (𝑥𝑇 ∗ )𝜂. Thus, we have (𝑥𝑇)∗ ⊂ 𝑇 ∗ 𝑥∗ . Consequently, (𝑥𝑇)∗ = 𝑇 ∗ 𝑥∗ .

Q.E.D.

9.3. In order to study more thoroughly the adjoint operator, we consider the unitary operator 𝑉H K ∶ H ⊕ K ∋ (𝜉, 𝜂) ↦ (𝜂, −𝜉) ∈ K ⊕ H . Obviously, we have −1 𝑉H K = −𝑉H K .

If H = K , we shall denote 𝑉H = 𝑉H K . It is easily verified that for any densely defined linear operator 𝑇, from H into K , we have G 𝑇 ∗ = (𝑉H K G 𝑇 )⟂ . In particular, G 𝑇 ∗ is closed. Proposition. If 𝑇 is a preclosed linear operator from H into K , then 𝑇 ∗ is closed and 𝑇 ∗∗ = 𝑇. Proof. Let 𝜂 ∈ K , 𝜂 ⟂ D 𝑇 ∗ . Then (0, 𝜂) ∈ (G 𝑇 ∗ )⟂ = ((𝑉H K G 𝑇 )⟂ )⟂ = 𝑉H K ((G 𝑇 )⟂⟂ ) = 𝑉H K G 𝑇 ; hence (0, 𝜂) ∈ G 𝑇 , whence 𝜂 = 0. Thus, D 𝑇 ∗ is dense in K . Since G 𝑇 ∗ is closed, 𝑇 ∗ is closed.

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Unbounded linear operators in Hilbert spaces Finally, we have G 𝑇 ∗∗ = (𝑉K H G 𝑇 ∗ )⟂ = 𝑉K H (𝑉H K G 𝑇 )⟂ )⟂ = G 𝑇 ; hence, 𝑇 ∗∗ = 𝑇.

Q.E.D.

It is easy to verify that r(𝑇)∗ = l(𝑇). 9.4. A linear operator 𝑇 in H is said to be symmetric if it is densely defined and 𝑇 ⊂ 𝑇 ∗ . In other words, 𝑇 is symmetric iff it is densely defined and 𝜉, 𝜂 ∈ D 𝑇 .

(𝑇𝜉|𝜂) = (𝜉|𝑇𝜂),

Obviously, any symmetric operator is preclosed. If 𝑇 is symmetric, then, for any 𝜉 ∈ D 𝑇 , the number (𝑇𝜉|𝜉) is real. A symmetric operator 𝑇 is said to be lower (resp., upper) semibounded if it is densely defined and there exists a real number 𝛼, such that (𝑇𝜉|𝜉) ≥ 𝛼(𝜉|𝜉), (resp., (𝑇𝜉|𝜉) ≤ 𝛼(𝜉|𝜉),

𝜉 ∈ D 𝑇, 𝜉 ∈ D 𝑇 ).

In this case, the greatest (resp., the smallest) 𝛼 ∈ ℝ with this property is called the greatest lower bound (resp., least upper bound) of 𝑇. A linear operator 𝑇 in H is said to be positive if it is densely defined and (𝑇𝜉|𝜉) ≥ 0,

𝜉 ∈ D 𝑇.

It is easy to see that any positive operator is symmetric and lower semibounded, with greatest lower bound ≥ 0. A linear operator 𝑇 in H is said to be self-adjoint if it is densely defined and 𝑇 = 𝑇 ∗ . Obviously, any self-adjoint operator is closed and symmetric. It is easily verified that any symmetric operator everywhere defined is self-adjoint. If 𝑇 is a self-adjoint operator in H , then one denotes s(𝑇) = r(𝑇) = l(𝑇), and the projection s(𝑇) is called the support of 𝑇. 9.5. Let 𝐴 be a positive linear operator in H . Then, for any 𝜉 ∈ D 𝐴 , we have ‖(1 + 𝐴)𝜉‖2 = ‖𝜉‖2 + 2(𝐴𝜉|𝜉) + ‖𝐴𝜉‖2 ≥ ‖𝜉‖2 ; hence the mapping 1 + 𝐴 is injective. Therefore, (1 + 𝐴)−1 is a linear operator in H , whose domain of definition is (1 + 𝐴)D 𝐴 . Moreover, (1 + 𝐴)−1 is bounded, of norm ≤ 1 and obviously positive.

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Lemma. Let 𝐴 be a positive linear operator in H . Then 𝐴 is self-adjoint iff (1 + 𝐴)D 𝐴 = H . Proof. Let us first assume that 𝐴 is self-adjoint. Then 𝐴 is closed. If {𝜉𝑛 } ⊂ D 𝐴 , and (1 + 𝐴)𝜉𝑛 → 𝜂0 , then, from the inequality ‖𝜉𝑛 − 𝜉𝑚 ‖ ≤ ‖(1 + 𝐴)(𝜉𝑛 − 𝜉𝑚 )‖ = ‖(1 + 𝐴)𝜉𝑛 − (1 + 𝐴)𝜉𝑚 ‖, we infer that the sequence {𝜉𝑛 } converges to a vector 𝜉0 ∈ H . Since 𝜉𝑛 → 𝜉, 𝐴𝜉𝑛 → 𝜂0 − 𝜉0 and 𝐴 is closed, we get that 𝜉0 ∈ D 𝐴 , and 𝜂0 −𝜉0 = 𝐴𝜉0 , i.e. 𝜂0 = (1+𝐴)𝜉0 . Consequently, (1 + 𝐴)D 𝐴 is a closed vector subspace of H . Let now 𝜂 ∈ H , 𝜂 ⟂ (1 + 𝐴)D 𝐴 . Then (0, 𝜂) ∈ (G 1+𝐴 )⟂ = (G (1+𝐴)∗ )⟂ = 𝑉H G 1+𝐴 , hence (𝜂, 0) ∈ G 1+𝐴 , i.e. 𝜂 ∈ D 𝐴 and (1 + 𝐴)𝜂 = 0. But 1 + 𝐴 is an injective mapping, hence 𝜂 = 0. Thus, (1 + 𝐴)D 𝐴 is dense in H . Consequently, we have (1 + 𝐴)D 𝐴 = H . Conversely, let us assume that (1 + 𝐴)D 𝐴 = H and let us consider an element, (𝜂0 , 𝜉0 ) ∈ G 𝐴∗ . For any 𝜉 ∈ D 𝐴 , we have (𝐴𝜉|𝜂0 ) = (𝜉|𝜉0 ), ((1 + 𝐴)𝜉|𝜂0 ) = (𝜉|𝜉0 + 𝜂0 ); hence, for any 𝜂 ∈ H , we have (𝜂|𝜂0 ) = ((1 + 𝐴)−1 𝜂|𝜉0 + 𝜂0 ). Since (1 + 𝐴)−1 is everywhere defined and symmetric, it is self-adjoint. Thus for any 𝜂 ∈ H , we have (𝜂|𝜂0 ) = (𝜂|(1 + 𝐴)−1 (𝜉0 + 𝜂0 )), whence we infer that 𝜂0 = (1 + 𝐴)−1 (𝜉0 + 𝜂0 ). Consequently, 𝜂0 ∈ D 𝐴 , and (1 + 𝐴)𝜂0 = 𝜉0 + 𝜂0 , 𝐴𝜂0 = 𝜉0 , i.e. (𝜂0 , 𝜉0 ) ∈ G 𝐴 . Therefore, 𝐴 is self-adjoint.

Q.E.D.

9.6. If a linear operator 𝐴 in H has a symmetric extension, then 𝐴 is symmetric and any symmetric extension of 𝐴 is a restriction of 𝐴∗ . These assertions immediately follow, if we take into account the implication 𝑇 ⊃ 𝑆 ⇒ 𝑇 ∗ ⊂ 𝑆∗. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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An interesting problem in the theory of symmetric operators is that of the existence and the classification of the self-adjoint extensions. The semibounded symmetric operators have a canonical self-adjoint extension, with the conservation of the bound (lower, or upper, as the case may be). In the following theorem, we describe the corresponding construction for positive operators: Theorem. Let 𝐵 be a positive operator in H . We define a linear operator 𝐴 in H by the relations D𝐴 = {

𝜉 ∈ D 𝐵 ∗ ; there exists a {𝜉𝑛 } ⊂ D 𝐵 , such that 𝜉𝑛 → 𝜉 and (𝐵(𝜉𝑛 − 𝜉𝑚 )|𝜉𝑛 − 𝜉𝑚 ) → 0, 𝐴𝜉 = 𝐵∗ 𝜉,

}

𝜉 ∈ D 𝐴.

Then 𝐴 is a positive self-adjoint extension of 𝐵. Proof. We consider, on D 𝐵 , the scalar product (𝜉|𝜂)𝐵 = ((1 + 𝐵)𝜉|𝜂), and we define D = {𝜉 ∈ H ; there exists a {𝜉𝑛 } ⊂ D 𝐵 such that 𝜉𝑛 → 𝜉 and (𝜉𝑛 − 𝜉𝑚 |𝜉𝑛 − 𝜉𝑚 )𝐵 → 0.} Then D is a vector subspace of H , D ⊃ D 𝐵 and D 𝐴 = D ∩ D 𝐵 ∗ . If 𝜉 ∈ D and {𝜉𝑛 } ⊂ D 𝐵 , 𝜉𝑛 → 𝜉, (𝜉𝑛 −𝜉𝑚 |𝜉𝑛 −𝜉𝑚 )𝐵 → 0, then the sequence {(𝜉𝑛 |𝜉𝑛 )𝐵 } converges and its limit does not depend on the choice of the sequence {𝜉𝑛 }. We denote this limit by (𝜉|𝜉)𝐵 , and for any 𝜉, 𝜂 ∈ D , we define 1 1 i i (𝜉|𝜂)𝐵 = (𝜉 + 𝜂|𝜉 + 𝜂)𝐵 − (𝜉 − 𝜂|𝜉 − 𝜂)𝐵 + (𝜉 + i𝜂|𝜉 + i𝜂)𝐵 − (𝜉 − i𝜂|𝜉 − i𝜂)𝐵 . 4 4 4 4 We easily verify that (𝜉|𝜂)𝐵 is a scalar product on D and that D , endowed with this scalar product, is a Hilbert space. By definition, D 𝐵 is dense in the Hilbert space D . We can now easily prove the following relations: (𝜉|𝜉)𝐵 ≥ ‖𝜉‖2 ,

𝜉 ∈ D,

(𝜉|𝜂)𝐵 = ((1 + 𝐵)𝜉|𝜂),

𝜉 ∈ D 𝐵, 𝜂 ∈ D .

Let 𝜉 ∈ D 𝐴 = D ∪ D 𝐵 ∗ and let {𝜉𝑛 } ⊂ D 𝐵 be a sequence, such that (𝜉 − 𝜉𝑛 |𝜉 − 𝜉𝑛 )𝐵 → 0. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Then we have (𝐴𝜉|𝜉) = (𝐵∗ 𝜉|𝜉) = lim (𝐵∗ 𝜉|𝜉𝑛 ) = lim (𝜉|𝐵𝜉𝑛 ) 𝑛→∞

𝑛→∞

= lim ((𝜉|(1 + 𝐵)𝜉𝑛 ) − (𝜉|𝜉𝑛 )) = lim ((𝜉|𝜉𝑛 )𝐵 − (𝜉|𝜉𝑛 )) 𝑛→∞

𝑛→∞

2

= (𝜉|𝜉)𝐵 − ‖𝜉‖ ≥ 0. Consequently, 𝐴 is positive. Let 𝜂0 ∈ H be arbitrary. The mapping 𝜉 ↦ (𝜉|𝜂0 ) is a bounded form on the Hilbert space D ; hence there exists a 𝜉0 ∈ D , such that (𝜉|𝜂0 ) = (𝜉|𝜉0 )𝐵 ,

𝜉 ∈ D.

In particular, for any 𝜉 ∈ D 𝐵 , we have (𝜉|𝜂0 ) = (𝜉|𝜉0 )𝐵 = ((1 + 𝐵)𝜉|𝜉0 ), (𝜉|𝜂0 − 𝜉0 ) = (𝐵𝜉|𝜉0 ); hence 𝜉0 ∈ D 𝐵 ∗ ,

𝜂0 − 𝜉0 = 𝐵∗ 𝜉0 ,

𝜉0 ∈ D 𝐴 ,

𝜂0 = (1 + 𝐴)𝜉0 .

Consequently, we have (1 + 𝐴)D 𝐴 = H , and according to lemma 9.5, 𝐴 is self-adjoint. Obviously, 𝐴 is an extension of 𝐵. Q.E.D. The positive self-adjoint extension of the positive operators, which has just been constructed, is called the Friedrichs extension. 9.7. Let M ⊂ B (H ) be a von Neumann algebra and 𝑇 a linear operator in H . One says that 𝑇 is affiliated to M if, for any unitary operator 𝑢′ ∈ M ′ , one has 𝑢′∗ 𝑇𝑢′ = 𝑇. From Corollary 3.4, we infer that if 𝑇 is everywhere defined and bounded, then 𝑇 is affiliated to M iff 𝑇 ∈ M . If 𝑇 is densely defined and affiliated to M , then 𝑇 ∗ is affiliated to M . If 𝑇 is preclosed and affiliated to M , then 𝑇 is affiliated to M . ′ ˜ )2 ⊂ In what follows we shall use the notations Mat2 (M ) ⊂ B (H ⊕ H ) and (M B (H ⊕ H ), already introduced in Section 2.32. In accordance with Lemma 3.16, the ′ ˜ )2 . commutant of Mat2 (M ) is (M Let 𝑇 be a closed linear operator in H . Then the graph G 𝑇 is a closed linear subspace of H ⊕ H , and in order not to complicate the notations, the orthogonal projection on G 𝑇 will be denoted again by G 𝑇 . Thus, we have G 𝑇 ⊂ H ⊕ H , and at the same time, G 𝑇 ∈ B (H ⊕ H ).

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Lemma. Let M ⊂ B (H ) be a von Neumann algebra and 𝑇 a closed linear operator in H . Then 𝑇 is affiliated to M iff G 𝑇 ∈ Mat2 (M ). Proof. It is easy to verify the following equivalences: 𝑇 is affiliated to M ⇔ for any 𝑥′ ∈ ′ ˜ )2 , we have 𝑥̃′ (G 𝑇 ) ⊂ M ′ and (𝜉, 𝜂) ∈ G 𝑇 , we have (𝑥 ′ 𝜉, 𝑥 ′ 𝜂) ∈ G 𝑇 ⇔ for any 𝑥̃′ ∈ (M ′ ˜ )2 )′ = 𝑀𝑎𝑡2 (M ). G 𝑇 ⇔ G 𝑇 ∈ ((M Q.E.D. 9.8. We recall (E.4.20) that if M is finite, then Mat2 (M ) is also finite. In this case, we shall denote by ♮ the canonical central trace (7.12) on Mat2 (M ). Lemma. Let M ⊂ B (H ) be a finite von Neumann algebra and 𝑇 a closed linear operator in H , affiliated to M . Then (G 𝑇 )♮ = 1/2. Proof. We consider the projections 𝑃1 = (

1

0

0

0

),

𝑃2 = (

0

0

0

1

)

in Mat2 (M ). It is easy to see that 𝑃1 and 𝑃2 are equivalent orthogonal projections, whose sum is 1. Consequently, ♮ ♮ 𝑃1 = 𝑃2 = 1/2. On the other hand, it is easy to verify that n(𝑃1 G 𝑇 ) = (G 𝑇 )⟂ ,

n(G 𝑇 𝑃1 ) = 𝑃2 ,

whence r(𝑃1 G 𝑇 ) = G 𝑇 ,

l(𝑃1 G 𝑇 ) = 𝑃1 .

Consequently, in accordance with Theorem 4.3, we have G 𝑇 ∼ 𝑃1 . As a conclusion, we have

♮

(G 𝑇 )♮ = 𝑃1 = 1/2. Q.E.D. Theorem. Let M ⊂ B (H ) be a finite von Neumann algebra. If 𝑇 and 𝑆 are closed linear operators, affiliated to M , and if 𝑇 ⊂ 𝑆,

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then 𝑇 = 𝑆. Proof. The theorem follows from the preceding lemma and from the properties of the mapping ♮ (7.11). Q.E.D. For a symmetric operator, affiliated to a finite von Neumann algebra, the above theorem completely solves the problem of the existence and the classification of self-adjoint extensions: Corollary. Let 𝑇 be a symmetric operator in H , affiliated to a finite von Neumann algebra. Then 𝑇 is self-adjoint, and it is the unique self-adjoint extension of 𝑇. Proof. Since the operator 𝑇 is symmetric, we have 𝑇 ⊂ 𝑇 ∗ . The theorem implies that 𝑇 = 𝑇 ∗ ; hence 𝑇 is self-adjoint. On the other hand, if 𝑆 is a self-adjoint extension of 𝑇, then 𝑇 ⊂ 𝑆 ⊂ 𝑇 ∗ = 𝑇, whence 𝑆 = 𝑇. Q.E.D. 9.9. In this section, we describe the operational calculus for positive self-adjoint operators, with the help of Lemma 9.5, of the operational calculus for bounded self-adjoint operators (2.20) and of a natural passage to the limit process. Let 𝐴 be a positive self-adjoint linear operator in the Hilbert space H . From Lemma 9.5, we infer that 𝑎 = (1 + 𝐴)−1 ∈ B (H ), 0 ≤ 𝑎 ≤ 1,

s(𝑎) = 1.

For any natural number 𝑛, let 𝜒𝑛 be the characteristic function of the set ((𝑛 + 1)−1 , +∞). Let us define 𝑒𝑛 = 𝜒𝑛 (𝑎) ∈ R ({𝑎}). There exists a unique 𝑎𝑛 ∈ R ({𝑎𝑛 }), such that 𝑒𝑛 ≤ 𝑎𝑛 ≤ (𝑛 + 1)𝑒𝑛 𝑒𝑛 = 𝑎𝑎𝑛 . Since 𝑎H = (1 + 𝐴)−1 H = D 𝐴 , Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

Unbounded linear operators in Hilbert spaces

225

it follows that, for any 𝑛, we have 𝑒𝑛 H = 𝑎𝑎𝑛 H ⊂ D 𝐴 . Hence, the operator 𝐴𝑒𝑛 is everywhere defined. Moreover, we have 𝐴𝑒𝑛 = 𝐴(1 + 𝐴)−1 𝑎𝑛 = (1 − (1 + 𝐴)−1 )𝑎𝑛 = (1 − 𝑎)𝑎𝑛 = 𝑎𝑛 − 𝑒𝑛 . In particular, 𝐴𝑒𝑛 ∈ R ({𝑎}) ⊂ B (H ), 0 ≤ 𝐴𝑒𝑛 ≤ 𝑛𝑒𝑛 . It is easy to verify that 𝑒𝑛 𝐴 ⊂ 𝐴𝑒𝑛 . We shall denote by B ([0, +∞)) the *-algebra of all Borel measurable complex functions, which are defined on [0, +∞) and bounded on compact sets. For any 𝑓 ∈ B ([0, +∞)), we have 𝑓|𝜎(𝐴𝑒𝑛 ) ∈ B (𝜎(𝐴𝑒𝑛 )); hence we can consider the operator 𝑓(𝐴𝑒𝑛 ) ∈ B (H ). For any 𝑓 ∈ B ([0, +∞)), we define a linear operator 𝑓(𝐴) in H , by the relations D 𝑓(𝐴) = {𝜉 ∈ H ; the sequence { 𝑓(𝐴𝑒𝑛 )𝜉}𝑛 converges} 𝜉 ∈ D 𝑓(𝐴) .

𝑓(𝐴)𝜉 = lim 𝑓(𝐴𝑒𝑛 )𝜉, 𝑛→∞

It is easy to see that 𝑓(0)(1 − 𝑒𝑛 ) + 𝑓(𝐴)𝑒𝑛 = 𝑓(𝐴𝑒𝑛 ). On the other hand, if 𝑓 ∈ B ([0, +∞)), 𝑓(0) = 0, then for any 𝜉 ∈ D 𝑓(𝐴) and any natural number 𝑛, 𝑒𝑛 𝑓(𝐴)𝜉 = lim 𝑒𝑛 𝑓(𝐴𝑒𝑘 )𝜉 = 𝑓(𝐴𝑒𝑛 )𝜉, 𝑘→∞

hence 𝑒𝑛 𝑓(𝐴) ⊂ 𝑓(𝐴𝑒𝑛 ). We now introduce the notation ∞

S𝐴 =

⋃

𝑒𝑛 H .

𝑛=1

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Then it is easy to verify that, for any 𝑓 ∈ B ([0, +∞)), we have S 𝐴 ⊂ D 𝑓(𝐴) , 𝑓(𝐴)S 𝐴 ⊂ S 𝐴 . If 𝐴 is an everywhere defined, positive and bounded linear operator, then 𝑒𝑛 = 1, for 𝑛 sufficiently great. Hence, for any 𝑓 ∈ B ([0, +∞)), the operator 𝑓(𝐴), we have just defined, coincides with that introduced by Theorem 2.20. If 𝐴 is a positive self-adjoint linear operator in H , and if 𝑓 ∈ B ([0, +∞)), then the linear operator 𝑓(𝐴), we have just defined, is affiliated to the von Neumann algebra R ({𝑎}) generated by 𝑎 = (1 + 𝐴)−1 . Thus, if 𝐴 is affiliated to a von Neumann algebra M , then for any 𝑓 ∈ B ([0, +∞)), 𝑓(𝐴) is affiliated to M . 9.10. Let 𝐴, 𝑎 and 𝑒𝑛 be as in the preceding section. For any bounded 𝑓 ∈ B ([0, +∞)), we consider the function 𝐹𝑓 ∈ B ([0, 1]), defined by 𝐹𝑓 (𝜆) = {

0

if 𝜆 = 0,

𝑓((1 − 𝜆)/𝜆)

if 𝜆 ∈ (0, 1].

Let 𝑛 be a fixed natural number. We recall that 𝐴𝑒𝑛 ∈ B (H ). It is easy to verify that the mapping B (𝜎(𝐴𝑒𝑛 )) ∋ 𝑓 ↦ 𝐹𝑓 (𝑎𝑒𝑛 ) ∈ B (H ) has the properties (i) and (ii) from Theorem 2.20, relatively to 𝑥 = 𝐴𝑒𝑛 . From Theorem 2.20, it then follows that for any bounded 𝑓 ∈ B ([0, +∞)), we have 𝑓(𝐴𝑒𝑛 ) = 𝐹𝑓 (𝑎𝑒𝑛 ) = 𝐹𝑓 (𝑎)𝑒𝑛 . By using the definition of 𝑓(𝐴) and the fact that 𝑒𝑛 ↑ s(𝑎) = 1, we infer that for any bounded 𝑓 ∈ B ([0, +∞)), we have the relation 𝑓(𝐴) = 𝐹𝑓 (𝑎). In particular, we found that if 𝑓 is bounded, then 𝑓(𝐴) ∈ B (H ), and we have ‖𝑓(𝐴)‖ = ‖𝐹𝑓 (𝑎)‖ ≤ sup{|𝐹𝑓 (𝜆)|; 𝜆 ∈ (0, 1]} = sup {|𝑓(𝜆)|; 𝜆 ∈ [0, +∞)}. 9.11. The following theorem states the main rules of the operational calculus for positive self-adjoint operators.

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Theorem. Let 𝐴 be a positive self-adjoint linear operator in the Hilbert space H . Then (i) for 𝑓0 (𝜆) = 𝑐 ∈ ℂ, 𝜆 ∈ [0, +∞), we have 𝑓0 (𝐴) = 𝑐; for 𝑓1 (𝜆) = 𝜆, 𝜆 ∈ [0, +∞), we have 𝑓1 (𝐴) = 𝐴; (ii) for any 𝑓 ∈ B ([0, +∞)), we have D 𝑓(𝐴) = {𝜉 ∈ H ; sup ‖𝑓(𝐴𝑒𝑛 )𝜉‖ < +∞}, 𝑛

the linear operator 𝑓(𝐴) is closed and 𝑓(𝐴)S 𝐴 = 𝑓(𝐴); (iii) for any 𝑓 ∈ B ([0, +∞)), we have 𝑓(𝐴)∗ = 𝑓(𝐴); (iv) for any 𝑓, 𝑔 ∈ B ([0, +∞)), we have the linear operator 𝑓(𝐴) + 𝑔(𝐴) is preclosed and 𝑓(𝐴) + 𝑔(𝐴) = (𝑓 + 𝑔)(𝐴); (v) for any 𝑓, 𝑔 ∈ B ([0, +∞)), we have the linear operator 𝑓(𝐴)𝑔(𝐴) is preclosed, D 𝑓(𝐴)𝑔(𝐴) = D (𝑓𝑔)(𝐴) ∩ D 𝑔(𝐴) and 𝑓(𝐴)𝑔(𝐴) = (𝑓𝑔)(𝐴); (vi) for any sequence {𝑓𝑘 }𝑘 ⊂ B ([0, +∞)), which is uniformly bounded on compact sets and pointwise convergent to 𝑓0 ∈ B ([0, +∞)), we have 𝑓0 (𝐴)𝜉 = lim 𝑓𝑘 (𝐴)𝜉, 𝑘→∞

𝜉 ∈ S 𝐴.

Proof. (i) For any 𝑛, we have 𝑓0 (𝐴𝑒𝑛 ) = 𝑐; hence, indeed, 𝑓0 (𝐴) = 𝑐. For any 𝑛, we have 𝑓1 (𝐴𝑒𝑛 ) = 𝐴𝑒𝑛 . If 𝜉 ∈ D 𝐴 , then 𝑓1 (𝐴𝑒𝑛 )𝜉 = 𝑒𝑛 𝐴𝜉 → 𝐴𝜉;

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hence 𝜉 ∈ D 𝑓1 (𝐴) and 𝑓1 (𝐴)𝜉 = 𝐴𝜉. If 𝜉 ∈ D 𝑓1 (𝐴) , then 𝑒𝑛 𝜉 → 𝜉, 𝐴𝑒𝑛 𝜉 = 𝑓1 (𝐴𝑒𝑛 )𝜉 → 𝑓1 (𝐴)𝜉; hence, since 𝐴 is closed, 𝜉 ∈ D 𝐴 and 𝐴𝜉 = 𝑓1 (𝐴)𝜉. Thus, we have 𝑓1 (𝐴) = 𝐴. If 𝑓 ∈ B ([0, +∞)) and 𝑓0 (𝜆) = 𝑓(0), 𝜆 ∈ [0, +∞), it is easy to verify that 𝑓(𝐴) = (𝑓 − 𝑓0 )(𝐴) + 𝑓(0). Since (𝑓 − 𝑓0 )(0) = 0, this remark will enable us to assume, without any essential loss of generality, that the considered functions vanish at 0. (ii) We can assume that 𝑓(0) = 0. Let 𝛼 = sup ‖𝑓(𝐴𝑒𝑛 )𝜉‖ < +∞. 𝑛

Since 𝑓(0) = 0, for any 𝑛, we have ‖𝑓(𝐴𝑒𝑛 )𝜉‖ = ‖𝑒𝑛 𝑓(𝐴𝑒𝑛+1 )𝜉‖ ≤ ‖𝑓(𝐴𝑒𝑛+1 )𝜉‖. Consequently, we have ‖𝑓(𝐴𝑒𝑛 )𝜉‖2 ↑ 𝛼 2 . On the other hand, for any 𝑛, 𝑘, we have ‖𝑓(𝐴𝑒𝑛+𝑘 )𝜉 − 𝑓(𝐴𝑒𝑛 )𝜉‖2 = ‖(𝑒𝑛+𝑘 − 𝑒𝑛 )𝑓(𝐴𝑒𝑛+𝑘 )𝜉‖2 = ‖𝑒𝑛+𝑘 𝑓(𝐴𝑒𝑛+𝑘 )𝜉‖2 − ‖𝑒𝑛 𝑓(𝐴𝑒𝑛+𝑘 )𝜉‖2 = ‖𝑓(𝐴𝑒𝑛+𝑘 )𝜉‖2 − ‖𝑓(𝐴𝑒𝑛 )𝜉‖2 . Consequently, the sequence {𝑓(𝐴𝑒𝑛 )𝜉} is fundamental; hence convergent. Therefore we have D 𝑓(𝐴) = {𝜉 ∈ H ; sup ‖𝑓(𝐴𝑒𝑛 )𝜉‖ < +∞}. 𝑛

Since 𝑒𝑛 ↑ s(𝑎) = 1, S 𝐴 ⊂ D 𝑓(𝐴) is a dense subset of H ; hence 𝑓(𝐴) is densely defined. If {𝜉𝑘 } ⊂ D 𝑓(𝐴) , 𝜉𝑘 → 𝜉0 and 𝑓(𝐴)𝜉𝑘 → 𝜂0 , then, for any 𝑛, we have 𝑓(𝐴𝑒𝑛 )𝜉0 = lim 𝑓(𝐴𝑒𝑛 )𝜉𝑘 = lim 𝑒𝑛 𝑓(𝐴)𝜉𝑘 = 𝑒𝑛 𝜂0 . 𝑘→∞

𝑘→∞

It follows that 𝑓(𝐴𝑒𝑛 )𝜉0 → 𝜂0 ; hence 𝜉0 ∈ D 𝑓(𝐴) and 𝑓(𝐴)𝜉0 = 𝜂0 . Consequently, 𝑓(𝐴) is closed. If (𝜉, 𝑓(𝐴)𝜉) ∈ G 𝑓(𝐴) and is orthogonal to the graph of the operator 𝑓(𝐴)|S 𝐴 , then, for any 𝑛, we have (𝜉|𝑒𝑛 𝜉) = (𝑓(𝐴)𝜉|𝑓(𝐴𝑒𝑛 )𝜉) = 0. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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By tending to the limit for 𝑛 → +∞, we find that ‖𝜉‖2 + ‖𝑓(𝐴)𝜉‖2 = 0; hence 𝜉 = 𝑓(𝐴)𝜉 = 0. Therefore, we have 𝑓(𝐴)S 𝐴 = 𝑓(𝐴). (iii) By taking into account the remark we made in (i) and Proposition 9.2, we can assume that 𝑓(0) = 0. It is then easy to verify that 𝑓(𝐴)|S 𝐴 ⊂ 𝑓(𝐴)∗ , whence, in accordance with (ii), 𝑓(𝐴) ⊂ 𝑓(𝐴)∗ . Let now 𝜂 ∈ D 𝑓(𝐴)∗ . For any 𝜉 ∈ H and any 𝑛, we have (𝜉|𝑓(𝐴𝑒𝑛 )𝜂) = (𝑓(𝐴𝑒𝑛 )𝜉|𝜂) = (𝑓(𝐴)𝑒𝑛 𝜉|𝜂) = (𝜉|𝑒𝑛 𝑓(𝐴)∗ 𝜂); hence 𝑓(𝐴𝑒𝑛 )𝜂 = 𝑒𝑛 𝑓(𝐴)∗ 𝜂. It follows that 𝑓(𝐴𝑒𝑛 )𝜂 → 𝑓(𝐴)∗ 𝜂, i.e. 𝜂 ∈ D 𝑓(𝐴) and 𝑓(𝐴)𝜂 = 𝑓(𝐴)∗ 𝜂. Consequently, we have 𝑓(𝐴)∗ = 𝑓(𝐴). (iv) It is easy to verify that 𝑓(𝐴) + 𝑔(𝐴) ⊂ (𝑓 + 𝑔)(𝐴); hence 𝑓(𝐴) + 𝑔(𝐴) is preclosed. Since on S 𝐴 the operators 𝑓(𝐴) + 𝑔(𝐴) and (𝑓 + 𝑔)(𝐴) coincide, from (ii) we infer that 𝑓(𝐴) + 𝑔(𝐴) = (𝑓 + 𝑔)(𝐴). (v) We can assume that 𝑓(0) = 0. It is easy to verify that 𝑓(𝐴)𝑔(𝐴) ⊂ (𝑓𝑔)(𝐴); hence 𝑓(𝐴)𝑔(𝐴) is preclosed. Since on S 𝐴 the operators 𝑓(𝐴)𝑔(𝐴) and (𝑓𝑔)(𝐴) coincide from (ii), we infer that 𝑓(𝐴)𝑔(𝐴) = (𝑓𝑔)(𝐴). In accordance with the preceding results, we have D 𝑓(𝐴)𝑔(𝐴) ⊂ D (𝑓𝑔)(𝐴) ∩ D 𝑔(𝐴) . In order to prove the reversed inclusion, we must prove the implication 𝜉 ∈ D (𝑓𝑔)(𝐴) ∩ D 𝑔(𝐴) ⇒ 𝑔(𝐴)𝜉 ∈ D 𝑓(𝐴) . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Indeed, since 𝑓(0) = 0, for any 𝑛, we have 𝑓(𝐴𝑒𝑛 )𝑔(𝐴)𝜉 = 𝑓(𝐴𝑒𝑛 )𝑔(𝐴𝑒𝑛 )𝜉 = (𝑓𝑔)(𝐴𝑒𝑛 )𝜉; hence the sequence {𝑓(𝐴𝑒𝑛 )𝑔(𝐴)𝜉} is convergent. (vi) We can assume that 𝑓𝑘 (0) = 0, for any 𝑘 ≥ 0. Let 𝜉 ∈ S 𝐴 ; then there exists an 𝑛, such that 𝜉 ∈ 𝑒𝑛 H . Then, for any 𝑘 ≥ 0, we have 𝑓𝑘 (𝐴)𝜉 = 𝑓1 (𝐴)𝑒𝑛 𝜉 = 𝑓𝑘 (𝐴𝑒𝑛 )𝜉. From Theorem 2.20, we infer that 𝑓𝑘 (𝐴𝑒𝑛 )𝜉 → 𝑓0 (𝐴𝑒𝑛 )𝜉, i.e. 𝑓𝑘 (𝐴)𝜉 → 𝑓0 (𝐴)𝜉. Q.E.D. Corollary 9.12. Let 𝐴 be a positive self-adjoint linear operator in H . If 𝑓, 𝑔 ∈ B ([0, +∞)) and |𝑓| ≤ |𝑔|, then D 𝑔(𝐴) ⊂ D 𝑓(𝐴) , ‖𝑓(𝐴)𝜉‖ ≤ ‖𝑔(𝐴)𝜉‖,

𝜉 ∈ D 𝑔(𝐴) .

In particular, if 𝑓 is bounded, then 𝑓(𝐴) ∈ B (H ) ‖𝑓(𝐴)‖ ≤ sup{|𝑓(𝜆)|; 𝜆 ∈ [0, +∞)}. Proof. If |𝑓| ≤ |𝑔|, then for any 𝜉 ∈ D 𝑔(𝐴) and any natural number 𝑛, we have ‖𝑓(𝐴𝑒𝑛 )𝜉‖2 = (|𝑓|2 (𝐴𝑒𝑛 )𝜉|𝜉) ≤ (|𝑔|2 (𝐴𝑒𝑛 )𝜉|𝜉) = ‖𝑔(𝐴𝑒𝑛 )𝜉‖2 ; hence sup ‖𝑓(𝐴𝑒𝑛 )𝜉‖ ≤ sup ‖𝑔(𝐴𝑒𝑛 )𝜉‖ = ‖𝑔(𝐴)𝜉‖ < +∞. 𝑛

𝑛

In accordance with Theorem 9.11 (ii), we infer that 𝜉 ∈ D 𝑓(𝐴) ; obviously, we have ‖𝑓(𝐴)𝜉‖ ≤ ‖𝑔(𝐴)𝜉‖, for any 𝜉 ∈ D 𝑔(𝐴) . If 𝑓 is bounded, we have |𝑓| ≤ 𝑐 = sup{|𝑓(𝜆)|; 𝜆 ∈ [0, +∞)}; then D 𝑓(𝐴) ⊃ D 𝑐 = H and ‖𝑓(𝐴)‖ = sup{‖𝑓(𝐴)𝜉‖; ‖𝜉‖ ≤ 1} ≤ sup{‖𝑐 𝜉‖; ‖𝜉‖ ≤ 1} = 𝑐. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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If 𝑓, 𝑔 ∈ B ([0, +∞)) and at least one of the functions 𝑓, 𝑔 is bounded, then it is easy to show that we have (𝑓 + 𝑔)(𝐴) = 𝑓(𝐴) + 𝑔(𝐴), (𝑓𝑔)(𝐴) = 𝑓(𝐴)𝑔(𝐴). If the sequence {𝑓𝑘 } ⊂ B ([0, +∞)) converges uniformly to 𝑓0 ∈ B ([0, +∞)), then, for a sufficiently great 𝑘, we have D 𝑓𝑘 (𝐴) = D 𝑓0 (𝐴) , 𝑓0 (𝐴) − 𝑓𝑘 (𝐴) ⊂ (𝑓0 − 𝑓𝑘 )(𝐴) is bounded and ‖𝑓0 (𝐴) − 𝑓𝑘 (𝐴)‖ → 0. Corollary 9.13. Let 𝐴 be a positive self-adjoint linear operator in H . Then (i) for any real 𝑓 ∈ B ([0, +∞)), 𝑓(𝐴) is self-adjoint. (ii) for any positive 𝑓 ∈ B ([0, +∞)), 𝑓(𝐴) is self-adjoint and positive. (iii) for any characteristic function 𝑓 ∈ B ([0, +∞)), 𝑓(𝐴) ∈ B (H ) is a projection; moreover, s(𝐴) = 𝜒(0,+∞) (𝐴). (iv) for any 𝑓 ∈ B ([0, +∞)), such that |𝑓| = 1, 𝑓(𝐴) ∈ B (H ) is unitary. A linear operator 𝑇 in H is said to be normal if 𝑇 is closed and 𝑇𝑇 ∗ = 𝑇 ∗ 𝑇. It is easy to see that 𝑇 is normal iff 𝐷𝑇 = 𝐷𝑇 ∗ and ‖𝑇𝜉‖ = ‖𝑇 ∗ 𝜉‖, 𝜉 ∈ D 𝑇 . For any positive self-adjoint linear operator 𝐴 in H and any 𝑓 ∈ B ([0, +∞)), the linear operator 𝑓(𝐴) is normal. Corollary 9.14. For any positive self-adjoint linear operator 𝐴 in H , there exists a unique positive self-adjoint linear operator B in H , such that 𝐵2 = 𝐴. Proof. We consider the continuous functions 𝑓, 𝑔, defined on [0, +∞) by the formulas 𝑓(𝜆) = 𝜆1/2 , 𝑔(𝜆) = 1 + 𝜆. According to Corollary 9.13, 𝑓(𝐴) is self-adjoint and positive. Since 0 ≤ 𝑓 ≤ 𝑔, from Corollary 9.12, we infer that D 𝐴 = D 𝑔(𝐴) ⊂ D 𝑓(𝐴) . By taking into account Theorem 9.11 (v), we get D 𝑓(𝐴)2 = D 𝐴 ∩ D 𝑓(𝐴) = D 𝐴 and 𝑓(𝐴)2 = 𝐴. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Let now 𝐵 be a positive self-adjoint operator in H , such that 𝐵2 = 𝐴. We denote 𝑏 = (1 + 𝐵)−1 , 𝑓𝑛 = 𝜒((1+𝑛)−1 ,+∞) (𝑏). We consider the continuous functions ℎ, 𝑘, defined on [0, 1] by the formulas ℎ(𝜆) = 𝜆2 /(𝜆2 + (1 − 𝜆)2 ), 𝑘(𝜆) = 𝜆1/2 /(𝜆1/2 + (1 − 𝜆)1/2 ). It is easy to verify that, for any 𝑛, we have ℎ(𝑏)𝑓𝑛 = ℎ(𝑏𝑓𝑛 ) = (1 + 𝐵2 )−1 𝑓𝑛 = (1 + 𝐴)−1 𝑓𝑛 , whence ℎ(𝑏) = (1 + 𝐴)−1 . Since (𝑘 ∘ ℎ)(𝜆) = 𝜆, 𝜆 ∈ [0, 1], by taking into account Corollary 2.7, we get 𝑘((1 + 𝐴)−1 ) = 𝑘(ℎ(𝑏)) = (𝑘 ∘ ℎ)(𝑏) = 𝑏 = (1 + 𝐵)−1 . From this equality, we infer that 𝐵 is determined by 𝐴 in a unique manner.

Q.E.D.

This unique positive self-adjoint linear operator 𝐵 in H , such that 𝐵2 = 𝐴, will be denoted 𝐵 = 𝐴1/2 . 9.15. Let 𝛼 ∈ ℂ, Re 𝛼 ≥ 0. We now consider the mapping 𝑓𝛼 ∶ [0, +∞) ∋ 𝜆 ↦ 𝜆𝛼 ∈ ℂ as in Section 2.30. Then 𝑓𝛼 ∈ B ([0, +∞)). For any positive self-adjoint linear operator 𝐴 in H , we define the operator 𝐴𝛼 = 𝑓𝛼 (𝐴). Corollary. Let 𝐴 be a positive self-adjoint linear operator in H , 𝜉 ∈ H and 𝜀 > 0. The following assertions are equivalent: (i) 𝜉 ∈ D 𝐴𝜀 . (ii) 𝜉 ∈ D 𝐴𝛼 for any 𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀, and the mapping 𝛼 ↦ 𝐴𝛼 𝜉 is continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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(iii) the mapping i𝑡 ↦ 𝐴i𝑡 𝜉, defined on the imaginary axis, has a continuous extension to the set {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}. Proof. (i) ⇒ (ii). We define the continuous functions 𝑓𝛼 and 𝑔 on [0, +∞) by the relations 𝑓𝑛 (𝜆) = 𝜆𝛼 , 𝑔(𝜆) = 1 + 𝜆𝜀 . Then, for any 𝛼 ∈ ℂ, such that 0 ≤ Re 𝛼 ≤ 𝜀, we have |𝑓𝛼 | ≤ 𝑔. In accordance with Corollary 9.12, we have D 𝐴𝜀 = D 𝑔(𝐴) ⊂ D 𝑓𝛼 (𝐴) = D 𝐴𝛼 . For any natural number 𝑛, we consider the projection 𝑒𝑛 , defined in Section 9.9. If 𝜉 ∈ 𝑒𝑛 H , then, in accordance with Corollary 2.30, the mapping 𝛼 ↦ 𝐴𝛼 𝜉 = (𝐴𝑒𝑛 )𝛼 𝜉 is continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}. Let 𝜉 ∈ D 𝐴𝛼 be arbitrarily chosen. We denote 𝜉𝑛 = 𝑒𝑛 𝜉. By taking into account Corollary 9.12, for any 𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀 and any natural number 𝑛, we get ‖𝐴𝛼 𝜉 − 𝐴𝛼 𝜉𝑛 ‖ = ‖𝐴𝛼 (𝜉 − 𝜉𝑛 )‖ ≤ ‖(1 + 𝐴𝜀 )(𝜉 − 𝜉𝑛 )‖ ≤ ‖𝜉 − 𝜉𝑛 ‖ + ‖𝐴𝜀 𝜉 − 𝐴𝜀 𝜉𝑛 ‖. Thus, the mappings 𝛼 ↦ 𝐴𝛼 𝜉𝑛 are uniformly convergent on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} to the mapping 𝛼 ↦ 𝐴𝛼 𝜉. It follows that the mapping 𝛼 ↦ 𝐴𝛼 𝜉 is continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}. (ii) ⇒ (iii). Obvious. (iii) ⇒ (i). We shall denote by 𝐹 a continuous extension on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}, of the mapping iℝ ∋ i𝑡 ↦ 𝐴i𝑡 𝜉. In accordance with the implication (i) ⇒ (ii), for any 𝜉, 𝜂 ∈ S 𝐴 , the mapping 𝛼 ↦ (𝐴𝛼 𝜉|𝜂) = (𝜉|𝐴𝛼 𝜂) Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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is continuous on {𝛼 ∈ ℂ; Re 𝛼 ≥ 0} and analytic in {𝛼 ∈ ℂ; Re 𝛼 > 0}. Let {𝜉𝑘 } ⊂ S 𝐴 , 𝜉𝑘 → 𝜉 and 𝜂 ∈ S 𝐴 . For any 𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀, in accordance with Corollary 9.12, we have |(𝜉𝑘 |𝐴𝛼 𝜂) − (𝜉|𝐴𝛼 𝜂)|2 ≤ ‖𝜉𝑘 − 𝜉‖2 ‖𝐴Re 𝛼 𝜂‖2 ≤ ‖𝜉𝑘 − 𝜉‖2 ‖(1 + 𝐴𝜀 )𝜂‖2 . It follows that the mappings 𝛼 ↦ (𝜉𝑘 |𝐴𝛼 𝜂) converge uniformly on {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀} to the mapping 𝛼 ↦ (𝜉|𝐴𝛼 𝜂). Thus, the mapping 𝛼 ↦ (𝜉|𝐴𝛼 𝜂) is continuous on {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀} and analytic in {𝛼 ∈ ℂ, 0 < Re 𝛼 < 𝜀}. For any 𝜂 ∈ S 𝐴 , the mappings 𝛼 ↦ (𝐹(𝛼)|𝜂), 𝛼 ↦ (𝜉|𝐴𝛼 𝜂), which are continuous on {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀} and analytic in {𝛼 ∈ ℂ, 0 < Re 𝛼 < 𝜀}, coincide on the imaginary axis iℝ. From the symmetry principle, we infer that they coincide on {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀}. In particular, (𝐹(𝜀)|𝜂) = (𝜉|𝐴𝜀 𝜂),

𝜂 ∈ S 𝐴,

whence, in accordance with Theorem 9.11 (ii), we get (𝐹(𝜀)|𝜂) = (𝜉|𝐴𝜀 𝜂),

𝜂 ∈ D 𝐴𝜀 .

Consequently, we have 𝜉 ∈ D (𝐴𝜀 )∗ = D 𝐴𝜀 . Q.E.D. 9.16. One calls a (one-parameter) group of unitary operators in a Hilbert space H a family {𝑢𝑡 ; 𝑡 ∈ ℝ} ⊂ B (H ) of unitary operators, such that 𝑢0 = 1, 𝑢𝑡+𝑠 = 𝑢𝑡 𝑢𝑠 ,

𝑡, 𝑠 ∈ ℝ.

One says that the group {𝑢𝑡 } is so-continuous (resp., wo-continuous) if the mapping ℝ ∋ 𝑡 ↦ 𝑢𝑡 ∈ B (H ) is so-continuous (wo-continuous). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Let 𝐴 be a positive self-adjoint linear operator in H , such that s(𝐴) = 1. Then, by taking into account Theorem 9.11 and Corollary 9.13, it follows that (𝐴i𝑡 )∗ 𝐴i𝑡 = 𝐴i𝑡 (𝐴i𝑡 )∗ = 𝜒(0,+∞) (𝐴) = s(𝐴) = 1; hence the operators 𝐴i𝑡 , 𝑡 ∈ ℝ, are unitary. By taking into account Theorem 9.11 and Corollary 9.15, it follows that {𝐴i𝑡 } is a so-continuous group of unitary operators. Conversely, we shall prove in what follows that any wo-continuous group of unitary operators is of the preceding form (The representation theorem of M. H. Stone). In particular, we shall infer that any wo-continuous group of unitary operators is so-continuous. 9.17. A mapping into H is said to be weakly continuous (resp., weakly analytic; resp., weakly entire) if it is continuous (resp., analytic; resp., entire) for the weak topology in H . Lemma. Let {𝑢𝑡 } be a wo-continuous group of unitary operators in H . Then the set {𝜉 ∈ H ; the mapping i𝑡 → 𝑢𝑡 𝜉 has a weakly entire extension} is a dense vector subspace of H . Proof. Let 𝜉 ∈ H and let 𝑛 be a natural number. For any 𝛼 ∈ ℂ, the mapping +∞

H ∋ 𝜂 ↦ (𝑛/𝜋)

1/2

∫

exp(−𝑛(𝑠 + i𝛼)2 )(𝑢𝑠 𝜉|𝜂)𝑑𝑠

−∞

is a bounded antilinear form on H ; hence, there exists an 𝐹𝜉,𝑛 (𝛼) ∈ H , such that +∞

(𝐹𝜉,𝑛 (𝛼)|𝜂) = (𝑛/𝜋)1/2 ∫

exp(−𝑛(𝑠 + i𝛼)2 )(𝑢𝑠 𝜉|𝜂)𝑑𝑠,

𝜂∈H.

−∞

It is obvious that the mapping 𝛼 ↦ 𝐹𝜉,𝑛 (𝛼) is weakly entire. Let 𝜉𝑛 = 𝐹𝜉,𝑛 (0). For any 𝜂 ∈ H and any 𝑡 ∈ ℝ, we have +∞

(𝐹𝜉,𝑛 (i𝑡)|𝜂) = (𝑛/𝜋)

1/2

∫

exp(−𝑛(𝑠 − 𝑡)2 )(𝑢𝑠 𝜉|𝜂)𝑑𝑠

−∞ +∞

= (𝑛/𝜋)1/2 ∫

exp(−𝑛𝑠2 )(𝑢𝑡+𝑠 𝜉|𝜂)𝑑𝑠

−∞ +∞

= (𝑛/𝜋)1/2 ∫

exp(−𝑛𝑠2 )(𝑢𝑠 𝜉|𝑢𝑡∗ 𝜂)𝑑𝑠 = (𝜉𝑛 |𝑢𝑡∗ 𝜂) = (𝑢𝑡 𝜉𝑛 |𝜂);

−∞

hence 𝐹𝜉,𝑛 (i𝑡) = 𝑢𝑡 𝜉𝑛 . Consequently, the mapping i𝑡 ↦ 𝑢𝑡 𝜉𝑛 has a weakly entire extension. It is easy to verify that 𝜉𝑛 ↦ 𝜉 weakly.

Q.E.D.

Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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9.18. Let {𝑢𝑡 } be a wo-continuous group of unitary operators in H . For any 𝜀 ≥ 0, we denote D 𝜀 = {𝜉 ∈ H ; the mapping i𝑡 ↦ 𝑢𝑡 𝜉 has a weakly continuous extension to {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}} For any 𝜉 ∈ D 𝜀 , the weakly continuous extension to the set {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}, which is weakly analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}, of the mapping i𝑡 ↦ 𝑢𝑡 𝜉, is determined in a unique manner and will be denoted by 𝐹𝜉 . It is easy to verify that for any 𝜉1 , 𝜉2 ∈ D 𝑠 , we have 𝐹𝜉1 +𝜉2 = 𝐹𝜉1 + 𝐹𝜉2 and that for any 𝜉 ∈ D 𝜀 and any 𝜆 ∈ ℂ, we have 𝐹𝜆𝜉 = 𝜆𝐹𝜉 . Lemma. Let {𝑢𝑡 } be a wo-continuous group of unitary operators in H and 𝜀 ≥ 0. Then (i) for any 𝜉 ∈ D 𝜀 and any 𝛽 ∈ ℂ, 0 ≤ Re 𝛽 ≤ 𝜀, we have 𝐹𝜉 (𝛽) ∈ D 𝜀−Re 𝛽 , 𝐹𝐹𝜉 (𝛽) (𝛼) = 𝐹𝜉 (𝛼 + 𝛽),

𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀 − Re 𝛽.

(ii) for any 𝜉, 𝜂 ∈ D 𝜀 , and any 𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀, we have (𝐹𝜉 (𝛼)|𝜂) = (𝜉|𝐹𝜂 (𝛼 )). Proof. (i) Let 𝑡 ∈ ℝ. The mappings 𝛾 ↦ 𝑢𝑡 𝐹𝜉 (𝛾), 𝛾 ↦ 𝐹𝜉 (i𝑡 + 𝛾), are both weakly continuous extensions, to the set {𝛾 ∈ ℂ, 0 ≤ Re 𝛾 ≤ 𝜀}, of the mapping i𝑠 ↦ 𝑢𝑡+𝑠 𝜉, and they are weakly analytic on {𝛾 ∈ ℂ, 0 < Re 𝛾 < 𝜀}; hence, they coincide. Thus, for any 𝑡 ∈ ℝ, we have 𝑢𝑡 𝐹𝜉 (𝛽) = 𝐹𝜉 (i𝑡 + 𝛽). It follows that the mapping 𝛼 ↦ 𝐹𝜉 (𝛼 + 𝛽) Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

Unbounded linear operators in Hilbert spaces

237

is a weakly continuous extension to {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀 − Re 𝛽} of the mapping i𝑡 ↦ 𝑢𝑡 𝐹𝜉 𝛽), and it is weakly analytic in {𝛼 ∈ ℂ, 0 < Re 𝛼 < 𝜀 −Re 𝛽}. Consequently, 𝐹𝜉 (𝛽) ∈ D 𝜀−Re 𝛽 , and, for 𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀 − Re 𝛽, we have 𝐹𝐹𝜉 (𝛽) (𝛼) = 𝐹𝜉 (𝛼 + 𝛽). (ii) The mappings 𝛼 ↦ (𝐹𝜉 (𝛼)|𝜂), 𝛼 ↦ (𝜉|𝐹𝜂 (𝛼 )) = (𝐹𝜂 (𝛼 )|𝜉), are continuous extensions to {𝛼 ∈ ℂ, 0 ⩽ Re 𝛼 ⩽ 𝜀} of the mapping i𝑡 ↦ (𝑢𝑡 𝜉|𝜂), and they are analytic in {𝛼 ∈ ℂ, 0 < Re 𝛼 < 𝜀}; hence they coincide.

Q.E.D.

9.19. Let {𝑢𝑡 } be a wo-continuous group of unitary operators in H . For any 𝜀 ≥ 0, we define a linear operator 𝐴𝜀 in H by the relations D 𝐴𝜀 = D 𝜀 , 𝐴𝜀 𝜉 = 𝐹𝜉 (𝜀),

𝜉 ∈ D 𝐴𝜀 .

Lemma. For any 𝜀 ≥ 0, the operator 𝐴𝜀 is self-adjoint and positive. For any 𝜀1 , 𝜀2 ≥ 0, the relation 𝐴𝜀1 +𝜀2 = 𝐴𝜀1 + 𝐴𝜀2 holds. Proof. Let 𝜀 > 0. In accordance with Lemma 9.17, the operator 𝐴𝜀 is densely defined. By taking into account Lemma 9.18 (ii), for any 𝜉, 𝜂 ∈ D 𝐴𝜀 we get (𝐴𝜀 𝜉|𝜂) = (𝐹𝜉 (𝜀)|𝜂) = (𝜉|𝐹𝜂 (𝜀)) = (𝜉|𝐴𝜀 𝜂); hence the linear operator 𝐴𝜀 is symmetric. Let 𝜂 ∈ D (𝐴𝜀 )∗ and 𝜉 ∈ D 𝐴𝜀 . The mapping 𝛼 ↦ (𝐹𝜉 (𝛼)|𝜂) is continuous on {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀} and analytic in {𝛼 ∈ ℂ, 0 < Re 𝛼 < 𝜀}. With the help of Lemma 9.18 (i), it is easy to see that 𝐹𝜉 is bounded on all lines parallel to the imaginary axis; hence it is bounded. For any 𝑡 ∈ ℝ, we have |(𝐹𝜉 (i𝑡)|𝜂)| = |(𝑢𝑡 𝜉|𝜂)| ≤ ‖𝜉‖ ⋅ ‖𝜂‖, Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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and by applying Lemma 9.18 (i), we get |(𝐹𝜉 (𝜀 + i𝑡)|𝜂)| = |(𝐹𝐹𝜉 (i𝑡) (𝜀)|𝜂)| = |(𝐴𝜀 𝐹𝜉 (i𝑡)|𝜂)| = |(𝑢𝑡 𝜉|(𝐴𝜀 )∗ 𝜂)| ≤ ‖𝜉‖ ⋅ ‖(𝐴𝜀 )∗ 𝜂‖. In accordance with Phragmen–Lindelöf principle (see Dunford and Schwartz [1958, 1963, 1970], III.14), for any 𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀, we have |(𝐹𝜉 (𝛼)|𝜂)| ≤ ‖𝜉‖ max{‖𝜂‖, ‖(𝐴𝜀 )∗ 𝜂‖}. From the preceding results, we infer that, for any 𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀, the mapping D 𝐴𝜀 ∋ 𝜉 ↦ (𝐹𝜉 (𝛼)|𝜂) is a bounded form on H , whose norm is ≤ max{‖𝜂‖, ‖(𝐴𝜀 )∗ 𝜂‖}. Thus, there exists a 𝐺𝜂 (𝛼 ) ∈ H , such that ‖𝐺𝜂 (𝛼 )‖ ≤ max{‖𝜂‖, ‖(𝐴𝜀 )∗ 𝜂‖}, (𝜉|𝐺𝜂 (𝛼 )) = (𝐹𝜉 (𝛼)|𝜂),

𝜉 ∈ D 𝐴𝜀 .

It is easy to verify that the mapping 𝛼 ↦ 𝐺𝜂 (𝛼) is a weakly continuous extension to {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀} of the mapping i𝑡 ↦ 𝑢𝑡 𝜂, and it is weakly analytic in {𝛼 ∈ ℂ, 0 < Re 𝛼 < 𝜀}; hence 𝜂 ∈ D 𝐴𝜀 . Consequently, 𝐴𝜀 is self-adjoint. Let now 𝜀1 , 𝜀2 ≥ 0. By taking into account Lemma 9.18 (i), it follows that if 𝜉 ∈ 𝐷𝐴𝜀 +𝜀 , 1 2 then 𝐴𝜀2 𝜉 = 𝐹𝜉 (𝜀2 ) ∈ D (𝜀1 +𝜀2 )−𝜀3 = D 𝜀1 , 𝐴𝜀1 𝐴𝜀2 (𝜉) = 𝐹𝐹𝜉 (𝜀2 ) (𝜀1 ) = 𝐹𝜉 (𝜀1 + 𝜀2 ) = 𝐴𝜀1 +𝜀2 (𝜉). Hence 𝐴𝜀1 +𝜀2 ⊂ 𝐴𝜀1 𝐴𝜀2 . Let now 𝜉 ∈ D 𝐴𝜀

1

𝐴𝜀2 .

For any 𝜂 ∈ D 𝐴𝜀

1 +𝜀2

, we have

(𝐴𝜀1 𝐴𝜀2 𝜉|𝜂) = (𝜉|𝐴𝜀2 𝐴𝜀1 𝜂) = (𝜉|𝐴𝜀1 +𝜀2 𝜂); hence 𝜉 ∈ D (𝐴𝜀

1 +𝜀2

)∗

= D 𝐴𝜀

1 +𝜀2

,

𝐴𝜀1 +𝜀2 𝜉 = (𝐴𝜀1 +𝜀2 )∗ 𝜉 = 𝐴𝜀1 𝐴𝜀2 𝜉. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Unbounded linear operators in Hilbert spaces Thus, we have 𝐴𝜀1 +𝜀2 = 𝐴𝜀1 𝐴𝜀2 . Finally, for any 𝜀 ≥ 0 and any 𝜉 ∈ D 𝐴𝜀 , we have (𝐴𝜀 𝜉|𝜉) = (𝐴𝜀/2 𝐴𝜀/2 𝜉|𝜉) = ‖𝐴𝜀/2 𝜉‖2 ≥ 0; hence 𝐴𝜀 is positive.

Q.E.D.

9.20. We now prove the representation theorem of M. H. Stone: Theorem. Let {𝑢𝑡 ; 𝑡 ∈ ℝ} ⊂ B (H ). The following assertions are equivalent (i) {𝑢𝑡 } is a wo-continuous group of unitary operators. (ii) {𝑢𝑡 } is a so-continuous group of unitary operators. (iii) There exists a positive self-adjoint linear operator 𝐴 in H , such that s(𝐴) = 1 and 𝑢𝑡 = 𝐴i𝑡 ,

𝑡 ∈ ℝ;

𝐴 is given by the equivalence (𝜉, 𝜂) ∈ G 𝐴 ⇔ [the mapping i𝑡 ↦ 𝑢𝑡 𝜉 has a weakly continuous extension to {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 1}, which is weakly analytic in {𝛼 ∈ ℂ, 0 < Re 𝛼 < 1}, and has the value 𝜂 at 1.] Moreover, the relation 𝑢𝑡 = 𝐴i𝑡 ,

𝑡∈ℝ

establishes a one-to-one correspondence between the so-continuous groups {𝑢𝑡 } of unitary operators on H and the positive self-adjoint linear operators in H , such that s(𝐴) = 1. Proof. The implication (iii) ⇒ (ii) follows from Corollary 9.15, whereas the implication (ii) ⇒ (i) is obvious. Let us now assume that {𝑢𝑡 } is a wo-continuous group of unitary operators on H . By using the notations from Section 9.19, we define 𝐴 = 𝐴1 . In accordance with Lemma 9.19, 𝐴 is a positive self-adjoint linear operator. If 𝜉 ∈ D 𝐴 = D 1 and 𝐹𝜉 (1) = 𝐴𝜉 = 0, then by taking into account Lemma 9.18 (i), we get 𝐹𝜉 (1 + i𝑡) = 𝐹𝐹𝜉 (1) (i𝑡) = 𝑢𝑡 𝐹𝜉 (1) = 0,

𝑡 ∈ ℝ.

Consequently, we have 𝐹𝜉 = 0, whence 𝜉 = 𝐹𝜉 (0) = 0. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Therefore, we have n(𝐴) = 0, i.e. s(𝐴) = 1. In accordance with Lemma 9.19 and from the uniqueness part of Corollary 9.14, we successively obtain 𝐴1/2 = 𝐴1/2 , 𝐴1/4 = 𝐴1/4 , 𝐴3/4 = 𝐴1/4 𝐴1/2 = 𝐴1/4 𝐴1/2 = 𝐴3/4 , 𝑒𝑡𝑐. Thus for any natural number 𝑛 and any integer 𝑘, 0 ≤ 𝑘 ≤ 2𝑛 , we have 𝑛

𝐴𝑘/2𝑛 = 𝐴𝑘/2 ; hence

𝑛

𝐹𝜉 (𝑘/2𝑛 ) = 𝐴𝑘/2𝑛 𝜉 = 𝐴𝑘/2 𝜉,

𝜉 ∈ D 𝐴.

By taking into account Corollary 9.15, we infer that for any 𝜉 ∈ D 𝐴 , the mappings 𝛼 ↦ 𝐹𝜉 (𝛼), 𝛼 ↦ 𝐴𝛼 𝜉, coincide on {𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜀}. In particular, we have 𝑢𝑡 𝜉 = 𝐹𝜉 (i𝑡) = 𝐴i𝑡 𝜉,

𝑡 ∈ ℝ.

The implication (i) ⇒ (iii) is thus established. The second part of the theorem immediately follows from the first part and from Corollary 9.15. Q.E.D. 9.21. Let 𝐴 be a positive self-adjoint operator in H , such that s(𝐴) = 1. As we have seen in the preceding section, 𝐴i𝑡 is a unitary operator, and (𝐴i𝑡 )−1 = 𝐴−i𝑡 . On the other hand, 𝐴−1 is also a positive self-adjoint operator, such that s(𝐴−1 ) = 1. Stone’s theorem leads to the following: Proposition. For any positive self-adjoint operator 𝐴, such that s(𝐴) = 1, we have (𝐴−1 )i𝑡 = 𝐴−i𝑡 ,

𝑡 ∈ ℝ.

Proof. Since {𝐴−i𝑡 } is a wo-continuous group of unitary operators, from Theorem 9.20, we infer that there exists a positive self-adjoint operator 𝐵, such that s(𝐵) = 1 and 𝐵i𝑡 = 𝐴−i𝑡 ,

𝑡 ∈ ℝ.

Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

241

Unbounded linear operators in Hilbert spaces In order to prove the proposition, it is sufficient to show that 𝐵 = 𝐴−1 . Let 𝜉 ∈ D 𝐵 and 𝜂 = 𝐵𝜉. In accordance with Corollary 9.15, the mapping 𝐹 ∶ {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1} ∋ 𝛼 ↦ 𝐵1−𝛼 𝜉 ∈ H

is continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1} and analytic in {𝛼 ∈ ℂ, 0 < Re 𝛼 < 1}. Since 𝐵i𝑡 𝐹(i𝑡) = 𝐵i𝑡 𝐵1−i𝑡 𝜉 = 𝐵𝜉 = 𝜂,

𝑡 ∈ ℝ,

it follows that 𝐹(i𝑡) = 𝐵−i𝑡 𝜂 = 𝐴i𝑡 𝜂,

𝑡 ∈ ℝ.

Obviously, we have 𝐹(1) = 𝜉. If we use again Corollary 9.15, we infer that 𝜂 ∈ D 𝐴 and 𝐴𝜂 = 𝜉, i.e. 𝜉 ∈ D (𝐴−1 ) and 𝐴−1 𝜉 = 𝜂. Consequently, we have 𝐵 ⊂ 𝐴−1 . Conversely, let 𝜉 ∈ D (𝐴−1 ) and 𝜂 = 𝐴−1 𝜉. Then 𝜂 ∈ D 𝐴 and 𝜉 = 𝐴𝜂. In accordance with Corollary 9.15, the mapping 𝐺 ∶ {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1} ∋ 𝛼 ↦ 𝐴1−𝛼 𝜂 ∈ H is continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1} and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1}. Since we have 𝐴i𝑡 𝐺(i𝑡) = 𝐴i𝑡 𝐴1−i𝑡 𝜂 = 𝐴𝜂 = 𝜉, 𝑡 ∈ ℝ, it follows that 𝐺(i𝑡) = 𝐴−i𝑡 𝜉 = 𝐵i𝑡 𝜉,

𝑡 ∈ ℝ.

Since 𝐺(1) = 𝜂, using again Corollary 9.15, we infer that 𝜉 ∈ D 𝐵 and 𝐵𝜉 = 𝜂. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Consequently, we have 𝐴−1 ⊂ 𝐵, and we infer that 𝐵 = 𝐴−1 . Q.E.D. Thus far we have defined (9.15) the operator 𝐴𝛼 for any positive self-adjoint 𝐴 and 𝛼 ∈ ℂ, Re 𝛼 ≥ 0. If s(𝐴) = 1, then it is natural to define 𝐴𝛼 , for any 𝛼 ∈ ℂ. The preceding proposition allows the formulation of the following definition: 𝐴𝛼 = {

𝐴𝛼

if Re 𝛼 ≥ 0,

−1 −𝛼

(𝐴 )

if Re 𝛼 ≤ 0.

Indeed, if Re 𝛼 = 0, then from this proposition, we infer that 𝐴𝛼 = (𝐴−1 )−𝛼 . It is clear that this relation holds for any 𝛼 ∈ ℂ. With this definition, Corollary 9.15 can be extended in the following manner: Corollary. Let 𝐴 be a positive self-adjoint linear operator in H , such that s(𝐴) = 1, 𝜉 ∈ H and 𝜀1 ≤ 0, 𝜀2 ≥ 0. The following assertions are equivalent: (i) 𝜉 ∈ D (𝐴𝜀1 ) ∩ D (𝐴𝜀2 ) ; (ii) 𝜉 ∈ D 𝐴𝛼 for any 𝛼 ∈ ℂ, 𝜀1 ≤ Re 𝛼 ≤ 𝜀2 , and the mapping 𝛼 ↦ 𝐴𝛼 𝜉 is continuous on {𝛼 ∈ ℂ; 𝜀1 ≤ Re 𝛼 ≤ 𝜀2 } and analytic in {𝛼 ∈ ℂ; 𝜀1 < Re 𝛼 < 𝜀2 }; (iii) the mapping i𝑡 → 𝐴i𝑡 𝜉, defined on the imaginary axis, has a continuous extension to the set {𝛼 ∈ ℂ; 𝜀1 ≤ Re 𝛼 ≤ 𝜀2 }, analytic in {𝛼 ∈ ℂ; 𝜀1 < Re 𝛼 < 𝜀2 }. Proof. The implication (ii) ⇒ (iii) is trivial, whereas the implication (iii) ⇒ (i) directly follows from Corollary 9.15. Let us now assume that 𝜉 ∈ D (𝐴𝜀1 ) ∩ D (𝐴𝜀2 ) . From Corollary 9.15, we infer that 𝜉 ∈ D 𝐴𝛼 , for any 𝛼 ∈ ℂ, such that 𝜀1 ≤ Re 𝛼 ≤ 𝜀2 , whereas the mapping 𝛼 ↦ 𝐴𝛼 𝜉 Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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is continuous on {𝛼 ∈ ℂ; 𝜀1 ≤ Re 𝛼 ≤ 𝜀2 } and analytic in {𝛼 ∈ ℂ; 𝜀1 < Re 𝛼 < 𝜀2 ; Re 𝛼 ≠ 0}. With the help of a classical argument, based on the Cauchy integral, we can infer that the preceding mapping is analytic in the set {𝛼 ∈ ℂ; 𝜀1 < Re 𝛼 < 𝜀2 }. Q.E.D. 9.22. Proposition 9.21 indicated a connection between the operational calculus for 𝐴 and that for 𝐴−1 , a connection that we shall now explain: Proposition. Let 𝐴 be a positive self-adjoint operator, such that s(𝐴) = 1, and let 𝑓, 𝑔 ∈ B ([0, +∞)) be bounded functions. If 𝑓(𝜆) = 𝑔(𝜆−1 ),

𝜆 ∈ (0, +∞),

then 𝑓(𝐴) = 𝑔(𝐴−1 ). Proof. We denote 𝑎 = (1 + 𝐴)−1 , 𝑏 = (1 + 𝐴−1 )−1 . It is easy to see that 𝑎 + 𝑏 = 1. Since s(𝑎) = 1 = s(𝑏), we infer that 𝜒(0,1) (𝑎) = 1 = 𝜒(0,1) (𝑏). With the notations from Section 9.10, we denote 𝐹 = 𝐹𝑓 ,

𝐺 = 𝐹𝑔 .

Then, for any 𝜆 ∈ (0, 1), we have 𝐹(𝜆) = 𝑓((1 − 𝜆)/𝜆) = 𝑔(𝜆/(1 − 𝜆)) = 𝐺(1 − 𝜆). By taking into account Section 9.10, we infer that 𝑓(𝐴) = 𝐹(𝑎) = (𝐹𝜒(0,1) )(𝑎) = (𝐺𝜒(0,1) )(1 − 𝑎) = (𝐺𝜒(0,1) (𝑏) = 𝐺(𝑏) = 𝑔(𝐴−1 ). Q.E.D. In accordance with the preceding proposition, if 𝐴 is a positive self-adjoint operator, such that s(𝐴) = 1 and if 0 ≤ 𝜆1 < 𝜆2 ≤ +∞, then 𝜒(𝜆1 ,𝜆2 ) (𝐴) = 𝜉(𝜆2−1 ,𝜆2−1 ) (𝐴−1 ). In particular, for any 𝜆, 1 < 𝜆 ≤ +∞, we have 𝜒(𝜆−1 ,𝜆) (𝐴) = 𝜒(𝜆−1 ,𝜆) (𝐴−1 ). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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9.23. In this section, we present an integral formula, which will be one of the main instruments for the development of Tomita’s theory, in Chapter 10. If 𝐹 ∶ ℝ → B (H ) is a wo-continuous mapping, and if the function 𝑡 ↦ ‖𝐹(𝑡)‖ is dominated by a Lebesgue integrable function, then +∞

(𝜉, 𝜂) ↦ ∫

(𝐹(𝑡)𝜉|𝜂)𝑑𝑡

−∞

is a bounded sesquilinear form on H × H . With the Riesz theorem, we infer that there exists a (unique) operator 𝑥𝐹 ∈ B (H ), such that +∞

(𝑥𝐹 𝜉|𝜂) = ∫

(𝐹(𝑡)𝜉|𝜂)𝑑𝑡,

𝜉, 𝜂 ∈ H .

−∞

In what follows we shall denote +∞

𝑥𝐹 = ∫

𝐹(𝑡) 𝑑𝑡.

−∞

Let now 𝐴 and 𝐵 be positive self-adjoint operators in H , such that s(𝐴) = s(𝐵) = 1, and 𝜆 > 0. For any 𝑥 ∈ B (H ), we shall denote +∞

Φ𝜆 (𝑥) = ∫ −∞

i𝑡−

1

𝜆 2 𝐴i𝑡 𝑥𝐵−i𝑡 𝑑𝑡. 𝜋𝑡 e + e−𝜋𝑡

Proposition. Let 𝐴 and 𝐵 be positive self-adjoint operators in H , such that s(𝐴) = s(𝐵) = 1 and 𝜆 > 0. For any 𝑥 ∈ B (H ), there exists a unique 𝑦 ∈ B (H ), such that (𝑥𝜂|𝜉) = 𝜆(𝑦𝐵−1/2 𝜂|𝐴1/2 𝜉) + (𝑦𝐵1/2 𝜂|𝐴−1/2 𝜉), 𝜉 ∈ D (𝐴1/2 ) ∩ D (𝐴−1/2 ) ,

𝜂 ∈ D (𝐵 1/2 ) ∩ D (𝐵 −1/2 ) ,

and it is given by 𝑦 = Φ𝜆 (𝑥). Proof. Let us define, for any natural number 𝑛, 𝑒𝑛 = 𝜒(1/𝑛,𝑛) (𝐴) = 𝜒(1/𝑛,𝑛) (𝐴−1 ), 𝑓𝑛 = 𝜒(1/𝑛,𝑛) (𝐵) = 𝜒(1/𝑛,𝑛) (𝐵−1 ), where by 𝜒(1/𝑛,𝑛) we denoted the characteristic function of the interval (1/𝑛, 𝑛). We now consider the mapping 𝛼 ↦ 𝐹𝑛,𝑚 (𝛼) =

−i𝜆i𝛼 𝐴i𝛼 𝑒𝑛 𝑥𝐵−i𝛼 𝑓𝑚 , 𝑒𝜋𝛼 − 𝑒−𝜋𝛼

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which is analytic in the set {𝛼 ∈ ℂ; 𝛼 ≠ i𝑘, 𝑘 ∈ ℤ} for the norm topology. At the point 𝛼 = 0, the function 𝐹𝑛,𝑚 has a first-order pole, with the residue lim 𝛼𝐹𝑛,𝑚 (𝛼) = −

𝛼→0

i 𝑒 𝑥𝑓 . 2𝜋 𝑛 𝑚

From the Cauchy residue theorem, we infer that +∞

∫ −∞

+∞

i i 𝐹𝑛,𝑚 (𝑡 − )𝑑𝑡 − ∫ 𝐹𝑛,𝑚 (𝑡 + )𝑑𝑡 = 𝑒𝑛 𝑥𝑓𝑚 . 2 2 −∞

Thus +∞

∫ −∞

i𝑡+

1

𝜆 2 𝐴1/2 e𝑛 (𝐴i𝑡 𝑥𝐵−i𝑡 )𝐵−1/2 𝑓𝑚 𝑑𝑡 𝜋𝑡 e + 𝑒−𝜋𝑡

+∞

+∫ −∞

i𝑡−

1

𝜆 2 𝐴−1/2 𝑒𝑛 (𝐴i𝑡 𝑥𝐵−i𝑡 )𝐵1/2 𝑓𝑚 𝑑𝑡 = 𝑒𝑛 𝑥𝑓𝑚 , 𝑒𝜋𝑡 + 𝑒−𝜋𝑡

whence 𝜆𝐴1/2 𝑒𝑛 Φ𝜆 (𝑥)𝐵−1/2 𝑓𝑚 + 𝐴−1/2 𝑒𝑛 Φ𝜆 (𝑥)𝐵1/2 𝑓𝑚 = 𝑒𝑛 𝑥𝑓𝑚 . Let 𝜉 ∈ D (𝐴1/2 ) ∩ D (𝐴−1/2 ) and 𝜂 ∈ D (𝐵 1/2 ) ∩ D (𝐵 −1/2 ) . From the above equality, we infer that, for any 𝑛 and 𝑚, 𝜆(Φ𝜆 (𝑥)𝐵−1/2 𝑓𝑚 𝜂|𝐴1/2 𝑒𝑛 𝜉) + (Φ𝜆 (𝑥)𝐵1/2 𝑓𝑚 𝜂|𝐴−1/2 𝑒𝑛 𝜉) = (𝑥𝑓𝑚 𝜂|𝑒𝑛 𝜉). Tending to the limit for 𝑛 and 𝑚 → ∞, we get 𝜆(Φ𝜆 (𝑥)𝐵−1/2 𝜂|𝐴1/2 𝜉) + (Φ𝜆 (𝑥)𝐵1/2 𝜂|𝐴−1/2 𝜉) = (𝑥𝜂|𝜉). Let now 𝑦 ∈ B (H ) be an arbitrary operator, which satisfies the relations in the statement of the proposition. By denoting 𝑧 = Φ𝜆 (𝑥) − 𝑦; for any 𝑛 and 𝑚, we have 𝜆𝐴1/2 𝑒𝑛 𝑧𝐵−1/2 𝑓𝑚 + 𝐴−1/2 𝑒𝑛 𝑧𝐵1/2 𝑓𝑚 = 0. If we multiply this equality by 𝑓𝑚 𝑧∗ 𝐴1/2 𝑒𝑛 to the left and by 𝐵1/2 𝑓𝑚 to the right, we get 𝜆𝑓𝑚 𝑧∗ 𝐴𝑒𝑛 𝑧𝑓𝑚 + 𝑓𝑚 𝑧∗ 𝑒𝑛 𝑧𝐵𝑓𝑚 = 0, i.e. (𝑓𝑚 𝑧∗ 𝑒𝑛 𝑧𝑓𝑚 )(𝐵𝑓𝑚 ) = −𝜆𝑓𝑚 𝑧∗ 𝐴𝑒𝑛 𝑧𝑓𝑚 ≤ 0. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Thus, the operators 𝑓𝑚 𝑧∗ 𝑒𝑛 𝑧𝑓𝑚 ≥ 0 and 𝐵𝑓𝑚 ≥ 0 commute; hence (𝑓𝑚 𝑧∗ 𝑒𝑛 𝑧𝑓𝑚 )(𝐵𝑓𝑚 ) ≥ 0. Consequently, 𝑓𝑚 𝑧∗ 𝑒𝑛 𝑧𝐵𝑓𝑚 = 0, whence we successively infer that 𝑓𝑚 𝑧∗ 𝑒𝑛 𝑧𝑓𝑚 = (𝑓𝑚 𝑧∗ 𝑒𝑛 𝑧𝐵𝑓𝑚 )(𝐵−1 𝑓𝑚 ) = 0, 𝑒𝑛 𝑧𝑓𝑚 = 0. By tending to the limit for 𝑛 and 𝑚 → ∞, we infer that 𝑧 = 0, i.e. 𝑦 = Φ𝜆 (𝑥). Q.E.D. With the help of Lemma 9.5, it is easy to see that if 𝐴 is a positive self-adjoint operator and 𝜆 > 0, then (𝜆 + 𝐴)−1 ∈ B (H ). Corollary. Let 𝐴 be a positive self-adjoint operator in H , such that s(𝐴) = 1 and 𝜆 > 0. Then +∞ −1/2

𝐴

−1 −1

(𝜆 + 𝐴 )

=∫ −∞

i𝑡−

1

𝜆 2 𝐴i𝑡 𝑑𝑡. 𝜋𝑡 𝑒 + 𝑒−𝜋𝑡

Proof. Let 𝑦 = 𝐴−1/2 (𝜆 + 𝐴−1 )−1 ∈ B (H ). It is easy to verify that 𝑦 satisfies the relation from the statement of the preceding proposition, with 𝐵 = 1 and 𝑥 = 1, i.e. (𝜂|𝜉) = 𝜆(𝑦𝜂|𝐴1/2 𝜉) + (𝑦𝜂|𝐴−1/2 𝜉), 𝜉 ∈ D (𝐴1/2 ) ∩ D (𝐴−1/2 ) , 𝜂 ∈ H . Thus the assertion in the corollary obviously follows from the preceding proposition. Q.E.D. The above Proposition and its Corollary remain valid if instead of 𝜆 > 0 we put any 𝜔 ∈ ℂ, |𝜔| = 1, 𝜔 ≠ −1. The proof is the same as above with the only difference that in order to get (𝑓𝑚 𝑧∗ 𝑒𝑛 𝑧𝑓𝑚 )𝐵𝑓𝑚 = 0,

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247

we use the following simple device: If 𝑎, 𝑏, 𝑐 ∈ B (H ) are positive operators and 𝜔 ∈ ℂ, |𝜔| = 1, 𝜔 ≠ −1, and 𝑎𝑏 = −𝜔𝑐, then 𝑐 = 0. Indeed, 𝑎𝑏 = −𝜔𝑐; hence 𝑏𝑎 = −𝜔𝑐 and hence their spectra are 𝜎(𝑎𝑏) ⊂ −𝜔ℝ+ ,

𝜎(𝑏𝑎) ⊂ −𝜔ℝ+ ,

and since 𝜎(𝑎𝑏) = 𝜎(𝑏𝑎), it follows that there are 𝑠, 𝑡 ∈ ℝ+ , with 𝜔𝑠 = 𝜔𝑡 = 𝜔−1 𝑡 so that 𝜔2 𝑠 = 𝑡 and then 𝑠 = |𝜔2 𝑠| = |𝑡| = 𝑡 ⇒ 𝜔2 = 1 ⇒ 𝜔 = 1. Finally, 𝑎𝑏 = −𝑐 ⇒ 𝑏𝑎 = −𝑐 = 𝑎𝑏 ⇒ 𝑎𝑏 ≥ 0, while − 𝑐 ≤ 0; hence 𝑐 = 𝑎𝑏 = 0. 9.24. In this section, we shall prove an analogue of Corollary 9.21, which we shall use in Chapter 10. We shall first state the following: Lemma. Let Ω ⊂ ℂ an open set and 𝐹 ∶ Ω → B (H ). Then the following assertions are equivalent: (i) 𝐹 is analytic for the norm topology. (ii) For any 𝜉, 𝜂 ∈ H , the function Ω ∋ 𝛼 ↦ (𝐹(𝛼)𝜉|𝜂) is analytic. Proof. Obviously, (i) ⇒ (ii). Let us now assume that assertion (ii) is true. Let 𝛼 ∈ Ω and 𝑉 ⊂ 𝑉 ⊂ Ω be a relatively compact neighbourhood of 𝛼. For any 𝛽, 𝛾 ∈ 𝑉, 𝛽 ≠ 𝛼, 𝛾 ≠ 𝛼, 𝛽 ≠ 𝛾, we define 𝐺(𝛼; 𝛽, 𝛾) =

1 1 1 (𝐹(𝛽) − 𝐹(𝛼)) − (𝐹(𝛾) − 𝐹(𝛼))] . [ 𝛾−𝛼 𝛽−𝛾 𝛽−𝛼

According to the hypothesis, for any 𝜉, 𝜂 ∈ H , we have sup {|(𝐺(𝛼; 𝛽, 𝛾)𝜉|𝜂)|; 𝛽, 𝛾 ∈ 𝑉, 𝛽 ≠ 𝛼, 𝛾 ≠ 𝛼, 𝛽 ≠ 𝛾} < +∞. From the Banach–Steinhaus theorem, we infer that 𝑐 = sup {‖𝐺(𝛼; 𝛽, 𝛾)‖; 𝛽, 𝛾 ∈ 𝑉, 𝛽 ≠ 𝛼, 𝛾 ≠ 𝛼, 𝛽 ≠ 𝛾} < +∞. Consequently, for any 𝛽, 𝛾 ∈ 𝑉, 𝛽 ≠ 𝛼, 𝛾 ≠ 𝛼, 𝛽 ≠ 𝛾, we have ‖

1 1 (𝐹(𝛽) − 𝐹(𝛼)) − (𝐹(𝛾) − 𝐹(𝛼))‖ ≤ 𝑐|𝛽 − 𝛾|. 𝛾−𝛼 𝛽−𝛼

With the help of the Cauchy criterion, it follows that 𝐹 is differentiable with respect to the norm topology. Consequently, assertion (i) follows from assertion (ii). Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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A mapping 𝐹, which satisfies the equivalent assertions of this lemma, will be called, briefly, an analytic mapping. Proposition. Let 𝐴 and 𝐵 be positive self-adjoint operators in H, such that s(𝐴) = s(𝐵) = 1, 𝑥 ∈ B (H ) and 𝜀1 ≤ 0 ≤ 𝜀2 . Then the following assertions are equivalent: (i) There exist vector subspaces D 1 ⊂ D (𝐴𝜀1 𝑥𝐵 −𝜀1 ) , D 2 ⊂ D (𝐴𝜀2 𝑥𝐵 −𝜀2 ) , such that 𝐵−𝜀1 |D 1 = 𝐵−𝜀1 ,

𝐵−𝜀2 |D 2 = 𝐵−𝜀2

and the operators 𝐴𝜀1 𝑥𝐵−𝜀1 |D 1 ,

𝐴𝜀2 𝑥𝐵−𝜀2 |D 2

are bounded. (ii) For any 𝛼 ∈ ℂ, 𝜀1 < Re 𝛼 < 𝜀2 , we have D (𝐴𝛼 𝑥𝐵 −𝛼 ) = D 𝐵 −𝛼 , the operator 𝐴𝛼 𝑥𝐵−𝛼 is bounded and the mapping 𝛼 ↦ 𝐴𝛼 𝑥𝐵−𝛼 is so-continuous on {𝛼 ∈ ℂ; 𝜀1 ≤ Re 𝛼 ≤ 𝜀2 } and analytic in {𝛼 ∈ ℂ; 𝜀1 < Re 𝛼 < 𝜀2 }. (iii) The mapping i𝑡 ↦ 𝐴i𝑡 𝑥𝐵−i𝑡 ,

𝑡 ∈ ℝ,

has a wo-continuous extension to the set {𝛼 ∈ ℂ; 𝜀1 ≤ Re 𝛼 ≤ 𝜀2 }, which is analytic in {𝛼 ∈ ℂ; 𝜀1 < Re 𝛼 < 𝜀2 }. Proof. It is obvious that we can consider only the case in which 𝜀1 = 0, 𝜀2 = 𝜀 (see the proof of Corollary 9.21). Let us assume that assertion (i) is true. We denote D = D 2 and 𝑐 = ‖𝐴𝜀 𝑥𝐵−𝜀 |D ‖. For any 𝜉 ∈ D and any 𝜂 ∈ D 𝐴𝜀 , we have |(𝑥𝐵−𝜀 𝜉|𝐴𝜀 𝜂)| = |(𝐴𝜀 𝑥𝐵−𝜀 𝜉|𝜂)| ≤ 𝑐‖𝜉‖‖𝜂‖. Since 𝐵−𝜀 |D = 𝐵−𝜀 , we infer that, for any 𝜉 ∈ D 𝐵 −𝜀 , 𝜂 ∈ D 𝐴𝜀 , we have |(𝑥𝐵−𝜀 𝜉|𝐴𝜀 𝜂)| ≤ 𝑐‖𝜉‖‖𝜂‖. If we replace 𝜉 by 𝐵−i𝑡 𝜉 and 𝜂 by 𝐴−i𝑡 𝜂, we obtain that for any 𝜉 ∈ D 𝐵 −𝜀 and any 𝜂 ∈ D 𝐴𝜀 , we have |(𝑥𝐵−𝜀−i𝑡 𝜉|𝐴𝜀−i𝑡 𝜂)| ≤ 𝑐‖𝜉‖‖𝜂‖, 𝑡 ∈ ℝ. On the other hand, for any 𝜉 and 𝜂, we trivially have |(𝑥𝐵−i𝑡 𝜉|𝐴−i𝑡 𝜂)| ≤ ‖𝑥‖‖𝜉‖‖𝜂‖,

𝑡 ∈ ℝ.

Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Unbounded linear operators in Hilbert spaces Let 𝜉 ∈ D 𝐵 −𝜀 and 𝜂 ∈ D 𝐴𝜀 . Then 𝛼 ↦ (𝑥𝐵−𝛼 𝜉|𝐴𝛼 𝜂)

is a bounded continuous function on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}. By taking into account the results that we have already obtained, with the help of the Phragmen–Lindelöf principle (see Dunford and Schwartz [1958/1963/1970], Chap. III, 14), we get that, for any 𝛼 ∈ ℂ such that 0 ≤ Re 𝛼 ≤ 𝜀, we have |(𝑥𝐵−𝛼 𝜉|𝐴𝛼 𝜂)| ≤ max{𝑐, ‖𝑥‖}‖𝜉‖‖𝜂‖. Obviously, this inequality extends for any 𝜉 ∈ D 𝐵 −𝛼 and 𝜂 ∈ D 𝐴𝛼 . Let now 𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀. From the preceding relation, it follows that for any 𝜉 ∈ D 𝐵 −𝛼 , we have 𝑥𝐵−𝛼 𝜉 ∈ D (𝐴𝛼 )∗ = D 𝐴𝛼 and |(𝐴𝛼 𝑥𝐵−𝛼 𝜉|𝜂)| ≤ max{𝑐, ‖𝑥‖}‖𝜉‖‖𝜂‖,

𝜂 ∈ D 𝐴𝛼 .

In other words, we have D (𝐴𝛼 𝑥𝐵 −𝛼 ) = D 𝐵 −𝛼 , the operator 𝐴𝛼 𝑥𝐵−𝛼 is bounded, and its norm is uniformly bounded with respect to 𝛼: ‖𝐴𝛼 𝑥𝐵−𝛼 ‖ ≤ max{𝑐, ‖𝑥‖}. With the help of the equality, ((𝐴𝛼 𝑥𝐵−𝛼 )𝜉|𝜂) = (𝑥𝐵−𝛼 𝜉|𝐴𝛼 𝜂),

0 ≤ Re 𝛼 ≤ 𝜀, 𝜉 ∈ D 𝐵 −𝜀 , 𝜂 ∈ D 𝐴𝜀 ,

it is easy to infer that the mapping 𝛼 ↦ 𝐴𝛼 𝑥𝐵−𝛼 is wo-continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}. It is easy to verify that for any natural number 𝑛, any vector 𝜉 ∈ 𝜒[1/𝑛,𝑛) (𝐵)H and any 𝛼 such that 0 ≤ Re 𝛼 ≤ 𝜀, we have 𝑥𝐵−𝛼 𝜉 ∈ D 𝐴𝜀 , whence, for any 𝛽, such that 0 ≤ Re 𝛼 ≤ 𝜀, we have ‖(𝐴𝛽 𝑥𝐵−𝛽 )𝜉 − (𝐴𝛼 𝑥𝐵−𝛼 )𝜉‖ ≤ ‖𝐴𝛽 𝑥𝐵−𝛽 𝜉 − 𝐴𝛽 𝑥𝐵−𝛼 𝜉‖ + ‖𝐴𝛽 𝑥𝐵−𝛼 𝜉 − 𝐴𝛼 𝑥𝐵−𝛼 𝜉‖ = ‖𝐴𝛽 𝑥𝐵−𝛽 (𝜉 − 𝐵𝛽−𝛼 𝜉)‖ + ‖𝐴𝛽 𝑥𝐵−𝛼 𝜉 − 𝐴𝛼 𝑥𝐵−𝛼 𝜉‖ ≤ max{𝑐, ‖𝑥‖}‖𝜉 − 𝐵𝛽−𝛼 𝜉‖ + ‖𝐴𝛽 (𝑥𝐵−𝛼 𝜉) − 𝐴𝛼 (𝑥𝐵−𝛼 𝜉)‖. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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With the help of Corollary 9.21, we infer from here that lim ‖(𝐴𝛽 𝑥𝐵−𝛽 )𝜉 − (𝐴𝛼 𝑥𝐵−𝛼 )𝜉‖ = 0.

𝛽→𝛼

Since ∪𝑛 𝜒[1/𝑛,𝑛) (𝐵)H is dense in H and since sup{‖𝐴𝛾 𝑥𝐵−𝛾 ‖; 0 ≤ Re 𝛾 ≤ 𝜀} < +∞, the preceding equality extends for any 𝜉 ∈ H . Consequently, the mapping 𝛼 ↦ 𝐴𝛼 𝑥𝐵−𝛼 is so-continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}. We have thus proved that (i) ⇒ (ii). The implication (ii) ⇒ (iii) is trivial. Finally, let us assume that assertion (iii) is true. We denote by 𝐹 the wo-continuous extension to {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} of the mapping i𝑡 ↦ 𝐴i𝑡 𝑥𝐵−i𝑡 ,

𝑡 ∈ ℝ,

which is analytic in {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}. For any 𝜉 ∈ D 𝐵 −𝜀 and 𝜂 ∈ D 𝐴𝜀 , the functions 𝛼 ↦ (𝐹(𝛼)𝜉|𝜂), 𝛼 ↦ (𝑥𝐵−𝛼 𝜉|𝐴𝛼 𝜂) are continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}; moreover, they coincide on the imaginary axis. Hence they coincide on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}. In particular, we have (𝑥𝐵−𝜀 𝜉|𝐴𝜀 𝜂) = (𝐹(𝜀)𝜉|𝜂),

𝜉 ∈ D 𝐵 −𝜀 ,

𝜂 ∈ D 𝐴𝜀 ,

whence, for any 𝜉 ∈ D 𝐵 −𝜀 , we have 𝑥𝐵−𝜀 𝜉 ∈ D (𝐴𝜀 )∗ = D 𝐴𝜀 and 𝐴𝜀 𝑥𝐵−𝜀 𝜉 = 𝐹(𝜀)𝜉. Hence 𝐷(𝐴𝜀 𝑥𝐵 −𝜀 ) = 𝐷𝐵 −𝜀 , and the operator 𝐴𝜀 𝑥𝐵−𝜀 is bounded. Consequently, (iii) ⇒ (i).

Q.E.D.

9.25. The following proposition shows that the operational calculus is invariant with respect to *-isomorphisms and provides a natural method of transfer by *-isomorphism of the positive self-adjoint operators, which are affiliated to a von Neumann algebra. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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251

Proposition. Let 𝜋 be a *-isomorphism of the von Neumann algebra M ⊂ B (H ) onto the von Neumann algebra N ⊂ B (K ). Obviously, 𝜋 establishes a one-to-one correspondence M + ∋ 𝑎 ↦ 𝑏 = 𝜋(𝑎) ∈ N + . This correspondence extends to a one-to-one correspondence 𝐴 ↦ 𝐵 = 𝜋(𝐴) between the positive self-adjoint operators 𝐴 in H , affiliated to M , and the positive self-adjoint operators 𝐵 in K , affiliated to N , which is unique, subject to the condition 𝜋((1 + 𝐴)−1 ) = (1 + 𝐵)−1 . Moreover, for any positive 𝑓 ∈ B ([0, +∞)), we have 𝜋(𝑓(𝐴)) = 𝑓(𝜋(𝐴)). Proof. Let 𝑎 ∈ M , 𝑎 ≥ 0. By taking into account Corollary 5.13, it is easy to verify that the mapping B (𝜎(𝑎)) ∋ 𝑓 ↦ 𝜋(𝑓(𝑎)) satisfies conditions (i) and (ii) from Theorem 2.20, for 𝑥 = 𝜋(𝑎). According to Theorem 2.20, it follows that for any 𝑓 ∈ B (𝜎(𝑎)), we have 𝜋(𝑓(𝑎)) = 𝑓(𝜋(𝑎)). Let 𝐴 be a positive self-adjoint operator in H , which is affiliated to M . We write 𝑎 = (1 + 𝐴)−1 . Then 𝑎 ∈ M , 0 ≤ 𝑎 ≤ 1, s(𝑎) = 1. Consequently, if we define 𝑏 = 𝜋(𝑎), we have 𝑏 ∈ N , 0 ≤ 𝑏 ≤ 1, s(𝑏) = 1. The operator 𝐵 = 𝑏−1 − 1 is a positive self-adjoint operator in K , affiliated to N , and 𝑏 = (1 + 𝐵)−1 . From the preceding results, it follows that the correspondence 𝐴 ↦ 𝐵 = 𝜋(𝐴), we have thus defined, is a one-to-one correspondence between the positive self-adjoint operators 𝐴 in H , which are affiliated to M , and the positive self-adjoint operators 𝐵 in K , which are affiliated to N , and moreover, 𝜋((1 + 𝐴)−1 ) = (1 + 𝐵)−1 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Let now 𝑓 ∈ B ([0, +∞)) be positive and 𝑔 = (1 + 𝑓)−1 ∈ B ([0, +∞)), which obviously is positive and bounded. By taking into account Section 9.10 and by using the notations, we have introduced there, we have 𝜋((1 + 𝑓(𝐴))−1 ) = 𝜋(𝑔(𝐴)) = 𝜋(𝐹𝑔 (𝑎)) = 𝐹𝑔 (𝜋(𝑎)) = 𝐹𝑔 (𝑏) = 𝑔(𝐵) = (1 + 𝑓(𝐵))−1 = (1 + 𝑓((𝜋(𝐴)))−1 ; hence 𝜋(𝑓(𝐴)) = 𝑓(𝜋(𝐴)). Q.E.D. With the preceding notations, if we also write 𝑒𝑛 = 𝜒[0,𝑛) (𝐴),

𝑓𝑛 = 𝜒[0,𝑛) (𝐵),

from the preceding proposition, we also obtain, in particular, that 𝜋(𝐴𝑒𝑛 ) = 𝐵𝑓𝑛 . 9.26. Let 𝑇 be a linear operator in H . Its resolvent set is defined by 𝜌(𝑇) = {𝜆 ∈ ℂ; (𝜆 − 𝑇)−1 ∈ B (H ) exists} and its spectrum by 𝜎(𝑇) = ℂ\𝜌(𝑇). As in the case in which 𝑇 ∈ B (H ), it is easy to prove that 𝜌(𝑇) is an open set, whereas the function 𝜌(𝑇) ∋ 𝜆 ↦ (𝜆 − 𝑇)−1 ∈ B (H ) is analytic for the norm topology in B (H ). In contrast to the case in which 𝑇 ∈ B (H ), in general 𝜎(𝑇) can be either the empty set or it can coincide with ℂ. Proposition. Let 𝐴 be a self-adjoint linear operator in H . Then 𝜎(𝐴) ⊂ ℝ, and for any 𝜆 ∈ ℂ\ℝ, we have ‖(𝜆 − 𝐴)−1 ‖ ≤ 1/|Im 𝜆|. Proof. Let 𝜆 ∈ ℂ\ℝ. For any 𝜉 ∈ D 𝐴 , we have ‖(𝜆 − 𝐴)𝜉‖2 = ((𝜆 − 𝐴)𝜉|(𝜆 − 𝐴)𝜉) = |Im 𝜆|2 ‖𝜉‖2 + ((Re 𝜆 − 𝐴)𝜉|(Re 𝜆 − 𝐴)𝜉) ≥ |Im 𝜆|2 ‖𝜉‖2 ; hence 𝜆 − 𝐴 is injective, its range is closed and (𝜆 − 𝐴)−1 is bounded of norm ≤ 1/Im 𝜆|. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Unbounded linear operators in Hilbert spaces Let 𝜂 ∈ H be orthogonal to (𝜆 − 𝐴)D 𝐴 . Then (0, 𝜂) ∈ (G (𝜆−𝐴) )⟂ = (G (𝜆−𝐴)∗ )⟂ = 𝑉H G (𝜆−𝐴) ; hence (𝜂, 0) ∈ G (𝜆−𝐴) , i.e. 𝜂 ∈ D 𝐴,

(𝜆 − 𝐴)𝜂 = 0.

Since Im 𝜆 = Im 𝜆 ≠ 0, from the first part of the proof, we infer that the operator 𝜆 − 𝐴 is injective; hence 𝜂 = 0. Consequently, (𝜆 − 𝐴)−1 is everywhere defined and bounded, i.e. 𝜆 ∈ 𝜌(𝐴) and ‖(𝜆 − 𝐴)−1 ‖ ≤ 1/Im 𝜆‖. Q.E.D. For any 𝜆 ∈ ℂ, we define 𝑑(𝜆, ℝ+ ) = inf {|𝜆 − 𝜇|;

𝜇 ∈ ℝ+ }.

Corollary. Let 𝐴 be a positive self-adjoint linear operator in H . Then 𝜎(𝐴) ⊂ ℝ+ , and for any 𝜆 ∈ ℂ\ℝ+ , we have ‖(𝜆 − 𝐴)−1 ‖ ≤ 1/𝑑(𝜆, ℝ+ ). Proof. From the preceding proposition and from Lemma 9.5, we infer that 𝜎(𝐴) ⊂ ℝ+ . Let 𝜆 ∈ ℂ\ℝ+ . For any 𝜉 ∈ D 𝐴 , we have ‖(𝜆 − 𝐴)𝜉‖2 = ((𝜆 − 𝐴)𝜉|(𝜆 − 𝐴)𝜉) = |Im 𝜆|2 ‖𝜉‖2 + ((Re 𝜆 − 𝐴)𝜉|(Re 𝜆 − 𝐴)𝜉) ≥{

|Im 𝜆|2 ‖𝜉‖2 , if Re 𝜆 ≥ 0 |Im 𝜆|2 ‖𝜉‖2 + | Re 𝜆|2 ‖𝜉‖2 , 𝑖𝑓 Re 𝜆 ≤ 0

= 𝑑(𝜆, ℝ+ )2 ‖𝜉‖2 ; hence ‖(𝜆 − 𝐴)−1 ‖ ≤ 1/𝑑(𝜆, ℝ+ ). Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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9.27. Proposition Let 𝐴 be a positive self-adjoint linear operator in H , 𝑓 a bounded analytic function, defined on an open convex neighbourhood of the interval [0, +∞) and Γ ∶ ℝ → ℂ a locally rectifiable Jordan curve, contained in the domain of definition of 𝑓, which contains in its ‘interior’ the interval [0, +∞) with respect to which it is positively oriented. We assume that ∫ |𝑓(𝜆)|𝑑(𝜆, ℝ+ )−1 𝑑𝜆 < +∞. Γ

Then, for any 𝜉 ∈ H , we have 𝑓(𝐴)𝜉 = (2𝜋i)−1 ∫ 𝑓(𝜆)(𝜆 − 𝐴)−1 𝜉 𝑑𝜆. Γ

(Some extra-condition are necessary in order to insure that the integral over the segment [Γ(𝑡𝑛 ), Γ(𝑡−𝑛 )] converges to zero for 𝑛 → ∞. This is automatically satisfied for the specific application of Proposition 9.27 in the proof of Lemma 1 from Section 10.19 [Translator’s Note].) Proof. Let 𝑒𝑛 = 𝜒[0,𝑛) (𝐴) (see 9.9). It is easy to verify that for any 𝜆 ∈ ℂ\ℝ+ and any 𝑛, we have (𝜆 − 𝐴)−1 𝑒𝑛 = (𝜆 − 𝐴𝑒𝑛 )−1 𝑒𝑛 . Since Γ contains the interval [0, +∞) in its ‘interior’ with respect to which it is positively oriented, there exist real numbers −∞ ← … < 𝑡−𝑛 < … < 𝑡−1 < 𝑡1 < … < 𝑡𝑛 < … → +∞, such that the curve Γ𝑛 , obtained by composing the restriction of Γ to [𝑡−𝑛 , 𝑡𝑛 ] with the segment [Γ(𝑡𝑛 ), Γ(𝑡−𝑛 )] ⊂ ℂ, should ‘contain’ the interval [0, 𝑛] in its interior. By taking into account Theorem 2.29 and Cauchy’s integral theorem, we infer that, for any 𝜉 ∈ 𝑒𝑛 H , we have 𝑓(𝐴)𝜉 = 𝑓(𝐴𝑒𝑛 )𝜉 = (2𝜋i)−1 ∫ 𝑓(𝜆)(𝜆 − 𝐴𝑒𝑛 )−1 𝜉 𝑑𝜆 Γ𝑛

= (2𝜋i)−1 ∫ 𝑓(𝜆)(𝜆 − 𝐴𝑒𝑛 )−1 𝜉 𝑑𝜆 Γ

= (2𝜋i)−1 ∫ 𝑓(𝜆)(𝜆 − 𝐴)−1 𝜉 𝑑𝜆. Γ

Let now 𝜉 ∈ H be arbitrary. We denote 𝜉𝑛 = 𝑒𝑛 𝜉. Then ‖𝜉 − 𝜉𝑛 ‖ → 0; Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Unbounded linear operators in Hilbert spaces hence ‖𝑓(𝐴)𝜉 − 𝑓(𝐴)𝜉𝑛 ‖ → 0, ‖(𝜆 − 𝐴)−1 𝜉 − (𝜆 − 𝐴)−1 𝜉𝑛 ‖ → 0. Since, for any 𝑛, we have 𝑓(𝐴)𝜉𝑛 = (2𝜋i)−1 ∫ 𝑓(𝜆)(𝜆 − 𝐴)−1 𝜉𝑛 𝑑𝜆, Γ

with the help of the Lebesgue dominated convergence theorem, we obtain 𝑓(𝐴)𝜉 = (2𝜋i)−1 ∫ 𝑓(𝜆)(𝜆 − 𝐴)−1 𝜉 𝑑𝜆, Γ

Q.E.D. 9.28. Proposition If 𝑇 is a closed linear operator from H into K , then 𝑇 ∗ 𝑇 is self-adjoint and positive. Moreover, 𝑇|D 𝑇 ∗ 𝑇 = 𝑇. Proof. Let 𝜁 ∈ H . Since G 𝑇 is closed in H ⊕ K , we can write (𝜁, 0) = (𝜉, 𝜂) + (𝜉0 , 𝜂0 ),

(𝜉, 𝜂) ∈ G 𝑇 ,

(𝜉0 , 𝜂0 ) ∈ (G 𝑇 )⟂ .

Then 𝜉 ∈ D 𝑇 , 𝑇𝜉 = 𝜂 𝑎𝑛𝑑 𝜂0 ∈ D 𝑇 ∗ , 𝑇 ∗ 𝜂0 = −𝜉0 ; hence 𝜁 = 𝜉 − 𝑇 ∗ 𝜂0 ,

0 = 𝑇𝜉 + 𝜂0 ,

whence 𝑇𝜉 = −𝜂0 ∈ D 𝑇 ∗ ,

𝜁 = 𝜉 + 𝑇 ∗ 𝑇𝜉 = (1 + 𝑇 ∗ 𝑇)𝜉.

Thus (1 + 𝑇 ∗ 𝑇)D 𝑇 ∗ 𝑇 = H . Let now 𝜁 ∈ H , 𝜁 ⟂ D 𝑇 ∗ 𝑇 . In accordance with what we have already proved, there exists a 𝜉 ∈ D 𝑇 ∗ 𝑇 , such that 𝜁 = (1 + 𝑇 ∗ 𝑇)𝜉. Then 0 = (𝜁|𝜉) = (𝜉 + 𝑇 ∗ 𝑇𝜉|𝜉) = ‖𝜉‖2 + ‖𝑇𝜉‖2 , whence 𝜉 = 0 and 𝜁 = 0. It follows that D 𝑇 ∗ 𝑇 is dense in H . For any 𝜉 ∈ D 𝑇 ∗ 𝑇 , we have (𝑇 ∗ 𝑇𝜉|𝜉) = ‖𝑇𝜉‖2 ≥ 0. Thus, 𝑇 ∗ 𝑇 is positive. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Since (1 + 𝑇 ∗ 𝑇)D 𝑇 ∗ 𝑇 = H , from Lemma 9.5, we infer that 𝑇 ∗ 𝑇 is self-adjoint. Finally, let (𝜁, 𝑇𝜁) ∈ G 𝑇 be orthogonal to the graph of the operator 𝑇|D 𝑇 ∗ 𝑇 . Then, for any 𝜉 ∈ 𝐷𝑇 ∗ 𝑇 , (𝜁|(1 + 𝑇 ∗ 𝑇)𝜉) = (𝜁|𝜉) + (𝑇𝜁|𝑇𝜉) = 0, and since (1 + 𝑇 ∗ 𝑇)D 𝑇 ∗ 𝑇 = H , it follows that 𝜁 = 0. Consequently, we have 𝑇|D 𝑇 ∗ 𝑇 = 𝑇.

Q.E.D.

For any closed linear operator 𝑇, its absolute value (or modulus) |𝑇| is defined by |𝑇| = (𝑇 ∗ 𝑇)1/2 . 9.29. The notion of partial isometry extends to the case of two different Hilbert spaces: it is a bounded linear operator 𝑣 ∶ H → K, such that ‖𝑣𝜉‖ = ‖𝜉‖, 𝜉 ∈ r(𝑣)H . If 𝑣 is a partial isometry, then 𝑣∗ is also a partial isometry, 𝑣∗ 𝑣 = r(𝑣), 𝑣𝑣 ∗ = l(𝑣). A bounded linear operator 𝑣 ∶ H → K is a partial isometry iff 𝑣∗ 𝑣 is a projection. The following theorem extends Theorem 2.14: Theorem (of polar decomposition). Let 𝑇 be a closed linear operator from H into K . Then there exists a positive self-adjoint linear operator 𝐴 in H and a partial isometry 𝑣 ∶ H → K , such that 𝑇 = 𝑣𝐴, 𝑣∗ 𝑣 = s(𝐴). These conditions determine in a unique manner the operators 𝐴 and 𝑣. Moreover 𝐴 = |𝑇|, 𝑣∗ 𝑣 = r(𝑇), 𝑣𝑣 ∗ = l(𝑇). Proof. In accordance with Corollary 9.14, we have |𝑇|2 = 𝑇 ∗ 𝑇. Hence, for any 𝜉 ∈ D 𝑇 ∗ 𝑇 , ‖|𝑇|𝜉‖2 = (|𝑇|2 𝜉|𝜉) = (𝑇 ∗ 𝑇𝜉‖) = ‖𝑇𝜉‖2 . Consequently, the relations 𝑣(|𝑇|𝜉) = 𝑇𝜉, 𝜉 ∈ D 𝑇 ∗ 𝑇 , 𝑣(𝜂) = 0,

𝜂 ∈ (|𝑇|D 𝑇 ∗ 𝑇 )⟂

determine a partial isometry 𝑣 ∶ H → K . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Let 𝜉 ∈ D |𝑇| . By taking into account Proposition 9.28, we infer that there exists a sequence {𝜉𝑛 } ⊂ D 𝑇 ∗ 𝑇 , such that 𝜉𝑛 → 𝜉,

|𝑇|𝜉𝑛 → |𝑇|𝜉.

Since 𝑇 is closed and 𝑇𝜉𝑛 = 𝑣(|𝑇|𝜉𝑛 ) → 𝑣(|𝑇|𝜉), it follows that 𝜉 ∈ D 𝑇,

𝑇𝜉 = 𝑣(|𝑇|𝜉).

Moreover, since ‖𝑣(|𝑇|𝜉)‖ = ‖|𝑇|𝜉‖, it follows that |𝑇|𝜉 ∈ 𝑣 ∗ 𝑣H . Thus, 𝑣∗ 𝑣H = |𝑇|D 𝑇 ∗ 𝑇 ⊂ |𝑇|D |𝑇| ⊂ 𝑣 ∗ 𝑣H , 𝑣∗ 𝑣H = |𝑇|D |𝑇| , i.e. 𝑣∗ 𝑣 = s(|𝑇|). Let now 𝜉 ∈ D 𝑇 . In accordance with Proposition 9.28, there exists a sequence {𝜉𝑛 } ⊂ D 𝑇 ∗ 𝑇 , such that 𝜉𝑛 → 𝜉, 𝑇𝜉𝑛 → 𝑇𝜉. Since ‖|𝑇|𝜉𝑛 − |𝑇|𝜉𝑚 ‖ = ‖𝑇𝜉𝑛 − 𝑇𝜉𝑚 ‖, the sequence {|𝑇|𝜉𝑛 } is convergent. Since |𝑇| is closed, 𝜉 ∈ D |𝑇| . We have thus already proved that D |𝑇| = D 𝑇 , 𝑇𝜉 = 𝑣(|𝑇|𝜉), 𝜉 ∈ D 𝑇 . Consequently, we have 𝑇 = 𝑣|𝑇|. Let 𝐴′ be a positive self-adjoint operator in H and 𝑣′ ∶ H → K a partial isometry, such that 𝑇 = 𝑣 ′ 𝐴′ , 𝑣′∗ 𝑣′ = s(𝐴′ ). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Then 𝑣′∗ 𝑣′ 𝐴′ = 𝐴′ , and therefore, |𝑇|2 = 𝑇 ∗ 𝑇 = 𝐴′ 𝑣′∗ 𝑣′ 𝐴′ = (𝐴′ )2 . From Corollary 9.14, we infer that 𝐴′ = |𝑇|. On the other hand, the relations 𝑣′ (|𝑇|𝜉) = 𝑣 ′ 𝐴′ 𝜉 = 𝑇𝜉, 𝑣′ (𝜂) = 0,

𝜉 ∈ D 𝑇,

𝜂 ∈ (|𝑇|D |𝑇| )⟂ = (𝐴′ D 𝐴′ )⟂

show that 𝑣′ is determined by 𝑇 in a unique manner.

Q.E.D.

Let 𝑇 be a closed linear operator in H and let 𝑇 = 𝑣|𝑇|,

𝑣 ∗ 𝑣 = s(|𝑇|)

be its polar decomposition. If 𝑇 is affiliated to a von Neumann algebra M ⊂ B (H ), then 𝑣 ∈ M and |𝑇| is affiliated to M . In particular, r(𝑇), l(𝑇) ∈ M . 9.30. Let 𝑇 be a closed linear operator from H into K and let 𝑇 = 𝑣|𝑇|,

𝑣 ∗ 𝑣 = s(|𝑇|)

be its polar decomposition. Since the operator 𝑣|𝑇|𝑣 ∗ is self-adjoint and positive, from 𝑇 ∗ = 𝑣 ∗ (𝑣|𝑇|𝑣∗ ), 𝑣𝑣 ∗ = s(𝑣|𝑇|𝑣 ∗ ), we infer that these two relations yield the polar decomposition of 𝑇 ∗ . In particular, |𝑇 ∗ | = 𝑣|𝑇|𝑣 ∗ , and 𝑇 = |𝑇 ∗ |𝑣,

𝑣𝑣 ∗ = s(|𝑇 ∗ |).

Corollary 9.31. Let 𝐴 be a self-adjoint linear operator in H . Then there exist two positive, self-adjoint linear operators 𝐴+ , 𝐴− in H , such that 𝐴 = 𝐴+ − 𝐴− , s(𝐴+ )s(𝐴− ) = 0. Moreover, these conditions determine the operators 𝐴+ , 𝐴− in a unique manner. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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259

Proof. Let 𝐴 = 𝑣|𝐴| the polar decomposition of 𝐴. In accordance with Section 9.30, the polar decomposition of 𝐴∗ is 𝐴∗ = 𝑣 ∗ (𝑣|𝐴|𝑣 ∗ ). Since 𝐴 = 𝐴∗ , from the uniqueness of the polar decomposition, we infer that 𝑣 = 𝑣∗ |𝐴| = 𝑣|𝐴|𝑣 ∗ . Consequently, 𝑣 = 𝑒 − 𝑓, where 𝑒, 𝑓 ∈ B (H ) are projections, 𝑒𝑓 = 0 and 𝐴 = (𝑒 − 𝑓)|𝐴| = |𝐴|(𝑒 − 𝑓). Since 𝑒 + 𝑓 = 𝑣 ∗ 𝑣 = s(𝐴), it follows that |𝐴| = (𝑒 + 𝑓)|𝐴| = |𝐴|(𝑒 + 𝑓). Consequently, 1 (|𝐴| + 𝐴) = 𝑒|𝐴| = |𝐴|𝑒|D 𝐴 , 2 1 (|𝐴| − 𝐴) = 𝑓|𝐴| = |𝐴|𝑓|D 𝐴 . 2 In particular, 𝑒D 𝐴 ⊂ D 𝐴 ,

𝑓D 𝐴 ⊂ D 𝐴 .

We define 𝐴+ = |𝐴|𝑒, 𝐴− = |𝐴|𝑓. It is easy to see that 𝐴+ and 𝐴− are positive operators. For any 𝜂 ∈ H , there exists a 𝜉 ∈ D 𝐴 , such that 𝜂 = 𝜉 + |𝐴|𝜉. Then 𝜁 = 𝑒𝜉 + (1 − 𝑒)𝜂 ∈ D 𝐴+ and 𝜁 + 𝐴+ 𝜁 = 𝑒𝜉 + (1 − 𝑒)𝜂 + |𝐴|𝑒𝜉 = 𝑒(𝜉 + |𝐴|𝜉) + (1 − 𝑒)𝜂 = 𝜂. Then, in accordance with Lemma 9.5, 𝐴+ is self-adjoint. In an analogous manner, one shows that 𝐴− is self-adjoint. Obviously, 𝐴 ⊂ 𝐴+ − 𝐴− . On the other hand, D (𝐴+ −𝐴− ) = {𝜉 ∈ H ; 𝑒𝜉 ∈ D 𝐴 , 𝑓𝜉 ∈ D 𝐴 } ⊂ {𝜉 ∈ H ; (𝑒 − 𝑓)𝜉 ∈ D 𝐴 } = D (|𝐴|(𝑒−𝑓)) = D 𝐴 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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Thus 𝐴 = 𝐴+ − 𝐴− . Finally, since s(𝐴+ ) ≤ 𝑒 and s(𝐴− ) ≤ 𝑓, we have s(𝐴+ ) s (𝐴− ) = 0. The uniqueness part of the corollary can be easily obtained from the uniqueness of the polar decomposition of 𝐴. Q.E.D. If 𝐴 is a self-adjoint linear operator in H , which is affiliated to a von Neumann algebra M ⊂ B (H ), then 𝐴+ and 𝐴− are affiliated to M . 9.32. In this section, we indicate a method for extending the operational calculus, defined for positive self-adjoint operators (9.9), to arbitrary self-adjoint operators. We shall denote by B ((−∞, +∞)) the *-algebra of all Borel measurable complex functions, defined on (−∞, +∞), which are bounded on compact subsets. For any 𝑓 ∈ B ((−∞, +∞)), we define the function f ̂ ∈ B ((−∞, +∞)) by the relation ̂ = 𝑓(−𝜆), f(𝜆)

𝜆 ∈ ℝ.

Let 𝐴 be a self-adjoint linear operator in H . For any 𝑓 ∈ B ((−∞, +∞)), we define ̂ (0,+∞) )(𝐴− ) + 𝑓(0)(1 − s(𝐴)). 𝑓(𝐴) = (𝑓𝜒(0,+∞) )(𝐴+ ) + (f𝜒 For the operational calculus with self-adjoint operators, defined in this manner, it is easy to verify that properties, analogous to those already established for the case of the positive self-adjoint operators, hold, too (see 9.11–9.13). 9.33. In the final sections of this chapter, we shall study the tensor product of linear operators. Let 𝑇𝑗 be a linear operator from H 𝑗 into K 𝑗 , 𝑗 = 1, 2. We define a linear operator 𝑇1 ⊗𝑇2 from H 1 ⊗H 2 into K 1 ⊗K 2 , by the following relations: D 𝑇1 ⊗𝑇2 = the vector subspace of H 1 ⊗H 2 generated by {𝜉1 ⊗𝜉2 ; 𝜉1 ∈ D 𝑇1 , 𝜉2 ∈ D 𝑇2 }, (𝑇1 ⊗ 𝑇2 )(𝜉1 ⊗ 𝜉2 ) = (𝑇1 𝜉1 ) ⊗ (𝑇2 𝜉2 ),

𝜉1 ∈ D 𝑇1 , 𝜉2 ∈ D 𝑇2 .

If the operator 𝑇1 ⊗ 𝑇2 is preclosed, then one denotes 𝑇1 ⊗𝑇2 = 𝑇1 ⊗ 𝑇2 , and the operator 𝑇1 ⊗𝑇2 is called the tensor product of the operators 𝑇1 and 𝑇2 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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For example, let us assume that the operators 𝑇1 and 𝑇2 are closed. Then the operators 𝑇1 , 𝑇2 , 𝑇1∗ , 𝑇2∗ are densely defined, and for any 𝜉1 ∈ D 𝑇1 , 𝜉2 ∈ D 𝑇2 , 𝜂1 ∈ D 𝑇1∗ , 𝜂2 ∈ D 𝑇2∗ , we have ((𝑇1 ⊗ 𝑇2 )(𝜉1 ⊗ 𝜉2 )|𝜂1 ⊗ 𝜂2 ) = (𝜉1 ⊗ 𝜉2 |(𝑇1∗ ⊗ 𝑇2∗ )(𝜂1 ⊗ 𝜂2 )), whence we immediately infer that the operator 𝑇1 ⊗ 𝑇2 is preclosed, and therefore, it makes sense to consider the closed operator 𝑇1 ⊗𝑇2 . Obviously, if 𝑇1 and 𝑇2 are bounded operators, then 𝑇1 ⊗𝑇2 is bounded, whereas if 𝑇1 ∈ B (H 1 ) and 𝑇2 ∈ B (H 2 ), then 𝑇1 ⊗𝑇2 coincides with the tensor product already defined in Section 2.33. Proposition. Let 𝑇𝑗 be a closed linear operator from H 𝑗 into K 𝑗 , 𝑗 = 1, 2. Then (𝑇1 ⊗𝑇2 )∗ = 𝑇1∗ ⊗𝑇2∗ . Proof. From the preceding argument, which allowed the definition of 𝑇1 ⊗𝑇2 , we also infer the relation: 𝑇1∗ ⊗𝑇2∗ ⊂ (𝑇1 ⊗𝑇2 )∗ . Let (𝜎, 𝜏) ∈ G (𝑇1 ⊗𝑇2 )∗ ,

(𝜎, 𝜏) ⟂ G (𝑇 ∗ ⊗𝑇 ∗ ) . 1

2

Then, for any 𝜉1 ∈ D 𝑇1 , 𝜉2 ∈ D 𝑇2 , we have (𝑇1 𝜉1 ⊗ 𝑇2 𝜉2 |𝜎) = (𝜉1 ⊗ 𝜉2 |𝜏), and for any 𝜂1 ∈ D 𝑇1∗ , 𝜂2 ∈ D 𝑇2∗ , (𝜂1 ⊗ 𝜂2 |𝜎) + (𝑇1∗ 𝜂1 ⊗ 𝑇2∗ 𝜂2 |𝜏) = 0. Consequently, for any 𝜉1 ∈ D 𝑇1∗ 𝑇1 , 𝜉2 ∈ D 𝑇2∗ 𝑇2 , we have ((1 + (𝑇1∗ 𝑇1 ⊗𝑇2∗ 𝑇2 ))(𝜉1 ⊗ 𝜉2 )|𝜏) = (𝜉1 ⊗ 𝜉2 |𝜏) + (𝑇1∗ 𝑇1 𝜉1 ⊗𝑇2∗ 𝑇2 𝜉2 |𝜏) = (𝑇1 𝜉1 ⊗𝑇2 𝜉2 |𝜎) + (𝑇1∗ (𝑇1 𝜉1 )⊗𝑇2∗ (𝑇2 𝜉2 )|𝜏) = 0. Consequently, we have 𝜏 ⟂ (1 + (𝑇1∗ 𝑇1 ⊗𝑇2∗ 𝑇2 ))D 𝑇 ∗ 𝑇1 ⊗𝑇 ∗ 𝑇2 . 1

In accordance with Proposition 9.28, the operators self-adjoint. If we write, for any natural number 𝑛,

2

𝑇1∗ 𝑇1

and 𝑇2∗ 𝑇2 are positive and

𝑒𝑛1 = 𝜒(1/(1+𝑛),+∞) ((1 + 𝑇1∗ 𝑇1 )−1 ), 𝑒𝑛2 = 𝜒(1/(1+𝑛),+∞) ((1 + 𝑇2∗ 𝑇2 )−1 ), Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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then the operators (𝑇1∗ 𝑇1 ⊗𝑇2∗ 𝑇2 )(𝑒𝑛1 ⊗𝑒𝑛2 ) = (𝑒𝑛1 ⊗𝑒𝑛2 )(𝑇1∗ 𝑇1 ⊗𝑇2∗ 𝑇2 )(𝑒𝑛1 ⊗𝑒𝑛2 ) = (𝑇1∗ 𝑇1 𝑒𝑛1 )⊗(𝑇2∗ 𝑇2 𝑒𝑛2 ) are defined everywhere, bounded and positive. Hence ∞

⋃

(𝑒𝑛1 ⊗𝑒𝑛2 )(H 1 ⊗H 2 )

⋃

(𝑒𝑛1 ⊗𝑒𝑛2 )(1 + (𝑇1∗ 𝑇1 ⊗𝑇2∗ 𝑇2 ))(𝑒𝑛1 ⊗𝑒𝑛2 )(H 1 ⊗H 2 )

⋃

(1 + (𝑇1∗ 𝑇1 ⊗𝑇2∗ 𝑇2 ))(𝑒𝑛1 ⊗𝑒𝑛2 )(H 1 ⊗H 2 )

𝑛=1 ∞

=

𝑛=1 ∞

=

𝑛=1

⊂ (1 + (𝑇1∗ 𝑇1 ⊗𝑇2∗ 𝑇2 ))D 𝑇 ∗ 𝑇1 ⊗𝑇 ∗ 𝑇2 . 1

2

Consequently, (1 + (𝑇1∗ 𝑇1 ⊗𝑇2∗ 𝑇2 ))D 𝑇 ∗ 𝑇1 ⊗𝑇 ∗ 𝑇2 1

2

is a dense vector subspace of H . From the preceding argument, we infer that 𝜏 = 0. Since, for any 𝜂1 ∈ D 𝑇1∗ , 𝜂2 ∈ D 𝑇2∗ , (𝜂1 ⊗𝜂2 |𝜎) = −(𝑇1∗ 𝜂1 ⊗𝑇2∗ 𝜂2 |𝜏) = 0, it follows that 𝜎 = 0. Consequently, we have G (𝑇1 ⊗𝑇2 )∗ = D (𝑇 ∗ ⊗𝑇 ∗ ) ; hence 1

2

(𝑇1 ⊗𝑇2 )∗ = 𝑇1∗ ⊗𝑇2∗ . Q.E.D. 9.34. By taking into account Proposition 9.33 and Corollary 9.14, one obtains the following: Corollary. If 𝑇1 and 𝑇2 are self-adjoint (resp., positive self-adjoint) linear operators, then 𝑇1 ⊗𝑇2 is a self-adjoint (resp., positive self-adjoint) linear operator. 9.35. Let H and K be Hilbert spaces. One says that 𝑆 is an antilinear operator from H into K if 𝑆 is a mapping from a vector subspace D 𝑆 ⊂ H into K , such that 𝑆(𝜉 + 𝜂) = 𝑆𝜉 + 𝑆𝜂, 𝜉, 𝑆(𝜆𝜉) = 𝜆𝑆𝜉,

𝜂 ∈ D 𝑆,

𝜆 ∈ ℂ, 𝜉 ∈ D 𝑆 .

Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:47:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.012

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The notions and the results from Sections 9.1–9.3, 9.28, 9.30, and 9.33 obviously extend to the antilinear operators. The antilinear operators will be used often in Chapter 10. For this reason, we formulate below some statements about antilinear operators, which we have proved for linear operators. The product of two antilinear operators is linear. The product of an antilinear operator by a linear operator is antilinear. The adjoint 𝑆 ∗ of a densely defined antilinear operator 𝑆 is an antilinear operator, and it is defined by the relations (𝑆𝜉|𝜂) = (𝑆 ∗ 𝜂|𝜉),

𝜉 ∈ D 𝑆,

𝜂 ∈ D 𝑆∗ .

If 𝑆 is preclosed, then 𝑆 ∗ is closed and 𝑆 ∗∗ = 𝑆. If 𝑆 is closed, then 𝑆 ∗ 𝑆 is a positive self-adjoint linear operator. If 𝑆 is a closed antilinear operator from H into K , then there exist a positive self-adjoint linear operator 𝐴, in H , and an antilinear partial isometry 𝑣 ∶ H → K , such that 𝑆 = 𝑣𝐴, ∗

𝑣 𝑣 = s(𝐴). These conditions determine in a unique manner the operators 𝐴 and 𝑣, and the above relations give the polar decomposition of 𝑆. An antilinear operator 𝐽 ∶ H → H is called a conjugation if 𝐽 = 𝐽 ∗ = 𝐽 −1 . Conjugations are the antilinear analogues of the self-adjoint unitary operators. EXERCISES In the exercises in which the symbols H and K are not explained they will denote Hilbert spaces. E.9.1 Let 𝑇 be a closed linear operator in H , and D a vector subspace of D 𝑇 . The following assertions are equivalent: (i) 𝑇|D = 𝑇. (ii) (1 + |𝑇|)D is dense in H . E.9.2 Let 𝑇1 and 𝑇2 be linear operators from H into K 1 and K 2 , respectively. Show that if 𝑇1 is closed, 𝑇2 is preclosed and D 𝑇1 ⊂ D 𝑇2 , then there exists a constant 𝑐 > 0, such that ‖𝑇2 𝜉‖2 ≤ 𝑐(‖𝑇1 𝜉‖2 + ‖𝜉‖2 ),

𝜉 ∈ D 𝑇1 . ▶

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E.9.3 Let 𝑇 be a closed linear operator from H into K . With the help of the Hahn–Banach and Banach–Steinhauss theorems, show that the following assertions are equivalent: (i) 𝑇(D 𝑇 ) = K . (ii) there exists a constant 𝑐 > 0, such that ‖𝜂‖ ≤ 𝑐‖𝑇 ∗ 𝜂‖,

𝜂 ∈ D 𝑇∗ .

By assuming that condition (ii) is satisfied, show that for any 𝜉 ∈ H , 𝜉 ⟂ n(𝑇), there exists an 𝜂 ∈ D 𝑇 ∗ , such that 𝑇 ∗ 𝜂 = 𝜉 and ‖𝜂‖ ≤ 𝑐‖𝜉‖. E.9.4 Let 𝑇 be a linear operator in H . Show that if D 𝑇 = H , and D 𝑇 ∗ = H , then 𝑇 ∈ B (H ). E.9.5 Let 𝑇 and 𝑆 be normal linear operators in H . Show that if 𝑇 ⊂ 𝑆, then 𝑇 = 𝑆. Infer from this result that any normal symmetric operator is self-adjoint. E.9.6 Let 𝐴 be a symmetric linear operator in H . Then 𝐴 is self-adjoint iff (𝑖 + 𝐴)D 𝐴 = H . E.9.7 Let 𝐴 be a linear operator in H . Show that the following assertions are equivalent: (1) 𝐴 is self-adjoint. (2) For any non-zero 𝑡 ∈ ℝ, we have i𝑡 ∈ 𝜌(𝐴) and 1 ‖(i𝑡 + 𝐴)−1 ‖ ≤ . |𝑡| (Hint: by assuming that (2) is satisfied, show that ‖𝐴𝜉‖2 ≥ 2𝑡 Im (𝐴𝜉|𝜉),

𝑡 ∈ ℝ,

whence (𝐴𝜉|𝜉) ∈ ℝ; if 𝜁 ∈ D 𝐴 and 𝜉 = (i + 𝐴)−1 𝜁, then ‖𝜁‖2 = ‖𝐴𝜉‖2 + i(𝐴𝜉|𝜉), whence 𝜁 = 0). E.9.8 Let 𝐴 be a symmetric linear operator in H . Then the linear operator (𝐴 + i) is injective; hence one can define a linear operator 𝑉(𝐴) in H by 𝑉(𝐴)𝜉 = (𝐴 − i)(𝐴 + i)−1 𝜉,

𝜉 ∈ D 𝑉(𝐴) = (𝐴 + i)D 𝐴 .

Show that 𝑉(𝐴) is an isometric linear operator, i.e. ‖𝑉(𝐴)𝜉‖ = ‖𝜉‖,

𝜉 ∈ D 𝑉(𝐴) ,

and that (1 − 𝑉(𝐴))D 𝑉(𝐴) is a dense vector subspace of H . ▶

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Let 𝑉 be an isometric linear operator in H , such that the vector subspace (1 − 𝑉)D 𝑉 , is dense in H . Then the linear operator (1 − 𝑉) is injective; hence one can define a linear operator 𝐴(𝑉) in H by 𝐴(𝑉)𝜉 = i(1 + 𝑉)(1 − 𝑉)−1 𝜉,

𝜉 ∈ D 𝑉(𝐴) = (1 − 𝑉)D 𝑉 .

Show that 𝐴(𝑉) is a symmetric linear operator. With the foregoing notations, we have (1) 𝐴(𝑉(𝐴)) = 𝐴, 𝑉(𝐴(𝑉)) = 𝑉. (2) 𝐴 is closed iff 𝑉(𝐴) is closed, 𝑉 is closed iff 𝐴(𝑉) is closed. (3) 𝐴 is self-adjoint iff 𝑉(𝐴) is unitary, 𝑉 is unitary iff 𝐴(𝑉) is self-adjoint. (4) 𝐴1 ⊂ 𝐴2 iff 𝑉(𝐴1 ) ⊂ 𝑉(𝐴2 ), 𝑉1 ⊂ 𝑉2 iff 𝐴(𝑉1 ) ⊂ 𝐴(𝑉2 ). The operator 𝑉(𝐴) is called the Cayley transform of the symmetric linear operator 𝐴. E.9.9 Let 𝑆 be a closed symmetric operator in H . Show that there exist an isometry 𝑣 of H into a Hilbert space K , and a self-adjoint operator 𝐴 in K , such that D 𝑆 = {𝜉 ∈ H ; 𝑣𝜉 ∈ D 𝐴 }, 𝑆 = 𝑣 ∗ 𝐴𝑣𝜉,

𝜉 ∈ D 𝑆.

E.9.10 Let 𝐴 be a positive self-adjoint operator in H . For any 𝜆 ∈ (0, +∞), we define the spectral projection 𝑒𝜆 = 𝜒[0,𝜆) (𝐴). Show that (i) (ii) (iii) (iv) (v)

𝑒𝜆 ∈ R ({(1 + 𝐴)−1 }). 𝜆1 ≤ 𝜆2 ⇒ 𝑒𝜆1 ≤ 𝑒𝜆2 . 𝜆𝑛 ↑ 𝜆 ⇒ 𝑒𝜆𝑛 ↑ 𝑒𝜆 . 𝑒𝜆 ↓𝜆→0 1 − s(𝐴), 𝑒𝜆 ↑𝜆→∞ 1. 𝐴𝑒𝜆 ≤ 𝜆𝑒𝜆 , 𝐴(1 − 𝑒𝜆 ) ≥ 𝜆(1 − 𝑒𝜆 ). ▶

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∞

(vi) D 𝐴 = {𝜉 ∈ H ; ∫0 𝜆2 𝑑(𝑒𝜆 𝜉|𝜉) < +∞}, and for any 𝜉 ∈ D 𝐴 , we have ∞

𝐴𝜉 = ∫ 𝜆 𝑑𝑒𝜆 𝜉, 0

with norm convergent vector Stieltjes integral. These assertions make up the contents of the so-called spectral theorem for the positive self-adjoint operator 𝐴, and the family of projections {𝑒𝜆 }𝜆∈[0,+∞) is called the spectral scale of 𝐴. E.9.11 Let 𝐴 be a positive self-adjoint operator in H and 𝑓 ∈ B ([0, +∞)). Show that ∞ D 𝑓(𝐴) = {𝜉 ∈ H ; ∫ |𝑓(𝜆)|2 𝑑(𝑒𝜆 𝜉|𝜉) < ∞} 0

and that for any 𝜉 ∈ D 𝑓(𝐴) , ∞

𝑓(𝐴)𝜉 = ∫ 𝑓(𝜆) 𝑑𝑒𝜆 𝜉, 0

with a norm convergent vector Stieltjes integral. E.9.12 Let 𝐴 be a self-adjoint operator in H and 𝐴 = 𝐴+ − 𝐴− the decomposition given by Corollary 9.31. We define the spectral scale of 𝐴 by + if 𝜆 > 0, ⎧ 𝜒[0,𝜆] (𝐴 ), − 𝑒𝜆 = s(𝐴 ), if 𝜆 = 0, ⎨ − ⎩ 𝜒[−𝜆,+∞] (𝐴 ), if 𝜆 < 0

Extend to this case the assertions from exercises E.9.10 and E.9.11. In this manner, one obtains the spectral theorem and the integral formula of the operational calculus for self-adjoint operators. E.9.13 Let 𝐴 be a positive self-adjoint operator in H . For any Borel measurable subset 𝐷 of the spectrum of 𝐴, we define the spectral projection of 𝐴, which corresponds to 𝐷, by the formula 𝑒(𝐷) = 𝜒𝐷 (𝐴). Then 𝑒(𝜎(𝐴)) = 1. Show that for any 𝑓 ∈ B ([0, +∞)), we have ‖𝑓(𝐴)‖ =

inf

sup |𝑓(𝜆)|

𝐷⊂𝜍(𝐴), 𝑒(𝐷)=1 𝜆∈𝐷

▶

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and 𝜎(𝑓(𝐴)) =

{𝑓(𝜆); 𝜆 ∈ 𝐷} ⊂ {𝑓(𝜆); 𝜆 ∈ 𝜎(𝐴)}.

⋂

𝐷⊂𝜍(𝐴), 𝑒(𝐷)=1

E.9.14 Let 𝐴 be a positive self-adjoint operator in H and 𝑓 ∈ B ([0, +∞)), real and positive. Then, for any 𝑔 ∈ B ([0, +∞]), we have 𝑔(𝑓(𝐴)) = (𝑔 ∘ 𝑓)(𝐴). Infer from this result a new proof for the uniqueness of the positive square root of 𝐴 (cf. Corollary 9.14). E.9.15 Let 𝐴 be a positive self-adjoint operator in H and H 𝜆 the range of the projection 𝜒[0,𝜆] (𝐴), 𝜆 ∈ [0, +∞). Show that, for any 𝜆 ∈ [0, +∞), ∞

H 𝜆 = {𝜉 ∈

⋂

D 𝐴𝑛 ; lim ‖𝐴𝑛 𝜉‖1/𝑛 ≤ 𝜆} . 𝑛→∞

𝑛=1

Infer from this result a new proof for the uniqueness of the positive square root of 𝐴. E.9.16 Let 𝐴 be a positive self-adjoint operator in H , such that s(𝐴) = 1, and let ∞ 0 ≠ 𝜉 ∈ ⋂𝑛=−∞ D 𝐴𝑛 . Then lim ‖𝐴𝑛 𝜉‖1/𝑛 lim ‖𝐴−𝑚 𝜉‖1/𝑚 ≥ 1,

𝑛→∞

𝑚→∞

and the equality holds iff 𝐴𝜉 = ( lim ‖𝐴𝑛 𝜉‖1/𝑛 )𝜉. 𝑛→∞

(Hint: one can use Corollary 9.15 and the ‘theorem of the three lines’, from Dunford and Schwartz (1958/1963/1970), VI, 10.3.) E.9.17 For any function 𝑓 ∈ L 1 (ℝ), we denote by 𝑓 ̂ the ‘inverse Fourier transform’ of 𝑓: +∞

̂ =∫ 𝑓(𝑠)

𝑓(𝑡)𝑒i𝑡s 𝑑𝑡,

s ∈ ℝ.

−∞

With the same hypotheses and notations as in E.9.15, show that, for any 𝜆 ∈ [0, +∞), H 𝜆 = {𝜉 ∈ H ; for 𝑓 ∈ L 1 (ℝ), such that support 𝑓 ̂ ⊂ (ln 𝜆, +∞), +∞

we have ∫

𝑓(𝑡)𝐴i𝑡 𝑑𝑡 = 0} .

−∞

▶

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E.9.18 For any linear operator 𝑇 in H and any 𝜉 ∈ D 𝑇 , we denote 𝐸𝜉 (𝑇) = (𝑇𝜉|𝜉), 𝜎𝜉 (𝑇) = ‖(𝑇 − 𝐸𝜉 (𝑇)𝜉‖. Let 𝐴 and 𝐵 be self-adjoint operators in H , such that the intersection D = D 𝐴𝐵 ∩ D 𝐵𝐴 be dense in H . Show that 1 𝜎𝜉 (𝐴)𝜎𝜉 (𝐵) ≥ |𝐸𝜉 (𝐴𝐵 − 𝐵𝐴)|, 2

𝜉 ∈ D.

This inequality is a variant of Heisenberg’s ‘uncertainty principle’. E.9.19 Let 𝐴 be a self-adjoint operator in H . Show that the following assertions are equivalent: (i) 𝐴 is positive. (ii) 𝜎(𝐴) ⊂ [0, +∞). E.9.20 One says that a linear operator 𝑇 in H commutes with an operator 𝑥 ∈ B (H ) if 𝑥𝑇 ⊂ 𝑇𝑥. Show that if 𝑇 is closed, then the set {𝑇}′ of all operators 𝑥 ∈ B (H ), which commute with 𝑇 is a wo-closed subalgebra of B (H ) and ({𝑇}′ )∗ = {𝑇 ∗ }′ . In particular, if 𝑇 is self-adjoint, then {𝑇}′ is a von Neumann algebra (we mention the fact that {𝑇}′ is a von Neumann algebra even if 𝑇 is normal, this being an extension of Fuglede’s theorem 2.31). E.9.21 Let 𝑇 be a closed linear operator in H and 𝑥 ∈ B (H ) an operator that commutes with 𝑇. Show that 𝑥𝑇 = 𝑇𝑥. E.9.22 Let 𝑇 be a closed linear operator and 𝑇 = 𝑣|𝑇| its polar decomposition. Show that if 𝑇 is normal, then l(𝑇) = r(𝑇) and 𝑣|𝑇| = |𝑇|𝑣. Conversely, if l(𝑇) = r(𝑇) and if |𝑇| commutes with 𝑣, then 𝑇 is normal. ▶

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E.9.23 Let 𝐴 be a positive self-adjoint operator in H and 𝑥 ∈ B (H ). Then the following assertions are equivalent: (i) (ii) (iii) (iv)

𝐴 commutes with 𝑥. (1 + 𝐴)−1 commutes with 𝑥. 𝑒𝜆 commutes with 𝑥, 𝜆 ∈ (0, +∞). 𝐴i𝑡 commutes with 𝑥, 𝑡 ∈ ℝ.

Show that if 𝐴 commutes with 𝑥, then, for any 𝑓 ∈ B ([0, +∞)), the operator 𝑥𝑓(𝐴) is preclosed and 𝑥𝑓(𝐴) = 𝑓(𝐴)𝑥. Show that if 𝑒 is a projection that commutes with 𝐴 and if 𝑓 ∈ B ([0, +∞)), 𝑓(0) = 0, then 𝑓(𝐴𝑒) = 𝑓(𝐴)𝑒. E.9.24 Let 𝐴 and 𝐵 be positive self-adjoint operators in H and {𝑒𝜆 } (resp., {𝑓𝜇 }) the spectral scale of 𝐴 (resp., 𝐵). Show that the following assertions are equivalent: (i) (1 + 𝐴)−1 commutes with (1 + 𝐵)−1 . (ii) 𝑒𝜆 commutes with 𝑓𝜇 , 𝜆, 𝜇 ∈ (0, +∞). (iii) 𝐴i𝑡 commutes with 𝐵i𝑠 ∶ 𝑡, 𝑠 ∈ ℝ. In this case, one says that 𝐴 and 𝐵 commute. !E.9.25 Let M ⊂ B (H ) be a von Neumann algebra and 𝐴 a positive self-adjoint operator in H . Show that the following assertions are equivalent: (i) (ii) (iii) (iv) (v) (vi)

𝐴 is affiliated to M . (1 + 𝐴)−1 ∈ M . 𝑒𝜆 ∈ M , 𝜆 ∈ (0, +∞). 𝐴i𝑡 ∈ M , 𝑡 ∈ ℝ. 𝐴 commutes with any 𝑥 ′ ∈ M ′ . For any 𝑥 ′ ∈ M ′ , we have 𝑥 ′ (D 𝐴 ) ⊂ D 𝐴 and (𝐴𝑥 ′ 𝜉|𝐴𝜂) = (𝐴𝜂|𝐴𝑥 ′∗ 𝜂),

𝜉, 𝜂 ∈ D 𝐴 .

E.9.26 Let M ⊂ B (H ) be a finite von Neumann algebra and O (𝐴) the set of all closed linear operators in H , which are affiliated to M . ▶

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Let 𝑇, 𝑆 ∈ O (M ). With the help of the polar decomposition (9.28, 9.29) and of Corollary 7.6, show that 𝑇 + 𝑆 and 𝑇𝑆 are densely defined. Then, with the help of the inclusions 𝑇 + 𝑆 ⊂ (𝑇 ∗ + 𝑆 ∗ )∗ , 𝑇𝑆 ⊂ (𝑆 ∗ 𝑇 ∗ )∗ , show that 𝑇 + 𝑆 and 𝑇𝑆 are preclosed. Infer from these results and from Theorem 9.8, that O (M ) is a *-algebra for the operations (𝑇, 𝑆) ↦ (𝑇 + 𝑆),

(𝜆, 𝑇) ↦ 𝜆𝑇,

(𝑇, 𝑆) ↦ 𝑇𝑆 𝑇 ↦ 𝑇 ∗. !E.9.27 Let M ⊂ B (H ) be a von Neumann algebra and 𝐵 a positive operator in H , which is affiliated to M . Show that the Friedrichs extension of 𝐵 is affiliated to M . E.9.28 Let 𝐴 be a positive operator in H . Then the following assertions are equivalent: (i) 𝐴 is self-adjoint. (ii) 𝐵 is a positive linear operator in H , 𝐵 ⊃ 𝐴 ⇒ 𝐵 = 𝐴. E.9.29 Let 𝑇 be a closed linear operator in H . Then, with the scalar product (𝜉|𝜂)𝑇 = (𝜉|𝜂) + (𝑇𝜉|𝑇𝜂), D 𝑇 becomes a Hilbert space. Show that D 𝑇 = D ((1+|𝑇|2 )1/2 ) , and that (𝜉|𝜂)𝑇 = ((1 + |𝑇|2 )1/2 𝜉|(1 + |𝑇|2 )1/2 )𝜂),

𝜉, 𝜂 ∈ D 𝑇 .

E.9.30 Let 𝐵 be a positive operator in H , 𝐴 its Friedrichs extension and D ={

𝜉 ∈ H ; there exists {𝜉𝑛 } ⊂ D 𝐵 , such that 𝜉𝑛 → 𝜉 and (𝐵(𝜉𝑛 − 𝜉𝑚 )|𝜉𝑛 − 𝜉𝑚 ) → 0

}

Show that D = D (𝐴1/2 ) . (Hint: D (𝐴1/2 ) = D ((1+𝐴)1/2 ) is a Hilbert space for the scalar product (𝜉, 𝜂) ↦ ((1 + 𝐴)1/2 𝜉|(1 + 𝐴)1/2 𝜂), and D is a closed vector subspace of the latter.) ▶

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E.9.31 Let 𝐵 be a positive operator in H and 𝐴 its Friedrichs extension. Show that any self-adjoint extension of 𝐵, whose domain of definition is included in D (𝐴1/2 ) , coincides with 𝐴. E.9.32 Let M ⊂ B (H ) be a von Neumann algebra and 𝜉, 𝜂 ∈ H . The following assertions are equivalent: (i) There exists a closed linear operator 𝑇 ′ in H , which is affiliated to M ′ , such that 𝜂 = 𝑇 ′ 𝜉. (ii) There exists a positive self-adjoint operator 𝐴′ in H , which is affiliated to M ′ , such that 𝜔𝜂 = 𝜔𝐴′ 𝜉 . Moreover, the operator 𝐴′ from condition (ii) can be chosen in such a manner that s(𝐴′ ) ≤ 𝑝𝜉′ . E.9.33 Let M be a von Neumann algebra and 𝜑, 𝜓 normal forms on M . We denote by 𝜋𝜑 ∶ M ↦ B (H 𝜑 ) (resp., 𝜋𝜓 ∶ M ↦ B (H 𝜓 )) the *-representation associated to the form 𝜑 (resp., to the form 𝜓), whose corresponding cyclic vector is 1𝜑 (resp., 1𝜓 ) (see 5.18). (1) The correspondence H 𝜓 ⊃ 𝜋𝜓 (M )1𝜓 ∋ 𝜋𝜓 (𝑥)1𝜓 ↦ 𝜋𝜑 (𝑥)1𝜑 ∈ H 𝜑 is a correctly defined linear operator iff s(𝜑) ≤ s(𝜓). (2) The correspondence H 𝜓 ⊃ 𝜋𝜓 (M )1𝜓 ∋ 𝜋𝜓 (𝑥)1𝜓 ↦ 𝜋𝜑 (𝑥)1𝜑 ∈ H 𝜑 is a preclosed linear operator iff the implication 𝑥𝑛 ∈ M ,

𝜓(|𝑥𝑛 |2 ) → 0,

𝜑(|𝑥𝑛 − 𝑥𝑚 |2 ) → 0 ⇒ 𝜑(|𝑥𝑛 |2 ) → 0

holds. In this case, one says that 𝜑 is almost dominated by 𝜓. (3) The correspondence H 𝜓 ⊃ 𝜋𝜓 (M )1𝜓 ∋ 𝜋𝜓 (𝑥)1𝜓 ↦ 𝜋𝜑 (𝑥)1𝜑 ∈ H 𝜑 ▶

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is a bounded linear operator iff there exists a 𝜆 > 0, such that 𝜑 ≤ 𝜆𝜓. In this case, one says that 𝜑 is dominated by 𝜓. E.9.34 Let {𝜑𝑘 }𝑘 be a sequence of normal forms on the von Neumann algebra M , which are almost dominated by the normal form 𝜓 on M , such that ∞ ∞ ∑𝑘=1 𝜑𝑘 (1) < +∞. Show that 𝜑 = ∑𝑘=1 𝜑𝑘 is a normal form on M , which is almost dominated by 𝜓. E.9.35 Let M ⊂ B (H ) be a von Neumann algebra, 𝜑 a normal form on M , and 𝜉 ∈ H . Then the following assertions are equivalent: (i) There exists a positive self-adjoint operator 𝐴′ in H , which is affiliated to M ′ , such that 𝜑 = 𝜔𝐴′ 𝜉 . (ii) 𝜑 is almost dominated by 𝜔𝜉 . (Hint for the implication (ii) ⇒ (i): one can assume 𝜉 to be cyclic and separating, and in this case, H 𝜔𝜉 identifies with [M 𝜉] = H ; one denotes by 𝑇 ′ the closure of the operator that one obtains from E.9.33 (2) for 𝜓 = 𝜔𝜉 ; if 𝑇 ′ = 𝑣 ′ 𝐴′ is the polar decomposition of 𝑇 ′ , show that 𝐴′ is affiliated to M ′ , with the help of exercise E.9.25 (vi)). E.9.36 Let 𝐴 and 𝐵 be positive self-adjoint operators in H , such that s(𝐴) = s(𝐵) = 1. For any 𝜀 > 0, we consider the set D 𝜀 of all operators 𝑥 ∈ B (H ), such that the mapping i𝑡 ↦ 𝐴i𝑡 𝑥𝐵−i𝑡 has a w-continuous extension to {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝜀}. If 𝑥 ∈ D 𝜀 , the preceding extension is unique, and we denote it by 𝐹𝑥 . We define the operator 𝑇 in B (H ) by 𝑇(𝑥) = 𝐹𝑥 (1),

𝑥 ∈ D 𝑇 = D 1.

Show that, for any 𝜆 > 0, the operator 𝜆 + 𝑇 is injective. Show that, for any 𝑥 ∈ D 𝑇 and any 𝜆 > 0, 𝑇𝑥 ∈ D (𝜆+𝑇)−1 and 𝑐+i∞

1 𝜆−𝛼 (𝜆 + 𝑇) 𝑇𝑥 = ∫ 𝐹 (𝛼) 𝑑𝛼, 2i 𝑐−i∞ sin 𝜋𝛼 𝑥 −1

0 < 𝑐 < 1.

By making in this formula 𝑐 = 1/2, infer Proposition 9.23. ▶

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Prove that for 𝑥 ∈ D 𝑇 and 𝛼 ∈ ℂ; 0 < Re 𝛼 < 1, the following inversion formula holds: ∞

𝐹𝑥 (𝛼) =

sin 𝜋𝛼 ∫ 𝜆𝛼−1 (𝜆 + 𝑇)−1 𝑇𝑥 𝑑𝜆. 𝜋 0

E.9.37 Let 𝐴 be a positive self-adjoint operator in H , such that s(𝐴) = 1. Show that for any 𝜉 ∈ D 𝐴 and 𝛼 ∈ ℂ; 0 < Re 𝛼 < 1, we have ∞

𝐴𝛼 𝜉 =

sin 𝜋𝛼 ∫ 𝜆𝛼−1 (𝜆 + 𝐴)−1 𝐴𝜉 𝑑𝜆. 𝜋 0

COMMENTS C.9.1 In our presentation of the theory, we have developed the operational calculus only for positive self-adjoint operators, since this is the essential tool in Chapter 10 and in most of operator algebra theory. In the case of arbitrary self-adjoint operators, one can make an analogous construction, by replacing the transform 𝑎 = (1 + 𝐴)−1 (9.9) by the Cayley transform (E.9.8) and using the operational calculus for unitary operators. Another method is that which was indicated in Section 9.32 (see also exercise E.9.12). As in the case of bounded operators, one can develop an operational calculus for normal operators, for which we refer the reader to C. Ionescu Tulcea (1956), F. Riesz and B. Sz.-Nagy (1952/1953/1954/1965), N. Dunford and J. Schwartz (1958/1963/1970), R. P. Halmos (1951), B. Sz.-Nagy (1942/1967), M. A. Naimark (1956/1968). An analytic operational calculus for closed operators in Banach spaces was also developed (see Dunford and Schwartz [1958/1963/1970], Chap. VII, § 9). From the theory of self-adjoint extensions of symmetric operators, we presented only the theorem of Friedrichs. This theory has important applications in the theory of differential operators, and its basic results are exposed in F. Riesz and B. Sz.-Nagy (1952/1953/1954/1965), N. Dunford and J. Schwartz (1958/1963/1970), M. G. Krein (1947), as well as in M. A. Naimark’s book on linear differential operators.

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C.9.2 The aim of the theory of generators of one-parameter groups of operators is to characterize these groups with the help of a single mathematical object, which is usually an unbounded operator. The theory of generators uses differential and integral calculus techniques, Fourier analysis and complex analysis and is usually developed for operators in Banach spaces. Let {ᵆ𝑡 }𝑡∈ℝ be a strongly continuous one-parameter group of operators in a Banach space H . The classical infinitesimal generator 𝐺 of {ᵆ𝑡 } is defined in the following manner: 1 D 𝐺 = {𝜉 ∈ H ; lim (ᵆ𝜀 𝜉 − 𝜉) exists} 𝜀→0 𝜀 1 𝐺𝜉 = = lim (ᵆ𝜀 𝜉 − 𝜉), 𝜉 ∈ D 𝐺 . 𝜀→0 𝜀 One proves that 𝐺 is a closed linear operator, and in a certain sense, ᵆ𝑡 is the exponential of 𝑡𝐺, 𝑡 ∈ ℝ. If H is a Hilbert space and if ᵆ𝑡 are unitary operators, then 𝐵 = −i𝐺 is a self-adjoint operator and ᵆ𝑡 = exp(i𝑡𝐵),

𝑡 ∈ ℝ,

where the exponential is meant in the sense of the operational calculus. This is the form in which Stone’s representation theorem is usually stated. For details regarding the theory of the infinitesimal generator, which also applies to the one-parameter semigroups, see E. Hille and R. Phillips (1957) and N. Dunford and J. Schwartz (1958/1963/1970). Another type of generator, which is more suitable for the applications we have in mind in Chapter 10, can be defined with the help of the analytic continuation. More precisely, we define the analytic generator 𝐴 of ᵆ𝑡 by D 𝐴 = {𝜉 ∈ H | the mapping i𝑡 ↦ ᵆ𝑡 𝜉 has an extension to 𝐹𝜉 , which is continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1} and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1}. } 𝐴𝜉 = 𝐹𝜉 (1), 𝜉 ∈ D 𝐴 . One shows that 𝐴 is a closed linear operator and that ᵆ𝑡 is the (i𝑡)th power of 𝐴, in the sense of V. Balakrishnan’s ‘fractional powers’. If H is a Hilbert space and if ᵆ𝑡 are unitary operators, then Theorem 9.20 shows that 𝐴 is self-adjoint and positive and ᵆ𝑡 = 𝐴i𝑡 ,

𝑡 ∈ ℝ.

For details regarding the theory of the analytic generator, see I. Ciorǎnescu and L. Zsidó (1976), and L. Zsidó (1977).

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We note that Propositions 9.23 and 9.24 and exercises E.9.36 and E.9.37 are particular cases of some results of the theory of the analytic generator. C.9.3 Bibliographical comments. Corollary 9.15 was first mentioned by G. K. Pedersen and M. Takesaki (1973, Lemma 3.2), and it lies at the basis of the methods of analytic continuation that we use in Chapters 9 and 10. Proposition 9.23 is due to A. van Daele (1974), and it is the principal argument in the proof given by him to the fundamental theorem of M. Tomita (10.12). The other material, concerning the theory of operators, is classic. Theorem 9.8 and exercise E.9.26 are due to F. J. Murray and J. von Neumann (1936).

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THE THEORY OF STANDARD VON NEUMANN ALGEBRAS

10

In the preceding chapters, we have presented that part of the theory of operator algebras which is based on the original ideas of F. J. Murray and J. von Neumann. A turning point in the development of the theory of von Neumann algebras was produced in 1967 by M. Tomita. In this chapter, we present Tomita’s theory, which enables us to obtain canonical forms for the von Neumann algebras, which are called standard von Neumann algebras. Introduction. The simplest general example is considered in Section 10.6 where it is assumed that the von Neumann algebra M ⊂ B (H ) has a cyclic and separating vector 𝜉0 ∈ H . For this reason, we begin with an Introduction where this case is considered separately with several simplifications with respect to the general case and we also prove a specific result for this situation. 1∘ . Consider the conjugate linear operators 𝑆0 ∶ M 𝜉0 ∋ 𝑥𝜉0 ↦ 𝑥 ∗ 𝜉0 ∈ M 𝜉0 ⊂ H ; D 𝑆0 = M 𝜉0 , 𝐹0 ∶ M ′ 𝜉0 ∋ 𝑥 ′ 𝜉0 ↦ 𝑥 ′∗ 𝜉0 ∈ M ′ 𝜉0 ⊂ H ; D 𝐹0 = M ′ 𝜉0 . Then 𝑆0 ⊂ 𝐹0∗ , i.e. (𝑆0 𝜉 ∣ 𝜂) = (𝐹0 𝜂 ∣ 𝜉),

(∀)𝜉 ∈ D 𝑆0 , 𝜂 ∈ D 𝐹0 ,

i.e. (𝑆0 𝑥𝜉0 ∣ 𝑥 ′ 𝜉0 ) = (𝐹0 𝑥 ′ 𝜉0 ∣ 𝑥𝜉0 ),

(∀)𝑥 ∈ M , 𝑥 ′ ∈ M ′ ,

which is obvious: (𝑆0 𝑥𝜉0 ∣ 𝑥 ′ 𝜉0 ) = (𝑥 ∗ 𝜉0 ∣ 𝑥 ′ 𝜉0 ) = (𝜉0 ∣ 𝑥𝑥 ′ 𝜉0 ) = (𝜉0 ∣ 𝑥 ′ 𝑥𝜉0 ) = (𝑥 ′∗ 𝜉0 ∣ 𝑥𝜉0 ) = (𝐹0 𝑥 ′ 𝜉0 ∣ 𝑥𝜉0 ). 276

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Since every adjoint operator is closed, it follows that 𝑆0 is preclosed. Let 𝑆 = 𝑆0 denote its closure: 𝑆 ∶ D 𝑆 → H , 𝑆|M 𝜉0 = 𝑆, 𝑆𝑥𝜉0 = 𝑥 ∗ 𝜉0 (𝑥 ∈ M ). Similarly, 𝐹 ∶ D 𝐹 → H , 𝐹|M ′ 𝜉0 = 𝐹, 𝐹𝑥 ′ 𝜉0 = 𝑥 ′∗ 𝜉0

(𝑥′ ∈ M ′ )

and from 𝑆0 ⊂ 𝐹0∗ , 𝐹0 ⊂ 𝑆0∗ , it follows that 𝑆 ⊂ 𝐹∗,

𝐹 ⊂ 𝑆∗.

Also, it is obvious that 𝑆02 ⊂ 𝐼 and this yields 𝑆 2 ⊂ 𝐼, i.e. 𝜉 ∈ D 𝑆 ⇒ 𝑆𝜉 ∈ D 𝑆 and 𝑆𝑆𝜉 = 𝜉. Let us recall the standard proof. Since 𝜉 ∈ D (𝑆) and 𝑆 = 𝑆|M 𝜉0 , there is a sequence {𝑥𝑛 } ⊂ M , such that 𝑥𝑛 𝜉0 → 𝜉 and 𝑥𝑛∗ 𝜉0 = 𝑆𝑥𝑛 𝜉0 → 𝑆𝜉. We read this as 𝑥𝑛∗ 𝜉0 → 𝑆𝜉 and 𝑆𝑥𝑛∗ 𝜉0 = 𝑥𝑛 𝜉0 → 𝜉, and since 𝑆 is closed, it follows that 𝑆𝜉 ∈ D 𝑆 and 𝑆(𝑆𝜉) = 𝜉. Moreover, 𝑆 is invertible and 𝑆 −1 = 𝑆. Consider now the modular operator Δ = 𝑆 ∗ 𝑆. By the Theorem 9.28, Δ is a positive selfadjoint linear operator and 𝑆|D ∆ = 𝑆. This enables us to consider the polar decomposition (9.29): 𝑆 = 𝐽Δ1/2 . Since 𝑆 = 𝑆 −1 , Δ is a nonsingular operator and 𝐽Δ1/2 = 𝑆 = 𝑆 −1 = Δ−1/2 𝐽 −1 ; hence Δ−1/2 = 𝐽Δ1/2 𝐽 = 𝐽 2 (𝐽 ∗ Δ1/2 𝐽) and by the uniqueness of the polar decomposition, 𝐽 2 = 𝐼, i.e. 𝐽 = 𝐽 ∗ = 𝐽 −1 , i.e. 𝐽 is a conjugation, the canonical conjugation. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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We also have 𝑆 = 𝐽Δ1/2 = Δ−1/2 𝐽, 𝑆 ∗ = 𝐽Δ−1/2 = Δ1/2 𝐽,

D 𝑆 = D ∆1/2 , D 𝑆 ∗ = D ∆−1/2 .

Notice that 𝑆𝜉0 = 𝜉0 = 𝑆 ∗ 𝜉0 , Δ𝜉0 = 𝜉0 so Δ1/2 𝜉0 = 𝜉0 ; hence 𝐽𝜉0 = 𝜉0 . Moreover, for any Borel function 𝑓 ∶ [0, ∞) → ℂ, we have 𝐽𝑓(Δ)𝐽 = 𝑓(Δ−1 ) since 𝐽Δ𝐽 = Δ−1 and 𝐽(𝜆Δ)𝐽 = 𝜆 𝐽Δ𝐽 = 𝜆Δ−1 , and we conclude by functional calculus. In particular, 𝐽Δi𝑡 = Δi𝑡 𝐽 (𝑡 ∈ ℝ, i2 = −1). Tomita’s Theorem: 𝐽M 𝐽 = M ′ and Δi𝑡 M Δ−i𝑡 = M , (∀)𝑡 ∈ ℝ. Both statements will be proved simultaneously by a method – as in the general case – due to Alfons van Daele (1976c). 2∘ . The problem at this primary stage is to construct some ‘bridge’ between M 𝜉0 and M ′ 𝜉0 (once we prove 𝐽M 𝐽 = M ′ , it follows that M ′ 𝜉0 = 𝐽M 𝐽𝜉0 = 𝐽M 𝜉0 , so 𝐽 is the ‘good’ bridge, but first we shall find a more primitive ‘wood’ bridge). This is based on the following: Lemma (S. Sakai). Let M be a von Neumann algebra and 𝜑, 𝜓 ∈ M + ∗ normal positive forms, such that 𝜓 ≤ 𝜑, and let 𝜆 ∈ ℂ with 𝜆 + 𝜆 = 1. There exists 𝑎 ∈ 𝑀, 0 ≤ 𝑎 ≤ 1, such that 𝜓(𝑥) = 𝜑 (𝜆𝑎𝑥 + 𝜆𝑥𝑎), (∀)𝑥 ∈ M . Proof. The set X = {𝜑(𝜆𝑎 ⋅ +𝜆 ⋅ 𝑎); 𝑎 = 𝑎∗ ∈ M , ‖𝑎‖ ≤ 1} is a convex subset of M ℎ∗ and 𝜎(M ∗ , M )-compact. If 𝜓 ∈ X , then by Hahn–Banach theorem, there exist 𝑏 ∈ M , 𝑏 = 𝑏∗ and 𝑡 ∈ ℝ with 𝜓(𝑏) > 𝑡 while 𝜑(𝜆𝑎𝑏 + 𝜆𝑏𝑎) ≤ 𝑡, (∀)𝑎 = 𝑎∗ ∈ M , ‖𝑎‖ ≤ 1. Let 𝑏 = 𝑣|𝑏| be the polar decomposition. Then 𝑣 ∈ M , 𝑣 = 𝑣 ∗ , ‖𝑣‖ ≤ 1 and |𝑏| = 𝑣𝑏 = 𝑏𝑣; hence 𝑡 < 𝜓(𝑏) ≤ 𝜓(|𝑏|) ≤ 𝜑(|𝑏|) = 𝜑(𝜆𝑣𝑏 + 𝜆𝑏𝑣) ≤ 𝑡, Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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The theory of standard von Neumann algebras which is a contradiction. Thus, there exists 𝑎1 ∈ M , 𝑎1∗ = 𝑎1 , ‖𝑎1 ‖ ≤ 1, such that (∀)𝑥 ∈ M .

𝜓(𝑥) = 𝜑(𝜆𝑎1 𝑥 + 𝜆𝑥𝑎1 ), Since 𝜑 ⩾ 0, 𝜓 ⩾ 0, it follows that 2

𝜓(𝑎1− ) = 𝜑 (− (𝑎1− ) ) = 0; hence 𝜑(𝑎1− 𝑥) = 0 = 𝜑(𝑥𝑎1− ) by the Cauchy–Schwarz inequality, and consequently 𝑎 = 𝑎1+ ∈ M , 0 ≤ 𝑎 ≤ 1 and 𝜓(𝑥) = 𝜑(𝜆𝑎𝑥 + 𝜆𝑥𝑎),

(∀)𝑥 ∈ M . Q.E.D.

3∘ . Lemma. For 𝜔 ∈ ℂ, |𝜔| = 1, 𝜔 ≠ −1, we have (Δ + 𝜔)−1 M ′ 𝜉0 ⊂ M 𝜉0 . Proof. It suffices to consider 𝑎′ ∈ M ′ , 0 ≤ 𝑎′ ≤ 1 and to show that (Δ + 𝜔)−1 𝑎′ 𝜉0 ∈ M 𝜉0 . To this end, consider 𝜆 = (1 + 𝜔)−1 ∈ ℂ, notice that 𝜆 + 𝜆 = 1 and consider 𝜑, 𝜓 ∈ M + ∗ defined by 𝜑(𝑥) = (𝑥𝜉0 ∣ 𝜉0 ), 𝜓(𝑥) = (𝑥𝜉0 ∣ 𝑎′ 𝜉0 ), (𝑥 ∈ M ). It is clear that 0 ≤ 𝜓 ≤ 𝜑 so, by the Sakai’s Lemma, there exists 𝑎 ∈ M , 0 ≤ 𝑎 ≤ 1, such that, for all 𝑥 ∈ M , (𝑥𝜉0 ∣ 𝑎′ 𝜉0 ) = 𝜓(𝑥) = 𝜑(𝜆𝑎𝑥 + 𝜆𝑥𝑎) = (𝜆𝑎𝑥𝜉0 ∣ 𝜉0 ) + (𝜆𝑥𝑎𝜉0 ∣ 𝜉0 ) = (𝑥𝜉0 ∣ 𝜆𝑎𝜉0 ) + (𝑎𝜉0 ∣ 𝜆𝑥 ∗ 𝜉0 ), i.e. (𝑥𝜉0 ∣ 𝑎′ 𝜉0 − 𝜆𝑎𝜉0 ) = (𝜆𝑎𝜉0 ∣ 𝑆𝑥𝜉0 ),

(∀)𝑥 ∈ M ,

[read it as (𝜉 ∣ 𝜁) = (𝜂 ∣ 𝑆𝜉), for all 𝜉 ∈ D 𝑆 ], which means that [𝜂 ∈ D 𝑆 ∗ , 𝑆 ∗ 𝜂 = 𝜁]: 𝑎′ 𝜉0 − 𝜆𝑎𝜉0 = 𝑆 ∗ (𝜆𝑎𝜉0 ) = 𝜆𝑆 ∗ (𝑎𝜉0 ) = 𝜆𝑆 ∗ 𝑆𝑎𝜉0 = 𝜆Δ𝑎𝜉0 , i.e. 𝑎′ 𝜉0 = (𝜆 + 𝜆Δ)𝑎𝜉0 = (Δ + 𝜔)(1 + 𝜔)−1 𝑎𝜉0 , (Δ + 𝜔)−1 𝑎′ 𝜉0 = (1 + 𝜔)−1 𝑎𝜉0 ∈ M 𝜉0 . Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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4∘ . We shall also need the following approximation result: Lemma. Let 𝜉 ∈ D ∆1/2 ∩ D ∆−1/2 . There is a sequence 𝜉𝑛 = (Δ + 1)−1 M ′ 𝜉0 ⊂ M 𝜉0 ∩ (D ∆1/2 ∩ D ∆−1/2 ) , such that 𝜉𝑛 → 𝜉,

Δ1/2 𝜉𝑛 → Δ1/2 𝜉,

Δ−1/2 𝜉𝑛 → Δ−1/2 𝜉.

Proof. Since 𝐽Δ−1/2 M ′ 𝜉0 = 𝑆 ∗ M ′ 𝜉0 = 𝐹M ′ 𝜉0 = M ′ 𝜉0 is dense in H , we see that Δ−1/2 M ′ 𝜉0 is dense in H ; hence there exist 𝜂𝑛 ∈ M ′ 𝜉0 , such that Δ−1/2 𝜂𝑛 → Δ1/2 𝜉 + Δ−1/2 𝜉. Consider then 𝜉𝑛 = (1 + Δ)−1 𝜂𝑛 ∈ (1 + Δ)−1 M ′ 𝜉0 ⊂ M 𝜉0

(by Lemma 3∘ ).

Notice that 𝜉𝑛 ∈ 𝑀𝜉0 ⊂ D 𝑆 = D ∆1/2 and M ′ 𝜉0 ⊂ D 𝑆 ∗ = D ∆−1/2 , so that 𝜉𝑛 = (1 + Δ)−1 𝜂𝑛 = Δ−1/2 (1 + Δ−1 )−1 Δ−1/2 𝜂𝑛 and 𝜉𝑛 ∈ D ∆−1/2 since Δ−1 (1 + Δ−1 )−1 is bounded. We have 𝜉𝑛 = (1 + Δ)−1 𝜂𝑛 = Δ1/2 (1 + Δ−1 )Δ−1/2 𝜂𝑛 → Δ1/2 (1 + Δ−1 )(Δ1/2 𝜉 + Δ−1/2 𝜉) = ... = 𝜉, Δ−1/2 𝜉𝑛 = (1 + Δ)−1 Δ−1/2 𝜂𝑛 → (1 + Δ)−1 (Δ1/2 𝜉 + Δ−1/2 𝜉) = ... = Δ−1/2 𝜉, Δ1/2 𝜉𝑛 = Δ(1 + Δ−1 )Δ−1/2 𝜂𝑛 → Δ(1 + Δ)−1 (Δ1/2 𝜉 + Δ−1/2 𝜉) = ... = Δ1/2 𝜉. Q.E.D. 5∘ . Lemma. Let 𝑥 ′ ∈ M ′ and 𝜔 ∈ ℂ, |𝜔| = 1, 𝜔 ≠ −1. Then (1)

(𝑥′ 𝜂 ∣ 𝜁) = (𝐽𝑥 ∗ 𝐽Δ−1/2 𝜂 ∣ Δ1/2 𝜁) + 𝜔(𝐽𝑥 ∗ 𝐽Δ1/2 𝜂 ∣ Δ−1/2 𝜁)

for any 𝜂, 𝜁 ∈ D ∆1/2 ∩ D ∆−1/2 where 𝑥 ∈ M is given by 𝑥 ′ 𝜉0 = (Δ + 𝜔)𝑥𝜉0

(by Lemma 3∘ ).

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281

Proof. By the Approximation Lemma 4∘ , it suffices to consider 𝜂 = (Δ + 1)−1 𝑦 ′ 𝜉0 = 𝑦𝜉0

(𝑦 ′ ∈ M ′ , 𝑦 ∈ M ),

𝜁 = (Δ + 1)−1 𝑧′ 𝜉0 = 𝑧𝜉0

(𝑧′ ∈ M ′ , 𝑧 ∈ M ).

Since 𝑥 ′ 𝜉0 = Δ𝑥𝜉0 + 𝜔𝑥𝜉0 = 𝑆 ∗ 𝑆𝑥𝜉0 + 𝜔𝑥𝜉0 , we have (𝑥′ 𝜂 ∣ 𝜁) = (𝑥 ′ 𝑦𝜉0 ∣ 𝑧𝜉0 ) = (𝑦𝑥 ′ 𝜉0 ∣ 𝑧𝜉0 ) = (𝑦𝑆 ∗ 𝑆𝑥𝜉0 ∣ 𝑧𝜉0 ) + 𝜔(𝑦𝑥𝜉0 ∣ 𝑧𝜉0 ) = (𝑆𝑦 ∗ 𝑧𝜉0 ∣ 𝑆𝑥𝜉0 ) + 𝜔(𝑦𝑥𝜉0 ∣ 𝑧𝜉0 ) = (𝑧∗ 𝑦𝜉0 ∣ 𝑥 ∗ 𝜉0 ) + 𝜔(𝑦𝑥𝜉0 ∣ 𝑧𝜉0 ). But we want to get back 𝜂 = 𝑦𝜉0 and 𝜁 = 𝑧𝜉0 . Thus (𝑧∗ 𝑦𝜉0 ∣ 𝑥 ∗ 𝜉0 ) = (𝑦𝜉0 ∣ 𝑧𝑥 ∗ 𝜉0 ) = (𝑦𝜉0 ∣ 𝑆𝑥𝑧∗ 𝜉0 ) = (𝑦𝜉0 ∣ 𝑆𝑥𝑆𝑧𝜉0 ) = (𝑦𝜉0 ∣ Δ−1/2 𝐽𝑥𝐽Δ1/2 𝑧𝜉0 ) = (𝐽𝑥 ∗ 𝐽Δ−1/2 𝑦𝜉0 ∣ Δ1/2 𝑧𝜉0 ) = (𝐽𝑥 ∗ 𝐽Δ−1/2 𝜂 ∣ Δ1/2 𝜁), (𝑦𝑥𝜉0 ∣ 𝑧𝜉0 ) = (𝑆𝑥 ∗ 𝑦 ∗ 𝜉0 ∣ 𝑧𝜉0 ) = (𝑆𝑥 ∗ 𝑆𝑦𝜉0 ∣ 𝑧𝜉0 ) = (Δ−1/2 𝐽𝑥 ∗ 𝐽Δ1/2 𝑦𝜉0 ∣ 𝑧𝜉0 ) = (𝐽𝑥 ∗ 𝐽Δ1/2 𝑦𝜉0 ∣ Δ−1/2 𝑧𝜉0 ) = (𝐽𝑥 ∗ 𝐽Δ1/2 𝜂 ∣ Δ−1/2 𝜁). These prove (1). 6∘ . Thus, we started with arbitraries 𝑥 ′ ∈ M ′ and 𝜔 ∈ ℂ, |𝜔| = 1, 𝜔 ≠ −1, and we obtained 5∘ (1) with 𝑥 ∈ M , such that 𝑥′ 𝜉0 = (Δ + 𝜔)𝑥𝜉0 . Consider now another arbitrary 𝑦 ′ ∈ M ′ . By Proposition 9.23 (see also the remark at the end of Section 9.23) from Lemma 5∘ , we get ∞

i𝑡−

1

𝜔 2 𝐽𝑥 ∗ 𝐽 = ∫ 𝜋𝑡 Δ−i𝑡 𝑥 ′ Δi𝑡 𝑑𝑡, e + e−𝜋𝑡 −∞

hence

∞

i𝑡−

1

𝜔 2 𝑥 = ∫ 𝜋𝑡 𝐽Δ−i𝑡 𝑥 ′ Δi𝑡 𝐽 𝑑𝑡, e + e−𝜋𝑡 ∗

−∞

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i𝑡−

1

𝜔 2 𝑥 ∗ 𝑦 ′ 𝜉0 = ∫ 𝜋𝑡 𝐽Δ−i𝑡 𝑥 ′ Δi𝑡 𝐽𝑦 ′ 𝜉0 𝑑𝑡. e + e−𝜋𝑡 −∞

In contrast, by using the Corollary 9.23, 𝑥 ∗ 𝑦 ′ 𝜉0 = 𝑦 ′ 𝑥 ∗ 𝜉0 = 𝑦 ′ 𝑆𝑥𝜉0 = 𝑦 ′ 𝐽Δ1/2 (Δ + 𝜔)−1 𝑥 ′ 𝜉0 ∞

i𝑡−

1

𝜔 2 = ∫ 𝜋𝑡 𝑦 ′ 𝐽Δ−i𝑡 𝑥 ′ 𝜉0 𝑑𝑡. e + e−𝜋𝑡 −∞

It follows that ∞

i𝑡−

1

𝜔 2 𝐹(𝜔) ∶= ∫ 𝜋𝑡 (𝐽Δ−i𝑡 𝑥 ′ Δi𝑡 𝐽𝑦 ′ 𝜉0 − 𝑦 ′ Δ−i𝑡 𝑥 ′ 𝜉0 ) 𝑑𝑡 = 0 e + e−𝜋𝑡 −∞

for every 𝜔 ∈ ℂ, |𝜔| = 1, 𝜔 ≠ −1. This equality extends by analyticity to all 𝜔 ∈ ℂ\ (−∞; 0), so it is valid for all 𝜔 ∈ [0; ∞). But this means that the Fourier transform of the continuous 𝐿1 -function 𝑡↦

e𝜋𝑡

1 (𝐽Δ−i𝑡 𝑥 ′ Δi𝑡 𝐽𝑦 ′ 𝜉0 − 𝑦 ′ Δ−i𝑡 𝑥 ′ 𝜉0 ) + e−𝜋𝑡

vanishes and then, by the uniqueness theorem in Fourier Analysis, we get 𝐽Δ−i𝑡 𝑥 ′ Δi𝑡 𝐽𝑦 ′ 𝜉0 = 𝑦 ′ Δ−i𝑡 𝑥 ′ 𝜉0 ,

(1)

(∀)𝑥 ′ , 𝑦 ′ ∈ M ′ .

7∘ . We now exploit this equality. For 𝑥 ′ , 𝑦 ′ , 𝑧′ ∈ M ′ , we get: 𝐽Δ−i𝑡 𝑥 ′ Δi𝑡 𝐽𝑦 ′ 𝑧′ 𝜉0 = 𝑦 ′ 𝑧′ Δ−i𝑡 𝑥 ′ 𝜉0 = 𝑦 ′ 𝐽Δ−i𝑡 𝑥′ Δi𝑡 𝐽𝑧′ 𝜉0 , and since M ′ 𝜉0 is dense in H , it follows that (𝐽Δ−i𝑡 𝑥 ′ Δi𝑡 𝐽)𝑦 ′ = 𝑦 ′ (𝐽Δ−i𝑡 𝑥′ Δi𝑡 𝐽),

(∀)𝑦 ′ ∈ M ′ ,

which, as (M ′ )′ = M , means that (1)

𝐽Δ−i𝑡 𝑥′ Δi𝑡 𝐽 ∈ M ,

(∀)𝑥 ′ ∈ M , (∀)𝑡 ∈ ℝ,

in particular (𝑡 = 0) (2)

𝐽M ′ 𝐽 ⊂ M .

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283

We want to show that also 𝐽M 𝐽 ⊂ M ′ . Let 𝑦, 𝑧 ∈ M and 𝑥 ′ ∈ M ′ . Then, using the above conclusion (2), we get (𝑧𝐽𝑦𝜉0 ∣ 𝑥 ′ 𝜉0 ) = (𝐽𝑦𝜉0 ∣ 𝑧∗ 𝑥′ 𝜉0 ) = (𝐽𝑦𝜉0 ∣ 𝑥 ′ 𝑧∗ 𝜉0 ) = (𝑥 ′∗ 𝐽𝑦𝜉0 ∣ 𝐽Δ1/2 𝑧𝜉0 ) = (Δ1/2 𝑧𝜉0 ∣ 𝐽𝑥 ′∗ 𝐽𝑦𝜉0 ) ! = (Δ1/2 𝑧𝜉0 ∣ 𝑆𝑦 ∗ 𝐽𝑥 ′ 𝐽𝜉0 ) = (Δ1/2 𝑧𝜉0 ∣ Δ−1/2 𝐽𝑦 ∗ 𝐽𝑥 ′ 𝜉0 ) = (𝑧𝜉0 ∣ 𝐽𝑦 ∗ 𝐽𝑥 ′ 𝜉0 ) = (𝐽𝑦𝐽𝑧𝜉0 ∣ 𝑥 ′ 𝜉0 ), and as M ′ 𝜉0 is dense in H , we conclude (∀)𝑦, 𝑧 ∈ M ;

𝑧 𝐽𝑦𝜉0 = 𝐽𝑦𝐽𝑧𝜉0 , hence, for 𝑥, 𝑦, 𝑧 ∈ M ,

𝐽𝑦𝐽𝑥𝑧𝜉0 = 𝑥𝑧 𝐽𝑦𝜉0 = 𝑥𝐽𝑦𝐽𝑧𝜉0 so that (𝐽𝑦𝐽)𝑥 = 𝑥(𝐽𝑦𝐽), which means that 𝐽𝑦𝐽 ∈ M ′ and therefore 𝐽M 𝐽 ⊂ M ′ . We thus get the first main conclusion: (3)

𝐽M 𝐽 = M ′

𝐽M ′ 𝐽 = M .

Note that if 𝑧 ∈ Z M = M ∩ M ′ and 𝑥 ∈ M is arbitrary, then 𝐽𝑧∗ 𝐽𝑥𝜉0 = 𝐽𝑧∗ 𝑥 ∗ 𝜉0 = 𝑥𝑧𝜉0 = 𝑧𝑥𝜉0 ; hence 𝐽𝑧∗ 𝐽 = 𝑧 for any 𝑧 ∈ Z M . Since 𝐽Δi𝑡 = Δi𝑡 𝐽(𝑡 ∈ ℝ), and 𝐽M ′ 𝐽 = M , from (1), we get now also the second main conclusion: (4)

Δi𝑡 M Δ−i𝑡 = M ,

(𝑡 ∈ ℝ).

8∘ . The Tomita’s Theorem (6.9) is now proved. The map 𝑗 ∶ M ∋ 𝑥 ↦ 𝑗(𝑥) = 𝐽𝑥 ∗ 𝐽 ∈ M ′ is a ∗-anti-isomorphism of M onto M ′ acting identically on the center Z M , and the mappings 𝜎𝑡 ∶ M ∋ 𝑥 ↦ 𝜎𝑡 (𝑥) = Δi𝑡 𝑥Δ−i𝑡 ∈ M Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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constitute a one-parameter group of ∗-isomorphisms of M , called the modular automorphism group of M associated to 𝜉0 . 9∘ . Assume again that M ⊂ B (H ) is a von Neumann algebra with a cyclic and separating vector 𝜉 ∈ H . With the notation introduced in 1∘ , we saw that 𝑆 ⊂ 𝐹 ∗ . The following result is specific for the case of the cyclic and separating vector 𝜉0 ∈ H : Theorem. 𝐹 ∗ = 𝑆. Proof. We know that 𝑆 ⊂ 𝐹 ∗ so we have to prove that 𝐹 ∗ ⊂ 𝑆, i.e. that every 𝜉 ∈ D 𝐹 ∗ belongs to D 𝑆 and 𝑆𝜉 = 𝐹 ∗ 𝜉. Since 𝑆 = 𝑆|M 𝜉0 , we have to show that there exists a sequence 𝑥𝑛 ∈ M , such that (1)

𝑥𝑛 𝜉0 → 𝜉,

𝑥𝑛∗ 𝜉0 = 𝑆𝑥𝑛 𝜉0 → 𝐹 ∗ 𝜉 =∶ 𝜂.

(a) We can define two linear operators 𝑋0 , 𝑌0 in H as follows: 𝑋0 ∶ M ′ 𝜉0 ∋ 𝑥 ′ 𝜉0 ↦ 𝑥 ′ 𝜉 ∈ H ,

D 𝑋0 = M ′ 𝜉0 ,

𝑌0 ∶ M ′ 𝜉0 ∋ 𝑥 ′ 𝜉0 ↦ 𝑥 ′ 𝜂 ∈ H ,

D 𝑌0 = M ′ 𝜉0 .

We check that 𝑋0 ⊂ 𝑌0∗ , i.e. (𝑋0 𝛼 ∣ 𝛽) = (𝛼 ∣ 𝑌0 𝛽),

(∀)𝛼 ∈ D 𝑋0 , 𝛽 ∈ D 𝑌0 ,

i.e. (𝑋0 𝑎′ 𝜉0 ∣ 𝑏′ 𝜉0 ) = (𝑎′ 𝜉0 ∣ 𝑌0 𝑏′ 𝜉0 ),

(∀)𝑎′ , 𝑏′ ∈ M ′ ,

and indeed, since 𝜂 = 𝐹 ∗ 𝜉, we have (𝑋0 𝑎′ 𝜉0 ∣ 𝑏′ 𝜉0 ) = (𝑎′ 𝜉 ∣ 𝑏′ 𝜉0 ) = (𝜉 ∣ 𝑎′∗ 𝑏′ 𝜉0 ) = (𝜉 ∣ 𝐹(𝑏′∗ 𝑎′ 𝜉0 )) = (𝑏′∗ 𝑎′ 𝜉0 ∣ 𝐹 ∗ 𝜉) = (𝑎′ 𝜉0 ∣ 𝑏′ 𝜂) = (𝑎′ 𝜉0 ∣ 𝑌0 𝑏′ 𝜉0 ). Thus 𝑋0 is preclosed and we may consider its closure: 𝑋 = 𝑋0 . On the other hand, 𝑋0 commutes with each 𝑥 ′ ∈ M ′ : 𝛼 ∈ D 𝑋0 ⇒ 𝑥 ′ 𝛼 ∈ D 𝑋0 and 𝑋0 𝑥′ 𝛼 = 𝑥 ′ 𝑋0 𝛼

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The theory of standard von Neumann algebras since for every 𝑎′ ∈ M ′ , we have 𝑥 ′ 𝑎′ 𝜉0 ∈ M ′ 𝜉0 and 𝑋0 𝑥 ′ 𝑎′ 𝜉0 = 𝑥 ′ 𝑎′ 𝜉 = 𝑥 ′ 𝑋0 𝑎′ 𝜉0 .

It follows that also the closure 𝑋 of 𝑋0 commutes with each 𝑥 ′ ∈ M ′ , i.e. 𝑋 is affiliated to (M ′ )′ = M . Clearly, 𝜉0 ∈ D 𝑋0 , 𝑋𝜉0 = 𝜉.

(2)

Also 𝜉0 ∈ D 𝑌0 and 𝑌0 ⊂ 𝑋0∗ = 𝑋 ∗ , so that 𝑋 ∗ 𝜉 = 𝜂.

(3)

Thus, (1) is ‘almost proved’ the only weakeness being that 𝑋 can be unbounded. (b) Consider the polar decompositions of the closed operator 𝑋: 𝑋 = 𝑢𝐴; 𝐴 = (𝑋 ∗ 𝑋)1/2 , 𝑢∗ 𝑢 = 𝑠(𝐴) = Range (𝑋 ∗ ) ∋ 𝜂, 𝑋 = 𝐵𝑢; 𝐵 = (𝑋𝑋 ∗ )1/2 , 𝑢𝑢∗ = 𝑠(𝐵) = Range (𝑋) ∋ 𝜉. Of course, 𝐵 = 𝑢𝐴𝑢∗ ; hence, for any Borel function 𝜑 ∶ [0; ∞) → ℂ, 𝑢𝜑(𝐴) = 𝜑(𝐵)𝑢. Since 𝑋 is affiliated to M , we have 𝑢 ∈ M and 𝜑(𝐴), 𝜑(𝐵) ∈ M whenever 𝜑 is bounded. Note also that 𝑋 ∗ = 𝑢∗ 𝐵. Let 𝑒𝑛 = 𝜒( 1 ;𝑛) (𝐴), 𝑓𝑛 = 𝜒( 1 ;𝑛) (𝐵) and 𝜑(𝑡) = 𝑡𝜒( 1 ;𝑛) (𝑡) (𝑡 ≥ 0). Since 𝜉0 ∈ D 𝑋 , 𝑛

𝑋𝜉0 = 𝜉 and 𝑋 = 𝑢𝐴, we have

𝑛

𝑛

𝑢𝜑𝑛 (𝐴)𝜉0 = 𝑢𝐴𝑒𝑛 𝜉0 → 𝑢𝐴𝜉0 = 𝑋𝜉0 = 𝜉. Since 𝜉0 ∈ D 𝑋 ∗ , 𝑋 ∗ 𝜉0 = 𝜂 and 𝑋 ∗ = 𝑢∗ 𝐵, we have (𝑢𝜑𝑛 (𝐴))∗ 𝜉0 = 𝜑𝑛 (𝐴)𝑢∗ 𝜉0 = 𝑢∗ 𝜑𝑛 (𝐵)𝜉0 = 𝑢∗ 𝐵𝑓𝑛 𝜉0 → 𝑢∗ 𝐵𝜉0 = 𝑋 ∗ 𝜉0 = 𝜂. Hence (1) is satisfied with 𝑥𝑛 = 𝑢𝜑𝑛 (𝐴) ∈ M .

Q.E.D.

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The logical dependence of the sections in Chapter 10. 10.1 10.2

10.7

10.10

10.6

10.9

10.3

10.8

10.4

10.5

10.11 10.12

10.1

10.14

10.20

10.17

10.19 10.23 10.24 10.25

10.26

10.21 10.27

10.16(1−9)

10.22

10.15

10.16(10−11) 10.18

10.28 10.29

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10.1. Let 𝔄 be a complex algebra with involution, which is also endowed with a scalar product (⋅|⋅). We denote by 𝜉 ↦ 𝜉 ♯ the involution in 𝔄 and by H the Hilbert space obtained by the completion of 𝔄. We denote by 𝔄2 the vector space generated by the elements of the form 𝜉𝜂, 𝜉, 𝜂 ∈ 𝔄. One says that 𝔄 is a left Hilbert algebra if (i) 𝔄 ∋ 𝜂 ↦ 𝜉𝜂 ∈ 𝔄 is continuous, for any 𝜉 ∈ 𝔄. (ii) (𝜉𝜂1 |𝜂2 ) = (𝜂1 |𝜉 ♯ 𝜂2 ), for any 𝜉, 𝜂1 , 𝜂2 ∈ 𝔄. (iii) 𝔄2 is dense in 𝔄. (iv) H ⊃ 𝔄 ∋ 𝜉 ↦ 𝜉 ♯ ∈ H is a preclosed antilinear operator. In accordance with (i), for any 𝜉 ∈ 𝔄, one defines a 𝐿𝜉 ∈ B (H ) by the formula 𝐿𝜉 (𝜂) = 𝜉𝜂,

𝜂 ∈ 𝔄.

With the help of (ii), it is easy to see that the mapping 𝐿 ∶ 𝔄 ∋ 𝜉 ↦ 𝐿𝜉 ∈ B (H ) is a *-representation of 𝐴, i.e. 𝐿𝜉1 𝜉2 = 𝐿𝜉1 𝐿𝜉2 , (𝐿𝜉 )∗ = 𝐿𝜉 ♯ , We define

𝜉1 , 𝜉1 ∈ 𝔄,

𝜉 ∈ 𝔄.

𝔏(𝔄) = R ({𝐿𝜉 ; 𝜉 ∈ 𝔄}) = {𝐿𝜉 ; 𝜉 ∈ 𝔄}″ .

With the help of (iii), it is easy to see that 𝑠𝑜

𝔏(𝔄) = {𝐿𝜉 ; 𝜉 ∈ 𝔄} . Conditions (iii) and (iv) allow the definition of the closed antilinear operator 𝑆, as being the closure of the operator H ⊃ 𝔄2 ∋ ∑ 𝜉i 𝜂i ↦ (∑ 𝜉i 𝜂i )♯ ∈ H . i

i ∗

We recall that the adjoint antilinear operator 𝑆 is defined by (𝑆 ∗ 𝜂|𝜉) = (𝑆𝜉|𝜂),

𝜉 ∈ D 𝑆 , 𝜂 ∈ D 𝑆∗ .

Since ♯ is an involution, we have 𝜉 ∈ D 𝑆 ⇒ 𝑆𝜉 ∈ D 𝑆 , 𝜂 ∈ D 𝑆∗ ⇒ 𝑆∗𝜂 ∈ D 𝑆∗ ,

𝑆𝑆𝜉 = 𝜉, hence 𝑆 2 ⊂ 1, 𝑆 ∗ 𝑆 ∗ 𝜂 = 𝜂, hence (𝑆 ∗ )2 ⊂ 1.

In other words, 𝑆 = 𝑆 −1 ,

𝑆 ∗ = (𝑆 ∗ )−1 .

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We consider the positive self-adjoint linear operator Δ = 𝑆 ∗ 𝑆, which is called the modular operator associated to the left Hilbert algebra 𝔄. Then s(Δ) = 1 and Δ−1 = 𝑆𝑆 ∗ . We also consider the polar decomposition of 𝑆 𝑆 = 𝐽Δ1/2 . Since 𝑆 = 𝑆 −1 , we get successively 𝐽Δ1/2 = Δ−1/2 𝐽 −1 , Δ−1/2 = 𝐽Δ1/2 𝐽 = 𝐽 2 (𝐽 ∗ Δ1/2 𝐽). whence, with the help of the polar decomposition, we obtain 𝐽 2 = 1,

𝐽 = 𝐽 ∗ = 𝐽 −1 .

Consequently, 𝐽 is a conjugation in H , which is called the canonical conjugation associated to the left Hilbert algebra 𝔄. We have thus obtained the following relations: 𝑆 = 𝐽Δ1/2 = Δ−1/2 𝐽, 𝑆 ∗ = 𝐽Δ−1/2 = Δ1/2 𝐽,

D 𝑆 = D (∆1/2 ) , D 𝑆 ∗ = D (∆−1/2 ) .

The data concerning 𝐽 are synthetized in the following relations: 𝐽 2 = 1, (𝐽𝜂|𝜉) = (𝐽𝜉|𝜂),

𝜉, 𝜂 ∈ H .

Finally, we mention the fact that for any 𝑓 ∈ B ([0, ∞)), we have ̄ −1 ); 𝐽𝑓(Δ)𝐽 = 𝑓(Δ this equality being a consequence of the relations 𝐽Δ𝐽 = Δ−1 ̄ ̄ −1 . 𝐽(𝜆Δ)𝐽 = 𝜆𝐽Δ𝐽 = 𝜆Δ We also mention the following equivalent forms of the preceding formula: ̄ −1 ), 𝑓(Δ)𝐽 = 𝐽 𝑓(Δ ̄ −1 )𝐽. 𝐽𝑓(Δ) = 𝑓(Δ Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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The theory of standard von Neumann algebras In particular, 𝐽Δi𝑡 = Δi𝑡 𝐽,

𝑡 ∈ ℝ.

i𝑡

The so-continuous group {Δ } of unitary operators is called the modular group associated to the left Hilbert algebra 𝔄. 10.2. In this section, we consider some operators which are naturally associated to a left Hilbert algebra 𝔄 ⊂ H . Let 𝜂 ∈ H . We define the linear operator 𝑅0𝜂 by the relations 𝑅0𝜂 (𝜉) = 𝐿𝜉 (𝜂),

𝜉 ∈ D 𝑅𝜂0 = 𝔄.

If 𝑅0𝜂 is preclosed, we denote its closure by 𝑅𝜂 . Lemma 1. Let 𝜂 ∈ H . If 𝑅0𝜂 is preclosed, then 𝑅𝜂 is affiliated to the von Neumann algebra 𝔏(𝔄)′ . Proof. For any 𝜉 ∈ 𝔄, we have 𝐿𝜉 𝑅0𝜂 ⊂ 𝑅0𝜂 𝐿𝜉 , 𝐿𝜉 𝑅𝜂 ⊂ 𝑅𝜂 𝐿𝜉 , whence, for any 𝑥 ∈ 𝔏(𝔄), 𝑥𝑅𝜂 ⊂ 𝑅𝜂 𝑥. Q.E.D. Lemma 2. If 𝜂 ∈ D 𝑆 ∗ , then 𝑅0𝜂 is preclosed and 𝔄 ⊂ D (𝑅𝜂 )∗ (𝑅𝜂 )∗ 𝜉 = 𝐿𝜉 𝑆 ∗ 𝜂,

𝜉 ∈ 𝔄.

Proof. Since 𝑅0𝜂 ⊂ (𝑅0𝑆 ∗ 𝜂 )∗ , the operator 𝑅0𝜂 is preclosed. A trivial argument now completes the proof.

Q.E.D.

10.3. In this section, we consider the ‘multiplications to the right’, which are associated to a left Hilbert algebra 𝔄 ⊂ H . We define 𝔄′ = {𝜂 ∈ D 𝑆 ∗ ; 𝑅𝜂 is bounded}. Lemma 1 from 10.2 now implies that 𝜂 ∈ 𝔄′ ⇒ 𝑅𝜂 ∈ 𝔏(𝔄)′ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Proposition. Let 𝜂 ∈ H , 𝜁 ∈ H and 𝑥 ′ ∈ B (H ). Then the following assertions are equivalent: (i) 𝜂 ∈ 𝔄′ and 𝑆 ∗ (𝜂) = 𝜁, 𝑅𝜂 = 𝑥 ′ . (ii) For any 𝜉 ∈ 𝔄, we have 𝐿𝜉 (𝜂) = 𝑥 ′ 𝜉, 𝐿𝜉 (𝜁) = 𝑥 ′∗ 𝜉. Proof. Assuming (i) to be true, for any 𝜉 ∈ 𝔄, we have 𝐿𝜉 (𝜂) = 𝑅𝜂 (𝜉) = 𝑥 ′ 𝜉 and from Lemma 2, Section 10.2, we get 𝐿𝜉 (𝜁) = 𝐿𝜉 𝑆 ∗ 𝜂 = (𝑅𝜂 )∗ 𝜉 = 𝑥 ′∗ 𝜉. Let us now assume that (ii) is true. For any 𝜉𝑙 , 𝜉2 ∈ 𝔄, we have ♯ ♯

♯

♯

(𝜂|𝑆(𝜉1 𝜉2 )) = (𝜂|𝜉2 𝜉1 ) = (𝜂|𝐿𝜉 ♯ (𝜉1 )) = (𝜂|(𝐿𝜉2 )∗ 𝜉1 ) 2

♯

♯

♯

= (𝐿𝜉2 (𝜂)|𝜉1 ) = (𝑥 ′ 𝜉2 |𝜉1 ) = (𝜉2 |𝑥 ′∗ 𝜉1 ) = (𝜉2 |𝐿𝜉 ♯ (𝜁)) = (𝐿𝜉1 (𝜉2 )|𝜁) = (𝜉1 𝜉2 |𝜁), 1

whence 𝜂 ∈ D 𝑆 ∗ and 𝑆 ∗ (𝜂) = 𝜁. Obviously, 𝑅𝜂 = 𝑥 ′ is bounded; hence 𝜂 ∈ 𝔄′ .

Q.E.D.

Let 𝜂 ∈ D 𝑆 ∗ and let 𝑅𝜂 = 𝑢𝜂 𝐴𝜂 = 𝐵𝜂 𝑢𝜂 be the polar decompositions of 𝑅𝜂 . For any 𝑓 ∈ B ([0, +∞)), we have the relation 𝑢𝜂 𝑓(𝐴𝜂 ) = 𝑓(𝐵𝜂 )𝑢𝜂 . We observe that 𝜂 ∈ 𝑅𝜂 D 𝑅𝜂 = s(𝐵𝜂 )H , 𝑆𝜂∗ ∈ (𝑅𝜂 )∗ D (𝑅𝜂 )∗ = s(𝐴𝜂 )H . 𝑠𝑜

Indeed, in accordance with Section 10.1, there exists a net {𝜉𝑖 } ⊂ 𝔄, such that 𝐿𝜉𝑖 → 1, hence, from Lemma 2, Section 10.2, we get 𝑅𝜂 D 𝑅𝜂 ∋ 𝑅𝜂 (𝜉𝑖 ) = 𝐿𝜉𝑖 (𝜂) → 𝜂, (𝑅𝜂 )∗ D (𝑅𝜂 )∗ ∋ (𝑅𝜂 )∗ (𝜉𝑖 ) = 𝐿𝜉𝑖 (𝑆 ∗ 𝜂) → 𝑆 ∗ 𝜂. With the help of the preceding proposition, it is easy to prove the following assertions: Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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The theory of standard von Neumann algebras Corollary 1. If 𝜂 ∈ 𝔄′ , then 𝑆 ∗ 𝜂 ∈ 𝔄′ , and 𝑆 ∗ (𝑆 ∗ 𝜂) = 𝜂,

𝑅𝑆 ∗ 𝜂 = (𝑅𝜂 )∗ .

Corollary 2. If 𝜂1 , 𝜂2 ∈ 𝔄′ , then 𝑅𝜂1 (𝜂2 ) ∈ 𝔄′ , and 𝑆 ∗ 𝑅𝜂1 (𝜂2 ) = 𝑅𝑆 ∗ 𝜂2 (𝑆 ∗ 𝜂1 ),

𝑅 𝑅𝜂

1

(𝜂2 )

= 𝑅𝜂1 𝑅𝜂2 .

More generally, we have the Corollary 3. If 𝜂1 , 𝜂2 ∈ 𝔄′ and 𝑥′ ∈ 𝔏(𝔄)′ , then 𝑅𝜂1 𝑥 ′ (𝜂2 ) ∈ 𝔄′ , and 𝑆 ∗ 𝑅𝜂1 𝑥 ′ (𝜂2 ) = 𝑅𝑆 ∗ 𝜂2 (𝑥 ′ )∗ 𝑆 ∗ 𝜂1 ,

𝑅𝑅𝜂

1

𝑥′ (𝜂2 )

= 𝑅 𝜂1 𝑥 ′ 𝑅 𝜂2 .

The following two corollaries enable us to obtain elements in 𝔄′ . Corollary 4. If 𝜂 ∈ D 𝑆 ∗ and 𝑓 ∈ B ([0, +∞)), sup{𝜆2 |𝑓(𝜆)|; 𝜆 ∈ [0, ∞)} < +∞, then 𝑅𝜂 𝑓(𝐴𝜂 )𝑆 ∗ 𝜂 ∈ 𝔄′ , and ̄ 𝜂 )𝑆 ∗ 𝜂, 𝑆 ∗ (𝑅𝜂 𝑓(𝐴𝜂 )𝑆 ∗ 𝜂) = 𝑅𝜂 𝑓(𝐴

𝑅𝑅𝜂 𝑓(𝐴𝜂 )𝑆 ∗ 𝜂 = 𝐵𝜂2 𝑓(𝐵𝜂 ).

Corollary 5. If 𝜂 ∈ D 𝑆 ∗ and 𝑓 ∈ B ([0, +∞)), sup{𝜆|𝑓(𝜆)|; 𝜆 ∈ [0, ∞)} < +∞, then 𝑓(𝐵𝜂 )𝜂 ∈ 𝔄′ , and ̄ 𝜂 )𝑆 ∗ 𝜂, 𝑆 ∗ (𝑓(𝐵𝜂 )𝜂) = 𝑓(𝐴

𝑅𝑓(𝐵𝜂 )𝜂 = 𝐵𝜂 𝑓(𝐵𝜂 )𝑢𝜂 .

Finally, the last corollary in this series shows that 𝔄′ contains enough elements. Corollary 6. If 𝜂 ∈ D 𝑆 ∗ , then there exists sequences {𝜂𝑛 }, {𝜁𝑛 } ⊂ 𝔄′ , such that 𝑅𝜂𝑛 (𝜁𝑛 ) → 𝜂, 𝑆 ∗ 𝑅𝜂𝑛 (𝜁𝑛 ) → 𝑆 ∗ 𝜂. Proof. Let us consider the functions 𝑓𝑛 ∈ B ([0, +∞))), 𝑛 ∈ ℕ, defined by 𝑓𝑛 (𝜆) = 𝜆−1 𝜒(𝑛−1 ,𝑛) (𝜆). We define 𝜂𝑛 = 𝑅𝜂 𝑓𝑛 (𝐴𝜂 )𝑆 ∗ 𝜂,

𝜁𝑛 = 𝑓𝑛 (𝐵𝜂 )𝜂.

With the help of Corollaries 2, 4 and 5, it is easy to see that, for any natural number 𝑛, we have 𝜂𝑛 , 𝜁𝑛 , 𝑅𝜂𝑛 (𝜁𝑛 ) ∈ 𝔄′ , and 𝑅𝜂𝑛 (𝜁𝑛 ) = 𝐵𝜂2 𝑓𝑛2 (𝐵𝜂 )𝜂,

𝑆 ∗ 𝑅𝜂𝑛 (𝜁𝑛 ) = 𝐴2𝜂 𝑓𝑛2 (𝐴𝜂 )𝑆 ∗ 𝜂.

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Since 𝜂 ∈ s(𝐵𝜂 )H and 𝑆 ∗ 𝜂 ∈ s(𝐴𝜂 )H , by taking into account Theorem 9.11 (vi) and Corollary 9.13 (iii), it follows that 𝑅𝜂𝑛 (𝜁𝑛 ) → 𝜂, 𝑆 ∗ 𝑅𝜂𝑛 (𝜁𝑛 ) → 𝑆 ∗ 𝜂. Q.E.D. 10.4. Let 𝔅 be a complex algebra with involution, endowed also with a scalar product (⋅|⋅). We denote by 𝜂 ↦ 𝜂 ♭ the involution in 𝔅 and by H the Hilbert space obtained by the completion of 𝔅. One says that 𝔅 is a right Hilbert algebra if (i) 𝔅 ∋ 𝜉 ↦ 𝜉𝜂 ∈ 𝔅 is continuous for any 𝜂 ∈ 𝔅. (ii) (𝜉1 𝜂|𝜉2 ) = (𝜉1 |𝜉2 𝜂 ♭ ), for any 𝜂, 𝜉1 , 𝜉2 ∈ 𝔅. (iii) 𝔅2 is dense in 𝔅. (iv) H ⊃ 𝔅 ∋ 𝜂 ↦ 𝜂 ♭ ∈ H is a preclosed antilinear operator. Let 𝔄 ⊂ H be a left Hilbert algebra. Then 𝔄′ , endowed with the operations 𝜂1 𝜂2 = 𝑅𝜂2 (𝜂1 ), 𝜂 ♭ = 𝑆 ∗ 𝜂,

𝜂1 , 𝜂2 ∈ 𝔄 ′ ,

𝜂 ∈ 𝔄′ ,

and with the scalar product of H , is a right Hilbert algebra (see Corollaries 1, 2 and 6 from Section 10.3). If 𝔅 ⊂ H is a right Hilbert algebra, then the closed antilinear operator 𝐹 is defined to be the closure of the operator H ⊃ 𝔅2 ∋ ∑ 𝜂i 𝜉i ↦ (∑ 𝜂i 𝜉i )♭ ∈ H . i

i

Let 𝔄 ⊂ H be a left Hilbert algebra and 𝔅 = 𝔄′ . Then, in accordance with Corollary 6 from Section 10.3, we have (1)

𝑆 ∗ = 𝑆 ∗ |(𝔄′ )2 ,

i.e. 𝐹 = 𝑆∗. For a right Hilbert algebra 𝔅 ⊂ H , the operators 𝑅𝜂 ∈ B (H ), 𝜂 ∈ 𝔅 are defined by the formula: 𝑅𝜂 (𝜉) = 𝜉𝜂, 𝜉 ∈ 𝔅. If 𝔅 = 𝔄′ , then, for any 𝜂 ∈ 𝔅, 𝑅𝜂 is just the operator defined in Sections 10.2 and 10.3. If 𝔅 ⊂ H is a right Hilbert algebra, then the mapping 𝑅 ∶ 𝔅 ∋ 𝜂 ↦ 𝑅𝜂 ∈ B (H ) Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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The theory of standard von Neumann algebras is a *-antirepresentation of 𝔅, i.e. 𝑅𝜂1 𝜂2 = 𝑅𝜂2 𝑅𝜂1 , ∗

(𝑅𝜂 ) = 𝑅𝜂♭ ,

𝜂1 , 𝜂2 ∈ 𝔅,

𝜂 ∈ 𝔅.

One defines ℜ(𝔅) = R ({𝑅𝜂 ; 𝜂 ∈ 𝔅}) = {𝑅𝜂 ; 𝜂 ∈ 𝔅}″ = {𝑅𝜂 ; 𝜂 ∈ 𝔅}

𝑠𝑜

If 𝔅 = 𝔄′ , then (2)

ℜ(𝔄′ ) = 𝔏(𝔄)′ .

Indeed, the inclusion ℜ(𝔄′ ) ⊂ 𝔏(𝔄)′ is obvious. Conversely, let 𝑥 ′ ∈ 𝔏(𝔄)′ and {𝜂𝑖 } ⊂ 𝔄′ 𝑠𝑜

be a net, such that ‖𝑅𝜂𝑖 ‖ ≤ 1 and 𝑅𝜂𝑖 → 1. Then, in accordance with Corollary 3 from Section 10.3, we have 𝑅𝜂𝑖 𝑥 ′ (𝜂𝑖 ) ∈ 𝔄′

and

𝑅𝑅𝜂 𝑥′ (𝜂𝑖 ) = 𝑅𝜂𝑖 𝑥′ 𝑅𝜂𝑖 .

Thus, we have

𝑖

𝑠𝑜

ℜ(𝔄′ ) ∋ 𝑅𝜂𝑖 𝑥′ 𝑅𝜂𝑖 → 𝑥 ′ , and relation (2) is proved. Let 𝔅 ⊂ H be a right Hilbert algebra. For any 𝜉 ∈ D 𝐹 ∗ , one defines the closed linear operator 𝐿𝜉 in H by analogy with Section 10.2, as being the closure of the operator H ⊃ 𝔅 ∋ 𝜂 ↦ 𝑅𝜂 (𝜉) ∈ H . Then 𝐿𝜉 is affiliated to ℜ(𝔅)′ , 𝔅 ⊂ D (𝐿𝜉 )∗ and (𝐿𝜉 )∗ 𝜂 = 𝑅𝜂 𝐹 ∗ 𝜉,

𝜂 ∈ 𝔅.

One also defines 𝔅′ = {𝜉 ∈ D 𝐹 ∗ ; 𝐿𝜉 is bounded} and a proposition analogous to Proposition 10.3 holds. By a dualization of the preceding discussion, 𝔅′ canonically becomes a left Hilbert algebra and the following relation holds 𝔏(𝔅′ ) = ℜ(𝔅)′ . If 𝔅 = 𝔄′ , we have 𝐹 ∗ = 𝑆. Thus, for any 𝜉 ∈ D 𝑆 , we defined a closed linear operator 𝐿𝜉 , which is affiliated to ℜ(𝔄′ )′ = 𝔏(𝔄), as beign the closure of the operator H ⊃ 𝔄′ ∋ 𝜂 ↦ 𝑅𝜂 (𝜉) ∈ H .

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We mention that 𝔄′ ⊂ D (𝐿𝜉 )∗ and we have the relation (𝐿𝜉 )∗ (𝜂) = 𝑅𝜂 𝑆𝜉.

(3) We denote

𝔄″ = 𝔅′ = {𝜉 ∈ D 𝑆 ; 𝐿𝜉 is bounded}. Then 𝔄″ becomes a left Hilbert algebra, with the operations 𝜉1 𝜉2 = 𝐿𝜉1 (𝜉2 ), 𝜉 ♯ = 𝑆𝜉,

𝜉1 , 𝜉2 ∈ 𝔄″ ,

𝜉 ∈ 𝔄″

(We shall see later that these notations are compatible with those already introduced for the operations in 𝔄). The right Hilbert algebras were only introduced in order to systematize the discussions about 𝔄, 𝔄′ , 𝔄″ . Thus, if 𝔄 ⊂ H is a left Hilbert algebra, then the proposition analogous to Proposition 10.3 is the following: Proposition. Let 𝜉 ∈ H , 𝜁 ∈ H and 𝑥 ∈ B (H ). Then the following assertions are equivalent: (i) 𝜉 ∈ 𝔄″ and 𝑆(𝜉) = 𝜁, 𝐿𝜉 = 𝑥; (ii) For any 𝜂 ∈ 𝔄′ , we have 𝑅𝜂 (𝜉) = 𝑥𝜂, 𝑅𝜂 (𝜁) = 𝑥 ∗ 𝜂. With the help of the preceding proposition, it is easy to obtain the following: Corollary 1. If 𝜉 ∈ 𝔄, then 𝜉 ∈ 𝔄″ and 𝑆𝜉 = 𝜉 ♯ , 𝐿𝜉 = the operator defined in Section 10.1. In particular, 𝔄 ⊂ D 𝑆 and 𝑆𝜉 = 𝜉 ♯ , for any 𝜉 ∈ 𝔄. Thus, 𝑆 is the closure of the operator H ⊃ 𝔄 ∋ 𝜉 ↦ 𝜉♯ ∈ H . Let 𝔄1 ⊂ H be a left Hilbert algebra. One calls a left Hilbert subalgebra of 𝔄1 any involutive subalgebra 𝔄2 of 𝔄1 , which, endowed with the scalar product of H , becomes a left Hilbert algebra, which is dense in H . It is easy to verify the following: Corollary 2. 𝔄 is a left Hilbert subalgebra of 𝔄″ and the following relations hold: (𝔄″ )′ = 𝔄′ , (𝔄″ )″ = 𝔄″ , 𝔏(𝔄″ ) = 𝔏(𝔄).

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The theory of standard von Neumann algebras We conclude this section with a ‘dominated convergence’ result:

Corollary 3. Let 𝑥 ∈ 𝔏(𝔄) and {𝜉𝑖 } ⊂ 𝔄″ be a net having the following properties: 𝑤𝑜

𝐿𝜉𝑖 → 𝑥,

sup ‖𝜉𝑖 ‖ < +∞,

sup ‖𝑆𝜉𝑖 ‖ < +∞.

𝑖

𝑖

Then there exists a 𝜉 ∈ 𝔄″ , such that 𝑥 = 𝐿𝜉 , 𝜉𝑖 → 𝜉 weakly, 𝑆𝜉𝑖 → 𝑆𝜉 weakly. Proof. We consider the linear form 𝜑 defined on (𝔄′ )2 by the formula 𝜑(𝑅𝑆 ∗ 𝜂 (𝜁)) = (𝜁|𝑥𝜂) = lim(𝜁|𝐿𝜉𝑖 (𝜂)) = lim(𝑅𝑆 ∗ 𝜂 (𝜁)|𝜉𝑖 ), 𝑖

𝑖

𝜂, 𝜁 ∈ 𝔄′ .

Since sup𝑖 ‖𝜉𝑖 ‖ < +∞, 𝜑 is bounded; hence, there exists 𝜉 ∈ H , such that (𝜁|𝑥𝜂) = (𝑅𝑆 ∗ 𝜂 (𝜁)|𝜉) = (𝜁|𝑅𝜂 (𝜉)),

𝜂, 𝜁 ∈ 𝔄′ ,

whence 𝑅𝜂 (𝜉) = 𝑥(𝜂),

𝜂 ∈ 𝔄′ .

Since (𝑅𝑆 ∗ 𝜂 (𝜁)|𝜉𝑖 ) → (𝑅𝑆 ∗ 𝜂 (𝜁)|𝜉),

𝜂, 𝜁 ∈ 𝔄′ ,

from the facts that (𝔄′ )2 is dense in H and that sup𝑖 ‖𝜉𝑖 ‖ < +∞, it follows that 𝜉𝑖 → 𝜉, weakly. 𝑤𝑜

Using the fact that 𝐿𝑆𝜉𝑖 = (𝐿𝜉𝑖 )∗ → 𝑥 ∗ , one can analogously show that there exists a 𝜁 ∈ H , such that 𝑅𝜂 (𝜁) = 𝑥 ∗ (𝜂),

𝜂 ∈ 𝔄′ .

𝑆𝜉𝑖 → 𝜁, weakly. By applying the preceding proposition, it follows that 𝜉 ∈ 𝔄″ ,

𝑆𝜉 = 𝜁,

𝐿𝜉 = 𝑥. Q.E.D.

10.5. Let 𝔄 ⊂ H be a left Hilbert algebra. By taking into account the definition of the Hilbert algebras, 𝔄2 is dense in 𝔄. More precisely, for any 𝜉 ∈ 𝔄, we have 𝜉 ∈ 𝔄𝜉. 𝑠𝑜

Indeed, if {𝜉𝑖 } ⊂ 𝔄 is a net, such that 𝐿𝜉𝑖 → 1, then 𝔄𝜉 ∋ 𝐿𝜉𝑖 (𝜉) → 𝜉. Actually, for any 𝜉 ∈ 𝔄, we have 𝜉 ∈ 𝜉𝔄𝜉, as a consequence of the following lemma. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Lemma 1. If 𝜉 ∈ 𝔄, 𝜉 ≠ 0, then 𝐿𝜉𝜉 ♯ ≠ 0, and 𝑝𝑛 (‖𝐿𝜉𝜉 ♯ ‖−1 𝜉𝜉 ♯ )𝜉 → 𝜉, where 𝑝𝑛 is the polynomial defined by 𝑝𝑛 (𝜆) = 1 − (1 − 𝜆)𝑛 . 𝑠𝑜

Proof. Let {𝜂𝑖 } ⊂ 𝔄′ be a net, such that 𝑅𝜂𝑖 → 1. Then 𝐿𝜉 (𝜂𝑖 ) = 𝑅𝜂𝑖 (𝜉) → 𝜉; hence 𝜉 ∈ 𝐿𝜉 (H ) = l(𝐿𝜉 )H = s(𝐿𝜉𝜉 ♯ )H . In particular, 𝐿𝜉𝜉 ♯ ≠ 0. On the other hand, according to a remark we made in Section 2.22, we have 𝑠𝑜

𝑝𝑛 (‖𝐿𝜉𝜉 ♯ ‖−1 𝐿𝜉𝜉 ♯ ) → s(‖𝐿𝜉𝜉 ♯ ‖−1 𝐿𝜉𝜉 ♯ ) = s(𝐿𝜉𝜉 ♯ ); hence 𝑝𝑛 (‖𝐿𝜉𝜉 ♯ ‖−1 𝜉𝜉 ♯ )𝜉 = 𝑝𝑛 (‖𝐿𝜉𝜉 ♯ ‖−1 𝐿𝜉𝜉 ♯ )𝜉 → s(𝐿𝜉𝜉 ♯ )𝜉 = 𝜉. Q.E.D. In particular, from what we have just proved, it follows that the *-representation 𝐿 ∶ 𝔄 → B (H ) is injective. In order to avoid any confusion, in what follows we shall mark the symbols 𝐿, 𝑆, …, which correspond to 𝔄, in the following manner: 𝐿𝔄 , 𝑆 𝔄 , … . Lemma 2. Let 𝔄1 ⊂ H be a left Hilbert algebra and 𝔄2 an involutive subalgebra of 𝔄1 , which is dense in H . Then 𝔄2 is a left Hilbert subalgebra of 𝔄1 . Proof. If is sufficient to prove that (𝔄2 )2 is dense in 𝔄2 . In accordance with Lemma 1, for any 𝜉 ∈ 𝔄2 , 𝜉 ≠ 0, we have 𝔄

(𝔄2 )2 ∋ 𝑝𝑛 (‖𝐿𝜉𝜉1 ♯ ‖−1 𝜉𝜉 ♯ )𝜉 → 𝜉, where 𝑝𝑛 is the polynomial from the statement of Lemma 1.

Q.E.D.

Lemma 3. Let 𝔄2 be a left Hilbert subalgebra of a left Hilbert algebra 𝔄1 ⊂ H . Then the following assertions are equivalent: (i) 𝑆 𝔄1 = 𝑆 𝔄2 . (ii) (𝔄1 )′ = (𝔄2 )′ . (iii) (𝔄1 )″ = (𝔄2 )″ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Proof. The proofs of the implications (i) ⇒ (ii) ⇒ (iii) are immediate, on the basis of the corresponding definitions; the implication (iii) ⇒ (i) follows from the considerations we have just made in Section 10.4. Q.E.D. In particular, if 𝔄 is a left Hilbert algebra, then 𝔄2 is a left Hilbert sub-algebra of 𝔄, and (𝔄2 )″ = 𝔄″ . 10.6. In this section, we consider an important example of a left Hilbert algebra. Let M ⊂ B (H ) be a von Neumann algebra and 𝜉0 ∈ H a separating cyclic vector for M (see 3.8). Then 𝔄 = M 𝜉0 = {𝑥𝜉0 ; 𝑥 ∈ M }, endowed with the operations (𝑥𝜉0 )(𝑦𝜉0 ) = 𝑥𝑦𝜉0 , (𝑥𝜉0 )♯ = 𝑥 ∗ 𝜉0 . and with the scalar product of H , becomes a left Hilbert algebra. Indeed, the first two conditions are immediate, the third one is trivially satisfied because 1 ∈ M , whereas the fourth one follows from the relation: (𝑥𝜉0 |𝑥 ′ 𝜉0 ) = (𝑥 ′∗ 𝜉0 |(𝑥𝜉0 )♯ ),

(1)

𝑥 ∈ M , 𝑥′ ∈ M ′ ,

and by taking into account the fact that M ′ 𝜉0 = 𝐻. For 𝜉 = 𝑥𝜉0 ∈ 𝔄, we obviously have 𝐿𝜉 = 𝑥, hence 𝔏(𝔄) = {𝐿𝜉 ; 𝜉 ∈ 𝔄} = M . If 𝜂 ∈ 𝔄′ , then 𝑅𝜂 ∈ 𝔏(𝔄)′ = M ′ , and 𝑅𝜂 (𝜉0 ) = 𝐿𝜉0 (𝜂) = 𝜂. Thus, 𝔄′ ⊂ M ′ 𝜉0 = {𝑥 ′ 𝜉0 ;

𝑥 ′ ∈ M ′ }.

With the help of relation (1), it is easy to see that, for any 𝑥 ′ ∈ M ′ , we have 𝑥 ′ 𝜉0 ∈ 𝔄′ , and 𝑆 ∗ (𝑥′ 𝜉0 ) = 𝑥 ′∗ 𝜉0 ,

𝑅𝑥′ 𝜉0 = 𝑥 ′ .

Consequently, we have 𝔄′ = M ′ 𝜉0 , and the operations of a right Hilbert algebra in 𝔄′ are the following ones: (𝑥′ 𝜉0 )(𝑦 ′ 𝜉0 ) = 𝑅𝑦′ 𝜉0 (𝑥′ 𝜉0 ) = 𝑦 ′ 𝑥 ′ 𝜉0 , (𝑥′ 𝜉0 )♭ = 𝑆 ∗ (𝑥′ 𝜉0 ) = 𝑥 ′∗ 𝜉0 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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By analogy with the preceding argument, it follows that 𝔄″ = M ″ 𝜉0 = M 𝜉0 = 𝔄. Because of their relevance to the following sections, we restate the following facts, which have already been proved: The closure 𝑆 of the antilinear operator H ⊃ M 𝜉0 ∋ 𝑥𝜉0 ↦ 𝑥 ∗ 𝜉0 ∈ H has as its adjoint 𝑆 ∗ the closure of the antilinear operator H ⊃ M ′ 𝜉0 ∋ 𝑥 ′ 𝜉0 ↦ 𝑥 ′∗ 𝜉0 ∈ H . Also, if 𝜂 ∈ D 𝑆 ∗ , then the operator H ⊃ M 𝜉0 ∋ 𝑥𝜉0 ↦ 𝑥𝜂 ∈ H is preclosed and its closure is affiliated to M ′ . 10.7. This section contains an important application of the theory already developed to the commutation theorem for tensor products. Lemma 1. Let M 1 ⊂ B (H 1 ) and M 2 ⊂ B (H 2 ) be von Neumann algebras with separating cyclic vectors 𝜉1 ∈ H 1 and 𝜉2 ∈ H 2 , respectively. Then (M 1 ⊗M 2 )′ = M ′1 ⊗M ′2 . Proof. Let 𝜉 = 𝜉1 ⊗ 𝜉2 ∈ H 1 ⊗H 2 = H . Then 𝜉 is a cyclic and separating vector for M 1 ⊗M 2 . We now consider the operators 𝑆1 = the closure of [M 1 𝜉1 ∋ 𝑥1 𝜉1 ↦ 𝑥1∗ 𝜉1 ∈ H 1 ] 𝑆2 = the closure of [M 2 𝜉2 ∋ 𝑥2 𝜉2 ↦ 𝑥2∗ 𝜉2 ∈ H 2 ] 𝑆 = the closure of [(M 1 ⊗M 2 )𝜉 ∋ 𝑥𝜉 ↦ 𝑥 ∗ 𝜉 ∈ H ]. Using the fact that M 1 ⊗M 2 is 𝑤-dense in M 1 ⊗M 2 , it is easy to verify that 𝑆 = 𝑆1 ⊗𝑆2 . In accordance with Proposition 9.33, we now infer that 𝑆 ∗ = 𝑆1∗ ⊗𝑆2∗ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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The theory of standard von Neumann algebras According to the last statement made in the preceding section, we get 𝑆 ∗ = the closure of [(M 1 ⊗M 2 )′ 𝜉 ∋ 𝑥 ′ 𝜉 ↦ 𝑥 ′∗ 𝜉 ∈ H ] 𝑆1∗ = the closure of [M ′1 𝜉1 ∋ 𝑥1′ 𝜉1 ↦ 𝑥1′∗ 𝜉1 ∈ H 1 ] 𝑆2∗ = the closure of [M ′2 𝜉2 ∋ 𝑥2′ 𝜉2 ↦ 𝑥2′∗ 𝜉2 ∈ H 2 ] 𝑆1∗ ⊗𝑆2∗ = the closure of [(M ′1 ⊗M ′2 )𝜉 ∋ 𝑦 ′ 𝜉 ↦ 𝑦 ′∗ 𝜉 ∈ H ]. Therefore, we have 𝑆 (M 1 ⊗M 2 ) = 𝑆 ∗ = 𝑆1∗ ⊗𝑆2∗ = 𝑆 (M 1 ⊗M 2 ) . ′

′

′

From the inclusion (M 1 ⊗M 2 )′ 𝜉 ⊃ (M ′1 ⊗M ′2 )𝜉, with the help of Lemma 3 from Section 10.5, we obtain (M 1 ⊗M 2 )′ 𝜉 = (M ′1 ⊗M ′2 )𝜉, and this equality implies that (M 1 ⊗M 2 )′ = M ′1 ⊗M ′2 .

Q.E.D.

Lemma 2. Let M 1 , M 2 , N 1 , N 2 be von Neumann algebras, M 1 being *-isomorphic to N 1 and M 2 being *-isomorphic to N 2 . If (N 1 ⊗N 2 )′ = N ′1 ⊗N ′2 , then (M 1 ⊗M 2 )′ = M ′1 ⊗M ′2 . Proof. According to exercise E.8.8, we can separately consider the cases of the spatial isomorphism, induction and amplification. (I) The case of the spatial isomorphism is trivial. (II) The case of the induction: if M 1 = (N 1 )𝑒1′ ,

𝑒1′ ∈ P N ′ , M 2 = (N 2 )𝑒2′ , 1

𝑒2′ ∈ P N ′ , 2

then, by taking into account exercise E.3.15 and Theorem 3.13, we have (M 1 ⊗M 2 )′ = ((N 1 )𝑒1′ ⊗(N 2 )𝑒2′ )′ = ((N 1 ⊗N 2 )𝑒′ ⊗𝑒′ )′ 1

=

(N 1 ⊗N 2 )′𝑒′ ⊗𝑒′ 1 2

= (N

2

′ ′ 1 ⊗N 2 )𝑒1′ ⊗𝑒2′

= (N ′1 )𝑒1′ ⊗(N ′2 )𝑒2′ = M ′1 ⊗M ′2 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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(III) The case of the amplification: let N 1 ⊂ B (H 1 ),

N 2 ⊂ B (H 2 )

M 1 = N 1 ⊗ C (K 1 ),

M 2 = N 2 ⊗ C (K 2 ),

where H 1 , H 2 , K 1 , K 2 are Hilbert spaces. We define a unitary operator 𝑢 ∶ H 1 ⊗K 1 ⊗H 2 ⊗K 2 → H 1 ⊗H 2 ⊗K 1 ⊗K 2 by the formula 𝑢(𝜉1 ⊗ 𝜂1 ⊗ 𝜉2 ⊗ 𝜂2 ) = 𝜉1 ⊗ 𝜉2 ⊗ 𝜂1 ⊗ 𝜂2 . Then, with the help of Proposition 3.17, we infer that (M 1 ⊗M 2 )′ = (N 1 ⊗ C (K 1 ) ⊗ N 2 ⊗ C (K 2 ))′ = (𝑢∗ (N 1 ⊗N 2 ⊗ C (K 1 )⊗ C (K 2 ))𝑢)′ = 𝑢∗ ((N 1 ⊗N 2 )⊗ C (K 1 )⊗ C (K 2 ))′ 𝑢 = 𝑢∗ ((N 1 ⊗N 2 )′ ⊗B (K 1 )⊗B (K 2 ))𝑢 = 𝑢∗ (N ′1 ⊗N ′2 ⊗B (K 1 )⊗B (K 2 ))𝑢 = N ′1 ⊗B (K 1 )⊗N ′2 ⊗B (K 2 ) = (N 1 ⊗ C (K 1 ))′ ⊗(N 2 ⊗ C (K 2 ))′ = M ′1 ⊗M ′2 . Q.E.D. We now prove the commutation theorem for tensor products: Theorem. For any pair of von Neumann algebra M 1 ⊂ B (H 1 ) and M 2 ⊂ B (H 2 ), the following relation holds: (M 1 ⊗M 2 )′ = M ′1 ⊗M ′2 . Proof. In accordance with Lemma 1, the assertion is true if M 1 and M 2 have separating cyclic vectors. With the help of exercise E.5.6, Proposition 5.18 and Lemma 2, the assertion of the theorem extends to the case in which M 1 and M 2 are of countable type. Then the assertion of the theorem trivially extends to the case in which the unit projections in M 1 and M 2 are piecewise of countable type (7.2). By taking into account Lemma 7.2, Proposition 3.17 and Theorem 4.22, the assertion of the theorem easily obtains for the case in which M 1 is finite, or uniform (8.4), and M 2 is finite, or uniform, too. Finally, with the help of Theorem 4.17 and of Proposition 8.4, the assertion of the theorem obtains in the general case. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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301

Alternate Proof. (Rousseau, van Daele, and van Heeswijck 1976). If we show that (M 1 ⊗M 2 )′ and (M 1′ ⊗M ′2 )′ commute, then it follows that (M 1 ⊗M 2 )′ ⊂ (M ′1 ⊗M ′2 )″ = M 1′ ⊗M ′2 by the von Neumann bicommutant theorem, which is the desired conclusion, since the reverse inclusion is obvious. To this end, consider: 𝑇 ′ ∈ (M 1 ⊗M 2 )′ , 𝑇 ∈ (M ′1 ⊗M ′2 )′ and 𝜉 ∈ H 1 , 𝜂 ∈ H 2 , where M 1 ⊂ B (H 1 ), M 2 ⊂ B (H 2 ). It is sufficient to show that (4)

(𝑇𝑇 ′ (𝜉⊗𝜂) ∣ 𝜉⊗𝜂) = (𝑇 ′ 𝑇(𝜉⊗𝜂) ∣ 𝜉⊗𝜂).

Consider the cyclic projections 𝑒 = M ′1 𝜉 ∈ M 1 , 𝑒′ = M 1 𝜉 ∈ M ′1 , 𝑓 = M ′2 𝜂 ∈ M 2 , 𝑓 ′ = M 2 𝜂 ∈ M ′2 . Since 𝜉 = 𝑒𝜉 = 𝑒′ 𝜉, 𝜂 = 𝑓𝜂 = 𝑓 ′ 𝜂 and 𝑒⊗𝑓 ∈ M 1 ⊗M 2 , 𝑒′ ⊗𝑓 ′ ∈ M ′1 ⊗M ′2 , while 𝑇 ′ ∈ (M 1 ⊗M 2 )′ , 𝑇 ∈ (M ′1 ⊗M ′2 )′ , we are led to the following transformations: (𝑒⊗𝑓)(𝑒′ ⊗𝑓 ′ )𝑇𝑇 ′ (𝑒⊗𝑓)(𝑒′ ⊗𝑓 ′ ) = (𝑒⊗𝑓)(𝑒′ ⊗𝑓 ′ )𝑇(𝑒′ ⊗𝑓 ′ )(𝑒⊗𝑓)𝑇 ′ (𝑒⊗𝑓)(𝑒′ ⊗𝑓 ′ ) = (𝑒𝑒′ ⊗𝑓𝑓 ′ )𝑇(𝑒𝑒′ ⊗𝑓𝑓 ′ )(𝑒𝑒′ ⊗𝑓𝑓 ′ )𝑇 ′ (𝑒𝑒′ ⊗𝑓𝑓 ′ ), so that (4) will follow if we show that the operators (𝑒𝑒′ ⊗𝑓𝑓 ′ )𝑇(𝑒𝑒′ ⊗𝑓𝑓 ′ ) and (𝑒𝑒′ ⊗𝑓𝑓 ′ )𝑇 ′ (𝑒𝑒′ ⊗𝑓𝑓 ′ ) commute. They belong, respectively, to the following algebras: (𝑒𝑒′ ⊗𝑓𝑓 ′ )(M ′1 ⊗M ′2 )′ (𝑒𝑒′ ⊗𝑓𝑓 ′ ) = (𝑒𝑒′ M 1′ 𝑒′ 𝑒⊗𝑓𝑓 ′ M ′2 𝑓 ′𝑓)′ , (𝑒𝑒′ ⊗𝑓𝑓 ′ )(M 1 ⊗M 2 )′ (𝑒𝑒′ ⊗𝑓𝑓 ′ ) = (𝑒𝑒′ M ′1 𝑒′ 𝑒⊗𝑓𝑓 ′ M ′2 𝑓 ′𝑓)′ . These equalities refer to the elementary operations of reduction, induction, commutant and tensor product. Now 𝑒𝑒′ M 1 𝑒′ 𝑒 ⊂ B (𝑒𝑒′ H 1 ) and 𝑓𝑓 ′ M 2 𝑓 ′𝑓 ⊂ B (𝑓𝑓 ′ H 2 ) are von Neumann algebras with cyclic and separating vectors 𝜉 and 𝜂, respectively. In this case, we know that the Commutation theorem for tensor products holds, so we get (2) and hence (M 1 ⊗M 2 )′ = M ′1 ⊗M ′2 , for any von Neumann algebras M 1 ⊂ B (H 1 ), M 2 ⊂ B (H 2 ).

Q.E.D.

Corollary. Let M 1 , M 2 be von Neumann algebras, Z 1 , Z 2 their centers, respectively. Then the center of M 1 ⊗M 2 is Z 1 ⊗Z 2 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Proof. We denote by Z the center of M 1 ⊗M 2 . Obviously, Z 1 ⊗Z 2 ⊂ Z . On the other hand, the inclusions M 1 ⊗M 2 ⊂ Z ′ , M ′1 ⊗M ′2 ⊂ (M 1 ⊗M 2 )′ ⊂ Z

′

imply that Z ′1 ⊗Z

′ 2

= R (M 1 , M ′1 )⊗R (M 2 , M ′2 ) ⊂ Z ′ .

With the help of the preceding theorem, we obtain Z ⊂ (Z ′1 ⊗Z ′2 )′ = Z 1 ⊗Z 2 . Q.E.D. 10.8. In this section, we exhibit a class of ‘positive’ elements in D 𝑆 (respectively, in D 𝑆 ∗ ). Let 𝔄 ⊂ H be a left Hilbert algebra. For any vector 𝜉 ∈ H , we define the linear operator 𝐿0𝜉 by 𝐿0𝜉 (𝜂) = 𝑅𝜂 (𝜉), 𝜂 ∈ D (𝐿0 ) = 𝔄′ . 𝜉

We observe that 𝜉 ∈ 𝐿0𝜉 (𝔄′ ). 𝑠𝑜

Indeed, if {𝜂𝑖 } ⊂ 𝔄′ and 𝑅𝜂𝑖 → 1, then 𝐿0𝜉 (𝔄′ ) ∋ 𝐿0𝜉 𝜂𝑖 = 𝑅𝜂𝑖 𝜉 → 𝜉. If 𝐿0𝜉 is preclosed, we denote its closure by 𝐿𝜉 . By analogy with Lemma 1 from Section 10.2, one can show that if 𝐿0𝜉 is preclosed, then 𝐿𝜉 is affiliated to ℜ(𝔄′ )′ = 𝔏(𝔄). In Section 10.4, we saw that if 𝜉 ∈ D 𝑆 , then 𝐿0𝜉 is preclosed. Also, if 𝐿0𝜉 is symmetric, then 𝐿0𝜉 is preclosed (9.4). We write 𝔓𝑆 = {𝜉 ∈ H ; 𝐿0𝜉 is positive}. Let 𝜉 ∈ 𝔓𝑆 and let 𝐴 be the Friedrichs extension of 𝐿0𝜉 (9.6). In accordance with exercise E.9.27, 𝐴 is affiliated to 𝔏(𝔄). With the help of Proposition 10.4, it immediately follows that, for any function 𝑓 ∈ B ([0, +∞)), such that sup{𝜆2 |𝑓(𝜆)|; 𝜆 ∈ [0, +∞)} < +∞, we have 𝐴𝑓(𝐴)𝜉 ∈ 𝔄″ and ̄ 𝑆(𝐴𝑓(𝐴)𝜉) = 𝐴𝑓(𝐴)𝜉,

𝐿𝐴𝑓(𝐴)𝜉 = 𝐴2 𝑓(𝐴).

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Analogously, we denote 𝔓𝑆 ∗ = {𝜂 ∈ H ; 𝑅0𝜂 is positive}. Proposition. Let 𝔄 ⊂ H be a left Hilbert algebra. The following assertions regarding a vector 𝜉0 ∈ H are equivalent: (i) 𝜉0 ∈ 𝔓𝑆 . (ii) 𝜉0 ∈ D 𝑆 , 𝑆𝜉0 = 𝜉0 and 𝐿𝜉0 is positive. (iii) 𝜉0 belongs to the norm closure of the set {𝐿𝜉 𝑆𝜉; 𝜉 ∈ 𝔄″ }. (iv) (𝜉0 |𝜂) ≥ 0, for any 𝜂 ∈ 𝔓𝑆 ∗ . The following assertions regarding a vector 𝜂0 ∈ H are equivalent: (j) 𝜂0 ∈ 𝔓𝑆 ∗ . (jj) 𝜂0 ∈ D 𝑆 ∗ , 𝑆 ∗ 𝜂0 = 𝜂0 and 𝑅𝜂0 is positive. (jjj) 𝜂0 belongs to the norm closure of the set {𝑅𝜂 𝑆 ∗ 𝜂; 𝜂 ∈ 𝔄′ }. (jw) (𝜉|𝜂0 ) ≥ 0, for any 𝜉 ∈ 𝔓𝑆 . Proof. We first observe that, for any 𝜂 ∈ 𝔄′ , we have (1)

(𝜉0 |𝑅𝑆 ∗ 𝜂 𝜂) = (𝑅𝜂 𝜉0 |𝜂) = (𝐿0𝜉0 𝜂|𝜂). We prove that (i) ⇒ (ii). From condition (i) and relation (1), we get, for any 𝜂 ∈ 𝔄′ , (𝜉0 |𝑅𝑆 ∗ 𝜂 𝜂) ≥ 0.

Hence, for any 𝜂1 , 𝜂2 ∈ 𝔄′ , we have 3

1 (𝜉0 |𝑅𝑆 ∗ 𝜂1 𝜂2 ) = ∑ i𝑘 (𝜉0 |𝑅𝑆∗(𝜂1 +i𝑘 𝜂2 ) (𝜂1 + i𝑘 𝜂2 )) 4 𝑘=0 3

=

1 ∑ i𝑘 (𝑅𝑆∗(𝜂1 +i𝑘 𝜂2 ) (𝜂1 + i𝑘 𝜂2 )|𝜉0 ) 4 𝑘=0

= (𝑅𝑆 ∗ 𝜂2 𝜂1 |𝜉0 ) = (𝑆 ∗ 𝑅𝑆 ∗ 𝜂1 (𝜂2 )|𝜉0 ). Since 𝑆 ∗ |(𝔄′ )2 = 𝑆 ∗ (see 10.4), it follows that 𝜉0 ∈ D 𝑆 and 𝑆𝜉0 = 𝜉0 , thereby proving assertion (ii). We now show that (ii) ⇒ (iii). Let 𝐴 be the Friedrichs extension of 𝐿𝜉0 . For any natural number 𝑛, we consider the function 𝑓𝑛 ∈ B ([0, +∞)), defined by 𝑓𝑛 (𝜆) = 𝜆−3/2 𝜒(1/𝑛,𝑛) (𝜆). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Since lim 𝜆3 𝑓𝑛2 (𝜆) = 𝜒(0,∞) (𝜆),

𝑛→∞

from Theorem 9.11 (vi) and Corollary 9.13, (iii), it follows that 𝑠𝑜

𝐴3 𝑓𝑛2 (𝐴) → s(𝐴). On the other hand, we have 𝜉𝑛 = 𝐴𝑓𝑛 (𝐴)𝜉0 ∈ 𝔄″ and 𝑆𝜉𝑛 = 𝐴𝑓𝑛 (𝐴)𝜉0 ,

𝐿𝜉𝑛 = 𝐴2 𝑓𝑛 (𝐴) ≥ 0.

Since 𝜉0 ∈ 𝐿𝜉0 (𝔄′ ) ⊂ s(𝐴)H , we have 𝐿𝜉𝑛 𝑆𝜉𝑛 = 𝐴3 𝑓𝑛2 (𝐴)𝜉0 → 𝜉0 , and assertion (iii) is thus proved. The implications (j) ⇒ (jj) ⇒ (jjj) can be proved similary. Let 𝜉 ∈ 𝔄″ , 𝜂 ∈ 𝔄′ . Then (𝐿𝜉 𝑆𝜉|𝑅𝜂 𝑆 ∗ 𝜂) = (𝑅𝑆 ∗ 𝜂 𝐿𝜉 𝑆𝜉|𝑆 ∗ 𝜂) = (𝐿𝜉 𝑅𝑆 ∗ 𝜂 𝑆𝜉|𝑆 ∗ 𝜂) = (𝐿𝜉 𝐿∗𝜉 𝑆 ∗ 𝜂|𝑆 ∗ 𝜂) = ‖𝐿∗𝜉 𝑆 ∗ 𝜂‖2 ≥ 0. The implication (iii) ⇒ (iv) now follows by tending to the limit, and by taking into account the implication (j) ⇒ (jjj). Since {𝑅𝑆 ∗ 𝜂 𝜂; 𝜂 ∈ 𝔄′ } ⊂ 𝔓𝑆 ∗ , the implication (iv) ⇒ (i) obviously follows from relation (1). The implications (jjj) ⇒ (jw) ⇒ (j) can be proved similarly. Q.E.D. Corollary. Let 𝔄 be a left Hilbert algebra. Then (1) 𝔓𝑆 is a closed convex cone, included in D 𝑆 ; D 𝑆 is the linear hull of 𝔓𝑆 . (2) 𝔓𝑆 ∗ is a closed convex cone, included in D 𝑆 ∗ ; D 𝑆 ∗ is the linear hull of 𝔓𝑆 ∗ . Proof. The fact that 𝔓𝑆 is a closed convex cone, included in D 𝑆 , obviously follows from the preceding proposition. Let 𝜉 ∈ D 𝑆 . We must prove that 𝜉 is a linear combination of elements in 𝔓𝑆 . We can assume that 𝑆𝜉 = 𝜉. Then, by taking into account the fact that 𝑆|𝔄″ = 𝑆, it follows that for any 𝑛, there exists 𝜉𝑛 ∈ 𝔄″ , 𝑆𝜉𝑛 = 𝜉𝑛 , ‖𝜉𝑛 ‖ ≤ 1/2𝑛 , Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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∞

∑ 𝜉𝑛 = 𝜉. 𝑛=1

The operators 𝑎𝑛 = 𝐿𝜉𝑛 ∈ 𝔏(𝔄) are self-adjoint. In accordance with Corollary 2.10, we consider the decompositions 𝑎𝑛 = 𝑎𝑛+ − 𝑎𝑛− ; then 𝑎𝑛+ , 𝑎𝑛− ∈ 𝔏(𝔄), |𝑎𝑛 | = 𝑎𝑛+ + 𝑎𝑛− ,

𝑎𝑛+ , 𝑎𝑛− ≥ 0,

s(𝑎𝑛 ) = s(𝑎𝑛+ ) + s(𝑎𝑛− ),

s(𝑎𝑛+ )s(𝑎𝑛− ) = 0.

We define 𝜉𝑛+ = s(𝑎𝑛+ )𝜉𝑛 ,

𝜉𝑛− = s(𝑎𝑛− )𝜉𝑛 .

‖𝜉𝑛+ ‖ ≤ 1/2𝑛 ,

‖𝜉𝑛− ‖ ≤ 1/2𝑛 ,

We obviously have

𝜉𝑛 = s(𝑎𝑛 )𝜉𝑛 = s(𝑎𝑛+ )𝜉𝑛 + s(𝑎𝑛− )𝜉𝑛 = 𝜉𝑛+ − 𝜉𝑛− . It is easy to verify that 𝐿𝜉𝑛+ ⊂ s(𝑎𝑛+ )𝑎𝑛 = s(𝑎𝑛+ )|𝑎𝑛 | ≥ 0, 𝐿𝜉𝑛− ⊂ −s(𝑎𝑛− )𝑎𝑛 = s(𝑎𝑛− )|𝑎𝑛 | ≥ 0; hence 𝜉𝑛+ ∈ 𝔓𝑆 , We denote

𝜉𝑛− ∈ 𝔓𝑆

∞ +

𝜉 = ∑

∞

𝜉𝑛+ ,

−

𝜉 = ∑ 𝜉𝑛− .

𝑛=1

𝑛=1

Since 𝔓𝑆 is closed, we have 𝜉 + ∈ 𝔓𝑆 ,

𝜉 − ∈ 𝔓𝑆 ,

and obviously, 𝜉 = 𝜉+ − 𝜉−. We have thus proved assertion (1). Assertion (2) can be proved analogously.

Q.E.D.

The assertions (iv) and (jw) from the statement of the proposition show that 𝔓𝑆 and 𝔓𝑆 ∗ are cones ‘polar’ to one another. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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10.9. Let M ⊂ B (H ), be a von Neumann algebra and 𝜉0 a separating cyclic vector for M . In accordance with Section 10.6, 𝔄 = M 𝜉0 ⊂ H is canonically endowed with a structure of a left Hilbert algebra and 𝔄′ = M ′ 𝜉0 ,

𝔄″ = 𝔄.

′ For 𝜉, 𝜂 ∈ H , we consider the forms 𝜔𝜉,𝜂 ∈ M ∗ , 𝜔𝜉,𝜂 ∈ (M ′ )∗ given by

𝑥 ∈ M,

𝜔𝜉,𝜂 (𝑥) = (𝑥𝜉|𝜂),

𝑥′ ∈ M ′ ,

′ 𝜔𝜉,𝜂 (𝑥′ ) = (𝑥 ′ 𝜉|𝜂),

In particular, we have (see 5.22) 𝜔𝜉,𝜉 = 𝜔𝜉 ,

′ 𝜔𝜉,𝜉 = 𝜔𝜉′ ,

𝜉∈H.

It is obvious that, for any 𝜉, 𝜂 ∈ H , we have 𝑎 ∈ M,

𝑅𝑎 𝜔𝜉,𝜂 = 𝜔𝑎𝜉,𝜂 , ′ 𝑅𝑎′ 𝜔𝜉,𝜂 = 𝜔𝑎′ ′ 𝜉,𝜂 ,

𝑎′ ∈ M ′ .

For any 𝜉 ∈ H , the operator 𝐿0𝜉 is given by 𝐿0𝜉 (𝑥 ′ 𝜉0 ) = 𝑥 ′ 𝜉,

𝑥 ′ 𝜉0 ∈ D (𝐿0 ) = M ′ 𝜉0 . 𝜉

This operator is positive iff ′ 𝜔𝜉,𝜉 ≥ 0. 0

Consequently, with the notations from Section 10.8, we have 𝔓𝑆 = {𝜉 ∈ H ;

′ 𝜔𝜉,𝜉 ≥ 0}. 0

If 𝜉 ∈ 𝔓𝑆 and if we denote by 𝐴 the Friedrichs extension of 𝐿0𝜉 , then 𝐴 is a positive self-adjoint operator in H , affiliated to M , and 𝜉0 ∈ D 𝐴 ,

𝐴𝜉0 = 𝜉.

Conversely, if 𝐴 is a positive self-adjoint operator in H , affiliated to M , and if 𝜉0 ∈ D 𝐴 , then 𝐴𝜉0 ∈ 𝔓𝑆 , 𝐿𝐴𝜉0 ⊂ 𝐴. Thus 𝔓𝑆 = {𝐴𝜉0 ; A positive self-adjoint, affiliated to M , 𝜉0 ∈ D 𝐴 }.

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Lemma. Let M ⊂ B (H ) be a von Neumann algebra, with the separating cyclic vector to 𝜉0 ∈ H . For any normal form 𝜑 on M , there exists a unique vector 𝜉 ∈ 𝔓𝑆 , such that 𝜑 = 𝜔𝜉 . In particular, there exists a positive self-adjoint operator 𝐴 in H , affiliated to M , such that 𝜉0 ∈ D 𝐴 , 𝜑 = 𝜔𝐴𝜉0 . Proof. In order to prove the uniqueness of the vector 𝜉, let 𝜉1 , 𝜉2 ∈ 𝔓𝑆 be such that 𝜔𝜉1 = 𝜔𝜉2 . Then ‖𝑥𝜉1 ‖ = ‖𝑥𝜉2 ‖, 𝑥 ∈ M ; hence, there exists a partial isometry 𝑣′ ∈ M ′ , such that 𝑣′ 𝑥𝜉1 = 𝑥𝜉2 ,

𝑥 ∈ M,

𝑣′ ([M 𝜉1 ]⟂ ) = {0}. In particular, 𝑣′ 𝜉1 = 𝜉2 ,

𝑣 ′∗ 𝜉2 = 𝜉1 ,

and therefore 𝜔𝜉′ 2 ,𝜉0 = 𝜔𝑣′ ′ 𝜉1 ,𝜉0 = 𝑅𝑣′ 𝜔𝜉′ 1 ,𝜉0 , 𝜔𝜉′ 1 ,𝜉0 = 𝜔𝑣′ ′∗ 𝜉2 ,𝜉0 = 𝑅𝑣′∗ 𝜔𝜉′ 2 ,𝜉0 . Since 𝜉1 , 𝜉2 ∈ 𝔓𝑆 , we have 𝜔𝜉′ 1 ,𝜉0 ≥ 0, 𝜔𝜉′ 2 ,𝜉0 ≥ 0, and by taking into account the uniqueness of the polar decomposition (5.16), it follows that 𝜔𝜉′ 1 ,𝜉0 = 𝜔𝜉′ 2 ,𝜉0 . Since M ′ 𝜉0 = H , we infer that 𝜉1 = 𝜉2 . We now prove the existence of the vector 𝜉. In accordance with Theorem 5.23, there exists a vector 𝜁 ∈ H , such that 𝜑 = 𝜔𝜁 . Let now, in accordance with Theorem 5.16, ′ 𝜔𝜁,𝜉 = 𝑅𝑣′ 𝜓 ′ 0 ′ be the polar decomposition of the form 𝜔𝜁,𝜉 ∈ (M ′ )∗ . We define 0

𝜉 = 𝑣 ′∗ 𝜁. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Then ′ ′ 𝜔𝜉,𝜉 = 𝑅𝑣′∗ 𝜁,𝜉0 = 𝑅𝑣′∗ 𝜔𝜁,𝜉 = 𝜓′ ≥ 0 0 0

and consequently, we have 𝜉 ∈ 𝔓𝑆 . On the other hand, ′ ′ 𝜔𝜁,𝜉 = 𝑅𝑣′ 𝜓′ = 𝑅𝑣′ 𝜔𝜉,𝜉 = 𝜔𝑣′ ′ 𝜉,𝜉0 , 0 0

whence 𝜁 = 𝑣 ′ 𝜉. Thus, for any 𝑥 ∈ M , we have 𝜑(𝑥) = 𝜔𝜁 (𝑥) = (𝑥𝜁|𝜁) = (𝑥𝜁|𝑣 ′ 𝜉) = (𝑣 ′∗ 𝑥𝜁|𝜉) = (𝑥𝑣 ′∗ 𝜁|𝜉) = (𝑥𝜉|𝜉) = 𝜔𝜉 (𝑥). Consequently, we have 𝜑 = 𝜔𝜉 . Q.E.D. 10.10. Another important application of the theory developed so far is a general Radon–Nikodym-type theorem, which we shall present in this section. Let M ⊂ B (H ) be a von Neumann algebra and 𝜑 a normal form on M . Let 𝐴 be a positive self-adjoint operator in H , affiliated to M . We write (cf. 9.9): 𝑒𝑛 = 𝜒[0,𝑛] (𝐴) ∈ M . One says that 𝐴 is of summable square with respect to 𝜑 if 𝑐 = sup 𝜑(𝐴2 𝑒𝑛 ) < +∞; 𝑛

then 𝜑(𝐴2 𝑒𝑛 ) ↑ 𝑐. In this case, for any 𝑥 ∈ M and any 𝑚 > 𝑛, we have |𝜑(𝐴𝑒𝑚 𝑥𝐴𝑒𝑚 ) − 𝜑(𝐴𝑒𝑛 𝑥𝐴𝑒𝑛 )| ≤ |𝜑(𝐴𝑒𝑚 𝑥𝐴(𝑒𝑚 − 𝑒𝑛 ))| + |𝜑(𝐴(𝑒𝑚 − 𝑒𝑛 )𝑥𝐴𝑒𝑛 )| ≤ 𝜑(𝐴𝑒𝑚 𝑥𝑥 ∗ 𝐴𝑒𝑚 )1/2 𝜑(𝐴2 (𝑒𝑚 − 𝑒𝑛 ))1/2 + 𝜑(𝐴2 (𝑒𝑚 − 𝑒𝑛 ))1/2 𝜑(𝐴𝑒𝑛 𝑥 ∗ 𝑥𝐴𝑒𝑛 )1/2 ≤ 2‖𝑥‖𝑐1/2 [𝜑(𝐴2 𝑒𝑚 ) − 𝜑(𝐴2 𝑒𝑛 )]1/2 . Consequently, the sequence {𝜑(𝐴𝑒𝑛 𝑥𝐴𝑒𝑛 )} is fundamental. We write 𝐿𝐴 𝑅𝐴 𝜑(𝑥) = lim 𝜑(𝐴𝑒𝑛 𝑥𝐴𝑒𝑛 ), 𝑛→∞

𝑥 ∈ M.

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Therefore, we get a linear form 𝐿𝐴 𝑅𝐴 𝜑 on M . Tending to the limit for 𝑚 → ∞, in the preceding inequality, we get |(𝐿𝐴 𝑅𝐴 𝜑)(𝑥) − (𝐿𝐴𝑒𝑛 𝑅𝐴𝑒𝑛 𝜑)(𝑥)| ≤ 2𝑐1/2 [𝑐 − 𝜑(𝐴2 𝑒𝑛 )]1/2 ‖𝑥‖. Since 𝐿𝐴𝑒𝑛 𝑅𝐴𝑒𝑛 𝜑 ∈ (M ∗ )+ , it follows that 𝐿𝐴 𝑅𝐴 𝜑 ∈ (M ∗ )+ . By taking into account Theorem 9.11 (ii), it is obvious that if 𝜑 = 𝜔𝜉 ,

𝜉∈H,

then the operator 𝐴 is of summable square with respect to 𝜑 iff 𝜉 ∈ D 𝐴 . In this case, 𝐿𝐴 𝑅𝐴 𝜑 = 𝜔𝐴𝜉 . Theorem. Let 𝜑 and 𝜓 be normal forms on the von Neumann algebra M ⊂ B (H ), such that s(𝜑) ≤ s(𝜓). Then there exists a positive self-adjoint operator 𝐴 in H , affiliated to M , such that s(𝐴) ≤ s(𝜓) 𝜑 = 𝐿𝐴 𝑅𝐴 𝜓. Moreover, if the projection s(𝜓) is finite in M , then the operator 𝐴 is uniquely determined by the required properties. Proof. Without any loss of generality, we can assume that s(𝜓) = 1. From Proposition 5.18, we infer that there exists a *-isomorphism 𝜋 of M onto a von Neumann algebra N ⊂ B (H ), 𝜋(1) = 1, and a separating cyclic vector 𝜁0 ∈ K for N , such that 𝜓 = 𝜔𝜁0 ∘ 𝜋. Then 𝜑 = 𝜃 ∘ 𝜋, where 𝜃 is a normal form on the von Neumann algebra N ⊂ B (H ), uniquely determined by this relation. From Lemma 10.9, we infer that there exists a positive self-adjoint operator 𝐵 in K , affiliated to N , such that 𝜃 = 𝜔𝐵𝜁0 , Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Let 𝐴 = 𝜋 −1 (𝐵) be the positive self-adjoint operator in H , affiliated to M , which is obtained by transporting the operator 𝐵 by the *-isomorphism 𝜋 −1 (see Section 9.25). If we denote 𝑒𝑛 = 𝜒[0,𝑛] (𝐴), 𝑒𝑛 = 𝜒[0,𝑛] (𝐵), then we have 𝜓(𝐴2 𝑒𝑛 ) = 𝜓(𝜋 −1 (𝐵2 𝑓𝑛 )) = 𝜔𝜁0 (𝐵2 𝑓𝑛 ) = ‖𝑓𝑛 𝐵𝜁0 ‖2 ≤ ‖𝐵𝜁0 ‖2 , hence 𝐴 is of summable square with respect to 𝜓. Then (𝐿𝐴 𝑅𝐴 𝜓)(𝑥) = lim 𝜓(𝐴𝑒𝑛 𝑥𝐴𝑒𝑛 ) = lim 𝜓(𝜋 −1 (𝐵𝑒𝑛 )𝑥𝜋 −1 (𝐵𝑒𝑛 )) 𝑛→∞

𝑛→∞

= lim 𝜔𝜁0 (𝐵𝑒𝑛 𝜋(𝑥)𝐵𝑒𝑛 ) = 𝜔𝐵𝜁0 (𝜋(𝑥)) = 𝜃(𝜋(𝑥)) = 𝜑(𝑥), 𝑛→∞

𝑥 ∈ M,

hence 𝐿𝐴 𝑅𝐴 𝜓 = 𝜑. We have proved the first part of the theorem. Let us now assume that s(𝜓) is finite. We can assume again that s(𝜓) = 1, hence M is finite. Moreover, by considering the *-isomorphism 𝜋 and the possibility of transporting by 𝜋 the positive self-adjoint operators that are affiliated to M (see 9.25), we can assume that M has a separating cyclic vector 𝜉0 ∈ H and 𝜓 = 𝜔𝜁0 . Therefore let 𝐴 be a positive self-adjoint operator in H , which is affiliated to M , such that 𝜑 = 𝐿𝐴 𝑅𝐴 𝜓 = 𝜔𝐴𝜉0 . Then the vector 𝐴𝜉0 is uniquely determined by 𝜑 (see 10.9). On the other hand, 𝐴 is an extension of the positive operator 𝐿𝐴𝜉0 (see 10.9). Since M is finite, Corollary 9.8 implies that 𝐴 is, indeed, determined in a unique manner by 𝜑. Q.E.D. Another case in which the uniqueness of the operator 𝐴 holds is that in which the normal form 𝜑 is dominated (see E.9.33) by the normal form 𝜓. Indeed, in this case, it is easy to verify that the operator 𝐴 is bounded and one applies Theorem 5.21. 10.11. This section contains the main technical result for the subsequent development of the theory; with its help, a bridge between 𝔄″ and 𝔄′ is established. Proposition. Let 𝔄 ⊂ H be a left Hilbert algebra and 𝜆 ∈ ℂ\ℝ+ . Then, for any 𝜉 ∈ 𝔄″ , we have (𝜆 − Δ−1 )−1 𝜉 ∈ 𝔄′ and ‖𝑅(𝜆−∆−1 )−1 𝜉 ‖ ≤ 2−1/2 (|𝜆| − Re 𝜆)−1/2 ‖𝐿𝜉 ‖. Similarly, for any 𝜂 ∈ 𝔄′ , we have (𝜆 − Δ)−1 𝜂 ∈ 𝔄″ and ‖𝐿(𝜆−∆)−1 𝜂 ‖ ≤ 2−1/2 (|𝜆| − Re 𝜆)−1/2 ‖𝑅𝜂 ‖. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

The theory of standard von Neumann algebras

311

Proof. We shall prove only the first assertion, the proof of the second one being similar. We denote 𝜂 = (𝜆 − Δ−1 )−1 𝜉. Then 𝜂 ∈ D (∆−1 ) ⊂ D (∆−1/2 ) = D 𝑆 ∗ . We consider the polar decompositions 𝑅𝜂 = 𝑢𝐴 = 𝐵𝑢. With Lemma 1 from 10.2, we infer that 𝐴 and 𝐵 are affiliated to 𝔏(𝔄)′ . By taking into account Corollary 5 from 10.3, we infer that, for any function 𝑓 ∈ B ([0, +∞)), assumed only to be positive and with compact support, we have the following sequence of relations: ‖𝐿𝜉 ‖2 ‖𝑓(𝐴)𝑆 ∗ 𝜂‖2 ≥ ‖𝐿𝜉 𝑓(𝐴)𝑆 ∗ 𝜂‖2 = ‖𝑓(𝐴)𝐿𝜉 𝑆 ∗ 𝜂‖2 = ‖𝑓(𝐴)(𝑅𝜂 )∗ 𝜉‖2 = ‖𝑓(𝐴)𝐴𝑢∗ 𝜉‖2 = ‖𝐴𝑓(𝐴)𝑢∗ (𝜆 − Δ−1 )𝜂‖2 = |𝜆|2 ‖𝐴𝑓(𝐴)𝑢∗ 𝜂‖2 + ‖𝐴𝑓(𝐴)𝑢∗ Δ−1 𝜂‖2 − 2Re [𝜆(𝐴𝑓(𝐴)𝑢∗ 𝜂|𝐴𝑓(𝐴)𝑢∗ Δ−1 𝜂)] ≥ 2|𝜆|‖𝐴𝑓(𝐴)𝑢∗ 𝜂‖‖𝐴𝑓(𝐴)𝑢∗ Δ−1 𝜂‖ − 2Re [𝜆(𝐴𝑓(𝐴)𝑢∗ 𝜂|𝐴𝑓(𝐴)𝑢∗ Δ−1 𝜂)] ≥ 2|𝜆||(𝐴𝑓(𝐴)𝑢∗ 𝜂|𝐴𝑓(𝐴)𝑢∗ Δ−1 𝜂)| − 2Re [𝜆(𝐴𝑓(𝐴)𝑢∗ 𝜂|𝐴𝑓(𝐴)𝑢∗ Δ−1 𝜂)] = 2|𝜆||(𝐵2 𝑓 2 (𝐵)𝜂|𝑆𝑆 ∗ 𝜂)| − 2Re [𝜆(𝐵2 𝑓 2 (𝐵)𝜂|𝑆𝑆 ∗ 𝜂)] = 2|𝜆||(𝐴2 𝑓 2 (𝐴)𝑆 ∗ 𝜂|𝑆 ∗ 𝜂)| − 2Re [𝜆(𝐴2 𝑓 2 (𝐴)𝜂𝑆 ∗ 𝜂|𝑆 ∗ 𝜂)] = 2|𝜆|‖𝐴𝑓(𝐴)𝑆 ∗ 𝜂‖2 − 2Re [𝜆‖𝐴𝑓(𝐴)𝑆 ∗ 𝜂‖2 ] = 2(|𝜆| − Re 𝜆)‖𝐴𝑓(𝐴)𝑆 ∗ 𝜂‖2 . Consequently, 2−1/2 (|𝜆| − Re 𝜆)−1/2 ‖𝐿𝜉 ‖‖𝑓(𝐴)𝑆 ∗ 𝜂‖ ≥ ‖𝐴𝑓(𝐴)𝑆 ∗ 𝜂‖. We denote 𝑐 = 2−1/2 (|𝜆| − Re 𝜆)−1/2 ‖𝐿𝜉 ‖ and 𝑓𝑛 = 𝜒(𝑐+𝑛−1 ,𝑛) . From the above inequality, we successively obtain 𝑐‖𝑓𝑛 (𝐴)𝑆 ∗ 𝜂‖ ≥ ‖𝐴𝑓𝑛 (𝐴)𝑆 ∗ 𝜂‖ ≥ (𝑐 + 𝑛−1 )‖𝑓𝑛 (𝐴)𝑆 ∗ 𝜂‖, 𝑓𝑛 (𝐴)𝑆 ∗ 𝜂 = 0, 𝐴𝑓𝑛 (𝐴)𝑢∗ 𝜉 = 𝑓𝑛 (𝐴)(𝑅𝜂 )∗ 𝜉 = 𝐿𝜉 𝑓𝑛 (𝐴)𝑆 ∗ 𝜂 = 0, 𝜉 ∈ 𝔄, 𝐴𝑓𝑛 (𝐴)𝑢∗ = 0, 𝐴𝑓𝑛 (𝐴) = 0, 𝑓𝑛 (𝐴) = 0. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Thus 𝜒(𝑐,+∞) (𝐴) = so-lim 𝑓𝑛 (𝐴) = 0; 𝑛→∞

hence 𝐴 is bounded and ‖𝐴‖ ≤ 𝑐. It follows that 𝑅𝜂 is bounded; hence 𝜂 ∈ 𝔄′ , and ‖𝑅𝜂 ‖ ≤ 𝑐.

Q.E.D.

10.12. In this section, we present the fundamental theorem of Tomita. Lemma 1. Let 𝔄 ⊂ H be a left Hilbert algebra, 𝜉 ∈ 𝔄″ , 𝜆 > 0 and 𝜂 = (𝜆 + Δ−1 )−1 𝜉. Then 𝜂 ∈ 𝔄′ and the relation (𝐿𝜉 (𝜁1 )|𝜁2 ) = 𝜆(𝐽(𝑅𝜂 )∗ 𝐽Δ−1/2 𝜁1 |Δ1/2 𝜁2 ) + (𝐽(𝑅𝜂 )∗ 𝐽Δ1/2 𝜁1 |Δ−1/2 𝜁2 ) holds for any 𝜁1 , 𝜁2 ∈ D (∆1/2 ) ∩ D (∆−1/2 ) . Proof. Proposition 10.11 obviously implies that 𝜂 ∈ 𝔄′ . We shall first assume that 𝜁1 , 𝜁2 ∈ (1 + Δ−1 )−1 𝔄″ . Since 𝔄″ ⊂ D 𝑆 = D (∆1/2 ) , we have 𝜁1 , 𝜁2 ∈ Δ1/2 (Δ + 1)−1 (Δ1/2 𝔄″ ) ⊂ D (∆1/2 ) . On the other hand, from Proposition 10.11, we infer that 𝜁1 , 𝜁2 ∈ 𝔄′ ⊂ D 𝑆 ∗ = D (∆−1/2 ) . With the help of Corollary 2 from 10.3, we infer that the following sequence of equalities holds: (𝐿𝜉 (𝜁1 )|𝜁2 ) = (𝑅𝜁1 (𝜉)|𝜁2 ) = (𝜉|(𝑅𝜁1 )∗ |𝜁2 ) = ((𝜆 + Δ−1 )𝜂|(𝑅𝜁1 )∗ 𝜁2 ) = 𝜆(𝜂|(𝑅𝜁1 )∗ 𝜁2 ) + (𝑆𝑆 ∗ 𝜂|(𝑅𝑆 ∗ 𝜁1 (𝜁2 )) = 𝜆(𝜂|(𝑅𝜁1 )∗ 𝜁2 ) + (𝑆 ∗ 𝑅𝑆 ∗ 𝜁1 (𝜁2 )|𝑆 ∗ 𝜂) = 𝜆(𝜂|(𝑅𝜁1 )∗ 𝜁2 ) + (𝑅𝑆 ∗ 𝜁2 (𝜁1 )|𝑆 ∗ 𝜂) = 𝜆(𝑅𝜁1 (𝜂)|𝜁2 ) + (𝜁1 |𝑅𝜁2 𝑆 ∗ 𝜂) = 𝜆(𝑅𝜁1 (𝜂)|𝑆𝑆𝜁2 ) + (𝑆𝑆𝜁1 |𝑅𝜁2 𝑆 ∗ 𝜂) = 𝜆(𝑆𝜁2 |𝑆 ∗ 𝑅𝜁1 (𝜂)) + (𝑆 ∗ 𝑅𝜁2 𝑆 ∗ 𝜂|𝑆𝜁1 ) = 𝜆(𝑆𝜁2 |𝑅𝑆 ∗ 𝜂 𝑆 ∗ 𝜁1 ) + (𝑅𝜂 𝑆 ∗ 𝜁2 |𝑆𝜁1 ) = 𝜆(𝑆𝜁2 |(𝑅𝜂 )∗ 𝑆 ∗ 𝜁1 ) + (𝑆 ∗ 𝜁2 |(𝑅𝜂 )∗ 𝑆𝜁1 ) = 𝜆(𝐽Δ1/2 𝜁2 |(𝑅𝜂 )∗ 𝐽Δ−1/2 𝜁1 ) + (𝐽Δ−1/2 𝜁2 |(𝑅𝜂 )∗ 𝐽Δ1/2 𝜁1 ) = 𝜆(𝐽(𝑅𝜂 )∗ 𝐽Δ−1/2 𝜁1 |Δ1/2 𝜁2 ) + (𝐽(𝑅𝜂 )∗ 𝐽Δ1/2 𝜁1 |Δ−1/2 𝜁2 ). If we can prove that, for any 𝜁 ∈ D (∆1/2 ) ∩ D (∆−1/2 ) , there exists a sequence {𝜁𝑛 } ⊂ (1 + Δ−1 )−1 𝔄″ , such that 𝜁𝑛 → 𝜁,

Δ1/2 𝜁𝑛 → Δ1/2 𝜁,

Δ−1/2 𝜁𝑛 → Δ−1/2 𝜁,

then the assertion immediately follows from the preceding equalities. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Let then 𝜁 ∈ D (∆1/2 ) ∩ D (∆−1/2 ) . Since the set Δ1/2 𝔄″ = 𝐽𝑆𝔄″ = 𝐽𝔄″ is dense in H , there exists a sequence {𝜉𝑛 } ⊂ 𝔄″ , such that Δ1/2 𝜉𝑛 → Δ1/2 𝜁 + Δ−1/2 𝜁. If we write 𝜁𝑛 = (1 + Δ−1 )−1 𝜉𝑛 ∈ (1 + Δ−1 )−1 𝔄″ , we have 𝜁𝑛 = Δ−1/2 (1 + Δ−1 )−1 (Δ1/2 𝜉𝑛 ) → Δ−1/2 (1 + Δ−1 )−1 (Δ1/2 𝜁 + Δ−1/2 𝜁) = 𝜁, Δ1/2 𝜁𝑛 = (1 + Δ−1 )−1 (Δ1/2 𝜉𝑛 ) → (1 + Δ−1 )−1 (Δ1/2 𝜁 + Δ−1/2 𝜁) = Δ1/2 𝜁, Δ−1/2 𝜁𝑛 = Δ−1 (1 + Δ−1 )−1 (Δ1/2 𝜉𝑛 ) → Δ−1 (1 + Δ−1 )−1 (Δ1/2 𝜁 + Δ−1/2 𝜁) = Δ−1/2 𝜁 Q.E.D. Lemma 2. Let 𝔄 ⊂ H be a left Hilbert algebra and 𝜉 ∈ 𝔄″ . Then, for any 𝑡 ∈ ℝ, we have 𝐽Δi𝑡 𝜉 ∈ 𝔄′ and 𝑆 ∗ 𝐽Δi𝑡 𝜉 = 𝐽Δi𝑡 𝑆𝜉,

𝑅𝐽∆i𝑡 𝜉 = 𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽.

Proof. By taking into account Lemma 1 and Proposition 9.23, in which we make 𝐴 = 𝐵 = Δ, we infer that, for any 𝜆 > 0, we have (𝜆 + Δ−1 )−1 𝜉 ∈ 𝔄′ and +∞ ∗

𝐽(𝑅(𝜆+∆−1 )−1 𝜉 ) 𝐽 = ∫ −∞

i𝑡−

i.e. +∞

−i𝑡−

𝜆

∗

(𝑅(𝜆+∆−1 )−1 𝜉 ) = ∫ −∞

1

𝜆 2 Δi𝑡 𝐿𝜉 Δ−i𝑡 𝑑𝑡, 𝜋𝑡 e + e−𝜋𝑡 1 2

e𝜋𝑡 + e−𝜋𝑡

𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽 𝑑𝑡.

On the other hand, from Corollary 9.23, we infer that for any 𝜆 > 0 and any 𝜁 ∈ 𝔄, we have the equalities (𝑅(𝜆+∆−1 )−1 𝜉 )∗ 𝜁 = 𝐿𝜁 𝑆 ∗ (𝜆 + Δ−1 )−1 𝜉 +∞ −1/2

= 𝐿𝜁 𝐽Δ

−1 −1

(𝜆 + Δ ) 𝜉 = ∫ −∞

−i𝑡−

𝜆

1 2

e𝜋𝑡 + e−𝜋𝑡

𝐿𝜁 𝐽Δi𝑡 𝜉 𝑑𝑡.

Let 𝜁 ∈ 𝔄. From the preceding formulas, we infer that the equality +∞

∫ −∞

𝜆−i𝑡

1 (𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽𝜁 − 𝐿𝜁 𝐽Δi𝑡 𝜉) 𝑑𝑡 = 0 e𝜋𝑡 + e−𝜋𝑡

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holds for any 𝜆 > 0, i.e. the Fourier transform of the mapping 𝑡↦

1 (𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽𝜁 − 𝐿𝜁 𝐽Δi𝑡 𝜉) e𝜋𝑡 + e−𝜋𝑡

vanishes identically. Since the Fourier transform is injective, it follows that 𝐿𝜁 𝐽Δi𝑡 𝜉 = 𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽𝜁,

𝑡 ∈ ℝ.

Thus, for any 𝑡 ∈ ℝ and any 𝜁 ∈ 𝔄, we have 𝐿𝜁 (𝐽Δi𝑡 𝜉) = (𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽)𝜁; substituting 𝑆𝜉 for 𝜉, we find that 𝐿𝜁 (𝐽Δi𝑡 𝑆𝜉) = (𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽)∗ 𝜁. If we now apply Proposition 10.3, we infer that, for any 𝑡 ∈ ℝ, the following relations hold: 𝐽Δi𝑡 𝜉 ∈ 𝔄′ and 𝑆 ∗ 𝐽Δi𝑡 𝜉 = 𝐽Δi𝑡 𝑆𝜉,

𝑅𝐽∆i𝑡 𝜉 = 𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽. Q.E.D.

We now prove the fundamental theorem of Tomita: Theorem. Let 𝔄 ⊂ H be a left Hilbert algebra. Then (1) 𝐽𝔄″ = 𝔄′ and for any 𝜉 ∈ 𝔄″ , we have 𝑆 ∗ 𝐽𝜉 = 𝐽𝑆𝜉,

𝑅𝐽𝜉 = 𝐽𝐿𝜉 𝐽.

(2) Δi𝑡 𝔄″ = 𝔄″ , 𝑡 ∈ ℝ, and for any 𝜉 ∈ 𝔄″ , we have 𝑆Δi𝑡 𝜉 = Δi𝑡 𝑆𝜉,

𝐿∆i𝑡 𝜉 = Δi𝑡 𝐿𝜉 Δ−i𝑡 .

Similarly, (1′ ) 𝐽𝔄′ = 𝔄″ and for any 𝜂 ∈ 𝔄′ , we have 𝑆𝐽𝜂 = 𝐽𝑆 ∗ 𝜂,

𝐿𝐽𝜂 = 𝐽𝑅𝜂 𝐽;

(2′ ) Δi𝑡 𝔄′ = 𝔄′ , 𝑡 ∈ ℝ, and for any 𝜂 ∈ 𝔄′ , we have 𝑆 ∗ Δi𝑡 𝜂 = Δi𝑡 𝑆 ∗ 𝜂,

𝑅∆i𝑡 𝜂 = Δi𝑡 𝑅𝜂 Δ−i𝑡 .

Proof. If we apply Lemma 2 for 𝑡 = 0, it follows that 𝐽𝔄″ ⊂ 𝔄′ and for any 𝜉 ∈ 𝔄″ we have 𝑆 ∗ 𝐽𝜉 = 𝐽𝑆𝜉, 𝑅𝐽𝜉 = 𝐽𝐿𝜉 𝐽. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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315

Let 𝜂 ∈ 𝔄′ . Then, for any 𝜁 ∈ 𝔄′ and 𝜉 ∈ 𝔄″ , we have (𝑅𝜁 𝐽𝜂|𝜉) = (𝐽𝜂|𝑅𝑆 ∗ 𝜁 (𝜉)) = (𝐽𝜂|𝐿𝜉 𝑆 ∗ 𝜁) = (𝐿𝑆𝜉 𝐽𝜂|𝑆 ∗ 𝜁) = (𝐽𝑅𝐽𝑆𝜉 (𝜂)|𝑆 ∗ 𝜁) = (𝐽𝑅𝑆 ∗ 𝐽𝜉 (𝜂)|𝑆 ∗ 𝜁) = (𝐽𝑆 ∗ 𝑅𝑆 ∗ 𝜂 𝐽𝜉|𝐽Δ−1/2 𝜁) = (Δ−1/2 𝜁|Δ1/2 𝐽(𝑅𝜂 )∗ 𝐽𝜉) = (𝜁|𝐽(𝑅𝜂 )∗ 𝐽𝜉) = (𝐽𝑅𝜂 𝐽𝜁|𝜉). Thus, for any 𝜁 ∈ 𝔄′ , we have 𝑅𝜁 (𝐽𝜂) = (𝐽𝑅𝜂 𝐽)𝜁, and substituting 𝑆 ∗ 𝜂 for 𝜂, we get 𝑅𝜁 (𝐽𝑆 ∗ 𝜂) = (𝐽𝑅𝜂 𝐽)∗ 𝜁. If we now apply Proposition 10.4, we infer that 𝐽𝜂 ∈ 𝔄″ and 𝑆𝐽𝜂 = 𝐽𝑆 ∗ 𝜂,

𝐿𝐽𝜂 = 𝐽𝑅𝜂 𝐽.

Consequently, 𝐽𝔄″ ⊂ 𝔄′ = 𝐽(𝐽𝔄′ ) ⊂ 𝐽𝔄″ , i.e. 𝐽𝔄″ = 𝔄′ . Now let 𝜉 ∈ 𝔄″ and 𝑡 ∈ ℝ. From Lemma 2, we infer that 𝜂 = 𝐽Δi𝑡 𝜉 ∈ 𝔄′ and 𝑆 ∗ 𝐽Δi𝑡 𝜉 = 𝐽Δi𝑡 𝑆𝜉,

𝑅𝐽∆i𝑡 𝜉 = 𝐽Δi𝑡 𝐿𝜉 Δ−i𝑡 𝐽.

From the first part of the proof, we now infer that Δi𝑡 𝜉 = 𝐽𝜂 ∈ 𝔄″ and 𝑆Δi𝑡 𝜉 = 𝑆𝐽𝜂 = 𝐽𝑆 ∗ 𝜂 = Δi𝑡 𝑆𝜉, 𝐿∆i𝑡 𝜉 = 𝐿𝐽𝜂 = 𝐽𝑅𝜂 𝐽 = Δi𝑡 𝐿𝜉 Δ−i𝑡 . Obviously Δi𝑡 𝔄″ ⊂ 𝔄″ = Δi𝑡 (Δ−i𝑡 𝔄″ ) ⊂ Δi𝑡 𝔄″ , i.e. Δi𝑡 𝔄″ = 𝔄″ . We have thus proved assertions (1) and (2). Assertions (1′ ) and (2′ ) readily follow from these. On the other hand, it is clear that assertions (1′ ) and (2′ ) have direct proofs, analogous to those given for assertions (1) and (2). Q.E.D. According to the theorem, 𝐽 is the natural bridge linking 𝔄″ to 𝔄′ . Another such bridge is Δ1/2 : Δ1/2 𝔄″ = 𝐽𝐽Δ1/2 𝔄″ = 𝐽𝑆𝔄″ = 𝐽𝔄″ = 𝔄″ . The following corollary allows the construction of useful elements in 𝔄′ and 𝔄″ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Corollary. Let 𝔄 ⊂ H be a left Hilbert algebra and 𝑓 ∈ L 1 (ℝ). Then, for any 𝜉 ∈ 𝔄″ , +∞ we have 𝜉𝑓 = ∫−∞ 𝑓(𝑡)Δi𝑡 𝜉 𝑑𝑡 ∈ 𝔄″ and +∞

+∞ i𝑡 ̄ 𝑓(𝑡)Δ 𝑆𝜉 𝑑𝑡,

𝑆𝜉𝑓 = ∫

𝑓(𝑡)Δi𝑡 𝐿𝜉 Δ−i𝑡 𝑑𝑡.

𝐿𝜉𝑓 = ∫

−∞

−∞

+∞

Similarly, for any 𝜂 ∈ 𝔄′ , we have 𝜂𝑓 = ∫−∞ 𝑓(𝑡)Δi𝑡 𝜂 𝑑𝑡 ∈ 𝔄′ and +∞

+∞ i𝑡 ∗ ̄ 𝑓(𝑡)Δ 𝑆 𝜂 𝑑𝑡,

𝑆 ∗ 𝜂𝑓 = ∫ −∞

𝑅𝜂𝑓 = ∫

𝑓(𝑡)Δi𝑡 𝑅𝜂 Δ−i𝑡 𝑑𝑡.

−∞

Proof. Let

+∞

𝑓(𝑡)Δi𝑡 𝑆𝜉 𝑑𝑡.

𝜁=∫ −∞

From the preceding theorem, we infer that, for any 𝜃 ∈ 𝔄′ , we have +∞

+∞ i𝑡

𝑅𝜃 (𝜉𝑓 ) = ∫

𝑓(𝑡)𝑅𝜃 Δ 𝜉 𝑑𝑡 = ∫

−∞

+∞

𝑓(𝑡)Δi𝑡 𝐿𝜉 Δ−i𝑡 𝑑𝑡] (𝜃)

𝑓(𝑡)𝐿∆i𝑡 𝜉 (𝜃) 𝑑𝑡 = [∫

−∞

−∞

and +∞

𝑅𝜃 (𝜁) = ∫

+∞

̄ 𝑓(𝑡)𝑅 𝜃 𝑆Δ 𝜉 𝑑𝑡 = ∫ i𝑡

−∞

∗

+∞

̄ 𝑓(𝑡)(𝐿 ∆i𝑡 𝜉 ) (𝜃) 𝑑𝑡 = [∫ ∗

−∞

i𝑡

−i𝑡

𝑓(𝑡)Δ 𝐿𝜉 Δ

𝑑𝑡] (𝜃).

−∞

If we now apply Proposition 10.4, the first assertion follows. The second assertion can be proved similarly.

Q.E.D.

10.13. We now consider a left Hilbert algebra 𝔄 ⊂ H . Let 𝑧 ∈ 𝔏(𝔄) ∩ 𝔏(𝔄)′ and 𝜉 ∈ D 𝑆 . With the help of Corollary 3 from 10.3, we infer that, for any 𝜂1 , 𝜂2 ∈ 𝔄′ , we have (𝑧𝜉|𝑅𝜂1 (𝜂2 )) = (𝜉|𝑅𝜂1 𝑧∗ (𝜂2 )) = (𝑆𝑆𝜉|𝑅𝜂1 𝑧∗ (𝜂2 )) = (𝑆 ∗ 𝑅𝜂1 𝑧∗ (𝜂2 )|𝑆𝜉) = (𝑅𝑆 ∗ 𝜂2 𝑧𝑆 ∗ 𝜂1 |𝑆𝜉) = (𝑧𝑅𝑆 ∗ 𝜂2 𝑆 ∗ 𝜂1 |𝑆𝜉) = (𝑆 ∗ 𝑅𝜂1 (𝜂2 )|𝑧∗ 𝑆𝜉). If we now take into account relation 10.4 (1), we infer that (𝑧𝜉|𝜂) = (𝑆 ∗ 𝜂|𝑧∗ 𝑆𝜉),

𝜂 ∈ D 𝑆∗ ;

consequently, we have 𝑧𝜉 ∈ D 𝑆 and 𝑆𝑧𝜉 = 𝑧∗ 𝑆𝜉. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Let 𝑢 ∈ 𝔏(𝔄) ∩ 𝔏(𝔄)′ be a unitary element and 𝜉 ∈ D ∆ ⊂ D (∆1/2 ) = D 𝑆 . From the preceding results, we infer that 𝑢𝜉 ∈ D 𝑆 = D (∆1/2 ) and Δ1/2 𝑢𝜉 = 𝐽𝑆𝑢𝜉 = 𝐽𝑢∗ 𝑆𝜉 = 𝐽𝑢∗ 𝐽Δ1/2 𝜉. Thus, for any 𝜁 ∈ D (∆1/2 ᵆ) = D (∆1/2 ) , we have (Δ𝜉|𝜁) = (Δ1/2 𝜉|Δ1/2 𝜁) = (𝐽𝑢∗ 𝐽Δ1/2 𝜉|𝐽𝑢∗ 𝐽Δ1/2 𝜁) = (Δ1/2 𝑢𝜉|Δ1/2 𝑢𝜁); hence Δ1/2 𝑢𝜉 ∈ D (∆1/2 ᵆ)∗ = D (ᵆ∗ ∆1/2 ) and 𝑢∗ Δ𝑢𝜉 = Δ𝜉. Consequently, the modular operator Δ is affiliated to the von Neumann algebra (𝔏(𝔄) ∩ 𝔏(𝔄)′ )′ = R (𝔏(𝔄), 𝔏(𝔄)′ ). Corollary 1. Let 𝔄 ⊂ H be a left Hilbert algebra. Then 𝔏(𝔄) ∋ 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 ∈ B (H ) is a *-anti-isomorphism of 𝔏(𝔄) onto 𝔏(𝔄)′ , which acts identically on the center. Proof. From Theorem 10.12, we infer that, for any 𝜉 ∈ 𝔄″ , we have 𝐽𝑆𝜉 ∈ 𝔄′ and 𝐽(𝐿𝜉 )∗ 𝐽 = 𝐽𝐿𝑆𝜉 𝐽 = 𝑅𝐽𝑆𝜉 ∈ R (𝔄′ ). On the other hand, from the same theorem, we infer that for any 𝜂 ∈ 𝔄′ , we have 𝑆𝐽𝜂 ∈ 𝔄″ and 𝐽(𝐿𝑆𝐽𝜂 )∗ 𝐽 = 𝐽𝐿𝐽𝜂 𝐽 = 𝑅𝜂 . Thus, the mapping in the statement of the corollary is a *-anti-isomorphism of 𝔏(𝔄) = 𝔏(𝔄″ ) onto ℜ(𝔄′ ) = 𝔏(𝔄)′ . From the discussion at the beginning of this section, we infer that, for any 𝑧 ∈ 𝔏(𝔄) ∩ 𝔏(𝔄)′ and any 𝜉 ∈ D 𝑆 , we have 𝐽𝑧∗ 𝐽𝜉 = 𝐽𝑧∗ Δ1/2 𝑆𝜉 = 𝐽Δ1/2 𝑧∗ 𝑆𝜉 = 𝑆𝑧∗ 𝑆𝜉 = 𝑧𝜉, and this shows that the considered mapping acts identically on the center.

Q.E.D.

Corollary 2. Let 𝔄 ⊂ H be a left Hilbert algebra. Then the formula 𝜎𝑡 (𝑥) = Δi𝑡 𝑥Δ−i𝑡 yields a so-continuous group of *-automorphisms {𝜎𝑡 }𝑡∈ ℝ of 𝔏(𝔄), which acts identically on the center. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Proof. According to Theorem 10.12, for any 𝜉 ∈ 𝔄″ and any 𝑡 ∈ ℝ, we have Δi𝑡 𝜉 ∈ 𝔄″ and 𝜎𝑡 (𝐿𝜉 ) = Δi𝑡 𝐿𝜉 Δ−i𝑡 = 𝐿∆i𝑡 𝜉 ∈ 𝔏(𝔄). Thus, {𝜎𝑡 }𝑡∈ ℝ is a so-continuous group of *-automorphisms of 𝔏(𝔄). At the beginning of this section, we saw that Δ is affiliated to the commutant of the center of 𝔏(𝔄), whence it obviously follows that any *-automorphism 𝜎𝑡 acts identically on the center. Q.E.D. The so-continuous group {𝜎𝑡 }𝑡∈ ℝ of *-automorphisms of 𝔏(𝔄) is called the group of the modular automorphisms of 𝔏(𝔄), associated to the left Hilbert algebra 𝔄. 10.14. In Section 10.6 to any von Neumann algebra M , with a separating cyclic vector, we associated a left Hilbert algebra, such that M = 𝔏(𝔄). By taking into account Section 5.18, this association can be described in the following equivalent manner: to any von Neumann algebra M , of countable type, and to any faithful normal form 𝜑 on M , we associated a left Hilbert algebra 𝔄𝜑 ⊂ H 𝜑 , such that 𝜋𝜑 (M ) = 𝔏(𝔄𝜑 ). In this section, we extend the above association to the case of arbitrary von Neumann algebras, and to some ‘unbounded forms’, called weights. Let M be a von Neumann algebra. A mapping 𝜑 ∶ M + → [0, +∞] = ℝ+ ∪ {+∞} is called a weight if 𝜑(𝑎 + 𝑏) = 𝜑(𝑎) + 𝜑(𝑏),

(1) (2)

𝑎, 𝑏 ∈ M + ,

𝜑(𝜆𝑎) = 𝜆𝜑(𝑎), 𝑎 ∈ M + , 𝜆 ≥ 0. [With the convention that 0(+∞) = 0.]

From condition (1), it follows that 𝑎, 𝑏 ∈ M + ,

𝑎 ≤ 𝑏 ⇒ 𝜑(𝑎) ≤ 𝜑(𝑏).

For any weight 𝜑 on M + , one defines 𝔉𝜑 = {𝑎 ∈ M + ; 𝜑(𝑎) < +∞}. It is obvious that 𝔉𝜑 is a face (see 3.21) and therefore, by taking into account Proposition 3.21, it follows that 𝔑𝜑 = {𝑥 ∈ M ; 𝜑(𝑥 ∗ 𝑥) < +∞} is a left ideal of M , and 𝔐𝜑 = 𝔑∗𝜑 𝔑𝜑 ⊂ 𝔑𝜑 ∩ 𝔑∗𝜑 Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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319

is a *-subalgebra of M , such that 𝔐+ 𝜑 = 𝔉𝜑 , 𝔐𝜑 = the linear hull of 𝔉𝜑 . From the latter property, it easily follows that 𝜑 uniquely extends to a positive linear form on 𝔐𝜑 , which is also denoted by 𝜑. The weight 𝜑 is said to be semifinite if (3)

𝔐𝜑 is w-dense in M . The weight 𝜑 is said to be faithful if

(4)

𝑎 ∈ M +,

𝜑(𝑎) = 0 ⇒ 𝑎 = 0.

We shall say that the weight 𝜑 is normal if (5)

there exists a family {𝜑i } of normal forms on M , such that 𝑎 ∈ M +.

𝜑(𝑎) = ∑ 𝜑i (𝑎), i

It is easy to see that if 𝜑 is normal, then 𝜑 is lower w-semicontinuous. In particular, for any family {𝑎𝛼 } ⊂ M + , which is increasingly directed and bounded, we have 𝜑(sup 𝑎𝛼 ) = sup 𝜑(𝑎𝛼 ). 𝛼

𝛼

The construction by which to any positive linear form one can associate a *-representation (see 5.18) can be extended to weights. Let 𝜑 be a faithful semifinite normal weight on M + . Since 𝜑 is faithful, the positive sesquilinear form (𝑥|𝑦)𝜑 = 𝜑(𝑦 ∗ 𝑥), 𝑥, 𝑦 ∈ 𝔑𝜑 , is a scalar product on 𝔑𝜑 . We shall denote by H 𝜑 the Hilbert space obtained by the completion of 𝔑𝜑 and for any 𝑥 ∈ 𝔑𝜑 , we shall denote by 𝑥𝜑 ∈ H 𝜑 the image of 𝑥 through the canonical embedding of 𝔑𝜑 in H 𝜑 . Any element 𝑥 ∈ M determines an operator 𝜋𝜑 (𝑥) ∈ B (H 𝜑 ), given by the relations 𝜋𝜑 (𝑥)𝑦𝜑 = (𝑥𝑦)𝜑 ,

𝑦 ∈ 𝔑𝜑 .

It is easy to verify that 𝜋𝜑 ∶ M → B (H 𝜑 ) is a *-representation. Since 𝜑 is normal, the same argument as that used in the proof of Proposition 5.18 shows that the *-representation 𝜋𝜑 is w-continuous. It is clear that 𝜋𝜑 (1) = 1; hence 𝜋𝜑 (M ) is a von Neumann algebra. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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From the faithfulness and the semifiniteness of the weight 𝜑, it follows that 𝜋𝜑 is injective; hence 𝜋𝜑 is a *-isomorphism of the von Neumann algebra M onto the von Neumann algebra 𝜋𝜑 (M ). Since 𝜑 is semifinite, Proposition 3.21 shows that there exists a family {𝑢𝛼 } ⊂ 𝔐𝜑 , such that 𝑢𝛼 ↑ 1. Then, for any 𝑥 ∈ ℜ𝜑 , we have ‖𝑥𝜑 − 𝜋𝜑 (𝑢𝛼 )𝑥𝜑 ‖2𝜑 = ‖𝑥𝜑 − (𝑢𝛼 𝑥)𝜑 ‖2𝜑 = 𝜑((𝑥 − 𝑢𝛼 𝑥)∗ (𝑥 − 𝑢𝛼 𝑥)) (*) ≤ 2[𝜑(𝑥 ∗ 𝑥) − 𝜑(𝑥 ∗ 𝑢𝛼 𝑥)] → 0. We hence infer that 𝜋𝜑 (𝑢𝛼 ) ↑ 1. ∗ On the other hand, since 𝑥 ∈ 𝔑𝜑 and 𝑢𝛼 ∈ 𝔐+ 𝜑 imply that 𝑢𝛼 𝑥 ∈ 𝔑𝜑 𝔑𝜑 = 𝔐𝜑 , from relation (*), we infer that 𝔐𝜑 is densely embedded in H 𝜑 . Similarly, since 𝑥 ∈ 𝔐𝜑 and 𝑢𝛼 ∈ 𝔐𝜑 , imply that 𝑢𝛼 𝑥 ∈ 𝔐2𝜑 , from the same relation, we infer that 𝔐2𝜑 is dense in 𝔐𝜑 , with respect to the topology corresponding to the scalar product we have just defined. Consequently 𝔐2𝜑 is densely embedded in H 𝜑 . Theorem. Let M be a von Neumann algebra and 𝜑 a faithful, semifinite, normal weight on M + . Then 𝔑𝜑 ∩ 𝔑∗𝜑 , endowed with the structure of *-algebra induced by M , and with the scalar product induced by that of H 𝜑 , is a left Hilbert algebra 𝔄𝜑 ⊂ H 𝜑 , and the following relations hold: 𝔄𝜑 = 𝔄″𝜑 , 𝜋𝜑 (M ) = 𝔏(𝔄𝜑 ), 2

⎧ ‖𝜉‖ , if there exists 𝑎 𝜉 ∈ 𝔄, such that 𝜑(𝑎) = 𝜋𝜑 (𝑎)1/2 = 𝐿𝜉 . ;𝑎 ∈ M+ ⎨ ⎩ +∞, in the contrary case. Proof. Conditions (i) and (ii) from 10.1 are easy to verify. Since 𝔐𝜑 ⊂ 𝔑𝜑 ∩ 𝔑∗𝜑 , and since 𝔐2𝜑 is densely embedded in H 𝜑 , it follows that condition (iii) from 10.1 is also satisfied. In order to verify condition (iv) from 10.1, we must prove that for any net {𝑥𝛼 } ⊂ 𝔑𝜑 ∩ ∗ 𝔑𝜑 , the following implication holds: 𝜑(𝑥𝛼∗ 𝑥𝛼 ) → 0 𝜑((𝑥𝛼 − 𝑥𝛽 )(𝑥𝛼 − 𝑥𝛽 )∗ ) →𝛼,𝛽 0

} ⇒ 𝜑(𝑥𝛼 𝑥𝛼∗ ) →𝛼 0.

From Sections 5.18 and 10.6, we infer that such an implication is true for faithful normal forms. Since 𝜑 is normal, there exists an increasingly directed family {𝜑𝜈 } of normal Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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The theory of standard von Neumann algebras forms on M , such that 𝜑(𝑎) = sup 𝜑𝜈 (𝑎),

𝑎 ∈ M +.

𝜈

If we denote by 𝑒𝜈 ∈ M the support of 𝜑𝜈 , the restriction of 𝜑𝜈 to 𝑒𝜈 M 𝑒𝜈 is faithful. From the hypotheses of the implication we must prove, it follows that 𝜑𝜈 (𝑒𝜈 𝑥𝛼 𝑒𝜈 )∗ (𝑒𝜈 𝑥𝛼 𝑒𝜈 )) ⟶ 0, 𝛼 𝜑𝜈 ((𝑒𝜈 𝑥𝛼 𝑒𝜈 − 𝑒𝜈 𝑥𝛽 𝑒𝜈 )(𝑒𝜈 𝑥𝛼 𝑒𝜈 − 𝑒𝜈 𝑥𝛽 𝑒𝜈 )∗ ) ⟶ 0, 𝛼, 𝛽 and therefore, 𝜑𝜈 (𝑥𝛼 𝑒𝜈 𝑥𝛼∗ ) = 𝜑𝜈 ((𝑒𝜈 𝑥𝛼 𝑒𝜈 )(𝑒𝜈 𝑥𝛼 𝑒𝜈 )∗ ) ⟶ 0. 𝛼 Since 𝜑 is faithful, we have 𝑒𝜈 ↑𝜈 1. For any 𝛼, 𝛽 and 𝜈 ≤ 𝜇, we have 𝜑𝜈 (𝑥𝛼 𝑒𝜇 𝑥𝛼∗ )1/2 ≤ 𝜑𝜇 (𝑥𝛼 𝑒𝜇 𝑥𝛼∗ )1/2 ≤ 𝜑𝜇 ((𝑥𝛼 − 𝑥𝛽 )𝑒𝜇 (𝑥𝛼 − 𝑥𝛽 )∗ )1/2 + 𝜑𝜇 (𝑥𝛽 𝑒𝜇 𝑥𝛽∗ )1/2 ≤ 𝜑((𝑥𝛼 − 𝑥𝛽 )(𝑥𝛼 − 𝑥𝛽 )∗ )1/2 + 𝜑𝜇 (𝑥𝛽 𝑒𝜇 𝑥𝛽∗ )1/2 . Let 𝜀 > 0. Then there exists an 𝛼𝜀 , such that for any 𝛼, 𝛽 ≥ 𝛼𝜀 , we have 𝜑((𝑥𝛼 − 𝑥𝛽 )(𝑥𝛼 − 𝑥𝛽 )∗ )1/2 ≤ 𝜀. Then 𝜑𝜈 (𝑥𝛼 𝑒𝜇 𝑥𝛼∗ )1/2 ≤ 𝜀 + 𝜑𝜇 (𝑥𝛽 𝑒𝜇 𝑥𝛽∗ )1/2 ,

𝛼, 𝛽 ≥ 𝛼𝜀 ,

𝜈 ≤ 𝜇,

By tending to the limit with respect to 𝛽, from this relation, we get 𝜑𝜈 (𝑥𝛼 𝑒𝜇 𝑥𝛼∗ )1/2 ≤ 𝜀,

𝛼 ≥ 𝛼𝜀 ,

𝜈 ≤ 𝜇,

and, by tending to the limit with respect to 𝜇, we obtain 𝜑𝜈 (𝑥𝛼 𝑥𝛼∗ )1/2 ≤ 𝜀,

𝛼 ≥ 𝛼𝜀 , any 𝜈.

Finally, if we now compute the least upper bound, from this inequality, we get 𝜑(𝑥𝛼 𝑥𝛼∗ )1/2 ≤ 𝜀,

𝛼 ≥ 𝛼𝜀 .

Consequently, condition (iv) from 10.1 is verified. Since 𝔑𝜑 ∩ 𝔑∗𝜑 ⊃ 𝔐𝜑 is w-dense in M , 𝜋𝜑 (𝔑𝜑 ∩ 𝔑∗𝜑 ) = {𝐿𝑥𝜑 ; 𝑥𝜑 ∈ 𝔄𝜑 } is w-dense in 𝜋𝜑 (M ). Therefore, 𝜋𝜑 (M ) = 𝔏(𝔄𝜑 ). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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By considering on 𝔑𝜑 the positive sesquilinear form (𝑥𝜑 , 𝑦𝜑 ) ↦ 𝜑𝜈 (𝑦 ∗ 𝑥) and by observing that it is bounded from above by the scalar product in H 𝜑 , it follows that there exists an 𝑎𝜈′ ∈ B (H 𝜑 ), 0 ≤ 𝑎𝜈′ ≤ 1, such that (𝑎𝜈′ 𝑥𝜑 |𝑦𝜑 )𝜑 = 𝜑𝜈 (𝑦 ∗ 𝑥),

𝑥, 𝑦 ∈ ℜ𝜑 .

It is easy to verify that 𝑎𝜈′ ∈ (𝜋𝜑 (M )′ ). 𝑠𝑜

Let {𝑢𝛼 } ⊂ 𝔐𝜑 be such that 𝑢𝛼 → 1 and ‖𝑢𝛼 ‖ ≤ 1. Then ‖(𝑎𝜈′ )1/2 (𝑢𝛼 )𝜑 − (𝑎𝜈′ )1/2 (𝑢𝛽 )𝜑 ‖2𝜑 = 𝜑𝜈 ((𝑢𝛼 − 𝑢𝛽 )∗ (𝑢𝛼 − 𝑢𝛽 )) →𝛼,𝛽 0; hence, there exists a vector 𝜂𝜈 ∈ H , such that (𝑎𝜈′ )1/2 (𝑢𝛼 )𝜑 → 𝜂𝜈 . For any 𝑥, 𝑦 ∈ 𝔑𝜑 , we have (𝜋𝜑 (𝑥)𝜂𝜈 |(𝑎𝜈′ )1/2 𝑦𝜑 )𝜑 = lim(𝜋𝜑 (𝑥)(𝑎𝜈′ )1/2 (𝑢𝛼 )𝜑 |(𝑎𝜈′ )1/2 𝑦𝜑 )𝜑 𝛼

= lim((𝑎𝜈′ ((𝑥𝑢𝛼 )𝜑 |𝑦𝜑 )𝜑 = lim 𝜑𝜈 (𝑦 ∗ 𝑥𝑢𝛼 ) 𝛼

𝛼

= 𝜑𝜈 (𝑦 ∗ 𝑥) = ((𝑎𝜈′ )1/2 𝑥𝜑 |(𝑎𝜈′ )1/2 𝑦𝜑 )𝜑 . Since 𝜋𝜑 (𝑥)𝜂𝜈 = lim𝛼 (𝑎𝜈′ )1/2 ((𝑥𝑢𝛼 )𝜑 ) ∈ (𝑎𝜈′ )1/2 H 𝜑 , we infer that 𝜋𝜑 (𝑥)𝜂𝜈 = (𝑎𝜈′ )1/2 (𝑥𝜑 ),

𝑥 ∈ 𝔑𝜑 .

With the help of Proposition 10.3, one shows that 𝜂𝜈 ∈ (𝔄𝜑 )′ , 𝑆 ∗ 𝜂𝜈 = 𝜂𝜈 and 𝑅𝜂𝜈 = (𝑎𝜈′ )1/2 . Let 𝜉 ∈ (𝔄𝜑 )″ . Since 𝐿𝜉 ∈ 𝔏(𝔄𝜑 ) = 𝜋𝜑 (M ), there exists an 𝑥 ∈ M , such that 𝐿𝜉 = 𝜋𝜑 (𝑥). For any 𝜈, we have 𝜑𝜈 (𝑥∗ 𝑥) = ‖𝜋𝜑 (𝑥)𝜂𝜈 ‖2 = ‖𝐿𝜉 (𝜂𝜈 )‖2 = ‖𝑅𝜂𝜈 (𝜉)‖2 ≤ ‖𝜉‖2 ; hence, 𝜑(𝑥 ∗ 𝑥) ≤ ‖𝜉‖2 < +∞. Similarly, one can prove that 𝜑(𝑥𝑥 ∗ ) = ‖𝑆𝜉‖2 < +∞. Consequently, 𝑥 ∈ 𝔑𝜑 ∩ 𝔑∗𝜑 , and therefore, 𝑥𝜑 ∈ 𝔄𝜑 . Since 𝐿𝜉 = 𝜋𝜑 (𝑥) = 𝐿𝑥𝜑 , we infer that 𝜉 = 𝑥𝜑 ∈ 𝔄𝜑 . The proof of the formula, given in the statement of the theorem, for 𝜑, offers no difficulties. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

The theory of standard von Neumann algebras

323

We remark that any von Neumann algebra M has a faithful, semifnite normal weight. Indeed, if {𝜑i } is a maximal family of normal forms on M , whose supports are mutually orthogonal, then the formula 𝜑(𝑎) = ∑ 𝜑i (𝑎),

𝑎 ∈ M +,

i

yields a faithful, semifinite normal weight on M + . Thus, any von Neumann algebra is *-isomorphic to a von Neumann algebra of the form 𝔏(𝔄), where 𝔄 is a left Hilbert algebra. 10.15. A von Neumann algebra M ⊂ B (H ) is said to be standard if there exists a conjugation 𝐽 ∶ H → H , such that the mapping 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 be a *-anti-isomorphism of M onto M ′ , which acts identically on the center. For particular cases of standard von Neumann algebras, the reader is referred to theorems 6.7, 6.8, 6.10, 6.11, to exercises E.7.15, E.7.16, E.7.17, E.7.18, E.7.19 and E.8.5. To the same end, exercises E.3.9, E.3.10, E.6.9 are also useful. In accordance with Section 10.6 and Corollary 1 from Section 10.13, any von Neumann algebra with a separating cyclic vector is standard. Conversely, let M ⊂ B (H ) be a standard von Neumann algebra of countable type and 𝐽 the corresponding conjugation. In accordance with Lemma 7.18, there exist 𝜉, 𝜂 ∈ H , such that the projections 𝑝𝜉 and 𝑝𝜂′ be central, and 𝑝𝜉 + 𝑝𝜂′ = 1. From exercise E.6.9, we infer that ′ 𝑝𝐽𝜉 = 𝐽𝑝𝜉 𝐽 = 𝑝𝜉 ,

𝑝𝐽𝜂 = 𝐽𝑝𝜂′ 𝐽 = 𝑝𝜂′ ;

hence 𝐽𝜉 + 𝜂 is a cyclic vector, whereas 𝜉 + 𝐽𝜂 is a separating vector for M . In accordance with exercise E.6.3, it follows that M has a separating cyclic vector. Thus, the standard von Neumann algebras of countable type are precisely the von Neumann algebras with a separating cyclic vector. In accordance with Proposition 5.18, any von Neumann algebra of countable type is *-isomorphic to a standard von Neumann algebra, whereas, in accordance with Corollary 5.25, any *-isomorphism between two standard von Neumann algebras of countable type is spatial. The following result extends these statements to the general case: Corollary. Any von Neumann algebra is *-isomorphic to a standard von Neumann algebra, and any *-isomorphism between two standard von Neumann algebras is spatial. Proof. In accordance with the remark at the end of section 10.14, and with Corollary 1 from Section 10.13, any von Neumann algebra is *-isomorphic to a standard von Neumann algebra. By taking into account Theorem 4.17 and Proposition 8.5, the second assertion of the statement can be considered separately, for the finite, respectively the uniform, von Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Lectures on von Neumann algebras

Neumann algebras. In the first case, using Lemma 7.2, the assertion is reduced to the case of the von Neumann algebras of countable type, whereas in the second case one applies Theorem 8.6, and one uses the fact that the uniformity (8.6) of a standard von Neumann algebra is equal to the uniformity of its commutant. Q.E.D. 10.16. In this section, we begin a construction inverse to that developed in Section 10.14. More precisely, to any left Hilbert algebra 𝔄, we associate a function 𝜑 𝔄 ∶ 𝔏(𝔄)+ → ℝ+ ∪ {+∞}, which measures the ‘weight’ of 𝔄″ in the operators belonging to 𝔏(𝔄): 2 ″ ⎧ ‖𝜉‖ , if there exists a 𝜉 ∈ 𝔄 , such that 𝜑 𝔄 (𝑎) = (𝑎)1/2 = 𝐿𝜉 . ; 𝑎 ∈ 𝔏(𝔄)+ ⎨ ⎩ +∞, in the contrary case.

We first prove that 𝜑 𝔄 is increasing: (1)

𝑎, 𝑏 ∈ 𝔏(𝔄)+ ,

𝑎 ≤ 𝑏 ⇒ 𝜑 𝔄 (𝑎) ≤ 𝜑 𝔄 (𝑏).

If 𝜑 𝔄 (𝑏) = +∞, then the implication is trivially true. We now assume that 𝑏1/2 = 𝐿𝜁 , 𝜁 ∈ 𝔄″ . It is easy to see that the relations 𝑥(𝑏1/2 𝜂) = 𝑎1/2 𝜂, 𝑥(𝜃) = 0,

𝜂∈H,

𝜃 ∈ [𝑏1/2 H ]⟂ ,

determine an operator 𝑥 ∈ 𝔏(𝔄), ‖𝑥‖ ≤ 1, such that 𝑎1/2 = 𝑥𝑏1/2 . We denote 𝜉 = 𝑥𝜁. Since, for any 𝜂 ∈ 𝔄′ , we have 𝑅𝜂 (𝜉) = 𝑅𝜂 𝑥(𝜁) = 𝑥𝑅𝜂 (𝜁) = 𝑥𝐿𝜁 (𝜂) = 𝑥𝑏1/2 (𝜂) = 𝑎1/2 (𝜂), Proposition 10.4 implies that 𝜉 = 𝔄″ and 𝐿𝜉 = 𝑎1/2 . Thus 𝜑 𝔄 (𝑎) = ‖𝜉‖2 ≤ ‖𝑥‖2 ‖𝜁‖2 ≤ ‖𝜁‖2 = 𝜑 𝔄 (𝑏). We now prove that 𝜑 𝔄 is additive (2)

𝜑 𝔄 (𝑎 + 𝑏) = 𝜑 𝔄 (𝑎) + 𝜑 𝔄 (𝑏),

𝑎, 𝑏 ∈ 𝔏(𝔄)+ .

Since 𝜑 𝔄 is increasing, if 𝜑 𝔄 (𝑎) = +∞ or 𝜑 𝔄 (𝑏) = +∞, then relation (2) is obviously satisfied. Thus we can assume that 𝑎1/2 = 𝐿𝛾 ,

𝑏1/2 = 𝐿𝛿 ,

𝛾, 𝛿 ∈ 𝔄″ .

Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

325

The theory of standard von Neumann algebras The equalities 𝑥((𝑎 + 𝑏)1/2 𝜂) = 𝑎1/2 𝜂, 𝑥(𝜃) = 0,

𝜃 ∈ [(𝑎 + 𝑏)

𝑦((𝑎 + 𝑏)1/2 𝜂) = 𝑏1/2 𝜂, 𝑦(𝜃) = 0,

𝜂∈H, 1/2

H ]⟂ ,

𝜂∈H,

𝜃 ∈ [(𝑎 + 𝑏)1/2 H ]⟂ ,

determine the operators 𝑥, 𝑦 ∈ 𝔏(𝔄), ‖𝑥‖, ‖𝑦‖ ≤ 1, such that 𝑎1/2 = 𝑥(𝑎 + 𝑏)1/2 and 𝑏1/2 = 𝑦(𝑎 + 𝑏)1/2 . Since (𝑎 + 𝑏)1/2 (𝑥∗ 𝑥 + 𝑦 ∗ 𝑦)(𝑎 + 𝑏)1/2 = 𝑎 + 𝑏, the positive operator (𝑥 ∗ 𝑥 + 𝑦 ∗ 𝑦)1/2 is isometric on [(𝑎 + 𝑏)1/2 ]H and obviously, vanishes on [(𝑎 + 𝑏)1/2 H ]⟂ . Thus 𝑥 ∗ 𝑥 + 𝑦 ∗ 𝑦 = s(𝑎 + 𝑏). We denote 𝜉 = 𝑥 ∗ 𝛾 + 𝑦 ∗ 𝛿. For any 𝜂 ∈ 𝔄′ , we have 𝑅𝜂 (𝜉) = 𝑥 ∗ 𝑅𝜂 (𝛾) + 𝑦 ∗ 𝑅𝜂 (𝛿) = (𝑥 ∗ 𝐿𝛾 + 𝑦 ∗ 𝐿𝛿 ) (𝜂) = (𝑥 ∗ 𝑎1/2 + 𝑦 ∗ 𝑏1/2 )(𝜂) = (𝑥 ∗ 𝑥 + 𝑦 ∗ 𝑦)(𝑎 + 𝑏)1/2 (𝜂) = (𝑎 + 𝑏)1/2 (𝜂). With the help of Proposition 10.4, we infer that 𝜉 ∈ 𝔄″ ,

𝑆𝜉 = 𝜉,

𝐿𝜉 = (𝑎 + 𝑏)1/2 .

For any 𝜂 ∈ 𝔄′ , we have 𝑅𝜂 (𝑥𝜉) = 𝑥𝑅𝜂 (𝜉) = 𝑥𝐿𝜉 (𝜂) = 𝑥(𝑎 + 𝑏)1/2 (𝜂) = 𝑎1/2 (𝜂) = 𝐿𝛾 (𝜂) = 𝑅𝜂 (𝛾); 𝑠𝑜

hence, making 𝑅𝜂 → 1, it follows that 𝑥𝜉 = 𝛾. Similarly, 𝑦𝜉 = 𝛿. Since 𝜉 ∈ 𝐿𝜉 H = s(𝑎 + 𝑏)H (see 10.5), we have 𝜑 𝔄 (𝑎 + 𝑏) = ‖𝜉‖2 = ((𝑥 ∗ 𝑥 + 𝑦 ∗ 𝑦)𝜉|𝜉) = ‖𝑥𝜉‖2 + ‖𝑦𝜉‖2 = ‖𝛾‖2 + ‖𝛿‖2 = 𝜑 𝔄 (𝑎) + 𝜑 𝔄 (𝑏). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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It is obvious that (3)

𝜑 𝔄 (𝜆𝑎) = 𝜆𝜑 𝔄 (𝑎),

𝑎 ∈ 𝔏(𝔄)+ ,

𝜆 > 0.

From relations (2) and (3), we infer that 𝜑 𝔄 is a weight on 𝔏(𝔄)+ , which is called the weight associated to the left Hilbert algebra 𝔄. With the help of Theorem 10.14 it is easy to verify that if M is a von Neumann algebra and 𝜑 is a faithful, semifinite, normal weight on M + , and if we consider the weight 𝜑𝔄𝜑 , associated to the left Hilbert algebra 𝔄𝜑 , then 𝜑 = 𝜑𝔄𝜑 ∘ 𝜋𝜑 . For the von Neumann algebra 𝔏(𝔄) and the weight 𝜑 𝔄 , we shall denote briefly (see Section 10.14): 𝔉𝔄 = 𝔉𝜑𝔄 , 𝔑𝔄 = 𝔑𝜑𝔄 , 𝔐𝔄 = 𝔐𝜑𝔄 . We now show that 𝜑 𝔄 measures indeed the ‘weight’ of 𝔄″ in the operators belonging to 𝔏(𝔄). More precisely, (4)

for any operator 𝑥 ∈ 𝔏(𝔄), we have the equivalence: there exists a 𝜉 ∈ 𝔄″ , such that 𝑥 = 𝐿𝜉 ⇔ 𝑥 ∈ 𝔑 ∩ 𝔑∗ ; moreover, if 𝜉, 𝜁 ∈ 𝔄″ , then (𝐿𝜁 )∗ 𝐿𝜉 ∈ 𝔐𝔄 and 𝜑 𝔄 ((𝐿𝜁 )∗ 𝐿𝜉 ) = (𝜉|𝜁).

Indeed, let 𝑥 = 𝑣|𝑥|, 𝑥 ∗ = 𝑣 ∗ |𝑥 ∗ | be the polar decompositions. Assuming that 𝑥 = 𝐿𝜉 , 𝜉 ∈ 𝔄″ , for any 𝜂 ∈ 𝔄′ , we have 𝑅𝜂 (𝑣 ∗ 𝜉) = 𝑣 ∗ 𝑅𝜂 (𝜉) = 𝑣 ∗ 𝐿𝜉 (𝜂) = |𝑥|(𝜂), 𝑅𝜂 (𝑣𝑆𝜉) = 𝑣𝑅𝜂 (𝑆𝜉) = 𝑣(𝐿𝜉 )∗ (𝜂) = |𝑥 ∗ |(𝜂), and with the help of Proposition 10.4, we infer that 𝑣 ∗ 𝜉 ∈ 𝔄″ , 𝑣𝑆𝜉 ∈ 𝔄″ ,

𝑆(𝑣 ∗ 𝜉) = 𝑣 ∗ 𝜉, 𝑆(𝑣𝑆𝜉) = 𝑣𝑆𝜉,

𝐿𝑣∗ 𝜉 = |𝑥|, 𝐿𝑣𝑆𝜉 = |𝑥 ∗ |.

Thus 𝜑 𝔄 (𝑥∗ 𝑥) = 𝜑 𝔄 (|𝑥|2 ) = ‖𝑣 ∗ 𝜉‖2 = ‖𝜉‖2 < +∞, 𝜑 𝔄 (𝑥𝑥 ∗ ) = 𝜑 𝔄 (|𝑥∗ |2 ) = ‖𝑣𝑆𝜉‖2 = ‖𝑆𝜉‖2 < +∞, i.e. 𝑥 ∈ 𝔑𝔄 ∩ 𝔑∗𝔄 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Conversely, if 𝑥 ∈ 𝔑𝔄 ∩ 𝔑∗𝔄 , then there exist 𝜉, 𝜁 ∈ 𝔄″ , such that |𝑥| = 𝐿𝜉 , |𝑥 ∗ | = 𝐿𝜁 . For any 𝜂 ∈ 𝔄′ , we have 𝑅𝜂 (𝑣𝜉) = 𝑣𝑅𝜂 (𝜉) = 𝑣|𝑥|(𝜂) = 𝑥(𝜂), 𝑅𝜂 (𝑣∗ 𝜉) = 𝑣 ∗ 𝑅𝜂 (𝜁) = 𝑣 ∗ |𝑥∗ |(𝜂) = 𝑥 ∗ (𝜂), and with the help of Proposition 10.4, we infer that 𝑣𝜉 ∈ 𝔄″ ,

𝑆(𝑣𝜉) = 𝑣 ∗ 𝜉,

𝐿𝑣𝜉 = 𝑥.

Let 𝜉, 𝜁 ∈ 𝔄″ . From the first part of the proof, it follows that 𝐿𝜉 and 𝐿𝜁 belong to 𝔑𝔄 ∩ 𝔑∗𝔄 ; hence (𝐿𝜁 )∗ 𝐿𝜉 ∈ 𝔐𝜂 . With the help of the polarization identity, 3

(𝐿𝜁 )∗ 𝐿𝜉 = 4−1 ∑ i𝑘 (𝐿𝜁+i𝑘 𝜉 )∗ 𝐿𝜁+i𝑘 𝜉 , 𝑘=0

one easily obtains the equality 𝜑 𝔄 ((𝐿𝜁 )∗ 𝐿𝜁 ) = (𝜉|𝜁). From assertion (4), one easily infers that 𝜑 𝔄 is semifinite, i.e. (5)

+ + 𝔐+ 𝔄 = {𝑎 ∈ 𝔏(𝔄) , 𝜑 𝔄 (𝑎) < +∞} is w-dense in 𝔏(𝔄) .

It is immediately verified that 𝜑 𝔄 is faithful, i.e. (6)

𝑎 ∈ 𝔏(𝔄)+ , 𝜑 𝔄 (𝑎) = 0 ⇒ 𝑎 = 0.

The normality of 𝜑 𝔄 is a more difficult problem and it will be proved later (see 10.18). Here we shall prove only that 𝜑 𝔄 is lower w-semicontinuous. To this end it is sufficient to prove that the set {𝑎 ∈ 𝔏(𝔄)+ ; ‖𝑎‖ ≤ 𝑐, 𝜑 𝔄 (𝑎) ≤ 𝜆} ̆ is so-closed, where 𝑐 and 𝜆 are arbitrary positive constants (one uses the Krein–Smulian theorem, see C.l.l, and Corollary 1.5). Let {𝑎𝑖 } be a net in the above set, which is so-convergent to 𝑎 ∈ 𝔏(𝔄)+ . For any 𝑖, there exists a 𝜉𝑖 ∈ 𝔄″ , such that 𝑎𝑖1/2 = 𝐿𝜉𝑖 , and ‖𝜉𝑖 ‖2 ≤ 𝜆. Since 𝐿𝑆𝜉𝑖 = (𝐿𝜉𝑖 )∗ = 𝐿𝜉𝑖 , and since the representation 𝐿 is injective (10.5), it follows that 𝑆𝜉𝑖 = 𝜉𝑖 . The set 𝑠𝑜

{𝑓 ∈ C ([0, 𝑐]); 𝑓(𝑎𝑖 ) → 𝑓(𝑎)} is closed in the norm topology and contains all polynomials; hence, it coincides with C ([0, 𝑐]). Consequently, 𝑠𝑜

𝐿𝜉𝑖 = 𝑎𝑖1/2 → 𝑎1/2 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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With the help of Corollary 3 from Section 10.4, it follows that there exists a 𝜉 ∈ 𝔄″ , such that 𝑎𝑙/2 = 𝐿𝜉 and 𝜉𝑖 → 𝜉, weakly. Thus, 𝜑 𝔄 (𝑎) = ‖𝜉‖2 ≤ lim ‖𝜉𝑖 ‖2 ≤ 𝜆. 𝑖

In particular, we have proved that (7)

for any bounded, increasingly directed family {𝑎𝑖 } ⊂ 𝔏(𝔄)+ one has 𝜑 𝔄 (sup 𝑎𝑖 ) = sup 𝜑 𝔄 (𝑎𝑖 ). 𝑖

𝑖

With the help of Theorem 10.12, one easily infers that 𝜑 𝔄 is invariant with respect to the modular automorphisms 𝜎𝑡 (⋅) = Δi𝑡 ⋅ Δ−i𝑡 , 𝑡 ∈ ℝ: (8)

𝑎 ∈ 𝔏(𝔄)+ , 𝑡 ∈ ℝ.

𝜑 𝔄 (𝜎𝑡 (𝑎)) = 𝜑 𝔄 (𝑎),

At the end of this section, we shall prove two other useful properties of the pair (𝔏(𝔄), 𝜑 𝔄 ), which are important for their own sake. In accordance with a remark we have made above, these will be properties shared by any pair (M , 𝜑), consisting of a von Neumann algebra M , and a faithful, semifinite, normal weight 𝜑 on M + . We denote by 𝔏(𝔄)∞ the set of all elements 𝑥 ∈ 𝔏(𝔄), such that the mapping i𝑡 ↦ 𝜎𝑡 (𝑥) = Δi𝑡 𝑥Δ−i𝑡 has an entire analytic continuation. It is easy to see that this continuation takes values in 𝔏(𝔄). Obviously, any fixed element for the group of modular automorphisms belongs to 𝔏(𝔄)∞ . In particular, 𝔏(𝔄)∞ contains the center of 𝔏(𝔄). It is easy to verify that 𝔏(𝔄)∞ is a subalgebra of 𝔏(𝔄). If 𝐹 is the entire analytic continuation of i𝑡 ↦ 𝜎𝑡 (𝑥), then 𝛼 ↦ 𝐹(−𝛼)̄ ∗ is an entire analytic continuation of i𝑡 ↦ 𝜎𝑡 (𝑥∗ ). It follows that 𝔏(𝔄)∞ is self-adjoint. On the other hand, if 𝑥 ∈ 𝔏(𝔄) and if we define +∞ 2

e−𝑛𝑡 𝜎𝑡 (𝑥) 𝑑𝑡,

𝑥𝑛 = √𝑛/𝜋 ∫

𝑛 ∈ ℕ,

−∞

then the mapping +∞

𝛼 ↦ √𝑛/𝜋 ∫

2

e−𝑛(𝑡+i𝛼) 𝜎𝑡 (𝑥) 𝑑𝑡,

𝑛 ∈ ℕ,

−∞

is an entire analytic continuation of the mapping is ↦ 𝜎𝑠 (𝑥𝑛 ); hence 𝑥𝑛 ∈ 𝔏(𝔄)∞ ,

𝑛 ∈ ℕ.

By taking into account the so-continuity of the mapping 𝑡 ↦ 𝜎𝑡 (𝑥) and by applying the Lebesgue dominated convergence theorem, one easily infers that 𝑠𝑜

𝑥𝑛 → 𝑥. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

329

The theory of standard von Neumann algebras Consequently, 𝔏(𝔄)∞ is a so-dense *-subalgebra 𝔏(𝔄).

We observe that the structure of the entire analytic continuation of the mapping i𝑡 ↦ 𝜎𝑡 (𝑥), 𝑥 ∈ 𝔏(𝔄)∞ , is given by Proposition 9.24. We now prove that 𝜉 ∈ 𝔄″ , 𝑥 ∈ 𝔏(𝔄)∞ ⇒ 𝑥𝜉 ∈ 𝔄″ ,

𝐿𝑥𝜉 = 𝑥𝐿𝜉 .

Indeed, since the mapping it i𝑡 ↦ Δi𝑡 𝑥Δ−i𝑡 has an entire analytic continuation, whereas the mapping i𝑡 ↦ Δi𝑡 𝜉 has a continuous extension to the set {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1/2}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1/2} (in accordance with Corollary 9.21, because 𝜉 ∈ D (∆1/2 ) ), it follows that the mapping i𝑡 ↦ Δi𝑡 𝑥𝜉 = (Δi𝑡 𝑥Δ−i𝑡 )Δi𝑡 𝜉 has a continuous extension to {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1/2}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1/2}. With the help of Corollary 9.21, we infer that 𝑥𝜉 ∈ D (∆1/2 ) = D 𝑆 . On the other hand, for any 𝜂 ∈ 𝔄′ , we have ‖𝐿𝑥𝜉 (𝜂)‖ = ‖𝑅𝜂 𝑥𝜉‖ = ‖𝑥𝑅𝜂 (𝜉)‖ = ‖𝑥𝐿𝜉 (𝜂)‖ ≤ ‖𝑥𝐿𝜉 ‖‖𝜂‖, and therefore, 𝑥𝜉 ∈ 𝔄″ . Obviously, 𝐿𝑥𝜉 = 𝑥𝐿𝜉 . We are now able to prove that 𝔏(𝔄)∞ 𝔐𝔄 𝔏(𝔄)∞ = 𝔐𝔄 .

(9)

To this end, it is sufficient to verify the inclusion 𝔏(𝔄)∞ 𝔉𝔄 ⊂ 𝔐𝔄 . Let then 𝑥 ∈ 𝔏(𝔄)∞ , and 𝑎 ∈ 𝔉𝔄 . There exists a 𝜉 ∈ 𝔄″ , such that 𝑎1/2 = 𝐿𝜉 . From the implication we have proved above, we infer that 𝑥𝜉 ∈ 𝔄″ and 𝑥𝑎1/2 = 𝑥𝐿𝜉 = 𝐿𝑥𝜉 ∈ 𝔑∗𝔄 . Since 𝑎1/2 = 𝐿𝜉 ∈ 𝔑𝔄 , it follows that 𝑥𝑎 = (𝑥𝑎1/2 )𝑎1/2 ∈ 𝔑∗𝔄 𝔑𝔄 = 𝔐𝔄 . The invariance property expressed by relation (9) allows the construction of an increasingly directed family of elements in 𝔐+ 𝔄 , whose supremum is 1, of a very particular nature: (10) there exists a family{𝑒𝑖 }𝑖∈𝐼 ⊂ 𝔏(𝔄) of mutually orthogonal projections of countable type, such that 𝜎𝑡 (𝑒𝑖 ) = 𝑒𝑖 , 𝑖 ∈ 𝐼, 𝑡 ∈ ℝ, and such that, for any 𝑖 ∈ 𝐼, (∗) there exists a sequence {𝑎𝑖,𝑛 }𝑛≥1 ⊂ 𝔐+ 𝔄 , such that 𝑎𝑖,𝑛 ↑ 𝑒𝑖 and for any 𝑛 ≥ 1 and any rational 𝑟, there exists an integer 𝑚(𝑛, 𝑟) ≥ 1, such that 𝜎𝑟 (𝑎𝑖,𝑛 ) ≤ 𝑎𝑖,𝑚(𝑛,𝑟) . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Indeed, let {𝑒𝑖 }𝑖∈𝐼 ⊂ 𝔏(𝔄) be a maximal family with the above property. We assume that 𝑒 = 1 − ∑𝑖 𝑒𝑖 ≠ 0. From assertion (5), we infer that there exists a 𝑏 ∈ 𝔐+ 𝔄 , such that 𝑎 = 𝑒𝑏𝑒 ≠ 0. Since 𝜎𝑡 (𝑒) = 𝑒, 𝑡 ∈ ℝ, we have 𝑒 ∈ 𝔏(𝔄)∞ , and from assertion (9), we infer that 𝑎 ∈ 𝔐+ 𝔄 . Let {𝑟𝑛 }𝑛≥1 be an enumeration of the set of rational numbers. We define −1

𝑛

𝑎0,𝑛 = (𝑛−1 + ∑ 𝜎𝑟𝑗 (𝑎)) 𝑗=1

𝑛

∑ 𝜎𝑟𝑗 (𝑎),

𝑛 ≥ 1.

𝑗=1

It is easy to verify that the sequence {𝑎0,𝑛 } is increasing and so-convergent to the projection ∞ 𝑒0 = ⋁𝑗=1 s(𝜎𝑟𝑗 (𝑎)). If 𝑛 ≥ 1 and 𝑟 is rational, then we choose 𝑚(𝑛, 𝑟) ≥ 1, such that {𝑟𝑗 }1≤ 𝑗≤𝑚(𝑛,𝑟) ⊃ {𝑟 + 𝑟𝑗 }1≤ 𝑗≤𝑛 . Then, for any rational 𝑟 and any 𝑛 ≥ 1, we have 𝜎𝑟 (𝑎0,𝑛 ) ≤ 𝑎0,𝑚(𝑛,𝑟) . It follows that for any rational 𝑟, we have 𝜎𝑟 (𝑒0 ) ≤ 𝑒0 . Consequently, for any 𝑡 ∈ ℝ, 𝜎𝑟 (𝑒0 ) = 𝑒0 . Let {𝑓𝛼 }𝛼∈Γ be a family of mutually orthogonal projections in 𝔏(𝔄), such that ∑𝛼 𝑓𝛼 ≤ 𝑒0 . For any 𝑛 ≥ 1, we have, in accordance with assertion (7), ∑ 𝜑 𝔄 ((𝑎0,𝑛 )1/2 𝑓𝛼 𝑛(𝑎0,𝑛 )1/2 ) = 𝜑 𝔄 (∑(𝑎0,𝑛 )1/2 𝑓𝛼 (𝑎0,𝑛 )1/2 ) = 𝜑 𝔄 (𝑎0,𝑛 ) < +∞. 𝛼

𝛼

With the help of assertion (6), we infer that there exists an at most countable subset Γ𝑛 ⊂ Γ, such that 𝛼 ∉ Γ𝑛 ⇒ (𝑎0,𝑛 )1/2 𝑓𝛼 (𝑎0,𝑛 )1/2 = 0. If 𝛼 ∉ ⋃𝑛 Γ𝑛 , then

𝑓𝛼 = wo-lim(𝑎0,𝑛 )1/2 𝑓𝑛 (𝑎0,𝑛 )1/2 = 0. 𝑛

We have thus proved that 𝑒0 is a projection of countable type. Consequently, 0 ≠ 𝑒0 ≤ 1 − ∑𝑖 𝑒𝑖 has property (*), thus contradicting the maximality of the family {𝑒𝑖 }𝑖∈𝐼 . It follows that ∑ 𝑒𝑖 = 1. 𝑖∈𝐼

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331

The theory of standard von Neumann algebras In the last part of this section, we shall prove the following assertion: (11)

there exists a faithful, semifinite normal weight 𝜑, on 𝔏(𝔄)+ , such that (i) 𝜑 ≤ 𝜑 𝔄 . (ii)

𝜑(𝑎) = 𝜑 𝔄 (𝑎), for any 𝑎 ∈ 𝔐+ 𝔄.

(iii) 𝜑(𝜎𝑡 (𝑎)) = 𝜑(𝑎), for any 𝑎 ∈ 𝔏(𝔄)+ , 𝑡 ∈ ℝ. The proof of the fact that 𝜑 𝔄 is normal will consist in showing that conditions (i)–(iii) from (11) imply that 𝜑 = 𝜑 𝔄 (see 10.18). In order to prove assertion (11), we shall first observe that to 𝔄′ one can also associate a weight 𝜑𝔄′ on ℜ(𝔄′ )+ , namely 2 ′ ⎧ ‖𝜂‖ , if there exists an 𝜂 ∈ 𝔄 , such that 𝜑𝔄′ (𝑎′ ) = (𝑎′ )1/2 = 𝑅𝜂 , ; 𝑎′ ∈ ℜ(𝔄′ )+ . ⎨ ⎩ +∞, in the contrary case,

It is easy to verify that 𝜑𝔄′ has properties analogous to properties (1)–(10), already proved for 𝜑 𝔄 . In what follows we shall actually prove that the pair (ℜ(𝔄′ ), 𝜑𝔄′ ) has the property analogous to property (11). We shall denote by {𝜎𝑡′ } the group of the modular automorphisms of ℜ(𝔄′ ): 𝜎𝑡′ (𝑥′ ) = Δ−i𝑡 𝑥 ′ Δi𝑡 ,

𝑥 ′ ∈ ℜ(𝔄′ ),

𝑡 ∈ ℝ.

Let {𝑎𝑖,𝑛 } ∈ 𝔐𝔄+ be the elements from (10). For any 𝑖 and 𝑛, there exists a 𝜉𝑖,𝑛 ∈ 𝔄″ , such that (𝑎𝑖,1 )1/2 = 𝐿𝜉𝑖,1 ; (𝑎𝑖,𝑛 − 𝑎𝑖,𝑛−1 )1/2 = 𝐿𝜉𝑖,𝑛 , 𝑛 ≥ 2. Obviously, ∑ ∑(𝐿𝜉𝑖,𝑛 )2 = 1. 𝑖 ′

𝑛

′ +

We define a weight 𝜑 on ℜ(𝔄 ) by the formula 𝜑′ (𝑎′ ) = ∑ ∑ 𝜔𝜉𝑖,𝑛 (𝑎′ ), 𝑖

𝑎′ ∈ ℜ(𝔄′ )+ .

𝑛

If 𝑎′ ∈ ℜ(𝔄′ )+ and 𝜑𝔄′ (𝑎′ ) < +∞, then there exists an 𝜂 ∈ 𝔄′ , such that (𝑎′ )1/2 = 𝑅𝜂 ; hence 𝜔𝜉𝑖,𝑛 (𝑎′ ) = ‖𝑅𝜂 (𝜉𝑖,𝑛 )‖2 = ‖𝐿𝜉𝑖,𝑛 (𝜂)‖2 = 𝜔𝜂 ((𝐿𝜉𝑖,𝑛 )2 ). and therefore 𝜑′ (𝑎′ ) = ∑ ∑ 𝜔𝜉𝑖,𝑛 (𝑎′ ) = 𝜔𝜂 (1) = ‖𝜂‖2 = 𝜑𝔄′ (𝑎′ ). 𝑖

𝑛

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Lectures on von Neumann algebras

Consequently, 𝜑′ is a semifinite, normal weight on ℜ(𝔄′ )+ , which coincides with 𝜑𝔄′ on {𝑎′ ∈ ℜ(𝔄′ )+ ; 𝜑𝔄′ (𝑎′ ) < +∞}. Let 𝑛 ≥ 1, 𝑟 be rational and 𝑚(𝑛, 𝑟) ≥ 1, as in (10). For any 𝜂 ∈ 𝔄′ , such that 𝑅𝜂 ≥ 0, we have 𝑛

𝑛

𝑛

∑ 𝜔𝜉𝑖,𝑗 (𝜎𝑟′ ((𝑅𝜂 )2 )) = ∑ 𝜔𝜉𝑖,𝑗 ((𝑅∆−i𝑟 𝜂 )2 ) = ∑ 𝜔∆−i𝑟 𝜂 ((𝐿𝜉𝑖,𝑗 )2 ) 𝑗=1

𝑗=1

𝑗=1 𝑚(𝑛,𝑟)

= 𝜔∆−i𝑟 𝜂 (𝑎𝑖,𝑛 ) = 𝜔𝜂 (𝜎𝑟 (𝑎𝑖,𝑛 )) ≤ 𝜔𝜂 (𝑎𝑖,𝑚(𝑛,𝑟) ) = ∑ 𝜔𝜉𝑖,𝑗 ((𝑅𝜂 )2 ). 𝑗=1

Consequently, for any 𝑎′ ∈ ℜ(𝔄′ )+ , we have 𝑛

∑

𝑚(𝑛,𝑟)

𝜔𝜉𝑖,𝑗 (𝜎𝑟′ (𝑎′ ))

≤ ∑ 𝜔𝜉𝑖,𝑗 (𝑎′ ) = 𝜑′ (𝑎′ ).

𝑗=1

𝑗=1

Hence we infer that 𝜑′ (𝜎𝑟′ (𝑎′ )) ≤ 𝜑′ (𝑎′ ), for any 𝑎′ ∈ ℜ(𝔄′ )+ and any rational 𝑟. With the help of the lower w-semicontinuity of 𝜑′ , it is easy to infer that 𝜑′ (𝜎𝑡′ (𝑎′ )) = 𝜑′ (𝑎′ ),

𝑎′ ∈ ℜ(𝔄′ )+ ,

𝑡 ∈ ℝ.

We have still to prove that 𝜑′ is faithful. As for normal forms (see 5.15), by a similar argument one easily shows that the set {𝑎′ ∈ ℜ(𝔄′ );

0 ≤ 𝑎′ ≤ 1, 𝜑′ (𝑎′ ) = 0}

has a greatest element 𝑒′ , which is a projection. Since 𝜑′ is invariant with respect to the group of modular automorphisms {𝜎𝑡′ }𝑡 , it follows that 𝜎𝑡′ (𝑒′ ) = 𝑒′ ,

𝑡 ∈ ℝ.

Let 𝑎′ ∈ ℜ(𝔄′ )+ be such that 𝜑𝔄′ (𝑎′ ) < +∞. With the help of the assertion similar to assertion (9), it follows that 𝜑𝔄′ (𝑒′ 𝑎′ 𝑒′ ) < +∞. Thus 𝜑𝔄′ (𝑒′ 𝑎′ 𝑒′ ) = 𝜑′ (𝑒′ 𝑎′ 𝑒′ ) ≤ ‖𝑎′ ‖𝜑′ (𝑒′ ) = 0; hence 𝑒′ 𝑎′ 𝑒′ = 0. Since the set {𝑎′ ∈ ℜ(𝔄′ )+ ; 𝜑𝔄′ (𝑎′ ) < +∞} is w-dense in ℜ(𝔄′ )+ , we infer that 𝑒′ = 0. 10.17. In this section, we show that the weight 𝜑 𝔄 determines the group of the modular automorphisms {𝜎𝑡 } only in terms of the von Neumann algebra 𝔏(𝔄). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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The theory of standard von Neumann algebras

Let M be a von Neumann algebra, 𝜑 a weight on M + and {𝜋𝑡 }𝑡∈ ℝ a group of *-automorphisms of M . We assume that the group {𝜋𝑡 } leaves invariant the weight 𝜑, i.e. 𝜑(𝜋𝑡 (𝑎)) = 𝜑(𝑎),

𝑎 ∈ M + , 𝑡 ∈ ℝ.

One says that 𝜑 satisfies the Kubo–Martin–Schwinger condition (briefly, the KMScondition) for the elements 𝑥, 𝑦 ∈ 𝔑𝜑 ∩ 𝔑∗𝜑 , with respect to {𝜋𝑡 }, if there exists a bounded continuous function {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1} ∋ 𝛼 ↦ 𝑓𝑥,𝑦 (𝛼) ∈ ℂ, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1}, and such that 𝑓𝑥,𝑦 (i𝑡) = 𝜑(𝑥𝜋𝑡 (𝑦)),

𝑡 ∈ ℝ,

𝑓𝑥,𝑦 (1 + i𝑡) = 𝜑(𝜋𝑡 (𝑦)𝑥),

𝑡 ∈ ℝ.

Theorem. Let 𝔄 ⊂ H be a left Hilbert algebra, M = 𝔏(𝔄), {𝜎𝑡 }𝑡∈ ℝ the group of modular automorphisms and 𝜑 = 𝜑 𝔄 . Then 𝜑 satisfies the KMS-condition with respect to {𝜎𝑡 } for any pair of elements in 𝔑𝜑 ∩ 𝔑∗𝜑 . Conversely, if {𝜋𝑡 }𝑡∈ ℝ is a group of *-automorphisms of M , which leaves invariant the weight 𝜑 and with respect to which 𝜑 satisfies the KMS-condition, for any pair of elements in (𝔐𝜑 )2 , then 𝜋𝑡 = 𝜎𝑡 , 𝑡 ∈ ℝ. Proof. Let 𝑥, 𝑦 ∈ 𝔑𝜑 ∩ 𝔑∗𝜑 . In accordance with assertion (4) from Section 10.16, there exist 𝜉, 𝜁 ∈ 𝔄″ , such that 𝑥 = 𝐿 𝜉 , 𝑦 = 𝐿𝜁 . Then the function 𝛼 ↦ (Δ𝛼 𝜁|𝑆𝜉) is bounded and continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1/2} and it is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1/2}; also, the function 𝛼−

𝛼 ↦ (𝜉|𝐽Δ

1 2

𝜁)

is bounded and continuous on {𝛼 ∈ ℂ; 1/2 ≤ Re 𝛼 ≤ 1}, and analytic in {𝛼 ∈ ℂ; 1/2 < Re 𝛼 < 1}. Since, for any 𝑡 ∈ ℝ, we have 1

(Δ 2

+i𝑡

1

𝜁|𝑆𝜉) = (Δ 2

+i𝑡

−

𝜁|Δ

1 2

𝐽𝜉) = (Δi𝑡 𝜁|𝐽𝜉) 1 2

( +i𝑡)−

= (𝜉|𝐽Δi𝑡 𝜁) = (𝜉|𝐽Δ the two functions coincide on the line 𝛼 =

1 2

1 2

𝜁),

+ i𝑡, 𝑡 ∈ ℝ.

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Thus, we can define 𝑓𝑥,𝑦 (𝛼) = {

(Δ𝛼 𝜁|𝑆𝜉), 1 𝛼− 2

(𝜉|𝐽Δ

if 0 ≤ Re 𝛼 ≤ 1/2,

𝜁), if 1/2 ≤ Re 𝛼 ≤ 1.

The function 𝑓𝑥,𝑦 thus defined is bounded and continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1} and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1}. With the help of assertion (4) from Section 10.16, we infer that for any 𝑡 ∈ ℝ, we have 𝑓𝑥,𝑦 (i𝑡) = (Δi𝑡 𝜁|𝑆𝜉) = 𝜑(𝐿𝜉 𝐿∆i𝑡 𝜁 ) = 𝜑(𝑥𝜎𝑡 (𝑦)), 1

𝑓𝑥,𝑦 (1 + i𝑡) = (𝜉|𝐽Δ 2

+i𝑡

𝜁) = (𝜉|𝑆Δi𝑡 𝜁) = 𝜑(𝐿∆i𝑡 𝜁 𝐿𝜉 ) = 𝜑(𝜎𝑡 (𝑦)𝑥).

We have thus proved the first part of the theorem. Let now {𝜋𝑡 } be a group of *-automorphisms of M , which leaves invariant the weight 𝜑 and with respect to which 𝜑 satisfies the KMS-condition, for any pair of elements in (𝔐𝜑 )2 . We denote 𝔄0 = {𝜉 ∈ 𝔄″ ; 𝐿𝜉 ∈ (𝔐𝜑 )2 }. Then 𝔄0 is a *-subalgebra of 𝔄″ . With the help of assertion (4) from Section 10.16, it is easy to see that 𝔄0 ⊂ (𝔄″ )4 and therefore, in accordance with the last remark in Section 10.5, 𝔄0 is a left Hilbert subalgebra of 𝔄″ and (𝔄0 )″ = 𝔄″ . In particular, 𝔏(𝔄0 ) = M . From the invariance of 𝜑 with respect to {𝜋𝑡 }, it is easy to infer that for any 𝜉 ∈ 𝔄0 and any 𝑡 ∈ ℝ, there exists a unique element in 𝔄0 , denoted by 𝑢𝑡 𝜉, such that 𝐿ᵆ𝑡 𝜉 = 𝜋𝑡 (𝐿𝜉 ); also, for any 𝑡 ∈ ℝ, the mapping 𝔄0 ∋ 𝜉 ↦ 𝑢𝑡 𝜉 extends to a unitary operator 𝑢𝑡 ∈ B (H ). Obviously, {𝑢𝑡 } is a one-parameter group. From the KMS-condition we infer that, for any 𝜉, 𝜁 ∈ 𝔄0 , the mapping 𝑡 ↦ 𝜑(𝐿𝑆𝜉 𝜋𝑡 (𝐿𝜁 )) = (𝑢𝑡 𝜁|𝜉) is continuous. From this result, it is easy to infer that the group {𝑢𝑡 } is wo-continuous. For any 𝜉 ∈ 𝔄0 , we have 𝐿𝑆ᵆ𝑡 𝜉 = (𝜋𝑡 (𝐿𝜉 ))∗ = 𝜋𝑡 (𝐿∗𝜉 ) = 𝐿ᵆ𝑡 𝑆𝜉 ,

𝑡 ∈ ℝ;

hence 𝑆𝑢𝑡 𝜉 = 𝑢𝑡 𝑆𝜉,

𝑡 ∈ ℝ.

Since 𝑆|𝔄0 = 𝑆, the preceding equality is true for any 𝜉 ∈ D 𝑆 . Thus, for any 𝜉 ∈ D (∆1/2 ) = D 𝑆 , we have ‖Δ1/2 𝑢𝑡 𝜉‖ = ‖𝑆𝑢𝑡 𝜉‖ = ‖𝑢𝑡 𝑆𝜉‖ = ‖𝑆𝜉‖ = ‖Δ1/2 𝜉‖,

𝑡 ∈ ℝ;

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The theory of standard von Neumann algebras hence, for any 𝜉 ∈ D ∆ , 𝜁 ∈ D (∆1/2 ) , we have (Δ1/2 𝑢𝑡 𝜉|Δ1/2 𝑢𝑡 𝜁) = (Δ1/2 𝜉|Δ1/2 𝜁) = (Δ𝜉|𝜁),

𝑡 ∈ ℝ.

From these equalities, it is easy to see that, for any 𝜉 ∈ D ∆ , 𝑡 ∈ ℝ, we have Δ1/2 𝑢𝑡 𝜉 ∈ D (∆1/2 )∗ = D (∆1/2 ) , i.e. 𝑢𝑡 𝜉 ∈ D ∆ , and Δ𝑢𝑡 𝜉 = 𝑢𝑡 Δ𝜉. From the KMS-condition, we infer that, for any 𝜉, 𝜁 ∈ 𝔄0 , there exists a bounded and continuous function 𝑓𝜉,𝜂 on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1}, such that, for any 𝑡 ∈ ℝ, 𝑓𝜉,𝜁 (i𝑡) = 𝜑(𝐿𝑆𝜉 𝜋𝑡 (𝐿𝜁 )) = (𝑢𝑡 𝜁|𝜉), 𝑓𝜉,𝜁 (1 + i𝑡) = 𝜑(𝜋𝑡 (𝐿𝜁 )𝐿𝑆𝜉 ) = (𝑆𝜉|𝑆𝑢𝑡 𝜁) = (𝑆𝜉|𝑢𝑡 𝑆𝜁). From the equality 𝑆|𝔄0 = 𝑆, and with the help of the Phragmén–Lindelöf principle, it is easy to see that the preceding assertion extends for any 𝜉, 𝜁 ∈ D 𝑆 . If 𝜁 ∈ D ∆ , then, for any 𝜉 ∈ D 𝑆 and any 𝑡 ∈ ℝ, we have 𝑓𝜉,𝜁 (1 + i𝑡) = (𝐽Δ1/2 𝜉|𝐽Δ1/2 𝑢𝑡 𝜁) = (Δ1/2 𝑢𝑡 𝜁|Δ1/2 𝜉) = (𝑢𝑡 Δ𝜁|𝜉). From the fact that D 𝑆 = H , and with the help of the Phragmén–Lindelöf principle, it is easy to see that, for any 𝜁 ∈ D ∆ and any 𝜉 ∈ H , there exists a bounded continuous function 𝑓𝜉,𝜁 on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝑙}, and such that, for any 𝑡 ∈ ℝ, 𝑓𝜉,𝜁 (i𝑡) = (𝑢𝑡 𝜁|𝜉), 𝑓𝜉,𝜁 (1 + i𝑡) = (𝑢𝑡 Δ𝜁|𝜉). Moreover, for any 𝛼 ∈ ℂ; 0 < Re 𝛼 ≤ 1, we have |𝑓𝜉,𝜁 (𝛼)| ≤ max {‖𝜁‖, ‖Δ𝜁‖}‖𝜉‖. From this inequality, it is easy to infer that, for any 𝜁 ∈ D ∆ , the mapping i𝑡 ↦ 𝑢𝑡 𝜁 has a weakly continuous extension to {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, which is weakly analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 𝑙}. Moreover, the value of this extension at 1 is Δ𝜁. As a result of the Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Stone representation theorem (9.20), there exists a positive self-adjoint operator 𝐴 in H , such that 𝑢𝑡 = 𝐴i𝑡 , 𝑡 ∈ ℝ. From the preceding results, we infer that, for any 𝜁 ∈ D ∆ , we have Δ𝜁 = 𝐴𝜁. Consequently, we have the following relations: Δ⊂𝐴 𝐴 = 𝐴∗ ⊂ Δ∗ = Δ, which imply that 𝐴 = Δ. Therefore, for any 𝜉 ∈ 𝔄0 and any 𝑡 ∈ ℝ, we have 𝜋𝑡 (𝐿𝜉 ) = 𝐿ᵆ𝑡 𝜉 = 𝐿∆i𝑡 𝜉 = Δi𝑡 𝐿𝜉 Δ−i𝑡 = 𝜎𝑡 (𝐿𝜉 ). whence we infer that, for any 𝑥 ∈ 𝔏(𝔄0 ) = M , we have 𝜋𝑡 (𝑥) = 𝜎𝑡 (𝑥),

𝑡 ∈ ℝ. Q.E.D.

Let 𝜑 be a faithful, semifinite, normal weight on a von Neumann algebra M , and let 𝔄𝜑 be the left Hilbert algebra we have constructed in Section 10.15. If {𝜎𝑡 } is the group of the modular automorphism of 𝔏(𝔄𝜑 ), which is associated to 𝔄𝜑 , we shall denote 𝜑

𝜎𝑡 = 𝜋𝜑−1 ∘ 𝜎𝑡 ∘ 𝜋𝜑 ,

𝑡 ∈ ℝ,

𝜑

and we shall say that {𝜎𝑡 } is the group of the modular automorphisms of M , which is associated to 𝜑. 𝜑 Obviously, the assertions made in the preceding theorem are true for M , 𝜑 and {𝜎𝑡 }. 10.18. Let 𝔄 ⊂ H be a left Hilbert algebra, and 𝜑 𝔄 the associated weight on 𝔏(𝔄)+ . In this section, we shall prove that 𝜑 𝔄 is normal. In accordance with assertion (11) from Section 10.16, there exists a faithful, semifinite, normal weight 𝜑 ≤ 𝜑 𝔄 on 𝔏(𝔄)+ , which is invariant with respect to the group {𝜎𝑡 } and which coincides with 𝜑 𝔄 on 𝔐+ 𝔄. + Let 𝑎 ∈ 𝔐𝜑 . We define +∞

𝑎𝑛 = √𝑛/𝜋 ∫

2

e−𝑛𝑡 𝜎𝑡 (𝑎) 𝑑𝑡,

𝑛 ∈ ℕ.

−∞

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From Proposition 3.21 and Corollary 3.20, we infer that there exists a net {𝑢𝜈 } ⊂ 𝔐+ 𝜑 , such that 𝑢𝜈 ↑ 1. From assertion (9) from Section 10.16, and the discussion preceding it, we have + 𝑎𝑛 𝑢𝜈 𝑎𝑛 ∈ 𝔐𝔄 ; hence 𝜑 𝔄 (𝑎𝑛 𝑢𝜈 𝑎𝑛 ) = 𝜑(𝑎𝑛 𝑢𝜈 𝑎𝑛 ) ≤ 𝜑((𝑎𝑛 )2 ) ≤ ‖𝑎𝑛 ‖𝜑(𝑎𝑛 ). By taking into account the normality of 𝜑 and the fact that {𝜎𝑡 } leaves invariant the weight 𝜑, we have +∞

2

e−𝑛𝑡 𝜑(𝜎𝑡 (𝑎)) 𝑑𝑡 = 𝜑(𝑎),

𝜑(𝑎𝑛 ) = √𝑛/𝜋 ∫

𝑛 ∈ ℕ;

−∞

hence, for any 𝜈, we have 𝜑 𝔄 (𝑎𝑛 𝑢𝜈 𝑎𝑛 ) ≤ 𝜑(𝑎), 𝑛 ∈ ℕ; hence 𝜑 𝔄 ((𝑎𝑛 )2 ) ≤ 𝜑(𝑎),

𝑛∈ℕ

𝑤

Since (𝑎𝑛 )2 → 𝑎2 , with the help of the lower w-semicontinuity of the weight 𝜑 𝔄 , we infer that 𝜑 𝔄 (𝑎2 ) ≤ 𝜑(𝑎) < +∞, i.e. 𝑎2 ∈ 𝔐+ 𝔄. From what we have just proved, it follows that (𝔐𝜑 )2 ⊂ 𝔐𝔄 . In accordance with Theorem 10.17, 𝜑 𝔄 satisfies the KMS-condition with respect to {𝜎𝑡 }, for any pair of elements in (𝔐𝜑 )2 ⊂ 𝔐𝔄 . Since on (𝔐𝜑 )2 ⊂ 𝔐𝔄 the weights 𝜑 and 𝜑 𝔄 coincide, it follows that 𝜑 satisfies the KMS-condition, for any pair of elements in (𝔐𝜑 )2 . If we now apply the uniqueness part of Theorem 10.17, we infer that 𝜑

𝜎𝑡 = 𝜎𝑡 ,

𝑡 ∈ ℝ.

We also consider the left Hilbert algebra 𝔄𝜑 and we denote by 𝑆𝜑 , Δ𝜑 , 𝐽𝜑 , the corresponding operators. Let +∞

𝑣𝜈 = 1/√𝜋 ∫

2

e−𝑡 𝜎𝑡 (𝑢𝑣 ) 𝑑𝑡.

−∞

Since the mapping i𝑠 ↦ 𝜎𝑠 (𝑣𝜈 ) has an entire analytic continuation, from Proposition 9.24 we infer that, for any 𝛼 ∈ ℂ, we have D ((∆𝜑 )𝛼 𝜋𝜑 (𝑣𝜈 )(∆𝜑 )−𝛼 ) = D (∆𝜑 )−𝛼 and the operator (Δ𝜑 )𝛼 𝜋𝜑 (𝑣𝜈 )(Δ𝜑 )−𝛼 is bounded. We denote +∞

𝐹𝑣𝜈 (𝛼) = (Δ𝜑 )𝛼 𝜋𝜑 (𝑣𝜈 )(Δ𝜑 )−𝛼 ) = 1/√𝜋 ∫

2

e−(𝑡+i𝛼) 𝜎𝑡 (𝑣𝜈 ) 𝑑𝑡.

−∞

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It is easy to verify that, for any 𝛼 ∈ ℂ, 𝑠𝑜

𝐹𝑣𝜈 (𝛼) → 1, 2

‖𝐹𝑣𝜈 (𝛼)‖ ≤ e(Re 𝛼) . For any 𝑎 ∈ 𝔐+ 𝜑 and any 𝜈, we have 𝜑 𝔄 (𝑣𝜈 𝑎𝑣𝜈 ) ≤ 𝜑 𝔄 ((𝑣𝜈 )2 ) < +∞; hence 𝜑 𝔄 (𝑣𝜈 𝑎𝑣𝜈 ) = 𝜑(𝑣𝜈 𝑎𝑣𝜈 ). With the help of the lower w-semicontinuity of 𝜑 𝔄 , we get 𝜑 𝔄 (𝑎) ≤ sup 𝜑 𝔄 (𝑣𝜈 𝑎𝑣𝜈 ) = sup 𝜑(𝑣𝜈 𝑎𝑣𝜈 ) = sup ‖(𝑎1/2 𝑣𝜈 )𝜑 ‖2𝜑 𝜈

𝜈

= sup ‖𝑆𝜑 𝜋𝜑 (𝑣𝜈 )𝑆𝜑 ((𝑎 𝜈

𝜈

1/2

)𝜑 )‖2𝜑

= sup ‖𝐽𝜑 (Δ𝜑 )1/2 𝜋𝜑 (𝑣𝜈 )(Δ𝜑 )−1/2 𝐽𝜑 ((𝑎1/2 )𝜑 )‖2𝜑 𝜈

= sup ‖𝜋𝜑 (𝐹𝑣𝜈 (1/2))𝐽𝜑 ((𝑎1/2 )𝜑 )‖2𝜑 𝜈

≤ e1/4 ‖(𝑎1/2 )𝜑 ‖2𝜑 ≤ e1/4 𝜑(𝑎) < +∞; hence, 𝑎 ∈ 𝔐+ 𝜑. Consequently, we have + 𝔐+ 𝜑 = 𝔐𝔄 ,

whence 𝜑 = 𝜑 𝔄. From the results in Section 10.16 and from those we have just obtained, we infer the following: Theorem. Let 𝔄 ⊂ H be a left Hilbert algebra. We define the mapping 𝜑 𝔄 ∶ 𝔏(𝔄)+ → ℝ+ ∪ {+∞} by the formula ‖𝜉‖2 , if there exists 𝑎 𝜉 ∈ 𝔄″ , such that ⎧ ⎪ 𝜑 𝔄 (𝑎) = 𝑎1/2 = 𝐿𝜉 , ; 𝑎 ∈ 𝔏(𝔄)+ . ⎨ ⎪ ⎩ +∞, in the contrary case, Then 𝜑 𝔄 is a faithful, semifinite, normal weight on 𝔏(𝔄)+ and the mapping 𝜉 ↦ 𝐿𝜉 is a *-isomorphism of 𝔄″ onto 𝔑𝜑 𝔄 ∩ 𝔑∗𝜑 𝔄 , such that (𝜉|𝜁) = 𝜑 𝔄 ((𝐿𝜁 )∗ 𝐿𝜉 ),

𝜉, 𝜁 ∈ 𝔄″ .

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The theory of standard von Neumann algebras

339

Two pairs (M 𝑗 , 𝜑𝑗 ), 𝑗 = 1, 2, where 𝜑𝑗 is a faithful, semifinite, normal weight on the von Neumann algebra M 𝑗 , are said to be equivalent if there exists a *-isomorphism 𝜋 ∶ M 1 → M 2 , such that 𝜑1 = 𝜑2 ∘ 𝜋. Two left Hilbert algebras 𝔄𝑗 ⊂ H 𝑗 , 𝑗 = 1, 2, are said to be equivalent if there exists a unitary operator 𝑢 ∶ H 1 → H 2 , such that the restriction of 𝑢 to 𝔄1 be a *-isomorphism of 𝔄1 onto 𝔄2 . From the above theorem and from Theorem 10.14, we infer that the associations (M , 𝜑) ↦ 𝔄𝜑 , 𝔄 ↦ (𝔏(𝔄), 𝜑 𝔄 ), establish bijections, inverse to one another, between the classes of equivalence of pairs (M , 𝜑), where 𝜑 is a faithful, semifinite normal weight on the von Neumann algebra M , and the classes of equivalence of left Hilbert algebras 𝔄, such that 𝔄 = 𝔄″ . 10.19. The aim of the following sections is to exhibit some ‘suitable’ elements in 𝔄″ ∩ 𝔄′ and to prove their ‘abundance’. Let 𝔄 ⊂ H be a left Hilbert algebra. We consider the vector space 𝔗0 , generated by the set {Δ𝑛 exp(−𝑟Δ) exp(−𝑠Δ−1 )𝜉; 𝜉 ∈ 𝔄″ , 𝑟, 𝑠 > 0, 𝑛 ∈ ℤ}. Lemma 1. 𝔗0 is contained in 𝔄″ ∩ 𝔄′ . Proof. Let 𝜉 ∈ 𝔄″ , 𝑟, 𝑠 > 0 and 𝑛 ∈ ℤ. We consider the curve Γ ∶ ℝ → ℂ, given by the formula −𝑡 − 1 + i, if 𝑡 ≤ −1, ⎧ ⎪ i𝜋𝑡 Γ(𝑡) = −e 2 , if − 1 ≤ 𝑡 ≤ 1, ⎨ ⎪ ⎩ 𝑡 − 1 − i, if 𝑡 ≥ 1. We shall assume that 𝑛 ≥ 0. From Proposition 9.27, we infer that exp(−𝑠Δ−1 )𝜉 = (2𝜋i)−1 ∫ exp(−𝑠𝜆)(𝜆 − Δ−1 )−1 𝜉𝑑𝜆. Γ

With the help of Proposition 10.11, it is easy to verify that exp(−𝑠Δ−1 )𝜉 ∈ 𝔄′ . A similar argument now shows that Δ𝑛 exp(−𝑟Δ) exp(−𝑠Δ−1 )𝜉 ∈ 𝔄″ . Assuming that 𝑛 ≤ 0 and by repeating the preceding argument, one obtains successively Δ𝑛 exp(−𝑠Δ−1 )𝜉 = (Δ−1 )−𝑛 exp(−𝑠Δ−1 )𝜉 ∈ 𝔄′ , Δ𝑛 exp(−𝑟Δ) exp(−𝑠Δ−1 )𝜉 = exp(−𝑟Δ)(Δ𝑛 exp(−𝑠Δ−1 )𝜉) ∈ 𝔄″ . Hence, 𝔗0 ⊂ 𝔄″ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Let 𝜉 ∈ 𝔄″ , 𝑟, 𝑠 ≥ 0, 𝑛 ∈ ℤ again. From the preceding argument, we infer that 𝑠 Δ𝑛 exp(−𝑟Δ) exp(− Δ−1 )𝜉 ∈ 𝔄″ . 2 The argument used in the first part of the proof shows that 𝑠 𝑠 Δ𝑛 exp(−𝑟Δ) exp(−𝑠Δ−1 )𝜉 = exp(− Δ−1 )(Δ𝑛 exp(−𝑟Δ) exp(− Δ−1 )𝜉) ∈ 𝔄′ . 2 2 Consequently, 𝔗0 ⊂ 𝔄′ .

Q.E.D.

It is obvious that Δ𝔗0 = 𝔗0 and Δ−1 𝔗0 = 𝔗0 . With the help of Lemma 1, we get 𝜉 ∈ 𝔗0 ⇒ 𝑆𝜉 = 𝑆(𝑆𝑆 ∗ )Δ𝜉 = 𝑆 ∗ Δ𝜉 ∈ 𝑆 ∗ 𝔄′ ⊂ 𝔄′ , 𝜉 ∈ 𝔗0 ⇒ 𝑆 ∗ 𝜉 = 𝑆 ∗ (𝑆 ∗ 𝑆)Δ−1 𝜉 = 𝑆Δ−1 𝜉 ∈ 𝑆𝔄″ ⊂ 𝔄″ Lemma 2. 𝔗0 is contained in ∩𝛼∈ℂ D ∆𝛼 and for any 𝛼 ∈ ℂ, Δ𝛼 |𝔗0 = Δ𝛼 . Proof. Since 𝔗0 ⊂ ⋂𝑛∈ℤ D ∆𝑛 , with Corollary 9.21, we infer that 𝔗0 ⊂

⋂

D ∆𝛼 .

𝛼∈ℂ

Let 𝛼 ∈ ℂ and assume that (𝜉, Δ𝛼 𝜉) ∈ G ∆𝛼 is orthogonal to the graph of the operator Δ |𝔗0 . Then, for any 𝜁 ∈ 𝔄″ , we have 𝛼

(𝜉| exp(−Δ) exp(−Δ−1 )𝜁) + (Δ𝛼 𝜉|Δ𝛼 exp(−Δ) exp(−Δ−1 )𝜁) = 0, (𝜉|(1 + Δ2Re 𝛼 ) exp(−Δ) exp(−Δ−1 )𝜁) = 0. Since the operator (1 + Δ2Re 𝛼 ) exp(−Δ) exp(−Δ−1 ) ∈ B (H ) is positive and injective, its range is dense in H . Since 𝔄″ is dense in H , it follows that (1 + Δ2Re 𝛼 ) exp(−Δ) exp(−Δ−1 )𝔄″ is dense in H . Consequently, 𝜉 = 0. We have thus proved that Δ𝛼 |𝔗0 = Δ𝛼 .

Q.E.D.

Lemma 3. For any 𝜉 ∈ 𝔗0 and any 𝑛 ∈ ℤ, we have D (∆𝑛 𝐿𝜉 ∆−𝑛 ) = D (∆−𝑛 ) and Δ𝑛 𝐿𝜉 Δ−𝑛 ⊂ 𝐿∆𝑛 𝜉 , D (∆𝑛 𝑅𝜉 ∆−𝑛 ) = D (∆−𝑛 ) and Δ𝑛 𝑅𝜉 Δ−𝑛 ⊂ 𝑅∆𝑛 𝜉 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Proof. It is obvious that it is sufficient to prove the assertions in the lemma only for 𝑛 = 1 and 𝑛 = −1. We consider the case 𝑛 = 1. From Lemma 2 and Proposition 9.24, it is sufficient to prove that, for any 𝜁 ∈ 𝔗0 , we have 𝜁 ∈ D (∆𝐿𝜉 ∆−1 ) and Δ𝐿𝜉 Δ−1 (𝜁) = 𝐿∆𝜉 (𝜁), 𝜁 ∈ D (∆𝑅𝜉 ∆−1 ) and Δ𝑅𝜉 Δ−1 (𝜁) = 𝑅∆𝜉 (𝜁). Indeed, by taking into account the remark made just after Lemma 1, we have 𝐿𝜉 (Δ−1 (𝜁)) = 𝑆𝐿𝑆∆−1 𝜁 𝑆𝜉 = 𝑆𝐿𝑆 ∗ 𝜁 𝑆𝜉 = 𝑆𝑅𝑆𝜉 𝑆 ∗ 𝜁 = 𝑆𝑆 ∗ 𝑅𝜁 𝑆 ∗ 𝑆𝜉 = Δ−1 𝐿∆𝜉 (𝜁); hence 𝐿𝜉 Δ−1 (𝜁) ∈ D 𝐴 and Δ𝐿𝜉 Δ−1 (𝜁) = 𝐿∆𝜉 (𝜁). Similarly 𝑅𝜉 Δ−1 (𝜁) = 𝐿∆−1 𝜁 (𝜉) = 𝑆𝐿𝑆𝜉 𝑆Δ−1 𝜁 = 𝑆𝐿𝑆𝜉 𝑆 ∗ 𝜁 = 𝑆𝑅𝑆 ∗ 𝜁 𝑆𝜉 = 𝑆𝑆 ∗ 𝑅𝑆 ∗ 𝑆𝜉 (𝜁) = Δ−1 𝑅∆𝜉 (𝜁); hence 𝑅𝜉 Δ−1 (𝜁) ∈ D 𝐴 and Δ𝑅𝜉 Δ−1 (𝜁) = 𝑅∆𝜉 (𝜁). The case 𝑛 = −1 can be treated similarly.

Q.E.D.

We have not as yet used the fundamental theorem of Tomita (10.12). In what follows we shall use it, in order to extend Lemma 3, by replacing, in its statement, 𝑛 ∈ ℤ by 𝛼 ∈ ℂ. Lemma 4. For any 𝜉 ∈ 𝔗0 and any 𝛼 ∈ ℂ, we have Δ𝛼 𝜉 ∈ 𝔄″ ∩ 𝔄′ , D (∆𝛼 𝐿𝜉 ∆−𝛼 ) = D (∆−𝛼 ) and Δ𝛼 𝐿𝜉 Δ−𝛼 ⊂ 𝐿∆𝛼 𝜉 , D (∆𝛼 𝑅𝜉 ∆−𝛼 ) = D (∆−𝛼 ) and Δ𝛼 𝑅𝜉 Δ−𝛼 ⊂ 𝑅∆𝛼 𝜉 . Proof. Let 𝜉 ∈ 𝔗0 . In accordance with Lemma 2 and Corollary 9.21, for any 𝛼 ∈ ℂ, we have 𝜉 ∈ D ∆𝛼 , whereas the mapping 𝛼 ↦ Δ𝛼 𝜉 is entire analytic. On the other hand, in accordance with Lemma 2 and Proposition 9.24, for any 𝛼 ∈ ℂ, we have D (∆𝛼 𝐿𝜉 ∆−𝛼 ) = D (∆−𝛼 ) , the operator Δ𝛼 𝐿𝜉 Δ−𝛼 is bounded and the mapping 𝛼 ↦ 𝐹(𝛼) = Δ𝛼 𝐿𝜉 Δ−𝛼 ∈ B (H ) is entire analytic. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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For any 𝛼 ∈ ℂ, we have Δ𝛼 𝜉 ∈ D (∆1/2 ) = D 𝑆 ; hence the closed operator 𝐿∆𝛼 𝜉 is defined. Theorem 10.12 shows that, for any 𝑡 ∈ ℝ, we have Δi𝑡 𝜉 ∈ 𝔄″ and 𝐿∆i𝑡 𝜉 = Δi𝑡 𝐿𝜉 Δ−i𝑡 = 𝐹(i𝑡). Thus, for any 𝜂 ∈ 𝔄′ , the entire analytic functions 𝛼 ↦ 𝑅𝜂 (Δ𝛼 𝜉) = 𝐿∆𝛼 𝜉 (𝜂), 𝛼 ↦ 𝐹(𝛼)(𝜂) coincide on the imaginary axis, and therefore, they coincide on the entire complex plane ℂ. Therefore, for any 𝛼 ∈ ℂ, we have ‖𝐿∆𝛼 𝜉 (𝜂)‖ = ‖𝐹(𝛼)(𝜂)‖ ≤ ‖𝐹(𝛼)‖‖𝜂‖,

𝜂 ∈ 𝔄′ ;

hence Δ𝛼 𝜉 ∈ 𝔄″ and 𝐿∆𝛼 𝜉 = 𝐹(𝛼) ⊃ Δ𝛼 𝐿𝜉 Δ−𝛼 . Similarly one can show that for any 𝛼 ∈ ℂ, we have Δ𝛼 𝜉 ∈ 𝔄′ , and D (∆𝛼 𝑅𝜉 ∆−𝛼 ) = D (∆−𝛼 ) and Δ𝛼 𝑅𝜉 Δ−𝛼 ⊂ 𝑅∆𝛼 𝜉 . Q.E.D. 10.20. Let 𝔄 ⊂ H be a left Hilbert algebra. We observe that, for 𝜉, 𝜂 ∈ 𝔄″ ∩ 𝔄′ , we have 𝐿𝜉 (𝜂) = 𝑅𝜂 (𝜉). We can, therefore, use the notation 𝜉𝜂 both in the sense of the product in 𝔄″ and in the sense of the product in 𝔄′ . Obviously, 𝔄′ ∩ 𝔄″ is an algebra. We now consider the vector space ⎧ ⎪ ⎪

for any 𝛼 ∈ ℂ, we have 𝛼

″

′

⎫ ⎪ ⎪

Δ 𝜉 ∈𝔄 ∩𝔄 , 𝔗= 𝜉∈ D ∆𝛼 ⋂ ⎨ D (∆𝛼 𝐿𝜉 ∆−𝛼 ) = D (∆−𝛼 ) and Δ𝛼 𝐿𝜉 Δ−𝛼 ⊂ 𝐿∆𝛼 𝜉 , ⎬ 𝛼∈ℂ ⎪ ⎪ ⎪ ⎪ 𝛼 −𝛼 D (∆𝛼 𝑅𝜉 ∆−𝛼 ) = D (∆−𝛼 ) and Δ 𝑅𝜉 Δ ⊂ 𝑅∆𝛼 𝜉 . ⎭ ⎩ The following theorem of ‘calculus in 𝔗’ is the culminating point of Tomita’s theory. Theorem. Let 𝔄 ⊂ H be a left Hilbert algebra. Then 𝔗 is a left Hilbert sub-algebra of 𝔄″ and 𝔗′ = 𝔄 ′ ,

𝔗″ = 𝔄″ .

Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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(1)

Δ𝛼 𝔗 = 𝔗 and Δ𝛼 |𝔗 = Δ𝛼 , 𝛼 ∈ ℂ. 𝐽𝔗 = 𝔗.

(2)

Δ𝛼 𝐽𝜉 = 𝐽Δ−𝛼 𝜉,

(3)

𝜉 ∈ 𝔗,

Δ𝛼 (𝜉𝜂) = (Δ𝛼 𝜉) (Δ𝛼 𝜂),

(4)

𝜉, 𝜂 ∈ 𝔗,

𝐽(𝜉𝜂) = 𝐽(𝜂) 𝐽(𝜉),

(5)

𝛼 ∈ ℂ. 𝛼 ∈ ℂ.

𝜉, 𝜂 ∈ 𝔗.

Proof. We first prove assertion (1). Let 𝛼 ∈ ℂ. From the definition of 𝔗, we have 𝔗 ⊂ D ∆𝛼 . Let 𝜉 ∈ 𝔗 and 𝛽 ∈ ℂ. It is easy to see that if 𝜂 ∈ ⋂𝛾∈ℂ D ∆𝛾 , then 𝜂 belongs to the domains of definition of the operators Δ𝛼+𝛽 𝐿𝜉 Δ−𝛼−𝛽 and Δ 𝛽 𝐿∆𝛼 𝜉 Δ−𝛽 and the following equality holds Δ 𝛽 𝐿∆𝛼 𝜉 Δ−𝛽 (𝜂) = Δ𝛼+𝛽 𝐿𝜉 Δ−𝛼−𝛽 (𝜂) = 𝐿∆𝛼+𝛽 𝜉 (𝜂). Since Δ−𝛽 |

⋂

D ∆𝛾 = Δ−𝛽 ,

𝛾∈ℂ

from the preceding equality and from Proposition 9.24, we infer that D (∆ 𝛽 𝐿∆𝛼 𝜉 ∆−𝛽 ) = D (∆−𝛽 ) and Δ 𝛽 𝐿∆𝛼 𝜉 Δ−𝛽 ⊂ 𝐿∆ 𝛽 (∆𝛼 𝜉) . Similarly D (∆ 𝛽 𝑅∆𝛼 𝜉 ∆−𝛽 ) = D (∆−𝛽 ) and Δ 𝛽 𝑅∆𝛼 𝜉 Δ−𝛽 ⊂ 𝑅∆ 𝛽 (∆𝛼 𝜉) . Consequently, Δ𝛼 𝔗 = 𝔗. In accordance with Lemma 4 from Section 10.19, the vector space 𝔗0 is contained in 𝔗. Thus, Lemma 2 from Section 10.19 shows that Δ𝛼 |𝔗 = Δ𝛼 . ̄ −1 ) from We now prove assertion (3). With the help of the formula 𝑓(Δ)𝐽 = 𝐽 𝑓(Δ Section 10.1, we infer that, for any 𝜉 ∈ 𝔗 and any 𝛼 ∈ ℂ, we have 𝐽𝜉 ∈ D ∆𝛼 and Δ𝛼 𝐽𝜉 = 𝐽Δ−𝛼 𝜉. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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We now prove assertion (2). Let 𝜉 ∈ 𝔗. Then, as we have seen, 𝐽𝜉 ∈ ⋂𝛼∈ℂ D ∆𝛼 , and from Theorem 10.12, we infer that Δ𝛼 𝐽𝜉 = 𝐽(Δ−𝛼 𝜉) ∈ 𝐽(𝔄″ ∩ 𝔄′ ) ⊂ 𝔄″ ∩ 𝔄′ ,

𝛼 ∈ ℂ.

Using again Theorem 10.12, we infer that for any 𝛼 ∈ ℂ, we have the relations Δ𝛼 𝐿𝐽𝜉 Δ−𝛼 = Δ𝛼 𝐽𝑅𝜉 𝐽Δ−𝛼 = 𝐽Δ−𝛼 𝑅𝜉 Δ𝛼 𝐽 ⊂ 𝐽𝑅∆−𝛼 𝜉 𝐽 = 𝐿𝐽∆−𝛼𝜉 = 𝐿∆𝛼 𝐽𝜉 and D (∆𝛼 𝐿𝐽𝜉 ∆−𝛼 ) = D (𝐽∆−𝛼 𝑅𝜉 ∆𝛼 𝐽) = D (∆𝛼 𝐽) = D (𝐽∆−𝛼 ) = D (∆−𝛼 ) . Similarly, one can show that, for any 𝛼 ∈ ℂ, we have D (∆𝛼 𝑅𝐽𝜉 ∆−𝛼 ) = D (∆−𝛼 ) and Δ𝛼 𝑅𝐽𝜉 Δ−𝛼 ⊂ 𝑅∆𝛼 𝐽𝜉 . We now prove assertion (4). Let 𝜉, 𝜂 ∈ 𝔗 and 𝛼 ∈ ℂ. Then Δ𝛼 𝜂 ∈ D (∆−𝛼 ) = D (∆𝛼 𝐿𝜉 ∆−𝛼 ) ; hence 𝜉𝜂 = 𝐿𝜉 (𝜂) = 𝐿𝜉 Δ−𝛼 (Δ𝛼 𝜂) ∈ D (∆𝛼 ) and Δ𝛼 (𝜉𝜂) = Δ𝛼 𝐿𝜉 Δ−𝛼 (Δ𝛼 𝜂) = 𝐿∆𝛼 𝜉 (Δ𝛼 𝜂) = (Δ𝛼 𝜉)(Δ𝛼 𝜂). We also have D (∆𝛼 𝐿𝜉𝜂 ∆−𝛼 ) = D (∆𝛼 𝐿𝜉 ∆−𝛼 )(∆𝛼 𝐿𝜂 ∆−𝛼 ) = D (∆−𝛼 ) and Δ𝛼 𝐿𝜉𝜂 Δ−𝛼 = (Δ𝛼 𝐿𝜉 Δ−𝛼 )(Δ𝛼 𝐿𝜂 Δ−𝛼 ) ⊂ 𝐿∆𝛼 𝜉 𝐿∆𝛼 𝜂 = 𝐿∆𝛼 (𝜉𝜂) . Similarly, one proves the following relations: D (∆𝛼 𝑅𝜉𝜂 ∆−𝛼 ) = D (∆−𝛼 ) and Δ𝛼 𝑅𝜉𝜂 Δ−𝛼 ⊂ 𝑅∆𝛼 (𝜉𝜂) . We have thus proved that, if 𝜉, 𝜂 ∈ 𝔗, then 𝜉𝜂 ∈ 𝔗, whereas from assertions (1) and (2), we infer that, if 𝜉 ∈ 𝔗, then 𝑆𝜉 = 𝐽Δ1/2 𝜉 ∈ 𝔗. Since 𝔗 ⊃ 𝔗0 , we infer that 𝔗 is dense in H . From what we have already proved and from Lemma 2, Section 10.5, it follows that 𝔗 is a left Hilbert sub-algebra of 𝔄″ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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From assertion (1), we infer that Δ1/2 |𝔗 = Δ1/2 , and therefore, 𝑆|𝔗 = 𝑆. With the help of Lemma 3, from Section 10.5, we deduce the equalities 𝔗′ = 𝔄 ′ ,

𝔗″ = 𝔄″ . Q.E.D.

We shall call 𝔗 the Tomita algebra associated to the left Hilbert algebra 𝔄. 10.21. We shall now prove a criterion with the help of which we can establish that an element belongs to 𝔗. Corollary 1. Let 𝔄 ⊂ H be a left Hilbert algebra, 𝔗 the associated Tomita algebra, 𝜉 ∈ ∩𝛼∈ℂ D ∆𝛼 and {𝜀𝑛 }𝑛∈ℤ ⊂ ℝ a family of real numbers, such that lim 𝜀𝑛 = −∞,

𝑛→−∞

lim 𝜀𝑛 = +∞.

𝑛→∞

Then the following assertions are equivalent: (i) 𝜉 ∈ 𝔗. (ii) for any 𝑛 ∈ ℤ, we have Δ𝜀𝑛 𝜉 ∈ 𝔄″ . (iii) for any 𝑛 ∈ ℤ, we have Δ𝜀𝑛 𝜉 ∈ 𝔄′ . (iv) for any 𝑛 ∈ ℤ, we have 𝔗 ⊂ D (∆𝜀𝑛 𝐿𝜉 ∆−𝜀𝑛 ) and the operator Δ𝜀𝑛 𝐿𝜉 Δ−𝜀𝑛 |𝔗 is bounded. (v) for any 𝑛 ∈ ℤ, we have 𝔗 ⊂ D (∆𝜀𝑛 𝑅𝜉 ∆−𝜀𝑛 ) and the operator Δ𝜀𝑛 𝑅𝜉 Δ−𝜀𝑛 |𝔗 is bounded. Proof. We shall prove the implications (i) ⇒ (ii) ⇒ (iv) ⇒ (i); the proofs of the implications (i) ⇒ (iii) ⇒ (v) ⇒ (i) are completely similar. The implication (i) ⇒ (ii) is obvious. We now assume that Δ𝜀𝑛 𝜉 ∈ 𝔄″ , for any 𝑛 ∈ ℤ. Then, for any 𝜂 ∈ 𝔗 and any 𝑛 ∈ ℤ, we have 𝐿𝜉 Δ−𝜀𝑛 (𝜂) = 𝑅∆−𝜀𝑛 (𝜂) (𝜉) = Δ−𝜀𝑛 𝑅𝜂 Δ𝜀𝑛 𝜉 = Δ−𝜀𝑛 𝐿∆𝜀𝑛 𝜉 (𝜂); hence 𝐿𝜉 Δ−𝜀𝑛 (𝜂) ∈ D (∆𝜀𝑛 ) and Δ𝜀𝑛 𝐿𝜉 Δ−𝜀𝑛 (𝜂) = 𝐿∆𝜀𝑛 (𝜂). Thus, 𝔗 ⊂ D (∆𝜀𝑛 𝐿𝜉 ∆−𝜀𝑛 ) and Δ𝜀𝑛 𝐿𝜉 Δ−𝜀𝑛 |𝔗 ⊂ 𝐿∆𝜀𝑛 𝜉 ∈ B (H ). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Finally, let us assume that 𝔗 ⊂ D (∆𝜀𝑛 𝐿𝜉 ∆−𝜀𝑛 ) and also that the operator Δ𝜀𝑛 𝐿𝜉 Δ−𝜀𝑛 |𝔗 is bounded, for any 𝑛 ∈ ℤ. We denote 𝑥0 = Δ𝜀0 𝐿𝜉 Δ−𝜀0 |𝔗 ∈ B (H ). Then, for any 𝑛 ∈ ℤ, we have 𝔗 ⊂ D (∆𝜀𝑛 −𝜀0 𝑥0 ∆𝜀0 −𝜀𝑛 ) and the operator Δ𝜀𝑛 −𝜀0 𝑥0 Δ𝜀0 −𝜀𝑛 |𝔗 = Δ𝜀𝑛 𝐿𝜉 Δ−𝜀𝑛 |𝔗 is bounded. With the help of Theorem 10.20(1), from Proposition 9.24, we infer that, for any 𝛼 ∈ ℂ, we have D (∆𝛼 𝑥0 ∆−𝛼 ) = D (∆−𝛼 ) , and the operator Δ𝛼 𝑥0 Δ−𝛼 is bounded. In particular, the operator Δ−𝜀0 𝑥0 Δ𝜀0 is bounded. On the other hand, if 𝜂 ∈ 𝔗, then Δ𝜀0 𝜂 ∈ 𝔗 ⊂ D (∆𝜀0 𝐿𝜉 ∆−𝜀0 ) ; hence 𝐿𝜉 (𝜂) = Δ−𝜀0 (Δ𝜀0 𝐿𝜉 Δ−𝜀0 )Δ𝜀0 (𝜂) = Δ−𝜀0 𝑥0 Δ𝜀0 (𝜂). Thus, 𝐿𝜉 |𝔗 ⊂ Δ−𝜀0 𝑥0 Δ𝜀0 is bounded. It follows that the closed operator 𝐿𝜉 is bounded, hence 𝜉 ∈ 𝔄″ . If we now again apply Proposition 9.24, we infer that, for any 𝛼 ∈ ℂ, we have D (∆𝛼 𝐿𝜉 ∆−𝛼 ) = D (∆−𝛼 ) and the operator Δ𝛼 𝐿𝜉 Δ−𝛼 is bounded, whereas the mapping 𝛼 ↦ 𝐹(𝛼) = Δ𝛼 𝐿𝜉 Δ−𝛼 ∈ B (H ) is entire analytic. Theorem 10.12 now shows that, for any 𝑡 ∈ ℝ, we have Δi𝑡 𝜉 ∈ 𝔄″ and 𝐿∆i𝑡 𝜉 = Δi𝑡 𝐿𝜉 Δ−i𝑡 = 𝐹(i𝑡). Thus, for any 𝜂 ∈ 𝔄′ , the entire analytic functions 𝛼 ↦ 𝑅𝜂 (Δ𝛼 𝜉) = 𝐿∆𝛼 𝜉 (𝜂), 𝛼 ↦ 𝐹(𝛼)(𝜂), coincide on the imaginary axis; hence, everywhere. Consequently, for any 𝛼 ∈ ℂ, we have ‖𝐿∆𝛼 𝜉 (𝜂)‖ = ‖𝐹(𝛼)(𝜂)‖ ≤ ‖𝐹(𝛼)‖‖𝜂‖, 𝜂 ∈ 𝔄′ ; therefore Δ𝛼 𝜉 ∈ 𝔄″ and 𝐿∆𝛼 𝜉 = 𝐹(𝛼) ⊃ Δ𝛼 𝐿𝜉 Δ−𝛼 . For any 𝛼 ∈ ℂ, we have Δ𝛼−(1/2) 𝜉 ∈ 𝔄″ ; hence Δ𝛼 𝜉 = Δ1/2 (Δ𝛼−(1/2) 𝜉) ∈ Δ1/2 𝔄″ ⊂ 𝔄′ .

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347

The theory of standard von Neumann algebras Finally, for any 𝛼 ∈ ℂ and any 𝜁 ∈ 𝔗, we have 𝑅𝜉 Δ−𝛼 (𝜁) = 𝐿∆−𝛼 𝜁 (𝜉) = Δ−𝛼 𝐿𝜁 Δ𝛼 𝜉 = Δ−𝛼 𝑅∆𝛼 𝜉 (𝜁) ∈ D ∆𝛼 and Δ𝛼 𝑅𝜉 Δ−𝛼 (𝜁) = 𝑅∆𝛼 𝜉 (𝜁). If we now apply Proposition 9.24, it follows that, for any 𝛼 ∈ ℂ, we have D (∆𝛼 𝑅𝜉 ∆−𝛼 ) = D (∆−𝛼 ) and Δ𝛼 𝑅𝜉 Δ−𝛼 ⊂ 𝑅∆𝛼 𝜉 .

Q.E.D. Let us now consider a vector 𝜉 ∈ H . For any 𝜀 > 0, we shall define +∞ 2

e−𝜀𝑡 Δi𝑡 𝜉 𝑑𝑡 ∈ H .

𝜉𝜀 = √𝜀/𝜋 ∫ −∞

Then the mapping +∞ 2

e−𝜀(𝑡+i𝛼) Δi𝑡 𝜉 𝑑𝑡

𝛼 ↦ √𝜀/𝜋 ∫ −∞

is an entire analytic continuation of the mapping i𝑠 ↦ Δi𝑠 𝜉𝜀 ; hence, in accordance with Corollary 9.21, we have 𝜉𝜀 ∈

⋂

D ∆𝛼 ,

𝛼∈ℂ +∞ 2

Δ𝛼 𝜉𝜀 = √𝜀/𝜋 ∫

e−𝜀(𝑡+i𝛼) Δi𝑡 𝜉 𝑑𝑡,

𝛼 ∈ ℂ.

−∞

Therefore, by taking into account Corollary 10.12 and the above corollary, it follows that if 𝜉 belongs to 𝔄″ , or to 𝔄′ , then, for any 𝜀 > 0, the element 𝜉𝜀 belongs to 𝔗. We observe that 𝜉𝜀 →𝜀→+∞ 𝜉, 𝜉 ∈ H , a fact that is easy to establish, by taking into account the continuity of the mapping 𝑡 ↦ Δi𝑡 𝜉 at 0 and using the Lebesgue dominated convergence theorem. One can similarly prove that +∞

𝐿𝜉𝜀 = √𝜀/𝜋 ∫ −∞ +∞

𝑅𝜉𝜀 = √𝜀/𝜋 ∫ −∞

2

𝑠𝑜

2

𝑠𝑜

e−𝜀𝑡 Δi𝑡 𝐿𝜉 Δ−i𝑡 𝑑𝑡 ⃗ 𝐿𝜉 , 𝜀→+∞ e−𝜀𝑡 Δi𝑡 𝑅𝜉 Δ−i𝑡 𝑑𝑡 ⃗ 𝑅𝜉 , 𝜀→+∞

𝜉 ∈ 𝔄″ , 𝜉 ∈ 𝔄′ .

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From the preceding arguments, we retain the following: Corollary 2. Let 𝔄 ⊂ H be a left Hilbert algebra and 𝔗 the associated Tomita algebra. Then, for any 𝜉 ∈ 𝔄″ , there exists a sequence {𝜉𝑛 } ⊂ 𝔗, such that (1) 𝜉𝑛 → 𝜉, 𝑆𝜉𝑛 → 𝑆𝜉. 𝑠𝑜

𝑠𝑜

(2) 𝐿𝜉𝑛 → 𝐿𝜉 , (𝐿𝜉𝑛 )∗ → (𝐿𝜉 )∗ . (3) ‖𝐿𝜉𝑛 ‖ ≤ ‖𝐿𝜉 ‖, for any 𝑛 ∈ ℕ. Similarly, for any 𝜂 ∈ 𝔄′ , there exists a sequence {𝜂𝑛 } ⊂ 𝔗, such that (1) 𝜂𝑛 → 𝜂, 𝑆 ∗ 𝜂𝑛 → 𝑆 ∗ 𝜂. 𝑠𝑜

𝑠𝑜

(2) 𝑅𝜂𝑛 → 𝑅𝜂 , (𝑅𝜂𝑛 )∗ → (𝑅𝜂 )∗ . (3) ‖𝑅𝜂𝑛 ‖ ≤ ‖𝑅𝜂 ‖, for any 𝑛 ∈ ℕ. 10.22. In this section, we show that 𝔗 contains sufficiently many elements 𝜉, such that ‖Δ𝛼 𝜉‖, ‖𝐿∆𝛼 𝜉 ‖ and ‖𝑅∆𝛼 𝜉 ‖ have exponential upper bounds depending on Re 𝛼. Let 𝔄 ⊂ H be a left Hilbert algebra and 𝔗 the associated Tomita algebra. We define the set 𝔖, consisting of all elements 𝜉 ∈ 𝔗, such that there exist 𝜆1 , 𝜆2 ∈ (0, ∞), 𝜆1 ≤ 𝜆2 , such that 𝛼 ‖Δ𝛼 𝜉‖ ≤ 𝜆Re , 2

𝛼 ‖𝐿∆𝛼 𝜉 ‖ ≤ 𝜆Re ‖𝐿𝜉 ‖, 2

𝛼 ‖𝑅∆𝛼 𝜉 ‖ ≤ 𝜆Re ‖𝑅𝜉 ‖, 2

if Re 𝛼 ≥ 0,

𝛼 ‖Δ𝛼 𝜉‖ ≤ 𝜆Re , 1

𝛼 ‖𝐿∆𝛼 𝜉 ‖ ≤ 𝜆Re ‖𝐿𝜉 ‖, 1

𝛼 ‖𝑅∆𝛼 𝜉 ‖ ≤ 𝜆Re ‖𝑅𝜉 ‖, 1

if Re 𝛼 ≤ 0.

We recall that if 𝑓 ∈ L 1 (ℝ) and if its inverse Fourier transform 𝑓,̂ defined by +∞

̂ =∫ 𝑓(𝑠)

𝑓(𝑡)ei𝑡𝑠 𝑑𝑡,

𝑠 ∈ ℝ,

−∞

belongs to L 1 (ℝ), then the following inversion formula holds: +∞

1 −i𝑡𝑠 ̂ ∫ 𝑓(𝑠)e 𝑓(𝑡) = 𝑑𝑠, 2𝜋 −∞

𝑡 ∈ ℝ.

If 𝑓 ∈ L 1 (ℝ), 𝑓 ̂ ∈ C 2 (ℝ) and supp 𝑓 ̂ ⊂ [𝑐1 , 𝑐2 ], then 𝑓 has an entire analytic continuation 𝑐 2

−i𝛼𝑠 ̂ 𝛼 ↦ (2𝜋)−1 ∫ 𝑓(𝑠)e 𝑑𝑠, 𝑐1

which is also denoted by 𝑓, and the following equality holds 𝑐2

𝛼 2 𝑓(𝛼) = −(2𝜋)−1 ∫ 𝑓 ″̂ (𝑠)e−i𝛼𝑠 𝑑𝑠. 𝑐1

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349

The theory of standard von Neumann algebras Consequently, if Im 𝛼 ≥ 0, we have |𝑓(𝛼)| ≤

𝑐2 − 𝑐1 1 ̂ (sup |𝑓(𝑠)| + sup |𝑓 ″̂ (𝑠)|) e𝑐2 Im 𝛼 , 2𝜋 𝑠∈ℝ 1 + |𝛼|2 𝑠∈ℝ

whereas if Im 𝛼 ≤ 0, we have |𝑓(𝛼)| ≤

𝑐2 − 𝑐1 1 ̂ (sup |𝑓(𝑠)| + sup |𝑓 ″̂ (𝑠)|) e𝑐1 Im 𝛼 . 2 2𝜋 𝑠∈ℝ 1 + |𝛼| 𝑠∈ℝ

Corollary. Let 𝔄 ⊂ H be a left Hilbert algebra and 𝔗 the associated Tomita algebra. Then 𝔖 is a left Hilbert subalgebra of 𝔄″ and 𝔖′ = 𝔄 ′ ,

𝔖″ = 𝔄″ .

Moreover, (1) 𝔖 = 𝔗 ∩ S ∆ ∩ S ∆−1 +∞

𝑓(𝑡)Δi𝑡 𝜉 𝑑𝑡; 𝜉 ∈ 𝔗, 𝑓 ∈ L 1 (ℝ), 𝑠𝑢𝑝𝑝 𝑓 ̂ 𝑖𝑠 𝑐𝑜𝑚𝑝𝑎𝑐𝑡} ;

= {∫ −∞

(2) Δ𝛼 𝔖 = 𝔖 and Δ𝛼 |𝔖 = Δ𝛼 , for any 𝛼 ∈ ℂ; (3) 𝐽𝔖 = 𝔖. Proof. Let 𝜉 ∈ 𝔖. Then there exist 𝜆1 , 𝜆2 ∈ (0, +∞), 𝜆1 ≤ 𝜆2 , such that 𝛼 𝛼 ‖Δ𝛼 𝜉‖ ≤ 𝜆Re ‖𝜉‖, Re 𝛼 ≥ 0; ‖Δ𝛼 𝜉‖ ≤ 𝜆Re ‖𝜉‖, Re 𝛼 ≤ 0. 1 2

Then, for any 𝜀 > 0, we have (𝜆2 + 𝜀)𝑛 ‖𝜒(𝜆2 +𝜀,+∞) (Δ)𝜉‖ ≤ ‖Δ𝛼 𝜉‖ ≤ 𝜆𝑛2 ‖𝜉‖,

𝑛 ≥ 0,

whence 𝜒(𝜆2 +𝜀,+∞) (Δ)𝜉 = 0. Since 𝜀 > 0 is arbitrary, it follows that 𝜉 ∈ 𝜒[0,𝜆2 ] (Δ)H ⊂ S ∆ . One can similarly prove that 𝜉 ∈ 𝜒[𝜆1 ,+∞] (Δ)H = 𝜒[0,𝜆1−1 ] (Δ−1 )H ⊂ S ∆−1 . Consequently, we have proved that (*)

𝔖 ⊂ 𝔗 ∩ S ∆ ∩ S ∆−1 .

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350

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Let 𝜉 ∈ 𝔗 ∩ S ∆ ∩ S ∆−1 . There exist 𝜆1 , 𝜆2 ∈ (0, +∞), 𝜆1 ≤ 𝜆2 , such that 𝜉 ∈ 𝜒[𝜆1 ,𝜆2 ] (Δ)H . ̂ = 1, for any 𝑠 ∈ We now consider a function 𝑓 ∈ L 1 (ℝ), such that 𝑓 ̂ ∈ C 2 (ℝ), 𝑓(𝑠) ̂ [ln 𝜆1 , ln 𝜆2 ], such that supp 𝑓 be compact. Then +∞

𝑓(𝑡)𝜆i𝑡 𝜒[𝜆1 ,𝜆2 ] (𝜆) 𝑑𝑡 = 𝜒[𝜆1 ,𝜆2 ] (𝜆),

∫

𝜆 ∈ (0, +∞).

−∞

With the help of Theorem 9.11 (vi), it is easy to prove that +∞

𝑓(𝑡)𝜆i𝑡 𝜒[𝜆1 ,𝜆2 ] (Δ) 𝑑𝑡 = 𝜒[𝜆1 ,𝜆2 ] (Δ);

∫ −∞

hence +∞

∫

+∞ i𝑡

𝑓(𝑡)Δi𝑡 𝜒[𝜆1 ,𝜆2 ] (Δ)𝜉 𝑑𝑡 = 𝜒[𝜆1 ,𝜆2 ] (Δ)𝜉 = 𝜉.

𝑓(𝑡)Δ 𝜉 𝑑𝑡 = ∫

−∞

−∞

Consequently, +∞

𝑓(𝑡)Δi𝑡 𝜉 𝑑𝑡; 𝜉 ∈ 𝔗, 𝑓 ∈ L 1 (ℝ), supp 𝑓 ̂ is compact}.

(* *) 𝔗 ∩ S ∆ ∩ S ∆−1 ⊂ {∫ −∞

Finally, let us consider an element +∞

𝑓(𝑡)Δi𝑡 𝜉 𝑑𝑡,

𝜁=∫ −∞

where 𝜉 ∈ 𝔗, 𝑓 ∈ L 1 (ℝ), supp 𝑓 ̂ ⊂ [𝑐1 , 𝑐2 ], 𝑐1 , 𝑐2 ∈ ℝ, 𝑐1 ≤ 𝑐2 . Then +∞

𝛼↦∫

𝑓(𝑡)Δi𝑡 (Δ𝛼 𝜉) 𝑑𝑡

−∞

is an entire analytic continuation of the mapping +∞

𝑓(𝑡)Δi𝑡 (Δi𝑠 𝜉) 𝑑𝑡 = Δi𝑠 𝜁.

𝑖𝑠 ↦ ∫ −∞

If we now apply Corollary 9.21, we get +∞

𝜁∈

⋂

𝛼∈ℂ

D ∆𝛼 and Δ 𝜁 = ∫ 𝛼

𝑓(𝑡)Δi𝑡 (Δ𝛼 𝜉) 𝑑𝑡,

𝛼 ∈ ℂ.

−∞

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In accordance with Corollary 10.12 and Corollary 1 from Section 10.21, we infer that 𝜁 ∈ 𝔗. Let 𝑔 ∈ L 1 (ℝ), be such that 𝑔 ̂ ∈ C 2 (ℝ), 𝑔(𝑠) ̂ = 1 for any 𝑠 ∈ [𝑐1 , 𝑐2 ], and supp 𝑔 ̂ ⊂ [𝑐1 − 𝜀, 𝑐2 + 𝜀], 𝜀 > 0. Then, if we denote by ‘∗’ the convolution product in L 1 (ℝ), we have +∞

+∞

i𝑡

∫

(𝑔 ∗ 𝑓)(𝑡)Δi𝑡 𝜉 𝑑𝑡.

𝑔(𝑡)Δ 𝜁 𝑑𝑡 = ∫

−∞

−∞

Since (𝑔 ∗ 𝑓) = 𝑔𝑓 ̂ ̂ = 𝑓,̂ we infer that 𝑔 ∗ 𝑓 = 𝑓 and therefore, ∧

+∞

+∞ i𝑡

∫

𝑔(𝑡)Δ 𝜁 𝑑𝑡 = ∫

−∞

𝑓(𝑡)Δi𝑡 𝜉 𝑑𝑡 = 𝜁.

−∞

With the help of Corollary 9.21, as above we can prove that +∞

Δ𝛼 𝜁 = ∫

𝑔(𝑡)Δi𝑡 (Δ𝛼 𝜁) 𝑑𝑡,

𝛼 ∈ ℂ.

−∞

We denote 𝑐=

𝑐2 − 𝑐1 + 2𝜀 (sup |𝑔(𝑠)| ̂ + sup |𝑔″̂ (𝑠)|). 2𝜋 𝑠∈ℝ 𝑠∈ℝ

With the help of the Cauchy integral formula and the remarks at the beginning of the present section, we infer that, for any 𝑛 ≥ 0, +∞ 𝑛

+∞ i𝑡+𝑛

‖Δ 𝜁‖ = ‖ ∫

𝑔(𝑡)Δ

𝜁 𝑑𝑡‖ = ‖ ∫

−∞

𝑔(𝑡 + i𝑛)Δi𝑡 𝜁 𝑑𝑡‖

−∞ +∞

≤ 𝑐‖𝜁‖ ∫ −∞

1 𝑑𝑡 ⋅ e𝑐2 𝑛 = 𝜋𝑐‖𝜁‖e𝑐2 𝑛 . 1 + 𝑡2

If we now apply the ‘three lines’ theorem (see Dunford and Schwartz 1958/1963/1970, VI.10.3), we obtain, for any 𝛼 ∈ ℂ, Re 𝛼 ≥ 0, and any 𝑛 ≥ Re 𝛼, ‖Δ𝛼 𝜁‖ ≤ ‖𝜁‖

1−

Re 𝛼 𝑛

⋅ ‖Δ𝑛 𝜁‖

Re 𝛼 𝑛

≤ ‖𝜁‖

1−

Re 𝛼 𝑛

⋅ (𝜋𝑐‖𝜁‖)

Re 𝛼 𝑛

⋅ e𝑐2 Re 𝛼 .

Tending to the limit for 𝑛 → +∞, we get ‖Δ𝛼 𝜁‖ ≤ ‖𝜁‖e𝑐2 Re 𝛼 ,

Re 𝛼 ≥ 0.

Similarly, one can prove that ‖Δ𝛼 𝜁‖ ≤ ‖𝜁‖e𝑐1 Re 𝛼 ,

Re 𝛼 ≤ 0.

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On the other hand, in accordance with Corollary 10.12, +∞

𝐿𝜁 = ∫

𝑔(𝑡)𝐿∆i𝑡 𝜁 𝑑𝑡.

−∞

With the help of Proposition 9.24, it is easy to verify that +∞

𝐿∆𝛼 𝜁 = ∫

𝑔(𝑡)𝐿∆i𝑡 (∆𝛼 𝜁) 𝑑𝑡,

𝛼 ∈ ℂ.

−∞

If we argue as above, we first obtain ‖𝐿∆𝑛 𝜁 ‖ ≤ 𝜋𝑐‖𝐿𝜁 ‖e𝑐2 𝑛 ,

𝑛 ≥ 0,

and then ‖𝐿∆𝛼 𝜁 ‖ ≤ ‖𝐿𝜁 ‖e𝑐2 Re 𝛼 ,

Re 𝛼 ≥ 0;

‖𝐿∆𝛼 𝜁 ‖ ≤ ‖𝐿𝜁 ‖e𝑐1 Re 𝛼 ,

Re 𝛼 ≤ 0,

similarly, one obtains If we now repeat the above arguments for 𝑅, instead of 𝐿, we obtain ‖𝑅∆𝛼 𝜁 ‖ ≤ ‖𝑅𝜁 ‖e𝑐2 Re 𝛼 ,

Re 𝛼 ≥ 0,

‖𝑅∆𝛼 𝜁 ‖ ≤ ‖𝑅𝜁 ‖e𝑐1 Re 𝛼 ,

Re 𝛼 ≤ 0.

Thus 𝜁 ∈ 𝔖. Consequently, we have just proved the inclusion +∞

(***)

𝑓(𝑡)Δi𝑡 𝜉 𝑑𝑡; 𝜉 ∈ 𝔗, 𝑓 ∈ L 1 (ℝ), supp 𝑓 ̂ is compact} ⊂ 𝔖.

{∫ −∞

From relations (*), (**), (***), we infer that assertion (1) is true. If 𝜉 ∈ 𝔗 and 𝑓 ∈ L 1 (ℝ) is such that supp 𝑓 ̂ is compact, then, by taking into account Theorem 10.20, we get +∞

Δ𝛼 ∫

+∞

𝑓(𝑡)Δi𝑡 𝜉 𝑑𝑡 = ∫

−∞ +∞

𝐽∫

𝑓(𝑡)Δi𝑡 (Δ𝛼 𝜉) 𝑑𝑡 ∈ 𝔖,

𝛼 ∈ ℂ,

−∞ +∞

𝑓(𝑡)Δi𝑡 𝜉 𝑑𝑡 = ∫

−∞

i𝑡 ̄ 𝑓(𝑡)Δ (𝐽𝜉) 𝑑𝑡 ∈ 𝔖.

−∞

Thus, we have proved that Δ𝛼 𝔖 = 𝔖,

𝛼 ∈ ℂ;

𝐽𝔖 = 𝔖.

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We consider a function 𝑓0 ∈ L 1 (ℝ), such that 𝑓0̂ (𝑠) = 1 for any 𝑠 ∈ [−1, 1] and such that supp 𝑓0̂ be compact. We write 𝑓𝑛 (𝑡) = 𝑛𝑓0 (𝑛𝑡),

𝑛 ≥ 1.

Then 𝑓𝑛̂ (𝑠) = 𝑓0̂ (𝑠/𝑛), 𝑠 ∈ ℝ; hence +∞

𝑓𝑛 (𝑡)𝜆i𝑡 𝑑𝑡) 𝜒[e−𝑛 ,𝑒𝑛 ] (𝜆) = 𝜒[e−𝑛 ,𝑒𝑛 ] (𝜆).

(∫ −∞

With the help of Theorem 9.11 (vi), it is easy to prove that +∞

𝑓𝑛 (𝑡)Δi𝑡 𝑑𝑡) 𝜒[e−𝑛 ,𝑒𝑛 ] (Δ) = 𝜒[e−𝑛 ,𝑒𝑛 ] (Δ).

(∫ −∞

Since, for any 𝑛, we have +∞ +∞ ‖ +∞ ‖ ‖∫ 𝑓𝑛 (𝑡)Δi𝑡 𝑑𝑡 ‖ ≤ ∫ |𝑓𝑛 (𝑡)| 𝑑𝑡 = ∫ |𝑓0 (𝑡)| 𝑑𝑡, ‖ −∞ ‖ −∞ −∞

from the preceding equality, we infer that +∞

∫

𝑠𝑜

𝑓𝑛 (𝑡)Δi𝑡 𝑑𝑡 → 1.

−∞

Thus, for any 𝜉 ∈ 𝔗 and any 𝛼 ∈ ℂ, we get +∞

Δ𝛼 ∫ −∞

+∞

𝑓𝑛 (𝑡)Δi𝑡 𝜉 𝑑𝑡 = ∫

𝑓𝑛 (𝑡)Δi𝑡 (Δ𝛼 𝜉) 𝑑𝑡 → Δ𝛼 𝜉.

−∞

With the help of assertion (1) and of Theorem 10.20(1), it follows that Δ𝛼 |𝔖 = Δ𝛼 |𝔗 = Δ𝛼 ,

𝛼 ∈ ℂ.

Thus, assertions (2) and (3) are also true. Finally, if we now use the definition of 𝔖, it is easy to see that 𝜉1 , 𝜉2 ∈ 𝔖 ⇒ 𝐿𝜉1 (𝜉2 ) ∈ 𝔖. With the help of assertions (2) and (3), we get 𝜉 ∈ 𝔖 ⇒ 𝑆𝜉 = 𝐽Δ1/2 𝜉 ∈ 𝔖. From assertion (2), we infer that Δ1/2 𝔖 = Δ1/2 ; hence 𝑆|𝔖 = 𝑆. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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If we apply Lemmas 2 and 3 from Section 10.5, we infer that 𝔖 is a left Hilbert subalgebra of 𝔄″ , and 𝔖′ = 𝔄 ′ , 𝔖 ″ = 𝔄 ″ . Q.E.D. 10.23. Let 𝔄 ⊂ H be a left Hilbert algebra and 𝔗 the Tomita algebra associated to 𝔄 (see Section 10.20). In Section 10.8, we introduced two cones, polar to one another: 𝔓𝑆 ⊂ D 𝑆 = D (∆1/2 ) ,

𝔓𝑆 ∗ ⊂ D 𝑆 ∗ = D (∆−1/2 ) .

With the help of Proposition 10.8 and of Theorem 10.12, it is easy to verify that 𝐽𝔓𝑆 = 𝔓𝑆 ∗ . Since 𝑆𝜉 = 𝜉 for any 𝜉 ∈ 𝔓𝑆 , we infer that Δ1/2 𝔓𝑆 = 𝔓𝑆 ∗ . Consequently, Δ1/4 𝔓𝑆 = Δ−1/4 𝔓𝑆 ∗ . In the present section, we shall study the set 𝔓 = Δ1/4 𝔓𝑆 = Δ−1/4 𝔓𝑆 ∗ . Since 𝔓𝑆 and 𝔓𝑆 ∗ are convex cones, it follows that 𝔓 is a closed convex cone. In order to make the notations as simple and as expressive as possible, we shall use the following abbreviations, already introduced above (see Section 10.4): 𝜉𝜂 = 𝐿𝜉 (𝜂), for 𝜉, 𝜂 ∈ 𝔄″ ; 𝜉𝜂 = 𝑅𝜉 (𝜂), for 𝜉, 𝜂 ∈ 𝔄′ . We recall that, if 𝜉, 𝜂 ∈ 𝔄′ ∩ 𝔄″ , then 𝐿𝜉 (𝜂) = 𝑅𝜂 (𝜉). With these notations, we have the equalities 𝔓𝑆 = {𝜉(𝑆𝜉); 𝜉 ∈ 𝔄″ },

𝔓𝑆 ∗ = {𝜂(𝑆 ∗ 𝜂); 𝜂 ∈ 𝔄′ }.

Let 𝜉 ∈ 𝔄″ and let {𝜉𝑛 } ⊂ 𝔗 be a sequence having the properties from Corollary 2, Section 10.21. Then it is easy to prove that 𝜉𝑛 (𝑆𝜉𝑛 ) = 𝐿𝜉𝑛 (𝑆𝜉𝑛 ) → 𝐿𝜉 (𝑆𝜉) = 𝜉(𝑆𝜉). Consequently, (1)

𝔓𝑆 = 𝔓0𝑆 , where 𝔓0𝑆 = {𝜉(𝑆𝜉); 𝜉 ∈ 𝔗}.

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355

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𝔓𝑆 ∗ = 𝔓0𝑆 ∗ , where 𝔓0𝑆 ∗ = {𝜂(𝑆 ∗ 𝜂); 𝜂 ∈ 𝔗}.

By taking into account Theorem 10.20, it follows that for any 𝜉 ∈ 𝔗, we have Δ1/4 (𝜉(𝑆𝜉)) = Δ1/4 (𝜉)Δ1/4 (𝑆𝜉) = (Δ1/4 (𝜉))(𝐽(Δ1/4 (𝜉))). Since Δ1/4 𝔗 = 𝔗, we hence infer that, if we denote 𝔓0 = {𝜁(𝐽𝜁); 𝜁 ∈ 𝔗}, then we have Δ1/4 𝔓0𝑆 = 𝔓0 . Similarly, Δ−1/4 𝔓0𝑆 ∗ = 𝔓0 . If 𝜉 ∈ 𝔓𝑆 , then there exists a sequence {𝜉𝑛 } ⊂ 𝔓0𝑆 , such that 𝜉𝑛 → 𝜉. Since 𝑆𝜉𝑛 = 𝜉𝑛 , 𝑆𝜉 = 𝜉 and Δ1/2 = 𝐽𝑆, it follows that Δ1/2 𝜉𝑛 → Δ1/2 𝜉. Therefore, ‖Δ1/4 𝜉 − Δ1/4 𝜉𝑛 ‖2 = (Δ1/2 (𝜉 − 𝜉𝑛 )|𝜉 − 𝜉𝑛 ) → 0, whence Δ1/4 𝜉 ∈ Δ1/4 𝔓0𝑆 = 𝔓0 . Consequently, 𝔓 ⊂ 𝔓0 . On the other hand, it is obvious that 𝔓0 = Δ1/4 𝔓0𝑆 ⊂ Δ1/4 𝔓𝑆 ⊂ 𝔓. Therefore, (2)

𝔓 = 𝔓0 = {𝜁(𝐽𝜁); 𝜁 ∈ 𝔗}.

With the help of relation (2) and of Corollary 2 from Section 10.21, it is easy to verify that (3)

𝔓 = {𝜉(𝐽𝜉); 𝜉 ∈ 𝔄″ } = {(𝐽𝜂)𝜂; 𝜂 ∈ 𝔄′ }. From relation (2), it obviously follows that 𝜁 ∈ 𝔓 ⇒ 𝐽𝜁 = 𝜁.

(4)

For any 𝑡 ∈ ℝ and any 𝜁 ∈ 𝔗, in accordance with Theorem 10.20, we have Δi𝑡 (𝜁(𝐽𝜁)) = Δi𝑡 (𝜁)Δi𝑡 (𝐽𝜁) = (Δi𝑡 (𝜁))(𝐽(Δi𝑡 (𝜁))). Thus, Δi𝑡 𝔓0 ⊂ 𝔓0 . By taking into account relation (2), we get Δi𝑡 𝔓 = 𝔓,

(5)

𝑡 ∈ ℝ.

Since 𝔓 is a closed convex cone, from (5), we infer that +∞

(6)

𝜁 ∈ 𝔓, 𝑓 ∈ L (ℝ), 𝑓 ≥ 0 ⇒ ∫ 1

𝑓(𝑡)Δi𝑡 𝜁 𝑑𝑡 ∈ 𝔓.

−∞ Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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If 𝜉 ∈ 𝔄″ and 𝜁 ∈ 𝔗, then, by taking into account Corollary 1 from Section 10.13, we get [𝐿𝜉 (𝐽𝐿𝜉 𝐽)](𝜁(𝐽𝜁)) = 𝐿𝜉 𝐽𝐿𝜉 (𝐽𝐿𝜁 𝐽)(𝜁) = 𝐿𝜉 𝐽(𝐽𝐿𝜁 𝐽)𝐿𝜉 (𝜁) = 𝐿𝜉 𝐿𝜁 𝐽𝐿𝜉 (𝜁) = (𝜉𝜁)(𝐽(𝜉𝜁)); hence [𝐿𝜉 (𝐽𝐿𝜉 𝐽)]𝔓0 ⊂ 𝔓, and therefore, [𝐿𝜉 (𝐽𝐿𝜉 𝐽)]𝔓 ⊂ 𝔓. With the help of Kaplansky’s density theorem (3.10), we now infer that 𝑥 ∈ 𝔏(𝔄) ⇒ [𝑥(𝐽𝑥𝐽)]𝔓 ⊂ 𝔓.

(7)

For any 𝜉 ∈ 𝔓𝑆 and any 𝜂 ∈ 𝔓𝑆 ∗ , we have (Δ1/4 𝜉|Δ−1/4 𝜂) = (𝜉|𝜂) ≥ 0, because the cones 𝔓𝑆 and 𝔓𝑆 ∗ are polar to one another (10.8). From this result and from the definition of 𝔓, it follows that 𝜁, 𝜃 ∈ 𝔓 ⇒ (𝜁|𝜃) ≥ 0. On the other hand, let 𝜁 ∈ H be such that (𝜁|𝜃) ≥ 0, for any 𝜃 ∈ 𝔓. For any 𝑛 ∈ ℕ, we denote +∞

+∞

2

e−𝑛𝑡 Δi𝑡 𝜁 𝑑𝑡,

𝜁𝑛 = √𝑛/𝜋 ∫

𝜃𝑛 = √𝑛/𝜋 ∫

−∞

2

e−𝑛𝑡 Δi𝑡 𝜃 𝑑𝑡.

−∞

If 𝜃 ∈ 𝔓, then, in accordance with relation (6), we have 𝜃𝑛 ∈ 𝔓 and therefore, (𝜁𝑛 |𝜃) = (𝜁|𝜃𝑛 ) ≥ 0,

𝑛 ∈ ℕ.

In particular, if 𝜉 ∈ 𝔓𝑆 , then Δ1/4 𝜉 ∈ 𝔓 and (Δ1/4 𝜁𝑛 |𝜉) = (𝜁𝑛 |Δ1/4 𝜉) ≥ 0,

𝑛 ∈ ℕ;

hence, in accordance with Proposition 10.8, we infer that Δ1/4 𝜁𝑛 ∈ 𝔓𝑆 ∗ . Thus 𝜁𝑛 ∈ Δ−1/4 𝔓𝑆 ∗ ⊂ 𝔓. Since 𝜁𝑛 → 𝜁 and since 𝔓 is closed, it follows that 𝜁 ∈ 𝔓. Consequently, the cone 𝔓 is selfpolar, i.e. for any 𝜁 ∈ H , we have the equivalence (8)

𝜁 ∈ 𝔓 ⇔ (𝜁|𝜃) ≥ 0, for any 𝜃 ∈ 𝔓.

If 𝜁 ∈ 𝔓 ∩ (−𝔓), then, from relation (8), we infer that (𝜁| − 𝜁) ≥ 0, whence 𝜁 = 0. Consequently, (*)

𝔓 ∩ (−𝔓) = {0}.

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357

Since 𝔓 is a convex cone having property (*), we infer that 𝔓 determines an order relation ‘≤’ in the set {𝜁 ∈ H ; 𝐽𝜁 = 𝜁}: 𝜁 ≤ 𝜃 ⇔ 𝜃 − 𝜁 ∈ 𝔓. We shall now prove that for any 𝜁 ∈ H , such that 𝐽𝜁 = 𝜁, there exist 𝜁 + , 𝜁 − ∈ 𝔓, such that (**)

𝜁 = 𝜁+ − 𝜁− ,

𝜁+ ⟂ 𝜁− ,

and these elements are uniquely determined by these conditions. Indeed, let 𝜁 ∈ H be such that 𝐽𝜁 = 𝜁. Since 𝔓 is a closed convex subset of the Hilbert space H , there exists a unique element 𝜁 + ∈ 𝔓, such that ‖𝜁 + − 𝜁‖ = inf{‖𝜃 − 𝜁‖; 𝜃 ∈ 𝔓}. We denote 𝜁 − = 𝜁 + − 𝜁. For any 𝜃 ∈ 𝔓 and any 𝑡 ≥ 0, we have 0 ≤ ‖(𝜁 + + 𝑡𝜃) − 𝜁‖2 − ‖𝜁 + − 𝜁‖2 = 𝑡 2 ‖𝜃‖2 + 2𝑡Re (𝜁 − |𝜃), whence we infer that Re (𝜁 − |𝜃) ≥ 0. Since 𝐽𝜁 − = 𝜁 − and 𝐽𝜃 = 𝜃, we have (𝜁 − |𝜃) = Re (𝜁 − |𝜃) ≥ 0. From relation (8), we infer that 𝜁 − ∈ 𝔓. On the other hand, for any 𝑡 ∈ (0, 1), we have 0 ≤ ‖(1 − 𝑡)𝜁 + − 𝜁‖2 − ‖𝜁 + − 𝜁‖2 = 𝑡 2 ‖𝜁 + ‖2 − 2𝑡Re (𝜁 + |𝜁 − ), whence we infer that Re (𝜁 + |𝜁 − ) ≤ 0. Since 𝜁 + , 𝜁 − ∈ 𝔓, from relation (8), we infer that (𝜁 + |𝜁 − ) ≥ 0. Consequently, we have 𝜁 + ⟂ 𝜁 − . Let us now assume that 𝜁 = 𝜉 − 𝜂, 𝜉, 𝜂 ∈ 𝔓, 𝜉 ⟂ 𝜂. Then 𝜁 + − 𝜉 = 𝜁 − − 𝜂 and therefore, ‖𝜁 + − 𝜉‖2 = (𝜁 + − 𝜉|𝜁 − − 𝜂) = −(𝜁 + |𝜂) − (𝜉|𝜁 − ) ≤ 0, whence 𝜉 = 𝜁 + , 𝜂 = 𝜁 − . In particular, from the proved assertion, we infer that H coincides with the linear hull of 𝔓: H = (𝔓 − 𝔓) + i(𝔓 − 𝔓). We observe that any selfpolar convex cone 𝔓 in a Hilbert space H determines a unique conjugation 𝐽 in H , such that 𝐽𝜁 = 𝜁, for any 𝜁 ∈ 𝔓. Indeed, if 𝜁 ∈ H and 𝜁 ⟂ 𝔓, then 𝜁 ∈ 𝔓, because 𝔓 is selfpolar and therefore, (𝜁|𝜁) = 0, whence 𝜁 = 0. Thus 𝔓 is a total subset of H ; hence (𝔓 − 𝔓) + i(𝔓 − 𝔓) is a dense subset of H . Consequently, the conjugation 𝐽 is uniquely determined by the formula 𝐽(𝜉 + i𝜂) = 𝜉 − i𝜂,

𝜉, 𝜂 ∈ 𝔓 − 𝔓.

The decomposition (**) is valid in this more general situation, with the same proof. In particular, in this general situation H coincides with the linear hull of 𝔓, too. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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We shall say that a von Neumann algebra M ⊂ B (H ) is hyperstandard if there exists a conjugation 𝐽 ∶ H → H and a selfpolar convex cone 𝔓 ⊂ H , such that (1) the mapping 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 is a *-anti-isomorphism of M onto M ′ , which acts identically on the center. (2) 𝜁 ∈ 𝔓 ⇒ 𝐽𝜁 = 𝜁. (3) 𝑥 ∈ M ⇒ [𝑥(𝐽𝑥𝐽)]𝔓 ⊂ 𝔓. From the above results, we infer that, for any left Hilbert algebra 𝔄 ⊂ H , the von Neumann algebra 𝔏(𝔄) ⊂ B (H ) is a hyperstandard von Neumann algebra. 10.24. We now consider a hyperstandard von Neumann algebra M ⊂ B (H ), with the conjugation 𝐽 and the selfpolar cone 𝔓. We denote by Z the center of M . Let 𝑒 be a projection in M . Then 𝐽𝑒𝐽 is a projection in M ′ and z(𝐽𝑒𝐽) = z(𝑒). From Corollary 3.9, we easily infer that 𝑒 ≠ 0 ⇔ 𝑒(𝐽𝑒𝐽) ≠ 0. We denote by 𝑞 the projection 𝑒(𝐽𝑒𝐽) = (𝐽𝑒𝐽)𝑒 ∈ B (H ). By taking into account Sections 3.13, 3.14 and 3.15, we infer that M 𝑞 ⊂ B (𝑞H ) is a von Neumann algebra, whose commutant is (M ′ )𝑞 and whose center is Z 𝑞 ; also, the mapping 𝑥𝑒 ↦ 𝑥𝑞 is a *-isomorphism of the reduced von Neumann algebra M 𝑒 onto the von Neumann algebra M 𝑞 . Since 𝐽𝑞 = 𝑞𝐽, and 𝑞𝔓 = [𝑒(𝐽𝑒𝐽)]𝔓 ⊂ 𝔓, it is easy to prove that M 𝑞 ⊂ B (𝑞H ) is a hyperstandard von Neumann algebra, whose conjugation is 𝑞𝐽𝑞 and whose selfpolar cone is 𝑞𝔓. This remark will enable us to reduce some of the problems to the case of the hyperstandard von Neumann algebras of countable type. Lemma 1. For any projection 𝑒 ∈ M , there exists a family {𝜉𝑛 }𝑛∈𝐼 ⊂ 𝔓, such that the projections 𝑝𝜉𝑛 be non-zero, mutually orthogonal and such that 𝑒 = ∑ 𝑝𝜉𝑛 . 𝑛∈𝐼

If 𝑒 is of countable type, then there exists a 𝜉0 ∈ 𝔓, such that 𝑒 = 𝑝𝜉0 . In particular, if M is of countable type, then M has a separating cyclic vector to 𝜉0 ∈ 𝔓. Proof. If 𝑒 ≠ 0, then 𝑒(𝐽𝑒𝐽) ≠ 0 and since H is the linear hull of 𝔓, there exists a non-zero vector 𝜉 ∈ [𝑒(𝐽𝑒𝐽)]𝔓 ⊂ 𝔓. Obviously, 𝑝𝜉 ≤ 𝑒. Thus, the first assertion follows with a familiar argument based on the Zorn lemma. If 𝑒 is of countable type, then the set 𝐼 is at most countable and we can assume that 1 ‖𝜉𝑛 ‖ = 1. If we define to 𝜉0 = ∑𝑛 𝑛 𝜉𝑛 , it follows that 𝜉0 ∈ 𝔓 and 𝑒 = 𝑝𝜉0 . 2

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The last assertion of the lemma follows from the remark that 𝐽𝜉0 = 𝜉0 , for any 𝜉0 ∈ 𝔓, ′ and from the fact that 𝐽𝑝𝜉0 𝐽 = 𝑝𝐽𝜉 (see E.6.9). Q.E.D. 0 If the von Neumann algebra M is of countable type and if 𝜉0 ∈ 𝔓 is a separating cyclic vector, then we can consider the left Hilbert algebra 𝔄 = M 𝜉0 ⊂ H and we have M = 𝔏(𝔄) (see Section 10.6). On the one hand, by hypothesis, M is a hyperstandard von Neumann algebra. On the other hand, as we have seen in Section 10.23, 𝔏(𝔄) is endowed with a natural structure of a hyperstandard von Neumann algebra. We shall denote by 𝑆 𝔄 , 𝐽 𝔄 the operators which are associated to the left Hilbert algebra 𝔄 = M 𝜉0 , and by 𝔓𝔄 the selfpolar convex cone, which is associated to the left Hilbert algebra 𝔄 = M 𝜉0 ; it is easy to verify that 𝔓𝔄 = {[𝑥(𝐽 𝔄 𝑥𝐽 𝔄 )]𝜉0 ;

𝑥 ∈ M }.

Under these assumptions, we have the following: Lemma 2. 𝐽 𝔄 = 𝐽 and 𝔓𝔄 = 𝔓. Proof. For any 𝑥 ∈ M , we have 𝐽𝑥𝐽 ∈ M ′ . Consequently, [𝐽(𝑆 𝔄 )∗ 𝐽]𝑥𝜉0 = 𝐽(𝑆 𝔄 )∗ (𝐽𝑥𝐽)𝜉0 = 𝐽(𝐽𝑥𝐽)∗ 𝜉0 = 𝑥 ∗ 𝜉0 . From these equalities, it is easy to infer that 𝑆 𝔄 ⊂ 𝐽(𝑆 𝔄 )∗ 𝐽, whereas a similar argument shows that (𝑆 𝔄 )∗ ⊂ 𝐽𝑆 𝔄 𝐽. Thus, 𝐽𝑆 𝔄 = (𝑆 𝔄 )∗ 𝐽. By taking into account Proposition 9.2, we infer that (𝐽𝑆 𝔄 )∗ = (𝑆 𝔄 )∗ 𝐽 = 𝐽𝑆 𝔄 ; hence the linear operator 𝐽𝑆 𝔄 is self-adjoint. For any 𝑥 ∈ M , we have ((𝐽𝑆 𝔄 )(𝑥𝜉0 )|𝑥𝜉0 ) = (𝐽𝑥 ∗ 𝜉0 |𝑥𝜉0 ) = (𝑥𝐽𝑥𝜉0 |𝜉0 ) = ([𝑥(𝐽𝑥𝐽)]𝜉0 |𝜉0 ) ≥ 0, because 𝜉0 and [𝑥(𝐽𝑥𝐽)]𝜉0 belong to 𝔓. Since 𝐽𝑆 𝔄 = 𝐽𝑆 𝔄 |M 𝜉0 , it follows that the operator 𝐽𝑆 𝔄 is positive. Since 𝑆 𝔄 = 𝐽(𝐽𝑆 𝔄 ), by taking into account Section 10.1 and the uniqueness of the polar decomposition, we get 𝐽 = 𝐽𝔄. Then 𝔓𝔄 = {[𝑥(𝐽 𝔄 𝑥𝐽 𝔄 )]𝜉0 ; 𝑥 ∈ M } = {[𝑥(𝐽𝑥𝐽)]𝜉0 ; 𝑥 ∈ M } ⊂ 𝔓, and since 𝔓𝔄 and 𝔓 are, both, selfpolar, we infer that 𝔓𝔄 = 𝔓. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Thus, if the hyperstandard von Neumann algebra M is of countable type, then 𝐽 and 𝔓 are naturally derived from a left Hilbert algebra 𝔄 = M 𝜉0 , where to 𝜉0 ∈ 𝔓 is a separating cyclic vector for M . Consequently, in this case, we can avail ourselves of the powerful instrument of the left Hilbert algebras. We shall continue to assume that M is of countable type and we choose a separating cyclic vector 𝜉0 ∈ 𝔓. We shall denote by Δ the modular operator corresponding to 𝔄 = M 𝜉0 . 1 We obviously have Δ𝜉0 = 𝑆 ∗ 𝑆𝜉0 = 𝜉0 , whence (1 + Δ)−1 𝜉0 = 𝜉0 . It is easy to verify, 2 first for polynomials and then, by tending to the limit, that for any function 𝐹 ∈ B ([0, 1]), we have 𝐹((1 + Δ)−1 )𝜉0 = 𝐹(1/2)𝜉0 . By taking into account Section 9.10, we infer that, for any bounded 𝑓 ∈ B ([0, +∞)), we have 𝑓(Δ)𝜉0 = 𝐹𝑓 ((1 + Δ)−1 )𝜉0 = 𝐹𝑓 (1/2)𝜉0 = 𝑓(1)𝜉0 . In particular, Δi𝑡 𝜉0 = 𝜉0 ,

𝑡 ∈ ℝ,

and by analytic continuation, we obtain Δ𝛼 𝜉0 = 𝜉0 ,

𝛼 ∈ ℂ.

Lemma 3. The mapping Φ ∶ 𝑎 ↦ Δ1/4 𝑎𝜉0 is an order isomorphism of the set of all self-adjoint operators 𝑎 ∈ M onto the set of all vectors 𝜉 ∈ H , such that 𝐽𝜉 = 𝜉 and having the property that there exists a 𝜆 > 0, for which −𝜆𝜉0 ≤ 𝜉 ≤ 𝜆𝜉0 . Proof. Since Δ1/4 is injective, and since 𝜉0 is separating for M , the mapping Φ is injective. In accordance with Section 10.9, and with the definition of 𝔓 (Section 10.23), we infer that for 𝑎 ∈ M , 𝑎 = 𝑎∗ , we have 𝑎 ≥ 0 ⇔ 𝑎𝜉0 ∈ 𝔓𝑆 ⇔ Δ1/4 𝑎𝜉0 ∈ 𝔓. Thus, the mapping Φ is an order isomorphism and we have still to show that Φ is surjective. For 𝑎 ∈ M , 𝑎 = 𝑎∗ , we have (1 + Δ1/2 )(𝑎𝜉0 ) = 𝑎𝜉0 + 𝐽𝑎𝜉0 , whence Δ1/4 (𝑎𝜉0 ) = (Δ1/4 + Δ−1/4 )−1 (𝑎𝜉0 + 𝐽𝑎𝜉0 ). Since the operator (Δ1/4 + Δ−1/4 )−1 is bounded, it follows that 𝑤

𝑎𝑖 → 𝑎 ⇒ Φ(𝑎𝑖 ) → Φ(𝑎) weakly. Since the set {𝑎 ∈ M ; 0 ≤ 𝑎 ≤ 1} is w-compact, it follows that the set Φ({𝑎 ∈ M ; 0 ≤ 𝑎 ≤ 1}) is weakly compact in H . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Let now 𝜉 ∈ H , 𝐽𝜉 = 𝜉, 0 ≤ 𝜉 ≤ 𝜉0 . With the help of relation (6) from Section 10.23, we obtain +∞

0 ≤ 𝜉𝑛 = √𝑛/𝜋 ∫

+∞

e

−𝑛𝑡 2 i𝑡

Δ 𝜉 𝑑𝑡 ≤ √𝑛/𝜋 ∫

−∞

2

e−𝑛𝑡 Δi𝑡 𝜉0 𝑑𝑡 = 𝜉0 .

−∞

For any 𝜂 ∈ 𝔓𝑆 ∗ , we have Δ−1/4 𝜂 ∈ 𝔓; hence (Δ−1/4 𝜉𝑛 |𝜂) = (𝜉𝑛 |Δ−1/4 𝜂) ≥ 0. Thus, Δ−1/4 𝜉𝑛 ∈ 𝔓𝑆 (see Section 10.8). Similarly, 𝜉0 − Δ−1/4 𝜉𝑛 = Δ−1/4 (𝜉0 − 𝜉𝑛 ) ∈ 𝔓𝑆 . From Proposition 10.8, we infer that the operators 𝐿0∆−1/4 𝜉𝑛 and 1 − 𝐿0∆−1/4 𝜉𝑛 = 𝐿0𝜉 −∆−1/4𝜉𝑛 0 are positive. Consequently, the operator 𝑎𝑛 = 𝐿∆−1/4 𝜉𝑛 is bounded and 0 ≤ 𝑎𝑛 ≤ 1. Obviously, 𝜉𝑛 = Φ(𝑎𝑛 ). Since Φ(𝑎𝑛 ) = 𝜉𝑛 → 𝜉 and since the set Φ({𝑎 ∈ M ; 0 ≤ 𝑎 ≤ 1}) is closed, it follows that there exists an 𝑎 ∈ M , 0 ≤ 𝑎 ≤ 1, such that 𝜉 = Φ(𝑎). Hence we easily infer that Φ is surjective. Q.E.D. We now return to the general situation in which M ⊂ B (H ) is a hyperstandard von Neumann algebra, with the conjugation 𝐽 and the selfpolar convex cone 𝔓. The remarks we made so far enable us to prove the following important result. Proposition. For any 𝜉, 𝜂 ∈ 𝔓, we have ‖𝜉 − 𝜂‖2 ≤ ‖𝜔𝜉 − 𝜔𝜂 ‖ ≤ ‖𝜉 − 𝜂‖‖𝜉 + 𝜂‖. Proof. The second inequality follows from the relation 1 (𝜔𝜉 − 𝜔𝜂 )(𝑥) = [(𝑥(𝜉 + 𝜂)|𝜉 − 𝜂) + (𝑥(𝜉 − 𝜂)|𝜉 + 𝜂)], 2

𝑥 ∈ M,

and holds for any 𝜉, 𝜂 ∈ H . Let 𝑒 = s(𝜔𝜉 ) ∨ s(𝜔𝜂 ) and 𝑞 = 𝑒(𝐽𝑒𝐽). Then 𝑒 is of countable type and 𝜉, 𝜂 ∈ 𝑞𝔓. In accordance with the remark we made at the beginning of the present section, M 𝑞 is hyperstandard and *-isomorphic with M 𝑒 . Consequently, in order to prove the first inequality, we can assume that M is of countable type. We shall first consider the case in which the vector 𝜉 + 𝜂 is separating. Since 𝐽(𝜉 + 𝜂) = 𝜉 + 𝜂, it follows that 𝜉 + 𝜂 is also cyclic (see. E.6.9). We shall denote by Δ the modular operator associated to the left Hilbert algebra 𝔄 = M (𝜉 +𝜂). In accordance with Lemma 2, 𝔓 is the selfpolar convex cone associated to 𝔄. Since −(𝜉 + 𝜂) ≤ 𝜉 − 𝜂 ≤ 𝜉 + 𝜂, from Lemma 3, we infer that there exists an a 𝑎 ∈ M ; −1 ≤ 𝑎 ≤ 1, such that 𝜉 − 𝜂 = Δ1/4 𝑎(𝜉 + 𝜂). Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Then ‖𝜔𝜉 − 𝜔𝜂 ‖ ≥ (𝜔𝜉 − 𝜔𝜂 )(𝑎) = Re (𝑎(𝜉 + 𝜂)|𝜉 − 𝜂) = (Δ−1/4 (𝜉 − 𝜂)|𝜉 − 𝜂). Since 𝐽Δ1/4 = Δ−1/4 𝐽, it follows that (Δ−1/4 (𝜉 − 𝜂)|𝜉 − 𝜂) = (Δ1/4 (𝜉 − 𝜂)|𝜉 − 𝜂); hence

1 ‖𝜔𝜉 − 𝜔𝜂 ‖ ≥ ( (Δ1/4 + Δ−1/4 )(𝜉 − 𝜂)|𝜉 − 𝜂) ≥ ‖𝜉 − 𝜂‖2 . 2 If the vector 𝜉 + 𝜂 is not separating, then 1 − 𝑝(𝜉+𝜂) ≠ 0. In accordance with Lemma 1, there exists a 𝜁 ∈ 𝔓, such that 𝑝𝜁 = 1 − 𝑝(𝜉+𝜂) . We now consider the vectors 𝜉𝑛 = 𝜉 +

1 𝜁 ∈ 𝔓, 𝑛

1 𝜂𝑛 = 𝜂 + 𝜁 ∈ 𝔓, 𝜂

𝑛 ∈ ℕ.

In accordance with the remark just made after the Corollary 3.8, it follows that ′ 𝑝(𝜉 = 𝑝′ 𝑛 +𝜂𝑛 )

2 𝑛

(𝜉+𝜂+ 𝜁)

= 1,

because 𝑝(𝜉+𝜂) and 𝑝 2 𝜁 are orthogonal. Consequently, we have 𝑝(𝜉𝑛 +𝜂𝑛 ) = 1. 𝑛

In accordance with the first case, just considered, we have ‖𝜔𝜉𝑛 − 𝜔𝜂𝑛 ‖ ≥ ‖𝜉𝑛 − 𝜂𝑛 ‖2 ,

𝑛 ∈ ℕ.

If we tend to the limit, in this inequality, for 𝑛 → ∞, we obtain ‖𝜔𝜉 − 𝜔𝜂 ‖ ≥ ‖𝜉 − 𝜂‖2 . Q.E.D. Corollary 1. For 𝜉, 𝜂 ∈ 𝔓, we have (i) 𝜉 ⟂ 𝜂 ⇔ 𝑝𝜉 ⟂ 𝑝𝜂 . (ii) if 𝜁 ∈ 𝔓 and 𝜁 ⟂ 𝜂 imply that 𝜁 ⟂ 𝜉, then 𝑝𝜉 ≤ 𝑝𝜂 . (iii) 𝜉 ≤ 𝜂 ⇒ 𝑝𝜉 ≤ 𝑝𝜂 . Proof. (i) If 𝜉 ⟂ 𝜂 then ‖𝜔𝜉 − 𝜔𝜂 ‖ ≥ ‖𝜉 − 𝜂‖2 = ‖𝜉‖2 + ‖𝜂‖2 = ‖𝜔𝜉 ‖ + ‖𝜔𝜂 ‖; hence, in accordance with exercise E.5.15, 𝑝𝜉 = s(𝜔𝜉 ) ⟂ s(𝜔𝜂 ) = 𝑝𝜂 . The converse is obvious. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

The theory of standard von Neumann algebras

363

(ii) In accordance with Lemma 1, there exists a family {𝜁𝑛 }𝑛∈𝐼 ⊂ 𝔓, such that 1 − 𝑝𝜂 = ∑ 𝑝𝜁𝑛 . 𝑛∈𝐼

Now the assertion (ii) is easy to prove, by using assertion (i). (iii) if 0 ≤ 𝜉 ≤ 𝜂, then, for any 𝜁 ∈ 𝔓, we have 0 ≤ (𝜉|𝜁) ≤ (𝜂|𝜁), and assertion (iii) follows from assertion (ii). Q.E.D. Corollary 2. The mapping 𝜉 → 𝜔𝜉 is a homeomorphism of 𝔓 onto a closed subset of (M ∗ )+ = {𝜑 ∈ M ∗ ; 𝜑 ≥ 0}, with respect to the norm topologies. Proof. From the preceding proposition, it obviously follows that the mapping in the statement of the proposition is a homeomorphism of 𝔓 onto {𝜔𝜉 ; .𝜉 ∈ 𝔓} ⊂ (M ∗ )+ . If {𝜔𝜉𝑛 }, 𝜉𝑛 ∈ 𝔓, is a Cauchy sequence, then the same proposition shows that {𝜉𝑛 } is a Cauchy sequence in 𝔓; hence, there exists a 𝜉 ∈ 𝔓, such that 𝜔𝜉𝑛 → 𝜔𝜉 . Q.E.D. 10.25. In this section, we shall present a Radon–Nikodym-type theorem, which is similar to Theorem 5.23, for normal forms on hyperstandard von Neumann algebras. We consider a hyperstandard von Neumann algebra M ⊂ B (H ), whose conjugation is 𝐽 and whose selfpolar cone is 𝔓. Lemma 1. Let 𝜉0 ∈ 𝔓. For any normal form 𝜑 on M , such that 𝜑 ≤ 𝜔𝜉0 , there exists an 𝜂 ∈ 𝔓, such that 1 𝜑 = 𝜔𝜉0 ,𝜂 + 𝜔𝜂,𝜉0 , and 𝜂 ≤ 𝜉0 . 2 Proof. Let 𝑒 = s(𝜔𝜉0 ) and 𝑞 = 𝑒(𝐽𝑒𝐽). Then 𝜉0 ∈ 𝑞𝔓 is a separating cyclic vector for the hyperstandard von Neumann algebra M 𝑞 ⊂ B (𝑞H ) which is *-isomorphic to M 𝑒 (see Section 10.24). Consequently, we can assume that 𝜉0 is a separating cyclic vector for M . In this case, we shall denote by 𝑆 and Δ, the operators that are associated to the left Hilbert algebra 𝔄 = M 𝜉0 . In accordance with Lemma 2 from Section 10.24, 𝐽 and 𝔓 are also associated to 𝔄. By taking into account Lemma 5.19, it follows that there exists an operator 𝑎′ ∈ M ′ , 0 ≤ 𝑎′ ≤ 1, such that 𝜑(𝑥) = (𝑥𝜉0 |𝑎′ 𝜉0 ), 𝑥 ∈ M . From Section 10.9, we infer that 𝑎′ 𝜉0 , (1 − 𝑎′ )𝜉0 ∈ 𝔓𝑆 ∗ . Thus, 𝜁 = Δ−1/4 (𝑎′ 𝜉0 ) ∈ 𝔓,

𝜉0 − 𝜁 = Δ−1/4 ((1 − 𝑎′ )𝜉0 ) ∈ 𝔓.

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We define 𝜂 = (1 + Δ1/2 )−1 𝑎′ 𝜉0 = (1 + Δ1/2 )−1 Δ1/4 𝜁. If we denote by 𝑓 the function 𝑡 ↦ 2/(e2𝜋𝑡 + e−2𝜋𝑡 ), then, by applying Corollary 9.23, for 𝐴 = Δ−1/2 , we infer that +∞

𝜂=∫

𝑓(𝑡)Δi𝑡 𝜁 𝑑𝑡.

−∞

Since 0 ≤ 𝜁 ≤ 𝜉0 , with help of relation (6) from Section 10.23, we infer that +∞

0≤∫

+∞

𝑓(𝑡)Δi𝑡 𝜁𝑑𝑡 ≤ ∫

−∞

𝑓(𝑡)Δi𝑡 𝜉0 𝑑𝑡,

−∞

1

i.e. 0 ≤ 𝜂 ≤ 𝜉0 . 2 On the other hand, by taking into account 𝐽𝜂 = 𝜂, we have 𝑎′ 𝜉0 = (1 + Δ1/2 )𝜂 = 𝜂 + 𝑆 ∗ 𝐽𝜂 = 𝜂 + 𝑆 ∗ 𝜂, whence, for any 𝑥 ∈ M , 𝜑(𝑥) = (𝑥𝜉0 |𝑎′ 𝜉0 ) = (𝑥𝜉0 |𝜂) + (𝑥𝜉0 |𝑆 ∗ 𝜂) = (𝑥𝜉0 |𝜂) + (𝜂|𝑆(𝑥𝜉0 )) = (𝑥𝜉0 |𝜂) + (𝜂|𝑥 ∗ 𝜉0 ) = (𝜔𝜉0 ,𝜂 + 𝜔𝜂,𝜉0 )(𝑥). Q.E.D. Lemma 2. Let 𝜉0 ∈ 𝔓. For any normal form 𝜑 on M , such that 𝜑 ≤ 𝜔𝜉0 , there exists a 𝜉 ∈ 𝔓, such that 𝜑 = 𝜔𝜉 . Proof. If we apply Lemma 1 to the normal form 𝜑 = 𝜔𝜉0 − 𝜑, then it follows that there exists an 𝜂1 ∈ 𝔓, such that 𝜑1 = 𝜔𝜉0 ,𝜂1 + 𝜔𝜂1 ,𝜉0 and 𝜂1 ≤ 𝜉0 . We denote 𝜉1 = 𝜉0 − 𝜂1 . It easy to prove that 𝜔𝜉1 − 𝜑 = 𝜔𝜂1 . Also, ‖𝜑1 ‖ = 𝜑1 (1) = 2 Re(𝜉0 |𝜂1 ) = 2(𝜉0 |𝜂1 ). Since 𝜂1 ∈ 𝔓 and 𝜂1 ≤ 𝜉0 , we have (𝜂1 |𝜂1 ) ≤ (𝜉0 |𝜂1 ). Consequently, 1 1 ‖𝜔𝜉1 − 𝜑‖ = ‖𝜔𝜂1 ‖ = ‖𝜂1 ‖2 ≤ ‖𝜑1 ‖ = ‖𝜔𝜉0 − 𝜑‖. 2 2 Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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The theory of standard von Neumann algebras By induction, we find a sequence {𝜉𝑛 } ∈ 𝔓, such that

1 ‖𝜔𝜉𝑛+1 − 𝜑‖ ≤ ‖𝜔𝜉𝑛 − 𝜑‖, 𝑛 ∈ ℕ. 2 Then ‖𝜔𝜉𝑛 − 𝜑‖ → 0. In accordance with Corollary 2 from Section 10.24, it follows that there exists a 𝜉 ∈ 𝔓, such that 𝜑 = 𝜔𝜉 . Q.E.D. Theorem. Let M ⊂ B (H ) be a hyperstandard von Neumann algebra, with the conjugation 𝐽 and the selfpolar convex cone 𝔓. Then the mapping 𝜉 → 𝜔𝜉 is a homeomorphism of 𝔓 onto (M ∗ )+ , with respect to the norm topologies. Proof. By taking into account Corollary 2 from Section 10.24, we must prove only that the set {𝜔𝜉 ; 𝜉 ∈ 𝔓} is dense in (M ∗ )+ . It is easy to see that, without any loss of generality, we can assume that M is of countable type. In this case, in accordance with Lemma 1 from Section 10.24, there exists a separating cyclic vector 𝜉0 ∈ 𝔓. In accordance with Lemma 2 from above, the set {𝜔𝜉 ; 𝜉 ∈ 𝔓} contains the set {𝜑 ∈ (M ∗ )+ ; there exists a 𝜆 > 0, such that 𝜑 ≤ 𝜆𝜔𝜉0 }; hence it is sufficient to show that this set is dense in (M ∗ )+ . Let 𝜑 be a normal form on M . Then, in accordance with Corollary 5.24, there exists an 𝜂 ∈ H = [M ′ 𝜉0 ], such that 𝜑 = 𝜔𝜂 . Consequently, there exists a sequence {𝑥𝑛′ } ⊂ M ′ , such that 𝑥𝑛′ 𝜉0 → 𝜂. Then 𝜔𝑥𝑛′ 𝜉0 → 𝜔𝜂 = 𝜑 and 0 ≤ 𝜔𝑥𝑛′ 𝜉0 ≤ ‖𝑥𝑛′ ‖2 𝜔𝜉0 . Thus, the theorem is proved.

Q.E.D.

For any normal form 𝜑 on M , we shall denote by 𝜑1/2 the vector 𝜉 ∈ 𝔓, uniquely determined by the condition 𝜑 = 𝜔𝜉 . If 𝜑, 𝜓 are normal forms on M , and 𝜑 ≤ 𝜓, then 𝜔𝜑1/2 ≤ 𝜔𝜓1/2 . Since 𝐽(𝜑1/2 ) = 𝜑1/2 and 𝐽(𝜓 1/2 ) = 𝜓 1/2 , we infer that 𝜔𝜑′ 1/2 ≤ 𝜔𝜓′ 1/2 . With the help of this relation, it is easy to see that there exists a uniquely determined. operator 𝑥 ∈ M , such that (*)

𝜑1/2 = 𝑥𝜓 1/2 and s(|𝑥|) ≤ s(𝜓);

namely, 𝑥(𝑥 ′ 𝜓1/2 ) = 𝑥 ′ 𝜑1/2 , 𝑥(𝜂) = 0,

𝑥′ ∈ M ′ ,

𝜂 ∈ [M ′ 𝜓1/2 ].

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In particular, 𝜑 = 𝐿𝑥∗ 𝑅𝑥 𝜓 and s(|𝑥|) ≤ s(𝜓), an assertion of the Radon-Nikodym type, similar to the Sakai theorem (5.21), but different from it, since 𝑥 is not assumed to be positive. The operator 𝑥 ∈ M , which is uniquely determined by the conditions (*). will be 𝑑𝜑 denoted by . 𝑑𝜓

If 𝜃, 𝜑, 𝜓 are normal forms on M and if 𝜃 ≤ 𝜑 ≤ 𝜓, then it is easy to prove the ‘chain rule’ 𝑑𝜃 𝑑𝜑 𝑑𝜃 ⋅ = . 𝑑𝜑 𝑑𝜓 𝑑𝜓 10.26. From Sections 10.14 and 10.23, we infer that any von Neumann algebra is *-isomorphic to a hyperstandard von Neumann algebra. It is obvious that any hyperstandard von Neumann algebra is standard. Consequently, in accordance with Corollary 10.15, any *-isomorphism between two hyperstandard von Neumann algebras is spatial. Theorem 10.25 implies the following much more precise result. Corollary. Let M 𝑘 ⊂ B (H 𝑘 ) be a hyperstandard von Neumann algebra, 𝐽𝑘 its conjugation and 𝔓𝑘 its selfpolar cone, 𝑘 = 1, 2. If 𝜋 ∶ M1 → M2 is a *-isomorphism, then there exists a unitary operator 𝑢 ∶ H1 → H2 uniquely determined by the conditions (1) 𝜋(𝑥1 ) = 𝑢 ∘ 𝑥1 ∘ 𝑢∗ , for any 𝑥1 ∈ M 1 . (2) 𝐽2 = 𝑢 ∘ 𝐽1 ∘ 𝑢∗ . (3) 𝔓2 = 𝑢(𝔓1 ). Proof. If the unitary operator 𝑢 has the required properties, then, for any vector 𝜉1 ∈ 𝔓1 , we have 𝜔ᵆ𝜉1 = 𝜔𝜉1 ∘ 𝜋 −1 and 𝑢𝜉1 ∈ 𝔓2 . From Theorem 10.25, we infer that 𝑢𝜉1 is uniquely determined. Since H 1 is the linear hull of 𝔓1 , we infer that the unitary operator 𝑢 is uniquely determined by the stated properties. In order to prove the existence of 𝑢, we first consider the case in which M 1 is of countable type. Then M 1 has a separating cyclic vector 𝜉1 ∈ 𝔓1 (see Lemma 1 from Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Section 10.24). In accordance with Theorem 10.25, there exists a vector 𝜉2 ∈ 𝔓2 , such that 𝜔𝜉2 = 𝜔𝜉1 ∘ 𝜋 −1 . It follows that 𝜉2 is separating for M 2 ; since 𝐽2 𝜉2 = 𝜉2 , we infer that 𝜉2 is also cyclic. As in the proof of Corollary 5.25, it is possible to prove that by the relations 𝑢(𝑥1 𝜉1 ) = 𝜋(𝑥1 )𝜉2 ,

𝑥1 ∈ M 1 ,

one determines a unitary operator 𝑢 ∶ H 1 → H 2 , which satisfies condition (1) from the statement of the theorem. If 𝑆𝑘 is the operator associated to the left Hilbert algebra 𝔄𝑘 = M 𝑘 𝜉𝑘 , 𝑘 = 1, 2, then it is easy to prove that 𝑆2 = 𝑢 ∘ 𝑆1 ∘ 𝑢∗ . By taking into account Lemma 2 from Section 10.24, we infer that 𝑢 satisfies condition (2) from the statement of the theorem. Finally, if we now use again Lemma 2 from Section 10.24, we obtain 𝔓2 = {[𝑥2 (𝐽2 𝑥2 𝐽2 )]𝜉2 ; 𝑥2 ∈ M 2 } = {[(𝑢𝑥1 𝑢∗ )(𝑢𝐽1 𝑢∗ )(𝑢𝑥1 𝑢∗ )(𝑢𝐽1 𝑢∗ )]𝜉2 ; 𝑥1 ∈ M 1 } = 𝑢({[𝑥1 (𝐽1 𝑥1 𝐽1 )]𝜉1 ; 𝑥1 ∈ M 1 }) = 𝑢(𝔓1 ). In the general case, there exists an increasingly directed family {𝑒1,𝑖 }𝑖∈𝐼 , consisting of projections of countable type in M 1 whose least upper bound is equal to 1. For any 𝑖 ∈ 𝐼, we denote 𝑒2,𝑖 = 𝜋(𝑒1,𝑖 ), 𝑞1,𝑖 = 𝑒1,𝑖 (𝐽1 𝑒1,𝑖 𝐽1 ), 𝑞2,𝑖 = 𝑒2,𝑖 (𝐽2 𝑒2,𝑖 𝐽2 ). By taking into account the remark made at the beginning of Section 10.24, it follows that, for any 𝑖 ∈ 𝐼, there exists a uniquely determined *-isomorphism 𝜋𝑖 ∶ (M 1 )𝑞1,𝑖 → (M 2 )𝑞2,𝑖 , such that 𝜋𝑖 ((𝑥1 )𝑞1,𝑖 ) = (𝜋(𝑥1 ))𝑞2,𝑖 ,

𝑥1 ∈ M 1 .

In accordance with the first part of the proof, for any 𝑖 ∈ 𝐼, there exists a unitary operator 𝑢𝑖 ∶ 𝑞1,𝑖 (H 1 ) → 𝑞2,𝑖 (H 2 ), which is uniquely determined by the properties similar to (1), (2), (3), from the statement of the theorem, but applied to the hyperstandard von Neumann algebras (M 1 )𝑞1,𝑖 , (M 2 )𝑞2,𝑖 and to the *-isomorphism 𝜋𝑖 . It follows that if 𝑖 ≤ 𝑘, then 𝑢𝑖 ⊂ 𝑢𝑘 . Consequently, there exists a uniquely determined unitary operator 𝑢 ∶ H 1 → H 2 , which is an extension of all the operators 𝑢𝑖 , 𝑖 ∈ 𝐼, and it is easy to verify that it satisfies conditions (1), (2), (3) in the statement of the theorem. Q.E.D. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Let M ⊂ B (H ) be a hyperstandard von Neumann algebra, 𝐽 its conjugation and 𝔓 its selfpolar cone. By taking into account Sections 10.14 and 10.23, from the above corollary, we infer that there exists a left Hilbert algebra 𝔄 ⊂ H , such that 𝐽 = 𝐽 𝔄 and 𝔓 = 𝔓𝔄 . Consequently, in any hyperstandard von Neumann algebra, we have at our disposal the tool of the left Hilbert algebras. We remark that there exist standard von Neumann algebras, which are not hyperstandard (see, e.g., U. Haagerup, preprint of [1975b], Proposition 5.3). 10.27. We now return to the study of the faithful semifinite, normal weights on von Neumann algebras. Let M be a von Neumann algebra, Z its center and 𝜑 a faithful, semifinite, normal weight on M + . We write 𝜑

𝜑

M ∞ = {𝑥 ∈ M ; i𝑡 ↦ 𝜎𝑡 (𝑥) has an entire analytic continuation}. 𝜑

In Section 10.16, we showed that M ∞ is a so-dense *-subalgebra and 𝜑

𝜑

M ∞ ⋅ 𝔐𝜑 ⋅ M ∞ = 𝔐 𝜑 . We now write 𝜑

M 0 = {𝑥 ∈ M ;

𝜑

𝜎𝑡 (𝑥) = 𝑥,

𝑡 ∈ ℝ}.

Obviously, 𝜑

𝜑

Z ⊂ M 0 ⊂ M ∞. 𝜑

The set M 0 is a von Neumann algebra and it is called the centralizer of 𝜑. Theorem. Let M be a von Neumann algebra, 𝜑 a faithful, semifinite normal weight on M + and 𝑥 ∈ M . Then the following assertions are equivalent 𝜑

(i) 𝑥 ∈ M 0 ; (ii) 𝑥𝔐𝜑 ⊂ 𝔐𝜑 , 𝔐𝜑 𝑥 ⊂ 𝔐𝜑 and 𝜑(𝑥𝑦) = 𝜑(𝑦𝑥), 𝑦 ∈ 𝔐𝜑 . Proof. In accordance with Theorems 10.15 and 10.18, we can assume that M = 𝔏(𝔄), where 𝔄 is a left Hilbert algebra and 𝜑 = 𝜑 𝔄 . 𝜑 Let 𝑥 ∈ M 0 . By taking into account the remarks made at the beginning of the section, we have 𝑥𝔐𝜑 ⊂ 𝔐𝜑 ,

𝔐𝜑 𝑥 ⊂ 𝔐𝜑 .

1/2 Thus, if 𝑎 ∈ 𝔐+ = 𝐿𝜉 , 𝜉 ∈ 𝔄″ , then 𝜑 and 𝑎

𝑥𝜉 ∈ 𝔄″ , 𝐿𝑥𝜉 = 𝑥𝐿𝜉 and 𝑥 ∗ 𝜉 ∈ 𝔄″ , 𝐿𝑥∗ 𝜉 = 𝑥 ∗ 𝐿𝜉 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

369

The theory of standard von Neumann algebras By taking into account Proposition 9.24, we have 𝜑(𝑥𝑎) = 𝜑(𝐿𝑥𝜉 𝐿𝑆𝜉 ) = (𝑆𝜉|𝑆𝑥𝜉) = (𝐽Δ1/2 𝜉|𝐽Δ1/2 𝑥𝜉) = ((Δ1/2 𝑥Δ−1/2 )Δ1/2 𝜉|Δ1/2 𝜉) = (𝑥Δ1/2 𝜉|Δ1/2 𝜉) = (Δ1/2 𝜉|(Δ1/2 𝑥 ∗ Δ−1/2 )Δ1/2 𝜉) = (𝐽Δ1/2 𝑥∗ 𝜉|𝐽Δ1/2 𝜉) = (𝑆𝑥 ∗ 𝜉|𝑆𝜉) = 𝜑(𝐿𝜉 (𝐿𝑥∗ 𝜉 )∗ ) = 𝜑(𝐿𝜉 (𝑥 ∗ 𝐿𝜉 )∗ ) = 𝜑(𝑎𝑥).

Thus, we have shown that (i) ⇒ (ii). Conversely, let us now assume that 𝑥 satisfies condition (ii). Then, obviously, the elements 𝜎𝑡 (𝑥), 𝑡 ∈ ℝ, satisfy condition (ii), too. For any 𝜉, 𝜂 ∈ 𝔗, let us consider the function 𝑓𝜉,𝜂 , which is bounded and continuous on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, and analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1}, and given by the formula 𝑓𝜉,𝜂 (𝛼) = (𝑥Δ−𝛼+1 𝜂|Δ𝛼 𝜉). If we use the first part of condition (ii), then, for any 𝑡 ∈ ℝ, we obtain 𝜎𝑡 (𝑥)𝜂 ∈ 𝔄″ , 𝐿𝜍𝑡 (𝑥)𝜂 = 𝜎𝑡 (𝑥)𝐿𝜂 and 𝜎𝑡 (𝑥∗ )𝜉 ∈ 𝔄″ , 𝐿𝜍𝑡 (𝑥∗ )𝜉 = 𝜎𝑡 (𝑥∗ )𝐿𝜉 . With the help of the second part of condition (ii), we obtain 𝑓𝜉,𝜂 (1 + i𝑡) = (𝑥Δ−i𝑡 𝜂|Δ−i𝑡 𝑆 ∗ 𝑆𝜉) = (𝜎𝑡 (𝑥)𝜂|𝑆 ∗ 𝑆𝜉) = (𝑆𝜉|𝑆𝜎𝑡 (𝑥)𝜂) = 𝜑(𝜎𝑡 (𝑥)𝐿𝜂 𝐿𝑆𝜉 ) = 𝜑(𝐿𝜂 𝐿𝑆𝜉 𝜎𝑡 (𝑥)) = 𝜑(𝐿𝜂 (𝜎𝑡 (𝑥∗ )𝐿𝜉 )∗ ) = (𝑆𝜎𝑡 (𝑥∗ )𝜉|𝑆𝜂) = (𝑆 ∗ 𝑆𝜂|Δi𝑡 𝑥∗ Δ−i𝑡 𝜉) = (𝑥Δ−i𝑡+1 𝜂|Δ−i𝑡 𝜉) = 𝑓𝜉,𝜂 (i𝑡). Thus, 𝑓𝜉,𝜂 can be extended, by periodicity, to a bounded entire analytic function. Liouville’s theorem now implies that 𝑓𝜉,𝜂 is a constant. In particular, (𝜎𝑡 (𝑥)𝜂|Δ𝜉) = 𝑓𝜉,𝜂 (i𝑡) = 𝑓𝜉,𝜂 (1) = (𝑥𝜂|Δ𝜉),

𝑡 ∈ ℝ.

Since the vectors 𝜉, 𝜂 ∈ 𝔗 are arbitrary, we hence infer that 𝜎𝑡 (𝑥) = 𝑥, We have thus proved the implication (ii) ⇒ (i).

𝑡 ∈ ℝ. Q.E.D.

Corollary. Let M be a von Neumann algebra, 𝜑 a faithful, semifinite, normal weight on M + and 𝑢 ∈ M a unitary element. Then the following assertions are equivalent: 𝜑

(i) 𝑢 ∈ M 0 . (ii) 𝜑(𝑢∗ 𝑎𝑢) = 𝜑(𝑢𝑎𝑢∗ ) = 𝜑(𝑎), 𝑎 ∈ 𝔐+ 𝜑. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Proof. According to the theorem, we have the implication (i) ⇒ (ii). ∗ + Let us now assume that 𝑢 satisfies condition (ii). For any 𝑎 ∈ 𝔐+ 𝜑 , we have 𝑢 𝑎𝑢 ∈ 𝔐𝜑 , 1/2 1/2 ∗ hence 𝑎 𝑢 ∈ 𝔑𝜑 . Since we obviously have 𝑎 ∈ 𝔑𝜑 , it follows that 𝑎𝑢 = 𝑎1/2 (𝑎1/2 𝑢) ∈ 𝔑∗𝜑 𝔑𝜑 = 𝔐𝜑 . Consequently, we have 𝔐𝜑 𝑢 ⊂ 𝔐𝜑 . Similarly, one can show that 𝑢𝔐𝜑 ⊂ 𝔐𝜑 . Now one can easily prove that 𝑢 satisfies condition (ii) in the statement of the theorem, hence 𝜑 𝑢 ∈ M0. Q.E.D. In particular, it follows that 𝜑 is a trace iff the centralizer of 𝜑 coincides with M , i.e. iff 𝜑 the group {𝜎𝑡 } acts identically on M . 10.28. We now prove a remarkable Radon-Nikodym type property, due to A. Connes, which establishes a link between the groups of modular automorphisms, associated to any pair of faithful, semifinite, normal weights on a von Neumann algebra. Theorem. Let M be a von Neumann algebra and 𝜑, 𝜓 two faithful, semifinite, normal weights on M + . Then, there exists a so-continuous mapping ℝ ∋ 𝑡 ↦ 𝑢𝑡 ∈ M , such that (1) 𝑢𝑡 is unitary, 𝑡 ∈ ℝ. 𝜑

(2) 𝑢𝑡+𝑠 = 𝑢𝑡 𝜎𝑡 (𝑢𝑠 ), 𝑠, 𝑡 ∈ ℝ. 𝜓

𝜑

(3) 𝜎𝑡 (𝑥) = 𝑢𝑡 𝜎𝑡 (𝑥)𝑢𝑡∗ , 𝑥 ∈ M , 𝑡 ∈ ℝ. Proof. We denote N = Mat2 (M ) (see 2.32 and 3.16). For any 𝑎 = (𝑎i𝑗 ) ∈ N , 𝑎 ≥ 0, we define 𝜃(𝑎) = 𝜑(𝑎11 ) + 𝜓(𝑎22 ) ∈ ℝ+ ∪ {+∞}. It is easy to see that 𝜃 is a faithful, normal weight on N + . If 𝑥11 , 𝑥21 ∈ 𝔑𝜑 and 𝑥12 , 𝑥22 ∈ 𝔑𝜓 , then (𝑥i𝑗 ) ∈ 𝔑𝜃 . We hence infer that 𝔑𝜃 is so-dense in N . Consequently, 𝜃 is also semifinite. We denote by 𝑒𝑖𝑗 , 𝑖 = 1, 2; 𝑗 = 1, 2, the ‘matrix units’ of N : 𝑒11 = ( 𝑒21 = (

1

0

0

0

0

0

1

0

) , 𝑒12 = ( ) , 𝑒22 = (

0

1

0

0

0

0

0

1

),

).

Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

371

The theory of standard von Neumann algebras If we denote 𝑢 = 𝑒11 − 𝑒22 . then 𝑢 is a self-adjoint unitary operator and 𝑢(

𝑎11

𝑎12

𝑎21

𝑎22

)𝑢 = (

𝑎11

−𝑎12

−𝑎21

𝑎22

(𝑎i𝑗 ) ∈ N +

),

It follows that, for any 𝑎 ∈ N + , we have 𝜃(𝑢𝑎𝑢) = 𝜃(𝑎). If we now apply Corollary 10.27, we infer that 𝑢 ∈ N 𝜃0 . Since we obviously have 1 ∈ N 𝜃0 , it follows that 1 (1 + 𝑢) ∈ N 𝜃0 , 2 1 = (1 − 𝑢) ∈ N 𝜃0 . 2

𝑒11 = 𝑒22

Since 𝑒11 ∈ N 𝜃0 , for any 𝑥 ∈ M and any 𝑡 ∈ ℝ, we have 𝑒11 𝜎𝑡𝜃 ((

hence 𝜎𝑡𝜃 ((

𝑥

0

0

0

𝑥

0

0

0

)) 𝑒11 = 𝜎𝑡𝜃 (𝑒11 (

𝑥

0

0

0

) 𝑒11 ) = 𝜎𝑡𝜃 ((

𝑥

0

0

0

))

)) is of the form

𝜎𝑡𝜃 ((

𝑥

0

0

0

)) = (

𝜋𝑡 (𝑥)

0

0

0

)

It is easy to verify that the above relation determines a group {𝜋𝑡 } of *-automorphisms of M , which leaves 𝜑 invariant. Since, for any 𝑥, 𝑦 ∈ 𝔐𝜑 and any 𝑡 ∈ ℝ, we have (

𝑥

0

0

0

𝜃 ((

), (

𝑥

0

0

0

𝜃 (𝜎𝑡𝜃 ((

𝑦

0

0

0

) ∈ 𝔐𝜃

) 𝜎𝑡𝜃 ((

𝑦

0

0

0

𝑦

0

0

0

)) (

))) = 𝜑(𝑥𝜋𝑡 (𝑦)),

𝑥

0

0

0

)) = 𝜑(𝜋𝑡 (𝑦)𝑥),

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372

Lectures on von Neumann algebras

from Theorem 10.17, we infer that 𝜑 satisfies the KMS-condition, with respect to {𝜋𝑡 }, for any pair of elements in 𝔐𝜑 . From the uniqueness part of Theorem 10.17, we infer that 𝜑

𝜋𝑡 = 𝜎𝑡 ,

𝑡 ∈ ℝ.

Consequently, we have 𝜎𝑡𝜃

((

𝑥

0

0

0

𝜑

)) = (

𝜎𝑡 (𝑥)

0

0

0

),

𝑥 ∈ M.

), 𝜓 𝜎𝑡 (𝑦)

𝑦 ∈ M.

One can similarly prove that 𝜎𝑡𝜃 ((

0

0

0 𝑦

)) = (

0

0

0

By taking into account the fact that 𝑒11 , 𝑒22 ∈ N 𝜃0 , it is easy to see that for any 𝑡 ∈ ℝ, we have 𝑒11 𝜎𝑡𝜃 (𝑒21 ) = 𝜎𝑡𝜃 (𝑒21 )𝑒22 = 0; hence, 𝜎𝑡𝜃 (𝑒21 ) is of the form 𝜎𝑡𝜃 (𝑒21 ) = (

0

0

𝑢𝑡

0

𝑢𝑡 ∈ M .

),

Since the group {𝜎𝑡𝜃 } is so-continuous, the mapping ℝ ∋ 𝑡 ↦ 𝑢𝑡 ∈ M is so-continuous. Then, since 𝜎𝑡𝜃 (𝑒21 )∗ 𝜎𝑡𝜃 (𝑒21 ) = 𝜎𝑡𝜃 (𝑒12 𝑒21 ) = 𝜎𝑡𝜃 (𝑒11 ) = 𝑒11 , 𝜎𝑡𝜃 (𝑒21 )𝜎𝑡𝜃 (𝑒21 )∗ = 𝜎𝑡𝜃 (𝑒21 𝑒12 ) = 𝜎𝑡𝜃 (𝑒22 ) = 𝑒22 , it follows that the operators 𝑢𝑡 are unitary. For any 𝑥 ∈ M and any 𝑡 ∈ ℝ, we have (

0 0

0

) 𝜓 𝜎𝑡 (𝑥)

= 𝜎𝑡𝜃 (( =(

hence

0

0

0 𝑥

0

0

𝑢𝑡

0

)(

)) = 𝜎𝑡𝜃 (𝑒21 ( 𝜑 𝜎𝑡 (𝑥)

0

0

0 𝜓

)(

𝑥

0

0

0

0 𝑢𝑡∗ 0

0

) 𝑒12 ) = 𝜎𝑡𝜃 (𝑒21 )𝜎𝑡𝜃 ((

)=(

0 0

𝑥

0

0

0

)) 𝜎𝑡𝜃 (𝑒12 )

0

); 𝜑 𝑢𝑡 𝜎𝑡 (𝑥)𝑢𝑡∗

𝜑

𝜎𝑡 (𝑥) = 𝑢𝑡 𝜎𝑡 (𝑥)𝑢𝑡∗ . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

373

The theory of standard von Neumann algebras Finally, for any 𝑡, 𝑠 ∈ ℝ, we have (

0

0

𝑢𝑡+𝑠

0

𝜃 (𝑒21 ) = 𝜎𝑡𝜃 (( ) = 𝜎𝑡+𝑠

=(

0 𝑢𝑡

0 0

0

0

𝑢𝑠

0

)) = 𝜎𝑡𝜃 (𝑒21 (

𝜑

)(

𝜎𝑡 (𝑢𝑠 ) 0 0

0

hence

)=(

𝑢𝑠

0

0

0

0

0

𝜑 𝑢𝑡 𝜎𝑡 (𝑢𝑠 )

0

))

);

𝜑

𝑢𝑡+𝑠 = 𝑢𝑡 𝜎𝑡 (𝑢𝑠 ). Q.E.D. The preceding theorem enables us to define the notion of ‘commutation’, relatively to weights. Corollary. Let M be a von Neumann algebra and 𝜑, 𝜓 faithful, semifinite, normal weights on M + . Then the following assertions are equivalent: 𝜓

(i) {𝜎𝑡 }𝑡∈ ℝ leaves invariant the weight 𝜑. 𝜑

(ii) {𝜎𝑡 }𝑡∈ ℝ leaves invariant the weight 𝜓. 𝜓

𝜑

(iii) there exists a so-continuous group {𝑢𝑡 }𝑡∈ ℝ of unitary operators in M 0 ∩ M 0 , such that 𝜓 𝜑 𝜎𝑡 (𝑥) = 𝑢𝑡 𝜎𝑡 (𝑥)𝑢𝑡∗ , 𝑥 ∈ M , 𝑡 ∈ ℝ. Proof. The implication (iii) ⇒ (i) obviously follows from Corollary 10.27. 𝜓 Let us now assume that {𝜎𝑡 }𝑡∈ ℝ leaves invariant the weight 𝜑 and let us consider a so-continuous mapping 𝑡 ↦ 𝑢𝑡 , as in the statement of Theorem 10.28. For any 𝑡 ∈ ℝ, we have 𝜑 𝜓 𝜑 𝜑(𝑎) = 𝜑(𝜎−𝑡 (𝑎)) = 𝜑(𝜎𝑡 (𝜎−𝑡 (𝑎))) = 𝜑(𝑢𝑡 𝑎𝑢𝑡∗ ), 𝑎 ∈ M + ; hence, in accordance with Corollary 10.27, 𝜑

𝑢𝑡 ∈ M 0 . Then, for any 𝑡, 𝑠 ∈ ℝ, we have 𝜑

𝑢𝑡+𝑠 = 𝑢𝑡 𝜎𝑡 (𝑢𝑠 ) = 𝑢𝑡 𝑢𝑠 ; hence, {𝑢𝑡 } is a one-parameter group. Finally, for any 𝑡 ∈ ℝ, we have 𝜓

𝜑

𝜎𝑡 (𝑢𝑡 ) = 𝑢𝑠 𝜎𝑠 (𝑢𝑡 )𝑢𝑠∗ = 𝑢𝑠 𝑢𝑡 𝑢𝑠∗ = 𝑢𝑡 , hence

𝑠 ∈ ℝ;

𝜓

𝑢𝑡 ∈ M 0 . Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

374

Lectures on von Neumann algebras

Consequently, (i) ⇔ (iii). Similarly, one proves that (ii) ⇔ (iii).

Q.E.D.

If the faithful, semifinite, normal weights 𝜑 and 𝜓 satisfy the equivalent conditions from the statement of the preceding corollary, we shall say that 𝜑 and 𝜓 commute. 10.29. At the end of this chapter, we present a criterion of semifiniteness for the von Neumann algebras, expressed in terms of the group of modular automorphisms. Theorem. For any von Neumann algebra M ⊂ B (H ), the following assertions are equivalent: (i) M is semifinite. (ii) There exists a faithful, semifinite, normal weight 𝜑 on M + , and a so-continuous group {𝑢𝑡 }𝑡∈ℝ of unitary operators in M , such that 𝜑

𝜎𝑡 (𝑥) = 𝑢𝑡 𝑥𝑢𝑡∗ ,

𝑥 ∈ M,

𝑡 ∈ ℝ.

(iii) For any faithful, semifinite, normal weight 𝜑 on M there exists a so-continuous group {𝑢𝑡 }𝑡∈ℝ of unitary operators in M , such that 𝜑

𝜎𝑡 (𝑥) = 𝑢𝑡 𝑥𝑢𝑡∗ ,

𝑥 ∈ M,

𝑡 ∈ ℝ.

Proof. If M is semifinite, then, in accordance with Corollary 7.15, there exists a faithful, semifinite, normal trace 𝜇 on M + . With the help of Corollary 10.27, we have 𝜇

𝑥 ∈ M,

𝜎𝑡 (𝑥) = 𝑥,

𝑡 ∈ ℝ. 𝜇

If 𝜑 is any faithful, semifinite, normal weight on M + , then {𝜎𝑡 } leaves the weight 𝜑 invariant. Thus, if we now apply Corollary 10.28, it follows that there exists a so-continuous group {𝑢𝑡 }𝑡∈ ℝ of unitary operators in M , such that 𝜑

𝜇

𝜎𝑡 (𝑥) = 𝑢𝑡 𝜎𝑡 (𝑥)𝑢𝑡∗ = 𝑢𝑡 𝑥𝑢𝑡∗ ,

𝑥 ∈ M,

𝑡 ∈ ℝ.

Since on M + there exists a faithul, semifinite, normal weight (see Section 10.14), the implication (iii) ⇒ (ii) is trivial. Finally, let us assume that assertion (ii) is true. According to the Stone representation theorem (9.20), there exists a positive self-adjoint operator 𝐴 in H , such that s(𝐴) = 1, which is affiliated to M , and such that 𝑢𝑡 = 𝐴i𝑡 ,

𝑡 ∈ ℝ.

For any natural 𝑛, we denote 𝑒𝑛 = 𝜒( 1 ,𝑛) (𝐴) ∈ M . 𝑛

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375

The theory of standard von Neumann algebras Then

𝜑

𝑒𝑛 ∈ M 0 ,

𝑛 ∈ ℕ,

and 𝑒𝑛 ↑ s(𝐴) = 1. In order to prove the semifiniteness of M , it is sufficient to show that the reduced von Neumann algebras M 𝑒𝑛 are semifinite. Let us choose a natural number 𝑛. According to Theorem 10.27, we have 𝑒𝑛 𝔐𝜑 𝑒𝑛 ⊂ 𝔐𝜑 ; hence, the weight 𝜑𝑛 , defined on (M 𝑒𝑛 )+ by the formula 𝜑𝑛 (𝑒𝑛 𝑎|𝑒𝑛 H ) = 𝜑(𝑒𝑛 𝑎𝑒𝑛 ),

𝑎 ∈ M +,

is semifinite. It is easy to see that 𝜑𝑛 is normal and faithful. If we denote 𝑎𝑛 = 𝐴𝑒𝑛 ∈ M 𝑒𝑛 , we have 𝜑 𝜎𝑡 𝑛 (𝑥) = 𝑎𝑛i𝑡 𝑥𝑎𝑛−i𝑡 , 𝑥 ∈ (M 𝑒𝑛 )+ , 𝑡 ∈ ℝ. Thus, our problem reduced to the following one: to show that, if 𝔄 ⊂ H is a left Hilbert algebra and if there exists an invertible 𝑎 ∈ 𝔏(𝔄)+ , such that 𝜎𝑡 (𝑥) = 𝑎i𝑡 𝑥𝑎−i𝑡 ,

𝑥 ∈ 𝔏(𝔄),

𝑡 ∈ ℝ,

then 𝔏(𝔄) is semifinite. Indeed, let us define on 𝔏(𝔄)+ the weight 𝜇(𝑏) = 𝜑 𝔄 (𝑎−1/2 𝑏𝑎1/2 ),

𝑏 ∈ 𝔏(𝔄)+ . 𝜑

It is easy to see that 𝜇 is normal and faithful. Since 𝑎 ∈ 𝔏(𝔄)0 𝔄 , it follows that 𝜇 is semifinite and M 𝜇 = M 𝜑𝔄 . For any 𝜉, 𝜁 ∈ 𝔄″ and any 𝑡 ∈ ℝ, with the help of Proposition 9.24, we infer that (𝑎

1 2

−( +i𝑡) 1

−

= (Δ 2 [Δ 1

= (Δ 2 𝑎 = (𝑎

1 2

1

Δ2

𝑎

+i𝑡

𝜉|𝐽𝑎

1 2

(− +i𝑡)

1

−( +i𝑡) i𝑡 2

Δi𝑡 𝜉|𝐽𝑎

1

Δ 2 𝜁)

1

1

−

Δ 2 ]Δi𝑡 𝜉|𝐽Δ 2 [Δ −

Δ 𝜉|Δ

1 −( +i𝑡) 2

1 2

(− +i𝑡)

1 2

𝐽𝑎

1 2

(− +i𝑡)

1 (− +i𝑡) 2

1 2

𝑎

1 2

(− +i𝑡)

1

Δ 2 ]𝜁)

𝜁)

𝜁).

Thus, by the formula 𝑓𝜉,𝜁 (𝛼) =

⎧ (𝑎

−𝛼 𝛼

1

1

Δ 𝜉|𝐽𝑎𝛼−1 Δ 2 𝜁), if 0 ≤ Re 𝛼 ≤ ,

⎨ −𝛼 𝛼− 12 𝜉|𝐽𝑎𝛼−1 𝜁), if ⎩ (𝑎 Δ

2

1 2

≤ Re 𝛼 ≤ 1,

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376

Lectures on von Neumann algebras

we define a bounded and continuous function 𝑓𝜉,𝜁 on {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, which is analytic in {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1}. For any 𝑡 ∈ ℝ, we have 1

−

𝑓𝜉,𝜁 (i𝑡) = (𝑎−i𝑡 Δi𝑡 𝜉|𝐽Δ 2 [Δ

1 2

1

𝑎i𝑡−1 Δ 2 ]𝜁)

= (𝑎−i𝑡 Δi𝑡 𝜉|𝑆𝑎i𝑡−1 𝜁) = 𝜑 𝔄 (𝑎−1 𝑎i𝑡 𝐿𝜁 𝑎−i𝑡 𝐿∆i𝑡 𝜉 ) = 𝜑 𝔄 (𝑎−1 𝜎𝑡 (𝐿𝜁 )𝜎𝑡 (𝐿𝜉 )) = 𝜇(𝐿𝜁 𝐿𝜉 ), 1

𝑓𝜉,𝜁 (1 + i𝑡) = (𝑎−1−i𝑡 Δ 2 1

−

= (Δ 2 [Δ

1 2

+i𝑡

𝜉|𝐽𝑎i𝑡 𝜁) 1

𝑎−1−i𝑡 Δ 2 ]Δi𝑡 𝜉|𝐽𝑎i𝑡 𝜁)

= (𝑎i𝑡 𝜁|𝑆𝑎−1−i𝑡 Δi𝑡 𝜉) = 𝜑 𝔄 (𝑎−1 𝑎−i𝑡 𝐿∆i𝑡 𝜉 𝑎i𝑡 𝐿𝜁 ) = 𝜑 𝔄 (𝑎−1 𝜎−𝑡 (𝜎𝑡 (𝐿𝜉 ))𝐿𝜁 ) = 𝜇(𝐿𝜉 𝐿𝜁 ). Consequently, 𝑓𝜉,𝜁 is constant on the imaginary axis; hence, everywhere. We hence infer that 𝜇(𝐿𝜁 𝐿𝜉 ) = 𝜇(𝐿𝜉 𝐿𝜁 ), 𝜉, 𝜁 ∈ 𝐴″ . In particular, we have 𝜇(𝑥𝑦) = 𝜇(𝑦𝑥),

𝑥, 𝑦 ∈ 𝔐𝜇 = 𝔐𝜑𝔄 ;

consequently, 𝜇 satisfies the KMS-condition with respect to the identity group for any pair of elements in 𝔐𝜇 . With the help of Theorem 10.17, we infer that 𝜇

𝜎𝑡 (𝑥) = 𝑥,

𝑥 ∈ 𝔏(𝔄),

𝑡 ∈ ℝ;

hence, according to Corollary 10.27, 𝜇 is a trace. Thus, with the help of corollary 7.15, we infer that 𝔏(𝔄) is semifinite. Q.E.D. EXERCISES E.10.1 Let 𝔄 be a complex algebra endowed with an involution ♯ and a scalar product (⋅|⋅). We suppose that 𝔄 satisfies conditions (i)–(iii) from 10.1, and that H ⊃ 𝔄2 ∋ 𝜉1 𝜉2 ↦ (𝜉1 𝜉2 )♯ ∈ H is a preclosed antilinear operator. Show that the operator H ⊃ 𝔄 ∋ 𝜉 ↦ 𝜉♯ ∈ H is preclosed, hence 𝔄 is a left Hilbert algebra. ▶

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The theory of standard von Neumann algebras

377

E.10.2 Let M ⊂ B (H ) be a von Neumann algebra with a separating cyclic vector to 𝜉0 ∈ H . Give direct proofs to the following assertions: (1) The adjoint 𝑆 ∗ of the closure 𝑆 of the antilinear operator H ⊃ M 𝜉0 ∋ 𝑥𝜉0 ↦ 𝑥 ∗ 𝜉0 ∈ H is the closure of the antilinear operator H ⊃ M ′ 𝜉0 ∋ 𝑥 ′ 𝜉0 ↦ 𝑥 ′∗ 𝜉0 ∈ H (2) If 𝜂 ∈ D 𝑆 ∗ , then the operator H ⊃ M 𝜉0 ∋ 𝑥𝜉0 ↦ 𝑥𝜂 ∈ H is preclosed and its closure is affiliated to M ′ . E.10.3 Let 𝐺 be a locally compact topological group, 𝑑𝑔 a left invariant Haar measure on 𝐺 and 𝜗 ∶ 𝐺 ↦ ℝ+ , the modular function. Show that the set 𝔄𝐺 of all continuous complex functions, which are defined on 𝐺 and whose supports are compact, is a left Hilbert algebra with respect to the operations: (𝜉𝜂)(𝑔) = ∫ 𝜉(ℎ)𝜂(ℎ−1 𝑔)𝑑ℎ, 𝜉 ♯ (𝑔) = 𝜗(𝑔)−1 𝜉(𝑔−1 ) and the scalar product (𝜉|𝜂) = ∫ 𝜉(𝑔)𝜂(𝑔) 𝑑𝑔 Determine, in this case, the objects H , Δ, 𝐽, 𝔖. E.10.4 Let H be a Hilbert space. Show that the set 𝔄 = F (H ) of all operators in B (H ), whose ranges are finitely dimensional, is a left Hilbert algebra with respect to the *-algebra operations induced by those of B (H ) and with the scalar product (𝑥|𝑦) = tr(𝑦 ∗ 𝑥) Determine, in this case, the objects H , Δ, 𝐽, 𝔄″ . ▶

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378

Lectures on von Neumann algebras

E.10.5 A left Hilbert algebra 𝔄 ⊂ H is said to be unimodular if Δ = 1. Show that for a left Hilbert algebra 𝔄 ⊂ H , the following assertions are equivalent: (i) 𝔄 is unimodular. (ii) 𝑆 is isometric. (iii) 𝜑 𝔄 is a trace. Show that if 𝐺 is a locally compact topological group, then the left Hilbert algebra 𝔄𝐺 (E.10.3) is unimodular iff the group 𝐺 is unimodular. E.10.6 Prove that any von Neumann algebra is *-isomorphic to a standard von Neumann algebra along the lines of the proof for Theorem 10.7. E.10.7 Let M be a standard von Neumann algebra and 𝐽 its conjugation. Show that R (M , M ′ ) is the w-closed linear hull of the set {𝑥(𝐽𝑥𝐽);

𝑥 ∈ M}

(Hint: use a polarization relation). In the following three exercises, M ⊂ B (H ) is a von Neumann algebra with the separating cyclic vector 𝜉0 ∈ H , whereas 𝐽𝜉0 is the canonical conjugation associated to 𝔄 = M 𝜉0 . E.10.8 For 𝜂 ∈ H , the following assertions are equivalent: (i) 𝜂 ∈ 𝔄′ . (ii) 𝑅0𝜂 is bounded. (iii) The form 𝜔𝜂 is dominated (E.9.33) by the form 𝜔𝜉0 . E.10.9 For 𝑥 ∈ M , the following assertions are equivalent: (i) 𝜔𝑥𝜉0 ≤ 𝜔𝜉0 . (ii) ‖Δ1/2 𝑥Δ−1/2 ‖ ≤ 1. E.10.10 Show that if 𝐽 is a conjugation in H , with the properties (1) the mapping 𝑥 ↦ 𝐽𝑥 ∗ 𝐽 is a *-anti-isomorphism of M onto M ′ , which acts identically on the center. (2) 𝐽𝜉0 = 𝜉0 . (3) (𝜉0 |[𝑥(𝐽𝑥𝐽)]𝜉0 ) ≥ 0, for any 𝑥 ∈ M . then 𝐽 = 𝐽𝜉0 . (Hint: prove that 𝐽𝑆 = 𝐽𝐽𝜉0 Δ1/2 is a positive self-adjoint operator.) ▶

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In the following two exercises, M ⊂ B (H ) is a von Neumann algebra, Z is its center; 𝜉0 , 𝜁0 ∈ H are separating cyclic vectors, 𝐽𝜉0 , 𝐽𝜁0 are the corresponding canonical conjugations and 𝔓𝜉0 , 𝔓𝜁0 are the selfpolar convex cones associated, respectively, to the left Hilbert algebras M 𝜉0 , M 𝜁0 . E.10.11 Show that the following assertions are equivalent: (i) 𝜁0 ∈ 𝔓𝜉0 . (ii) 𝜉0 ∈ 𝔓𝜁0 . (iii) 𝐽𝜉0 = 𝐽𝜁0 , and (𝑧𝜉0 |𝑧𝜁0 ) ≥ 0, for any 𝑧 ∈ Z , 𝑧 ≥ 0. (Hint for the proof of the implication (iii) ⇒ (i): in accordance with Section 10.23, 𝜁0 can be written 𝜁0 = 𝜁0+ − 𝜁0− , 𝑝𝜁0+ ⟂ 𝑝𝜁0− , with respect to 𝔓𝜉0 ; show that (𝜁0+ |𝑥(𝐽𝜉0 𝑥𝐽𝜉0 )𝜁0− ) = 0, 𝑥 ∈ M ; from exercise E.10.7, we infer that 𝜁0+ ⟂ [M M ′ 𝜁0− ], whence 0 ≤ (𝜉0 |[M M ′ 𝜁0− ]𝜁0 ) = −(𝜉0 |𝜁0− ) ≤ 0; hence, (𝜉0 |𝜁0− ) = 0, 𝜁0− = 0.) E.10.12 Show that there exists a unitary 𝑢′ ∈ M ′ , such that 𝐽𝜁0 = 𝑢′ ∘ 𝐽𝜉0 ∘ 𝑢′∗ . Infer that the *-automorphism M ∋ 𝑥 ↦ 𝐽𝜉0 𝐽𝜁0 𝑥𝐽𝜁0 𝐽𝜉0 ∈ M is inner. In the following six exercises, 𝔄 ⊂ H is a left Hilbert algebra, the other notations corresponding to those introduced in the main text. E.10.13 We define the set ⎧ ⎪

for any 𝑛 ∈ 𝑍, we have +∞

Δ𝑛 𝜉 ∈ 𝔄″ ∩ 𝔄′

⎫ ⎪

𝔗1 = 𝜉 ∈ D ∆𝑛 ⋂ ⎨ D (∆𝑛 𝐿𝜉 ∆−𝑛 ) = D (∆−𝑛 ) and Δ𝑛 𝐿𝜉 Δ−𝑛 ⊂ 𝐿∆𝑛 𝜉 ⎬ 𝑛=−∞ ⎪ ⎪ D (∆𝑛 𝑅𝜉 ∆−𝑛 ) = D (∆−𝑛 ) and Δ𝑛 𝑅𝜉 Δ−𝑛 ⊂ 𝑅∆𝑛 𝜉 ⎭ ⎩ With the help of Lemmas 1–3 from 10.19, show that 𝔗1 is a left Hilbert subalgebra of 𝔄″ and 𝔗″1 = 𝔄″ . ▶

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We now consider the operator 𝑇 on B (H ), defined as in E.9.36, with 𝐴 = 𝐵 = Δ. Show that for any 𝜉 ∈ 𝔗1 , and any 𝜆 > 0, we have (𝜆 + Δ)−1 𝜉 ∈ 𝔗1 and 𝐿(𝜆+∆)−1 𝜉 = (𝜆 + 𝑇)−1 (𝐿𝜉 ). With the help of exercises E.9.36 and E.9.37, infer from this result that 𝔗 = 𝔗1 . Show that the preceding assertions imply Theorem 10.12. E.10.14 For any 𝜉 ∈ H , the following assertions are equivalent: (i) 𝜉 ∈ 𝔖. (ii) 𝜉 ∈ 𝔄″ ∩ S ∆ ∩ S ∆−1 . (iii) 𝜉 ∈ 𝔄″ and the mapping i𝑡 ↦ Δi𝑡 𝐿𝜉 Δ−i𝑡 has an entire analytic continuation 𝐹, such that lim ‖𝐹(𝑛)‖1/𝑛 < +∞,

𝑛→∞

lim ‖𝐹(−𝑛)‖1/𝑛 < +∞,

𝑛→∞

E.10.15 Show that for any 𝜉 ∈ 𝔄″ , there exists a sequence {𝜉𝑛 } ⊂ 𝔖, such that (i) 𝜉𝑛 → 𝜉, 𝑆𝜉𝑛 → 𝑆𝜉. (ii) 𝐿𝜉𝑛 → 𝐿𝜉 , (𝐿𝜉𝑛 )∗ → (𝐿𝜉)∗ . (iii) sup ‖𝐿𝜉𝑛 ‖ < +∞. E.10.16 Show that if 𝜉 ∈ 𝔄″ ∩ D (∆−𝛼 ) and 𝜂 ∈ 𝔄′ ∩ D (∆𝛼 ) , then 𝑅𝜂 Δ−𝛼 𝜉 ∈ D (∆𝛼 ) , and Δ𝛼 𝑅𝜂 Δ−𝛼 𝜉 = 𝐿𝜉 Δ𝛼 𝜂. E.10.17 Show that the set {𝜉 ∈ 𝔄′ ∩ D ∆ ;

Δ𝜉 ∈ 𝔄′ }

is a left Hilbert subalgebra of 𝔄″ and that, for any 𝜉1 , 𝜉2 , belonging to this set, the following relation holds: Δ(𝜉1 𝜉2 ) = Δ(𝜉1 )Δ(𝜉2 ). E.10.18 For any 𝜆 ∈ [0, 1/2], one defines the set 𝔓𝜆 = Δ𝜆 𝔓𝑆 . ▶

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Show that (i) Δi𝑡 𝔓𝜆 = 𝔓𝜆 . (ii) 𝐽𝔓𝜆 = 𝔓( 1 −𝜆) . 2

(iii) 𝔓𝜆 is a convex cone, polar to 𝔓( 1 −𝜆) . 2

1 ( −2𝜆) 2

(iv) 𝔓𝜆 = {𝜉(𝐽Δ

𝜉); 𝜉 ∈ 𝔗}.

In the following four exercises, M ⊂ B (H ) is a hyperstandard von Neumann algebra, whose conjugation is 𝐽 and whose selfpolar convex cone is 𝔓. E.10.19 Let Aut (M ) be the group of the *-automorphisms of M and 𝑈(M ) the group of all unitary elements in M . Show that there exists a group homomorphism Aut (M ) ∋ 𝜋 ↦ 𝑢𝜋 ∈ 𝑈(M ), which is uniquely determined by the following conditions: (1) 𝜋(𝑥) = 𝑢𝜋 𝑥𝑢𝜋∗ , 𝜋 ∈ Aut (M ), 𝑥 ∈ M . (2) 𝑢𝜋 (𝔓) = 𝔓, 𝜋 ∈ Aut (M ). E.10.20 Show that the von Neumann algebra M ′ ⊂ B (H ) is also hyperstandard, with the same conjugation 𝐽 and the same self polar convex cone 𝔓. With the help of Theorem 10.25, infer from this result that for any vector 𝜁 ∈ H , there exists a vector 𝜉 ∈ 𝔓 and a partial isometry 𝑣 ∈ M , which are uniquely determined by the properties 𝜁 = 𝑣𝜉,

𝑣 ∗ 𝑣 = 𝑝𝜉 .

The vector 𝜉 is denoted by |𝜁| and it is called the modulus of 𝜁, whereas the equalities 𝜁 = 𝑣|𝜁|, 𝑣 ∗ 𝑣 = 𝑝|𝜁| yield the polar decomposition of 𝜁. E.10.21 Let 𝜁 = 𝑣|𝜁| be the polar decomposition of a vector 𝜁 ∈ H . Show that the polar decomposition of 𝐽𝜁 is 𝐽𝜁 = 𝑣 ∗ ([𝑣(𝐽𝑣𝐽)]|𝜁|). With the help of Corollary 1 from Section 10.24, infer from this result that if 𝐽𝜁 = 𝜁, then |𝜁| = 𝜁 + + 𝜁 − , 𝑣 = 𝑝𝜁 + − 𝑝𝜁 − , where 𝜁 = 𝜁 + − 𝜁 − is the decomposition (**) from Section 10.23. ▶

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E.10.22 Let 𝜑 and 𝜓 be normal forms on M . Show that 𝜑 ≤ 𝜓 ⇒ 𝜑1/2 ≤ 𝜓 1/2 . E.10.23 Let 𝜑 be a faithful, semifinite weight on the von Neumann algebra M . Show that if there exists an increasingly directed family {𝜑𝑣 } of normal forms on M , such that 𝜑(𝑎) = sup 𝜑𝑣 (𝑎), 𝑎 ∈ M + , 𝑣

then 𝜑 is normal. E.10.24 Let M be a von Neumann algebra, 𝜑 a faithful, semifinite normal weight on M + and 𝜋 a *-automorphism of M . Show that 𝜑∘𝜋

𝜎𝑡

𝜑

= 𝜋 −1 ∘ 𝜎𝑡 ∘ 𝜋,

𝑡 ∈ ℝ. 𝜑

In particular, if 𝜋 leaves invariant the weight 𝜑, then 𝜋 commutes with 𝜎𝑡 for any 𝑡 ∈ ℝ. The case of a trace shows that the converse is not true. E.10.25 Let M be a von Neumann algebra and 𝜑 a faithful, semifinite, normal weight 𝜑 on M + , such that the restriction of 𝜑 to (M 0 )+ be semifinite. Then the von 𝜑 Neumann algebra M 0 is semifinite. E.10.26 Let M be a von Neumann algebra, Aut(M ) the group of all the *-automorphisms of M and Int(M ) the group of all the inner *-automorphisms of M . Show that Int(M ) is an invariant subgroup of Aut(M ). One denotes by Out(M ) the quotient group Aut(M )/Int(M ) and by 𝔠 the canonical homomorphism 𝔠 ∶ Aut(M ) → Out(M ). Show that if 𝜑 and 𝜓 are faithful, semifinite, normal weights on M + , then 𝜑 𝜓 𝔠(𝜎𝑡 ) = 𝔠(𝜎𝑡 ), 𝑡 ∈ ℝ. Show that the mapping

𝜑

ℝ ∋ 𝑡 ↦ 𝔠(𝜎𝑡 ) is a homomorphism of the additive group ℝ into the center of the group Out (M ), which does not depend on the faithful, semifinite, normal weight 𝜑 on M + . 𝜑 The kernel of the mapping 𝑡 ↦ 𝔠(𝜎𝑡 ) is denoted by 𝑇(M ). Show that if M is semifinite, then 𝑇(M ) = ℝ.

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COMMENTS C.10.1 The theory of the left Hilbert algebras was devised by M. Tomita (1967a, c) and it became known through M. Takesaki’s lessons (1970). In M. Takesaki’s book (1970), the left Hilbert algebras appear as generalized Hilbert algebras, whereas Tomita’s algebra is introduced axiomatically as modular Hilbert algebra (see C.10.7). Although this terminology is still in use, the terminology we have introduced in our text is becoming more common in the literature. In contrast, M. Takesaki’s notations from 1970, which differ from those used in our text, are currently used in the literature, and therefore, we indicate their correspondence with those introduced by us: Our notation

M. Takesaki’s notations

𝔄, 𝔄′ , 𝔄″

𝔄, 𝔄′ , 𝔄″

𝔏(𝔄), 𝔎(𝔄′ )

𝔏(𝔄), 𝔎(𝔄′ )

D𝑆

D♯

𝑆𝜉

𝜉♯

D 𝑆∗

D♭

𝑆 ∗𝜂

𝜂♭

𝐿𝜉

𝜋(𝜉)

𝐿𝜉 (𝜁)

𝜉𝜁

𝑅𝜂

𝜋 ′ (𝜂)

𝑅𝜂 (𝜁)

𝜁𝜂

𝑆∗

𝐹

𝔓𝑆

𝔓 (or P ♯ )

𝔓𝑆∗

𝔓♭ (or P ♭ )

♯

We give two examples of formulas that correspond to one another under these different notations 𝐿𝜉 (𝜂) = 𝑆𝐿𝑆𝜂 𝑆𝜉

𝜉𝜂 = (𝜂 ♯ 𝜉 ♯ )♯

𝔓𝑆∗ = {𝑅𝜂 𝑆 ∗ 𝜂; 𝜂 ∈ 𝔄′ }

𝔓♭ = {𝜂 ♭ 𝜂; 𝜂 ∈ 𝔄′ }

The left Hilbert algebras 𝔄 ⊂ H , such that 𝔄 = 𝔄″ , are called maximal (or ‘achieved’, ‘full’; ‘achevée’, in French).

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Lectures on von Neumann algebras For a left Hilbert algebras 𝔄 ⊂ H , F. Perdrizet (1971) also introduced the sets F ♯ = {𝜉 ∈ H ;

𝐿0𝜉 is preclosed},

B ♯ = {𝜉 ∈ H ;

𝐿0𝜉 is bounded}

and showed by examples that the inclusions, indicated in the following diagram by arrows  𝔄

B♯

@

@

″

@ @

@ R F♯ ⟶ H @ 

@ @ RD♯

are, in general, strict inclusions. If, nevertheless, 𝔄 is of the form 𝔄 = M 𝜉0 (see Section 10.6), then we obviously have 𝔄 = B ♯ . F. Perdrizet (1971) also introduced the sets ♯

𝔓𝑎 = {𝜉 ∈ 𝔓♯ ; 𝐿𝜉 is self-adjoint}, 𝔄+ = {𝜉 ∈ 𝔄″ ; +

𝐿𝜉 ≥ 0},

♯

ℑ = {𝜉𝜉 ; 𝜉 ∈ 𝔄″ }, and has shown, by examples, that the inclusions ♯

ℑ+ → 𝔄 → 𝔓 𝑎 → 𝔓 ♯ are, in general, strict. If 𝔄 = M 𝜉0 , then it is obvious that ℑ+ = 𝔄+ . Similar considerations can be made for 𝔄′ , endowed with the involution ♭. C.10.2 The unimodular Hilbert algebras (E.10.5) have been known for a long time as Hilbert algebras, or unitary algebras and were the basis for obtaining the standard forms of the semifinite von Neumann algebras. Important contributions to this theory have been obtained by W. Ambrose (1945, 1949), J. Dixmier (1952a), H. A. Dye (1952), R. Godement (1949, 1951, 1954), F. J. Murray and J. von Neumann (1936, 1937), H. Nakano (1950), R. Pallu de la Barrière (1953), L. Pukánszky (1955), V. Rokhlin (1948), I. E. Segal (1947, 1950), O. Takenouchi (1951, 1952, 1960), M. Tomita (1953), H. Umegaki (1952), and others. We mention the fact that J. Dixmier (1952a) extended the notion of a (unimodular) Hilbert algebra to that of a quasi-Hilbert algebra

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(or quasi-unitary algebra) and showed that the set of all continuous complex functions with compact supports, defined on a locally compact topological group, can be canonically endowed with a quasi-Hilbert algebra structure (cf. E.10.3). The results concerning the (quasi-)Hilbert algebras and the standard forms for semifinite von Neumann algebras are set out in the book by I. Dixmier (1957/1969) (see also Loomis [1955] and Rieffel [1969]). Let 𝔄 ⊂ H be a unimodular Hilbert algebra. We shall use some notations from C.10.1. From exercise E.10.5, we infer that 𝑆 = 𝐽; hence D♯ = D♭ = H and for any 𝜁 ∈ H , we have 𝜁 ♯ = 𝜁 ♭ = 𝐽𝜁. For any 𝜉 ∈ H (= D ♯ ), the operator 𝐿0𝜉 is preclosed and we have (𝐿𝜉 )∗ ⊃ 𝐿𝐽𝜉 . In fact, in this particular case, we have the equality (𝐿𝜉 )∗ = 𝐿𝐽𝜉 . Indeed, let 𝜂 ∈ D (𝐿𝜉 )∗ . Since 𝜂 ∈ H = D ♭ , the closed operator 𝑅𝜂 makes sense. With the help of Corollary 5 from Section 10.3, we infer the existence of a sequence {𝑒𝑘 } ⊂ 𝔏(𝔄)′ of projections, such that 𝑒𝑘 ↑ 1, and 𝑒𝑘 𝜂 ∈ 𝔄′ , 𝑘 = 1, 2, …. Then 𝑒𝑘 𝜂 ∈ D 𝐿𝐽𝜉 and since (𝐿𝜉 )∗ is affiliated to 𝔏(𝔄), we have 𝐿𝐽𝜉 𝑒𝑘 𝜂 = (𝐿𝜉 )∗ 𝑒𝑘 𝜂 = 𝑒𝑘 (𝐿𝜉 )∗ 𝜂; 𝑘 = 1, 2, … For 𝑘 → ∞, we obtain 𝑒𝑘 𝜂 → 𝜂 and 𝐿𝐽𝜉 𝑒𝑘 𝜂 → (𝐿𝜉 )∗ 𝜂; hence 𝜂 ∈ D 𝐿𝐽𝜉 and 𝐿𝐽𝜉 𝜂 = (𝐿𝜉 )∗ 𝜂, thereby proving the asserted equality. It is now easy to verify that ♯

𝔓𝑎 = 𝔓♭𝑎 = 𝔓♯ = 𝔓♭ = 𝔓. On the other hand, from the equality 𝐽 = 𝑆, and from Theorem 10.12, we infer that 𝔄′ = 𝐽𝔄″ = 𝑆𝔄″ = 𝔄″ ; thus, by taking into account relation 10.4.2, we obtain the ‘commutation theorem’: 𝔏(𝔄)′ = 𝔎(𝔄) (for a simpler direct proof of this equality, see Dixmier [1957/1969] or Rieffel [1969]).

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Lectures on von Neumann algebras Since D ♯ = H it is easy to see that 𝜉 ∈ 𝔄″ ,

𝑥 ∈ 𝔏(𝔄) ⇒ 𝑥𝜉 ∈ 𝔄″ ,

𝐿𝑥𝜉 = 𝑥𝐿𝜉 .

The weight 𝜑 𝔄 , associated with the unimodular Hilbert algebra 𝔄, is a trace (E.10.5), which is called the natural trace associated to 𝔄 and it is usually denoted by 𝜇𝔄 . Conversely, if 𝜇 is a faithful semifinite normal trace on the von Neumann algebra M , then the left Hilbert algebra 𝔄𝜇 = 𝔑𝜇 = 𝔑∗𝜇 (see Section 10.14) is unimodular. If M = B (H ) and 𝜇 = 𝑡𝑟 (see E.10.4) then the operators in 𝔄𝑡𝑟 = 𝔑𝑡𝑟 are called the Hilbert-Schmidt operators in the Hilbert space H . We mention that the unimodular Hilbert algebra 𝔄𝑡𝑟 is complete with respect to the scalar product. More precisely, we have the following result, from T. Ogasawara and K. Yoshinaga (1955), whose proof can be found in J. Dixmier (1957/1969, Prop. 6, Chap. I, §8.5): Proposition. Let 𝔄 be a maximal unimodular Hilbert algebra, such that 𝔏(𝔄) is a factor. Then the following assertions are equivalent: (i) 𝔄 is complete. (ii) 𝔏(𝔄) is of type I. (iii) Up to a multiplication of the norm in 𝔄 by a suitable constant, 𝔄 is isomorphic to the unimodular Hilbert algebra of all Hilbert–Schmidt operators on a Hilbert space. In accordance with Theorem 10.25, the mapping 𝜉 ↦ 𝜔𝜉 is a bijection of 𝔓 onto (𝔏(𝔄)∗ )+ (this result also has a simpler direct proof; see Perdrizet 1971, Prop. 3.3). Consequently, given a normal form 𝜑 on 𝔏(𝔄), there exists ♯ a uniquely determined element 𝜉 ∈ 𝔓 = 𝔓𝑎 , such that 𝜑 = 𝜔𝜉 . Then 𝐴 = 𝐿𝜉 is a positive self-adjoint operator in H , which is affiliated to 𝔏(𝔄). If we denote 𝑒𝑛 = 𝜒(𝑛−1 ,𝑛) (𝐴), it is easy to infer that 𝑒𝑛 𝜉 ∈ 𝔄+ , 𝐿𝑒𝑛 𝜉 = 𝑒𝑛 𝐴 = 𝐴𝑒𝑛 and 𝑒𝑛 𝜉 → 𝜉. We have 𝜇𝔄 (𝐴2 𝑒𝑛 ) = 𝜇𝔄 ((𝐿𝑒𝑛 𝜉 )∗ (𝐿𝑒𝑛 𝜉 )) = ‖𝑒𝑛 𝜉‖2 ≤ ‖𝜉‖2 < +∞; hence the operator 𝐴 is of summable square with respect to 𝜇𝔄 . Moreover, for any 𝑥 ∈ 𝔏(𝔄)+ , we have 𝐿𝐴 𝑅𝐴 𝜇𝔄 (𝑥) = lim 𝜇𝔄 (𝐴𝑒𝑛 𝑥𝐴𝑒𝑛 ) = lim 𝜇𝔄 ((𝐿𝑥1/2 𝑒𝑛 𝜉 )∗ (𝐿𝑥1/2 𝑒𝑛 𝜉 )) 𝑛→∞

𝑛→∞

= lim ‖𝑥 1/2 𝑒𝑛 𝜉‖2 = ‖𝑥 1/2 𝜉‖2 = 𝜑(𝑥). 𝑛→∞

Thus, 𝜑 = 𝐿𝐴 𝑅𝐴 𝜇𝔄 .

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Conversely, let 𝐴 be a positive self-adjoint operator in H , which is affiliated to 𝔏(𝔄), of summable square with respect to 𝜇𝔄 and such that 𝜑 = 𝐿𝐴 𝑅𝐴 𝜇𝔄 . Since 𝜇𝔄 (𝐴2 𝑒𝑛 ) < +∞, there exists 𝜉𝑛 ∈ 𝔄+ , such that 𝐿𝜉𝑛 = 𝐴𝑒𝑛 . For 𝑛 > 𝑚 we have ‖𝜉𝑛 − 𝜉𝑚 ‖2 = 𝜇𝔄 ((𝐴𝑒𝑛 − 𝐴𝑒𝑚 )2 ) = 𝜇𝔄 (𝐴(𝑒𝑛 − 𝑒𝑚 )𝐴) = 𝜑(𝑒𝑛 − 𝑒𝑚 ); ♯

hence {𝜉𝑛 } is a Cauchy sequence. Let 𝜉 = lim𝑛→∞ 𝜉𝑛 ∈ 𝔓 = 𝔓𝑎 . For any 𝜂 ∈ 𝔄′ , we have 𝐿𝜉 𝜂 = 𝑅𝜂 𝜉 = lim 𝑅𝜂 𝜉𝑛 = lim 𝐿𝜉𝑛 𝜂 = lim 𝐴𝑒𝑛 𝜂 = 𝐴𝜂. 𝑛→∞

𝑛→∞

𝑛→∞

Since the operators 𝐿𝜉 and 𝐴 are self-adjoint, it follows that 𝐴 = 𝐿𝜉 . Thus, for any 𝑥 ∈ 𝔏(𝔄)+ , we have 𝜑(𝑥) = 𝐿𝐴 𝑅𝐴 𝜇𝔄 (𝑥) = 𝜇𝔄 (𝑥)((𝐿𝑥1/2 𝜉 )∗ (𝐿𝑥1/2 𝜉 )) = ‖𝑥 1/2 𝜉‖2 = 𝜔𝜉 (𝑥); hence, 𝜑 = 𝜔𝜉 . Consequently, 𝜉 is uniquely determined by 𝜑 and 𝐴 = 𝐿𝜉 is also uniquely determined by 𝜑. If we now take into account the possibility of using a *-isomorphism (see Section 9.26), from the preceding results, we infer the following: Theorem. Let M ⊂ B (H ) be a semifinite von Neumann algebra and 𝜇 a faithful, semifinite, normal trace on M + . For any normal form 𝜑 on M , there exists a unique positive self-adjoint operator 𝐴 in H , which is affiliated to M and of summable square with respect to 𝜇, such that 𝜑 = 𝐿𝐴 𝑅𝐴 𝜇. This is the Radon–Nikodym-type theorem, with respect to a semifinite, normal trace, obtained by I. E. Segal (1953, Th. 14) and L. Pukánszky (1954, Th. 1). The above proof belongs to F. Perdrizet (1971, Cor. 3.6). We mention the fact that if the trace 𝜇 is finite, then the theorem is a particular case of Theorem 10.10; in this case, the theorem has been obtained by H. A. Dye (1952, Cor. 5.1). In Section C.10.4, we shall present an extension of this theorem for weights. C.10.3 For a weight 𝜑 on the von Neumann algebra M , we consider the following properties: (𝑁1) {𝑎𝑖 } ⊂ M + , a w-summable family ⇒ 𝜑(∑ 𝑎𝑖 ) = ∑ 𝜑(𝑎𝑖 ). (𝑁2) {𝑎𝑖 } ⊂ M + , 𝑎𝑖 ↑ 𝑎 ⇒ 𝜑(𝑎𝑖 ) ↑ 𝜑(𝑎). (𝑁3) 𝜑 is lower w-semicontinuous.

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Lectures on von Neumann algebras (𝑁4) there exists a family {𝜑𝑣 } of normal forms on M , such that 𝜑(𝑎) = sup 𝜑𝑣 (𝑎),

𝑎 ∈ M +.

𝑣

(𝑁5) there exists a family {𝜑i } of normal forms on M , such that 𝜑(𝑎) = ∑ 𝜑i (𝑎),

𝑎 ∈ M +.

In our text (see Section 10.14), we said that 𝜑 is normal if it satisfies property (𝑁5). It is obvious that (𝑁5) ⇒ (𝑁4) ⇒ (𝑁3) ⇒ (𝑁2) ⇒ (𝑁1), and it is natural to inquire about the equivalence of these properties (see Dixmier 1957/1969: 52–53, 2nd ed.) Theorem 10.14 retains its validity for the faithful semifinite weights having property (𝑁4). More precisely, F. Combes (1971a, Th. 2.13) showed that if 𝜑 is a faithful semifinite weight on M + , which has property (𝑁3), then 𝔑𝜑 ∩𝔑∗𝜑 , endowed with the structure of a *-algebra induced by that of M and with the scalar product induced by that of H 𝜑 , is a left Hilbert algebra 𝔄𝜑 ⊂ H 𝜑 , and 𝜋𝜑 (M ) = 𝔏(𝔄𝜑 ); if 𝜑 has property (𝑁4), then 𝔄𝜑 = 𝔄″𝜑 and 𝜑 = 𝜑 𝔄 ∘ 𝜋𝜑 . A variant of proof for this fact can be found in M. Takesaki’s course (1969–1970, 13.5–13.12). At the basis of the proof lies a result about the ‘𝜀-filtration’ of the normal forms, which are majorized by a weight, result which is due to F. Combes (1968, Lemma 1.9) (see, also, Takesaki 1969–1970, Th. 13.8, for a simpler form of this result, that which is actually used). Since the weight that is associated with a left Hilbert algebra has property (N5) (in accordance with 10.18), it follows that (𝑁4) ⇔ (𝑁5). This equivalence has been established by G. K. Pedersen and M. Takesaki (1973, Th. 7.2); in our exposition of the results in Section 10.16.(9)–10.16.(11) and 10.18, we used the main arguments contained in this article. U. Haagerup (1972) completely solved the problem of the equivalence of the above properties, by showing that (𝑁1) ⇔ (𝑁4).

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We mention the fact that the elegant arguments in the article of U. Haagerup can be easily read and the equivalence (𝑁3) ⇔ (𝑁4) is proved in a more general case. Also, U. Haagerup (1972, 1.12) shows by an example, in the commutative case, that the property (𝑁0 ) {𝑒𝑖 } ⊂ M family of orthogonal projections ⇒ 𝜑(∑ 𝑒𝑖 ) = ∑ 𝜑(𝑒𝑖 ), is not equivalent to the normality (compare with Theorem 5.11) and the problem arises whether a result, analogous to Corollary 5.12, is true for weights. We mention that the equivalence of the above properties, in the case of traces, is well known since a long time (see Dixmier 1957/1969, Cor. Prop. 2, Chap. I, §6.1; see also E.8.10). C.10.4 With the help of a technique similar to that used in Section 10.18, one can prove the following: Proposition. Let 𝜑 and 𝜓 be faithful, semifinite, normal weights on the von 𝜑 Neumann algebra M . If 𝜑 and 𝜓 commute and are equal on a 𝜍𝑡 -invariant *-subalgebra of 𝔐𝜑 , which is w-dense in M , then 𝜑 = 𝜓. For the details of the proof, we refer to the article of G. K. Pedersen and M. Takesaki (1973, Lemma 5.2, Prop 5.9; see also Prop. 7.8, loc. cit.). In what follows we choose two faithful, semifinite, normal weights 𝜑 and 𝜓 on M + . We shall use the notations from the proof of Theorem 10.28, and we shall also denote (in accordance with A. Connes [1973c]) by 𝐷𝜓 𝐷𝜓 ∶𝑡↦ (𝑡) 𝐷𝜑 𝐷𝜑 the mapping 𝑡 ↦ ᵆ𝑡 that was obtained there. We shall show that if

𝐷𝜓 (𝑡) = 1, for any t ∈ ℝ, then 𝜑 = 𝜓. 𝐷𝜑

Indeed, from the hypothesis, we infer that 𝜍𝑡𝜃 (𝑒21 ) = 𝑒21 , for any 𝑡 ∈ ℝ, i.e. 𝑒21 ∈ N 𝜃0 . In accordance with Theorem 10.27, it follows that 𝑥 ∈ 𝔐𝜃 ⇒ 𝑥𝑒21 , 𝑒21 𝑥 ∈ 𝔐𝜃

and

𝜃(𝑥𝑒21 ) = 𝜃(𝑒21 𝑥);

hence (𝑥𝑖𝑗 ) ∈ 𝔐𝜃 ⇒ 𝑥12 ∈ 𝔐𝜑 ∩ 𝔐𝜓

and

𝜑(𝑥12 ) = 𝜓(𝑥12 ).

If 𝑦11 , 𝑦21 ∈ 𝔑𝜑 and 𝑦12 , 𝑦22 ∈ 𝔑𝜓 , then 𝑦 = (𝑦𝑖𝑗 ) ∈ 𝔑𝜃 , whence 𝑎 = 𝑦11 𝑦12 + 𝑦21 𝑦22 ∈ 𝔐𝜑 ∩ 𝔐𝜓

and 𝜑(𝑎) = 𝜓(𝑎).

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390

Lectures on von Neumann algebras In particular, for 𝑦11 = 𝑦12 = 0, 𝑦21 = ᵆ ∈ 𝔑𝜑 , 𝑦22 = 𝑣 ∈ 𝔑𝜓 , we obtain (1)

ᵆ ∈ 𝔑 𝜑 , 𝑣 ∈ 𝔑𝜓 ⇒ ᵆ ∗ 𝑣 ∈ 𝔐 𝜑 ∩ 𝔐 𝜓

and 𝜑(ᵆ∗ 𝑣) = 𝜓(ᵆ∗ 𝑣).

If we make ᵆ run over an approximate unit of 𝔑𝜑 , it follows that 𝔑𝜓 ⊂ 𝑤

𝔐𝜑 ∩ 𝔐𝜓 , hence (𝔐𝜑 ∩ 𝔐𝜓 )+ is a w-dense face of M + (see Sections 3.20 and 3.21). If 𝑎 ∈ (𝔐𝜑 ∩ 𝔐𝜓 )+ , then 𝑎1/2 ∈ 𝔑𝜑 ∩ 𝔑𝜓 , and by applying the relation (1), in which we make ᵆ = 𝑣 = 𝑎1/2 , we deduce that 𝑎 ∈ 𝔐𝜑 ∩ 𝔐𝜓 ⇒ 𝜑(𝑎) = 𝜓(𝑎). On the other hand, from the hypothesis, it easily follows that the weights 𝜑 and 𝜓 commute. Thus, if we now apply the above proposition, we obtain 𝜑 = 𝜓. By using the fact (𝑒12 )∗ = 𝑒21 , it is easy to prove that 𝐷𝜑 𝐷𝜓 (𝑡) = [ (𝑡)] 𝐷𝜑 𝐷𝜓

−1

,

𝑡 ∈ ℝ.

Also, if 𝜑1 , 𝜑2 , 𝜑3 are faithful, semifinite, normal weights on M + , then 𝐷𝜑3 𝐷𝜑3 𝐷𝜑2 (𝑡) = (𝑡) ⋅ (𝑡), 𝐷𝜑1 𝐷𝜑2 𝐷𝜑1

𝑡 ∈ ℝ.

Indeed, let {𝑒𝑖𝑗 , 𝑖, 𝑗 = 1, 2, 3} be the matrix units in Mat3 (M ), and let 𝜔 be the weight on Mat3 (M )+ , given by 𝜔(𝑥) = 𝜑1 (𝑥11 ) + 𝜑2 (𝑥22 ) + 𝜑3 (𝑥33 ),

𝑥 = (𝑥i𝑗 ).

The foregoing equality then follows by an argument similar to that used in the proof of Theorem 10.28, by taking into account the fact that 𝑒31 = 𝑒32 𝑒21 . Let 𝜑 be a faithful, semifinite, normal weight on M + , and let 𝐴 be a 𝜑 positive self-adjoint operator in H , which is affiliated to M 0 ; we denote 𝑒𝑛 = 𝜒(𝑛−1 ,𝑛) (𝐴). One defines a semifinite, normal weight 𝜑𝐴 on M + by the relations 𝜑𝐴 (𝑥) = lim 𝜑((𝐴𝑒𝑛 )1/2 𝑥(𝐴𝑒𝑛 )1/2 ), 𝑛→∞

𝑥 ∈ M +.

If s(𝐴) = 1, then 𝜑𝐴 is faithful and 𝜑

𝜑

𝜍𝑡 𝐴 = 𝐴i𝑡 𝜍𝑡 𝐴−i𝑡 ,

𝑡 ∈ ℝ,

𝑥 ∈ M.

For the proofs, we refer the reader to G. K. Pedersen and M. T. Takesaki (1973, §4).

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391

The theory of standard von Neumann algebras We define on Mat2 (M + ) two weights 𝜏 and 𝜔 by the relations 𝜏(𝑎) = 𝜑(𝑎11 ) + 𝜑𝐴 (𝑎22 ), 𝜔(𝑎) = 𝜑(𝑎11 ) + 𝜑(𝑎22 ), Then 𝜏 = 𝜔𝐵 , where 𝐵 = (

1

0

0

𝐴

𝜍𝑡𝜏 (𝑥) = 𝐵 i𝑡 𝜍𝑡𝜔 𝐵 −i𝑡 ,

𝑎 = (𝑎i𝑗 ), 𝑎 = (𝑎i𝑗 ).

); hence 𝑡 ∈ ℝ,

𝑥 ∈ Mat2 (M ).

In particular, if we make 𝑥 = 𝑒21 , we obtain 𝐷𝜑𝐴 (𝑡) = 𝐴i𝑡 , 𝐷𝜑

𝑡 ∈ ℝ.

From the preceding results, we infer the following Radon–Nikodym-type theorem for weights, due to G. K. Pedersen and M. Takesaki (1973, Th. 5.12). Theorem. Let 𝜑 and 𝜓 be two faithful, semifinite, normal weights on the von Neumann algebra M . If 𝜑 and 𝜓 commute, then there exists a uniquely 𝜑 determined positive self-adjoint operator 𝐴 in H , which is affiliated to M 0 , such that s(𝐴) = 1 and 𝜓 = 𝜑𝐴 . Indeed, since 𝜑 and 𝜓 commute, from the proof of Corollary 10.28, we infer that {

𝐷𝜓 𝐷𝜑

𝜑

(𝑡)} 𝑡∈ ℝ

is a so-continuous group of unitary operators in M 0 .

From the Stone theorem (see 9.20), we infer that there exists a positive self-adjoint operator 𝐴 in H , such that s(𝐴) = 1 and 𝐷𝜓 (𝑡) = 𝐴i𝑡 , 𝐷𝜑

𝑡 ∈ ℝ. 𝜑

In accordance with exercise E.9.25, 𝐴 is affiliated to M 0 ; hence we can define the weight 𝜑𝐴 and we have 𝐷𝜑𝐴 (𝑡) = 𝐴i𝑡 , 𝐷𝜑 Thus,

𝑡 ∈ ℝ.

𝐷𝜑𝐴 𝐷𝜑𝐴 𝐷𝜑 (𝑡) = (𝑡) (𝑡) = 𝐴i𝑡 (𝐴i𝑡 )−1 , 𝐷𝜑 𝐷𝜓 𝐷𝜓

𝑡 ∈ ℝ;

Consequently, we have 𝜓 = 𝜑𝐴 . The uniqueness of the operator 𝐴 immediately follows from the uniqueness of the analytic generator in Stone’s theorem. [ This method of proving the theorem of G. K. Pedersen and M. Takesaki was communicated to us by

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392

Lectures on von Neumann algebras Gr. Arsene. In a recent paper, G. A. Elliott (1975) gives a yet simpler proof and also indicates technical simplifications for the proofs of the results in Sections 10.18, 10.27 and 10.28.] It is obvious that any weight commutes with any trace; hence, the theorem of I. E. Segal and L. Pukánszky, we have stated in section C.10.2, is a particular case of the theorem of G. K. Pedersen and M. Takesaki. 𝜑 For a faithful, semifinite, normal weight 𝜑 on M + , {𝜍𝑡 }𝑡∈ ℝ is the only group of *-automorphisms of M , with respect to which 𝜑 satisfies the KMS-conditions (see Section 10.17). Consequently, if 𝜓 is another faithful, semifinite, normal weight, which satisfies the KMS-conditions with respect 𝜑 𝜓 𝜑 to {𝜍𝑡 }𝑡∈ℝ , then 𝜍𝑡 = 𝜍𝑡 , 𝑡 ∈ ℝ. We hence infer that 𝜑 and 𝜓 commute; hence, in accordance with the above theorem, there exists a positive 𝜑 self-adjoint operator 𝐴 in H , which is affiliated to M 0 , such that s(𝐴) = 1 and 𝜓 = 𝜑𝐴 . Since 𝜑

𝜓

𝜑

𝜑

𝜍𝑡 (𝑥) = 𝜍𝑡 (𝑥) = 𝜍𝑡 𝐴 (𝑥) = 𝐴i𝑡 𝜍𝑡 (𝑥)𝐴−i𝑡 ,

𝑥 ∈ M.

it follows that 𝐴i𝑡 belongs to the center Z of M , for any 𝑡 ∈ ℝ; hence 𝐴 is affiliated to Z . We thus obtain the following Corollary 1. Let 𝜑 and 𝜓 be faithful, semifinite, normal weights on M + . The following assertions are then equivalent: 𝜑

(i) 𝜓 satisfies the KMS-condition with respect to {𝜍𝑡 }𝑡∈ ℝ . 𝜓

𝜑

(ii) 𝜍𝑡 = 𝜍𝑡 , 𝑡 ∈ ℝ. (iii) There exists a positive self-adjoint operator 𝐴 in H , which is affiliated to Z , such that s(𝐴) = 1 and 𝜓 = 𝜑𝐴 . The assertions in this corollary are true, for example, if 𝜑 and 𝜓 are faithful semifinite normal traces on M + ; if, moreover, 𝜓 ≤ 𝜑, then 𝐴 ∈ Z , 0 ≤ 𝐴 ≤ 1. In particular, we have the following: Corollary 2. Let M be a factor and 𝜑, 𝜓 two semifnite normal traces on M + . Then there exists 𝜆 ≥ 0, such that 𝜓 = 𝜆𝜑. Indeed, the corollary follows from the facts that the support of a trace is a central projection and 𝜑 + 𝜓 is also a semifinite normal trace, whereas 𝜑 ≤ 𝜑 + 𝜓, 𝜓 ≤ 𝜑 + 𝜓, etc. The preceding results concerning the traces have direct and simpler proofs (see Dixmier 1957/1969, Chap. I., §6.4). For finite traces, they can easily be obtained from the Radon–Nikodym-type theorem of Sakai (see E.7.14).

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The problem now arises whether the theorem in this section can be extended for weights that do not commute. Theorem 10.10 is such an extension for the case in which 𝜑 and 𝜓 are normal forms. In the article of G. K. Pedersen and M. Takesaki (1973) another partial extension is given (loc. cit., Prop. 7.6), as well as a negative result (loc. cit., Prop. 7.7). Another theorem of the Radon–Nikodym type, for weights, was obtained by A. van Daele (1976b), who generalized a theorem of S. Sakai for normal forms (see C.5.5). C.10.5 Let 𝜑 and 𝜓 be faithful normal forms on the von Neumann algebra M ⊂ B (H ). We recall that by |𝜑 ± i𝜓|, we denote the modulus of the w-continuous linear form 𝜑 ± i𝜓, in accordance with Theorem 5.16. M. Takesaki (1970, Th. 15.2) and R. H. Herman and M. Takesaki (1970, Th. 1, Th. 2) proved the following results: Proposition 1. The following assertions are equivalent: (i) 𝜑 and 𝜓 commute. 𝜑

𝜓

𝜑

𝜓

𝜓

𝜑

(ii) {𝜍𝑡 } and {𝜍𝑡 } commute. 𝜍𝑡 ∘ 𝜍𝑡 = 𝜍𝑡 ∘ 𝜍𝑡 , 𝑡, 𝑠 ∈ ℝ; (iii) |𝜑 + i𝜓| = |𝜑 − i𝜓|. Proposition 2. If 𝜋 is a *-automorphism of M , which acts identically on the center, then the following assertions are equivalent: (i) 𝜑 is 𝜋-invariant: 𝜑 ∘ 𝜋 = 𝜑; 𝜑

𝜑

𝜑

(ii) 𝜋 commutes with {𝜍𝑡 } ∶ 𝜋 ∘ 𝜍𝑡 = 𝜍𝑡 ∘ 𝜋,

𝑡 ∈ ℝ.

The proofs of these propositions can also be found in the course of M. Takesaki (1969–1970, 15.14–15.18). Let now 𝜑 and 𝜓 be faithful, semifinite, normal weights on M + . The problem now arises whether the equivalences (i) ⇔ (ii) in the two above propositions remain true. If 𝜋 is an arbitrary *-automorphism of M , then, with the help of the KMS-conditions, it is easy to prove that 𝜑∘𝜋

𝜍𝑡

𝜑

= 𝜋 −1 ∘ 𝜍𝑡 ∘ 𝜋,

𝑡 ∈ ℝ.

Thus it is obvious that the implications (i) ⇒ (ii) in both propositions remain true for weights, too. Nevertheless, the converse implications are not true, in general, as G. K. Pedersen and M. Takesaki have shown (1973, Prop. 5.11). In the presence of some additional hypotheses, the equivalence (i) ⇔ (ii) from Proposition 1 retains its validity for weights, too (see Pedersen and Takesaki 1973, Lemma 5.8, Prop. 6.1, Cor. 6.4, Th. 6.6). We also mention that from

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Lectures on von Neumann algebras condition (ii) (in Proposition 2), it follows that the weight 𝜑 + 𝜓 is semifinite (in accordance with loc. cit., Prop. 5.10). Let us now assume that 𝜑 and 𝜓 are faithful forms on the hyperstandard von Neumann algebra M ⊂ B (H ) and that 𝜓 ≤ 𝜑. On the one hand, we 𝑑𝜓 have the ‘derivative’ , introduced in Section 10.25, and also, we have the 𝑑𝜑

‘derivative’ given by the theorem of Sakai (5.21). It is obvious that the two 𝑑𝜓 ‘derivatives’ coincide iff ≥ 0. One can prove the following result (see 𝑑𝜑

Araki 1974b, Th. 13): Proposition 3. The following assertions are equivalent: ∗

𝑑𝜓 𝑑𝜓 . ) = 𝑑𝜑 𝑑𝜑 𝑑𝜑 (ii) ≥ 0. 𝑑𝜓 𝑑𝜓 𝜑 𝑑𝜓 (iii) 𝜍𝑡 ( , 𝑡 ∈ ℝ. )= 𝑑𝜑 𝑑𝜑 (iv) 𝜓 and 𝜑 commute. (i) (

𝜑

C.10.6 The problem of the continuous dependence of the group {𝜍𝑡 } of modular automorphisms with respect to the faithful normal form 𝜑 has been solved by A. Connes (1972b, Th. 1) by the following: Theorem. Let 𝜑𝑛 and 𝜑 be faithful normal forms on the von Neumann algebra M . If ‖𝜑𝑛 − 𝜑‖ → 0, then 𝜑

𝑠𝑜

𝜑

𝜍𝑡 𝑛 (𝑥) → 𝜍𝑡 (𝑥),

𝑡 ∈ ℝ,

𝑥 ∈ M,

and the convergence is uniform with respect to 𝑡, for |𝑡| < 𝑡0 . A proof of this theorem, based on the methods developed in Sections 10.23–10.25, can be found in H. Araki (1974b, Th. 10). See also A. Connes (1980b) for another result of this kind. C.10.7 One calls a modular Hilbert algebra (or a Tomita algebra) a complex algebra 𝔄 with an involution 𝜉 ↦ 𝜉 ♯ , endowed also with a scalar product (⋅|⋅) and with a group of algebra automorphisms {∆(𝛼)}𝛼∈ℂ , depending on a complex parameter, which satisfies axioms (i)–(iii) from Section 10.1, and also the axioms: (iv) (∆(𝛼)𝜉)♯ = ∆(−𝛼)𝜉 ♯ , for any 𝜉 ∈ 𝔄, 𝛼 ∈ ℂ. (v) (∆(𝛼)𝜉|𝜂) = (𝜉|∆(𝛼)𝜂), for any 𝜉, 𝜂 ∈ 𝔄, 𝛼 ∈ ℂ. (vi) (∆(1)𝜉 ♯ |𝜂 ♯ ) = (𝜂|𝜉), for any 𝜉, 𝜂 ∈ 𝔄. (vii) ℂ ∋ 𝛼 ↦ (∆(𝛼)𝜉|𝜂) ∈ ℂ is an entire analytic function, for any 𝜉, 𝜂 ∈ 𝔄. (viii) (1 + ∆(𝑡))𝔄 is dense in 𝔄, for any 𝑡 ∈ ℝ. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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It is easy to see that the Tomita algebra, associated to a left Hilbert algebra, is, in a natural manner, a modular Hilbert algebra, with ∆(𝛼) = ∆𝛼 |𝔄,

𝛼 ∈ ℂ.

Let 𝔄 now be a modular Hilbert algebra and H the Hilbert space obtained by the completion of 𝔄. From axiom (v), one infers that ∆(i𝑡) is an isometric mapping; hence, by denoting by ᵆ𝑡 the closure of ∆(i𝑡), we obtain a one-parameter group {ᵆ𝑡 }𝑡∈ℝ of unitary operators on H . In accordance with the Stone theorem, there exists a positive self-adjoint operator ∆ in H , such that s(∆) = 1 and ᵆ𝑡 = ∆i𝑡 , 𝑡 ∈ ℝ. One can then prove that ∆𝛼 is the closure of ∆(𝛼),

𝛼 ∈ ℂ.

The operator ∆ is called the modular operator associated to the modular Hilbert algebra 𝔄. For any 𝑡 ∈ ℝ, the mapping 𝔄 ∋ 𝜉 ↦ ∆𝑡 𝜉 ♯ ∈ 𝔄 is an involution in 𝔄, which is compatible with its algebra structure. It is easy to prove that the involution corresponding to 𝑡 = 1/2 extends to a conjugation 𝐽 of H , called the canonical conjugation associated to the modular Hilbert algebra 𝔄. In contrast, the involution corresponding to 𝑡 = 1, i.e. 𝜂 ↦ 𝜂 ♭ = ∆𝜂 ♯ , is called the adjoint involution and it has the property that (𝜉|𝜂) = (𝜂 ♭ |𝜉 ♯ ),

𝜉, 𝜂 ∈ 𝔄.

Hence one can immediately infer that 𝔄 also satisfies axiom (iv) from Section 10.1; hence 𝔄 is a left Hilbert algebra. It is easy to prove that ∆ and 𝐽 are associated to the structure of a left Hilbert algebra of 𝔄, as in Section 10.1, i.e. 𝑆 = 𝐽∆1/2 . For details, we refer to M. Takesaki (1969–1970, 1970). The modular Hilbert algebras are a useful tool for the computations (see, e.g. Takesaki 1973c). C.10.8 Bibliographical comments. In Sections 10.1–10.6, which contain the ‘elementary’ part of the Tomita theory, we followed the lessons of M. Takesaki (1970), but the systematic use of Proposition 10.3, exhibited by A. van Daele (1974, Lemma 2.6) allowed the simplification of the exposition given by M. Takesaki (1970). The commutation theorem for tensor products (10.7) has been known for a long time, for semifinite von Neumann algebras, and conjectured in general (J. Dixmier 1957/1969, Chap. I, §2.4, §6.9). S. Sakai (1968c) proved this theorem by assuming that only one of the two von Neumann algebras is semifinite. The general case was obtained by M. Tomita (1967a, c), as Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

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Lectures on von Neumann algebras a corollary of his main results. The direct proof we presented here was obtained by I. Cuculescu (1971) and S. Sakai (1971) (see, also, Takesaki [1972a], and L. Zsidó [1970]). Recently, M. A. Rieffel and A. von Daele (1975) obtained a simple proof of another nature, which does not use the theory of unbounded operators. See also R. Rousseau, A. van Daele, and L. van Heeswijck (1976). The cones 𝔓𝑆 and 𝔓𝑆∗ (§10.9) were introduced by M. Takesaki (1970), for the case 𝔄 = M 𝜉0 , and by F. Perdrizet (1971) in the general case. Lemma 10.9, which is the main argument in the proof of the general Radon–Nikodym-type theorem for normal forms (10.10), is due to M. Takesaki (1970). We mention that M. Takesaki (1970) also gives a new proof to the Radon–Nikodym-type theorem of Sakai, which is based on elementary results from the theory of Tomita. Tomita’s fundamental theorem (10.12) allows the conclusion that any von Neumann algebra is *-isomorphic to a standard von Neumann algebra (10.15), a result that concludes a long series of efforts in the development of the operator algebras theory, highlighted by the works of F. J. Murray and J. von Neumann (1936, 1937), H. A. Dye (1952), I. E. Segal [11], J. Dixmier (1952a), L. Pukánszky (1955), M. Tomita (1960), and others (see C.10.2). The proof of Theorem 10.12, given in the text, is due to A. van Daele (1974) and it is different and simpler from the proofs given by M. Tomita, in 1978a, and by M. Takesaki, in 1976b. Another proof, given by L. Zsidó (1975a), is indicated in exercise E.10.13. The culminating point in Tomita’s theory, and the main technical instrument for handling the left Hilbert algebras, is the theorem on the existence of the Tomita algebra (10.20). For the exposition given in Sections 10.19–10.21, we developed the ideas from the article of L. Zsidó (1975a), the analytic continuation methods we use (Sections 9.15, 9.24) originating in the article by G. K. Pedersen and M. Takesaki (1973), which also suggested to us a part of the criterion 10.21. The proof of Theorem 10.20, thus obtained, is different from that given by M. Takesaki (1970). The algebra 𝔖 (Section 10.22) was first considered by L. Zsidó (1975a). The weights were introduced by F. Combes (1968) and G. K. Pedersen (1966/1968/1969), whereas the link existing between the weights and the left Hilbert algebras was established by F. Combes (1971a) and M. Tomita (1967a). For the exposition given by us in Sections 10.16 and 10.18, we used the articles by F. Combes (1971a), G. K. Pedersen and M. Takesaki (1973), as well as the course by M. Takesaki (1969–1970). The KMS-condition originates in theoretical physics, and it was framed into the theory of operator algebras by R. Haag, N. M. Hugenholtz and

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The theory of standard von Neumann algebras

397

M. Winnink (1967), who showed that, given a 𝐶 ∗ -algebra, endowed with a one-parameter group of *-automorphisms, the cyclic representation associated to a positive form, which satisfies the KMS-condition, is standard. Another application of the KMS-conditions was given by N. M. Hugenholtz (1967). These papers appeared at the same time as M. Tomita’s papers (1967a, c) whereas M. Takesaki (1970, §13) found the deep link between the Tomita theory and the KMS-condition, by proving Theorem 10.17 for the case of the faithful normal forms. Subsequently, F. Combes (1971b) and M. Takesaki ([17], Th. 14.6) proved a variant of Theorem 10.17 for the case of the weights. The KMS-condition for weights is similar to the condition 𝜑(𝑥𝑦) = 𝜑(𝑦𝑥), which is characteristic for the traces. For various results on the KMS-condition, we refer to H. Araki (1968), H. Araki and H. Miyata (1968), F. Combes (1971a, b), N. M. Hugenholtz (1972), O. Bratteli, A. Kishimoto, and D. W. Robinson (1976), D. W. Robinson (1970), F. Rocca, M. Sirugue, and D. Testard (1968, 1969), M. Sirugue and M. Winnink (1970, 1971), M. Takesaki (1969–1970, 1970, 1971/1972, 1973f ), M. Winnink (1970). We also mention the works which gave a name to the KMS-condition: R. Kubo, J. Phys. Soc. Japan, 12(1957), p. 570, and P. C. Martin, J. Schwinger, Phys. Rev., 115 (1959), p. 1342. The results in Section 10.27 are due to G. K. Pedersen and M. Takesaki (1973, §3), whereas Theorem 10.28 is due to A. Connes (1972c, 1973c). In our exposition, we have used these sources. The characterization of the semifinite von Neumann algebras in terms of the group of the modular automorphisms (Section 10.29) was obtained by M. Takesaki (1970, §14) by a very intricate proof. A simpler proof was given by G. K. Pedersen and M. Takesaki (1973, Th. 7.4), on the basis of their theorem of the Radon-Nikodym type for weights (see C.10.4). The proof given in our text does not explicitly use this theorem, but the theorem of A. Connes. The results in Sections 10.23–10.26 are from H. Araki (1974b), A. Connes (1972c, 1973c) and U. Haagerup (1975b). In our exposition, we followed the preprint of U. Haagerup (1975b). The sets 𝔓𝜆 (E.10.18) were introduced by H. Araki (1974a, b), who also proved Radon–Nikodym-type theorems relatively to these sets. A. Connes (1974b) found a characterization of the von Neumann algebras as ordered vector spaces, thus giving an answer to a problem posed by S. Sakai (1956a). U. Haagerup (1975b) made a comprehensive study of the hyperstandard von Neumann algebras. One of the most important applications of Tomita’s theory concerns the classification of the factors of type III. In this direction, we mention the wealth and depth of the results obtained by A. Connes (1971, 1972a, b, c, d, 1973a, Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:45:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.013

398

Lectures on von Neumann algebras b, c, 1974a, b, c, 1975d, e, 1976a, c, d, 1980b). Other important results were obtained by M. Takesaki (1973e, g) and E. Størmer (1971, 1971/1972/1975). A remarkable application of Tomita’s theory to the structure of type III von Neumann algebras was found by M. Takesaki (1973a, b, c, d), who showed that any type III von Neumann algebra is, in a unique manner, the cross-product of a type II∞ von Neumann algebra by a one-parameter group of *-automorphisms. Particular cases of this theorem were previously proved by A. Connes (1973c) and M. Takesaki (1973e, g). For other results and applications concerning the Tomita theory, we refer the reader to the Proceedings of some International Conferences on Operator Algebras such as C*-algebras and their applications to statistical mechanics and quantum field theory, North-Holland, 1976; Symposia Mathematica, vol. XX, Academic Press, 1975; Operator Algebras and their Applications to Mathematical Physics (Conference held in Marseille, June, 1977 CNRS).

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APPENDIX

Fixed-point theorems In this Appendix, we shall prove Ryll-Nardzewski’s fixed-point theorem, which we used in Chapter 7. A.l. Let X be a vector space and K ⊂ X a convex set. A mapping 𝑇 ∶ K → K is said to be affine if for any 𝑥1 , 𝑥∈ K and any 𝜆 ∈ [0, 1], we have 𝑇(𝜆𝑥1 + (1 − 𝜆)𝑥2 ) = 𝜆𝑇(𝑥1 ) + (1 − 𝜆)𝑇(𝑥2 ). Lemma (Markov–Kakutani). Let X be a Hausdorff topological vector space, K ⊂ X a non-empty, compact, convex subset and J a family of continuous, affine mappings of K into K , which are mutually commuting. Then there exists an 𝑥0 ∈ K , such that 𝑇𝑥0 = 𝑥0 , 𝑇 ∈ J . Proof. For any 𝑇 ∈ J and any natural number 𝑛, we write 𝑇𝑛 =

1 (𝐼 + 𝑇 + … + 𝑇 𝑛−1 ). 𝑛

The sets 𝑇𝑛 (K ) are compact subsets of K . For any 𝑇 (1) , 𝑇 (2) , … , 𝑇 (𝑘) ∈ J and any natural numbers 𝑛1 , … , 𝑛𝑘 , we have 𝑘

(𝑇 (1) )𝑛1 … (𝑇 (𝑘) )𝑛𝑘 K ⊂

⋂

(𝑇 (𝑖) )𝑛i K ;

𝑖=1

hence

𝑘

(𝑇 (𝑖) )𝑛i K ≠ ∅. ⋂ 𝑖=1

399

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Consequently, there exists an 𝑥0 ∈

𝑇𝑛 (K ) ⊂ K .

⋂

𝑇∈J , 𝑛∈ℕ

Let now 𝑇 ∈ J be arbitrary. Since the set K − K is compact, for any neighbourhood U of the origin, there exists a natural number 𝑛, such that 1 (K − K ) ⊂ U . 𝑛 Since 𝑥0 ∈ 𝑇𝑛 (K ), there exists an 𝑥 ∈ K , such that 𝑥0 = 𝑇𝑛 𝑥; consequently 𝑇𝑥0 − 𝑥0 =

1 𝑛 1 (𝑇 𝑥 − 𝑥) ∈ (K − K ) ⊂ U . 𝑛 𝑛

Since X is separated and U was an arbitrary neighbourhood of the origin, 𝑇𝑥0 = 𝑥0 . Q.E.D. A.2. Let 𝑝 be a seminorm on the vector space X . For any subset S ⊂ X , let us denote 𝑝-diam(S ) = sup 𝑝(𝑥 − 𝑦). 𝑥, 𝑦∈S

Lemma (Namioka–Asplund 1967). Let X be a Hausdorff locally convex vector space, K ⊂ X a non-empty, separable, weakly compact, convex subset, p a continuous seminorm on X and 𝜀 > 0. Then there exists a closed convex subset C ⊂ X , such that C ≠ K and 𝑝-diam(K \C ) ≤ 𝜀. Proof. Let

U = {𝑥; 𝑥 ∈ X , 𝑝(𝑥) ≤ 𝜀/4}.

Since K is separable, there exists a sequence {𝑥𝑖 } ⊂ K , such that ∞

K ⊂

⋃

(𝑥𝑖 + U ).

𝑖=1

We denote by E the weak closure of the set of all extreme points of K . Since E is weakly compact, and the sets 𝑥𝑖 + U are weakly closed, from ∞

E ⊂

⋃

(𝑥𝑖 + U ).

𝑖=1

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and by taking into account the theorem of Baire, we infer that there exists an index 𝑖0 and a weakly open subset D ⊂ X , such that ∅ ≠ E ∩ D ⊂ E ∩ (𝑥𝑖0 + U ). Let K 1 be the closed convex hull of E \D and K 2 the closed convex hull of E ∩ D . From the Krein–Milman theorem, K is the convex hull of K 1 ∪ K 2 . Since the set E \D is weakly compact, it contains all the extreme points of K 1 by virtue of the converse Milman theorem (see Phelps 1966, Chap. 1). Since E ∩D ≠ ∅, it follows that K 1 ≠ K . Obviously, 𝜀 𝑝-diam(K 2 ) ≤ 𝑝-diam(𝑥𝑖0 + U ) ≤ . 2 We denote 𝑑 = 𝑝-diam(K ), and we consider a number 𝛿, such that 0 < 𝛿 < min{1, 𝜀/4𝑑}. The set C = {𝜆𝑦1 + (1 − 𝜆)𝑦2 ; 𝑦1 ∈ K 1 , 𝑦2 ∈ K 2 , 𝛿 ≤ 𝜆 ≤ 1} is a closed convex subset of K . Let us assume that C = K . Let 𝑥 be an extreme point of K . Then there exist 𝑦1 ∈ K 1 , 𝑦2 ∈ K 2 , 𝜆 ∈ [𝛿, 1], such that 𝑥 = 𝜆𝑦1 + (1 − 𝜆)𝑦2 . If 𝜆 = 1, then 𝑥 = 𝑦1 ; if 𝜆 < 1, since 𝑥 is extreme, it follows that 𝑥 = 𝑦1 = 𝑦2 . In both cases, 𝑥 = 𝑦1 ∈ K 1 . Since 𝑥 is an arbitrary extreme point of K , we obtain K ⊂ K 1 , thus contradicting the fact that K 1 ≠ K . Consequently, C ≠ K . Since K is the convex hull of K 1 ∪ K 2 , for any 𝑦 ∈ K \C , there exist 𝑦1 ∈ K 1 , 𝑦2 ∈ K 2 , 𝜆 ∈ [0, 𝛿], such that 𝑦 = 𝜆𝑦1 + (1 − 𝜆)𝑦2 . Thus, 𝑝(𝑦 − 𝑦2 ) = 𝜆𝑝(𝑦1 − 𝑦2 ) ≤ 𝛿𝑑. Since 𝑝-diam(K 2 ) ≤ 𝜀/2, it follows that 𝑝-diam(K \C ) ≤ 2𝛿𝑑 + 𝜀/2 ≤ 𝜖. Q.E.D. A.3. Let X be a locally convex vector space, S ⊂ X and E a semigroup of mappings of S into S . One says that E is non-contracting if, for any 𝑥, 𝑦 ∈ S , 𝑥 ≠ 𝑦, there exists a continuous seminorm 𝑝 on X , such that inf 𝑝(𝑇𝑥 − 𝑇𝑦) > 0.

𝑇∈E

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Theorem (Ryll–Nardzewski 1967). Let X be a Hausdorff locally convex vector space, K ⊂ X a non-empty, weakly compact, convex subset and E a non-contracting semigroup of weakly continuous affine mappings of K into K . Then there exists an 𝑥0 ∈ K , such that 𝑇𝑥0 = 𝑥0 , 𝑇 ∈ E . Proof. Let 𝑇1 , … , 𝑇𝑛 ∈ E and let 𝑇0 =

1 (𝑇 + … + 𝑇𝑛 ). 𝑛 1

In accordance with Lemma A.1, there exists an 𝑥0 ∈ K , such that 𝑇0 𝑥0 = 𝑥0 . Let us assume that there exists an index 𝑖, 1 ≤ 𝑖 ≤ 𝑛, such that 𝑇𝑖 𝑥0 ≠ 𝑥0 . We can assume that 𝑇𝑖 𝑥0 ≠ 𝑥0 , 𝑇𝑖 𝑥0 = 𝑥0 ,

for 1 ≤ 𝑖 ≤ 𝑚, for 𝑖 > 𝑚.

1

If we denote 𝑇0′ = (𝑇1 + … + 𝑇𝑚 ), we have 𝑇0′ 𝑥0 = 𝑥0 . 𝑚 Since E is non-contracting, there exists a continuous seminorm 𝑝 on X and an 𝜀 > 0, such that (∗) 𝑝(𝑇𝑇𝑖 𝑥0 − 𝑇𝑥0 ) ≥ 2𝜀, 𝑇 ∈ E , 1 ≤ 𝑖 ≤ 𝑚. Let E 0 be the subsemigroup of E , generated by 𝑇1 , … , 𝑇𝑚 , and let K 0 be the weakly closed convex hull of the set {𝑇𝑥0 ; 𝑇 ∈ E 0 }. Obviously, K 0 is a nonempty, separable, weakly compact, convex subset of K . In accordance with Lemma A.2, there exists a closed convex subset C 0 ⊂ K 0 , such that C 0 ≠ K 0 and 𝑝-diam(K 0 \C 0 ) ≤ 𝜀. Since C 0 ≠ K 0 , there exists a 𝑆0 ∈ E 0 , such that 𝑆0 𝑥0 ∈ K 0 \C 0 . From the equality ′ 𝑇0 𝑥0 = 𝑥0 , we infer that 𝑆0 𝑥0 =

1 (𝑆 𝑇 𝑥 + … + 𝑆0 𝑇𝑚 𝑥0 ). 𝑚 0 1 0

Thus, there exists an index 𝑖, 1 ≤ 𝑖 ≤ 𝑛, such that 𝑆0 𝑇i 𝑥0 ∈ K 0 \C 0 . Then 𝑝(𝑆0 𝑇i 𝑥0 − 𝑆0 𝑥0 ) ≤ 𝑝-diam(K 0 \C 0 ) ≤ 𝜀, and this contradicts the relation (*). Consequently, any finite subset of E has a common fixed point. A familiar compactness argument shows that E has a fixed point. Q.E.D.

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———. 1972b. ‘Sur un theorème de Misonou.’ Revue Roumaine de Mathématique Pures et Appliquées 17: 309–10. ———. 1975/1976. ‘A Non-Commutative Weyl-von Neumann Theorem.’ Comptes Rendus Mathématiques Académie des Sciences, Paris 281: 735–36; Revue Roumaine de Mathématiques Pures et Appliquées 21: 97–113. ———. 1976. ‘Représentations factorielles de type II1 de U(∞).’ Journal de Mathématiques Pures et Appliquées 55: 1–20. ———. 1991. ‘Limit Laws for Random Matrices and Free Products.’ Inventiones Mathematicae 104 (1): 201–20. ———. 1996. ‘The Analogues of Entropy and of Fisher’s Information Measure in Free Probability Theory. III. The Absence of Cartan Subalgebras.’ Geometric and Functional Analysis 6 (1): 172–99. Voiculescu, D. V., K. J. Dykema, and A. Nica. 1992. Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups. CRM Universite de Montreal Series. AMS, Providence RI. von Neumann, J. 1932. Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. ———. 1936. ‘On a Certain Topology for Rings of Operators.’ Annals of Mathematics 37: 111–15. ———. 1936/1937/1960. ‘Continuous Geometry.’ Proceedings of the National Academy of Sciences of the United States of America 22; Lecture Notes I, II, III, Michigan. Princeton University Press, Princeton, NJ. ———. 1938. ‘On Infinite Direct Products.’ Compositio Mathematica 6: 1–77. ———. 1940. ‘On Rings of Operators, III.’ Annals of Mathematics 41: 94–161. ———. 1949. ‘On Rings of Operators. Reduction Theory.’ Annals of Mathematics 50: 401–85. ———. 1961. Collected Works, Vol. III (On Rings of Operators). Oxford: Pergamon Press. Vowden, J. 1967. ‘On the Gelfand-Naimark Theorem.’ Journal of the London Mathematical Society 42: 725–31. ———. 1969. ‘A New Proof in the Spatial Theory of von Neumann Algebras.’ Journal of the London Mathematical Society 44: 429–32. Walter, M. E. 1972. ‘W*-algebras and Non-Abelian Harmonic Analysis.’ Journal of Functional Analysis 11: 17–38. ———. 1974. ‘A Duality between Locally Compact Groups and Certain Banach Algebras.’ Journal of Functional Analysis 17: 131–60. Wassermann, S. 1977. ‘Injective W*-algebras.’ Mathematical Proceedings of the Cambridge Philosophical Society 82: 39–48. Weil, A. 1964. ‘Sur certains groups d’opérateurs unitaires.’ Acta Mathematica 111: 143–211. Wenzl, H. 1987. ‘On Sequences of Projections.’ Comptes Rendus de l’Académie des Sciences du Canada. 9 (1): 5–9. Winnink, M. 1970. ‘Algebraic Aspects of the KMS Condition.’ In Cargèse Lectures in Phyisics, vol. 4, 235–55. Editor Daniel Kastler, New York, Gordon an Breach. Yeadon, F. J. 1971. ‘A new proof of the existence of a trace in finite von Neumann algebras’ Bulletin of the American Mathematical Society, 77: 257–260.

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SUBJECT INDEX

algebra 𝐶 ∗ -algebra (2.2) 𝐶 ∗ -algebra of operators (2.2) continuous, discrete, finite, properly infinite, purely infinite, semifinite, von Neumann ∼ (4.16, 4.21) homogeneous von Neumann ∼ of type I (E.4.14, 7.16, 8.5) hyperstandard von Neumann ∼ (10.23) induced von Neumann ∼ (3.14) left Hilbert ∼ (10.1) maximal abelian von Neumann ∼ (E.3.10) modular Hilbert ∼ (C.10.7) von Neumann ∼ (2.2) von Neumann ∼ of countable type (E.3.5) von Neumann ∼ of type I, II, III, Ifin , I∞ , II1 , II∞ (4.16, 4.21) reduced von Neumann ∼ (3.14) right Hilbert ∼ (10.4) standard von Neumann ∼ (E.7.15, 10.15) Tomita ∼ (C.10.7) Tomita ∼ associated to a left Hilbert ∼ (10.20) uniform von Neumann ∼ of type 𝑆𝛾 , of type 𝛾 (E.4.14, 8.5) unimodular Hilbert ∼ (E.10.5) 𝑊 ∗ -algebra (C.5.3) *-algebra (2.1) amplification (3.17) bicommutant (3.1) center of a von Neumann algebra (3.9) centralizer of a weight (10.27) commutant (3.1) (KMS)-condition (10.17) conjugation (E.3.9, 9.35) canonical ∼ associated to a left Hilbert algebra (10.1) element coupling ∼ (7.20) extension Friedrichs ∼ (9.6) face (3.21) invariant ∼ (3.21)

422

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Subject index

423

factor (3.9) family sufficient ∼ of traces (7.13) form adjoint ∼ (5.1) central ∼ (7.9) completely additive ∼ (5.6) faithful ∼ (5.15) normal ∼ (5.15) positive ∼ (5.1) function coupling ∼ (7.20) normalized dimension ∼ (7.12) geometry continuous ∼ (7.7) group of modular automorphisms (10.13, 10.16, 10.17) modular ∼ associated to a left Hilbert algebra (10.1) one-parameter ∼ of unitary operators (9.16) so-continuous one-parameter ∼ of unitary operators (9.16) wo-continuous one-parameter ∼ of unitary operators (9.16) homogeneity of M ′ (8.7) inequality Schwarz ∼ (5.3) isometry partial ∼ (2.13, 9.29) isomorphism spatial ∼ (4.22, 8.1) mapping analytic ∼ Ω → B (H ) (9.24) weakly analytic ∼ Ω → H (9.17) weakly continuous ∼ Ω → H (9.17) ♯-mapping (7.12) *-operation (2.1) operator adjoint linear ∼ (2.1, 9.2) antilinear ∼ (9.35) bounded linear ∼ (2.1, 9.1) closed linear ∼ (9.1) compact linear ∼ (E.2.19) densely defined linear ∼ (9.1) linear ∼ affiliated to a von Neumann algebra (9.7, E.9.25) linear ∼ (domain, equality, extension, graph, range, restriction of a ∼) (9.1) linear ∼ summable with respect to a normal form (10.10) lower semi-bounded linear ∼ (9.4) modular ∼ associated to a left Hilbert algebra (10.1) normal linear ∼ (2.5, 9.13) positive linear ∼ (2.9, 2.12, 9.4) preclosed linear ∼ (9.1) self-adjoint linear ∼ (2.5, 9.4) symmetric linear ∼ (9.4)

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424

Lectures on von Neumann algebras

unitary linear ∼ (2.24, 4.22) upper semi-bounded linear ∼ (9.4) commuting linear ∼ (E.9.20, E.9.23, E.9.24) predual (1.11) product cross ∼ of von Neumann algebras (C.3.7) tensor ∼ of Hilbert spaces (2.33) tensor ∼ of linear operators (2.33, 9.33) tensor ∼ of von Neumann algebras (3.17) projection (2.13) abelian ∼ (E.4.11) continuous ∼ (E.4.11) cyclic ∼ (3.8) final ∼ of a partial isometry (2.13) finite ∼ (4.8) initial ∼ of a partial isometry (2.13) maximally cyclic ∼ (E.8.1) ∼ of countable type (4.13) ∼ piecewise of countable type (7.2) properly infinite ∼ (4.8) spectral ∼ of a self-adjoint operator (E.2.17, E.9.10, E.9.13) equivalent ∼ (4.1) radius spectral ∼ (2.3) rule parallelogram ∼ (4.5) scale spectral ∼ of a self-adjoint operator (2.19, E.9.10) set resolvent ∼ of a linear operator (2.3, 9.26) spectrum of a linear operator (2.3, 9.26) subalgebra left Hilbert ∼ (10.4) sum direct ∼ of von Neumann algebras (C.3.7) support central ∼ of an element in a von Neumann algebra (3.9) left ∼ of a linear operator (2.13, 9.1) right ∼ of a linear operator (2.13, 9.1) ∼ of a normal linear form (5.15) ∼ of a self-adjoint operator (2.13, 9.4) ∼ of a trace (7.13) symmetry (E.2.5) theorem comparison ∼ (4.6) coupling ∼ (7.19) density ∼ of I. Kaplansky (3.10) density ∼ of J. von Neumann (3.2) fundamental ∼ of M. Tomita (10.12) Jordan type decomposition ∼ for w-continuous forms (5.17)

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Subject index

425

Radon–Nikodym type ∼ of S. Sakai (5.21) representation ∼ of M. H. Stone (9.20) Schröder–Bernstein type ∼ of J. von Neumann (4.7) spectral ∼ (2.19, E.9.10) ∼ of H. A. Dye (C.6.2) ∼ of operational calculus for analytic functions (2.25) ∼ of operational calculus for Borel function (2.20, 9.11) ∼ of operational calculus for continuous functions (2.6) ∼ of polar decomposition for linear operators (2.14, 9.29) ∼ of polar decomposition for w-continuous forms (5.16, E.5.10) topology strong operator (so-) ∼ (1.3) s-topology (E.5.5) s*-topology (C.5.1) ultrastrong operator ∼ (E.1.4) ultraweak operator (w-) ∼ (1.3, E.1.3, E.7.9, 8.17) weak operator (wo-) ∼ (1.3) 𝜏-topology (C.5.1) trace canonical central ∼ on a finite von Neumann algebra (7.12) canonical ∼ on B(H) (E 7.6) faithful ∼ (7.13) finite ∼ (7.13) normal ∼ (7.13) semifinite ∼ (7.13) transform Cayley ∼ (E 9.8) transport by *-isomorphism (9.25) uniformity of M ′ (8.6) unit approximate ∼ (3.20) value absolute ∼ of a linear operator (2.14, 9.28) eigen ∼ (E.2.20) vector cyclic ∼ (3.8) eigen ∼ (E.2.20) separating ∼ (3.8) totalizing ∼ (3.8) trace ∼ (E.3.9) weight (10.14) faithful ∼ (10.14) normal ∼ (10.14) semifinite ∼ (10.14) ∼ associated to a left Hilbert algebra (10.16) ∼ invariant with respect to a group of *-automorphisms (10.17) commuting ∼ (10.28, C.10.5)

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NOTATION INDEX

𝐴+ , 𝐴− (9.31); 𝐴𝛼 (9.15); 𝐴𝜀 (9.19); 𝔄 (10.1), 𝔄′ (10.3), 𝔄″ (10.4); 𝔄𝐺 (E.10.3); 𝔄𝜑 (10.15).

B (H ), B (H )∗ , B (H )∼ , B (H )∗ (1.3); B (H )ℎ , B (H )+ (2.16); B (Ω) (2.20); B ([0, +∞) (9.9); B ((−∞, +∞)) (9.32). ∗ C (X ) (2.2); C (H ) (3.17); C (Ω) (2.6); 𝔠M , M ′ (7.20). D 𝑇 (9.1); D 𝜀 (9.18). 𝑒(𝐷) (E.2.17, E.9.13); 𝑒𝜆 (2.19, E.9.10); 𝑒𝜉, 𝜂 (2.19). 𝐹𝑓 (9.10); 𝐹𝜉 (9.18); 𝑓(𝐴) (9.9); 𝑓(𝑥) (2.6); 𝑓A (𝑥) (2.25); 𝔉𝜑 (10.14).

G 𝑇 (9.1). ˜ (2.32); H (5.18, 10.14). H 𝛾 𝜑 𝐽 (10.1); 𝐽 𝔄 (10.24).

K (H ) (E.2.19). 𝐿𝑎 𝜑 (5.15); 𝐿𝐴 𝑅𝐴 𝜑 (10.10); 𝐿𝜉 (10.1, 10.4, 10.8); 𝐿0𝜉 (10.8); l(𝑥) (2.13); l(𝑇) (9.1); 𝔏(𝔄) (10.1); 𝔏(𝔄)∞ (10.16). Mat𝛾 (X ) (2.32); 𝑀(𝑥), 𝑚(𝑥) (2.19); M ∗ , M ∼ , M ∗ (1.10); M 𝑒 , M ′𝑒 (3.13); M ∞ , M 0 (10.27); 𝜑

𝜑

𝔐𝜑 (10.14). n(𝑥) (2.13); n(𝑇) (9.1); 𝔑𝜑 (10.14). 𝔬M ′ (8.7).

P B (H ) (2.13); PM (3.7); 𝑝𝜉 , 𝑝𝜉′ (3.8); 𝔓, 𝔓0 , 𝔓𝑆0 , 𝔓0𝑆∗ (10.23); 𝔓𝑆 , 𝔓𝑆∗ (10.8); 𝔓𝔄 (10.24); 𝔓𝜆 (E.10.18). 𝑅𝑎 𝜑 (5.15);

𝑅𝜂0 , 𝑅𝜂

(10.2); r(𝑥) (2.13); r(𝑇) (9.1); R (X ) (2.2); R (M , M ′ ) (3.9); 𝔎(𝔄′ ) (10.4).

𝑆 (10.1); s(𝑥) (2.13); s(𝑇) (9.4); s(𝜑) (5.15); s(𝜇) (7.13); S 𝐴 (9.9); 𝔖 (10.22). |𝑇| (9.28); 𝑇𝑎 𝜑 (5.15); 𝑡𝑟 (E.7.6); T 𝑟(H ) (E.7.7); 𝔗 (10.20); 𝔗0 (10.19); 𝔗1 (E.10.13). 𝔲M ′ (8.6). 𝑉H , 𝑉H K (9.3).

426

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427

Notation index

𝑥 + , 𝑥 − (2.10); |𝑥| (2.14); 𝑥̃ (2.32); 𝑥𝑒 (3.13); 𝑥𝛼 (2.28, 2.30); 𝑥𝜑 (5.18, 10.14); (𝑥i𝑘 ) (2.32); X ′ , X ″ (3.1); ˜ (2.32); X S , [X S ] (3.1). X (3.13); X 𝑒

z(𝑥) (3.9); X

𝛾

˜ (7.20). (3.9); Z

∆ (10.1). 𝜋𝜑 (5.18, 10.14). 𝜌(𝑥) (2.3); 𝜌(𝑇) (9.26). 𝜍(E , F ) (1.2); 𝜍(𝑥), |𝜍(𝑥)| (2.3); 𝜍(𝑇) (9.26); 𝜍𝑡 (10.13); 𝜍𝑡 (10.17). 𝜑

|𝜑| (5.16); 𝜑∗ (5.1); 𝜑𝔄 (10.16); 𝜑1/2 (10.25). 𝜒𝐷 (2.19). ′ 𝜔𝜉,𝜂 , 𝜔𝜉 (1.3); 𝜔𝜉 , 𝜔𝜉′ (5.22); 𝜔𝜉,𝜂 , 𝜔𝜉,𝜂 (10.9).

𝑒 ∼ 𝑓, 𝑒 ≺ 𝑓 (4.1).

H ⊗K , 𝑥⊗𝑦 (2.33); M ⊗N (3.17); 𝑇1 ⊗𝑇2 (9.33). ⋁i∈𝐼 𝑒i , ⋀i∈𝐼 𝑒i (2.17). 𝑥 ≤ 𝑦 (2.16); 𝑒1 ≤ 𝑒2 (2.17); 𝜑 ≤ 𝜓 (5.19); 𝜁 ≤ 𝜃 (10.23). ♮ (7.11); ♯ (10.1); (10.4, C.10.1).

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