MODULAR THEORY IN OPERATOR ALGEBRAS. [2 ed.] 9781108489607, 1108489605

Table of contents :
Copyright
Contents
Preface to the second edition
Preface to the first edition
Chapter I Normal Weights
1 Characterizations of Normality
2 The Standard Representation
3 The Balanced Weight
4 The Pedersen–Takesaki Construction
5 The Converse of the Connes Theorem
6 Equality and Majorization of Weights
7 The Spatial Derivative
8 Tensor Products
Chapter II Conditional Expectations and Operator-Valued Weights
9 Conditional Expectations
10 Existence and Uniqueness of Conditional Expectations
11 Operator-Valued Weights
12 Existence and Uniqueness of Operator-Valued Weights
Chapter III Groups of Automorphisms
13 Groups of Isometries on Banach Spaces
14 Spectra and Spectral Subspaces
15 Continuous Actions on W∗-algebras
16 The Connes Invariant Γ(?)
17 Outer Automorphisms
Chapter IV Crossed Products
18 Hopf–von Neumann Algebras
19 Crossed Products
20 Comparison of cocycles
21 Abelian Groups
22 Discrete Groups
Chapter V Continuous Decompositions
23 Dominant Weights and Continuous Decompositions
24 The Flow of Weights
25 The Fundamental Homomorphism
26 The Extension of the Modular Automorphism Group
Chapter VI Discrete Decompositions
27 The Connes Invariant T(ℳ)
28 The Connes Invariant S(ℳ)
29 Factors of Type III? (0 ≤ ? < 1)
30 The Discrete Decomposition of Factors of Type III? (0 ≤ ? < 1)
Appendix
References
Notation Index
Subject Index

Citation preview

Modular Theory in Operator Algebras

The first edition of this book appeared in 1981 as a direct continuation of Lectures of von Neumann Algebras (Ş. V. Str̆atil̆a and L. Zsidó) and, until 2003, was the only comprehensive monograph on the subject. Addressing students of mathematics and physics and researchers interested in operator algebras, noncommutative geometry and free probability, this revised edition covers the fundamentals and some latest developments in the field of operator algebras. The intent is to make modular theory accessible, with complete proofs, to readers having elementary training in operator algebras The text provides detailed discussion of normal weights, conditional expectations and U. Haagerup’s operator-valued weights, groups of automorphisms and their spectral theory, duality theory for noncommutative groups, and crossed products of von Neumann algebras by actions of groups and duals of groups, which enables the extension of M. Takesaki duality for noncommutative groups of automorphisms. It also contains detailed discussion on the groupmeasure space construction of factors and information about ITPFI factors and Krieger factors. The core of the book is the continuous decomposition of A. Connes and M. Takesaki and discrete decomposition of A. Connes for type III factors. It also explores new results, such as the A. Ocneanu’s result on the actions of amenable groups on the hyperfinite factor, H. Kosaki’s extension of the V. Jones index to arbitrary factors and F. Rădulescu’s examples of non-hyperfinite factors of type III𝜆 , 𝜆 ∈ (0, 1) and of type III1 . Şerban Strătilă is Professor at the Institute of Mathematics, Romanian Academy, and at the Department of Mathematics, University of Bucharest, Romania. His current research areas include operator algebras and representation theory. He received the 1975 Simion Stoilow Prize for Mathematics from the Romanian Academy. He has published Lectures on von Neumann Algebras, 2nd edition (2019) with Cambridge University Press.

CAMBRIDGE–IISc SERIES Cambridge–IISc Series aims to publish the best research and scholarly work in different areas of science and technology with emphasis on cutting-edge research. The books aim at a wide audience including students, researchers, academicians and professionals and are being 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, 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 • Lectures on von Neumann Algebras, 2nd Edition by Serban Valentin Str̆atil̆a and László Zsidó • Biomaterials Science and Tissue Engineering: Principles and Methods by Bikramjit Basu • Knowledge Driven Development: Bridging Waterfall and Agile Methodologies by Manoj Kumar Lal • Partial Differential Equations: Classical Theory with a Modern Touch by A. K. Nandakumaran and P. S. Datti

Cambridge–IISc Series

Modular Theory in Operator Algebras Second Edition

Şerban Valentin Strătilă

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www.cambridge.org education, learning and research at the highest international levels of excellence. Information on this title: www.cambridge.org/9781108839808

www.cambridge.org © A. K. Nandakumaran and P. S. Datti 2020 Information on this title: www.cambridge.org/9781108489607

This publication is in copyright. Subject to statutory exception and to the provisions relevant 2020 collective licensing agreements, © Şerban ValentinofStrătilă no reproduction of any part may take place without the written This publication is in University copyright.Press. Subject to statutory exception permission of Cambridge

and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written Printed in India permission of Cambridge University Press. First published 2020

A catalogue record for this publication is available from the Library First edition published by Editura Academiei andBritish Abacus Press, 1981

Secondof edition by Cambridge University Press, 2020 Library Congresspublished Cataloging-in-Publication Data Names: A. K., author. | Datti, P. S., author. PrintedNandakumaran, in India Title: Partial differential equations : classical theory with a modern Atouch catalogue record for thisP.S. publication is available from the British Library / A.K. Nandakumaran, Datti. Description: Cambridge, United Kingdom ; New York, NY : Cambridge ISBN 978-1-108-48960-7 Hardback University Press, 2020. | Includes bibliographical references and index. Identifiers: LCCN 2020001143 (print) LCCN 2020001144 (ebook) ISBN Cambridge University Press has| no responsibility for the |persistence or accuracy (hardback) | ISBNparty 9781108885171 (ebook) referred to in this publication, of9781108839808 URLs for external or third internet websites Subjects: Differential that equations, Partial–Textbooks. and doesLCSH: not guarantee any content on such websites is, or will remain, Classification: LCC QA374 .N365 2020 (print) | LCC QA374 (ebook) | DDC accurate or appropriate. 515/.353–dc23 LC record available at https://lccn.loc.gov/2020001143 LC ebook record available at https://lccn.loc.gov/2020001144 ISBN 978-1-108-83980-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To the memory of my wife Sanda Strătilă, the writer Alexandra Stănescu (1944–2011)

Contents

Preface to the second edition Preface to the first edition

ix xi

Chapter I Normal Weights

1

1 2 3 4 5 6 7 8

Characterizations of Normality The Standard Representation The Balanced Weight The Pedersen–Takesaki Construction The Converse of the Connes Theorem Equality and Majorization of Weights The Spatial Derivative Tensor Products

1 10 33 49 63 67 77 93

Chapter II Conditional Expectations and Operator-Valued Weights 9 10 11 12

Conditional Expectations Existence and Uniqueness of Conditional Expectations Operator-Valued Weights Existence and Uniqueness of Operator-Valued Weights

Chapter III Groups of Automorphisms 13 14 15 16 17

Groups of Isometries on Banach Spaces Spectra and Spectral Subspaces Continuous Actions on W∗ -algebras The Connes Invariant Γ(𝜎) Outer Automorphisms

Chapter IV Crossed Products 18 19 20 21 22

99 99 111 129 142

159 159 165 181 196 208

229

Hopf–von Neumann Algebras Crossed Products Comparison of cocycles Abelian Groups Discrete Groups

229 261 288 302 313

vii

viii

Contents

Chapter V Continuous Decompositions 23 24 25 26

Dominant Weights and Continuous Decompositions The Flow of Weights The Fundamental Homomorphism The Extension of the Modular Automorphism Group

Chapter VI Discrete Decompositions 27 28 29 30

The Connes Invariant T(ℳ) The Connes Invariant S(ℳ) Factors of Type III𝜆 (0 ≤ 𝜆 < 1) The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

Appendix References Notation Index Subject Index

331 331 348 356 360

369 369 374 383 396

415 429 443 445

Preface to the Second Edition

The first edition of this book appeared in 1981 as a direct continuation of Lectures of von Neumann Algebras by Ş.V. Strătilă and L. Zsidó and, until 2003, was the only comprehensive monograph on the subject. The book Lectures on von Neumann Algebras, 2nd Edition, Cambridge University Press, 2019, will be always referred to as [L]. It is assumed that the reader is familiar with the material contained in this book, including the terminology and notation. The present book contains the continuous decomposition and the discrete decomposition for factors of type III and all the necessary results such as the extensive theory of normal weights including the U. Haagerup characterization of normality, the A. Connes theorem of Radon–Nikodym type and the Pedersen–Takesaki construction, the conditional expectations and the operator-valued weights, a detailed consideration of groups of automorphisms and their spectral theory, and the theory of crossed products. In order to include our extension of the Takesaki duality theorem to noncommutative groups of automorphisms, we considered a simultaneous generalization of groups and duals of groups, namely the Kac algebras, and their actions on von Neumann algebras, as well as the corresponding crossed products. Instead of Kac algebras we can also consider quantum groups, the arguments being exactly the same as for Kac algebras. In this second edition we added information and references for several results, which appeared after 1981, untill now, including A. Ocneanu’s theorem concerning actions of amenable groups and its extensions, H. Kosaki’s extension of the index to arbitrary factors and F. Rădulescu’s examples of non-hyperfinite factors of type III𝜆 , 𝜆 ∈ (0, 1) and of type III1 relying on D.-V. Voiculescu’s theory of free probability. For subjects such as the equivalence of injectivity and hyperfiniteness (originally due to A. Connes) we indicate references for more recent shorter proofs. This guarantees the uniqueness of the hyperfinite factor of type II∞ and of type III𝜆 factors, 𝜆 ∈ (0, 1). Another important result which could not be considered in detail, due to its length, is the proof of U. Haagerup of the A. Connes conjecture that the hyperfinite factor of type III1 is unique. For the same reason we restricted ourselves to only a few results concerning Krieger factors and ITPFI factors. However, we considered in detail the “group-measure space construction” which produced many examples of factors, among them the first example of a type III factor by Murray and von Neumann, the Pukanszky non-hyperfinite factor of type III and also the famous Powers factors which are actually the only hyperfinite factors of type III𝜆 , 𝜆 ∈ (0, 1). I am grateful to Gadadhar Misra and V. Sunder for proposing this second edition, to Gadadhar Misra for his considerable help with the Latex conversion of the book, to Ms Rajitha Reddy for her excellent typing, to Florin Rădulescu for his help for Ocneanu’s theorem and his own theorems, and to Alexandru Negrescu for his help in typing the new material for the second edition. I am also grateful to Cambridge University Press and to Ms Taranpreet Kaur for editing this second edition. Şerban Strătilă

Bucharest, January 20, 2020

Preface to the First Edition

The discovery of the modular operator and the modular automorphism group associated with a normal semifinite faithful weight has led to a powerful theory – the modular theory – which is nowadays essential to the consideration of many problems concerning operator algebras. This theory has been developed in close association with the effort to understand the structure and produce examples and refined classifications of factors. Thus, the crossed product construction, which gave rise to the first non-trivial examples of factors, has been shown to play a fundamental role in the structure theory as well, by reducing the study of the purely infinite algebras to the study of the more familiar semifinite algebras and their automorphisms. Moreover, several algebraic invariants, previously defined only in some special cases, have been introduced via the modular theory for arbitrary factors and the corresponding classification has been proved to be almost complete for approximately finite dimensional factors. The present book is a unified exposition of the technical tools of the modular theory and of its applications to the structure and classification of factors. It is based on several works recently published in periodicals or just circulated as preprints. The main sources used in writing this book are the works of W. B. Arveson, A. Connes, U. Haagerup, M. Landstad, G. K. Pedersen, M. Takesaki, and J. Tomiyama. The general treatment of crossed products follows an article by Ş. Strătilă, D. Voiculescu and L. Zsidó. Due to the wealth and variety of results recently obtained, it has not been possible to include here a detailed exposition of the classification of injective factors and their automorphisms; these topics and several others are just mentioned in the Notes sections, together with appropriate references. The reader is assumed to have a good knowledge of the general theory of von Neumann algebras, including the standard forms. Actually, the present book can be viewed as a sequel to a previous book, Ş. Strătilă and L. Zsidó – Lectures on von Neumann algebras, Editura Academiei & Abacus Press, 1979, which is often quoted here and referred to as [L]. There is also an Appendix which contains some supplementary results on positive self-adjoint operators and introduces the terminology connected with W∗ -algebras. The list of references in the present book contains only those items which have been used, quoted or consulted. A more extensive bibliography is contained in [L] (and in the Preprint Series, INCREST, Bucharest) and the new preprints are periodically recorded in C∗ –News (issued by CPT/CNRS, Marseille). I am very indebted to Zoia Ceauşescu, Alain Connes, Sanda Strătilă and Dan Voiculescu for the moral support they offered me in writing this book. I am grateful to my colleagues Constantin Apostol, Grigore Arsene, Zoia Ceauşescu, Radu Gologan, Adrian Ocneanu, Cornel Pasnicu, Mihai Pimsner, Sorin Popa and Dan Timotin for several useful discussions and a critical reading of various parts of the manuscript. During his short visit in Romania, Alain Connes kindly informed me of the most recent developments of the theory. Thanks are also due to the National Institute for Scientific and Technical Creation, for the technical assistance.

xii

Preface to the First Edition

It is a pleasure for me to acknowledge the most efficient and understanding cooperation of the Publishing House of the Romanian Academy (Editura Academiei) and Abacus Press, especially Mrs Sorana Gorjan, who edited this book, and Dr Simon Wassermann of Glasgow University, whose comments on the original translation were most helpful. Şerban Strătilă

Bucureşti, Romania, October 1979

CHAPTER I

Normal Weights

1 Characterizations of Normality In this section, we prove the Theorem of Haagerup asserting that every normal weight on a W ∗ -algebra is the pointwise least upper bound of the normal positive forms it majorizes. 1.1. Let 𝒜 be a C∗ -algebra. A weight on 𝒜 is a mapping 𝜑 ∶ 𝒜 + → [0, +∞] with the properties 𝜑(x + y) = 𝜑(x) + 𝜑( y), 𝜑(𝜆x) = 𝜆𝜑(x)

(x, y ∈ 𝒜 + , 𝜆 ∈ ℝ+ ).

The set 𝔉𝜑 = {x ∈ 𝒜 + ; 𝜑(x) < +∞} is a face of 𝒜 + , the set 𝔑𝜑 = {x ∈ 𝒜 ; 𝜑(x∗ x) < +∞} is a left ideal of 𝒜 , and the set 𝔐𝜑 = 𝔑∗𝜑 𝔑𝜑 = lin 𝔉𝜑 is a facial subalgebra of 𝒜 with 𝔐𝜑 ∩ 𝒜 + = 𝔉𝜑 ([L], 3.21), hence 𝜑 can be extended uniquely to a positive linear form, still denoted by 𝜑, on the *-algebra 𝔐𝜑 . A family ℱ of weights on 𝒜 is called sufficient if x ∈ 𝒜 and 𝜑(a∗ x∗ xa) = 0 for all 𝜑 ∈ ℱ , a ∈ 𝔑𝜑 ⇒ x = 0 and is called separating if x ∈ 𝒜 and 𝜑(x∗ x) = 0 for all 𝜑 ∈ ℱ ⇒ x = 0. In particular, the weight 𝜑 is called faithful if x ∈ 𝒜 and 𝜑(x∗ x) = 0 ⇒ x = 0.

1

2

Normal Weights

1.2. Let 𝜑 be a weight on the C∗ -algebra 𝒜 . The formula (a|b)𝜑 = 𝜑(b∗ a)

(a, b ∈ 𝔑𝜑 )

defines a prescalar product on 𝔑𝜑 with the properties: (xa|xa)𝜑 ≤ ‖x‖2 (a|a)𝜑 (x ∈ 𝒜 , a ∈ 𝔑𝜑 ), (xa|b)𝜑 = (a|x∗ b)𝜑 (x ∈ 𝒜 , a, b ∈ 𝔑𝜑 ). Let ℋ𝜑 be the Hilbert space associated with 𝔑𝜑 with the scalar product (⋅|⋅)𝜑 . It follows that there exists a *-representation 𝜋𝜑 ∶ 𝒜 → ℬ(ℋ𝜑 ), uniquely determined, such that (𝜋𝜑 (x)a𝜑 |b𝜑 )𝜑 = 𝜑(b∗ xa)

(x ∈ 𝒜 , a, b ∈ 𝔑𝜑 ),

(1)

where 𝔑𝜑 ∋ a ↦ a𝜑 ∈ ℋ𝜑 denotes the canonical mapping. The *-representation 𝜋𝜑 is called the GNS representation or the standard representation associated with 𝜑. We remark that 𝜑(x∗ ) = 𝜑(x) (x ∈ 𝔐𝜑 ), |𝜑(b a)| ≤ 𝜑(a a)𝜑(b b) ∗

2

∗

∗

(a, b ∈ 𝔑𝜑 ).

(2) (3)

1.3. Let ℳ be a W ∗ -algebra. A weight 𝜑 on ℳ is called normal if 𝜑(sup xi ) = sup 𝜑(xi ) i

i

for every norm-bounded increasing net {xi }i ⊂ ℳ + , and lower w-semicontinuous if the convex sets {x ∈ ℳ + ; 𝜑(x) ≤ 𝜆}

(𝜆 ∈ ℝ+ )

are w-closed. An important result concerning weights on W ∗ -algebras is the following characterization of normality: Theorem (U. Haagerup). Let 𝜑 be a weight on the W ∗ -algebra ℳ. The following statements are equivalent: (i) 𝜑 is normal; (ii) 𝜑 is lower w-semicontinuous; (iii) 𝜑(x) = sup{f (x); f ∈ ℳ∗+ , f ≤ 𝜑} for all x ∈ ℳ + . Later (2.10, 5.8) we shall see that 𝜑 is normal if and only if it is a sum of normal positive forms, in accordance with the definition used in ([L], 10.14). In Sections 1.4–1.7, we present some general results that will be used in the proof of the theorem; Sections 1.6–1.12 contain the main steps of the proof. 1.4 Proposition. If x, x1 , … , xn ∈ ℬ(ℋ ) and x∗ x = x∗1 x1 + … + x∗n xn , then there exist z1 , … , zn ∈ ℛ{x, x1 , … , xn } such that z∗1 z1 + … + z∗n zn = s(xx∗ ) and xk = zk x for all 1 ≤ k ≤ n. ⨁ Proof. The equations zk (x𝜉) = xk 𝜉(𝜉 ∈ ℋ ) and zk 𝜂 = 0(𝜂 ∈ ℋ xℋ ) define operators zk ∈ ℬ(ℋ ), ‖zk ‖ ≤ 1, with xk = zk x and zk (ℋ ⊖ xℋ ) = 0. Using the double commutant theorem

Characterizations of Normality

3

∑ ([L], 3.2) it is easy to check that zk ∈ ℛ{x, xk }. Also the relation k z∗k zk = s(xx∗ ) follows, since the ∑ ∑ positive operator ( k z∗k zk )1∕2 vanishes on ℋ ⊖ xℋ and x∗ ( k z∗k zk )x = x∗ x. In particular if x, y ∈ ℬ(ℋ ) and y∗ y ≤ x∗ x, there exists z ∈ ℛ{x, y} such that z∗ z ≤ s(xx∗ ) and y = zx. 1.5. For each 𝛼 > 0 we shall consider the function f𝛼 ∶ (−𝛼 −1 , +∞) → ℝ defined by f𝛼 (t) = t(1 + 𝛼t)−1 = a−1 (1 − (1 + 𝛼t)−1 ). These functions have the following properties: f𝛼 (t) ≤ min{t, 𝛼 −1 } (t ∈ (−𝛼 −1 , +∞)) 𝛼 ≤ 𝛽 ⇒ f𝛼 (t) ≥ f𝛽 (t) (t ∈ (−𝛼 −1 , +∞)) 𝛼 ≤ 𝛽 ⇒ 𝛼f𝛼 (t) ≤ 𝛽f𝛽 (t) (t ∈ (0, +∞))

(1) (2) (3)

f𝛼 ( f𝛽 (t)) = f𝛼+𝛽 (t) (t ∈ (−(𝛼 + 𝛽)−1 , +∞)) lim f𝛼 (t) = t uniformly on compact subsets of ℝ

(4) (5)

lim 𝛼f𝛼 (t) = 1 uniformly on compact subsets of ℝ+ .

(6)

𝛼→0

𝛼→∞

A continuous function f ∶ I → ℝ is called operator monotone on the interval I ⊂ ℝ if for every x, y ∈ ℬ(ℋ ), x = x∗ , y = y∗ , with Sp(x) ⊂ I, Sp( y) ⊂ I, we have x ≤ y ⇒ f (x) ≤ f ( y). For instance, it is easy to see that the functions f𝛼 are operator monotone (𝛼 ∈ ℝ+ ).

(7)

Also, we recall that (see Pedersen, 1973–1977 or Strătilă & Zsidó, 1977, 1979) the functions t → t𝛾 are operator monotone (0 < 𝛾 < 1).

(8)

On the other hand, using (A.2) we see that the functions f𝛼 are operator continuous (𝛼 ∈ ℝ+ ).

(9)

1.6. Let 𝒳 be a locally convex Hausdorff real vector space with a partial ordering defined by a convex cone 𝒳 + ⊂ 𝒳 such that 𝒳 + ∩ (−𝒳 + ) = {0} and 𝒳 = (𝒳 + − 𝒳 + ). The dual cone 𝒳+∗ = {f ∈ 𝒳 ∗ ; f (x) ≥ 0 for all x ∈ 𝒳 + } defines a partial ordering on 𝒳 ∗ . A subset ℰ of 𝒳 + is called hereditary if x ∈ ℰ , y ∈ 𝒳 +, x − y ∈ 𝒳 + ⇒ y ∈ ℰ . For ℰ ⊂ 𝒳 + and ℱ ⊂ 𝒳+∗ , we define ℰ ∧ and ℱ ∧ by ℰ ∧ = {f ∈ 𝒳+∗ ; f (x) ≤ 1 for all x ∈ ℰ }, ℱ ∧ = {x ∈ 𝒳 + ; f (x) ≤ 1 for all f ∈ ℱ }.

4

Normal Weights

Proposition. For 𝒳 as above the following statements are equivalent: (i) ℰ = (ℰ − 𝒳 + ) ∩ 𝒳 + for every closed hereditary convex subset ℰ of 𝒳 + ; (ii) ℰ = ℰ ∧∧ for every closed hereditary convex subset ℰ of 𝒳 + ; (iii) every subadditive, positively homogeneous, increasing and lower semicontinuous function 𝜑 ∶ 𝒳 + → [0, +∞] has the property 𝜑(x) = sup{f (x); f ∈ 𝒳+∗ , f ≤ 𝜑}

(x ∈ 𝒳 + ).

Proof. We shall denote by 𝒮 0 the polar of a subset 𝒮 of 𝒳 or 𝒳 ∗ . (i) ⇒ (ii). The sets ℱ = ℰ ∧ and ℱ ′ = −(ℰ − 𝒳 + )0 = {f ∈ 𝒳 ∗ ; f (x) ≤ 1 for all x ∈ ℰ − 𝒳 + } are equal. Indeed, it is clear that ℱ ⊂ ℱ ′ . Let f ∈ ℱ ′ and x ∈ 𝒳 + . Since 0 ∈ ℰ , we have f (−𝜆x) ≤ 1 for all 𝜆 ≥ 0, whence f (x) ≥ 0. Thus ℱ ′ ⊂ 𝒳+∗ and so ℱ ′ ⊂ ℱ . By the bipolar theorem it follows that (ℰ − 𝒳 + ) = (ℰ − 𝒳 + )00 = (−ℱ )0 = {x ∈ 𝒳 ; f (x) ≤ 1 for allf ∈ ℱ } and, using (i), we get ℰ = (ℰ − 𝒳 + ) ∩ 𝒳 + = {x ∈ 𝒳 + ; f (x) ≤ 1 for all f ∈ ℱ } = ℰ ∧∧ . (ii) ⇒ (iii). If 𝜑 satisfies the conditions required in (iii), then the set ℰ = {x ∈ 𝒳 + ; 𝜑(x) ≤ 1} is closed, hereditary and convex. Also, ℱ = ℰ ∧ = {f ∈ 𝒳+∗ ; f (x) ≤ 𝜑(x) for all x ∈ 𝒳 + } and, by (ii), {x ∈ 𝒳 + ; 𝜑(x) ≤ 1} = ℰ = ℱ ∧ = {x ∈ 𝒳 + ; supf∈ℱ f (x) ≤ 1}. It follows that 𝜑(x) = sup{f (x); f ∈ ℱ }, for all x ∈ 𝒳 + . (iii)⋃⇒ (i). Let ℰ ⊂ 𝒳 + be closed, hereditary and convex. Define 𝜑(x) = inf{𝜆 > 0; x ∈ 𝜆ℰ } if x ∈ 𝜆>0 𝜆ℰ and 𝜑(x) = +∞ otherwise. Then 𝜑 satisfies the hypotheses in (iii) and therefore 𝜑(x) = sup{f (x); f ∈ ℱ }(x ∈ 𝒳 + ), where ℱ = {f ∈ 𝒳+∗ ; f (x) ≤ 𝜑(x) for all x ∈ 𝒳 + }. It follows that ℰ − 𝒳 + ⊂ {x ∈ 𝒳 ; f (x) ≤ 1 for all f ∈ ℱ } and, since the latter set is closed, we get (ℰ − 𝒳 + ) ∩ 𝒳 + ⊂ {x ∈ 𝒳 + ; f (x) ≤ 1 for all f ∈ ℱ } ⊂ ℰ , hence (ℰ − 𝒳 + ) ∩ 𝒳 + = ℰ . 1.7 Proposition. Let ℳ be a W ∗ -algebra and ℰ ⊂ ℳ + a w-closed hereditary convex set. Then ℰ = (ℰ − ℳ + )w ∩ ℳ + . Proof. We shall use the properties of the functions f𝛼 from 1.5. For x ∈ ℳh let 𝛼x = sup{𝛼 > 0; −𝛼 −1 ≤ x}. Consider the set 𝒮 = {x ∈ ℳh ; f𝛼 (x) ∈ ℰ − ℳ + for all 𝛼 ∈ (0, 𝛼x )}, and let ℳ𝜆 ; . = {x ∈ ℳ; ‖x‖ ≤ 𝜆}. We first show that for every 𝜆 > 0 the set 𝒮 ∩ ℳ𝜆 is s-closed. s

s

Indeed, let x ∈ 𝒮 ∩ ℳ𝜆 . There is a net {xi }i∈I ⊂ 𝒮 such that ‖xi ‖ ≤ 𝜆 and xi → x. Then 𝛼xi ≥ 1∕𝜆, hence f𝛼 (xi ) ∈ ℰ − ℳ + for every 𝛼 ∈ (0, 1∕2𝜆) and every i ∈ I. Let 𝛼 ∈ (0, 1∕2𝜆) be fixed. There is a net {yi }i∈I ⊂ ℰ such that f𝛼 (xi ) ≤ yi

(i ∈ I).

Since f𝛼 is operator monotone, f2𝛼 (xi ) = f𝛼 ( f𝛼 (xi )) ≤ f𝛼 ( yi ) (i ∈ I).

Characterizations of Normality

5

Since f2𝛼 is operator continuous on [−𝜆, +𝜆], s

f2𝛼 (xi ) → f2𝛼 (x). Since 0 ≤ f𝛼 ( yi ) ≤ 𝛼 −1 and ℳ1 is w-compact, we may assume that there is y ∈ ℳ such that w

f𝛼 ( yi ) → y. Since 0 ≤ f𝛼 ( yi ) ≤ yi ∈ ℰ and ℰ is hereditary, f𝛼 ( yi ) ∈ ℰ and, since ℰ is w-closed, it follows that y ∈ ℰ . Then y − f2𝛼 (x) = w- lim( f𝛼 ( yi ) − f2𝛼 (xi )) ≥ 0, i

hence f2𝛼 (x) ∈ ℰ − ℳ + . We have thus proved that f𝛼 (x) ∈ ℰ − ℳ + for every 𝛼 ∈ (0, 1∕𝜆). Consider now 𝛼 ∈ [1∕𝜆, 𝛼x ) and 𝛽 ∈ (0, 1∕𝜆). Then f𝛼 (x) ≤ f𝛽 (x), hence f𝛼 (x) ∈ (ℰ − ℳ + ) − ℳ + = ℰ − ℳ + . We conclude that x ∈ 𝒮 ∩ ℳ𝜆 . We now show that 𝒮 is convex. Indeed, it is sufficient to show that each 𝒮 ∩ ℳ𝜆 is convex, and this will follow from the equality 𝒮 ∩ ℳ𝜆 = ((ℰ − ℳ + ) ∩ ℳ𝜇 )s ∩ ℳ𝜆

for 𝜇 > 𝜆.

If x ∈ 𝒮 ∩ ℳ𝜆 , then f𝛼 (x) ∈ ℰ − ℳ + for 𝛼 ∈ (0, 𝛼x ) and f𝛼 (x) ∈ ℳ𝜇 for small 𝛼 > 0, hence x = s- lim f𝛼 (x) ∈ ((ℰ − ℳ + ) ∩ ℳ𝜇 )s ∩ ℳ𝜆 . 𝛼→0

Conversely, since ℰ is hereditary and f𝛼 (x) ≤ x for all 𝛼 ∈ (0, 𝛼x ), we have ℰ − ℳ + ⊂ 𝒮 , hence s (ℰ − ℳ + ) ∩ ℳ𝜇 ⊂ 𝒮 ∩ ℳ𝜇 . Using the first part of the proof we get ((ℰ − ℳ + ) ∩ ℳ𝜇 ) ⊂ 𝒮 ∩ ℳ𝜇 , and the desired inclusion follows. ̆ Using the Krein-Smulian theorem ([L], C.1.1; Dunford & Schwartz, 1958, 1963, V.5.7), from the earlier it follows that 𝒮 is w-closed. We have seen that ℰ − ℳ + ⊂ 𝒮 . Since x = w− lim𝛼→0 f𝛼 (x), w w we obtain 𝒮 ⊂ (ℰ − ℳ + ) . Consequently, 𝒮 = (ℰ − ℳ + ) . w Finally, let x ∈ (ℰ − ℳ + ) ∩ ℳ + = 𝒮 ∩ ℳ + . For every 𝛼 > 0 we have f𝛼 (x) ∈ (ℰ − ℳ + ) ∩ ℳ + , hence f𝛼 (x) ∈ ℰ , as ℰ is hereditary. It follows that x = w- lim𝛼→0 f𝛼 (x) ∈ ℰ . From Propositions 1.6 and 1.7, we obtain the equivalence (ii) ⇒ (iii) in Theorem 1.3, as the implication (iii) ⇒ (ii) is obvious. In Sections 1.8–1.12, we assume that 𝜑 is a fixed normal weight on the W ∗ -algebra ℳ. 1.8 Lemma. There exists a linear mapping Φ ∶ 𝔐𝜑 → 𝜋𝜑 (ℳ)′∗ , uniquely determined, such that Φ(b∗ a)(T ′ ) = (T ′ a𝜑 |b𝜑 )𝜑

(T ′ ∈ 𝜋𝜑 (ℳ)′ , a, b ∈ 𝔑𝜑 ).

(1)

6

Normal Weights

Moreover, for every x ∈ 𝔐𝜑 ∩ ℳh we have ‖Φ(x)‖ = inf{𝜑( y) + 𝜑(z); y, z ∈ 𝔐𝜑 ∩ ℳ + , x = y − z}.

(2)

Proof. The uniqueness of Φ follows from the relation 𝔐𝜑 = 𝔑∗𝜑 𝔑𝜑 . If a, b, c ∈ 𝔑𝜑 , c∗ = c and c∗ c = a∗ a + b∗ b, then, by Proposition 1.4, there exist x, y ∈ ℳ such that a = xc, b = yc and x∗ x + y∗ y = s(cc∗ ) = s(c), and for every T ′ ∈ 𝜋𝜑 (ℳ)′ we have (T ′ c𝜑 |c𝜑 )𝜑 = (T ′ 𝜋𝜑 (x∗ x + y∗ y)c𝜑 |c𝜑 )𝜑 = (T ′ 𝜋𝜑 (x)c𝜑 |𝜋𝜑 (x)c𝜑 )𝜑 + (T ′ 𝜋𝜑 ( y)c𝜑 |𝜋𝜑 ( y)c𝜑 )𝜑 = (T ′ a𝜑 |a𝜑 )𝜑 + (T ′ b𝜑 |b𝜑 )𝜑 . It follows that the mapping Φ0 ∶ 𝔐𝜑 ∩ ℳ + ∋ a∗ a ↦ 𝜔a𝜑 |𝜋𝜑 (ℳ)′ ∈ 𝜋𝜑 (ℳ)′∗ . is well defined and additive. Clearly, Φ0 is positively homogeneous. Since 𝔐𝜑 = lin(𝔐𝜑 ∩ℳ + ), Φ0 has a unique linear extension Φ to 𝔐𝜑 and (1) follows using the polarization relation ([L], p. 75). The function 𝜌 defined on 𝔐𝜑 ∩ ℳh by the right-hand side of (2) is a semi-norm on 𝔐𝜑 ∩ ℳh . If x ∈ 𝔐𝜑 ∩ ℳ + , then ‖Φ(x)‖ = Φ(x)(1) = ((x1∕2 )𝜑 |(x1∕2 )𝜑 )𝜑 = 𝜑(x) = 𝜌(x). Consequently, for x = y − z, with y, z ∈ 𝔐𝜑 ∩ ℳ + , we have ‖Φ(x)‖ ≤ ‖Φ( y)‖ + ‖Φ(z)‖ = 𝜑( y) + 𝜑(z). Hence ‖Φ(x)‖ ≤ 𝜌(x) for all x ∈ 𝔐𝜑 ∩ ℳh . Let x0 ∈ 𝔐𝜑 ∩ ℳh . By the Hahn–Banach theorem, there exists a real linear form f on 𝔐𝜑 ∩ ℳh such that f (x0 ) = 𝜌(x0 ) and |f (x)| ≤ 𝜌(x) for every x ∈ 𝔐𝜑 ∩ ℳh . Then f can be extended to a complex linear form, still denoted by f, on 𝔐𝜑 . Since −𝜑(x) ≤ f (x) ≤ 𝜑(x) for any x ∈ 𝔐𝜑 ∩ ℳ + , we may consider 𝜑 + f and 𝜑 − f as weights on ℳ. Consequently, using the Schwarz inequality 1.2.(3), for a, b ∈ 𝔑𝜑 we obtain f (b∗ a) ≤ 2−1 [|(𝜑 + f )(b∗ a)| + |(𝜑 − f )(b∗ a)|] ≤ 2−1 [(𝜑 + f )(a∗ a)1∕2 (𝜑 + f )(b∗ b)1∕2 + (𝜑 − f )(a∗ a)1∕2 (𝜑 − f )(b∗ b)1∕2 ] ≤ 2−1 [(𝜑 + f )(a∗ a) + (𝜑 − f )(a∗ a)]1∕2 [(𝜑 + f )(b∗ b) + (𝜑 − f )(b∗ b)]1∕2 = 𝜑(a∗ a)1∕2 𝜑(b∗ b)1∕2 = ‖a𝜑 ‖𝜑 ‖b𝜑 ‖𝜑 . Thus, there exists an operator T ′ ∈ ℬ(ℋ𝜑 ), ‖T ′ ‖ ≤ 1, such that f (b∗ a) = (T ′ a𝜑 ∕b𝜑 )𝜑 for all a, b ∈ 𝔑𝜑 . Moreover T ′ ∈ 𝜋𝜑 (ℳ)′ , since for every x ∈ ℳ and every a, b ∈ 𝔑𝜑 we have (T ′ 𝜋𝜑 (x)a𝜑 |b𝜑 )𝜑 = f (b∗ xa) = (𝜋𝜑 (x)T ′ a𝜑 |b𝜑 )𝜑 . It follows that 𝜌(x0 ) = |f (x0 )| = |Φ(x0 )(T ′ )| ≤ ‖Φ(x0 )‖‖T ′ ‖ ≤ ‖Φ(x0 )‖. 1.9 Lemma. Let {xn } be a norm-bounded sequence in 𝔐𝜑 ∩ ℳ + such that the sequence {Φ(xn )} is norm-convergent in 𝜋𝜑 (ℳ)′∗ . Then s

xn → x ∈ ℳ ⇒ x ∈ 𝔐𝜑 ∩ ℳ + , s

xn → 0 ⇒ ‖Φ(xn )‖ → 0.

(1) (2)

Characterizations of Normality

7

Proof. Let 𝜀 > 0 and 𝜓 = limn Φ(xn ) ∈ 𝜋𝜑 (ℳ)′∗ . Without loss of generality we may assume that ‖Φ(xn ) − 𝜓‖ < 𝜀∕2n , so that ‖Φ(xn+1 − xn )‖ < 𝜀∕2n−1 for all n ∈ ℕ. By Lemma 1.8, there exist sequences {yn } and {zn } in 𝔐𝜑 ∩ ℳ + such that xn+1 − xn = yn − zn and 𝜑( yn ) + 𝜑(zn ) < 𝜀∕2n−1

(n ∈ ℕ).

We shall again use the functions f𝛼 from Section 1.5. s ∑n (1) Since xn+1 ≤ x1 + k=1 yk and xn+1 → x, we obtain ( f𝛼 (x) = s- lim f𝛼 (xn+1 ) ≤ sup f𝛼 n

x1 +

n

n ∑

) yk

k=1

and then, using the normality of 𝜑, ( 𝜑( f𝛼 (x)) ≤ supn 𝜑

( f𝛼

x1 +

n ∑

))

( ≤ supn 𝜑 x1 +

yk

k=1

≤ 𝜑(x1 ) +

∞ ∑

n ∑

) yk

k=1

𝜑( yk ) ≤ 𝜑(x1 ) +

∞ ∑

k=1

𝜀∕2k−1 = 𝜑(x1 ) + 2𝜀.

k=1

Since f𝛼 (x) ↑ x, again using the normality of 𝜑 we get 𝜑(x) = sup 𝜑( f𝛼 (x)) ≤ 𝜑(x1 ) + 2𝜀 < +∞, 𝛼>0

hence x ∈ 𝔐𝜑 ∩ ℳ + . ∑n (2) Since − supn ‖xn ‖ ≤ x1 − xn+1 ≤ k=1 zk , for 𝛼 > (supn ‖xn ‖)−1 we obtain ( f𝛼 (x1 − xn+1 ) ≤ sup fn n

n ∑

) .

zk

k=1

s

Since x1 − xn+1 → x1 , it follows that ( f𝛼 (x1 ) = s- lim f𝛼 (x1 − xn+1 ) ≤ sup f𝛼 n

n

n ∑

) zk

.

k=1

Using the normality of 𝜑 we infer that ( 𝜑(x1 ) = sup 𝜑( f𝛼 (x1 )) ≤ sup sup 𝜑 𝛼>0

𝛼>0

n

( f𝛼

n ∑

)) zk

k=1

( n ) ∞ ∑ ∑ ≤ sup 𝜑 zk ≤ 𝜀∕2k−1 = 2𝜀. n

k=1

k=1

Consequently, ‖𝜓‖ ≤ ‖𝜓 − Φ(x1 )‖ + ‖Φ(x1 )‖ ≤ 𝜀∕2 + 2𝜀 = 3𝜀∕2. We conclude that 𝜓 = 0.

8

Normal Weights

1.10. Let 𝒢𝜑 = {(x, x𝜑 ); x ∈ 𝔑𝜑 } ⊂ ℳ ×ℋ𝜑 . Since every Hilbert space is a reflexive Banach space, ℳ × ℋ𝜑 is the dual of the Banach space ℳ∗ × ℋ𝜑 . For 𝜆, 𝜇 > 0, let ℳ𝜆 = {x ∈ ℳ; ‖x‖ ≤ 𝜆} and (ℋ𝜑 )𝜇 = {𝜉 ∈ ℋ𝜑 ; ‖𝜉‖ ≤ 𝜇}. Lemma. If ℳ is countably decomposable, then 𝒢𝜑 ∩ (ℳ𝜆 × (ℋ𝜑 )𝜇 ) is 𝜎(ℳ × ℋ𝜑 , ℳ∗ × ℋ𝜑 )compact, for ever 𝜇 > 0. Proof. Since 𝒢𝜑 ∩(ℳ𝜆 ×(ℋ𝜑 )𝜇 ) is convex and bounded, it is sufficient to prove that it is closed with respect to the product topology 𝜏 on ℳ × (ℋ𝜑 ) of the s∗ -topology on ℳ and the norm-topology on ℋ𝜑 . Since ℳ is countably decomposable, ℳ𝜆 is s∗ -metrizable ([L], E.5.7; Strătilă & Zsidó, 1977, 1979, 8.12). If (x, 𝜉) ∈ ℳ × ℋ𝜑 is 𝜏-adherent to 𝒢𝜑 ∩ (ℳ𝜆 × (ℋ𝜑 )𝜇 ), then there exists a sequence {xn } in ℳ𝜆 s∗

s∗

such that xn → x, ‖(xn )𝜑 ‖𝜑 ≤ 𝜇 and ‖(xn )𝜑 − 𝜉‖ → 0. Then x∗n xn → x∗ x and Φ(x∗n xn ) = 𝜔(xn )𝜑 is norm s

convergent to 𝜔𝜉 , whence x ∈ 𝔑𝜑 by Lemma 1.8.(1). On the other hand, (xn − x)∗ (xn − x) → 0 and Φ((xn − x)∗ (xn − x)) = 𝜔(xn )𝜑 −x𝜑 → 𝜔𝜉−x𝜑 so that 𝜔𝜉−x𝜑 = 0 by Lemma 1.8.(2). Thus 𝜉 = x𝜑 and (x, 𝜉) ∈ 𝒢𝜑 . 1.11. If ℳ is not countably decomposable, we consider the set 𝒫0 of all countably decomposable projections of ℳ and put ⋃ ℳ0 = pℳp. p∈𝒫0

It is easy to check that ℳ0 is a self-adjoint ideal in ℳ. Lemma. Let ℰ be a hereditary convex subset of ℳ0 ∩ ℳ + . Then ℰ is w-closed in ℳ0 if and only if ℰ ∩ pℳp is w-closed for every p ∈ 𝒫0 . Proof. Assume that ℰ ∩ pℳp is w-closed for every p ∈ 𝒫0 . The set ℱ = {x ∈ ℳ; x∗ x ∈ ℰ } is convex and aℱ ⊂ ℱ for all a ∈ ℳ1 . ̆ We first show that pℱ , or equivalently ℱ ∗ p, is w-closed for any p ∈ 𝒫0 . Using the Krein-Smulian theorem and the fact that any s-closed convex set is also w-closed, it is sufficient to show that ℱ ∗ p ∩ ℳ𝜆 is s-closed for every 𝜆 > 0. Let x ∈ ℳ be such that x∗ is s-adherent to ℱ ∗ p ∩ ℳ𝜆 . Since s

ℳp ∩ ℳ𝜆 is s-metrizable, there exists a sequence {xn } in pℱ , ‖xn ‖ ≤ 𝜆, with x∗n → x∗ . There exists a projection q ∈ 𝒫0 such that xn ∈ qℳq for all n ∈ ℕ. Thus xn ∈ ℱ ∩ qℳq = {y ∈ qℳq; y∗ y ∈ ℰ ∩ qℳq} (n ∈ ℕ). By assumption, ℰ ∩ qℳq is w-closed, hence ℱ ∩ qℳq is s-closed. It follows that ℱ ∩ qℳq is w-closed and, consequently, x ∈ ℱ ∩ qℳq. Since px = x and ‖x‖ ≤ 𝜆, we get x∗ ∈ ℱ ∗ p ∩ ℳ𝜆 . Hence pℱ is w-closed.

Characterizations of Normality

9 s

Let x ∈ ℳ0 be w-adherent to ℰ . There exists a net {xi }i∈I ⊂ ℰ such that xi → x. Then 1∕2

s

p = l(x) ∈ 𝒫0 and pxi → px1∕2 = x1∕2 . By the earlier paragraph, we know that pℱ is s-closed, hence x1∕2 ∈ pℱ ⊂ ℱ , that is, x ∈ ℰ . Hence ℰ is w-closed in ℳ0 . The converse is obvious. 1.12. Proof of Theorem 1.3. As we have already seen (1.7), (ii) ⇔ (iii). The implication (ii) ⇒ (i) is obvious. To show that (i) ⇒ (ii), we have to prove that the set ℰ = {x ∈ ℳ + ; 𝜑(x) ≤ 1} is w-closed. Clearly, ℰ is hereditary and convex. Assume first that ℳ is countably decomposable. As in the last part of the proof of Lemma 1.11, it is sufficient to show that the set ℱ = {x ∈ ℳ; 𝜑(x∗ x) ≤ 1} is w-closed. Since ℱ ∩ ℳ𝜆 is the image of 𝒢𝜑 ∩ (ℳ𝜆 × (ℋ𝜑 )1 ) by the canonical projection mapping (x, 𝜉) ↦ x, from Lemma 1.10 it follows that ℱ ∩ ℳ𝜆 is w-compact for every 𝜆 > 0. Since ℱ is convex, we infer that ℱ is w-closed. Consider now the general case. By the earlier argument and by Lemma 1.11, it follows that ℰ ∩ℳ0 s

is w-closed in ℳ0 . Let x ∈ ℳ + be w-adherent to ℰ . There exists a net {xi }i∈I ⊂ ℰ such that x1 → x. Also, there exists an increasing net {pk }k∈K ⊂ 𝒫0 with pk ↑ 1. Since ℳ0 is a two-sided ideal in ℳ, for every k ∈ K we have 1∕2

1∕2 w

ℰ ∩ ℳ0 ∋ xi pk xi

→i∈I x1∕2 pk x1∕2 ∈ ℳ0 ,

hence x1∕2 pk x1∕2 ∈ ℰ ∩ ℳ0 . Since x1∕2 pk x1∕2 ↑ x, using the normality of 𝜑 we infer that 𝜑(x) = supk∈K 𝜑(x1∕2 pk x1∕2 ) ≤ 1, that is, x ∈ ℰ . 1.13. We recall that a positive form 𝜑 on the W ∗ -algebra ℳ is normal if and only if it is completely additive on projections ([L], 5.6, 5.11). This statement cannot be extended to weights, as the following example shows. Let 𝓁 ∞ (ℕ) be the W ∗ -algebra of all bounded complex sequences. The weight 𝜑 defined on 𝓁 ∞ (ℕ) ∑ by 𝜑({an }) = n an if the set {n ∈ ℕ; an ≠ 0} is finite, and 𝜑({an }) = +∞ otherwise, is completely additive on projections, but is not normal. 1.14 Proposition. Let 𝜑 be a normal weight on the W ∗ -algebra ℳ and a, b ∈ 𝔑𝜑 . Then the mapping 𝜑(b∗ ⋅ a) ∶ ℳ ∋ x ↦ 𝜑(b∗ xa) ∈ ℂ is a w-continuous linear form on ℳ. Proof. Since a, b ∈ 𝔑𝜑 , for any x ∈ ℳ we have b∗ xa ∈ ℛ𝜑∗ 𝔑𝜑 = 𝔐𝜑 , hence 𝜑(b∗ ⋅ a) is well defined. If xi ↑ x in ℳ + , then a∗ xi a ↑ a∗ xa in ℳ + , and hence 𝜑(a∗ xi a) ↑ 𝜑(a∗ xa), since 𝜑 is normal. It follows that 𝜑(a∗ ⋅ a) is w-continuous ([L], 5.11) and the general case is obtained using a polarization relation ([L], 3.21). 1.15. Notes. The main result (Thm. 1.3) of this section is due to Haagerup (1975a). For our exposition we have used Haagerup (1975a) and [L].

10

Normal Weights

2 The Standard Representation In this section, we prove that every normal semifinite weight is the supremum of an upward directed family of normal positive forms; also, we review and complete the results in ([L], Chapter 10) concerning the associated standard representation. 2.1. Let 𝜑 be a normal weight on the W ∗ -algebra ℳ. Using ([L], 2.22) and the normality of 𝜑 it is easy to see that x ∈ ℳ + , 𝜑(x) = 0 ⇒ 𝜑(s(x)) = 0.

(1)

If e, f ∈ ℳ are projections and 𝜑(e) = 𝜑( f ) = 0, then 𝜑(e ∨ f ) = 𝜑(s(e + f )) = 0. Thus the family ℰ = {e ∈ Proj(ℳ); 𝜑(e) = 0} is upward directed. Let e0 = sup ℰ . By the normality of 𝜑, it follows that 𝜑(e0 ) = 0, so that e0 is the greatest projection in ℳ annihilated by 𝜑. The projection s(𝜑) = 1 − e0 is called the support of 𝜑. Using (1), we obtain 𝜑(x∗ x) = 0 ⇔ xs(𝜑) = 0

(x ∈ ℳ).

(2)

In particular, 𝜑 is faithful (1.1) if and only if s(𝜑) = 1. Also 𝜑(x) = 𝜑(s(𝜑)xs(𝜑)) (x ∈ ℳ + ). w

(3) w

On the other hand, the w-closure 𝔑𝜑 of 𝔑𝜑 is a w-closed left ideal of ℳ, hence ℛ 𝜑 = ℳe for w

some projection e ∈ ℳ and 𝔐𝜑 = eℳe ([L], 3.20, 3.21). The weight 𝜑 is called semifinite if e = 1, that is, if 𝔑𝜑 , or equivalently, 𝔐𝜑 , is w-dense in ℳ. In this case, there exists an increasing net {ui }i∈I in 𝔉𝜑 = 𝔐𝜑 ∩ ℳ + such that ui ↑ 1 ([L], 3.20, 3.21). We abbreviate the words “normal semifinite faithful” to n.s.f. Recall that on every W ∗ -algebra there exists an n.s.f. weight, while the countably decomposable W ∗ -algebras are characterized by the existence of a normal faithful positive form ([L], 10.14, E.5.6). 2.2 Theorem. Let 𝜑 be a normal weight on the W ∗ -algebra ℳ. Then the associated GNS representation 𝜋𝜑 ∶ ℳ → ℬ(ℋ𝜑 ) is normal and nondegenerate. If 𝜑 is semifinite, then ((𝔐𝜑 )n )𝜑 is dense in ℋ𝜑

(n ∈ ℕ).

(1)

If 𝜑 is an n.s.f. weight, then 𝜋𝜑 is a *-isomorphism of ℳ onto the von Neumann algebra 𝜋𝜑 (ℳ) ⊂ ℬ(ℋ𝜑 ). Proof. Clearly, 𝜋𝜑 (1) = 1, hence 𝜋𝜑 is nondegenerate. To show that 𝜋𝜑 is normal, that is, w-continuous, we have to check that 𝜔 ◦ 𝜋𝜑 ∈ ℳ∗ for every 𝜔 ∈ ℬ(ℋ𝜑 )∗ . Since the vector forms are total in ℬ(ℋ𝜑 )∗ ([L], 1.3) and 𝔑𝜑 is dense in ℋ𝜑 , it is sufficient to do this only for 𝜔 = 𝜔a𝜑 ,b𝜑 with a, b ∈ 𝔑𝜑 . In this case, we have 𝜔a𝜑 ,b𝜑 ◦ 𝜋𝜑 = 𝜑(b∗ ⋅ a) ∈ ℳ∗ , by Proposition 1.14. Since 𝜋𝜑 is normal and nondegenerate, 𝜋𝜑 (ℳ) ⊂ ℬ(ℋ𝜑 ) is a von Neumann algebra ([L], 3.12). If 𝜑 is semifinite, then there exists an increasing net {ui }i∈I in 𝔉𝜑 = 𝔐𝜑 ∩ ℳ + with ui ↑ 1. For a ∈ 𝔑𝜑 , we have ‖a𝜑 − (ui a)𝜑 ‖2𝜑 = 𝜑((a − ui a)∗ (a − ui a)) ≤ 2[𝜑(a∗ a) − 𝜑(a∗ ui a)] → 0.

(2)

The Standard Representation

11

Since ui ∈ 𝔉𝜑 ⊂ 𝔑∗𝜑 and a ∈ 𝔑𝜑 , we have ui a ∈ 𝔑∗𝜑 𝔑𝜑 = 𝔐𝜑 and from (2) it follows that (𝔐𝜑 )𝜑 is dense in 𝔑𝜑 , hence also in ℋ𝜑 . Statement (1) follows now using (2) repeatedly. Assume that 𝜑 is an n.s.f. weight. If x ∈ ℳ and 𝜋𝜑 (x) = 0, then 𝜑((xa)∗ (xa)) = ‖𝜋𝜑 (x)a𝜑 ‖2𝜑 = 0 for all a ∈ 𝔑𝜑 . Since 𝜑 is faithful it follows that x𝔑𝜑 = 0 and hence x = 0, as 𝜑 is semifinite. Consequently, 𝜋𝜑 is a *-isomorphism. In the next three sections, we study the majorization relation between weights in terms of the associated GNS representation. 2.3 Proposition. Let 𝜑, 𝜓 be weights on the C∗ -algebra 𝒜 such that 𝜓 ≤ 𝜑, that is, 𝜓(x) ≤ 𝜑(x) for all x ∈ 𝒜 + . There exists a unique operator T ′ ∈ 𝜋𝜑 (𝒜 )′ , 0 ≤ T ′ ≤ 1, such that 𝜓(b∗ a) = (T ′ a𝜑 |T ′ b𝜑 )𝜑

(a, b ∈ 𝔑𝜑 ).

(1)

Proof. Since 𝜓 ≤ 𝜑 we have 𝔑𝜑 ⊂ 𝔑𝜓 and, for every a ∈ 𝔑𝜑 , ‖a𝜓 ‖2𝜓 = 𝜓(a∗ a) ≤ 𝜑(a∗ a) = ‖a𝜑 ‖2𝜑 . It follows that there exists a unique linear operator S′ ∶ ℋ𝜑 → ℋ𝜓 , ‖S′ ‖ = 1, such that S′ a𝜑 = a𝜓 for all a ∈ 𝔑𝜑 . Then T ′ = (S′∗ S′ )1∕2 ∈ ℬ(ℋ𝜑 ), 0 ≤ T ′ ≤ 1. For every a, b ∈ 𝔑𝜑 and every x ∈ 𝒜 we have 𝜓(b∗ a) = (a𝜓 |b𝜓 )𝜓 = (S′ a𝜑 |S′ b𝜑 )𝜓 = (T ′2 a𝜑 |b𝜑 )𝜑 = (T ′ a𝜑 |T ′ b𝜑 )𝜑 (S′∗ S′ 𝜋𝜑 (x)a𝜑 |b𝜑 )𝜑 = 𝜓(b∗ xa) = 𝜓((x∗ b)∗ a) = (𝜋𝜑 (x)S′∗ S′ a𝜑 |b𝜑 )𝜑 , hence T ′2 = S′∗ S′ ∈ 𝜋𝜑 (𝒜 )′ . Since 𝜋𝜑 (𝒜 )′ is a C∗ -algebra, we infer that T ′ = (T ′2 )1∕2 ∈ 𝜋𝜑 (𝒜 )′ . From (1) it follows that the numbers (T ′2 a𝜑 |b𝜑 )𝜑 (a, b ∈ 𝔑𝜑 ) are uniquely determined by 𝜓 and 𝜑, and this implies the uniqueness of T ′ . If 𝜑 and 𝜓 are finite and 𝒜 is unital, then from (1) it follows that 𝜓(x) = (𝜋𝜑 (x)T ′ 1𝜑 |T ′ 1𝜑 )𝜑

(x ∈ 𝒜 ).

(2)

2.4 Corollary. Let 𝜑 be a weight on the C∗ -algebra 𝒜 and denote by 𝒯𝜑′ the set of all T ′ ∈ 𝜋𝜑 (𝒜 )′ such that there exists some 𝜆T ′ > 0 with the property that ‖T ′ a𝜑 ‖𝜑 ≤ 𝜆T ′ ‖a‖ for all a ∈ 𝔑𝜑 . Then 𝒯𝜑′ is a left ideal of the W ∗ -algebra 𝜋𝜑 (𝒜 )′ and: (1) for every positive form f on 𝒜 with f ≤ 𝜑 there exist a unique T ′ ∈ 𝒯𝜑′ , 0 ≤ T ′ ≤ 1, and a unique 𝜂 ∈ 𝜋𝜑 (𝔑∗𝜑 )ℋ𝜑 such that f (b∗ a) = (T ′ a𝜑 |T ′ b𝜑 )𝜑 (a, b ∈ 𝔑𝜑 ); f (x) = (𝜔𝜂 ◦ 𝜋𝜑 )(x) (x ∈ 𝔐𝜑 ). (2) for every T ′ ∈ 𝒯𝜑′ , 0 ≤ T ′ ≤ 1, there exist a positive form f ≤ 𝜑 on 𝒜 and a unique 𝜂 ∈ 𝜋𝜑 (𝔑∗𝜑 )ℋ𝜑 such that f (b∗ a) = (T ′ a𝜑 |T ′ b𝜑 )𝜑 (a, b ∈ 𝔑𝜑 ); T ′ a𝜑 = 𝜋𝜑 (a)𝜂 (a ∈ 𝔑𝜑 ).

12

Normal Weights

Proof. It is clear that 𝒯𝜑′ is a left ideal of 𝜋𝜑 (𝒜 )′ . Also, if f ≤ 𝜑 is a positive form on 𝒜 , we infer from 2.3.(1) that ‖T ′ a𝜑 ‖𝜑 ≤ ‖f‖1∕2 ‖a‖(a ∈ 𝔑𝜑 ), hence T ′ ∈ 𝒯𝜑′ . Let {ui }i∈I be a right approximate unit for the left ideal 𝔑𝜑 of 𝒜 ([L], 3.20). For T ′ ∈ 𝒯𝜑′ and a ∈ 𝔑𝜑 it follows that 𝜋𝜑 (a)T ′ (ui )𝜑 = T ′ (aui )𝜑 → T ′ a𝜑 . Thus, if ak ∈ 𝔑𝜑 , 𝜉k ∈ ℋ𝜑 (1 ≤ k ≤ n), and 𝜁 =

∑n k=1

𝜋𝜑 (a∗k )𝜉k ∈ 𝜋k (𝔑∗𝜑 )ℋ𝜑 , then

n |∑ | | | | (𝜉k |T ′ (ak )𝜑 )𝜑 | = lim |(𝜁|T ′ (ui )𝜑 )𝜑 | ≤ 𝜆T ′ ‖𝜁‖𝜑 . | | i | k=1 | ∑n It follows that the mapping 𝜁 ↦ k=1 (𝜉k |T ′ (ak )𝜑 )𝜑 defines a bounded linear form on 𝜋𝜑 (𝔑∗𝜑 )ℋ𝜑

and, consequently, there exists a unique vector 𝜂 ∈ 𝜋𝜑 (𝔑∗𝜑 )ℋ𝜑 such that (𝜉|T ′ a𝜑 )𝜑 = (𝜉|𝜋𝜑 (a)𝜂)𝜑

(a ∈ 𝔑𝜑 , 𝜉 ∈ ℋ𝜑 ),

that is, T ′ a𝜑 = 𝜋𝜑 (a)𝜂 for all a ∈ 𝔑𝜑 . In particular, f = 𝜔𝜂 ◦ 𝜋𝜑 is a positive form on 𝒜 and f (b∗ a) = (T ′ a𝜑 |T ′ b𝜑 )𝜑 for all a, b ∈ 𝔑𝜑 . 2.5 Corollary. Let 𝜑 be a normal weight on the W ∗ -algebra ℳ and f0 a positive form on ℳ. If f0 ≤ 𝜑, then there exists a normal positive form f on ℳ such that f ≤ 𝜑 and f|𝔐𝜑 = f0 |𝔐𝜑 . Proof. By Corollary 2.4.(1), there exists a vector 𝜂 ∈ ℋ𝜑 such that f0 (x) = (𝜔𝜂 ◦ 𝜋𝜑 )(x) for every x ∈ 𝔐𝜑 . Since 𝜑 is normal, 𝜋𝜑 is normal (2.2), so we can take f = 𝜔𝜂 ◦ 𝜋𝜑 . 2.6. By Haagerup’s theorem (1.3), every normal weight 𝜑 on the W ∗ -algebra ℳ is the pointwise supremum of the family 𝔉𝜑 = {f ∈ ℳ∗+ ; f ≤ 𝜑}; thus 𝜑 is also the pointwise supremum of the family {f ∈ ℳ∗+ ; there exists 𝜀 > 0 such that (1 + 𝜀)f ≤ 𝜑}. The next result shows, in particular, that every normal semifinite weight on a W ∗ -algebra is the pointwise supremum of an upward directed family of normal positive forms. Theorem (F. Combes). Let 𝜑 be a normal semifinite weight on the W ∗ -algebra ℳ. Then the family {f ∈ ℳ∗+ ; there exists 𝜀 > 0 such that (1 + 𝜀)f ≤ 𝜑} is upward directed. We first prove two general results of independent interest. 2.7 Proposition. Let 𝔑 be a left ideal of the W ∗ -algebra ℳ. Then ℱ = (𝔑∗ 𝔑) ∩ ℳ + is a face of ℳ + , 𝔑 = {x ∈ ℳ; x∗ x ∈ ℱ } and 𝔑∗ 𝔑 = lin ℱ . In particular, 𝔑∗ 𝔑 is a facial subalgebra of ℳ. Proof. Clearly, 𝔑 ⊂ {xℳ; x∗ x ∈ ℱ } and, by the polarization relation ([L], 3.21), 𝔑∗ 𝔑 = lin ℱ . Let x ∈ ℳ be such that x∗ x ≤ b ∈ ℱ . Since b is self-adjoint, using again the polarization relation we can find xk , yk ∈ 𝔑, (1 ≤ k ≤ n) such that x∗ x ≤ b =

n ∑ k=1

x∗k xk −

n ∑ k=1

y∗k yk ≤

n ∑ k=1

x∗k xk = a

The Standard Representation

13

By Proposition 1.4, there exist z, zk ∈ ℳ(1 ≤ k ≤ n) such that x = za1∕2 , xk = zk a1∕2 and s(a). It follows that x = za

1∕2

=z

( n ∑

) z∗k zk

a1∕2 =

k=1

n ∑

∑n

∗ k=1 zk zk

=

zz∗k xk ∈ 𝔑.

k=1

Hence ℱ is a face of ℳ + and {x ∈ ℳ; x∗ x ∈ ℱ } ⊂ 𝔑. 2.8 Proposition. Let 𝒜 be a C∗ -algebra and 𝔐 a facial subalgebra of 𝒜 . Then the set {x ∈ 𝔐∩𝒜 + ; ‖x‖ < 1} is upward directed. Proof. Let x, y ∈ 𝔐 ∩ 𝒜 + , ‖x‖ < 1, ‖y‖ < 1, and let u = x(1 − x)−1 , v = y(1 − y)−1 , z = (u + v) (1 + u + v)−1 . Then u, v, z ∈ 𝒜 + , ‖z‖ < 1 and x = u(1 + u)−1 , y = v(1 + v)−1 . Since the function f1 defined in Section 1.5 is operator monotone, we have x ≤ z, y ≤ z. Since 𝔐 is a facial subalgebra in 𝒜 , 𝔐 ∩ 𝒜 + is a face of 𝒜 + . We have u ≤ ‖(1 − x)−1 ‖x ∈ 𝔐 ∩ 𝒜 + , so that u ∈ 𝔐 ∩ 𝒜 + and, similarly, v ∈ 𝔐 ∩ 𝒜 + and z ∈ 𝔐 ∩ 𝒜 + . 2.9. Proof of Theorem 2.6. We have to show that for every f1 , f2 ∈ 𝔉𝜑 and every 𝜀 > 0 there exists f ∈ 𝔉𝜑 such that (1 − 𝜀)f1 ≤ f and (1 − 𝜀)f2 ≤ f. Let f1 , f2 ∈ 𝔉𝜑 and 𝜀 > 0. By Corollary 2.4, the set 𝒯𝜑′ of all the operators T ′ ∈ 𝜋𝜑 (ℳ)′ such that there exists some 𝜆T ′ > 0 with the property ‖T ′ a𝜑 ‖𝜑 ≤ 𝜆T ′ ‖a‖

(a ∈ 𝔑𝜑 )

is a left ideal of the von Neumann algebra 𝜋𝜑 (ℳ)′ ; there exist T1′ , T2′ ∈ 𝒯𝜑′ such that 0 ≤ Tj′ ≤ 1 and fj (b∗ a) = (Tj′ a𝜑 |Tj′ b𝜑 )𝜑

(a, b ∈ 𝔑𝜑 , j = 1, 2).

(1)

By Proposition 2.7, it follows that (𝒯𝜑′ )∗ 𝒯𝜑′ is a facial subalgebra of 𝜋𝜑 (ℳ)′ . Using Proposition 2.8, we obtain an element X′ ∈ (𝒯𝜑′ )∗ 𝒯𝜑′ such that (1 − 𝜀)Tj′∗ Tj′ ≤ X′ ≤ 1, (j = 1, 2). Let T ′ = (X′ )1∕2 . Using again Proposition 2.7, we see that T ′ ∈ 𝒯𝜑′ , 0 ≤ T ′ ≤ 1 and (1 − 𝜀)Tj′∗ Tj′ ≤ T ′∗ T ′

(j = 1, 2).

(2)

By Corollaries 2.4 and 2.5, there exists a normal positive form f ≤ 𝜑, that is, f ∈ 𝔉𝜑 , such that f (b∗ a) = (T ′ a𝜑 |T ′ b𝜑 )𝜑

(a, b ∈ 𝔑𝜑 ).

(3)

From (1), (2), and (3), it follows that (1 − 𝜀)fj (x) ≤ f (x)

(x ∈ 𝔐𝜑 , j = 1, 2).

(4)

Since 𝜑 is semifinite, 𝔐𝜑 is w-dense in ℳ. As f, f1 , f2 are w-continuous, it follows that inequalities (4) remain valid for every x ∈ ℳ, that is, (1 − 𝜀)f1 ≤ f, (j = 1, 2).

14

Normal Weights

2.10 Corollary. Let 𝜑 be a normal semifinite weight on the W ∗ -algebra ℳ. For each w-continuous ∞ function ℝ ∋ t ↦ x(t) ∈ ℳ + such that ∫−∞ ‖x(t)‖dt < +∞ we have ( 𝜑

)

+∞

∫−∞

x(t)dt

∞

=

∫−∞

𝜑(x(t))dt.

Proof. Since ℳ = (ℳ∗ )∗ ([L], 1.10; A.16), the properties of the function t ↦ x(t) show that there exists a unique element x = ∫ x(t)dt ∈ ℳ + such that f (x) = ∫ f (x(t))dt for all f ∈ ℳ∗ . By Theorem 2.6, there exists an increasing net {fi }i∈I ⊂ ℳ∗+ such that 𝜑 = supi fi . For each i ∈ I, we have fi (x) = ∫ fi (x(t))dt. Since fi (x(t)) ↑i 𝜑(x(t))(t ∈ ℝ), using the classical Beppo-Levi theorem we obtain 𝜑(x) = supi fi (x) = supi ∫ fi (x(t))dt = ∫ supi fi (x(t))dt = ∫ 𝜑(x(t))dt. 2.11. Using the theorems of Haagerup (1.3) and Combes (2.6), we can now extend, without any modification, the statement and the proof of the standard respresentation theorem ([L], 10.14) for weights that are normal in the sense defined in Section 1.3: Theorem. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ. Then 𝔑𝜑 ∩ 𝔑∗𝜑 , endowed with the *-algebra structure inherited from ℳ and the scalar product of ℋ𝜑 , is a left Hilbert algebra 𝔄𝜑 ⊂ ℋ𝜑 such that 𝔄𝜑 = 𝔄′′𝜑 , 𝜋𝜑 (ℳ) = 𝔏(𝔄𝜑 ) and { 𝜑(a) =

‖𝜉‖2 if there exists 𝜉 ∈ 𝔄𝜑 with 𝜋𝜑 (a)1∕2 = L𝜉 +∞, otherwise. (a ∈ ℳ + ).

Indeed, from Theorems 1.3 and 2.6, it follows that every normal semifinite weight is the supremum of an upward directed family of normal positive forms and it is exactly this definition of normality which is used in the proof of ([L], Thm. 10.14). Consequently, every n.s.f. weight on a W ∗ -algebra is the natural weight associated with a left Hilbert algebra ([L], 10.16). Since these natural weights are sums of normal positive forms ([L], 10.18), any n.s.f. weight has the same property. In Section 5.8, we shall give a simpler proof of this result. 2.12. We shall use the notation and results of ([L], Chapter 10) for left Hilbert algebras and the objects associated with them. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and 𝔄𝜑 ⊂ ℋ𝜑 the associated left Hilbert algebra (2.11). In this section, we recall some of the notation and results just used, together with some new results. Since the weight 𝜑 is faithful, the left ideal 𝔑𝜑 ⊂ ℳ will be also considered as a linear subspace of ℋ𝜑 via the mapping 𝔑𝜑 ∋ x ↦ x𝜑 ∈ ℋ𝜑 . The closed antilinear operator S = S𝜑 in ℋ𝜑 is the closure of the preclosed antilinear operator S0𝜑 ∶ 𝔄𝜑 ∋ x𝜑 ↦ (x𝜑 )∗ = (x∗ )𝜑 ∈ ℋ𝜑 . For each 𝜂 ∈ ℋ𝜑 , one defines a linear operator R0𝜂 on ℋ𝜑 affiliated to 𝜋𝜑 (ℳ)′ , with domain D(R0𝜂 ) = 𝔄𝜑 = 𝔑𝜑 ∩ 𝔑∗𝜑 and R0𝜂 x𝜑 = 𝜋𝜑 (x)𝜂

(x ∈ 𝔑𝜑 ∩ 𝔑∗𝜑 ).

The Standard Representation

15

If 𝜂 ∈ D(S∗ ), then R0𝜂 is preclosed and its closure R𝜂 = R0𝜂 satisfies R∗𝜂 ⊃ RS∗ 𝜂 ∗ ). The right Hilbert algebra 𝔄′𝜑 ⊂ ℋ𝜑 is the set 𝔄′𝜑 = {𝜂 ∈ D(S∗ ); R𝜂 ∈ ℬ(ℋ𝜑 )} with the scalar product of ℋ𝜑 and with the operations 𝜂 b = S∗ 𝜂, 𝜂1 𝜂2 = R𝜂2 𝜂1

(𝜂, 𝜂1 , 𝜂2 ∈ 𝔄′𝜑 ).

We have so

𝜋𝜑 (ℳ)′ = ℜ(𝔄′𝜑 ) = {R𝜂 ; 𝜂 ∈ 𝔄′𝜑 } . If R0𝜂 (𝜂 ∈ ℋ𝜑 ) is bounded, then R𝜂 x𝜑 = 𝜋𝜑 (x)𝜂

(x ∈ 𝔑𝜑 ).

(1)

s∗

Indeed, if {yi }i∈I ⊂ 𝔑𝜑 is a norm-bounded net with yi → 1, then for every x ∈ 𝔑𝜑 we have y∗i x ∈ 𝔑𝜑 ∩ 𝔑∗𝜑 and, by the definition of R0𝜂 , R𝜂 x𝜑 = lim 𝜋𝜑 ( y∗i )R𝜂 x𝜑 = lim R𝜂 𝜋𝜑 ( y∗i )x𝜑 i

i

= lim R0𝜂 ( y∗i x)𝜑 = lim 𝜋𝜑 ( y∗i x)𝜂 = 𝜋𝜑 (x)𝜂. i

i

Similarly, for each 𝜉 ∈ ℋ𝜑 one defines a linear operator L0𝜉 on ℋ𝜑 , affiliated to 𝜋𝜑 (ℳ), with domain D(L0𝜉 ) = 𝔄′𝜑 and L0𝜉 𝜂 = R𝜂 𝜉

(n ∈ 𝔄′𝜑 ).

If 𝜉 ∈ D(S), then L0𝜉 is preclosed and its closure L𝜉 = L0𝜉 satisfies the relation: L∗𝜉 ⊃ LS𝜉 ∗ ). By Theorem 2.11, we have 𝔄𝜑 = 𝔄′′𝜑 = {𝜉 ∈ D(S); L𝜉 ∈ ℬ(ℋ𝜑 )} so

𝜋𝜑 (ℳ) = 𝔏(𝔄𝜑 ) = {L𝜉 ; 𝜉 ∈ 𝔄𝜑 } . For 𝜉 ∈ ℋ𝜑 , we have L0𝜉 is bounded ⇔ there exists x ∈ 𝔑𝜑 with 𝜉 = x𝜑 ; in this case L𝜉 = 𝜋𝜑 (x).

*The equality is not always true (cf. [L], C.10.1 and an example by M. Pimsner).

(2)

16

Normal Weights

Indeed, let x ∈ 𝔑𝜑 . By (1), for every 𝜂 ∈ 𝔄′𝜑 we have L0x 𝜂 = R𝜂 x𝜑 = 𝜋𝜑 (x)𝜂. 𝜑

Conversely, assume that L0𝜉 is bounded. Then L𝜉 ∈ 𝜋𝜑 (ℳ), hence there exists x ∈ ℳ with 𝜋𝜑 (x) = L𝜉 . Let x = v|x| be the polar decomposition of x in ℳ. It is easy to check that L𝜋𝜑 (v∗ )𝜉 = 𝜋𝜑 (v∗ )L𝜉 = 𝜋𝜑 (v∗ x) = 𝜋𝜑 (|x|) = |L𝜉 | ≥ 0. Using ([L], 10.8) it follows that 𝜋𝜑 (v∗ )𝜉 ∈ 𝔄′′𝜑 and then, using Theorem 2.11 we get 𝜑(x∗ x) = ‖𝜋𝜑 (v∗ )𝜉‖2𝜑 = ‖𝜉‖2𝜑 < +∞, hence x ∈ 𝔑𝜑 . Since Lx𝜑 = 𝜋𝜑 (x) = L𝜉 , we conclude that 𝜉 = x𝜑 . Also, note that x ∈ 𝔑𝜑 , x𝜑 ∈ D(S𝜑 ) ⇒ x𝜑 ∈ 𝔄𝜑 , that is, 𝔑𝜑 ∩ D(S𝜑 ) = 𝔑𝜑 ∩ 𝔑∗𝜑 .

(3)

Indeed, the inclusion 𝔑𝜑 ∩ 𝔑∗𝜑 = 𝔄𝜑 ⊂ 𝔑𝜑 ∩ D(S𝜑 ) is obvious. Conversely, if x ∈ 𝔑𝜑 and x𝜑 ∈ D(S𝜑 ), it follows from (2) that x𝜑 = 𝔄′′𝜑 = 𝔄𝜑 hence x ∈ 𝔑𝜑 ∩ 𝔑∗𝜑 . 1∕2

From the polar decomposition S𝜑 = J𝜑 Δ𝜑 of S𝜑 , one obtains the modular operator Δ𝜑 = S∗𝜑 S𝜑 and the canonical conjugation J𝜑 = J∗𝜑 = J−1 𝜑 , associated with 𝔄𝜑 ⊂ ℋ𝜑 . Since J𝜑 is antilinear and −1 J𝜑 Δ𝜑 J𝜑 = Δ𝜑 , it follows that J𝜑 f (Δ𝜑 )J𝜑 = f (Δ−1 𝜑 ) 1∕2

−1∕2

for every Borel function f. In particular, J𝜑 Δit𝜑 = Δit𝜑 J𝜑 (t ∈ ℝ), and S𝜑 = J𝜑 Δ𝜑 = Δ𝜑

−1∕2 J𝜑 Δ𝜑

1∕2 Δ𝜑 J𝜑 .

J𝜑 , S∗𝜑 =

= By Tomita’s fundamental theorem ([L], 10.12), we have Δit𝜑 𝔄𝜑 = 𝔄𝜑 , Δit𝜑 𝔄′𝜑 = 𝔄′𝜑 (t ∈ ℝ), J𝜑 𝔄𝜑 = 𝔄′𝜑 and LΔ𝜑it 𝜉 = Δit𝜑 L𝜉 Δ−it 𝜑 , RJ𝜑 𝜉 = J𝜑 L𝜉 J𝜑

(𝜉 ∈ 𝔄𝜑 ),

RΔit𝜑 𝜂 = Δit𝜑 R𝜂 Δ−it 𝜑 , LJ𝜑 𝜂 = J𝜑 R𝜂 J𝜑

(𝜂 ∈ 𝔄′𝜑 ).

(4)

Using the definition of the operators L0 , R0 the validity of (4) can be extended to arbitrary vectors 𝜉, 𝜂 ∈ ℋ𝜑 , replacing the operators L, R by L0 , R0 , respectively. If the operators L0𝜉 , R0𝜂 are preclosed, these identities can be extended to their closures, that is, they remain valid in the earlier form. From Tomita’s fundamental theorem it follows that the mapping j𝜑 ∶ x ↦ J𝜑 𝜋𝜑 (x∗ )J𝜑 defines a *-antiisomorphism j𝜑 of ℳ onto 𝜋𝜑 (ℳ)′ , which coincides with 𝜋𝜑 on the center of ℳ ([L], 10.13). We note that J𝜑 𝜋𝜑 (x)J𝜑 y𝜑 = 𝜋𝜑 ( y)J𝜑 x𝜑

(x, y ∈ 𝔑𝜑 ).

(5)

The Standard Representation

17

Indeed, using the extension of (4) as well as (1) and (2); we obtain J𝜑 𝜋𝜑 (x)J𝜑 y𝜑 = J𝜑 Lx𝜑 J𝜑 y𝜑 = RJ𝜑 x𝜑 y𝜑 = 𝜋𝜑 ( y)J𝜑 x𝜑 . Using the isometric character of J𝜑 it is easy to check that the n.s.f. weight 𝜑′ on 𝜋𝜑 (ℳ)′ defined by 𝜑′ (j𝜑 (x)) = 𝜑(x) (x ∈ ℳ + )

(6)

is just the natural weight on 𝜋𝜑 (ℳ)′ = ℜ(𝔄′𝜑 ) associated with the right Hilbert algebra 𝔄′𝜑 ⊂ ℋ𝜑 ([L], p. 331), that is, 𝜑′ (R∗𝜁 R𝜂 ) = (𝜂|𝜁)𝜑

(7)

for every 𝜂, 𝜁 ∈ ℋ𝜑 , such that R𝜂 , R𝜁 are bounded, in particular for every 𝜂, 𝜁 ∈ 𝔄′𝜑 . We note that the standard representation of 𝜋𝜑 (ℳ)′ associated with the n.s.f. weight 𝜑′ is unitarily equivalent to the identity representation 𝜋𝜑 (ℳ)′ ⊂ ℬ(ℋ𝜑 ) and we have S𝜑′ = S∗𝜑 ,

Δ𝜑′ = Δ−1 𝜑 ,

J𝜑′ = J𝜑 .

(8)

On the other hand, it follows from Tomita’s fundamental theorem that the relation 𝜋𝜑 (𝜎t𝜑 (x)) = Δit𝜑 𝜋𝜑 Δ−it 𝜑

(x ∈ ℳ, t ∈ ℝ)

defines an s∗ -continuous group {𝜎t𝜑 }t∈ℝ of *-automorphisms of ℳ which act identically on the center of ℳ ([L], 10.13). With an argument similar to that used in proving (5), we obtain R𝜂 (𝜎t𝜑 (x))𝜑 = 𝜋𝜑 (𝜎t𝜑 (x))𝜂 = R𝜂 Δit𝜑 x𝜑 so

for every x ∈ 𝔑𝜑 and every 𝜂 ∈ 𝔄′𝜑 . Letting R𝜂 → 1, we conclude (𝜎t𝜑 (x))𝜑 = Δit𝜑 x𝜑

(x ∈ 𝔑𝜑 , t ∈ ℝ).

(9)

′ ′ Since Δ𝜑′ = Δ−1 𝜑 for the weight 𝜑 on 𝜋𝜑 (ℳ) we have 𝜑 (x)) (x ∈ ℳ, t ∈ ℝ). 𝜎t𝜑 (j𝜑 (x)) = j𝜑 (𝜎−t ′

(10)

The weight 𝜑 is invariant with respect to the modular automorphism group {𝜎t𝜑 }t∈ℝ , that is, 𝜑 ◦ 𝜎t𝜑 = 𝜑(t ∈ ℝ), and satisfies the KMS condition with respect to {𝜎t𝜑 }t∈ℝ in any two elements x, y ∈ 𝔑𝜑 ∩ 𝔑∗𝜑 , that is, there exists a function f = fx,y defined, continuous and bounded on the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, analytic in the interior of this strip and such that f (it) = 𝜑(x𝜎t𝜑 ( y)),

f (1 + it) = 𝜑(𝜎t𝜑 ( y)x) (t ∈ ℝ).

18

Normal Weights

These properties characterize the modular automorphism group associated with 𝜑. More exactly, in ([L], 10.17) one actually proves the following uniqueness statement: if {𝜎t }t∈ℝ is a group of ∗ -automorphisms of ℳ with the properties ∶ (a) 𝜑 ◦ 𝜎t = 𝜑 for all t ∈ ℝ; (b) there exists a ∗ -subalgebra 𝒳 ⊂ 𝔄𝜑 such that S𝜑 |𝒳 = S𝜑 and 𝜑 satisfies the KMS condition with respect to {𝜎t }t∈ℝ in any two elements of 𝒳 ; then 𝜎t = 𝜎t𝜑 for t ∈ ℝ.

(11)

If the weight 𝜑 is finite, that is, if 𝜑 ∈ ℳ∗+ , then s

xi → x in ℳ ⇒ ‖(xi )𝜑 − x𝜑 ‖𝜑 → 0.

(12)

Thus, in this case, we can replace 𝒳 in condition (b) of (11) by any w-dense *-subalgebra of ℳ. If 𝜑 is not necessarily finite, we still have the following convergence result weakly

w

𝔑𝜑 ∋ xi → x ∈ ℳ, (xt )𝜑 → 𝜉 ∈ ℋ𝜑 ⇒ x ∈ 𝔑𝜑 , 𝜉 = x𝜑 .

(13)

Indeed, for 𝜂 ∈ 𝔄′𝜑 and 𝜁 ∈ ℋ𝜑 , we have (L0𝜉 𝜂|𝜁)𝜑 = (R𝜂 𝜉|𝜁)𝜑 = limi (R𝜂 (xi )𝜑 |𝜁)𝜑 = limi (𝜋𝜑 (xi )𝜂|𝜁) = (𝜋𝜑 (x)𝜂|𝜁)𝜑 and using (2) we conclude x ∈ 𝔑𝜑 and 𝜉 = x𝜑 . Using (13) and the w-compactness of the closed unit ball of ℳ, we obtain also the following result: if {xi } ⊂ 𝔑𝜑 is a norm-bounded net and if {(xi )𝜑 } is weakly convergent to some 𝜉 ∈ ℋ𝜑 , then there exists x ∈ 𝔑𝜑 such that

(14)

s

xi → x and x𝜑 = 𝜉. An important technical tool in the standard representation associated with 𝜑 is the Tomita algebra ([L], 10.20, 10.21) 𝔗𝜑 ⊂ 𝔄𝜑 ∩ 𝔄′𝜑 ∩

⋂ 𝛼∈ℂ

D(Δ𝜑𝛼 ).

Recall that 𝔗𝜑 is a left Hilbert subalgebra of 𝔄𝜑 , equivalent to 𝔄𝜑 , and J𝜑 𝔗𝜑 = 𝔗𝜑 , Δ𝛼𝜑 𝔗𝜑 = 𝔗𝜑 Δ𝛼𝜑 |𝔗𝜑 = Δ𝛼𝜑 , (𝛼 ∈ ℂ). The identities Δ𝛼𝜑 (𝜉𝜂) = (Δ𝛼𝜑 𝜉)(Δ𝛼𝜑 𝜂), J𝜑 (𝜉𝜂) = ( J𝜑 𝜂)( J𝜑 𝜉) (𝜉, 𝜂 ∈ 𝔗𝜑 ) are straightforward consequences of (9) and (5). The arguments in ([L], 10.21) prove that for 𝜉 ∈ ℋ𝜑 we have 𝜉 ∈ 𝔗𝜑 ⇔ 𝜉 ∈

⋂ 𝛼∈ℂ

D(Δ𝛼𝜑 ) and Δn𝜑 𝜉 ∈ 𝔑𝜑 ⊂ ℋ𝜑 for all n ∈ ℤ.

(15)

The Standard Representation

19

Using this criterion and arguing as in ([L], p. 302), it is easy to obtain the following approximation result: for every x ∈ 𝔑𝜑 there exists a sequence {xn } ⊂ 𝔗𝜑 such that s

‖xn ‖ ≤ ‖x‖, xn → x and ‖(xn )𝜑 − x𝜑 ‖𝜑 → 0;

(16)

in fact we can take xn =

√

+∞

n∕𝜋

∫−∞

e−nt 𝜎t𝜑 (x)dt. 2

s

If x ∈ 𝔑𝜑 ∩𝔑∗𝜑 , then the approximation is stronger, namely we have also ([L], Cor. 2/10.21) x∗n → x∗ and ‖(x∗n )𝜑 − (x∗ )𝜑 ‖𝜑 → 0. In the next sections, we define the translation of a weight by certain elements (2.13), consider analytic elements with respect to a weight (2.14–2.16), give some useful reformulations of the KMS condition (2.17–2.10), study the centralizer of a weight (2.21, 2.22), and use the standard representation in order to introduce a natural topology on the group of all *-automorphisms (2.23–2.26). 2.13 Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ. If a ∈ 𝔗2𝜑 , then the linear form x ↦ 𝜑(xa) (resp. x ↦ 𝜑(ax)) is defined and w-continuous on the set {x ∈ ℳ; x𝔗𝜑 ⊂ 𝔄𝜑 }

(resp. {x ∈ ℳ; 𝔗𝜑 x ⊂ 𝔄𝜑 })

which contains 𝔗𝜑 and can therefore be extended to a w-continuous linear form on ℳ, denoted by 𝜑(⋅a) (resp. 𝜑(a⋅)). The sets {𝜑(⋅a); a ∈ 𝔗2𝜑 } and {𝜑(a⋅); a ∈ 𝔗2𝜑 } are norm-dense linear subspaces of ℳ. Proof. Let a = bc∗ with b, c ∈ 𝔗𝜑 and x ∈ ℳ with x𝔗𝜑 ⊂ 𝔄𝜑 . Then xb ∈ 𝔄𝜑 ⊂ 𝔑∗𝜑 , c∗ ∈ 𝔑𝜑 , hence xa ∈ 𝔐𝜑 and we have 𝜑(xa) = (𝜋𝜑 (x)b𝜑 |Δ𝜑 c𝜑 )𝜑 .

(1)

Indeed, 𝜑(xbc∗ ) = ((c∗ )𝜑 |(b∗ x∗ )𝜑 )𝜑 = (S𝜑 c𝜑 |S𝜑 (xb)𝜑 )𝜑 1∕2 1∕2 1∕2 = ( J𝜑 Δ1∕2 𝜑 c𝜑 | J𝜑 Δ𝜑 (xb)𝜑 )𝜑 = (Δ𝜑 (xb)𝜑 |Δ𝜑 c𝜑 )𝜑 = ((xb)𝜑 |Δ𝜑 c𝜑 )𝜑 = (𝜋𝜑 (x)b𝜑 |Δ𝜑 c𝜑 )𝜑 .

Similarly, 𝜑(ax) = (𝜋𝜑 (x)Δ𝜑 b𝜑 |c𝜑 )𝜑 .

(2)

20

Normal Weights

This proves the first part of the proposition. Moreover, (1) and (2) give the explicit form of the extensions 𝜑(⋅a) and 𝜑(a⋅). Assume that the linear subspace {𝜑(⋅a); a ∈ 𝔗2𝜑 } is not norm-dense in ℳ∗ . Then, by the Hahn– Banach theorem, there exists x ∈ ℳ, x ≠ 0, such that 𝜑(xa) = 0 for all a ∈ 𝔗2𝜑 . Using (1) we infer that (𝜋𝜑 (x)𝜉|𝜂)𝜑 = 0 for all 𝜉, 𝜂 ∈ 𝔗𝜑 , hence 𝜋𝜑 (x) = 0, contradicting x ≠ 0. Note that, conversely, if 0 ≤ a ∈ 𝔑𝜑 and 𝔑∗𝜑 ∋ x ↦ 𝜑(xa) is w-continuous, then a ∈ 𝔐𝜑 .

(3)

Indeed, there exists a sequence {en } of spectral projections of a such that aen ≥ n−1 en and en ↑ s(a). We have en ≤ n2 aen a ∈ 𝔐𝜑 , so that 𝔑∗𝜑 ∋ en ↑ 1 and hence, by assumption, supn 𝜑(en a) < +∞. On the other hand, we have en a = a1∕2 en a1∕2 ↑ a, hence 𝜑(a) = supn 𝜑(en a) < +∞, that is a a ∈ 𝔐𝜑 . 2.14. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and a ∈ ℳ. The element a is called analytic in the vertical strip {𝛼 ∈ ℂ; −𝜀1 ≤ Re 𝛼 ≤ 𝜀2 }, (0 ≤ 𝜀1 , 𝜀2 < +∞), if there exists an ℳ-valued function F, defined and w-continuous on this strip and analytic in the interior of the strip, such that F(it) = 𝜎t𝜑 (a)

(t ∈ ℝ).

In this case, for each 𝛼 ∈ ℂ, −𝜀1 ≤ Re 𝛼 ≤ 𝜀2 , we let 𝜑 (a) = F(𝛼). 𝜎−i𝛼

For 𝛼 ∈ ℂ, we shall write a ∈ D(𝜎𝛼𝜑 ) if the element a is analytic in some vertical strip containing i𝛼. The following statements are easily verified: a ∈ D(𝜎𝛼𝜑 ) ⇒ a∗ ∈ D(𝜎𝛼𝜑̄ ), 𝜎𝛼𝜑̄ (a∗ ) = 𝜎𝛼𝜑 (a)∗ ; a, b ∈ D(𝜎𝛼𝜑 ) ⇒ ab ∈ D(𝜎𝛼𝜑 ), 𝜎𝛼𝜑 (ab) = 𝜎𝛼𝜑 (a)𝜎𝛼𝜑 (b).

(1) (2)

Using ([L], 9.21), from the relation 𝜋𝜑 (𝜎t𝜑 (a))Δit𝜑 𝜉 = Δit𝜑 𝜋𝜑 (a)𝜉 we infer that 𝜑 𝜑 a ∈ D(𝜎−i𝛼 ), 𝜉 ∈ D(Δ𝛼𝜑 ) ⇒ 𝜋𝜑 (a)𝜉 ∈ D(Δ𝛼𝜑 ), Δ𝛼𝜑 𝜋𝜑 (a)𝜉 = 𝜋𝜑 (𝜎−i𝛼 (a))Δ𝛼𝜑 𝜉

(3)

or, using ([L], 9.24) and replacing 𝛼 by −i𝛼, −i𝛼 |D(Δ−i𝛼 ). a ∈ D(𝜎𝛼𝜑 ) ⇒ 𝜋𝜑 (𝜎𝛼𝜑 (a)) = Δi𝛼 𝜑 𝜑 𝜋𝜑 (a)Δ𝜑

(4)

It follows that if the element a is analytic in the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} then the function 𝜑 (a) is norm-continuous and norm-bounded on this strip. 𝛼 ↦ 𝜎−i𝛼 Also, we have 𝜑 𝜑 a ∈ D(𝜎𝛽𝜑 ), 𝜎𝛽𝜑 (a) ∈ D(𝜎𝛼𝜑 ) ⇒ a ∈ D(𝜎𝛼+𝛽 ), 𝜎𝛼+𝛽 (a) = 𝜎𝛼𝜑 (𝜎𝛽𝜑 (a));

(5)

𝜑 𝜑 a ∈ D(𝜎𝛼𝜑 ) ⇒ 𝜎𝛼𝜑 (a) ∈ D(𝜎−𝛼 ), 𝜎−𝛼 (𝜎𝛼𝜑 (a)) = a.

(6)

The Standard Representation

21

Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ, a ∈ ℳ and 𝜆 ∈ (0, +∞). The following statements are equivalent: (i) 𝜑(ax∗ xa∗ ) ≤ 𝜆2 𝜑(x∗ x) for every x ∈ ℳ; (ii) x ∈ 𝔑𝜑 ⇒ xa∗ ∈ 𝔑𝜑 and ‖(xa∗ )𝜑 ‖𝜑 ≤ 𝜆‖x𝜑 ‖𝜑 ; 𝜑 𝜑 (iii) a ∈ D(𝜎−i∕2 ) and ‖𝜎−i∕2 (a)‖ ≤ 𝜆. 𝜑 If a ∈ D(𝜎−i∕2 ), then 𝜑 (xa∗ )𝜑 = J𝜑 𝜋𝜑 (𝜎−i∕2 (a))J𝜑 x𝜑

(x ∈ 𝔑𝜑 )

(7)

and if moreover 𝜎t𝜑 (aa∗ ) = aa∗ (t ∈ ℝ), then 𝜑 𝜑 𝜑(𝜎−i∕2 (a)∗ x∗ x𝜎−i∕2 (a)) = 𝜑(aa∗ x∗ x) (x ∈ 𝔑𝜑 ).

(8)

Proof. It is clear that (i) ⇔ (ii). (ii) ⇒ (iii). From (ii) it follows that there exists T ∈ ℬ(ℋ𝜑 ), ‖T‖ ≤ 𝜆, such that Tx𝜑 = (xa∗ )𝜑 (x ∈ 𝔑𝜑 ). For x ∈ 𝔄𝜑 , we have −1∕2 Tx𝜑 = (xa∗ )𝜑 = S𝜑 (ax∗ )𝜑 = S𝜑 𝜋𝜑 (a)S𝜑 x𝜑 = J𝜑 Δ1∕2 J𝜑 x𝜑 𝜑 𝜋𝜑 (a)Δ𝜑 1∕2

−1∕2

−1∕2

so the operator Δ𝜑 𝜋𝜑 (a)Δ𝜑 |𝔄′𝜑 is bounded with norm ≤ 𝜆. Since Δ𝜑 𝜑 𝜑 ) and ‖𝜎−i∕2 (a)‖ ≤ 𝜆. ([L], 9.24) we infer that a ∈ D(𝜎−i∕2

−1∕2

|𝔄′𝜑 = Δ𝜑

, using

1∕2

(iii) ⇒ (i). Let x ∈ 𝔄𝜑 = 𝔑𝜑 ∩ 𝔑∗𝜑 . Then (x∗ )𝜑 ∈ D(Δ𝜑 ) and, using (3) with 𝛼 = 1∕2, we get 𝜑 ∗ ∗ (a))Δ1∕2 Δ1∕2 𝜑 𝜋𝜑 (a)(x )𝜑 = 𝜋𝜑 (𝜎−i∕2 𝜑 (x )𝜑 . 𝜑 S𝜑 (ax∗ )𝜑 = J𝜑 𝜋𝜑 (𝜎−i∕2 (a))J𝜑 S𝜑 (x∗ )𝜑 ,

hence 𝜑 (xa∗ )𝜑 = J𝜑 𝜋𝜑 (𝜎−i∕2 (a))J𝜑 x𝜑

(x ∈ 𝔄𝜑 ).

(9)

It follows that 𝜑 (a)‖𝜑(z) 𝜑(aza∗ ) ≤ ‖𝜎−i∕2

(z ∈ ℳ + ),

and this proves (i). Indeed, if 𝜑(z) = +∞ the inequality is obvious and if 𝜑(z) < +∞ then x = z1∕2 ∈ 𝔄𝜑 and we use (9). Consider now x ∈ 𝔑𝜑 . There is a sequence {xn } ⊂ 𝔄𝜑 such that ‖(xn )𝜑 − x𝜑 ‖𝜑 → 0. From (ii) it follows that xa∗ ∈ 𝔑𝜑 and ‖(xn a∗ )𝜑 − (xa∗ )𝜑 ‖𝜑 → 0. Thus, (7) follows from (9) in the limit. 𝜑 𝜑 𝜑 Finally, we prove (8). Let b = 𝜎−i∕2 (a). Using (1) and (6) it follows that b∗ ∈ D(𝜎−i∕2 ) and 𝜎−i∕2 𝜑 (aa∗ ) = aa∗ . Using (b∗ ) = a∗ . On the other hand, since 𝜎t𝜑 (aa∗ ) = aa∗ (t ∈ ℝ), it is obvious that 𝜎−i∕2

22

Normal Weights

(7) we obtain 𝜑(b∗ x∗ xb) = ‖(xb)𝜑 ‖2𝜑 = ‖ J𝜑 𝜋𝜑 (a∗ )J𝜑 x𝜑 ‖2𝜑 = (x𝜑 | J𝜑 𝜋𝜑 (aa∗ )J𝜑 x𝜑 )𝜑 = (x𝜑 |(xaa∗ )𝜑 )𝜑 = 𝜑(aa∗ x∗ x). 2.15. An element a ∈ ℳ such that a ∈ D(𝜎𝜑𝛼 ) for all 𝛼 ∈ ℂ is called an entire analytic element. We put 𝜑 ℳ∞ = {a ∈ ℳ; a is an entire analytic element}.

Using ([L], 10.20, 9.24) we see that 𝜑 a ∈ 𝔗𝜑 ⇒ a ∈ ℳ∞ and 𝜎𝛼𝜑 (a) ∈ 𝔗𝜑 for all 𝛼 ∈ ℂ.

(1)

𝜑 From Section 2.14, it follows that ℳ∞ is an s∗ -dense *-subalgebra of ℳ. Moreover, the sets 𝜑 𝔑𝜑 , 𝔄𝜑 , 𝔐𝜑 are all invariant under left or right multiplications by elements of ℳ∞ . Note also that 𝜑 a ∈ 𝔗𝜑 ⇒ Δi𝛼 = 𝜋𝜑 (𝜎𝛼𝜑 (a)). 𝜑 a𝜑 = (𝜎𝛼 (a))𝜑 and LΔi𝛼 𝜑 a𝜑

(2)

Indeed, for 𝜉 ∈ 𝔗𝜑 we have i𝛼 −i𝛼 a = Δi𝛼 LΔi𝛼𝜑 a𝜑 𝜉 = R𝜉 Δi𝛼 𝜑 a𝜑 = Δ𝜑 RΔ−i𝛼 𝜑 La𝜑 Δ𝜑 𝜉 𝜑 𝜉 𝜑 −i𝛼 𝜑 𝜑 = Δi𝛼 𝜑 𝜋𝜑 (a)Δ𝜑 𝜉 = 𝜋𝜑 (𝜎𝛼 (a))𝜉 = L(𝜎𝛼 (a))𝜑 𝜉.

In particular, RΔi𝜑𝛼̄ J𝜑 a𝜑 = RJ𝜑 Δ𝜑i𝛼 a𝜑 = J𝜑 LΔi𝛼𝜑 a𝜑 J𝜑 = J𝜑 𝜋𝜑 (𝜎𝛼𝜑 (a))J𝜑 .

(3)

2.16 Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ, {xk }k∈K ⊂ 𝔄𝜑 a net such that s∗

xk → 1, supk ‖xk ‖ ≤ 1 and ak =

√

+∞

1∕𝜋

∫−∞

e−t 𝜎t𝜑 (xk )dt 2

(k ∈ K).

(1)

𝜑 and for every 𝛼 ∈ ℂ, k ∈ K, we have Then {ak }k∈K ⊂ 𝔗𝜑 ⊂ ℳ∞ s∗

𝜎𝛼𝜑 (ak ) → 1,

(2)

‖𝜎𝛼𝜑 (ak )‖ ≤ exp((Im 𝛼)2 ). Proof. Arguing as in ([L], p. 347) we see that ak ∈ 𝔗𝜑 and 𝜎𝛼𝜑 (ak ) =

√ 1∕𝜋

+∞

∫−∞

e−(t−𝛼) 𝜎t𝜑 (xk )dt. 2

(3)

The Standard Representation

23

Let r = Re 𝛼, s = Im 𝛼. Then (t − 𝛼)2 = −s2 + (t − r)2 − 2is(t − r), so ‖𝜎𝛼𝜑 (ak )‖ ≤ es

2

√

+∞

1∕𝜋

∫−∞

2

2

e−(t−r) ‖𝜎t (xk )‖dt ≤ es

√

+∞

1∕𝜋

∫−∞

2

2

e−t dt = es .

Let ℳ ⊂ ℬ(ℋ ) be realized as a von Neumann algebra. For 𝜉 ∈ ℋ , we have ‖𝜉 − 𝜎𝛼𝜑 (ak )𝜉‖ ≤ es

+∞

2

∫−∞ +∞

2

= es

∫−∞

e−(t−r) ‖𝜎t𝜑 (1 − xk )𝜉‖dt 2

e−t ‖𝜎t𝜑 (1 − xk )𝜉‖dt → 0, 2

since limk ‖𝜎t𝜑 (1 − xk )𝜉‖ = 0 (t ∈ ℝ), using the Lebesgue dominated convergence theorem. so

so

Consequently, 𝜎𝛼𝜑 (ak ) → 1 and, similarly, 𝜎𝛼𝜑 (ak )∗ → 1.

2.17 Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ, x, y ∈ ℳ and 𝛼 ∈ ℂ. If x ∈ 𝜑 𝜑 𝔑∗𝜑 ∩ D(𝜎𝛼−i ), 𝜎𝛼−i (x) ∈ 𝔑𝜑 and y ∈ 𝔑𝜑 ∩ D(𝜎𝛼𝜑 ), 𝜎𝛼𝜑 ( y) ∈ 𝔑∗𝜑 , then 𝜑 𝜑(xy) = 𝜑(𝜎𝛼𝜑 ( y)𝜎𝛼−i (x)).

(1) s∗

Proof. By Proposition 2.16, there exists a net {ak } ⊂ 𝔗𝜑 such that 𝜎𝛽𝜑 (ak ) → 1 for all 𝛽 ∈ ℂ. Using Properties 2.14.(1), 2.14.(4), 2.14.(6) and 2.15.(3), we obtain 𝜋𝜑 (ak )y𝜑 = (ak y)𝜑 = S𝜑 ( y∗ a∗k )𝜑 = S𝜑 𝜋𝜑 ( y∗ )S𝜑 (ak )𝜑

𝜑 ∗ 𝛼̄ i𝛼̄ = S𝜑 𝜋𝜑 (𝜎−𝜑𝛼̄ (𝜎𝛼𝜑̄ ( y∗ )))S𝜑 (ak )𝜑 = S𝜑 Δ−i 𝜑 𝜋𝜑 (𝜎𝛼̄ ( y ))Δ𝜑 S𝜑 (ak )𝜑

̄ 𝛼+(1∕2) ̄ J𝜑 (ak )𝜑 = J𝜑 Δ−i 𝜋𝜑 (𝜎𝛼𝜑 ( y)∗ )Δi𝜑𝛼−(1∕2) 𝜑 𝛼+(1∕2) ̄ ̄ = J𝜑 Δ−i RΔi𝛼−(1∕2) J 𝜑 𝜑

𝜑 (ak )𝜑

(𝜎𝛼𝜑 ( y)∗ )𝜑

𝜑 𝛼+(1∕2) ̄ = J𝜑 Δ−i J𝜑 𝜋𝜑 (𝜎𝛼−(i∕2) (ak ))J𝜑 (𝜎𝛼𝜑 ( y)∗ )𝜑 𝜑

and taking the limit over k it follows that 𝛼+(1∕2) ̄ y𝜑 = J𝜑 Δ−i (𝜎𝛼𝜑 ( y)∗ )𝜑 . 𝜑

Similarly, we obtain 𝜑 (x∗ )𝜑 = J𝜑 Δ−i𝛼−(1∕2) (𝜎𝛼−i (x))𝜑 . 𝜑

Consequently, 𝜑(xy) = ( y𝜑 |(x∗ )𝜑 )𝜑 𝜑 𝛼+(1∕2) ̄ = ( J𝜑 Δ−i (𝜎𝛼𝜑 ( y)∗ )𝜑 | J𝜑 Δ𝜑−i𝛼−(1∕2) (𝜎𝛼−i (x))𝜑 )𝜑 𝜑 𝜑 𝜑 = (𝜎𝛼−i (x)𝜑 |(𝜎𝛼𝜑 ( y)∗ )𝜑 )𝜑 = 𝜑(𝜎𝛼𝜑 ( y)𝜎𝛼−i (x)).

24

Normal Weights

In particular, for 𝛼 = 0, 𝛼 = i, and 𝛼 = i∕2, we have 𝜑 𝜑 𝜑 𝜑(xy) = 𝜑( y𝜎−i (x)) = 𝜑(𝜎i𝜑 ( y)x) = 𝜑(𝜎1∕2 ( y)𝜎−1∕2 (x)),

(2)

whenever x, y ∈ 𝔗𝜑 . These identities replace for weights the relation 𝜑(xy) = 𝜑( yx), which is valid only for traces. 2.18. Another similar result, which ( formally) follows from 2.17.(1), is contained in the next statement: 𝜑 𝜑 (a)z) for all 𝛼 ∈ ℂ. a ∈ ℳ∞ , z ∈ 𝔐𝜑 ⇒ 𝜑(z𝜎𝛼𝜑 (a)) = 𝜑(𝜎𝛼+i

(1)

We give a direct proof here. Since z ∈ 𝔐𝜑 = 𝔑∗𝜑 𝔑𝜑 , we may assume z = y∗ x with x, y ∈ 𝔑𝜑 . Using Proposition 2.14, we get 𝜑 𝜑( y∗ x𝜎𝛼𝜑 (a)) = ((x𝜎𝛼𝜑 (a))𝜑 |y𝜑 )𝜑 = ( J𝜑 𝜋𝜑 (𝜎−i∕2 (𝜎𝛼𝜑 (a)∗ )J𝜑 x𝜑 |y𝜑 )𝜑

𝜑 𝜑 = ( J𝜑 𝜋𝜑 (𝜎𝛼+(i∕2) (a))∗ J𝜑 x𝜑 |y𝜑 )𝜑 = (x𝜑 |J𝜑 𝜋𝜑 (𝜎𝛼+(i∕2) (a))J𝜑 y𝜑 )𝜑 𝜑 𝜑 𝜑 = (x𝜑 |J𝜑 𝜋𝜑 (𝜎−i∕2 (a)))J𝜑 y𝜑 )𝜑 = (( y𝜎𝛼+i (a)∗ )𝜑 |x𝜑 )𝜑 (𝜎𝛼+i 𝜑 = 𝜑(𝜎𝛼+i (a)y∗ x).

𝜑 As 𝛼 ↦ (x𝜑 | J𝜑 𝜋𝜑 (𝜎𝛼+(i∕2) (a))J𝜑 y𝜑 )𝜑 is an entire analytic function, bounded on horizontal strips, it follows that the functions

𝛼 ↦ 𝜑(z𝜎𝛼𝜑 (a)) and 𝛼 ↦ 𝜑(𝜎𝛼𝜑 (a)z) 𝜑 are entire analytic and bounded on horizontal strips, for all a ∈ ℳ∞ and all z ∈ 𝔐𝜑 . 𝜑 + On the other hand, if a ∈ ℳ∞ , and z ∈ 𝔐𝜑 ∩ ℳ , then for all 𝛼 ∈ ℂ we have 𝜑 𝜑(𝜎𝛼𝜑 (a)z𝜎𝛼𝜑 (a)∗ ) ≤ ‖𝜎𝛼−(i∕2) (a)‖2 𝜑(z), 𝜑 (a)‖2 𝜑(z). 𝜑(𝜎𝛼𝜑 (a)∗ z𝜎𝛼𝜑 (a)) ≤ ‖𝜎𝛼+(i∕2)

(2)

Indeed, z = x∗ x with x = z1∕2 ∈ 𝔑𝜑 and using Proposition 2.14, we get 𝜑(𝜎𝛼𝜑 (a)z𝜎𝛼𝜑 (a)∗ ) = ‖(x𝜎𝛼𝜑 (a)∗ )𝜑 ‖2𝜑 𝜑 𝜑 ≤ ‖ J𝜑 𝜋𝜑 (𝜎−i∕2 (𝜎𝛼𝜑 (a)))J𝜑 ‖2 ‖x𝜑 ‖2𝜑 = ‖𝜎𝛼−(i∕2) (a)‖2 𝜑(z)

and the other inequality is verified in a similar way. We note that the right-hand side of (2) depends just on |Im 𝛼|, as the 𝜎t𝜑 (t ∈ ℝ) being *-automorphisms, are isometric. Properties (1) and (2) characterize the entire analytic functions of the type 𝛼 ↦ 𝜎𝛼𝜑 (a) with 𝜑 a ∈ ℳ∞ as we shall see in Theorem 2.19. The identity (1) is meaningful also for certain other a and z. Indeed, since 𝔐𝜑 is contained and w-dense in the set {z ∈ ℳ; z𝔗𝜑 ⊂ 𝔄𝜑 , 𝔗𝜑 z ⊂ 𝔄𝜑 }, using Proposition 2.13, we infer from (1),

The Standard Representation

25

taking the limit, the following statement: if a ∈ 𝔗2𝜑 and z ∈ ℳ; z𝔗𝜑 ⊂ 𝔄𝜑 , 𝔗𝜑 z ⊂ 𝔄𝜑 , then

𝜑 𝜑(z𝜎𝛼𝜑 (a)) = 𝜑(𝜎𝛼+i (a)z) for all 𝛼 ∈ ℂ.

(3)

Under the same conditions as in statement (3), the functions 𝛼 ↦ 𝜑(z𝜎𝛼𝜑 (a)) and 𝛼 ↦ 𝜑(𝜎𝛼𝜑 (a)z) are entire analytic and bounded on horizontal strips. Indeed, if a = bc∗ with b, c ∈ 𝔗𝜑 , then, using 2.13.(1) and 2.13.(2), we get i𝛼+1 ̄ c𝜑 )𝜑 , 𝜑(z𝜎𝛼𝜑 (a)) = 𝜑(z𝜎𝛼𝜑 (b)𝜎𝛼𝜑̄ (c)∗ ) = (𝜋𝜑 (z)Δi𝛼 𝜑 b𝜑 |Δ

𝜑(𝜎𝛼𝜑 (a)z) = 𝜑(𝜎𝛼𝜑 (b)𝜎𝛼𝜑̄ (c)∗ z) = (𝜋𝜑 (z)Δi𝛼+1 b𝜑 |Δi𝛼̄ c𝜑 )𝜑 . 2.19 Theorem (A. Connes). Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and consider a function F ∶ ℂ → ℳ such that (a) for every z ∈ 𝔐𝜑 ∩ ℳ + we have F(𝛼)z ∈ 𝔐𝜑 , zF(𝛼) ∈ 𝔐𝜑 the functions 𝛼 ↦ 𝜑(F(𝛼)z) and 𝛼 ↦ 𝜑(zF(𝛼)) are entire analytic and 𝜑(zF(𝛼)) = 𝜑(F(𝛼 + i)z) for all 𝛼 ∈ ℂ; (b) for every 𝜀 > 0 there exists 𝛿 > 0 such that if 𝛼 ∈ ℂ, |Im 𝛼| ≤ 𝛿, and z ∈ 𝔐𝜑 ∩ ℳ + , then 𝜑(F(𝛼)zF(𝛼)∗ ) ≤ 𝜀𝜑(z) and 𝜑(F(𝛼)∗ zF(𝛼)) ∈ 𝜀𝜑(z). 𝜑 Then F(0) ∈ ℳ∞ and F(𝛼) = 𝜎𝛼𝜑 (F(0)) for all 𝛼 ∈ ℂ. Proof. We have to show that F is an entire analytic function and F(t) = 𝜎𝛼𝜑 (F(0)) for all t ∈ ℝ. By assumption, 𝔑𝜑 F(𝛼) ⊂ 𝔑𝜑 , 𝔑𝜑 F(𝛼)∗ ⊂ 𝔑𝜑 , 𝔐𝜑 F(𝛼) ⊂ 𝔐𝜑 , F(𝛼)𝔐𝜑 ⊂ 𝔐𝜑 and, for x ∈ 𝔑𝜑 and 𝛼 ∈ ℂ, |Im 𝛼| ≤ 𝛿, we have ‖(xF(𝛼))𝜑 ‖𝜑 ≤ 𝜀1∕2 ‖x𝜑 ‖𝜑 , ‖(xF(𝛼)∗ )𝜑 ‖𝜑 ≤ 𝜀1∕2 ‖x𝜑 ‖𝜑 .

(1)

Let a, b ∈ 𝔗𝜑 . For every 𝛾 ∈ ℂ, we have 𝜎𝛾𝜑 (ab) = 𝜎𝛾𝜑 (a)𝜎𝛾𝜑 (b) ∈ 𝔗𝜑 𝔗𝜑 ⊂ 𝔑∗𝜑 𝔑𝜑 = 𝔐𝜑 . We define a function G of two complex variables by G(𝛼, 𝛽) = 𝜑(F(𝛼)𝜎𝛽𝜑 (ab)) = ((𝜎𝛽𝜑 (b))𝜑 |(𝜎𝛽𝜑 (a)∗ F(𝛼)∗ )𝜑 )𝜑 .

(2)

By assumption and by the first equation in (2) it follows that 𝛼 ↦ G(𝛼, 𝛽) is an entire analytic function and G(𝛼 + i, 𝛽) = 𝜑(𝜎𝛽𝜑 (ab)F(𝛼)).

(3)

Using (1) and the second equation in (2) it follows that 𝛽 ↦ G(𝛼, 𝛽) is also an entire analytic function. Consequently, G is an entire analytic function in both variables, by the Hartogs theorem (Hormander, 1966, p. 2.2.8). On the other hand, using (3) and 2.18.(3), we obtain 𝜑 G(𝛼 + i, 𝛽 + i) = 𝜑(𝜎𝛽+i (ab)F(𝛼)) = 𝜑(F(𝛼)𝜎𝛽𝜑 (ab)) = G(𝛼, 𝛽).

26

Normal Weights

Thus, 𝛼 ↦ g(𝛼) = G(𝛼, 𝛼) is an entire analytic function and g(𝛼 + i) = g(𝛼). Using (1) with 𝛿 = 1 and (2), for 𝛼 ∈ ℂ with |Im 𝛼| ≤ 1 we get |g(𝛼)| = |((𝜎𝛼𝜑 (b))𝜑 |(𝜎𝛼𝜑 (a)∗ F(𝛼)∗ )𝜑 )𝜑 i𝛼+(1∕2) a𝜑 ‖𝜑 . ≤ 𝜀1∕2 ‖(𝜎𝛼𝜑 (b))𝜑 ‖𝜑 ‖(𝜎𝛼𝜑 (a)∗ )𝜑 ‖𝜑 = 𝜀1∕2 ‖Δi𝛼 𝜑 b𝜑 ‖𝜑 ‖Δ𝜑

Therefore, the entire analytic function g is bounded, and hence constant, by the Liouville theorem. 𝜑 In particular, for t ∈ ℝ we have 𝜑(𝜎−t (F(t))ab) = 𝜑(F(t)𝜎t𝜑 (ab)) = g(t) = g(0) = 𝜑(F(0)ab). Since 𝔗𝜑 is dense in ℋ𝜑 it follows that F(t) = 𝜎t𝜑 (F(0)) for all t ∈ ℝ. In order to prove that F is an entire analytic function, it is sufficient to show that F is bounded on each compact subset of ℂ, as the set {𝜑(⋅z); z ∈ 𝔗2𝜑 } is norm-dense in ℳ∗ (2.13) and the functions 𝛼 ↦ 𝜑(F(𝛼)z)(z ∈ 𝔗2𝜑 ) are, by assumption, entire analytic (use the Montel theorem and [L], Lemma 9.24). According to (1), the boundedness of F on compact subsets of ℂ will follow once we establish the following identity (compare with 2.14.(7)): J𝜑 𝜋𝜑 (F(𝛼))J𝜑 a𝜑 = (aF(𝛼 + (i∕2))∗ )𝜑

(a ∈ 𝔗𝜑 , 𝛼 ∈ ℂ).

(4)

Since the assumptions are stable under translations 𝛼 ↦ 𝛼 + 𝛼0 , it is sufficient to prove (4) only for 𝛼 = 0. To this end, consider a, b ∈ 𝔗𝜑 . For 𝛽 ∈ ℂ let ̄

∗ −i𝛽 f1 (𝛽) = (𝜋𝜑 (F(0))Δ−i𝛽 𝜑 (a )𝜑 |Δ𝜑 b𝜑 )𝜑 ,

b𝜑 |(aF(𝛽)∗ )𝜑 )𝜑 . f2 (𝛽) = ( J𝜑 Δ−1∕2 𝜑 The function f1 is obviously entire analytic and the function f2 is entire analytic by the assumption (b). For t ∈ ℝ, it is easy to check that f1 (t) = f2 (t), since F(t) = 𝜎t𝜑 F(0). Hence f1 = f2 . In particular, f1 (i∕2) = f2 (i∕2), that is, ( J𝜑 𝜋𝜑 (F(0))J𝜑 a𝜑 | J𝜑 Δ−1∕2 b𝜑 )𝜑 = (aF(i∕2)∗ )𝜑 | J𝜑 Δ−1∕2 b𝜑 )𝜑 . 𝜑 𝜑 Since b ∈ 𝔗𝜑 was arbitrary, we obtain (4) for 𝛼 = 0.

□

2.20. The results presented in Section 2.18 involve several variants of the KMS condition. 𝜑 For instance, if a ∈ ℳ∞ and z ∈ 𝔐𝜑 , then from 2.18.(1) it follows that the equation 𝜑 f (𝛼) = 𝜑(z𝜎−i𝛼 (a))

(𝛼 ∈ ℂ)

defines an entire analytic function f, bounded on vertical strips, such that f (it) = 𝜑(z𝜎t𝜑 (a)), f (1 + it) = 𝜑(𝜎t𝜑 (a)z)

(t ∈ ℝ).

Also, if a ∈ 𝔗2𝜑 and z ∈ ℳ, z𝔗𝜑 ⊂ 𝔄𝜑 , 𝔗𝜑 z ⊂ 𝔄𝜑 , then the same conclusion is obtained from 2.18.(3). We record here one more variant of the KMS condition, where the similarity to, as well as the contrast with, the trace property is very striking.

The Standard Representation

27

Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and let x ∈ 𝔑𝜑 . There exists a bounded regular positive Borel measure 𝜇 on (0, +∞) such that 𝜑(x∗ 𝜎t𝜑 (x)) =

∞

∫0

∞

𝜆it d𝜇(𝜆),

𝜑(xx∗ ) =

∫0

𝜆 d𝜇(𝜆)

(t ∈ ℝ).

Proof. Let {e𝜆 = 𝜒[0,𝜆) (Δ𝜑 )}𝜆>0 be the spectral scale of Δ𝜑 ([L], E.9.10). Since x ∈ 𝔑𝜑 , we obtain a bounded regular positive Borel measure 𝜇 on (0, +∞) setting d𝜇(𝜆) = d(e𝜆 x𝜑 |x𝜑 )𝜑 , that is, 𝜇 is “the spectral measure associated with Δ𝜑 and x𝜑 ∈ ℋ𝜑 .” According to ([L], E.9.11) we get 𝜑(x∗ 𝜎t𝜑 (x)) = (Δit𝜑 x𝜑 |x𝜑 )𝜑 =

∞

∫0

𝜆it d𝜇(𝜆)

and if x𝜑 ∈ 𝔑𝜑 ∩ 𝔑∗𝜑 , ∞ 2 1∕2 2 𝜑(xx∗ ) = ‖(x∗ )𝜑 ‖2𝜑 = ‖S𝜑 x𝜑 ‖2𝜑 = ‖ J𝜑 Δ1∕2 𝜑 x𝜑 ‖𝜑 = ‖Δ𝜑 x𝜑 ‖𝜑 = 1∕2

∫0

𝜆d𝜇(𝜆).

∞

The proof is completed by the remark that x ∈ D(S𝜑 ) = D(Δ𝜑 ) ⇔ ∫0 𝜆 d𝜇(𝜆) < +∞ and 𝔑𝜑 ∩ D(S𝜑 ) = 𝔑𝜑 ∩ 𝔑∗𝜑 g (2.12.(3)). 2.21. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ. The centralizer of 𝜑 is the W ∗ -subalgebra of ℳ defined by ℳ 𝜑 = {a ∈ ℳ; 𝜎t𝜑 (a) = a for all t ∈ ℝ}. 𝜑 Since 𝜋𝜑 (𝜎t𝜑 (a)) = Δit𝜑 𝜋𝜑 (a)Δ−it 𝜑 (t ∈ ℝ), it follows that a ∈ ℳ if and only if 𝜋𝜑 (a) commutes with Δ𝜑 ([L], E.9.20, E.9.23). Clearly, 𝜑 a ∈ ℳ 𝜑 ⇒ a ∈ ℳ∞ and 𝜎𝛼𝜑 (a) = a for all 𝛼 ∈ ℂ.

Also, it follows from statement (7) of 2.14 that a ∈ ℳ 𝜑 , x ∈ 𝔑𝜑 ⇒ xa∗ ∈ 𝔑𝜑 and (xa∗ )𝜑 = J𝜑 𝜋𝜑 (a)J𝜑 x𝜑 .

(1)

The Pedersen–Takesaki theorem ([L], 10.27) shows that if a ∈ ℳ, then a ∈ ℳ 𝜑 ⇔ a𝔐𝜑 ⊂ 𝔐𝜑 , 𝔐𝜑 a ⊂ 𝔐𝜑 and 𝜑(ax) = 𝜑(xa) for x ∈ 𝔐𝜑 .

(2)

The implication (⇒) follows obviously from 2.15 and 2.18.(1). Conversely, we have a𝔗𝜑 ⊂ 𝜑 𝔄𝜑 , 𝔗𝜑 a ⊂ 𝔄𝜑 , hence (2.20, 2.18(3)) for every x ∈ 𝔗2𝜑 the function fx (𝛼) = 𝜑(a𝜎−i𝛼 (x)) is an

28

Normal Weights

entire analytic function, bounded on vertical strips and such that fx (1 + it) = 𝜑(𝜎t𝜑 (x)a) = 𝜑(a𝜎t𝜑 (x)) = fx (it) (t ∈ ℝ). By the Liouville theorem, it follows that fx is constant and hence 𝜑 𝜑(𝜎t𝜑 (a)x) = 𝜑(a𝜎−t (x)) = fx (−it) = fx (0) = 𝜑(ax)

(t ∈ ℝ).

Since x ∈ 𝔗2𝜑 was arbitrary, using Proposition 2.13 we infer that 𝜎t𝜑 (a) = a(t ∈ ℝ), so that a ∈ ℳ 𝜑 . Let v ∈ ℳ be a partial isometry such that vv∗ ∈ ℳ 𝜑 . We define a normal semifinite weight 𝜑v on ℳ by 𝜑v (x) = 𝜑(vxv∗ )

(x ∈ ℳ + ).

It is easy to check that s(𝜑v ) = v∗ v. In particular, for every projection e ∈ ℳ 𝜑 , we have defined a subweight 𝜑e on ℳ with s(𝜑e ) = e. Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and v ∈ ℳ a partial isometry such that e = v∗ v ∈ ℳ 𝜑 and f = vv∗ ∈ ℳ 𝜑 . Then: 𝜑v = 𝜑e ⇔ v ∈ ℳ 𝜑 . Proof. We assume first that v ∈ ℳ 𝜑 . By (2) we have 𝔑𝜑 v ⊂ 𝔑𝜑 and 𝔑𝜑 v∗ ⊂ 𝔑𝜑 . Since 𝔑𝜑v = {x ∈ ℳ; xv∗ ∈ 𝔑𝜑 } and 𝔑𝜑e = {x ∈ ℳ; xe ∈ 𝔑𝜑 }, and since e = v∗ v, v∗ = ev∗ , it follows that 𝔑𝜑v = 𝔑𝜑e and hence 𝔐𝜑v = 𝔐𝜑e . For x ∈ 𝔐𝜑v = 𝔐𝜑e we obtain vxv∗ ∈ 𝔐𝜑 , exe ∈ 𝔐𝜑 and, since v = ve ∈ ℳ 𝜑 , 𝜑(vxv∗ ) = 𝜑(vexv∗ ) = 𝜑(exv∗ v) = 𝜑(exe). We conclude that 𝜑v = 𝜑e . Conversely, assume that 𝜑v = 𝜑e . For every x ∈ ℳ + we have 𝜑(vxv∗ ) = 𝜑(exe). Replacing x by v∗ xv here, we get 𝜑(v∗ xv) = 𝜑( fxf ). If x ∈ 𝔑𝜑 , then xf ∈ 𝔑𝜑 , as f ∈ ℳ 𝜑 ; consequently, 𝜑(v∗ x∗ xv) = 𝜑( fx∗ xf ) < +∞, that is, xv ∈ 𝔑𝜑 . Thus, 𝔑𝜑 v ⊂ 𝔑𝜑 and, similarly, 𝔑𝜑 v∗ ⊂ 𝔑𝜑 . It follows that v𝔐𝜑 ⊂ 𝔐𝜑 and 𝔐𝜑 v ⊂ 𝔐𝜑 . Then, for x ∈ 𝔐𝜑 we have 𝜑(vx) = 𝜑( fvx) = 𝜑(vxf ) = 𝜑(vxvv∗ ) = 𝜑(exve) = 𝜑(xve) = 𝜑(xv). Using (2) we conclude that v ∈ ℳ 𝜑 . In particular, for a unitary element u ∈ ℳ we have 𝜑u = 𝜑 if and only if u ∈ ℳ 𝜑 . 2.22. If 𝜑 is a normal, semifinite, but not necessarily faithful, weight on the W ∗ -algebra ℳ, then we shall denote by {𝜎t𝜑 }t∈ℝ the modular automorphism group associated with the n.s.f. weight on 𝜑 the W ∗ -algebra s(𝜑)ℳs(𝜑) defined by the restriction of 𝜑; also, we shall denote by ℳ∞ , and ℳ 𝜑 the *-subalgebra of all entire analytic elements and the centralizer of 𝜑|s(𝜑)ℳs(𝜑) respectively. The results obtained for n.s.f. weights, in particular the characterization of the modular automorphism group with the aid of the KMS condition, Proposition 2.20, and so on, can easily be extended to normal semifinite weights. For instance, if 𝜑 is a normal semifinite weight on ℳ and v ∈ ℳ is a partial isometry with vv∗ ∈ ℳ 𝜑 , then we get a new normal semifinite weight 𝜑v on ℳ, with s(𝜑v ) = v∗ v and, using the KMS condition it follows that 𝜑

𝜎t v (x) = v∗ 𝜎t𝜑 (vxv∗ )v

(x ∈ v∗ vℳv∗ v, t ∈ ℝ),

(1)

The Standard Representation

29

hence ℳ 𝜑v = v∗ ℳ 𝜑 v.

(2)

In particular, if e ∈ ℳ 𝜑 is a projection, then 𝜑

𝜎t e (x) = 𝜎t𝜑 (x) (x ∈ eℳe, t ∈ ℝ), ℳ 𝜑e = eℳ 𝜑 e.

(3) (4)

We mention also that if 𝜑 is an n.s.f. weight on ℳ and 𝜎 is a *-automorphism of ℳ, then, using the KMS condition we get 𝜎t𝜑 ◦ 𝜎 = 𝜎 −1 ◦ 𝜎t𝜑 ◦ 𝜎

(t ∈ ℝ),

ℳ 𝜑 ◦ 𝜎 = 𝜎 −1 (ℳ 𝜑 ).

(5) (6)

2.23. Recall ([L), 10.23) that a von Neumann algebra ℳ ⊂ ℬ(ℋ ) is called hyperstandard if there exists a conjugation J ∶ ℋ → ℋ and a self-polar convex cone 𝔓 ⊂ ℋ with the following properties: (a) the mapping x ↦ Jx∗ J is a *-antiisomorphism of ℳ onto ℳ ′ , which acts identically on the center; (b) J𝜉 = 𝜉 for every 𝜉 ∈ 𝔓; c) [x( JxJ)]𝔓 ⊂ 𝔓 for every x ∈ ℳ. For any n.s.f. weight 𝜑 on the W ∗ -algebra ℳ, the von Neumann algebra 𝜋𝜑 (ℳ) ⊂ ℬ(ℋ𝜑 ) with the conjugation J𝜑 and the self-polar convex cone 𝔓𝜑 = {𝜋𝜑 (x)J𝜑 x𝜑 ; x ∈ 𝔗𝜑 } = {𝜋𝜑 (x)J𝜑 x𝜑 ; x ∈ 𝔄𝜑 } is a hyperstandard form of ℳ ([L], 10.23). Using statement (16) of 2.12 and arguing as in ([L], 10.23), one shows that 𝔓𝜑 = {𝜋𝜑 (x)J𝜑 x𝜑 ; x ∈ 𝔑𝜑 }.

(1)

Also, any two hyperstandard forms of ℳ are spatially isomorphic by a unique unitary operator that preserves the self-polar convex cones and hence also the conjugations ([L], 10.26, 10.23). Let ℳ be a W ∗ -algebra and (ℳ, ℋ , J, 𝔓) a hyperstandard form of ℳ. As usual, we denote by Aut(ℳ) the group of *-automorphisms of ℳ and by U(ℋ ) the group of unitary operators on ℋ . Also, for u ∈ U(ℋ ) with uℳu∗ = ℳ, we denote by Ad(u) ∈ Aut(ℳ) the *-automorphism defined by [Ad(u)](x) = uxu∗ (x ∈ ℳ). It follows that there exists an injective group homomorphism Aut(ℳ) ∋ 𝜎 ↦ u𝜎 ∈ U(ℋ ), uniquely determined, such that 𝜎 = Ad(u𝜎 ) and u𝜎 (𝔓) = 𝔓(𝜎 ∈ Aut(ℳ)). Clearly. {u𝜎 ; 𝜎 ∈ Aut(ℳ)} = {u ∈ U(ℋ ); uℳu∗ = ℳ, u(𝔓) = 𝔓}. The mapping 𝜎 ↦ u𝜎 is called the canonical implementation of Aut(ℳ).

30

Normal Weights

On the linear space ℬw (ℳ) of all w-continuous linear mappings T ∶ ℳ → ℳ, we may consider several locally convex topologies, for instance: the n-topology, defined by the seminorms T ↦ ‖Tx‖, the p-topology, defined by the seminorms T ↦ ‖𝜑(Tx)‖, the u-topology, defined by the seminorms T ↦ ‖𝜑 ◦ T‖, where x ∈ ℳ and 𝜑 ∈ ℳ∗ . Clearly, the u-topology and the n-topology are stronger than the p-topology. By restriction, we obtain the topologies n, p, and u on Aut(ℳ) ⊂ ℬw (ℳ). On the other hand, on U(ℋ ) the topologies wo, so, so∗ , w, s, s∗ , and also the Mackey topologies 𝜏w0 , 𝜏w all coincide and, with this topology, U(ℋ ) is a topological group (on the unit ball of ℬ(ℋ ), the Mackey topology 𝜏w coincides with the s∗ -topology by the theorem of Akemann (Aarnes, 1968; Akemann, 1967). If ℋ is separable, then U(ℋ ) with the two-sided uniform structure associated with this topology is a complete separable metric space, that is, a polish group. Theorem (U. Haagerup). The canonical implementation 𝜎 ↦ u𝜎 of Aut(ℳ) is an isomorphism of topological groups between Aut(ℳ) with the u-topology and a closed subgroup of U(ℋ ). Proof. Let 𝜎 ∈ Aut(ℳ) and let {𝜎i } be a net in Aut(ℳ); also let u = u𝜎 ∈ U(ℋ ) and ui = u𝜎i ∈ u

U(ℋ ). We have 𝜎i → 𝜎 in Aut(ℳ) if and only if ‖𝜑 ◦ 𝜎i −𝜑 ◦ 𝜎‖ → 0 for every 𝜑 ∈ ℳ∗+ , that is, ([L], 10.25) if and only if ‖𝜔𝜉 ◦ 𝜎i −𝜔𝜉 ◦ 𝜎‖ → 0 for every 𝜉 ∈ 𝔓, that is, if and only if ‖𝜔u∗ 𝜉 −𝜔u∗ 𝜉 ‖ → 0 i for every 𝜉 ∈ 𝔓. According to ([L], Proposition 10.24), this means that ‖u∗t 𝜉 − u∗ 𝜉‖ → 0 for every wo

𝜉 ∈ 𝔓 and, since ℋ is the linear span of 𝔓 ([L], 10.23), it follows that ui → u. Thus, the canonical implementation is a homeomorphism of Aut(ℳ), with the u-topology, onto the set {u ∈ U(ℳ) ∶ uℳu∗ = ℳ, u(𝔓) = 𝔓}, which is clearly a closed subgroup of U(ℋ ). In particular, Aut(ℳ) with the u-topology is a topological group. In view of the above Theorem, we shall consider the u-topology as the natural topology on Aut(ℳ). In general, the u-topology does not coincide either with the p-topology, or with the n-topology. For instance, if ℳ is the von Neumann algebra ℒ ∞ ([0, 1]) ⊂ ℬ(ℒ 2 ([0, 1])), then the u-topology is not comparable with the n-topology and hence both of them are strictly stronger than the p-topology (Haagerup, 1975b, p. 3.14). Note that if the W ∗ -algebra ℳ has a separable predual (i.e., if ℋ is separable), then Aut(ℳ) with the u-topology is a polish group. p Finally, we note that if {𝜎n }n≥0 ⊂ Aut(ℳ) is a sequence such that 𝜎n → 𝜎0 , then, for every compact set K ⊂ U(ℳ), we have s∗

𝜎n (u) → 𝜎0 (u) uniformly for u ∈ K. w

Clearly, it is sufficient to show that 𝜎n (u) → 𝜎0 (u) uniformly for u ∈ K. Let 𝜑 ∈ ℳ∗ and 𝜀 > 0 be fixed. The set {𝜑 ◦ 𝜎n ; n ≥ 0} ⊂ ℳ∗ is norm-compact and hence also 𝜎(ℳ∗ , ℳ)-compact. Since the topology on U(ℳ) is equal, in particular, to the topology induced by the Mackey topology 𝜏w on ℳ, it follows that the function u ↦ sup{|𝜑(𝜎n (u))|; n ≥ 0} is continuous on U(ℳ). Thus, there exist u1 , … , um ∈ K such that inf{|𝜑(𝜎n (u − uj ))|; 1 ≤ j ≤ m} < 𝜀∕3 for all u ∈ K and all n ≥ 0. p

On the other hand, as 𝜎n → 𝜎0 , there exists n0 ≥ 1 such that |𝜑(𝜎n (uj )) − 𝜑(𝜎0 (uj ))| < 𝜀∕3 for all 1 ≤ j ≤ m and all n ≥ n0 . It follows that |𝜑(𝜎n (u)) − 𝜑(𝜎0 (u))| < 𝜀 for all u ∈ K and all n ≥ n0 .

The Standard Representation

31

2.24. Let 𝜎 be an action of the locally compact group G on the W ∗ -algebra ℳ, that is a group homomorphism 𝜎 ∶ G → Aut(ℳ). In Section 13.5, we prove that the homomorphism 𝜎 is p-continuous if and only if it is u-continuous; in this case we say that 𝜎 is a continuous action. The following result is an obvious consequence of Theorem 2.23: Corollary. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact group G on the W ∗ -algebra ℳ, let (ℳ, ℋ , J, 𝔓) be a hyperstandard form of ℳ and denote by u(g) = u𝜎g (g ∈ G) the canonical implementation. Then G ∋ g ↦ u(g) ∈ U(ℋ ) is an so-continuous unitary representation and 𝜎g = Ad(u(g)), (g ∈ G). 2.25. We now show that in certain cases the u-topology coincides with the p-topology. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and let Aut𝜑 (ℳ) = {𝜎 ∈ Aut(ℳ); 𝜑 ◦ 𝜎 = 𝜑}. If 𝜎 ∈ Aut𝜑 (ℳ), then the canonical implementation of 𝜎 in the hyperstandard form (𝜋𝜑 (ℳ), ℋ𝜑 , J𝜑 , 𝔓𝜑 ) has an explicit form, namely u𝜎 a𝜑 = (𝜎(a))𝜑

(a ∈ 𝔑𝜑 ).

(1)

Indeed, for a ∈ 𝔑𝜑 we have ‖(𝜎(a))𝜑 ‖2𝜑 = 𝜑(𝜎(a∗ a)) = 𝜑(a∗ a) = ‖a𝜑 ‖2𝜑 and hence (1) defines a unitary operator u𝜎 ∈ U(ℋ𝜑 ). For every x ∈ ℳ, a ∈ 𝔑𝜑 , we have u𝜎 𝜋𝜑 (x)u∗𝜎 a𝜑 = u𝜎 𝜋𝜑 (x)(𝜎 −1 (a))𝜑 = u𝜎 (x𝜎 −1 (a))𝜑 = 𝜋𝜑 (𝜎(x))a𝜑 . On the other hand, for a ∈ 𝔄𝜑 we have u𝜎 S𝜑 a𝜑 = u𝜎 (a∗ )𝜑 = (𝜎(a∗ ))𝜑 = (𝜎(a)∗ )𝜑 = S𝜑 (𝜎(a))𝜑 = S𝜑 u𝜎 a𝜑 , hence u𝜎 commutes with S𝜑 . Consequently, u𝜎 commutes with J𝜑 and with Δ𝜑 . Moreover, we have u𝜎 𝜋𝜑 (a)J𝜑 a𝜑 = 𝜋𝜑 (𝜎(a))u𝜎 J𝜑 a𝜑 = 𝜋𝜑 (𝜎(a))J𝜑 u𝜎 a𝜑 = 𝜋𝜑 (𝜎(a))J𝜑 (𝜎(a))𝜑 hence u𝜎 (𝔓𝜑 ) ⊂ 𝔓𝜑 by 2.23.(1). We conclude that u𝜎 is the canonical implementation of 𝜎. Corollary. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ. Then the p-topology coincides with the u-topology on the u-closed subgroup Aut𝜑 (ℳ) of Aut(ℳ). p

Proof. Let 𝜎i → 𝜎 in Aut𝜑 (ℳ). Let 𝜉 = a𝜑 ∈ 𝔑𝜑 and 𝜂 = R∗𝜂 𝜂2 with 𝜂1 , 𝜂2 ∈ 𝔄′𝜑 . Using (1) we 1 obtain (u𝜎t 𝜉|𝜂)𝜑 = ((𝜎t (a))𝜑 |R∗𝜂 𝜂2 )𝜑 = (R𝜂1 (𝜎i (a))𝜑 |𝜂2 )𝜑 1

= (𝜋𝜑 (𝜎i (a))𝜂1 |𝜂2 )𝜑 → (𝜋𝜑 (𝜎(a))𝜂1 |𝜂2 )𝜑 = (R𝜂1 (𝜎(a))𝜑 |𝜂2 )𝜑 = ((𝜎(a))𝜑 |R∗𝜂 𝜂2 )𝜑 = (u𝜎 𝜉|𝜂)𝜑 . 1

wo

u

It follows that u𝜎t → u and hence, by Theorem 2.23, 𝜎i → 𝜎 in Aut𝜑 (ℳ).

32

Normal Weights

In particular, if ℳ is a factor of type I, or a factor of type II1 , then the p-topology coincides with the u-topology on Aut(ℳ), because in these cases every *-automorphism preserves the trace on ℳ. 2.26. Finally, we consider the particular case of a von Neumann algebra, ℳ ⊂ ℬ(ℋ ) with a cyclic and separating vector 𝜉 ∈ ℋ . Then 𝜑 = 𝜑𝜉 |ℳ is a faithful normal positive on ℳ and we can identify ℋ𝜑 with ℋ via the mapping x𝜑 ↦ x𝜉(x ∈ ℳ); let S = S𝜑 , J = J𝜑 , Δ = Δ𝜑 . We have the convex cones ([L], 10.8) 𝔓S = {x𝜉 ∶ x ∈ ℳ + },

𝔓S∗ = {x′ 𝜉 ∶ x′ ∈ ℳ ′+ }

polar to one another, and the self-polar convex cone ([L], 10.23) 𝔓 = Δ1∕4 𝔓S = Δ−1∕4 𝔓S∗ = {xJxJ𝜉 ∶ x ∈ ℳ}. Proposition. 𝔓 = {xJxJ𝜉; x ∈ ℳ + }. Proof. It is sufficient to show that for every x ∈ ℳ + there exists a sequence {xn } ⊂ ℳ + such that xn Jxn J𝜉 ∈ Δ1∕4 x𝜉. Moreover, we may assume that x is invertible. In this case, both 𝜉 and x1∕2 𝜉 are cyclic and separating vectors for ℳ ′ ⊂ ℬ(ℋ ). By the uniqueness of the hyperstandard form of a von Neumann algebra, there exists a unitary operator u ∈ (ℳ ′ )′ = ℳ such that ux1∕2 𝜉 ∈ 𝔓. Thus, y = ux1∕2 ∈ ℳ and y𝜉 ∈ 𝔓. If yn =

√ n∕𝜋

+∞

∫−∞

e−nt 𝜎t𝜑 ( y)dt 2

(n ∈ ℕ),

s∗

𝜑 then (2.12.(16)) ℳ∞ ∋ yn ↦ y and ([L], 10.23.(6))

yn 𝜉 =

√ n∕𝜋

+∞

∫−∞

2

e−nt Δit y𝜉 dt ∈ 𝔓

(n ∈ ℕ).

𝜑 Let xn = 𝜎i∕4 ( yn ) = Δ−1∕4 yn Δ1∕4 ∈ ℳ(n ∈ ℕ). Then Δ1∕4 xn 𝜉 = yn 𝜉 ∈ 𝔓, that is, xn 𝜉 ∈ Δ−1∕4 𝔓 ⊂ 𝜑 𝜑 𝔓S , and hence xn ≥ 0. In particular xn = x∗n = 𝜎i∕4 ( yn )∗ = 𝜎−i∕4 ( y∗n ), by 2.14.(1), and using this identity it is easy to check that

xn Jxn 𝜉 = Δ1∕4 y∗n yn 𝜉 → Δ1∕4 y∗ y𝜉 = Δ1∕4 x𝜉.

2.27. Notes. Theorem 2.6 and the technical results (2.3, 2.4) used for its proof are due to Combes (1968). Proposition 2.15 is due to Connes (1973b). Proposition 2.16 and the various forms of the KMS condition (2.17, 2.18, 2.20) are due to Pedersen and Takesaki (1973) and Connes (1980). Theorem 2.19 is due to Connes (1973a). The canonical implementation of *-automorphisms (2.23–2.25) is explicitly stated by Haagerup (1975b), having been obtained also by Araki (1974) and Connes (1974c). Proposition 2.26 is due to Connes (1974c).

The Balanced Weight

33

For our exposition we have used Combes (1968, 1971a, 1971b); Connes (1973a, 1972, 1973b, 1980); Connes and Takesaki (1977); Van Daele (1973); Digernes (1975); Haagerup (1975b); Pedersen (1973–1977); Strătilă, Voiculescu and Zsidó (1976, 1977); [L]; Takesaki (1969–1970); Takesaki (1970). In the framework of quantum field theory, Bisognano and Wichmann (1975) explicitly computed the modular operator and the canonical conjugation for some von Neumann algebras associated with a hermitian scalar field. Recent developments concerning geometric aspects of standard representations are due to Haagerup and Skau (see Skau, 1980). Other related references are Haagerup (1979c) and Woronowicz (1979).

3 The Balanced Weight This section contains a detailed study, with applications, of the balanced weight, which is the main technical idea in the proof of the Connes cocycle theorem. 3.1. Let ℳ be a W ∗ -algebra, 𝜑 an n.s.f. weight on ℳ and 𝜓 a normal semifinite (but not necessarily faithful) weight on ℳ. We define the balanced weight 𝜃 = 𝜃(𝜑, 𝜓) on the W ∗ -algebra 𝒩 = Mat2 (ℳ) of all 2×2 matrices over ℳ, by ( 𝜃

x11 x21

x12 x22

)

(( = 𝜑(x11 ) + 𝜓(x12 )

x11 x21

x12 x22

)

) ∈𝒩+ .

It is easy to check that 𝜃 is a normal weight on 𝒩 . We have (

x11 x21

x12 x22

) ∈ 𝔑𝜃 ⇔ x11 , x12 ∈ 𝔑𝜑 and x11 , x12 ∈ 𝔑𝜓 ,

hence (

) ( ∗ ) 𝔑𝜑 𝔑∗𝜑 𝔑𝜓 ∗ 𝔑𝜃 = , 𝔑𝜃 = ; 𝔑𝜓 𝔑∗𝜓 𝔑∗𝜓 ( ) 𝔑𝜑 ∩ 𝔑∗𝜑 𝔑𝜓 ∩ 𝔑∗𝜑 𝔑𝜃 ∩ 𝔑∗𝜃 = ; 𝔑𝜑 ∩ 𝔑∗𝜓 𝔑𝜓 ∩ 𝔑∗𝜓 ( ∗ ) ( ) 𝔑𝜑 𝔑𝜑 𝔑∗𝜑 𝔑𝜓 𝔐𝜑 𝔑∗𝜑 𝔑𝜓 ∗ 𝔐𝜃 = 𝔑𝜃 𝔑𝜃 = = . 𝔑∗𝜓 𝔑𝜑 𝔑∗𝜓 𝔑𝜓 𝔑∗𝜓 𝔑𝜑 𝔐𝜓 𝔑𝜑 𝔑𝜑

(1) (2) (3)

In particular, 𝜃 is semifinite. Also ( s(𝜃) =

1 0 0 s(𝜓)

) .

(4)

In the proof of the Connes cocycle theorem ([L], 10.28), we have already considered the case when both 𝜑 and 𝜓 are faithful. Taking into account the remarks made in Section 2.22, we can extend the Connes theorem to the more general case considered here.

34

Normal Weights

Using Proposition 2.21, we obtain (

1 0 0 s(𝜓)

) ( ) ( ) ( ) 1 0 1 0 0 0 , ∈ 𝒩 𝜃 and hence , ∈ 𝒩 𝜃. 0 −s(𝜓) 0 0 0 s(𝜓)

Using, as in ([L], 10.28), the KMS condition, we infer that (( )) ( 𝜑 ) x 0 𝜎t (x) 0 (x ∈ ℳ), 𝜎t𝜃 = 0 0 0 0 (( )) ( ) 0 0 0 0 𝜎t𝜃 = (x ∈ s(𝜓)ℳs(𝜓)). 𝜓 0 y 0 𝜎t ( y) ( Since

0 s(𝜓)

0 0

(6)

)

(

𝜎t𝜃

(5)

1 0 ((

∈ s(𝜃)𝒩 s(𝜃) and since 0 0

)

((

0 s(𝜓) )) ( 0 0 0 s(𝜓) 0 0 𝜎t𝜃

0 0

))

0 s(𝜓)

= )

𝜎t𝜃

= 𝜎t𝜃

(( ((

1 0

0 0

0 s(𝜓)

)(

)) 0 0 = 0, s(𝜓) 0 )( )) 0 0 0 = 0, 0 0 s(𝜓)

it follows that there exists ut ∈ ℳ such that 𝜎t𝜃

((

0 0 s(𝜓) 0

))

( =

0 ut

0 0

) .

Thus, we get an s∗ -continuous mapping ℝ ∋ t ↦ ut ∈ ℳ such that ut u∗t = s(𝜓) = u0 , u∗t ut = 𝜎t𝜑 (s(𝜓)), ut+s = ut 𝜎t𝜑 (us ), 𝜑 ∗ u−t = 𝜎−t (ut ), 𝜑 𝜓 ∗ 𝜎t (x) = ut 𝜎t (x)ut (x ∈ s(𝜓)ℳs(𝜓)).

(7) (8) (9) (10)

The arguments for checking (7), (8), and (10) are similar to those given in ([L], 10.28) and (9) is an easy consequence of (7) and (8). For X = [xij ] ∈ 𝒩 , we have X ∈ s(𝜃)𝒩 s(𝜃) if and only if x12 = x12 s(𝜓), x21 = s(𝜓)x21 , x22 = s(𝜓)x22 = x22 s(𝜓); in this case 𝜎t𝜃

((

x11 x21

x12 x22

))

( =

𝜎t𝜑 (x11 ) ut 𝜎t𝜑 (x21 )

𝜎t𝜑 (x12 )u∗t 𝜎t𝜓 (x22 )

) .

(11)

On the other hand, the weight 𝜃 satisfies the KMS condition with respect to {𝜎t𝜃 }t∈ℝ . In order to avoid notational complications, we assume that s(𝜓) = 1. Then, for any X, Y ∈ 𝔑𝜃 ∩ 𝔑∗𝜃 , there exists a function F defined, continuous and bounded on the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, analytic in the interior of this strip, and such that F(it) = 𝜃(X𝜎t𝜃 (Y )), F(1 + it) = 𝜃(𝜎t𝜃 (Y )X ) for all t ∈ ℝ.

The Balanced Weight

35

(

) ( ) 0 x 0 0 In particular, if X = ,Y = with x ∈ ℛ𝜓 ∩ 𝔑∗𝜑 , y ∈ 𝔑𝜑 ∩ ℛ𝜓∗ , then F(it) = 0 0 y 0 𝜑(xut 𝜎t𝜑 ( y)), F(1 + it) = 𝜓(𝜎t𝜓 ( y)ut x) for all t ∈ ℝ. We have thus proved the existence part of the following Theorem (A. Connes). Let ℳ be a W ∗ -algebra, 𝜑 an n.s.f. weight on ℳ and 𝜓 a normal semifinite weight on ℳ. There exists an s∗ -continuous mapping ℝ ∋ t ↦ ut ∈ ℳ, uniquely determined, such that ut u∗t = s(𝜓) = u0 , u∗t ut = 𝜎t𝜑 (s(𝜓)) ut+s = ut 𝜎t𝜑 (us ) 𝜑 ∗ u−t = 𝜎−t (ut ) 𝜓 𝜑 ∗ 𝜎t (x) = ut 𝜎t (x)ut (x ∈ s(𝜓)ℳs(𝜓)),

(1) (2) (3) (4)

for every x = xs(𝜓) ∈ 𝔑𝜓 ∩ 𝔑∗𝜑 and every y = s(𝜓) ∈ 𝔑𝜑 ∩ 𝔑∗𝜓 , there exists a function F defined, continuous and bounded on the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, analytic in the interior of this strip, such that F(it) = 𝜑(xut 𝜎t𝜑 ( y)), F(1 + it) = 𝜓(𝜎t𝜓 ( y)ut x) for all t ∈ ℝ.

(5)

To prove the uniqueness part of the theorem, we consider an s∗ -continuous mapping ℝ ∋ t ↦ vt ∈ ℳ with the same properties and we assume that s(𝜓) = 1. For each t ∈ ℝ, we define a mapping 𝜎t ∶ 𝒩 → 𝒩 by putting (( 𝜎t

x11 x21

x12 x22

))

( =

𝜎t𝜑 (x11 ) vt 𝜎t𝜑 (x21 )

v∗t 𝜎t𝜓 (x12 ) 𝜎t𝜓 (x22 )

) .

Conditions (1)–(4) insure that {𝜎t }t∈ℝ is an s∗ -continuous one-parameter group of *-automorphisms of 𝒩 , which preserve the weight 𝜃. For X = [xij ] ∈ 𝔑𝜃 ∩ 𝔑∗𝜃 and Y = [yij ] ∈ 𝔑𝜃 ∩ 𝔑∗𝜃 we have 𝜃(X𝜎t (Y )) = 𝜑(x11 𝜎t𝜑 ( y11 )) + 𝜑(x12 ut 𝜎t𝜑 ( y211 )) + 𝜓(x21 u∗t 𝜎t𝜓 ( y12 )) + 𝜓(x22 𝜎t𝜓 ( y22 )), 𝜃(𝜎t (Y )X ) = 𝜑(𝜎t𝜑 ( y11 )x11 ) + 𝜓(𝜎t𝜓 ( y21 )ut x12 ) + 𝜑(𝜎t𝜑 ( y12 )u∗t x21 ) + 𝜓(𝜎t𝜓 ( y22 )x22 ). Using the KMS condition satisfied by 𝜑 and 𝜓, we find two function F11 , F22 defined, continuous and bounded on the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, analytic in the interior of this strip, such that F11 (it) = 𝜑(x11 𝜎t𝜑 ( y11 )), F11 (1 + it) = (𝜎t ( y11 )x11 ), F22 (it) = 𝜓(x22 𝜎t𝜓 ( y22 )), F22 (1 + it) = 𝜓(𝜎t𝜓 ( y22 )x22 ). By condition (5), there exist two functions F12 , G21 defined, continuous and bounded on the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, analytic in the interior of this strip, such that F12 (it) = 𝜑(x12 ut 𝜎t𝜑 ( y21 )), F12 (1 + it) = 𝜓(𝜎t𝜓 ( y21 )ut x12 ), G21 (it) = 𝜑(x∗21 ut 𝜎t𝜑 ( y∗12 )), G21 (1 + it) = 𝜓(𝜎t𝜓 ( y∗12 )ut x∗21 ).

36

Normal Weights

Then the function F21 (𝛼) = G21 (−𝛼 + 1) satisfies the equalities F21 (it) = G21 (it + 1) = 𝜓(𝜎t𝜓 ( y∗21 )ut x∗21 ) = 𝜓(x21 u∗t 𝜎t𝜓 ( y12 )), F21 (1 + it) = G21 (it) = 𝜑(x∗21 ut 𝜎t𝜑 ( y∗12 )) = 𝜑(𝜎t𝜑 ( y12 )u∗t x21 ). Putting F = F11 + F12 + F21 + F22 , it follows that F(it) = 𝜃(X𝜎t (Y )), F(1 + it) = 𝜃(𝜎t (Y )X )

(t ∈ ℝ).

We have proved that 𝜃 satisfies the KMS condition with respect to {𝜎t }t∈ℝ . Consequently, 𝜎t = 𝜎t𝜃 and hence vt = ut for all t ∈ ℝ. The mapping ℝ ∋ t ↦ ut ∈ ℳ, uniquely determined by the above theorem, is called the Connes cocycle associated with the normal semifinite weight 𝜓 with respect to the n.s.f. weight 𝜑, and is denoted by [D𝜓 ∶ D𝜑], that is, [D𝜓 ∶ D𝜑]t = ut

(t ∈ ℝ).

3.2. Let ℳ be a W ∗ -algebra. Let U(ℳ) = {u ∈ ℳ; u∗ u = uu∗ = 1}, Int(ℳ) = {Ad(u); u ∈ U(ℳ)} ⊂ Aut(ℳ). Then Int(ℳ) is a normal subgroup of Aut(ℳ): 𝜎 ⋅ Ad(u) ⋅ 𝜎 −1 = Ad(𝜎(u))

(𝜎 ∈ Aut(ℳ), u ∈ U(ℳ)) .

Let Out(ℳ) = Aut(ℳ)∕Int(ℳ) and let 𝔞ℳ ∶ Aut(ℳ) → Out(ℳ) be the canonical quotient mapping. An obvious and important consequence of Theorem 3.1 is that the mapping 𝛿ℳ ∶ ℝ → Out(ℳ) defined by 𝛿ℳ (t) = 𝔞ℳ (𝜎t𝜑 ), (t ∈ ℝ), is a group homomorphism, independent of the choice of the n.s.f. weight 𝜑 on ℳ. The mapping 𝛿ℳ ∶ ℝ → Out(ℳ) is called the modular homomorphism of ℳ. Using 2.22.(5), we see that 𝛿ℳ (ℝ) is contained in the center of the group Out(ℳ). 3.3. Let 𝜑1 , 𝜑2 , … , 𝜑n be n.s.f. weights on the W ∗ -algebra ℳ. The equation 𝜃([xij ]) =

n ∑

𝜑k (xkk ) ([xij ] ∈ Matn (ℳ))

k=1

defines an n.s.f. weight 𝜃 on the W ∗ -algebra 𝒩 = Matn (ℳ) of n × n matrices over ℳ. For each i, j ∈ {1, … , n} we denote by eij ∈ Matn (ℂ) the matrix eij = [𝛿hi 𝛿kj ]1 𝜑(uyu∗ ) + 𝜑(vyv∗ ) = 𝜑(a1∕2 xa1∕2 ) + 𝜑(b1∕2 xb1∕2 ). Thus, again by 2.21.(2), we get 𝜑a (x) + 𝜑b (x) = 𝜑( y(u∗ u + v∗ v)) = 𝜑( y) = 𝜑a+b (x). Conversely let x ∈ 𝔐𝜑a +𝜑b ∩ ℳ + . Then a1∕2 xa1∕2 ∈ 𝔐𝜑 ∩ ℳ + , b1∕2 xb1∕2 ∈ 𝔐𝜑 ∩ ℳ + and by a further application of 2.21.(2), we infer that u∗ a1∕2 xa1∕2 u ∈ 𝔐𝜑 ∩ ℳ + , v∗ b1∕2 xb1∕2 v ∈ 𝔐𝜑 ∩ ℳ + . As (a + b)1∕2 x(a + b)1∕2 = w- lim(a + b + 𝜀)−1∕2 (a + b)x(a + b)(a + b + 𝜀)−1∕2 𝜀→0

≤ w- lim 2(a + b + 𝜀)−1∕2 (axa + bxb)(a + b + 𝜀)−1∕2 𝜀→0 ∗ 1∕2

= 2(u a

xa1∕2 u + v∗ b1∕2 xb1∕2 v),

it follows that x ∈ 𝔐𝜑a+b ∩ ℳ + . Thus, (1) is completely proved. From (1) we infer that a ≤ b ⇒ 𝜑a ≤ 𝜑b .

(2)

Finally, if a ∈ ℳ 𝜑 , a ≥ 0, and {ai }i∈I ⊂ ℳ 𝜑 is a net of positive elements, then ai ↑ a ⇒ 𝜑ai ↑ 𝜑a .

(3)

Indeed, let x ∈ ℳ + . From (2) it follows that supi 𝜑ai (x) ≤ 𝜑a (x). On the other hand, since 𝜑 is 1∕2

1∕2 w

normal (1.3) and ai xai supi 𝜑ai (a).

1∕2

1∕2

→ a1∕2 xa1∕2 , we have 𝜑a (x) = 𝜑(a1∕2 xa1∕2 ) ≤ lim infi 𝜑(ai xai ) ≤

4.4. Let 𝜑 be a normal semifinite weight on the W ∗ -algebra ℳ and A a positive self-adjoint operator affiliated to ℳ 𝜑 (A.16).

52

Normal Weights

We shall consider the bounded positive operators A𝜀 = A(1 + 𝜀A)−1 ∈ ℳ 𝜑

(𝜀 > 0);

Recall (A.5) that A𝜀 ↑ A for 𝜀 ↓ 0. Also, for each n ∈ ℕ, n ≥ 1, let en = 𝜒[1∕n,n] (A) ∈ ℳ 𝜑 ; we recall ([L], 9.9) that Aen is a bounded positive operator, invertible in en ℳ 𝜑 en , and en ↑ s(A). In view of 4.3.(2), a normal weight 𝜑A on ℳ is defined by 𝜑A (x) = sup 𝜑A𝜀 (x) = lim 𝜑A𝜀 (x) (x ∈ ℳ + ). 𝜀→0

𝜀→0

If A is bounded, then by 4.3.(3), we see that the weight 𝜑A defined here coincides with the weight defined in Section 4.1. If x ∈ ℳ + and s(x)s(A) = 0, then clearly 𝜑A (x) = 0. On the other hand, if x ∈ en (𝔐𝜑 ∩ ℳ + )en ⊂ 1∕2 1∕2 1∕2 1∕2 𝔐𝜑 ∩ ℳ + ⊂ 𝔐𝜑Ae ∩ ℳ + , then 𝜑A (x) = lim𝜀→0 𝜑(A𝜀 en xen A𝜀 ) = lim𝜀→0 𝜑((Aen )𝜀 x(Aen )𝜀 ) = n 𝜑Aen (x) < +∞. It follows that the weight 𝜑A is finite and s(𝜑A ) ≤ s(A). Actually, we have s(𝜑A ) = s(A).

(1) 1∕2

1∕2

1∕2

1∕2

Indeed, if x ∈ ℳ + and 𝜑A (x) = 0, then for every 𝜀 > 0 we have 𝜑(A𝜀 xA𝜀 ) = 0, A𝜀 xA𝜀 s(𝜑) = 0, xA𝜀 = 0, xs(A𝜀 ) = 0, and hence xs(A) = 0. We remark that for every x ∈ 𝔑𝜑 we have 𝜑A (x∗ x) = ‖𝜋𝜑 (A)1∕2 J𝜑 x𝜑 ‖2𝜑 ≤ +∞.

(2)

Indeed, using 2.21.(1) and (A.5), we obtain ∗ 1∕2 ∗ 𝜑A (x∗ x) = lim 𝜑(A1∕2 𝜀 x xA𝜀 ) = lim 𝜑(A𝜀 x x) = lim(x𝜑 |(xA𝜀 )𝜑 )𝜑 𝜀→0

𝜀→0

= lim(x𝜑 | J𝜑 𝜋𝜑 (A𝜀 )J𝜑 x𝜑 )𝜑 = ‖𝜋𝜑 (A)

𝜀→0

1∕2

𝜀→0

J𝜑 x𝜑 ‖2𝜑 .

4.5 Proposition. Let 𝜑 be a normal semifinite weight on the W ∗ -algebra ℳ and let A, B be positive self-adjoint operators affiliated to ℳ 𝜑 . Then: A ≤ B ⇔ 𝜑A ≤ 𝜑B . Proof. If A ≤ B, then (A.4) for every 𝜀 > 0 we have A𝜀 ≤ B𝜀 so 𝜑A𝜀 ≤ 𝜑B𝜀 by 4.3.(2), and hence 𝜑A (x) = lim𝜀 𝜑A𝜀 (x) ≤ lim𝜀 𝜑B𝜀 (x) = 𝜑B (x) for all x ∈ ℳ + , that is, 𝜑A ≤ 𝜑B . Conversely, assume that 𝜑A ≤ 𝜑B . Let 𝜀 > 0 and fn = 𝜒[0,n] (B) ∈ ℳ 𝜑 for each n ∈ ℕ. Then for every x ∈ 𝔑𝜑 we have (2.21.(1)): (𝜋𝜑 ( fn A𝜀 fn )J𝜑 x𝜑 | J𝜑 x𝜑 )𝜑 = 𝜑A𝜀 ( fn x∗ xfn ) ≤ 𝜑B ( fn x∗ xfn ) = (𝜋𝜑 (Bfn )J𝜑 x𝜑 | J𝜑 x𝜑 )𝜑

The Pedersen–Takesaki Construction

53

so that fn A𝜀 fn ≤ Bfn . Since fn ↑ s(B) = s(𝜑B ) ≥ s(𝜑A ) = s(A), it follows that A𝜀 ≤ B. Since A𝜀 ↑ A, we conclude A ≤ B. 4.6 Proposition. Let 𝜑 be a normal semifinite weight on the W ∗ -algebra ℳ and let A, {Ai }i∈I be positive self-adjoint operators affiliated to ℳ 𝜑 . Then: Ai ↑ A ⇔ 𝜑Ai (x) ↑ 𝜑A (x) for all x ∈ ℳ + . Proof. Assume that Ai ↑ A and let x ∈ ℳ + . From Proposition 4.5, it follows that supi 𝜑Ai (x) ≤ 𝜑A (x). On the other hand, since Ai ↑ A, we have (Ai )𝜀 ↑ A𝜀 , for all 𝜀 > 0 (A.5) and using 4.3.(3) we deduce supi 𝜑(Ai )𝜀 (x) = 𝜑A𝜀 (x) for all 𝜀 > 0. Consequently, 𝜑A (x) = sup𝜀 𝜑A𝜀 (x) = sup𝜀 supi 𝜑(Ai )𝜀 (x) ≤ supi 𝜑Ai (x). Conversely, if 𝜑Ai ↑ 𝜑A , then, by Proposition 4.5, {Ai }i∈I is an increasing net bounded above by A. By (4.5) there exists a positive self-adjoint operator B such that Ai ↑ B. It follows that B is affiliated to ℳ 𝜑 and, by the first part of the proof, 𝜑Ai ↑ 𝜑B . Consequently, 𝜑B = 𝜑A . Using Proposition 4.5 again we obtain A ≤ B and B ≤ A, that is, (A.4) A = B. In particular, with the notation of Section 4.4, we have the following equivalent definition of 𝜑A : 𝜑A (x) = sup 𝜑Aen (x) = lim 𝜑Aen (x) (x ∈ ℳ + ). n

n

(1)

4.7 Proposition. Let 𝜑 be a normal semifinite weight on the W ∗ -algebra ℳ and A a positive selfadjoint operator affiliated to ℳ 𝜑 . Then 𝜑

𝜎t A (x) = Ait 𝜎t𝜑 (x)A−it

(x ∈ s(A)ℳs(A), t ∈ ℝ).

Proof. Assume the W ∗ -algebra ℳ realized as a von Neumann algebra ℳ ⊂ ℬ(ℋ ). Then A|s(A)ℋ is a nonsingular positive self-adjoint operator on the Hilbert space s(A)ℋ , so that for each t ∈ ℝ we can form the unitary operator Ait = (A|s(A))it on s(A)ℋ . Of course, the operator Ait can also be regarded as a partial isometry in ℳ. We shall use the notation introduced in Section 4.4. Let n ∈ ℕ be fixed. Then 𝜑en is an n.s.f. weight on the W ∗ -algebra en ℳen and Aen is a bounded invertible positive operator in en ℳ 𝜑 en , which is the centralizer of the weight 𝜑en . Using 4.6.(1), we see that (𝜑A )en = (𝜑en )Aen as n.s.f. weights on en ℳen . Thus, taking into account 2.22.(3) and 4.2.(4), for x ∈ en ℳen we obtain 𝜑

𝜑

𝜎t A (x) = 𝜎 (𝜑A )en (x) = 𝜎 (𝜑en )Aen (x) = (Aen )it 𝜎t en (x)(Aen )−it = Ait en 𝜎t𝜑 (x)en A−it = Ait 𝜎t𝜑 (x)A−it . Since en ↑ s(A), the set follows.

⋃

n∈ℕ en ℳen

is w-dense in s(A)ℳs(A). Thus, the assertion of the proposition

4.8 Corollary. Let 𝜑 be a normal semifinite weight on the W ∗ -algebra ℳ and A a positive selfadjoint operator affiliated to ℳ. Then [D𝜑A ∶ D𝜑]t = Ait

(t ∈ ℝ).

54

Normal Weights

Proof. ( Consider ) the balanced weights 𝜃 = 𝜃(𝜑, 𝜑) and 𝜏 = 𝜃(𝜑, 𝜑A ) on Mat2 (ℳ). Then 𝜏 = 𝜃B with 1 0 B= . By Proposition 4.7, we have 𝜎t𝜏 (X ) = Bit 𝜎t𝜃 (X )B−it for all X ∈ s(B)[Mat2 (ℳ)]s(B). 0 A ( ) 0 0 In particular, for X = we obtain the desired result. s(A) 0 From the above corollary and from Corollary 3.6, it follows again (see 4.5) that 𝜑A = 𝜑B ⇔ A = B. 4.9 Corollary. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and let A, B be two commuting positive self-adjoint operators affiliated to ℳ 𝜑 . Then the weights 𝜑AB , (𝜑A )B , (𝜑B )A are defined and equal: (𝜑A )B = 𝜑AB = (𝜑B )A Proof. Since A and B commute, the closure AB of AB is a positive self-adjoint operator and (AB)it = Ait Bit (t ∈ ℝ) (A.6). It is easy to check that AB (resp. A, B) is affiliated to the centralizer of 𝜑 (resp. 𝜑B , 𝜑A ), hence the weights 𝜑AB , (𝜑A )B , (𝜑B )A are defined. Using Corollaries 3.5 and 4.8, we see that the Connes cocycles of these weights with respect to 𝜑 coincide and hence, by Corollary 3.6, these weights are equal. 4.10 Theorem (Pedersen & Takesaki, 1973). Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and 𝜓 a normal semifinite weight on M. The following conditions are equivalent: (i) (ii) (iii) (iv) (v)

𝜓 ◦ 𝜎t𝜑 = 𝜓 for all t ∈ ℝ; [D𝜓 ∶ D𝜑]t ∈ ℳ 𝜓 for all t ∈ ℝ; [D𝜓 ∶ D𝜑]t ∈ ℳ 𝜑 for all t ∈ ℝ; {[D𝜓 ∶ D𝜑]t }t∈ℝ is an s-continuous group of unitary elements of s(𝜓)ℳs(𝜓); there exists a positive self-adjoint operator A affiliated to ℳ 𝜑 such that 𝜓 = 𝜑A .

If moreover 𝜓 is faithful, then also the following statement is equivalent to those earlier: (vi) 𝜑 ◦ 𝜎t𝜓 = 𝜑 for all t ∈ ℝ. Proof. Let ut = [D𝜓 ∶ D𝜑], (t ∈ ℝ). (i) ⇒ (ii). From (i) it follows that s(𝜓) and ℳ 𝜓 are 𝜎 𝜑 -invariant. In particular, for every t ∈ ℝ we have 𝜑 𝜑 𝜑 𝜑 𝜎−t (ut )𝜎−t (ut )∗ = s(𝜓) = 𝜎−t (ut )∗ 𝜎−t (ut ).

Then, for every x ∈ (s(𝜓)ℳs(𝜓))+ and every t ∈ ℝ we obtain 𝜓(s(𝜓)xs(𝜓)) = 𝜓(x) = 𝜓(𝜎t𝜓 (x)) = 𝜓(ut 𝜎t𝜑 (x)u∗t ) 𝜑 𝜑 ∗ 𝜑 𝜑 (ut )x𝜎−t (ut ))) = 𝜓(𝜎−t (ut )x𝜎−t (ut )∗ ). = 𝜓(𝜎t𝜑 (𝜎−t 𝜑 𝜑 By Proposition 2.21, it follows that 𝜎−t (ut ) ∈ ℳ 𝜓 , and hence ut = 𝜎t𝜑 (𝜎−t (ut )) ∈ ℳ 𝜓 . 𝜓 𝜑 ∗ ∗ (ii) ⇒ (iv). From (ii) it follows that ut = 𝜎s (ut ) = us 𝜎s (ut )us = us+t us and therefore us+t = ut us for all s, t ∈ ℝ. (iii) ⇒ (iv). If ut ∈ ℳ 𝜑 , then us+t = us 𝜎s𝜑 (ut ) = us ut . Conversely, if us ut = us+t = us 𝜎s𝜑 (ut ), then ut = 𝜎s𝜑 (ut ), hence ut ∈ ℳ 𝜑 .

The Pedersen–Takesaki Construction

55

(iv) ⇒ (v). If the condition (iv) holds then, by Stone’s theorem ([L], 9.20), there exists a positive self-adjoint operator A affiliated to ℳ 𝜑 , with s(A) = s(𝜓), such that [D𝜓 ∶ D𝜑]t = Ait = [D𝜑A ∶ D𝜑]t (t ∈ ℝ). By Corollary 3.6, it follows that 𝜓 = 𝜑A . (v) ⇒ (i). This follows obviously from the definition of 𝜑A . If 𝜓 is faithful, then (vi) is equivalent to the other conditions owing to the symmetry between (ii) and (iii). If the above equivalent conditions are satisfied, we shall say that the normal semifinite weight 𝜓 commutes with the n.s.f. weight 𝜑 (see also [L], Cor. 10.28). Thus, the weights of the form 𝜑A are exactly the weights commuting with 𝜑. Clearly, if 𝜓 commutes with 𝜑, then s(𝜓) ∈ ℳ 𝜑 . 4.11 Corollary. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ with center 𝒵 (ℳ) and 𝜓 a normal semifinite weight on ℳ. The following conditions are equivalent: (i) 𝜓 satisfies the KMS condition with respect to {𝜎t𝜑 }t∈ℝ ; (ii) s(𝜓) ∈ 𝒵 (ℳ) and 𝜎t𝜓 = 𝜎t𝜑 |ℳs(𝜓) for all t ∈ ℝ; (iii) there exists a positive self-adjoint operator A affiliated to 𝒵 (ℳ) such that 𝜓 = 𝜑A . Proof. Let q = s(𝜓) and p = 1 − q. (i) ⇒ (ii). Since 𝜓 ◦ 𝜎t𝜑 = 𝜓, we have p, q ∈ ℳ 𝜑 . Let x, y ∈ 𝔑𝜓 ∩ 𝔑∗𝜓 . Then 𝜓((xp)∗ (xp)) = 𝜓(px∗ xp) = 0 and 𝜓((py)∗ (py)) = 𝜓( y∗ py) ≤ 𝜓( y∗ y) < +∞, hence xp, py ∈ 𝔑𝜓 ∩ 𝔑∗𝜓 . By assumption, there exists a function f, defined, continuous and bounded on the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, analytic in the interior of this strip, such that f (it) = 𝜓(𝜎t𝜑 (xp)py) = 𝜓(𝜎t𝜑 (x)py) and f (1+ it) = 𝜓(py𝜎t𝜑 (xp)) = 0 for all t ∈ ℝ. It follows that f is identically zero, in particular 𝜓(𝜎t𝜑 (x)py) = 0 (t ∈ ℝ). Consequently, 𝜓(xpx∗ ) = 0 for every x ∈ 𝔑𝜓 ∩ 𝔑∗𝜓 , so that xpx∗ = pxpx∗ p for all x ∈ ℳ. In particular, for x = v ∈ ℳ, unitary, we have vpv∗ ≤ p, vpv∗ = p and therefore p ∈ 𝒵 (ℳ) and s(𝜓) = q = 1 − p ∈ 𝒵 (ℳ). Now, the restriction of 𝜓 to ℳq is an n.s.f. weight on ℳq which satisfies the KMS condition with respect to {𝜎t𝜑 |ℳq}t∈ℝ , so that 𝜎t𝜑 |ℳq = 𝜎t𝜓 (t ∈ ℝ), by 2.12.(11). (ii) ⇒ (iii). From (ii) it follows that 𝜓 ◦ 𝜎t𝜑 = 𝜓, (t ∈ ℝ). By Theorem 4.10, we have 𝜓 = 𝜑A for some positive self-adjoint operator A affiliated to ℳ 𝜑 . Then for every x ∈ ℳq and every t ∈ ℝ, we have 𝜎t𝜑 (x) = 𝜎t𝜓 (x) = Ait 𝜎t𝜑 (x)A−it , hence A is affiliated to 𝒵 (ℳ)q ⊂ 𝒵 (ℳ). Finally, the implication (iii) ⇒ (ii) follows from Proposition 4.7, as s(𝜓) = s(A) ∈ 𝒵 (ℳ), while the implication (ii) ⇒ (i) is obvious. The equivalent conditions in the above corollary hold, for instance, if 𝜑, 𝜓, are n.s.f. traces on ℳ. 4.12. From ([L], 10.29), we know that a W ∗ -algebra ℳ is semifinite if and only if there exist an n.s.f weight 𝜑 on ℳ and an s-continuous one-parameter group {ut }t∈ℝ of unitary operators in ℳ such that 𝜎t𝜑 (x) = ut xu∗t (x ∈ ℳ, t ∈ ℝ). The preceding results allow a simple proof of this theorem. If 𝜇 is an n.s.f. trace on ℳ and 𝜑 is any n.s.f. weight on ℳ, then ut = [D𝜑 ∶ D𝜇]t (t ∈ ℝ) satisfies the required condition because any trace commutes with any weight. Conversely, if this condition is satisfied and A is the unique positive self-adjoint operator such that A−it = ut (t ∈ ℝ), then 𝜇 = 𝜑A is an n.s.f. trace on ℳ because 𝜎t𝜇 (x) = x(x ∈ ℳ, t ∈ ℝ). 4.13 Corollary. Let 𝜑 be an n.s.f weight on the W ∗ -algebra ℳ of type III and 𝛽 ∈ ℝ, 𝛽 ≠ 1. Then 𝜑 no n.s.f. weight on ℳ satisfies the KMS condition with respect to {𝜎𝛽t }t∈ℝ .

56

Normal Weights

𝜑 Proof. Assume to the contrary; then there exists an n.s.f. weight 𝜓 on ℳ such that 𝜎𝛽t = 𝜎t𝜓 (t ∈ ℝ), in particular 𝜓 commutes with 𝜑. Consequently, there exists a positive self-adjoint operator 𝜑 A affiliated to ℳ 𝜑 such that 𝜓 = 𝜑A . It follows that 𝜎𝛽t (x) = Ait 𝜎t𝜑 (x)A−it (x ∈ ℳ, t ∈ ℝ). Putting 𝜑 𝛼 = (𝛽 − 1)−1 , we obtain 𝜎t (x) = (A𝛼 )it x(A𝛼 )−it (x ∈ ℳ, t ∈ ℝ), contradicting the fact that ℳ is not semifinite (4.12).

4.14. Let 𝜑, 𝜓 be n.s.f. weights on the W ∗ -algebra ℳ and 𝜎 ∈ Aut(ℳ). By 2.22.(5), we get 𝜑 ◦ 𝜎 = 𝜑 ⇒ 𝜎t𝜑 ◦ 𝜎 = 𝜎 ◦ 𝜎t𝜑 for all t ∈ ℝ.

(1)

𝜓 commutes with 𝜑 ⇒ 𝜎t𝜑 ◦ 𝜎s𝜓 = 𝜎s𝜓 ◦ 𝜎t𝜑 for all s, t ∈ ℝ.

(2)

In particular,

In general, the converse of (1) is not true. For instance, there exist measurable but not measurepreserving transformations on measure spaces. Even with the supplementary assumption that 𝜎 acts identically on the center 𝒵 (ℳ) of ℳ, the converse of (1) need not hold, as there exist *-automorphisms of type II∞ -factors which do not preserve the trace. Also, the converse of statement (2) is not valid in general, as we shall see in the next section in an important example. Following this example, we shall say that the weights 𝜑, 𝜓 anticommute if they do not commute but the corresponding modular automorphism groups commute. However, there are certain special cases in which the converses of (1) and (2) are true. These cases are considered in Sections 4.17–4.20. 4.15. Consider the Hilbert space ℋ = ℒ 2 (ℝ) and the operators ut , vs ∈ ℬ(ℋ ) defined by (ut 𝜉)(r) = 𝜉(r + t), (vs 𝜉)(r) = eisr 𝜉(r)

(𝜉 ∈ ℋ , r, s, t ∈ ℝ),

Then {ut }t∈ℝ and {vs }s∈ℝ are so-continuous one-parameter groups of unitary operators on ℋ , which satisfy the following anticommutation relations: vs ut = e−its ut vs

(s, t ∈ ℝ).

(1)

By Stone’s theorem ([L], 9.20), there exist positive self-adjoint operators A, B in ℋ , uniquely determined, such that ut = Ait (t ∈ ℝ), and vs = Bis (s ∈ ℝ). Thus Bis Ait = e−its Ait Bis (s, t ∈ ℝ), and using the definition of the operator A ([L], 9.20) we infer that B−is ABis = es A (s ∈ ℝ).

(2)

Consider the W ∗ -algebra ℳ = ℬ(ℋ ) with the canonical trace tr and the n.s.f. weights 𝜑 = trA , 𝜓 = trB on ℳ, Then, by Proposition 4.7, we have 𝜎t𝜑 = Ad(ut ), 𝜎s𝜓 = Ad(vs )(s, t ∈ ℝ), and it follows from (1) that 𝜎t𝜑 ◦ 𝜎s𝜓 = 𝜎s𝜓 ◦ 𝜎t𝜑

(s, t ∈ ℝ).

(3)

The Pedersen–Takesaki Construction

57

However, 𝜑 and 𝜓 do not commute, more precisely we have 𝜑 ◦ 𝜎s𝜓 = es 𝜑

(s ∈ ℝ).

(4)

Indeed, let x ∈ ℳ + and A𝜀 = A(1 + 𝜀A)−1 (𝜀 > 0). From (2), it follows that B−is A𝜀 Bis = (es A)𝜀 (s ∈ ℝ, 𝜀 > 0), and therefore is −is 1∕2 𝜑(𝜎s𝜓 (x)) = trA (Bis xB−is ) = lim tr(A1∕2 𝜀 B xB A𝜀 )

= lim tr(x

1∕2 −is

B

𝜀→0

s

= lim tr((e 𝜀→0

𝜀→0 A𝜀 Bis x1∕2 )

s 1∕2 A)1∕2 𝜀 x(e A)𝜀 )

= lim tr(x1∕2 (es A)𝜀 x1∕2 ) 𝜀→0

= tres A (x) = es 𝜑(x).

4.16. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ, 𝜎 ∈ Aut(ℳ) and A a positive self-adjoint operator affiliated to ℳ. The operator A is affiliated to ℳ 𝜑 ◦ 𝜎 if and only if the operator 𝜎(A) is affiliated to ℳ 𝜑 (see 2.22.(6) and [L], 9.25) and in this case we have (𝜑 ◦ 𝜎)A = 𝜑𝜎(A) ◦ 𝜎.

(1)

Using Corollary 4.8, it follows that if A is affiliated to ℳ 𝜑 , then 𝜑 ◦ 𝜎 = 𝜑, 𝜑A ◦ 𝜎 = 𝜑A ⇒ 𝜎(A) = A.

(2)

Note that the set of all nonsingular positive self-adjoint operators affiliated to the center 𝒵 (ℳ) of ℳ is a group with respect to the operation (A, B) ↦ AB. Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and 𝜎 ∶ G → Aut(ℳ) an action of the group G on ℳ. We assume that each *-automorphism 𝜎g (g ∈ G), acts identically on 𝒵 (ℳ). Then the following statements are equivalent: (i) 𝜎g ◦ 𝜎t𝜑 = 𝜎t𝜑 ◦ 𝜎g for every g ∈ G and every t ∈ ℝ; (ii) there exists a homomorphism g ↦ Ag of the group G into the group of all nonsingular positive self-adjoint operators affiliated to 𝒵 (ℳ) such that 𝜑 ◦ 𝜎g = 𝜑Ag for every g ∈ G. If moreover G = ℝ and the action 𝜎 ∶ ℝ → Aut(ℳ) is continuous, then the following statement is equivalent to (i) and (ii). (iii) there exists a nonsingular positive self-adjoint operator A affiliated to 𝒵 (ℳ) such that 𝜑 ◦ 𝜎s = 𝜑As for every s ∈ ℝ. Proof. It is clear that (iii) ⇒ (ii), and (ii) → (i) follows by using Corollary 4.11 and 2.22.(5). With the same arguments, from (i) it follows that for each g ∈ G there exists a unique Ag such that 𝜑 ◦ 𝜎g = 𝜑Ag . Since each 𝜎g acts identically on 𝒵 (ℳ), for g, h ∈ G we obtain 𝜑Agh = (𝜑 ◦ 𝜎g ) ◦ 𝜎h = 𝜑Ag ◦ 𝜎h = (𝜑 ◦ 𝜎h )Ag = (𝜑Ah )Ag = 𝜑A A

g h

by Corollary 4.9. Hence (i) ⇒ (ii).

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Normal Weights

We now show that (ii) ⇒ (iii). By assumption, there exists a homomorphism ℝ ∋ s ↦ As such that 𝜑 ◦ 𝜎s = 𝜑As (s ∈ ℝ). Let A = A1 . Since As+t = As At , it follows that As = As for every rational number s ∈ ℝ. Let en = 𝜒[1∕n,n] (A) and x ∈ 𝔑𝜑 . The function ℝ ∋ s ↦ 𝜑(As en x∗ x) = ‖𝜋𝜑 (As en )1∕2 x𝜑 ‖2𝜑

(3)

is continuous and bounded. On the other hand, we have 𝜑(𝜎s (en x∗ x)) = lim 𝜑((As )𝜀 en x∗ x) 𝜀→0

2 1∕2 x𝜑 ‖𝜑2 . = lim ‖𝜋𝜑 (As )1∕2 𝜀 𝜋𝜑 (en )x𝜑 ‖𝜑 = ‖𝜋𝜑 (As en ) 𝜀→0

Since the weight 𝜑 is normal, it follows that the function ℝ ∋ s ↦ 𝜑(𝜎s (en x∗ x)) = ‖𝜋𝜑 (As en )1∕2 x𝜑 ‖2𝜑 ∈ [0, +∞]

(4)

is lower w-semicontinuous. As the functions (3) and (4) coincide for every rational s ∈ ℝ and every x ∈ 𝔑𝜑 , we infer that As en ≤ As en

(s ∈ ℝ).

Consequently, s ↦ (As en )(As en )−1 is a one-parameter group of positive operators with norm ≤ 1 on en ℋ . Since the only such group is the trivial one, it follows that As en = As en , and, since en ↑ 1, we conclude that As = As for all s ∈ ℝ. Let 𝜎 ∶ G → Aut(ℳ) be an action of G on ℳ which is trivial on 𝒵 (ℳ) and satisfies the equivalent conditions (i) and (ii) of the above proposition. Then G0 = {g ∈ G; 𝜑 ◦ 𝜎g = 𝜑} is the kernel of the homomorphism g ↦ Ag and hence a normal subgroup of G. Since {Ag }g∈G is an abelian group, it follows that the quotient group G∕G0 is abelian. 4.17 Corollary. Let 𝜑 be a faithful normal state on the W ∗ -algebra ℳ and 𝜎 ∶ G → Aut(ℳ) an action of the group G on ℳ which is trivial on 𝒵 (ℳ). Then the following statements are equivalent: (i) 𝜑 ◦ 𝜎g = 𝜑 for all g ∈ G; (ii) 𝜎t𝜑 ◦ 𝜎g = 𝜎g ◦ 𝜎t𝜑 for all g ∈ G and all t ∈ ℝ. Proof. By 4.14.(1), we know that (i) ⇒ (ii). If (ii) holds, then we can write 𝜑 ◦ 𝜎g = 𝜑Ag (g ∈ G), as in Proposition 4.16. Let g ∈ G, 𝜀 > 0 and let e𝜀 = 𝜒[1+𝜀,+∞) (Ag ). Then for every k ∈ ℕ we get ‖𝜑‖ ≥ 𝜑(𝜎gk (e𝜀 )) = 𝜑(Akg e𝜀 ) ≥ (1 + 𝜀)k 𝜑(e𝜀 ), hence e𝜀 = 0. It follows that Ag ≤ 1, so 1 ≤ A−1 g = Ag−1 ≤ 1 and hence Ag = 1 for all g ∈ G. 4.18 Corollary. Let 𝜑, 𝜓 be n.s.f. weights on the W ∗ -algebra ℳ. If 𝜑 is finite, or if 𝜑 ≤ 𝜓, then the following are equivalent: (i) 𝜑 commutes with 𝜓; (ii) 𝜎t𝜑 ◦ 𝜎s𝜓 = 𝜎s𝜓 ◦ 𝜎t𝜑 for all s, t ∈ ℝ. Proof. By 4.14.(2), we know that (i) ⇒ (ii). Also, if 𝜑 is finite, then (ii) ⇒ (i) by Corollary 4.17.

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59

Assume that (ii) holds and 𝜑 ≤ 𝜓. By Proposition 4.16, there exists a nonsingular positive self-adjoint operator A affiliated to the center 𝒵 (ℳ) such that 𝜑 ◦ 𝜎s𝜓 = 𝜑As (s ∈ ℝ). Let e𝜀 = 𝜒[1+𝜀,+∞) (A). For x ∈ 𝔐𝜓 ∩ ℳ + and s ∈ ℝ, s ≥ 0, we have +∞ > 𝜓(x) ≥ 𝜓(e𝜀 x) = 𝜓(es 𝜎s𝜓 (x)) ≥ 𝜑(es 𝜎s𝜓 (x)) = 𝜑(As es x) ≥ (1 + 𝜀)s 𝜑(e𝜀 x). Consequently, 𝜑(e𝜀 x) = 0. Since 𝜑 is normal and semifinite, it follows that 𝜑(e𝜀 ) = 0; since 𝜑 is faithful we conclude e𝜀 = 0. Thus, A ≤ 1. Similarly, using f = 𝜒[0,1−𝜀] (A), we get A ≥ 1. Hence A = 1 and 𝜑 commutes with 𝜓. 4.19. Let ℳ be a W ∗ -algebra and 𝜑 ∈ ℳ∗ . Put 𝔏𝜑 = {x ∈ ℳ; 𝜑(ax) = 0 for all a ∈ ℳ}, ℜ𝜑 = {x ∈ ℳ; 𝜑(xa) = 0 for all a ∈ ℳ}. Then 𝔏𝜑 (resp. ℜ𝜑 ) is a w-closed left (resp. right) ideal of ℳ, hence ([L], 3.20) there exists a unique projection e ∈ ℳ (resp. f ∈ ℳ) such that 𝔏𝜑 = ℳe (resp. ℜ𝜑 = fℳ). The projection r(𝜑) = 1 − e (resp. l(𝜑) = 1 − f) is called the right support of 𝜑 (resp. the left support of 𝜑) in ℳ. Thus, 𝔏𝜑 = ℳ(1 − r(𝜑)), ℜ𝜑 = (1 − l(𝜑))ℳ.

(1)

Since 1 − r(𝜑) ∈ 𝔏𝜑 and 1 − l(𝜑) ∈ ℜ𝜑 , we get 𝜑 = 𝜑(⋅r(𝜑)) = 𝜑(l(𝜑)⋅).

(2)

Also, using the Hahn–Banach theorem, we infer from (1) that {𝜑(a⋅); a ∈ ℳ} is norm-dense in ℳ∗ ⋅ r(𝜑), {𝜑(⋅a); a ∈ ℳ} is norm-dense in l(𝜑) ⋅ ℳ∗ .

(3)

Since 𝔏𝜑 = (ℜ𝜑∗ )∗ , we have r(𝜑) = l(𝜑∗ ). In particular, if 𝜑 = 𝜑∗ , then r(𝜑) = l(𝜑) is called the support of 𝜑 and is denoted by s(𝜑). If 𝜑 is positive, then the Schwarz inequality implies {x ∈ ℳ; 𝜑(x∗ x) = 0} = 𝔏𝜑 = ℳ(1 − s(𝜑)), hence s(𝜑) is the usual (2.1) support of 𝜑. Consider now 𝜑, 𝜓 ∈ ℳ∗+ . It is easy to check that if 𝜑 is faithful, then the left and right supports of the forms 𝜑 + i𝜓 and 𝜑 − i𝜓 are all equal to 1, hence their absolute values |𝜑 + i𝜓| and |𝜑 − i𝜓| are faithful normal positive forms and the partial isometries from the corresponding polar decompositions are unitary elements ([L], 5.16, E.5.10). Proposition. Let 𝜑, 𝜓 be normal positive forms on the W ∗ -algebra ℳ and s(𝜑) = 1. Then the following conditions are equivalent: (i) 𝜓 commutes with 𝜑; (ii) |𝜑 + i𝜓| = |𝜑 − i𝜓|; (iii) [assume also s(𝜓) = 1] 𝜎t𝜑 ◦ 𝜎s𝜓 = 𝜎s𝜓 ◦ 𝜎t𝜑 for all s, t ∈ ℝ. Proof. The equivalence (i) ⇔ (iii) follows from 4.18. (i) ⇒ (ii). By assumption (4.10), we have 𝜓 = 𝜑A for some positive self-adjoint operator A affiliated to ℳ 𝜑 . It follows that (𝜑 + i𝜓)(⋅(1 + iA)−1 ) = 𝜑 = (𝜑 − i𝜓)(⋅(1 − iA)−1 ), hence u = (1 − iA)(1 + iA)−1 ∈ ℳ 𝜑 is a unitary operator and we have (𝜑 + i𝜓)(⋅u) = 𝜑 − i𝜓.

60

Normal Weights

If 𝜑 + i𝜓 = 𝜔(⋅v) is the polar decomposition of 𝜑 + i𝜓, then 𝜑 − i𝜓 = 𝜔(⋅uv) is the polar decomposition of 𝜑 − i𝜓, and hence |𝜑 + i𝜓| = 𝜔 = |𝜑 − i𝜓|. (ii) ⇒ (i). Let 𝜑 + i𝜓 = 𝜔(⋅v) be the polar decomposition of 𝜑 + i𝜓, with 𝜔 = |𝜑 + i𝜓| and v ∈ ℳ unitary. By assumption, we have 𝜔 = |𝜑 − i𝜓| = |(𝜑 + i𝜓)∗ | = 𝜔(v∗ ⋅ v). It follows that 𝜑 + i𝜓 = (𝜑 + i𝜓)(v∗ ⋅ v), that is, 𝜑 = 𝜑(v∗ ⋅ v) and 𝜓 = 𝜓(v∗ ⋅ v). By Proposition 2.21, we infer that v ∈ ℳ 𝜑 . On the other hand, we have 𝜑 + i𝜓 = 𝜔(⋅v) = 𝜔(⋅v2 v∗ ) = (𝜑 − i𝜓)(⋅v2 ), that is, 𝜑(⋅(v2 − 1)) = i𝜓(⋅(v2 + 1)). Since v ∈ ℳ 𝜑 , it follows that 𝜓(⋅(v2 + 1)) is 𝜎t𝜑 -invariant (t ∈ ℝ). For fixed t ∈ ℝ let e = s(𝜓 − 𝜓 ◦ 𝜎t𝜑 ). By the above arguments, we have (v2 + 1)e = 0, hence e = −v2 e. Consequently, 𝜑(e) − i𝜓(e) = −𝜑(v2 e) + i𝜓(v2 e) = −(𝜑 − i𝜓)(ev2 ) = −(𝜑 + i𝜓)(e) = −𝜑(e)−i𝜓(e), so that 𝜑(e) = 0 and e = 0. Thus, 𝜓 ◦ 𝜎t𝜑 = 𝜓(t ∈ ℝ), that is, 𝜓 commutes with 𝜑. 4.20 Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and 𝜎 ∶ G → Aut(ℳ) a continuous action of the compact group G on ℳ which is trivial on the center 𝒵 (ℳ) of ℳ. The following conditions are equivalent: (i) 𝜑 ◦ 𝜎g = 𝜑 for all g ∈ G; (ii) 𝜎t𝜑 ◦ 𝜎g = 𝜎g ◦ 𝜎t𝜑 for all g ∈ G and all t ∈ ℝ. Proof. By 4.14.(1), we know that (i) ⇒ (ii). If (ii) holds, then by Proposition 4.16, we can write 𝜑 ◦ 𝜎g = 𝜑Ag

(g ∈ G).

Let g ∈ G be fixed. For the proof, we may assume that G is topologically generated by the element g. If G is discrete, G is finite so that there exists n ∈ ℕ, n ≥ 1, such that 𝜎gn is the identity automorphism. It follows that Ang = 1, hence Ag = 1 and 𝜑 ◦ 𝜎g = 𝜑. If G is not discrete, then there exists a sequence nk → +∞ of positive integers such that {gnk }k converges to the neutral element of G. Let 𝜀 > 0, e𝜀 = 𝜒[1+𝜀,+∞) (Ag ) and x ∈ 𝔐𝜑 ∩ ℳ + . Since 𝜎 is −n a continuous action, the sequence {𝜎g k (e𝜀 x)}k is w-convergent to e𝜀 x, hence −n

−n

𝜑(e𝜀 x) ≤ lim inf 𝜑(𝜎g k (e𝜀 x)) = lim inf 𝜑(Ag k e𝜀 x) k→∞

k→∞

≤ lim inf(1 + 𝜀)−nk 𝜑(e𝜀 x) = 0, k→∞

so that 𝜑(e𝜀 x) = 0 and e𝜀 x = 0. Since 𝔐𝜑 is s-dense in ℳ we get e𝜀 = 0. Consequently, Ag ≤ 1. Similarly we obtain Ag ≥ 1, hence Ag = 1 and 𝜑 ◦ 𝜎g = 𝜑. 4.21 Proposition. Let ℳ be a semifinite factor and 𝜎 ∶ G → Aut(ℳ) an action of the group G on ℳ. If there exists some nonzero 𝜎-invariant normal state 𝜑 on ℳ, then the trace 𝜇 on ℳ is 𝜎-invariant. Proof. By Theorem 4.10, there exists a positive self-adjoint operator A affiliated to ℳ such that 𝜑 = 𝜇A and by Proposition 4.16, there exists a group homomorphism G ∋ g ↦ 𝜆g ∈ ℝ+ such that 𝜇 ◦ 𝜎g = 𝜆g 𝜇(g ∈ G). Since 𝜑 = 𝜇A , is 𝜎-invariant, it follows that 𝜎g (A) = 𝜆g A (g ∈ G). Let g ∈ G be fixed and assume that 𝜆g < 1. For each 𝜀 > 0 and each k ∈ N, we have 𝜑(𝜒(0,𝜀) (A)) = 𝜑(𝜎gk (𝜒(0,𝜀) (A))) = 𝜑(𝜒(0,𝜀) (𝜎gk (A))) = 𝜑(𝜒(0,𝜀) (𝜆kg (A))).

The Pedersen–Takesaki Construction

61

Since 𝜆g < 1, we have limk→∞ 𝜒(0,+∞) (𝜆kg t) = 𝜒(0,+∞) (t) for every t ∈ ℝ. It follows that 𝜑(𝜒(0,𝜀) (A)) = 𝜑(𝜒(0,+∞) (A)), hence A = As(𝜑) ≤ 𝜀. As 𝜀 > 0 was arbitrary, we obtain A = 0, contradicting the fact that 𝜑 ≠ 0. Consequently, 𝜆g ≥ 1 and, similarly, 𝜆g ≤ 1, that is, 𝜆g = 1. 4.22. Even though the commutation of the modular automorphism groups does not insure the commutation of the n.s.f. weights, it does imply that the sum of the two weights is still semifinite. Proposition. Let 𝜑, 𝜓 be n.s.f. weights on the W ∗ -algebra ℳ such that 𝜎t𝜑 ◦ 𝜎s𝜓 = 𝜎s𝜓 ◦ 𝜎t𝜑 for all s, t ∈ ℝ. Then the normal faithful weight 𝜑 + 𝜓 is semifinite. Proof. By Proposition 4.16, there exists a nonsingular positive self-adjoint operator A affiliated to the center of ℳ such that 𝜓 ◦ 𝜎t𝜑 = 𝜓At

(t ∈ ℝ).

Putting em = 𝜒(0,m] (A) (m ∈ ℕ), we have em ↑ s(A) = 1. Let x ∈ ℳ and n ∈ ℕ. Consider the elements +∞

xn =

+∞

2 2 n e−n(t +s ) 𝜎t𝜑 (𝜎s𝜓 (x)) dt ds ∈ ℳ. 𝜋 ∫−∞ ∫−∞

s

𝜑 𝜓 It is easy to check that xn → x, xn ∈ ℳ∞ ∩ ℳ∞ and +∞

+∞

𝜎𝛼𝜑 (xn ) =

2 2 n e−n((t−𝛼) +s ) 𝜎t𝜑 (𝜎s𝜓 (x)) dt ds 𝜋 ∫−∞ ∫−∞

𝜎𝛼𝜓 (xn ) =

2 2 n e−n(t +(s−𝛼) ) 𝜎t𝜑 (𝜎s𝜓 (x)) dt ds 𝜋 ∫−∞ ∫−∞

+∞

(𝛼 ∈ ℂ),

+∞

(𝛼 ∈ ℂ).

If x ∈ 𝔐𝜓 ∩ ℳ + , then +∞

+∞

2 2 n e−nt e−ns 𝜓(𝜎s𝜓 (𝜎t𝜑 (x)em ) dt ds 𝜋 ∫−∞ ∫−∞ √ √ +∞ +∞ 2 n n 𝜑 −nt2 = e 𝜓(𝜎t (xem ))dt = e−nt 𝜓(At em x)dt ∫ ∫ 𝜋 −∞ 𝜋 −∞ √ +∞ 2 n ≤ mt e−nt 𝜓(x)dt = mt 𝜓(x) < +∞, 𝜋 ∫−∞

𝜓(xn em ) =

s

𝜑 𝜓 hence xn em ∈ ℳ∞ ∩ ℳ∞ ∩ 𝔐𝜓 . Since xn em → x, it follows that 𝜑 𝜓 ℳ∞ ∩ ℳ∞ ∩ 𝔐𝜓 is s-dense in ℳ.

Similarly, 𝜑 𝜓 ℳ∞ ∩ ℳ∞ ∩ 𝔐𝜑 is s-dense in ℳ.

62

Normal Weights

Consequently, the product of these two sets is w-dense in ℳ. On the other hand, as we noted in Section 2.15, this product is contained in 𝔐𝜑 ∩ 𝔐𝜓 ⊂ 𝔐𝜑+𝜓 , hence 𝜑 + 𝜓 is semifinite. 4.23. Let tr be the canonical trace on ℬ(ℋ ), A a positive self-adjoint operator in ℋ and 𝜑 = trA . For 𝜉, 𝜂 ∈ ℋ , we shall use the notation 𝜉 ⊗ 𝜂̄ for the operator ℋ ∋ 𝜁 ↦ (𝜁|𝜂)𝜉 ∈ ℋ . It is clear that ̄ (𝜉 ⊗ 𝜂)(𝜉 (𝜉 ⊗ 𝜂) ̄ ∗ = 𝜂 ⊗ 𝜉, ̄ ′ ⊗ 𝜂̄ ′ ) = (𝜂|𝜉 ′ )𝜉 ⊗ 𝜂̄ ′

(1)

x(𝜉 ⊗ 𝜂) ̄ = x𝜉 ⊗ 𝜂, ̄ (𝜉 ⊗ 𝜂)x ̄ = 𝜉 ⊗ x∗ 𝜂.

(2)

and, for x ∈ ℬ(ℋ ),

On the other hand, we have 𝜉 ⊗ 𝜂̄ ∈ 𝔑𝜑 ⇔ 𝜂 ∈ D(A1∕2 ), 𝜉 ⊗ 𝜂̄ ∈ 𝔐𝜑 ⇔ 𝜉 ⊗ 𝜂̄ ∈ 𝔑𝜑 ∩

𝔑∗𝜑

⇔ 𝜉, 𝜂 ∈ D(A

(3) 1∕2

)

𝜉, 𝜂 ∈ D(A1∕2 ) ⇒ 𝜑(𝜉 ⊗ 𝜂) ̄ = (A1∕2 𝜉|A1∕2 𝜂).

(4) (5)

Indeed, let x = 𝜉 ⊗ 𝜂. ̄ Then x∗ x = ‖𝜉‖2 (𝜂 ⊗ 𝜂) ̄ and so, by (2), 1∕2

∗ 1∕2 𝜂) A1∕2 = ‖𝜉‖2 (A1∕2 𝜀 x xA𝜀 𝜀 𝜂 ⊗ A𝜀

is a multiple of the orthogonal projection onto the linear subspace spanned by the vector A1∕2 𝜂. It follows that 1∕2

∗ 1∕2 2 1∕2 𝜑(x∗ x) = sup tr(A1∕2 𝜂) 𝜀 x xA𝜀 ) = ‖𝜉‖ sup tr(A𝜀 𝜂 ⊗ A𝜀 𝜀>0

𝜀>0 2 1∕2

2 = ‖𝜉‖2 sup ‖A1∕2 𝜀 𝜂‖ = ‖𝜉‖ ‖A 𝜀>0

𝜂‖2 .

From this we obtain (3) and then, by polarization, (4) and (5). 4.24. Consider again the canonical trace tr on ℬ(ℋ ) and two positive self-adjoint operators A and ̂ is defined if and only if B on ℋ . We recall (A.11) that the weak sum A+B D = D(A1∕2 ) ∩ D(B1∕2 ) is dense in ℋ

(1)

̂ is determined by and, in this case, A+B ̂ 1∕2 𝜉‖2 , 𝜉 ∈ D = D((A+B) ̂ 1∕2 ). ‖A1∕2 𝜉‖2 + ‖B1∕2 𝜉‖2 = ‖(A+B)

(2)

The Converse of the Connes Theorem

63

On the other hand, if the normal weight trA + trB is semifinite, then (4.10) there exists a unique positive self-adjoint operator C on ℋ such that trA +trB = trC . If A and B are bounded, then C = A+B by 4.3.(1). In the general case, we have the following result: ̂ is defined. Proposition. The normal weight trA + trB is semifinite if and only if the weak sum A+B In this case, we have trA + trB = trA+B ̂ . Proof. Assume first that the weight trA + trB is semifinite and write trA + trB = trC as above. Since trA ≤ trC , trB ≤ trC , we have (4.5) A ≤ C, B ≤ C. Thus, D = D(A1∕2 ) ∩ D(B1∕2 ) ⊃ D(C1∕2 ) is dense ̂ is defined. If 𝜉 ∈ D, then (4.23.(5)) in ℋ that is A+B ‖C1∕2 𝜉‖2 = trC (𝜉 ⊗ 𝜉) = trA (𝜉 ⊗ 𝜉) + trB (𝜉 ⊗ 𝜉) = ‖A1∕2 𝜉‖2 + ‖B1∕2 𝜉‖2 , ̂ hence C = A+B. ̂ is defined and consider the increasing sequence {A𝜀 + B𝜀 }𝜀>0 of Conversely, assume that A+B bounded positive operators. It is clear that lim((A𝜀 + B𝜀 )𝜉|𝜉) < +∞ ⇔ 𝜉 ∈ D = D(A1∕2 ) ∩ D(B1∕2 ). 𝜀

Using (A.5) it follows that there exists a unique positive self-adjoint operator C in ℋ such that D(C1∕2 ) = D and A𝜀 + B𝜀 ↑ C. By Proposition 4.6, we have trA𝜀 +B𝜀 ↑ trC . Since trA𝜀 +B𝜀 = trA𝜀 + trB𝜀 , we conclude that trA + trB = trC is semifinite. 4.25. Notes. The construction of the weight 𝜑A , the definition of commutation for weights and almost all the results contained in this section are due to Pedersen and Takesaki (1973). Corollary 4.17 is due to Takesaki (1970) and Proposition 4.19 is due to Herman and Takesaki (1970). Several simplifications of the proofs given in Pedersen and Takesaki (1973) were made possible by the use of the Connes cocycle theorem. For our exposition we have used Elliott (1975); Pedersen and Takesaki (1973); Takesaki (1969– 1970) and Zsidó (1978). Further results related to Proposition 4.21 are contained in Pedersen and Takesaki (1973) and Størmer (1971a).

5 The Converse of the Connes Theorem In this section, we state and prove a converse to Theorem 3.1 and give some applications. 5.1. Let ℳ be a W ∗ -algebra, G a locally compact group and 𝜎 ∶ G → Aut(ℳ) a continuous action of G on ℳ. We denote by e ∈ G the neutral element of G. A 𝜎-cocycle (of degree 1) is an s∗ -continuous function w ∶ G → ℳ with the properties: w(gh) = w(g)𝜎g (w(h)), w(g−1 ) = 𝜎g−1 (w(g)∗ )

(g, h ∈ G).

64

Normal Weights

In this case w(g) are partial isometries and w(g)w(g)∗ = w(e),

w(g)∗ w(g) = 𝜎g (w(e))

(g, h ∈ G).

The set of 𝜎-cocycles is denoted by Z𝜎 (G; ℳ). A detailed study of Z𝜎 (G; ℳ) is contained in Section 20. If 𝜑 is an n.s.f. weight on ℳ, the modular automorphism group {𝜎t𝜑 }t∈ℝ is a continuous action 𝜑 𝜎 ∶ ℝ → Aut(ℳ) and, by the Connes theorem (3.1), for every normal semifinite weight 𝜓 on ℳ the Connes cocycle [D𝜓 ∶ D𝜑] is a 𝜎 𝜑 -cocycle. Conversely, we have the following result: Theorem (A. Connes). Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ. For every 𝜎 𝜑 -cocycle w ∈ Z𝜎 𝜑 (ℝ; ℳ) there exists a unique normal semifinite weight 𝜓 on ℳ such that [D𝜓 ∶ D𝜑] = w. The proof is contained in Sections 5.2–5.7. We shall consider the W ∗ -algebra ℳ realized as a von Neumann algebra ℳ ⊂ ℬ(ℋ ). Also, we ̄ ℱ∞ and Mat∞ (ℳ) denote by ℱ∞ the discrete factor ℬ(ℒ 2 (ℝ)) and identify the W ∗ -algebras ℳ ⊗ ̄ ℱ∞ is determined by a certain matrix [xij ] with as in ([L], 3.17). Thus, every element x ∈ ℳ ⊗ ̄ ℱ∞ be the element corresponding to the matrix [xij ] with xij = 0 if elements in ℳ. Let en ∈ ℳ ⊗ i ≠ j or if i > n, and x11 = x22 = ⋯ = xnn = 1. ̄ ℱ∞ and weights 𝜓 ′ , 𝜓 on ℳ. We shall successively construct weights Φ, Φ′ , Ψ, Ψ′ on ℳ ⊗ ̄ ℱ∞ by 5.2. We first define the n.s.f. weight Φ on ℳ ⊗ ∑ ̄ ℱ∞ )+ ). 𝜑(xkk ) (x = [xij ] ∈ (ℳ ⊗ Φ(x) = k

Then we have ̄ id 𝜎tΦ = 𝜎t𝜑 ⊗

(t ∈ ℝ).

(1)

Indeed, using 2.21.(2), it is easy to check that the projection en belongs to the centralizer of Φ and using 2.22.(3) and Proposition 3.3 we obtain ̄ id)(x) 𝜎tΦ (x) = (𝜎t𝜑 ⊗

̄ ℱ∞ )en ). (x ∈ en (ℳ ⊗

Since en ↑ 1, this proves (1). ̄ tr where tr denotes the canonical trace on ℱ∞ Actually, Φ is nothing but the tensor product 𝜑 ⊗ (see 8.2). 5.3. Let {ut }t∈ℝ ⊂ ℬ(ℒ 2 (ℝ)) be the so-continuous unitary group defined in Section 4.15. Using Stone’s theorem ([L], 9.20) and the Pedersen–Takesaki construction (4.7), we obtain a new n.s.f. ̄ ℱ∞ such that weight Φ′ on ℳ ⊗ ′ ̄ Ad(ut ) (t ∈ ℝ). 𝜎tΦ = Ad(1⊗ut ) ◦ 𝜎tΦ = 𝜎t𝜑 ⊗

̄ ℱ∞ acts on the Hilbert space ℋ ⊗ ̄ ℒ 2 (ℝ) which can be 5.4. The von Neumann algebra ℳ ⊗ 2 identified with ℒ (ℝ, ℋ ) via the mapping ̄ ℒ 2 (ℝ) ∋ 𝜉 ⊗ f ↦ {t ↦ f (t)𝜉} ∈ ℒ 2 (ℝ, ℋ ). ℋ⊗

The Converse of the Connes Theorem

65

Using the given cocycle w ∈ Z𝜎 𝜑 (ℝ; ℳ), we define a partial isometry W on this Hilbert space by (W𝜁)(t) = w(t)𝜁 (t) (𝜁 ∈ ℒ 2 (ℝ, ℋ ), t ∈ ℝ).

(1)

̄ ℂ of ℳ ⊗ ̄ ℱ∞ and hence W ∈ It is easy to check that W commutes with the commutant ℳ ′ ⊗ ̄ ℳ ⊗ ℱ∞ by the von Neumann double commutant theorem. Also, we have ′

W𝜎tΦ (W ∗ ) = w(t) ⊗ 1

(t ∈ ℝ).

(2)

Indeed, W is defined in (1) by the function s ↦ w(s) and, similarly, 𝜎tΦ (W ∗ ) is defined by the function ′ s ↦ 𝜎t𝜑 (w(s−t)∗ ), so that W𝜎tΦ (W ∗ ) is defined by the constant function s ↦ w(s)𝜎t𝜑 (w(s−t)∗ ) = w(t) and hence it is equal to w(t) ⊗ 1. ̄ ℱ∞ . Using (2) and 5.3.(1) and We define the normal semifinite weight Ψ = Φ′ (W ∗ ⋅ W) on ℳ ⊗ Corollary 3.7, we obtain ′

̄ Ad(ut ) 𝜎tΨ = (Ad(w(t)) ◦ 𝜎t𝜑 ) ⊗

(t ∈ ℝ).

(3)

5.5. Using the Pedersen–Takesaki construction again, we obtain from Ψ a new normal semifinite ̄ ℱ∞ such that weight Ψ′ on ℳ ⊗ ̄ id 𝜎tΨ = (Ad(w(t)) ◦ 𝜎t𝜑 ) ⊗ ′

(t ∈ ℝ).

(1)

5.6. Let p be a minimal projection of ℱ∞ . Using the mapping x ↦ x⊗p we identify ℳ with ̄ p)(ℳ ⊗ ̄ ℱ∞ )(1 ⊗ ̄ p). It is clear that 1 ⊗ ̄ p belongs to the centralizer of Ψ′ , hence 𝜓 ′ = Ψ′ ̄ (1 ⊗ (1 ⊗ p) ̄ p)(ℳ ⊗ ̄ ℱ∞ )(1 ⊗ ̄ p). For x ∈ s(𝜓 ′ )ℳs(𝜓 ′ ), we have is a normal semifinite weight on ℳ = (1 ⊗ (2.22.(3)) ̄ p) = w(t)𝜎t𝜑 (x)w(t)∗ ⊗ ̄ p = w(t)𝜎t𝜑 (x)w(t)∗ , 𝜎t𝜓 (x) = 𝜎tΨ (x ⊗ ′

′

hence 𝜎t𝜓 = Ad(w(t)) ◦ 𝜎t𝜑 ′

(t ∈ ℝ).

(1)

5.7. Let w′ (t) = [D𝜓 ′ ∶ D𝜑]t and a(t) = w′ (t)∗ w(t), (t ∈ ℝ). Since Ad(w(t)) ◦ 𝜎t𝜑 = 𝜎t𝜓 = Ad(w′ (t)) ◦ 𝜎t𝜑 , it follows that {a(t)}t∈ℝ is a so-continuous group of unitary operators in the center of the von Neumann algebra s(𝜓 ′ )ℳs(𝜓 ′ ). By Stone’s theorem, there exists a positive self-adjoint operator A in ℋ , affiliated to the center of ℳ, such that a(t) = Ait , (t ∈ ℝ). Let 𝜓 = 𝜓A′ . Then 𝜓 is a normal semifinite weight on ℳ and, for every t ∈ ℝ, we have [D𝜓 ∶ D𝜑]t = [D𝜓A′ ∶ D𝜓 ′ ]t [D𝜓 ′ ∶ D𝜑]t = a(t)w′ (t) = w′ (t)a(t) = w(t). Thus, the existence assertion of Theorem 5.1 is proved. The uniqueness part has been already considered in Corollary 3.6. ′

5.8 Corollary (Pedersen & Takesaki, 1973). For every normal weight 𝜑 on a W ∗ -algebra ℳ there exists a family {𝜑i }i∈I of normal positive forms on ℳ such that ∑ 𝜑(x) = 𝜑i (x) (x ∈ ℳ + ). i∈I

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Normal Weights

Proof. We shall say that a weight 𝜑 has property S if 𝜑 is a sum of normal positive forms. If {𝜔i }i∈I is a maximal family of normal positive forms on ℳ with mutually orthogonal supports, ∑ then 𝜔 = i∈I 𝜔i is an n.s.f. weight on ℳ with property S. Let 𝜓 be an arbitrary normal semifinite weight on ℳ and w(t) = [D𝜓 ∶ D𝜔]t (t ∈ ℝ). By the proof of Theorem 5.1, the weight 𝜓 is obtained from 𝜔 by a repeated application of the following operations: ̄ tr, (see 5.2); 𝜑↦Φ=𝜑⊗ 𝜑 ↦ 𝜑e (e ∈ ℳ 𝜑 ); 𝜑 ↦ 𝜑 ◦ 𝜎, (𝜎 ∈ Aut(ℳ)); 𝜑 ↦ 𝜑A , (see 4.4).

(2) (3) (4) (5)

The point is that all these operations preserve property S. This is obvious for (1)–(3) and also for (4) if A is bounded. In the general case, there exists a sequence {en } of mutually orthogonal spectral pro∑ jections of A such that Aen are bounded and 𝜑A = n 𝜑Aen . Thus every normal semifinite weight has property S. Finally, let 𝜑 be an arbitrary normal weight and denote by e the unique projection in ℳ such that w 𝔑𝜑 = ℳe. Then 𝜑e is a normal semifinite weight on eℳe, hence 𝜑e has property S. On the other hand, 𝜑1−e takes only the values 0 and +∞ so that it is obvious that 𝜑1−e has property S. Finally, it is easy to check that 𝜑 = 𝜑e + 𝜑1−e . 5.9 Corollary. Let 𝜑 be a normal weight on the von Neumann algebra ℳ ⊂ ℬ(ℋ ). Then there exists a family {𝜉i }i∈I ⊂ ℋ such that ∑ (x𝜉i |𝜉i ) (x ∈ ℳ + ).

𝜑(x) =

i∈I

Proof. Every positive normal form on ℳ has the stated property ([L], 8.17), so this result follows from the preceding corollary. In particular it follows that the function ( ℳ ∋ x ↦ 𝜑(x x) ∗

1∕2

=

∑

)1∕2 ‖x𝜉i ‖

2

∈ [0, +∞]

i∈I

is subadditive and lower w-semicontinuous (see also 2.12). 5.10. Recall the notation (3.2) 𝔬ℳ ∶ Aut(ℳ) → Out(ℳ) for the canonical quotient mapping and 𝛿ℳ ∶ ℝ → Out(ℳ) for the modular homomorphism of the W ∗ -algebra ℳ. Another consequence of Theorem 5.1 is the following Corollary. Let ℳ be a W ∗ -algebra with separable predual and 𝜎 ∶ ℝ → Aut(ℳ) a continuous action of ℝ on ℳ. Then 𝜎 is the modular automorphism group of some n.s.f. weight on ℳ if and only if 𝔬ℳ (𝜎t ) = 𝛿ℳ (t) for all t ∈ ℝ. We give here only a sketch of the proof (Connes (1973a)). Let 𝜑 be any n.s.f. weight on ℳ. Since 𝔬ℳ (𝜎t ) = 𝛿ℳ (t) = 𝔬ℳ (𝜎t𝜑 ), (t ∈ ℝ), there is a mapping v ∶ ℝ → U(ℳ) such that 𝜎t = Ad(v(t)) ◦ 𝜎t𝜑 ,

Equality and Majorization of Weights

67

that is, 𝜎t (x) = v(t)𝜎t𝜑 (x)v(t)∗

(x ∈ ℳ, t ∈ ℝ).

Moreover, it is possible to choose a Borel mapping v ∶ ℝ → U(ℳ) with this property (Kallman, 1971; Moore, 1976). Then, for s, t ∈ ℝ and x ∈ ℳ, we have 𝜑 𝜑 (x)𝜎s𝜑 (v(t)∗ )v(s)∗ = 𝜎s+t (x) = v(s + t)𝜎s+t (x)v(s + t)∗ v(s)𝜎s𝜑 (v(t))𝜎s+t

so that we obtain a Borel function a ∶ ℝ2 ∋ (s, t) ↦ a(s, t) = v(s)𝜎s𝜑 (v(t))v(s + t)∗ ∈ U(𝒵 (ℳ)). Since a(s, t) ∈ 𝒵 (ℳ), we have 𝜎r𝜑 (a(s, t)) = a(s, t) and an easy computation based on this remark gives (r, s, t) ∈ ℝ.

a(r, s)a(r + s, t) = a(s, t)a(r, s + t)

Now, by arguments of Borel cohomology (Kadison, 1965; Moore, 1976) it follows that there exists a Borel mapping b ∶ ℝ → U(𝒵 (ℳ)) such that a(s, t) = b(s + t)b(s)−1 b(t)−1

(s, t ∈ ℝ).

Let w(t) = b(t)v(t)(t ∈ ℝ). Then w ∶ ℝ → U(ℳ) is a Borel mapping and w(s + t) = w(s)𝜎s𝜑 (w(t))(s, t ∈ ℝ), so that (see Connes, 1973a) this mapping is continuous, that is, w ∈ Z𝜎 𝜑 (ℝ; ℳ). Also, we have 𝜎t = Ad(w(t)) ◦ 𝜎t𝜑 (t ∈ ℝ). By Theorem 5.1, there exists an n.s.f. weight 𝜓 on ℳ such that [D𝜓 ∶ D𝜑] = w and it follows that 𝜎t = 𝜎t𝜓 (t ∈ ℝ). 5.11. Notes. Theorem 5.1 and Corollary 5.10 are due to Connes (1973a). Corollary 5.8 is due to Pedersen and Takesaki (1973) and the proof given here is that of Elliott (1975). For our exposition, we have used Connes (1973a) and Elliott (1975).

6 Equality and Majorization of Weights In this section, we present an important criterion that insures the equality of two weights, study various order relations between weights, and give some examples. 6.1 Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ, a ∈ ℳ 𝜑 , a ≥ 0, and 𝜓 a normal semifinite weight on ℳ. If there exists a *-subalgebra ℬ ⊂ 𝔑𝜑 , 𝜎 𝜑 -invariant and w-dense in ℳ such that 𝜓( y∗ y) = 𝜑a ( y∗ y) ( y ∈ ℬ), then 𝜓 ≤ 𝜑a .

68

Normal Weights

Proof. By the Kaplansky density theorem ([L], 3.10), there exists a net {bj }j∈J ⊂ ℬ such that s

0 ≤ bj → 1 and supj ‖bj ‖ ≤ 1. Then √ 0 ≤ aj =

+∞

2 1 𝜑 e−t 𝜎t𝜑 (bj ) dt ∈ 𝔗𝜑 ⊂ ℳ∞ 𝜋 ∫−∞

s

and 𝜎𝛼𝜑 (aj ) → 1 for all a ∈ ℂ, by Proposition 2.16. Let y ∈ ℬ. By Proposition 1.14, 𝜑a ( y∗ ⋅ y) and 𝜓( y∗ ⋅ y) are normal positive forms on ℳ and 𝜓( y∗ ⋅ y) = 𝜑a ( y∗ ⋅ y). In particular, this equality is valid for ys,t;j,k = 𝜎s𝜑 (bj ) + ik 𝜎t𝜑 (bj ) ∈ ℬ

(s, t ∈ ℝ, j ∈ J, k = 0, 1, 2, 3).

Since for every x ∈ ℳ we have 3 +∞ +∞ 2 2 1∑ k aj xaj = i e−(s +t ) y∗s,t;j,k xys,t;j,k ds dt, 𝜋 k=0 ∫−∞ ∫−∞

by Corollary 2.10 we infer that 𝜓(aj xaj ) = 𝜑a (aj xaj ) (x ∈ ℳ + ). w

s

𝜑 Let x ∈ 𝔑𝜑 . Since 𝜓 is normal, aj x∗ xaj → x∗ x and 𝜎−i∕2 (aj ) → 1, using Proposition 2.14 we obtain

𝜓(x∗ x) ≤ lim inf 𝜓(aj x∗ xaj ) ≤ lim inf 𝜑a (aj x∗ xaj ) j

j

= lim inf 𝜑(a1∕2 aj x∗ xaj a1∕2 ) = lim inf ‖(xaj a1∕2 )𝜑 ‖2𝜑 j

j

𝜑 = lim inf ‖ J𝜑 𝜋𝜑 (a1∕2 𝜎−i∕2 (aj ))J𝜑 x𝜑 ‖2𝜑 j

= ‖ J𝜑 𝜋𝜑 (a1∕2 )J𝜑 x𝜑 ‖2𝜑 = ‖(xa1∕2 )𝜑 ‖2𝜑 = 𝜑(a1∕2 x∗ xa1∕2 ) = 𝜑a (x∗ x). In particular, it follows that s(𝜓) ≤ s(a). Consider now x ∈ 𝔑𝜑a , that is xa1∕2 ∈ 𝔑𝜑 . There exists a sequence {en } ⊂ ℳ 𝜑 , of spectral projections of a such that en ↑ 1 and aen ≥ n−1 e𝛼 , (n ∈ ℕ). Since a ∈ ℳ 𝜑 and aen is invertible in en ℳ 𝜑 en , we see again by Proposition 2.14 that xen ∈ 𝔑𝜑 . Consequently, we have 𝜓(x∗ x) = 𝜓(s(a)x∗ xs(a)) ≤ lim inf 𝜓(en x∗ xen ) n

≤ lim inf 𝜑a (en x∗ xen ) ≤ 𝜑a (x∗ x); n

the last inequality is obtained by applying Proposition 2.14 once more. Hence 𝜓(z) ≤ 𝜑a (z) for every z ∈ 𝔐𝜑a ∩ ℳ + , that is, 𝜓 ≤ 𝜑a .

Equality and Majorization of Weights

69

6.2 Theorem. Let 𝜑 be an n.s.f weight on the W ∗ -algebra ℳ, a ∈ ℳ 𝜑 , a ≥ 0, and 𝜓 a normal semifinite weight on ℳ. If 𝜓 commutes with 𝜑 and there exists a ∗-subalgebra ℬ ⊂ 𝔑𝜑 , 𝜎 𝜑 invariant and w-dense in ℳ such that 𝜓( y∗ y) = 𝜑a ( y∗ y) ( y ∈ ℬ), then 𝜓 = 𝜑a . Proof. Since 𝜓 commutes with 𝜑, there exists a positive self-adjoint operator A affiliated to ℳ such that 𝜓 = 𝜑A (4.10). By Proposition 6.1, it follows that 𝜑A = 𝜓 ≤ 𝜑a and then, using Proposition 4.5, we deduce that A ≤ a, so that A ∈ ℳ 𝜑 is bounded. Again by Proposition 6.1, with 𝜑a instead of 𝜓 and A instead of a, we get 𝜑a ≤ 𝜑A = 𝜓. Hence 𝜓 = 𝜑a . The particular case a = 1 of the preceding theorem is known as “the Pedersen–Takesaki theorem on the equality of weights.” 6.3. By the Radon–Nikodym type theorem of Sakai ([L], 5.21), if 𝜑, 𝜓 are normal positive forms on the W ∗ -algebra ℳ and 𝜓 ≤ 𝜑, there exists a ∈ ℳ (actually, 0 ≤ a ≤ 1) such that 𝜓 = 𝜑(a ⋅ a∗ ). The extension of this result to weights is contained in Corollary 3.13. We remark that Proposition 2.14 gives necessary and sufficient conditions for the inequality 𝜑(a ⋅ a∗ ) ≤ 𝜑 to hold. 6.4. We now consider another form of Radon–Nikodym type theorems, which has been pointed out by S. Sakai (see [L], C.5.5). We begin with the case of normal positive forms. Let ℳ be a W ∗ -algebra, 𝜑, 𝜓 ∈ ℳ∗+ such that 𝜓 ≤ 𝜑, and 𝜆 ∈ ℂ with 𝜆 + 𝜆̄ = 1. The set 𝒳 = {𝜑(𝜆a ⋅ +𝜆̄ ⋅ a); a ∈ ℳ, a = a∗ , ‖a‖ ≤ 1} ⊂ ℳ h is convex and 𝜎(ℳ∗ , ℳ)-compact. If 𝜓 ∉ 𝒳 , then by the Hahn–Banach theorem there exists b ∈ ̄ ℳ, b = b∗ and t ∈ ℝ such that 𝜓(b) > t while 𝜑(𝜆ab + 𝜆ba) ≤ t for every aℳ, a = a∗ , ‖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, a contradiction. Thus, there exists a1 ∈ ℳ, ∗ ̄ 1 ), (x ∈ ℳ). Since 𝜑 ≥ 0, it follows that with a1 = a1 , ‖a1 ‖ ≤ 1, such that 𝜓(x) = 𝜑(𝜆a1 x + 𝜆xa + − 𝜓(a1 ) = 0, hence a = a1 ∈ ℳ, 0 ≤ a ≤ 1 and ̄ 𝜓(x) = 𝜑(𝜆ax + 𝜆xa)(x ∈ ℳ).

(1)

Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and 𝜓 ∈ ℳ∗+ , 𝜓 ≤ 𝜑. There exists a unique a ∈ 𝔐𝜑 , 0 ≤ a ≤ 1, such that 𝜓(x) = 𝜑(ax + xa)∕2

(x ∈ 𝔐𝜑 ).

Proof. By Corollary 2.4, there exists a vector 𝜂 ∈ ℋ𝜑 and an operator T ′ ∈ 𝜋𝜑 (ℳ)′ , 0 ≤ T ′ ≤ 1, such that 𝜓(x) = (𝜋𝜑 (x)𝜂|𝜂)𝜑 T x𝜑 = 𝜋𝜑 (x)𝜂 ′

(x ∈ ℳ𝜑 ), (x ∈ 𝔄𝜑 ).

(2) (3)

70

Normal Weights

Let 𝜁 = T ′ 𝜂 and 𝜉 = 2(Δ𝜑 + 1)−1 𝜁. From (3), it follows that 𝜂 ∈ 𝔄′𝜑 , S∗𝜑 𝜂 = 𝜂 and R𝜂 = T ′ . Hence 𝜁 ∈ 𝔄′𝜑 , S∗𝜑 𝜁 = 𝜁 and R𝜁 = T ′2 . Using ([L], 10.11), we infer that 𝜉 ∈ 𝔄𝜑 , and, since S∗𝜑 𝜁 = 𝜁 and −1 J𝜑 Δ𝜑 J−1 𝜑 = Δ𝜑 , we have S𝜑 𝜉 = 𝜉. Using ([L], Cor. 9,23), we get +∞ −1 𝜉 = S𝜑 𝜉 = 2J𝜑 Δ1∕2 𝜑 (Δ𝜑 + 1) 𝜁 =

∫−∞

2(e𝜋t + e−𝜋t )−1 Δit𝜑 J𝜑 𝜁 dt.

Since (2.12.(4)) it ′2 −it LΔit𝜑 J𝜑 𝜁 = Δit𝜑 J𝜑 R𝜁 J𝜑 Δ−it 𝜑 = Δ𝜑 J𝜑 T J𝜑 Δ𝜑 ,

it follows that +∞

L𝜉 =

∫−∞

2(e𝜋t + e−𝜋t )−1 Δit𝜑 J𝜑 T ′2 J𝜑 Δ−it 𝜑 dt.

Consequently, a = L𝜉 ∈ ℳ, 0 ≤ a ≤ 1. Also, since J𝜑 T ′ J𝜑 = J𝜑 R𝜂 J𝜑 = LJ𝜑 𝜂 , we get (2.11) 𝜑(a) = 𝜑( J𝜑 T ′2 J𝜑 ) = ‖ J𝜑 𝜂‖2𝜑 < +∞, hence a ∈ 𝔐𝜑 . For x ∈ 𝔐𝜑 , we have 2𝜓(x) = 2(𝜋𝜑 (x)𝜂|𝜂)𝜑 = 2(T ′ x𝜑 |𝜂)𝜑 = 2(x𝜑 |T ′ 𝜂)𝜑 = 2(x𝜑 |𝜁)𝜑 = (x𝜑 |(Δ𝜑 + 1)𝜉)𝜑 = (x𝜑 |𝜉)𝜑 + (x𝜑 |S∗𝜑 S𝜑 𝜉)𝜑 = (x𝜑 |𝜉)𝜑 + (𝜉|S𝜑 x𝜑 )𝜑 = (x𝜑 |a𝜑 )𝜑 + (a𝜑 |(x∗ )𝜑 )𝜑 = 𝜑(ax + xa). To prove the uniqueness assertion, assume that a ∈ 𝔐𝜑 , 0 ≤ a ≤ 1, and 𝜑(ax + xa) = 0 for every x ∈ 𝔐𝜑 . Then 𝜉 = a𝜑 ∈ 𝔄𝜑 , S𝜑 𝜉 = 𝜉, and for every 𝜂 ∈ 𝔗𝜑 we have (𝜂| − 𝜉)𝜑 = (𝜉|S𝜑 𝜂)𝜑 = (S𝜑 𝜉|S𝜑 𝜂)𝜑 = (Δ𝜑 𝜂|𝜉)𝜑 . Since Δ𝜑 = Δ∗𝜑 = [Δ𝜑 |𝔗𝜑 ]∗ , it follows that 𝜉 ∈ D(Δ𝜑 ) and Δ𝜑 𝜉 = −𝜉, which is possible only if 𝜉 = 0, as Δ𝜑 is positive. Hence a = 0. 6.5. In order to treat the general case of two weights, we recall ([L], 10.22) that for any n.s.f. weight 𝜑 on the W ∗ -algebra ℳ, the set 𝔊𝜑 = 𝔗𝜑 ∩ 𝒮 (Δ𝜑 ) ∩ 𝒮 (Δ−1 𝜑 ) is a left Hilbert subalgebra of 𝔄𝜑 , equivalent with 𝔄𝜑 , and J𝜑 𝔊𝜑 = 𝔊𝜑 , Δ𝛼𝜑 𝔊𝜑 = 𝔊𝜑 and Δ𝛼𝜑 |𝔊𝜑 = Δ𝛼𝜑 . Note that 𝔊𝜑 ⊂ D(ln Δ𝜑 ).

Equality and Majorization of Weights

71

Also, the following approximation result holds: −1∕2 for every 𝜉 ∈ D(Δ1∕2 ) there exists a 𝜑 ) ∩ D(Δ𝜑 sequence {𝜉n } ⊂ 𝔊𝜑 such that Δ𝛼𝜑 𝜉n → Δ𝛼𝜑 𝜉 for each 𝛼 ∈ ℂ with − 1∕2 ≤ Re 𝛼 ≤ 1∕2

(1)

−1∕2

−1∕2

Indeed, since Δ𝜑 𝔊𝜑 = 𝔊𝜑 is dense in ℋ𝜑 , there exists a sequence {𝜁n } ⊂ 𝔊𝜑 such Δ𝜑 𝜁n → 1∕2 −1∕2 𝛼+(1∕2) Δ𝜑 𝜉 + Δ𝜑 𝜉. If −1∕2 ≤ Re 𝛼 ≤ 1∕2, then the operator Δ𝜑 (1 + Δ𝜑 )−1 is bounded and sends 1∕2 −1∕2 the vector Δ𝜑 𝜉 + Δ𝜑 𝜉 into the vector Δ𝛼𝜑 𝜉. Thus (1) follows with 𝜉n = (1 + Δ𝜑 )−1 𝜁n . Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and 𝜓 any weight on ℳ with 𝜓 ≤ 𝜑. There exists a unique element a ∈ ℳ, 0 ≤ a ≤ 1, such that 𝔊𝜑 a ⊂ 𝔑𝜑 and x ∈ 𝔊2𝜑 ⇒ ax + xa ∈ 𝔐𝜑 and 𝜓(x) = 𝜑(ax + xa)∕2. Proof. By Proposition 2.3, there exists an operator T ′ ∈ 𝜋𝜑 (ℳ)′ , 0 ≤ T ′ ≤ 1, such that 𝜓(x∗ x)(T ′ x𝜑 |T ′ x𝜑 )𝜑

(x ∈ 𝔑𝜑 ).

Since J𝜑 T ′2 J𝜑 ∈ 𝜋𝜑 (ℳ), there exists b ∈ ℳ, 0 ≤ b ≤ 1, such that 𝜋𝜑 (b) = J𝜑 T ′2 J𝜑 . Define an element a ∈ ℳ, 0 ≤ a ≤ 1, by +∞

a=

∫−∞

2(e𝜋t + e−𝜋t )−1 Δit𝜑 Δ−it 𝜑 dt.

−1∕2

Let x ∈ 𝔊𝜑 . Then J𝜑 x𝜑 ∈ 𝔊𝜑 ⊂ D(Δ𝜑

) ∩ D(ln Δ𝜑 ). Using Proposition A.13, it follows that

(ax∗ )𝜑 = 𝜋𝜑 (a)Δ−1∕2 J𝜑 x𝜑 ∈ D(Δ1∕2 𝜑 𝜑 ) = D(S𝜑 ), hence (2.12.(1)) xa ∈ 𝔑𝜑 and −1∕2 (xa)𝜑 = S𝜑 𝜋𝜑 (a)S𝜑 x𝜑 = J𝜑 Δ1∕2 J𝜑 x𝜑 𝜑 𝜋𝜑 (a)Δ𝜑 +∞

= J𝜑 𝜋𝜑 (b)J𝜑 x𝜑 − 2i PV

∫−∞

(e𝜋t + e−𝜋t )−1 J𝜑 Δit𝜑 𝜋𝜑 (b)Δ−it 𝜑 J𝜑 x𝜑 dt.

Since 𝜋𝜑 (b) ≥ 0, we get Re ((xa)𝜑 |x𝜑 )𝜑 = ( J𝜑 𝜋𝜑 (b)J𝜑 x𝜑 |x𝜑 )𝜑 = (T ′2 x𝜑 |x𝜑 )𝜑 , that is 𝜑(ax∗ x + x∗ xa)∕2 = 𝜓(x∗ x). The desired conclusion follows now by polarization. To prove the uniqueness part we consider an element a = a∗ ∈ ℳ, such that 𝔊𝜑 a ⊂ 𝔑𝜑 and 𝜑(ay∗ x + y∗ xa) = 0 for every x, y ∈ 𝔊𝜑 .

72

Normal Weights

Let X = J𝜑 𝜋𝜑 (a)J𝜑 . Then, for x, y ∈ 𝔊𝜑 , we get 0 = (x𝜑 |( ya)𝜑 )𝜑 + ((xa)𝜑 |y𝜑 )𝜑 = (x𝜑 |S𝜑 𝜋𝜑 (a)S𝜑 y𝜑 )𝜑 + (S𝜑 𝜋𝜑 (a)S𝜑 x𝜑 |y𝜑 )𝜑 1∕2 −1∕2 = (XΔ−1∕2 x𝜑 |Δ1∕2 y𝜑 )𝜑 . 𝜑 𝜑 y𝜑 )𝜑 + (XΔ𝜑 x𝜑 |Δ𝜑 1∕2

−1∕2

−1∕2

Using the approximation result (1), for every 𝜉, 𝜂 ∈ D(Δ𝜑 )∩D(Δ𝜑 ) we obtain (XΔ𝜑 1∕2 −1∕2 (XΔ𝜑 𝜉|Δ𝜑 𝜂)𝜑 = 0 and by ([L], 9.23) it follows that X = 0, that is, a = 0.

1∕2

𝜉|Δ𝜑 𝜂)𝜑 +

We now consider some examples. 6.6. Let K be an infinite dimensional separable Hilbert space and consider the von Neumann ̄ ℬ(𝒦 ) ⊗ 1𝒦 acting on the Hilbert space algebras ℳ = ℬ(𝒦 ) ⊗ 1𝒦 ⊗ 1𝒦 and 𝒩 = ℬ(𝒦 ) ⊗ ̄ 𝒦⊗ ̄ 𝒦 . By ([L], 8.15) we know that each of the von Neumann algebras ℳ, ℳ ′ , 𝒩 , 𝒩 ′ ℋ =𝒦⊗ has a separating vector. By the Dixmier–Marechal theorem (Dixmier & Marechal, 1971; [L], C.3.5), the set of separating vectors of any von Neumann algebra is either empty or is a dense G𝛿 set. We conclude that there exists a vector in ℋ which is simultaneously cyclic and separating for both ℳ and 𝒩 . Thus, there exist von Neumann algebras ℳ ⊂ 𝒩 ⊂ ℬ(ℋ ), ℳ ≠ 𝒩 , having a common cyclic and separating vector 𝜉0 ∈ ℋ ∗) . Let A = Δ(ℳ, 𝜉0 ) and B = Δ(𝒩 , 𝜉0 ) be the associated modular operators ([L], 10.6). Then A and B are nonsingular positive self-adjoint operators in ℋ such that D(A1∕2 ) ⊂ D(B1∕2 ) and ‖B1∕2 𝜉‖ = ‖A1∕2 𝜉‖ for all 𝜉 ∈ D(A1∕2 )

(1)

B ≠ A.

(2)

but

Indeed, for the corresponding S operators we have S(ℳ, 𝜉0 ) ⊂ S(𝒩 , 𝜉0 ) and, consequently, for 𝜉 ∈ D(A1∕2 ) we get ‖A1∕2 𝜉‖ = ‖ J(ℳ, 𝜉0 )S(ℳ, 𝜉0 )𝜉‖ = ‖S(ℳ, 𝜉0 )𝜉)‖ = ‖S(𝒩 , 𝜉0 )𝜉‖ = ‖ J(𝒩 , 𝜉0 )S(𝒩 , 𝜉0 )𝜉)‖ = ‖B1∕2 𝜉‖, hence assertion (1) holds. Also (2) holds, for if A = B then J(ℳ, 𝜉0 ) = J(𝒩 , 𝜉0 ) = J and, since ℳ ⊂ 𝒩 , we obtain 𝒩 ′ ⊂ ℳ ′ and also ℳ ′ = JℳJ ⊂ J𝒩 J = 𝒩 ′ , that is ℳ = 𝒩 , a contradiction. Consider now the canonical trace tr on ℬ(ℋ ) and the n.s.f. weights 𝜑 = trA , 𝜓 = trB on ℬ(ℋ ) (cf. 4.4). From assertion (1), it follows that B ≤ A and hence (4.5) 𝜓 ≤ 𝜑.

(3)

𝜓 ≠ 𝜑.

(4)

while from assertion (2) we obtain (4.5, 4.8)

∗) Using

the T-theorem ([L]; C.6.1), it is easy to see that this situation can occur only if ℳ, 𝒩 , ℳ ′ , 𝒩 ′ are all infinite.

Equality and Majorization of Weights

73

Also, we have 𝜓(x∗ x) = 𝜑(x∗ x) for every x ∈ 𝔑𝜑 .

(5)

Indeed, from (1) it follows that the operator B1∕2 A−1∕2 is defined and isometric on D(A−1∕2 ) so, by ([L], Proposition 9.24), the function 𝛼 ↦ F(𝛼) = B𝛼 A−𝛼 satisfies condition 3.14.(ii). This example shows that the equivalent conditions of Corollary 3.14 do not insure that 𝜓 = 𝜑, and that in Theorem 6.2 we cannot omit the condition that 𝜓 commutes with 𝜑. Also, the example shows that Proposition 6.1 is no longer valid if the operator a is not assumed to be bounded. 6.7. Consider again the canonical trace tr on ℬ(ℋ ), a nonsingular positive self-adjoint operator A in ℋ and the n.s.f. weight 𝜑 = trA on ℬ(ℋ ). Let a ∈ 𝔑𝜑 and x0 = 𝜉 ⊗ 𝜂̄ with 𝜉 ∈ D(A1∕2 ) and 𝜂 ∉ D(A1∕2 ). Then (4.23) x0 ∈ 𝔑∗𝜑 hence x0 a ∈ 𝔐𝜑 , but ax0 = a𝜉 ⊗ 𝜂̄ ∉ 𝔐𝜑 hence ax0 + x0 a ∉ 𝔐𝜑 . It follows that the assertion of Proposition 6.4 cannot be extended to all x ∈ ℳ. In what follows we show that nor can the assertion of Proposition 6.5 be extended to all x ∈ 𝔐𝜑 . By (A.14) there exist be b ∈ ℬ(ℋ ), 0 ≤ b ≤ 1, a nonsingular positive self-adjoint operator A in ℋ and a vector 𝜁 ∈ D(A−1∕2 ) such that, putting +∞

a=

∫−∞

2(e𝜋t + e−𝜋t )−1 Ait bA−it dt ∈ ℬ(ℋ ),

(1)

we have 0 ≤ a ≤ 1 and aA−1∕2 𝜁 ∉ D(A1∕2 ). Then, for 𝜂 = A−1∕2 𝜁, we have 𝜂 ∈ D(A1∕2 ), a𝜂 ∉ D(A1∕2 ). On the other hand, for an arbitrary vector 𝜉0 ∈ D(A−1∕2 ) ∩ D(ln A), the vector 𝜉 = A−1∕2 𝜉0 has the properties (A.13): 𝜉 ∈ D(A1∕2 ), a𝜉 ∈ D(A1∕2 ). Consider the n.s.f. weight 𝜑 = trA on ℬ(ℋ ) and the operator x0 = 𝜉 ⊗ 𝜂. ̄ Then (4.23) x0 ∈ 𝔐𝜑 and ax0 = a𝜉 ⊗ 𝜂̄ ∈ 𝔐𝜑 but x0 a = 𝜉 ⊗ a𝜂 ∉ 𝔐𝜑 , hence ax0 + x0 a ∉ 𝔐𝜑 . Using the same arguments as in ([L], 10.16.(1), (2), (3)) one shows that a weight 𝜓 on ℬ(ℋ ) is defined by { 𝜓( y) =

‖ J𝜑 𝜋𝜑 (b1∕2 )J𝜑 ( y1∕2 )𝜑 ‖2𝜑 , if y ∈ 𝔐𝜑 ( y ∈ ℬ(ℋ )+ ). +∞, in the contrary case

Since b ≤ 1, we have 𝜓 ≤ 𝜑. The element a defined by (1) is exactly the element given by the proof of Proposition 6.5, that is, the unique element a ∈ ℳ, 0 ≤ a ≤ 1, such that 𝔊𝜑 a ⊂ 𝔑𝜑 and x ∈ 𝔊2𝜑 ⇒ ax + xa ∈ 𝔐𝜑 and 𝜓(x) = 𝜑(ax + xa)∕2, but x0 ∈ 𝔐𝜑 and ax0 + x0 a ∉ 𝔐𝜑 .

74

Normal Weights

6.8. We now give an example connected with the Radon–Nikodym type theorem of Sakai ([L], 5.21), namely we show that on ℬ(ℋ ) there exist an n.s.f. weight 𝜑 and a normal positive form 𝜓 ≤ 𝜑 such that 𝜓 ≠ 𝜑(a ⋅ a) for all a ∈ ℳ + . Indeed, let {𝜉n }n∈ℕ be an orthonormal basis of the separable infinite dimensional Hilbert space ℋ . Let A be the positive self-adjoint operator in ℋ , diagonalizable with respect to {𝜉n }n∈ℕ , such that A𝜉n = n𝜉n

(n ∈ ℕ),

and let e ∈ ℬ(ℋ ) be the orthogonal projection onto the linear subspace spanned by the vector ∑ 𝜉 = n n−1 𝜉n ∈ ℋ . Define 𝜑 = trA and 𝜓 = tre . Then 𝜑 is an n.s.f. weight on ℬ(ℋ ), 𝜓 is a normal positive form on ℬ(ℋ ), and 𝜓 ≤ 𝜑 as e ≤ 1 ≤ A. Assume that there exists a ∈ ℬ(ℋ )+ such that 𝜓 = 𝜑(a ⋅ a). Let 𝜂 ∈ ℋ , ‖𝜂‖ = 1 and f = 𝜂 ⊗ 𝜂. ̄ Then 1∕2 lim(aA𝜀 a𝜂|𝜂) = lim tr( faA𝜀 af ) = lim tr(A1∕2 𝜀 afaA𝜀 ) = 𝜑(afa)

𝜀→0

𝜀→0

𝜀→0

= 𝜓( f ) = tr(efe) = tr( fef ) = (e𝜂|𝜂), wo

hence aA𝜀 a → e. Since A𝜀 ↑ A, it follows that aA𝜀 a ↑ e. As A ≥ 1, we have A𝜀 = A(1 + 𝜀A)−1 ≥ (1 + 𝜀)−1 , hence a2 ≤ e. Since e is a minimal projection, we conclude a = 𝜆e with 0 ≤ 𝜆 ≤ 1, so that ∑ ‖𝜉‖2 = (e𝜉|𝜉) ≥ (aA𝜀 a𝜉|𝜉) = 𝜆2 (A𝜀 𝜉|𝜉) = 𝜆2 n−1 (1 + 𝜀n)−1 n

tends to +∞ when 𝜀 → 0, a contradiction. 6.9. Let us denote by Wnsf (ℳ) the set of all n.s.f. weights on the W ∗ -algebra ℳ. For 𝜆 ∈ [0, +∞] and 𝜑1 , 𝜑2 ∈ Wnsf (ℳ) write 𝜑2 ≤ 𝜑1 (𝜆) if there exists an ℳ-valued function F, defined and w-continuous on the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜆}, analytic in the interior of this strip, such that F(it) = [D𝜑2 ∶ D𝜑1 ] for every t ∈ ℝ and ‖F(𝛼)‖ ≤ 1 for every 𝛼 ∈ ℂ, 0 ≤ Re 𝛼 ≤ 𝜆. Proposition. Let ℳ be a W ∗ -algebra. For each 𝜆 ∈ [0, +∞], the relation “𝜑2 ≤ 𝜑1 (𝜆)” is an order relation on Wnsf (ℳ). Proof. Let S(𝜆) = {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜆} and D(𝜆) = {𝛼 ∈ ℂ; −𝜆 ≤ Re 𝛼 ≤ 𝜆}. It is obvious that 𝜑 ≤ 𝜑(𝜆) for all 𝜑 ∈ Wnsf (ℳ). Let 𝜑1 , 𝜑2 , 𝜑3 ∈ Wnsf (ℳ) be such that 𝜑3 ≤ 𝜑2 (𝜆), 𝜑2 ≤ 𝜑1 (𝜆) and denote by F32 ∶ S(𝜆) → ℳ, F21 ∶ S(𝜆) → ℳ the corresponding functions. Using the chain rule (3.5), it is easy to check that the properties of the function F31 ∶ S(𝜆) → ℳ defined by F31 (𝛼) = F32 (𝛼)F21 (𝛼), (𝛼 ∈ S(𝜆)), yield the relation 𝜑3 ≤ 𝜑1 (𝜆). Let 𝜑1 , 𝜑2 ∈ Wnsf (ℳ) be such that 𝜑2 ≤ 𝜑1 (𝜆) and 𝜑1 ≤ 𝜑2 (𝜆), and denote by F21 ∶ S(𝜆) → ℳ, F12 ∶ S(𝜆) → ℳ the corresponding functions. Let 𝜓 be an arbitrary normal state on ℳ and f21 = 𝜓 ◦ F21 , f12 = 𝜓 ◦ F12 . By Corollary 3.4, we have F21 (it) = F12 (it)∗ , hence f21 (it) = f12 (it) = f12 (−it)

Equality and Majorization of Weights

75

for all t ∈ ℝ. It follows that the function f21 can be extended to an analytic function, still denoted by f21 , defined on D(𝜆), and such that |f21 (𝛼)| ≤ 1(𝛼 ∈ D(𝜆)). Since f21 (0) = 1, by the maximum modulus principle we get f21 (𝛼) = 1(𝛼 ∈ D(𝜆)). In particular, 𝜓(F21 (it)) = f21 (it) = 1 for all t ∈ ℝ. Since 𝜓 was an arbitrary normal state on ℳ we infer that [D𝜑2 ∶ D𝜑1 ]t = F21 (it) = 1(t ∈ ℝ), that is, 𝜑2 = 𝜑1 by Corollary 3.6. By Corollary 3.13, the ordering 𝜑2 ≤ 𝜑1 (1∕2) is the usual pointwise ordering 𝜑2 (x) ≤ 𝜑1 (x), (x ∈ ℳ)+ . On the other hand, the ordering 𝜑2 ≤ 𝜑1 (1∕4) for faithful normal positive forms is studied in Proposition 3.18. It is clear that if 𝜑2 ≤ 𝜑1 (𝜆) and 𝜇 < 𝜆, then 𝜑2 ≤ 𝜑1 (𝜇). In particular, 𝜑2 ≤ 𝜑1 (∞) ⇒ 𝜑2 ≤ 𝜑1 .

(1)

6.10. With the help of the ordering corresponding to 𝜆 = ∞, we can define a metric d on the set Wnsf (ℳ) by putting, for 𝜑1 , 𝜑2 ∈ Wnsf (ℳ), d(𝜑1 , 𝜑2 ) = inf{𝛿 > 0, 𝜑2 ≤ e𝛿 𝜑1 (∞), 𝜑1 ≤ e𝛿 𝜑2 (∞)}. By Proposition 6.9, d is indeed a metric on Wnsf (ℳ). By remark 6.9.(1), we see that if d(𝜑1 , 𝜑2 ) ≤ 𝛿, then e−𝛿 𝜑1 (x) ≤ 𝜑2 (x) ≤ e𝛿 𝜑1 (x) (x ∈ ℳ + ). It follows that for every x ∈ ℳ + the function Wnsf (ℳ) ∋ 𝜑 ↦ 𝜑(x) is d-continuous.

(1)

Also, if 𝜑1 , 𝜑2 are faithful normal positive forms on ℳ and d(𝜑1 , 𝜑2 ) ≤ 𝛿, then ‖𝜑2 − 𝜑1 ‖ ≤ 2 sup{|𝜑2 (x) − 𝜑1 (x)|; x ∈ ℳ + , ‖x‖ ≤ 1} ≤ 2 max{e𝛿 − 1, 1 − e−𝛿 } ≤ 4𝜀, hence ‖𝜑1 − 𝜑2 ‖ ≤ 4d(𝜑1 , 𝜑2 ).

(2)

Let 𝜑1 , 𝜑2 ∈ Wnsf (ℳ) and assume that d(𝜑1 , 𝜑2 ) ≤ 𝛿 < +∞. Then there exist ℳ-valued functions G21 , G12 defined and w-continuous on {𝛼 ∈ ℂ; Re 𝛼 ≥ 0}, analytic in {𝛼 ∈ ℂ; Re 𝛼 > 0}, such that ‖G21 (𝛼)‖ ≤ 1, ‖G12 (𝛼)‖ ≤ 1 and G21 (it) = [D𝜑2 ∶ D(e𝛿 𝜑1 )]t = e−it𝛿 [D𝜑2 ∶ D𝜑1 ]t , G12 (it) = [D𝜑1 ∶ D(e𝛿 𝜑2 )]t = e−it𝛿 [D𝜑1 ∶ D𝜑2 ]t . Taking into account Corollary 6.4, we see that the function F21 ∶ ℂ → ℳ defined by { F21 (𝛼) =

e𝛿𝛼 G21 (𝛼), if Re 𝛼 ≥ 0 e−𝛿𝛼 G12 (−𝛼) ̄ ∗ , if Re 𝛼 ≤ 0

76

Normal Weights

is entire analytic and we have ‖F21 (𝛼)‖ ≤ e𝛿|Re 𝛼| (𝛼 ∈ ℂ), F21 (it) = [D𝜑2 ∶ D𝜑1 ]t (t ∈ ℝ).

(3) (4)

Thus, the relation d(𝜑1 , 𝜑2 ) ≤ 𝛿 is equivalent to the existence of an entire analytic ℳ-valued function F21 satisfying (3) and (4). To continue the study of the metric d we need the following Lemma. For every 𝜀 > 0 and every r > 0, there exists 𝛿 = 𝛿(𝜀, r) > 0 such that if F is a Banach space valued entire analytic function with the property ‖F(𝛼)‖ ≤ e𝛿|Re 𝛼| , (𝛼 ∈ ℂ), then ‖F(𝛼) − F(0)‖ ≤ 𝜀 for all 𝛼 ∈ ℂ with |𝛼| < r. Proof. Let 𝒜 be the linear space of all entire analytic complex valued functions equipped with the compact-open topology and denote by 𝒦 (𝛿) the set of those f ∈ 𝒜 such that |f (𝛼)| ≤ e𝛿|Re 𝛼| for all ⋂ 𝛼 ∈ ℂ. Then each 𝒦 (𝛿) is a compact subset of 𝒜 and 𝛿>0 𝒦 (𝛿) = {𝛼 ∈ ℂ; |𝛼| ≤ 1} = 𝒦 (0). On the other hand, 𝒟 (𝜀, r) = {f ∈ 𝒜 ; |f (𝛼) − f (0)| < 𝜀 for all 𝛼 ∈ ℂ with |𝛼| ≤ r} is an open subset of 𝒜 and 𝒟 (𝜀, r) ⊃ 𝒦 (0). It follows that there exists 𝛿 = 𝛿(𝜀, r) > 0 such that 𝒟 (𝜀, r) ⊃ 𝒦 (𝛿) and it is easy to check that this 𝛿 = 𝛿(𝜀, r) satisfies the requirements of the lemma. Thus, if d(𝜑1 , 𝜑2 ) ≤ 𝛿 = 𝛿(𝜀, r), then ‖F21 (𝛼) − 1‖ ≤ 𝜀 for all 𝛼 ∈ ℂ with |𝛼| ≤ r.

(5)

Moreover, for any other 𝜑0 ∈ Wnsf (ℳ) such that d(𝜑1 , 𝜑0 ) = 𝛾 < +∞ we have F20 (𝛼) = F21 (𝛼)F10 (𝛼)(𝛼 ∈ ℂ), and using (3) and (5) we obtain ‖F20 (𝛼) − F10 (𝛼)‖ ≤ 𝜀e𝛾|Re 𝛼| for all 𝛼 ∈ ℂ with |𝛼| ≤ r.

(6)

Proposition. The metric space (Wnsf (ℳ), d) is complete. Proof. Let {𝜑n }n≥0 be a d-Cauchy sequence in Wnsf (ℳ). By (6) it follows that the sequence {Fn0 }n≥0 converges uniformly on compact subsets of ℂ to a function F∞0 ∶ ℂ → ℳ. Consequently, the function F∞0 is entire analytic and the mapping ℝ ∋ t ↦ F∞0 (it) ∈ U(ℳ) is a 𝜎 𝜑0 -cocycle. By Theorem 5.1, there exists 𝜑∞ ∈ Wnsf (ℳ) such that F∞0 (it) = [D𝜑∞ ∶ D𝜑0 ]t for every t ∈ ℝ. Let 𝜀 > 0. Since {𝜑n } is a d-Cauchy sequence there exists m ∈ ℕ such that ‖Fnm (𝛼)‖ ≤ e𝜀|Re 𝛼| (𝛼 ∈ ℂ) for every n ≥ m. It follows that ‖F∞m (𝛼)‖ = ‖F∞0 (𝛼)F0m (𝛼)‖ = lim ‖Fn0 (𝛼)F0m (𝛼)‖ n

= lim ‖Fnm (𝛼)‖ ≤ e𝜀|Re𝛼| n

for all 𝛼 ∈ ℂ, that is d(𝜑∞ , 𝜑m ) ≤ 𝜀. Hence d(𝜑∞ , 𝜑m ) → 0. 6.11. Notes. Theorem 6.2 is a refinement, given in Zsidó (1978), of the Pedersen–Takesaki theorem on the equality of weights (Pedersen & Takesaki, 1973). Propositions 6.4, 6.5 and the examples in

The Spatial Derivative

77

Section 6.7 are due to Van Daele (1976). The examples in Sections 6.6 and 6.8 are from Connes (1973b) and Pedersen and Takesaki (1973). The material contained in Sections 6.9 and 6.10 is due to Connes and Takesaki (1977). For our exposition, we have used Connes (1973b); Connes and Takesaki (1977); Van Daele (1976); Pedersen and Takesaki (1973) and Zsidó (1978).

7 The Spatial Derivative In this section, we introduce a positive self-adjoint operator called the spatial derivative of a weight 𝜑 on a von Neumann algebra ℳ ⊂ ℬ(ℋ ) with respect to a weight 𝜑′ on its commutant ℳ ′ ⊂ ℬ(ℋ ), as a generalization of the modular operator. By way of applications we give several continuity properties. 7.1. Let 𝜓 be an n.s.f. weight on the von Neuman algebra 𝒩 ⊂ ℬ(ℋ ). A vector 𝜂 ∈ ℋ is called 𝜓-bounded if there exists a constant 0 < 𝜆 < +∞ such that ‖b𝜂‖ ≤ 𝜆‖b𝜓 ‖𝜓 for every b ∈ 𝔑𝜓 . Let D(ℋ , 𝜓) = {𝜂 ∈ ℋ ; 𝜂 is 𝜓-bounded}. We show that D(ℋ , 𝜓) is a dense linear subspace of ℋ .

(1)

∑ Indeed, by Corollary 5.9 there is a family {𝜂k }k∈K ⊂ ℋ such that 𝜓( y) = k ( y𝜂k |𝜂k ) for all y ∈ 𝒩 + . For each k ∈ K, we have 𝜂k ∈ D(ℋ , 𝜓) as ‖b𝜂k ‖2 = (b∗ b𝜂k |𝜂k ) ≤ 𝜓(b∗ b) = ‖b𝜓 ‖2𝜓 (b ∈ 𝔑𝜓 ). On the other hand, it is clear that D(ℋ , 𝜓) is an 𝒩 -invariant linear subspace of ℋ , so that the orthogonal projection f ∈ ℬ(ℋ ) onto the closure of D(ℋ , 𝜓) belongs to 𝒩 . Since f𝜂k = 𝜂k (k ∈ K), it follows that 𝜓(1 − f ) = 0, hence f = 1 since 𝜓 is faithful. Each vector 𝜂 ∈ D(ℋ , 𝜓) defines a bounded operator R𝜂𝜓 ∶ ℋ𝜓 → ℋ , uniquely determined, such that R𝜂𝜓 b𝜓 = b𝜂, (b ∈ 𝔑𝜓 ), and its adjoint (R𝜂𝜓 )∗ ∶ ℋ → ℋ𝜓 , is uniquely determined by ((R𝜂𝜓 )∗ 𝜉|b𝜓 )𝜓 = (𝜉|b𝜂) for every b ∈ 𝔑𝜓 , 𝜉 ∈ ℋ . It is easy to check that 𝜂 ∈ D(ℋ , 𝜓) and y′ ∈ 𝒩 ′ ⇒ y′ 𝜂 ∈ D(ℋ , 𝜓) and Ry𝜓′ 𝜂 = y′ R𝜂𝜓

(2)

and that the operators R𝜂𝜓 ∶ ℋ𝜓 → ℋ intertwine the standard representation 𝜋𝜓 of 𝒩 and the identity representation of 𝒩 : 𝜂 ∈ D(ℋ , 𝜓) and y ∈ 𝒩 ⇒ yR𝜂𝜓 = R𝜂𝜓 𝜋𝜓 ( y).

(3)

𝜂 ∈ D(ℋ , 𝜓) ⇒ 𝜂 ∈ R𝜂𝜓 (ℋ𝜓 ) = s(R𝜂𝜓 (R𝜂𝜓 )∗ )ℋ

(4)

Also,

so

since if 𝔑𝜓 ∋ bk → 1 then R𝜂𝜓 (bk )𝜓 = bk 𝜂 → 𝜂.

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Normal Weights

Let 𝒮 (ℋ , 𝜓) be the linear subspace spanned in ℬ(ℋ ) by the operators of form R𝜂𝜓 (R𝜁𝜓 )∗ with 𝜂, 𝜁 ∈ D(ℋ , 𝜓). Then 𝒮 (ℋ , 𝜓) is an s∗ -dense two-sided ideal in 𝒩 ′ .

(5)

Indeed, the fact that 𝒮 (ℋ , 𝜓) is a two-sided ideal in 𝒩 ′ follows from (2) and (3). Let y′ ∈ 𝒩 ′ , y′ ≠ 0. From (1), it follows that there exists 𝜂 ∈ D(ℋ , 𝜓) with y′ 𝜂 ≠ 0. Using (4) and (2), we get 0 ≠ Ry𝜓′ 𝜂 (Ry𝜓′ 𝜂 )∗ = y′ R𝜂𝜓 (R𝜂𝜓 )∗ y′∗ ≤ ‖R𝜂𝜓 ‖2 y′ y′∗ . Thus, for every positive element b′ = y′ y′∗ ≠ 0 of 𝒩 ′ , there exists a positive element c′ = Ry𝜓′ 𝜂 (Ry𝜓′ 𝜂 )∗ ≠ 0 of the two-sided ideal 𝒮 (ℋ , 𝜓) such that c′ ≤ b′ . By ([L], 3.20) it follows that 𝒮 (ℋ , 𝜓) is s∗ -dense in 𝒩 ′ . Also, every positive element in 𝒮 (ℋ , 𝜓) is of the form

∑

R𝜂𝜓 (R𝜂𝜓 )∗ k

k=1

k

(6)

with 𝜂k ∈ D(ℋ , 𝜓) and 𝜂 ∈ ℕ. Indeed, let b′ =

∑n k=1

R𝜂𝜓k (R𝜁𝜓 )∗ ∈ J(ℋ , 𝜓), b′ ≥ 0. Then b′ = (b′ + b′∗ )∕2, so

0 ≤ 2−1

k

n ∑ k=1

(R𝜂𝜓 (R𝜁𝜓 )∗ + R𝜁𝜓 (R𝜂𝜓 )∗ ) = b′ ≤ c′ = 2−1 k

k

n ∑

k

k

R𝜂𝜓+𝜁 (R𝜂𝜓+𝜁 )∗ .

k=1

k

k

k

k

Since 0 ≤ b′ ≤ c′ , it follows (1.4) that there exists y′ ∈ 𝒩 ′ such that b′ = y′ c′ y′∗ . Consequently, b′ = 2−1

n ∑

Ry𝜓′ (𝜂 +𝜁 ) (Ry𝜓′ (𝜂 +𝜁 ) )∗ .

k=1

k

k

k

k

Finally, we show that there exists a family {𝜂k }k ⊂ D(ℋ , 𝜓) such that s∗ -

∑ k

R𝜂𝜓 (R𝜂𝜓 )∗ = 1. k

k

(7)

Indeed, since the two-sided ideal 𝒮 (ℋ , 𝜓) is s∗ -dense in 𝒩 ′ , there exists a series of positive elements in 𝒮 (ℋ , 𝜓), which is s∗ -convergent to 1. Thus (7) follows using (6). 7.2. In the particular case when 𝒩 = 𝜋𝜓 (𝒩 ) ⊂ ℬ(ℋ𝜓 ), the operators R𝜂𝜓 are just the operators R𝜂 ∈ 𝜋𝜓 (𝒩 )′ considered in Section 2.12. If 𝜓 ′ is the natural weight on 𝜋𝜓 (𝒩 )′ , then (2.12.(7)) 𝜓 ′ (R∗𝜁 R𝜂 ) = (𝜂|𝜁 )𝜓

(𝜂, 𝜁 ∈ D(ℋ𝜓 , 𝜓)).

In the general case considered in Section 7.1, the following similar result holds: 𝜂, 𝜁 ∈ D(ℋ , 𝜓) ⇒ (R𝜁𝜓 )∗ R𝜂𝜓 ∈ 𝔐𝜓 ′ and 𝜓 ′ ((R𝜁𝜓 )∗ R𝜂𝜓 ) = (𝜂|𝜁).

(1)

Indeed, due to the polarization relation ([L], 3.21) it is sufficient to consider only the case 𝜁 = 𝜂. Then let (R𝜂𝜓 )∗ = V(R𝜂𝜓 (R𝜂𝜓 )∗ )1∕2 be the polar decomposition of (R𝜂𝜓 )∗ . Using 7.1.(4) and 7.1.(3),

The Spatial Derivative

79

we see that the partial isometry V ∶ ℋ → ℋ𝜓 has the properties V ∗ V𝜂 = 𝜂 and 𝜋𝜓 ( y)V = Vy for every y ∈ 𝒩 . Since V𝜂 ∈ ℋ𝜓 and 𝜋𝜓 (b)V𝜂 = Vb𝜂 = VR𝜂𝜓 b𝜓 (b ∈ 𝔑𝜓 ), it follows that V𝜂 ∈ D(ℋ𝜓 , 𝜓) and RV𝜂 = VR𝜂𝜓 . Consequently, R𝜂𝜓 = V ∗ RV𝜂 , (R𝜂𝜓 )∗ R𝜂𝜓 = R∗V𝜂 RV𝜂 and 𝜓 ′ ((R𝜂𝜓 )∗ R𝜂𝜓 ) = 𝜓 ′ (R∗V𝜂 RV𝜂 ) = ‖V𝜂‖2𝜓 = ‖𝜂‖2 . 7.3. Consider now a von Neumann algebra ℳ ⊂ ℬ(ℋ ) with commutant 𝒩 = ℳ ′ ⊂ ℬ(ℋ ); let 𝜑 be a normal semifinite weight on ℳ and 𝜓 an n.s.f. weight on 𝒩 . We show that the function q ∶ D(ℋ , 𝜓) ∋ 𝜂 ↦ q(𝜂) = 𝜑(R𝜂𝜓 (R𝜂𝜓 )∗ ) ∈ [0, +∞] is lower semicontinuous and has the following properties: D(q) = {𝜂 ∈ D(ℋ , 𝜓); q(𝜂) < +∞} is dense in ℋ ;

(1)

q(𝜆𝜂) = |𝜆| q(𝜂) for all 𝜂 ∈ D(ℋ , 𝜓) and 𝜆 ∈ ℂ; q(𝜂 + 𝜁) + q(𝜂 − 𝜁) = 2q(𝜂) + 2q(𝜁) for all 𝜂, 𝜁 ∈ D(ℋ , 𝜓).

(2) (3)

2

Indeed, by Corollary 5.9, there exists a family {𝜉i }i∈I ⊂ ℋ such that 𝜑(x) = Then, for 𝜂 ∈ D(ℋ , 𝜓), we get ∑ ∑ q(𝜂) = ‖(R𝜂𝜓 )∗ 𝜉i ‖2𝜓 = sup |((R𝜂𝜓 )∗ 𝜉i |b𝜓 )𝜓 | i

=

∑ i

i

sup

b∈𝔑𝜓 ‖b𝜓 ‖𝜓 ≤1

∑

i (x𝜉i |𝜉i )(x

∈ ℳ + ).

b∈𝔑𝜓 ‖b𝜓 ‖𝜓 ≤1

|(b 𝜉i |𝜂)|, ∗

hence q is lower semicontinuous. Since 𝜑 is semifinite, 𝔑∗𝜑 is s-dense in ℳ so that the set {x𝜂; x ∈ 𝔑∗𝜑 , 𝜂 ∈ D(ℋ , 𝜓)} is total 𝜓 𝜓 ∗ (Rx𝜂 ) )= in ℋ and contained in D(ℋ , 𝜓). For x ∈ 𝔑∗𝜑 and 𝜂 ∈ D(ℋ , 𝜓) we have q(x𝜂) = 𝜑(Rx𝜂 𝜓 𝜓 ∗ ∗ 𝜓 2 ∗ 𝜑(xR𝜂 (R𝜂 ) x ) ≤ ‖R𝜂 ‖ 𝜑(xx ) < +∞, so that x𝜂 ∈ D(q). This proves (1), while (2) and (3) are obvious. By (A.10) it follows that there exists a greatest positive self-adjoint operator Δ(𝜑∕𝜓) in ℋ such that D(Δ(𝜑∕𝜓)1∕2 ) ⊃ D(q), 𝜂 ∈ D(ℋ , 𝜓) ⇒ ‖Δ(𝜑∕𝜓)

1∕2

𝜂‖ = 2

𝜑(R𝜂𝜓 (R𝜂𝜓 )∗ ),

(4) (5)

and we have Δ(𝜑∕𝜓)1∕2 |D(q) = Δ(𝜑∕𝜓)1∕2 .

(6)

The operator Δ(𝜑∕𝜓) is called the spatial derivative of the normal semifinite weight 𝜑 on ℳ ⊂ ℬ(ℋ ) with respect to the n.s.f. weight 𝜓 on 𝒩 = ℳ ′ ⊂ ℬ(ℋ ).

80

Normal Weights

Note that if 𝜑 ∈ ℳ∗+ , then D(q) = D(ℋ , 𝜓) and hence Δ(𝜑∕𝜓)1∕2 |D(ℋ , 𝜓) = Δ(𝜑∕𝜓)1∕2 .

(7)

Using (5) and 7.1(2), it is easy to check that s(Δ(𝜑∕𝜓)) ≤ s(𝜑),

(8)

but we shall see that in fact equality holds (7.4). Let us compute the spatial derivative in a very simple case, namely ℳ = ℬ(ℋ ), so that 𝒩 = ℳ ′ = ℂ ⋅ 1ℋ . On 𝒩 we consider the weight 𝜓 ∶ 𝜆 ⋅ 1ℋ ↦ 𝜆. An arbitrary normal semifinite weight 𝜑 on ℳ is of the form 𝜑 = trA with A a positive self-adjoint operator in ℋ (4.10). It is clear that ℋ𝜓 = ℂ and ‖𝜆 ⋅ 1ℋ ‖𝜓 = |𝜆| (𝜆 ⋅ 1ℋ ∈ 𝒩 ). Also, D(ℋ , 𝜓) = ℋ and for 𝜂 ∈ ℋ we have R𝜂𝜓 ∶ ℂ ∋ 𝜆 ↦ 𝜆𝜂 ∈ ℋ , hence (R𝜂𝜓 )∗ ∶ ℋ ∋ 𝜉 ↦ (𝜉|𝜂) ∈ ℂ. Thus, R𝜂𝜓 (R𝜂𝜓 )∗ = 𝜂 ⊗ 𝜂̄ and q(𝜂) = 𝜑(R𝜂𝜓 (R𝜂𝜓 )∗ ) = 𝜑(𝜂 ⊗ 𝜂) ̄ = ‖A1∕2 𝜂‖2 (see 4.23), hence Δ(𝜑∕𝜓) = A. Denoting the weight 𝜓 by 1, we can write the conclusion as follows: Δ(trA ∕1) = A

(ℳ = ℬ(ℋ ), 𝒩 = ℳ ′ = ℂ ⋅ 1ℋ ).

(9)

7.4. The main result concerning the spatial derivative is the following Theorem (A. Connes). Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra with commutant 𝒩 = ℳ ′ ⊂ ℬ(ℋ ), 𝜑 an n.s.f. weight on ℳ and 𝜓 an n.s.f. weight on 𝒩 . Then s(Δ(𝜑∕𝜓)) = 1 and Δ(𝜑∕𝜓)−1 = Δ(𝜓∕𝜑); 𝜎t𝜑 (x) = Δ(𝜑∕𝜓)it xΔ(𝜑∕𝜓)−it 𝜎t𝜓 ( y) = Δ(𝜑∕𝜓)−it yΔ(𝜑∕𝜓)it

(1)

(x ∈ ℳ, t ∈ ℝ);

(2)

( y ∈ 𝒩 , t ∈ ℝ);

(3)

and for every normal semifinite 𝜑1 on ℳ we have s(Δ(𝜑1 ∕𝜓)) = s(𝜑1 ); it

(4) it

Δ(𝜑1 ∕𝜓) = [D𝜑1 ∶ D𝜑]t Δ(𝜑∕𝜓)

(t ∈ ℝ).

(5)

The proof is given in Sections 7.5–7.10. 7.5. For the W ∗ -algebra 𝒩 , we have the identity representation id ∶ 𝒩 → ℬ(ℋ ) and the standard representation 𝜋𝜓 ∶ 𝒩 → ℬ(ℋ𝜓 ) associated with 𝜓. We consider the direct sum of these representations 𝜋 ∶ 𝒩 → ℬ(ℋ ⊕ ℋ𝜓 ) and the von Neumann algebra 𝜋(𝒩 )′ ⊂ ℬ(ℋ ⊕ ℋ𝜓 ).

The Spatial Derivative

81

Every operator T ∈ ℬ(ℋ ⊕ ℋ𝜓 ) is a matrix ( T11 T= T21

T12 T22

) ,

with T11 ∈ ℬ(ℋ ), T12 ∈ ℬ(ℋ𝜓 , ℋ ), T21 ∈ ℬ(ℋ , ℋ𝜓 ), T22 ∈ ℬ(ℋ𝜓 ) such that for any vector 𝜁 ∈ ℋ ⊕ ℋ𝜓 , ( ) 𝜁1 𝜁= , 𝜁2 with 𝜁1 ∈ ℋ , 𝜁2 ∈ ℋ𝜓 , we have

( T𝜁 =

T11 𝜁1 + T12 𝜁2 T21 𝜁1 + T22 𝜁2

) .

Define ℐ (𝜋𝜓 , id) = {X ∈ ℬ(ℋ𝜓 , ℋ ); yX = X𝜋𝜓 ( y) for all y ∈ 𝒩 } ℐ (id, 𝜋𝜓 ) = {X ∈ ℬ(ℋ , ℋ𝜓 ); Xy = 𝜋𝜓 ( y)X for all y ∈ 𝒩 }. It is easy to check that T ∈ 𝜋(𝒩 )′ ⇔ T11 ∈ 𝒩 ′ = ℳ, T12 ∈ ℐ (𝜋𝜓 , id), T21 ∈ ℐ (𝜋𝜓 , id), T22 ∈ 𝜋𝜓 (𝒩 )′ Let 𝜓 ′ be the natural weight on 𝜋𝜓 (𝒩 )′ (2.12.(6)). Then the weight 𝜓 ′ on 𝜋𝜓 (𝒩 )′ and the weight 𝜑 on ℳ define an n.s.f. weight 𝜃 = 𝜃(𝜑, 𝜓 ′ ) on 𝜋(𝒩 )′ 𝜃(T) = 𝜑(T11 ) + 𝜓 ′ (T22 ) (T ∈ 𝜋(𝒩 )′+ ). As for the balanced weight (3.1, 3.10) one shows that ( ) ( 𝜑 ) 𝜎t (T11 ) 0 T11 0 𝜃 ′ 𝜎t = 0 T22 0 𝜎t𝜓 (T22 )

(T11 ∈ ℳ, T22 ∈ 𝜋𝜓 (𝒩 )′ )

and that there exists a group of isometrics {S𝜑t }t∈ℝ on the Banach space ℐ (𝜋𝜓 , id), uniquely determined, such that 𝜎t𝜃

(

0 0

T12 0

)

( =

0 S𝜑t (T12 ) 0 0

) (T12 ∈ ℐ (𝜋𝜓 , id)).

Recall (7.1) that R𝜂𝜓 ∈ ℐ (𝜋𝜓 , id) for all 𝜂 ∈ D(ℋ , 𝜓). 7.6. With the assumptions of Theorem 7.4 and the notation of Section 7.5, we have the following Lemma. There exists an so-continuous unitary representation ℝ ∋ t ↦ u𝜑t ∈ ℬ(ℋ ),

82

Normal Weights

uniquely determined, such that 𝜎t𝜑 (x) = u𝜑t xu𝜑−t

𝜎t𝜓 (x) = u𝜑−t yu𝜑t S𝜑t (R𝜂𝜓 ) = Ru𝜓𝜑 𝜂 (𝜂 t

(x ∈ ℳ)

(1)

(y ∈ 𝒩 )

(2)

∈ D(ℋ , 𝜓)).

(3)

Proof. The uniqueness of u𝜑t follows obviously from (3). The proof of the existence statement is divided into three steps. (I) If the lemma is valid for a certain n.s.f. weight 𝜑0 on ℳ, then it is valid for any other n.s.f. weight 𝜑 on ℳ. Indeed, it is easy to check that the mapping 𝜑

ℝ ∋ t ↦ u𝜑t = [D𝜑 ∶ D𝜑0 ]t ut 0 ∈ ℬ(ℋ ) is an so-continuous unitary representation with properties (1) and (2). Also (3) follows if we note that Ru𝜓𝜑 𝜂 = [D𝜑 ∶ D𝜑0 ]t R 𝜓𝜑0

ut 𝜂

t

(𝜂 ∈ D(ℋ , 𝜓))

and that for 𝜃0 = 𝜃(𝜑0 , 𝜓 ′ ) we have ( [D𝜃 ∶ D𝜃0 ]t =

[D𝜑 ∶ D𝜑0 ]t 0

0 1

) (t ∈ ℝ).

(II) If the lemma is valid for some particular realization of 𝒩 (i.e., of ℳ) as a von Neumann algebra, then it remains valid for any other realization. Indeed, if we have two different realizations of 𝒩 , then the corresponding von Neumann algebras are *-isomorphic. Since any *-isomorphism between von Neumann algebras is the composition of an amplification, an injective induction and a spatial isomorphism ([L], E.8.8), we may consider separately these three cases. The case of a spatial isomorphism is trivial. The case of an amplification: we assume the lemma is valid for the von Neumann algebras 𝒩 ⊂ ℬ(ℋ ), ℳ = 𝒩 ′ ⊂ ℬ(ℋ ); we have to prove it for the von Neumann algebras 𝒩 ⊗ 1 ⊂ ℬ(ℋ )⊗ℬ(𝓁 2 (I)), ℳ⊗ℬ(𝓁 2 (I)) = (𝒩 ⊗ 1)′ ⊂ ℬ(ℋ )⊗ℬ(𝓁 2 (I)), where I is an arbitrary set. A vector 𝜂 = {𝜂i }i∈I in ℋ ⊗𝓁 2 (I) = 𝓁(I, ℋ ) is 𝜓-bounded whenever each 𝜂i ∈ ℋ (i ∈ I) is ∑ 𝜓-bounded and i ‖R𝜂𝜓i ‖2 < +∞. In this case, the operator R𝜂𝜓 ∶ ℋ𝜓 → ℋ ⊗𝓁 2 (I) acts as follows: (R𝜂𝜓 𝜁)i = R𝜂𝜓 𝜁 i

(𝜁 ∈ ℋ ).

If 𝜑 is any n.s.f. weight on ℳ and tr is the canonical trace on ℬ(𝓁 2 (I)), then Φ = 𝜑⊗tr (see 5.2 or 8.2) is an n.s.f. weight on ℳ⊗ℬ(𝓁 2 (I)) and 𝜎tΦ = 𝜎t𝜑 ⊗ t(t ∈ ℝ). It is now easy to check that if {u𝜑t }t satisfies the requirements of the lemma for (𝒩 , 𝜓; ℳ, 𝜑), 𝜑 2 then {UΦ t = ut ⊗1}t satisfies those requirements for (𝒩 ⊗ 1, 𝜓; ℳ⊗ℬ(𝓁 (I)), Φ).

The Spatial Derivative

83

Taking into account step (I) of the proof, it follows that the lemma is true for the von Neumann algebras 𝒩 ⊗ 1, ℳ⊗ℬ(𝓁 2 (I)). The case of an injective induction: we assume the lemma is valid for the von Neumann algebras 𝒩 ⊂ ℬ(ℋ ), ℳ = 𝒩 ′ ⊂ ℬ(ℋ ) and we have to prove it for the von Neumann algebras 𝒩 e ⊂ ℬ(eℋ ), eℳe = (𝒩 e)′ ⊂ ℬ(eℋ ), where e ∈ ℳ is a projection with the central support z(e) = 1. It is easy to see that D(eℋ , 𝜓) = eD(ℋ , 𝜓) and that for 𝜂 ∈ D(eℋ , 𝜓) ⊂ D(ℋ , 𝜓) the operator R𝜂𝜓 ∶ ℋ𝜓 → eℋ is the corestriction of the operator R𝜂𝜓 ∶ ℋ𝜓 → ℋ . There exists an n.s.f. weight 𝜑 on ℳ such that e ∈ ℳ 𝜑 . Indeed, if ℳ is semifinite, we can take 𝜑 equal to any n.s.f. trace on ℳ; if ℳ is of type III, then ℳ ≈ Mat2 (eℳe) and we can take 𝜑 equal to the balanced weight corresponding to an arbitrary n.s.f. weight on eℳe with respect to itself. With this choice of 𝜑, the restriction of the weight 𝜑e (2.22) to eℳe is an n.s.f. weight on eℳe; its modular automorphism group is the restriction of the modular automorphism group of 𝜑. If {u𝜑t }t satisfies the requirements of the lemma for (𝒩 , 𝜓; ℳ, 𝜑), then u𝜑t e = eu𝜑t as e ∈ ℳ 𝜑 and it is easy to check that {u𝜑t e}, satisfies the requirements of the lemma for (𝒩 e, 𝜓; eℳe, 𝜑e ). Taking into account step (I) of the proof, it follows that the lemma is true for the von Neumann algebras 𝒩 e, eℳe. (III) By (I) and (II), it is sufficient to prove the lemma assuming 𝒩 = 𝜋𝜓 (𝒩 ) ⊂ ℬ(ℋ𝜓 ), ℳ = 𝜋𝜓 (𝒩 )′ , and 𝜑 = 𝜓 ′ . In this case, the mapping ℝ ∋ t ↦ u𝜑t = Δ−it 𝜓 ∈ ℬ(ℋ𝜓 ) is an so-continuous unitary representation satisfying (1) and (2). Also, the weight 𝜃 = 𝜃(𝜑, 𝜓 ′ ) is just the balanced weight, so that S𝜑t = 𝜎t𝜑 by 3.10.(1), and consequently it 𝜑 S𝜑t (R𝜂 ) = 𝜎t𝜑 (R𝜂 ) = Δ−it 𝜓 R𝜂 Δ𝜓 = RΔ𝜓−it 𝜂 = Rut 𝜂 ,

□

which proves (3).

7.7. By Stone’s theorem ([L], 9.20), there exists a unique positive self-adjoint operator A𝜑 in ℋ with s(A𝜑 ) = 1 such that u𝜑t = Ait𝜑 (t ∈ ℝ). In this section, we show that Δ(𝜑∕𝜓) = A𝜑 thus proving statements (2) and (3) of Theorem 7.4, as well as showing that s(Δ(𝜑∕𝜓)) = 1. By Proposition 2.20, we obtain for every T ∈ 𝜋(𝒩 )′ with 𝜃(T ∗ T) < +∞, a bounded regular positive Borel measure 𝜇 on (0, +∞) such that 𝜃(T ∗ 𝜎t𝜃 (T)) =

∞

∫0

∞

𝜆it d𝜇(𝜆), 𝜃(TT ∗ ) =

𝜆 d𝜇(𝜆) ≤ +∞.

∫0

Let 𝜂 ∈ D(ℋ , 𝜓) and T ∈ 𝜋(𝒩 )′ with T12 = R𝜂𝜓 and T11 = T21 = T22 = 0. Then, using 7.6.(3) and 7.2.(1), we get ∞

∫0

𝜆it d𝜇(𝜆) = 𝜓 ′ ((R𝜂𝜓 )∗ S𝜑t (R𝜂𝜓 )) = 𝜓 ′ ((R𝜂𝜓 )∗ Ru𝜓𝜑 𝜂 ) = (u𝜑t 𝜂|𝜂) = (Ait𝜑 𝜂|𝜂) t

84

Normal Weights

hence 𝜇 is the spectral measure associated with A𝜑 and 𝜂 ∈ ℋ . Consequently, ∞ 2 ‖A1∕2 𝜑 𝜂‖ =

∫0

𝜆 d𝜇(𝜆) = 𝜑(R𝜂𝜓 (R𝜂𝜓 )∗ ) = ‖Δ(𝜑|𝜓)1∕2 𝜂‖2

(𝜂 ∈ D(ℋ , 𝜓)).

(1)

On the other hand, we infer from 7.6.(1) and 7.6.(2) that the *-automorphisms ℬ(ℋ ) ∋ z ↦ u𝜑t zu𝜑−t ∈ ℬ(ℋ )(t ∈ ℝ) preserve ℳ, 𝒩 , 𝜑, 𝜓, hence u𝜑t Δ(𝜑∕𝜓)u𝜑−t = Δ(𝜑∕𝜓)(t ∈ ℝ), that is, the operators Δ(𝜑∕𝜓) and A𝜑 commute.

(2)

Now, the conclusions (1) and (2) imply that Δ(𝜑∕𝜓) = A𝜑 (see, for instance, Theorem 6.2). 7.8. We prove that for two n.s.f. weights 𝜓1 , 𝜓2 on 𝒩 Δ(𝜑∕𝜓2 )−it Δ(𝜑∕𝜓1 )it = [D𝜓2 ∶ D𝜓1 ]t

(t ∈ ℝ).

(1)

To this end consider the balanced weight 𝜓 = 𝜃(𝜓1 , 𝜓2 ) on the von Neumann algebra Mat2 (𝒩 ) ⊂ ̃ ⊂ ℬ(ℋ ⊕ ℋ ) isomorphic to ℳ by amplification, and the n.s.f. ℬ(ℋ ⊕ ℋ ) with commutant ℳ ̃ weight 𝜑̃ on ℳ corresponding to 𝜑 under this amplification. Also, as in 3.11.(1), identify the Hilbert spaces ℋ𝜓 = ℋ𝜓1 ⊕ ℋ𝜓2 ⊕ ℋ𝜓1 ⊕ ℋ𝜓2 . It is easy to check that the vector 𝜂 = 𝜂1 ⊕ 𝜂2 ∈ ℋ ⊕ ℋ is 𝜓-bounded if and only if 𝜂1 ∈ D(ℋ , 𝜓1 ), 𝜂2 ∈ D(ℋ , 𝜓2 ) and in this case R𝜂𝜓 =

(

𝜓

R𝜂11 0

𝜓

R𝜂22 0

0 𝜓 R𝜂11

0 𝜓 R𝜂22

) ∶ ℋ𝜓 → ℋ ⊕ ℋ ,

hence 𝜑(R ̃ 𝜂𝜓 (R𝜂𝜓 )∗ ) = 𝜑(R𝜓𝜂 1 (R𝜓𝜂 1 )∗ ) + 𝜑(R𝜓𝜂 2 (R𝜓𝜂 2 )∗ ). 1

1

2

2

Consequently, Δ(𝜑∕𝜓) ̃ = Δ(𝜑∕𝜓1 ) ⊕ Δ(𝜑∕𝜓2 ). Using 7.4.(3) which has already been proved, we get (

0 0 [D𝜓2 ∶ D𝜓1 ]t 0

) =

𝜎t𝜓 (

= ( =

(

0 0 1 0

)

Δ(𝜑∕𝜓1 )−it 0

0 Δ(𝜑∕𝜓2 )−it

0 Δ(𝜑∕𝜓2 )−it Δ(𝜑∕𝜓1 )it

0 0

)( )

0 0 1 0

)(

Δ(𝜑∕𝜓1 )it 0

0 Δ(𝜑∕𝜓2 )it

)

.

7.9. We now prove that Δ(𝜓, 𝜑) = Δ(𝜑∕𝜓)−1 . Taking into account the argument in 7.6.(I) and 7.8.(1), we see that it is sufficient to prove the earlier identity only for a particular pair of weights 𝜑, 𝜓. Then by the arguments of 7.6.(II), it follows

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85

that it is sufficient for the proof to assume that 𝒩 = 𝜋𝜓 (𝒩 ) ⊂ ℬ(ℋ𝜓 ), ℳ = 𝜋𝜓 (𝒩 )′ ⊂ ℬ(ℋ𝜓 ), and 𝜑 = 𝜓 ′ . In this last case, equality is obvious. 7.10. To complete the proof of Theorem 7.4, we still have to check 7.4.(4) and 7.4.(5). If the weight 𝜑1 is faithful, this has already been done in Sections 7.7 and 7.6.(I). Since, by 7.3.(8), s(Δ(𝜑1 ∕𝜓)) ≤ s(𝜑1 ), the general case follows by considering the induction of 𝒩 by s(𝜑1 ), as the restriction of 𝜑1 to s(𝜑1 )ℳs(𝜑1 ) is faithful. 7.11 Corollary. Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra with commutant 𝒩 = ℳ ′ ⊂ ℬ(ℋ ), 𝜑 an n.s.f. weight on ℳ and 𝜓 an n.s.f. weight on 𝒩 . For every 𝜉 ∈ D(ℋ , 𝜑) and every 𝜂 ∈ D(ℋ , 𝜓), we have |(𝜉|𝜂)|2 ≤ 𝜑(R𝜂𝜓 (R𝜂𝜓 )∗ )𝜓(R𝜑𝜉 (R𝜑𝜉 )∗ ) ≤ +∞. Proof. Let Δ = Δ(𝜑∕𝜓), hence Δ−1 = Δ(𝜓∕𝜑). Using Definition 7.3.(5), we see that if either 𝜉 ∉ D(Δ−1∕2 ) or 𝜂 ∉ D(Δ1∕2 ) then the right-hand side of the inequality is equal to +∞, while if 𝜉 ∈ D(Δ−1∕2 ) and 𝜂 ∈ D(Δ1∕2 ) the inequality follows from the estimate |(𝜉|𝜂)|2 = |(Δ−1∕2 𝜉|Δ1∕2 𝜂)|2 ≤ ‖Δ−1∕2 𝜉‖2 ‖Δ1∕2 𝜂‖2 . 7.12 Corollary (Haagerup). Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra with commutant 𝒩 = ℳ ′ ⊂ ℬ(ℋ ) and ℝ ∋ t ↦ ut ∈ ℬ(ℋ ) an so-continuous unitary representation. Then the following conditions are equivalent: (i) there exists 𝜑 ∈ Wnsf (ℳ) such that 𝜎t𝜑 = Ad(ut )t , (t ∈ ℝ); (ii) there exists 𝜓 ∈ Wnsf (𝒩 ) such that 𝜎t𝜓 = Ad(u∗t )t , (t ∈ ℝ). Proof. By symmetry it is sufficient to prove (i) ⇒ (ii). Let 𝜓0 be an arbitrary n.s.f. weight on 𝒩 . Using (i) it follows that ℝ ∋ t ↦ wt = u∗t Δ(𝜑∕𝜓0 )it ∈ ℳ ′ = 𝒩 is a 𝜎 𝜓0 -cocycle hence, by Theorem 5.1, there exists an n.s.f. weight 𝜓 on 𝒩 such that [D𝜓 ∶ D𝜓0 ]t = wt (t ∈ ℝ). Then 𝜓

𝜎t𝜓 ( y) = wt 𝜎t 0 ( y)w∗t = u∗t yut

( y ∈ 𝒩 , t ∈ ℝ).

7.13. Some computation rules for the spatial derivative are contained in the following Proposition. Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra with commutant 𝒩 = ℳ ′ ⊂ ℬ(ℋ ), 𝜑, 𝜑1 , 𝜑2 normal semifinite weights on ℳ, 𝜓 an n.s.f. weight on 𝒩 and a ∈ ℳ an invertible element. Then 𝜑1 ≤ 𝜑2 ⇔ Δ(𝜑1 ∕𝜓) ≤ Δ(𝜑2 ∕𝜓) Δ(𝜑(a ⋅ a∗ )∕𝜓) = a∗ Δ(𝜑∕𝜓)a

(1) (2)

̂ Δ(𝜑1 + 𝜑2 ∕𝜓) = Δ(𝜑1 ∕𝜓)+Δ(𝜑 2 ∕𝜓).

(3)

and, if 𝜑1 , 𝜑2 ∈ ℳ∗+ ,

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Normal Weights

Proof. (1) Let q1 , q2 be the functions associated as in Section 7.3 with the weights 𝜑1 , 𝜑2 , respectively. If 𝜑1 ≤ 𝜑2 , then we obviously have D(q2 ) ⊂ D(q1 ) and q1 (𝜂) ≤ q2 (𝜂) for any 𝜂 ∈ D(q2 ), so it follows that Δ(𝜑1 ∕𝜓) ≤ Δ(𝜑2 ∕𝜓). Conversely, if Δ(𝜑1 ∕𝜓) ≤ Δ(𝜑2 ∕𝜓), then by 7.1.(6) we infer that 𝜑1 (x) ≤ 𝜑2 (x) for every positive element x ∈ 𝒮 (ℋ , 𝜓). Since 𝒮 (ℋ , 𝜓) is an s∗ -dense two-sided ideal of ℳ = 𝒩 ′ (7.1.(5)), any element x ∈ ℳ + is the s∗ -limit of an increasing net of positive elements in 𝒮 (ℋ , 𝜓) ([L], 3.21). Thus, the inequality 𝜑1 (x) ≤ 𝜑2 (x) remains valid for all x ∈ ℳ + , since 𝜑1 , 𝜑2 are normal. (2) By 7.1.(2), for every 𝜂 ∈ D(ℋ , 𝜓) we have a𝜂 ∈ D(ℋ , 𝜓) and 𝜓 𝜓 ∗ ‖(a∗ Δ(𝜑∕𝜓)a)1∕2 𝜂‖2 = ‖Δ(𝜑∕𝜓)1∕2 a𝜂‖2 = 𝜑(Ra𝜂 (Ra𝜂 ) ) = 𝜑(aR𝜂𝜓 (R𝜂𝜓 )∗ a∗ ).

Since a is invertible, it follows from the definition of Δ(𝜑∕𝜓) that a∗ Δ(𝜑∕𝜓)a is the greatest positive self-adjoint operator which satisfies the above equation; hence it coincides with the operator Δ(𝜑(a ⋅ a∗ )∕𝜓). (3) This follows from the definition of the weak sum (A.11), using 7.3.(7). 7.14. The operators of form Δ(𝜑∕𝜓) are characterized by the following Theorem (A. Connes). Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra with commutant 𝒩 = ℳ ′ ⊂ ℬ(ℋ ) and 𝜓 an n.s.f. weight on 𝒩 . For a positive self-adjoint operator A in ℋ , the following statements are equivalent: (i) there exists a normal semifinite weight 𝜑 on ℳ such that A = Δ(𝜑∕𝜓); 𝜓 (ii) Ait y = 𝜎−t ( y)Ait for all y ∈ 𝒩 and all t ∈ ℝ; (iii) A1∕2 |D(A1∕2 ) ∩ D(ℋ , 𝜓) = A1∕2 and, for any 𝜂1 , … , 𝜂n ∈ D(ℋ , 𝜓), the number ∑n 1∕2 𝜂 ‖2 depends only on ∑n R 𝜓 (R 𝜓 )∗ . 𝜂k k k=1 ‖A k=1 𝜂k Proof. The implications (i) ⇒ (ii) and (i) ⇒ (iii) are obvious. (ii) ⇒ (i). If 𝜑0 is any n.s.f. weight on ℳ, then by (ii) the mapping ℝ ∋ t ↦ Ait Δ(𝜑0 ∕𝜓)−it is a 𝜎 𝜑0 -cocycle; thus by Theorem 5.1, there exists a unique normal semifinite weight 𝜑 on ℳ such that [D𝜑 ∶ D𝜑0 ]t = Ait Δ(𝜑0 ∕𝜓)−it (t ∈ ℝ). By 7.4.(5), we get Δ(𝜑∕𝜓)it = Ait (t ∈ ℝ), whence A = Δ(𝜑∕𝜓). (iii) ⇒ (i). Assuming (iii), we infer from 7.1.(6), 7.1.(5) that there exists a unique weight 𝜑0 on 𝒮 (ℋ , 𝜓) such that 𝜑0 (a) =

n ∑

‖A1∕2 𝜂k ‖2 for 0 ≤ a =

k=1

n ∑ k=1

R𝜂𝜓 (R𝜂𝜓 )∗ ∈ 𝒮 (ℋ , 𝜓). k

(1)

k

s

If {zi } ⊂ ℳ is a net such that zi → 1, then 𝜑0 (a) ≤ lim inf 𝜑0 (zi az∗i ),

(2)

i

since the mapping ℋ ∋ 𝜁 ↦ ‖A1∕2 𝜁‖2 is lower semicontinuous and 𝜑0 (zi az∗i ) = For x ∈ ℳ + , let 𝜑(x) = sup{𝜑0 (a); a ∈ 𝒮 (ℋ , 𝜓), 0 ≤ a ≤ x}.

∑n k=1

‖A1∕2 zi 𝜂k ‖2 .

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87

We thus obtain a function 𝜑 ∶ ℳ + → [0, +∞] with the properties 𝜑(𝜆x) = 𝜆𝜑(x),

𝜑(x + y) ≥ 𝜑(x) + 𝜑( y) (x, y ∈ ℳ + , 𝜆 ≥ 0).

(3)

Assume that xi ↑ x in ℳ + . By Proposition 1.4, there exist zi ∈ ℳ, with ‖zi ‖ ≤ 1, such that xi = zi xz∗i s

and zi → 1. If a ∈ 𝒮 (ℋ , 𝜓) and 0 ≤ a ≤ x, then 0 ≤ zi az∗i ≤ zi xz∗i = xi and 𝜑0 (a) ≤ lim inf 𝜑0 (zi az∗i ) ≤ sup 𝜑(xi ). i

i

It follows that 𝜑(x) ≤ sup 𝜑(xi ), and hence 𝜑 is normal. Since 𝒮 (ℋ , 𝜓) is an s∗ -dense two-sided ideal in ℳ, the elements of ℳ + can be approximated by increasing positive nets in 𝒮 (ℋ , 𝜓). Since 𝜑 is normal and superadditive (cf. (3)) and since 𝜑|𝒮 (ℋ , 𝜓) = 𝜑0 is additive, it follows that 𝜑 is a normal weight on ℳ such that 𝜑(R𝜂𝜓 (R𝜂𝜓 )∗ ) = ‖A1∕2 𝜂‖2

(𝜂 ∈ D(ℋ , 𝜓)).

(4)

Now, by 7.1.(7) we see that 𝜑 is semifinite. Thus (4) is equivalent to ‖Δ(𝜑∕𝜓)1∕2 𝜂‖ = ‖A1∕2 𝜂‖ (𝜂 ∈ D(ℋ , 𝜓)). Since each of the operators Δ(𝜑∕𝜓)1∕2 and A1∕2 is equal to the closure of its restriction to the intersection of D(ℋ , 𝜓) with its domain of definition, it follows that A = Δ(𝜑∕𝜓). 7.15. The following corollary characterizes the operators of the form Δ(𝜑∕𝜓) with 𝜑 ∈ ℳ∗+ , the assumptions being as in Theorem 7.14. Corollary. For a positive self-adjoint operator A in ℋ , the following statements are equivalent: (i) there exists 𝜑 ∈ ℳ∗+ such that A = Δ(𝜑∕𝜓); 𝜓 (ii) Ait y = 𝜎−t ( y)Ait for all y ∈ 𝒩 , t ∈ ℝ, and there exists a family {𝜂k } in D(ℋ , 𝜓) such that ∑ k

R𝜂𝜓 (R𝜂𝜓 )∗ = 1 and k

k

∑

‖A1∕2 𝜂k ‖2 < +∞.

k

(iii) D(ℋ , 𝜓) ⊂ D(A1∕2 ), A1∕2 |D(ℋ , 𝜓) = A1∕2 and there exists a constant 0 < 𝜆 < +∞ such that for any 𝜂1 , 𝜁1 , … , 𝜂n , 𝜁n ∈ D(ℋ , 𝜓) we have n n |∑ | ‖∑ ‖ | | ‖ 𝜓 ‖ | (A1∕2 𝜂k |A1∕2 𝜁k )| ≤ 𝜆 ‖ R𝜂𝜓k (R𝜁 )∗ ‖ . k | | ‖ ‖ | k=1 | ‖ k=1 ‖

Proof. (i) ⇔ (iii). If A = Δ(𝜑, 𝜓) with 𝜑 ∈ ℳ∗+ , then (iii) is obviously satisfied with 𝜆 = ‖𝜑‖. Conversely, if (iii) holds, then 7.14.(iii) holds also, and so there exists a normal semifinite weight 𝜑 on ℳ with A = Δ(𝜑∕𝜓). By the construction of 𝜑 and by (iii), it follows that |𝜑(a)| ≤ 𝜆‖a‖ for every element 0 ≤ a ∈ 𝒮 (ℋ , 𝜓), hence 𝜑 ∈ ℳ∗+ and ‖𝜑‖ = 𝜑(1) ≤ 𝜆. (i) and (iii) ⇒ (ii). This follows from 7.3.(7) and 7.1.(7).

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Normal Weights

(ii) ⇒ (i). By Theorem 7.14, there is a normal semifinite weight 𝜑 on ℳ such that Δ(𝜑∕𝜓) = A and we have ∑ ∑ 𝜑(1) = 𝜑(R𝜂𝜓 (R𝜂𝜓 )∗ ) = ‖A1∕2 𝜂k ‖2 < +∞. k

k

k

k

7.16. Let 𝜓 be an n.s.f. weight on the von Neumann algebra 𝒩 ⊂ ℬ(ℋ ). A closed linear operator T in ℋ , with polar decomposition T = u|T|, is called homogeneous of degree s ∈ ℝ with respect to 𝜓, if u ∈ 𝒩 ′ and |T|it y = 𝜎st𝜓 ( y)|T|it

( y ∈ 𝒩 , t ∈ ℝ).

Thus, the operators of form A = Δ(𝜑∕𝜓) are characterized by Theorem 7.14 as the only positive self-adjoint operators homogeneous of degree s = −1 with respect to 𝜓. For such an operator, it is possible to define “the integral with respect to 𝜓” by choosing any family {𝜂k } ⊂ D(ℋ , 𝜓) with ∑ 𝜓 𝜓 ∗ k R𝜂k (R𝜂k ) = 1 (7.1.(7)) and putting ∫

A d𝜓 =

∑

‖A1∕2 𝜂k ‖2 = 𝜑(1).

(1)

k

A is called “𝜓-integrable” if 𝜑(1) < +∞, that is if A satisfies the equivalent conditions of Corollary 7.15. Exploiting these ideas, it is possible to develop a “noncommutative integration theory” by defining spaces ℒ p (ℋ , 𝜓), where ℒ p (ℋ , 𝜓) is the set of closed linear operators T on ℋ which are homogeneous of degree s = −1∕p with respect to 𝜓 and such that |T|p is 𝜓-integrable. As an example, we show that T ∈ ℒ 2 (ℋ , 𝜓) ⇒ T ∗ ∈ ℒ 2 (ℋ , 𝜓) and

∫

TT ∗ d𝜓 =

∫

T ∗ T d𝜓.

(2)

Indeed, let T = uA = Bu be the polar decompositions of T with u∗ u = s(A), uu∗ = s(B), A = ∑ |T|, B = |T ∗ | = uAu∗ ([L], 9.30). If {𝜂k } is a family in D(ℋ , 𝜓) with k R𝜂𝜓k (R𝜂𝜓k )∗ = s(A), then ∑ {𝜁k = u𝜂k } is a family in D(ℋ , 𝜓) with k R𝜁k (R𝜁k )∗ = s(B) and ∫

TT ∗ d𝜓 =

∫

B2 d𝜓 =

∑

‖B𝜁k ‖2 =

k

∑

‖A𝜂k ‖2 =

k

∫

A2 d𝜓 =

∫

T ∗ T d𝜓.

We also record the following form of (1): ∫

Δ(𝜑∕𝜓) d𝜓 = 𝜑(1).

(3)

7.17. Another important application of Theorem 7.14 is contained in the following Proposition. Let 𝜑, {𝜑k }k∈K be normal semifinite weights on the von Neumann algebra ℳ ⊂ ℬ(ℋ ) with commutant 𝒩 = ℳ ′ ⊂ ℬ(ℋ ) and 𝜓 an n.s.f. weight on 𝒩 . If 𝜑k (x) ↑ 𝜑(x) for all x ∈ ℳ + ,

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89

then 𝜑

s∗

𝜎t k (x) → 𝜎t𝜑 (x) for all x ∈

⋃

s(𝜑k )ℳs(𝜑k ) and all t ∈ ℝ;

(1)

k s∗

[D𝜑k ∶ D𝜏] → [D𝜑 ∶ D𝜏]t for all 𝜏 ∈ Wnsf (ℳ) and all t ∈ ℝ;

(2)

Δ(𝜑k ∕𝜓) ↑ Δ(𝜑∕𝜓).

(3)

Moreover, the convergence in (1) and (2) is uniform for |t| ≤ t0 . Proof. Let Δk = Δ(𝜑k ∕𝜓), Δ = Δ(𝜑∕𝜓). Using 7.13.(1) and (A.5), we see that there exists a unique positive self-adjoint operator A in ℋ such that Δk ↑ A ≤ Δ, hence (A.3) s∗

Δitk → Ait uniformly for |t| ≤ t0 . Since Δk are homogeneous of degree s = −1 with respect to 𝜓, it follows that A enjoys the same property and hence, by Theorem 7.14, there exists a unique normal semifinite weight 𝜇 on ℳ such that A = Δ(𝜇∕𝜓). Since A ≤ Δ, it follows (7.13.(1)) that 𝜇 ≤ 𝜑 and since Δk ≤ A we have 𝜑k ≤ 𝜇. Hence 𝜇 = 𝜑. s∗

We have thus proved that (3) holds and that Δitk → Ait uniformly for |t| ≤ t0 . Now (1) follows on applying Theorem 7.4, and (2) is obtained by applying (1) to the balanced weights 𝜃(𝜑k , 𝜏) ↑ 𝜃(𝜑, 𝜏) on Mat2 (ℳ) and to the element e21 ∈ Mat2 (ℳ). 7.18. A similar result holds for norm convergence of normal positive forms: Proposition. Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra with commutant 𝒩 = ℳ ′ ⊂ ℬ(ℋ ), 𝜑, {𝜑n }n∈ℕ faithful normal positive forms on ℳ and 𝜓 an n.s.f. weight on 𝒩 . If ‖𝜑n −𝜑‖ → 0, then s∗

𝜑

𝜎t n (x) → 𝜎t𝜑 (x) for all x ∈ ℳ and all t ∈ ℝ; s∗

[D𝜑n ∶ D𝜏]t → [D𝜑 ∶ D𝜏]t for all 𝜏 ∈ Wnsf (ℳ) and all t ∈ ℝ; s∗

Δ(𝜑n ∕𝜓)it → Δ(𝜑∕𝜓)it for all t ∈ ℝ.

(1) (2) (3)

Moreover, in each case convergence is uniform for |t| ≤ t0 . Proof. It is sufficient to prove (1). Indeed, (2) for 𝜏 ∈ ℳ∗+ follows from (1) by considering the balanced forms ‖𝜃(𝜑n , 𝜏) − 𝜃(𝜑, 𝜏)‖ → 0 and then, for an arbitrary 𝜏 ∈ Wnsf (ℳ), we can use the chain rule (3.5). Also, (3) follows from (2) using 7.4.(5): s∗

Δ(𝜑n ∕𝜓)it = [D𝜑n ∶ D𝜑]t Δ(𝜑∕𝜓)it → Δ(𝜑∕𝜓)it . To prove (1) we may assume that 𝜑 = 𝜔𝜉0 where 𝜉0 ∈ ℋ is a cyclic and separating vector for ℳ. Then ([L], Thm. 10.25) for each n ∈ ℕ there exists a unique vector 𝜉n ∈ 𝔓𝜉0 , cyclic and separating

90

Normal Weights

for ℳ, such that 𝜑n = 𝜔𝜉n . By ([L], Proposition 10.24), we have ‖𝜉n − 𝜉‖ ≤ ‖𝜑n − 𝜑‖1∕2 . Let Δ0 = Δ𝜉0 , Δn = Δ𝜉n be the corresponding modular operators, and J = J𝜉0 = J𝜉n ([L], Lemma 2/10.24) the canonical conjugation. For every x ∈ ℳ, we have 1∕2

1∕2

1∕2 ∗ ∗ ‖Δ1∕2 n x𝜉n − Δ0 x𝜉0 ‖ = ‖ JΔn x𝜉n − JΔ0 x𝜉0 ‖ = ‖x 𝜉n − x 𝜉0 ‖

≤ ‖x‖‖𝜑n − 𝜑‖1∕2 , 1∕2

1∕2

hence ‖(Δn + 1)x𝜉n − (Δ0 + 1)x𝜉0 ‖ ≤ 2‖x‖‖𝜑n − 𝜑‖1∕2 and 1∕2

1∕2

−1 ‖[(Δ1∕2 − (Δ0 + 1)−1 ](Δ0 + 1)x𝜉0 ‖ n + 1) 1∕2

−1 1∕2 ≤ ‖(Δ1∕2 n + 1) [(Δ0 + 1)x𝜉0 − (Δn + 1)x𝜉n ] + (x𝜉n − x𝜉0 ‖

≤ 3‖x‖‖𝜑n − 𝜑‖1∕2 . 1∕2

Since Δ0

1∕2

1∕2

= Δ0 |ℳ𝜉0 , the linear subspace (Δ0 s 1∕2 (Δn + 1)−1 →

1∕2 ‖(Δn + 1)−1 ‖

≤ 1, it follows that for |t| ≤ t0 . Consequently, for x ∈ ℳ we have

+ 1)ℳ𝜉0 is dense in ℋ ([L], E.9.1) and since s

1∕2

(Δ0 + 1)−1 and hence (A.3) Δitn → Δit0 uniformly

s∗

𝜑

𝜑 it −it 𝜎t n (x) = Δitn xΔ−it t → Δ0 xΔ0 = 𝜎t (x),

uniformly for |t| ≤ t0 . 7.19. A related result is contained in the next proposition. Proposition. For each t ∈ ℝ, there exists a constant 0 < Ct < +∞ such that for every W ∗ -algebra ℳ and every pair 𝜑, 𝜓 of normal states on ℳ with s(𝜓) ≤ s(𝜑) we |1 − 𝜑([D𝜓 ∶ D𝜑]t )| ≤ Ct ‖𝜓 − 𝜑‖. Proof. If the statement is not true, then there exist W ∗ -algebras ℳn and normal states 𝜑n , 𝜓n on ℳn with s(𝜑n ) = 1 such that ‖𝜑n − 𝜓n ‖ ≤ 2−n |1 − 𝜑n ([D𝜓n ∶ D𝜑n ]t )|

(n ∈ ℕ).

We have ∑

‖𝜑n − 𝜓n ‖ < +∞

(1)

n

and, by repeating some pairs 𝜑n , 𝜓n if necessary, we may further assume that ∑ n

|1 − 𝜑n ([D𝜓n ∶ D𝜑n ]t )| = +∞.

(2)

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91

We may assume ℳn realized as a von Neumann algebra ℳn ⊂ ℬ(ℋn ) with cyclic and separating vector 𝜉n ∈ ℋn such that 𝜑n = 𝜔𝜉n . By ([L], 10.24, 10.25) there exists 𝜂n ∈ ℋn such that 𝜓n = 𝜔𝜂n , (𝜂n |𝜉n ) > 0 and ‖𝜂n − 𝜉n ‖2 ≤ ‖𝜑n − 𝜓n ‖

(n ∈ ℕ).

(3)

The infinite tensor product W ∗ -algebra ℳ = ⊗n (ℳn , 𝜑n ) (see A.17) can be realized as an infinite tensor product von Neumann algebra acting on the Hilbert space ℋ = ⊗n (ℋn , 𝜉n ), and on this von Neumann algebra we have vector states Φ = ⊗n 𝜑n and Φk = ⊗kn=1 𝜓n ⊗ ⊗∞ m=k+1 𝜑m defined, respectively, by the vectors 𝜉 = ⊗n 𝜉n ∈ ℋ and 𝜂 (k) = ⊗kn=1 𝜂n ⊗ ⊗∞ m=k+1 𝜉m ∈ ℋ . From (1) and (3), it follows that the sequence {𝜂 (k) } is norm-convergent in ℋ , so that the sequence {Φk } is norm-convergent in ℳ∗ . Taking into account the results of Section 3.9 and Proposition 7.18, it follows that the sequence [DΦk ∶ DΦ]t = [D𝜓1 ∶ D𝜑1 ) ⊗ … ⊗ [D𝜓k ∶ D𝜑k ]t ⊗ 1 is s-convcrgent in ℳ. In particular, the sequence k ∏

([D𝜓n ∶ D𝜑n ]t 𝜉n |𝜉n ) = ([DΦk ∶ DΦ]t 𝜉|𝜉)

n=1

is convergent, and hence ∑

|1 − ([D𝜓n ∶ D𝜑n ]t 𝜉n |𝜉n )| < +∞.

n

contradicting (2). 7.20. Let 𝜑 be a normal positive form on the W ∗ -algebra ℳ. Besides the notation ‖x‖𝜑 = ‖x𝜑 ‖𝜑 =

𝜑(x∗ x)1∕2 , we shall also write ‖x‖♯𝜑 = 𝜑(x∗ x + xx∗ )1∕2 , (x ∈ ℳ). For a ∈ ℳ, we shall consider the commutator [a, 𝜑] = 𝜑(⋅a) − 𝜑(a⋅).

Corollary. There exists an absolute constant 0 < C < +∞ such that if ℳ is a W ∗ -algebra, 𝜑 a normal state on ℳ and a ∈ ℳ, with ‖a‖ ≤ 1, we have ‖𝜎t𝜑 (a) − a‖∗𝜑 ≤ C(1 + |t|)‖[a, 𝜑]‖1∕2

(t ∈ ℝ).

Proof. We divide the proof into several steps. (I) First, let a = v ∈ ℳ be unitary. Then, using Proposition 7.19 for 𝜑 and 𝜓 = 𝜑(v ⋅ v∗ ), and the identity [D𝜓 ∶ D𝜑]t = v∗ 𝜎t𝜑 (v) (cf. 3.7), we obtain ‖𝜎t𝜑 (v) − v‖♯𝜑 ≤ 2Ct ‖[v, 𝜑]‖1∕2 1∕2

(t ∈ ℝ).

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Normal Weights

(II) If 0 ≤ a ≤ 1∕2, then ‖[(1 − a2 )1∕2 , 𝜑]‖ ≤ 32 ‖[a, 𝜑]‖. Indeed, by induction over n it is easy to check that ‖[an , 𝜑]‖ ≤ n‖a‖n−1 ‖[a, 𝜑]‖ ≤ n2−n+1 ‖[a, 𝜑)‖ and the desired inequality follows using the expansion (1 − a2 )1∕2 =

∞ ∑ n=0

(−1)n

2−1 (2−1 − 1) … (2−1 − n + 1) 2n a . n!

(III) If a = a∗ and ‖a‖ ≤ 1, then ‖𝜎t𝜑 (a) − a‖∗𝜑 ≤ 8Ct ‖[a, 𝜑]‖1∕2 1∕2

(t ∈ ℝ).

Indeed, let ‖[a, 𝜑]‖ = 𝜀. We have 0 ≤ (1+a)∕4 ≤ 1∕2. Putting v = (1+a)∕4+i(1−((1+a)∕4)2 )1∕2 , it follows from (II) that ‖[v, 𝜑]‖ ≤ 𝜀∕2 and ‖[v∗ , 𝜑]‖ ≤ 𝜀∕2; and, since v is unitary, we get from (I) 1∕2 1∕2 that ‖𝜎t𝜑 (v) − v‖♯𝜑 ≤ 2Ct 𝜀1∕2 , ‖𝜎t𝜑 (v∗ ) − v∗ ‖♯𝜑 ≤ 2Ct 𝜀1∕2 . Thus, the desired conclusion follows, ∗ as (1 + a)∕2 = v + v . (IV) Consider now a ∈ ℳ, ‖a‖ ≤ 1. By applying (III) for (a + a∗ )∕2 and (a − a∗ )∕2i, we get ‖𝜎t𝜑 (a) − a‖♯𝜑 ≤ 16Ct ‖[a, 𝜑]‖1∕2 1∕2

(t ∈ ℝ).

For each t ∈ ℝ, we define K(t) as the least upper bound of the numbers 𝜆 > 0 such that ‖𝜎t𝜑 (a) − a‖♯𝜑 ≤ 𝜆‖[a, 𝜑]‖1∕2 for every ℳ, 𝜑, a as in the statement of the corollary. Then the function ℝ ∋ t ↦ K(t) is lower semicontinuous, K(0) = 0, K(−t) = K(t) and K(t + s) ≤ K(t) + K(s), (t, s ∈ ℝ). With these conditions, it is easy to show that C = sup{K(t)∕(1 + |t|) ∶ t ∈ ℝ} < +∞ and this proves the corollary. In particular, we obtain, as an immediate consequence, the implication (⇐) of 2.21.(2) for 𝜑 ∈ ℳ∗+ . Moreover, if {ai }i ⊂ ℳ is a norm-bounded net, then ‖[ai , 𝜑]‖ → 0 ⇒ ‖𝜎 𝜑 (ai ) − ai ‖∗𝜑 → 0

(1)

uniformly for |t| ≤ t0 . 7.21. Notes. The results in this section are due to Connes (1980). An important part of the motivation of Connes (1980) was the work of Haagerup (1979a), which included Corollary 7.12 and several arguments used in the proof of Lemma 7.6. The proof of Proposition 7.18 is due to Araki (1974). For our exposition, we have used Araki (1974); Connes (1980) and Haagerup (1979a). Noncommutative integration theory, initiated and developed in the semifinite case by Dixmier (1953) and Segal (1953) has been extended to the general case by Haagerup (1979b). Introducing the spaces ℒ P (ℋ , 𝜑) (7.16), Connes (1980) proposed the problem of establishing the properties of these spaces and their connection with the earlier theory of Haagerup (1979b); this has been done by Hilsum (1981). Another general approach is contained in the recent work of Connes (1979a).

Tensor Products

93

8 Tensor Products In this section, we introduce the tensor product of weights, starting with the tensor product of left Hilbert algebras, and study its properties. 8.1 Proposition. Let 𝔄k ⊂ ℋk be a left Hilbert algebra with associated operators Sk , S∗k , Jk , Δk (k = 1, 2). Then ̄ ℋ2 = ℋ , 𝔄 = 𝔄1 ⊗ 𝔄2 ⊂ ℋ1 ⊗ equipped with the tensor product involutive algebra structure and with the scalar product of ℋ is a left Hilbert algebra with associated operators ̄ S2 , S∗ = S∗ ⊗ ̄ S∗ , J = J1 ⊗ ̄ J2 , Δ = Δ1 ⊗ ̄ Δ2 ; S = S1 ⊗ 1 2

(1)

we have ̄ L𝜉 L𝜉1 ⊗𝜉2 = L𝜉1 ⊗ 2 ̄ R𝜂 ⊗𝜂 = R𝜂 ⊗ R𝜂

(𝜉1 ∈ D(S1 ), 𝜉2 ∈ D(S2 ))

(2)

(𝜂1 ∈ D(S∗1 ), 𝜂2 ∈ D(S∗2 )) ̄ 𝔏(𝔄2 ), ℜ(𝔄′ ) = ℜ(𝔄′ ) ⊗ ̄ ℜ(𝔄′ ). 𝔏(𝔄) = 𝔏(𝔄1 ) ⊗ 1

2

1

2

1

2

(3) (4)

Proof. We begin by checking the axioms of a left Hilbert algebra ([L], 10.1. (i)–(iv)) for 𝔄. In order to avoid notational complications, we shall write the sums which define the elements of 𝔄 = 𝔄1 ⊗𝔄2 without specifying summation indices. Let 𝜉1 , 𝜂1 , 𝜁1 ∈ 𝔄1 and 𝜉2 , 𝜂2 , 𝜁2 ∈ 𝔄2 . ∑ ∑ ∑ ∑ ∑ ∑ (i) We have ( 𝜉1 ⊗𝜉2 )( 𝜂1 ⊗𝜂2 ) = 𝜉1 𝜂1 ⊗𝜉2 𝜂2 = L𝜉1 𝜂1 ⊗L𝜉2 𝜂2 = ( L𝜉1 ⊗L𝜉2 )( 𝜂1 ⊗𝜂2 ), ∑ ∑ ∑ hence the mapping ( 𝜂1 ⊗ 𝜂2 ) ↦ ( 𝜉1 ⊗ 𝜉2 )( 𝜂1 ⊗ 𝜂2 ) is continuous and ̄ L𝜉 L𝜉1 ⊗𝜉2 = L𝜉1 ⊗ 2

(𝜉1 ∈ 𝔄1 , 𝜉2 ∈ 𝔄2 ).

(ii) We have ((

) |(∑ )) ∑ 𝜂1 ⊗ 𝜂2 || 𝜁1 ⊗ 𝜁2 = (𝜉1 𝜂1 |𝜁1 )(𝜉2 𝜂2 |𝜁2 ) | (( ) |(∑ )∗ (∑ )) ∑ ∑ = (𝜂1 |𝜉1∗ 𝜁1 )(𝜂2 |𝜉2∗ 𝜁2 ) = 𝜂1 ⊗ 𝜂2 || 𝜉1 ⊗ 𝜉2 𝜁1 ⊗ 𝜁2 | ∑

𝜉1 ⊗ 𝜉2

) (∑

(iii) It is clear that 𝔄2 is dense in 𝔄 and hence 𝔏(𝔄) = {L𝜉1 ⊗𝜉2 ; 𝜉1 ∈ 𝔄1 , 𝜉2 ∈ 𝔄2 }so ̄ L𝜉 ; 𝜉1 ∈ 𝔄1 , 𝜉2 ∈ 𝔄2 }so = 𝔏(𝔄1 ) ⊗ ̄ 𝔏(𝔄2 ). = {L𝜉1 ⊗ 2 ∑ ∑ ∑ (iv) We have ( 𝜉1 𝜂1 ⊗ 𝜉2 𝜂2 )∗ = (𝜉1 𝜂1 )∗ ⊗ (𝜉2 𝜂2 )∗ = S1 (𝜉1 𝜂1 ) ⊗ S2 (𝜉2 𝜂2 ) = (S1 ⊗ ∑ S2 )( 𝜉1 𝜂1 ⊗ 𝜉2 𝜂2 ). Since S1 and S2 are preclosed, it follows that S1 ⊗ S2 is preclosed ([L], 9.33) and ̄ S2 . S = S1 ⊗

94

Normal Weights

̄ 𝔏(𝔄2 ) and S = S1 ⊗ ̄ S2 . From ([L], Thus, 𝔄 is indeed a left Hilbert algebra, 𝔏(𝔄) = 𝔏(𝔄1 ) ⊗ ̄ S∗ . Then 9.34) it follows that S∗ = S∗1 ⊗ 2 ̄ S2 = J1 Δ1∕2 ⊗ ̄ J2 Δ1∕2 = ( J1 ⊗ ̄ J2 )(Δ1∕2 ⊗ ̄ Δ1∕2 ), JΔ1∕2 = S = S1 ⊗ 1 2 1 2 ̄ Δ is a positive self-adjoint operator ([L], the last equality being an easy exercise. Since Δ1 ⊗ 2 ̄ J2 and Δ1∕2 = 9.34), it follows by the uniqueness of the polar decomposition that J = J1 ⊗ 1∕2 ̄ 1∕2 ̄ Δ2 . The other assertions in the statement are now easily verified. Δ1 ⊗ Δ2 ; hence Δ = Δ1 ⊗ Using Stone’s theorem ([L], 9.20) we obtain 1∕2

1∕2

̄ Δit Δit = Δit1 ⊗ 2

(t ∈ ℝ)

(5)

̄ Δ𝛼 Δ𝛼 = Δ𝛼1 ⊗ 2

(𝛼 ∈ ℂ).

(6)

and then, by analytic continuation,

̄ 𝔄2 the maximal left Hilbert algebra 𝔄′′ associated with 𝔄. We shall denote by 𝔄1 ⊗ From (3), it follows that 𝔄′1 ⊗ 𝔄′2 ⊂ 𝔄′ . On the other hand, we have S∗ |𝔄′1 ⊗ 𝔄′2 = S∗1 ⊗ S∗2 |𝔄′1 ⊗ 𝔄′2 = (S∗1 |𝔄′1 ) ⊗ (S∗2 |𝔄′2 ) ̄ (S∗ |𝔄′ ) = S∗ ⊗ ̄ S∗ = S∗ = (S∗1 |𝔄′1 ) ⊗ 1 2 2 2 and consequently ([L], Lemma 3/10.5) ̄ 𝔄′ = 𝔄′ , 𝔄′1 ⊗ 2

(7)

̄ 𝔄′ ⊂ that is, 𝔄′ is the maximal right Hilbert algebra associated with the right Hilbert algebra 𝔄′1 ⊗ 2 ℋ . Similarly, we get ̄ 𝔄′′ = 𝔄1 ⊗ ̄ 𝔄2 . 𝔄′′1 ⊗ 2

(8)

By ([L], 10.4.(2)) we have 𝔏(𝔄)′ = ℜ(𝔄′ ), so that from (4) we once again obtain the result ([L], Thm. 10.7) ̄ 𝔏(𝔄2 ))′ = 𝔏(𝔄1 )′ ⊗ ̄ 𝔏(𝔄2 )′ . (𝔏(𝔄1 ) ⊗ 8.2 Theorem. Let 𝜑k be a normal semifinite weight on the W ∗ -algebra ℳk , (k = 1, 2). There exists ̄ 𝜑2 on the W ∗ -algebra ℳ = ℳ1 ⊗ ̄ ℳ2 such that a unique normal semifinite weight 𝜑 = 𝜑1 ⊗ ̄ x2 ∈ 𝔐𝜑 and 𝜑(x1 ⊗ ̄ x2 ) = 𝜑1 (x1 )𝜑2 (x2 ) x1 ∈ 𝔐𝜑1 , x2 ∈ 𝔐𝜑2 ⇒ x1 ⊗ ̄ s(𝜑2 ) and s(𝜑) = s(𝜑1 ) ⊗

𝜎t𝜑

=

𝜑 𝜎t 1

̄ ⊗

𝜑 𝜎t 2

(t ∈ ℝ).

(1) (2)

Proof. Assume first that 𝜑1 and 𝜑2 are n.s.f. weights. To prove the existence assertion, we consider the standard representations 𝜋k ∶ ℳk → ℬ(ℋ𝜑k ) associated with 𝜑k and the maximal left Hilbert

Tensor Products

95

̄ 𝔄𝜑 ⊂ ℋ𝜑 ⊗ ̄ ℋ𝜑 = ℋ be the tensor product algebras 𝔄𝜑k ⊂ ℋ𝜑k , (k = 1, 2). Let 𝔄 = 𝔄𝜑1 ⊗ 2 1 2 ̄ 𝜋2 is a *-isomorphism of the W ∗ -algebra ℳ onto the left Hilbert algebra (8.1). Then 𝜋 = 𝜋1 ⊗ von Neumann algebra 𝔏(𝔄). If 𝜑𝔄 denotes the natural weight on 𝔏(𝔄), then 𝜑 = 𝜑𝔄 ⋅ 𝜋 is an n.s.f. weight on ℳ and 𝜋 ∶ ℳ → ℬ(ℋ ) can be identified with the standard representation associated with 𝜑. 1∕2 1∕2 Let 0 ≤ x1 ∈ 𝔐𝜑1 ; 0 ≤ x2 ∈ 𝔐𝜑2 . Then x1 ∈ 𝔄𝜑1 , x2 ∈ 𝔄𝜑2 and ̄ x2 )1∕2 = L 1∕2 𝜋(x1 ⊗ (x ) 1

𝜑1

̄ (x1∕2 )𝜑 . ⊗ 2 2

Hence ̄ x2 ) = 𝜑𝔄 (𝜋(x1 ⊗ ̄ x2 )) = ‖(x )𝜑 ⊗(x )𝜑 ‖2 𝜑(x1 ⊗ 1 2 1 2 1∕2

1∕2

1∕2

1∕2

= ‖(x1 )𝜑1 ‖2𝜑 ‖(x2 )𝜑2 ‖2𝜑 = 𝜑1 (x1 )𝜑2 (x2 ). 1

2

On the other hand, using (8.1.(5)) we obtain ̄ x2 )) = Δit 𝜋(x1 ⊗ ̄ x2 )Δ−it 𝜋(𝜎t𝜑 (x1 ⊗ 𝜑 𝜑 it −it ̄ = (Δit𝜑 𝜋1 (x1 )Δ−it 𝜑 ) ⊗ (Δ𝜑 𝜋2 (x2 )Δ𝜑 ) 1

1

2

2

𝜑 ̄ 𝜋2 (𝜎t𝜑2 (x2 )) = 𝜋((𝜎t𝜑1 ⊗ ̄ 𝜎t𝜑2 )(x1 ⊗ ̄ x2 )). = 𝜋1 (𝜎t 1 (x1 )) ⊗

Thus, 𝜑 satisfies conditions (1) and (2). If 𝜓 is another n.s.f. weight on ℳ satisfying (1) and (2), 𝜓 commutes with 𝜑 and coincides with 𝜑 on the 𝜎 𝜑 -invariant and w-dense *-subalgebra 𝔐𝜑1 ⊗𝔐𝜑2 of ℳ, so that 𝜓 = 𝜑 by the Pedersen–Takesaki theorem (6.2). ̄ 𝜑2 as an n.s.f. weight on the W ∗ If 𝜑1 and 𝜑2 are not necessarily faithful, we define 𝜑 = 𝜑1 ⊗ ̄ algebra eℳe, where e = s(𝜑1 ) ⊗ s(𝜑2 ), and then consider 𝜑 as a weight on ℳ, that is, 𝜑(x) = 𝜑(exe)(x ∈ ℳ + ). If 𝜑1 and 𝜑2 are normal positive forms on the W ∗ -algebras ℳ1 and ℳ2 , respectively, the normal ̄ 𝜑2 on ℳ1 ⊗ ̄ ℳ2 is determined uniquely by condition 8.2.(1) alone (see 3.9). positive form 𝜑1 ⊗ Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and let tr be the canonical trace on a factor ℱn of ̄ ℱn can be identified with Matn (ℳ) such that (see 5.2) type In . Then ℳ ⊗ ̄ tr)(x) = (𝜑 ⊗

∑

𝜑(xkk )

(x = [xij ] ∈ Matn (ℳ)+ ).

k

8.3. Recall (2.6) that for any normal semifinite weight 𝜑 on the W ∗ -algebra ℳ there exists an increasing net {𝜑i }i∈I of normal positive forms on ℳ such that 𝜑i ↑ 𝜑, that is, 𝜑i (x) ↑ 𝜑(x) for all x ∈ ℳ+. Proposition. Let 𝜑, 𝜓 be normal semifinite weights and {𝜑i }i∈I , {𝜓j }j∈J increasing nets of normal ̄ 𝜓j ↑ 𝜑 ⊗ ̄ 𝜓. positive forms on the W ∗ -algebra ℳ, 𝒩 , respectively. If 𝜑i ↑ 𝜑 and 𝜓j ↑ 𝜓, then 𝜑i ⊗ ̄ 𝜓i }i∈I,j∈J is an increasing net of normal positive forms -on ℳ ⊗ ̄ 𝒩 and Proof. Indeed, {𝜑i ⊗ ̄ 𝜓j ≤ 𝜑 ⊗ ̄ 𝜓 for all i ∈ I, j ∈ J. Consequently, we define a normal weight 𝜔 on ℳ ⊗ ̄ 𝒩 𝜑i ⊗ by ̄ 𝜓j )(z) = lim(𝜑i ⊗ ̄ 𝜓j )(z) 𝜔(z) = sup(𝜑i ⊗ ij

ij

̄ 𝒩 )+ ). (z ∈ (ℳ ⊗

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Normal Weights

For a ∈ 𝔐𝜑 ∩ ℳ + , b ∈ 𝔐𝜓 ∩ 𝒩 + we have ̄ b) = sup 𝜑i (a)𝜓j (b) = sup 𝜑i (a) sup 𝜓j (b) = 𝜑(a)𝜓(b). 𝜔(a ⊗ ij

i

j

̄ 𝜓j ) ↑ s(𝜔), whence ⊗ On the other hand, it is easy to see that s(𝜑i ) ↑ s(𝜑), s(𝜓j ) ↑ s(𝜓), and s(𝜑i⋃ ̄ s(𝜔) = s(𝜑) ⊗ s(𝜓). Moreover, by Proposition 7.17, for t ∈ ℝ, x ∈ i∈I s(𝜑i )ℳs(𝜑i ), y ∈ ⋃ j∈J s(𝜓j )𝒩 s(𝜓j ) we have 𝜑

s

𝜓

s

̄ 𝜓j 𝜑i ⊗

𝜎t i (x) → 𝜎t𝜑 (x), 𝜎t j ( y) → 𝜎t𝜓 ( y), 𝜎t

s

̄ y) → 𝜎 𝜔 (x ⊗ ̄ y), (x ⊗ t

̄ y) = 𝜎t𝜑 (x) ⊗ ̄ 𝜎t𝜓 ( y). Since s(𝜑i ) ↑ s(𝜑), s(𝜓j ) ↑ s(𝜓), this equality holds for every hence 𝜎t𝜔 (x ⊗ x ∈ s(𝜑)𝒩 s(𝜑), y ∈ s(𝜓)ℳs(𝜓). Thus, the weight 𝜔 satisfies 8.2.(1) and 8.2.(2), which determine ̄ 𝜓; hence 𝜔 = 𝜑 ⊗ ̄ 𝜓, that is, 𝜑i ⊗ ̄ 𝜓j ↑ 𝜑 ⊗ ̄ 𝜓. 𝜑⊗ 8.4. As a first application we obtain the distributive law for the tensor product with respect to addition: Corollary. Let 𝜑1 , 𝜑2 be normal semifinite weights on the W ∗ -algebra ℳ and 𝜓 a normal semifinite ̄ 𝜓 = 𝜑1 ⊗ ̄ 𝜓 + 𝜑2 ⊗ ̄ 𝜓. weight on the W ∗ -algebra 𝒩 . If 𝜑1 + 𝜑2 is semifinite, then (𝜑1 + 𝜑2 ) ⊗ Proof. Let {𝜓k } be an increasing net of normal positive forms on 𝒩 such that 𝜓k ↑ 𝜓. Assume that 𝜑1 , 𝜑2 are normal positive forms. Since the distributive law is obvious for normal ̄ 𝜓 = supk (𝜑1 ⊗ ̄ 𝜑2 ) ⊗ ̄ 𝜓k = supk 𝜑1 ⊗ ̄ 𝜓k + positive forms, we have by Proposition 8.3, (𝜑1 +𝜑2 ) ⊗ ̄ 𝜓k = 𝜑1 ⊗ ̄ 𝜓 + 𝜑2 ⊗ ̄ 𝜓. supk 𝜑2 ⊗ In the general case, let {𝜑1i }, {𝜓2j } be increasing nets of normal positive forms on ℳ such that 𝜑1i ↑ 𝜑1 , 𝜓2j ↑ 𝜓2 . It is then obvious that 𝜑1i + 𝜑2j ↑ 𝜑1 + 𝜑2 . We have, again by Proposition 8.3 ̄ 𝜓 = supij (𝜑1i + 𝜑2j ) ⊗ ̄ 𝜓 = supij (𝜑1i ⊗ ̄ 𝜓 + 𝜑2j ⊗ ̄ 𝜓) = and the first part of the proof, (𝜑1 + 𝜑2 ) ⊗ ̄ 𝜓 + supj 𝜑2j ⊗ ̄ 𝜓 = 𝜑1 ⊗ ̄ 𝜓 + 𝜑2 ⊗ ̄ 𝜓. supi 𝜑1i ⊗ 8.5. Another application concerns the relation between the tensor product and the balanced weight: Corollary. Let 𝜑1 , 𝜑2 be normal semifinite weights on the W ∗ -algebra ℳ and 𝜓 a normal semifinite ̄ 𝜓, 𝜑2 ⊗ ̄ 𝜓) = 𝜃(𝜑1 , 𝜑2 ) ⊗ ̄ 𝜓 as weights on the W ∗ weight on the W ∗ -algebra 𝒩 . Then 𝜃(𝜑1 ⊗ ̄ 𝒩 ) ≈ (ℳ ⊗ ̄ 𝒩 ) ⊗ Mat2 (ℂ) ≈ Mat2 (ℳ) ⊗ ̄ 𝒩. algebra Mat2 (ℳ ⊗ Proof. Let {𝜑1i }, {𝜑2j }, {𝜓k } be increasing nets of normal positive forms such that 𝜑1i ↑ 𝜑1 , 𝜑2j ↑ 𝜑2 , 𝜓k ↑ 𝜓. For the balanced weights is obvious that 𝜃(𝜑1i , 𝜑2j ) ↑ 𝜃(𝜑1 , 𝜑2 ); using Proposition 8.3, ̄ 𝜓k ↑ 𝜃(𝜑1 , 𝜑2 ) ⊗ ̄ 𝜓. we get 𝜃(𝜑1i , 𝜑2j ) ⊗ ̄ 𝜓 k ↑ 𝜑1 ⊗ ̄ 𝜓, 𝜑2j ⊗ ̄ 𝜓k ↑ 𝜑2 ⊗ ̄ 𝜓, hence 𝜃(𝜑1i ⊗ ̄ 𝜓k , 𝜑2j ⊗ ̄ 𝜓k ) ↑ Also, we have 𝜑1i ⊗ ̄ 𝜓, 𝜑2 ⊗ ̄ 𝜓). Since 𝜃(𝜑1i ⊗ ̄ 𝜓k , 𝜑2j ⊗ ̄ 𝜓k ) = 𝜃(𝜑1i , 𝜑2j ) ⊗ ̄ 𝜓k for all i, j, k (see 3.9), it 𝜃(𝜑1 ⊗ ̄ 𝜓, 𝜑2 ⊗ ̄ 𝜓) = 𝜃(𝜑1 , 𝜑2 ) ⊗ ̄ 𝜓. follows that 𝜃(𝜑1 ⊗ 8.6. If s(𝜑2 ) ≤ s(𝜑1 ), then, arguing as in Section 3.9, we infer from Corollary 8.5 that ̄ 𝜓) ∶ D(𝜑1 ⊗ ̄ 𝜓)]t = [D𝜑2 ∶ D𝜑1 ]t ⊗ ̄ s(𝜓) (t ∈ ℝ). [D(𝜑2 ⊗ Using the chain rule (3.5), we obtain the following general result:

Tensor Products

97

Corollary. Let 𝜑1 , 𝜑2 be normal semifinite weights on ℳ with s(𝜑2 ) ≤ s(𝜑1 ) and 𝜓1 , 𝜓2 normal semifinite weights on 𝒩 with s(𝜓2 ) ≤ s(𝜓1 ). Then ̄ 𝜓2 ) ∶ D(𝜑1 ⊗ ̄ 𝜓1 )]t = [D𝜑2 ∶ D𝜑1 ] ⊗ ̄ [D𝜓2 ∶ D𝜓1 ]t [D(𝜑2 ⊗

(t ∈ ℝ).

8.7. In particular, using Corollary 4.8, we obtain: Corollary. Let 𝜑, 𝜓 be normal semifinite weights on the W ∗ -algebras ℳ, 𝒩 , respectively, and ̄ B is a positive A, B positive self-adjoint operators affiliated to ℳ 𝜑 , 𝒩 𝜓 , respectively. Then A ⊗ ̄ 𝒩 )𝜑 ⊗̄ 𝜓 and self-adjoint operator affiliated to (ℳ ⊗ ̄ 𝜓)A ⊗̄ B = 𝜑A ⊗ ̄ 𝜓B . (𝜑 ⊗ 8.8. Let 𝜑, 𝜑1 , 𝜑2 , {𝜑i } be normal semifinite weights on the W ∗ -algebra ℳ and 𝜓, 𝜓1 , 𝜓2 , {𝜓j } normal semifinite weights on the W ∗ -algebra 𝒩 . By Corollary 8.6 and Corollary 3.13, ̄ 𝜓1 ≤ 𝜑2 ⊗ ̄ 𝜓2 . 𝜑1 ≤ 𝜑2 , 𝜓1 ≤ 𝜓2 ⇒ 𝜑1 ⊗

(1)

Also, arguing as in the proof of Proposition 8.3, ̄ 𝜓j ↑ 𝜑 ⊗ ̄ 𝜓, 𝜑i ↑ 𝜑, 𝜓j ↑ 𝜓 ⇒ 𝜑i ⊗

(2)

and, by Corollary 8.4, )

(

∑ i

𝜑i

̄ ⊗

( ∑ j

) 𝜓j

=

∑

̄ j. 𝜑i ⊗𝜓

(3)

ij

8.9. If 𝜑, 𝜓 are n.s.f. weights on the W ∗ -algebras ℳ, 𝒩 , respectively, and 𝜎 ∈ Aut(ℳ), 𝜏 ∈ Aut(𝒩 ), by the definition (8.2) of the tensor product weight and 2.22.(5), we get ̄ 𝜓) ◦ (𝜎 ⊗ ̄ 𝜏) = (𝜑 ◦ 𝜎) ⊗ ̄ (𝜓 ◦ 𝜏). (𝜑 ⊗ More generally, arguing as in the proofs of Corollaries 8.4 and 8.5, we obtain the following Corollary. Let Φ ∶ ℳ1 → ℳ, Ψ ∶ 𝒩1 → 𝒩 be normal completely positive linear mappings between W ∗ -algebras and 𝜑, 𝜓 normal semifinite weights on ℳ, 𝒩 respectively. If the weights 𝜑 ◦ Φ, 𝜓 ◦ Ψ are semifinite, then ̄ 𝜓) ◦ (Φ ⊗ ̄ Ψ) = (𝜑 ◦ Φ) ⊗ ̄ (𝜓 ◦ Ψ). (𝜑 ⊗ Recall (Sakai, 1971; Strătilă & Zsidó, 1977–2005, p. 8.8) that the algebraic tensor product Φ⊗Ψ of two normal completely positive linear mappings extends to a normal completely positive linear ̄ Ψ ∶ ℳ1 ⊗ ̄ 𝒩1 → ℳ ⊗ ̄ 𝒩. mapping Φ ⊗

98

Normal Weights

8.10. While the equality of normal forms is equivalent to their equality on w-dense subsets, the equality of normal weights is, as we have seen (6.2, 6.6), a more delicate problem. The next result concerns the equality of certain tensor product weights. Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and f a normal positive form on the ̄ ℱ such that 𝜓 ◦ (𝜎t𝜑 ⊗ ̄ 1ℱ ) = 𝜓, factor ℱ of type I. If 𝜓 is a normal semifinite weight on ℳ ⊗ 𝜑 (t ∈ ℝ), and there exists 𝜎 -invariant w-dense *-subalgebra ℬ of 𝔐𝜑 such that ̄ y∗ y) = 𝜑(x∗ x)f ( y∗ y) (x ∈ ℬ, y ∈ ℱ ), 𝜓(x∗ x ⊗ ̄ f. then 𝜓 = 𝜑 ⊗ Proof. Let tr be the canonical trace on ℱ . By ([L], E.7.8) there exists a ∈ ℱ , with a ≥ 0, such that ̄ tr is an n.s.f. weight on ℳ ⊗ ̄ ℱ , 𝜎t𝜑 ⊗̄ tr = 𝜎t𝜑 ⊗ ̄ idℱ , (t ∈ ℝ), 𝜑 ⊗ ̄ f = (𝜑 ⊗ ̄ tr)1 ⊗̄ a f = tra . Then 𝜑 ⊗ ̄ tr 𝜑 ⊗ ̄ and ℬ ⊗ 𝔐tr is a 𝜎 -invariant w-dense *-subalgebra of ℳ𝜑 ⊗̄ tr such that ̄ tr)1 ⊗̄ a (z∗ z) (z ∈ ℬ ⊗ ̄ 𝔐tr ). 𝜓(z∗ z) = (𝜑 ⊗ ̄ tr)1 ⊗̄ a = 𝜑 ⊗ ̄ f. Since 𝜓 is 𝜎 𝜑 ⊗̄ tr -invariant, we conclude by Theorem 6.2 that 𝜓 = (𝜑 ⊗ Notes. Proposition 8.1 is due to Tomita and Takesaki (1970). The definition of the tensor product weight appears in Takesaki (1970) and Connes (1973a). The other results in this section are from Digernes (1975); Katayama (1974); Strătilă (1977); Zsidó (1978). For our exposition we have used Connes (1973a); Strătilă (1979); Takesaki (1970) and Zsidó (1978).

CHAPTER II

Conditional Expectations and Operator-Valued Weights

9 Conditional Expectations In this section, we introduce a special kind of positive linear mapping, called a conditional expectation, and give some applications to tensor products. 9.1. Let 𝒜 be a C ∗ -algebra and ℬ ⊂ 𝒜 a C ∗ -subalgebra. A linear mapping Φ ∶ 𝒜 → ℬ is called a projection if Φ(b) = b for every b ∈ ℬ. In this case Φ ◦ Φ = Φ and ‖Φ‖ ≥ 1. A linear mapping Φ ∶ 𝒜 → ℬ is called ℬ-linear if Φ(ab) = Φ(a)b and Φ(ba) = bΦ(a) for every a ∈ 𝒜 , b ∈ ℬ. A ℬ-linear projection Φ ∶ 𝒜 → ℬ which is also a positive mapping, that is, Φ(𝒜 + ) ⊂ ℬ + , is called a conditional expectation. Theorem (J. Tomiyama). Every projection of norm 1 of the C ∗ -algebra 𝒜 onto the C ∗ -subalgebra ℬ ⊂ 𝒜 is a conditional expectation. Proof. Consider first a norm 1 projection Φ ∶ ℳ → 𝒩 of a W ∗ -algebra ℳ onto a W ∗ -subalgebra 𝒩 ⊂ ℳ. Then 𝒩 is a W ∗ -algebra and its unit element is a projection e𝒩 ∈ ℳ. Let e be a projection in ℳ, put f = 1 − e, and assume that either e or f is in 𝒩 . For any x ∈ 𝒩 , both ex and f x are then in 𝒩 and for x, y ∈ ℳ we have ‖ex + f y‖2 = ‖(ex + f y)∗ (ex + f y)‖ = ‖x∗ ex + y∗ f y‖ ≤ ‖ex‖2 + ‖ f y‖2 . So for 𝜆 ∈ ℝ, (𝜆 + 1)2 ‖ f Φ(ex)‖2 = ‖ f Φ(ex + 𝜆f Φ(ex))‖2 ≤ ‖ex + 𝜆f Φ(ex)‖2 ≤ ‖ex‖2 + ‖𝜆f Φ(ex)‖2 = ‖ex‖2 + 𝜆2 ‖ f Φ(ex)‖2 . As 𝜆 is arbitrary, we have (1 − e)Φ(ex) = f Φ(ex) = 0, that is, Φ(ex) = e Φ(ex). Interchanging e and f, we have e Φ(x − ex) = e Φ( f x) = 0, or e Φ(x) = e Φ(ex). Thus e Φ(x) = Φ(ex)

99

(x ∈ ℳ).

(1)

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Conditional Expectations and Operator-Valued Weights

Putting x = 1, e𝒩 = Φ(e𝒩 ) = e𝒩 Φ(1) = Φ(1). Let 𝜓 be any positive form on 𝒩 and 𝜑 = 𝜓 ◦ Φ. Since ‖𝜑‖ ≤ ‖𝜓‖ = 𝜓(e𝒩 ) = 𝜑(1) ≤ ‖𝜑‖, by ([L], 5.4.) it follows that 𝜑 is positive. Hence Φ is positive and so self-adjoint. Taking adjoints in (1) we have Φ(x)e = Φ(xe)

(x ∈ ℳ).

(2)

Since the W ∗ -algebra 𝒩 is the closed linear span of its projections ([L], 2.23), by (1) and (2) it follows that Φ(yx) = yΦ(x) and Φ(xy) = Φ(x)y for x ∈ ℳ, y ∈ 𝒩 . In the general case, when Φ ∶ 𝒜 → ℬ is a projection of norm 1 of the C ∗ -algebra 𝒜 onto its C ∗ -subalgebra ℬ, we consider the second transpose Ψ = Φtt , mapping of the second dual W ∗ -algebra ℳ = 𝒜 ∗∗ onto the second dual W ∗ -algebra 𝒩 = ℬ ∗∗ (A.15, A.16). Since 𝒜 is w-densely imbedded in 𝒜 ∗∗ and ℬ is w-densely imbedded in ℬ ∗∗ , we may identify 𝒩 with a W ∗ -subalgebra of ℳ and then Ψ ∶ ℳ → 𝒩 is obviously a projection of norm 1. By the first part of the proof, Ψ is a conditional expectation, which implies that its restriction Φ = Ψ|𝒜 is also a conditional expectation. 9.2. Let 𝒜 , ℬ be C ∗ -algebras. A linear mapping Φ ∶ 𝒜 → ℬ is called a Schwarz mapping if Φ(a)∗ Φ(a) ≤ Φ(a∗ a) for all a ∈ 𝒜 . Note that if Φ ∶ 𝒜 → ℬ is a Schwarz mapping and if a ∈ 𝒜 satisfies Φ(a)∗ Φ(a) = Φ(a∗ a), then for x ∈ 𝒜 we have Φ(x∗ a) = Φ(x)∗ Φ(a),

Φ(a∗ x) = Φ(a)∗ Φ(x).

Indeed, for x ∈ 𝒜 , t ∈ ℝ we have t(Φ(a)∗ Φ(x) + Φ(x)∗ Φ(a)) = Φ(ta + x)∗ Φ(ta + x) − t2 Φ(a)∗ Φ(a) − Φ(x)∗ Φ(x) ≤ Φ((ta + x)∗ (ta + x)) − t2 Φ(a∗ a) − Φ(x)∗ Φ(x) = tΦ(a∗ x + x∗ a) + (Φ(x∗ x) − Φ(x)∗ Φ(x)). Dividing this inequality by t ≷ 0 and letting |t| → +∞, we get Φ(a)∗ Φ(x) + Φ(x)∗ Φ(a) = Φ(a∗ x) + Φ(x∗ a). Replacing a by −ia here and then multiplying by i, we obtain Φ(a)∗ Φ(x) − Φ(x)∗ Φ(a) = Φ(a∗ x) − Φ(x∗ a); our assertion follows from the last two equations. In particular, it follows that a projection Φ ∶ 𝒜 → ℬ ⊂ 𝒜 which is also a Schwarz mapping, is a conditional expectation. Proposition. Every conditional expectation Φ ∶ 𝒜 → ℬ ⊂ 𝒜 is a Schwarz mapping and a projection of norm 1. If 𝒜 is unital, then ℬ is also unital and Φ(1𝒜 ) = 1ℬ . Proof. Indeed, for any a ∈ ℳ we have 0 ≤ Φ((Φ(a) − a)∗ (Φ(a) − a)) = −Φ(a)∗ Φ(a) + Φ(a∗ a) and, since a∗ a ≤ ‖a‖2 ⋅ 1𝒜 ∗∗ , we obtain ‖Φ(a)‖2 = ‖Φ(a)∗ Φ(a)‖ ≤ ‖Φ(a∗ a)‖ ≤ ‖a‖2 , hence ‖Φ‖ = 1.

Conditional Expectations

101

Also, by the preceding remark, Φ(1𝒜 )Φ(a) = Φ(a) = Φ(a)Φ(1𝒜 ), so that Φ(1𝒜 ) is the unit element of ℬ. A positive linear mapping Φ ∶ 𝒜 → ℬ between C ∗ -algebras is called faithful if for a ∈ 𝒜 , Φ(a∗ a) = 0 ⇒ a = 0. If Φ ∶ ℳ → 𝒩 is a Schwarz mapping between W ∗ -algebras, it is easy to check that Φ is w-continuous if and only if Φ is s∗ -continuous; recall that in this case Φ is called normal. 9.3. Recall ([L], C.5.2; Araki & Masuda, 1982; Str̆atil̆a & Zsidó, 1977, 2005) that a linear mapping Φ ∶ 𝒜 → ℬ between the C ∗ -algebras 𝒜 and ℬ is said to be completely positive if, for each n ∈ ℕ, the natural extension Φn ∶ Matn (𝒜 ) → Matn (ℬ) is a positive mapping. If ℬ = ℬ(ℋ ), then Matn (ℬ) = ℬ(ℋ̃ n ) where ℋ̃ n is the Hilbert space direct sum of n copies of ℋ . An element X ∈ ℬ(ℋ̃ n ) is positive if and only if (X𝜉|𝜉) > 0 for any 𝜉 = [𝜉1 , … , 𝜉n ] ∈ ℋ̃ n . On the other hand, it is easy to check that every positive element of the C ∗ -algebra Matn (𝒜 ) is a finite sum of matrices [aij ] ∈ Matn (𝒜 ) with aij = a∗i aj (1 ≤ i, j ≤ n), where a1 , … , an ∈ 𝒜 . Consequently, ∑ a linear mapping Φ ∶ 𝒜 → ℬ(ℋ ) is completely positive if and only if ij (Φ(a∗i aj )𝜉j |𝜉i ) ≥ 0 for n-tuples a1 , … an ∈ 𝒜 , and 𝜉1 , … , 𝜉n ∈ ℋ (n = 1, 2, …). It is easy to see that if ℋ = ⊕i∈I ℋi and Φ(𝒜 )ℋi ⊂ ℋi (i ∈ I), then Φ is completely positive if and only if each of the mappings Φi ∶ 𝒜 ∋ a ↦ Φ(a)|ℋi ∈ ℬ(ℋi ) is completely positive. Every C ∗ -algebra ℬ may be regarded as a concrete C ∗ -algebra ℬ ⊂ ℬ(ℋ ) such that ℬℋ = ℋ ; in this case, there exist a direct sum decomposition ℋ = ⊕i∈I ℋi and vectors 𝜉i ∈ ℋi such that ℬ𝜉i = ℋi (i ∈ I). Thus, in proving that a linear mapping Φ ∶ 𝒜 → ℬ is completely positive we may assume, without loss of generality, that ℬ ⊂ ℬ(ℋ ) and ℬ𝜉 = ℋ for some 𝜉 ∈ ℋ . Proposition. Every conditional expectation is completely positive. Proof. Let Φ ∶ 𝒜 → ℬ ⊂ 𝒜 be a conditional expectation and assume that ℬ ⊂ ℬ(ℋ ) with ℬ𝜉 = ℋ for some 𝜉 ∈ ℋ . Let ak ∈ 𝒜 , xk ∈ 𝒜 and 𝜉k = Φ(xk )𝜉(1 ≤ k ≤ n). We have ∑

(Φ(a∗i aj )𝜉j |𝜉i ) =

ij

∑

(Φ(xi )∗ Φ(a∗i aj )Φ(xj )𝜉|𝜉)

ij

=

∑

( ( (Φ(Φ(xi )∗ a∗i aj Φ(xj ))𝜉|𝜉) =

ij

Φ

∑

) Φ(xi )∗ a∗i aj Φ(xj ) 𝜉|𝜉

) ≥ 0,

ij

∑ as ij Φ(xi )∗ a∗i aj Φ(xj ) ≥ 0 and Φ is positive. Since ℬ𝜉 = ℋ , the above inequality holds for arbitrary vectors 𝜉1 , … , 𝜉n ∈ ℋ , hence Φ is completely positive. 9.4. If Φ ∶ 𝒜 → ℬ(ℋ ) is a completely positive linear mapping, the Stinespring theorem ([L], C.5.2; Arveson (1969); Str̆atil̆a & Zsidó, 1977, 2005) shows that there exist a Hilbert space 𝒦 , a *-representation 𝜋 ∶ 𝒜 → ℬ(𝒦 ) and a bounded linear mapping V ∶ ℋ → 𝒦 such that ‖V‖ = ‖Φ‖1∕2 , 𝒦 is the closed linear space generated by 𝜋(𝒜 )Vℋ , and Φ(a) = V∗ 𝜋(a)V

(a ∈ 𝒜 ).

The triple {𝜋, V, 𝒦 } is uniquely determined by these conditions and is called the Stinespring dilation of Φ.

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Conditional Expectations and Operator-Valued Weights

By means of the Stinespring dilation, we obtain Φ(a)∗ Φ(a) ≤ ‖Φ‖Φ(a∗ a)

(a ∈ 𝒜 );

(1)

so if ‖Φ‖ = 1, Φ is a Schwarz mapping. If 𝒜 is unital, then Φ(1) = V∗ V and ‖Φ‖ = ‖Φ(1)‖. Again, using the Stinespring dilation, we see that if Φ1 ∶ 𝒜1 → ℬ1 and Φ2 ∶ 𝒜2 → ℬ2 are (faithful) completely positive linear mappings, there exists a unique (faithful) completely positive linear mapping Φ1 ⊗C∗ Φ2 ∶ 𝒜1 ⊗C∗ 𝒜1 → ℬ1 ⊗C∗ ℬ2 such that (Φ1 ⊗C∗ Φ2 )(a1 ⊗ a2 ) = Φ1 (a1 ) ⊗ Φ2 (a2 ) for a1 ∈ 𝒜1 , a2 ∈ 𝒜2 , and moreover ‖Φ1 ⊗C∗ Φ2 ‖ = ‖Φ1 ‖‖Φ2 ‖.

(2)

It is easy to check that if Φ1 and Φ2 are conditional expectations, then Φ1 ⊗C∗ Φ2 is also a conditional expectation. If Φ ∶ ℳ → ℬ(ℋ ) is a normal completely positive linear mapping of the W ∗ -algebra ℳ, then the Stinespring dilation 𝜋 is also a normal *-representation. It follows that if Φ1 ∶ ℳ1 → 𝒩1 and Φ2 ∶ ℳ2 → 𝒩2 are (faithful) normal completely positive linear mappings between W ∗ -algebras, then there exists a unique (faithful) normal completely ̄ Φ2 ∶ ℳ1 ⊗ ̄ ℳ2 → 𝒩1 ⊗ ̄ 𝒩2 which extends the algebraic tensor positive linear mapping Φ1 ⊗ product mapping Φ1 ⊗ Φ2 ∶ ℳ1 ⊗ ℳ2 → 𝒩1 ⊗ 𝒩2 . Note that ℳ1 ⊗C∗ ℳ2 is a w-dense ̄ ℳ2 and the restriction of Φ1 ⊗ ̄ Φ2 to ℳ1 ⊗C∗ ℳ2 is just Φ1 ⊗C∗ Φ2 , C ∗ -subalgebra of ℳ1 ⊗ so that ̄ Φ2 ‖ = ‖Φ1 ‖‖Φ2 ‖. ‖Φ1 ⊗

(3)

̄ Φ2 is also a normal conditional If Φ1 and Φ2 are normal conditional expectations, Φ1 ⊗ expectation. ̄ ℳ2 is The extension of the algebraic tensor product mapping Φ1 ⊗ Φ2 to the W ∗ -algebra ℳ1 ⊗ possible even if Φ1 , Φ2 are not normal: Proposition (J. Tomiyama). Let Φ1 ∶ ℳ1 → 𝒩1 , Φ2 ∶ ℳ2 → 𝒩2 be completely positive linear mappings between W ∗ -algebras. There exists a completely positive linear mapping Φ ∶ ̄ ℳ2 → 𝒩1 ⊗ ̄ 𝒩2 , ‖Φ‖ = ‖Φ1 ‖‖Φ2 ‖, such that ℳ1 ⊗ ̄ x2 ) = Φ1 (x1 ) ⊗ ̄ Φ2 (x2 ) Φ(x1 ⊗

(x1 ∈ ℳ1 , x2 ∈ ℳ2 ).

If Φ1 and Φ2 are conditional expectations, we can choose Φ to be a conditional expectation. Before we give the proof (in Section 9.7) some preparation is needed. 9.5 Lemma. Let Φ ∶ ℳ → 𝒩 be a completely positive linear mapping between the W ∗ -algebras ℳ, 𝒩 . There exists a completely positive linear mapping Ψ ∶ ℳ → 𝒩 with Ψ(1) = 1 such that Φ(x) = Φ(1)1∕2 Ψ(x)Φ(1)1∕2

(x ∈ ℳ).

Conditional Expectations

103

Proof. Let b = Φ(1) ∈ 𝒩 + , f = s(b) and let 𝜑 be any state of ℳ. The equation Ψn (x) = (b + n−1 )−1∕2 Φ(x)(b + n−1 )−1∕2 + 𝜑(x)(1 − f ) (x ∈ ℳ) defines completely positive linear mappings Ψn ∶ ℳ → 𝒩 (n ∈ ℕ). Let a ∈ ℳ, 0 ≤ a ≤ 1. Since 0 ≤ Φ(a) ≤ b, there exists ([L], E.2.6) y ∈ 𝒩 , ‖y‖ ≤ 1, such that s∗

s∗

Φ(a)1∕2 = yb1∕2 . As b(b + n−1 )−1 → f, it follows that Φ(a)1∕2 (b + n−1 )−1∕2 → yf, and the sequence {Ψn (a)} is s∗ -convergent to the element Ψ(a) = f y∗ yf + 𝜑(a)(1 − f ). We have b1∕2 Ψ(a)b1∕2 = b1∕2 y∗ yb1∕2 = Φ(a). If a = 1, then y = 1 and 𝜑(a) = 1, hence Ψ(1) = 1. Since every element x ∈ ℳ is a linear combination of elements 0 ≤ a ≤ 1, it follows that the mappings Ψn are pointwise convergent, with respect to the s∗ -topology on 𝒩 , to a completely positive linear mapping Ψ ∶ ℳ → 𝒩 with the required properties. 9.6. Let I be a directed set. The space of all bounded nets {f (i)}i∈I of complex numbers is just the C ∗ -algebra ℬ(I) of bounded functions on I. For each i ∈ I, the equation Λi ( f ) = f (i)

( f ∈ ℬ(I))

defines a positive form Λi ∈ ℬ(I)∗ such that ‖Λi ‖ = 1. Since the closed unit ball of ℬ(I)∗ is 𝜎(ℬ(I)∗ , ℬ(I))-compact, it follows that the intersection ⋂

{Λj ∶ j ≥ i} ⊂ ℬ(I)∗

i∈I

is not empty. An arbitrary element Λ in this intersection will be called a Banach limit with respect to I. For f ∈ ℬ(I), we shall write Λ( f ) = LIMi f (i). The properties of the Banach limits follow immediately from the fact that Λ is a positive form of norm 1 on ℬ(I). Moreover, we have LIMi f (i) = limi f (i) whenever limi f (i) exists. Consider now a norm-bounded net {xi }i∈I ⊂ ℬ(ℋ ). The mapping ℋ ×ℋ ∋ (𝜉, 𝜂) ↦ LIMi (xi 𝜉|𝜂) is a bounded sesquilinear form on ℋ , so there exists a unique operator x = LIMi xi ∈ ℬ(ℋ ), such that (x𝜉|𝜂) = LIMi (xi 𝜉|𝜂) for 𝜉, 𝜂 ∈ ℋ . It is easy to check the following properties: LIMi (xi + yi ) = LIMi xi + LIMi yi ,

(1)

= (LIMi xi ) ,

(2)

LIMi (x∗i )

∗

LIMi (axi b) = a(LIMi xi )b, wo

xi → x ⇒ LIMi xi = x.

(3) (4)

104

Conditional Expectations and Operator-Valued Weights

Also, using the Hahn–Banach theorem we see that if 𝒳 ∶ ℬ(ℋ ) is a wo-closed convex set and xi ∈ 𝒳 whenever i ≥ i0 , then LIMi xi ∈ 𝒳 .

(5)

9.7. Proof of Proposition 9.4. We divide the proof into three steps. (I) Let Φ ∶ ℳ → 𝒩 be a completely positive linear mapping between the W ∗ -algebras ℳ, 𝒩 and 𝜄 ∶ ℱ → ℱ the identity mapping on a factor ℱ of type I. Let {ek }k∈K be a family of minimal ∑ ∑ projections in ℱ with k ek = 1, put I = {J ⊂ K; J finite}, eJ = k∈J ek for J ∈ I, and let LIM be any Banach limit with respect to I. Since eJ ℱ eJ is finite dimensional, we have ([L], 3.17) ̄ eJ )(ℳ ⊗ ̄ ℱ )(1 ⊗ ̄ eJ ) = ℳ ⊗ ̄ eJ ℱ eJ = ℳ ⊗ eJ ℱ eJ (1 ⊗ and so we can define a completely positive linear mapping ̃J ∶ ℳ ⊗ ̃ J ‖ ≤ ‖Φ‖, by ̄ ℱ →𝒩 ⊗ ̄ ℱ , ‖Φ Φ ̃ J (x) = (Φ ⊗ 𝜄J )((1 ⊗ ̄ eJ )x(1 ⊗ ̄ eJ )) Φ

̄ ℱ ). (x ∈ ℳ ⊗

̃ ∶ Now, using the results of Section 9.6, we obtain a completely positive linear mapping Φ ̃ ≤ ‖Φ‖, by putting ̄ ℱ →𝒩 ⊗ ̄ ℱ , ‖Φ‖ ℳ⊗ ̃ = LIMJ Φ ̃ J (x) Φ(x)

̄ ℱ ). (x ∈ ℳ ⊗

Since eJ ↑ 1 in ℱ , for a ∈ ℳ, b ∈ ℱ we have ̃ ⊗ ̄ b) = LIMJ (Φ(a) ⊗ ̄ eJ beJ ) = w- lim(Φ(a) ⊗ ̄ eJ beJ ) = Φ(a) ⊗ ̄ b; Φ(a J

̃ extends the algebraic tensor product mapping Φ ⊗ 𝜄. thus Φ ̃ = ‖Φ‖ = 1 and, as 1 ⊗ ̄ eJ ↑ 1 ⊗ ̄ 1, for y ∈ 𝒩 ⊗ ̄ ℱ ⊂ If Φ is a conditional expectation, then ‖Φ‖ ̄ ℳ ⊗ ℱ we obtain ̃ = w- lim(Φ ⊗ 𝜄J )((1 ⊗ ̄ eJ )y(1 ⊗ ̄ eJ ) = w- lim(1 ⊗ ̄ eJ )y(1 ⊗ ̄ eJ ) = y; Φ(y) J

J

̃ is a conditional expectation. it follows by Theorem 9.1, that Φ (II) We assume that Φ1 (1) = 1, Φ2 (1) = 1 and assume ℳ1 ⊂ ℬ(ℋ1 ), ℳ2 ⊂ ℬ(ℋ2 ), 𝒩1 ⊂ ℬ(𝒦1 ), 𝒩2 ⊂ ℬ(𝒦2 ) realized as von Neumann algebras. By (I) there exist completely positive linear mappings ̃ 1 ∶ ℳ1 ⊗ ̃ 2 ∶ ℬ(𝒦1 ) ⊗ ̄ ℬ(ℋ2 ) → 𝒩1 ⊗ ̄ ℬ(ℋ2 ) and Φ ̄ ℳ2 → ℬ(𝒦1 ) ⊗ ̄ 𝒩2 Φ which extend the algebraic tensor product mappings Φ1 ⊗ 𝜄2 and 𝜄1 ⊗ Φ2 , respectively. Let us show that ̃ 1 (x) ∈ 𝒩1 ⊗ ̄ ℳ2 ⇒ Φ ̄ ℳ2 . x ∈ ℳ1 ⊗

Conditional Expectations

105

Indeed, let x′1 ∈ ℳ2′ . Since Φ1 (1) = 1, we have ̃ 1 (1 ⊗ ̃ 1 ((1 ⊗ ̃ 1 (1 ⊗ ̄ x′ )∗ Φ ̄ x′ ) = Φ ̄ x′ )∗ (1 ⊗ ̄ x′ )). Φ 2 2 2 2 Using 9.4. (1) and the first remark in Section 9.2, we obtain ̃ 1 (x)Φ ̃ 1 (1 ⊗ ̃ 1 (x(1 ⊗ ̃ 1 (x)(1 ⊗ ̄ x′ ) = Φ ̄ x′ ) = Φ ̄ x′ )) Φ 2 2 2 ′ ′ ̃ ̃ ̃ 1 (x). ̃ ̄ ̄ ̄ x′ )Φ = Φ1 ((1 ⊗ x )x) = Φ1 (1 ⊗ x )Φ1 (x) = (1 ⊗ 2

2

2

̃ 1 (x) ∈ 𝒩1 ⊗ ̄ ℳ2 , as asserted. It follows that Φ Similarly, ̃ 2 (y) ∈ 𝒩1 ⊗ ̄ ℳ2 ⇒ Φ ̄ 𝒩2 . y ∈ 𝒩1 ⊗ ̃2 ◦ Φ ̃ 1 ∶ ℳ1 ⊗ ̄ ℳ2 → 𝒩1 ⊗ ̄ 𝒩2 is a completely positive linear mapping which Thus, Φ = Φ extends the algebraic tensor product mapping Φ1 ⊗ Φ2 . If Φ1 and Φ2 are conditional expectations, ̃ 1 and Φ ̃ 2 are also conditional expectations, and so Φ is a conditional expectation. Φ (III) The general case of Proposition 9.4 now follows easily using Lemma 9.5. □ ̄ 𝒩 , there 9.8. Let ℳ and 𝒩 be W ∗ -algebras and 𝜑 ∈ ℳ∗ . Since 𝒩 = (𝒩∗ )∗ , for every x ∈ ℳ ⊗ exists a unique element E𝒩𝜑 (x) ∈ 𝒩 such that ̄ 𝜓)(x) 𝜓(E𝒩𝜑 (x)) = (𝜑 ⊗

(𝜓 ∈ 𝒩∗ ).

̄ 𝒩 → 𝒩 is a w-continuous linear mapping, ‖E 𝜑 ‖ = ‖𝜑‖, and It is easy to check that E𝒩𝜑 ∶ ℳ ⊗ 𝒩 ̄ b) = 𝜑(a)b E𝒩𝜑 (a ⊗

(a ∈ ℳ, b ∈ 𝒩 ).

(1)

̄ 𝒩 , from (1) we infer that Using the w-continuity of E𝒩𝜑 and the w-density of ℳ ⊗ 𝒩 in ℳ ⊗ ̄ b)x(1 ⊗ ̄ c)) = bE 𝜑 (x)c E𝒩𝜑 ((1 ⊗ 𝒩

̄ 𝒩 , b, c ∈ 𝒩 ). (x ∈ ℳ ⊗

(2)

If 𝜑 ∈ ℳ∗ is positive, E𝒩𝜑 is also positive. Identifying 𝒩 with 1 ⊗ 𝒩 by amplification, it follows that if 𝜑 is a normal state on ℳ, then E𝒩𝜑 is a normal conditional expectational of ℳ ⊗ 𝒩 onto 𝒩 . The family of conditional expectations {E𝒩𝜑 ; 𝜑 normal state on ℳ} is separating in the following sense: ̄ 𝒩 there exists a normal state for every 0 ≠ x ∈ ℳ ⊗ 𝜑 𝜑 on ℳ such that E𝒩 (x) ≠ 0.

(3)

From (1) it follows that ̄ 𝜄𝒩 E𝒩𝜑 = 𝜑 ⊗

(4)

in the sense defined in Section 9.4, where 𝜄𝒩 denotes the identity mapping on 𝒩 . In view of their similarity to “partial integration” procedures, the mappings E𝒩𝜑 are also called Fubini mappings.

106

Conditional Expectations and Operator-Valued Weights

If 𝜑 ∈ ℳ∗+ , then for every normal semifinite weight 𝜓 on 𝒩 we have ̄ 𝜓)(x) 𝜓(E𝒩𝜑 (x)) = (𝜑 ⊗

̄ 𝒩 )+ ). (x ∈ (ℳ ⊗

(5)

This can be easily verified from the definition of E𝒩𝜑 , using Corollary 5.8 and Proposition 8.3. The next result is equivalent to the commutation theorem for tensor products ([L], 10.7): ̄ 𝒦) Proposition. Let ℳ ⊂ ℬ(ℋ ), 𝒩 ⊂ ℬ(ℋ ) be von Neumann algebras, and let 𝒮 ⊂ ℬ(ℋ ⊗ 𝜑 𝜓 ̄ be a von Neumann algebra such that ℳ ⊗ 𝒩 ⊂ 𝒮 and Eℬ(𝒦 ) (𝒮 ) ⊂ 𝒩 , Eℬ(ℋ ) (𝒮 ) ⊂ ℳ for every ̄ 𝒩. 𝜑 ∈ ℬ(ℋ )∗ , 𝜓 ∈ ℬ(𝒦 )∗ . Then 𝒮 = ℳ ⊗ Proof. Let x ∈ 𝒮 and a′ ∈ ℳ ′ . For 𝜑 ∈ ℬ(ℋ )∗ , 𝜓 ∈ ℬ(𝒦 )∗ , we have ̄ 𝜓)((a′ ⊗ ̄ 1)x) = 𝜑(E𝜓 ((a′ ⊗ ̄ 1)x) = 𝜑(a′ E𝜓 (x)) (𝜑 ⊗ ℬ(ℋ ) ℬ(ℋ )

𝜓 ̄ 1))) = (𝜑 ⊗ ̄ 𝜓)(x(a′ ⊗ ̄ 1)), = 𝜑(E𝜓ℬ(ℋ ) (x)a′ ) = 𝜑(Eℬ(ℋ (x(a′ ⊗ )

̄ 1. Similarly, for b′ ∈ 𝒩 ′ , x commutes with 1 ⊗ ̄ b′ . Thus, x ∈ hence x commutes with a′ ⊗ ′ ⊗ ′ ′ ̄ ̄ 𝒩 ) =ℳ⊗𝒩. (ℳ 9.9 Corollary. Let 𝜎 ∶ G → Aut(ℳ), 𝜏 ∶ ℋ → Aut(𝒩 ) be actions of the groups G, H on the ̄ 𝜏 ∶ G × H → Aut(ℳ ⊗ ̄ 𝒩 ) be the tensor product W ∗ -algebras ℳ, 𝒩 , respectively, and let 𝜎 ⊗ ̄ 𝜏)g,h = 𝜎g ⊗ ̄ 𝜏h (g ∈ G, h ∈ H). Then (ℳ ⊗ ̄ 𝒩 )𝜎 ⊗̄ 𝜏 = ℳ 𝜎 ⊗ ̄ 𝒩 𝜏. action, that is, (𝜎 ⊗ ̄ 𝒩 𝜏 ⊂ (ℳ ⊗ ̄ 𝒩 )𝜎 ⊗̄ 𝜏 . Now let x ∈ (ℳ ⊗ ̄ 𝒩 )𝜎 ⊗̄ 𝜏 , 𝜑 ∈ ℳ∗ and h ∈ H, and Proof. Clearly, ℳ 𝜎 ⊗ denote by e ∈ G the neutral element of G. For 𝜓 ∈ 𝒩∗ , we have ̄ 𝜓 ◦ 𝜏h )(x) = (𝜑 ⊗ ̄ 𝜓)((𝜎e ⊗ ̄ 𝜏h )(x)) 𝜓(𝜏h (E𝒩𝜑 (x))) = (𝜑 ⊗ ̄ 𝜓)(x) = 𝜓(E 𝜑 (x)), = (𝜑 ⊗ 𝒩

hence 𝜏h (E𝒩𝜑 (x)) = E𝒩𝜑 (x). Thus, E𝒩𝜑 (x) ∈ 𝒩 𝜏 . Similarly, E𝜓ℳ (x) ∈ ℳ 𝜎 for 𝜓 ∈ 𝒩∗ . By ̄ 𝒩 𝜏. Proposition 9.8, it follows that x ∈ ℳ 𝜎 ⊗ We obtain the next two results from Proposition 9.8, in a similar manner. 9.10 Corollary. Let 𝒜 , ℬ be maximal abelian *-subalgebras of the W ∗ -algebras ℳ, 𝒩 , respec̄ ℬ is a maximal abelian *-subalgebra of the W ∗ -algebra ℳ ⊗ ̄ 𝒩. tively. Then 𝒜 ⊗ 9.11 Corollary. Let ℳ1 ⊂ ℬ(ℋ ), ℳ2 ⊂ ℬ(ℋ ), 𝒩1 ⊂ ℬ(𝒦 ), 𝒩2 ⊂ ℬ(𝒦 ) be von Neumann algebras. Then ̄ 𝒩1 ∩ ℳ2 ⊗ ̄ 𝒩2 = (ℳ1 ∩ ℳ2 ) ⊗ ̄ (𝒩1 ∩ 𝒩2 ). ℳ1 ⊗ 9.12. The next result is a characterization of tensor product W ∗ -algebras: Theorem. Let ℛ be a W ∗ -algebra and ℳ, 𝒩 ⊂ ℛ W ∗ -subalgebras with the properties: (i) ℛ is the W ∗ -algebra generated by ℳ and 𝒩 ; (ii) ab = ba for every a ∈ ℳ, b ∈ 𝒩 ;

Conditional Expectations

107

(iii) there exists a family {Ei }i∈I of w-continuous 𝒩 -linear mappings Ei ∶ ℛ → 𝒩 , such that Ei (a) ∈ ℂ ⋅ 1 for a ∈ ℳ, i ∈ I, x ∈ ℛ, Ei (x∗ x) = 0 for all i ∈ I ⇒ x = 0.

(1) (2)

̄ 𝒩 → ℛ such that Then there exists a *-isomorphism Φ ∶ ℳ ⊗ ̄ b) = ab (a ∈ ℳ, b ∈ 𝒩 ). Φ(a ⊗ Proof. From the assumptions, it follows that the family {Ei (a⋅); a ∈ ℳ, i ∈ I} consists of w-continuous 𝒩 -linear mappings satisfying (1), and the condition x ∈ ℛ, Ei (ax) = 0 for a ∈ ℳ, i ∈ I ⇒ x = 0. Thus, we may assume that the family {Ei }i∈I is separating, that is, x ∈ ℛ, Ei (x) = 0 for all i ∈ I ⇒ x = 0. In this case, the set {𝜓 ◦ Ei ; i ∈ I, 𝜓 ∈ 𝒩∗+ } is total in ℛ∗ . From condition (1), it follows that for each i ∈ I there exists 𝜑i ∈ ℳ∗ such that Ei (a) = 𝜑i (a) ⋅ 1(a ∈ ℳ). Since the family {Ei }i∈I is separating, the set {𝜑i ; i ∈ I} is total in ℳ∗ and hence the set ̄ 𝜓; i ∈ I, 𝜓 ∈ 𝒩 + } is total in (ℳ ⊗ ̄ 𝒩 )∗ . {𝜑i ⊗ ∗ ∑ Let ℱ ⊂ ℳ∗ be the linear subspace generated by {𝜑i ; i ∈ I}. For 𝜑 = i 𝜆i 𝜑i ∈ ℱ let E 𝜑 = ∑ + 𝜑 + ̄ ̄ i 𝜆i Ei . Then the sets {𝜑 ⊗ 𝜓; 𝜑 ∈ ℱ , 𝜓 ∈ 𝒩∗ } ⊂ (ℳ ⊗ 𝒩 )∗ and {𝜓 ◦ E ; 𝜑 ∈ ℱ , 𝜓 ∈ 𝒩∗ } ⊂ + ℛ∗ are dense linear subspaces and, for every 𝜑 ∈ ℱ , 𝜓 ∈ 𝒩∗ , we have ‖𝜓 ◦ E 𝜑 ‖ ≥ ‖(𝜓 ◦ E 𝜑 )|ℳ‖ = sup{|(𝜓 ◦ E 𝜑 )(a)|; a ∈ ℳ, ‖a‖ ≤ 1} ̄ 𝜓‖; = sup{|𝜓(1)𝜑(a)|; a ∈ ℳ, ‖a‖ ≤ 1} = ‖𝜑‖‖𝜓‖ = ‖𝜑 ⊗

(3)

moreover, if ak ∈ ℳ, bk ∈ 𝒩 (1 ≤ k ≤ n), then 𝜑

(𝜓 ◦ E )

( ∑

) ak bk

(

∑

̄ 𝜓) = (𝜑 ⊗

k

) ̄ bk ak ⊗

.

(4)

k

Using (ii) and (4), we see that the equation Φ0

( ∑ k

) ̄ bk ak ⊗

=

∑

ak bk

k

̄ 𝒩 onto the *-subalgebra of ℛ generated by ℳ ∪ 𝒩 . defines a *-isomorphism of the *-algebra ℳ ⊗ Furthermore, using (3) and (4) we have ‖Φ0 ‖ ≤ 1. Since Φ0 is bounded and (𝜓 ◦ E 𝜑 ) ◦ Φ0 = ̄ 𝜓(𝜑 ∈ ℱ , 𝜓 ∈ 𝒩 + ), it follows that for any 𝜃 ∈ ℛ∗ the linear form 𝜃 ◦ Φ on ℳ ⊗ 𝒩 is 𝜑⊗ ∗ w-continuous. It follows that Φ0 is w-continuous and so can be extended to a normal ̄ 𝒩 onto the whole of ℛ, by assumption (i). If x ∈ ℳ ⊗ ̄ 𝒩 and *-homomorphism Φ of ℳ ⊗

108

Conditional Expectations and Operator-Valued Weights

̄ 𝜓)(x) = (𝜓 ◦ E 𝜑 )(Φ(x)) = 0 for all 𝜑 ∈ ℱ , 𝜓 ∈ 𝒩 + , so that x = 0. We Φ(x) = 0, then (𝜑 ⊗ ∗ conclude that Φ ∶ ℳ ⊗ 𝒩 → ℛ is the required *-isomorphism. □ 9.13. If, in Theorem 9.12, ℛ is a factor, the conclusion remains valid assuming (i), (ii) and the weaker condition (iii0 ) there exists a nonzero w-continuous 𝒩 -linear mapping E ∶ ℛ → 𝒩 . Indeed, let I = ℳ × ℳ and, for i = (a1 , a2 ) ∈ I, define Ei (x) = E(a1 xa2 )

(x ∈ ℛ).

We thus get a family {Ei }i∈I of w-continuous 𝒩 -linear mappings Ei ∶ ℛ → 𝒩 . If a ∈ ℳ, E(a) ∈ 𝒩 commutes with the elements of ℳ and for any b ∈ 𝒩 , we have E(a)b = E(ab) = E(ba) = bE(a), hence E(a) belongs to the center of the factor ℛ. It follows that the family {Ei }i∈I satisfies condition 9.12.(1). On the other hand, it is easy to check that the set ℐ = {x ∈ ℛ; Ei (x) = 0 for all i ∈ I} is a w-closed two-sided ideal of ℛ. Since E ≠ 0 we have ℐ ≠ ℛ and hence ℐ = {0}, as ℛ is a factor. Thus, the family {Ei }i∈I also satisfies condition 9.12.(2). Hence, our assertion follows from Theorem 9.12. 9.14. For a subset 𝒩 of the W ∗ -algebra ℳ, we shall denote by 𝒩 ′ ∩ ℳ = {x ∈ ℳ; xy = yx for all y ∈ 𝒩 } the relative commutant of 𝒩 in ℳ. If 𝒩 = 𝒩 ∗ , 𝒩 ′ ∩ ℳ is a W ∗ -subalgebra of ℳ. We record here an obvious consequence of Theorem 9.12: Corollary. Let ℳ be a W ∗ -algebra and 𝒩 ⊂ ℳ a W ∗ -subfactor of ℳ, 1𝒩 = 1ℳ , such that ℳ is generated, as a W ∗ -algebra, by 𝒩 and 𝒩 ′ ∩ ℳ, and such that there exists a faithful normal ̄ (𝒩 ′ ∩ℳ) → ℳ conditional expectation E ∶ ℳ → 𝒩 . Then there exists a *-isomorphism Φ ∶ 𝒩 ⊗ such that ̄ b) = ab Φ(a ⊗

(a ∈ 𝒩 , b ∈ 𝒩 ′ ∩ ℳ).

9.15. Let ℳ be a W ∗ -algebra and ℱ ⊂ ℳ a unital W ∗ -subalgebra which is a factor of type I. Since ℱ is *-isomorphic with some ℬ(ℋ ), there exists a system of matrix units {eij }i,j∈I in ℱ , that is, eij = e∗ji , eij ehk = 𝛿jh eik

(i, j, h, k ∈ I),

the eii (i ∈ I) are mutually orthogonal minimal projections in ℱ , and ∑

eii = 1.

i∈J

Let 1 ∈ I be a fixed index. Every element a ∈ ℱ determines a scalar matrix [aij ] satisfying e1i aej1 = aij e11

(i, j ∈ I),

Conditional Expectations

109

and we have a=

∑

eii aejj =

∑

ei1 e1i aej1 e1j =

∑

aij eij .

ij

ij

ij

On the other hand, w-continuous linear mappings Eij ∶ ℳ → ℱ ′ ∩ ℳ are defined by Eij (x) =

∑

eki xejk

(x ∈ ℳ, i, j ∈ I)

k

with the properties: Eij (xb) = Eij (x)b, Eij (bx) = bEij (x) (x ∈ ℳ, b ∈ ℱ ′ ∩ ℳ),

(1)

Eij (a) = aij ⋅ 1 (a ∈ ℱ ), ∑ eij Eij (x) = x (x ∈ ℳ).

(2) (3)

ij

It follows that ℳ is the W ∗ -algebra generated by ℱ and ℱ ′ ∩ ℳ. Moreover, by Theorem 9.12 we have a canonical *-isomorphism ̄ (ℱ ′ ∩ ℳ) → ℳ Φ∶ℱ ⊗

(4)

̄ b) = ab for every a ∈ ℱ , b ∈ ℱ ′ ∩ ℳ. such that Φ(a ⊗ ̄ (ℱ ′ ∩ ℳ) → ℱ give rise to a separating family of normal Note that the Fubini mappings ℱ ⊗ conditional expectations of ℳ onto ℱ (see 9.8.(3)). Also, the mappings Eii ∶ ℳ → ℱ ′ ∩ ℳ(i ∈ I) are normal conditional expectations. If ℱ is of finite type In (n ∈ ℕ) then I = {1, … , n}, and the mapping 1∑ E ∶ ℳ → ℱ′ ∩ℳ n i=1 ii n

E=

(5)

is a faithful normal conditional expectation. Identifying ℳ with ℱ ⊗ (ℱ ′ ∩ ℳ) via Φ, it is easy to ̄ 𝜄, where 𝜇 = 1 tr is the normalized trace on ℱ and 𝜄 is the identity mapping on check that E = 𝜇 ⊗ n ′ ℱ ∩ ℳ. Also, for every finite trace 𝜏 on ℳ we have 𝜏 ◦ E = 𝜏.

(6)

9.16. As an application, we obtain the following factorization result: Corollary. Let 𝜎 ∶ G → Aut(ℳ) be an action of the group G on the W ∗ -algebra ℳ such that the centralizer ℳ 𝜎 is properly infinite and let ℱ be the countably decomposable infinite factor of type I. Then there exists a *-isomorphism ̄ ℱ,𝜎 ⊗ ̄ 𝜄ℱ ). (ℳ, 𝜎) ≈ (ℳ ⊗

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Proof. By assumption, we may consider ℱ as a unital W ∗ -subalgebra of ℳ 𝜎 ⊂ ℳ. By the preceding ̄ (ℱ ′ ∩ ℳ) via the *-isomorphism Φ (9.15.(4)). Since ℱ ⊂ Section, we may identify ℳ with ℱ ⊗ 𝜎 ′ ′ ̄ (ℱ ′ ∩ℳ), 𝜄ℱ ⊗ ̄ (𝜎|ℱ ′ ∩ ℳ , we have also 𝜎g (ℱ ∩ℳ) = ℱ ∩ℳ(g ∈ G), and hence (ℳ, 𝜎) ≈ (ℱ ⊗ ̄ ̄ ̄ ̄ ℳ)). Since (ℱ , 𝜄ℱ ) ≈ (ℱ ⊗ ℱ , 𝜄ℱ ⊗ 𝜄ℱ ), it follows that (ℳ, 𝜎) ≈ (ℳ ⊗ ℱ , 𝜎 ⊗ 𝜄ℱ ). 9.17. A related factorization result concerning weights on W ∗ -algebras is the following Proposition. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ such that the centralizer ℳ 𝜑 contains a unital W ∗ -subfactor ℱ of type I. Then there exists an n.s.f. weight 𝜓 on ℱ ′ ∩ ℳ such that 𝜎t𝜓 = 𝜎t𝜑 |ℱ ′ ∩ ℳ(t ∈ ℝ), and there exists a *-isomorphism ̄ (ℱ ′ ∩ ℳ), tr ⊗ ̄ 𝜓). (ℳ, 𝜑) ≈ (ℱ ⊗ ̄ (ℱ ′ ∩ℳ) via the *-isomorphism Φ (9.15.(4)). Proof. By Section 9.15, we may identify ℳ with ℱ ⊗ Also, we shall use the notation {eij } ⊂ ℱ and Eij ∶ ℳ → ℱ ′ ∩ ℳ of Section 9.15. The mapping 𝜓 ∶ (ℱ ′ ∩ ℳ)+ ∋ b ↦ 𝜑(e11 b) ∈ [0, +∞] is a normal weight on ℱ ′ ∩ ℳ. If b ∈ (ℱ ′ ∩ ℳ)+ and 𝜓(b) = 0, then e11 b = 0 because 𝜑 is faithful. It follows that eii b = ∑ ei1 e11 be1i = 0 for all i ∈ I and hence b = i eii b = 0. Thus 𝜓 is faithful. Since 𝜑 is semifinite, w

w

there exists a net {xs } ⊂ 𝔐𝜑 ∩ ℳ + such that xs → 1. Then bs = E11 (xs ) ∈ (ℱ ′ ∩ ℳ)+ , bs → 1 and 𝜓(bs ) = 𝜑(e11 E11 (xs )e11 ) = 𝜑(e11 xs e11 ) < +∞, as e11 ∈ ℱ ⊂ ℳ 𝜑 (2.21.(2)). Hence 𝜓 is an n.s.f. weight on ℱ ′ ∩ ℳ. For each b ∈ (ℱ ′ ∩ℳ)+ , the mapping 𝜏b ∶ ℱ + ∋ a ↦ 𝜑(ab) ∈ [0, +∞] is a normal weight on ℱ . Since ℱ ⊂ ℳ 𝜑 , by 2.21.(2) we see that 𝜏b is actually a normal trace on ℱ , and hence 𝜏b = 𝜆(b) ⋅ tr, with 0 ≤ 𝜆(b) = 𝜏b (e11 ) = 𝜑(e11 b) = 𝜓(b) ≤ +∞. Consequently, for a ∈ ℱ + , b ∈ (ℱ ′ ∩ ℳ)+ and t ∈ ℝ, we have ̄ 𝜓)(ab) 𝜑(ab) = 𝜓(b)tr(a) = (tr ⊗ and ̄ 𝜓)(𝜎t𝜑 (ab)) = (tr ⊗ ̄ 𝜓)(a𝜎t𝜑 (b)) = tr(a)𝜓(𝜎t𝜑 (b)) (tr ⊗ = tr(a)𝜑(e11 𝜎t𝜑 (b)) = tr(a)𝜑(𝜎t𝜑 (e11 b)) = tr(a)𝜑(e11 b) = tr(a)𝜓(b) ̄ 𝜓)(ab). = (tr ⊗ Using the Pedersen–Takesaki theorem on the equality of weights (6.2), we conclude that ̄ 𝜓. 𝜑 = tr ⊗ ̄ 𝜎t𝜓 (8.2), it follows that 𝜎t𝜓 = 𝜎t𝜑 |ℱ ′ ∩ ℳ for t ∈ ℝ. Since 𝜎t𝜑 = 𝜎ttr⊗𝜓 = 𝜄ℱ ⊗ 9.18. A normal semifinite weight 𝜑 on the W ∗ -algebra ℳ is called of infinite multiplicity if the centralizer ℳ 𝜑 is properly infinite. Corollary. If 𝜑 is a normal semifinite weight of infinite multiplicity on the W ∗ -algebra ℳ, then there exists a *-isomorphism: ̄ ℱ,𝜑 ⊗ ̄ tr) (ℳ, 𝜑) ≈ (ℳ ⊗ where ℱ is the countably decomposable infinite type I factor.

Existence and Uniqueness of Conditional Expectations

111

̄ ℬ with 𝒜 ≈ ℬ ≈ ℱ . By Proposition Proof. Since ℳ 𝜑 is properly infinite, we have ℳ 𝜑 ⊃ 𝒜 ⊗ ̄ (𝒜 ′ ∩ 9.17, there exists a normal semifinite weight 𝜓 on 𝒜 ′ ∩ ℳ such that (ℳ, 𝜑) ≈ (𝒜 ⊗ ′ 𝜓 ∗ ̄ 𝜓) and (𝒜 ∩ℳ) ⊃ ℬ. Thus, there exists a W -algebra 𝒩 and a normal semifinite weight ℳ), tr ⊗ ̄ 𝒩 , tr ⊗ ̄ 𝜓). With the same argument, we 𝜓 of infinite multiplicity on 𝒩 such that (ℳ, 𝜑) ≈ (ℱ ⊗ ̄ 𝒫 , tr ⊗ ̄ 𝜃). find a W ∗ -algebra 𝒫 and a normal semifinite weight 𝜃 on 𝒫 such that (𝒩 , 𝜓) ≈ (ℱ ⊗ ̄ ℱ , tr ⊗ ̄ tr) ≈ (ℱ , tr), it follows that (ℳ, 𝜑) ≈ (ℱ ⊗ ̄ ℱ ⊗ ̄ 𝒫 , tr ⊗ ̄ tr ⊗ ̄ 𝜃) ≈ Since (ℱ ⊗ ̄ 𝒫 , tr ⊗ ̄ 𝜃) and therefore (ℳ, 𝜑) ≈ (ℱ ⊗ ̄ ℳ, tr ⊗ ̄ 𝜑). (ℱ ⊗ 9.19. Notes. Conditional expectations in a noncommutative setting were introduced by Dixmier (1949, 1952, 1953) and Umegaki (1954, 1956, 1959, 1962). The main results (9.1, 9.4, 9.6) concerning projections of norm one are due to Tomiyama (1957–1959, 1959, 1969, 1971, 1972, 1981). Theorem 9.12 and its consequences (9.13, 9.14) appeared in Nakamura (1954); Takesaki (1958); Tomiyama (1971). The simple proof of Theorem 9.1 appeared also in a course by E.C. Lance. Thanks are due to Simon Wassermann for mentioning this fact and for further simplifications of our proof. For our exposition, we have used Connes and Takesaki (1977a); Palmer (1974); Størmer (1974a); Str̆atil̆a and Zsidó (1977, 2005); and Tomiyama (1971). Theorem 9.1 allows an easy proof of Sakai’s characterization of von Neumann algebras as W ∗ -algebras (A.16). Further properties of conditional expectations are contained in Tomiyama (1971).

10 Existence and Uniqueness of Conditional Expectations In this section, we give several criteria for the existence and uniqueness of conditional expectations, and some applications to the type theory of W ∗ -algebras. 10.1 Theorem (M. Takesaki). Let 𝒩 be a unital W ∗ -subalgebra of the W ∗ -algebra ℳ and 𝜑 an n.s.f. weight on ℳ. The following conditions are equivalent: (i) the faithful normal weight 𝜑|𝒩 + is semifinite and 𝜎t𝜑 (𝒩 ) = 𝒩 for every t ∈ ℝ; (ii) there exists a faithful normal conditional expectation E ∶ ℳ → 𝒩 such that 𝜑(x) = 𝜑(E(x))

(x ∈ ℳ + ).

(1)

Condition (1) determines uniquely the faithful normal conditional expectation E ∶ ℳ → 𝒩 . The proof is given in Sections 10.2–10.3. 10.2. Assume that the weight 𝜑|𝒩 + is semifinite. Thus, we have an n.s.f. weight 𝜑 on ℳ and an n.s.f. weight 𝜓 = 𝜑|𝒩 + on 𝒩 . In this section, we study the relationship between the standard representations associated with 𝜑 and 𝜓. It is obvious that 𝔑𝜓 = 𝔑𝜑 ∩ 𝒩 and that the scalar products (⋅|⋅)𝜓 , (⋅|⋅)𝜑 coincide on 𝔑𝜓 , hence ℋ𝜓 can be identified with a closed linear subspace of ℋ𝜑 such that b𝜓 = b𝜑

(b ∈ 𝔑𝜓 ).

Let P be the orthogonal projection of ℋ𝜑 onto ℋ𝜓 .

(1)

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Conditional Expectations and Operator-Valued Weights

For y ∈ 𝒩 , b ∈ 𝔑𝜓 , we have 𝜋𝜑 (y)b𝜓 = 𝜋𝜑 (y)b𝜑 = (yb)𝜑 = (yb)𝜓 = 𝜋𝜓 (y)b𝜓 , hence ℋ𝜓 is 𝜋𝜑 (𝒩 )-invariant, that is, P ∈ 𝜋𝜑 (𝒩 )′ , and 𝜋𝜑 (y)|ℋ𝜓 = 𝜋𝜓 (y)

(y ∈ 𝒩 ).

(2)

Also, 𝔄𝜓 = 𝔄𝜑 ∩ 𝒩 , so that 𝔄𝜓 ⊂ 𝔄𝜑 ∩ ℋ𝜓 ,

(3)

and 𝔄𝜓 is a *-subalgebra of 𝔄𝜑 with the same scalar product. It follows that D(S𝜓 ) ⊂ D(S𝜑 ) ∩ ℋ𝜓 and S𝜑 𝜉 = S𝜓 𝜉 for 𝜉 ∈ D(S𝜓 ).

(4)

Assume moreover that 𝜎t𝜑 (𝒩 ) = 𝒩 (t ∈ ℝ). Using the KMS condition (2.12.(11)), we see that = 𝜎t𝜓 (y) for y ∈ 𝒩 , t ∈ ℝ. For b ∈ 𝔑𝜓 , we have (2.12.(9)) Δit𝜑 b𝜓 = Δit𝜑 b𝜑 = (𝜎t𝜑 (b))𝜑 = (𝜎t𝜓 (b))𝜓 = Δit𝜓 b𝜓 (t ∈ ℝ), hence ℋ𝜓 is Δit𝜑 -invariant, that is, Δit𝜑 P = PΔit𝜑 , and 𝜎t𝜑 (y)

Δit𝜑 |ℋ𝜓 = Δit𝜓

(t ∈ ℝ).

(5)

Thus, Δ𝜑 commutes with P and, using ([L], 9.21), we infer from (5) that for any 𝛼 ∈ ℂ D(Δ𝜓𝛼 ) = D(Δ𝛼𝜑 ) ∩ ℋ𝜓 and Δ𝛼𝜑 𝜉 = Δ𝛼𝜓 𝜉 for 𝜉 ∈ D(Δ𝛼𝜓 ).

(6)

For 𝛼 = 1∕2, we get from (6) that D(S𝜓 ) = D(S𝜑 ) ∩ ℋ𝜓 ; 1∕2

(7) 1∕2

1∕2

1∕2

and, using also (4), we obtain J𝜓 Δ𝜓 𝜉 = S𝜓 𝜉 = S𝜑 𝜉 = J𝜑 Δ𝜑 𝜉 = J𝜑 Δ𝜓 𝜉 for 𝜉 ∈ D(Δ𝜓 ). It follows that ℋ𝜓 is J𝜑 -invariant, that is, J𝜑 P = PJ𝜑 , and J𝜑 𝜉 = J𝜓 𝜉

(𝜉 ∈ ℋ𝜓 ).

(8)

Also, by (6) for 𝛼 = −1∕2 and (8), we get D(S∗𝜓 ) = D(S∗𝜑 ) ∩ ℋ𝜓 and S∗𝜑 𝜂 = S∗𝜓 𝜂 for 𝜂 ∈ D(S∗𝜓 ).

(9)

If 𝜂 ∈ 𝔄′𝜑 ∩ ℋ𝜓 , then 𝜂 ∈ D(S∗𝜑 ) ∩ ℋ𝜓 = D(S∗𝜓 ) and for b ∈ 𝔄𝜓 ⊂ 𝔄𝜑 we have 𝜋𝜑 (b)𝜂 = 𝜋𝜓 (b)𝜂 = R𝜑𝜂 b𝜑 = R𝜑𝜂 b𝜓 . Consequently, 𝔄′𝜑 ∩ ℋ𝜓 ⊂ 𝔄′𝜓

(10)

and, for 𝜂 ∈ 𝔄′𝜑 ∩ ℋ𝜓 we have R𝜓𝜂 = R𝜑𝜂 |ℋ𝜓 and also R𝜓S∗ 𝜂 = R𝜑S∗ 𝜂 |ℋ𝜓 = (R𝜑𝜂 )∗ |ℋ𝜓 . Since ℋ𝜓 is invariant with respect to R𝜑𝜂 and (R𝜑𝜂 )∗ , it follows that PR𝜑𝜂 = R𝜑𝜂 P = R𝜓𝜂 P

𝜑

𝜑

(𝜂 ∈ 𝔄′𝜑 ∩ ℋ𝜓 ).

(11)

Existence and Uniqueness of Conditional Expectations

113

Applying J𝜑 to (10) and using (8), (3) and Tomita’s fundamental theorem (2.12), we get 𝔄𝜓 = 𝔄𝜑 ∩ ℋ𝜓 .

(12)

𝔄′𝜓 = 𝔄′𝜑 ∩ ℋ𝜓 .

(13)

Then, applying J𝜑 to (12), we obtain

1∕2

1∕2

If a ∈ 𝔄𝜑 , then a𝜑 ∈ D(S𝜑 ) = D(Δ𝜑 ), and hence Pa𝜑 ∈ D(Δ𝜑 ) = D(S𝜑 ); for 𝜂 ∈ 𝔄′𝜓 = ′ 𝔄𝜑 ∩ ℋ𝜓 , we have L𝜓Pa 𝜂 = R𝜓𝜂 Pa𝜑 = PR𝜑𝜂 a𝜑 = P𝜋𝜑 (a)𝜂, so that 𝜑

𝔄𝜓 = P 𝔄𝜑 and L𝜓Pa = P𝜋𝜑 (a)P for a ∈ 𝔄𝜑 . 𝜑

(14)

Thus, 𝜋𝜓 (𝒩 ) = 𝔏(𝔄𝜓 ) = P𝜋𝜑 (ℳ)P and so there exists a w-continuous positive linear mapping E ∶ ℳ → 𝒩 , uniquely determined, such that 𝜋𝜓 (E(x)) = P𝜋𝜑 (x)P

(x ∈ ℳ).

(15)

Using (2), for x ∈ ℳ, y ∈ 𝒩 and 𝜉 ∈ ℋ𝜓 we get 𝜋𝜓 (E(xy))𝜉 = P𝜋𝜑 (xy)P𝜉 = P𝜋𝜑 (x)𝜋𝜑 (y)𝜉 = P𝜋𝜑 (x)P𝜋𝜓 (y)𝜉 = 𝜋𝜓 (E(x))𝜋𝜓 (y)𝜉 = 𝜋𝜓 (E(x)y)𝜉, hence E(xy) = E(x)y. If x ∈ ℳ and E(x∗ x) = 0, for b ∈ 𝔑𝜓 we have 𝜑((xb)∗ (xb)) = ‖𝜋𝜑 (x)b𝜑 ‖2𝜑 = (P𝜋𝜑 (x)Pb𝜑 |b𝜑 )𝜑 = (𝜋𝜓 (E(x))b𝜓 |b𝜓 )𝜓 = 0, so that xb = 0. Since 𝔑𝜓 is w-dense in 𝒩 , it follows that x = 0. Thus, E ∶ ℳ → 𝒩 is a faithful normal conditional expectation. Moreover, we have E ◦ 𝜎t𝜑 = 𝜎t𝜓 ◦ E

(t ∈ ℝ).

(16)

Indeed, using (5), for x ∈ ℳ and 𝜉 ∈ ℋ𝜓 , we get 𝜋𝜓 (E(𝜎t𝜑 (x)))𝜉 = P𝜋𝜑 (𝜎t𝜑 (x))P𝜉 it −it = PΔit𝜑 𝜋𝜑 (x)Δ−it 𝜑 P𝜉 = Δ𝜓 P𝜋𝜑 (x)PΔ𝜓 𝜉 𝜓 = Δit𝜓 𝜋𝜓 (E(x))Δ−it 𝜓 𝜉 = 𝜋𝜓 (𝜎t (E(x))𝜉.

It follows that 𝜓 ◦ E is a 𝜎 𝜑 -invariant n.s.f. weight on ℳ, that is, 𝜓 ◦ E commutes with 𝜑. On the other hand, ℬ = 𝔑∗𝜓 ℳ𝔑𝜓 ⊂ 𝔑∗𝜑 𝔑𝜑 = 𝔐𝜑 is a 𝜎 𝜑 -invariant and w-dense *-subalgebra of ℳ and for every x ∈ ℳ, b ∈ 𝔑𝜓 we have (𝜓 ◦ E)(b∗ xb) = 𝜓(b∗ E(x)b) = (𝜋𝜓 (E(x)b𝜓 |b𝜓 )𝜓 = (P𝜋𝜑 (x)Pb𝜑 |b𝜑 )𝜑 = (𝜋𝜑 (x)b𝜑 |b𝜑 )𝜑 = 𝜑(b∗ xb). By the Pedersen–Takesaki theorem on the equality of weights (6.2), it follows that 𝜓 ◦ E = 𝜑, that is, 𝜑(E(x)) = 𝜑(x) for x ∈ ℳ + . Thus, we have proved the implication (i) ⇒ (ii) of Theorem 10.1. Let us note that by analogy with (14) we have 𝔄′𝜓 = P𝔄′𝜑 and R𝜓P𝜂 = PR𝜑𝜂 P for 𝜂 ∈ 𝔄′𝜑 .

(17)

Using (2.12.(15) or [L], 10.21), we obtain 𝔗𝜓 = 𝔗𝜑 ∩ ℋ𝜓 = P 𝔗𝜑 .

(18)

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Conditional Expectations and Operator-Valued Weights

Also, since the inclusion 𝒩 ↪ ℳ is an isometric normal *-homomorphism, it follows that the mapping 𝜌 ∶ ℜ(𝔄′𝜓 ) → ℜ(𝔄′𝜑 ) defined by 𝜌(J𝜓 𝜋𝜓 (y)J𝜓 ) = J𝜑 𝜋𝜑 (y)J𝜑 (y ∈ 𝒩 ) is an isometric s

s

normal *-homomorphism. In particular, for 𝜂 ∈ 𝔄′𝜓 , we have ‖R𝜓𝜂 ‖ = ‖R𝜑𝜂 ‖ and R𝜓𝜂 → 1 ⇔ R𝜑𝜂 → 1. 10.3. Assume now that there exists a normal conditional expectation E ∶ ℳ → 𝒩 such that 𝜑 ◦ E = 𝜑; then it follows that E is also faithful. w Let 𝜓 = 𝜑|𝒩 + . If xi ∈ 𝔐𝜑 ∩ ℳ + and xi → 1, then 𝜓(E(xi )) = 𝜑(xi ) < +∞, hence E(xi ) ∈ w

𝔐𝜓 ∩ 𝒩 + and E(xi ) → 1. Consequently, 𝜓 is an n.s.f. weight on 𝒩 . For x ∈ ℳ and b ∈ 𝔑𝜓 , we have (𝜋𝜓 (E(x))b𝜓 |b𝜓 )𝜓 = 𝜓(b∗ E(x)b) = 𝜑(E(b∗ xb)) = 𝜑(b∗ xb). Thus, the normal conditional expectation E ∶ ℳ → 𝒩 is uniquely determined by the condition 𝜑 ◦ E = 𝜑. In what follows we shall use the notation introduced in the first part of Section 10.2. For a ∈ 𝔑𝜑 , we have 𝜓(E(a)∗ E(a)) ≤ 𝜓(E(a∗ a)) = 𝜑(a∗ a), hence E(a) ∈ 𝔑𝜓 . Since for every b ∈ 𝔑𝜓 , we have (a𝜑 |b𝜑 )𝜑 = 𝜑(b∗ a) = 𝜓(E(b∗ a)) = 𝜓(b∗ E(a)) = ((E(a))𝜓 |b𝜓 )𝜓 , it follows that a ∈ 𝔑𝜑 ⇒ E(a) ∈ 𝔑𝜓 and (E(a))𝜓 = Pa𝜑 .

(1)

Since E ∶ ℳ → 𝒩 is a self-adjoint mapping, we further obtain P 𝔄𝜑 = 𝔄𝜓 ⊂ 𝔄𝜑 , and S𝜑 Pa𝜑 = PS𝜑 a𝜑 for a ∈ 𝔄𝜑 .

(2)

As S𝜑 = S𝜑 |𝔄𝜑 , it follows that PS𝜑 ⊂ S𝜑 P, that is, (1 − 2P)S𝜑 (1 − 2P) = S𝜑 , where 1 − 2P is a unitary operator. Then (1 − 2P)S∗𝜑 (1 − 2P) = S∗𝜑 , (1 − 2P)Δ𝜑 (1 − 2P) = Δ𝜑 , so that P commutes with Δ𝜑 , that is, Δit𝜑 P = PΔit𝜑

(t ∈ ℝ).

(3)

From (2) and (3), we infer that Δit𝜑 𝔄𝜓 = Δit𝜑 P𝔄𝜑 = PΔit𝜑 𝔄𝜑 = P 𝔄𝜑 = 𝔄𝜑 .

(4)

Let b ∈ 𝔄𝜑 and t ∈ ℝ. By (4), there exists an element y ∈ 𝔄𝜓 such that Δit𝜑 b𝜓 = y𝜓 . Then = Δit𝜑 b𝜑 = Δit𝜑 b𝜓 = y𝜓 = y𝜑 , hence 𝜎t𝜑 (b) = y. Thus, for every t ∈ ℝ, we have 𝜎t𝜑 (𝔄𝜓 ) ⊂ 𝔄𝜓 and so 𝜎t𝜑 (𝒩 ) = 𝒩 , as 𝔄𝜓 is w-dense in 𝒩 . We have proved the implication (ii) ⇒ (i) in Theorem 10.1. (𝜎t𝜑 (b))𝜑

10.4. Let 𝒩 be a unital W ∗ -subalgebra of the W ∗ -algebra ℳ. For every n.s.f. weight 𝜑 on ℳ such that 𝜑|𝒩 + is semifinite and 𝜎t𝜑 (𝒩 ) = 𝒩 (t ∈ ℝ), we shall denote by E𝒩 𝜑 ∶ℳ →𝒩 the unique faithful normal conditional expectation such that 𝜑 ◦ E𝒩 𝜑 = 𝜑. Corollary. In the above situation, for 𝜎 ∈ Aut(ℳ) we have ) E𝜑𝜎 ◦ (𝒩 = 𝜎 −1 ◦ E𝒩 𝜎 𝜑 ◦ 𝜎. −1

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115

10.5 Corollary. Let E ∶ ℳ → 𝒩 be a faithful normal conditional expectation of the W ∗ -algebra ℳ onto its W ∗ -subalgebra 𝒩 . If 𝜓 is any n.s.f. weight on 𝒩 , then 𝜑 = 𝜓 ◦ E is an n.s.f. weight on ℳ and 𝜎t𝜓 (E(x)) = 𝜎t𝜑 (E(x)) = E(𝜎t𝜑 (x))

(x ∈ ℳ, t ∈ ℝ).

(1)

If 𝜓1 , 𝜓2 are n.s.f. weights on 𝒩 and 𝜑1 = 𝜓1 ◦ E, 𝜑2 = 𝜓2 ◦ E, then [D𝜑2 ∶ D𝜑1 ]t = [D𝜓2 ∶ D𝜓1 ]t ∈ 𝒩

(t ∈ ℝ).

(2)

Proof. Indeed, 𝜑 is an n.s.f. weight on ℳ, 𝜓 = 𝜑|𝒩 + is semifinite, and from Theorem 10.1 it follows that 𝒩 is 𝜎 𝜑 -invariant and E = E𝒩 𝜑 . Now (1) follows using the KMS condition and Corollary 10.4. ̄ Mat2 (ℂ) onto Then E ⊗ 𝜄 is a faithful normal conditional expectation of Mat2 (ℳ) = ℳ ⊗ Mat2 (𝒩 ) = 𝒩 ⊗ Mat2 (ℂ) (see 9.4) and for the balanced weights (see 3.1) we have 𝜃(𝜑1 , 𝜑2 ) = ̄ 𝜄), so that (2) follows from the first equation in (1) applied to the balanced weights 𝜃(𝜓2 , 𝜓2 ) ◦ (E ⊗ ( ) 0 0 and to the element ∈ Mat2 (𝒩 ). 1 0 (1) and (2) can also be expressed in the form: 𝜓2 ,𝜓1

𝜎t

𝜑2 ,𝜑1

(E(x)) = 𝜎t

𝜑2 ,𝜑1

(E(x)) = E(𝜎t

(x))

(x ∈ ℳ, t ∈ ℝ).

(3)

Let 𝜓 and 𝜑 be as in the statement of the corollary and 𝜔 an arbitrary n.s.f. weight on ℳ. An easy application of the corollary and Theorem 5.1 shows that if [D𝜔 ∶ D𝜑]t ∈ 𝒩 for all t ∈ ℝ, then 𝜔|𝒩 + is semifinite.

(4)

Indeed, since 𝜎t𝜓 = 𝜎t𝜑 |𝒩 , there exists an n.s.f. weight 𝜏 on 𝒩 such that [D(𝜏 ◦ E) ∶ D𝜑]t = [D𝜏 ∶ D𝜓]t = [D𝜔 ∶ D𝜑]t (t ∈ ℝ), hence 𝜔 = 𝜏 ◦ E, and 𝜔|𝒩 + = 𝜏 is semifinite. 10.6. From Theorem 10.1, it follows that if 𝜏 is an n.s.f. trace on the W ∗ -algebra ℳ and 𝒩 ⊂ ℳ is a unital W ∗ -subalgebra such that 𝜏|𝒩 + is semifinite, there exists a unique faithful normal conditional 𝒩 expectation E𝒩 𝜏 ∶ ℳ → 𝒩 such that 𝜏 ◦ E𝜏 = 𝜏. In particular: Corollary. Let 𝜏 be a faithful normal finite trace on the W ∗ -algebra ℳ. For every unital W ∗ -subalgebra 𝒩 ⊂ ℳ, there exists a unique faithful normal conditional expectation E𝒩 𝜏 ∶ ℳ → 𝒩 such that 𝜏 ◦ E𝒩 𝜏 = 𝜏. If ℳ is a countably decomposable finite W ∗ -algebra and 𝒩 = 𝒵 (ℳ) is its center, then all the (ℳ) coincide with the canonical central trace ♮ ∶ ℳ → 𝒵 (ℳ) ([L], conditional expectations E𝒵 𝜏 7.11). Also, if ℱ ⊂ ℳ is a finite type I factor and 𝒩 = ℱ ′ ∩ ℳ, then all the E𝒩 𝜏 coincide with the conditional expectation defined in 9.15.(5). 10.7 Corollary. Let ℳ be a W ∗ -algebra, 𝜑 a faithful normal state on ℳ and 𝒩 ⊂ ℳ a 𝜎 𝜑 invariant unital W ∗ -subfactor such that ℳ generated as a W ∗ -algebra by 𝒩 and 𝒩 ′ ∩ ℳ. There ̄ (𝒩 ′ ∩ ℳ) → ℳ such that Φ(a ⊗ ̄ b) = ab(a ∈ 𝒩 , b ∈ 𝒩 ′ ∩ ℳ), exists a *-isomorphism Φ ∶ 𝒩 ⊗ ′ ̄ (𝜑|𝒩 ∩ ℳ), that is, and 𝜑 ◦ Φ = (𝜑|𝒩 ) ⊗ 𝜑(ab) = 𝜑(a)𝜑(b) (a ∈ 𝒩 , b ∈ 𝒩 ′ ∩ ℳ).

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Proof. By Theorem 10.1, there exists a unique faithful normal conditional expectation E ∶ ℳ → 𝒩 with 𝜑 ◦ E = 𝜑. Thus, the existence of Φ follows from Corollary 9.14. For b ∈ 𝒩 ′ ∩ ℳ, the element E(b) belongs to the center of the factor 𝒩 , and hence 𝜑(b) = 𝜑(E(b)) = E(b); if a ∈ 𝒩 , then 𝜑(ab) = 𝜑(E(ab)) = 𝜑(aE(b)) = 𝜑(a)E(b) = 𝜑(a)𝜑(b). 10.8. Let 𝜎 ∶ G → Aut(ℳ) be an action of the group G on the W ∗ -algebra ℳ. We shall say that ℳ is 𝜎-finite if for every x ∈ ℳ, x ≠ 0, there exists a 𝜎-invariant normal state 𝜑 on ℳ such that 𝜑(x) ≠ 0. Corollary (Kovács & Szücs, 1966). Let 𝜎 ∶ G → Aut(ℳ) be an action of the group G on the W ∗ -algebra ℳ. The following statements are equivalent: (i) ℳ is 𝜎-finite; (ii) there exists a 𝜎-invariant faithful normal conditional expectation E ∶ 𝒩 → ℳ 𝜎 . Proof. The implication (ii) ⇒ (i) is obvious. Conversely, if ℳ is 𝜎-finite, there exists a family ∑ ∑ {𝜑i }i∈I of 𝜎-invariant normal states on ℳ with i s(𝜑1 ) = 1, and 𝜑 = i 𝜑i is a 𝜎-invariant n.s.f. weight on ℳ. Then each 𝜎t𝜑 (t ∈ ℝ) commutes with each 𝜎g (g ∈ G), in particular ℳ 𝜎 is 𝜎 𝜑 -invariant. Since s(𝜑i ) ∈ ℳ 𝜎 and 𝜑(s(𝜑i )) = 𝜑i (s(𝜑i )) < +∞, it follows that 𝜑|(ℳ 𝜎 )+ is semifinite. By Theorem 10.1, there exists a unique faithful normal conditional expectation E ∶ ℳ → ℳ 𝜎 with 𝜑 ◦ E = 𝜑. For g ∈ G, we have 𝜑 ◦ 𝜎g = 𝜑 and 𝜎g−1 (ℳ 𝜎 ) = ℳ 𝜎 , hence E ◦ 𝜎g = 𝜎g ◦ E = E, by Corollary l0.4. 10.9 Corollary (F. Combes). Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ. The following statements are equivalent: (i) 𝜑|(ℳ 𝜑 )+ is an n.s.f. trace on ℳ 𝜑 ; (ii) there exists a faithful normal conditional expectation E ∶ ℳ → ℳ 𝜑 such that 𝜑 = 𝜑 ◦ E; (iii) there exists a 𝜎-invariant faithful normal conditional expectation E ∶ ℳ → ℳ 𝜑 such that 𝜑 = 𝜑 ◦ E; ∑ ∑ (iv) there exists a family {𝜑i }i∈I ⊂ ℳ∗+ with i s(𝜑i ) = 1 and 𝜑 = i 𝜑i ; (v) ℳ is 𝜎-finite. Proof. (i) ⇒ (ii), by Theorem 10.1. (ii) ⇒ (iii), by Corollary 10.4. (iii) ⇒ (i). By Theorem 10.1, it follows, assuming (iii), that 𝜑|(ℳ 𝜑 )+ is an n.s.f. weight on ℳ 𝜑 and the trace property follows from 2.21.(2). (i) ⇒ (iv). Since 𝜓 = 𝜑|(ℳ 𝜑 )+ is an n.s.f. trace on ℳ 𝜑 , by ([L], E.7.11) we know that there exists a family {𝜓i }i∈I of normal positive forms on ℳ 𝜑 with mutually orthogonal supports such that ∑ 𝜓 = i 𝜓i . If E is the conditional expectation given by (iii) (cf. (i) ⇒ (iii)), then 𝜑i = 𝜓i ◦ E ∈ ∑ ∑ ∑ ℳ∗+ (i ∈ I), i s(𝜑i ) = s(𝜓i ) = 1, and 𝜑 = 𝜓 ◦ E = i 𝜑i . (iv) ⇒ (v). Since 𝜑(⋅s(𝜑i )) = 𝜑(s(𝜑i )⋅) = 𝜑i , it follows from 2.21.(2) that s(𝜑i ) ∈ ℳ 𝜑 and from 2.22.(3) that 𝜑i is 𝜎 𝜑 -invariant (i ∈ I); hence ℳ is 𝜎-finite. (v) ⇒ (iii), by Corollary 10.8. If the equivalent conditions of Corollary 10.9 are satisfied, then the n.s.f. weight 𝜑 is called strictly semifinite and we shall abbreviate this by saying that 𝜑 is an n.ss.f. weight. With obvious modifications, Corollary 10.9 can be extended to weights which are not necessarily faithful and we get the notion of normal strictly semifinite weight. Note that any normal semifinite trace and any normal positive form are strictly semifinite.

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If there exists t0 ∈ ℝ, t0 > 0, such that 𝜎t𝜑0 = 𝜄ℳ , the identity mapping on ℳ, then the weight 𝜑 is strictly semifinite, as the mapping E∶ℳ∋x↦

t

0 1 𝜎 𝜑 (x) dt ∈ ℳ 𝜎 t0 ∫0

is then a faithful normal conditional expectation with 𝜑 ◦ E = 𝜑. Finally, we note that the tensor product of two normal strictly semifinite weights is again strictly semifinite (8.8.(3)). 10.10. On every W ∗ -algebra ℳ, there exists an n.ss.f. weight. Indeed, there exists a family {𝜑i }i∈I ∑ ∑ of normal states on ℳ with supports ei = s(𝜑i ) mutually orthogonal and i ei = 1. Then 𝜑 = i 𝜑i is an n.ss.f. weight on ℳ. Let 𝜉i = (ei )𝜑 ∈ ℋ𝜑 . Then the set {𝜉i }i∈I ∈ ℋ𝜑 is cyclic and separating for 𝜋𝜑 (ℳ) and, for i ≠ j, we have (𝜉i |𝜉j )𝜑 = ((ei )𝜑 |(ej )𝜑 )𝜑 = 𝜑(ej ei ) = 0. Since every *-isomorphism between two standard von Neumann algebras is spatial ([L], 10.15), it follows that if a von Neumann algebra ℳ ⊂ ℬ(ℋ ) is standard, then there exists a family {𝜉i }i∈I ⊂ ℋ of mutually orthogonal vectors, cyclic and separating for ℳ. The converse of this assertion is also true (Rousseau, Van Daele & Van Heeswijck, 1977). 10.11 Proposition. Let 𝜑 be an n.ss.f. weight on the W ∗ -algebra ℳ. If 𝜎 ∶ ℝ → Aut(ℳ) is an action of ℝ on ℳ such that 𝜎t (𝔐𝜑 ) = 𝔐𝜑 , (t ∈ ℝ), and 𝜑 satisfies the KMS condition with respect to {𝜎t }t∈ℝ in any two elements of 𝔐𝜑 , then 𝜑 is 𝜎-invariant and hence 𝜎t = 𝜎t𝜑 (t ∈ ℝ). Proof. Since 𝜓 = 𝜑|(ℳ 𝜑 )+ is an n.s.f. trace on ℳ 𝜑 , the w-dense two-sided ideal of 𝔐𝜓 of ℳ 𝜑 has an increasing approximate unit uk ↑ 1. Let x ∈ 𝔐𝜑 ∩ ℳ + and let fk be a complex function, defined, continuous and bounded on the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, analytic in the interior of this strip, such that fk (it) = 𝜑(𝜎t (x)uk ) and fk (1 + it) = 𝜑(uk 𝜎t (x))(t ∈ ℝ). Since uk ∈ ℳ 𝜑 , using 2.21.(2) it follows that fk (it) = fk (1 + it)(t ∈ ℝ), and hence fk is constant, that is, 𝜑(𝜎t (x)uk ) = 𝜑(xuk )(t ∈ ℝ). Let E ∶ ℳ → ℳ 𝜑 , be the faithful normal conditional expectation with 𝜑 = 𝜓 ◦ E. We have 1∕2

1∕2

𝜓(E(𝜎t (x))1∕2 uk E(𝜎t (x))1∕2 ) = 𝜓(uk E(𝜎t (x))uk ) 1∕2

1∕2

1∕2

1∕2

= 𝜓(E(uk 𝜎t (x)uk )) = 𝜑(uk 𝜎t (x)uk ) = 𝜑(𝜎t (x)uk ) = 𝜑(xuk ) 1∕2

1∕2

1∕2

1∕2

1∕2

1∕2

= 𝜑(uk xuk ) = 𝜓(E(uk xuk )) = 𝜓(uk E(x)uk ) = 𝜓(E(x)1∕2 uk E(x)1∕2 ). Since uk ↑ 1 we get 𝜑(𝜎t (x)) = 𝜓(E(𝜎t (x)) = 𝜓(E(x)) = 𝜑(x), for every y ∈ 𝔐𝜑 ∩ ℳ + , and because of the assumption 𝜎t (𝔐𝜑 ) = 𝔐𝜑 it follows that 𝜑 ◦ 𝜎t = 𝜑. The above result holds, in particular, for normal positive forms. In this case, the proof is simpler because we can take uk = 1 and the assumption 𝜎t (𝔐𝜑 ) = 𝔐𝜑 (t ∈ ℝ) is automatically satisfied. 10.12. Let G be a locally compact group. For a function f defined on G and an element g ∈ G, we define the function g f on G by (g f )(h) = f (g−1 h)(h ∈ G). A left invariant mean on ℒ ∞ (G) is a positive linear form m on the C ∗ -algebra ℒ ∞ (G) with m(1) = 1 and 𝔪(g f ) = 𝔪( f ) ( f ∈ ℒ ∞ (G), g ∈ G).

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The locally compact group G is called amenable if there exists a left invariant mean on ℒ ∞ (G). There are many other equivalent definitions of amenability for which we refer to (Day, 1957; Greenleaf, 1969). Clearly, every compact group is amenable, the invariant mean being given by the normalized Haar measure of G. Also, every commutative discrete group is amenable (Day, 1957; Greenleaf, 1969). If the group G is discrete, then every left invariant mean 𝔪 on 𝓁 ∞ (G) is also right invariant (Greenleaf, 1969), that is, 𝔪( fg ) = 𝔪( f ) ( f ∈ 𝓁 ∞ (G), g ∈ G), where fg (h) = f (hg) (h ∈ G). We record the following criterion concerning the existence of (not necessarily normal) conditional expectations: Proposition. Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the W ∗ -algebra ℳ. If G is amenable, then there exists a 𝜎-invariant conditional expectation E ∶ ℳ → ℳ 𝜎 . Proof. Let 𝔪 be an invariant mean on 𝓁 ∞ (G). For each x ∈ ℳ, the mapping ℳ∗ ∋ 𝜑 ↦ 𝔪(g ↦ 𝜑(𝜎g (x)) ∈ ℂ is a bounded linear form on ℳ∗ , with norm ≤ ‖x‖, hence there exists a unique element E(x) ∈ ℳ, ‖E(x)‖ ≤ ‖x‖, such that 𝜑(E(x)) = 𝔪(g ↦ 𝜑(𝜎g (x)) for every 𝜑 ∈ ℳ∗ . Since 𝔪 is invariant, it follows that E(x) ∈ ℳ 𝜎 and E(𝜎g (x)) = E(x) for all x ∈ ℳ, g ∈ G. If x ∈ ℳ, then clearly E(x) = x. Thus, E ∶ ℳ → ℳ 𝜎 is a 𝜎-invariant projection of norm 1, and hence a conditional expectation, by the Tomiyama theorem (9.1). 10.13. Even if the group G is not necessarily amenable, the following result is known: Theorem (J. T. Schwartz). Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the W ∗ -algebra ℳ with the property co ({𝜎g (x); g ∈ G}) ∩ ℳ 𝜎 ≠ ∅ (x ∈ ℳ). w

(1)

For every x0 ∈ ℳ and every a0 ∈ co ({𝜎g (x0 ); g ∈ G}) ∩ ℳ 𝜎 , there exists a conditional expectation E ∶ ℳ → ℳ 𝜎 such that E(x0 ) = a0 . w

Proof. Let ℰ be the set of all bounded linear mappings E ∶ ℳ → ℳ with the following properties: w (a) E(x0 ) = a0 ; (b) E(x) ∈ co ({𝜎g (x); g ∈ G}) for x ∈ ℳ; (c) E(y) = y for y ∈ ℳ 𝜎 . ∑ Let Λ be the set of all finitely supported functions 𝜆 ∶ G → [0, 1] such that g∈G 𝜆(g) = 1. For ∑ 𝜆 ∈ Λ and x ∈ ℳ, we shall write 𝜆[x] = g∈G 𝜆(g)𝜎g (x). w

w

Since a0 ∈ co ({𝜎g (x); g ∈ G}), there exists a net {𝜆i }i∈I ⊂ Λ such that 𝜆i [x0 ] → a0 . If LIM is a Banach limit with respect to I and E0 (x) = LIMi 𝜆i [x](x ∈ ℳ), then E0 ∈ ℰ , and hence ℰ is not empty. We define a preorder relation “≤” on ℰ by writing E1 ≤ E2 for E1 , E2 ∈ ℰ if and only if w

w

co ({𝜎g (E1 (x)); g ∈ G}) ⊇ co ({𝜎g (E2 (x)); g ∈ G})

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119

for every x ∈ ℳ. If {Ek }k∈K is an increasing net in ℰ and LIM is a Banach limit with respect to K, then the equation E(x) = LIMk Ek (x), (x ∈ ℳ), defines an upper bound E ∈ ℰ of {Ek }k∈K . Thus, ℰ is inductively ordered. Let E be a maximal element of ℰ . We show that E(ℳ) = ℳ 𝜎 . If this is not the case, there exists w x1 ∈ ℳ such that E(x1 ) ∉ ℳ 𝜎 . By assumption (1), there exists a1 ∈ co ({𝜎g (x1 ); g ∈ G}) ∩ ℳ 𝜎 . w

Let {𝜆j }j∈J ⊂ Λ be a net such that 𝜆j [x1 ] → a1 and consider a Banach limit LIM with respect to J. Then the equation E0 (x) = LIM𝜆j [E(x)](x ∈ ℳ) defines an element E0 ∈ ℰ , E0 ≥ E such that w w E0 ≠ E, as E(x1 ) ∈ co ({𝜎g (E(x1 )); g ∈ G}) but co ({𝜎g (E0 (x1 )); g ∈ G}) = {a1 } ∋ ∕ E(x1 ), and this contradicts the maximality of E. It follows that the maximal element E of ℰ is a projection of norm 1 of ℳ onto ℳ 𝜎 , and hence a conditional expectation, by the Tomiyama theorem (9.1). If there exists a 𝜎-invariant normal conditional expectation E ∶ ℳ → ℳ 𝜎 , then from (1) it follows that co ({𝜎g (x); g ∈ G}) ∩ ℳ 𝜎 = {E(x)} (x ∈ ℳ), w

(2)

so that E is uniquely determined and faithful. ∑ To see that E is faithful, choose a family {𝜓i }i of normal states on ℳ with i s(𝜓i ) = 1. Then each 𝜓i ◦ E is a 𝜎-invariant normal state on ℳ, hence s(𝜓i ◦ E) ∈ ℳ 𝜎 . It follows that s(𝜓i ◦ E) = s(𝜓i ), (i ∈ I), hence ℳ is 𝜎-finite and E is faithful (10.8). Note that if the W ∗ -algebra ℳ is 𝜎-finite, then using the Ryll-Nardzewski fixed point theorem ([L], A.3) one can show that (1), and hence (2) hold (Abdalla & Szücs, 1974; Kovács & Szücs, 1966). 10.14 Proposition. Let 𝒩 be a unital W ∗ -subalgebra of the W ∗ -algebra ℳ and 𝜏 a faithful normal trace on ℳ. If x ∈ 𝔑𝜏 , then w

co ({vxv∗ ; v ∈ U(𝒩 )}) ⊂ 𝔑𝜏 , w co ({vxv∗ ; v ∈ U(𝒩 )}) ∩ (𝒩 ′ ∩ ℳ) ≠ ∅.

(1) (2)

Proof. If a = vxv∗ , then 𝜏(a∗ a) = 𝜏(x∗ x) < +∞. Since the function ℳ ∋ a ↦ 𝜏(a∗ a)1∕2 is a lower w-semicontinuous seminorm (5.9), statement (1) follows. w The set 𝒦 = co ({vxv∗ ; v ∈ U(𝒩 )}) is norm bounded (by ‖x‖) and w-closed, thus 𝒦 is w-compact and the lower w-semicontinuous function a ↦ 𝜏(a∗ a) attains its greatest lower bound on 𝒦 , in some a0 ∈ 𝒦 ; let 𝜆 = 𝜏(a∗0 a0 ). Clearly, va0 v∗ ∈ 𝒦 and 𝜏((va0 v∗ )∗ (va0 v∗ )) = 𝜏(a∗0 a0 ) = 𝜆 for every v ∈ U(𝒩 ). To prove (2), it is therefore sufficient to show that the set 𝒦0 = {a ∈ 𝒦 ; 𝜏(a∗ a) = 𝜆} reduces to the singleton {a0 }. If a, b ∈ 𝒦0 , then (a + b)∕2 ∈ 𝒦 and hence 𝜆 ≤ 𝜏((a + b)∗ (a + b))∕4 = 𝜏(a∗ a)∕2 + 𝜏(b∗ b)∕2 − 𝜏((a − b)∗ (a − b))∕4 = 𝜆 − 𝜏((a − b)∗ (a − b))∕4, so that 𝜏((a − b)∗ (a − b)) = 0 and a = b, as 𝜏 is faithful. Statement (2) is a fixed point theorem. If 𝒩 is abelian, then (2) follows by the Markov–Kakutani theorem ([L], A.1) without any restriction on x ∈ ℳ. Also, if 𝒩 = ℳ, then 𝒩 ′ ∩ ℳ = 𝒵 (ℳ) and (2) holds for all x ∈ ℳ by a theorem of Dixmier ([L], C.4.4).

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Note that 𝒩 ′ ∩ ℳ = ℳ G , where G = {Ad(v); v ∈ U(𝒩 )} ⊂ Aut(ℳ). Thus, by Theorem 10.13 we obtain existence criteria for conditional expectations E ∶ ℳ → 𝒩 ′ ∩ ℳ. 10.15. We shall say that a von Neumann algebra 𝒫 ⊂ ℬ(ℋ ) has the property P of J. T. Schwartz if, for every x ∈ ℬ(ℋ ), w

co ({vxv∗ ; v ∈ U(𝒫 ′ )}) ∩ 𝒫 ≠ ∅.

(1)

In this case, there exists by Theorem 10.13 a conditional expectation E ∶ ℬ(ℋ ) → 𝒫 . In particular, by the Markov–Kakutani fixed point Theorem (see [L] Appendix), it follows that every von Neumann algebra 𝒫 ⊂ ℬ(ℋ ) with an abelian commutant 𝒫 ′ ⊂ ℬ(ℋ ) has property P. Consequently, for every von Neumann algebra 𝒫 ⊂ ℬ(ℋ ) with abelian commutant there exists a conditional expectation E ∶ ℬ(ℋ ) → 𝒫 .

(2)

If 𝒜 is a maximal abelian ∗-subalgebra of the W ∗ -algebra ℳ, then 𝒜 ′ ∩ ℳ = 𝒜 , and again using w the Markov–Kakutani theorem, we see that co ({vxv∗ ; v ∈ U(𝒜 )}) ∩ 𝒜 ≠ ∅(x ∈ ℳ). Thus, by Theorem 10.13, it follows that for every maximal abelian ∗ -subalgebra 𝒜 of the W ∗ -algebra ℳ there exists a conditional expectation E ∶ ℳ → 𝒜 .

(3)

Also, by the remark made in Section 10.13, we see that if there exists a normal conditional expectation E ∶ ℳ → 𝒜 , then E is uniquely determined and faithful.

(4)

10.16. Note that for every C ∗ -subalgebra 𝒜 of an abelian W ∗ -algebra ℳ there exists a *-homomorphism 𝜋 ∶ ℳ → 𝒜 such that 𝜋(a) = a for a ∈ 𝒜 (Gleason, 1958; Hasumi, 1958; Str̆atil̆a & Zsidó, 1977–1979; 9.27). Combined with 10.15.(3), this result shows that for every abelian C ∗ -subalgebra 𝒜 of an arbitrary W ∗ -algebra ℳ there exists a conditional expectation E ∶ ℳ → 𝒜. We record the following result concerning the existence of normal conditional expectations: Proposition. Let ℳ be a W ∗ -algebra and 𝒞 any unital W ∗ -subalgebra of the center 𝒵 (ℳ) of ℳ. For every x0 ∈ ℳ, x0 ≠ 0, there exists a normal conditional expectation E ∶ ℳ → 𝒞 such that E(x0 ) ≠ 0. Proof. Let 𝜑 be a normal state on ℳ with 𝜑(x0 ) ≠ 0. Let x ∈ ℳ + . For every z ∈ 𝒞 + we have 𝜑(xz) ≤ ‖x‖𝜑(z) so, by the Radon–Nikodym theorem ([L], 5.21), there exists a unique E(x) ∈ 𝒞 , 0 ≤ E(x) ≤ ‖x‖, such that 𝜑(xz) = 𝜑(E(x)z)(z ∈ 𝒞 ), and E(x) = E(x)s(𝜑|𝒞 ). It is easy to check that the mapping ℳ + ∋ x ↦ E(x) ∈ 𝒞 can be uniquely completed to a normal conditional expectation E ∶ ℳ → 𝒞 such that 𝜑(xz) = 𝜑(E(x)z) for all x ∈ ℳ, z ∈ 𝒞 . In particular, 𝜑(E(x0 )) = 𝜑(x0 ) ≠ 0.

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10.17. The uniqueness of the normal conditional expectation onto a maximal commutative *subalgebra follows also from the next result: Proposition. Let 𝒩 be a unital W ∗ -subalgebra of the W ∗ -algebra ℳ such that 𝒩 ′ ∩ ℳ ⊂ 𝒩 . If there exists a normal conditional expectation E ∶ ℳ → 𝒩 , then this is uniquely determined and faithful. Moreover, for u ∈ U(ℳ) we have u𝒩 u∗ = 𝒩 ⇔ E(uxu∗ ) = uE(x)u∗ for all x ∈ ℳ.

(1)

Proof. As for weights (2.1), we can define the support s(E) of E as the complement of the greatest projection p ∈ ℳ with E(p) = 0. For every v ∈ U(𝒩 ) we have E(vpv∗ ) = vE(p)v∗ = 0, so that vpv∗ = p and hence p ∈ 𝒩 ′ ∩ ℳ ⊂ 𝒩 . Consequently, p = E(p) = 0, s(E) = 1 and E is faithful. Let E1 ∶ ℳ → 𝒩 be another faithful normal conditional expectation, 𝜓 an n.s.f. weight on 𝒩 , 𝜑 = 𝜓 ◦ E, 𝜑1 = 𝜓 ◦ E1 n.s.f. weights on ℳ and ut = [D𝜑1 ∶ D𝜑]t (t ∈ ℝ). 𝜑 For any y ∈ 𝒩 we have (10.5.(1)) 𝜎t𝜑 (y) = 𝜎t𝜓 (y) = 𝜎t 1 (y) = ut 𝜎t𝜑 (y)u∗t , hence ut ∈ 𝒩 ′ ∩ 𝜑 ℳ ⊂ 𝒩 (t ∈ ℝ). Then, for x ∈ ℳ + and t ∈ ℝ we obtain 𝜑1 (𝜎t𝜑 (x)) = 𝜓(E1 (u∗t 𝜎t 1 (x)ut )) = 𝜑 𝜑 𝜓(u∗t E1 (𝜎t 1 (x))ut ) = 𝜓(E1 (𝜎t 1 (x))) = 𝜑1 (x), so that 𝜑1 commutes with 𝜑, that is, (4.10) there exists a positive self-adjoint operator A affiliated to ℳ 𝜑 such that 𝜑1 = 𝜑A . We have Ait = ut ∈ 𝒩 ′ ∩ ℳ = 𝒵 (𝒩 ) and 𝜓 = 𝜓A , hence A = 1 and 𝜑1 = 𝜑. We have proved that 𝜓 ◦ E = 𝜓 ◦ E1 for every n.s.f. weight 𝜓 on 𝒩 . It follows that E = E1 . If u ∈ U(ℳ) and u𝒩 u∗ = 𝒩 , then it is easy to check that the mapping E1 ∶ ℳ → 𝒩 defined by E1 (x) = u∗ E(uxu∗ )u(x ∈ ℳ), is a faithful normal conditional expectation. By the first part of the proof, we infer that E1 = E and statement (1) follows. For any conditional expectation E ∶ ℳ → 𝒩 ⊂ ℳ define the normalizer  (E) of E by  (E) = {u ∈ U(ℳ); E(uxu∗ ) = uE(x)u∗ for x ∈ ℳ}. If E is normal and 𝒩 ′ ∩ ℳ ⊂ ℳ, the above proposition shows that  (E) = {u ∈ U(ℳ); u𝒩 u∗ = 𝒩 }. ̄ 𝒜 10.18 Corollary. Let 𝒩 be a factor and 𝒜 an abelian W ∗ -algebra. If ℳ is a subfactor of 𝒩 ⊗ and ℳ ⊃ 𝒩 ⊗ 1, then ℳ = 𝒩 ⊗ 1. Proof. Consider 𝒩 realized as a von Neumann algebra 𝒩 ⊂ ℬ(ℋ ) and 𝒜 realized as a maximal ̄ 𝒜 ⊂ ℬ(ℋ ⊗ ̄ 𝒦 ) and, by abelian von Neumann algebra 𝒜 ⊂ ℬ(ℋ ). Then 𝒩 ⊗ 1 ⊂ ℳ ⊂ 𝒩 ⊗ Corollary 9.11, ̄ 𝒜 ) = (𝒩 ′ ∩ 𝒩 ) ⊗ ̄ (ℬ(𝒦 ) ∩ 𝒜 ) = 1 ⊗ 𝒜 , (𝒩 ⊗ 1)′ ∩ (𝒩 ⊗ since 𝒩 is a factor. Consequently, (𝒩 ⊗ 1)′ ∩ ℳ = (1 ⊗ 𝒜 ) ∩ ℳ = ℂ ⋅ 1 ⊂ 𝒩 ⊗ 1, ̄ 𝒜. as ℳ is also a factor and 1 ⊗ 𝒜 is the center of 𝒩 ⊗

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Thus, by Proposition 10.17, it follows that all the Fubini mappings (9.8) E𝜔𝒩 |ℳ ∶ ℳ → 𝒩 (𝜔 ∈ = 1), are the same faithful normal “conditional expectation” E ∶ ℳ → 𝒩 . Let x ∈ ℳ. For 𝜓 ∈ 𝒩∗ and 𝜔 ∈ 𝒜∗+ , 𝜔(1) = 1, we have

𝒜∗+ , 𝜔(1)

̄ 𝜔)(E(x) ⊗ ̄ 1) = 𝜓(E(x)) = 𝜓(E𝜔 (x)) = (𝜓 ⊗ ̄ 𝜔)(x), (𝜓 ⊗ 𝒩 ̄ 1 ∈ 𝒩 ⊗ 1. so that x = E(x) ⊗ 10.19. If 𝒜 ⊂ ℬ(ℋ ) is a maximal abelian von Neumann algebra such that every nonzero projection of 𝒜 dominates a minimal projection of 𝒜 , then there exists a system of matrix units (9.15) {eij }i,j∈I in ℬ(ℋ ) such that 𝒜 is the von Neumann algebra generated by {eii ; i ∈ I} and it is easy to see that the mapping ∑ ∑ ℬ(ℋ ) ∋ a ↦ eii aeii = aii eii ∈ 𝒜 i∈I

i∈I

is a faithful normal conditional expectation. On the other hand, let ℋ = ℒ 2 ([0, 1]) with respect to Lebesgue measure and let 𝒜 ⊂ ℬ(ℋ ) be von Neumann algebra ℒ ∞ ([0, 1]) of multiplication operators. Since 1 ∈ ℋ is a cyclic vector for 𝒜 , it follows that 𝒜 is maximal abelian in ℬ(ℋ ) (see [L], E.3.9, E.3.10; Str̆atil̆a & Zsidó, 1977, 2005, Cor. 3/8.13, Prop. 3/9.37), but it is obvious that 𝒜 has no minimal projections. Consider now a general maximal abelian von Neumann algebra 𝒜 ⊂ ℬ(ℋ ) without minimal projections and let E ∶ ℬ(ℋ ) → 𝒜 be a conditional expectation. Let 𝛾 be a character of the abelian C ∗ -algebra 𝒜 . Then 𝛾 is a singular form ([A.16]) on the ∗ W -algebra 𝒜 . Indeed, let 0 ≠ p ∈ 𝒜 be a projection. Since 𝒜 has no minimal projections, there exists a projection 0 ≠ q ≤ p in A with r = p − q ≠ 0. If 𝛾(p) ≠ 0 and 𝛾(q) ≠ 0, then 𝛾(p) = 𝛾(q) = 1, hence 𝛾(r) = 0. Thus, every nonzero projection of 𝒜 dominates a nonzero projection of 𝒜 annihilated by 𝛾, that is, 𝛾 is singular. Then 𝛾 ◦ E is an extension to ℬ(ℋ ) of the singular form 𝛾 on 𝒜 and hence (Tomiyama, 1971, Lemma 4.3; Str̆atil̆a & Zsidó, 1977, 2005, p. 8.5) 𝛾 ◦ E is a singular form on ℬ(ℋ ). It follows that 𝛾(E(e)) = 0 for every minimal projection e ∈ ℬ(ℋ ). Since 𝛾 was an arbitrary character of 𝒜 we have E(e) = 0 for every minimal projection e ∈ ℬ(ℋ ). Hence, E is not normal, for the normality of E would imply that E(x) = 0 for all x ∈ ℬ(ℋ ), a contradiction. In conclusion, there exists a maximal abelian von Neumann algebra 𝒜 ⊂ ℬ(ℋ ) without minimal projections; for any such an algebra 𝒜 there is no normal conditional expectation of ℬ(ℋ ) onto 𝒜 . 10.20. As an application of the above remarks we establish a characterization of finiteness for W ∗ -algebras: Proposition. A W ∗ -algebra ℳ is finite if and only if for every maximal abelian *-subalgebra 𝒜 of ℳ there exists a normal conditional expectation of ℳ onto 𝒜 . Proof. If ℳ is finite, the desired conclusion follows from Corollary 10.6. Assume that ℳ is properly ̄ ℬ(ℋ ) with ℋ a separable infinite dimensional Hilbert space. Let 𝒞 be a infinite. Then ℳ ≈ ℳ ⊗ maximal abelian *-subalgebra of ℳ and 𝒜 ⊂ ℬ(ℋ ) a maximal abelian von Neumann algebra. By ̄ 𝒜 is a maximal abelian *-subalgebra of ℳ ⊗ ̄ ℬ(ℋ ). Corollary 9.10, 𝒞 ⊗

Existence and Uniqueness of Conditional Expectations

123

̄ ℬ(ℋ ) onto 𝒞 ⊗ ̄ 𝒜 and let Suppose that there exists a normal conditional expectation E of ℳ ⊗ 𝜑 be a normal state on ℳ. Then the mapping ̄ 𝜄)(E(1 ⊗ ̄ x)) ∈ 𝒜 ℬ(ℋ ) ∋ x ↦ (𝜑 ⊗ is a normal conditional expectation of ℬ(ℋ ) onto 𝒜 , contradicting the conclusion of Section 10.19. 10.21. The existence of a normal conditional expectation E ∶ ℳ → 𝒩 transfers certain properties of ℳ to 𝒩 . Recall ([L], E.6.11; Str̆atil̆a & Zsidó, 1977, 2005, 9.40) that the supremum of all the minimal projections (or atoms) of a W ∗ -algebra ℳ is a central projection of ℳ and that the W ∗ -algebra ℳ is called atomic if this central projection is equal to 1. Proposition. Let E ∶ ℳ → 𝒩 be a normal conditional expectation of the W ∗ -algebra ℳ onto its unital W ∗ -subalgebra 𝒩 . Then ℳ is semifinite ⇒ 𝒩 is semifinite ℳ is discrete ⇒ 𝒩 is discrete ℳ is atomic ⇒ 𝒩 is atomic

(1) (2) (3)

Proof. (1) Assume that ℳ is semifinite and 𝒩 is of type III. Let 𝜏 be an n.s.f. trace on ℳ. Since E ≠ 0 is normal, there exists a projection e ∈ ℳ with 𝜏(e) < +∞ and E(e) ≠ 0. Since E(e) ∈ 𝒩 + , there exists a projection 0 ≠ f ∈ 𝒩 and 𝜆 ∈ (0, +∞) such that 𝜆f ≤ E(e). We shall obtain a contradiction by showing that f𝒩 f is finite with the help of Sakai’s topological s criterion of finiteness ([L], 7.23). So, let {yi }i ⊂ f𝒩 f be a net such that ‖yi ‖ ≤ 1 and yi → 0. Since s

s

e ∈ ℳ is a finite projection and yi e → 0, we have ey∗i → 0 ([L], 7.23). By the last remark of s

Section 9.2, we infer that E(e)y∗i → 0 and therefore s

y∗i = ( fE(e)f + (1 − f))−1 fE(e)f y∗i → 0. Hence ([L], 7.23) f𝒩 f is finite, a contradiction. (2) If ℳ is discrete, then by (1), 𝒩 is semifinite and hence ([L], E.4.14) there exists a family {𝒩i }i∈I of finite W ∗ -algebras and a family {ℋ }i∈I of Hilbert spaces such that 𝒩 = ⨁ ̄ i∈I 𝒩i ⊗ ℬ(ℋi ). Taking into account the existence of Fubini mappings (9.8), we see that in order to prove (2) we may assume 𝒩 is finite. Moreover, if 𝒩 is not discrete, we may assume that 𝒩 is of type II1 . Let 𝜏 be an n.s.f. trace on ℳ and 𝜇 a nonzero normal finite trace on 𝒩 . Then 𝜑 = 𝜇 ◦ E is a nonzero normal positive form on ℳ and, by Theorem 4.10, there exists a nonzero positive self-adjoint operator A affiliated to ℳ such that 𝜑 = 𝜏A . For y ∈ 𝒩 and x ∈ ℳ, we have 𝜑(xy) = 𝜇(E(x)y) = 𝜇(yE(x)) = 𝜑(yx), and hence (by 2.21.(2) and 4.7) y = 𝜎t𝜑 (y) = Ait yA−it (t ∈ ℝ). Thus, A is affiliated to 𝒩 ′ ∩ ℳ. There exists a nonzero spectral projection e of A and an element 0 ≤ a ∈ eℳe such that Ae is bounded and Aea = e. Then e ∈ 𝒩 ′ ∩ ℳ and 𝜏(e) = 𝜏(Aea) = 𝜑(a) < +∞, hence e is finite in ℳ. The W ∗ -algebra eℳe is finite and discrete so that every W ∗ -subalgebra of eℳe is discrete (see [L], 7.16, 7.17). However, the induced algebra 𝒩 e = e𝒩 e ⊂ eℳe is of type ([L], E.6.10), a contradiction.

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(3) If ℳ is atomic and 𝒩 contains no minimal projections, then the desired contradiction can be obtained arguing as in Section 10.19, and replacing the characters by pure states of 𝒩 . ̄ 𝒩 → ℳ ⊗ 1 (9.8), from the above proposition Since there exist conditional expectations ℳ ⊗ we get the conclusions listed in ([L], C.7.4) concerning the type of the tensor product. Recall in particular that ̄ 𝒩 is of type III; if ℳ is of type III, then ℳ ⊗ ̄ 𝒩 is continuous. if ℳ is continuous, then ℳ ⊗

(4) (5)

10.22. A W ∗ -algebra ℳ is called injective if, whenever ℳ is imbedded as a C ∗ -subalgebra of a C ∗ -algebra 𝒜 , there exists a conditional expectation E ∶ 𝒜 → ℳ. Of course, we could similarly define the notions of an injective C ∗ -algebra, but it appears that the injective C ∗ -algebras so defined are almost W ∗ -algebras (more precisely, they are monotone complete AW ∗ -algebras, see Str̆atil̆a & Zsidó, 1977, 2005, §9). It is easy to check that injectivity is stable under *-isomorphisms. Theorem. Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra. The following statements are equivalent: (i) (ii) (iii) (iv)

ℳ is injective; there exists a conditional expectation ℬ(ℋ ) → ℳ; there exists a conditional expectation ℬ(ℋ ) → ℳ ′ ; ℳ ′ is injective.

Proof. Taking into account the structure of *-isomorphisms between von Neumann algebras ([L], E.8.8), the existence of Fubini mappings (9.8) and using Proposition 9.4, it is easy to check that properties (ii) and (iii) are stable under *-isomorphisms. We shall therefore assume that ℳ ⊂ ℬ(ℋ ) is a standard von Neumann algebra ([L], 10.15). Let J ∶ ℋ → ℋ be a conjugation such that JℳJ = ℳ ′ . (ii) ⇔ (iii). If E ∶ ℬ(ℋ ) → ℳ is a conditional expectation, then the mapping E ′ ∶ ℬ(ℋ ) → ℳ ′ defined by E ′ (x) = JE(JxJ)J(x ∈ ℬ(ℋ )) is a conditional expectation. (i) ⇔ (ii). It is clear that (i) ⇒ (ii). Conversely, let 𝒜 be any C ∗ -algebra containing ℳ as a C ∗ -subalgebra. Then ℳ ∗∗ is a W ∗ -subalgebra of 𝒜 ∗∗ and there exist a central projection p ∈ ℳ ∗∗ and a *-isomorphism 𝜋 ∶ ℳ ∗∗ p → ℳ (A.16). Since ℳ has property (ii), which is stable to *-isomorphisms, there exists a conditional expectation E0 ∶ 𝒜 ∗∗ p → ℳ ∗∗ p. The mapping E ∶ 𝒜 → ℳ defined by E(x) = 𝜋(E0 (xp)) (x ∈ 𝒜 ⊂ 𝒜 ∗∗ ), is then a conditional expectation. (iii) ⇔ (iv) follows from (i) ⇔ (ii). There are many other important characterizations of injective W ∗ -algebras (see 10.31). In what follows we give some examples of injective W ∗ -algebras. 10.23 Proposition. Every discrete W ∗ -algebra is injective. Proof. By ([L], 6.5) we can realize ℳ as a von Neumann algebra ℳ ⊂ ℬ(ℋ ) with an abelian commutant, so that there exists a conditional expectation E ∶ ℬ(ℋ ) → ℳ (10.15.(2)) and ℳ is injective by Theorem 10.22.

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10.24 Proposition. Let ℳ, 𝒩 be W ∗ -algebras. Then ̄ 𝒩 is injective ⇔ ℳ and 𝒩 are injective; ℳ⊗ ̄ 𝒩 is injective ⇔ ℳ ′ and 𝒩 ′ are injective. ℳ⊗

(1) (2)

Proof. Use Theorem 10.22, Proposition 9.4, and the Fubini mappings. 10.25 Proposition. Let ℳ be a W ∗ -algebra and {ℳi }i∈I an upward directed family of ⋃ W ∗ -subalgebras of ℳ such that ℳ is generated by i∈I ℳi . If each ℳi is injective, then ℳ is also injective. Proof. Let ℳ ⊂ ℬ(ℋ ) be realized as a von Neumann algebra and choose a Banach limit LIM with respect to I (9.6). Since the ℳi′ are injective (10.22), there exist conditional expectations ⋂ ′ Ei′ ∶ ℬ(ℋ ) → ℳi′ (i ∈ I). The mapping E ′ ∶ ℬ(ℋ ) → ℳ ′ = i∈I ℳi defined by ′ ′ ′ E (x) = LIMi Ei (x)(x ∈ ℬ(ℋ )) is a conditional expectation, so that ℳ is injective (10.22). 10.26. A W ∗ -algebra ℳ is called approximately finite dimensional if it is generated by an upward directed family of finite dimensional W ∗ -subalgebras. By Propositions 10.23 and 10.25, it follows that every approximately finite dimensional W ∗ -algebra is injective. Conversely, Connes (1976a) has proved that every injective factor with separable predual is approximately finite dimensional. Simpler proofs were obtained by Haagerup (1985) and Popa (1986). A W ∗ -algebra ℳ is called semidiscrete if the identity mapping on ℳ can be approximated, with respect to the p-topology (2.23), by normal, finite rank, completely positive linear contractions (Effors & Lance, 1977). It is known that ℳ is semidiscrete if and only if it is injective (Choi & Effros, 1976a; Connes, 1976a; Wassermann, 1977). On the other hand, it is easy to check directly that every approximately finite dimensional von Neumann algebra ℳ ⊂ ℬ(ℋ ) has property P of J .T. Schwartz, so that ℳ is injective (10.15.(1)). If ℳ is a W ∗ -algebra with separable predual, then the assertions: ℳ is injective, ℳ is semidiscrete, ℳ ⊂ ℬ(ℋ ) has property P, are all equivalent to saying that ℳ is generated by an increasing sequence of finite dimensional *-subalgebras. Another property equivalent to injectivity (in the case of separable predual) is amenability, which we shall consider in the next sections. 10.27. We first prove another existence criterion for conditional expectations. Theorem. Let 𝒜 be a unital C ∗ -algebra and ℳ a countably decomposable finite W ∗ -algebra contained as a unital C ∗ -subalgebra in 𝒜 . The following statements are equivalent: (i) there exists a conditional expectation E ∶ 𝒜 → ℳ; (ii) there exists an ℳ-linear projection P ∶ 𝒜 → ℳ; (iii) there exists a bounded linear form 𝜑 on 𝒜 such that 𝜑|ℳ is positive and faithful and 𝜑(x⋅) = 𝜑(⋅x) for every x ∈ ℳ; (iv) there exists a positive linear form 𝜑 on 𝒜 such that 𝜑|ℳ is faithful and 𝜑(x⋅) = 𝜑(⋅x) for every x ∈ ℳ. Proof. (i) ⇒ (ii) is obvious. (ii) ⇒ (iii). If 𝜏 is any faithful finite trace on ℳ, then 𝜑 = 𝜏 ◦ P is as required in (iii).

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(iii) ⇒ (iv). Replacing the bounded linear form 𝜑 given in (iii) by 𝜑 + 𝜑∗ , we may assume that 𝜑 is self-adjoint. Then let 𝜑 = 𝜑+ − 𝜑− be the Jordan decomposition of 𝜑 ([L], 5.17). For every u ∈ U(ℳ), we have 𝜑 = 𝜑(u⋅u∗ ) = 𝜑+ (u⋅u∗ )−𝜑− (u⋅u∗ ), so that 𝜑+ = 𝜑+ (u⋅u∗ ), by the uniqueness of the Jordan decomposition. Since 𝜑+ |ℳ ≥ 𝜑|ℳ, it follows that 𝜑+ |ℳ is faithful. Hence 𝜑+ is the required form. (iv) ⇒ (i). Recall first that for every bounded linear form 𝜓 on a W ∗ -algebra 𝒩 there exist unique normal and singular linear forms 𝜓nor and 𝜓sing such that 𝜓 = 𝜓nor + 𝜓sing ; if 𝜓 is positive and faithful, then 𝜓nor is also positive and faithful (A.16; Str̆atil̆a & Zsidó, 1977, 2005, p. 8.4). We define a linear mapping F ∶ 𝒜 → ℳ∗ by F(a) = (𝜑(a⋅)|ℳ)nor (a ∈ 𝒜 ). Then 𝜏 = F(1) = (𝜑|ℳ)nor is a faithful normal finite trace on ℳ. Let a ∈ 𝒜 with a ≥ 0. For x ∈ ℳ, we have 𝜑(ax∗ x) = 𝜑(a|x|2 ) = 𝜑(|x|a|x|) ≥ 0, hence F(a) ≥ 0. Since 0 ≤ a ≤ ‖a‖ ⋅ 1, we obtain 0 ≤ F(a) ≤ ‖a‖𝜏. By the Radon–Nikodym type theorem ([L], 5.21), there exists a unique element E(a) ∈ ℳ + , ‖E(a)‖ ≤ ‖a‖, such that F(a) = 𝜏(E(a)1∕2 ⋅ E(a)1∕2 ) = 𝜏(E(a)⋅). We thus get a positive linear mapping E ∶ 𝒜 → ℳ, uniquely determined, such that F(a) = 𝜏(E(a)⋅)(a ∈ 𝒜 ). It is easy to check that E is a projection of 𝒜 onto ℳ and that ‖E‖ = ‖E(1)‖ = ‖1‖ = 1, as E is positive. By Theorem 9.1, it follows that E is a conditional expectation. 10.28 Corollary. A von Neumann algebra ℳ ⊂ ℬ(ℋ ) is injective if and only if there exists an ℳ-linear projection P ∶ ℬ(ℋ ) → ℳ. Proof. If ℳ is finite and countably decomposable, then the corollary follows obviously from Theorem 10.27. Assume that ℳ is semifinite and there exists an ℳ-linear projection P ∶ ℬ(ℋ ) → ℳ. There exists an increasing net {ei }i∈I of countably decomposable finite projections in ℳ with ei ↑ 1 (see [L], 4.20, 7.2). For each i ∈ I, P defines by restriction an ei ℳei -linear projection Pi ∶ ℬ(ei ℋ ) = ei ℬ(ℋ )ei → ei ℳei . Hence each ei ℳei is injective so that ℳ is also injective, by Proposition 10.25. Assume that ℳ is properly infinite and that there exists an ℳ-linear projection P ∶ ℬ(ℋ ) → ℳ. In this case, we need the Connes–Takesaki continuous decomposition theorem, which will be proved later (23.6). According to this theorem, we may assume that there exists a semifinite von Neumann subalgebra 𝒩 ⊂ ℳ together with an so-continuous unitary representation ℝ ∋ t ↦ u(t) ∈ ℳ such that u(t)𝒩 u(t)∗ = 𝒩 (t ∈ ℝ), and ℳ = (𝒩 ∪u(ℝ))′′ . Moreover, 𝜎 ∶ ℝ ∋ t ↦ Ad(u(t))|𝒩 ∈ Aut(𝒩 ) is a continuous action of ℝ on 𝒩 which defines a dual action 𝜎̂ of ℝ on ℳ such that 𝒩 = ℳ 𝜎̂ . Since the discrete abelian group ℝ is amenable, by Proposition 10.12 there exists a conditional expectation Q ∶ ℳ → ℳ 𝜎̂ = 𝒩 . Then Q ◦ P ∶ ℬ(ℋ ) → 𝒩 is an 𝒩 -linear projection. Since 𝒩 is semifinite, by the first part of the proof it follows that 𝒩 is injective and so there exists a conditional expectation (10.22) F′ ∶ ℬ(ℳ) → 𝒩 ′ . ′ Since ℳ = (𝒩 ∪ u(ℝ))′′ , we have ℳ ′ = 𝒩 ′ ∩ u(ℝ)′ , that is, ℳ ′ = (𝒩 ′ )𝜎 where 𝜎 ′ ∶ ℝ ∋ ′ ′ ′ t ↦ Ad(u(t))|𝒩 ∈ Aut(𝒩 ) is an action of the discrete abelian group ℝ on 𝒩 . We obtain again by ′ Proposition 10.12 a conditional expectation E ′ ∶ 𝒩 ′ → (𝒩 ′ )𝜎 = ℳ ′ . Then E ′ ◦ F′ ∶ ℬ(ℋ ) → ℳ ′ is a conditional expectation and so ℳ is injective (10.22). In the general case, there exists a central projection p ∈ ℳ such that ℳp is semifinite and ℳ(1−p) is of type III, hence properly infinite. If there exists an ℳ-linear projection P ∶ ℬ(ℋ ) → ℳ, then the von Neumann algebras ℳp ⊂ ℬ(pℋ ), ℳ(1 − p) ⊂ ℬ((1 − p)ℋ ) enjoy the same property so that they are injective as well as their direct sum ℳ (10.24.(2)). 10.29. Let ℳ be a W ∗ -algebra. A normal dual Banach ℳ-bimodule is a Banach ℳ-bimodule 𝒳 ∗) such that the Banach space 𝒳 is the dual of a certain Banach space 𝒳∗ and, for f ∈ 𝒳∗ , a0 ∈ ℳ, x0 ∈

Existence and Uniqueness of Conditional Expectations

127

𝒳 , the mappings ℳ ∋ a ↦ f (a ⋅ x0 ) ∈ ℂ,

ℳ ∋ a ↦ f (x0 ⋅ a) ∈ ℂ

are w-continuous and the mappings 𝒳 ∋ x ↦ f (a0 ⋅ x) ∈ ℂ,

𝒳 ∋ x ↦ f (x ⋅ a0 ) ∈ ℂ

are 𝜎(𝒳 , 𝒳∗ )-continuous. A derivation of ℳ into 𝒳 is a bounded linear mapping 𝛿 ∶ ℳ → 𝒳 such that 𝛿(ab) = 𝛿(a)⋅b+a⋅ 𝛿(b)(a, b ∈ ℳ). Every element x ∈ 𝒳 determines an inner derivation 𝛿x ∶ ℳ ∋ a ↦ x⋅a−a⋅x ∈ 𝒳 . The W ∗ -algebra ℳ is called amenable if every derivation of ℳ into some normal dual Banach ℳ-bimodule 𝒳 is inner. Proposition. If ℳ is an amenable unital W ∗ -subalgebra of the W ∗ -algebra ℬ, then there exists an (ℳ ′ ∩ ℬ)-linear projection P ∶ ℬ → ℳ ′ ∩ ℬ. Proof. Starting from the inclusion ℳ ⊂ ℬ, we shall construct a normal dual Banach ℳ-bimodule 𝒳 . ̂ ℬ∗ be the Banach space completion of the algebraic tensor product ℬ ⊗ ℬ∗ with Let 𝒴∗ = ℬ ⊗ respect to the greatest cross-norm (Schatten, 1950) and let 𝒴 = (𝒴∗ )∗ . It is known (Schatten, 1950) that 𝒴 can be identified isometrically with the Banach space ℬ(ℬ) of all bounded linear mappings ℬ → ℬ in such a way that S(b ⊗ 𝜓) = 𝜓(S(b)) (S ∈ ℬ(ℬ), b ∈ ℬ, 𝜓 ∈ ℬ∗ ). If we define the left and right actions of ℳ on 𝒴 = ℬ(ℬ) by (a ⋅ S)(b) = aS(b), (S ⋅ a)(b) = S(b)a

(a ∈ ℳ, S ∈ ℬ(ℬ), b ∈ ℬ),

then 𝒴 becomes a normal dual Banach ℳ-bimodule. Now let 𝒳 be the set of those S ∈ 𝒴 with the property that for every a′ ∈ ℳ ′ ∩ℬ, b ∈ ℬ, 𝜓 ∈ ℬ∗ we have S(a′ ⊗ 𝜓) = 0, that is, S|(ℳ ′ ∩ ℬ) = 0; S(a′ b ⊗ 𝜓 − b ⊗ 𝜓(a′ ⋅)) = 0, that is, S(a′ b) = a′ S(b); S(ba′ ⊗ 𝜓 − b ⊗ 𝜓(⋅a′ )) = 0, that is, S(ba′ ) = S(b)a′ .

(1) (2) (3)

Then 𝒳 is a 𝜎(𝒴 , 𝒴∗ )-closed sub-ℳ-bimodule of 𝒴 and 𝒳 becomes a normal dual Banach ℳ-bimodule with the induced structure. Let 𝜄ℬ ∈ ℬ(ℬ) = 𝒴 be the identity mapping on ℬ. It is easy to check that the inner derivation of ℳ into 𝒴 determined by 𝜄ℬ ∈ 𝒴 takes values in 𝒳 , thus defining a derivation 𝛿 ∶ ℳ → 𝒳 .

*That is, there are bounded bilinear mappings ℳ × 𝒳 ∋ (a, x) ↦ a ⋅ x ∈ 𝒳 and 𝒳 × ℳ ∋ (x, a) ↦ x ⋅ a ∈ 𝒳 called the left and right actions of ℳ on 𝒳 , such that, for every a, b ∈ ℳ, x ∈ 𝒳 , we have 1 ⋅ x = x ⋅ 1 = x, a ⋅ (b ⋅ x) = (ab) ⋅ x, (x ⋅ a) ⋅ b = x ⋅ (ab), a ⋅ (x ⋅ b) = (a ⋅ x) ⋅ b.

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Since ℳ is amenable, there exists an element S ∈ 𝒳 such that 𝛿 is the inner derivation determined by S. Let P = 𝜄ℬ − S ∈ 𝒴 = ℬ(ℬ). Then P ⋅ a − a ⋅ P = 0 for all a ∈ ℳ, hence P ∶ ℬ → ℳ ′ ∩ ℬ is a bounded linear mapping. Since S ∈ 𝒳 , it follows from (1) that P is a projection of ℬ onto ℳ ′ ∩ ℬ while from (2) and (3) we deduce that P is (ℳ ′ ∩ ℬ)-linear. 10.30. Taking ℬ = ℬ(ℋ ) in Proposition 10.29 and using Corollary 10.28 and Theorem 10.22, we obtain the following Corollary. Every amenable W ∗ -algebra is injective. Conversely, if ℳ is an injective W ∗ -algebra with separable predual, then ℳ is generated by an increasing sequence of finite dimensional *-subalgebras (as mentioned in 10.26) and by Johnson, Kadison and Ringrose (1972) it follows that ℳ is amenable. The equivalence of the notions of amenability, semidiscreteness and injectivity in the case of separable predual, to the structural property of approximate finite dimensionality is one of the deepest and most important results in the theory of operator algebras to date. 10.31. Notes. Theorem 10.1 and Corollary 10.7 are due to Takesaki (1972a). The particular case of Theorem 10.1 contained in Corollary 10.6 is due to Dixmier (1949) and Umegaki (1956). Corollary 10.8 is due to Kovács and Szücs (1966); actually, they proved a better result, namely w that co ({𝜎g (x); g ∈ G}) ∩ ℳ 𝜎 = {E(x)} whenever ℳ is 𝜎-finite (this result can be obtained using the Ryll-Nardzewski fixed point theorem, see Abdalla & Szücs, 1974). The strictly semifinite weights were introduced by Combes (1971a, 1971b) who proved Corollary 10.9 and Proposition 10.11. Theorem 10.13 is due to Schwartz (1963); in the same article, Schwartz introduced property P (see also Choda & Echigo, 1963, 1964). Proposition 10.14 is due to Dixmier (1957, 1969), Proposition 10.16 appears in Halpern (1969) and Str̆atil̆a, Voiculescu, and Zsidó (1976, 1977), Proposition 10.17 is due to Connes [1973a] and Corollary 10.18 is due to Connes and Takesaki (1977). Proposition 10.20 is the answer to a problem posed by Kadison, given by Takesaki (1972a) and Tomiyama (1972). Proposition 10.21 is due to Sakai (1957) and Tomiyama (1957–1959, 1971). For the injectivity property for W ∗ -algebras or properties equivalent to injectivity, as well as for the results contained in Sections 10.22–10.26, we refer to Arveson (1969); Choda and Echigo (1963, 1964); Choi and Effros (1977a, 1976a, 1977b); Connes (1973a); Effors (1972); Effors and Lance (1977); Hakeda and Tomiyama (1967); Lance (1973); Schwartz (1963); and Wassermann (1977). Theorem 10.27 and Corollary 10.28 are due to Connes (1977a, 1978a) and Bunce and Paschke (1978). For the definition of an amenable W ∗ -algebra (10.29), we refer to Johnson (1972); Johnson, Kadison, and Ringrose (1972). Corollary 10.30 is due to Connes (1978a) and its proof, based on Proposition 10.29, is due to Bunce and Paschke (1978). For our exposition, we have used Bunce and Paschke (1978); Choi and Effros (1977a); Combes (1971a, 1971b); Diximer (1969); Takesaki (1972a) and Tomiyama (1971). Further results related to the Kovács–Szücs theorem are contained in Guichardet (1974); Størmer (1970, 1971a, 1972a, 1973, 1972b, 1974b). The property of a W ∗ -algebra of being approximately finite dimensional was considered for the first time by Murray and von Neumann (1936, 1937, 1943, p. IV) in the case of finite factors. They obtained several characterizations of this property and showed that all approximately finite dimensional factors of type II1 are *-isomorphic ∗) (see Dixmier, 1957, 1969). The approximately

∗) We

consider only W ∗ -algebras with separable preduals.

Operator-Valued Weights

129

finite dimensional factor of type II1 , usually called the hyperfinite II1 factor, will be denoted by ℛ. For properly infinite W ∗ -algebras, several characterizations of this property are given in Elliott and Woods (1976) and interesting stability results appeared in Golodec (1971); Connes et al. (1973). Connes (1976a) proved the fundamental result that all injective factors of type II1 are *-isomorphic to ℛ. In particular, it follows that any subfactor of ℛ is either finite dimensional, or *-isomorphic to ̄ ℱ∞ , where ℛ. Also, it follows that all injective factors of type II∞ , are *-isomorphic to ℛ∞ = ℛ ⊗ ℱ∞ is the only factor of type I∞ (see Kadison, 1971). Recently, Connes, Feldman, and Weiss (1981) proved that for any two maximal abelian *-subalgebras 𝒜1 , 𝒜2 of ℛ whose normalizers generate ℛ there exists 𝜎 ∈ Aut(ℛ) such that 𝜎(𝒜1 ) = 𝒜2 . Among the most important examples of injective W ∗ -algebras, we have the crossed products of abelian W ∗ -algebras by continuous actions of amenable locally compact groups, the von Neumann algebras generated by so-continuous unitary representations of connected locally compact groups and the von Neumann algebras generated by any *-representation of a nuclear C ∗ -algebra (Choi & Effros, 1976a, 1977b; Connes, 1976a; Connes, 1977b). Note that if G is a discrete ICC-group (22.6), then the factor ℒ (G) is injective, that is, ℒ (G) ≈ ℛ, if and only if G is amenable (see Connes, 1976a; Sakai, 1971; Schwartz, 1963; Tomiyama, 1971). Since the free group Fk on k generators, (k ≥ 2), is not amenable, it follows that ℒ (Fk ) is not *-isomorphic to ℛ(≈ ℒ (S(∞), see 22.6).

11 Operator-Valued Weights In this section, we introduce the extended positive part of a W ∗ -algebra and operator-valued weights, together with their main properties. 11.1. Let ℳ be a W ∗ -algebra. The extended positive part ℳ̄ + of ℳ is the set of all functions 𝔪 ∶ ℳ∗+ → [0, ∞] such that 𝔪(𝜑 + 𝜓) = 𝔪(𝜑) + 𝔪(𝜓) (𝜑, 𝜓 ∈ ℳ∗+ );

(1)

𝔪(𝜆𝜑) = 𝜆𝔪(𝜑) (𝜑 ∈ ℳ∗+ , 𝜆 ≥ 0);

(2)

𝔪 is lower semicontinuous.

(3)

If 𝔪, 𝔫 ∈ ℳ̄ + , x ∈ ℳ and 𝜆 > 0, then one defines the elements 𝔪 + 𝔫, 𝜆𝔪 and x∗ 𝔪x of ℳ̄ + by (𝔪 + 𝔫)(𝜑) = 𝔪(𝜑) + 𝔫(𝜑) (𝜑 ∈ ℳ∗+ ); (𝜆𝔪)(𝜑) = 𝜆𝔪(𝜑)

(𝜑 ∈ ℳ∗+ );

(x 𝔪x)(𝜑) = 𝔪(𝜑(x ⋅ x)) ∗

∗

(𝜑 ∈

ℳ∗+ ).

(4) (5) (6)

If z is a positive clement in the center 𝒵 (ℳ) of ℳ, then we write z𝔪 or 𝔪z instead of z1∕2 𝔪z1∕2 .

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For 𝔪1 , 𝔪2 ∈ ℳ̄ + , write 𝔪1 ≤ 𝔪2 if 𝔪1 (𝜑) ≤ 𝔪2 (𝜑) for all 𝜑 ∈ ℳ∗+ . If {𝔪i }i∈I ⊂ ℳ̄ + is an increasing net, then one defines an element 𝔪 = supi∈I 𝔪i ∈ ℳ̄ + by 𝔪(𝜑) = supi∈I 𝔪i (𝜑)(𝜑 ∈ ℳ∗+ ); in this case we also write 𝔪i ↑ 𝔪. In particular, for an arbitrary family {𝔪i }i∈I ⊂ ℳ̄ + , an ∑ ∑ element 𝔪 = 𝔪i ∈ ℳ̄ + is defined by 𝔪(𝜑) = 𝔪i (𝜑)(𝜑 ∈ ℳ + ). i∈I

i∈I

∗

11.2. Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra, e ∈ ℳ a projection and A a positive self-adjoint operator in the Hilbert space eℋ , affiliated to the von Neumann algebra eℳe ⊂ ℬ(eℋ ). Let en = 𝜒[n−1,n) (A) ≤ e

(n = 1, 2, …).

For each n, Aen is a bounded positive operator in eℳe and s(Aen ) ≤ en . Define an element 𝔪A ∈ ℳ̄ + by 𝔪A (𝜑) =

∞ ∑

𝜑(Aen ) + ∞ ⋅ 𝜑(1 − e) (𝜑 ∈ ℳ∗+ ).

n=1

If e = 1 and A = a ∈ ℳ + , then 𝔪a (𝜑) = 𝜑(a), (𝜑 ∈ ℳ∗+ ). The mapping ℳ + ∋ a ↦ 𝔪a ∈ ℳ̄ + is injective and preserves the operations introduced in Section 11.1, so that we can identify 𝔪a with a and write ℳ + ⊂ ℳ̄ + . If e = 1 − f and A = 0, then the element 𝔪A will be denoted by ∞ ⋅ f; for 𝜑 ∈ ℳ∗+ we have (∞ ⋅ f )(𝜑) = ∞ ⋅ 𝜑( f ), that is, this value is 0 if 𝜑( f ) = 0 and +∞ if 𝜑( f ) > 0. Thus, in the general case considered above we have 𝔪A =

∞ ∑

Aen + ∞ ⋅ (1 − e),

(1)

n=1

so that 𝔪A is a sum of elements in ℳ + with mutually orthogonal supports and a symbol ∞ on a projection which is orthogonal on all these supports. If 𝜉 ∈ ℋ , then 𝜔𝜉 ∈ ℳ∗+ and, by ([L], 9.9), we obtain { 𝔪A (𝜔𝜉 ) =

‖A1∕2 𝜉‖2 if 𝜉 ∈ D(A1∕2 ) ⊂ eℋ +∞, in the contrary case.

(2)

In particular, it follows that the element 𝔪A determines the projection e and the operator A uniquely: if f ∈ ℳ is another projection and B is a positive self-adjoint operator in fℋ affiliated to fℳf, then 𝔪A = 𝔪B ⇔ e = f and A = B.

(3)

s(A) = e ⇔ 𝔪A (𝜑) > 0 for all 0 ≠ 𝜑 ∈ ℳ∗+ ;

(4)

It is easy to check that

in this case we say that 𝔪 = 𝔪A ∈ ℳ̄ + is faithful or nonsingular. Also, e = 1 ⇔ {𝜑 ∈ ℳ∗+ ; 𝔪A (𝜑) < +∞} is dense in ℳ∗+ ;

(5)

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131

in this case we say that 𝔪 = 𝔪A ∈ ℳ̄ + is semifinite. Moreover, e = 1 and A is bounded ⇔ 𝔪A (𝜑) < +∞ for all 𝜑 ∈ ℳ∗+ .

(6)

If {e𝜆 }𝜆∈(0,+∞) is the spectral scale of A (see [L), E.9.10), then e𝜆 ↑ e for 𝜆 ↑ +∞ and ∞

𝔪A (𝜑) =

∫0

𝜆 d𝜑(e𝜆 ) + ∞ ⋅ 𝜑(1 − e)

(𝜑 ∈ ℳ∗+ ).

(7)

Indeed, this identity is easily verified for 𝜑 = 𝜔𝜉 (𝜉 ∈ ℋ ), using (2) and ([L], E.9.10), and then ∑ ∑ extended for any 𝜑 ∈ ℳ∗+ , since 𝜑 = n 𝜔𝜉n with n ‖𝜉n ‖2 < +∞ (see [L], E.7.9, 8.17). 11.3. The example considered in the preceding section is general, as is shown by the following Proposition. Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra and 𝔪 ∈ ℳ̄ + . There exists a projection e ∈ ℳ and a positive self-adjoint operator A in eℋ affiliated to eℳe, uniquely determined, such that 𝔪 = 𝔪A . Proof. The uniqueness follows from 11.2.(3). Since 𝔪 ∈ ℳ̄ + , the function q ∶ ℋ ∋ 𝜉 ↦ q(𝜉) = 𝔪(𝜔𝜉 ) ∈ [0, +∞] is lower semicontinuous and has the following properties: q(𝜆𝜉) = |𝜆|2 q(𝜉) (𝜉 ∈ ℋ , 𝜆 ∈ ℂ); q(𝜉 + 𝜂) + q(𝜉 − 𝜂) = 2q(𝜉) + 2q(𝜂) (𝜉, 𝜂 ∈ ℋ ); q(u′ 𝜉) = q(𝜉) (𝜉 ∈ ℋ , u′ ∈ ℳ ′ unitary). It follows that the set D(q) = {𝜉 ∈ ℋ ; q(𝜉) < +∞} is a linear subspace of ℋ , stable under ℳ ′ , so ̄ = eℋ . that there exists a unique projection e ∈ ℳ such that D(q) By (A.10) there exists a unique positive self-adjoint operator A in eℋ such that ‖A1∕2 𝜉‖2 = q(𝜉) = 𝔪(𝜔𝜉 ) for all 𝜉 ∈ D(A1∕2 ) = D(q). Using 11.2.(2), it follows that 𝔪(𝜔𝜉 ) = 𝔪A (𝜔𝜉 )(𝜉 ∈ ℋ ), and since any 𝜑 ∈ ℳ∗+ is a sum of vector forms we conclude that 𝔪 = 𝔪A . By this proposition and the results in Section 11.2, it follows that for each element m in the extended positive part of a W ∗ -algebra ℳ there exists a sequence {an } of positive elements in ℳ ∑ with mutually orthogonal supports and a projection e ∈ ℳ, e ≥ n s(an ), such that 𝔪=

∑

an + ∞ ⋅ (1 − e).

(1)

n

Also, by 11.2.(7), each element 𝔪 ∈ ℳ̄ + has a unique “spectral decomposition”: ∞

𝔪=

∫0

𝜆 de𝜆 + ∞ ⋅ (1 − e).

(2)

Finally, by 11.2.(6), for 𝔪 ∈ ℳ̄ + we have 𝔪(𝜑) < +∞ for all 𝜑 ∈ ℳ∗+ ⇔ m = a ∈ ℳ̄ + .

(3)

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Conditional Expectations and Operator-Valued Weights

Let 𝒵 be an abelian W ∗ -algebra. In ([L], 7.20) we defined an extension 𝒵̃ of 𝒵 . It is easy to check that 𝒵̃ + consists of those elements 𝔪 of the extended positive part 𝒵̄ + such that the set {𝜑 ∈ 𝒵∗+ ; m(𝜑) < +∞} is dense in 𝒵∗+ . Also, each element 𝔪 ∈ 𝒵̄ + is of the form 𝔪 = z + ∞ ⋅ p with z ∈ 𝒵̃ + , p ∈ 𝒵 , pz = 0.

(4) +

Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra. We say that an element 𝔪 ∈ ℬ(ℋ ) is affiliated to ℳ if u∗ 𝔪u = 𝔪 for all unitaries u ∈ ℳ ′ . Let e ∈ ℬ(ℋ ) and A be the projection and the positive self-adjoint operator in eℋ , respectively, such that 𝔪 = 𝔪A . Then 𝔪 = 𝔪A is affiliated to ℳ ⇔ e ∈ ℳ and A is affiliated to eℳe.

(5) +

Thus, there is a natural bijective correspondence between ℳ̄ + and the elements in ℬ(ℋ ) affiliated to ℳ. 11.4 Proposition. Let ℳ be a W ∗ -algebra. Every normal weight 𝜑 on ℳ has a unique extension to ℳ̄ + , still denoted by 𝜑, such that 𝜑(𝔪 + 𝔫) = 𝜑(𝔪) + 𝜑(𝔫) (𝔪, 𝔫 ∈ ℳ̄ + ) 𝜑(𝜆𝔪) = 𝜆𝜑(𝔪) (𝔪 ∈ ℳ̄ + , 𝜆 > 0) 𝔪i , 𝔪 ∈ ℳ̄ + , 𝔪i ↑ 𝔪 ⇒ 𝜑(𝔪i ) ↑ 𝜑(𝔪).

(1) (2) (3)

If 𝜑 ∈ ℳ∗+ , then 𝜑(𝔪) = 𝔪(𝜑)

(𝔪 ∈ ℳ̄ + ).

(4)

Proof. If 𝜑 ∈ ℳ∗+ , define the extension of 𝜑 to ℳ̄ + by (4); (1), (2), (3) then follow. ∑ By Corollary 5.8, every normal weight 𝜑 on ℳ can be written 𝜑 = i 𝜑i with 𝜑i ∈ ℳ∗+ , so that ∑ we can define 𝜑(𝔪) = i 𝜑i (𝔪)(𝔪 ∈ ℳ̄ + ), and (1), (2), (3) again follow easily. For any m ∈ ℳ̄ + , there exists an increasing sequence {an } ⊂ ℳ̄ + such that an ↑ 𝔪 (see 11.3.(1)); condition (3) implies that 𝜑(an ) ↑ 𝜑(𝔪). This proves the uniqueness of the extension of 𝜑. 11.5. Let ℳ be a W ∗ -algebra and 𝒩 ⊂ ℳ a W ∗ -subalgebra. An operator-valued weight on ℳ with values in 𝒩 , or an 𝒩 -valued weight on ℳ, is a mapping E ∶ ℳ + → 𝒩̄ + with the properties: E(a + b) = E(a) + E(b) (a, b ∈ ℳ + ) E(𝜆a) = 𝜆E(a) (a ∈ ℳ + , 𝜆 > 0) E(y∗ ay) = y∗ E(a)y (a ∈ ℳ + , y ∈ 𝒩 ).

(1) (2) (3) +

Any weight 𝜑 on ℳ can be regarded as an operator-valued weight 𝜑 ∶ ℳ + → (ℂ ⋅ 1ℳ ) . Every conditional expectation (9.1) E ∶ ℳ → 𝒩 is an operator-valued weight such that E(1ℳ ) = 1𝒩 . Conversely, every operator-valued weight E ∶ ℳ + → 𝒩̄ + such that E(1ℳ ) = 1𝒩 can be extended by linearity to a conditional expectation E ∶ ℳ → 𝒩 .

Operator-Valued Weights

133

For an operator-valued weight E ∶ ℳ + → 𝒩̄ + consider the sets 𝔉E = {a ∈ ℳ + ; E(a) ∈ 𝒩 + }, 𝔑E = {x ∈ ℳ; E(x∗ x) ∈ 𝒩 + }, 𝔐E = 𝔑∗E 𝔑E . 𝔉E is a face of ℳ + , 𝔑E is a left ideal of ℳ and 𝔐E is a facial subalgebra of ℳ; 𝔐E = lin 𝔉E , 𝔐E ∩ ℳ + = 𝔉E . Consequently, E can be extended uniquely up to a linear mapping E ∶ 𝔐E → 𝒩 . Also, it is easy to see that 𝔑E and 𝔐E are 𝒩 -bimodules, that is, 𝒩 ⋅ 𝔑E ⋅ 𝒩 ⊂ 𝔑E

𝒩 ⋅ 𝔐E ⋅ 𝒩 ⊂ 𝔐E

(4)

(x ∈ 𝔐E , y1 , y2 ∈ 𝒩 ).

(5)

and E(y1 xy2 ) = y1 E(x)y2 In particular, it follows from (5) that E(𝔐E ) is a two sided ideal in 𝒩 .

(6)

The face 𝔉E contains an increasing right approximate unit {ui }i∈I for the left ideal 𝔑E , which is ̄ w = ℳe and 𝔐 ̄ w = eℳe. If 𝒩 contains s∗ -convergent to the unique projection e ∈ ℳ such that 𝔑 E E ′ the unit element of ℳ, then from (4) it follows that e ∈ 𝒩 ∩ ℳ. We say that E is semifinite if e = 1. On the other hand, we say that E is faithful if for x ∈ ℳ, E(x∗ x) = 0 ⇒ x = 0. Note that w

if E is semifinite and faithful, then E(𝔐E ) = 𝒩 .

(7) w

Indeed, by (6) there exists a unique central projection q in 𝒩 such that E(𝔐E ) = 𝒩 q. Assume that q ≠ 1. Since E is semifinite, there exists x ∈ 𝔑E with x(1−q) ≠ 0, hence 0 ≠ (1−q)x∗ x(1−q) ∈ 𝔐E . Since E is faithful, it follows that 0 ≠ E((1 − q)x∗ x(1 − q)) = (1 − q)E(x∗ x)(1 − q) ∈ 𝒩 (1 − q), contradicting E(𝔐E ) ⊂ 𝒩 q. We say that E is normal if E(ai ) ↑ E(a) whenever ai , a ∈ ℳ + and ai ↑ a. In this case, there exists a unique projection s(E) ∈ 𝒩 ′ ∩ℳ, called the support of E, such that for x ∈ ℳ we have E(x∗ x) = 0 if and only if xs(E) = 0. Moreover, E is faithful if and only if s(E) = 1. If E is normal, then, by Proposition 11.4, for every 𝜓 ∈ 𝒩∗+ we get a normal weight 𝜓 ◦ E on ℳ. Again by Proposition 11.4, for each 𝔪 ∈ ℳ̄ + we can define an element E(𝔪) ∈ 𝒩̄ + such that E(𝔪)(𝜓) = (𝜓 ◦ E)(𝔪), (𝜓 ∈ 𝒩∗+ ). We thus get a unique extension of E to a normal, additive and positively homogeneous mapping E ∶ ℳ̄ + → 𝒩̄ + . Note that if E is normal semifinite and faithful, then E(ℳ̄ + ) = 𝒩̄ + .

(8)

Indeed, consider first b ∈ 𝒩 + ∩ E(𝔐E ). Then b is the image of a hermitian element of 𝔐E , and there exist a, a′ ∈ 𝔐E ∩ ℳ + such that b = E(a) − E(a′ ) ≤ E(a). By ((L], E.2.6) there exists y ∈ 𝒩 such that b = y∗ E(a)y = E(y∗ ay), hence b ∈ E(ℳ + ). Consider now b ∈ 𝒩 + and let {vi } be an increasing approximate unit for the w-dense two sided ideal E(𝔐E ) of 𝒩 (see (6), (7)). Since b ≥ b1∕2 vi b1∕2 ↑ b, it follows that there exists a family

134

Conditional Expectations and Operator-Valued Weights

∑ {bj } ⊂ E(𝔐E ) ∩ 𝒩 + such that b = j bj . For each j there exists aj ∈ ℳ + with E(aj ) = bj . Hence ∑ a = j aj ∈ ℳ̄ + and E(a) = b. Finally, any element b ∈ 𝒩̄ + is a sum of elements in 𝒩 + (11.3.(1)), and the same argument as above shows that b ∈ E(ℳ̄ + ). The set of all normal semifinite faithful 𝒩 -valued weights on ℳ will be denoted by P (ℳ, 𝒩 ). An 𝒩 -valued weight E on ℳ such that E(x∗ x) = E(xx∗ ) for all x ∈ ℳ will be called an operatorvalued trace. In this case, 𝔑E and 𝔐E are two sided ideals of ℳ, E(ab) = E(ba) for a, b ∈ 𝔑E and E(xa) = E(ax) for a ∈ 𝔐E , x ∈ ℳ. 11.6 Proposition. Let ℳ be a W ∗ -algebra, Q ⊂ 𝒩 W ∗ -subalgebras of ℳ and E ∈ P (ℳ, 𝒩 ), F ∈ P (𝒩 , Q). Then F ◦ E ∈ P (ℳ, Q). Proof. Clearly, F ◦ E is a normal and faithful operator-valued weight. Let x ∈ 𝔑E and let {vj }j∈J ⊂ 𝔑F ⊂ 𝒩 be such that 0 ≤ vj ↑ 1. Then (F ◦ E)(v∗j x∗ xvj ) = F(v∗j E(x∗ x)vj ) ≤ ‖E(x∗ x)‖F(v∗j vj ) < +∞, w

so that xvj ∈ 𝔑F ◦ E and xvj → x. Thus, 𝔑F ◦ E is w-dense in 𝔑E which is w-dense in ℳ. Consequently, F ◦ E is also semifinite. It is easy to check that (F ◦ E)(x) = F(E(x)) for all x ∈ lin(𝔐E ∩ 𝔐F ◦ E ∩ ℳ + ).

(1)

In particular, if E ∈ P (ℳ, 𝒩 ) and 𝜓 is an n.s.f. weight on 𝒩 , then 𝜓 ◦ E is an n.s.f. weight on ℳ and (𝜓 ◦ E)(x) = 𝜓(E(x)) for all x ∈ lin(𝔐E ∩ 𝔐𝜓 ◦ E ∩ ℳ + ).

(2)

11.7. Conversely, Proposition. Let E ∶ ℳ + → 𝒩̄ + be a normal operator-valued weight. If there exists a faithful normal weight 𝜓 on 𝒩 such that the normal weight 𝜑 = 𝜓 ◦ E on ℳ is faithful and semifinite, then E is faithful and semifinite. Proof. Indeed, for x ∈ ℳ we have E(x∗ x) = 0 ⇒ 𝜑(x∗ x) = 0 ⇒ x = 0; hence E is faithful. Let a ∈ 𝔐𝜑 ∩ℳ + and let {bn } be a family of elements in 𝒩 + with mutually orthogonal supports fn = s𝒩 (bn ) ∑ ∑ and f ≥ n fn such that E(a) = n bn + ∞ ⋅ (1 − f ) (see 11.3. (1)). Then 𝜓(E(a)) = 𝜑(a) < +∞, w ∑∞ hence f = 1. Putting ek = 1 − n=k+1 fn ∈ 𝒩 , we have ek ↑ 1, hence ek xek → x; also ek xek ∈ 𝔑E , as ∑k E(ek xek ) = ek E(x)ek = n=1 bn ∈ 𝒩 + . Consequently 𝔑E is w-dense in 𝔑𝜑 , which is w-dense in ℳ, that is, E is semifinite. 11.8. Let E ∶ ℳ + → 𝒩̄ + be an n.s.f. operator-valued weight, ℱ2 = Mat2 (ℂ) a factor of type I2 and 𝜄 ∶ ℱ2 → ℱ2 the identity mapping on ℱ2 . Consider the tensor product linear mapping ̃ = E ⊗ 𝜄 ∶ 𝔐E ⊗ ℱ2 → 𝒩 ⊗ ℱ2 . E Using the polarization relation it is easy to check that 𝔐E ⊗ ℱ2 = (𝔑E ⊗ ℱ2 )∗ (𝔑E ⊗ ℱ2 ) is a *-subalgebra of ℳ ⊗ ℱ2 , linearly spanned by its positive elements. Also, 𝔐E ⊗ ℱ2 is an

Operator-Valued Weights

135

(𝒩 ⊗ ℱ2 )-bimodule and we have ̃ 1 xy2 ) = y1 E(x)y ̃ E(y 2

(x ∈ 𝔐E ⊗ ℱ2 , y1 , y2 ∈ 𝒩 ⊗ ℱ2 ).

(1)

Note that ̃ ≥ 0. x ∈ 𝔐E ⊗ ℱ2 , x ≥ 0 ⇒ E(x)

(2)

̄ tr ∈ (𝒩 ⊗ ℱ2 )+ and 𝜃̃ = (𝜓 ◦ E) ⊗ ̄ tr be a normal semifinite Indeed, let 𝜓 ∈ 𝒩∗+ , 𝜃 = 𝜓 ⊗ ∗ ̃ = 𝜃(E(x)) ̃ weight on ℳ ⊗ ℱ2 . We have 𝔐E ⊗ ℱ ⊂ 𝔐𝜃̃ and 𝜃(x) for all x ∈ 𝔐E ⊗ ℱ2 . For ∗̃ ̃ ∗ ̃ ̃ ∗ every y ∈ 𝒩 ⊗ ℱ2 , we have (𝜋𝜃 (E(x))y𝜉 𝜃 |y𝜉𝜃 )𝜃 = 𝜃(y E(x)y) = 𝜃(E(y xy)) = 𝜃(y xy) ≥ 0, hence + ̃ ≥ 0. ̃ ≥ 0. Since 𝜓 ∈ 𝒩∗ was arbitrary, it follows that E(x) 𝜋𝜃 (E(x)) Lemma. Let E ∶ ℳ + → 𝒩̄ + be an n.s.f. operator-valued weight, let 𝜑 and 𝜓 be n.s.f. weights on 𝒩 and let 𝜑 ̃ = 𝜑 ◦ E, 𝜓 ̃ = 𝜓 ◦ E. Then x ∈ (𝔑𝜑̃ ∩ 𝔑E )∗ (𝔑𝜓̃ ∩ 𝔑E ) ⇒ E(x) ∈ 𝔑∗𝜑 𝔑𝜓 . Proof. We may assume that x = a∗ b with a ∈ 𝔑𝜑̃ ∩ 𝔑E , b ∈ 𝔑𝜓̃ ∩ 𝔑E , Then 0 ≤ ̃ x = ( )∗ ( ) ( ∗ ) a b a b a a x = ∈ 𝔐E ⊗ ℱ2 and hence, by (2), 0 0 0 0 x∗ b∗ b ̃ x) = E(̃

(

E(a∗ a) E(x∗ )

E(x) E(b∗ b)

) ≥ 0.

̃ x)) = 𝜑(E(a∗ a)) + 𝜓(E(b∗ b)) = Let 𝜃 = 𝜃(𝜑, 𝜓) be the balanced weight on 𝒩 ⊗ ℱ2 . Then 𝜃(E(̃ ∗ ∗ ̃ 𝜑 ̃ (a a) + 𝜓 ̃ (b b) < +∞, hence E(̃ x) ∈ 𝔐𝜃 and consequently (see 3.1.(3)) E(x) ∈ 𝔑∗𝜑 𝔑𝜓 . 11.9 Theorem (U. Haagerup). Let E ∶ ℳ + → 𝒩̄ + be an n.s.f. operator-valued weight and 𝜑, 𝜓 n.s.f. weights on 𝒩 . Then 𝜎t𝜓 ◦ E,𝜑 ◦ E (y) = 𝜎t𝜓,𝜑 (y)

(y ∈ 𝒩 , t ∈ ℝ),

(1)

that is, 𝜎t𝜑 ◦ E (y) = 𝜎t𝜑 (y) (y ∈ 𝒩 , t ∈ ℝ) [D(𝜓 ◦ E) ∶ D(𝜑 ◦ E)]t = [D𝜓 ∶ D𝜑]t

(t ∈ ℝ).

(2) (3)

In particular, it follows from this statement that 𝒩 is 𝜎 𝜑 ◦ E -invariant and [D(𝜓 ◦ E) ∶ D(𝜑 ◦ E)]t ∈ 𝒩 (t ∈ ℝ). The proof of the theorem will be given in Section 11.12. In Sections 11.10 and 11.11, we consider properties of s∗ -continuous one-parameter groups of isometries on W ∗ -algebras which are necessary for the proof. Note that if E(1ℳ ) = 1𝒩 , then the theorem follows from Corollary 10.5.

136

Conditional Expectations and Operator-Valued Weights

11.10. Let {𝜎t }t∈ℝ be an s∗ -continuous one-parameter group of isometries on the W ∗ -algebra ℳ. Define the analytic extensions 𝜎𝛼 (𝛼 ∈ ℂ) as in Sections 3.12 and 2.14. We say that an element a ∈ ℳ is of exponential type with respect to {𝜎t } if a ∈ D(𝜎𝛼 ) for all 𝛼 ∈ ℂ and there exist two constants 𝛾, 𝛿 > 0 such that ‖𝜎𝛼 (a)‖ ≤ 𝛾exp(−𝛿‖Im 𝛼|)(𝛼 ∈ ℂ). The set 𝛼 . Note that for a ∈ ℳ we have of all elements in ℳ of exponential type will be denoted by ℳexp 𝜎 a ∈ ℳexp ⇔ a ∈ D((𝜎−i )n ) for all n ∈ ℤ and there exist y, 𝛿 > 0

such that ‖(𝜎−i ) (a)‖ ≤ 𝛾 exp(𝛿|n|) n

(1)

(n ∈ ℤ).

Indeed, the implication (⇒) is obvious. Note that (𝜎−i )n = 𝜎−ni (n ∈ ℤ), and ‖𝜎t+is (a)‖ = ‖𝜎is (a)‖ = supt∈ℝ ‖𝜎t+is (a)‖(s, t ∈ ℝ), since all the 𝜎t (t ∈ ℝ) are isometries. Thus the converse implication (⇐) follows using the “three lines theorem” (Dunford & Schwartz, 1958, 1963, VI.10.3). 𝜎 is w-dense in ℳ. Lemma. ℳexp

Proof. For each 𝜆 > 0 consider the entire analytic function F𝜆 (𝛼) = (1 − cos 𝜆𝛼)∕𝜋𝜆𝛼 2 (𝛼 ∈ ℂ). It is well known that +∞

∫−∞

F𝜆 (t) dt = 1

(2)

and that for any bounded continuous function f ∶ ℝ → ℝ we have +∞

lim

𝜆→+∞ ∫−∞

F𝜆 (t)f (t) dt = f (0).

(3)

We shall show that +∞

∫−∞

|F𝜆 (t + is)| dt ≤ exp(𝜆|s|);

s ∈ ℝ.

To this end we consider the functions f𝜆 defined by { f𝜆 (r) =

𝜆−1∕2 if |r| ≤ 𝜆∕2 0 if |r| > 𝜆∕2

and their Fourier–Laplace transforms ̂f𝜆 (𝛼) = (2𝜋)−1∕2

+∞

∫−∞

f𝜆 (r)e−i𝛼r dr = (2∕𝜋𝜆x)−1∕2 sin(𝜆r∕2).

Since ̂f𝜆 (𝛼)2 = F𝜆 (𝛼) and ̂f𝜆 (t + is) = (2𝜋)−1∕2

+∞

∫−∞

f𝜆 (r)esr e−itr dr,

(4)

Operator-Valued Weights

137

we get using the Plancherel formula +∞

∫−∞

−∞

|F𝜆 (t + is)| dt =

∫+∞

|̂f𝜆 (t + is)|2 dt = +𝜆∕2

= 𝜆−1

∫−𝜆∕2

+∞

∫−∞

| f𝜆 (r)esr |2 dr

e2sr dr ≤ e𝜆|s| .

+∞

𝜎 and Consider now a ∈ ℳ and a𝜆 = ∫−∞ F𝜆 (r)𝜎r (a) dr (𝜆 > 0). We shall show that a𝜆 ∈ ℳexp w

a𝜆 → a when 𝜆 → +∞. +∞ Indeed, it is easy to check that a𝜆 is an entire analytic element and 𝜎𝛼 (a𝜆 ) = ∫−∞ F𝜆 (r − 𝛼)𝜎r (a) dr 𝜎 (𝛼 ∈ ℂ). Moreover, using (4) we obtain ‖𝜎𝛼 (a𝜆 )‖ ≤ ‖a‖ exp(𝜆|Im 𝛼|), hence a𝜆 ∈ ℳexp . On the other hand, using (2) and (3), we get, for every 𝜑 ∈ ℳ∗ , +∞

𝜑(a𝜆 − a) =

∫−∞

F𝜆 (r)𝜑(𝜎r (a) − a) dr → 0

w

hence a𝜆 → a when 𝜆 → +∞. 11.11 Proposition. Let ℳ be a W ∗ -algebra, 𝒩 ⊂ ℳ a W ∗ -subalgebra and {𝜎t }t∈ℝ , {𝜏t }t∈ℝ s∗ -continuous one-parameter groups of isometries ℳ, 𝒩 respectively. If b ∈ D(𝜏−i ) ⊂ 𝒩 ⇒ b ∈ D(𝜎−i ) and 𝜎−i (b) = 𝜏−i (b) then 𝜎t (y) = 𝜏t (y) (y ∈ 𝒩 , t ∈ ℝ). 𝜏 ⊂ ℳ 𝜎 . For b ∈ 𝒩 𝜏 we have 𝜎 Proof. By assumption and by 11.10. (1) it follows that 𝒩exp −ni (b) = exp exp 𝜏ni (b)(n ∈ ℕ), and there exist 𝛾, 𝛿 > 0 such that ‖𝜎n (b) − 𝜏n (b)‖ ≤ 𝛾 exp(𝛿|Im 𝛼|)(𝛼 ∈ ℂ). By a theorem of F. Carlson (Polya & Szegö, 1964, part 3, chapter 6, problem 328) it follows that 𝜎𝛼 (b) = 𝜏 , t ∈ ℝ. Since the isometrics 𝜎 and 𝜏 are 𝜏𝛼 (b)(𝛼 ∈ ℂ). Thus, 𝜎t (b) = 𝜏t (b) for all b ∈ 𝒩exp t t 𝜏 is w-dense in 𝒩 , we conclude that automatically w-continuous and since, by Lemma 11.10, 𝒩exp 𝜎t (y) = 𝜏t (y) for y ∈ 𝒩 , t ∈ ℝ.

11.12. Proof of Theorem 11.9. Let 𝜑 ̃ = 𝜑 ◦ E, 𝜓 ̃ = 𝜓 ◦ E be n.s.f. weights on ℳ. By Proposition 11.11, we have to show that, if 𝜓,𝜑 𝜓,𝜑 a ∈ D(𝜎−i ) ⊂ 𝒩 and b = 𝜎−i (a) ∈ 𝒩 ,

(1)

𝜓 ̃ ,̃ 𝜑 𝜓 ̃ ,̃ 𝜑 a ∈ D(𝜎−i ) and b = 𝜎−i (a).

(2)

then

By Theorem 3.15, this amounts to showing that a𝔑∗𝜑̃ ⊂ 𝔑∗𝜓̃ , 𝔑𝜓̃ b ⊂ 𝔑𝜑̃ 𝜓 ̃ (ax) = 𝜑 ̃ (xb) for all x ∈

(3) 𝔑∗𝜑̃ 𝔑𝜓̃ .

(4)

138

Conditional Expectations and Operator-Valued Weights

𝜓,𝜑 𝜑,𝜓 From (1) it follows that a ∈ D(𝜎−i∕2 ) and b∗ ∈ D(𝜎−i∕2 ) (see 3.12. (4)). Using Proposition 3.12 we ∗ 2 infer that there exists 𝜆 > 0 such that 𝜓(aya ) ≤ 𝜆 𝜑(y) and 𝜑(b∗ yb) ≤ 𝜆2 𝜓(y) for all y ∈ 𝒩 + . These inequalities remain valid for any y ∈ 𝒩̄ + and hence we get 𝜓 ̃ (axa∗ ) ≤ 𝜆2 𝜑 ̃ (x) and 𝜑 ̃ (b∗ xb) ≤ 𝜆2 𝜓 ̃ (x) + for all x ∈ ℳ . By Proposition 3.12 again, we obtain the required inclusions (3) as well as the inequalities

‖(xa∗ )𝜓̃ ‖𝜓̃ ≤ 𝜆‖x𝜑̃ ‖𝜑̃

(x ∈ 𝔑𝜑̃ )

(5)

‖(xb)𝜑̃ ‖𝜑̃ ≤ 𝜆‖x𝜓̃ ‖𝜓̃

(x ∈ 𝔑𝜓̃ ).

(6)

We now prove (4) in the particular case x = y∗ z with y ∈ 𝔑𝜑̃ ∩ 𝔑E , z ∈ 𝔑𝜓̃ ∩ 𝔑E . Using the first inclusion in (3) and the fact that 𝔑E is a right 𝒩 -module, we obtain ax = (ya∗ )∗ z ∈ (𝔑𝜓̃ ∩ 𝔑E )∗ (𝔑𝜓̃ ∩ 𝔑E ) ⊂ lin(𝔐E ∩ 𝔐𝜓̃ ∩ ℳ + ) and, similarly, xb ∈ lin(𝔐E ∩ 𝔐𝜑̃ ∩ ℳ + ). We have x ∈ 𝔐E and, by Lemma 11.7, E(x) ∈ 𝔑∗𝜑 𝔑𝜓 . Using 11.6.(2), assumption (1) and Theorem 3.15, we get 𝜓 ̃ (ax) = 𝜓(E(ax)) = 𝜓(aE(x)) = 𝜑(E(x)b) = 𝜑(E(xb)) = 𝜑 ̃ (xb). Finally, we consider the general case: x = y∗ z with y ∈ 𝔑𝜑̃ , z ∈ 𝔑𝜓̃ . Since 𝜑(E(y∗ y)) < +∞ and 𝜑 is faithful, from 11.3 and 11.2.(7) we infer that E(y∗ y) has a spectral ∞ decomposition of the form E(y∗ y) = ∫0 t det . For any t ≥ 0, we have yet ∈ 𝔑𝜑̃ ∩ 𝔑E

(7)

and for t → +∞ we get ‖(yet − y)𝜑̃ ‖2𝜑̃ = 𝜑(E((yet − y)∗ (yet − y)) = = 𝜑((1 − et )E(y y)(1 − et )) = 𝜑 ∗

(

)

∞

∫t

s des

→0

(8)

so that, using (5), we further deduce (yet a∗ )𝜓̃ → (ya∗ )𝜓̃ in ℋ𝜓̃ .

(9)

∞

Similarly, we have a spectral decomposition E(z∗ z) = ∫0 t dft such that, for any t ≥ 0 zft ∈ 𝔑𝜓̃ ∩ 𝔑E

(10)

(zft )𝜓̃ → z𝜓̃ in ℋ𝜓̃ ,

(11)

(zft b)𝜑̃ → (zb)𝜑̃ in ℋ𝜑̃ .

(12)

and, for t → +∞,

Operator-Valued Weights

139

Using (7)–(12) and the particular case of statement (4) proved above, we conclude that 𝜓 ̃ (ax) = (z𝜓̃ |(ya∗ )𝜓̃ )𝜓̃ = lim ((zft )𝜓̃ |(yet a∗ )𝜓̃ )𝜓̃ t→∞

̃ ((yet )∗ (zft )b) = lim 𝜓 ̃ (a(yet )∗ (zft )) = lim 𝜑 t→∞

t→∞

= lim ((zft b)𝜑̃ |(yet )𝜑̃ )𝜑̃ = ((zb)𝜑̃ |y𝜑̃ )𝜑̃ = 𝜑 ̃ (xb), t→∞

□

and this completes the proof of Theorem 11.9.

11.13 Corollary. Let E1 , E2 ∶ ℳ + → 𝒩̄ + be n.s.f. operator-valued weights. If there exists an n.s.f. weight 𝜑 on 𝒩 such that 𝜑 ◦ E1 = 𝜑 ◦ E2 , then E1 = E2 . Proof. Let 𝜓 be another n.s.f. weight on 𝒩 . By Theorem 11.9, we have [D(𝜓 ◦ E1 ) ∶ D(𝜑 ◦ E1 )]t = [D𝜓 ∶ D𝜑]t = [D(𝜓 ◦ E2 ) ∶ D(𝜑 ◦ E2 )]t (t ∈ ℝ), so that, using Corollary 3.6, it follows from the assumption 𝜑 ◦ E1 = 𝜑 ◦ E2 that 𝜓 ◦ E1 = 𝜓 ◦ E2 . Now let 𝜔 ∈ 𝒩∗+ and e = s𝒩 (𝜔). There exists an n.s.f. weight 𝜓 on 𝒩 such that 𝜔(y) = 𝜓(eye) for all y ∈ 𝒩 + and this identity extends to all y ∈ 𝒩̄ + . Thus, for x ∈ ℳ + we have E1 (x)(𝜔) = 𝜔(E1 (x)) = 𝜓(E1 (exe)) = 𝜓(E2 (exe)) = 𝜔(E2 (x)) = E2 (x)(𝜔). Since 𝜔 ∈ 𝒩∗+ was arbitrary, it follows that E1 (x) = E2 (x)(x ∈ ℳ + ). Note that if 𝜑1 , 𝜑2 are n.s.f. weights on 𝒩 , then [D(𝜑1 ◦ E1 ) ∶ D(𝜑2 ◦ E2 )]t = [D𝜑1 ∶ D𝜑2 ]t (t ∈ ℝ) ⇒ E1 = E2 .

(1)

Indeed, using 11.9.(3), we have by assumption [D(𝜑1 ◦ E1 ) ∶ D(𝜑2 ◦ E1 )]t = [D(𝜑1 ◦ E2 ) ∶ D(𝜑2 ◦ E2 )]t (t ∈ ℝ), so that 𝜑1 ◦ E1 = 𝜑1 ◦ E2 by Corollary 3.6 and hence E1 = E2 . Arguing as in the second part of the proof of the above corollary we also see that E1 ≤ E2 ⇔ 𝜑 ◦ E1 ≤ 𝜑 ◦ E2 for all 𝜑 ∈ Wnsf (𝒩 ).

(2)

11.14. Every normal positive linear mapping between W ∗ -algebras Φ ∶ ℳ → 𝒩 can be extended to a normal, additive and positively homogeneous mapping Φ ∶ ℳ̄ + → 𝒩̄ + by Φ(𝔪)(𝜓) = 𝔪(𝜓 ◦ Φ)

(𝜓 ∈ 𝒩∗+ , 𝔪 ∈ ℳ̄ + ).

In particular, every *-automorphism 𝜎 of ℳ can be extended to ℳ̄ + . Note that (ℳ̄ 𝜎 )+ = {𝔪 ∈ ℳ̄ + ; 𝜎(𝔪) = 𝔪}.

(1)

Indeed, it is clear that (ℳ̄ 𝜎 )+ ⊂ {𝔪 ∈ ℳ̄ + ; 𝜎(𝔪) = 𝔪}. Conversely, let 𝔪 ∈ ℳ̄ + , with 𝜎(𝔪) = 𝔪, have spectral decomposition ∞

𝔪=

∫0

∞

𝜆 de𝜆 + ∞ ⋅ (1 − e); then 𝜎(𝔪) =

∫0

𝜆 d𝜎(e𝜆 ) + ∞ ⋅ 𝜎(1 − e).

Since 𝜎(𝔪) = 𝔪, it follows that e𝜆 , e ∈ ℳ 𝜎 , hence 𝔪 ∈ (ℳ̄ 𝜎 )+ .

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Conditional Expectations and Operator-Valued Weights

Corollary. Let E ∶ ℳ + → 𝒩̄ + be an n.s.f. operator-valued weight and 𝜑 an n.s.f. weight on 𝒩 . Then E(𝜎t𝜑 ◦ E (x)) = 𝜎t𝜑 (E(x))

(x ∈ ℳ + , t ∈ ℝ).

(2)

In particular, 𝜎t𝜑 ◦ E (𝔐E ) = 𝔐E

(t ∈ ℝ).

(3)

Proof. For each t ∈ ℝ consider the mapping 𝜑 Et = 𝜎−t ◦ E ◦ 𝜎t𝜑 ◦ E ∶ ℳ + → 𝒩̄ + .

It is easy to check that Et is an n.s.f. operator-valued weight; condition 11.5.(3) follows using Theorem 11.9. Since 𝜑 ◦ Et = 𝜑 ◦ E ◦ 𝜎t𝜑 ◦ E = 𝜑 ◦ E, using Corollary 11.13 we infer that Et = E, the desired conclusion. 11.15. Let E ∶ ℳ + → 𝒩 + be an n.s.f. operator-valued weight and 𝜑, 𝜓 n.s.f. weights on 𝒩 . By 11.9.(2) we have 𝜎t𝜑 ◦ E (𝒩 ) = 𝜎t𝜑 (𝒩 ) = 𝒩 , so that 𝜎t𝜑 ◦ E (𝒩 ′ ∩ ℳ) = 𝒩 ′ ∩ ℳ

(t ∈ ℝ).

(1)

Since [D(𝜓 ◦ E) ∶ D(𝜑 ◦ E)]t = [D𝜓 ∶ D𝜑]t ∈ 𝒩 , for any z ∈ 𝒩 ′ ∩ ℳ we have 𝜎t𝜑 ◦ E (z) ∈ 𝒩 ′ ∩ ℳ, hence 𝜎t𝜓 ◦ E (z) = [D(𝜓 ◦ E) ∶ D(𝜑 ◦ E)]t 𝜎t𝜑 ◦ E (z)[D(𝜓 ◦ E) ∶ D(𝜑 ◦ E)]∗t = 𝜎t𝜑 ◦ E (z). Thus, 𝜎t𝜑 ◦ E |𝒩 ′ ∩ ℳ does not depend on the n.s.f. weight 𝜑 on 𝒩 , so that we can define 𝜎tE = 𝜎t𝜑 ◦ E |𝒩 ′ ∩ ℳ

(t ∈ ℝ),

(2)

by choosing an arbitrary n.s.f. weight 𝜑 on 𝒩 . The one-parameter group {𝜎tE }t∈ℝ ⊂ Aut(𝒩 ′ ∩ ℳ) is called the modular automorphism group associated with the operator-valued weight E ∶ ℳ + → 𝒩̄ + . Consider now two n.s.f. operator-valued weights E, F ∶ ℳ + → 𝒩̄ + and two n.s.f. weights 𝜑, 𝜓 on 𝒩 . We define an n.s.f. operator-valued weight Θ = Θ(E, F), Θ ∶ Mat2 (ℳ)+ → (𝒩 ⊗ 1)+ by (compare with 3.1) (( Θ

x11 x21

x12 x22

))

( =

E(x11 ) + F(x22 ) 0

0 E(x11 + F(x22 )

) ([xij ] ∈ Mat2 (ℳ)+ )

Consider also the weight 𝜑 ̃ = 𝜑 ⊗ 1 on 𝒩 ⊗ 1 and the balanced weight 𝜃 = 𝜃(𝜑 ◦ E, 𝜑 ◦ F) on Mat2 (ℳ). It is easy to check that 𝜑 ̃ ◦ Θ = 𝜃 and (𝒩 ⊗ 1)′ ∩ Mat2 (ℳ) = Mat2 (𝒩 ′ ∩ ℳ). By

Operator-Valued Weights

141

applying (1) in this case we get (

0 [D(𝜑 ◦ F) ∶ D(𝜑 ◦ E)]t

0 0

)

= 𝜎t𝜑̃ ◦ Θ

((

0 0 1 0

)) ∈ Mat2 (𝒩 ′ ∩ ℳ),

hence [D(𝜑 ◦ F) ∶ D(𝜑 ◦ E)]t ∈ 𝒩 ′ ∩ ℳ

(t ∈ ℝ).

(3)

Using (3), 3.4, 3.5, and 11.9.(3), we obtain [D(𝜓 ◦ F) ∶ D(𝜓 ◦ E)]t = [D(𝜓 ◦ F) ∶ D(𝜑 ◦ F)]t [D(𝜑 ◦ F) ∶ D(𝜑 ◦ E)]t [D(𝜑 ◦ E) ∶ D(𝜓 ◦ E)]t = [D𝜓 ∶ D𝜑]t [D(𝜑 ◦ F) ∶ D(𝜑 ◦ E)]t [D𝜑 ∶ D𝜓]t = [D(𝜑 ◦ F) ∶ D(𝜑 ◦ E)]t . Thus, [D(𝜑 ◦ F) ∶ D(𝜑 ◦ E)]t does not depend on the n.s.f. weight 𝜑 on 𝒩 , so that we can define [DF ∶ DE]t = [D(𝜑 ◦ F) ∶ D(𝜑 ◦ E)]t ∈ 𝒩 ′ ∩ ℳ

(t ∈ ℝ),

(4)

with an arbitrary n.s.f. weight 𝜑 on 𝒩 . The function t ↦ [DF ∶ DE]t is called the 𝜎 E -cocycle associated with F. Indeed, using the results in Sections 3.1–3.5, it is easy to check that 𝜎tF = Ad([DF ∶ DE]t ) ◦ 𝜎tE , [DF ∶ DE]s+t = [DF [DE ∶ DF]t = [DF ∶

∶ DE]s 𝜎sE ([DF DE]∗t .

(5) ∶ DE]t ),

(6) (7)

Also, the “chain rule” holds. Using 11.13.(2) and Corollary 3.13, we obtain ⎡ there exists a w-continuous function ⎢ f ∶ {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1} → ℳ, analytic in F≤E⇔⎢ {𝛼 ∈ ℂ; 0 < Re 𝛼 < 1} such that ⎢ ⎣ f (it) = [DF ∶ DE]t , (t ∈ ℝ), and ‖ f (1∕2)‖ ≤ 1.

(8)

Note that all the above results apply in particular for normal faithful conditional expectations. 11.16. Notes. Operator-valued weights appeared in the works of Connes and Takesaki (1977) and Landstad (1979), but their systematic study is due to Haagerup (1979). The modular automorphism group associated with an operator-valued weight was first considered in the case of conditional expectations by Combes and Delaroche (1975). For our exposition, we have used Haagerup (1979).

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Conditional Expectations and Operator-Valued Weights

12 Existence and Uniqueness of Operator-Valued Weights In this section, we give several criteria for the existence and uniqueness of operator-valued weights, together with some applications. 12.1. The main existence criterion for n.s.f. operator-valued weights is the following Theorem (U. Haagerup). Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ and 𝜓 an n.s.f. weight on the W ∗ -algebra 𝒩 of ℳ. If 𝜎t𝜑 (y) = 𝜎t𝜓 (y) (y ∈ 𝒩 , t ∈ ℝ), then there exists a unique n.s.f. operator-valued weight E ∶ ℳ + → 𝒩 + such that 𝜑 = 𝜓 ◦ E. The proof of this theorem is contained in Sections 12.2–12.5. After some preparation (12.2), we prove the theorem in the particular case when 𝜑 and 𝜓 are n.s.f. traces in Section 12.3. In Section 12.4, we present a brief review of some results concerning the crossed product of a W ∗ -algebra by the modular automorphism group associated with an n.s.f. weight; the exposition here is essentially self-contained although these results will be considered later in greater generality. Finally, using these results, we complete the proof of the theorem in Section 12.5. 12.2. Let 𝜏 be an n.s.f. trace on the semifinite W ∗ -algebra ℳ. Each semifinite element A ∈ ℳ̄ + (see 11.2.(5)) defines a normal semifinite weight 𝜏A on ℳ as in Sections 4.1, 4.4. By Theorem 4.10, the mapping A ↦ 𝜏A establishes a bijective correspondence between the semifinite elements A ∈ ℳ̄ + and the normal semifinite weights on ℳ. Note that for a, b ∈ ℳ + we have 𝜏a (b) = 𝜏b (a).

(1)

Indeed, 𝜏a (b) = 𝜏(a1∕2 ba1∕2 ) = 𝜏((b1∕2 a1∕2 )∗ (b1∕2 a1∕2 )) = 𝜏((b1∕2 a1∕2 )(b1∕2 a1∕2 )∗ ) = 𝜏(b1∕2 ab1∕2 ) = 𝜏b (a). Consider now A ∈ ℳ̄ + . Using 4.3.(1), 4.3.(3), and Proposition 11.4, we see that the equation 𝜏A (b) = 𝜏b (A)

(b ∈ ℳ + )

defines a normal weight 𝜏A on ℳ, which extends to ℳ̄ + . By the definition of 𝜏A and by Proposition 11.4, it follows that 𝜏A+B = 𝜏A + 𝜏B , 𝜏𝜆A = 𝜆𝜏A

(A, B ∈ ℳ̄ + , 𝜆 > 0).

(2)

Also, it is easy to check that ℳ + ∋ an ↑ A ∈ ℳ̄ + , ℳ + ∋ bm ↑ B ∈ ℳ̄ + ⇒ 𝜏an (bm ) ↑ 𝜏A (B).

(3)

Using (1) and (3), we infer that 𝜏A (B) = 𝜏B (A)

(A, B ∈ ℳ̄ + ),

(4)

Existence and Uniqueness of Operator-Valued Weights

143

and from (4) and 11.4. (3) it follows that Ai ↑ A, Bj ↑ B in ℳ̄ + ⇒ 𝜏Ai (Bj ) ↑ 𝜏A (B);

(5)

indeed, 𝜏A (B) = supj 𝜏A (Bj ) = supj 𝜏Bj (A) = supj supi 𝜏Bj (Ai ) = supij 𝜏Ai (Bj ). Note also that 𝜏xAx∗ (B) = 𝜏A (x∗ Bx)

(A, B ∈ ℳ̄ + , x ∈ ℳ);

(6)

indeed, if A = a ∈ ℳ + , B = b ∈ ℳ + , then 𝜏xax∗ (b) = 𝜏b (xax∗ ) = 𝜏(b1∕2 xax∗ b1∕2 ) = 𝜏(a1∕2 x∗ bxa1∕2 ) = 𝜏a (x∗ bx), and the general case is obtained using (3). It is easy to check that the normal weight 𝜏A is semifinite if and only if the element A ∈ ℳ̄ + is semifinite and that in this case the above definition of 𝜏A agrees with the definition given in Section 4.4. Also, the normal weight 𝜏A is faithful if and only if the clement A ∈ ℳ̄ + is faithful (11.2.(4)). Since any element of ℳ∗+ is of the form 𝜏A with A ∈ ℳ̄ + (by Theorem 4.10) and since every normal weight on ℳ is a sum of elements in ℳ∗+ (by Corollary 5.8), it follows that every normal weight on ℳ is of the form 𝜏A with A ∈ ℳ̄ + . If A, B ∈ ℳ̄ + , then again by Theorem 4.10 we get: 𝜏A = 𝜏B ⇔ 𝜏A (X) = 𝜏B (X) for all X ∈ ℳ̄ + ⇔ 𝜏X (A) = 𝜏X (B) for all X ∈ ℳ̄ + ⇔ 𝜑(A) = 𝜑(B) for all 𝜑 ∈ ℳ∗+ ⇔ A = B. Consequently, the mapping A ↦ 𝜏A establishes a bijective correspondence between the sets ̄ ℳ + and Wn (ℳ).

(7)

Similarly, for A, Ai , B ∈ ℳ̄ + , we obtain A ≤ B ⇔ 𝜏A ≤ 𝜏B , Ai ↑ A ⇔ 𝜏Ai ↑ 𝜏A .

(8) (9)

12.3. We now prove Theorem 12.1 in the case when 𝜑 and 𝜓 are n.s.f. traces on ℳ and 𝒩 , respectively. For each a ∈ ℳ + , 𝜑a |𝒩 + is a normal weight on 𝒩 and hence, by 12.2.(7), there exists a unique element E(a) ∈ 𝒩̄ + such that 𝜑a |𝒩 + = 𝜓E(a) .

(1)

We thus define a mapping E ∶ ℳ + → 𝒩 + . Using the results of Section 12.2, it is easy to check that E is a normal operator-valued weight and that 𝜑 = 𝜓 ◦ E. Moreover, using Proposition 11.7 it follows that E is faithful and semifinite. If F ∶ ℳ + → 𝒩̄ + is another operator-valued weight such that 𝜑 = 𝜓 ◦ F, then for any a ∈ ℳ + , b ∈ 𝒩 + we get 𝜑a (b) = 𝜑b (a) = 𝜑(b1∕2 ab1∕2 ) = 𝜓(b1∕2 F(a)b1∕2 ) = 𝜓b (F(a)) = 𝜓F(a) (b), hence F(a) = E(a). Actually, the uniqueness of E follows also from Corollary 11.13. 12.4. Let 𝜑 be an n.s.f. weight on the von Neumann algebra ℳ ⊂ ℬ(ℋ ). The modular automorphism group {𝜎t𝜑 }t∈ℝ determines a continuous action 𝜎 = 𝜎 𝜑 of the group ℝ on ℳ. Consider ̄ ℒ 2 (ℝ). the Hilbert space ℒ 2 (ℝ, ℋ ) = ℋ ⊗

144

Conditional Expectations and Operator-Valued Weights

The crossed product of ℳ by the action 𝜎 of ℝ is the von Neumann algebra ℛ(ℳ, 𝜎) ⊂ ℬ(ℒ 2 (ℝ, ℋ )) generated by the operators I(x) = I𝜑 (x)(x ∈ ℳ), (I(x)𝜉)(r) = 𝜎−r (x)𝜉(r) (𝜉 ∈ ℒ 2 (ℝ, ℋ ), r ∈ ℝ), and by the unitary operators 𝜆 (t)(t ∈ ℝ), 𝜆(t)𝜉)(r) = 𝜉(r − t) (𝜉 ∈ ℒ 2 (ℝ, ℋ ), r ∈ ℝ). (𝜆 The mapping I ∶ ℳ → ℛ(ℳ, 𝜎) is an injective normal *-homomorphism. We shall identify ℳ with I(ℳ) ⊂ ℛ(ℳ, 𝜎). With this identification, we have 𝜆(t∗ ) (x ∈ ℳ, t ∈ ℝ). 𝜎t (x) = 𝜆 (t)x𝜆

(1)

The unitary operators m(s)(s ∈ ℝ), m(s)𝜉(r) = e−isr 𝜉(r) (𝜉 ∈ ℒ 2 (ℝ, ℋ ), r ∈ ℝ), define a dual action 𝜃 = 𝜃 𝜑 = 𝜎̂ 𝜑 of ℝ on ℛ(ℳ, 𝜎): 𝜃s (X) = m(s)Xm(s)∗

(X ∈ ℛ(ℳ, 𝜎), s ∈ ℝ),

which is characterized by the equalities 𝜃s (x) = x 𝜆(t)) = e 𝜃s (𝜆

(x ∈ ℳ, s ∈ ℝ),

(2)

𝜆 (t)

(3)

−ist

(s, t ∈ ℝ).

Moreover, we shall see later (Proposition 19.3) that the centralizer of the dual action coincides with ℳ: ℛ(ℳ, 𝜎)𝜃 = ℳ.

(4)

We shall show that +∞

E = E𝜑 =

∫−∞

𝜃s ds ∶ ℛ(ℳ, 𝜎)+ → ℳ̄ +

is an n.s.f. ℳ-valued weight on ℛ(ℳ, 𝜎) and E(𝜃s (X)) = E(X) (X ∈ ℛ(ℳ, 𝜎)+ , s ∈ ℝ), 𝜆(t)X𝜆 𝜆(t)∗ ) = 𝜆 (t)E(X)𝜆 𝜆(t)∗ = 𝜎t (E(X)) (X ∈ ℛ(ℳ, 𝜎), t ∈ ℝ). E(𝜆 More precisely, E is defined by +∞

E(X)(𝜔) =

∫−∞

𝜔(𝜃s (X)) ds (X ∈ ℛ(ℳ, 𝜎)+ , 𝜔 ∈ ℛ(ℳ, 𝜎)+∗ )

(5) (6)

Existence and Uniqueness of Operator-Valued Weights

145

+

as a mapping E ∶ ℛ(ℳ, 𝜎)+ → ℛ(ℳ, 𝜎) . Let X ∈ ℛ(ℳ, 𝜎)+ . Using the translation invariance of the Lebesgue measure, we obtain 𝜃s (E(X)) = E(X)(s ∈ ℝ), and hence (see (4) and 11.14.(1)) 𝜃 E(X) ∈ (ℛ(ℳ, 𝜎) )+ = ℳ̄ + . It is now easy to check that E is a normal faithful ℳ-valued weight on ℛ(ℳ, 𝜎) which satisfies (5) and (6). To show that E is semifinite, we consider a continuous function f ∶ ℝ → ℂ with compact support and the operator 𝜆( f ) =

+∞

∫−∞

𝜆(r) dr ∈ ℛ(ℳ, 𝜎). f (r)𝜆

We have 𝜆 ( f )∗𝜆 ( f ) = 𝜆 ( f ∗ ∗ f ), where f ∗ ∗ f is the convolution of the function f ∗ (r) = f (−r) with the function f. Also, we consider the functions gn (s) √ = exp(s2 ∕2n2 )(s ∈ ℝ, n ∈ ℕ), and their +∞ −ist Fourier transforms hn = ĝ n , hn (t) = ∫−∞ gn (s)e dt = n 2 exp(n2 t2 ∕2)(t ∈ ℝ, n ∈ ℕ). Note that +∞ gn (s) ↑ 1 uniformly on compact sets, as n ↑ ∞, hn (t) ≥ 0 and ∫−∞ hn (t) dt = 2𝜋. 𝜆(( f ∗ ∗ f )hn )‖ ≤ ‖( f ∗ ∗ f )hn ‖1 ≤ ‖ f ∗ ∗ f‖∞ ‖hn ‖1 ≤ 2𝜋‖ f‖22 , it follows by Fatou’s Since ‖𝜆 lemma and Fubini’s theorem that 𝜆( f )∗𝜆 ( f )) = sup E(𝜆 n

+∞

∫−∞

𝜆( f ∗ ∗ f ))gn (s) ds 𝜃s (𝜆

+∞

= sup n

∫−∞

+∞

gn (s)

∫−∞

( f ∗ ∗ f )(t)e−ist𝜆 (t) dt ds

+∞

𝜆(t) dt ( f ∗ ∗ f )(t)hn (t)𝜆 ∫−∞ = sup 𝜆 (( f ∗ ∗ f )hn ) ∈ ℳ + , = sup n

n

that is, 𝜆 ( f ) ∈ 𝔑E . Since 𝔑E is a right ℳ-module, it follows that 𝜆 ( f )x ∈ 𝔑E for any x ∈ ℳ and any continuous 𝜆(t); t ∈ ℝ}, it follows function f with compact support. Since ℛ(ℳ, 𝜎) is generated by ℳ and {𝜆 that 𝔑E is dense in ℛ(ℳ, 𝜎), that is, E is semifinite. Thus, we can define an n.s.f. weight 𝜑̂ = 𝜑 ◦ E on ℛ(ℳ, 𝜎) called the dual weight of 𝜑. By (5) it follows that the dual weight is invariant with respect to the dual action: 𝜑̂ ◦ 𝜃s = 𝜑̂ (s ∈ ℝ).

(7)

By Theorem 11.9, we have 𝜆(t)∗ 𝜎t𝜑̂ (x) = 𝜎t𝜑 (x) = 𝜆 (t)x𝜆

(x ∈ ℳ, t ∈ ℝ).

(8)

On the other hand, by (6), for X ∈ ℛ(ℳ, 𝜎)+ and t ∈ ℝ we have 𝜆(s)∗ ) = 𝜑(𝜎s𝜑 (E(X))) = 𝜑(E(X)) = 𝜑(X), 𝜑(𝜆 ̂ 𝜆(s)X𝜆 ̂ so that, using Corollary 3.7 we deduce that 𝜆(s)) = 𝜆 (s) = 𝜆 (t)𝜆 𝜆(s)𝜆 𝜆(t)∗ 𝜎t𝜑̂ (𝜆

(s, t ∈ ℝ).

(9)

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Conditional Expectations and Operator-Valued Weights

𝜆(s); s ∈ ℝ}, from (8) and (9) it follows that 𝜎t𝜑̂ = Since ℛ(ℳ, 𝜎) is generated by ℳ and {𝜆 𝜆(t))(t ∈ ℝ). Ad(𝜆 There exists a unique positive self-adjoint operator A affiliated to ℛ(ℳ, 𝜎) such that 𝜆 (t) = A−it

(t ∈ ℝ).

(10)

Then the n.s.f. weight 𝜏 = 𝜏𝜑 = 𝜑̂ A , constructed as in Section 4.4, has a trivial modular automorphism group (see Corollary 4.8) and hence is an n.s.f. trace on ℛ(ℳ, 𝜎) (see, for instance, 2.18.(1)). Note that [D𝜑̂ ∶ D𝜏]t = 𝜆 (t) (t ∈ ℝ).

(11)

From (3) and (10), it follows that 𝜃s (A) = es A

(s ∈ ℝ),

(12)

(s ∈ ℝ),

(13)

and using (7) and (12), it is easy to check that 𝜏 ◦ 𝜃s = e−s 𝜏

12.5. Proof of Theorem 12.1. We may consider 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) realized as von Neumann algebras. Since 𝜎t𝜓 = 𝜎t𝜑 |𝒩 (t ∈ ℝ), it follows that ℛ(𝒩 , 𝜎 𝜓 ) is the von Neumann subalgebra 𝜆(t); t ∈ ℝ}, and of ℛ(ℳ, 𝜎 𝜑 ) generated by 𝒩 and {𝜆 𝜃s𝜓 = 𝜃s𝜑 |ℛ(𝒩 , 𝜎 𝜓 ) (s ∈ ℝ).

(1)

E𝜓 = E𝜑 |ℛ(𝒩 , 𝜎 𝜓 ).

(2)

Consequently,

Consider the n.s.f. traces 𝜏𝜑 and 𝜏𝜓 defined on ℛ(ℳ, 𝜎 𝜑 ) and ℛ(𝒩 , 𝜎 𝜓 ) as in Section 12.4. By the result of Section 12.3, there exists a unique n.s.f. operator-valued weight +

F ∶ ℛ(ℳ, 𝜎 𝜑 )+ → ℛ(𝒩 , 𝜎 𝜓 ) such that 𝜏𝜓 ◦ F = 𝜏𝜑 .

(3)

By (1) it follows that for each s ∈ ℝ 𝜓 Fs = 𝜃−s ◦ F ◦ 𝜃𝜑s ∶ ℛ(ℳ, 𝜎 𝜑 )+ → ℛ(𝒩 , 𝜎 𝜓 )

+

is an n.s.f. operator-valued weight. Since 𝜏𝜑 ◦ 𝜃s𝜑 = e−s 𝜏𝜑 and 𝜏𝜓 ◦ 𝜃s𝜓 = e−s 𝜏𝜓 , using equality (3) we obtain 𝜏𝜓 ◦ Fs = 𝜏𝜑 , hence Fs = F, that is, F ◦ 𝜃s𝜑 = 𝜃s𝜓 ◦ F(s ∈ ℝ). It follows that f ◦ E𝜑 = E𝜓 ◦ F.

(4)

Existence and Uniqueness of Operator-Valued Weights

147

From 11.5.(8), we infer that +

+

F(ℳ + ) ⊂ F(E𝜑 (ℛ(ℳ, 𝜎 𝜑 ) )) = E𝜓 (F(ℛ(ℳ, 𝜎 𝜑 ) )) ⊂ 𝒩̄ + hence E = F|ℳ + ∶ ℳ + → 𝒩̄ + is a normal 𝒩 -valued weight on ℳ. Recall (12.4.(11)) that for the dual weights 𝜑̂ = 𝜑 ◦ E𝜑 and 𝜓̂ = 𝜓 ◦ E𝜓 we have [D𝜑̂ ∶ D𝜏𝜑 ]t = 𝜆 (t) = [D𝜓̂ ∶ D𝜏𝜓 ]t (t ∈ ℝ). Using (3) and Theorem 11.9, we obtain (D(𝜓̂ ◦ F) ∶ D𝜏𝜑 ]t = [D(𝜓̂ ◦ F) ∶ D(𝜏𝜓 ◦ F)]t = [D𝜓̂ ∶ D𝜏𝜓 ]t = [D𝜑̂ ∶ D𝜏𝜑 ]t (t ∈ ℝ). By Corollary 3.6, we deduce + that 𝜑̂ = 𝜓̂ ◦ F, that is, 𝜑 ◦ E𝜑 = 𝜓 ◦ E𝜓 ◦ F = 𝜓 ◦ E ◦ E𝜑 . Since E𝜑 (ℛ(ℳ, 𝜎 𝜑 ) ) = ℳ̄ + (see 11.5.(8)), we conclude that 𝜑 = 𝜓 ◦ E. According to Proposition 11.7, it follows that the normal operator-valued weight E is semifinite and faithful. The uniqueness of E follows from Corollary 11.13. □ 12.6 Corollary. Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be von Neumann algebras. Then P (ℳ, 𝒩 ) ≠ ∅ if and only if there exist n.s.f. weights 𝜑 and 𝜓 on ℳ and 𝒩 , respectively, such that 𝜎t𝜓 = 𝜎t𝜑 |𝒩 (t ∈ ℝ). Proof. Follows from Theorems 11.9 and 12.1. 12.7 Corollary. Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be von Neumann algebras and let Wnsf (𝒩 ) ∋ 𝜑 ↦ 𝜑 ̃ ∈ Wnsf (ℳ) be a mapping with the following properties 𝜎t𝜑 = 𝜎t𝜑̃ |𝒩 (t ∈ ℝ, 𝜑 ∈ Wnsf (𝒩 )) [D̃ 𝜓 ∶ D̃ 𝜑]t = [D𝜓 ∶ D𝜑]t (t ∈ ℝ, 𝜑, 𝜓 ∈ Wnsf (𝒩 )). Then there exists a unique n.s.f. operator-valued weight E ∶ ℳ + → 𝒩 + such that 𝜑 ̃ = 𝜑 ◦ E for all 𝜑 ∈ Wnsf (𝒩 ). Proof. Let 𝜑 ∈ Wnsf (𝒩 ) be fixed. By Theorem 12.1, there exists a unique n.s.f. operator-valued weight E ∶ ℳ + → 𝒩̄ + such that 𝜑 ̃ = 𝜑 ◦ E. By our assumption and Theorem 11.9, for any 𝜓 ∈ Wnsf (𝒩 ) we have [D̃ 𝜓 ∶ D̃ 𝜑]t = [D𝜓 ∶ D𝜑]t = [D(𝜓 ◦ E) ∶ D(𝜑 ◦ E)]t = [D(𝜓 ◦ E) ∶ D̃ 𝜑]t (t ∈ ℝ); hence 𝜓 ̃ = 𝜓 ◦ E by Corollary 3.6. 12.8 Corollary. Let 𝒩1 ⊂ ℳ1 ⊂ ℬ(ℋ1 ), 𝒩2 ⊂ ℳ2 ⊂ ℬ(ℋ2 ) be von Neumann algebras and E1 ∈ ̄ E2 ∈ P (ℳ1 ⊗ ̄ ℳ2 , 𝒩1 ⊗ ̄ 𝒩2 ) P (ℳ1 , 𝒩1 ), E2 ∈ P (ℳ2 , 𝒩2 ). There exists a unique element E = E1 ⊗ such that ̄ 𝜓2 ) ◦ E = (𝜓1 ◦ E1 ) ⊗ ̄ (𝜓2 ◦ E2 ) for all 𝜓1 ∈ Wnsf (𝒩1 ), 𝜓2 ∈ Wnsf (𝒩2 ) (𝜓1 ⊗

(1)

Proof. Let 𝜑1 ∈ Wnsf (𝒩1 ), 𝜑2 ∈ Wnsf (𝒩2 ) be fixed. According to the definition of the tensor product of n.s.f. weights (8.2) and to Theorem 11.9, we have ̄ (𝜑2 ◦ E2 ) (𝜑1 ◦ E1 ) ⊗

𝜎t

̄ 𝜑2 𝜑1 ⊗

̄ 𝒩2 = 𝜎t |𝒩1 ⊗

(t ∈ ℝ).

148

Conditional Expectations and Operator-Valued Weights

̄ ℳ2 , 𝒩1 ⊗ ̄ 𝒩2 ) such that By Theorem 12.1, there exists a unique element E ∈ P (ℳ1 ⊗ ̄ 𝜑2 ) ◦ E = (𝜑1 ◦ E1 ) ⊗ ̄ (𝜑2 ◦ E2 ). (𝜑1 ⊗ For other n.s.f. weights 𝜓1 , 𝜓2 , (1) now follows again using Theorem 11.9, Corollary 8.6, and Corollary 3.6. ̄ E is called the tensor product of E and E . It is easy to check The operator-valued weight E1 ⊗ 2 1 2 ̄ a2 ∈ 𝔐E ⊗̄ E and that if a1 ∈ 𝔐E1 , a2 ∈ 𝔐E2 then a1 ⊗ 1 2 ̄ E2 )(a1 ⊗ ̄ a2 ) = E1 (a1 ) ⊗ ̄ E2 (a2 ), (E1 ⊗ ̄ E2 requires (1) for some 𝜓1 , 𝜓2 . but the uniqueness of E1 ⊗ From Section 11.15, it is clear that ̄ E2 E ⊗

𝜎t 1

E ̄ 𝜎tE2 = 𝜎t 1 ⊗

(t ∈ ℝ).

(2)

If E1 and E2 are normal faithful conditional expectations, then, by Corollary 8.7, the present ̄ E2 agrees with that considered in Section 9.4. definition of E1 ⊗ ̄ ℱ ⊂ 12.9. Let 𝒩0 ⊂ ℳ0 ⊂ ℬ(ℋ ) be von Neumann algebras, ℱ a type I factor and 𝒩 = 𝒩0 ⊗ ̄ ℱ = ℳ. Consider an n.s.f. operator-valued weight E ∶ ℳ + → 𝒩 + . ℳ0 ⊗ ̄ u ∈ 𝒩 and Let x ∈ ℳ0+ . For u ∈ ℱ unitary, we have 1 ⊗ ̄ u)E(x ⊗ ̄ 1)(1 ⊗ ̄ u)∗ = E((1 ⊗ ̄ u)(x ⊗ ̄ 1)(1 ⊗ ̄ u)∗ ) = E(x ⊗ ̄ 1), (1 ⊗ +

̄ 1) ∈ (𝒩0 ⊗ ̄ 1)+ . Thus, there exists E0 (x) ∈ 𝒩 such that hence E(x ⊗ 0 ̄ 1) = E0 (x) ⊗ ̄ 1. E(x ⊗

(1)

It is easy to see that E0 ∶ ℳ0+ → 𝒩̄0+ is a normal faithful operator-valued weight. ̄ u∈𝒩 Let 𝜓 be an n.s.f. weight on 𝒩0 and tr the canonical trace on ℱ . For u ∈ ℱ we have 1 ⊗ ̄ tr) ◦ E ̄ tr (𝜓 ⊗ 𝜓 ⊗ ̄ u) = 𝜎t ̄ u) = 1 ⊗ ̄ u. Thus, 1 ⊗ ̄ ℱ is contained in the and hence (11.9.(2)) 𝜎t (1 ⊗ (1 ⊗ ̄ tr) ◦ E on ℳ = ℳ0 ⊗ ̄ ℱ . By Proposition 9.17, there exists an centralizer of the n.s.f. weight (𝜓 ⊗ ̄ tr) ◦ E = 𝜑 ⊗ ̄ tr. Then, for any x ∈ ℳ + and any minimal n.s.f. weight 𝜑 on ℳ0 such that (𝜓 ⊗ 0 ̄ tr)(x ⊗ ̄ e) = (𝜓 ⊗ ̄ tr)(E(x ⊗ ̄ e)) = (𝜓 ⊗ ̄ tr)(E0 (x) ⊗ ̄ e) = projection e ∈ ℱ , we have 𝜑(x) = (𝜑 ⊗ 𝜓(E0 (x)), hence 𝜑 = 𝜓 ◦ E0 . Consequently, ̄ tr) ◦ E = (𝜓 ◦ E0 ) ⊗ ̄ tr. (𝜓 ⊗

(2)

From (2) and Proposition 11.7, it follows that E0 is semifinite. Then, by (2) and Corollary 12.8, we get ̄ 𝜄ℱ , E = E0 ⊗ where 𝜄ℱ is the identity mapping on ℱ . ̄ ℱ , 𝒩0 ⊗ ̄ ℱ ) is of the form (3), with E0 ∈ P (ℳ0 , 𝒩0 ). Thus, every E ∈ P (ℳ0 ⊗

(3)

Existence and Uniqueness of Operator-Valued Weights

149

12.10 Corollary. Let 𝜑 be an n.s.f. weight on the W ∗ -algebra ℳ. The centralizer ℳ 𝜑 of 𝜑 is semifinite if and only if there exists a 𝜎 𝜑 -invariant n.s.f. ℳ 𝜑 -valued weight on ℳ. Proof. Assume that ℳ 𝜑 is semifinite and let 𝜏 be an n.s.f. trace on ℳ. Since 𝜎t𝜑 |ℳ = 𝜄 = 𝜎t𝜏 (t ∈ ℝ), it follows by Theorem 12.1 that there exists a unique n.s.f. operator-valued weight E ∶ ℳ + → (ℳ̄ 𝜑 )+ such that 𝜑 = 𝜏 ◦ E. Since 𝜏 ◦ (E ◦ 𝜎t𝜑 ) = 𝜑 ◦ 𝜎t𝜑 = 𝜑 = 𝜏 ◦ E, it follows that E ◦ 𝜎t𝜑 = E, that is, E is 𝜎 𝜑 -invariant. Conversely, let E ∶ ℳ + → (ℳ̄ 𝜑 )+ be a 𝜎 𝜑 -invariant n.s.f. operator-valued weight and let 𝜓 be an n.s.f. weight on ℳ 𝜑 . Then the n.s.f. weights 𝜑 and 𝜓 ◦ E on ℳ commute and so, by Theorem 4.10, there exists a semifinite element A ∈ (ℳ̄ 𝜑 )+ such that 𝜓 ◦ E = 𝜑A . Using Theorem 11.9 and 𝜑 Corollary 4.8, we obtain 𝜎t𝜓 = 𝜎t𝜓 ◦ E = 𝜎t A = Ad(Ait ) (t ∈ ℝ). Consequently, 𝜏 = 𝜓A−1 is an n.s.f. trace on the W ∗ -algebra ℳ. 12.11. Another consequence of Theorem 12.1, as well as Theorems 7.4, 7.14, and 11.9, is the following Corollary. Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be von Neumann algebras. There is a bijection P (ℳ, 𝒩 ) ∋ E ↦ E ′ ∈ P (𝒩 ′ , ℳ ′ ), uniquely determined, with the property Δ(𝜓∕𝜑′ ◦ E ′ ) = Δ(𝜓 ◦ E∕𝜑′ ) (𝜓 ∈ Wnsf (𝒩 ), 𝜑′ ∈ Wnsf (ℳ ′ )),

(1)

where Δ(⋅∕⋅) stands for the spatial derivative (7.3). For any E, E1 , E2 ∈ P (ℳ, 𝒩 ) and any t ∈ ℝ we have ′

E 𝜎tE = 𝜎−t , ′ [DE1 ∶ DE2′ ]t = [DE1 ∶ DE2 ]−t E1 ≤ E2 ⇔ E2′ ≤ E1′ .

(2) (3) (4)

Proof. Let 𝜓 ∈ Wnsf (𝒩 ) and 𝜑′ ∈ Wnsf (ℳ ′ ) be fixed. Let E ∈ P (ℳ, 𝒩 ). Then 𝜑 = 𝜓 ◦ E ∈ Wnsf (ℳ). Writing ut = Δ(𝜑∕𝜑′ )it , by Theorem 7.4 we

have 𝜎t𝜑 = Ad(ut )|ℳ and 𝜎t𝜑 = Ad(u∗t )|ℳ ′ (t ∈ ℝ). By Theorem 11.9, we get 𝜎t𝜓 = 𝜎t𝜑 |𝒩 = Ad(ut )|𝒩 (t ∈ ℝ). Using 7.4.(1) and Theorem 7.14, we infer the existence of a unique weight 𝜓 ′ ∈ ′ ′ ′ Wnsf (𝒩 ′ ) such that ut = Δ(𝜓∕𝜓 ′ )it (t ∈ ℝ). Then 𝜎t𝜓 = Ad(u∗t )|𝒩 ′ , and hence 𝜎t𝜑 = 𝜎t𝜓 |ℳ ′ (t ∈ ℝ). By Theorem 12.1, there exists a unique operator-valued weight E ′ ∈ P (𝒩 ′ , ℳ ′ ) such that 𝜓 ′ = 𝜑′ ◦ E ′ and we have ′

Δ(𝜓∕𝜑′ ◦ E ′ ) = Δ(𝜓 ◦ E∕𝜑′ ).

(5)

Equation (5) determines the weight 𝜑′ ◦ E ′ and hence the operator-valued weight E ′ , uniquely (see 7.13.(1) and 11.13). Similarly, one can construct a mapping P (𝒩 ′ , ℳ ′ ) ∋ E ′ ↦ E ∈ P (ℳ, 𝒩 ), satisfying (5). It follows that the mappings E ↦ E ′ and E ′ ↦ E are reciprocal bijections.

150

Conditional Expectations and Operator-Valued Weights

If 𝜓̄ ∈ Wnsf (𝒩 ), 𝜑̄ ′ ∈ Wnsf (ℳ ′ ) are other weights, then, using Theorem 7.4.(5) and Theorem 11.9.(3), we deduce from (5) that Δ(𝜓∕ ̄ 𝜑̄ ′ ◦ E ′ ) = Δ(𝜓̄ ◦ E∕𝜑̄ ′ ), proving (1). ′ ′ ′ ′ For z ∈ 𝒩 ′ ∩ ℳ = (ℳ ′ )′ ∩ 𝒩 ′ , we have by definition (11.15.(2)) 𝜎tE (z) = 𝜎t𝜑 ◦ E (z) = 𝜎t𝜓 (z) = 𝜑 𝜓 ◦E E (z). This proves (2). u∗t zut = u∗−t zu∗−t = 𝜎−t (z) = 𝜎−t (z) = 𝜎−t Consider now E1 , E2 ∈ P (ℳ, 𝒩 ) and let 𝜑1 , u1t , 𝜓1′ , E1′ and 𝜑2 , u2t , 𝜓2′ , E2′ be associated, as in the first part of the proof, with E1 and E2 , respectively. By Theorem 7.4.(5), we have [D𝜓1′ ∶ D𝜓2′ ]t = u1−t u2t = [D𝜑1 ∶ D𝜑2 ]−t (t ∈ ℝ). Since 𝜓j′ = 𝜑′ ◦ Ej′ , 𝜑j = 𝜓 ◦ Ej (j = 1, 2), using Definition 11.15.(4) we obtain (3). Finally, (4) is an immediate consequence of (3) and 11.15.(8). 12.12. In particular, we have the following Corollary. Let ℳ ⊂ ℬ(ℋ ) be von Neumann algebra. There is a uniquely determined bijection Wnsf (ℳ ′ ) ∋ 𝜓 ↦ E𝜓 ∈ P (ℬ(ℋ ), ℳ) such that 𝜑 ◦ E𝜓 = trΔ(𝜑∕𝜓) for any 𝜑 ∈ Wnsf (ℳ).

(1)

̄ 𝜂) E𝜓 (𝜂 ⊗ ̄ = R𝜓𝜂 (R𝜓𝜂 )∗ for all 𝜂 ∈ D(ℋ , 𝜓).

(2)

Moreover, we have

Proof. The notation is as in Sections 4.23 and 7.1. We apply Corollary 12.11, replacing ℳ, 𝒩 , E, 𝜓, ℳ ′ , 𝒩 ′ , E ′ , 𝜑′ in 12.11 by ℳ ′ , ℂ ⋅ 1ℋ , 𝜓, t, ℳ, ℬ(ℋ ), E𝜓 , 𝜑, respectively, where ℳ ⊂ ℬ(ℋ ), 𝜓 ∈ Wnsf (ℳ ′ ), E𝜓 ∈ (ℬ(ℋ ), ℳ) are as in the statement of 12.12, 𝜑 is an n.s.f. weight on ℳ, and the positive real number t > 0 is regarded as the weight ℂ ⋅ 1ℳ ∋ 𝜆 ↦ t𝜆 ∈ ℂ. We thus deduce the existence of the required bijection 𝜓 ↦ E𝜓 , uniquely determined by the condition Δ(t∕𝜑 ◦ E𝜓 ) = Δ(t𝜓∕𝜑) for all 𝜑 ∈ Wnsf (ℳ) and all t > 0. According to 7.4.(1), 7.13.(2), and 7.3.(6), the above condition is equivalent to the condition Δ(𝜑 ◦ E𝜓 ∕1) = Δ(𝜑∕𝜓) = Δ(trΔ(𝜑∕𝜓) ∕1) for all 𝜑 ∈ Wnsf (ℳ). Using 7.13.(1), we get 𝜑 ◦ E𝜓 = trΔ(𝜑∕𝜓) for all 𝜑 ∈ Wnsf (ℳ). Thus, for any 𝜂 ∈ D(ℋ , 𝜓) and any 𝜑 ∈ Wnsf (ℳ) we have (see 4.23.(4) and 7.3.(2)) ̄ 𝜂)) ̄ 𝜂) 𝜑(E𝜓 (𝜂 ⊗ ̄ = trΔ(𝜑∕𝜓) (𝜂 ⊗ ̄ = ‖Δ(𝜑∕𝜓)1∕2 𝜂‖2 = 𝜑(R𝜓𝜂 (R𝜓𝜂 )∗ ) ̄ 𝜂) and hence E𝜓 (𝜂 ⊗ ̄ = R𝜓𝜂 (R𝜓𝜂 )∗ (𝜂 ∈ D(ℋ , 𝜓)).

Existence and Uniqueness of Operator-Valued Weights

151

12.12. bis As an application of Corollary 12.11, we get an extension of the V. Jones’ theory of index ([L], C.7.7) to arbitrary factors, due to Kosaki (1986). Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be arbitrary factors and assume that there is a normal conditional expectation E ∶ ℳ → 𝒩 (otherwise, 𝒩 is too small in ℳ, so that the index should be +∞). By Corollary 12.11, we get an operator-valued weight E ′ ∈ P (𝒩 ′ , ℳ ′ ). For any unitary u′ ∈ ℳ ′ we have u′ E ′ (1)u′∗ = E ′ (u′ ⋅ 1 ⋅ u′∗ ) = E ′ (1) and, since ℳ ′ is a factor, this means that E ′ (1) is a scalar, possibly +∞. By definition, Index E = E ′ (1) and, when E is self-understood, we denote [ℳ ∶ 𝒩 ] = Index E. It is shown that the number E ′ (1) does not depend on ℋ . If ℳ is a factor of type II1 and E ∶ ℳ → 𝒩 is the unique conditional expectation with respect to the normalized trace, then this index coincides with the V. Jones’ index. When Index E < +∞, the operator-valued weight E ′ ∈ P (𝒩 ′ , ℳ ′ ) is a scalar muultiple of a conditional expectation, namely, with 𝜏 = (E ′ (1))−1 , we have (𝜏E ′ )(x′ ) = (𝜏E ′ )(x′ ⋅ 1) = x′ (𝜏E ′ )(1) = x′ ,

(x′ ∈ ℳ ′ ).

By imitating the arguments of V. Jones, H. Kosaki obtained the same restrictions for the values of the index as V. Jones in the case of finite factors. 12.13. Another consequence of Theorem 12.l is an extension of Theorem 5.1 for operator-valued weights: Corollary. Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be von Neumann algebras and E ∈ P (ℳ, 𝒩 ) ≠ ∅. Then the mapping F ↦ {[DF ∶ DE]t }t∈ℝ establishes a bijection between P (ℳ, 𝒩 ) and the set of all unitary 𝜎 E -cocycles {wt }t∈ℝ ⊂ 𝒩 ′ ∩ ℳ. Proof. Let {wt }t∈ℝ ⊂ 𝒩 ′ ∩ℳ be a unitary 𝜎 E -cocyle and 𝜓 ∈ Wnsf (𝒩 ). Since 𝜎tE = 𝜎t𝜓 ◦ E |𝒩 ′ ∩ℳ (11.15.(2)), {wt }t∈ℝ is a unitary 𝜎 𝜓 ◦ E -cocycle so that, by Theorem 5.1, there exists 𝜑 ∈ Wnsf (ℳ) such that [D𝜑 ∶ D(𝜓 ◦ E)]t = wt (t ∈ ℝ).

(1)

Since wt ∈ 𝒩 ′ ∩ ℳ, for y ∈ 𝒩 we have (11.9.(2)) 𝜎t𝜑 (y) = wt 𝜎t𝜓 ◦ E (y)w∗t = wt 𝜎t𝜓 (y)w∗t = ∈ ℝ). By Theorem 12.1, there exists an operator-valued weight F ∈ P (ℳ, 𝒩 ) such that

𝜎t𝜓 (y)(t

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Conditional Expectations and Operator-Valued Weights

𝜑 = 𝜓 ◦ F. Then, from (1) and 11.15.(4) it follows that [DF ∶ DE]t = wt (t ∈ ℝ). Thus, the mapping considered in the statement is surjective. The injectivity of this mapping follows immediately from 3.6 and 11.13. 12.14. Let ℳ be a semifinite von Neumann algebra and 𝒵 a von Neumann subalgebra of the center 𝒵 (ℳ) of ℳ. Let us fix an n.s.f. trace 𝜏 on ℳ and an n.s.f. weight ν on 𝒵 . By Theorem 12.1 (actually, by the particular case considered in Section 12.3), there exists a unique n.s.f. operator-valued weight ♮ = ♮(𝒵 , 𝜏, ν) ∶ ℳ + ∋ x ↦ x♮ ∈ 𝒵 + such that 𝜏 = ν ◦ ♮, that is, 𝜏(x) = ν(x♮ )

(x ∈ ℳ + ).

(1)

(x ∈ ℳ + , a ∈ 𝒵 + ).

(2)

Since 𝒵 is contained in 𝒵 (ℳ), we have (ax)♮ = ax♮

For every x ∈ ℳ and every a ∈ 𝒵 + , we have ν(a(x∗ x)♮ ) = ν((ax∗ x)♮ ) = 𝜏(ax∗ x) = 𝜏(axx∗ ) = ν((axx∗ )♮ ) = ν(a(xx∗ )♮ ). Since ν is faithful and a ∈ 𝒵 + was arbitrary, it follows that (x∗ x)♮ = (xx∗ )♮

(x ∈ ℳ)

(3)

that is, ♮ is actually an operator-valued trace. Thus, 𝔐♮ and 𝔑♮ are w-dense two sided ideals of ℳ and, consequently, any nonzero element of ℳ + dominates a nonzero element of 𝔐♮ ∩ ℳ + . In what follows we assume that 𝒵 = 𝒵 (ℳ). Then, for two projections e, f ∈ ℳ we have e ≺ f ⇔ e♮ ≤ f ♮ ;

(4)

this follows easily, using (3), the faithfulness of ♮ and the comparison theorem ([L], 4.6). Consider now an arbitrary normal operator-valued trace E ∶ ℳ + → 𝒵̄ + . Then ν ◦ E is a normal trace on ℳ and hence (see 12.2.(7) and [L], E.7.14, C.10.4) there exists A ∈ 𝒵̄ + such that ν ◦ E = 𝜏A . Thus, for x ∈ ℳ + and a ∈ 𝒵 + we have ν(aE(x)) = ν(E(ax)) = 𝜏(Aax) = ν((Aax)♮ ) = ν(aAx♮ ). Consequently, E(x) = Ax♮

(x ∈ ℳ + ).

(5)

If ℳ is finite and countably decomposable and if 𝜏(1) = 1, ν(1) = 1, then ♮ ∶ ℳ → 𝒵 (ℳ) is just the canonical central trace ([L], 7.11). If ℳ is properly infinite, then ν(z) = +∞ for all 0 ≠ z ∈ 𝒵 + , hence z♮ = ∞ ◦ s(z)

(z ∈ 𝒵 + ).

(6)

Existence and Uniqueness of Operator-Valued Weights

153

Proposition. Let ℳ be a type II∞ von Neumann algebra with center 𝒵 , 𝜏 an n.s.f. trace on ℳ, ν an n.s.f. weight on 𝒵 and ♮ = ♮(𝒵 , 𝜏, ν) ∶ ℳ + → 𝒵 + . Then {e♮ ; e ∈ ℳ, projection} = 𝒵̄ + . Proof. Let 0 ≠ A ∈ 𝒵̄ + and let f ∈ ℳ be a projection such that A ≤ f ♮ . We first show that there exists a projection e ∈ ℳ, 0 ≠ e ≤ f, such that e♮ ≤ A. If A = ∞ ⋅ p with p a central projection, then, according to (6), we can take e = fp. Otherwise we may assume that A is bounded (11.3.(1)). Since ♮ is semifinite, there exists x ∈ ℳ + , 0 ≠ x ≤ f, with ‖x♮ ‖ ≤ 1. Then ([L], 2.21) there exists a spectral projection g ≠ 0 of xA and a positive integer n > 1 such that g ≤ nxA; note that g ≤ s(xA) ≤ f. Since ℳ is a continuous von Neumann algebra, it follows by ([L], 4.11, E.4.10) that there exist mutually orthogonal and equivalent projections e1 , … , en ∈ ℳ such that g = e1 + … + en . Then, for e = e1 , we have 0 ≠ e ≤ f and e♮ = n−1 f ≤ n−1 (nxA)♮ = x♮ A ≤ A. Consider now an arbitrary clement 0 ≠ A ∈ 𝒵̄ + . There exists a projection e ∈ ℳ, which is maximal with the property e♮ ≤ A. If A = ∞ ⋅ s(A), then we have e = s(A) and hence e♮ = A. Assume that A ≠ ∞ ⋅ s(A) and e♮ ≠ A. Then there is a central projection p ≤ s(A) such that Ap and e♮ p be bounded. Also, there is a central projection q ≤ p such that Aq − e♮ q ≤ (1 − e)♮ q and (1 − e)♮ (p − q) ≤ A(p − q) − e♮ (p − q). We have q ≠ 0 since q = 0 would imply ∞ ⋅ p = p♮ = e♮ p + (1 − e)♮ p ≤ e♮ p + Ap − e♮ p = Ap, contradicting the fact that Ap is bounded. By the first part of the proof, there exists a projection h ∈ ℳ, 0 ≠ h ≤ (1−e)q such that h♮ ≤ Aq−e♮ q. Then e+h ∈ ℳ is a projection, e + h ≥ e, e + h ≠ e and (e + h)♮ = e♮ + h♮ = e♮ (1 − q) + e♮ q + h♮ ≤ A(1 − q) + Aq = A, contradicting the maximality of e. Hence e♮ = A. Corollary. Let ℳ be a type II∞ von Neumann algebra with center 𝒵 , 𝜏 an n.s.f. trace on ℳ and 𝜔 a normal weight on 𝒵 . There exists a projection e ∈ ℳ such that 𝜔(z) = 𝜏(ez)(z ∈ 𝒵 + ). Proof. Let ν be an n.s.f. weight on 𝒵 and ♮ = ♮(𝒵 , 𝜏, ν). By 12.2.(7) there exists A ∈ 𝒵̄ + such that 𝜔 = νA and by the preceding proposition there exists a projection e ∈ ℳ such that e♮ = A. Then, for any z ∈ 𝒵 + , we have 𝜔(z) = ν(Az) = ν(e♮ z) = ν((ez)♮ ) = 𝜏(ez). 12.15. Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be von Neumann algebras and E ∶ ℳ + → 𝒩̄ + a normal operatorvalued weight. For x ∈ 𝒩 ′ ∩ ℳ and unitary v ∈ 𝒩 , we have v∗ E(x)v = E(v∗ xv) = E(x), + hence E(x) ∈ 𝒵 (𝒩 ) . Consequently, putting 𝒩 c = 𝒩 ′ ∩ ℳ, we obtain a normal operator-valued weight +

E c = E|(𝒩 c )+ ; (𝒩 c )+ → 𝒵 (𝒩 ) . Clearly, if E is faithful, then E c is also faithful. Theorem (U. Haagerup). Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be von Neumann algebras. The following statements are equivalent: (i) (ii) (iii) (iv)

there exists E ∈ P (ℳ, 𝒩 ) such that E c ∈ P (𝒩 c , 𝒵 (𝒩 )); P (ℳ, 𝒩 ) ≠ ∅ and E ∈ P (ℳ, 𝒩 ) ⇒ E c ∈ P (𝒩 c , 𝒵 (𝒩 )); there exists a separating family of bounded normal 𝒩 -valued weights on ℳ; there exists a separating family of normal conditional expectations of ℳ onto 𝒩 .

154

Conditional Expectations and Operator-Valued Weights

If these conditions are satisfied, then the mapping P (ℳ, 𝒩 ) ∋ E ↦ E c ∈ P (𝒩 c , 𝒵 (𝒩 )) is a bijection and, for any E, F ∈ P (ℳ, 𝒩 ) and any t ∈ ℝ, we have c

𝜎tE = 𝜎tE , [DF c ∶ DE c ]t = [DF ∶ DE]t .

(1) (2)

Proof. (1) We first assume that 𝒩 is countably decomposable. Consider a fixed faithful normal state 𝜑 on 𝒩 and put 𝜔 = 𝜑|𝒵 (𝒩 ). (i) ⇔ (ii). Let E, F ∈ P (ℳ, 𝒩 ). Assume that E c is semifinite, that is, E c ∈ P (𝒩 c , 𝒵 (𝒩 )). It is clear that (𝜑 ◦ E)|(𝒩 c )+ = 𝜔 ◦ E c is semifinite.

(3)

Since (11.15.(4)) [D(𝜑 ◦ F) ∶ D(𝜑 ◦ E)]t = [DF ∶ DE]t ∈ 𝒩 c (t ∈ ℝ), it follows also that 𝜔 ◦ F c = (𝜑 ◦ F)|(𝒩 c )+ is semifinite

(4)

and hence, by Proposition 11.7, F c is semifinite, that is, F c ∈ P (𝒩 c , 𝒵 (𝒩 )). (i) ⇔ (iii). Let E ∈ P (ℳ, 𝒩 ) and assume that E c is semifinite. Then there exists a net {vi }i∈I ⊂ 𝒩 c ∩ 𝔐F such that vi ↑ 1. It follows that the mappings Fi ∶ ℳ + ∋ x ↦ E(v∗i xvi ) ∈ 𝒩 +

(i ∈ I)

constitute a separating family of bounded normal operator-valued weights. (iii) ⇒ (iv). Let x0 ∈ ℳ + , x0 ≠ 0. Assuming (iii), there exists a bounded normal operator-valued weight F0 ∶ ℳ + → 𝒩 + such that F0 (x0 ) ≠ 0. Since 1 ∈ 𝒩 c , we have F0 (1) ∈ 𝒵 (𝒩 ) and we may assume that F0 (1) ≤ 1. Let {Fi }i∈I be a maximal family of bounded normal 𝒩 -valued weights on ℳ which contains F0 , such that Fi (1) ≤ 1 and the supports s(Fi (1)) ∈ 𝒵 (𝒩 ) are mutually orthogonal. ∑ ∑ In view of (iii) we get i∈I s(Fi (1)) = 1. Consequently, F = i∈I Fi is a bounded normal 𝒩 -valued +

weight on ℳ, F(x0 ) ≠ 0 and a = F(1) ∈ 𝒵 (𝒩 )+ , 0 ≤ a ≤ 1, s(a) = 1. Then a−1 ∈ 𝒵 (𝒩 ) is a semifinite element and the equation E(x) = a−1 F(x); x ∈ ℳ + defines a normal conditional expectation E of ℳ onto 𝒩 such that E(x0 ) ≠ 0. (iv) ⇒ (i). Let {Ei }i∈I be a maximal family of bounded normal 𝒩 -valued weights on ℳ with ∑ mutually orthogonal supports 0 ≠ s(Ei ) ∈ 𝒩 c and let e = 1 − i∈I s(Ei ) ∈ 𝒩 c . If e ≠ 0, then by (iv) there exists a normal conditional expectation F0 ∶ ℳ → 𝒩 with F0 (e) ≠ 0. Since e ∈ 𝒩 c it follows that the mapping E0 ∶ ℳ ∋ x ↦ F0 (exe) ∈ 𝒩 is a bounded normal operator-valued weight with 0 ≠ s(E0 ) ≤ e, contradicting the maximality of the family {Ei }i∈I . Hence e = 0, that is,

Existence and Uniqueness of Operator-Valued Weights ∑ i∈I

155

s(Ei ) = 1. Thus, the equation E(x) =

∑

Ei (x) (x ∈ ℳ + )

i∈I

defines a normal faithful operator-valued weight E ∶ ℳ + → 𝒩̄ + . Since s(Ei ) ∈ 𝒩 c , E(s(Ei )) = ∑ Ei (1) is bounded, and i∈I s(Ei ) = 1, it follows that E c and E are semifinite. Now consider E, F ∈ P (ℳ, 𝒩 ) and assume that E c , F c are semifinite. Using (3), 11.15.(2), 10.1, and 10.5, it follows that 𝜎tE = 𝜎t𝜔 ◦ E |𝒩 c = 𝜎t𝜑 ◦ E |𝒩 c = 𝜎tE c

c

(t ∈ ℝ).

̄ 1) defined in Consider also the operator-valued weight Θ = Θ(E, F) ∈ P (Mat2 (ℳ), 𝒩 ⊗ Section 11.15. It is easy to check that ̄ 1)′ ∩ Mat2 (ℳ) = Θ(E c , F c ) ∈ P (Mat2 (𝒩 c ), 𝒵 (𝒩 ) ⊗ ̄ 1), Θc = Θ|(𝒩 ⊗ that is, Θc is semifinite. It follows that 𝜎tΘ = 𝜎tΘ (t ∈ ℝ), and hence [DF c ∶ DE c ]t = [DF ∶ DE]t (t ∈ ℝ). We have thus proved statements (1) and (2). The injectivity of the mapping E ↦ E c follows from (2); its surjectivity can be easily proved using (2) and Corollary 12.13. (II) Assume now that 𝒩 is uniform of type 𝛾 ([L], 8.5). Then there exists a countably decompos̄ ℱ . Let ℳ = ℳ0 ⊗ ̄ ℱ able von Neumann algebra 𝒩0 and a type I factor ℱ such that 𝒩 = 𝒩0 ⊗ be the corresponding factorization of ℳ with ℳ0 ⊃ 𝒩0 (see 9.15). By 12.9, every E ∈ P (ℳ, 𝒩 ) ̄ 𝜄ℱ with E0 ∈ P (ℳ0 , 𝒩0 ). It is easy to check that E c is semifinite if and is of the form E = E0 ⊗ c only if E0 is semifinite and, in this case, c

c

c ̄ 𝜄ℱ , 𝜎 E c = 𝜎tE0 ⊗ ̄ 𝜄ℱ (t ∈ ℝ). 𝜎tE = 𝜎tE ⊗ t

Using these facts, it is easy to see that the proof of the theorem in this case reduces to the case considered in the first part of the proof. (III) In the general case, there exist a family Γ of distinct infinite cardinals and a family {q𝛾 }𝛾∈Γ ∑ of projections in 𝒵 (𝒩 ) such that 𝛾 q𝛾 = 1 and each 𝒩 q𝛾 is uniform of type 𝛾 ([L], 8.5). If E ∈ P (ℳ, 𝒩 ) and E𝛾 = E|q𝛾 ℳq𝛾 , E𝛾c = E|q𝛾 𝒩 c , (𝛾 ∈ Γ), then E c is semifinite if and only if each E𝛾c is semifinite and, in this case, c

Ec

E

𝜎tE = 𝜎tE (t ∈ ℝ) ⇔ 𝜎t 𝛾 = 𝜎t 𝛾 (t ∈ ℝ, 𝛾 ∈ Γ). Thus, the general case reduces to the case (II). 12.16. Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be von Neumann algebras. The equivalent conditions in Theorem 12.15 are satisfied in each of the following particular cases: (1) 𝒩 is a direct sum of type I factors; ̄ ℛ, where 𝒩 is identified with 𝒩 ⊗ 1 ⊂ ℳ; (2) ℳ = 𝒩 ⊗ (3) 𝒩 = 𝒵 (ℳ).

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Conditional Expectations and Operator-Valued Weights

Indeed, in all these cases there are separating families of normal conditional expectations of ℳ onto 𝒩 (see 10.23, 9.8.(3), 10.16). In particular, in all these cases we have P (ℳ, 𝒩 ) ≠ ∅. According to Corollary 12.11, we have P (ℳ, 𝒩 ) ≠ ∅ also in the case when ℳ is a direct sum of type I factors.

(4)

Finally, by Theorem 12.1 it follows that P (ℳ, 𝒩 ) ≠ ∅ whenever ℳ and 𝒩 are semifinite.

(5)

12.17. In this section we consider some results which show that the equivalent conditions in Theorem 12.15 are not always satisfied. We begin with two general remarks. Let 𝒩 ⊂ ℳ ⊂ ℬ(ℋ ) be von Neumann algebras. If ℳ and 𝒩 are semifinite, then, for every E ∈ P (ℳ, 𝒩 ), the modular automorphism group {𝜎tE }t∈ℝ ⊂ Aut(𝒩 ′ ∩ ℳ) is implemented by a unitary representation ℝ ↦ U(𝒩 ′ ∩ ℳ).

(1)

Indeed, let 𝜑 and 𝜓 be n.s.f. traces on ℳ and 𝒩 , respectively, and let E ∈ P (ℳ, 𝒩 ) be uniquely determined such that 𝜑 = 𝜓 ◦ E (see 12.3). Then, by 11.15.(2), 𝜎tE = 𝜎t𝜓 ◦ E |𝒩 ′ ∩ℳ = 𝜎t𝜑 = 𝜄(t ∈ ℝ), and hence, for any F ∈ P (ℳ, 𝒩 ), we have wt = [DF ∶ DE]t ∈ 𝒩 ′ ∩ ℳ and 𝜎tF = Ad(wt )(t ∈ ℝ) (see 11.15.(4), 11.15.(5)). If ℳ and 𝒩 satisfy the equivalent conditions in 12.15, then for every E ∈ P (ℳ, 𝒩 ) there exists 𝜑 ∈ Wnsf (𝒩 ′ ∩ ℳ) such that

𝜎tE

=

𝜎t𝜑 (t

(2)

∈ ℝ).

Indeed, let ν be an n.s.f. weight on 𝒵 (𝒩 ) and 𝜑 = ν ◦ E c . Then for every x ∈ 𝒩 ′ ∩ ℳ we have c 𝜓 ◦ Ec E E (12.15.(1)): 𝜎t (x) = 𝜎t (x) = 𝜎t (x) = 𝜎t𝜑 (x)(t ∈ ℝ). Haagerup (1977) showed that there exists an approximately finite dimensional type II∞ factor ℛ ⊂ ℬ(ℋ ), with ℋ a separable Hilbert space, and an abelian von Neumann algebra 𝒜 ⊂ ℛ such that the relative commutent 𝒜 c = 𝒜 ′ ∩ ℛ is of type III. Then ℛ and 𝒜 are semifinite, hence P (ℛ, 𝒜 ) ≠ ∅, but they do not satisfy the equivalent conditions in Theorem 12.15, as is easily seen using (1), (2) and ([L], 10.29). From this example of Haagerup, it also follows that there exists an n.s.f. weight 𝜑 on the von Neumann algebra ℳ = ℛ such, that the centralizer ℳ 𝜑 is of type III. Indeed, since 𝒜 ⊂ ℬ(ℋ ) is abelian and ℋ is separable, there exists a positive operator a ∈ 𝒜 , 0 ≤ a ≤ 1, s(a) = 1, such that 𝒜 is the von Neumann algebra generated by a, that is, 𝒜 = {a}′′ (Str̆atil̆a & Zsidó, 1977–1979, Prop. 8.14). Let 𝜏 be an n.s.f. trace on ℛ and 𝜑 = 𝜏a an n.s.f. weight on ℳ = ℛ. Since 𝜎t𝜑 = Ad(ait )(t ∈ ℝ), it follows that ℳ 𝜑 = ℛ ∩ {a}′ = 𝒜 ′ ∩ ℛ is of type III. By Corollary 12.10, we see that in this case there is no 𝜎 𝜑 -invariant n.s.f. ℳ 𝜑 -valued weight on ℳ. Another instance when P (ℳ, 𝒩 ) ≠ ∅ but ℳ and 𝒩 do not satisfy the equivalent conditions in Theorem 12.15 is the case of the continuous decomposition ℳ = ℛ(𝒩 , 𝜃) of a properly infinite

Existence and Uniqueness of Operator-Valued Weights

157

W ∗ -algebra ℳ (see 23.7). In this case there are no nonzero normal conditional expectations of ℳ onto 𝒩 (Haagerup, U. [1979a]). 12.18. Let ℳ, 𝒩 be W ∗ -algebras and 𝜑 a normal weight on ℳ. We shall identify 𝒩 with ̄ 𝒩 ⊂ ℳ ⊗ ̄ 𝒩 . As in Section 9.8, we define a normal operator-valued weight E 𝜑 ∶ 1ℳ ⊗ 𝒩 ̄ 𝒩 )+ → 𝒩̄ + by (ℳ ⊗ ̄ 𝜓)(x) (x ∈ (ℳ ⊗ ̄ 𝒩 )+ , 𝜓 ∈ 𝒩 + ); E𝒩𝜑 (x)(𝜓) = (𝜑 ⊗ ∗

(1)

E𝒩𝜑 is called the Fubini mapping associated with 𝜑. It is easy to check that E𝒩𝜑 is semifinite (resp. faithful) if and only if 𝜑 is semifinite (resp. faithful). Using Combes’ theorem (2.6), (1) has the extension ̄ 𝜓, 𝜓 ◦ E𝒩𝜑 = 𝜑 ⊗

(2)

valid for any normal weight 𝜓 on 𝒩 . If 𝜑 is an n.s.f. weight on ℳ, then equation (2) means that ̄ 𝜄𝒩 , E𝒩𝜑 = 𝜑 ⊗

(3)

where the tensor product is defined as in Corollary 12.8. 12.19. Notes. The results contained in this section are due to Haagerup (1979a). Proposition 12.12 contains an improvement due to Connes (1980) and Proposition 12.14 is a classical result. For our exposition, we have used Connes (1980) and Haagerup (1979a).

CHAPTER III

Groups of Automorphisms

13 Groups of Isometries on Banach Spaces In this section, we describe the general framework for the spectral analysis of groups of isometrics on Banach spaces. 13.1. Let 𝒳 be a Banach space and 𝒳∗ ⊂ 𝒳 ∗ a closed linear subspace. Besides the norm topologies, we shall also consider the weak topologies w = 𝜎(𝒳 , 𝒳∗ ) on 𝒳 and w∗ = 𝜎(𝒳∗ , 𝒳 ) on 𝒳∗ . Consider the following conditions on the pair (𝒳 , 𝒳∗ ): ‖x‖ = sup{|𝜌(x)|; 𝜌 ∈ 𝒳∗ , ‖𝜌‖ ≤ 1} for every x ∈ 𝒳 ; w if 𝒦 ⊂ 𝒳 is w-compact, then co (𝒦 ) ⊂ 𝒳 is also w-compact; w if𝒦 ⊂ 𝒳∗ is w∗ -compact, then co ∗ (𝒦 ) ⊂ 𝒳∗ is also w∗ -compact.

(1𝒳 ) (2𝒳 ) (3𝒳 )

Lemma 1. Let (𝒳 , 𝒳∗ ) be a pair-satisfying conditions (1𝒳 ), (2𝒳 ) and 𝜇 a bounded regular Borel measure on a separable locally compact Hausdorff space S. For every w-continuous norm-bounded function x (⋅) ∶ S → 𝒳 , there exists a unique element x ∈ 𝒳 such that 𝜌(x) =

∫S

𝜌(x (s)) d𝜇(s) (𝜌 ∈ 𝒳∗ ).

(1)

Proof. The equation f (𝜌) =

∫S

𝜌(x (s)) d𝜇(s) (𝜌 ∈ 𝒳∗ )

defines a linear form f on 𝒳∗ . To prove the lemma it is sufficient to show that f is 𝜎(𝒳∗ , 𝒳 )continuous or, equivalently, that f is continuous with respect to the Mackey topology 𝜏(𝒳∗ , 𝒳 ). Thus, we have to show that there exist an absolutely convex w-compact set ℒ ⊂ 𝒳 and c > 0 such that | f (𝜌)| ≤ c sup{|𝜌(x)|; x ∈ ℒ } (𝜌 ∈ 𝒳∗ ).

159

(2)

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We first assume that C = supp 𝜇 is compact. Then we have | f (𝜌)| ≤ ‖𝜇‖ sup{|𝜌(x (s))|; s ∈ C} (𝜌 ∈ 𝒳∗ ).

(3)

Since C is compact and x (⋅) is w-continuous, the set x (C ) ⊂ 𝒳 is w-compact. Then the set 𝒦 = w {𝜆⋅x (s); s ∈ C, |𝜆| = 1} is w-compact, and hence ℒ = co (𝒦 ) is absolutely convex and w-compact by condition (2𝒳 ). Since x (C ) ⊂ ℒ , inequality (2) follows from (3). In the general case, there exists an increasing sequence {Cn }n≥1 of compact subsets of S such that |𝜇|(S∖Cn ) → 0. By the first part or the proof, there exist xn ∈ 𝒳 such that 𝜌(xn ) =

∫Cn

𝜌(x (s)) d𝜇(s)

(𝜌 ∈ 𝒳∗ , n ≥ 1).

Then, for every 𝜌 ∈ 𝒳∗ , we have |𝜌(xn ) − f (x)| ≤ ‖𝜌‖ sup{‖x (s)‖; s ∈ S}|𝜇|(S∖Cn ) → 0

(4)

and using condition (1𝒳 ) it follows that {xn }n≥1 is a Cauchy sequence in 𝒳 . If x ∈ 𝒳 is the limit of this sequence, it then follows from (4) that 𝜌(x) = f (𝜌) for all 𝜌 ∈ 𝒳∗ . The uniqueness of the element x satisfying (1) follows obviously using (1𝒳 ). The unique element x ∈ 𝒳 satisfying (1) will be denoted by x=

∫S

x (s) d𝜇(s).

Consider now two pairs (𝒳 , 𝒳∗ ) and (𝒴 , 𝒴∗ ) satisfying conditions (1𝒳 ), (2𝒳 ) and (1𝒴 ), (2𝒴 ). Let ℬ(𝒳 , 𝒴 ) be the Banach space or all bounded linear operators 𝒳 → 𝒴 and ℬw (𝒳 , 𝒴 ) the linear space or all w-continuous linear operators 𝒳 → 𝒴 . Using the Banach–Steinhauss theorem, it is easy to check that ℬw (𝒳 , 𝒴 ) ⊂ ℬ(𝒳 , 𝒴 ) is a norm-closed linear subspace. In particular, ℬw (𝒳 , 𝒴 ) is a Banach space. For 𝜌 ∈ 𝒴∗ and x ∈ 𝒳 define a bounded linear form 𝜌(⋅x) on ℬw (𝒳 , 𝒴 ) by 𝜌(⋅x)(T ) = 𝜌(Tx) (T ∈ ℬw (𝒳 , 𝒴 )) and define the norm-closed linear subspace ℬw (𝒳 , 𝒴 )∗ by ℬw (𝒳 , 𝒴 )∗ = lin{𝜌(⋅x); 𝜌 ∈ 𝒴∗ , x ∈ 𝒳 } ⊂ ℬw (𝒳 , 𝒴 )∗ . Lemma 2. In the above situation, if the pair (𝒳 , 𝒳∗ ) also satisfies condition (3𝒳 ), then the pair (ℬw (𝒳 , 𝒴 ), ℬw (𝒳 , 𝒴 )∗ ) satisfies conditions (1ℬw (𝒳 ,𝒴 ) ), (2ℬw (𝒳 ,𝒴 ) ). Proof. Let T ∈ ℬw (𝒳 , 𝒴 ). By (1𝒴 ) we have ‖Tx‖ = sup{|𝜌(Tx)|; 𝜌 ∈ 𝒴∗ , ‖𝜌‖ ≤ 1} for all x ∈ 𝒳 and hence ‖T ‖ = sup{|𝜌(Tx)|; 𝜌 ∈ 𝒴∗ , x ∈ 𝒳 , ‖𝜌‖ ≤ 1, ‖x‖ ≤ 1} ≤ sup{|𝜑(T )|; 𝜑 ∈ ℬw (𝒳 , 𝒴 )∗ , ‖𝜑‖ ≤ 1‖ ≤ ‖T ‖. This proves that condition (1ℬw (𝒳 ,𝒴 ) ) is satisfied. Note that each separately (w, w∗ )-continuous bilinear form 𝒳 × 𝒴∗ ∋ (x, 𝜌) ↦ ⟨x, 𝜌⟩ ∈ ℂ defines a unique element T ∈ ℬw (𝒳 , 𝒴 ) such that 𝜌(Tx) = ⟨x, 𝜌⟩ for x ∈ 𝒳 and 𝜌 ∈ 𝒴∗ .

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161

Now let 𝒦 ⊂ ℬw (𝒳 , 𝒴 ) be a w-compact set and denote by 𝔗 the convex set of all regular Borel probability measures on 𝒦 . For each 𝜇 ∈ 𝔗, we can define a bilinear form ⟨., .⟩𝜇 on 𝒳 × 𝒴∗ by ⟨x, 𝜌⟩𝜇 =

∫𝒦

𝜌(Tx) d𝜇(T )

(x ∈ 𝒳 , 𝜌 ∈ 𝒴∗ ).

We show that ⟨., .⟩𝜇 is separately (w, w∗ )-continuous. Let 𝜌 ∈ 𝒴∗ be fixed. The mapping ℬw (𝒳 , 𝒴 ) ∋ T ↦ 𝜌◦T ∈ 𝒳∗ is continuous with respect to the w-topology on ℬw (𝒳 , 𝒴 ) and the w∗ -topology on 𝒳∗ , so that the set {𝜌◦T; T ∈ 𝒦 } ⊂ 𝒳∗ is w∗ -compact. According to condition (3𝒳 ), it follows that the w∗ -closed absolutely convex envelope ℒ ⊂ 𝒳 of {𝜌◦T; T ∈ 𝒦 } is w∗ -compact. We have |⟨x, 𝜌⟩𝜇 | =

∫𝒦

|𝜌(Tx)|d|𝜇|(T ) ≤ ‖𝜇‖ sup{|𝜑(x)|; 𝜑 ∈ ℒ }.

Thus, the mapping 𝒳 ∋ x ↦ ⟨x, 𝜌⟩𝜇 is continuous with respect to the Mackey topology 𝜏(𝒳 , 𝒳∗ ) and hence also with respect to the weak topology w = 𝜎(𝒳 , 𝒳∗ ). Similarly, for each fixed x ∈ 𝒳 , the mapping 𝒴∗ ∋ 𝜌 ↦ ⟨x, 𝜌⟩𝜇 is w∗ -continuous. Consequently, for every measure 𝜇 ∈ 𝔗 there exists a unique element T𝜇 ∈ ℬw (𝒳 , 𝒴 ) such that 𝜌(T𝜇 x) =

∫𝒦

𝜌(Tx) d𝜇(T )

(𝜌 ∈ 𝒴∗ , x ∈ 𝒳 ).

(5)

Since these equalities are valid in particular for the Dirac measures on 𝒦 , it follows that {T𝜇 ; 𝜇 ∈ 𝔗} is a convex set containing 𝒦 . Thus, in order to check condition (2ℬw (𝒳 ,𝒴 ) ), it is sufficient to show that the set {T𝜇 ; 𝜇 ∈ 𝔗} ⊂ ℬw (𝒳 , 𝒴 ) is w-compact. By the Alaoglu theorem, 𝔗 is a 𝜎(ℳ(𝒦 ), 𝒞 (𝒦 ))compact subset of ℳ(𝒦 ) = 𝒞 (𝒦 )∗ , hence it is enough to show that the mapping 𝔗 ∋ 𝜇 ↦ T𝜇 ∈ ℬw (𝒳 , 𝒴 ) is continuous with respect to the corresponding topologies. Thus, we have to show that for each F ∈ ℬw (𝒳 , 𝒴 )∗ the mapping 𝔗 ∋ 𝜇 ↦ F (T𝜇 ) is 𝜎(ℳ(𝒦 ), 𝒞 (𝒦 ))-continuous. Indeed, the function 𝒦 ∋ T ↦ F (T ) belongs to 𝒞 (𝒦 ) and therefore the mapping 𝔗 ∋ 𝜇 ↦ ∫𝒦 F (T ) d𝜇(T ) is 𝜎(ℳ(𝒦 ), 𝒞 (𝒦 ))-continuous, while from (5) it follows that F (T𝜇 ) = ∫𝒦 F (T ) d𝜇(T ). Note that x=

∫S

x (s) d𝜇(s) ∈ 𝒳 , T ∈ ℬw (𝒳 , 𝒴 ) ⇒ Tx =

∫S

Tx (s) d𝜇(s) ∈ 𝒴 .

(6)

13.2. Let (𝒳 , 𝒳∗ ) be a pair-satisfying conditions (1𝒳 ), (2𝒳 ) and let G be a separable locally compact group with neutral element e ∈ G. A continuous representation of G on 𝒳 is a mapping U ∶ G → ℬ𝜔 (𝒳 ) such that Ue = 𝜄x , Ust = Us Ut and ‖Ut ‖ = 1 for s, t ∈ G, and the functions G ∋ t ↦ 𝜌(Ut x) ∈ ℂ

(x ∈ 𝒳 , 𝜌 ∈ 𝒳∗ )

are continuous. Sometimes it is necessary to impose stronger continuity conditions, such as the norm-continuity of the functions G ∋ t ↦ Ut x ∈ 𝒳

(x ∈ 𝒳 )

(CU )

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or the norm-continuity of the functions G ∋ t ↦ 𝜌◦Ut ∈ 𝒳∗

(x ∈ 𝒳∗ ).

(C∗U )

Let ℳ(G) be the convolution Banach algebra of bounded regular Borel measures on G; the *-subalgebra of those measures that are absolutely continuous with respect to the Haar measure can be identified with ℒ 1 (G). Let 𝜇 ∈ ℳ(G). For each x ∈ 𝒳 , the function G ∋ t ↦ Ut x ∈ 𝒳 is norm-bounded and w-continuous, so that Lemma 1/13.1 assures us that there is a well-defined element U𝜇 x such that U𝜇 x = ∫G Ut x d𝜇(t). We thus obtain an element U𝜇 ∈ ℬ(𝒳 ) with ‖U𝜇 ‖ ≤ ‖𝜇‖. The mapping ℳ(G) ∋ 𝜇 ↦ U𝜇 ∈ ℬ(𝒳 ) is a Banach algebra homomorphism, in particular we have U𝜇∗ν = U𝜇 Uν (𝜇, ν ∈ ℳ(G)). Indeed, for any x ∈ 𝒳 we have ( U𝜇 Uν x = =

∫

Us Uν x d𝜇(s) =

∫ ∫

∫

Us

Ust x d𝜇(s) dν(t) =

∫ ∫

) Ut x dν(t) d𝜇(s)

Ur x d(𝜇 ∗ ν)(r) = U𝜇∗ν x.

For the Dirac measures 𝛿t , we obviously have U𝛿t = Ut (t ∈ G). On the other hand, the set {Uf x; f ∈ ℒ 1 (G), x ∈ 𝒳 } is w-dense in X. More precisely, we have the following result: Lemma. Let {Vi }i∈I be a fundamental system of neighborhoods of the neutral element of G. For each i ∈ I, let fi be a positive continuous function on G with compact support supp fi ⊂ Vi and w

∫ fi (t)dt = 1. Then Ufi x → x for all x ∈ 𝒳 . i∈I

Proof. Let x ∈ 𝒳 , 𝜌 ∈ 𝒳∗ , and 𝜀 > 0. Since U is a continuous representation, there exists i ∈ I such that |𝜌(Ut x − x)| < 𝜀 for t ∈ Vi . Then |𝜌(Ufi x − x)| = | ∫V 𝜌(Ut x − x)fi (t) dt| ≤ 𝜀. i

Note that the strong continuity condition (CU ) implies that ‖Ufi x − x‖ → 0 for all x ∈ 𝒳 . i∈I

If the pair (𝒳 , 𝒳∗ ) also satisfies condition (3𝒳 ), then U𝜇 ∈ ℬw (𝒳 ) for 𝜇 ∈ ℳ(G). Indeed, for any 𝜌 ∈ 𝒳∗ , the function G ∋ t ↦ 𝜌 ◦ Ut ∈ 𝒳∗ is norm-bounded and w∗ -continuous. Condition (3𝒳 ) shows that we can apply Lemma 1/13.1 to the pair (𝒳∗ , 𝒳 ) and obtain an element 𝜌′ ∈ 𝒳∗ such that 𝜌′ (x) = ∫ 𝜌(Ut x) d𝜇(t) = 𝜌(U𝜇 x) for x ∈ 𝒳 . It follows that 𝜌 ◦ U𝜇 = 𝜌′ ∈ 𝒳∗ for each 𝜌 ∈ 𝒳∗ , that is, U𝜇 ∈ ℬw (𝒳 ). 13.3 Lemma. Consider two pairs (𝒳 , 𝒳∗ ) and (𝒴 , 𝒴∗ ) satisfying conditions (1𝒳 ), (2𝒳 ), (3𝒳 ) and (1𝒴 ), (2𝒴 ). Consider also two continuous representations U ∶ G → ℬw (𝒳 ) and V ∶ G ↦ ℬw (𝒴 ) of the separable locally compact group G on 𝒳 and 𝒴 , respectively, and assume that at least one of the strong continuity conditions (CU ) or (C∗V ) holds. Then the equation 𝔖s T = Vs TU−1 s

(T ∈ ℬw (𝒳 , 𝒴 ), s ∈ G)

defines a continuous representation 𝔖 ∶ G ↦ ℬw (ℬw (𝒳 , 𝒴 )) and for every 𝜇 ∈ ℳ(G) we have 𝔊𝜇 ∈ ℬw (ℬw (𝒳 , 𝒴 )).

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163

Proof. By Lemma 2/13.1, the pair (ℬw (𝒳 , 𝒴 ), ℬw (𝒳 , 𝒴 )∗ ) satisfies conditions (1ℬw (𝒳 ,𝒴 ) ), (2ℬw (𝒳 ,𝒴 ) ). It is clear that 𝔖e is the identity mapping, 𝔖st = 𝔖s 𝔖t , ‖𝔖s ‖ ≤ ‖Vs ‖‖U−1 s ‖ = 1 and 𝔖s ∈ ℬw (ℬw (𝒳 , 𝒴 )) for s, t ∈ G. To check the continuity of 𝔖, we must show that for each T ∈ ℬw (𝒳 , 𝒴 ) and each F ∈ ℬw (𝒳 , 𝒴 )∗ the function G ∋ s ↦ F (𝔖s T ) is continuous. Clearly, we x). may assume that F = 𝜌(⋅x) with 𝜌 ∈ 𝒴∗ and x ∈ 𝒳 . Then F (𝔖s T ) = 𝜌(Vs TU−1 s x) = (𝜌◦Vs )(TUs−1 Using, for instance, condition (C∗V ), for s → s0 we obtain ‖F (𝔖s T ) − F (𝔖s0 T )‖ ≤ ‖𝜌 ◦ Vs − 𝜌 ◦ Vs0 ‖‖T ‖‖x‖ + |(𝜌◦Vs0 )(TUs−1 x − TUso−1 x)| → 0. Consider now 𝜇 ∈ ℳ(G). In order to check that 𝔖𝜇 ∈ ℬw (ℬw (𝒳 , 𝒴 )), we have to show that F◦𝔖𝜇 ∈ ℬw (𝒳 , 𝒴 )∗ for any F ∈ ℬw (𝒳 , 𝒴 )∗ , but it is sufficient to consider only F = 𝜌(⋅x) with 𝜌 ∈ 𝒴∗ and x ∈ 𝒳 . Then (F◦𝔖𝜇 )(T ) =

∫

(𝜌◦Vs )(TU−1 s x) d𝜇(s).

Assume first that supp 𝜇 = C is compact and let 𝜀 > 0. Using, for instance, the strong continuity condition (C∗V ), it follows that the set {𝜌◦Vs ; s ∈ C} ⊂ 𝒴∗ is norm-compact, so that there exists a finite set {𝜌1 , … , 𝜌n } ⊂ 𝒴∗ such that the union of the sets Ek = {s ∈ G; ‖𝜌◦Vs − 𝜌k ‖ ≤ 𝜀}, (k = 1, … , n), contains {𝜌◦Vs ; s ∈ C}.⋃Then S1 ⋃ = E1 and Sk = Ek ∖(E1 ∪ ⋯ ∪ Ek−1 )(k = 2, … , n) are mutually disjoint Borel sets and k Sk = k Ek ⊃ {𝜌◦Vs ; s ∈ C}. With xk = ∫S U−1 s x d𝜇(s) ∈ k 𝒳 , (k = 1, … , n), it is easy to see that n ‖ ‖ ∑ ‖ ‖ 𝜌k (⋅xk )‖ ≤ 𝜀‖x‖‖𝜇‖. ‖F◦𝔖𝜇 − ‖ ‖ k=1 ‖ ‖

Since x > 0 was arbitrary, it follows that F◦𝔖𝜇 ∈ ℬw (𝒳 , 𝒴 )∗ . In the general case, there exists an increasing sequence {Cn }n≥1 of compact sets in G with |𝜇|(G∖Cn ) → 0. By the preceding paragraph, there exist Fn ∈ ℬw (𝒳 , 𝒴 )∗ such that Fn (T ) = ∫C F (𝔖s T ) d𝜇(s)(n ≥ 1). It follows that the sequence {Fn }n≥1 is norm-convergent to F◦𝔖𝜇 , and so n F◦𝔖𝜇 ∈ ℬw (𝒳 , 𝒴 )∗ . 13.4. Let 𝒳 be a Banach space and 𝒳∗ = 𝒳 ∗ Then the pair (𝒳 , 𝒳 ∗ ) satisfies condition (1𝒳 ) by the Hahn–Banach theorem, and condition (2𝒳 ) by a theorem of Krein and Smulian (Dunford & Schwartz, 1958, 1963, V.6.4); condition (3𝒳 ) is an obvious consequence of the Alaoglu theorem. In this case, any continuous representation U ∶ G → ℬw (𝒳 ) also satisfies the strong continuity condition (CU ). More precisely, we have the following result: Lemma. Let U ∶ G → ℬw (𝒳 ) be a homomorphism of the locally compact group G into the group of all bounded linear bijections on the Banach space 𝒳 . The following statements are equivalent: (i) the mappings G ∋ t ↦ 𝜌(Ut x) ∈ ℂ are continuous for all x ∈ 𝒳 , 𝜌 ∈ 𝒳 ∗ ; (ii) the mappings G ∋↦ Ut x ∈ 𝒳 are norm-continuous for all x ∈ 𝒳 ; (iii) the mapping G × 𝒳 ∋ (t, x) ↦ Ut x ∈ 𝒳 is norm-continuous.

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Proof. It is clear that (iii) ⇒ (ii) ⇒ (i). (i) ⇒ (ii). Using the Banach–Steinhauss theorem, from (i) we infer that 𝜆K = sup{‖Ut ‖; t ∈ K} < +∞ for every compact set K ⊂ G. The set 𝒮 of those elements x ∈ 𝒳 such that the mapping G ∋ t ↦ Ut x ∈ 𝒳 is norm-continuous is a norm-dense linear subspace of 𝒳 and so also 𝜎(𝒳 , 𝒳∗ )closed. On the other hand, for x ∈ 𝒳 , f ∈ ℒ 1 (G) with support contained in a compact neighborhood K of 0 in G and t ∈ G, t → e, we have (see Hewitt & Ross, 1963, 1970, 20.4): ‖ ‖ ‖ ‖Ut (Uf x) − Uf x‖ = ‖ ‖Ut ∫ f (s)Us x ds − ∫ f (s)Us x ds‖ ‖ ‖ ‖ ‖ ‖ =‖ ‖∫ f (s)Uts x ds − ∫ f (s)Us x ds‖ ‖ ‖ ‖ ‖ −1 ‖ =‖ ‖∫ f (t s)Us x ds − ∫ f (s)Us x ds‖ ‖ ‖ ≤ 𝜆K ‖x‖

∫

| f (t−1 s) − f (s)| ds → 0.

Thus, 𝒮 contains the set {Uf x; x ∈ 𝒳 , f ∈ ℒ 1 (G), supp f compact} which, by (i) and Lemma 13.2, is 𝜎(𝒳 , 𝒳 ∗ )-dense in 𝒳 . Hence 𝒮 = 𝒳 . (ii) ⇒ (iii). Let K ⊂ G be a compact set. From (ii) it follows that 𝜆K = sup{‖Ut ‖; t ∈ K} < +∞. If K ∋ tn ↦ t ∈ G and xn → x in 𝒳 , then, again by assumption (ii), we get ‖Utn xn − Ut x‖ ≤ ‖Utn (xn − x)‖ + ‖Utn x − Ut x‖ ≤ 𝜆K ‖xn − x‖ + ‖Utn x − Ut x‖ → 0. 13.5. Let ℳ be a W∗ -algebra with predual ℳ∗ ⊂ ℳ ∗ . Then ℳ = (ℳ∗ )∗ , condition (1ℳ ) is clearly satisfied for the pair (ℳ, ℳ∗ ), (2ℳ ) follows from the Alaoglu theorem and (3ℳ ) follows from the ̆ Krein–Smulian theorem (Dunford & Schwartz, 1958, 1963, V.6.4). The topology w = 𝜎(ℳ, ℳ∗ ) is just the usual w-topology on ℳ. Recall ([L], C.5.1) that on ℳ we can consider also the topologies s, s∗ , as well as the Mackey topology 𝜏w associated with the w-topology. A representation of the locally compact group G by *-automorphisms of ℳ, or an action (2.24) of G on ℳ, is a group homomorphism 𝜎 ∶ G → Aut(ℳ). The weak continuity condition (13.2) for such a representation is just the continuity of the mapping 𝜎 with respect to the p-topology (2.23) on Aut(ℳ), while the strong continuity condition (C∗𝜎 ) amounts to continuity with respect to the u-topology (2.23) on Aut(ℳ). Actually, these two conditions are equivalent, as the following shows. Proposition. Let 𝜎 ∶ G → Aut(ℳ) be an action of the locally compact group G on the W∗ -algebra ℳ. The following statements are equivalent: the mappings G ∋ t ↦ 𝜎t (x) ∈ ℳ are w-continuous (x ∈ ℳ); the mappings G ∋ t ↦ 𝜎t (x) ∈ ℳ are s-continuous (x ∈ ℳ); the mappings G ∋ t ↦ 𝜎t (x) ∈ ℳ are s∗ -continuous (x ∈ ℳ); the mappings G ∋ t ↦ 𝜎t (x) ∈ ℳ are 𝜏w -continuous (x ∈ ℳ); for every 𝜎(ℳ∗ , ℳ)-compact subset ℒ of ℳ∗ , the mapping G × ℒ ∋ (t, 𝜑) ↦ 𝜑◦𝜎t ∈ ℳ∗ is continuous with respect to the 𝜎(ℳ∗ , ℳ)-topology on ℒ and ℳ∗ ; (vi) the mappings G ∋ t ↦ 𝜑◦𝜎t ∈ ℳ∗ are norm-continuous (𝜑 ∈ ℳ∗ ); (vii) the mapping G × ℳ∗ ∋ (t, 𝜑) ↦ 𝜑◦𝜎t ∈ ℳ∗ is norm-continuous. (i) (ii) (iii) (iv) (v)

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165

Proof. It is clear that (iv) ⇒ (iii) ⇒ (ii) ⇒ (i) and it is easy to check that (iv) ⇒ (v) ⇒ (i). Since (ℳ∗ )∗ = ℳ, it follows from Lemma 13.4 that (i)⇔ (vi) ⇔ (vii). By a result due to Akemann ((1967); Str̆atil̆a & Zsidó, 1977, 2005, Cor. 8.17) the restriction of the Mackey topology 𝜏w to the closed unit ball of ℳ coincides with the restriction of the s∗ -topology; since ‖𝜎t (x)‖ = ‖x‖, it follows that (iii) ⇔ (iv). For x ∈ ℳ, 𝜑 ∈ ℳ∗ , t ∈ G, we have 𝜑((𝜎t (x) − x)∗ (𝜎t (x) − x)) = 𝜑(𝜎t (x∗ x)) − 𝜑(𝜎t (x)∗ x) − 𝜑(x∗ 𝜎t (x)) + 𝜑(x∗ x). If t ∈ G converges to e ∈ G, it follows from (i) that the right hand side of this equation converges to 0. Thus, (i) ⇒ (ii) and (i) ⇒ (iii). Recall (2.24) that if 𝜎 satisfies the equivalent conditions of the above proposition, we say that 𝜎 is a continuous action of G on ℳ. 13.6. Notes. The exposition in this section is based on the article of Arveson (1974).

14 Spectra and Spectral Subspaces In this section we introduce the spectral subspaces associated to a continuous representation of a locally compact abelian group on a Banach space, together with their main properties and some applications. ̂ we shall ̂ For t ∈ G and 𝛾 ∈ G 14.1. Let G be a locally compact abelian group with dual group G. denote by ⟨t, 𝛾⟩ the value of the character 𝛾 at t. We shall denote the group operation by addition; in ̂ particular, 0 will denote the neutral element of G or of G. For 𝜇 ∈ ℳ(G), f ∈ ℒ 1 (G), we define the Fourier transforms 𝜇, ̂ ̂f by 𝜇(𝛾) ̂ =

∫

⟨t, 𝛾⟩ d𝜇(t), ̂f (𝛾) =

∫

f (t)⟨t, 𝛾⟩ dt

̂ (𝛾 ∈ G).

̂ ̂f (𝛾) = 0} and for an ideal ℐ of ℒ 1 (G) define its “hull” For f ∈ ℒ 1 (G), we write Z( f ) = {𝛾 ∈ G; ̂ Clearly, Z(ℐ̄ ) = Z(ℐ ). Recall the following to be the closed set Z(ℐ ) = ∩{Z( f ); f ∈ ℐ } ⊂ G. important result: The Maximal Tauberian Theorem. Let f ∈ ℒ 1 (G) and let ℐ ⊂ ℒ 1 (G) be a closed ideal such that Z(ℐ ) ⊂ Z( f ). If the intersection of the boundaries of the sets Z(ℐ ) and Z( f ) does not contain any nonempty perfect set, then f ∈ ℐ . In particular, for a closed ideal ℐ ⊂ ℒ 1 (G) we have f ∈ ℒ 1 (G), Z(ℐ ) ⊂ int Z( f ) ⇒ f ∈ ℐ .

(1)

Z(ℐ ) = ∅ ⇒ ℐ = ℒ (G).

(2)

1

Z(ℐ ) = {𝛾} ⇒ ℐ = {f ∈ ℒ (G); ̂f (𝛾) = 0}. 1

(3)

̂ be a closed set. Then the set of all closed ideals ℐ ⊂ ℒ 1 (G) with Z (ℐ ) = F has a Let F ⊂ G greatest element 𝒦 (F) = {f ∈ ℒ 1 (G); F ⊂ Z( f )}

166

Groups of Automorphisms

and a smallest element ℐ (F) which is the closure of each of the following ideals: ℐ0 (F) = {f ∈ ℒ 1 (G); F ⊂ int Z( f )} ℐ00 (F) = {f ∈ ℒ 1 (G); supp ̂f compact, F ⊂ int Z( f )}. In particular, the ideal {f ∈ ℒ 1 (G); supp ̂f compact} is dense in ℒ 1 (G). Note that every dense ideal ℐ of ℒ 1 (G) contains an approximate unit of ℒ 1 (G) with elements of norm ≤ 1. Indeed, if {fi }i∈I is an approximate unit of ℒ 1 (G) with ‖ fi ‖i = 1, then, for each i ∈ I and each n ≥ 1, there exists an element hi,n ∈ ℐ such that ‖ fi − hi,n ‖1 < 1∕n and the family {hi,n ∕‖hi,n ‖1 }i,n ⊂ ℐ is an approximate unit of ℒ 1 (G). Consequently, there exists an approximate unit {ki }i∈I of ℒ 1 (G) with ‖ki ‖1 ≤ 1 and supp k̂ i compact (i ∈ I). In particular, for every f ∈ ℒ 1 (G) and every 𝜀 > 0 there exists k ∈ ℒ 1 (G) with ‖k‖1 ≤ 1 and supp k̂ compact such that ‖ f − f ∗ k‖1 < 𝜀.

(4)

Moreover, ̂ and every open set D ⊂ G ̂ with C ⊂ D for every compact set C ⊂ G 1 there exists a function f ∈ ℒ (G) with supp ̂f ⊂ D such that ̂f (𝛾) = 1 for any 𝛾 ∈ C.

(5)

For all the above results, and for various other results in harmonic analysis which we shall use in the sequel, we refer to Hewitt and Ross (1963, 1970); Rudin (1962); Str̆atil̆a (1967, 1968). 14.2. Let (𝒳 , 𝒳∗ ) be a pair consisting of a Banach space 𝒳 and a closed linear subspace 𝒳∗ ⊂ 𝒳 ∗ which satisfies conditions (1𝒳 ) and (2𝒳 ) of Section 13.1. Let U ∶ G → ℬw (𝒳 ) be a continuous representation of the locally compact abelian group G on 𝒳 such that U𝜇 ∈ ℬw (𝒳 ) for all 𝜇 ∈ ℳ(G); this last condition is implied, for instance, by condition (3𝒳 ). For each element x ∈ 𝒳 , we consider the closed ideal ℐxU = {f ∈ ℒ 1 (G); Uf x = 0} ⊂ ℒ 1 (G) and define the spectrum of x with respect to U by ̂ f ∈ ℒ 1 (G), Uf x = 0 ⇒ ̂f (𝛾) = 0}. SpU (x) = Z(ℐxU ) = {𝛾 ∈ G; ̂ 0 ≠ 𝜆 ∈ ℂ we have Proposition. For x, y ∈ 𝒳 ; 𝜇, 𝜇1 , 𝜇2 ∈ ℳ(G); t ∈ G; 𝛾 ∈ G; SpU (𝜆x) = SpU (x) SpU (x + y) ⊂ SpU (x) ∪ SpU (y) SpU (U𝜇 x) ⊂ SpU (x) ∩ supp 𝜇̂

(1) (2) (3)

SpU (Ut x) = SpU (x) SpU (x) = ∅ ⇔ x = 0

(4) (5)

Spectra and Spectral Subspaces

167

SpU (x) = {𝛾} ⇔ x ≠ 0 and Us x = (s, 𝛾)x for s ∈ G SpU (x) = {0} ⇔ Us x = x ≠ 0 for s ∈ G ̂ 𝜇̂ 1 (𝜔) = 𝜇̂ 2 (𝜔)} ⇒ U𝜇 x = U𝜇 x SpU (x) ⊂ int {𝜔 ∈ G; 1 2 ̂ 𝜇(𝜔) SpU (x) ⊂ int {𝜔 ∈ G; ̂ = 1} ⇒ U𝜇 x = x.

(6) (7) (8) (9)

U . Proof. (1) follows from ℐxU = ℐ𝜆xU and (2) from ℐxU ∩ ℐyU ⊂ ℐx+y

Let us prove (3). If 𝛾 ∉ SpU (x), there exists f ∈ ℒ 1 (G) with Uf x = 0 and ̂f (𝛾) ≠ 0, so that Uf U𝜇 x = U𝜇 Uf x = 0 and ̂f (𝛾) ≠ 0, hence 𝛾 ∉ SpU (U𝜇 x). If 𝛾 ≠ supp 𝜇, ̂ there exists f ∈ ℒ 1 (G) with ̂f (𝛾) = 1 and supp ̂f ∩ supp 𝜇̂ = ∅ (see 14.1.(5)); we have ( f ∗ 𝜇)̂ = ̂f𝜇̂ = 0, hence f ∗ 𝜇 = 0 and Uf U𝜇 x = Uf∗𝜇 x = 0, but ̂f (𝛾) ≠ 0, so that 𝛾 ∉ SpU (U𝜇 x). ̂ Equation (4) follows from (3) since for the Dirac measure 𝛿t we have 𝛿̂t (𝛾) = ⟨t, 𝛾⟩, (𝛾 ∈ G). 1 Let us prove (5). If x ≠ 0, then by Lemma 13.2 there exists f ∈ ℒ (G) with Uf x ≠ 0, hence ℐxU ≠ ℒ 1 (G) and using 14.1.(2) we see that SpU (x) ≠ ∅. If x = 0, then ℐxU = ℒ 1 (G) and therefore (14.1.(5)) SpU (x) = ∅. Let us prove (6). If SpU (x) = {𝛾}, then (14.1.(3)) ℐxU = {f ∈ ℒ 1 (G); ̂f (𝛾) = 0}, that is, for f ∈ ℒ 1 (G) we have Uf x = 0 ⇔ ̂f (𝛾) = 0. Then, for 𝜇 ∈ ℳ(G) we get U𝜇 x = 0 ⇔ 𝜇(𝛾) ̂ = 0. Indeed, U𝜇 x = 0 ⇒ Uf U𝜇 x = 0 ⇒ Uf ∗ 𝜇 x = 0 ⇒ ( f ∗ 𝜇)̂(𝛾) = 0 ⇒ ̂f (𝛾)𝜇(𝛾) ̂ = 0 for any f ∈ ℒ 1 (G) and therefore (14.1.(5)) 𝜇(𝛾) ̂ = 0; conversely, if 𝜇(𝛾) ̂ = 0, then we obtain similarly Uf U𝜇 x = 0 for any f ∈ ℒ 1 (G), and, using Lemma 13.2, we conclude that U𝜇 x = 0. Since 𝜇 − 𝜇(𝛾)𝛿 ̂ 0 ∈ ℳ(G) and (𝜇 − 𝜇(𝛾)𝛿 ̂ )̂(𝛾) = 0, it follows that U x = 𝜇(𝛾)x ̂ for all 𝜇 ∈ ℳ(G). Taking in particular 𝜇 = 𝛿s 0 𝜇 ̂ we obtain Us x = ⟨s, 𝛾⟩x (s ∈ G). Conversely, if Us x = ⟨s, 𝛾⟩x (s ∈ G), then Uf x = f (𝛾)x for any f ∈ ℒ 1 (G) and if x ≠ 0 it follows that ℐxU = {f ∈ ℒ 1 (G); ̂f (𝛾) = 0}, hence SpU (x) = {𝛾}. Equation (7) follows obviously from (6). ̂ 𝜇(𝜔) Let us prove (8). For 𝜇 = 𝜇1 − 𝜇2 we have SpU (x) ⊂ int {𝜔 ∈ G; ̂ = 0} by assumption, hence SpU (x) ∩ supp 𝜇̂ = ∅. Using (3) and (5) it follows that U𝜇 x = 0, hence U𝜇1 x = U𝜇2 x. ̂ Finally, (9) follows from (8) since 𝛿̂0 (𝜔) = 1 (𝜔 ∈ G). In this proof, we have also shown that ̂ 𝜇 ∈ ℳ(G), U𝜇 x = 0 ⇒ 𝜇(𝛾) SpU (x) = {𝛾 ∈ G; ̂ = 0}.

(10)

̂ Note that SpU (x) is “the support of the vector distribution ̂f ↦ Uf x,” that is, G∖Sp U (x) is the 1 ̂ with the property: f ∈ ℒ (G), supp ̂f ⊂ D ⇒ Uf x = 0. greatest open set D ⊂ G 14.3. We continue with the notation of the previous section. ̂ define the spectral subspace 𝒳 (U; E) of 𝒳 associated with U and E to be the For each set E ⊂ G w-closure of the set 𝒳0 (U; E) = {x ∈ 𝒳 ; SpU (x) ⊂ E}.

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Groups of Automorphisms

Clearly, E1 ⊂ E2 ⇒ 𝒳 (U; E1 ) ⊂ 𝒳 (U; E2 ).

(1)

̂ the spectral subspace 𝒳 (U; E) is a w-closed U-invariant Proposition. For every subset E ⊂ G, linear subspace of 𝒳 and is equal to the w-closure of the set 𝒳00 (U; E) ⊂ {x ∈ 𝒳 ; SpU (x) ⊂ E compact}. ̂ we have For every closed subset F ⊂ G 𝒳 (U; F) = 𝒳0 (U; F) = {x ∈ X; ℐ0 (F) ⊂ ℐxU } = {x ∈ X; ℐ00 (F) ⊂ ℐxU }.

(2)

̂ 𝒳 (U; D) is the w-closure of the set For every open subset D ⊂ G, {Uf x; x ∈ 𝒳 , f ∈ ℒ 1 (G), supp ̂f ⊂ D compact}. ̂ then If {Fi } is any family of closed subsets of G, 𝒳 (U;

⋂

Fi ) =

⋂

i

𝒳 (U; Fi )

(3)

𝒳 (U; Di ).

(4)

i

̂ then and if {Di } is any family of open subsets of G, 𝒳 (U;

⋃

Di ) =

i

⋁ i

Proof. By Proposition 14.2, it follows that 𝒳0 (U; E) is a U-invariant linear subspace of 𝒳 , so that its w-closure 𝒳 (U; E) is a w-closed U-invariant linear subspace of 𝒳 . By Lemma 13.2 and Proposition 14.2.(3), the set of elements in 𝒳 (U; E) of the form Uf x is w-dense in 𝒳 (U; E) and hence, using 14.1.(4) and 14.2.(3), it follows that 𝒳00 (U; E) is w-dense in 𝒳 (U; E). Consequently, w

w

𝒳 (U; E) = 𝒳0 (U; E) = 𝒳00 (U; E) .

(5)

̂ and x ∈ 𝒳0 (U; F). If f ∈ ℐ0 (F), that is, F ∩ supp ̂f = ∅, then Consider now a closed set F ⊂ G (14.2.(3), 14.2.(5)) Uf x = 0, that is, f ∈ ℐxU . It follows that 𝒳0 (U; F) ⊂ {x ∈ 𝒳 ; ℐ0 (F) ⊂ ℐxU } ⊂ {x ∈ 𝒳 ; ℐ00 (F) ⊂ ℐxU }. Conversely, if ℐ00 (F) ⊂ ℐxU , then SpU (x) = Z(ℐxU ) ⊂ Z(ℐ00 (F)) = F (14.1), hence x ∈ 𝒳0 (U; F). Thus, the above inclusions are actually equalities and it follows that 𝒳0 (U; F) =

⋂ {Ker Uf ; f ∈ ℐ0 (F)}

is w-closed, since the Uf ’s are w-continuous.

(6)

Spectra and Spectral Subspaces

169

̂ we have (14.2.(3)) {Uf x; x ∈ 𝒳 , supp ̂f ⊂ D compact} ⊂ 𝒳0 (U; D). Assume For any set D ⊂ G that D is open and let x ∈ 𝒳00 (U; D). Then K = SpU (x) ⊂ D is compact and hence (14.1.(5)) there ̂ ̂f (𝛾) = 1}. It follows (14.1.(9)) that exists f ∈ ℒ 1 (G) with supp ̂f ⊂ D compact and K ⊂ int {𝛾 ∈ G; x = Uf x with f ∈ ℒ 1 (G), supp ̂f ⊂ D, and using (5) we obtain w

𝒳 (U; D) = {Uf x; x ∈ 𝒳 , f ∈ ℒ 1 (G), supp ̂f ⊂ D compact} .

(7)

Equation (3) follows easily using (2) and (6). ⋃ To prove (4) we note first that⋃𝒳 (U; Di ) ⊂ 𝒳 (U; i Di ), so that ⋃ the w-closed linear sub⋁ space ⋃i 𝒳 (U; Di ) generated by i 𝒳⋃ (U; Di ) is contained in 𝒳 (U; i Di ). Conversely, let x ∈ 𝒳 (U; i Di ). Then K = SpU (x) ⊂ i Di is compact and there exist i1 , … , in such that K ⊂ ⋃00 n D . Using 14.1.(5), we find functions f1 , … , fn ∈ ℒ 1 (G) such that supp ̂fk ⊂ Dik (k = 1, … , n), k=1 ik ∑ ∑n n ̂ ̂ and K ⊂ int {𝛾 ∈ G; k=1 fk (𝛾) = 1}. Then (14.2) x = k=1 Ufk x with Ufk x ∈ 𝒳 (U; Dik ), and hence ⋁ x ∈ i 𝒳 (U, Di ). ̂ be a closed set. From (3) and (1), it follows that if {Di } is any family of open sets in G ̂ Let F ⊂ G ⋂ ⋂ ̄ such that F = i Di = i Di then 𝒳 (U; F) =

⋂

𝒳 (U; Di ) =

i

⋂

𝒳 (U; Di ).

(8)

i

Thus, for every fundamental system {Ni }i∈I of open and relatively compact neighborhoods of 0 in ⋂ G with i Ni = {0}, we have 𝒳 (U; F) =

⋂

𝒳 (U; F + Ni ) =

i

⋂

𝒳 (U; F + Ni ).

(9)

i

In particular, 𝒳 (U; F) =

⋂

{𝒳 (U; D); D ⊃ F open}.

̂ we have On the other hand, by (5), for every set E ⊂ G ⋁ 𝒳 (U; E) = {𝒳 (U; K); K ⊂ E compact}.

(10)

(11)

Equations (10) and (11) prove the “regularity” of the family of spectral subspaces associated with U. We have also (14.2.(5)) ̂ = X. 𝒳 (U; ∅) = {0}, 𝒳 (U; G)

(12)

̂ the spectral subspace (14.2.(6)) For 𝛾 ∈ G 𝒳 (U; {𝛾}) = {x ∈ 𝒳 ; Ut x = ⟨t, 𝛾⟩x for all t ∈ G}

(13)

170

Groups of Automorphisms

̂ In particular, will be called the eigenspace of U corresponding to the eigenvalue 𝛾 ∈ G. 𝒳 U = 𝒳 (U; {0}) = {x ∈ 𝒳 ; Ut x = x for all t ∈ G}

(14)

will be called the centralizer of U. 14.4. We shall need the following extension of 14.3.(13): ̂ each compact set K ⊂ G, and 𝜀 > 0, there exists a compact neighborhood Lemma. For each 𝛾 ∈ G, ̂ such that W of 0 in G ‖Ut x − ⟨t, 𝛾⟩x‖ ≤ 𝜀‖x‖ for x ∈ 𝒳 (U; 𝛾 + W) (t ∈ K). ̂ and f ∈ ℒ 1 (G) be such that ̂f (W0 ) = {1}. Proof. Let W0 be a compact neighborhood of 𝛾 in G 1 For each t ∈ K, we define a function ft ∈ ℒ (G) by ft (s) = f (s − t) − ⟨t, 𝛾⟩f (s) (s ∈ G). Then ̂ft (𝛾) = 0 and hence (Rudin, 1962, 2.6.3) there exist a function kt ∈ ℒ 1 (G) and a neighborhood ̂ such that k̂ t (Wt ) = {1} and ‖ ft ∗ kt ‖1 < 𝜀. Since the mapping G ∋ s ↦ fs ∈ ℒ 1 (G) is Wt of 𝛾 in G norm-continuous, there exists a neighborhood Nt of t in G such that ‖ ⋃fns ∗ kt ‖1 < 𝜀 for all s ∈ Nt . Since K is compact, there exist t1 , … , tn ∈ K such that K ⊂ i=1 Nti . Let W be a compact ̂ such that W ⊂ int(W0 ∩ Wt ∩ … ∩ Wt ). For each t ∈ K, there exists a neighborhood of 𝛾 in G 1 n ̂ ĥ t (𝜔) = 1} and ‖ ft ∗ ht }1 < 𝜀. function ht ∈ {kt1 , … , ktn } ⊂ ℒ 1 (G) such that W ⊂ int{𝜔 ∈ G; Let x ∈ 𝒳 (U; W) and t ∈ K. Then (𝛿t − ⟨t, 𝛾⟩𝛿0 )̂ coincides with ( ft ∗ ht )̂ on a neighborhood of SpU (x) and hence (14.2.(8)) ‖Ut x − ⟨t, 𝛾⟩x‖ = ‖Uft ∗ht x‖ ≤ 𝜀‖x‖. 14.5. In the setting of Section 14.2, the set Ker U = {f ∈ ℒ 1 (G); Uf = 0} is a closed ideal of ℒ 1 (G). The spectrum of the representation U is defined as being the closed set ̂ f ∈ ℒ 1 (G), Uf = 0 ⇒ ̂f (𝛾) = 0}. Sp U = Z(Ker U) = {𝛾 ∈ G; Using 14.1.(5), it is easy to check that for every w-total set 𝒳0 in 𝒳 we have Sp U =

⋃ {SpU (x); x ∈ 𝒳0 }.

(1)

̂ It is also easy to check that Sp U is “the support of the vector distribution ̂f ↦ Uf ,” that is, G∖Sp U 1 ̂ with the property: f ∈ ℒ (G), supp ̂f ⊂ D ⇒ Uf = 0. is the greatest open set D in G ̂ the following statements are equivalent: Proposition. For 𝛾 ∈ G (i) 𝛾 ∈ Sp U; ̂ (ii) 𝒳 (U; 𝛾 + W) ≠ {0} for every neighborhood W of 0 in G; (iii) there exists a net {xi } ⊂ 𝒳 , ‖xi ‖ = 1, such that lim ‖Ut xi − ⟨t, 𝛾⟩xi ‖ = 0 i

uniformly for t in compact subsets of G;

Spectra and Spectral Subspaces

171

(iv) |𝜇(𝛾)| ̂ ≤ ‖U𝜇 ‖ for all 𝜇 ∈ ℳ(G); (v) |̂f (𝛾)| ≤ ‖Uf ‖ for all f ∈ ℒ 1 (G). ̂ By 14.1.(5), there exists f ∈ ℒ 1 (G) with Proof. (i) ⇒ (ii). Let W be a neighborhood of 𝛾 in G. ̂f (𝛾) = 1 and supp ̂f ⊂ W. Since 𝛾 ∈ Sp U, we have Uf ≠ 0 so that 0 ≠ Uf x ∈ 𝒳 (U; W) for some x ∈ X. (ii) ⇒ (iii). Using Lemma 14.4, we infer, assuming (ii), that for each compact set K ⊂ G and 𝜀 > 0 there exists an element xK,𝜀 ∈ 𝒳 , ‖xK,𝜀 ‖ = 1, such that ‖Ut xK,𝜀 − ⟨t, 𝛾⟩xK,𝜀 ‖ < 𝜀 for all t ∈ K. Then {xK,𝜀 }K,𝜀 is the required net. (iii) ⇒ (iv). Let 𝜇 ∈ ℳ(G) and 𝜀 > 0. There exists a compact set K ⊂ G with |𝜇|(G∖K) < 𝜀∕4. By assumption (iii) there exists x ∈ 𝒳 , ‖x‖ ≤ 1, such that ‖Ut x − ⟨t, 𝛾⟩x‖ < 𝜀∕2|𝜇|(K) for all t ∈ K. Then ‖U𝜇 x − 𝜇(𝛾)x‖ ̂ ≤ ≤

∫G ∫K

‖Ut x − ⟨t, 𝛾⟩x‖ d|𝜇| ‖Ut x − ⟨t, 𝛾⟩x‖ d|𝜇| + 2|𝜇|(G∖K) ≤

𝜀 𝜀 + = 𝜀. 2 2

It follows that |𝜇(𝛾)| ̂ = ‖𝜇(𝛾)x‖ ̂ ≤ ‖U𝜇 x‖ + 𝜀 ≤ ‖U𝜇 ‖ + 𝜀. Since 𝜀 > 0 was arbitrary, we get |𝜇(𝛾)x| ̂ ≤ ‖U𝜇 ‖. (iv) ⇒ (v). Obvious. (v) ⇒ (i). f ∈ ℒ 1 (G) and Uf = 0, then ̂f (𝛾) = 0 by (v). Hence 𝛾 ∈ Sp U. Corollary. Let 𝒜 be the Banach algebra defined as the norm-closure of the subalgebra {Uf ; f ∈ ℒ 1 (G)} of ℬ(𝒳 ) and denote by Ω𝒜 the Gelfand spectrum of 𝒜 . Then the mapping ̂ Ω𝒜 ∋ 𝜔 ↦ 𝜔◦U ∈ Ωℒ 1 (G) = G is a homeomorphism of Ω𝒜 onto Sp U. Proof. If 𝜔 is a continuous character of 𝒜 , then the mapping 𝛾𝜔 ∶ ℒ 1 (G) ∋ f ↦ 𝜔(Uf ) is a ̂ and | f (𝛾𝜔 )| = |𝜔(Uf )| ≤ ‖Uf ‖ for every f ∈ continuous character of ℒ 1 (G); hence 𝛾𝜔 ∈ G 1 ℒ (G). The above proposition shows that 𝛾𝜔 ∈ Sp U. The mapping 𝜔 ↦ 𝛾𝜔 = 𝜔◦U is clearly continuous. If 𝛾 ∈ Sp U, the above proposition shows that the mapping Uf ↦ ̂f (𝛾) can be extended to a continuous character 𝜔𝛾 of 𝒜 , which is uniquely determined by the condition 𝜔𝛾 (Uf ) = ̂f (𝛾) ( f ∈ ℒ 1 (G)). The mapping 𝛾 ↦ 𝜔𝛾 is continuous and its inverse is just the above-defined mapping 𝜔 ↦ 𝛾𝜔 . 14.6. With the help of the set Sp U, we can describe the usual spectrum Sp(Ut ) of the operator Ut ∈ ℬ(𝒳 ): Proposition. Sp(Ut ) = {⟨t, 𝛾⟩; 𝛾 ∈ Sp U} (t ∈ G).

172

Groups of Automorphisms

Proof. If 𝛾 ∈ Sp U, then, by Proposition 14.5, there exists a net {xi } ⊂ 𝒳 , ‖xi ‖ = 1, such that ‖(Ut − ⟨t, 𝛾⟩)xi ‖ → 0, hence ⟨t, 𝛾⟩ ∈ Sp(Ut ) for t ∈ G. Since ‖Ut ‖ = ‖U−1 t ‖ = 1, we have Sp(Ut ) ⊂ 𝕋 = {z ∈ ℂ; |z| = 1}. Assume that there exists 𝜆 ∈

𝕋 , 𝜆 ∉ {⟨t, 𝛾⟩; 𝛾 ∈ Sp U} and consider two open neighborhoods V ⊂ W of the set {⟨t, 𝛾⟩; 𝛾 ∈ Sp U} with 𝜆 ∉ W. There exists a C∞ -function 𝜑 on 𝕋 which is identically equal to 1 on V and supp 𝜑 ⊂ W. Then 𝕋 ∋ z ↦ f (z) = 𝜑(z)(z − 𝜆)−1 is a C∞ -function on 𝕋 , equal to (z − 𝜆)−1 on V and supp f ⊂ W. The Fourier series associated with f is absolutely convergent, and so we can write ∑ ∑ f (z) = an zn with |an | < +∞. n∈ℤ

n∈ℤ

We consider the operator T=

∑

an Unt = U∑n an 𝛿nt ∈ ℬ(𝒳 ).

n∈ℤ

Then T(Ut − 𝜆) = (Ut − 𝜆)T = U𝜇 with 𝜇 = (𝛿t − 𝜆𝛿0 ) ∗

∑

an 𝛿nt ,

n

and we have 𝜇(𝛾) ̂ = (⟨t, 𝛾⟩ − 𝜆)f (⟨t, 𝛾⟩) = 𝜑(⟨t, 𝛾⟩)

̂ (𝛾 ∈ G).

Since 𝜑 is equal to 1 on V, it follows that Sp U ⊂ int{𝛾 ∈ G; 𝜇(𝛾) ̂ = 1}, and so U𝜇 = the identity mapping on 𝒳 . Thus, Ut − 𝜆 is invertible in ℬ(𝒳 ), that is, 𝜆 ∉ Sp(Ut ). From the above proposition, it is easy to obtain the following expression for the spectral radius of Ut − I: ‖Ut − I‖Sp = sup{|1 − ⟨t, 𝛾⟩|; y ∈ Sp U}. ̂ is compact if and only if limt→0 sup{|1− On the other hand, it is easy to check that a closed set K ⊂ G ⟨t, 𝛾⟩|; 𝛾 ∈ K} = 0. Consequently, we have lim ‖Ut − I‖Sp = 0 ⇔ Sp U is compact. t→0

Corollary. The representation U ∶ G → ℬw (𝒳 ) is norm-continuous if and only if Sp U is compact. Proof. By the above remarks, if limt→0 ‖Ut − I‖ = 0, then Sp U is compact. Conversely, if Sp U is compact, then there exists a function f ∈ ℒ 1 (G) with Sp U ⊂ {𝛾 ∈ ̂ G; ̂f (𝛾) = 1}, hence Uf = I and lim ‖Ut − I‖ = lim ‖U𝛿t ∗f − Uf ‖ ≤ lim ‖𝛿t ∗ f − f ‖1 = 0. t→0

t→0

t→0

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173

14.7. Consider now two pairs (𝒳 , 𝒳∗ ) and (𝒴 , 𝒴∗ ) satisfying conditions (1𝒳 ), (2𝒳 ), (3𝒳 ) and (1𝒴 ), (2𝒴 ), (3𝒴 ) of Section 13.1. By Lemma 2/13.1, the pair (ℬw (𝒳 , 𝒴 ), ℬw (𝒳 , 𝒴 )∗ ) satisfies conditions (1ℬw (𝒳 ,𝒴 ) ), (2ℬw (𝒳 ,𝒴 ) ). Consider also a separable locally compact abelian group G and two continuous representations U ∶ G → ℬw (𝒳 ), V ∶ G → ℬw (𝒴 ) of G on 𝒳 and 𝒴 , respectively. We assume that V satisfies the strong continuity condition (C∗V ). By Lemma 13.3, we then obtain a continuous representation 𝔖 ∶ G → ℬw (ℬw (𝒳 , 𝒴 )) defined by 𝔖s T = Vs TU−1 s (T ∈ ℬw (𝒳 , 𝒴 ), s ∈ G), such that 𝔖𝜇 ∈ ℬw (ℬw (𝒳 , 𝒴 )) for all 𝜇 ∈ ℳ(G). We shall write ℬ = ℬw (𝒳 , 𝒴 ). ̂ a closed set and {Wi }i∈I a Theorem (W. B. Arveson). In the above situation, for T ∈ ℬ, Q ⊂ G ⋂ ̂ family of neighborhoods of 0 in G such that Q = i Q + Wi , the following statements are equivalent (i) (ii) (iii) (iv) (v) (vi)

T ∈ ℬ(𝔖; Q); ̂ T𝒳 (U; F) ⊂ 𝒴 (V; Q + F) for any closed set F ⊂ G; ̂ T𝒳 (U; E) ⊂ 𝒴 (V; Q + E) for any set E ⊂ G; ̂ T𝒳 (U; D) ⊂ 𝒴 (V; Q + D) for any open set D ⊂ G; ̂ T𝒳 (U; K) ⊂ 𝒴 (V; Q + K) for any compact set K ⊂ G; ̂ and any i ∈ I. T𝒳 (U; 𝛾 + Wi ) ⊂ 𝒴 (V; 𝛾 + Wi + Q) for any 𝛾 ∈ G

Proof. (i) ⇒ (ii). Let x0 ∈ 𝒳 (U; F). To show that Tx0 ∈ 𝒴 (V; Q + F), it is sufficient to show that ̂ we have Tx0 ∈ 𝒴 (V; Q + F + W0 ) (see 14.3.(8)). Let W be an for every neighborhood W0 of 0 in G ̂ such that W + W ⊂ W0 . open and relatively compact neighborhood of 0 in G Since x0 ∈ 𝒳 (U; F) ⊂ 𝒳 (U; F + W), it follows using 14.3.(7) that x0 is the w-limit of a net of elements of the form Ug x with x ∈ 𝒳 and g ∈ ℒ 1 (G), supp ĝ ⊂ F + W compact. Since T ∈ ℬw (𝒳 , 𝒴 ), it is sufficient to show that TUg x ∈ 𝒴 (V; Q + F + W0 ) for any x ∈ X and any g ∈ ℒ 1 (G) with supp ĝ ⊂ F + W compact. Similarly, T is the w-limit of a net of elements of form 𝔖f S with S ∈ ℬw (𝒳 , 𝒴 ) and f ∈ ℒ 1 (G), supp ̂f ⊂ Q + W compact; hence it is sufficient to show that (𝔖f S)Ug x ∈ 𝒴 (V; Q + F + W0 ) for every S ∈ ℬw (𝒳 , 𝒴 ), x ∈ 𝒳 , f ∈ ℒ 1 (G) with supp ̂f ⊂ Q + W compact and g ∈ ℒ 1 (G) with supp ĝ ⊂ F + W compact. It is easy to see that ( (𝔖f S)Ug x = =

∫ ∫ ∫

) f (t)Vt SU−t dt

∫

g(s)Us x ds

f (t)g(s)Vt SUs−t x ds dt =

( ∫

) ∫

f (t)g(s + t)Vt SUs x dt

ds; .

thus it is sufficient to show that for every s ∈ G we have Vfgs (SUs x) =

∫

f (t)g(s + t)Vt SUs x dt ∈ 𝒴 (V; Q + F + W0 ),

(1)

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Groups of Automorphisms

where gs (t) = g(s + t), (t ∈ G). The functions ̂f and ĝ s are continuous and compactly supported, so they belong to ℒ 2 (G). By the Plancherel theorem, it follows that f, gs ∈ ℒ 2 (G), hence fgs ∈ ℒ 1 (G), and supp ( fgs )̂ = supp ̂f ∗ ĝ s ⊂ supp ̂f + supp ĝ ⊂ Q + W + F + W ⊂ Q + F + W0 , thus proving (1). (ii) ⇒ (iii). By 14.3.(5), it is sufficient to show that T𝒳00 (U; E) ⊂ 𝒴 (V; Q+E). Let x ∈ 𝒳00 (U; E). Then F = SpU (x) ⊂ E is compact; hence Q + F = Q + F, and x ∈ 𝒳 (U; F). By assumption (ii) it follows that Tx ∈ 𝒴 (V; Q + F) ⊂ 𝒴 (V; Q + E). (iii) ⇒ (iv). Obvious. (iv) ⇒ (v). Let {Nj }j∈J be a fundamental system of open and relatively compact neighborhoods of ̂ Assuming (iv) it follows that T𝒳 (U; K) ⊂ T𝒳 (U; K + Nj ) ⊂ 𝒴 (V; Q + K + Nj ) for all j ∈ J, 0 in G. and using 14.3.(9) we get T𝒳 (U; K) ⊂ 𝒴 (V; Q + K), as Q + K is closed. (v) ⇒ (vi). This follows easily using 14.3.(5). (vi) ⇒ (i). To prove (i) it is sufficient (14.3.(3)) to show that T ∈ ℬ(𝔖; Q + Wi ) for every i ∈ I. ̂ such that Accordingly, consider a neighborhood W of 0 in G T𝒳 (U; 𝛾 + W) ⊂ 𝒴 (W; 𝛾 + Q + W)

̂ (𝛾 ∈ G)

(2)

and let us show that T ∈ ℬ(𝔖; Q + W). To this end it is sufficient (14.3.(2)) to show that 𝔖h T = 0 for all h ∈ ℐ00 (Q + W).

(3)

̂ we denote by 𝛾̄ f the Let f ∈ ℒ 1 (G) with ̂f (0) ≠ 0 and supp ̂f ⊂ W compact. For 𝛾 ∈ G G ∋ t ↦ ⟨t, 𝛾⟩f (t). Then supp(̄𝛾 f )̂ ⊂ 𝛾 + W, hence U𝛾̄ f ∈ 𝒳 (U; 𝛾 + W) for all x ∈ 𝒳 . Using (2) we get TU𝛾̄ f x ∈ 𝒴 (V; 𝛾 + Q + W) (x ∈ 𝒳 ). If h ∈ ℐ00 (Q + W), then 𝛾̄ h ∈ ℐ00 (𝛾 + Q + W) ̂ Consequently, and hence (14.3.(2)) V𝛾̄ h TU𝛾̄ f x = 0(x ∈ 𝒳 , 𝛾 ∈ G). ℒ 1 -function

⟨s + t, 𝛾⟩f (s)h(t)Vt TUs x ds dt ( ) ⟨t, 𝛾⟩ = f (s)h(t − s)Vt−s TUs x ds dt. ∫ ∫

0=

∫ ∫

(4)

For every 𝜌 ∈ 𝒴∗ , the formula k𝜌 (t) = ∫ f (s)h(t − s)𝜌(Vt−s TUs x) ds (t ∈ G) determines a function k𝜌 ∈ ℒ 1 (G); from (4) it follows that k̂ 𝜌 = 0, so that k𝜌 = 0 as an element of ℒ 1 (G). Since the functions ̂f and ĥ are compactly supported, and so belong to ℒ 2 (G), we have f, h ∈ ℒ 2 (G), while the function s ↦ 𝜌(Vt−s TUs x) is bounded and continuous. Thus, the function k𝜌 is continuous, and it follows that k𝜌 (0) = 0, that is, ∫ f (−s)h(s)𝜌(Vs TU−s x) ds = 0. Replacing the function f here by its translates we get ∫

f (r − s)h(s)𝜌((𝔖s T )x) ds = 0

(r ∈ G).

(5)

Furthermore, the formula g𝜌 (s) = h(s)𝜌((𝔖s T )x) (s ∈ G) determines a function g𝜌 ∈ ℒ 1 (G); from (5) it follows that f ∗ g𝜌 = 0, and 0 = ( f ∗ g𝜌 )̂(0) = ̂f (0)̂g𝜌 (0). Since ̂f (0) ≠ 0, we obtain ĝ 𝜌 (0) = 0, that is, 𝜌((𝔖h T )x) = ∫ h(s)𝜌((𝔖s T )x) ds = 0. Since 𝜌 ∈ 𝒴∗ and x ∈ 𝒳 were arbitrary, we obtain the required conclusion (3).

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175

̂ we have ℬ(𝔖; P)𝒳 (U; E) ⊂ 𝒴 (V; P + E). ̂ E⊂G Corollary 1. For any sets P ⊂ G, Proof. Indeed, if T ∈ ℬ0 (𝔖; P), then Q = Sp𝔖 (T ) ⊂ P is closed and T ∈ ℬ(𝔖; Q), hence T𝒳 (U; E) ⊂ 𝒴 (V; Q + E) ⊂ 𝒴 (V; P + E) by the previous theorem. Thus, Corollary 1 follows using 14.3.(5). Corollary 2. An operator T ∈ ℬw (𝒳 , 𝒴 ) intertwines representations U and V, that is, Vs T = TUs

(s ∈ G)

(6)

̂ which is closed (or open, or if and only if the inclusion T𝒳 (U; E) ⊂ 𝒴 (V; E) holds for any E ⊂ G compact, or of the form 𝛾 + Wi ). Proof. Equation (6) means that 𝔖s T = T (s ∈ G), that is, (14.4.(13)) T ∈ ℬ(𝔖; {0}) and the corollary follows from the theorem by taking Q = {0}. In particular, taking (𝒳 , 𝒳∗ ) = (𝒴 , 𝒴∗ ) and T = I in Corollary 2, we see that the spectral subspaces associated with a continuous representation of G determine the representation uniquely: Corollary 3. Given two continuous representations U ∶ G → ℬw (𝒳 ) and V ∶ G → ℬw (𝒳 ) of G ̂ on 𝒳 , we have U = V if and only if the inclusion 𝒳 (U; E) ⊂ 𝒳 (V; E) holds for every set E ⊂ G which is closed (or open, or compact, or of the form 𝛾 + Wi ). ̂ is ordered by a closed semigroup S ⊂ G ̂ with S ∩ (−S) = {0}, In Corollary 4, we assume that G ̊ ̂ S ∪ (−S) = G and that 0 is adherent to the interior S of S. ̂ The following statements are equivalent: Corollary 4. Let T ∈ ℬw (𝒳 , 𝒴 ) and 𝛾 ∈ G. (i) T ∈ ℬ(𝔖; 𝛾 + S); ̂ (ii) T𝒳 (U; 𝜔 + S) ⊂ 𝒴 (V; 𝛾 + 𝜔 + S) for 𝜔 ∈ G; ̊ ̊ ̂ (iii) T𝒳 (U; 𝜔 + S) ⊂ 𝒴 (V; 𝛾 + 𝜔 + S) for 𝜔 ∈ G. ̊ ⊂ 𝛾 + 𝜔 + S, ̊ this implication follows from Corollary 1. Proof. (i) ⇒ (iii). Since (𝛾 + S) + (𝜔 + S) ̊ ̂ ⋂ (iii) ⇒ (ii).̊ Let {𝛾i } be a net in S which converges to 0. Then, for every 𝜆 ∈ G we̊ have 𝜆 + S = (iii) we get T𝒳 (U; 𝜔 + S) ⊂ T𝒳 (U; 𝜔 − 𝛾i + S) ⊂ 𝒴 (V; 𝛾 + i (𝜆 − 𝛾i + S). Thus, using assumption ⋂ ̊ and hence T𝒳 (U; 𝜔 + S) ⊂ ̊ 𝜔 − 𝛾i + S) i 𝒴 (V; 𝛾 + 𝜔 − 𝛾i + S) = 𝒴 (V; 𝛾 + 𝜔 + S). ̊ (ii) ⇒ (i). Let Q = 𝛾 + S. For each 𝜆 ∈ S, we consider the set W𝜆 = (−𝜆 + S) ∩ (𝜆 − S). Since ̊ ̊ W𝜆 is a neighborhood of 0 in G ̊ = ̂ and we have ⋂{Q + W𝜆 ; 𝜆 ∈ S} W − S), ⋂𝜆 ⊃ (−𝜆 + S) ∩ (𝜆 ̊ ̂ {𝛾 − 𝜆 + S; 𝜆 ∈ S} = Q. Using assumption (ii), for every 𝜔 ∈ G we obtain T𝒳 (U; 𝜔 + W𝜆 ) ⊂ T𝒳 (U; 𝜔 − 𝜆 + S) ⊂ 𝒴 (V; 𝛾 + 𝜔 − 𝜆 + S) = 𝒴 (V; 𝜔 + W𝜆 + Q) and using the implication (vi) ⇒ (i) from the previous theorem we get the required conclusion (i). 14.8. In the context of Section 14.7, we also note the following result: Proposition. Let x ∈ 𝒳 and T ∈ ℬw (𝒳 , 𝒴 ). If T intertwines the representations U and V, that is, Vs T = TUs (s ∈ G), then SpV (Tx) ⊂ SpU (x), and if moreover T is injective, then SpV (Tx) = SpU (x). V V Proof. For f ∈ ℒ 1 (G), we get Vf Tx = TUf x, so it follows that ℐxU ⊂ ℐTx , and indeed ℐxU = ℐTx if T is injective. Thus, the proposition follows by the definition of the spectrum of an element (14.2).

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14.9. Let U ∶ G → ℬw (𝒳 ) be a continuous representation. Consider another separable locally compact abelian group H and a continuous homomorphism 𝜑 ∶ H → G. Then U◦𝜑 ∶ H → ℬw (𝒳 ) is a continuous representation and the mapping 𝛾 → 𝛾◦𝜑 defines the dual homomorphism ̂ → H. ̂ 𝜑̂ ∶ G Proposition. In the above situation we have SpU◦𝜑 (x) = 𝜑(Sp ̂ U (x)) (x ∈ 𝒳 ).

(1)

̂ 𝒳 (U◦𝜑; E) ⊂ 𝒳 (U; 𝜑̂ (E)) (E ⊂ H). ̂ closed). 𝒳 (U◦𝜑; F) ⊂ 𝒳 (U; 𝜑̂ −1 (F)) (F ⊂ H,

(2)

Sp U◦𝜑 = 𝜑(Sp ̂ U).

(4)

−1

(3)

Proof. For every ν ∈ ℳ(H), we denote by 𝜑(ν) ∈ ℳ(G) the image of the measure ν by the continuous homomorphism 𝜑. It is easy to check that (U◦𝜑)ν = U𝜑(ν)

(ν ∈ ℳ(H))

̂ 𝜑(ν)̂(𝛾) = ν̂ (𝜑(𝛾)) ̂ (ν ∈ ℳ(H), 𝛾 ∈ G).

(5) (6)

̂ If ν ∈ ℳ(H) and (U◦𝜑)ν x = 0, then U𝜑(ν) x = 0, Let us prove (1). Let 𝛾 ∈ SpU (x) and 𝜑(𝛾) ̂ ∈ H. hence ν̂ (𝜑(𝛾)) ̂ = 𝜑(ν)̂(𝛾) = 0, since 𝛾 ∈ SpU (x) (see 14.3.(10)). Consequently, 𝜑(𝛾) ̂ ∈ SpU◦𝜑 (x). We ̂ have thus proved that 𝜑(Sp ̂ ̂ U (x)) ⊂ SpU◦𝜑 (x). Conversely, assume that 𝜔 ∈ H and 𝜔 ∉ 𝜑(Sp U (x)). ̂ There exists a function h ∈ ℒ 1 (H) ⊂ ℳ(H) with h(𝜔) ≠ 0 and supp ĥ ∩ 𝜑(Sp ̂ (x)) = ∅. Then U ̂ 𝜑̂ vanishes on a neighborhood of SpU (x), so that (U◦𝜑)h x = U𝜇 x = 0. 𝜇 = 𝜑(h) ∈ ℳ(G) and 𝜇̂ = h◦ ̂ ≠ 0, it follows that 𝜔 ∉ SpU◦𝜑 (x). We have thus proved that SpU◦𝜑 (x) ⊂ 𝜑(Sp ̂ Since h(𝜔) U (x)). The proof of (4) is quite similar to the above proof. Finally, (2) and (3) follow obviously from (1) using also 14.3.(5). In what follows we consider some examples. 14.10. Let ℋ be a Hilbert space and G a separable locally compact abelian group. Then the pair (ℋ , ℋ ∗ ) satisfies conditions (1ℋ ), (2ℋ ), (3ℋ ) of Section 13.1 and every so-continuous unitary representation u ∶ G → ℬ(ℋ ) is a continuous representation of G on ℋ according to the definition given in Section 13.2; moreover, u satisfies the strong continuity conditions (Cu ) and (C∗u ). In this case, as is easily verified, the mapping ℒ 1 (G) ∋ f ↦ uf ∈ ℬ(ℋ ) is a *-representation of the involutive Banach algebra ℒ 1 (G), where the involution f ↦ f ∗ is defined by f ∗ (t) = f (−t) (t ∈ G): uf ∗ = u∗f

( f ∈ ℒ 1 (G)).

On the other hand, for each 𝜉 ∈ ℋ , the function G ∋ t ↦ (ut 𝜉|𝜉) is a positive definite function on ̂ such that G and hence, by Bochner’s theorem, there exists a positive measure ν𝜉 ∈ ℳ(G) (ut 𝜉|𝜉) =

∫G

⟨t, 𝛾⟩ dν𝜉 (𝛾)

(t ∈ G).

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177

Then, for f ∈ ℒ 1 (G), we obtain ‖uf 𝜉‖2 = (uf ∗ ∗f 𝜉|𝜉) =

∫G

| ̂f (𝛾)|2 dν𝜉 (𝛾) ≤ ‖ ̂f ‖2∞ ‖𝜉‖2 .

̂ which separates the points of G and The set 𝒜 (G) = {̂f; f ∈ ℒ 1 (G)} is a *-subalgebra of 𝒞0 (G) ̂ by the Stone–Weierstrass theorem. The above inequality shows that hence is norm-dense in 𝒞0 (G), ̂ → ℬ(ℋ ) of the the mapping 𝒜 (G) ∋ ̂f ↦ uf can be extended to a *-representation 𝜋u ∶ 𝒞0 (G) ∗ ̂ C -algebra 𝒞0 (G), uniquely determined, such that 𝜋u (̂f ) = uf

(̂f ∈ ℒ 1 (G)).

Furthermore (see, e.g., Str̆atil̆a & Zsidó, 1977, 2005, 7.14) the *-representation 𝜋u can be uniquely ̂ of all bounded Baire extended to a *-representation, still denoted by 𝜋u , of the C∗ -algebra Baire(G) so ̂ ̂ such that 𝜋u (𝜑n ) → 𝜋u (𝜑) for every norm-bounded sequence {𝜑n } ⊂ Baire(G) functions on G, ̂ and we have which is pointwise convergent to 𝜑 ∈ Baire(G), (𝜋u (𝜑)𝜉|𝜉) =

∫Ĝ

𝜑(𝛾) dν𝜉 (𝛾)

̂ 𝜉 ∈ ℋ ). (𝜑 ∈ Baire(G),

̂ we obtain a ℬ(ℋ )-valued spectral measure Then, putting pu (E) = 𝜋u (𝜒E ) for every Baire set E ⊂ G, ̂ pu (⋅) on G, uniquely determined, such that (ut 𝜉|𝜉) =

∫Ĝ

⟨t, 𝛾⟩ d(pu (𝛾)𝜉|𝜉)

(t ∈ G, 𝜉 ∈ ℋ ).

pu is called the Stone spectral measure associated with u. This spectral measure has regularity properties similar to 14.3.(10) and 14.3.(11). We are now in a position to state the next result: Proposition. Let u ∶ G → ℬ(ℋ ) be an so-continuous unitary representation of G on ℋ . For every ̂ we have Baire set E ⊂ G ℋ (u; E) = pu (E)ℋ . Sp u is the support of the Stone spectral measure pu (⋅). ̂ we have Proof. For every f ∈ ℒ 1 (G) and every Baire set E ⊂ G uf pu (E) = pu (E)uf =

∫E

̂f (𝛾) dpu (𝛾).

(1)

̂ be an open set. If f ∈ ℒ 1 (G) and supp ̂f ⊂ D, then uf ℋ ⊂ pu (D)ℋ , by (1). Hence Let D ⊂ G (14.3.(7)). ℋ (u; D) ⊂ pu (D)ℋ .

(2)

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̂ be a closed set. For every f ∈ ℐ0 (F), we have uf pu (F) = 0 again by (1). Hence (14.3.(2)) Let F ⊂ G pu (F)ℋ ⊂ ℋ (u; F). On the other hand, using (2), 14.3.(10), and the regularity of the Stone spectral measure, we obtain ⋂ ⋂ ℋ (u; F) = {ℋ (u; D); D ⊃ F open} ⊂ {pu (D)ℋ ; D ⊃ F open} = pu (F)ℋ . Hence ℋ (u; F) = pu (F)ℋ .

(3)

̂ Using (3), 14.3.(11), and the regularity of the Stone spectral Consider now any Bairc set E ⊂ G. measure, we obtain ⋁ ℋ (u; E) = {ℋ (u; K); K ⊂ E compact} ⋁ = {pu (K)ℋ ; K ⊂ E compact} = pu (E)ℋ . ̂ we have ℋ (u; D) = 0 if and only if pu (D)ℋ = 0, it follows that Since, for an open set D ⊂ G, Sp u is just the support of pu (⋅). ̂ has a countable basis of open sets, then any Recall (Str̆atil̆a & Zsidó, 1977, 2005, 7.14) that if G ̂ Borel set in G is also a Baire set. ̂ with ℝ so that for t ∈ ℝ = G and s ∈ ℝ = G ̂ we have If G = ℝ, then we can identify G ⟨t, s⟩ = eitx . In this case, the Stone theorem shows that there exists a unique self-adjoint operator A in ℋ such that ut = exp(itA)(t ∈ ℝ); A is called the infinitesimal generator of the so-continuous unitary ̂ we have representation u. For any Borel set E ⊂ ℝ = G pu (E) = 𝜒E (A).

(4)

Sp u = Sp A

(5)

Also,

̂f (A) =

∫ℝ

f (t)eitA dt

( f ∈ ℒ 1 (ℝ)).

(6)

If (𝒳 , 𝒳∗ ) = (𝒴 , 𝒴∗ ) = (ℋ , ℋ ∗ ), then ℬw (𝒳 , 𝒴 ) = ℬ(ℋ ) and the w-topology on ℬw (𝒳 , 𝒴 ) is just the w-topology on the W∗ -algebra ℬ(ℋ ), defined by the predual ℬ(ℋ )∗ = ℬw (𝒳 , 𝒴 )∗ of ℬ(ℋ ). Given an so-continuous unitary representation u ∶ G → ℬ(ℋ ), the representation 𝔖(u, u) of G on ℬ(ℋ ) is just the continuous action 𝜎 ∶ G → Aut(ℬ(ℋ )) defined by 𝜎t (x) = ut xu∗t

(x ∈ ℬ(ℋ ), t ∈ G).

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179

̂ and ̂ is ordered by a closed semigroup S ⊂ G ̂ with S ∩ (−S) = {0}, S ∪ (−S) = G Assume that G that 0 is adherent to the interior S̊ of S. Then, applying Corollary 4/14.7, we obtain Corollary. Let u∶ G → ℬ(ℋ ) be a so-continuous unitary representation and 𝜎∶ G → Aut(ℬ(ℋ )) ̂ the following the continuous action defined by 𝜎t = Ad(ut )(t ∈ G). For x ∈ ℬ = ℬ(ℋ ) and 𝛾 ∈ G, statements are equivalent: (i) x ∈ ℬ(𝜎; 𝛾 + S); ̂ and (ii) xpu (𝜔 + S)ℋ ⊂ pu (𝛾 + 𝜔 + S)ℋ for all 𝜔 ∈ G; ̊ ̊ ̂ (iii) xpu (𝜔 + S)ℋ ⊂ pu (𝛾 + 𝜔 + S)ℋ for all 𝜔 ∈ G. ̂ or S = (−∞, 0] ⊂ G. ̂ In particular, for G = ℝ, we can take S = [0, +∞) ⊂ G 14.11. Let 𝒳 be a Banach space and D ∈ ℬ(𝒳 ) be such that ‖ exp(itD)‖ = 1 for all t ∈ ℝ.

(1)

Since ‖ exp(itD) − I‖ ≤ exp(t‖D‖) − 1, it follows that the formula Ut = exp(itD)

(t ∈ ℝ)

(2)

defines a norm-continuous representation U ∶ ℝ → ℬ(ℋ ) which is also a continuous representation in the sense defined in Section 13.2, with respect to the pair (𝒳 , 𝒳∗ ). From (1), it follows that the usual spectrum Sp(D) = Spℬ(ℋ ) (D) of the operator D ∈ ℬ(ℋ ) is real, and, clearly, Sp(D) ⊂ [−‖D‖, ‖D‖]. We have Sp(exp(itD)) = {eits ; s ∈ Sp(D)} (t ∈ ℝ).

(3)

On the other hand, since the representation U is norm-continuous, the set Sp U ∈ ℝ is compact, by Corollary 14.5. Using Proposition 14.5, it follows that Sp(Ut ) = {eits ; s ∈ Sp U} (t ∈ ℝ).

(4)

Using (2), (3), and (4), we conclude that Sp U = Sp(D) ⊂ [−‖D‖, ‖D‖].

(5)

It is a classical result that for any norm-continuous representation U ∶ ℝ → ℬ(𝒳 ) of ℝ by isometries on 𝒳 , there exists a unique operator D ∈ ℬ(𝒳 ) which satisfies conditions (1) and (2). This result can easily be obtained using the compactness of Sp U. Indeed, one can take D = Uf for any compactly supported C∞ -function f such that Sp U ⊂ int{s ∈ ℝ; ̂f (s) = s}. Moreover, D = normlimt→0 1t (Ut − I). 14.12. Let (𝒳 , 𝒳∗ ) be a pair-satisfying conditions (1𝒳 ), (2𝒳 ), (3𝒳 ) of Section 13.1 and let T ∈ ℬw (𝒳 ) be an isometry on 𝒳 . Then we obtain a representation U ∶ ℤ → ℬw (𝒳 ) defined by Un = Tn

(n ∈ ℤ).

(1)

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̂ of ℤ can be identified with the one-dimensional torus 𝕋 = {𝜆 ∈ ℂ; |𝜆| = 1} in The dual group ℤ such a way that for n ∈ ℤ and 𝜆 ∈ 𝕋 if we have ⟨n, 𝜆⟩ = 𝜆n . Since Sp U ∈ 𝕋 is compact, using Proposition 14.5 we get Sp(T ) = Sp(U1 ) = {𝜆; 𝜆 ∈ Sp U}, that is, Sp U = Sp(T )

(2)

where Sp(T ) is the usual spectrum Spℬ(𝒳 ) (T ) of the operator T ∈ ℬ(𝒳 ). For 𝜆 ∈ ℂ, |𝜆| = 1 we have T = 𝜆I ⇔ Sp(T ) = {𝜆}.

(3)

Indeed, the implication (⇒) is obvious. Conversely, if Sp(T ) = {𝜆}, then Sp U = {𝜆}, hence 𝒳 (U; {𝜆}) = 𝒳 . Consequently (14.3.(13)), for every x ∈ 𝒳 we have Tn x = Un x = 𝜆n x (n ∈ ℤ), hence T = 𝜆I. 14.13. Recall that if a, b are two commuting elements of a unital Banach algebra 𝒜 , then Sp𝒜 (a + b) ⊂ {𝜆 + 𝜇; 𝜆 ∈ Sp𝒜 (a), 𝜇 ∈ Sp𝒜 (b)}, Sp𝒜 (ab) ⊂ {𝜆𝜇; 𝜆 ∈ Sp𝒜 (a), 𝜇 ∈ Sp𝒜 (b)}. Now, for arbitrary elements a, b ∈ 𝒜 , the mappings La ∶ 𝒜 ∋ x ↦ ax ∈ 𝒜 , Rb ∶ 𝒜 ∋ x ↦ xb ∈ 𝒜 define two commuting elements La , Rb of the Banach algebra ℬ(𝒜 ) of all bounded linear operators on 𝒜 , and it is clear that Spℬ(𝒜 ) (La ) ⊂ Sp𝒜 (a), Spℬ(𝒜 ) (Rb ) ⊂ Sp𝒜 (b). Consequently, Spℬ(𝒜 ) (La + Rb ) ⊂ {𝜆 + 𝜇; 𝜆 ∈ Sp𝒜 (a), 𝜇 ∈ Sp𝒜 (b)},

(1)

Spℬ(𝒜 ) (La Rb ) ⊂ {𝜆𝜇; 𝜆 ∈ Sp𝒜 (a), 𝜇 ∈ Sp𝒜 (b)}.

(2)

Proposition. Let ℳ be a W∗ -factor, and let u ∈ U(ℳ), 𝜎 = Ad(u) ∈ Aut(ℳ). Then Spℬ(ℳ) (𝜎) = {𝜆𝜇 −1 ; 𝜆, 𝜇 ∈ Spℳ (u)}. Proof. By the above remarks, it is clear that Spℬ(ℳ) (𝜎) ⊂ {𝜆𝜇−1 ; 𝜆, 𝜇 ∈ Spℳ (u)}. Now, let 𝜆, 𝜇 ∈ Spℳ (u) and 𝜀 > 0. There exist two nonzero spectral projections e, f ∈ ℳ of u such that ‖ue − 𝜆e‖ ≤ 𝜀∕2 and ‖ fu − 𝜇f ‖ ≤ 𝜀∕2. Since ℳ is a factor, there exists a nonzero partial isometry v ∈ ℳ such that vv∗ ≤ e and v∗ v ≤ f. We have ‖𝜎(v) − 𝜆𝜇 −1 v‖ = ‖uv − 𝜆𝜇−1 vu‖ = ‖𝜆−1 uv − 𝜇−1 vu‖ ≤ ‖𝜆−1 uv − v‖ + ‖v − 𝜇 −1 vu‖ = ‖uv − 𝜆v‖ + ‖vu − 𝜇v‖ = ‖(ue − 𝜆e)v‖ + ‖v( fu − 𝜇f )‖ ≤ 𝜀. Hence Spℬ(ℳ) (𝜎) = {𝜆𝜇 −1 ; 𝜆, 𝜇 ∈ Spℳ (u)}.

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14.14. Notes. The spectral theory of the action of a locally compact abelian group on a Banach space appeared in the works of Borchers, Colojoar̆a and Foiaş, Godement, Lyubic, Macaev and Feldman, Wightman, and others, but it was Arveson (1974) who, as well as giving a systematic, self-contained exposition of this subject with several new results, showed the relevance of the theory for the study of operator algebras. The main result of this section, Theorem 14.7, is due to Arveson (1974). Propositions 14.4, 14.5, 14.9, and 14.13 are due to Connes (1973a) (see also Olesen, 1976). Four our exposition, we have used Arveson (1974); Combes and Delaroche (1973–1974); Connes (1973a), and Olesen (1976).

15 Continuous Actions on W∗ -algebras In this section, we apply the spectral theory so far developed to continuous actions of locally compact abelian groups by *-automorphisms of W∗ -algebras. The main results concern the inner implementation of *-automorphisms and derivations. 15.1. Let u ∶ G → Aut(ℳ) be a continuous action (13.5) of the separable locally compact abelian group G on the W∗ -algebra ℳ. Since 𝜎 is u-continuous, we infer (see 13.2) that every 𝜑 ∈ ℳ∗ belongs to the norm-closure of the set {𝜑◦𝜎f ; f ∈ ℒ 1 (G)}. It follows that for every norm-bounded approximate unit {fi }i∈I of the Banach algebra ℒ 1 (G) we have ‖𝜑◦𝜎fi − 𝜑‖ → 0 (𝜑 ∈ ℳ∗ ).

(1)

Indeed, let 𝜑 ∈ ℳ∗ , ‖𝜑‖ ≤ 1, and 𝜀 > 0. We assume that ‖ fi ‖ ≤ 1 for each i ∈ I. There exists f ∈ ℒ 1 (G) with ‖𝜑◦𝜎f − 𝜑‖ < 𝜀∕3 and then i𝜀 ∈ I such that ‖ f ∗ fi − f ‖1 < 𝜀∕3 for every i ≥ i𝜀 . Thus, for i ≥ i𝜀 , we get ‖𝜑◦𝜎fi − 𝜑‖ ≤ ‖𝜑◦𝜎fi − 𝜑◦𝜎f ◦𝜎fi ‖ + ‖𝜑◦𝜎f ◦𝜎fi − 𝜑◦𝜎f ‖ + ‖𝜑◦𝜎f − 𝜑‖ ≤ ‖ fi ‖1 ‖𝜑 − 𝜑◦𝜎f ‖ + ‖𝜑‖‖ f ∗ fi − f ‖1 + ‖𝜑◦𝜎f − 𝜑‖ < 𝜀. Also (13.2), every x ∈ ℳ belongs to the w-closure of the set {𝜎f x; f ∈ ℒ 1 (G)}. Using (1) it follows that for every norm-bounded approximate unit {f}i∈I of the Banach algebra ℒ 1 (G) we have w

𝜎fi x → x

(2)

(x ∈ ℳ).

Indeed, let x ∈ ℳ, ‖x‖ ≤ 1, 𝜑 ∈ ℳ∗ , ‖𝜑‖ ≤ 1, and 𝜀 > 0. There exists f ∈ ℒ 1 (G) with |𝜑(x − 𝜎f x)| < 𝜀∕4 and there exists i𝜀 ∈ I such that ‖ fi ∗ f − f ‖ < 𝜀∕4 and ‖𝜑 − 𝜑◦𝜎fi ‖ < (𝜀∕4)‖x − 𝜎f x‖ for i ≥ i𝜀 . Thus, for i ≥ i𝜀 we get |𝜑(𝜎fi x−x)| ≤ |(𝜑◦𝜎fi )(x−𝜎f x)|+|𝜑(𝜎fi 𝜎f x−𝜎f x)|+|𝜑(𝜎f x−x)| ≤ 2|𝜑(x − 𝜎f x)| + |𝜑◦𝜎fi − 𝜑‖‖x − 𝜎f x‖ + ‖ fi ∗ f − f ‖1 ‖𝜑‖‖x‖ < 𝜀. In particular, for every x ∈ ℳ s∗

x ∈ {𝜎k x; k ∈ ℒ 1 (G), ‖k‖1 ≤ 1, supp k̂ compact} .

(3)

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Indeed, {𝜎k x; k ∈ ℒ 1 (G), ‖k‖1 ≤ 1, supp k̂ compact} is a convex set and, by (2) and 14.1.(4), x is w-adherent and hence also s∗ -adherent to this set. We shall also use the following result: ̂ There exist a directed set Λ and a family Lemma. Let {Di }i∈I be an open covering of G. 1 {fi,𝜆 }i∈I,𝜆∈Λ ⊂ ℒ (G) such that (a) for each 𝜆 ∈ Λ the set {i ∈ I; fi,𝜆 ≠ 0} is finite and (b) for each i ∈ I and each 𝜆 ∈ Λ we have supp ̂fi,𝜆 ⊂ Di and, moreover, for every x ∈ ℳ we have ∑

w

𝜎fi,𝜆 x → x.

i∈I

𝜆∈Λ

Proof. Indeed, the linear subspace of ℒ 1 (G) spanned by the set {f ∈ ℒ 1 (G); there exists i ∈ I with supp ̂f ⊂ D1 } is a norm-dense ideal of ℒ 1 (G) and hence (14.1) it contains a norm-bounded approximate unit of ℒ 1 (G), so that the lemma follows using statement (2). 15.2. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ. In this section, we study the relationship between the spectral subspaces associated with 𝜎 and the *-operation on ℳ. Let f ∈ ℒ 1 (G) and let ̄f ∈ ℒ 1 (G) be its complex conjugate. It is easy to see that for any x ∈ ℳ ̂ we have and any 𝛾 ∈ G (𝜎f (x))∗ = 𝜎̄f (x∗ ), (̄f)∧ (𝛾) = ̂f (−𝛾).

(1)

Sp𝜎 (x∗ ) = −Sp𝜎 (x).

(2)

It follows that

̂ we have Consequently, for any set E ⊂ G ℳ(𝜎; E)∗ = ℳ(𝜎; −E)

(3)

̂ that is, and Sp 𝜎 is a symmetric subset of G, Sp 𝜎 = −Sp 𝜎.

(4)

Let 𝜎 ′ ∶ G → Aut(ℳ) be another continuous action of G on ℳ. We consider the pair (ℬ, ℬ∗ ) consisting of the Banach space ℬ of all w-continuous linear mappings ℳ → ℳ and the normclosed linear subspace ℬ∗ ⊂ ℬ ∗ generated by {𝜑(⋅x); 𝜑 ∈ ℳ∗ , x ∈ ℳ} (cf. 13.1), together with the continuous representation 𝔖 of G on ℬ defined by (cf. 13.3) 𝔖s T = 𝜎s′ ◦ T◦𝜎−s

(T ∈ ℬ, s ∈ G).

Note that there is a natural *-operation T ↦ T∗ on the Banach space ℬ, namely T∗ (x) = T(x∗ )∗ (T ∈ ℬ, x ∈ ℳ), and it is easy to check that 𝔖 is a *-representation, that is, 𝔖s (T∗ ) = (𝔖s T )∗ (T ∈ ℬ,

Continuous Actions on W∗ -algebras

183

s ∈ G). The same argument as above shows that Sp𝔖 (T∗ ) = −Sp𝔖 (T )

(T ∈ ℬ),

(5)

̂ so that the spectrum of every self-adjoint element of ℬ is symmetric with respect to 0 ∈ G. ̂ ̂ ̂ and Assume that G is ordered by a closed semigroup S ⊂ G with S ∩ (−S) = {0}, S ∪ (−S) = G ̊ that 0 is adherent to the interior S of S. Using (5) and Corollary 4/14.7, we get the following result: Proposition. Let 𝜎 ∶ G → Aut(ℳ), 𝜎 ′ ∶ G → Aut(ℳ) be continuous actions of G on ℳ and 𝜃 ∈ Aut(ℳ). The following statements are equivalent: (i) 𝜎t′ ◦𝜃 = 𝜃◦𝜎t for all t ∈ G; ̂ and (ii) 𝜃(ℳ(𝜎; 𝛾 + S)) ⊂ ℳ(𝜎 ′ ; 𝛾 + S) for all 𝛾 ∈ G; ̊ ⊂ ℳ(𝜎 ′ ; 𝛾 + S) ̊ for all 𝛾 ∈ G. ̂ (iii) 𝜃(ℳ(𝜎; 𝛾 + S)) Proof. By Corollary 4/14.7, statements (ii) and (iii) are both equivalent to the fact that the *-automorphism 𝜃 ∈ Aut(ℳ) ⊂ ℬ belongs to the spectral subspace ℬ(𝔖; S). Since 𝜃 is a selfadjoint element of ℬ, the spectrum of 𝜃 is symmetric and so Sp𝔖 (𝜃) ⊂ S ∩ (−S) = {0}, that is, (14.3.(14)) 𝔖t (𝜃) = 𝜃 and 𝜎t′ ◦𝜃 = 𝜃◦𝜎t for all t ∈ G. Clearly, the previous proposition still holds for any self-adjoint element 𝜃 ∈ ℬ and can be extended to actions on different W∗ -algebras. If 𝜃 = the identity mapping on ℳ, we get spectral conditions that are equivalent to the equalities 𝜎t = 𝜎t′ (t ∈ G). The previous proposition is particularly useful in the case G = ℝ, when we can take S = [0, +∞) ⊂ ̂ or S = (−∞, 0] ⊂ G. ̂ G 15.3. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ. We now study the relationship between the spectral subspaces associated with 𝜎 and the multiplication operation on ℳ. Consider again the pair (ℬ, ℬ∗ ) as in Section 15.2 and the continuous representation 𝔖 of G on ℬ defined by 𝔖s T = 𝜎s ◦T◦𝜎−s (T ∈ ℬ, s ∈ G). We define an operator L ∶ ℳ → ℳ by putting La (x) = ax (a ∈ ℳ, x ∈ ℳ). The operator L is injective, w-continuous, and intertwines the representations 𝜎 and 𝔖, that is, 𝔖s ◦L = L◦𝜎s

(s ∈ G)

since for a, x ∈ ℳ we have (𝔖s (La ))(x) = 𝜎s (a𝜎−s (x)) = 𝜎s (a)x = (L𝜎s (a) )(x). By Proposition 14.8 it follows that Sp𝔖 (La ) = Sp𝜎 (a) (a ∈ ℳ). Thus, using Theorem 14.7, we obtain the following result: ̂ a closed set, Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ, a ∈ ℳ, Q ⊂ G ̂ and {Wi }i∈I a fundamental system of compact neighborhoods of 0 in G. The following statements are equivalent:

184 (i) (ii) (iii) (iv) (v) (vi)

Groups of Automorphisms a ∈ ℳ(𝜎; Q); ̂ aℳ(𝜎; F) ⊂ ℳ(𝜎; Q + F) for any closed set F ⊂ G; ̂ aℳ(𝜎; E) ⊂ ℳ(𝜎; Q + E) for any set E ⊂ G; ̂ aℳ(𝜎; D) ⊂ ℳ(𝜎; Q + D) for any open set D ⊂ G; ̂ and aℳ(𝜎; K) ⊂ ℳ(𝜎; Q + K) for any compact set K ⊂ G; ̂ aℳ(𝜎; 𝛾 + Wi ) ⊂ ℳ(𝜎; 𝛾 + Wi + Q) for any 𝛾 ∈ G and any i ∈ I.

In particular, for every a, b ∈ ℳ we have Sp𝜎 (ab) ⊂ Sp𝜎 (a) + Sp𝜎 (b)

(1)

ℳ(𝜎; E1 )ℳ(𝜎; E2 ) ⊂ ℳ(𝜎; E1 + E2 ).

(2)

̂ we have and for every E1 , E2 ⊂ G

We note also the following obvious identity: 𝜎𝜇 (axb) = a𝜎𝜇 (x)b

(a, b ∈ ℳ 𝜎 , x ∈ ℳ, 𝜇 ∈ ℳ(G)).

(3)

15.4. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ. If 0 ≠ e ∈ ℳ 𝜎 is a projection, then 𝜎 obviously defines a continuous action 𝜎 e ∶ G → Aut(eℳe) of G on the reduced W∗ -algebra eℳe ∶ 𝜎te = 𝜎t |eℳe (t ∈ G). It is easy to check that for every x ∈ eℳe we have Sp𝜎 e (x) = Sp𝜎 (x).

(1)

̂ we shall write ℳ(𝜎 e ; E) instead of eℳe(𝜎 e ; E). From (1) it follows that For every set E ⊂ G ℳ(𝜎 e ; E) = ℳ(𝜎; E) ∩ eℳe.

(2)

(eℳe)𝜎 = ℳ 𝜎 ∩ eℳe.

(3)

In particular, e

If e1 and e2 are both projections in ℳ 𝜎 , then 0 ≠ e1 ≤ e2 ⇒ Sp 𝜎 e1 ⊂ Sp 𝜎 e2 .

(4)

Let 0 ≠ e ∈ ℳ 𝜎 be a projection and denote by ē ∈ 𝒳 (ℳ 𝜎 ) the central support of e in ℳ 𝜎 . Then we have Sp 𝜎 ē = Sp 𝜎 e .

(5)

̂ be any set and suppose that there exists 0 ≠ x ∈ ℳ(𝜎 ē ; E) = ℳ(𝜎; E) ∩ ē ℳ ē . Indeed, let E ⊂ G Since ē = ∨{ueu∗ ; u ∈ U(ℳ 𝜎 )}, we can find u, v ∈ U(ℳ 𝜎 ) such that ueu∗ xvev∗ ≠ 0. Then 0 ≠ eu∗ xve ∈ eℳe and eu∗ xve ∈ ℳ(𝜎; E) as e, u, v ∈ ℳ 𝜎 , x ∈ ℳ(𝜎; E) (15.3.(2)), so that ℳ(𝜎 e ; E) = ℳ(𝜎; E) ∩ eℳe ≠ {0}. Thus, (5) follows using Proposition 14.5.

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185

15.5. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the von Neuman algebra ℳ ⊂ ℬ(ℋ ). ̂ be an arbitrary set. We consider the w-closed left ideal 𝒯 (𝜎; E) = {x ∈ ℳ; xℳ Let E ⊂ G (𝜎; E) = 0} of ℳ and denote by q(𝜎; E) the greatest projection in 𝒯 (𝜎; E), that is, 𝒯 (𝜎; E) = ℳq(𝜎; E). From this definition, it follows that 1 − q(𝜎; E) = ∨{l(x); x ∈ ℳ(𝜎; E)}

(1)

(1 − q(𝜎; E))ℋ = ℳ(𝜎; E)ℋ .

(2)

or, equivalently,

The spectral subspace 𝒯 (𝜎; E) is 𝜎-invariant (14.2.(4)) and also invariant under left or right multiplications by elements in ℳ 𝜎 (15.3.(2), 14.3.(14)). It follows that 𝒯 (𝜎; E) enjoys the same properties and hence q(𝜎; E) ∈ 𝒵 (ℳ 𝜎 ).

(3)

If 𝜃 ∈ Aut(ℳ) and 𝜃◦𝜎t = 𝜎t ◦𝜃 (t ∈ G), then, by Corollary 2/14.7, 𝜃(ℳ(𝜎; E)) = ℳ(𝜎; E), and hence 𝜃 ∈ Aut(ℳ), 𝜃◦𝜎t = 𝜎t ◦𝜃 ⇒ 𝜃(q(𝜎; E)) = q(𝜎; E).

(4)

15.6. We now restrict ourselves to the case G = ℝ, and consider a continuous action 𝜎 ∶ ℝ → ̂ with ℝ so that ⟨t, s⟩ = eist (t ∈ Aut(ℳ) of ℝ on the von Neumann algebra ℳ ⊂ ℬ(ℋ ) identifying G ̂ G = ℝ, s ∈ G = ℝ). ̂ = ℝ we have a projection By the construction in Section 15.5, for each t ∈ G q𝜎t = q(𝜎; (t, +∞)) ∈ 𝒵 (ℳ 𝜎 )

(1)

1 − q𝛼t = ∨{l(x); x ∈ ℳ(𝜎; (t, +∞))} = ℳ(𝜎; (t, +∞))ℋ .

(2)

such that

We also consider the projection q𝜎∞ =

⋁

q𝜎t ∈ 𝒵 (ℳ 𝜎 );

(3)

ℳ(𝜎; (t, +∞))ℋ .

(4)

t∈ℝ

clearly 1 − q𝜎∞ =

⋀ t∈ℝ

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Lemma. The mapping ℝ ∋ t ↦ q𝜎t ∈ Proj(𝒵 (ℳ 𝜎 )) has the following properties: t1 ≤ t2 ⇒ q𝜎t ≤ q𝜎t , 1

(a)

2

s

tn ≥ t, tn → t ⇒ q𝜎t → q𝜎t ,

(b)

n

s

tn → +∞ ⇒ q𝜎t → q𝜎∞ ,

(c)

t < 0 ⇒ q𝜎t = 0.

(d)

n

If there exists 𝜀 > 0 such that Sp 𝜎 ⊂ [−𝜀, 𝜀], then t > 𝜀 ⇒ q𝜎t = 1.

(e)

Proof. For the proof put ℳ(t) = ℳ(𝜎; (t, +∞)), Q(t) = Q(𝜎; (t, +∞)) the left annihilator of ℳ(t) in ℳ∗ , qt = q𝜎t and q∞ = q𝜎∞ . We have 𝒯 (t) = ℳqt and q∞ = ∨t qt . Statement (a) follows from the fact that the mapping t ↦ ℳ(t) is decreasing and hence the mapping t ↦ 𝒯 (t) is increasing. Statement (c) is now obvious as q∞ = ∨t qt . s

Let tn ≥ t, tn → t. Since the mapping t ↦ qt is increasing, in order to show that qtn → qt we may ⋂ assume that the sequence {tn } is decreasing. Then (14.3.(4)) ℳ(t) = ∨n ℳ(tn ), hence 𝒯 (t) = n 𝒯 (tn ) and qt = ∧n qtn = s- limn qtn . We have thus proved statement (b). If t < 0, then ℳ(t) ⊃ ℳ 𝜎 ∋ 1, hence 𝒯 (t) = 0 and qt = 0. If Sp 𝜎 ⊂ [−𝜀, 𝜀], then for t > 𝜀 we have ℳ(t) = {0}, hence 𝒯 (t) = ℳ and qt = 1. 15.7. Continuing with the notation of the previous section, we assume that q𝜎∞ = 1. Then Lemma 15.6 shows that the mapping ℝ ∋ t ↦ q𝜎t ∈ 𝒵 (ℳ 𝜎 ) defines a spectral resolution of the identity on the Hilbert space ℋ (see Dunford & Schwartz, 1958, 1963, XI.5) and hence a self-adjoint operator A in ℋ affiliated to the von Neumann algebra 𝒵 (ℳ 𝜎 ), +∞

A=

∫−∞

t dq𝜎t ,

(1)

and an so-continuous unitary representation u ∶ ℝ → 𝒵 (ℳ 𝜎 ) ⊂ ℳ, us = exp(isA) (s ∈ ℝ).

(2)

Since q𝜎t = 0 for t < 0, the self-adjoint operator A is positive. Let pu (⋅) be the Stone spectral measure associated with u (14.10). Using Proposition 14.10 and 14.10.(4), 14.10.(5), we get ℋ (u; (t, +∞)) = pu ((t, +∞))ℋ = 𝜒(t,+∞) (A)ℋ = (q𝜎∞ − q𝜎t )ℋ

(t ∈ ℝ).

Since, by assumption, q𝜎∞ = 1, we obtain using 15.6.(2) ℋ (u; (t, +∞)) = ℳ(𝜎; (t, +∞))ℋ

(t ∈ ℝ).

(3)

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187

Since (15.3.(2)) ℳ(𝜎; (s, +∞))ℳ(𝜎; (t, +∞)) ⊂ ℳ(𝜎; (s + t, +∞)), we infer from (3) that ℳ(𝜎; (s, +∞))ℋ (u; (t, +∞)) ⊂ ℋ (u; (s + t, +∞)) (s, t ∈ ℝ).

(4)

Using Corollary 14.10, we further deduce from (4) that ℳ(𝜎; (s, +∞)) ⊂ ℳ(Ad(u); (s, +∞))

(s ∈ ℝ).

(5)

Finally, with the help of Proposition 15.2 we conclude from (5) that 𝜎t = Ad(ut )

(t ∈ ℝ).

(6)

Recall that the so-continuous unitary representation u ∶ ℝ → ℳ defined above has positive spectrum, Sp u = Sp(A) ⊂ [0, +∞), that is, its infinitesimal generator A is positive. If Sp 𝜎 ⊂ [−𝜀, 𝜀], then (15.6.(e)) q𝜎t = 1 for every t > 𝜀, in particular q𝜎∞ = 1, and from the definition (1) of the operator A it follows that 0 ≤ A ≤ 𝜀.

(7)

15.8. As a conclusion to the above considerations we have the following remarkable result: Theorem (Borchers & Arveson). Let 𝜎 ∶ ℝ → Aut(ℳ) be a continuous action of ℝ on the von Neuman algebra ℳ ⊂ ℬ(ℋ ). The following statements are equivalent: (i) there exists an so-continuous unitary representation u ∶ ℝ → ℬ(ℋ ) with positive spectrum such that 𝜎t = Ad(ut )(t ∈ ℝ); (ii) there exists an so-continuous unitary representation u ∶ ℝ → ℳ with positive spectrum such that 𝜎t = Ad(ut )(t ∈ ℝ); (iii) q𝜎∞ = 1; and ⋂ (iv) t∈ℝ ℳ(𝜎; (t, +∞))ℋ = {0}. Proof. (i) ⇒ (iv). Since Sp u ⊂ [0, +∞) we have ℋ = ℋ (u; [0, +∞)). Let pu (⋅) be the Stone spectral measure associated with u (14.10). Since 𝜎 = Ad(u), using Corollary 1/14.7 and Proposition 14.10, we obtain, for every t ∈ ℝ, ℳ(𝜎; (t, +∞))ℋ = ℳ(𝜎; (t, +∞))ℋ (u; [0, +∞)) ⊂ ℋ (u; (t, +∞)) = pu ((t, +∞))ℋ hence ⋂ t∈ℝ

ℳ(𝜎; (t, +∞))ℋ ⊂

⋂

( pu ((t, +∞))ℋ =

t∈ℝ

(iv) ⇒ (iii). Follows from 15.6.(4). (iii) ⇒ (iii). This was proved in Section 15.7. (ii) ⇒ (i). Obvious.

⋀ t∈ℝ

) pu ((t, +∞)) ℋ = {0}.

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Groups of Automorphisms

Clearly, the theorem still remains valid if we replace the condition concerning the positivity of the spectrum by the requirement that the spectrum be left or right bounded. 15.9. Let 𝜎 ∶ ℝ → Aut(ℳ) be a continuous action of ℝ on the von Neumann algebra ℳ ⊂ ℬ(ℋ ) and assume that q𝜎∞ = 1. Then the so-continuous unitary representation u ∶ ℝ → ℳ constructed in Section 15.7 has the following minimality property: If v ∶ ℝ → ℬ(ℋ ) is any so-continuous unitary representation with positive spectrum such that 𝜎s = Ad(vs )(s ∈ ℝ), then ℋ (u; (t, +∞)) ⊂∶ ℋ (v; (t, +∞))

(t ∈ ℝ),

that is, for the corresponding infinitesimal generators A, B we have 𝜒(t,+∞) (A) ≤ 𝜒(t,+∞) (B)

(t ∈ ℝ).

Indeed, since ℋ = ℋ (v; [0, +∞)) and 𝜎 = Ad(v), using Corollary 1/14.7 we obtain ℳ(𝜎; (t, +∞))ℋ = ℳ(𝜎; (t, +∞))ℋ (v; [0, +∞)) ⊂ ℋ (v; (t, +∞)), so that the desired conclusion follows using 15.7.(3). 15.10. The arguments presented in Section 15.7 also give the following: Proposition. Let 𝜎 ∶ ℝ → Aut(ℳ) be a continuous action of ℝ on the W∗ -algebra ℳ. If there exists 𝜀 > 0 such that Sp 𝜎 ⊂ [−𝜀, 𝜀], then there exists a = a∗ ∈ 𝒵 (ℳ 𝜎 ), ‖a‖ ≤ 𝜀∕2, such that 𝜎t = Ad(exp(ita)) (t ∈ ℝ). Proof. By the last remark in Section 15.7, there exists A ∈ 𝒵 (ℳ 𝜎 ), 0 ≤ A ≤ 𝜀, such that 𝜎t = Ad(exp(itA))(t ∈ ℝ), and we can take a = A − 2𝜀 . 15.11. Let 𝜎 ∶ G → Aut(ℳ) and 𝜏 ∶ G → Aut(ℳ) be two continuous actions of G on ℳ. We shall say that the actions 𝜎 and 𝜏 are outer conjugate, and we shall write 𝜎 ∼ 𝜏, if there exists a unitary 𝜎-cocycle w ∈ Z𝛼 (G; U(ℳ)) (see 5.1) such that 𝜏t = Ad(wt )◦𝜎t (t ∈ ℝ). It is easy to check that “∼” is an equivalence relation. In this case, we can define the balanced action 𝜃 = 𝜃(𝜎, w) of G on the W∗ -algebra 𝒫 = Mat2 (ℳ) by putting ( 𝜃t

x11 x21

x12 x22

)

( =

𝜎t (x11 ) wt 𝜎t (x21 ) (

𝜎t (x12 )w∗t wt 𝜎t (x22 )w∗t

) ([xij ] ∈ 𝒫 ).

(1)

) 1 0 ̄ 1)-cocycle (t ∈ G), we define a unitary (𝜎 ⊗ 0 wt ̄ 𝜄) (t ∈ G). Thus, 𝜃 ∶ G → Aut(𝒫 ) is indeed a v ∈ Z𝜎 ⊗s ̄ (G, U(𝒫 )) such that 𝜃t = Ad(vt )◦(𝜎 ⊗ continuous action of G on 𝒫 . It is clear that for t ∈ G and x ∈ ℳ we have ( ) ( ) ( ) ( ) x 0 𝜎t (x) 0 0 0 0 0 𝜃t = , 𝜃t = . (2) 0 0 0 0 0 x 0 𝜏t (x) Since w ∈ Z𝜎 (G, U(ℳ)), putting vt =

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189

(

) ( ) 1 0 0 0 The projections e = ,f = are equivalent in 𝒫 and we have e, f ∈ 𝒫 𝜃 . 0 0 0 1 Moreover, from (2) it follows that (ℳ, 𝜎) ≈ (e𝒫 e, 𝜃 e ),

(ℳ, 𝜏) ≈ ( f𝒫 f, 𝜃 f ),

(3)

In particular, for every x ∈ ℳ we have ( Sp𝜎 (x) = Sp𝜃

x 0

0 0

)

( , Sp𝜏 (x) = Sp𝜃

0 0 0 x

) .

(4)

Note that, conversely, if there exists a continuous action 𝜃 ∶ G → Aut(Mat2 (ℳ)) which satisfies (2), then the actions 𝜎 and 𝜏 are outer conjugate. 15.12. Using the results of Sections 15.10 and 15.11, we now prove a result which we shall need later. Proposition. Let 𝜎 ∶ ℝ → Aut(ℳ) be a continuous action of ℝ on the W∗ -algebra ℳ and 𝜀 > 0 such that Sp 𝜎 ∩ {[−2𝜀, −𝜀] ∪ [𝜀, 2𝜀]} = ∅.

(1)

Then there exists a = a∗ ∈ 𝒵 (ℳ 𝜎 ), ‖a‖ ≤ 𝜀∕2, such that for the continuous action 𝜏 ∈ ℝ → Aut(ℳ) defined by 𝜏t = Ad(exp(ita))◦𝜎t (t ∈ ℝ), we have Sp 𝜏 ∩ (−𝜀, 𝜀) = {0}.

(2)

Proof. From (1) and statements 15.2.(3), 15.3.(2), it follows that the spectral subspace 𝒩 = ℳ(𝜎; [−𝜀, 𝜀]) is a 𝜎-invariant unital W∗ -subalgebra of ℳ. Let 𝜎t′ = 𝜎t |𝒩 (t ∈ ℝ). Then 𝜎 ′ ∶ ℝ → ′ Aut(𝒩 ) is a continuous action of ℝ on 𝒩 , 𝒩 𝜎 = ℳ 𝜎 and Sp 𝜎 ′ ⊂ [−𝜀, 𝜀]. By Proposition 15.10, ′ ∗ 𝜎 𝜎 there exists a = a ∈ 𝒵 (𝒩 ) = 𝒵 (ℳ ), ‖a‖ ≤ 𝜀∕2, such that 𝜎t′ = Ad(exp(ita))(t ∈ ℝ). Since a ∈ ℳ 𝜎 , the equation 𝜏t (x) = exp(ita)𝜎t (x) exp(−ita)(x ∈ ℳ, t ∈ ℝ), defines a continuous action 𝜏 ∶ ℝ → Aut(ℳ). By (1) and Lemma 15.1, we see that the set 𝒩 ∪ ℳ(𝜎; ℝ∖[−2𝜀, 2𝜀]) is w-total in ℳ. Thus, in order to prove (2), it is sufficient (14.5.(1)) to show that x ∈ 𝒩 ∪ ℳ(𝜎; ℝ∖[−2𝜀, 2𝜀]) ⇒ Sp𝜏 (x) ∩ (−𝜀, 𝜀) = {0}. Since 𝒩 ⊂ ℳ 𝜏 , for x ∈ 𝒩 we obviously have Sp𝜏 (x) = {0}. Consider now the balanced action 𝜃 = 𝜃(𝜎, w) of ℝ on 𝒫 = Mat2 (ℳ) constructed as in Section 15.11 with wt = exp(ita)(t ∈ ℝ). Then we have ( 𝜃t

0 1

0 0

)

( =

0 exp(ita)

0 0

) (t ∈ ℝ)

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Groups of Automorphisms

and hence for every f ∈ ℒ 1 (ℝ) we get (see 14.10.(6)) ( 𝜃f

0 1

0 0

)

( =

0 0 ̂f (a) 0

) .

It follows that ( Sp𝜃

0 0 1 0

) = Sp(a) ⊂ [−𝜀∕2, 𝜀∕2].

(3)

(

) ( )( )( ) 0 0 0 0 x 0 0 1 = , using 15.3.(2), 0 x 1 0 0 0 0 0 15.2.(2), 15.11.(4), and (3) above, we get Sp𝜏 (x) ⊂ [−𝜀∕2, 𝜀∕2] + {ℝ ⧵ [−2𝜀, 2𝜀]} + [−𝜀∕2, 𝜀∕2]; hence Sp𝜏 (x) ∩ (−𝜀, 𝜀) = {0}.

Let x ∈ ℳ(𝜎; ℝ, [−2𝜀, 2𝜀]). Since

15.13. We now give another application of the technique developed in Section 15.7. Let ℳ be a W∗ -algebra. A linear mapping 𝛿 ∶ ℳ → ℳ∗ such that 𝛿(xy) = x𝛿(y) + 𝛿(x) y (x, y ∈ ℳ) is called a derivation on ℳ. Every derivation on a W∗ -algebra is bounded (Sakai, 1971, 4.1.3). Let 𝛿 be a derivation on ℳ. Then the mappings 𝛿1 , 𝛿2 ∶ ℳ → ℳ defined by 𝛿1 (x) = (𝛿(x) − 𝛿(x∗ )∗ )∕2, 𝛿2 (x) = (𝛿(x) + 𝛿(x∗ )∗ )∕2i are antihermitian derivations, that is, 𝛿k (x∗ ) = −𝛿k (x)∗ (k = 1, 2) and 𝛿 = 𝛿1 + i𝛿2 . Every element a ∈ ℳ determines an inner derivation 𝛿a on ℳ defined by 𝛿a (x) = ax−xa(x ∈ ℳ). Theorem (Kadison & Sakai). Every derivation on a W∗ -algebra is inner. Proof. Without loss of generality, we consider an antihermitian derivation 𝛿 on the W∗ -algebra ℳ. It is then easy to check that the elements 𝜎t = exp(it𝛿)(t ∈ ℝ) of the Banach algebra ℬ(ℳ) are *-automorphisms of ℳ and that the mapping 𝜎 ∶ ℝ ∋ t ↦ 𝜎t ∈ Aut(ℳ) is a norm-continuous action of ℝ on ℳ. In Section 14.11, we proved that Sp 𝜎 = Spℬ(ℳ) (𝛿) ⊂ [−‖𝛿‖, ‖𝛿‖]. As we saw in Section 15.7, a = ∫ t dq𝜎t is a positive element of ℳ, ‖a‖ ≤ ‖𝛿‖ and 𝜎t = Ad(exp(ita)) (t ∈ ℝ), that is, (exp(it𝛿))(x) = exp(ita)x exp(−ita)

(x ∈ ℳ, t ∈ ℝ).

Taking the derivative here with respect to t and then putting t = 0, it follows that 𝛿(x) = ax − xa (x ∈ ℳ), that is, 𝛿 = 𝛿a . 15.14. For an antihermitian derivation 𝛿 of ℳ∗ , the element a ∈ ℳ constructed in the proof of Theorem 15.13 is the smallest positive element b ∈ ℳ such that 𝛿 = 𝛿b ; moreover, for every central projection e in ℳ we have ‖ae‖ = ‖𝛿|eℳe‖. Indeed, let e ∈ ℳ be a central projection. We have 𝛿(e) = 𝛿a (e) = 0. It follows that 𝛿(exe) = e𝛿(x)e (x ∈ ℳ), and 𝜎t (e) = e(t ∈ ℝ), that is, e ∈ ℳ 𝜎 . Consequently, the restriction of 𝛿 to eℳe is a derivation 𝛿 e on the W∗ -algebra eℳe and exp(it𝛿 e ) = 𝜎te (t ∈ ℝ). By the proof of Theorem 15.13, we

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191

have 𝜎te = Ad(exp(ita′ ))(t ∈ ℝ), where the positive operator a′ ∈ eℳe with ‖a′ ‖ ≤ ‖𝛿 e ‖ is defined e e by a′ = ∫ tdq𝜎t . It is easy to check that q𝜎t = eq𝜎t = q𝜎t e (t ∈ ℝ), hence a′ = ae and therefore ‖ae‖ ≤ ‖𝛿|eℳe‖. If c = ae − ‖ae‖∕2, then ‖c‖ ≤ ‖ae‖∕2 and 𝛿c = 𝛿ae = 𝛿 e , hence ‖c‖ > ‖𝛿 e ‖∕2. Consequently ‖ae‖ > ‖𝛿|eℳe‖.

(1)

Consider now 0 ≤ b ∈ ℳ with 𝛿 = 𝛿b . Then the mapping v ∶ ℝ → eℳe defined by vt = exp(itbe) (t ∈ ℝ) is an so-continuous unitary representation with positive spectrum and 𝜎te = Ad(vt )(t ∈ ℝ). By the minimality property established in Section 15.9, we have 𝜒(t,+∞) (ae) ≤ 𝜒(t,+∞) (be) (t ∈ ℝ). In particular, taking t = ‖be‖, we obtain ‖ae‖ ≤ ‖be‖

(e ∈ Proj(𝒵 (ℳ))).

(2)

Assume that (a − b)+ ≠ 0. Then there exist 𝜆 > 0 and a spectral projection e ≠ 0 of (a − b) such that (a − b)e ≥ 𝜆e.

(3)

Since 𝛿a = 𝛿b , we have a − b ∈ 𝒵 (ℳ), hence (a − b)+ ∈ 𝒵 (ℳ) and e ∈ Proj(𝒵 (ℳ)). Now from (3) we deduce that ‖be‖ < ‖be‖ + 𝜆 = ‖be + 𝜆‖ ≤ ‖ae‖, contradicting (2). Thus, (a − b)+ = 0 and a ≤ b. 15.15. Using Theorem 15.13, we can also obtain a similar result for automorphisms of W∗ -algebras. To this end it is necessary to consider, when possible, the logarithm of an automorphism and to show that it is a derivation. We shall denote by ln the principal branch of the logarithm defined on the domain ℂ∖(−∞, 0]. Then ln ∶ ℂ∖(−∞, 0] → {𝜆 ∈ ℂ; |Im 𝜆| < 𝜋} is an analytic function, namely the inverse function of exp ∶ {𝜆 ∈ ℂ; |Im 𝜆| < 𝜋} → ℂ∖(−∞, 0]. Let 𝒜 be a unital Banach algebra and 𝜎 ∈ 𝒜 such that Sp𝒜 (𝜎) ⊂ {𝜆 ∈ ℂ; Re 𝜆 > 0}. Then 𝜎 is an invertible element and so defines a bounded linear mapping Ad(𝜎) ∶ 𝒜 ∋ 𝛼 ↦ 𝜎𝛼𝜎 −1 ∈ 𝒜 .

(1)

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Groups of Automorphisms

Since Sp𝒜 (𝜎) ⊂ ℂ∖(−∞, 0], it is meaningful to consider 𝛿 = ln(𝜎) ∈ 𝒜 . We have 𝜎 = exp(𝛿) and, by (1), Sp𝒜 (𝛿) ⊂ {𝜆 ∈ ℂ; |Im 𝜆| < 𝜋∕2}.

(2)

𝛿 ∈ 𝒜 defines a bounded linear mapping ad(𝛿) ∶ 𝒜 ∋ 𝛼 ↦ 𝛿𝛼 − 𝛼𝛿 ∈ 𝒜 . Then Ad(𝜎) and ad(𝛿) are elements of the Banach algebra ℬ(𝒜 ) and, using 14.13.(1), 14.13.(2), we deduce from (1), (2) that Spℬ(𝒜 ) (Ad(𝜎)) ⊂ {𝜆𝜇−1 ; 𝜆, 𝜇 ∈ Sp𝒜 (𝜎)} ⊂ ℂ∖(−∞, 0]

(3)

Spℬ(𝒜 ) (ad(𝛿)) ⊂ {𝜆 − 𝜇; 𝜆, 𝜇 ∈ Sp𝒜 (𝛿)} ⊂ {𝜆 ∈ ℂ; |Im 𝜆| < 𝜋}.

(4)

Since 𝜎 = exp(𝛿), it is easy to check that Ad(𝜎) = exp(ad(𝛿));

(5)

ad(𝛿) = ln(Ad(𝜎)).

(6)

and using (3) and (4) we deduce that

Now let ℳ be a unital Banach algebra and 𝒜 = ℬ(ℳ). If 𝜎 ∈ 𝒜 is an automorphism of ℳ satisfying (1), then 𝛿 = ln(𝜎) ∈ 𝒜 is a derivation on ℳ. Indeed, consider the mapping L ∶ ℳ → 𝒜 defined by Lx (y) = xy (x, y ∈ ℳ). Then L is a bounded injective Banach algebra homomorphism and we have (Ad(𝜎))(Lx ) = L𝜎(x) (x ∈ ℳ), that is, Ad(𝜎)◦L = L◦𝜎. It follows that p(Ad(𝜎))◦L = L◦p(𝜎) for every polynomial p and, using Runge’s theorem (Hormander, 1966, 1.3.2), we also obtain ln(Ad(𝜎))◦L = L◦ ln(𝜎). By (6), ad(𝛿)◦L = L◦𝛿. Since ad(𝛿) is a derivation of 𝒜 and L is an injective homomorphism, we conclude that 𝛿 is a derivation on ℳ. If 𝛿 is an inner derivation, that is, 𝛿 = 𝛿a = ad(a) with a ∈ ℳ, then 𝜎 = exp(𝛿) = exp(ad(a)) = Ad(exp(a)). Using Theorem 3.13, we thus obtain the following: Corollary. Let 𝜎 be an automorphism of the W∗ -algebra ℳ such that Spℬ(ℳ) (𝜎) ⊂ {𝜆 ∈ ℂ; Re 𝜆 > 0}. There exists an invertible element v ∈ ℳ such that 𝜎 = Ad(v), that is, 𝜎(x) = vxv−1

(x ∈ ℳ).

If, moreover, 𝜎 is a *-automorphism, then there exists a unitary element u ∈ ℳ such that 𝜎 = Ad(u). Proof. The first statement has already been proved. Assume that 𝜎 = Ad(v) is a *-automorphism and let v = ua be the polar decomposition of v. Then u ∈ ℳ is a unitary element and a ∈ ℳ is an invertible positive element. For every x ∈ ℳ, we have uax∗ a−1 u∗ = 𝜎(x∗ ) = 𝜎(x)∗ = ua−1 x∗ au∗ . It follows that a2 ∈ 𝒵 (ℳ), so that a ∈ 𝒵 (ℳ) and 𝜎 = Ad(u).

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193

We note that if 𝜎 ∈ Aut(ℳ) and ‖𝜎 − 𝜄ℳ ‖ < 1, then the requirement Spℬ(ℳ) (𝜎) ⊂ {𝜆; Re 𝜆 > 0} is satisfied and hence 𝜎 ∈ Int(ℳ). Actually, Kadison and Ringrose (1967) proved that if 𝜎 ∈ Aut(ℳ) and ‖𝜎 − 𝜄ℳ ‖ < 2, then 𝜎 ∈ Int(ℳ) (see also Borchers, 1973, Thm. 5.4.A). Since for any *-automorphism 𝜎 we have ‖𝜎 − 𝜄ℳ ‖ ≤ 2, √ it follows that ‖𝜎 − 𝜄ℳ ‖ = 2 whenever 𝜎 is outer. A simple proof of the implication ‖𝜎 − 𝜄ℳ ‖ < 3 ⇒ 𝜎 ∈ Int(ℳ) can be found in Dixmier (1957, 1969). 15.16. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact group G on the W∗ -algebra ℳ such that 𝜎t ∈ Int(ℳ) for all t ∈ G. Without the assumption that the predual ℳ∗ of ℳ is separable, this property of 𝜎 does not imply the existence of an s-continuous unitary representation u ∶ G → ℳ such that 𝜏t = Ad(ut )(t ∈ G) (see Connes, 1973a, 1.5.8.c). In the case when ℳ∗ is separable several positive results are known (Dixmier, 1957, 1969; Hansen, 1977; Kadison, 1965; Kallman, 1971; Moore, 1976). We note here the following result: Theorem (Kallman & Moore). Let 𝜎 ∶ ℝ → Aut(ℳ) be a continuous action of ℝ on the W∗ -algebra ℳ. If the predual ℳ∗ is separable and 𝜎t ∈ Int(ℳ) for all t ∈ ℝ, then there exists an s-continuous unitary representation u ∶ ℝ → ℳ such that 𝜎t = Ad(ut ) for all t ∈ ℝ. For the proof of the full theorem, we refer to Kallman (1971) and Moore (1976). In this section, we present some arguments, due to Hansen (1977), which lead to a simple and elementary proof of the theorem when ℳ is a factor. Let 𝜎 ∶ ℝ → Aut(ℳ) be a continuous action of ℝ on the W∗ -algebra ℳ and {vt }t∈ℝ ⊂ U(ℳ) such that 𝜎t = Ad(vt )(t ∈ ℝ). Then vs vt = vt vs (s, t ∈ ℝ).

(1)

Since Ad(vs vt ) = 𝜎s+t = Ad(vt vs ), there exists a mapping c ∶ ℝ × ℝ → U(𝒵 (ℳ)) such that 𝜎t (vs ) = vt vs v∗t = c(t, s)vs for all s, t ∈ ℝ. It is clear that the mapping c is separately w-continuous, c(0, s) = c(t, t) = 1, c(s, t) = c(−t, s) = c(t, s)∗ and c(t + t′ , s) = c(t, s)c(t′ , s). It follows that c(2t, t) = c(t, t)2 = 1, hence c(q2−n t, t) = c(2−n t, t)q = 1 for any t ∈ ℝ, n ∈ ℕ, q ∈ ℤ and the continuity of c allows us to conclude that c(s, t) = 1 for all s, t ∈ ℝ. Assume now that ℳ is a factor. Then the elements vt ∈ U(ℳ) with 𝜎t = Ad(vt ) are uniquely determined modulo the normal subgroup 𝕋 = {𝜆 ⋅ 1ℳ ; 𝜆 ∈ ℂ, |𝜆| = 1} of U(ℳ). Consider the topological group U(ℳ) with the w-topology and the group Int(ℳ) with the p-topology (2.23). Then the mapping Ad ∶ U(ℳ) ∋ u ↦ Ad(u) ∈ Int(ℳ) is the composition of the canonical quotient mapping k ∶ U(ℳ) → U(ℳ)∕𝕋 with an injective homomorphism 𝛼 ∶ U(ℳ)∕𝕋 → Int(ℳ) of the quotient topological group U(ℳ)∕𝕋 into Int(ℳ). Then the equation νt = k(vt )(t ∈ ℝ) defines a group homomorphism ν ∶ ℝ → U(ℳ)∕𝕋 ,

uniquely determined, such that 𝛼◦ν = 𝜎. Finally assume also that the factor ℳ has separable predual. Then U(ℳ) and U(ℳ)∕𝕋 are polish groups. Using the measurable selection theorem of von Neumann (Dixmier, 1957, 1969) and the

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continuity of the action 𝜎, we find a Lebesgue measurable mapping v ∶ ℝ → U(ℳ) such that Ad(vt ) = 𝜎t (t ∈ ℝ). It follows that the mapping ν is a Lebesgue measurable homomorphism and hence ν is a continuous homomorphism. To prove the theorem, it is sufficient to show that there exists a continuous homomorphism u ∶ ℝ → U(ℳ) such that k(ut ) = vt for t ∈ ℝ. By statement (1), the set G = {(u, t) ∈ U(ℳ) × ℝ; k(u) = νt } is an abelian subgroup of U(ℳ) × ℝ. With the topology inherited from U(ℳ) × ℝ, G is a topological group. The mapping j ∶ G ∋ (u, t) ↦ t ∈ ℝ is an open, continuous and surjective homomorphism, whose kernel is isomorphic and homeomorphic to 𝕋 . Since 𝕋 and ℝ are locally compact groups, it follows (Hewitt & Ross, 1963, 1970, 5.25) that G is also locally compact. On the other hand, the mapping i ∶ 𝕋 ∋ 𝜆 ↦ (𝜆, 0) ∈ G is a homeomorphism of 𝕋 onto a compact subgroup of G. The character 𝜆 ↦ 𝜆 on 𝕋 can be extended (Hewitt & Ross, 1963, 1970, 24.12; Rudin, 1962, 2.1.4) to a continuous character 𝛾 on G. Thus, there exists a continuous homomorhism 𝛾 ∶ G → 𝕋 such that 𝛾◦i = 𝜄𝕋 . Thus, the short exact sequence i

j

0→𝕋 →G→ℝ→0 is split and therefore there exists a continuous homomorphism 𝛿 ∶ ℝ → G with j◦𝛿 = 𝜄ℝ , that is, a continuous homomorphism u ∶ ℝ → U(ℳ) with k(ut ) = vt (t ∈ ℝ). 15.17. In this final section, we prove a general result which we shall use later in some particular situations. Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the separable locally compact abelian group G on the countably decomposable W∗ -algebra ℳ. Let x ∈ ℳ, e = r(x), e′ = l(x) and let x = u|x| the polar decomposition of x. We assume that there exist projections f, f ′ ∈ ℳ 𝜎 , f ≥ e, f ′ ≥ e′ , such that ′

(Sp𝜎 (x) − Sp𝜎 (x)) ∩ (Sp 𝜎 f ∪ Sp 𝜎 f ) = {0}.

(1)

Sp𝜎 (u) = Sp𝜎 (x), |x| ∈ ℳ 𝜎 , e ∈ ℳ 𝜎 , e′ ∈ ℳ 𝜎 .

(2)

Then

If, moreover, ℳ is properly infinite, then there exists a partial isometry v ∈ ℳ such that Sp𝜎 (v) ⊂ Sp𝜎 (x) v∗ v = the central support of e in ℳ 𝜎 v∗ v = the central support of e′ in ℳ 𝜎 vℳ 𝜎 v∗ ⊂ ℳ 𝜎 , v∗ ℳ 𝜎 v ⊂ ℳ 𝜎 a = v∗ x ∈ ℳ 𝜎 x = va.

(3) (4) (5) (6) (7) (8)

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195

Proof. Let E = Sp𝜎 (x). We have x ∈ ℳ(𝜎; E), x∗ ∈ ℳ(𝜎; −E) hence x∗ x ∈ ℳ(𝜎; E − E) ∩ fℳf ⊂ ℳ 𝜎 , by (1). Thus, |x| ∈ ℳ 𝜎 and Sp𝜎 (x) ⊂ Sp𝜎 (u), as x = u|x|. There exists a sequence {an } ⊂ ℳ 𝜎 w

w

such that |x|an → s(|x|) = e. It follows that e ∈ ℳ 𝜎 and ℳ(𝜎; E) ∋ xan = u|x|an → u, hence Sp𝜎 (u) ⊂ Sp𝜎 (x). Consequently, Sp𝜎 (u) = Sp𝜎 (x) = E and e′ = uu∗ ∈ ℳ(𝜎; E − E) ∩ f ′ ℳf ′ ⊂ ℳ 𝜎 , by (1). We have thus proved (2). Let c = zℳ 𝜎 (e), c′ = zℳ 𝜎 (e′ ) ∈ 𝒵 (ℳ 𝜎 ) be the central supports of the projections e, e′ ∈ ℳ 𝜎 . There exist projections e0 , e1 ∈ 𝒵 (eℳ 𝜎 e), uniquely determined, such that e = e0 + e1 , e0 ℳ 𝜎 e0 is properly infinite, and e1 ℳ 𝜎 e1 is finite. Also, there exist projections e′0 , e′1 ∈ 𝒵 (e′ ℳ 𝜎 e′ ), uniquely determined, such that e′ = e′0 + e′1 , e′0 ℳ 𝜎 e′0 is properly infinite, and e′1 ℳ 𝜎 e′1 is finite. Then there exist projections c0 , c1 ∈ 𝒵 (ℳ 𝜎 ) with c0 + c1 = c, c0 e = e0 , c1 e = e1 an c′0 , c′1 ∈ 𝒵 (ℳ 𝜎 ) with c′0 + c′1 = c′ , c′0 e′ = e′0 , c′1 e′ = e′1 . ′ For y ∈ eℳ 𝜎 e, we have uyu∗ ∈ e′ ℳe′ and Sp𝜎 (uyu∗ ) ⊂ (E − E) ∩ Sp𝜎 e = {0} hence uyu∗ ∈ e′ ℳ 𝜎 e′ . Thus, the mappings eℳ 𝜎 e ∋ y ↦ uyu∗ ∈ e′ ℳ 𝜎 e′ , e′ ℳ 𝜎 e′ ∋ z ↦ u∗ zu ∈ eℳ 𝜎 e are reciprocal *-isomorphisms. By the uniqueness of the decompositions e = e0 + e1 , e′ = e′0 + e′1 , it follows that ue0 u∗ = e′0 ,

ue1 u∗ = e′1 ,

u∗ e′0 u = e0 , u∗ e′1 u = e1 .

By construction, e1 is a finite projection in ℳ 𝜎 with central support c1 . Since ℳ 𝜎 is properly infinite, we can write c1 =

∞ ∑

en with en ∈ Proj(ℳ 𝜎 ), en ∼ e1 in ℳ 𝜎 ,

n=1

so there exist wn ∈ ℳ 𝜎 with w∗n wn = e1 , wn w∗n = en and w1 = e1 . Similarly, we can write c′1

=

∞ ∑

e′n with e′n ∈ Proj(ℳ 𝜎 ), e′n ∼ e′1 in ℳ 𝜎

n=1 ′ ′ ′ ′ ′ ′∗ ′ and there exist w′n ∈ ℳ 𝜎 with w′∗ n wn = e1 , wn wn = en and w1 = e1 . ′ ∗ ∗ ∗ ′ Then un = wn uwn ∈ ℳ(𝜎; E), un un = en , un un = en , hence

v1 =

∞ ∑

un ∈ ℳ(𝜎; E), v∗1 v1 = c1 , v1 v∗1 = c′1 .

n=1

On the other hand, e0 is a properly infinite projection in ℳ 𝜎 with central support equal to c0 , hence e0 ∼ c0 in ℳ 𝜎 , that is, there exists w ∈ ℳ 𝜎 with w∗ w = e0 , ww∗ = c0 . Similarly, there exists w′ ∈ ℳ 𝜎 with w′∗ w′ = e′0 , w′ w′∗ = c′0 . Then v0 = w′ uw∗ ∈ ℳ(𝜎; E), v∗0 v0 = c0 , v0 v∗0 = c′0 . Thus, v = v0 + v1 ∈ ℳ(𝜎; E), v∗ v = c, vv∗ = c′ and this proves (3), (4), (5).

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From (1) and 15.4.(5) it follows that ′

(E − E) ∩ (Sp 𝜎 c ∪ Sp 𝜎 c ) = {0}. Thus, if y ∈ ℳ 𝜎 , then vyv∗ ∈ ℳ(𝜎; E − E) ∩ c′ ℳc′ = ℳ 𝜎 and v∗ yv ∈ ℳ(𝜎; E − E) ∩ cℳc = ℳ 𝜎 , proving (6). Also, a = v∗ x ∈ ℳ(𝜎; E − E) ∩ cℳc = ℳ 𝜎 and, as vv∗ = c′ ≥ e′ = l(x), we have va = vv∗ x = x, proving (7) and (8). Corollary. Let 𝜎 ∶ ℝ → Aut(ℳ) be a continuous action of ℝ on the factor ℳ such that ℳ 𝜎 is properly infinite. Let 𝜀 > 0 and let p1 , p2 ∈ 𝒵 (ℳ 𝜎 ) be nonzero projections such that 0 be an isolated point in Sp 𝜎 p1 ∪ Sp 𝜎 p2 . Then there exists a partial isometry 0 ≠ v ∈ ℳ with p1 ≥ v∗ v ∈ 𝒵 (ℳ 𝜎 ), p2 ≥ vv∗ ∈ 𝒵 (ℳ 𝜎 ), and Sp𝜎 (v) − Sp𝜎 (v) ⊂ [−𝜀, 𝜀]. Proof. We may assume that (Sp 𝜎 p1 ∪ Sp 𝜎 p2 ) ∩ [−𝜀, 𝜀] = {0}. Since ℳ is a factor, there exists y ∈ ℳ with p2 yp1 ≠ 0. By Lemma 15.1, there exists f ∈ ℒ 1 (ℝ) with supp ̂f − supp ̂f ⊂ [−𝜀, 𝜀] and x = p2 𝜎f (y)p1 = 𝜎f (p2 yp1 ) ≠ 0. We have p2 x = xp1 = x and Sp𝜎 (x) − Sp𝜎 (x) ⊂ [−𝜀, 𝜀]. From the proposition, we infer that e1 = r(x) ∈ ℳ 𝜎 , e2 = l(x) ∈ ℳ 𝜎 . Let q1 = zℳ 𝜎 (e1 ) ≤ p1 , q2 = zℳ 𝜎 (e2 ) ≤ p2 . Again by the proposition, there exists 0 ≠ v ∈ ℳ such that v∗ v = q1 , vv∗ = q2 , and Sp𝜎 (v) ⊂ Sp𝜎 (x), hence Sp𝜎 (v) − Sp𝜎 (v) ⊂ [−𝜀, 𝜀]. 15.18. Notes. The proofs of Borchers’ theorem (15.8, (i) ⇒ (ii); Borchers, 1969) and of the Kadison– Sakai theorem (15.13; Kadison, 1966; Sakai, 1966) given in this section are due to Arveson (1974). Theorem 15.15 appears in Sakai (1966) and the proof is based on a device due to Zeller–Meier (1967). The results contained in Sections 15.10–15.12 and 15.17 are due to Connes (1973a). For our exposition, we have used Arveson (1974), Combes and Delaroche (1975), Connes (1973a), Hansen (1977); and Olesen (1974). Different proofs and extensions of Borchers’ theorem are contained in Combes and Delaroche (1975), Kadison (1969), Olesen (1976), and Pedersen (1978). The following references also contain applications of the theory of spectral subspaces to groups of *-automorphisms on C∗ algebras: Akemann et al. (1976), Bisognano and Wichmann (1975), Borchers (1973), Olesen (1974, 1975), Olesen and Pedersen (1974), and Pedersen (1973–1977).

16 The Connes Invariant 𝚪(𝝈) In this section, we introduce an outer conjugacy invariant for continuous actions. 16.1. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the separable commutative locally compact ̂ by group G on the W∗ -algebra ℳ. We define a closed set Γ(𝜎) ⊂ G ⋂ Γ(𝜎) = {Sp 𝜎 e ; 0 ≠ e ∈ Proj(ℳ 𝜎 )}. (1) According to 15.4.(5), we also have ⋂ Γ(𝜎) = {Sp 𝜎 e ; 0 ≠ e ∈ Proj(𝒵 (ℳ 𝜎 ))}.

(2)

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197

In particular, if ℳ 𝜎 is a factor, then Γ(𝜎) = Sp 𝜎.

(3)

Clearly, for every nonzero projection e ∈ ℳ 𝜎 we have Γ(𝜎) ⊂ Γ(𝜎 e ).

(4)

̂ and for every nonzero projection e ∈ ℳ 𝜎 we have Proposition. Γ(𝜎) is a closed subgroup of G Γ(𝜎) + Sp 𝜎 e = Sp 𝜎 e .

(5)

Proof. We remark first that 0 ∈ Γ(𝜎), since for every nonzero projection e ∈ ℳ 𝜎 we have Sp 𝜎 e ⊃ Sp𝜎 e (e) = {0}. Therefore, Γ(𝜎) + Sp 𝜎 ⊃ Sp 𝜎. Conversely, let 𝛾1 ∈ Sp 𝜎 and 𝛾2 ∈ Γ(𝜎). In order to prove that 𝛾1 + 𝛾2 ∈ Sp 𝜎, we have to show (14.5) that ℳ(𝜎; V) ≠ {0} for every neighborhood V of 𝛾1 + 𝛾2 . Let V1 and V2 be neighborhoods of 𝛾1 and 𝛾2 , respectively, such that V1 + V2 ⊂ V. Since 𝛾1 ∈ Sp 𝜎, ⋁ there exists 0 ≠ x1 ∈ ℳ(𝜎; V1 ). Then e = t∈G r(𝜎t (x1 )) is a nonzero projection in ℳ 𝜎 . Since 𝛾2 ∈ Γ(𝜎) ⊂ Sp 𝜎 e , it follows (15.4.(2)) that there exists 0 ≠ x2 ∈ ℳ(𝜎; V2 ) ∩ eℳe. Since ex2 = x2 ≠ 0, there exists t ∈ G with x = 𝜎t (x1 )x2 ≠ 0 and (14.2.(4), 15.3.(1)) x ∈ ℳ(𝜎; V1 + V2 ) ⊂ ℳ(𝜎; V). Hence Γ(𝜎) + Sp 𝜎 = Sp 𝜎. Using this conclusion and (4) we see that Γ(𝜎) + Sp 𝜎 e = Sp 𝜎 e for every 0 ≠ e ∈ Proj(ℳ 𝜎 ). From (1) and (5) we infer that Γ(𝜎) + Γ(𝜎) = Γ(𝜎). On the other hand, from (1) and 15.2.(4) it ̂ follows that Γ(𝜎) = −Γ(𝜎). Hence Γ(𝜎) is a closed subgroup of G. 16.2. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ. Proposition. Let e1 , e2 ∈ ℳ 𝜎 be two projections with zℳ (e1 )zℳ (e2 ) ≠ 0 and V a compact ̂ There exist two nonzero projections f1 , f2 ∈ ℳ 𝜎 , f1 ≤ e1 , f2 ≤ e2 , such neighborhood of 0 in G. that Sp 𝜎 f1 ⊂ V + Sp 𝜎 f2 , Sp 𝜎 f2 ⊂ V + Sp 𝜎 f1 .

(1)

Proof. Since zℳ (e1 )zℳ (e2 ) ≠ 0, there exists 0 ≠ y ∈ ℳ such hat e1 y = y = ye2 ([L], 4.5). By Lemma 15.1, there exists h ∈ ℒ 1 (G) with supp ĥ − supp ĥ ⊂ V and x = 𝜎h (y) ≠ 0. Then (15.3.(3)) e1 x = e1 𝜎h (y) = 𝜎h (e1 y) = 𝜎h (y) = x = ⋯ = xe2 and (14.2.(3)) Sp𝜎 (x) − Sp𝜎 (x) ⊂ V. We define f1 =

⋁ t∈G

l(𝜎t (x)),

f2 =

⋁

r(𝜎t (x)).

t∈G

Then f1 , f2 ∈ ℳ 𝜎 , 0 ≠ f1 ≤ e1 , 0 ≠ f2 ≤ e2 . Let 𝛾1 ∈ Sp 𝜎 f1 . Since V is compact, the set V + Sp 𝜎 f2 is closed. Thus, in order to prove that 𝛾1 ∈ V + Sp 𝜎 f2 , it is sufficient to show that (V1 − V) ∩ Sp 𝜎 f2 ≠ ∅ for any compact neighborhood V1 of 𝛾1 . Since 𝛾1 ∈ Sp 𝜎 f1 , there exists 0 ≠ x1 ∈ ℳ(𝜎; V1 ) ∩ f1 ℳf1 . Since x1 f1 = x1 ≠ 0, there exists t ∈ G with x1 𝜎t (x) ≠ 0. Since 𝜎1 (x)∗ x∗1 f1 = 𝜎1 (x)∗ x∗1 ≠ 0, there exists s ∈ G such that 𝜎1 (x)∗ x∗1 𝜎s (x) ≠ 0. Therefore, x2 = 𝜎s (x∗ )x1 𝜎t (x) ≠ 0. We obviously have x2 f2 = x2 = f2 x2 and

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(15.2.(2), 15.3(1)) Sp𝜎 (x2 ) ⊂ Sp𝜎 (x1 ) + Sp𝜎 (x) − Sp𝜎 (x) ⊂ V1 − V, hence (V1 − V) ∩ Sp 𝜎 f2 ≠ ∅. We have thus proved the first inclusion in (1). The second one is proved similarly. Corollary 1. If e1 , e2 ∈ Proj(ℳ 𝜎 ) and zℳ (e1 ) = zℳ (e2 ) ≠ 0, then Γ(𝜎 e1 ) = Γ(𝜎 e2 ).

(2)

In particular, if e ∈ Proj(ℳ 𝜎 ) and zℳ (e) = 1, then Γ(𝜎 e ) = Γ(𝜎).

(3)

Proof. It is clear that if e ∈ ℳ 𝜎 , then zℳ (e) ∈ ℳ 𝜎 . Thus, it is sufficient to prove (2) only for e2 = zℳ (e1 ), that is, it is sufficient to prove (3). We have (16.1.(5)) Γ(𝜎) ⊂ Γ(𝜎 e ). To prove the opposite inclusion we note that ⋂ Γ(𝜎) = {V + Sp 𝜎 f } (4) ̂ The above proposition shows that with 0 ≠ f ∈ Proj(ℳ 𝜎 ) and V compact neighborhood of 0 in G. ′ for any f and any V there exists 0 ≠ e′ ∈ Proj(ℳ 𝜎 ) with e′ ≤ e and Sp 𝜎 e ⊂ V + Sp 𝜎 f , hence ′ e e f e Γ(𝜎 ) ⊂ Sp 𝜎 ⊂ V + Sp 𝜎 . Therefore, Γ(𝜎 ) ⊂ Γ(𝜎). Corollary 2. If ℳ is a factor, then the family ̂ 𝔉(𝜎) = {(V + Sp 𝜎 e ; 0 ≠ e ∈ Proj ℳ 𝜎 ), V a compact neighborhood of 0 ∈ G} is the basis of a filter with intersection equal to Γ(𝜎). Proof. We have already noticed (4) that the intersection of the family 𝔉(𝜎) is equal to Γ(𝜎). Now, ̂ There let e1 , e2 ∈ ℳ 𝜎 be two nonzero projections and V1 , V2 two compact neighborhoods of 0 in G. ̂ exists a compact neighborhood V of 0 in G such that V ⊂ V1 and V + V ⊂ V2 . By the previous proposition, there exist two nonzero projections f1 , f2 ∈ ℳ 𝜎 , f1 ≤ e1 , f2 ≤ e2 , which satisfy (1). Since V ⊂ V1 and f1 ≤ e1 , we have (15.4.(4)) V + Sp 𝜎 f1 ⊂ V1 + Sp 𝜎 e1 . Since V + V ⊂ V2 and f2 ≤ e2 , it follows from (1) that V + Sp 𝜎 f1 ⊂ V2 + Sp 𝜎 e2 . 16.3. Let 𝜎 ∶ G → Aut(ℳ) and 𝜏 ∶ G → Aut(ℳ) be two continuous actions of G on the W∗ -algebra ℳ. In Section 15.11, we defined the outer conjugacy relation 𝜎 ∼ 𝜏. Proposition. If 𝜎 ∼ 𝜏, then Γ(𝜎) = Γ(𝜏). ( ) ( ) 1 0 0 0 Proof. Let 𝒫 = Mat2 (ℳ) and e = ,f = ∈ 𝒫 . Then e and f are equivalent 0 0 0 1 projections in 𝒫 and, as proved in Section 15.11, there exists a continuous action 𝜃 ∶ G → Aut(𝒫 ) such that e, f ∈ 𝒫 𝜃 and (ℳ, 𝜎) ≈ (e𝒫 e, 𝜃 e ), (ℳ, 𝜏) ≈ ( f𝒫 f, 𝜃 f ). Using Corollary 1/16.2, we conclude that Γ(𝜎) = Γ(𝜃 e ) = Γ(𝜃 f ) = Γ(𝜏).

The Connes Invariant Γ(𝜎)

199

By the above proposition and by Corollary 2/16.2, it follows that if ℳ is a factor and 𝜎 ∼ 𝜏, then for every F ∈ 𝔉(𝜎) there exists F′ ∈ 𝔉(𝜏) with F′ ⊂ F. 16.4. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W∗ -algebra ℳ. Put Ker 𝜎 = {t ∈ G; 𝜎t = 𝜄ℳ }, Int 𝜎 = {t ∈ G; there exists u ∈ U(ℳ 𝜎 ) with 𝜎t = Ad(u)}. Note that Ker 𝜎 ⊂ Int 𝜎 and u ∈ U(ℳ 𝜎 ), 𝜎t = Ad(u) ⇒ u ∈ U(𝒵 (ℳ 𝜎 )).

(1)

̂ the set For each subset E ⊂ G, E⟂ = {t ∈ G; ⟨t, 𝛾⟩ = 1 for all 𝛾 ∈ E} ̂ containing E (Rudin, 1962, is a closed subgroup of G and E⟂⟂ is the smallest closed subgroup of G 2.1.3). Using 14.13.(1) and Proposition 14.6, we get Ker 𝜎 = (Sp 𝜎)⟂ ⊂ Γ(𝜎)⟂ ,

(2)

since, by definition, Γ(𝜎) ⊂ Sp 𝜎. It follows that Sp 𝜎 = Γ(𝜎) ⇔ Ker 𝜎 = Γ(𝜎)⟂ .

(3)

Recall (16.1) that these conditions are satisfied if ℳ 𝜎 is a factor and that they imply that Sp 𝜎 is a ̂ Conversely, closed subgroup of G. Proposition. If Γ(𝜎) is discrete, then Sp 𝜎 = Γ(𝜎) ⇒ 𝒵 (ℳ 𝜎 ) = 𝒵 (ℳ)𝜎 .

(4)

In particular, if ℳ is a factor and Γ(𝜎) is discrete, then Sp 𝜎 = Γ(𝜎) ⇔ ℳ 𝜎 is a factor.

(5)

Proof. Let e, f ∈ ℳ 𝜎 be projections with zℳ (e)zℳ ( f ) ≠ 0. Then ([L],4.5) there exists y ∈ ℳ with eyf ≠ 0. Let 𝛾 ∈ Sp𝜎 (eyf ). Since Sp 𝜎 = Γ(𝜎) is discrete, there exists (14.1.(5)) h ∈ ℒ 1 (G) ̂ ̂ such that h(𝛾) ≠ 0 and h(𝜔) = 0 for all 𝜔 ∈ Sp 𝜎∖{𝛾}. Then x = 𝜎h (eyf ) ≠ 0, Sp𝜎 (x) = {y}, and (15.3.(3)) ex = x = xf. Since Sp𝜎 (x) = {𝛾} is a singleton, using 15.2.(2), 15.3.(1), and 14.3.(14) we obtain x∗ x ∈ ℳ 𝜎 , xx∗ ∈ ℳ 𝜎 , hence r(x) = s(x∗ x) ∈ ℳ 𝜎 , l(x) = s(xx∗ ) ∈ ℳ 𝜎 ; note that l(x) ≤ e, r(x) ≤ f. Since 𝛾 ∈ Sp 𝜎 = Γ(𝜎), −𝛾 ∈ Γ(𝜎) (16.1), and since 0 ≠ r(x) ∈ Proj(ℳ 𝜎 ), there exists 0 ≠ z ∈ ℳ(𝜎; {−𝛾}) ∩ r(x)ℳr(x). Then 0 ≠ a = xz ∈ ℳ 𝜎 and eaf = a ≠ 0. Hence ([L], 4.5) zℳ 𝜎 (e)zℳ 𝜎 ( f ) ≠ 0.

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It is clear that 𝒵 (ℳ 𝜎 ) ⊂ 𝒵 (ℳ 𝜎 ). Conversely, e ∈ 𝒵 (ℳ 𝜎 ) be a projection and let f = 1−e. Since ef = 0, by the above arguments we infer that zℳ (e)zℳ ( f ) = 0, hence e = zℳ (e) ∈ 𝒵 (ℳ). ̂ is discrete This proposition can be applied, in particular, if the group G is compact, since then G (Rudin, 1962, 1.2.5). 16.5. The main duality result is the following: Theorem. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W∗ -algebra ℳ. Then Int 𝜎 ⊂ Γ(𝜎)⟂ .

(1)

̂ then If ℳ is a factor and the set Sp 𝜎∕Γ(𝜎) is compact in G∕Γ(𝜎), Int 𝜎 = Γ(𝜎)⟂ .

(2)

Proof. Let t ∈ Int 𝜎 and let u ∈ U(𝒵 (ℳ 𝜎 )) be such that 𝜎t = Ad(u). Let 𝜀 > 0. There exist 𝜆0 ∈ ℂ, with |𝜆0 | = 1, and a spectral projection e ∈ 𝒵 (ℳ 𝜎 ) of u, such that ‖ue − 𝜆0 e‖ < 𝜀. Then the spectrum of ue in eℳe is contained in {𝜆 ∈ ℂ; |𝜆 − 𝜆0 | < 𝜀}. Using 14.13.(2), it follows that the spectrum of the automorphism 𝜎te = Ad(ue) ∈ Aut(eℳe) as an element of ℬ(eℳe) is contained in the set {𝜆 ∈ ℂ; |𝜆 − 1| < 2𝜀}. On the other hand, by Proposition 14.6, the spectrum of 𝜎te in ℬ(eℳe) is equal to the closure of the set {⟨t, 𝛾⟩; 𝛾 ∈ Sp 𝜎 e }. Consequently, |⟨t, 𝛾⟩ − 1| < 2𝜀 for every 𝛾 ∈ Sp 𝜎 e in particular for every 𝛾 ∈ Γ(𝜎). Since 𝜀 > 0 was arbitrary, it follows that t ∈ Γ(𝜎)⟂ . ̂ We assume now that ℳ is a factor and that Sp 𝜎∕Γ(𝜎) is a compact subset of G∕Γ(𝜎). ̂ ̂ Let k ∶ G → G∕Γ(𝜎) be the canonical quotient mapping. Since, by assumption, k(Sp 𝜎) is compact, using Corollary 2/16.2 we see that the family {k(F); F ∈ 𝔉(𝜎)} is the basis of a filter ̂ on G∕Γ(𝜎) with intersection equal to {k(0)}. By Proposition 16.1, we have k−1 (k(F)) = F for every F ∈ 𝔉(𝜎). ̂ Re ⟨t, 𝛾⟩ > 1 − 𝜀} is an open neighborhood Let t ∈ Γ(𝜎)⟂ and 0 < 𝜀 < 1. The set De = {y ∈ G; ̂ of k(0) in G∕Γ(𝜎). Consequently, there exists F ∈ 𝔉(𝜎) such that k(F) ⊂ k(D𝜀 ). Since t ∈ Γ(𝜎)⟂ , we have k−1 (k(D𝜀 )) = D𝜀 and therefore F ⊂ D𝜀 . We conclude that there exists a nonzero projection e ∈ ℳ 𝜎 such that Sp 𝜎 e ⊂ D𝜀 . Using Proposition 14.6 it follows that Sp (𝜎te ) ⊂ {𝜆 ∈ ℂ; |𝜆| = 1, Re 𝜆 ≥ 1 − 𝜀},

(3)

where, we recall, Sp(𝜎te ) is the spectrum of 𝜎te in ℬ(eℳe). By Corollary 15.15 we infer that 𝜎te ∈ Int(eℳe). Since ℳ is a factor, using Proposition 17.1 we deduce that 𝜎t ∈ lnt(ℳ), that is, there exists u ∈ U(ℳ) such that 𝜎t = Ad(u). We still have to show that u ∈ ℳ 𝜎 . Note that ueu∗ = 𝜎t (e) = e, ue = eu, and so 𝜎te = Ad(ue). Since eℳe is a factor, it follows from Proposition 14.13 that Sp(𝜎te ) = {𝜆𝜇−1 ; 𝜆, 𝜇 ∈ Spℳ (ue)}.

(4)

Comparing (3) and (4) and using the spectral theorem, we deduce by some elementary computations that √ inf{‖ue − 𝜆e‖; 𝜆 ∈ ℂ, |𝜆| = 1} ≤ 2𝜀. (5)

The Connes Invariant Γ(𝜎)

201

Let f ∈ ℳ 𝜎 be an arbitrary nonzero projection. Since Γ(𝜎 f ) = Γ(𝜎), it follows from what has just been shown that there exists a nonzero projection e ∈ ℳ 𝜎 , e ≤ f, such that (3) holds. As above, we see that ue = eu and 𝜎te = Ad(ue); and we deduce inequality (5). Using a standard argument it follows that for each 𝜀 > 0 there exist a family {ei } of mutually ∑ orthogonal nonzero projections in ℳ 𝜎 with i ei = 1 and a family {𝜆i } of unimodular complex ∑ numbers such that ‖uei − 𝜆i ei ‖ < 𝜀, hence ‖u − i 𝜆i ei } < 𝜀. We conclude that u ∈ ℳ 𝜎 . Hence t ∈ Int 𝜎. A different proof of (1) is given in Section 21.6. 16.6. The second main result concerning the invariant Γ is the following: Theorem. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the factor ℳ. For every nonzero projection e ∈ ℳ 𝜎 , there exists a continuous action 𝜏 ∶ G → Aut(ℳ) such that 𝜏 ∼ 𝜎 and Sp 𝜏 ⊂ Sp 𝜎 e . In particular, ⋂ Γ(𝜎) = Sp 𝜏. (1) 𝜏∼𝜎

By Proposition 14.6 and by the definition (16.1) of Γ(𝜎), the inclusion ⋂ Γ(𝜎) ⊂ Sp 𝜏

(2)

𝜏∼𝜎

is valid in general, for any W∗ -algebra. If ℳ is a factor, then (1) follows obviously from the first assertion of the theorem and the definition of Γ(𝜎). Note that (1) is trivially true if ℳ 𝜎 is a factor (16.1.(3)). However, (1) is not true for all ∗ W -algebras, as can easily be seen by considering a direct sum (ℳ, 𝜎) = (ℳ1 , 𝜎1 ) ⊕ (ℳ2 , 𝜎2 ). The proof of Theorem 16.6 will be given in Section 16.11, while in Sections 16.7–16.10 we present some auxiliary results which are of independent interest. In fact, Propositions 16.7, 16.8, and 16.9 have stronger conclusions, but in special situations. 16.7 Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W∗ -algebra ℳ. For every projection e ∈ ℳ 𝜎 which is equivalent to 1 in ℳ there exists a continuous action 𝜏 ∶ G → Aut(ℳ) such that 𝜏 ∼ 𝜎 and Sp 𝜏 = Sp 𝜎 e . Proof. Let u ∈ ℳ with u∗ u = 1, uu∗ = e and put wt = u∗ 𝜎t (u), (t ∈ ℝ). It is easy to check that the mapping t ↦ wt is a unitary 𝜎-cocycle w ∈ Z𝜎 (G; U(ℳ)) which defines a continuous action 𝜏 ∼ 𝜎 where 𝜏t (x) = wt 𝜎t (x)w∗t = u∗ 𝜎t (uxu∗ )u (x ∈ ℳ, t ∈ ℝ). Thus, the mapping x ↦ uxu∗ defines a ∗-automorphism (ℳ, 𝜏) ≈ (eℳe, 𝜎 e ) and Sp 𝜏 = Sp 𝜎 e . This proposition already implies Theorem 16.6 for countably decomposable type III factors. 16.8 Lemma. Let 𝜎 ∶ G → Aut(ℳ) and 𝜏 ∶ G → Aut(ℳ) be two continuous actions of G on the W∗ -algebra ℳ. If there exists a projection e ∈ ℳ 𝜎 ∩ ℳ 𝜏 with zℳ (e) = 1 such that 𝜎 e ∼ 𝜏 e , then 𝜎 ∼ 𝜏. Proof. By assumption, there exists an s-continuous mapping t ↦ vt of G into U(eℳe) such that vs+t = vs 𝜎s (vt ) and 𝜏t 𝜎t−1 (x) = vt xv∗t for s, t ∈ G, x ∈ eℳe. By Proposition 17.1, for each t ∈ G there −1 (x) = exists a unique wt ∈ U(ℳ) such that wt e = vt = ewt and 𝜏t 𝜎t−1 = Ad(wt ) (t ∈ G). Since 𝜏s+t 𝜎s+t

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𝜏s 𝜏t 𝜎t−1 𝜎s−1 (x) = 𝜏s (wt 𝜎s−1 ((x)w∗t ) = 𝜏s 𝜎s−1 (𝜎s (wt )x𝜎s (w∗t )) = ws 𝜎s (wt )x𝜎s (w∗t )w∗t (x ∈ ℳ), and ws 𝜎s (wt )e = vs 𝜎s (vt ) = vs+t = ews 𝜎s (wt ), by uniqueness it follows that ws 𝜎s (wt ) = ws+t (s, t ∈ G). Also, the mapping t ↦ wt is s-continuous (17.1.(2)). Hence 𝜏 ∼ 𝜎. Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W∗ -algebra ℳ. If e ∈ ℳ 𝜎 is a projection such that 1 − e is the sum of a family of mutually orthogonal projections in ℳ, each equivalent to e, then there exists a continuous action 𝜏 ∶ G → Aut(ℳ) such that 𝜏 ∼ 𝜎 and Sp 𝜏 = Sp 𝜎 e . Proof. By assumption it follows that there exist a type I factor ℱ , a minimal projection p in ℱ and ̄ ℱ such that 𝜋(x) = x ⊗ ̄ p for x ∈ eℳe. We then define a *-isomorphism 𝜋 of ℳ onto (eℳe) ⊗ −1 e ̄ 𝜄ℱ )◦𝜋 (t ∈ G). We have 𝜏 e = a continuous action 𝜏 ∶ G → Aut(ℳ) by putting 𝜏t = 𝜋 ◦(𝜎t ⊗ t e e e ̄ 𝜄ℱ ) = Sp 𝜎 e 𝜎t (t ∈ G), hence 𝜏 ∼ 𝜎 and so 𝜏 ∼ 𝜎 by the previous lemma. Thus, Sp 𝜏 = Sp(𝜎 e ⊗ (see 16.16.(1)). 16.9 Lemma. Let G be a closed subgroup of the locally compact abelian group G′ and ℳ a W∗ -algebra. Every s-continuous unitary representation u ∶ G → U(ℳ) can be extended to an s-continuous unitary representation u′ ∶ G′ → U(ℳ). Proof. Without loss of generality, we may assume that the W∗ -algebra ℳ is generated by u(G). Then ℳ is abelian and can be written as a direct sum of countably decomposable W∗ -algebras ([L], 7.2), so we can also assume that ℳ is countably decomposable. Then ([L], 10.15) we can realize ℳ as a von Neumann algebra ℳ ⊂ ℬ(ℋ ) with a cyclic and separating vector 𝜉 ∈ ℋ . As we have ̂ → ℳ such that seen in Section 14.10, there exists a unique *-homomorphism 𝜋u ∶ 𝒞0 (G) 𝜋u (̂f) = uf

( f ∈ ℒ 1 (G)),

(1)

̂ such that and there exists a unique positive measure 𝜇 ∈ ℳ(G) (𝜋u (𝜑)𝜉|𝜉) =

∫

̂ 𝜑(𝛾) d𝜇(𝛾) (𝜑 ∈ 𝒞0 (G)).

(2)

̂ 𝜇), 𝜋u (𝒞0 (G)) ̂ ̂ is a w-dense C∗ s ubalgebra of the W∗ -algebra ℒ ∞ (G, is a w-dense 𝒞0 (G) C∗ -subalgebra of the W∗ -algebra ℳ and, using (2), it is easy to check that the *-homomorphism 𝜋u is ̂ 𝜇) → normal, that is, w-continuous. Therefore, 𝜋u can be extended to a *-isomorphism 𝜋u ∶ ℒ ∞ (G, ℳ. Using (1) and approximate units in ℒ 1 (G), it is easy to check that (𝜋u−1 (ut ))(𝛾) = ⟨t, 𝛾⟩

̂ (t ∈ G, 𝛾 ∈ G).

̂ 𝜇) and ut (𝛾) = ⟨t, 𝛾⟩(t ∈ G, 𝛾 ∈ G). ̂ Furthermore, by Thus, we can assume that ℳ = ℒ ∞ (G, decomposing ℳ into a direct sum, we may assume that the measure 𝜇 has compact support (see, e.g., Dinculeanu, 1967, Prop. 41, §15). ̂′ → G ̂ be the dual homomorphism of the inclusion G → G′ . There exists a compact set Let p ∶ G ′ ′ ̂ K ⊂ G such that p(K′ ) = supp 𝜇. Let 𝜇 ′ be an extreme point of the weakly compact convex set of all positive measures ν′ on K′ such that p(ν′ ) = 𝜇. Then (Mokobodsky, 1962) the mapping ̂ 𝜇) ∋ 𝜑 ↦ 𝜑◦p ∈ ℒ ∞ (G ̂ ′ , 𝜇′ ) Φ ∶ ℳ = ℒ ∞ (G,

The Connes Invariant Γ(𝜎)

203

̂ ′ , 𝜇 ′ ) by v′′ (𝛾 ′ ) = ⟨t′ , 𝛾 ′ ⟩(𝛾 ′ ∈ G ̂ ′ ), and u′ = is a *-isomorphism. For t′ ∈ G′ , we define v′t′ ∈ ℒ ∞ (G t t ̂ ′ we Φ−1 (v′t′ ). Then u′ ∶ G′ → U(ℳ) is an s-continuous unitary representation and for t ∈ G, 𝛾 ′ ∈ G have (Φ(u′t ))(𝛾 ′ ) = v′t (𝛾 ′ ) = ⟨t, 𝛾 ′ ⟩ = ut (p(𝛾 ′ )) = (Φ(ut ))(𝛾 ′ ), hence u′t = ut . 16.10 Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the factor ℳ. If 𝒵 (ℳ 𝜎 ) contains a minimal projection, then there exists a continuous action 𝜏 ∶ G → Aut(ℳ) such that 𝜏 ∼ 𝜎 and Sp 𝜏 = Γ(𝜏) = Γ(𝜎). Proof. Let e ∈ 𝒵 (ℳ 𝜎 ) be a minimal projection. Then ℳ 𝜎 = eℳ 𝜎 e is a factor, hence (16.1.(3)) Sp 𝜎 e = Γ(𝜎 e ) and therefore (16.4.(3)) Ker 𝜎 e = Γ(𝜎 e )⟂ . Since ℳ is a factor, we have Γ(𝜎 e ) = Γ(𝜎) (16.2.(3)). Thus, for every t ∈ Γ(𝜎)⟂ we have 𝜎te = 𝜄 ∈ Int(eℳe). Using Proposition 17.1, it follows that for every t ∈ Γ(𝜎)⟂ there exists a unique ut ∈ U(ℳ) such that ut e = eut = e and 𝜎t = Ad(ut ); since, for any s ∈ G, the element 𝜎s (ut ) satisfies the same conditions as ut , we have ut ∈ ℳ 𝜎 . Also, by the uniqueness of ut and 17.1.(2) we see that the mapping u ∶ Γ(𝜎)⟂ ∋ t ↦ ut ∈ U(ℳ 𝜎 ) is an s-continuous unitary representation. By Lemma 16.9, there exists an s-continuous unitary representation v ∶ G → U(ℳ 𝜎 ) such that vt = ut for t ∈ Γ(𝜎)⟂ . Then v is a unitary 𝜎-cocycle; v defines a continuous action 𝜏 ∶ G → Aut(ℳ), 𝜏 ∼ 𝜎, by the formula 𝜏t = Ad(v∗t )◦𝜎t (t ∈ G). If t ∈ Γ(𝜏)⟂ = Γ(𝜎)⟂ , then 𝜏t = Ad(v∗t ut ) = 𝜄, that is, t ∈ Ker 𝜏. Consequently, Γ(𝜏)⟂ = Ker 𝜏 and hence (16.4.(3)) Sp 𝜏 = Γ(𝜏) = Γ(𝜎). e

16.11. Proof of Theorem 16.6. If ℳ 𝜎 has a minimal projection, then its central support in ℳ 𝜎 is a minimal projection in 𝒵 (ℳ 𝜎 ). Taking into account Propositions 16.10 and 16.7, we see that, in order to prove the theorem, we may assume that ℳ 𝜎 has no minimal projections and that the nonzero projection e ∈ ℳ 𝜎 is not equivalent to 1 in ℳ. In this case, we shall show that there exists a nonzero projection f ∈ ℳ 𝜎 , f ≤ e, such that 1 − f is the sum of a family of mutually orthogonal projections in ℳ, each equivalent to f. According to Proposition 16.8, it will follow that there exists 𝜏 ∼ 𝜎 with Sp 𝜏 = Sp 𝜎 f ⊂ Sp 𝜎 e . Assume first that ℳ is properly infinite. Since, by assumption, e is not equivalent to 1, we can take in this case f = e. Assume now that ℳ is finite and let 𝜇 be the n.s.f. trace on ℳ with 𝜇(1) = 1. By assumption, for every nonzero projection p ∈ ℳ 𝜎 and every 𝜀 > 0 there exists a nonzero projection q ∈ ℳ 𝜎 with 𝜇(q) < 𝜀. Let n ∈ ℕ be such that 1∕n ≤ 𝜇(e) and let {fi } be a maximal family of mutually orthogonal ∑ ∑ nonzero projections in ℳ 𝜎 , majorized by e and such that i 𝜇( fi ) ≤ 1∕n. Then i 𝜇( fi ) = 1∕n. Let ∑ f = i fi . Then 0 ≠ f ∈ ℳ 𝜎 , f ≤ e, and 𝜇( f ) = 1∕n, so that there exist (n − 1) mutually orthogonal projections in ℳ, each equivalent to f, with sum equal to 1 − f. □ ̂ 16.12 Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the factor ℳ. If G∕Γ(𝜎) is compact, then there exists a continuous action 𝜏 ∶ G → Aut(ℳ) such that 𝜏 ∼ 𝜎 and Sp 𝜏 = Γ(𝜏) = Γ(𝜎). ̂ Proof. Since G∕Γ(𝜎) is compact, its dual Γ(𝜎)⟂ is a discrete, hence closed, subgroup of G. Moreover, by Theorem 16.5, we have Γ(𝜎)⟂ = Int 𝜎. Let k ∶ Int 𝜎 → Int 𝜎∕Ker 𝜎 be the canonical quotient mapping. Let U be the subgroup of U(𝒵 (ℳ 𝜎 )) consisting of those u ∈ U(𝒵 (ℳ 𝜎 )) with the property that there exists t = t(u) ∈ Int 𝜎 such that 𝜎t = Ad(u). For each u ∈ U we put j(u) = k(t(u)) ∈ Int 𝜎∕Ker 𝜎. Then the mapping j ∶ U → Int 𝜎∕Ker 𝜎 is a well-defined surjective homomorphism.

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Let 𝕋 = {𝜆 ∈ ℂ; |𝜆| = 1} and denote by i ∶ 𝕋 → U the injective homomorphism defined by i(𝜆) = 𝜆 ⋅ 1ℳ (𝜆 ∈ 𝕋 ). Since ℳ is a factor we have a short exact sequence i

i

0 → 𝕋 → U → Int 𝜎∕Ker𝜎 → 0, which is split as 𝕋 a divisible group (Hewitt & Ross, 1963, 1970, Thm. A.7). Hence there exists a homomorphism h ∶ Int 𝜎∕Ker 𝜎 → U such that j◦h = the identity mapping. Then u = h◦k ∶ Γ(𝜎)⟂ = Int 𝜎 → U ⊂ U(𝒵 (ℳ 𝜎 )) is a unitary representation of the discrete group Γ(𝜎)⟂ in 𝒵 (ℳ 𝜎 ) such that 𝜎t = Ad(ut ) for all t ∈ Γ(𝜎)⟂ . By Lemma 16.9, u can be extended to an s-continuous unitary representation u ∶ G → U(𝒵 (ℳ 𝜎 )). Then u is a unitary 𝜎-cocycle and the equation 𝜏t = Ad(u∗t )◦𝜎t (t ∈ G) defines a continuous action 𝜏 ∼ 𝜎 such that Γ(𝜎)⟂ ⊂ Ker 𝜏. It follows (16.3, 16.4.(2)) that Ker 𝜏 = Γ(𝜏)⟂ and hence (16.4.(3)) Sp 𝜏 = Γ(𝜏) = Γ(𝜎). In general, all the sets in the family {Ker 𝜏; 𝜏 ∼ 𝜎}

(1)

̂ are contained in Γ(𝜎)⟂ (16.3, 16.4.(2)). If ℳ is a factor and either G∕Γ(𝜎) is compact or 𝒵 (ℳ 𝜎 ) has a minimal projection, then, by the above proposition and by Proposition 16.10, family (1) has a greatest element, equal to Γ(𝜎)⟂ (16.4.(3)). 16.13. We now give two applications of Theorem 16.5 to *-automorphisms of W∗ -algebras. Let ℳ be a W∗ -algebra and 𝜎 ∈ Aut(ℳ). Then the mapping n ↦ 𝜎 n defines an action 𝜎 ∶ ℤ → ̂ of ℤ with 𝕋 in such a way that ⟨n, 𝜆⟩ = 𝜆n (n ∈ ℤ, 𝜆 ∈ 𝕋 ). Aut(ℳ). As usual, we identify the dual ℤ By Proposition 14.6, the spectrum of 𝜎 ∈ Aut(ℳ) in ℬ(ℳ) is equal to the spectrum Sp 𝜎 ⊂ 𝕋 of the action 𝜎 ∶ ℤ → Aut(ℳ). If ℳ is a factor, then Theorem 16.5 can be applied to the action 𝜎 since 𝕋 = ℤ is compact. Proposition 1. Let ℳ be a factor and 𝜎 ∈ Aut(ℳ). If 𝜆(Sp 𝜎) ≠ Sp 𝜎 for every 𝜆 ∈ 𝕋 , 𝜆 ≠ 1, then 𝜎 ∈ Int(ℳ). Proof. By assumption and Proposition 16.1, it follows that Γ(𝜎) = {1}, hence Γ(𝜎)⟂ = ℤ ∋ 1. By Theorem 16.5, we conclude that 𝜎 ∈ Int(ℳ). Proposition 2. Let ℳ be a factor and 𝜎 ∈ Aut(ℳ). If Sp 𝜎 ≠ 𝕋 , then there exists n ∈ ℤ, n ≠ 0, such that 𝜎 n ∈ Int(ℳ). Proof. Since Sp 𝜎 ≠ 𝕋 we have Γ(𝜎) ≠ 𝕋 and hence Γ(𝜎)⟂ ≠ {0}. If n ∈ Γ(𝜎)⟂ , n ≠ 0, then 𝜎 n ∈ Int(ℳ) by Theorem 16.5. The requirement that ℳ be a factor comes from Theorem 16.5, but Proposition 2 is valid for any W∗ -algebra, as shown by Borchers (1974) with rather similar methods. 16.14. We now give an example of two continuous actions 𝜎 ∶ G → Aut(ℳ), 𝜏 ∶ G → Aut(ℳ) such that 𝜏t 𝜎t−1 ∈ Int(ℳ) for every t ∈ G but 𝜎 and 𝜏 are not outer conjugate. Let ℋ = 𝓁 2 (ℤ), ℳ = ℬ(ℋ ), G = ℤ × ℤ, and 𝜎 ∶ G ∋ t ↦ 𝜄ℳ ∈ Aut(ℳ) be the trivial action. There exists 𝜆 ∈ 𝕋 such that the set {𝜆n ; n ∈ ℤ} is dense in 𝕋 . We define the unitary operators u, v ∈ ℬ(ℋ ) by (u𝜉)(n) = 𝜉(n − 1), (v𝜉)(n) = 𝜆n 𝜉(n)

(𝜉 ∈ 𝓁 2 (ℤ), n ∈ ℤ),

The Connes Invariant Γ(𝜎)

205

and for t = (p, q) ∈ G we define 𝜏t = Ad(up vq ). Since vu = 𝜆uv, it follows that 𝜏 ∶ G ∋ t ↦ 𝜏t ∈ Aut(ℳ) is an action. By construction, we have 𝜏t 𝜎t−1 ∈ Int(ℳ) for all t ∈ G. On the other hand, it is clear that Ker 𝜏 = {0}. Thus, if we can show that ℳ 𝜏 = ℂ, it will follow from 16.1.(1) that Γ(𝜏) = Sp 𝜏, so that (16.4.(3)) Γ(𝜏)⟂ = Ker 𝜏 = {0} and Γ(𝜏) = 𝕋 , while Γ(𝜎) = {0}, so that (16.3) the actions 𝜎 and 𝜏 cannot be outer conjugate. Let us show that ℳ 𝜏 = ℂ. Since v is the multiplication operator by the character 𝜆 ∈ 𝓁 ∞ (ℤ) and since the set {𝜆n ; n ∈ ℤ} is dense in 𝕋 , the commutant of v is just the von Neumann algebra 𝓁 ∞ (ℤ) ⊂ ℬ(ℋ ). Since the only translation invariant functions are the constants, the commutant {u, v}′ ⊂ ℬ(ℋ ) = ℳ reduces to scalar operators, that is, ℳ 𝜏 = ℂ. 16.15. Let ℳ be a W∗ -algebra and 𝜎, 𝜏 ∈ Aut(ℳ). Consider also the corresponding actions 𝜎 ∶ ℤ ∋ n ↦ 𝜎 n ∈ Aut(ℳ), 𝜏 ∶ ℤ ∋ n ↦ 𝜏 n ∈ Aut(ℳ). Then 𝜏 ∼ 𝜎 ⇔ 𝜏 ≡ 𝜎(mod Int(ℳ)).

(1)

Indeed, if there exists u ∈ U(ℳ) such that 𝜏 = Ad(u)◦𝜎, then equation u0 = 1 and un = u𝜎(u) … 𝜎 n−1 (u), u−n = 𝜎 −1 (u∗ ) … 𝜎 −n (u∗ ), (n ≥ 1)

(2)

defines a unitary 𝜎-cocycle with 𝜏 n = Ad(un )◦𝜎 n (n ∈ ℤ), and u1 = u. Conversely, any unitary 𝜎-cocycle is defined as in (2) by a unitary element of ℳ. 16.16. Let ℳ, ℱ be W∗ -algebras, 𝜎 ∶ G → Aut(ℳ) a continuous action of G on ℳ, and 𝜄 ∶ G → ̄ 𝜄)f = 𝜎f ⊗ ̄ 𝜄, it follows Aut(ℱ ) the trivial action of G on ℱ . Since for every f ∈ ℒ 1 (G) we have (𝜎 ⊗ (14.5) that ̄ 𝜄) = Sp 𝜎. Sp(𝜎 ⊗

(1)

̄ ℱ )𝜎 ⊗̄ 𝜄 ) = 𝒵 (ℳ 𝜎 ⊗ ̄ ℱ ) = 𝒵 (ℳ 𝜎 ) ⊗ ̄ 𝒵 (ℱ ). By Also, using Corollary 9.9, we obtain 𝒵 ((ℳ ⊗ statement 16.1.(2) it follows that if ℱ is a factor, then ̄ 𝜄) = Γ(𝜎). Γ(𝜎 ⊗

(2)

16.17. The next group of results is concerned with periodic automorphisms, that is, with actions of finite cyclic groups. Let ℳ be a W∗ -algebra. A *-automorphism 𝜎 ∈ Aut(ℳ) is called minimal periodic if there exists n ∈ ℤ, n ≥ 2, such that 𝜎 n = 1 and Γ(𝜎) ⊃ {𝜆 ∈ ℂ; 𝜆n = 1};

(1)

the number n is called the minimal period of 𝜎. In this case, for every nonzero projection e ∈ ℳ 𝜎 we have Sp 𝜎 e = Γ(𝜎 e ) = {𝜆 ∈ ℂ; 𝜆n = 1} = {1, 𝛾, … , 𝛾 n−1 }. where 𝛾 = exp(2𝜋i∕n).

(2)

206

Groups of Automorphisms

Indeed, if 𝜆 ∈ Sp 𝜎, then, by Proposition 14.5, there exists a net {xi } ⊂ ℳ, ‖xi ‖ = 1, such that ‖𝜎 k (xi ) − 𝜆k xi ‖ → 0 for all k ∈ ℤ; in particular, for k = n we have |1 − 𝜆n | = ‖xi − 𝜆n xi ‖ → 0, that is, 𝜆n = 1. Therefore, Sp 𝜎 ⊂ {𝜆 ∈ ℂ; 𝜆n = 1} ⊂ Γ(𝜎) ⊂ Sp 𝜎, proving (2) for e = 1. For an arbitrary projection e ∈ ℳ 𝜎 , the *-automorphism 𝜎 e also satisfies (1) since (16.1.(4)) Γ(𝜎 e ) ⊃ Γ(𝜎); hence (2) is valid in general. From (2) and Proposition 16.4 it follows that 𝒵 (eℳ 𝜎 e) = e𝒵 (ℳ)𝜎

(e ∈ Proj(ℳ 𝜎 )).

(3)

zℳ 𝜎 (e) = zℳ (e) = z(e) (e ∈ Proj(ℳ 𝜎 )).

(4)

In particular, for central supports we have

Let x ∈ ℳ and write 1 ∑ −kj j 𝛾 𝜎 (x) n j=0 n−1

x̂ (k) =

(k = 0, 1, … , n − 1).

(5)

Then ‖̂x(k)‖ ≤ ‖x‖, x̂ (k)∗ = x̂ ∗ (n − k) (k = 0, 1, … , n − 1), and x=

n−1 ∑

x̂ (k) and x̂ (k) ∈ ℳ(𝜎; {𝛾 k }) (k = 0, 1, … , n − 1).

(6)

j=0

Proposition. Let 𝜎 ∈ Aut(ℳ) be a minimal periodic *-automorphism on the properly infinite W∗ -algebra ℳ. Then the centralizer ℳ 𝜎 is also properly infinite. Proof. For every nonzero projection p ∈ 𝒵 (ℳ 𝜎 ) = 𝒵 (ℳ)𝜎 , 𝜎 P is also a minimal periodic *-automorphism on the properly infinite W∗ -algebra ℳp. Hence it is sufficient just to prove that ℳ 𝜎 is infinite. By assumption, there exists n ∈ ℤ, n ≥ 2, such that (2) holds. Let e, f ∈ ℳ 𝜎 be nonzero projections such that z(e) z( f ) ≠ 0. Then ([L], 4.5) there exists x ∈ ℳ with exf ≠ 0. By (6) there exist k ∈ {0, 1, … , n − 1} and xk ∈ ℳ(𝜎; {𝛾 k }) such that exk f ≠ 0. Let r = r(exk f ) ≤ f and j ∈ {0, 1, … , n − 1} with k + j = 1(mod n). Since 0 ≠ r ∈ Proj(ℳ 𝜎 ), there exists xj ∈ ℳ(𝜎; {𝛾 j }) such that 0 ≠ xj rxj r. Then y = exk fxj ∈ ℳ(𝜎; {𝛾}) and eyf = y ≠ 0. If y = v|y| is the polar decomposition of y, then v ∈ ℳ(𝜎; {𝛾}) is a nonzero partial isometry such that v∗ v ≤ f, vv∗ ≤ e. Now let u be a maximal partial isometry such that u ∈ ℳ(𝜎; {𝛾}). If z(1−uu∗ )z(1−u∗ u) ≠ 0, the preceding argument shows that there exists a nonzero partial isometry v ∈ ℳ(𝜎; {𝛾}) with v∗ v ≤ 1 − u∗ u, vv∗ ≤ 1 − uu∗ , and w = u + v ∈ ℳ(𝜎; {𝛾}) is a partial isometry, contradicting the maximality of u. Consequently, z(1−uu∗ )z(1−u∗ u) = 0, and there exists a nonzero central projection p such that either u∗ up = p or uu∗ p = p. Replacing (ℳ, 𝜎) by (ℳp, 𝜎 p ) we may

The Connes Invariant Γ(𝜎)

207

assume that either u∗ u = 1 or uu∗ = 1. If u∗ u = 1 but uu∗ ≠ 1, then un is a nonunitary isometry in ℳ(𝜎; {𝛾 n }) = ℳ 𝜎 , hence ℳ 𝜎 is infinite. If uu∗ = 1 but u∗ u ≠ 1, then (u∗ )n is a nonunitary isometry in ℳ 𝜎 , and ℳ 𝜎 is again infinite. Finally, consider the case u∗ u = uu∗ = 1 and assume, to the contrary, that ℳ 𝜎 is finite. Then ([L], 7.23) the *-operation is s-continuous on the unit ball of ℳ 𝜎 . We shall show that the same property is valid for ℳ; it will follow that ℳ is finite ([L], 7.23), a contradiction. s So, consider x ∈ ℳ, ‖x‖ ≤ 1, and a net {xi }i∈I ⊂ ℳ, ‖xi ‖ ≤ 1, such that xi → x. For each k ∈ {0, 1, … , n − 1}, the elements

x̂ (k)u−k , x̂

i

(k)u−k

belong to the closed unit ball of

i∈I ℳ𝜎 ,

s

and x̂ i (k)u−k → s

x̂ (k)u−k since 𝜎 is s-continuous. By our last assumption ℳ 𝜎 is finite, so (̂xi (k)u−k )∗ → s

s

i∈I s

i∈I

i∈I

i∈I

i∈I −k (̂x(k)u )∗ ,

that is, uk x̂ ∗i (n − k) → uk x̂ ∗ (n − k) and hence x̂ ∗i (n − k) → x̂ ∗ (n − k). It follows that x∗i → x∗ . Corollary. Let 𝜎 ∈ Aut(ℳ) be a minimal *-automorphism of the countably decomposable W∗ -algebra ℳ with minimal period n ≥ 2. Let e, f ∈ Proj(ℳ 𝜎 ), v ∈ U(ℳ), and 𝜆 ∈ Sp 𝜎. Then e ∼ f in ℳ ⇔ there exists w ∈ ℳ(𝜎; {𝜆}) with w∗ w = e, ww∗ = f;

(7)

v𝜎(v) … 𝜎

(8)

n−1

(v) = 1 ⇔ there exists u ∈ U(ℳ) with v = u 𝜎(u). ∗

In particular, e ∼ f in ℳ ⇒ e ∼ f in ℳ 𝜎 ; there exists u ∈ U(ℳ) with un = 1 and 𝜎(u) = 𝜆u; there exists a nonzero projection p ∈ ℳ with 𝜎(p)p = 0.

(9) (10) (11)

Proof. We first prove (9). In view of (3) we can consider separately the cases e finite and e properly infinite in ℳ. If e is finite, then f and e ∨ f are also finite, so that we may may assume that ℳ is finite. Let ♮ ∶ ℳ → 𝒵 (ℳ) be the canonical central trace on ℳ. Using (3) it is easy to check that ♮|ℳ 𝜎 ∶ ℳ 𝜎 → 𝒵 (ℳ 𝜎 ) is the canonical central trace on ℳ. Since e ∼ f in ℳ, we have e♮ = f ♮ , hence e ∼ f in ℳ 𝜎 also ([L], 7.11, 7.12). If e is properly infinite in ℳ, then so is f and, by the previous proposition, e and f are also properly infinite in ℳ 𝜎 . Since e ∼ f in ℳ, we have z(e) = z( f ). Therefore ([L], 4.13), e ∼ f in ℳ 𝜎 , since ℳ is assumed countably decomposable. We now prove (8). If v = u∗ 𝜎(u) with u ∈ U(ℳ), then v𝜎(v) … 𝜎 n−1 (v) = u∗ 𝜎(u)𝜎(u∗ )𝜎 2 (u) … n−1 𝜎 (u)∗ 𝜎 n (u) = u∗ 𝜎 n (u) = u∗ u ( = 1. Conversely, assume that v𝜎(v) … 𝜎 n−1 (v) = 1. Let 𝒫 = ) 1 0 ̄ 𝜄) ∈ Aut(𝒫 ). Since Mat2 (ℳ) = ℳ ⊗ Mat2 (ℂ), V = ∈ U(𝒫 ), and 𝔖 = Ad(V)◦(𝜎 ⊗ 0 v 𝜎 n = 𝜄 and v𝜎(v) … 𝜎 n−1 (v) = 1, we have 𝔖n = 𝜄. Also, using the results of Sections 16.3, 16.15, ̄ 𝜄) = Γ(𝜎) = {𝜔 ∈ ℂ; 𝜔n = 1}. Thus, 𝔖 is a minimal periodic and 16.16, we get Γ(𝔖) = Γ(𝜎 ⊗

208

Groups of Automorphisms (

) ( ) 1 0 0 0 *-automorphism. On the other hand, the projections and belong to 𝒫 𝔖 and are 0 0 0 1 𝔖 equivalent in 𝒫 so that, by (9), they are also(equivalent ( ) ) in 𝒫 . A partial isometry implementing 0 u 0 u is this equivalence is necessarily of the form with u ∈ U(ℳ); the fact that 0 0 0 0 𝔖-invariant means that u = 𝜎(u)v∗ , that is, v = u∗ 𝜎(u). In particular, for v = 𝜆 ⋅ 1ℳ with 𝜆n = 1 there exists u1 ∈ U(ℳ) such that 𝜎(u1 ) = 𝜆u1 . Then un1 ∈ ℳ 𝜎 , and there exists u0 ∈ U(ℳ 𝜎 ) a Borel function of un1 such that u−n = un1 . Since 0 un1 commutes with u1 and u0 is a Borel function of un1 , it follows that u0 commutes with u1 . Then u = u0 u1 ∈ U(ℳ), 𝜎(u) = 𝜆u, and un = 1, proving (10). It is now easy to prove (7). If e ∼ f in ℳ, then, by (9), there exists x ∈ ℳ 𝜎 with x∗ x = e and xx∗ = f. Since 𝜎 e is also minimal periodic with minimal period n, using (10) we obtain u ∈ ℳ with u∗ u = uu∗ = e and 𝜎(u) = 𝜆u. Then w = xu ∈ ℳ, 𝜎(w) = 𝜆w, and w∗ w = e, ww∗ = f. Finally, we prove (11). Let 𝛾 = exp(2𝜋i∕n). By (10) there exists u ∈ U(ℳ) with un = 1, and ∑n−1 𝜎(u) = 𝛾u. Since un = 1, the spectral decomposition of u is of the form u = k=0 𝛾 k pk with ∑n−1 ∑n−1 k p0 , … , pn−1 mutually orthogonal projections in ℳ and k=0 pk = 1. Then k=0 𝛾 𝜎(pk ) = 𝜎(u) = ∑n−1 𝛾u = k=0 𝛾 k+1 pk . Thus, if pk ≠ 0, 𝜎(pk ) = pk−1 and hence 𝜎(pk )pk = 0. 16.18. Notes. The results in this section are due to Connes (1973a, 1977a, 1976b). For our exposition, we have used Combes and Delaroche (1973–1974); Combes and Delaroche (1975) and Connes (1977a). For the proofs of Proposition 16.17 and Corollary 17.24 (which are not explicit in the literature) the author has benefited from several useful discussions with Apostol and Digernes. We record the following related references: Ikunishi and Nakagami (1976); Kishimoto and Takai (1978); Olesen (1975); Olesen and Pedersen (1978, 1980); Olesen et al. (1977); Pedersen (1973– 1977); Pedersen (1980).

17 Outer Automorphisms In this section, we derive a canonical decomposition of a *-automorphism into an inner part and a properly outer part. We give an important characterization of properly outer *-automorphisms and some applications. 17.1 Proposition. Let ℳ be a W∗ -algebra, 𝜎 ∈ Aut(ℳ) and let e ∈ ℳ 𝜎 be a projection with central support p = zℳ (e). If there exists u ∈ U(eℳe) such that 𝜎 e = Ad(u), then there exists a unique v ∈ U(ℳp) such that 𝜎 p = Ad(v) and ve = u = ev. If a ∈ ℳp, ae = u = ea, and ax = 𝜎(x)a for all x ∈ ℳp, then a = v. Proof. Since e ∈ ℳ 𝜎 , we also have zℳ (e) ∈ ℳ 𝜎 . Thus, without loss of generality, we may assume that p = zℳ (e) = 1.

Outer Automorphisms

209

Let ℳ ⊂ ℬ(ℋ ) be realized as a von Neumann algebra. We recall ([L], 3.9) that zℳ (e)ℋ = ℳeℋ . For x1 , … , xn ∈ ℳ and 𝜉1 , … , 𝜉n ∈ eℋ we have ‖∑ ‖2 ∑ ‖ ‖ (u∗ 𝜎(ex∗j xi e)u𝜉i |𝜉j ) ‖ 𝜎(xk )u𝜉k ‖ = ‖ ‖ i,j ‖ k ‖ ‖∑ ‖2 ∑ ‖ ‖ ∗ = (exj xi e𝜉i |𝜉j ) = ‖ xk 𝜉k ‖ . ‖ ‖ i,j ‖ k ‖ Since zℳ (e) = 1, it follows that there exists a unique unitary operator v ∈ ℬ(ℋ ) such that vxe𝜉 = 𝜎(x)ue𝜉

(x ∈ ℳ, 𝜉 ∈ ℋ ).

(1)

It is easy to check that v ∈ U(ℳ ′ )′ = ℳ, and v|eℋ = u. For every x, y ∈ ℳ and 𝜉 ∈ ℋ we have vxye𝜉 = 𝜎(xy)ue𝜉 = 𝜎(x)𝜎(y)ue𝜉 = 𝜎(x)vye𝜉, hence vx = 𝜎(x)v, that is, 𝜎 = Ad(v). If a ∈ ℳ, ae = u = ea, and ax = 𝜎(x)a for all x ∈ ℳ, then v∗ a ∈ 𝒵 (ℳ) and v∗ ae = e; hence ∗ v a = 1 and a = v. With the same assumptions as in the proposition, let {𝜎i }i∈I ⊂ Aut(ℳ) and {ui }i∈I ⊂ U(eℳe) be two nets with 𝜎ie = Ad(ui ), (i ∈ I). For each i ∈ I, let vt ∈ U(ℳp) be the unique element such that p 𝜎i = Ad(vi ) and vi e = ui = evi . Then w

w

w

𝜎i (x) → 𝜎(x) for all x ∈ ℳ and ui → u ⇒ vi → v.

(2)

Indeed, arguing as above we may assume that p = 1 and ℳ ⊂ ℬ(ℋ ). For x ∈ ℳ and 𝜉, 𝜂 ∈ ℳ we have, by (1), (vi xe𝜉|𝜂) = (𝜎i (x)ui e𝜉|𝜂) → (𝜎(x)ue𝜉|𝜂) = (vxe𝜉|𝜂), w

hence vi → v. 17.2. Let ℳ be a W∗ -algebra and 𝜎 ∈ Aut(ℳ). Let {pi }i∈I ⊂ 𝒵 (ℳ) be a maximal family of ∑ mutually orthogonal projections with the property 𝜎 pi ∈ Int(ℳpi ) (i ∈ I), and put p(𝜎) = i∈I pi ∈ p(𝜎) 𝒵 (ℳ). Then it is clear that 𝜎 ∈ Int(ℳp(𝜎)) and from Proposition 17.1 it follows that p(𝜎) is the greatest projection e ∈ ℳ 𝜎 such that 𝜎 e ∈ Int(eℳe). We call p(𝜎) the inner part of 𝜎. If p(𝜎) = 0, then the *-automorphism 𝜎 is called properly outer. It is clear that p(𝜎 1−p(𝜎) ) = 0. We call 1 − p(𝜎) the properly outer part of 𝜎. Thus: Corollary. Let ℳ be a W∗ -algebra and 𝜎 ∈ Aut(ℳ). There exist two central projections p, q ∈ 𝒵 (ℳ) with p+q = 1, such that 𝜎 p is inner and 𝜎 q is properly outer. p and q are uniquely determined by these conditions. Since p(𝜎) is a central projection, it is easy to see that p(Ad(u)◦𝜎) = p(𝜎) = p(𝜎◦Ad(u)) (𝜎 ∈ Aut(ℳ), u ∈ U(ℳ)).

(1)

210

Groups of Automorphisms

It follows that if there exists 0 ≠ e ∈ Proj(ℳ) and u ∈ U(ℳ) such that 𝜎(x) = uxu∗ for all x ∈ eℳe, then p(𝜎) ≥ zℳ (e) ≠ 0

(2)

Indeed, by assumption it follows that (Ad(u∗ )◦𝜎)(x) = x for all x ∈ eℳe, hence p(𝜎) = p(Ad(u∗ )◦𝜎) ≥ e. 17.3. Let ℳ be a W∗ -algebra and G ⊂ Aut(ℳ) a subgroup. Let ⋁ [G] = {𝜎 ∈ Aut(ℳ) ∶ p(g−1 𝜎) = 1}. g∈G

Let 𝜎 ∈ [G] and denote by 𝔉(𝜎, G) the set of all families of the form {pi , ui , gi }i∈I such that {pi ; i ∈ I} are mutually orthogonal nonzero projections in 𝒵 (ℳ); ui ∈ ℳ, u∗i ui = ui u∗i = pi ;

(1) (2)

−1 ∗ gi ∈ G, (g−1 i 𝜎)(pi ) = pi and (gi 𝜎)(x) = ui xui for all x ∈ ℳpi .

(3)

The set 𝔉(𝜎, G) is inductively ordered by inclusion and hence there exists a maximal element ∑ {pi , ui , gi }i∈I ∈ 𝔉(𝜎, G). Since 𝜎 ∈ [G], it follows that i∈I pi = 1. For x ∈ ℳ we have x=

∑

xpi and 𝜎(x) =

i∈I

∑

gi (ui xu∗i ).

(4)

i∈I

Thus, the maximal element {pi , ui , gi }i∈I of 𝔉(𝜎, G) determines completely the *-automorphism 𝜎 ∈ [G]. It is now easy to check that [G] is a subgroup of Aut(ℳ), G ⊂ [G] and [[G]] = [G]. The group [G] is called the full group associated with G. 17.4 Proposition. Let ℳ be a W∗ -algebra and 𝜎 ∈ Aut(ℳ). Then a ∈ ℳ, ax = 𝜎(x)a for all x ∈ ℳ ⇒ a = ap(𝜎).

(1)

In particular, 𝜎 is properly outer if and only if a ∈ ℳ, ax = 𝜎(x)a for all x ∈ ℳ ⇒ a = 0,

(2)

and 𝜎 is outer, that is, 𝜎 ∉ Int(ℳ), if and only if a ∈ ℳ, ax = 𝜎(x)a for all x ∈ ℳ ⇒ z(a) ≠ 1.

(3)

Proof. Let a ∈ ℳ be such that ax = 𝜎(x)a for all x ∈ ℳ. If x ∈ ℳ is unitary, then x∗ a∗ = a∗ 𝜎(x)∗ , a∗ a = a∗ 𝜎(x)∗ 𝜎(x)a = x∗ a∗ ax, aa∗ = axx∗ a∗ = 𝜎(x)aa∗ 𝜎(x)∗ , hence a∗ a ∈ 𝒵 (ℳ), aa∗ ∈ 𝒵 (ℳ), r(a) = s(a∗ a) ∈ 𝒵 (ℳ), l(a) = s(aa∗ ) ∈ 𝒵 (ℳ). In particular, |a| ∈ 𝒵 (ℳ) and r(a) = l(a) = z(a) = p ∈ 𝒵 (ℳ). Also, pa = ap = 𝜎(p)a, hence (p − 𝜎(p))p = 0 and 𝜎(p) = p, that is, p ∈ 𝒵 (ℳ)𝜎 .

Outer Automorphisms

211

Let a = u|a| be the polar decomposition of a. Then u∗ u = uu∗ = p and for x ∈ ℳp we have ux|a| = u|a|x = ax = 𝜎(x)a = 𝜎(x)u|a|, hence ux = uxp = 𝜎(x)up = 𝜎(x)pu = 𝜎(xp)u = 𝜎(x)u, that is, 𝜎 p = Ad(u). Consequently, p ≤ p(𝜎) and a = ap = ap(𝜎). If 𝜎 is properly outer, that is, p(𝜎) = 0, the (2) follows from (1). Conversely, if p = p(𝜎) ≠ 0 and u ∈ ℳ, u∗ u = uu∗ = p, 𝜎 p = Ad(u), it follows from (2) that u = 0, a contradiction. If 𝜎 ∉ Int(ℳ), then p(𝜎) ≠ 1 and (3) follows from (1). Conversely, if 𝜎 ∈ Int(ℳ) and u ∈ U(ℳ), 𝜎 = Ad(u), it follows from (3) that 1 = z(u) ≠ 1, a contradiction. The main assertion (1) is obviously equivalent to p(𝜎) = ∨{z(a);

a ∈ ℳ, ax = 𝜎(x)a for all x ∈ ℳ}.

(4)

An element a ∈ ℳ such that ax = 𝜎(x)a for all x ∈ ℳ will be called a 𝜎-dependent element. The above proof shows that if a ∈ ℳ is 𝜎-dependent and a = u|a| is its polar decomposition, then |a| = |a∗ | ∈ 𝒵 (ℳ)𝜎 , l(a) = r(a) = z(a) ∈ 𝒵 (ℳ)𝜎 , u∗ u = uu∗ = z(a) and 𝜎(x) = uxu∗ (x ∈ ℳz(a)).

(5) (6)

It is easy to check that, for a ∈ ℳ, we have a is 𝜎-dependent ⇔ a∗ is 𝜎 −1 -dependent.

(7)

Note that all the above statements remain valid if we replace the conditions of the form ax = 𝜎(x)a by xa = a𝜎(x), the only change appearing in (6), where u must be replaced by u∗ . 17.5. Let 𝒵 be an abelian W∗ -algebra and 𝜎 ∈ Aut(𝒵 ). In this case, p(𝜎) is the greatest projection p ∈ 𝒵 such that 𝜎 𝜌 = 𝜄 = the identity mapping on 𝒵 p. We shall say that 𝜎 acts freely on 𝒵 if for every nonzero projection p ∈ 𝒵 there exists a nonzero projection q ∈ 𝒵 , q ≤ p, such that q𝜎(q) = 0. The next proposition shows, in particular, that 𝜎 acts freely on 𝒵 if and only if p(𝜎) = 0. Proposition. Let ℳ be a W∗ -algebra and 𝜎 ∈ Aut(ℳ). The following statements are equivalent: (i) p(𝜎|𝒵 (ℳ)) = 0; (ii) 𝜎 acts freely on 𝒵 (ℳ); (iii) a ∈ ℳ and az = 𝜎(z)a for all z ∈ 𝒵 (ℳ) ⇒ a = 0. In particular, if 𝜎 acts freely on 𝒵 (ℳ), then 𝜎 is properly outer. Proof. (i) ⇒ (ii). Let p ∈ 𝒵 (ℳ) be a nonzero projection. We first show that there exists a projection r ∈ 𝒵 (ℳ) with 𝜎(r) ≠ r. Otherwise, for every projection z ∈ 𝒵 (ℳ) we have pz = 𝜎(pz) = 𝜎(p)𝜎(z) = p𝜎(z) = 𝜎(z)p and the identity pz = 𝜎(z)p holds for any z ∈ 𝒵 (ℳ); this, by (i) and 17.4.(2), implies that p = 0, a contradiction. Thus, there exists a projection r ∈ 𝒵 (ℳ) with r ≤ p and 𝜎(r) ≠ r. If q = r − r𝜎(r) ≠ 0, then we have 0 ≠ q ≤ r ≤ p and q𝜎(q) = 0. If r = r𝜎(r), then r ≤ 𝜎(r), 𝜎 −1 (r) ≤ r, hence q′ = r − 𝜎 −1 (r) ≠ 0, q′ ≤ r ≤ p and q′ 𝜎(q′ ) = 0. (ii) ⇒ (iii). Let a ∈ ℳ be such that az = 𝜎(z)a for all z ∈ 𝒵 (ℳ), and suppose that p = z(a) ∈ 𝒵 (ℳ). If a ≠ 0, then, by (ii), there exists a nonzero projection q ∈ 𝒵 (ℳ), q ≤ p, with q𝜎(q) = 0; it follows that aq = 𝜎(q)a = 0, hence 0 ≠ q = qp = 0, a contradiction. Thus, a = 0. (iii) ⇒ (i). Obvious.

212

Groups of Automorphisms

17.6 Proposition. Let ℳ, 𝒩 be W∗ -algebras, 𝜎 ∈ Aut(ℳ), 𝜏 ∈ Aut(𝒩 ). Then ̄ 𝜏) = p(𝜎) ⊗ ̄ p(𝜏). p(𝜎 ⊗

(1)

̄ 𝜏 is properly outer ⇔ either 𝜎 or 𝜏 is properly outer. 𝜎⊗ ̄ 𝜏 ∈ Int(ℳ ⊗ ̄ 𝒩 ) ⇔ 𝜎 ∈ Int(ℳ) and 𝜏 ∈ Int(𝒩 ). 𝜎⊗

(2) (3)

In particular,

̄ 𝜏 = Proof. It is clear that if 𝜎 = Ad(u) and 𝜏 = Ad(v) with u ∈ U(ℳ) and v ∈ U(𝒩 ), then 𝜎 ⊗ ̄ v) with u ⊗ ̄ v ∈ U(ℳ ⊗ ̄ 𝒩 ). Hence p(𝜎 ⊗ ̄ 𝜏) ≥ p(𝜎) ⊗ ̄ p(𝜏). Ad(u ⊗ ̄ ̄ 𝜏)(z)a for all z ∈ ℳ ⊗ ̄ 𝒩 . Let 𝜓 ∈ 𝒩∗ and Conversely, let a ∈ ℳ ⊗𝒩 be such that az = (𝜎 ⊗ x ∈ ℳ. We have (9.8) ̄ 1)) = E𝜓 ((𝜎(x) ⊗ ̄ 1)a) = 𝜎(x)E𝜓 (a). E𝜓ℳ (a)x = E𝜓ℳ (a(x ⊗ ℳ ℳ Using Proposition 17.4.(1), we infer that ̄ 1)). 0 = E𝜓ℳ (a) − E𝜓ℳ (a)p(𝜎) = E𝜓ℳ (a((1 − p(𝜎)) ⊗ ̄ 1). Similarly, we get a = Since 𝜓 ∈ 𝒩∗ , was arbitrary, it follows (9.8.(3)) that a = a(p(𝜎) ⊗ ̄ p(𝜏)). Hence a = a(p(𝜎) ⊗ ̄ p(𝜏)). Using 17.4.(4), we infer that p(𝜎 ⊗ ̄ 𝜏) ≤ p(𝜎) ⊗ ̄ p(𝜏). We a(1 ⊗ have thus proved (1). Equations (2) and (3) follow immediately from (1). 17.7. Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the W∗ -algebra ℳ. We shall say that the action 𝜎 is properly outer if for every t ∈ G, t ≠ the neutral element of G, the *-automorphism 𝜎t ∈ Aut(ℳ) is properly outer. We recall that the action is called ergodic if ℳ 𝜎 = ℂ ⋅ 1ℳ . Every *-automorphism 𝜎 ∈ Aut(ℳ) defines an action 𝜎 ∶ ℤ ∋ n ↦ 𝜎 n ∈ Aut(ℳ). Clearly, the action 𝜎 is ergodic if and only if the *-automorphism 𝜎 is ergodic. We shall say that the *-automorphism is aperiodic if the action 𝜎 is properly outer, that is, if p(𝜎 n ) = 0 for all n ∈ ℤ, n ≠ 0. Proposition. Let ℳ be a W∗ -algebra without minimal projections and 𝜎 ∈ Aut(ℳ). If 𝜎 is ergodic, then 𝜎 is aperiodic. Proof. The set of all states {𝜑 ∈ ℳ∗ ∶ 𝜑 ≥ 0, 𝜑(1) = 1} on ℳ is a 𝜎(ℳ ∗ , ℳ)-compact convex subset of ℳ ∗ , invariant under the transformations 𝜑 ↦ 𝜑◦𝜎 n , (n ∈ ℤ). By the Markov–Kakutani fixed point theorem ([L], A.1) it follows that there exists a state 𝜑 on ℳ such that 𝜑◦𝜎 = 𝜑. Let n ∈ ℤ, n > 0, be such that p = p(𝜎 n ) ≠ 0. There exists u ∈ ℳ, u∗ u = uu∗ = p such that n 𝜎 (px) = upxu∗ (x ∈ ℳ). Then v = u + (1 − p) ∈ U(ℳ) and 𝜎 n (px) = vpxv∗ (x ∈ ℳ). Let 𝒜 be a maximal abelian *-subalgebra of ℳ containing v. For every a ∈ 𝒜 , we have 𝜎 n (pa) = pa. It is easy to check that every minimal projection of 𝒜 is also a minimal projection of ℳ so that, by assumption, 𝒜 has no minimal projections. Thus there exists a nonzero projection q ∈ 𝒜 , q ≤ p, such that 𝜑(q) ≤ 1∕n. Since 𝜎 n (q) = q, we have a = q + 𝜎(q) + ... + 𝜎 n−1 (q) ∈ ℳ 𝜎 . Since 𝜎 is ergodic, there exists 𝜆 ∈ ℂ with a = 𝜆 ⋅ 1ℳ . We have a ≥ q hence 𝜆 ≥ 1. On the other hand, 𝜆 = 𝜑(a) = 𝜑(q) + 𝜑(𝜎(q)) + … + 𝜑(𝜎 n−1 (q)) < n∕n = 1, a contradiction. Hence p(𝜎 n ) = 0 for all n ∈ ℤ, n ≠ 0.

Outer Automorphisms

213

17.8. Let ℳ be a W∗ -algebra and G ↪ Aut(ℳ) an ergodic action of the discrete group G on ℳ. Then 𝜎 ∈ Aut(ℳ), 𝜎g = g𝜎 for all g ∈ G ⇒ either p(𝜎) = 0 or p(𝜎) = 1,

(1)

since it follows from the assumption that p(𝜎) is G-invariant. Assume moreover that G is commutative and let 𝜄 ∈ G be the neutral element of G. Then from (1) it follows that g ∉ Int(ℳ) for all g ∈ G, g ≠ 𝜄 ⇒ p(g) = 0 for all g ∈ G, g ≠ 𝜄.

(2)

𝜎 ∈ [G], 𝜎g = g𝜎 for all g ∈ G ⇒ there exists g ∈ G with g−1 𝜎 ∈ Int(ℳ).

(3)

Also,

Indeed, we have (g−1 𝜎)h = h(g−1 𝜎) for all g, h ∈ G so, by (1), either p(g−1 𝜎) = 0 for all g ∈ G, or p(g−1 𝜎) = 1 for some g ∈ G. Since 𝜎 ∈ [G], the desired conclusion follows. Finally, if the W∗ -algebra ℳ is commutative, it follows from (3) that 𝜎 ∈ [G], 𝜎g = g𝜎 for all g ∈ G ⇒ 𝜎 ∈ G,

(4)

that is, G is “maximal commutative” in [G]. 17.9. The next theorem is a remarkable noncommutative extension of the equivalence (i) ⇔ (ii) of Proposition 17.5. Theorem (A. Connes). Let ℳ be a countably decomposable W∗ -algebra and 𝜎 ∈ Aut(ℳ). Then p(𝜎) is the smallest central projection p in ℳ with the following property: for every nonzero projection e ∈ ℳ, e ≤ 1 − p, and every 𝜀 > 0 there exists a nonzero projection f ∈ ℳ, f ≤ e, such that ‖ f𝜎( f )‖ < 𝜀.

(1)

In particular, 𝜎 is properly outer if and only if for every nonzero projection e ∈ ℳ and every 𝜀 > 0 there exists a nonzero projection f ∈ ℳ, f ≤ e, such that ‖ f𝜎( f )‖ < 𝜀.

(2)

Also, if 𝜎 ∉ Int(ℳ), then for every 𝜀 > 0 there exists a nonzero projection f ∈ ℳ such that ‖ f𝜎( f )‖ < 𝜀. The proof is contained in Sections 17.10–17.16, which are also of independent interest.

(3)

214

Groups of Automorphisms

17.10. In this section, we show that if a central projection p ∈ 𝒵 (ℳ) satisfies condition 17.9.(1), then p ≥ p(𝜎). If p ≱ p(𝜎), then 0 ≠ q = p(𝜎) − p(𝜎)p ≤ 1 − p and there exists u ∈ ℳ, u∗ u = uu∗ = q such that 𝜎(x) = uxu∗ for all x ∈ ℳq. There exists 𝜆 ∈ ℂ, |𝜆| = 1 and a nonzero spectral projection e ∈ ℳq of u such that ‖ue − 𝜆e‖ ≤ 1∕4. Since e ∈ ℳ 𝜎 , it follows that 𝜎 e = Ad(ue) ∈ Aut(eℳe) and ‖𝜎 e − 𝜄‖ ≤ 1∕2. Then for every projection f ∈ ℳ, f ≤ e, we have ‖𝜎( f ) − f ‖ ≤ 1∕2. For f ≠ 0 we get 2‖ f𝜎( f )‖ ≥ ‖ f𝜎( f ) − 𝜎( f )f ‖ = ‖(𝜎( f ) + f ) − (𝜎( f ) − f )2 ‖ ≥ ‖𝜎( f ) + f ‖ − ‖(𝜎( f ) − f )2 ‖ > ‖ f ‖ − ‖𝜎( f ) − f ‖2 ≥ 1 −

1 3 = , 4 4

that is, ‖ f𝜎( f )‖ ≥ 3∕8, contradicting 17.9.(1). 17.11. In order to complete the proof of Theorem 17.9, we still have to show that p(𝜎) satisfies condition 17.9.(1). To this end we shall first prove (17.11–17.14) that every outer *-automorphism 𝜎 ∈ Aut(ℳ) satisfies condition 17.9.(3). If 𝜎 does not act identically on 𝒵 (ℳ), that is, if p(𝜎|𝒵 (ℳ)) ≠ 1, then 17.9.(3) follows obviously from Proposition 17.5. Therefore, we shall assume that 𝜎 acts identically on 𝒵 (ℳ). Then a standard maximality argument shows that for each n ∈ ℤ, n > 1, there exists a greatest central projection pn (𝜎) ∈ 𝒵 (ℳ) such that the following statement concerning 0 ≠ p ∈ Proj(𝒵 (ℳ)) and 1 ≤ k ∈ ℤ: there exists u ∈ ℳ 𝜎 , u∗ u = uu∗ = p, such that 𝜎 k (x) = uxu∗ for all x ∈ ℳp

(1)

is true for p = pn (𝜎) when k = n, but false for every p ≤ pn (𝜎) when k < n. The projections {pn (𝜎)}n>1 are mutually orthogonal and ∑

pn (𝜎) =

n≥1

⋁

p(𝜎 n ).

(2)

n≥1

More precisely, we shall show that 2

p(𝜎 ) ≤ n

n ∑

pk (𝜎)

(n ≥ 1).

(3)

k=1

Indeed, let p = p(𝜎 n ) and let v ∈ ℳ and v∗ v = vv∗ = p, such that 𝜎 n (x) = vxv∗ , (x ∈ ℳp). For x ∈ ℳp we have 𝜎(v)x𝜎(v)∗ = 𝜎(v𝜎 −1 (x)v∗ ) = 𝜎(𝜎 n (𝜎 −1 (x))) = 𝜎 n (x) = vxv∗ , hence v∗ 𝜎(v) = z ∈ 𝒵 (ℳp). Since v = 𝜎 n (v) = zn v, it follows that zn = p. Therefore, 𝜎(vn ) = 𝜎(v)n = zn vn = vn and 2 𝜎 n (x) = (vn )x (vn )∗ for all x ∈ ℳp. This proves (3) and hence also (2). Note that 𝜎 ∈ Int(ℳ) ⇔ p1 (𝜎) = 1. So, in order to prove that 𝜎 ∉ Int(ℳ) ⇒ 17.9.(3), we distinguish two cases: either there exists n ∈ ℤ, n ≥ 2, such that pn (𝜎) = 1

(a)

Outer Automorphisms

215

or pn (𝜎) = 0 for all n ≥ 1, that is, by (3), 𝜎 ∈ Aut(ℳ) is aperiodic.

(b)

̂ =𝕋 = 17.12. In each of the cases 17.11.(a) and 17.11.(b), we now compute the invariant Γ(𝜎) ⊂ ℤ n {𝛾 ∈ ℂ; |𝛾| = 1} of the action 𝜎 ∶ ℤ ∋ n ↦ 𝛾 ∈ Aut(ℳ). If ℳ is a factor, it is easy to see, using Theorem 16.5, that in case 17.11.(a) we have Γ(𝜎) = (nℤ)⟂ = {𝛾 ∈ ℂ; 𝛾 n = 1}, ⟂

in case 17.11.(b) we have Γ(𝜎) = {0} = {𝛾 ∈ ℂ; |𝛾| = 1}.

(1) (2)

Statements (1) and (2) are true in general, without the assumption that ℳ is a factor. However, in proving the general case we shall use another characterization of invariant Γ(a) and some elementary results concerning crossed products by discrete groups, which are given in Sections 21.1 and 22.6. ̂ to the centre By Theorem 21.1, Γ(𝜎) is the kernel of the restriction of the dual action 𝜎̂ of 𝕋 = ℤ 𝒵 (ℛ(ℳ, 𝜎)) of the crossed product ℛ(ℳ, 𝜎). An arbitrary element X ∈ ℛ(ℳ, 𝜎) is of the form (22.1) ∑ ̄ 𝜆 (k)) X= 𝜋𝜎 (a(k))(1 ⊗ (3) k∈ℤ

with a(k) ∈ ℳ (k ∈ ℤ), and for every 𝛾 ∈ 𝕋 we have (19.3) ∑ ̄ 𝜆 (k)). 𝜎̂ 𝛾 (X) = 𝛾 k 𝜋𝜎 (a(k))(1 ⊗

(4)

k∈ℤ

By Theorem 22.6, we have X ∈ 𝒵 (ℛ(ℳ, 𝜎)) if and only if for every k ∈ ℤ. a(k) ∈ ℳ 𝜎 and a(k)𝜎 k (x) = xa(k) for x ∈ ℳ.

(5)

In case 17.11.(b), we have p(𝜎 k ) = 0 for all k ≠ 0 so, using (5) and 17.4.(2), it follows that a(k) = 0 for all k ≠ 0 and a(0) ∈ 𝒵 (ℳ)𝜎 . Thus, in this case we have 𝒵 (ℛ(ℳ, 𝜎)) = 𝜋𝜎 (𝒵 (ℳ)𝜎 ) and so the restriction of the dual action to 𝒵 (ℛ(ℳ, 𝜎)) is just the trivial action, that is its kernel is the whole ̂ = 𝕋 ; hence Γ(𝜎) = {𝛾 ∈ ℂ; |𝛾| = 1}. dual group ℤ Consider now case 17.11.(a). For each k ∈ ℤ, a(k)∗ ∈ ℳ is a 𝜎 k -dependent element, so that (17.4.(5), 17.4.(6)) the partial isometry appearing in the polar decomposition of a(k)∗ is 𝜎-invariant and implements the restriction of 𝜎 k to the (central) support of a(k)∗ . It follows from condition 17.11.(a) that a(k) ≠ 0 ⇒ k ∈ nℤ. From 17.11.(a) it also follows that there exists a unitary element u ∈ ℳ 𝜎 such that 𝜎 n = Ad(u). Then 𝒵 (ℛ(ℳ, 𝜎)) consists of all elements of the form X=

∑ m∈ℤ

̄ 𝜆 (nm)) 𝜋𝜎 (zm u−m )(1 ⊗

216

Groups of Automorphisms

with zm ∈ 𝒵 (ℳ) (m ∈ ℤ), and for every 𝛾 ∈ 𝕋 we have 𝜎̂ 𝛾 (X) =

∑

̄ 𝜆 (nm)). 𝛾 nm 𝜋𝜎 (zm u−m )(1 ⊗

m∈ℤ

Consequently, 𝜎̂ 𝛾 (X) = X for X ∈ 𝒵 (ℛ(ℳ, 𝜎)) if and only if 𝛾 n = 1, that is, Γ(𝜏) = {𝛾 ∈ ℂ; 𝛾 n = 1}. 17.13. We show that 17.9.(3) holds true in case 17.11.(b). In this case, by 16.1.(1) and 17.12.(2), we have Sp 𝜎 = Γ(𝜎) = {𝛾 ∈ ℂ; |𝛾| = 1}, so that it is sufficient to prove the following: Lemma. Let 𝜎 ∈ Aut(ℳ) be such that −1 ∈ Sp 𝜎. Then, for every 𝜀 > 0 there exists a nonzero projection f ∈ ℳ such that ‖ f𝜎( f )‖ ≤ 𝜀. Proof. Since −1 ∈ Sp 𝜎, there exists, by Proposition 14.5. x ∈ ℳ, ‖x‖ = 1, such that ‖𝜎(x) + x‖ ≤ 𝜀∕4 = 𝛿. Write x = b + ic with b = b∗ ∈ ℳ, c = c∗ ∈ ℳ. Then ‖b‖ + ‖c‖ ≥ 1, ‖𝜎(b) + b‖ ≤ 𝛿 and ‖𝜎(c) + c‖ ≤ 𝛿. Therefore, we may assume that ‖b‖ ≥ 1∕2 and then, putting a = ±b∕‖b‖, we have a = a∗ ∈ ℳ, ‖a‖ = 1, 1 ∈ Sp(a), ‖𝜎(a) + a‖ ≤ 2𝛿. Since 1 ∈ Sp(a), we have f = 𝜒[1−𝛿,1] (a) ≠ 0 and af ≥ (1 − 𝛿)f. Consider ℳ ⊂ ℬ(ℋ ) realized as a standard von Neumann algebra; then ([L], 10.15) there exists a unitary operator v ∈ ℬ(ℋ ) such that 𝜎 = Ad(v)|ℳ. For every 𝜉 ∈ fℋ , ‖𝜉‖ = 1, we have ‖a𝜉 − 𝜉‖2 = ‖(af − f )𝜉‖2 = 𝜔𝜉 ((af − f )2 ) = 𝜔𝜉 (a2 f − 2af + f ) ≤ 𝜔𝜉 (af − 2af + f ) = 𝜔𝜉 ( f − af ) ≤ 𝜔𝜉 ( f − (1 − 𝛿)f ) = 𝛿. Then 𝜎( f )ℋ = vfv∗ ℋ = vfℋ and, for 𝜂 = v𝜉 ∈ 𝜎( f )ℋ , we have ‖𝜎(a)𝜂 − 𝜂‖ = ‖vav∗ 𝜂 − 𝜂‖ = ‖va𝜉 − v𝜉‖ = ‖a𝜉 − 𝜉‖ ≤ 𝛿, ‖a𝜂 + 𝜂‖ = ‖(a𝜂 + 𝜎(a)𝜂) + (𝜂 − 𝜎(a)𝜂)‖ ≤ ‖a + 𝜎(a)‖‖𝜂‖ + ‖𝜂 − 𝜎(a)𝜂‖ ≤ 3𝛿. Consequently, for 𝜉 ∈ fℋ , ‖𝜉‖ = 1, and 𝜂 ∈ 𝜎( f )ℋ , ‖𝜂‖ = 1, we get |(𝜉|𝜂) − (a𝜉|𝜂)| ≤ 𝛿, |(𝜉|a𝜂) + (𝜉|𝜂)| ≤ 3𝛿, and hence |(𝜉|𝜂)| ≤ 4𝛿 = 𝜀. Therefore ‖ fa( f )‖ ≤ 𝜀. 17.14. Let us show that statement 17.9.(3) holds also in case 17.11.(a). In this case we have (17.11, 17.12) Γ(𝜎) = {𝛾 ∈ ℂ; 𝛾 n = 1} and 𝜎 n = Ad(u) with u ∈ U(ℳ 𝜎 ). Since u ∈ U(ℳ 𝜎 ), there exists v ∈ U(ℳ 𝜎 ) such that vn = u. Then 𝜏 = Ad(v∗ )◦𝜎 ∈ Aut(ℳ) and, since v ∈ ℳ 𝜎 and vn = u, we have 𝜏 n = Ad(v∗ )n ◦𝜎 n = 𝜄. Note that the corresponding actions 𝜎 and 𝜏 of ℤ on ℳ are outer conjugate (16.15) and hence, by Proposition 16.3, Γ(𝜎) = Γ(𝜏).

Outer Automorphisms

217

Let 𝜀 > 0. There exist 𝜆 ∈ ℂ, |𝜆| = 1, and a nonzero spectral projection e ∈ ℳ 𝜎 ∩ ℳ 𝜏 of v such that ‖ve − 𝜆e‖ ≤ 𝜀∕2. Then ‖𝜎 e − 𝜏 e ‖ ≤ 𝜀.

(1)

On the other hand, it is clear that (𝜏 e )n = 𝜄 and (16.1.(4)) Γ(𝜏 e ) ⊃ Γ(𝜏) = Γ(𝜎) = {𝛾 ∈ ℂ; 𝛾 n = 1}, so that 𝜏 e is minimal periodic. By Corollary 16.17.(11), there exists a nonzero projection f ∈ ℳ, f ≤ e, such that f𝜏( f ) = 0. Finally, using (1), we obtain ‖ f𝜎( f )‖ ≤ 𝜀. 17.15. Before completing the proof of Theorem 17.9 we shall review some elementary facts about the “relative position” of two projections. Let ℋ be a Hilbert space, e and f two projections in ℬ(ℋ ), ℳ = ℛ{e, f} ⊂ ℬ(ℋ ) the von Neumann algebra generated by e, f, and let s(e, f ) = |e − f | ∈ ℳ, c(e, f ) = |e ∨ f − e − f | ∈ ℳ. It is clear that e and f are abelian projections in ℳ

(1)

since eℳe = ℛ{e, efe} and fℳf = ℛ{f, fef}. Also, p = e ∧ f + (1 − e) ∧ f + e ∧ (1 − f ) + (1 − e) ∧ (1 − f ) is the greatest projection q ∈ 𝒵 (ℳ) such that ℳq is abelian and ℳ(1 − p) is of type I2

(2)

Indeed, p is clearly a projection, p is central since it commutes with e and f, and ℳp = ℛ{ep, fp} is abelian as epfp = e ∧ f = fpep. If q ∈ ℳ is a central projection and ℳq is abelian, then eq commutes with fq and it follows that pq = q, that is, q ≤ p. If p = 0, then e ∨ f = (1 − e) ∨ f = e ∨ (1 − f ) = (1 − e) ∨ (1 − f ) = 1 and, using the parallelogram law ([L], 4.4), we deduce that e = 1 − (1 − e) = (1 − e) ∨ f − (1 − e) ∼ f − (1 − e) ∧ f = f and 1 − e = e ∨ f − e ∼ f − e ∧ f = f; hence e and 1 − e are equivalent abelian projections in ℳ, which means that in this case ℳ is of type I2 . On the other hand, we have s(e, f )2 = e + f − ef − fe, c(e, f )2 = e ∨ f − e − f + ef + fe, hence s(e, f )2 + c(e, f )2 = e ∨ f.

(3)

We have ec(e, f )2 = efe = c(e, f )2 e, hence c(e, f ) commutes with e and f: s(e, f ) ∈ 𝒵 (ℳ), c(e, f ) ∈ 𝒵 (ℳ).

(4)

zℳ (e) ∨ zℳ ( f ) = e ∨ f.

(5)

Since e ∨ f ∈ 𝒵 (ℳ), we have

218

Groups of Automorphisms

Also, sℳ (c(e, f )) ≤ zℳ (e)zℳ ( f ).

(6)

Indeed, let q ∈ 𝒵 (ℳ) be a projection such that qe = e. Then e ∨ f − f ≥ q(e ∨ f − f ) ∼ q(e − e ∧ f ) = e − e ∧ f ∼ e ∨ f − f, hence q(e ∨ f − f ) = e ∨ f − f, since by (2) ℳ is finite. Therefore, qc(e, f )2 = q(e ∨ f − f − e + ef + fe) = e ∨ f − f − e + ef + fe = c(e, f )2 , so that qc(e, f ) = c(e, f ). Thus, sℳ (c(e, f )) ≤ zℳ (e) and, similarly, sℳ (c(e, f )) ≤ zℳ ( f ). We now prove that if sℳ (c(e, f )) = e ∨ f and e ∨ f − (e + f ) = uc(e, f ) is the polar decomposition, then u = u∗ , u2 = e ∨ f, ueu = f, ufu = e.

(7)

Indeed, e ∨ f − (e + f ) is self-adjoint, hence u = u∗ . Also, u2 = u∗ u = sℳ (c(e, f )) = e ∨ f. Since e∨f = sℳ (c(e, f )) = sℳ (e∨f−e−f ), we have sℳ (efe) = e, sℳ ( fef ) = f, for if q ∈ ℳ is a projection with q ≤ e, efeq = 0, then q ≤ e ∨ f, eq = e, fq = 0, so that (e ∨ f − e − f )q = 0 and hence q = 0. We have e ∨ f − ueu − ufu = u(e ∨ f − e − f )u = uuc(e, f )u = e ∨ f − e − f, hence ueu + ufu = e + f. Finally, ueu ≥ uec(e, f )2 eu = (e ∨ f − e − f )e(e ∨ f − e − f ) = fef, hence ueu ≥ sℳ ( fef ) = f. Similarly, ufu ≥ e. Thus ueu = f and ufu = e. Note that ‖s(e, f )‖ = ‖e − f ‖, ‖c(e, f )‖ = ‖ef ‖.

(8)

The first identity is obvious. Since ec(e, f )2 e = efe, we have ‖ec(e, f )‖ = ‖ef ‖. Using (6) and the fact that any induction by a projection with central support equal to 1 is a *-isomorphism ([L], 3.14), it follows that ‖c(e, f )‖ = ‖e c(e, f )‖ = ‖ef ‖. For 𝜆 > 0 we show that if for every nonzero projections e′ , f ′ ∈ ℳ, e′ ≤ e, f ′ ≤ f, we have ‖e′ f ‖ ≥ 𝜆 and ‖ef ′ ‖ ≥ 𝜆, then c(e, f ) ≥ 𝜆(e ∨ f ).

(9)

We first prove that c(e, f ) ≥ 𝜆zℳ (e). Otherwise, using the Gelfand representation, we find 0 < 𝜇 < 𝜆 and a nonzero projection q ∈ 𝒵 (ℳ) such that qc(e, f ) ≤ 𝜇qzℳ (e) ≠ 0. Then qe ≠ 0 and qe ≤ e, hence c(qe, f ) = |qe ∨ f − qe − f | = q|e ∨ f − e − f | + (1 − q)| f − f | = qc(e, f ) ≤ 𝜇qzℳ (e). Thus, by (8) and the assumption, we get 𝜆 ≤ ‖qef ‖ = ‖c(qe, f )‖ ≤ ‖𝜇qzℳ (e)‖ ≤ 𝜇, a contradiction. Therefore, c(e, f ) ≥ 𝜆zℳ (e) and, similarly, c(e, f ) ≥ 𝜆zℳ ( f ), so the desired inequality follows using (5). In connection with the first equation in (8) we note that ‖e − f ‖ < 1 ⇒ e = lℳ (ef ) ∼ rℳ (ef ) = f.

(10)

Indeed, if q ∈ ℳ is a projection such that q ≤ e and qef = 0, then qe = q, qf = 0, hence ‖q‖ = ‖q(e − f )‖ < 1 and q = 0. Thus, e = lℳ (ef ) and, similarly, f = rℳ (ef ). Finally, in connection with the second equation in (8), we note that ‖ef ‖ < 1 ⇒ f ′ = e ∨ f − e ∼ f,

e ∨ f = e + f ′,

‖ f − f ′ ‖ = ‖ef ‖.

(11)

Outer Automorphisms

219

Indeed, ‖e ∧ f ‖ = ‖(e ∧ f )ef ‖ < 1, hence e ∧ f = 0, so that f ′ = e ∨ f − e ∼ f − e ∧ f = f. Then using (8), we obtain ‖ f − f ′ ‖ = ‖e ∨ f − e − f ‖ = ‖c(e, f )‖ = ‖ef ‖. 17.16. End of the proof of Theorem 17.9. To prove that p(𝜎) satisfies condition 17.9.(1) we may assume that p(𝜎) = 0. Let e ∈ ℳ be a nonzero projection and suppose that 𝜆 = inf{‖ f𝜎( f )‖; f ∈ Proj(ℳ), 0 ≠ f ≤ e} > 0.

(1)

Choose 𝜀 > 0 with (𝜆 + 1)𝜀 < 𝜆 and a nonzero projection f ∈ ℳ such that ‖ f𝜎( f )‖ ≤ 𝜆 + 𝜀. For every nonzero projection f ′ ≤ f we have ‖ f ′ 𝜎( f ′ )‖ ≥ 𝜆, hence ‖ f𝜎( f ′ )‖ ≥ 𝜆, ‖ f ′ 𝜎( f )‖ ≥ 𝜆. From 17.15.(9), we deduce that c( f, 𝜎( f )) ≥ 𝜆( f ∨ 𝜎( f )). Using 17.15.(8), it follows from the choice of f that c( f, 𝜎( f )) ≤ (𝜆 + 𝜀)( f ∨ 𝜎( f )). On the other hand, let u be the partial isometry appearing in the polar decomposition of f ∨ 𝜎( f ) − f − 𝜎( f ). Using 17.15.(7), we obtain u = u∗ , u2 = f ∨ 𝜎( f ), ufu = 𝜎( f ), u𝜎( f )u = f, and ‖𝜆u − uc( f, 𝜎( f ))‖ ≤ ‖𝜆( f ∨ 𝜎( f )) − c( f, 𝜎( f )))‖ ≤ 𝜀, that is, ‖𝜆u − ( f ∨ 𝜎( f ) − f − 𝜎( f ))‖ ≤ 𝜀.

(2)

The equation 𝜏(x) = u𝜎(x)u∗ (x ∈ fℳf ) defines a *-automorphism 𝜏 ∈ Aut( fℳf ). Since p(𝜎) = 0, we obtain using 17.2.(2), 𝜏 ∉ Int( fℳf ). By the last statement in Theorem 17.9, which has already been proved (17.11–17.14), there exists a nonzero projection h ∈ ℳ, h ≤ f, such that ‖h𝜏(h)‖ ≤ 𝜀. Then we have ‖hu𝜎(h)‖ = ‖hu𝜎(h)u∗ ‖ = ‖h𝜏(h)‖ ≤ 𝜀. Using (2) we infer that ‖h( f ∨ 𝜎( f ) − f − 𝜎( f ))𝜎(h)‖ ≤ 𝜀(𝜆 + 1). Since h ≤ f, we have h( f ∨ 𝜎( f ) − f − 𝜎( f ))𝜎(h) = −h𝜎(h), hence ‖h𝜎(h)‖ ≤ 𝜀(𝜆 + 1) < 𝜆, contradicting (1). We conclude that 𝜆 = 0, and this completes the proof of Theorem 17.9. □ 17.17. Let ℳ be a finite W∗ -algebra and 𝜇 a faithful normal trace on ℳ with 𝜇(1) = 1. For x ∈ ℳ we consider the norms ‖x‖1 = 𝜇(|x|) and ‖x‖2 = 𝜇(x∗ x)1∕2 . It is easy to see that ‖x‖1 ≤ ‖x‖2 , ‖x‖22 ≤ ‖x‖‖x‖1 (x ∈ ℳ). Recall that the closed unit ball ℳ1 of ℳ endowed with s-topology is a complete metrizable space; in fact the metric ‖x − y‖2 defines the s-topology on ℳ1 (Str̆atil̆a & Zsidó, 1977, 2005, 8.12). A family of mutually orthogonal nonzero projections in ℳ with sum equal to 1 will be called, briefly, a partition of unity in ℳ. Theorem 17.9 is the main technical instrument in proving the following important result: Theorem (V. Rokhlin, A. Connes). Let ℳ be a finite W∗ -algebra, 𝜇 a faithful normal trace on ℳ with 𝜇(1) = 1, and 𝜎 ∈ Aut(ℳ) an aperiodic *-automorphism such that 𝜇◦𝜎 = 𝜇. For each n ∈ ℤ, n ≥ 1, and 𝜀 > 0, there exists a partition of unity {e1 , … , en } in ℳ such that ‖𝜎(e1 ) − e2 ‖2 ≤ 𝜀, … , ‖𝜎(en−1 ) − en ‖2 ≤ 𝜀, ‖𝜎(en ) − e1 ‖2 ≤ 𝜀. The proof is given in Sections 17.20–17.23, where we use Lemmas 17.18, 17.19. In the present section, we divide the proof into three distinct cases. There exists a partition of unity {pk }k>0 in 𝒵 (ℳ) such that, if k > 1, we have (𝜎|𝒵 (ℳ)pk )k = 𝜄 and (𝜎|𝒵 (ℳ)p)j ≠ 𝜄 for every central projection 0 ≠ p ≤ pk and every 1 ≤ j < k; then we

220

Groups of Automorphisms

have (𝜎|𝒵 (ℳ)p)j ≠ 𝜄 for every central projection 0 ≠ p ≤ p0 and every j > 1. Thus, in proving Theorem 17.17, we can consider separately the cases p1 = 1, pk = 1 with k > 2, and p0 = 1, that is, (I) 𝜎 acts identically on 𝒵 (ℳ); (II) there exists k ≥ 2 such that (𝜎|𝒵 (ℳ))k = 𝜄 but (𝜎|𝒵 (ℳ)p)j ≠ 𝜄 for every central projection p ≠ 0 and every 1 ≤ j < k; and (III) 𝜎|𝒵 (ℳ) is aperiodic. 17.18 Lemma. Let ℳ be a W∗ -algebra, let n ∈ ℤ, n > 1, and choose 𝜀 > 0 so that n!𝜀 < 1. If {f1 , … , fn } ⊂ ℳ is a family of projections with ‖ fj fk ‖ ≤ 𝜀 for j ≠ k, then there exists a family ∑n ⋁n {e1 , … , e2 } ⊂ ℳ of mutually orthogonal projections such that k=1 ek = k=1 fk and ek ∼ fk , ‖ek − fk ‖ ≤ n!𝜀 for 1 ≤ k ≤ n. Proof. For n = 1 the lemma is obvious. Assume the lemma is true for n − 1 projections. Then there ∑n−1 exists a family {e1 , … , en−1 } ⊂ ℳ of mutually orthogonal projections such that e = k=1 ek = ⋁n−1 k=1 fk and ek ∼ fk , ‖ek − fk ‖ ≤ (n − 1)!𝜀 for 1 ≤ k ≤ n − 1. Then ‖ek fn − fk fn ‖ ≤ (n − 1)!𝜀(1 ≤ k ≤ n − 1), hence ‖efn ‖ ≤

n−1 ∑ k=1

‖ek fn − fk fn ‖ +

n−1 ∑

‖ fk fn ‖ ≤ (n − 1)(n − 1)!𝜀 + (n − 1)!𝜀,

k=1

that is, ‖efn ‖ ≤ n!𝜀 < 1. Using 17.15.(11) we obtain a projection en ∈ ℳ such that the family {e1 , … , en−1 , en } satisfies the requirements of the lemma. 17.19 Lemma. Let ℳ ⊂ ℬ(ℋ ) be a countably decomposable von Neumann algebra, 𝜉 ∈ ℋ and e, f ∈ ℳ projections. Put 𝜀 = ‖e𝜉 − f𝜉‖. If fe = wa is the polar decomposition of fe, that is, a = (efe)1∕2 , w∗ w = s(efe), then ‖(w − e)𝜉‖4𝜀, ‖(w − f )𝜉‖ ≤ 3𝜀, ‖(w − e)∗ 𝜉‖ ≤ 3𝜀, ‖(w − f )∗ 𝜉‖ ≤ 4𝜀.

(1)

If e ∼ f, then there exists u ∈ ℳ such that u∗ u = e, uu∗ = f and ‖(u − f )𝜉‖ ≤ 6𝜀, ‖(u − f )∗ 𝜉‖ ≤ 7𝜀.

(2)

Proof. We have a2 = efe ≤ e, a2 ≤ a ≤ e, so that ‖(w − fe)𝜉‖ = ‖(w(e − a)𝜉‖ ≤ ‖(e − a2 )𝜉‖ = ‖e(e − f )e𝜉‖ ≤ ‖(e − fe)𝜉‖. ‖(e − fe)𝜉‖ ≤ ‖(e − f )𝜉‖ + ‖ f ( f − e)𝜉‖ ≤ 2𝜀, hence ‖(w − e)𝜉‖ ≤ ‖(w − fe)𝜉‖ + ‖(e − fe)𝜉‖ ≤ 4𝜀 and ‖(w − f )𝜉‖ ≤ ‖(w − fe)𝜉‖ + ‖( fe − f )𝜉‖ ≤ 3𝜀. This proves the first two inequalities in (1). The last two inequalities in (1) then follow since ef = w∗ (waw∗ ) is the polar decomposition of ef. Assume now that e ∼ f and let 𝛿 > 0. There exists a projection p ∈ 𝒵 (ℳ) such that the projections e1 = pe, f1 = pf are finite while the projections e0 = (1 − p)e, f0 = (1 − p)f are properly infinite.

Outer Automorphisms

221

Since e0 , f0 are properly infinite, we can write e0 = ∑n ∑n Let e′0 = k=1 ek0 , f0′ = k=1 f0k with n so large that

∑∞

k ,f k=1 e0 0

=

∑∞

k k=1 f0

with ek0 ∼ e0 ∼ f0 ∼ f0k .

‖(e0 − e′0 )𝜉‖ < 𝛿, ‖( f0 − f0′ )𝜉‖ < 𝛿. Clearly, e0 − e′0 ∼ e0 ∼ f0 ∼ f0 − f0′ . We define e1 = e1 + e′0 , f 1 = f1 + f0′ and e2 = r( f 1 e1 ), f 2 = l( f 1 e1 ). Then e1 f 1 e1 ≤ e2 ≤ e1 , f 1 e1 f 1 ≤ f 2 ≤ f 1 , so that ‖(e1 − e2 )𝜉‖ ≤ ‖(e1 − e1 f 1 e1 )𝜉‖ ≤ 2‖(e1 − f 1 )𝜉‖ ≤ 2(‖(e − f )𝜉‖ + ‖(e0 − e′0 )𝜉‖ + ‖( f0 − f0′ )𝜉‖ ≤ 2𝜀 + 4𝛿,

(3)

and hence ‖(e1 − e2 )𝜉‖ ≤ 2𝜀 + 4𝛿, ‖( f 1 − f 2 )𝜉‖ ≤ 2𝜀 + 4𝛿. Also, we have e2 ∼ f 2 more precisely e2 = w∗ w, f 2 = ww∗ , where w is the partial isometry appearing in the polar decomposition of f 1 e1 . Using (1) and (3) we deduce that ‖(w − f 1 )𝜉‖ ≤ 3(𝜀 + 2𝛿), ‖(w − f 1 )∗ 𝜉‖ ≤ 4(𝜀 + 2𝛿). On the other hand, we have e1 ∼ f1 and e1 ≥ pe2 ∼ pf 2 ≤ f1 , hence ([L], E.4.9) e1 − pe2 ∼ f1 − pf 2 . Also, we have e0 − (1 − p)e2 ≥ e0 − e′0 , f0 − (1 − p)f 2 ≥ f0 − f0′ ; hence e0 − (1 − p)e2 and f0 − (1 − p)f 2 are countably decomposable properly infinite projections having equal central supports so that ([L], 4.13) they are equivalent: e0 − (1 − p)e2 ∼ f0 − (1 − p)f 2 . It follows that e − e2 ∼ f − f 2 , and there exists v ∈ ℳ with v∗ v = e − e2 , vv∗ = f − f 2 . Let u = w + v. Clearly, u∗ u = e, uu∗ = f. Then ‖v𝜉‖ = ‖vv∗ v𝜉‖ ≤ ‖v∗ v𝜉‖ = ‖(e − e2 )𝜉‖ ≤ ‖(e1 − e2 )𝜉‖ + ‖(e0 − e′0 )𝜉‖ ≤ 2𝜀 + 5𝛿, ‖v∗ 𝜉‖ = ‖v∗ vv∗ 𝜉‖ ≤ ‖vv∗ 𝜉‖ = ‖( f − f 2 )𝜉‖ ≤ ‖( f 1 − f 2 )𝜉‖ + ‖( f0 − f0′ )𝜉‖ ≤ 2𝜀 + 5𝛿, ‖(w − f )𝜉‖ ≤ ‖(w − f 1 )𝜉‖ + ‖( f0 − f0′ )𝜉‖ ≤ 3𝜀 + 7𝛿, ‖(w − f )∗ 𝜉‖ ≤ ‖(w − f 1 )∗ 𝜉‖ + ‖( f0 − f0′ )𝜉‖ ≤ 4𝜀 + 9𝛿, so (2) follows on choosing 𝛿 < 𝜀∕9. 17.20 Lemma. Let ℳ be a finite W∗ -algebra, 𝜎 ∈ Aut(ℳ) an aperiodic *-automorphism acting identically on 𝒵 (ℳ), and 𝜇 a faithful normal 𝜎-invariant trace on ℳ with 𝜇(1) = 1. For each n ∈ ℤ, n > 2, and 𝛿 > 0, there exists a unitary element v ∈ ℳ and a family {f1 , … , fn } ⊂ ℳ of mutually orthogonal nonzero projections such that ‖v − 1‖1 ≤ 𝛿𝜇( f1 + … + fn ), (Ad(v)◦𝜎)( fj ) = fj+1 (j = 1, … , n),

(1) (2)

where fn+1 = f1 . Proof. Let 𝛾 = 𝛿∕12(n + 1), m = nr with 2m−1∕2 ≤ 𝛾∕2 and 0 < 𝜀 < 1∕m! with 2mm|𝜀 ≤ 𝛾∕2. Since 𝜎 is aperiodic, using Theorem 17.9 we find projections p1 ≥ p2 ≥ … ≥ pm ≠ 0 in ℳ such that ‖pk 𝜎 k (pk )‖ ≤ 𝜀 (1 ≤ k ≤ m). Let p = pm . Since p ≤ pk , it follows that ‖p𝜎 k (p)‖ ≤ 𝜀 (1 ≤ k ≤ m), so that ‖𝜎 i (p)𝜎 j (p)‖ ≤ 𝜀 for i ≠ j, 1 ≤ i, j ≤ m.

222

Groups of Automorphisms

⋁m k Let e = 17.18, we find a family {e1 , … , em } ⊂ ℳ of mutually k=1 𝜎 (p). Using Lemma ∑m orthogonal projections such that e = k=1 ek and ek ∼ 𝜎 k (p), ‖ek − 𝜎 k (p)‖ ≤ m!𝜀 ≤ 𝛾∕4m(1 ≤ k ≤ m). It follows that ‖𝜎(ek ) − ek+1 ‖ ≤ 𝛾∕2m

(k = 1, … , m − 1).

(3)

Since 𝜎 acts identically on 𝒵 (ℳ), the canonical central trace ♮ ∶ ℳ → 𝒵 (ℳ) is 𝜎-invariant. By ([L], 7.12) it follows that 𝜎 k (p) ∼ p (1 ≤ k ≤ m), hence e1 ∼ e2 ∼ … ∼ em .

(4)

Define fj =

r−1 ∑

ens+j

(1 ≤ j ≤ n).

(5)

s=0

Clearly,

∑n

j=1 fj

= e and f1 ∼ f2 ∼ … ∼ fn .

Let f = e ∨ 𝜎(e) = e ∨ 𝜎 m+1 (p) and 𝒩 = fℳf. Then ek , 𝜎(ek ) ∈ 𝒩 (1 ≤ k ≤ m) and fj , 𝜎( fj ) ∈ 𝒩 (1 ≤ j ≤ n). Then 𝜇 ′ = (𝜇|𝒩 )∕𝜇( f ) is a faithful normal trace on 𝒩 with 𝜇′ ( f ) = 1 and for x ∈ 𝒩 ⊂ ℳ we have ‖x‖′2 = 𝜇′ (x∗ x)1∕2 = 𝜇( f )−1∕2 𝜇(x∗ x)1∕2 = 𝜇( f )−1∕2 ‖x‖2 .

(6)

Using (3), (4), and (6), we obtain ‖𝜎( fj ) − fj+1 ‖′2 ≤ r𝛾∕2m < 𝛾

(1 ≤ j ≤ n − 1).

On the other hand, from (4) it follows that 𝜇 ′ (ek ) ≤ 1∕m, hence 𝜇′ (𝜎(ek )) ≤ 1∕m (since ek ∼ 𝜎(ek )) for 1 ≤ k ≤ m. Thus, ‖e1 ‖′2 ≤ m−1∕2 , ‖𝜎(enr )‖′2 ≤ m−1∕2 and, using (5), (3), and (6), we deduce that ‖𝜎( fn ) − f1 ‖′2 ≤ (r − 1)𝛾∕2m + 2m−1∕2 < 𝛾. For 1 ≤ j ≤ n, we have 𝜎( fj ) ∼ fj ∼ fj+1 so, by Lemma 17.19, there exists wj ∈ 𝒩 such that w∗i wj = 𝜎( fj ), wj w∗j = fj+1 and ‖wj − fj+1 ‖′2 ≤ 6𝛾. Since f − 𝜎(e) ∼ f − e, there exists w0 ∈ 𝒩 such that w∗0 w0 = f − 𝜎(e), w0 w∗0 = f − e. Then w = w0 + w1 + … + wn ∈ 𝒩 is unitary and w𝜎( fj )w∗ = fj+1

(1 ≤ j ≤ n).

(7)

Outer Automorphisms

223

Since 𝜇 ′ ( f − e) ≤ 𝜇 ′ (𝜎 m+1 (p)) = 𝜇 ′ (p) ≤ 1∕m, we obtain ‖w − f ‖′2 = ‖w0 − ( f − e)‖′2 +

n ∑

‖wj − fj+1 ‖′2

j=1

≤

‖w0 ‖′2

+ ‖f −

e‖′2

+

n ∑

‖wj − fj+1 ‖′2

j=1

≤ 2m−1∕2 + 6n𝛾 = 6(n + 1)𝛾 = 𝛿∕2. Finally, v = w + (1 − f ) ∈ ℳ is unitary, satisfies condition (2) and ‖v − 1‖1 = 𝜇(|v − 1|) = 𝜇(|w − f |) = 𝜇( f )𝜇 ′ (|w − f |) < 2𝜇( f )‖w − f ‖′2 ≤ 2𝜇( f )𝛿∕2 = 𝛿𝜇( f1 + … + fn ). □ 17.21. Proof of Theorem 17.17 in case 17.17.(I). We assume that 𝜎 acts identically on 𝒵 (ℳ). Let n ∈ ℤ, n ≥ 1, and 𝛿 = 𝜀∕4 > 0 be fixed. Let ℰ be the set of all n + 1-tuples (e1 , … , en ; u), where {e1 , … , en } ⊂ ℳ consists of mutually orthogonal equivalent projections and u ∈ ℳ is a unitary element such that ‖u − 1‖1 < 𝛿𝜇(e1 + … + en ) and u𝜎(ej )u∗ = ej+1 for 1 ≤ j ≤ n. We define an order relation “≤” on ℰ writing (e1 , … , en ; u) ≤ (e′1 , … , e′n ; u′ ) if and only if ej ≤ ′ ej (1 ≤ j ≤ n), and ‖u − u′ ‖1 ≤ 𝛿𝜇((e′1 − e1 ) + … + (e′n − en )). The set ℰ is then inductively ordered. Indeed, let ℱ ⊂ ℰ be a totally ordered set. Since the mapping ℱ ∋ (e1 , … , en ; u) ↦ 𝜇(e1 + … + en ) ∈ [0, 1] is an order isomorphism of ℱ onto a subset of [0, 1], we may assume that ℱ = {(ek1 , … , ekn ; uk )}k>1 is an increasing sequence. Then, for each 1 ≤ j ≤ n, the increasing sequence of projections {ekj }k>1 ⋁ is s-convergent to the projection ej = k≥1 ekj . Therefore, the projections {e1 , … , en } are mutually orthogonal and equivalent, and ekj ≤ ej for all j and k. On the other hand, we have ( ‖uk − uk+1 ‖1 ≤ 𝛿𝜇

)

n ∑

(ek+1 j

−

ekj )

,

j=1

hence ‖uk − uk+1 ‖1 ≤ 𝛿 < +∞. By the remarks made at the beginning of Section 17.17, it follows s

that there exists a unitary operator u ∈ ℳ such that ‖uk − u‖1 → 0 and uk → u. It is now easy to check that (e1 , … , en ; u) ∈ ℰ is an upper bound for ℱ . By Zorn’s lemma, there exists a maximal element (e1 , … , en ; u) in ℰ . Assume that f = 1 − (e1 + … + en ) ≠ 0. Let 𝒩 = fℳf, 𝜇 ′ = (𝜇|𝒩 )∕𝜇( f ), and 𝜎 ′ = (Ad(u)◦𝜎)|𝒩 . Then 𝜎 ′ ∈ Aut(𝒩 ) is aperiodic, acts identically on 𝒵 (𝒩 ) and 𝜇 ′ ◦𝜎 ′ = 𝜇 ′ . By Lemma 17.20, there exists an n + 1-tuple ( f1 , … , fn ; v), where {f1 , … , fn } ⊂ 𝒩 is a set of mutually orthogonal and equivalent projections and v ∈ 𝒩 is a unitary element such that ‖v − f ‖′1 ≤ 𝛿𝜇 ′ ( f1 + … + fn ) ≠ 0 and v𝜎 ′ ( fj )v∗ = fj+1 for 1 ≤ j ≤ n. We now define e′j = ej + fj , (1 ≤ j ≤ n), and u′ = v + (1 − f )u. Then {e′1 , … , e′n } ⊂ ℳ consists of

224

Groups of Automorphisms

mutually orthogonal and equivalent projections, u ∈ ℳ is a unitary element, f ′ 𝜎(e′j )u′∗ = e′j+1 for 1 ≤ j ≤ n, and we have ( ‖(v + (1 − f )) − 1‖1 = ‖v − f ‖1 = 𝜇( f )‖v −

f ‖′1

≤ 𝛿𝜇( f )𝜇 (

‖u − u‖1 ≤ ‖u‖‖(v + (1 − f )) − 1‖1 ≤ 𝛿𝜇 ′

∑

∑

′

j

)

(

( = 𝛿𝜇

fj

∑

= 𝛿𝜇 (

)

‖u′ − 1‖1 ≤ ‖u′ − u‖1 + ‖u − 1‖1 ≤ 𝛿𝜇

) fj

,

j

) ∑ ′ (ej − ej ) ,

fj

j

)

j

∑

e′j

,

j

hence (e′1 , … , e′n ; u′ ) ∈ ℰ is strictly greater than {e1 , … , en ; u‖, a contradiction. Thus, if (e1 , … , en ; u) is a maximal element in ℰ , then (e1 , … , en ) is a partition of unity in ℳ, u ∈ U(ℳ), ‖u − 1‖1 < 𝛿, and u𝜎(ej )u∗ = ej+1 (1 ≤ j ≤ n). Therefore, ‖𝜎(ej ) − ej+1 ‖22 = ‖𝜎(ej ) − u𝜎(ej )u∗ ‖22 ≤ 2‖𝜎(ej ) − u𝜎(ej )u∗ ‖1 ≤ 4‖u − 1‖1 ≤ 4𝛿 = 𝜀. 17.22. Proof of Theorem 17.17 in case 17.17.(II). We assume that there exists k ≥ 2 such that (𝜎|𝒵 (ℳ))k = 𝜄, but (𝜎|𝒵 (ℳ)p)j ≠ 𝜄 for any central projection p ≠ 0 and j such that 1 ≤ j < k. If p ∈ 𝒵 (ℳ) is a maximal projection such that p, 𝜎(p), … , 𝜎 k−1 (p) are mutually orthogonal, then p + 𝜎(p) + … + 𝜎 k−1 (p) = 1. Thus, there exists a partition of unity {p1 , … , pk } in 𝒵 (ℳ) such that 𝜎(pi ) = pi+1

(1 ≤ i ≤ k),

where pk+1 = pi . Let n ∈ ℤ, n ≥ 1, and 𝛿 = 𝜀∕k > 0 be fixed. The *-automorphism 𝜎 ′ = 𝜎 k |ℳp1 ∈ Aut(ℳp1 ) is aperiodic, acts identically on 𝒵 (ℳp1 ), and preserves the faithful normal trace 𝜇′ = (𝜇|ℳp1 )∕𝜇(p1 ) = k𝜇|ℳp1 . By Section 17.21, there exists a partition of unity {f0 , … , fn−1 } in ℳp1 such that (with fn = f0 ) ‖𝜎 ′ ( fs ) − fs+1 ‖′2 ≤ 𝛿

(0 ≤ s ≤ n − 1).

The family {𝜎 i ( fs ); 1 ≤ i ≤ k, 0 ≤ s ≤ n − 1} is a partition of unity {h1 , … , hsk } in ℳ, where hsk+i = 𝜎 i ( fs ) (1 ≤ i ≤ k, 0 ≤ s ≤ n − 1). For 1 ≤ r ≤ nk we have (with hnk+1 = h1 ) ‖𝜎(hr ) − hr+1 ‖2 ≤ k−1∕2 𝛿 < 𝛿. Indeed, if r = sk + i with 0 ≤ s ≤ n − 1 and 1 ≤ i < k, then r + 1 = sk + (i + 1), hence 𝜎(hr ) = hr+1 ; if r = sk + k, then r + 1 = (s + 1)k + 1, hence ‖𝜎(hr ) − hr+1 ‖2 = ‖𝜎(𝜎 k ( fs )) − 𝜎( fs+1 )‖2 = ‖𝜎 k ( fs ) − fs+1 ‖2 = k−1∕2 ‖𝜎 ′ ( fs ) − fs+1 ‖′2 ≤ k−1∕2 𝛿.

Outer Automorphisms

225

Finally, the projections ej = hj + hj+n + … + hj+(k−1)n

(1 ≤ j ≤ n)

constitute a partition of unity in ℳ and we have (with en+1 = e1 ) ‖𝜎(ej ) − ej+1 ‖2 ≤ k𝛿 = 𝜀

(1 ≤ j ≤ n).

17.23. Proof of Theorem 17.17 in case 17.17. (III). In this case we may assume that ℳ is abelian. So, in this section we give essentially the proof of the classical Rokhlin theorem. Let n ∈ ℤ, n ≥ 1, and 𝜀 > 0. Choose r ∈ ℤ, r > 1, such that 1∕r ≤ 𝜀∕16, and write m = nr. Let p ∈ ℳ be a maximal projection such that p, 𝜎(p), … , 𝜎 m−1 (p) are mutually orthogonal and define

(1)

pk = 𝜎

(2)

−m+1

(p)𝜎 (p) (1 ≤ k ≤ m). k

Then p1 , … , pm are mutually orthogonal projections and the maximal choice of p with property (1) implies that 𝜎 −m+1 (p) = p1 + … + pm ; indeed, if q = 𝜎 −1 (𝜎 −m+1 (p) − Consider the projections

∑m k=1

(3)

pk ) ≠ 0, then p + q contradicts the maximality of p.

𝜎 −1 (p2 )

(4)

𝜎 −1 (p3 ) 𝜎 −2 (p3 ) ………… 𝜎 −1 (pm ) 𝜎 −2 (pm ) … 𝜎 −1 (p) 𝜎 −2 (p) …

𝜎 −m+1 (pm ), 𝜎 −m+1 (p)

p.

(5)

All the projections appearing in (4) and (5) are mutually orthogonal. Indeed, all the projections in (5) are mutually orthogonal, by (1). Moreover, the projections appearing in the same column of (4) are mutually orthogonal, by (2), and two projections appearing in different columns of (4) are orthogonal, by (3) and (1); hence all the projections in (4) are mutually orthogonal. Finally, using (2) and (1) it follows that 𝜎 −i (pj )𝜎 −k (p) = 0 for all 1 ≤ i < j ≤ m, 0 ≤ k ≤ m − 1. Let q be the sum of all the projections in (4) and (5). We claim that q = 1. To prove this it is sufficient to show that q is 𝜎-invariant since, as 𝜎 is aperiodic, the assumption 1 − q ≠ 0 would then contradict the choice of p maximal with property (1). Furthermore, to show that 𝜎 −1 (q) = q it is sufficient to show just that 𝜎 −1 (q) ≤ q, since 𝜇 is a 𝜎-invariant faithful state on ℳ. Looking at (4) and (5) we see that the only projections from the sum defining 𝜎 −1 (q) which do not appear explicitly in the sum defining q are 𝜎 −m (p), 𝜎 −2 (p2 ), … , 𝜎 −m (pm ). By (3) we have 𝜎 −m (p) = 𝜎 −1 (p1 + p2 + … + pm ) ≤ 𝜎 −1 (p1 ) ∨ q and from (2) it follows that 𝜎 −k (pk ) ≤ p ≤ q for all 1 ≤ k ≤ m. Hence 𝜎 −1 (q) ≤ q, and consequently q = 1.

226

Groups of Automorphisms

We now define the projection

e=

r−1 ∑

𝜎 −ns (p) +

s=0

m s(k) ∑ ∑

𝜎 −sn−1 (pk ),

(6)

k=n+1 s=0

where s(k) is the greatest s ∈ ℤ, s ≥ 0, such that sn + 1 < k − n. Looking at (4) and (5) we see that e, 𝜎 −1 (e), … , 𝜎 −n+1 (e) are mutually orthogonal.

(7)

‖e − 𝜎 −n (e)‖1 ≤ 𝜀∕4.

(8)

We have

Indeed, if we compute 𝜎 −n (e) using (6) and compare with (6), we obtain 𝜇(|e − 𝜎 −n (e)|) ≤ 𝜇(p + 𝜎 −nr (p) +

m ∑

𝜎 −1 (pk ) + 𝜎 −(s(k)+1)n−1 (pk )

k=n+1

≤ 2𝜇(p) + 2

m ∑

𝜇(pk ) ≤ 4𝜇(p) ≤ 4∕m ≤ 4∕r ≤ 𝜀∕4,

k=1

since 𝜇 is 𝜎-invariant and we have (3). We also have ‖1 −

n−1 ∑

𝜎 −i (e)‖1 ≤ 𝜀∕4.

(9)

i=0

∑n−1 Indeed, the sum q = i=0 𝜎 −i (e) contains all the projections in (5) and the projections in (4) except for at most n projections from each row of (4). Since the sum of the projections in (4) and (5) is 1, we obtain (m ) m ∑ ∑ 𝜇(1 − q) ≤ n𝜇(pk ) ≤ n𝜇 pk = n𝜇(p) ≤ n∕m < 𝜀∕4, k=2

k=1

using (3) again, and the 𝜎-invariance of 𝜇. From (7), (8), and (9), it follows that the projections ( ej = 𝜎

−n+j

(e) (1 ≤ j ≤ n − 1) and en = e +

1−

n−1 ∑

) 𝜎 (e) −i

i=0

satisfy the requirements of Theorem 17.17. 17.24. Finally, we note a useful consequence of Theorem 17.9.

□

Outer Automorphisms

227

Corollary. Let 𝜎 ∶ G → Aut(ℳ) be a properly outer action of the finite group G on the countably decomposable infinite W∗ -algebra ℳ. Then the centralizer ℳ 𝜎 is infinite. Proof. Let 𝜀 > 0 be such that 𝜀(card G)!(card G)2 < 1. Since every 𝜎g (e ≠ g ∈ G) is properly outer, a familiar maximality argument based on Theorem 17.9 shows that there exists a projection p ∈ ℳ such that |𝜎g (p)𝜎h (p)‖ < 𝜀 for g, h ∈ G(g ≠ h), and ⋁ g∈G 𝜎g (p) = 1. Since ℳ is properly infinite and G is finite, it follows that some 𝜎g (p) is properly infinite and we may therefore assume that p is properly infinite. Thus, there exists a partial isometry ∑ v ∈ pℳp with v∗ v = p and vv∗ ≠ p. Let u = g∈G 𝜎g (v) ∈ ℳ 𝜎 . From the choice of 𝜀 > 0 it is easy to check that r(u) = 1 while l(u) ≠ 1. Hence ℳ 𝜎 is infinite. Note that a particular case of this result (16.17) has been used (17.14) in proving Theorem 17.9. 17.25. Here we describe the Ocneanu’s Theorem on cocycle conjugacy of outer actions of countable discrete amenable groups. Let ℳ be a type II1 factor and G be a countable discrete group. An outer action of G on  is an injective group homomorphism 𝛼 ̃ ∶ G → Out (ℳ) = Aut (ℳ) ∕Int (ℳ) . ( ) Equivalently, chosing a lifting 𝛼 = 𝛼g g∈G to Aut (ℳ) (i.e., each 𝛼g is a noninner automorphism of ℳ whose class in Out (ℳ) is 𝛼 ̃g , g ∈ G), an outer action is determined by the condition that there ( ) exist unitaries us,t s,t∈G in ℳ such that ( ) 𝛼s ◦𝛼t = Ad us,t ◦𝛼st ,

(s, t ∈ G).

In this case, the condition that 𝛼 determines a group homomorphism into Out (ℳ) is that, for all g, h, k ∈ G, we have ( ) ug,h ugh,k = 𝜆g,h,k 𝛼g uh,k ug,hk , where 𝜆g,h,k ∈ 𝕋 = {z ∈ ℂ; |z| = 1}. One may always assume, by requiring that 𝛼e = id, that ue,g = ug,e = 1,

(g ∈ G).

( ) The scalars 𝜆g,h,k g,h,k∈G define a 3-cocycle for the group G. The class of this cocycle in H3 (G, 𝕋 ) ( ) depends only on 𝛼 ̃ and is denoted by Ob 𝛼 ̃ . The following result is a particular case of the Theorem of Ocneanu (1985). Theorem. Let ℛ be the hyperfinite type II1 factor, let G be a countable discrete amenable group. Let 𝛼 ̃, 𝛽̃ ∶ G → Out (ℛ) be two outer actions of G on ℛ. Then 𝛼 ̃ is conjugated to 𝛽̃ (i.e., there exists ̃ 𝜎 ∈ Aut (ℛ) such that 𝛼 ̃g = 𝜎 𝛽g 𝜎−1 in Out (ℛ) for all g ∈ G) if and only if ( ) ( ) Ob 𝛼 ̃ = Ob 𝛽̃ .

228

Groups of Automorphisms

( ) Moreover, every element in H3 (G, 𝕋 ) may be realized as the invariant Ob 𝛼 ̃ for some outer action 𝛼 ̃ ∶ G → Out (ℛ). Corollary. Let G be a countable discrete amenable group and let ℛ be the hyperfinite type II1 factor. Any two free actions of G on ℛ are outer conjugate. Here an action 𝛼 ∶ G → Aut (ℳ) is called free if 𝛼g ∉ Int (ℳ) for all g ∈ G, g ≠ e. And two actions 𝛼, 𝛽 ∶ G → Aut (ℳ) are called outer conjugate if there is a unitary cocycle u for 𝛼, that is, unitaries ug ∈ M, (g ∈ G), with ( ) ugh = ug 𝛼g uh , (g, h ∈ G), and 𝜃 ∈ Aut (ℳ) such that ( ) 𝛽g = 𝜃◦Ad ug ◦𝛼g ◦𝜃 −1 ,

(g ∈ G).

The Theorem of Ocneanu generalizes previous results of Connes (1975e) for G = ℤ and of Jones (1980) for finite groups. There is a more general form of the theorem which also works for the hyperfinite type II∞ factor, and a relative form of the Theorem for the McDuff factors. A converse to the Ocneanu’s Theorem was formulated and proved by Jones (1983). The theorem was extended by Kawahigashi et al. (1992) to the type III {case. In the most general } form of the theorem, one finds a classification also in the case that g ∈ G; 𝛼g ∈ Int () is nontrivial. See also Sutherland and Takesaki (1985, 1989). 17.26. Notes. Propositions 17.4–17.7 are due to Kallman (1969, 1970). The results in Section 17.8 are from Choda (1971, 1974) and Tam (1971). The main results, Theorems 17.9 and 17.17, are due to Connes (1975e). For our exposition, we have used Choda (1974), Combes and Delaroche (1973–1974), Connes (1975e), and Kallman (1969). An interesting extension of Theorem 17.17 for several commuting *-automorphisms has been obtained by Ocneanu (1981). The notion of a full group was introduced in the commutative case by Dye (1959, 1963) and in the general case by Haga and Takeda (1972) and Connes (1973a).

CHAPTER IV

Crossed Products

18 Hopf–von Neumann Algebras In this section, we consider a category of objects called Hopf–von Neumann algebras, which in a certain sense generalize locally compact groups, and also their actions on W ∗ -algebras. The main interest of these objects consists in their giving a natural framework for the duality theory of locally compact groups. 18.1. We first introduce certain notation and conventions that will be used frequently in what follows. Let ℋ , 𝒦 be Hilbert spaces and ℳ ⊂ ℬ(ℋ ), 𝒩 ⊂ ℬ(𝒦 ) von Neumann algebras. There exists a unique unitary operator ̄ 𝒦 →𝒦 ⊗ ̄ ℋ ∼∶ ℋ ⊗ such that ∼ (𝜉 ⊗ 𝜂) = 𝜂 ⊗ 𝜉 for all 𝜉 ∈ ℋ , 𝜂 ∈ 𝒦 . The mapping ̃ =∼ ◦X◦ ∼∈ 𝒩 ⊗ ̄ 𝒩 ∋X↦X ̄ ℳ ̃· ∶ ℳ ⊗ ̄ y)∼ = y ⊗ ̄ x for x ∈ ℳ, y ∈ 𝒩 . is then a *-isomorphism, uniquely determined, such that (x ⊗ ∗ The value 𝜑(x) of a linear form 𝜑 ∈ ℳ on a vector x ∈ ℳ will also be denoted by ⟨x, 𝜑⟩. ̄ ℒ 2 (G) = ℒ 2 (G, ℋ ) Let G be a locally compact group. The elements of the Hilbert space ℋ ⊗ will be identified with vector-valued functions on G. Also, the linear operators T on ℒ 2 (G, ℋ ) will usually be defined by specifying the elements (T𝜉)( g)(𝜉 ∈ ℒ 2 (G, ℋ ), g ∈ G ). The formal versions of these procedures are standard and well known. The identity mapping on ℳ will be denoted by 𝜄ℳ , or by 𝜄k , where k indicates a position in tensor products. The same convention will be used for the unit element 1ℳ . 18.2. A Hopf–von Neumann algebra is a pair (𝒜 , 𝛿𝒜 ), where 𝒜 is a W ∗ -algebra and 𝛿𝒜 ∶ 𝒜 → ̄ 𝒜 is an injective unital normal *-homomorphism, called comultiplication, which is coassocia𝒜⊗ tive, that is, ̄ 𝛿𝒜 ) ◦ 𝛿𝒜 = (𝛿𝒜 ⊗ ̄ 𝜄𝒜 ) ◦ 𝛿𝒜 . (𝜄𝒜 ⊗

229

(1)

230

Crossed Products

The Hopf–von Neumann algebra (𝒜 , 𝛿𝒜 ) is said to be commutative if the algebra 𝒜 is commutative and is called cocommutative if ̃· ◦ 𝛿𝒜 = 𝛿𝒜 .

(2)

A coinvolutive Hopf–von Neumann algebra is a triple (𝒜 , 𝛿𝒜 , j𝒜 ), where (𝒜 , 𝛿𝒜 ) is a Hopf–von Neumann algebra and j ∶ 𝒜 → 𝒜 is an involutive *-antiautomorphism (i.e., j𝒜 ◦ j𝒜 = 𝜄𝒜 ), called coinvolution, such that ̄ j𝒜 ) ◦ 𝛿𝒜 = ̃· ◦ 𝛿𝒜 ◦ j𝒜 . ( j𝒜 ⊗

(3)

An action of the Hopf–von Neumann algebra (𝒜 , 𝛿𝒜 ) on the W ∗ -algebra ℳ is an injective unital ̄ 𝒜 such that normal *-homomorphism 𝛿 ∶ ℳ → ℳ ⊗ ̄ 𝛿𝒜 ) ◦ 𝛿 = (𝛿 ⊗ ̄ 𝜄𝒜 ) ◦ 𝛿. (𝜄ℳ ⊗

(4)

In this case, we also say that ℳ is an 𝒜 -comodule via 𝛿. Note that 𝛿𝒜 is an action of (𝒜 , 𝛿𝒜 ) on 𝒜 . ̄ 𝒜 is the set The centralizer of the action 𝛿 ∶ ℳ → ℳ ⊗ ̄ 1𝒜 }. ℳ 𝛿 = {x ∈ ℳ; 𝛿(x) = x ⊗ Clearly, ℳ 𝛿 is a unital W ∗ -subalgebra of ℳ. ̄ 𝒜 , 𝛿2 ∶ 𝒜 → ℳ2 ⊗ ̄ 𝒜, Let ℳ1 , ℳ2 be 𝒜 -comodules via the actions 𝛿1 ∶ 𝒜 → ℳ1 ⊗ respectively. We shall say that a normal completely positive linear mapping 𝜎 ∶ ℳ1 → ℳ2 intertwines the actions 𝛿1 , 𝛿2 or that 𝜎 is an 𝒜 -comodule mapping if ̄ 𝜄 𝒜 ) ◦ 𝛿1 . 𝛿2 ◦ 𝜎 = (𝜎 ⊗

(5)

In particular, if there exists a *-isomorphism 𝜎 ∶ ℳ1 → ℳ2 intertwining 𝛿1 , 𝛿2 then we say that the actions 𝛿1 , 𝛿2 are isomorphic. ̄ 𝒜 is an action of 𝒜 on ℳ and 𝒩 ⊂ ℳ is a unital W ∗ -subalgebra such that If 𝛿 ∶ ℳ → ℳ ⊗ ̄ ̄ 𝒜 is an action of 𝒜 on 𝒩 and the canonical injection 𝛿(𝒩 ) ⊂ 𝒩 ⊗ 𝒜 , then 𝛿|𝒩 ∶ 𝒩 → 𝒩 ⊗ 𝒩 ↪ ℳ is an 𝒜 -comodule mapping. In this case, we shall say that 𝒩 is an 𝒜 -subcomodule of ℳ. ̄ 𝒜1 , 𝛿2 ∶ ℳ → ℳ ⊗ ̄ 𝒜2 actions Let 𝒜1 , 𝒜2 be Hopf–von Neumann algebras and 𝛿1 ∶ ℳ → ℳ ⊗ ∗ of 𝒜1 , 𝒜2 on the same W -algebra ℳ. We shall say that 𝛿1 and 𝛿2 commute if ̄ 𝜄𝒜 ) ◦ 𝛿2 = (𝜄ℳ ⊗̃ ̄ · ) ◦ (𝛿2 ⊗ ̄ 𝜄𝒜 ) ◦ 𝛿1 . (𝛿1 ⊗ 2 1

(6)

̄ 𝒜 is an action of the Hopf–von Neumann algebra Finally, we remark that if 𝛿 ∶ ℳ → ℳ ⊗ (𝒜 , 𝛿𝒜 ) on the W ∗ -algebra ℳ and 𝒩 is an arbitrary W ∗ -algebra, then ̄ 𝛿∶𝒩 ⊗ ̄ ℳ→𝒩 ⊗ ̄ ℳ⊗ ̄ 𝒜 𝜄𝒩 ⊗ ̄ ℳ and we have is an action of (𝒜 , 𝛿𝒜 ) on 𝒩 ⊗ ̄ ℳ)𝜄𝒩 (𝒩 ⊗

̄ 𝛿 ⊗

̄ ℳ𝛿 . =𝒩 ⊗

(7)

Hopf–von Neumann Algebras

231

̄ ℳ be such that (𝜄𝒩 ⊗ ̄ 𝛿)(x) = x ⊗ ̄ 1𝒜 . Indeed, the inclusion “⊃” is obvious. Conversely, let x ∈ 𝒩 ⊗ For 𝜓 ∈ 𝒩∗ , 𝜑 ∈ ℳ∗ , k ∈ 𝒜∗ , we have 𝜓 ̄ k⟩ = ⟨E 𝜓 (x), (𝜑 ⊗ ̄ k) ◦ 𝛿⟩ = ⟨x, 𝜓 ⊗ ̄ ((𝜑 ⊗ ̄ k) ◦ 𝛿)⟩ ⟨𝛿(Eℳ (x)), 𝜑 ⊗ ℳ ̄ 𝜑⊗ ̄ k) ◦ (𝜄𝒩 ⊗ ̄ 𝛿)⟩ = ⟨x ⊗ ̄ 1𝒜 , 𝜓 ⊗ ̄ 𝜑⊗ ̄ k⟩ = ⟨x, (𝜓 ⊗ 𝜓 ̄ 𝜑⟩⟨1𝒜 , k⟩ = ⟨E (x) ⊗ ̄ 1𝒜 , 𝜑 ⊗ ̄ k⟩, = ⟨x, 𝜓 ⊗ ℳ 𝜓 ̄ ℳ𝛿 . hence Eℳ (x) ∈ ℳ 𝛿 . By Proposition 9.8 it follows that x ∈ 𝒩 ⊗

18.3. Let (𝒜 , 𝛿𝒜 ) be a Hopf–von Neumann algebra. It is easy to check that the predual 𝒜∗ with its Banach space structure and multiplication 𝒜∗ × 𝒜∗ ∋ (h, k) ↦ h ⋅ k ∈ 𝒜∗ defined by ̄ k⟩ ⟨a, h ⋅ k⟩ = ⟨𝛿𝒜 (a), h ⊗

(a ∈ 𝒜 )

(1)

is a Banach algebra. Note that (h ⋅ k)∗ = h∗ ⋅ k∗ (h, k ∈ 𝒜∗ ).

(2)

The Banach algebra 𝒜∗ is commutative if and only if the Hopf–von Neumann algebra (𝒜 , 𝛿𝒜 ) is cocommutative. If (𝒜 , 𝛿𝒜 , j𝒜 ) is a coinvolutive Hopf–von Neumann algebra, then the Banach algebra 𝒜∗ with the involution 𝒜∗ ∋ k ↦ k0 = k∗ ◦ j𝒜 ∈ 𝒜∗ , that is, ⟨a, k0 ⟩ = ⟨j𝒜 (a∗ ), k⟩

(a ∈ 𝒜 ),

(3)

is an involutive Banach algebra. Note that (k0 )∗ = (k∗ )0

(k ∈ 𝒜 ).

(4)

̄ 𝒜 be an action of the Hopf–von Neumann algebra (𝒜 , 𝛿𝒜 ) on the W ∗ -algebra Let 𝛿 ∶ ℳ → ℳ ⊗ k ̄ 𝒜 → ℳ and define ℳ. For every k ∈ 𝒜∗ , we consider the Fubini mapping (9.8) Eℳ ∶ℳ ⊗ k k ⋅ x = Eℳ (𝛿(x)) (x ∈ ℳ),

that is, ̄ k⟩ ⟨k ⋅ x, 𝜑⟩ = ⟨𝛿(x), 𝜑 ⊗

(x ∈ ℳ, k ∈ 𝒜∗ , 𝜑 ∈ ℳ∗ ).

(5)

It is easy to check that the mapping (k, x) ↦ k ⋅ x is bilinear; the mappings x ↦ k ⋅ x are w-continuous (k ∈ 𝒜∗ ); ‖k ⋅ x‖ ≤ ‖k‖‖x‖ (x ∈ ℳ, k ∈ 𝒜∗ ); (k ⋅ x)∗ = k∗ ⋅ x∗ (x ∈ ℳ, k ∈ 𝒜∗ ); x ∈ ℳ + , k ∈ 𝒜∗+ ⇒ k ⋅ x ∈ ℳ + ; h ⋅ (k ⋅ x) = (h ⋅ k) ⋅ x (x ∈ ℳ, h, k ∈ 𝒜∗ );

(6) (7) (8) (9) (10) (11)

232

Crossed Products

In particular, the mapping 𝒜∗ × ℳ ∋ (k, x) ↦ k ⋅ x determines the structure of a left Banach 𝒜∗ -module on ℳ. On the other hand, the mapping ℳ∗ × 𝒜∗ ∋ (𝜑, k) ↦ 𝜑 ⋅ k ∈ ℳ∗ , defined by ̄ k⟩ = ⟨k ⋅ x, 𝜑⟩ ⟨x, 𝜑 ⋅ k⟩ = ⟨𝛿(x), 𝜑 ⊗

(x ∈ ℳ),

(12)

determines the structure of a right Banach 𝒜∗ -module on ℳ∗ ; definition (1) is just a particular case of this definition. Note that ‖𝜑 ⋅ k‖ ≤ ‖𝜑‖‖k‖ (𝜑 ∈ ℳ∗ , k ∈ 𝒜∗ ), (𝜑 ⋅ k)∗ = 𝜑∗ ⋅ k∗ (𝜑 ∈ ℳ∗ , k ∈ 𝒜∗ ).

(13) (14)

Since 𝛿 is injective and ℳ = (ℳ∗ )∗ , it follows using the Hahn–Banach theorem that the set {𝜑 ⋅ k; 𝜑 ∈ ℳ∗ , k ∈ 𝒜∗ } is total in ℳ∗ .

(15)

If 𝒩 and ℳ are two 𝒜 -comodules and 𝜎 ∶ 𝒩 → ℳ is an 𝒜 -comodule mapping, then 𝜎(k ⋅ y) = k ⋅ 𝜎( y)

( y ∈ 𝒩 , k ∈ 𝒜∗ ).

(16)

In particular, if 𝒩 is an 𝒜 -subcomodule of ℳ, then y ∈ 𝒩 ⊂ ℳ, k ∈ 𝒜∗ ⇒ k ⋅ y ∈ 𝒩 .

(17)

18.4. Let G be a locally compact group with neutral element e ∈ G, left Haar measure dg = d𝓁 g and modular function Δ = ΔG . Recall that Δ ∶ G → ℝ+ ∖{0} is a continuous group homomorphism and we have for k ∈ ℒ 1 (G), t ∈ G ∫ ∫ ∫

k(tg) dg = k( g) dg, k( gt) dg = Δ(t)−1 k( g−1 ) dg =

∫

∫

(1) k( g) dg,

k( g)Δ( g)−1 dg.

(2) (3)

In particular, dr g = Δ( g)−1 dg is the right Haar measure. Let 𝒦 (G) be the set of all continuous functions on G of compact support. The set 𝒦 (G) endowed with the scalar product from ℒ 2 (G) and with the operations of multiplication and complex conjugation : (𝜉𝜂)(s) = 𝜉(s)𝜂(s)

̄ = 𝜉(s) 𝜉(s)

(4)

is a commutative Hilbert algebra. It is easy to check that the associated maximal Hilbert algebra is ℒ ∞ (G) ∩ ℒ 2 (G) ⊂ ℒ 2 (G) with the same operations. The associated modular operator ∇G and canonical conjugation KG are given by ∇G 𝜉 = 𝜉

KG 𝜉 = 𝜉.

(5)

Hopf–von Neumann Algebras

233

The associated left and right von Neumann algebras both coincide with the von Neumann algebra ℒ ∞ (G) acting by multiplication on ℒ 2 (G): ( f 𝜉)(s) = f (s)𝜉(s).

(6)

By the commutation theorem ([L], 10.4.(2)), it follows that ℒ ∞ (G) is maximal commutative in ℬ(ℒ 2 (G)): ℒ ∞ (G)′ = ℒ ∞ (G).

(7)

The natural weight on ℒ ∞ (G) associated with this Hilbert algebra ([L], 10.16) is denoted by 𝜇G and is called the Haar weight on ℒ ∞ (G). We have 𝜇G ( f ) =

∫

f ( g) dg

( f ∈ ℒ ∞ (G)+ ).

(8)

On the other hand, the set 𝒦 (G) endowed with the scalar product from ℒ 2 (G) and with the operations of convolution and involution (𝜉 ∗ 𝜂)(s) =

∫

𝜉( g)𝜂( g−1 s) dg,

𝜉 ∗ (s) = Δ(s)−1 𝜉(s−1 )

(9)

is a left Hilbert algebra. The associated modular operator ΔG and canonical conjugation JG are given by (ΔG 𝜉)(s) = Δ(s)𝜉(s) (JG 𝜉)(s) = Δ(s)−1∕2 𝜉(s−1 ).

(10)

The associated left von Neumann algebra, denoted by 𝔏(G), is generated by the left regular representation 𝜆 ∶ G ∋ g ↦ 𝜆( g) ∈ ℬ(ℒ 2 (G)), 𝜆( g)𝜉)(s) = 𝜉( g−1 s) (𝜆

(11)

while the associated right von Neumann algebra, denoted by ℜ(G), is generated by the right regular representation 𝜌 ∶ G ∋ g ↦ 𝜌( g) ∈ ℬ(ℒ 2 (G)), (𝜌𝜌( g)𝜉)(s) = Δ( g)1∕2 𝜉(sg).

(12)

Thus, 𝜆( g); g ∈ G} 𝔏(G) = ℛ{𝜆

ℜ(G) = ℛ{𝜌𝜌( g); g ∈ G}

(13)

and, by the commutation theorem ([L], 10.4.(2)), we have 𝔏(G)′ = ℜ(G).

(14)

The natural weight on 𝔏(G) associated with this left Hilbert algebra ([L], 10.16) is denoted by 𝜔G and is called the Plancherel weight on 𝔏(G). The properties of the Plancherel weight will be studied later (18.17).

234

Crossed Products

Using the commutation relations (7) and (14) as well as the fact that the only translation invariant functions on G are the constant functions, we obtain ℬ(ℒ 2 (G)) = ℛ{ℒ ∞ (G), 𝔏(G)} = ℛ{ℒ ∞ (G), ℜ(G)}.

(15)

̄ 2 (G) by We define a unitary operator WG on ℒ 2 (G)⊗ℒ (WG 𝜁)(s, t) = 𝜁(s, st)

(16)

̃ ∗ = ∼ ◦ W ∗ ◦ ∼, (𝜁 ∈ ℒ 2 (G × G ); s, t ∈ G ). Consider also the unitary operator VG = W G G (VG 𝜁 )(s, t) = 𝜁(t−1 s, t).

(17)

Using the commutation relations, we get ̄ 𝔏(G), WG ∈ ℒ ∞ (G) ⊗

̄ ℒ ∞ (G). VG ∈ 𝔏(G) ⊗

(18)

18.5. We now consider the first example of a coinvolutive Hopf–von Neumann algebra associated with a locally compact group G, namely the triple G = (ℒ ∞ (G), 𝜋G , kG ) consisting of the von Neumann algebra ℒ ∞ (G) ⊂ ℬ(ℒ 2 (G)) with comultiplication 𝜋G ∶ ℒ ∞ (G) → ̄ ℒ ∞ (G) defined by ℒ ∞ (G) ⊗ ̄ 1G )VG . 𝜋G ( f ) = VG∗ ( f ⊗

(1)

that is, 𝜋G ( f ) is the multiplication operator on ℒ 2 (G × G ) given by the function (𝜋G ( f ))(s, t) = f (ts),

(2)

and coinvolution kG ∶ ℒ ∞ (G) → ℒ ∞ (G) defined by kG ( f ) = JG ̄fJG ,

(3)

that is, kG ( f ) is the multiplication operator on ℒ 2 (G) given by the function (kG ( f ))(s) = f (s−1 ).

(4)

It is easy to check that requirements 18.2.(1) and 18.2.(3) are satisfied. The predual ℒ ∞ (G)∗ of the von Neumann algebra ℒ ∞ (G) ⊂ ℬ(ℒ 2 (G)) is identified with the Banach space ℒ 1 (G) in the usual way: ⟨f, k⟩ =

∫

f ( g)k( g) dg ( f ∈ ℒ ∞ (G), k ∈ ℒ 1 (G)).

(5)

Hopf–von Neumann Algebras

235

Indeed, since ℒ ∞ (G) ⊂ ℬ(ℒ 2 (G)) is in standard form, every element k ∈ ℒ ∞ (G)∗ is of the form 𝜔𝜉,𝜂 with 𝜉, 𝜂 ∈ ℒ 2 (G), hence k will correspond to the function 𝜉 𝜂̄ ∈ ℒ 1 (G). By definitions 18.3.(1) and 18.3.(3), ℒ 1 (G) becomes an involutive Banach algebra. It is easy to check that for h, k ∈ ℒ 1 (G) we have h ⋅ k = k ∗ h,

(6)

k =k ,

(7)

0

∗

the convolution “*” and the involution “♯” being defined by 18.4.(9). The definition of 𝜋G can be extended to an action of G on ℬ(ℒ 2 (G)), still denoted by ̄ ℒ ∞ (G), 𝜋G ∶ ℬ(ℒ 2 (G)) → ℬ(ℒ 2 (G)) ⊗ namely ̄ 1G )VG 𝜋G (x) = VG∗ (x ⊗

(x ∈ ℬ(ℒ 2 (G))).

(8)

Note that ℬ(ℒ 2 (G))𝜋G = ℜ(G).

(9)

̄ 1G , then VG (x ⊗ ̄ 1G ) = (x ⊗ ̄ 1G )VG . By applying this to Indeed, if x ∈ ℬ(ℒ 2 (G)) and 𝜋G (x) = x ⊗ a vector of the form 𝜉 ⊗ 𝜂 with 𝜉, 𝜂 ∈ 𝒦 (G) and then taking the scalar product with another vector 𝜆(t) = 𝜆 (t)x(t ∈ G ). Hence x ∈ 𝔏(G)′ = ℜ(G). of the same form, it follows that x𝜆 ∗ More generally, for any W -algebra ℳ, the mapping ̄ 𝜋G ∶ ℳ ⊗ ̄ ℬ(ℒ 2 (G)) → ℳ ⊗ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ ℒ ∞ (G) 𝜄ℳ ⊗ ̄ ℬ(ℒ 2 (G)), determines an action of G on ℳ ⊗ ̄ 𝜋G )(X) = (1ℳ ⊗ ̄ V ∗ )(X ⊗ ̄ 1G )(1ℳ ⊗ ̄ VG ) (𝜄ℳ ⊗ G 2 ̄ ℬ(ℒ (G))) (X ∈ ℳ ⊗

(10)

and, by (9) and 18.2.(7), we have ̄ ℬ(ℒ 2 (G))𝜄ℳ ℳ⊗

̄ 𝜋G ⊗

̄ ℛ(G). =ℳ⊗

(11)

18.6. In this section, we show that the actions of the Hopf–von Neumann algebra G on the W ∗ -algebra ℳ actually correspond to the continuous actions of the locally compact group G on ℳ. Consider first a continuous action 𝜎 ∶ G ∋ g ↦ 𝜎g ∈ Aut(ℳ) of G on ℳ. For each x ∈ ℳ, the function G ∈ g ↦ 𝜎g−1 (x) ∈ ℳ is w-continuous and bounded by ‖x‖, hence (Lemma 1/13.1) it ̄ ℒ ∞ (G) such that defines a unique element 𝜋𝜎 (x) ∈ ℳ ⊗ ̄ k⟩ = ⟨𝜋𝜎 (x), 𝜑 ⊗

∫

𝜑(𝜎g−1 (x))k( g) dg

(1)

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Crossed Products

̄ ℒ ∞ (G) ⊂ ℬ(ℒ 2 (G, ℋ )) are realized as for 𝜑 ∈ ℳ∗ and k ∈ ℒ 1 (G). If ℳ ⊂ ℬ(ℋ ) and ℳ ⊗ von Neumann algebras, then (𝜋𝜎 (x)𝜉)( g) = 𝜎g−1 (x)𝜉( g)

(𝜉 ∈ ℒ 2 (G, ℋ ), g ∈ G ).

(2)

Using (1) or (2), it is easy to check that ̄ ℒ ∞ (G) 𝜋𝜎 ∶ ℳ → ℳ ⊗ is an injective unital normal *-homomorphism. We show that 𝜋𝜎 is an action of G on ℳ, that is, ̄ 𝜋G ) ◦ 𝜋𝜎 = (𝜋𝜎 ⊗ ̄ 𝜄 G ) ◦ 𝜋𝜎 . (𝜄ℳ ⊗

(3)

Indeed, for x ∈ ℳ, 𝜑 ∈ ℳ∗ , and h, k ∈ ℒ 1 (G) we have ̄ 𝜋G )(𝜋𝜎 (x)), 𝜑 ⊗ ̄ h⊗ ̄ k⟩ = ⟨𝜋𝜎 (x), 𝜑 ⊗ ̄ ((h ⊗ ̄ k) ◦ 𝜋G ) ⟨(𝜄ℳ ⊗ ̄ ̄ = ⟨𝜋𝜎 (x), 𝜑 ⊗ (h ⋅ k)⟩ = ⟨𝜋𝜎 (x), 𝜑 ⊗ (k ∗ h)⟩ =

∫

𝜑(𝜎g−1 (x))(k ∗ h)( g) dg =

∫ ∫

𝜑(𝜎g−1 (x))k(t)h(t−1 g) dt dg

and ̄ 𝜄G )(𝜋𝜎 (x)), 𝜑 ⊗ ̄ h⊗ ̄ k⟩ = ⟨𝜋𝜎 (x), ((𝜑 ⊗ ̄ h) ◦ 𝜋𝜎 ) ⊗ ̄ k⟩ ((𝜋𝜎 ⊗ ̄ h)k(t) dt = ⟨𝜋𝜎 (𝜎t−1 (x)), 𝜑 ⊗

∫ ∫

𝜑(𝜎s−1 (𝜎t−1 (x)))h(s)k(t) ds dt

𝜑(𝜎ts−1 (x))h(s)k(t) ds dt =

∫ ∫

𝜑(𝜎g−1 (x))k(t)h(t−1 g) dg dt,

=

∫

=

∫ ∫

and Fubini’s theorem insures that the two integrals are equal. We now show that ℳ 𝜋𝜎 = ℳ 𝜎 ,

(4)

that is, for x ∈ ℳ we have ̄ 1G ⇔ 𝜎g (x) = x for all g ∈ G. 𝜋𝜎 (x) = x ⊗ ̄ k⟩ = Indeed, if 𝜎g (x) = x for all g ∈ G, then for 𝜑 ∈ ℳ∗ and k ∈ ℒ 1 (G) we have ⟨𝜋𝜎 (x), 𝜑 ⊗ ̄ 1G , 𝜑 ⊗ ̄ k⟩, hence 𝜋𝜎 (x) = x ⊗ ̄ 1G . Conversely, if 𝜋𝜎 (x) = ∫ 𝜑(x)k( g) dg = ⟨x, 𝜑⟩⟨1G , k⟩ = ⟨x ⊗ ̄ 1G , then for 𝜑 ∈ ℳ∗ and k ∈ ℒ 1 (G) we have x⊗ ∫

𝜑(𝜎g−1 (x))k( g) dg =

∫

𝜑(x)k( g) dg.

It follows that the continuous function G ∋ g ↦ 𝜑(𝜎g−1 (x)) coincides almost everywhere with the constant function 𝜑(x). Consequently, 𝜑(𝜎g−1 (x)) = 𝜑(x) for all g ∈ G and 𝜑 ∈ ℳ∗ , that is, x ∈ ℳ 𝜎 .

Hopf–von Neumann Algebras

237

Note that ̄ 𝜆 ( g))𝜋𝜎 (x)(1ℳ ⊗ ̄ 𝜆 ( g))∗ 𝜋𝜎 (𝜎g (x)) = (1ℳ ⊗

(x ∈ ℳ, g ∈ G ).

(5)

𝜆( g)) ∈ ℒ ∞ (G)∗ , considered as an Indeed, for k ∈ ℒ 1 (G) = ℒ ∞ (G)∗ , the element k ◦ Ad(𝜆 1 ℒ -function on G, can be written 𝜆( g)))(t) = k( gt) (k ◦ Ad(𝜆

(t ∈ G ).

Consequently, for 𝜑 ∈ ℳ∗ , k ∈ ℒ 1 (G), we have ̄ 𝜆 ( g))𝜋𝜎 (x)(1 ⊗ ̄ 𝜆 ( g))∗ , 𝜑 ⊗ ̄ k⟩ = ⟨𝜋𝜎 (x), 𝜑 ⊗ ̄ (k ◦ Ad(𝜆 𝜆( g))) ⟨(1 ⊗ =

∫

𝜑(𝜎t−1 (x))k( gt) dt =

=

∫

̄ k⟩. 𝜑(𝜎t−1 (𝜎g (x)))k(t) dt = ⟨𝜋𝜎 (𝜎g (x)), 𝜑 ⊗

∫

𝜑(𝜎t−1 g (x))k(t) dt

Recall that for every k ∈ ℒ 1 (G) and every x ∈ ℳ, we have defined an element (13.2) 𝜎k (x) =

∫

𝜎g (x)k( g) dg

and also an element (18.3.(5)) k k ⋅ x = Eℳ (𝜋𝜎 (x)).

We have k ⋅ x = 𝜎k♯ (x) (x ∈ ℳ, k ∈ ℒ 1 (G)).

(6)

Indeed, for every 𝜑 ∈ ℳ∗ we obtain ̄ k⟩ = ⟨k ⋅ x, 𝜑⟩ = ⟨𝜋𝜎 (x), 𝜑 ⊗ 𝜑(𝜎g−1 (x))k( g) dg ∫ ⟨ ⟩ ⟨ ⟩ −1 −1 = 𝜎 −1 (x)k( g) dg, 𝜑 = 𝜎 (x)k( g )Δ( g ) dg, 𝜑 ∫ g ∫ g ⟨ ⟩ ♯ = 𝜎 (x)k ( g) dg, 𝜑⟩ = ⟨𝜎k♯ (x), 𝜑 . ∫ g Note also that, writing kt ( g) = k( gt), we have 𝜎t (k ⋅ x) = Δ(t)(kt ⋅ x) the verification being similar to the above computation.

(t ∈ G ),

(7)

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Crossed Products

Using 15.1.(2), we infer from (6) that for any norm-bounded approximate unit {ki }i∈I of ℒ 1 (G) and any x ∈ ℳ, we have w

ki ⋅ x → x.

(8)

Moreover, using (7), from (8) we also obtain w

Δ(t)(kti ⋅ x) → 𝜎t (x)

(t ∈ G ).

(9)

Conversely, we have the following: ̄ ℒ ∞ (G) be an action of G on ℳ. There exists a continuous action Proposition. Let 𝜋 ∶ ℳ → ℳ ⊗ 𝜎 ∶ G → Aut(ℳ) of G on ℳ, uniquely determined such that 𝜋 = 𝜋𝜎 . Proof. (I) Consider first the action ̄ ℒ ∞ (G) 𝜋 = 𝜋G ∶ ℬ(ℒ 2 (G)) → ℬ(ℒ 2 (G)) ⊗ of G on ℬ(ℒ 2 (G)). There exists a continuous action 𝜆) ∶ G → Aut(ℬ(ℒ 2 (G))) 𝜎 = Ad(𝜆 of G on ℬ(ℒ 2 (G)), defined by the left regular representation. Since ℬ(ℒ 2 (G)) = ℛ{ℒ ∞ (G), ℜ(G)}, in order to establish that 𝜋 = 𝜋𝜎 it is sufficient to show that 𝜋(x) = 𝜋𝜎 (x) separately for ̄ 1 = 𝜋𝜎 (x). Also, for x ∈ ℜ(G) and for x ∈ ℒ ∞ (G). If x ∈ ℜ(G), then it is clear that 𝜋(x) = x ⊗ ∞ x = f ∈ ℒ (G) it is easy to check that 𝜋𝜎 (x) = 𝜋(x) is the multiplication operator on ℒ 2 (G × G ) given by the function (s, t) ↦ f (ts). Hence 𝜋 = 𝜋𝜎 . (II) Consider next the action ̄ 𝜋G ∶ ℳ ⊗ ̄ ℬ(ℒ 2 (G)) → ℳ ⊗ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ ℒ ∞ (G) 𝜋 = 𝜄ℳ ⊗ ̄ ℬ(ℒ 2 (G)) and the continuous action of G on ℳ ⊗ ̄ Ad(𝜆 ̄ ℬ(ℒ 2 (G))) 𝜆) ∶ G → Aut(ℳ ⊗ 𝜎 = 𝜄ℳ ⊗ ̄ ℬ(ℒ 2 (G)), that is, of G on ℳ ⊗ ̄ 𝜆 ( g))X(1ℳ ⊗ ̄ 𝜆 ( g))∗ 𝜎g (X) = (1ℳ ⊗

̄ ℬ(ℒ 2 (G))). ( g ∈ G, X ∈ ℳ ⊗

Using the result proved in step (I), it is easy to see that in the present situation also, we have 𝜋 = 𝜋𝜎 . (III) It is clear that if the action 𝜋 of G on ℳ is isomorphic to an action of the type 𝜋𝜎 , then 𝜋 is also of the form 𝜋a . (IV) Now let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ and let 𝒩 ⊂ ℳ be a unital ̄ ℒ ∞ (G). Then 𝜋 = 𝜋𝜎 |𝒩 is an action of G on 𝒩 and 𝒩 W ∗ -subalgebra such that 𝜋𝜎 (𝒩 ) ⊂ 𝒩 ⊗ with the G-comodule structure defined by 𝜋 is a G-subcomodule of the G-comodule ℳ defined by

Hopf–von Neumann Algebras

239

𝜋𝜎 . Consequently, for y ∈ 𝒩 ⊂ ℳ we have (see 18.3.(17)) k⋅y∈𝒩

(k ∈ ℒ 1 (G))

and if {ki }i∈I is a norm-bounded approximate unit of ℒ 1 (G), then, according to (9), w

𝒩 ∋ Δ(t)(kti ⋅ y) → 𝜎t ( y)

(t ∈ G ).

It follows that 𝜎t (𝒩 ) = 𝒩 (t ∈ G ), and 𝜋 = 𝜋𝜎 |𝒩 = 𝜋𝜎|𝒩 . ̄ ℒ ∞ (G) be any action of G on ℳ. By the coassociativity condition (V) Finally, let 𝜋 ∶ ℳ → 𝒩 ⊗ ̄ ̄ (𝜄ℳ ⊗ 𝜋G ) ◦ 𝜋 = (𝜋 ⊗ 𝜄G ) ◦ 𝜋 it follows that 𝜋(ℳ) is a G-subcomodule of the G-comodule ̄ ℬ(ℒ 2 (G)), which is isomorphic to the G-comodule ℳ. Thus, the existence assertion of the ℳ⊗ statement follows from (II), (III), and (IV). (V) The uniqueness assertion follows easily using (9). 18.7. We now consider the second example of a coinvolutive Hopf–von Neumann algebra associated with a locally compact group G, namely the triple ̂ = (𝔏(G), 𝛿G , jG ) G consisting of the von Neumann algebra 𝔏(G) ⊂ ℬ(ℒ 2 (G)) with comultiptication 𝛿G ∶ 𝔏(G) → ̄ 𝔏(G) defined by 𝔏(G) ⊗ ̄ 1G )WG , 𝛿G (x) = WG∗ (x ⊗

(1)

̄ 𝜆 ( g), 𝜆( g)) = 𝜆 ( g) ⊗ 𝛿G (𝜆

(2)

so that, in particular,

and coinvolution jG ∶ 𝔏(G) → 𝔏(G) defined by jG (x) = KG x∗ KG ,

(3)

𝜆( g)) = 𝜆 ( g−1 ). jG (𝜆

(4)

so that, in particular,

It is easy to check that 18.2.(1) and 18.2.(3) are satisfied. Let 𝒜 (G) = 𝔏(G)∗ be the predual of 𝔏(G) endowed with the corresponding involutive Banach algebra structure (18.3.(1) and 18.3.(3)). For every k ∈ 𝒜 (G), we consider the function k(⋅) defined by 𝜆( g), k⟩ k( g) = ⟨𝜆

( g ∈ G ).

(5)

240

Crossed Products

Then k(⋅) is a continuous function in ℒ ∞ (G) and ‖k(⋅)‖∞ ≤ ‖k‖𝒜 (G) . Actually, the mapping 𝒜 (G) ∋ k ↦ k(⋅) ∈ ℒ ∞ (G) is an injective *-homomorphism. Indeed, this mapping is clearly linear and injective and, for h, k ∈ 𝒜 (G) and g ∈ G, we have ̄ k⟩ 𝜆( g), h ⋅ k⟩ = ⟨𝛿G (𝜆 𝜆( g)), h ⊗ (h ⋅ k)( g) = ⟨𝜆 ̄ 𝜆 ( g), h ⊗ ̄ k⟩ = h( g)k( g), 𝜆( g) ⊗ = ⟨𝜆 𝜆( g), k0 ⟩ = ⟨jG (𝜆 𝜆( g)), k∗ ⟩ = ⟨𝜆 𝜆( g−1 )∗ , k⟩ k0 ( g) = ⟨𝜆

(6) (7)

𝜆( g), k⟩ = k( g). = ⟨𝜆 Since 𝔏(G) ⊂ ℬ(ℒ 2 (G)) is in standard form, every k ∈ 𝒜 (G) is of form 𝜔𝜉,𝜂 with 𝜉, 𝜂 ∈ ℒ 2 (G). ̌ where 𝜉( ̌ g) = 𝜉( g−1 ) ( g ∈ G ). In this case, it is easy to check that k(⋅) = 𝜂 ∗ 𝜉, The involutive Banach algebra 𝒜 (G) is called the Eymard algebra. ̂ on ℬ(ℒ 2 (G)) The definition of 𝛿G can be extended with the aid of equality (1) to an action of G still denoted by ̄ 𝔏(G). 𝛿G ∶ ℬ(ℒ 2 (G)) → ℬ(ℒ 2 (G)) ⊗

(8)

ℬ(ℒ 2 (G))𝛿G = ℒ ∞ (G),

(9)

Note that

the proof being similar to the proof of 18.5.(9). More generally, for any W ∗ -algebra ℳ, the mapping ̄ 𝛿G ∶ ℳ ⊗ ̄ ℬ(ℒ 2 (G)) → ℳ ⊗ ̄ ℬ(ℒ 2 (G)) ⊗𝔏(G) ̄ 𝜄ℳ ⊗ ̂ on ℳ ⊗ ̄ ℬ(ℒ 2 (G)), determines an action of G ̄ 𝛿G )(X) = (1ℳ ⊗ ̄ W ∗ )(X ⊗ ̄ 1G )(1ℳ ⊗ ̄ WG ) (𝜄ℳ ⊗ G 2 ̄ ℬ(ℒ (G))), (X ∈ ℳ ⊗

(10)

and, by (9) and 18.2.(7), we have ̄ ℬ(ℒ 2 (G)))𝜄ℳ (ℳ ⊗

̄ 𝛿G ⊗

̄ ℒ ∞ (G). =ℳ⊗

(11)

̂ on ℬ(ℒ 2 (G)), Also, there exists another action of G ̄ 𝔏(G). 𝛿G∗ ∶ ℬ(ℒ 2 (G)) → ℬ(ℒ 2 (G)) ⊗ defined by 𝛿G∗ (x) = WG (x ⊗ 1G )WG∗

(x ∈ ℬ(ℒ 2 (G))),

(12)

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241

with centralizer ℬ(ℒ 2 (G)))𝛿G = ℒ ∞ (G) ∗

(13)

̂ on ℳ ⊗ ̄ 𝛿 ∗ of G ̄ ℬ(ℒ 2 (G))) with centralizer and, correspondingly, an action 𝜄ℳ ⊗ G ̄ ℬ(ℒ 2 (G)))𝜄ℳ (ℳ ⊗

̄ 𝛿∗ ⊗ G

̄ ℒ ∞ (G). =ℳ⊗

(14)

18.8. Assume now that the locally compact group G is abelian and consider the dual locally compact ̂ of all continuous characters on G, endowed with the compact-open topology. We abelian group G ̂ such that the inversion theorem for the Fourier choose the Haar measures dg on G and d𝛾 on G transform holds. By the Pontryagin duality theorem, G can be identified with the dual group G∧∧ of G. For g ∈ ̂ we shall denote by ⟨g, 𝛾⟩ the value of 𝛾 at g (or the value of g at 𝛾). G = G∧∧ and 𝛾 ∈ G, By the Plancherel theorem, the restriction of the Fourier transform to ℒ 1 (G) ∩ ℒ 2 (G) can be extended to a unitary operator ̂ F ∶ ℒ 2 (G) → ℒ 2 (G) called the Fourier–Plancherel isomorphism, whose inverse is obtained similarly from the inverse Fourier transform. Recall that if we denote by ℬ(G) the linear span of continuous positive definite functions on G, then 𝒟 (G) = 𝒦 (G) ∩ ℬ(G) is a dense translation invariant linear subspace of ℒ 2 (G) and for every 𝜉 ∈ 𝒟 (G) we have ̂ = (F𝜉)(𝛾) = 𝜉(𝛾)

∫G

𝜉( g)⟨g, 𝛾⟩ dg

̂ (𝛾 ∈ G).

(1)

̂ invariant under multiplication by characters, Also, F(𝒟 (G)) is a dense linear subspace of ℒ 2 (G), and for every 𝜂 ∈ F(𝒟 (G)) we have ̂ g) = (F−1 𝜂)( g) = 𝜂(

∫Ĝ

𝜂(𝛾)⟨g, 𝛾⟩ d𝛾

( g ∈ G ).

(2)

̂ are ℒ ∞ -functions. For the sake of clarity, the unitary operator The characters g ∈ G = G∧∧ on G ̂ on ℒ 2 (G) ̂ will be denoted by m( g): given by the multiplication with the function g ∈ ℒ ∞ (G) ̂ 𝛾 ∈ G). ̂ (m( g)𝜂)(𝛾) = ⟨g, 𝛾)𝜂(𝛾) (𝜂 ∈ ℒ 2 (G),

(3)

̂ Since the trigonometric polynomials (i.e., the linear combinations of characters) are dense in 𝒦 (G) with respect to the topology of uniform convergence on compact subsets, it follows that the von ̂ ⊂ ℬ(ℒ 2 (G)) ̂ is generated by the operators m( g)( g ∈ G ). Neumann algebra ℒ ∞ (G) We recall that the von Neumann algebra 𝔏(G) ⊂ ℬ(ℒ 2 (G)) is generated by the left translation operators 𝜆 ( g), ( g ∈ G ). Proposition. The Fourier–Plancherel isomorphism establishes an isomorphism between the coin̂ 𝜋Ĝ , 𝜋Ĝ ). volutive Hopf–von Neumann algebras (𝔏(G), 𝛿G , jG ) and (ℒ ∞ (G),

242

Crossed Products

̂ by Proof. We define a *-isomorphism Φ ∶ ℬ(ℒ 2 (G)) → ℬ(ℒ 2 (G)) Φ(x) = F ◦ x ◦ F−1

(x ∈ ℬ(ℒ 2 (G)))

(4)

̂ (Φ ⊗ ̄ Φ) ◦ 𝛿G = 𝜋Ĝ ◦ Φ, Φ ◦ jG = kĜ ◦ 𝜙. and show that Φ(𝔏(G)) = ℒ ∞ (G), ̂ we have For g ∈ G, 𝜉 ∈ 𝒟 (G), 𝛾 ∈ G (m( g)F𝜉)(𝛾) = ⟨g, 𝛾⟩(F𝜉)(𝛾) = ⟨g, 𝛾⟩ =

∫

=

∫

𝜉(t)⟨gt, 𝛾⟩ dt =

∫

∫

𝜉(t)⟨t, 𝛾⟩ dt

𝜉( g−1 s)⟨s, 𝛾⟩ ds

𝜆( g)𝜉)(s)⟨s, 𝛾⟩ds = (F𝜆 𝜆( g)𝜉)(𝛾), (𝜆

𝜆( g), and hence that is, m( g)F = F𝜆 𝜆( g)) = m( g) Φ(𝜆

( g ∈ G ),

(5)

̂ so that Φ(𝔏(G)) = ℒ ∞ (G). For g ∈ G, the operator 𝜋Ĝ (m( g)) is (by 18.5.(2)) the multiplication operator by the function ̂ ×G ̂ ∋ (𝛼, 𝛽) ↦ ⟨g, 𝛼𝛽⟩ = ⟨g, 𝛼⟩⟨g, 𝛽⟩ G ̂ × G), ̂ that is, 𝜋Ĝ (m( g)) = m( g) ⊗ ̄ m( g). According to (5) and 18.7.(2) we get on ℒ 2 (G ̄ Φ)(𝛿G (𝜆 ̄ Φ(𝜆 ̄ m( g) = 𝜋Ĝ (Φ(𝜆 𝜆( g))) = Φ(𝜆 𝜆( g)) ⊗ 𝜆( g)) = m( g) ⊗ 𝜆( g))), (Φ ⊗ ̄ Φ) ◦ 𝛿G = 𝜋Ĝ ◦ Φ. so that (Φ ⊗ Finally, using (5), 18.5.(4) and 18.7.(4), it is easy to see that 𝜆( g−1 )) = m( g−1 ) = kĜ (m( g)) = kĜ (Φ(𝜆 𝜆( g))), 𝜆( g))) = Φ(𝜆 Φ( jG (𝜆 so that Φ ◦ jG = kĜ ◦ Φ. ̂ introduced with two different meanings in Sections 18.5 and Thus, if G is abelian, the notation G 18.7, designates the same coinvolutive Hopf–von Neumann algebra. ̂ = (𝔏(G), 𝛿G , jG ) ̄ 𝔏(G) of G Also, according to Proposition 18.6, every action 𝛿 ∶ ℳ → ℳ ⊗ ∗ ̂ ̂ on ℳ, uniquely on a W -algebra ℳ corresponds to a continuous action 𝜎 ∶ G → Aut(ℳ) of G ̄ determined, such that 𝜋𝜎 = (𝜄ℳ ⊗ Φ) ◦ 𝛿. ̂ we denote by 18.9. We continue to assume that the locally compact group G is abelian. For 𝛾 ∈ G, m(𝛾) ∈ ℒ ∞ (G) ⊂ ℬ(ℒ 2 (G)), the unitary operator of multiplication with the function 𝛾 ∈ ℒ ∞ (G) on ℒ 2 (G). Then ̂ ∋ 𝛾 ↦ m(𝛾) ∈ ℬ(ℒ 2 (G)) G ̂ on ℒ 2 (G). is an so-continuous unitary representation of G

Hopf–von Neumann Algebras

243

It is easy to check that ̂ 𝜆( g) ( g ∈ G, 𝛾 ∈ G). 𝜆 ( g)m(𝛾) = ⟨g, 𝛾⟩m(𝛾)𝜆

(1)

̂ → Aut(ℬ(ℒ 2 (G))) by We define a continuous action 𝜃 ∶ G ̂ 𝜃𝛾 (x) = m(𝛾)∗ xm(𝛾) (x ∈ ℬ(ℒ 2 (G)), y ∈ G)

(2)

̂ = (ℒ ∞ (G), ̂ of G ̂ 𝜋Ĝ , kĜ ) on ̄ ℒ ∞ (G) and the corresponding action 𝜋𝜃 ∶ ℬ(ℒ 2 (G)) → ℬ(ℒ 2 (G)) ⊗ 2 ℬ(ℒ (G)). ̂ hence It is clear that 𝜃𝛾 ( f ) = f for every f ∈ ℒ ∞ (G) (𝛾 ∈ G), ̄ 1G 𝜋𝜃 ( f ) = f ⊗

( f ∈ ℒ ∞ (G)).

(3)

On the other hand, using the definition 18.6.(1) of 𝜋𝜃 and the commutation relations (1), it is easy to check that ̄ m( g) ( g ∈ G ). 𝜆( g)) = 𝜆( g) ⊗ 𝜋𝜃 (𝜆

(4)

̂ the Fourier– Let 𝜄G be the identity mapping on ℬ(ℒ 2 (G)) and Φ ∶ ℬ(ℒ 2 (G)) → ℬ(ℒ 2 (G)) Plancherel isomorphism. Using (3), (4), 18.7.(2), and 18.8.(4), we obtain ̄ Φ)(𝛿G (x)), 𝜋𝜃 (𝜄G (x)) = (𝜄G ⊗ valid for x = f ∈ ℒ ∞ (G) and for x = 𝜆( g) ∈ 𝔏(G) and hence (18.4.(15)) for all x ∈ ℬ(ℒ 2 (G)). We thus get the following: Proposition. Let G be a locally compact abelian group and ℳ a W ∗ -algebra. The action ̄ 𝛿G = ℳ ⊗ ̄ ℬ(ℒ 2 (G)) → ℳ ⊗ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ 𝔏(G) 𝜄ℳ ⊗ ̂ = (𝔏(G), 𝛿G , jG ) on ℳ corresponds, via the Fourier–Plancherel isomorphism to the continuous of G action ̂ → Aut(ℳ ⊗ ̄ ℬ(ℒ 2 (G))) 𝜃∶G ̂ on ℳ ⊗ ̄ ℬ(ℒ 2 (G)) defined by of G ̂ ̄ m(𝛾))∗ X(1ℳ ⊗ ̄ m(𝛾)) (X ∈ ℳ ⊗ ̄ ℬ(ℒ 2 (G)), 𝛾 ∈ G). 𝜃𝛾 (X) = (1ℳ ⊗ Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact group G on the W ∗ -algebra ̄ ℒ ∞ (G) the corresponding action of G on ℳ and 𝜑 a normal weight ℳ, 𝜋𝜎 ∶ ℳ → ℳ ⊗ on ℳ. Recall that 𝜑 is called 𝜎-invariant, if 𝜑 ◦ 𝜎g = 𝜑 ( g ∈ G ). This condition is equivalent to the following condition: ̄ k⟩ = ⟨x ⊗ ̄ 1G , 𝜑 ⊗ ̄ k⟩ ⟨𝜋𝜎 (x), 𝜑 ⊗

(x ∈ ℳ + , k ∈ ℒ 1 (G)+ ).

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In what follows, we consider a convenient notion of invariance of weights with respect to the action of a coinvolutive Hopf–von Neumann algebra. ̄ 𝒜 be an action of the coinvolutive Hopf–von Neumann algebra 18.10. Let 𝛿 ∶ ℳ → ℳ ⊗ (𝒜 , 𝛿𝒜 , j𝒜 ) on the W ∗ -algebra ℳ and let 𝜑 be a normal semifinite weight on ℳ. We shall say that 𝜑 is 𝛿-invariant if ̄ k⟩ = ⟨x ⊗ ̄ 1𝒜 , 𝜑 ⊗ ̄ k⟩ ⟨𝛿(x), 𝜑 ⊗

(x ∈ ℳ + , k ∈ 𝒜∗+ ).

(1)

Lemma. Let 𝜑 be a 𝛿-invariant normal semifinite weight on the 𝒜 -comodule ℳ. Then s(𝜑) ∈ ℳ 𝛿 ,

(2)

for every x ∈ 𝔑𝜑 and k ∈ 𝒜∗ we have k ⋅ x ∈ 𝔑𝜑 and ‖(k ⋅ x)𝜑 ‖𝜑 ≤ ‖k‖‖x𝜑 ‖𝜑 ,

(3)

̄ 1𝒜 )𝛿(x) ∈ 𝔐𝜑 ⊗̄ k , y∗ (k ⋅ x) ∈ 𝔐𝜑 and and for every x, y ∈ 𝔑𝜑 and k ∈ 𝒜∗+ we have ( y∗ ⊗ ̄ 1𝒜 )𝛿(x), 𝜑 ⊗ ̄ k⟩ = ⟨y∗ (k ⋅ x), 𝜑⟩. ⟨( y∗ ⊗

(4)

Proof. Let x ∈ 𝔑𝜑 , k ∈ 𝒜∗ with polar decomposition k = |k|(v⋅), v ∈ ℳ, and let f ∈ ℳ∗+ , f ≤ 𝜑. We have ̄ 1𝒜 )𝛿𝒜 (x), f ⊗ ̄ k⟩ ⟨(k ⋅ x)∗ (k ⋅ x), f ⟩ = ⟨((k ⋅ x)∗ ⊗ ∗ ̄ ̄ = ⟨((k ⋅ x) ⊗ v)𝛿(x), f ⊗ |k|⟩ ̄ |k|⟩1∕2 ≤ ⟨(k ⋅ x)∗ (k ⋅ x), f ⟩1∕2 ‖k‖1∕2 ⟨𝛿(x∗ x), f ⊗ ∗ 1∕2 1∕2 ∗ ̄ |k|⟩1∕2 ≤ ⟨(k ⋅ x) (k ⋅ x), f ⟩ ‖k‖ ⟨𝛿(x x), 𝜑 ⊗ = ⟨(k ⋅ x)∗ (k ⋅ x), f ⟩1∕2 ‖k‖1∕2 ⟨x∗ x, 𝜑⟩1∕2 ‖k‖1∕2 hence ⟨(k ⋅ x)∗ (k ⋅ x), f ⟩1∕2 ≤ ‖k‖‖x𝜑 ‖𝜑 . Since f ∈ ℳ∗+ , f ≤ 𝜑 was arbitrary, (3) follows. Consider now x, y ∈ 𝔑𝜑 and k ∈ 𝒜∗+ . From (3) it follows that y∗ (k ⋅ x) ∈ 𝔐𝜑 . Since ̄ k⟩ = ⟨x∗ x, 𝜑⟩⟨1𝒜 , k⟩ < +∞, we have 𝛿(x) ∈ 𝔑𝜑 ⊗̄ k , hence ( y∗ ⊗ ̄ 1𝒜 )𝛿(x) ∈ ⟨𝛿(x)∗ 𝛿(x), 𝜑 ⊗ 𝔐𝜑 ⊗̄ k . Let ̄ 1𝒜 )𝛿(x) = a1 − a2 + ia3 − ia4 with aj ∈ 𝔐𝜑 ⊗̄ k , aj ≥ 0 ( y∗ ⊗

(1 ≤ j ≤ 4).

(5)

Using 9.8.(5) we get k ̄ k⟩ < +∞ (1 ≤ j ≤ 4), ⟨Eℳ (aj ), 𝜑⟩ = ⟨aj , 𝜑 ⊗

(6)

Hopf–von Neumann Algebras

245

k (aj ) ∈ 𝔐𝜑 . Thus, hence, 0 ≤ Eℳ k ̄ 1𝒜 )𝛿(x)) = E k (a1 ) − E k (a2 ) + iE k (a3 ) − iE k (a4 ) (( y∗ ⊗ y∗ (k ⋅ x) = Eℳ ℳ ℳ ℳ ℳ

and, using (5) and (6), we obtain (4). Finally, we prove assertion (2). Clearly, (1ℳ − s(𝜑))𝜑 = 0 and, by (3), this implies that (k ⋅ (1ℳ − s(𝜑))𝜑 = 0, hence (k ⋅ (1ℳ − s(𝜑)))s(𝜑) = 0 for all k ∈ 𝒜∗ . Consequently, for k ∈ 𝒜∗ and f ∈ ℳ∗ , we have ̄ 1𝒜 ), f ⊗ ̄ k⟩ = ⟨(k ⋅ (1ℳ − s(𝜑)))s(𝜑), f ⟩ = 0, ⟨𝛿(1ℳ − s(𝜑))(s(𝜑) ⊗ ̄ 1𝒜 ) = 0. Thus, that is, 𝛿(1ℳ − s(𝜑))(s(𝜑) ⊗ ̄ 1𝒜 . 𝛿(s(𝜑)) ≤ s(𝜑) ⊗ ̄ 1𝒜 ) − 𝛿(s(𝜑)). Since ⟨𝛿(1ℳ − s(𝜑)), 𝜑 ⊗ ̄ k⟩ = ⟨(1ℳ − On the other hand, let e = (s(𝜑) ⊗ ̄ 1𝒜 , 𝜑 ⊗ ̄ k) = 0 (k ∈ 𝒜 + ), we have ⟨e, 𝜑 ⊗ ̄ k⟩ = 0, so e(s(𝜑) ⊗ ̄ s(k)) = 0 for all k ∈ 𝒜 + s(𝜑)) ⊗ ∗ ∗ ̄ 1𝒜 )e = 0. and hence e = e(s(𝜑) ⊗ k From (3) it follows that the mappings Eℳ ∶ x ↦ k ⋅ x define bounded linear operators 𝜋𝜑𝛿 (k) ∈ ℬ(ℋ𝜑 ),

𝜋𝜑𝛿 (k)x𝜑 = (k ⋅ x)𝜑 ‖𝜋𝜑𝛿 (k)‖

(x ∈ 𝔑𝜑 , k ∈ 𝒜∗ )

≤ ‖k‖

(k ∈ 𝒜∗ )

(7) (8)

and the mapping 𝜋𝜑𝛿 ∶ 𝒜∗ → ℬ(ℋ𝜑 ) is a contractive representation of the Banach algebra 𝒜∗ . We shall say that the weight 𝜑 is (𝛿, j𝒜 )-invariant, if 𝜑 is 𝛿-invariant and ̄ 1𝒜 )𝛿(x), 𝜑 ⊗ ̄ k⟩ = ⟨(𝛿( y∗ )(x ⊗ ̄ 1𝒜 ), 𝜑 ⊗ ̄ (k ⋅ j𝒜 )⟩ ⟨( y∗ ⊗

(9)

for all x, y ∈ 𝔑𝜑 and k ∈ A+∗ . Using (4), we see that both sides of (9) are well defined and that (9) is equivalent to the equation: ((k ⋅ x)𝜑 |y𝜑 )𝜑 = (x𝜑 |(k0 ⋅ y)𝜑 )𝜑

(x, y ∈ 𝔑𝜑 , k ∈ 𝒜∗ ),

(10)

that is, 𝜋𝜑𝛿 (k)∗ = 𝜋𝜑𝛿 (k0 )

(k ∈ 𝒜∗ ).

Consequently, if 𝜑 is (𝛿, j𝒜 )-invariant, then 𝜋𝜑𝛿 is a contractive *-representation of the involutive Banach algebra 𝒜∗ .

246

Crossed Products

̄ 𝒜 of the coinvolutive Hopf–von Neumann algebra 18.11. Consider again an action 𝛿 ∶ ℳ → ℳ ⊗ (𝒜 , 𝛿𝒜 , j𝒜 ) on the W ∗ -algebra ℳ and a normal semifinite weight 𝜑 on ℳ. If there exists 𝜆 > 0, such that x ∈ 𝔑𝜑 , k ∈ 𝒜∗ → k ⋅ x ∈ 𝔑𝜑 , ‖(k ⋅ x)𝜑 ‖𝜑 ≤ 𝜆‖k‖‖x𝜑 ‖𝜑 ,

(1)

x, y ∈ 𝔑𝜑 , k ∈ 𝒜∗ → ((k ⋅ x)𝜑 |y𝜑 )𝜑 = (x𝜑 (k ⋅ y)𝜑 )𝜑 ,

(2)

0

then we shall say that 𝜋𝜑𝛿 is a bounded *-representation of the involutive Banach algebra 𝒜∗ . In this case, we can indeed define by 18.10.(7) a bounded *-representation 𝜋𝜑𝛿 ∶ 𝒜∗ → ℬ(ℋ𝜑 ) with ‖𝜋𝜑𝛿 ‖ ≤ 𝜆. Also, with the same arguments as in the proof of 18.10.(2), we can show that ̄ 𝒜. 𝛿(s(𝜑)ℳs(𝜑)) ⊂ s(𝜑)ℳs(𝜑) ⊗

(3)

Lemma. If 𝜋𝜑𝛿 is a bounded *-representation of the involutive Banach algebra 𝒜∗ , then the action 𝛿 commutes with the modular automorphism group {𝜎t𝜑 }t∈ℝ of 𝜑: ̄ 𝜄𝒜 ) ◦ 𝛿 𝛿 ◦ 𝜎t𝜑 = (𝜎t𝜑 ⊗

(i ∈ ℝ).

Proof. Taking into account (3) we see that, without loss of generality but disregarding the assump̄ 1𝒜 , we may assume that 𝜑 is an n.s.f. weight on ℳ. tion 𝛿(1ℳ ) = 1ℳ ⊗ By assumption, it follows that for every k ∈ 𝒜∗ and every x𝜑 ∈ 𝔄𝜑 we have S𝜑 𝜋𝜑𝛿 (k)x𝜑 = S𝜑 (k ⋅ x)𝜑 = ((k ⋅ x)∗ )𝜑 = (k∗ ⋅ x∗ )𝜑 = 𝜋𝜑𝛿 (k∗ )S𝜑 x𝜑 . Since S𝜑 = S𝜑 |𝔄𝜑 , we obtain 𝜉 ∈ D(S𝜑 ) ⇒ 𝜋𝜑𝛿 (k)𝜉 ∈ D(S𝜑 ),

S𝜑 𝜋𝜑𝛿 (k)𝜉 = 𝜋𝜑𝛿 (k∗ )S𝜑 𝜉.

Let 𝜂 ∈ D(Δ𝜑 ) and 𝜉 ∈ D(S𝜑 ). Then 𝜂 ∈ D(S𝜑 ), S𝜑 𝜂 ∈ D(S∗𝜑 ) and (S𝜑 𝜋𝜑𝛿 (k)𝜂|S𝜑 𝜉)𝜑 = (𝜋𝜑𝛿 (k∗ )S𝜑 𝜂| S𝜑 𝜉)𝜑 = (S𝜑 𝜂|𝜋𝜑𝛿 (k∗0 )S𝜑 𝜉)𝜑 = (S𝜑 𝜂|S𝜑 𝜋𝜑𝛿 (k0 )𝜉)𝜑 = (𝜋𝜑𝛿 (k0 )𝜉|S∗𝜑 S𝜑 𝜂)𝜑 = (𝜉|𝜋𝜑𝛿 (k)Δ𝜑 𝜂)𝜑 . Therefore, S𝜑 𝜋𝜑𝛿 (k)𝜂 ∈ D(S∗𝜑 ) and S∗𝜑 S𝜑 𝜋𝜑𝛿 (k)𝜂 = 𝜋𝜑𝛿 (k)Δ𝜑 𝜂. Thus, 𝜂 ∈ D(Δ𝜑 ) ⇒ 𝜋𝜑𝛿 (k)𝜂 ∈ D(Δ𝜑 ),

Δ𝜑 𝜋𝜑𝛿 (k) = 𝜋𝜑𝛿 (k)Δ𝜑 𝜂,

that is, 𝜋𝜑𝛿 (k) commutes with Δ𝜑 . It follows that for x ∈ 𝔑𝜑 , k ∈ 𝒜∗ , t ∈ ℝ, we have (𝜎t𝜑 (k ⋅ x))𝜑 = Δit𝜑 (k ⋅ x)𝜑 = Δit𝜑 𝜋𝜑𝛿 (k)x𝜑 = 𝜋𝜑𝛿 (k)Δit𝜑 x𝜑 = (k ⋅ 𝜎t𝜑 (x))𝜑 . Consequently, 𝜎t𝜑 (k ⋅ x) = k ⋅ 𝜎t𝜑 (x) for all x ∈ ℳ, k ∈ 𝒜∗ , t ∈ ℝ and, for every f ∈ ℳ∗ , we deduce that ̄ k⟩ = ⟨k ⋅ 𝜎t𝜑 (x), f ⟩ = ⟨𝜎t𝜑 (k ⋅ x), f ⟩ = ⟨k ⋅ x, f ◦ 𝜎t𝜑 ⟩ ⟨𝛿(𝜎t𝜑 (x)), f ⊗ ̄ k⟩ = ⟨(𝜎t𝜑 ⊗ ̄ 1𝒜 )(𝛿(x)), f ⊗ ̄ k⟩, = ⟨𝛿(x), ( f ◦ 𝜎t𝜑 ) ⊗ thus proving the lemma.

Hopf–von Neumann Algebras

247

̄ 𝒜 be an action of the convolutive Hopf–von Neumann algebra 18.12 Theorem. Let 𝛿 ∶ ℳ → ℳ ⊗ (𝒜 , 𝛿𝒜 , j𝒜 ) on the W ∗ -algebra ℳ and 𝜑 a normal semifinite weight on ℳ. The following statements are equivalent: (i) (ii) (iii) (iv)

𝜑 is (𝛿, j𝒜 )-invariant. 𝜋𝜑𝛿 is a bounded *-representation of the involutive Banach algebra 𝒜∗ . 𝜋𝜑𝛿 is a contractive *-representation of the involutive Banach algebra 𝒜∗ . The following conditions are satisfied: (a) s(𝜑) ∈ ℳ 𝛿 ; ̄ 𝜄𝒜 )(𝛿(x)) for all x ∈ s(𝜑)ℳs(𝜑), t ∈ ℝ; (b) 𝛿(𝜎t𝜑 (x)) = (𝜎t𝜑 ⊗ 𝜑 (c) there exists a 𝜎 -invariant *-subalgebra ℬ of 𝔐𝜑 , w-dense in s(𝜑)ℳs(𝜑), such that ̄ k⟩ = ⟨x ⊗ ̄ 1𝒜 , 𝜑 ⊗ ̄ k⟩ ⟨𝛿(x), 𝜑 ⊗

(x ∈ ℬ ∩ ℳ + , k ∈ 𝒜∗+ );

(d) there exist a ‖ ⋅ ‖𝜑 -dense subset 𝒟 of 𝔑𝜑 and a norm-dense subset ℱ of 𝒜∗+ such that for every x, y ∈ 𝒟 and every k ∈ ℱ we have 𝛿(x), 𝛿( y) ∈ ℜ𝜑 ⊗̄ k and ̄ 1𝒜 )𝛿(x), 𝜑 ⊗ ̄ k⟩ = ⟨𝛿( y∗ )(x ⊗ ̄ 1𝒜 ), 𝜑 ⊗ ̄ k⟩. ⟨( y∗ ⊗ Proof. (i) ⇒ (iii). By Section 18.10. (iii) ⇒ (ii). Obvious. (ii) ⇒ (i). We have to show just that 𝜑 is 𝛿-invariant, that is, ̄ k) ◦ 𝛿 = (1𝒜 , k⟩𝜑 (k ∈ 𝒜 + ), (𝜑 ⊗ ∗

(1)

since then condition 18.10.(9) will follow from 18.11.(2) using 18.10.(4). Let k ∈ 𝒜∗+ . From (ii) it follows that (k ⋅ (1ℳ − s(𝜑)))s(𝜑) = 0 and then, using 9.8.(5), ̄ k⟩ = ⟨k ⋅ (1ℳ − s(𝜑)), 𝜑⟩ = 0, that is, s((𝜑 ⊗ ̄ k) ◦ 𝛿) ≤ we deduce that ⟨𝛿(1ℳ − s(𝜑)), 𝜑 ⊗ ̄ ̄ s(𝜑), and therefore, s((𝜑 ⊗ k) ◦ 𝛿) = s((𝜑 ⊗ k) ◦ 𝛿|s(𝜑)ℳs(𝜑)). On the other hand, it is clear that s(⟨1𝒜 , k⟩𝜑) = s((1𝒜 , k⟩𝜑|s(𝜑)ℳs(𝜑)). Thus, we may assume that 𝜑 is an n.s.f. weight on ℳ. Then ℬ = 𝔗2𝜑 is a w-dense 𝜎 𝜑 -invariant *-subalgebra of ℳ and ℬ = lin(ℬ ∩ ℳ + ) ⊂ 𝔐𝜑 . Let x ∈ ℬ ∩ ℳ + and k ∈ 𝒜∗+ . Then k ⋅ x ∈ 𝔐𝜑 ∩ ℳ + . Indeed, for every y ∈ 𝔑𝜑 , we have (18.11.(2)) ⟨y∗ (k ⋅ x), 𝜑⟩ = ⟨(k0 ⋅ y)∗ x, 𝜑⟩, so that the assertion follows using 2.13.(3). Consequently, there exist a, b ∈ 𝔑𝜑 ∩ ℳ + such that x = a2 ,

k ⋅ x = b2 .

By Proposition 2.16, there exists a net {vj } ∈ 𝔗𝜑 such that s∗

𝜎𝛼𝜑 (vj ) → 1ℳ for all 𝛼 ∈ ℂ.

248

Crossed Products

Using Lemma 18.11, we obtain (here y ∈ ℳ is identified with 𝜋𝜑 ( y) ∈ ℬ(ℋ𝜑 )): ̄ k) ◦ 𝛿⟩ = ⟨𝛿(x), 𝜑 ⊗ ̄ k⟩ = ⟨k ⋅ x, 𝜑⟩ = ⟨b2 , 𝜑⟩ = (b𝜑 |b𝜑 )𝜑 ⟨x, (𝜑 ⊗ 𝜑 ∗ 1∕2 = (b𝜑 |S𝜑 b𝜑 )𝜑 = (b𝜑 |J𝜑 Δ1∕2 𝜑 b𝜑 )𝜑 = lim(b𝜑 |J𝜑 𝜎−i∕2 (vj )Δ𝜑 b𝜑 )𝜑 j

=

∗ lim(b𝜑 |J𝜑 Δ1∕2 𝜑 vj b𝜑 )𝜑 j

= lim(b𝜑 |S𝜑 (v∗j b𝜑 )𝜑 )𝜑 = lim(b𝜑 |(bvj )𝜑 )𝜑 j

j

= lim((k ⋅ x)𝜑 |(vj )𝜑 )𝜑 = lim(x𝜑 |(k ⋅ vj )𝜑 )𝜑 = lim(a𝜑 |(a(k0 ⋅ vj ))𝜑 )𝜑 0

j

j

j

= lim(a𝜑 |S𝜑 ((k ⋅ vj )a)𝜑 )𝜑 = 0

j

0 lim(a𝜑 |J𝜑 Δ1∕2 𝜑 ((k j

⋅ vj )a)𝜑 )𝜑

𝜑 𝜑 0 1∕2 = lim(a𝜑 |J𝜑 𝜎−i∕2 ((k0 ⋅ vj )Δ1∕2 (v 𝜑 a𝜑 )𝜑 = lim(a𝜑 |J𝜑 (k ⋅ 𝜎−i∕2 j ))Δ𝜑 a𝜑 )𝜑 j

j

= (a𝜑 |J𝜑 (k ⋅ 0

1ℳ )Δ1∕2 𝜑 a𝜑 )𝜑

= (a𝜑 |S𝜑 a𝜑 )𝜑 ⟨1𝒜 , k⟩ = (a𝜑 |a𝜑 )𝜑 ⟨1𝒜 , k⟩

= ⟨a2 , 𝜑⟩⟨1𝒜 , k⟩ = ⟨x, ⟨1𝒜 , k⟩𝜑⟩. ̄ k) ◦ 𝛿 is Consequently, the two weights appearing in (1) are equal on ℬ. In particular, (𝜑 ⊗ semifinite. ̄ k) ◦ 𝛿 commutes with ⟨1𝒜 , k⟩𝜑 , hence (1) follows Again using Lemma 18.11, we see that (𝜑 ⊗ from Theorem 6.2. If the equivalent conditions (i), (ii), (iii) are satisfied, then statement (iv) results as follows: (a) by 18.10.(2), (b) by Lemma 18.11, (c) is clear with ℬ = 𝔐𝜑 , and (d) is clear with 𝒟 = 𝔑𝜑 and ℱ = 𝒜∗+ . (iv) ⇒ (iii). From condition (a) it follows that we may assume that 𝜑 is an n.s.f. weight. Using conditions (b) and (c) and Theorem 6.2, we obtain (1), hence 𝜑 is 𝛿-invariant. Then 𝜋𝜑𝛿 is defined and contractive and, using condition (d), we obtain 𝜋𝜑𝛿 (k)∗ = 𝜋𝜑𝛿 (k0 ), first for k ∈ ℱ and then, by passing to the limit, for every k ∈ 𝒜∗+ . Note that in Sections 18.10–18.12, we have not used all the conditions, which define a coinvolutive Hopf–von Neumann algebra and its action on a W ∗ -algebra, but only the following: ̄ 𝒜 is a unital normal *-homomorphism and ℳ and 𝒜 are W ∗ -algebras, 𝛿 ∶ ℳ → ℳ ⊗ j𝒜 ∶ 𝒜 → 𝒜 is a *-antiautomorphism with j𝒜 ◦ j𝒜 = 𝜄𝒜 . The comultiplication 𝛿𝒜 of 𝒜 appeared only in considering the multiplicative structure on A∗ , but this structure has not been used. ̄ 𝒜 be an action of the coinvolutive Hopf–von Neumann 18.13 Corollary. Let 𝛿 ∶ ℳ → ℳ ⊗ algebra (𝒜 , 𝛿𝒜 , j𝒜 ) on the W ∗ -algebra ℳ and let 𝜑, 𝜓 be (𝛿, j𝒜 )-invariant n.s.f. weights on ℋ . Then [D𝜓 ∶ D𝜑]t ∈ ℳ 𝛿

(t ∈ ℝ).

(1)

̄ 𝛿 is an action of 𝒜 on the W ∗ -algebra Mat2 (ℳ) = Mat2 (ℂ) ⊗ ̄ ℳ. Proof. The mapping 𝛿2 = 𝜄2 ⊗ Since 𝜑 and 𝜓 are (𝛿, j𝒜 )-invariant, the balanced weight 𝜃 = 𝜃(𝜑, 𝜓) is (𝛿2 , j𝒜 )-iovariant. By Theorem 18.12, it follows that 𝛿2 commutes with 𝜎t𝜃 (t ∈ ℝ).

Hopf–von Neumann Algebras

249

Let ut = [D𝜓 ∶ D𝜑]t (t ∈ ℝ). Then (

0 0 𝛿(ut ) 0

)

(

) ( (( ))) 0 0 0 = 𝛿2 𝜎t𝜃 0 1ℳ 0 ( (( ))) (( 0 0 0 ̄ 𝜄𝒜 ) 𝛿2 ̄ 𝜄𝒜 ) = (𝜎t𝜃 ⊗ = (𝜎t𝜃 ⊗ 1ℳ 0 1ℳ ( ) 0 0 = ̄ 1ℳ 0 ut ⊗

= 𝛿2

0 ut

0 0

)

̄ 1𝒜 ⊗

)

̄ 1𝒜 , that is, ut ∈ ℳ 𝛿 . hence 𝛿(ut ) = ut ⊗ 18.14 Proposition. Let (𝒜 , 𝛿, j) be a coinvolutive Hopf–von Neumann algebra. If there exists a nonzero 𝛿-invariant normal semifinite weight on 𝒜 , then this weight is faithful, 𝒜 𝛿 = ℂ⋅1𝒜 and any two nonzero (𝛿, j)-invariant normal semifinite weights 𝜔, 𝜏 on 𝒜 are proportional: 𝜏 = 𝜆𝜔(𝜆 > 0). Proof. Let 𝜔 be a nonzero 𝛿-invariant normal semifinite weight on 𝒜 . Let e ∈ 𝒜 𝛿 be a projection. ̄ j)(𝛿(e)) = 1𝒜 ⊗ ̄ j(e), hence for every x ∈ 𝔐𝜔 ∩ 𝒜 + and k ∈ 𝒜 + we Then (18.2.(3)) 𝛿( j(e)) = ̃·( j ⊗ ∗ obtain ̄ k⟩ ⟨j(e)xj(e), 𝜔⟩⟨1𝒜 , k⟩ = ⟨𝛿( j(e)xj(e)), 𝜔 ⊗ ̄ j(e))𝛿(x)(1𝒜 ⊗ ̄ j(e)), 𝜔 ⊗ ̄ k⟩ = ⟨(1𝒜 ⊗ ̄ k( j(e) ⋅ j(e))⟩ = ⟨(𝛿(x), 𝜔 ⊗ = ⟨x, 𝜔⟩⟨j(e), k⟩.

(1)

Since 𝜔 ≠ 0, we have s(𝜔) ≠ 0, so that there exists k ∈ 𝒜∗+ , k ≠ 0, with ⟨j(1𝒜 − s(𝜔)), k⟩ = 0. By 18.10.(2), we have s(𝜔) ∈ 𝒜 𝛿 so that, replacing e by 1𝒜 − s(𝜔) in (1), we get ⟨j(1𝒜 − s(𝜔))xj(1ℳ − s(𝜔)), 𝜔⟩ = 0

(x ∈ 𝔐𝜔 ∩ 𝒜 + ).

It follows that j(1𝒜 − s(𝜔))s(𝜔) = 0, that is, s(𝜔) = j(s(𝜔))s(𝜔).

(2)

Assume now that s(𝜔) ≠ 1𝒜 . Then there exists k ∈ 𝒜∗+ , k ≠ 0, with ⟨j(s(𝜔)), k⟩ = 0 and, again using (1), we obtain ⟨j(s(𝜔))xj(s(𝜔)), 𝜔⟩ = 0

(x ∈ 𝔐𝜔 ∩ 𝒜 + ),

that is, j(s(𝜔))s(𝜔) = 0. This, together with (2), implies that 𝜔 = 0, a contradiction. Hence, 𝜔 is faithful. If e ∈ 𝒜 𝛿 , e ≠ 1𝒜 , there exists k ∈ 𝒜∗+ , k ≠ 0, with ⟨j(e), k⟩ = 0 and using (1) we obtain ⟨j(e)xj(e), 𝜔⟩ = 0 for all x ∈ 𝔐𝜔 ∩ 𝒜 + , that is, j(e) = j(e)s(𝜔) = 0, hence e = 0. Consequently, 𝒜 𝛿 = ℂ ⋅ 1𝒜 . Consider now two nonzero (𝛿, j)-invariant normal semifinite weights 𝜔 and 𝜏 on 𝒜 . Then 𝜔 and 𝜏 are faithful and, by Corollary 18.13, we have [D𝜏 ∶ D𝜔]t ∈ 𝒜 𝛿 = ℂ ⋅ 1𝒜 (t ∈ ℝ). It follows that there exists 𝜆 > 0 with [D𝜏 ∶ D𝜔]t = 𝜆it (t ∈ ℝ), and hence 𝜏 = 𝜆𝜔.

250

Crossed Products

18.15. A Kac algebra is a quadruple (𝒜 , 𝛿, j, 𝜔), where (𝒜 , 𝛿, j) is a coinvolutive Hopf–von 𝜔◦j Neumann algebra and 𝜔 is a (𝛿, j)-invariant n.s.f. weight on 𝒜 such that 𝜎t = 𝜎t𝜔 (t ∈ ℝ), that is, 𝜔 𝜎t𝜔 ◦ j = j ◦ 𝜎−t

(t ∈ ℝ).

(1)

By Proposition 18.14, the invariant weight 𝜔 appearing in the definition of a Kac algebra is unique up to a positive multiplicative constant. The weight 𝜔 will be called the left Haar weight on the Kac algebra 𝒜 . There exists a canonical way to associate with every Kac algebra (𝒜 , 𝛿, j, 𝜔) a dual Kac algebra ̂ ̂j, 𝜔), (𝒜̂, 𝛿, ̂ a procedure that we now describe without proof. The von Neumann algebra 𝒜̂ is defined by 𝒜̂ = ℛ{𝜋𝜔𝛿 (𝒜∗ )} ⊂ ℬ(ℋ𝜔 ). ̄ y ↦ 𝛿(x)(1 ⊗ ̄ y) defines an isometry Using the 𝛿-invariance of 𝜔, one shows that the mapping x ⊗ ̄ on ℋ𝜔 ⊗ ℋ𝜔 and, moreover, the adjoint W of this isometry is again an isometry, that is, W ∈ ̄ ℋ𝜔 ) is a unitary operator. We have W ∈ 𝒜̂ ⊗ ̄ 𝒜 and ℬ(ℋ𝜔 ⊗ ̄ 1)W 𝛿(x) = W ∗ (x ⊗

(x ∈ 𝒜 ).

̂ = ∼ ◦ W ∗ ◦ ∼∈ 𝒜 ⊗ ̄ 𝒜̂ and one defines by One then considers the unitary operator W ̂ =W ̂ ∗ (x ⊗ ̂ ̄ 1)W 𝛿(x)

(x ∈ 𝒜̂)

a coassociative comultiplication 𝛿̂ on 𝒜̂. On the other hand, the equation ̂j(x) = J𝜔 x∗ J𝜔

(x ∈ 𝒜̂)

̂ defines a coinvolution ̂j on 𝒜̂, which is compatible with 𝛿. In order to define the weight 𝜔, ̂ one constructs a *-subalgebra 𝔇 of 𝒜 , which is contained in 𝔗2𝜔 and which, regarded as a subset of 𝒜∗ via Proposition 2.13, is an involutive subalgebra of the involutive Banach algebra 𝒜∗ ; moreover, S𝜔 |𝔇 = S𝜔 . With the involutive algebra structure inherited from 𝒜∗ and the scalar product of ℋ𝜔 . 𝔇 becomes a left Hilbert algebra whose associated von Neumann algebra is equal to 𝒜̂. The natural weight associated with the left Hilbert algebra 𝔇 is a ̂ ̂j)-invariant n.s.f. weight 𝜔̂ on 𝒜̂. (𝛿, Of course, 𝔇, with the *-algebra structure inherited from 𝒜 and the scalar product of ℋ𝜔 , is also a left Hilbert algebra, 𝒜 is the associated von Neumann algebra and 𝜔 is the corresponding natural weight. This structure of a “Tomita bi-algebra” on 𝔇 is at the heart of the duality theory for Kac algebras, ̂̂ ̂̂j, 𝜔) ̂̂ is isomorphic to (𝒜 , 𝛿, j, 𝜔). the fundamental result being that (𝒜̂̂, 𝛿, We now return to the examples considered above. 18.16. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact group G on the ̄ ℒ ∞ (G) the corresponding action of G = (ℒ ∞ (G), 𝜋G , kG ) W ∗ -algebra ℳ and 𝜋𝜎 ∶ ℳ → ℳ ⊗

Hopf–von Neumann Algebras

251

on ℳ. For any normal semifinite weight 𝜑 on ℳ it is easy to check that 𝜑 is 𝜋𝜎 -invariant ⇒ 𝜑 is 𝜎-invariant ⇒ 𝜑 is (𝜋𝜎 , kG )-invariant ⇒ 𝜑 is 𝜋𝜎 -invariant

(1)

hence all meaningful notions of invariance coincide. In particular, since the left Haar measure dg is invariant under left translation and 𝜋G = 𝜋Ad(𝜆𝜆) , it follows that the left Haar weight 𝜇G on ℒ ∞ (G) is (𝜋G , kG )-invariant and G = (ℒ ∞ (G), 𝜋G , kG , 𝜇G ) is a Kac algebra. With the notation introduced in 18.10.(7), 13.2, 18.4.(9), it is easy to check that 𝜋

𝜋𝜇GG (k) = 𝜆(k∗ ) ∈ 𝔏(G) ⊂ ℬ(ℒ 2 (G)) (k ∈ ℒ 1 (G) = ℒ ∞ (G)∗ ).

(2)

̂ = (𝔏(G), 𝛿G , jG ) associated 18.17. Consider now the coinvolutive Hopf–von Neumann algebra G with the locally compact group G and the Plancherel weight 𝜔G on 𝔏(G). Recall (18.4) that 𝜔G is the natural weight associated with the left Hilbert algebra 𝔄 = 𝒦 (G) ⊂ ℒ 2 (G) with the operations of convolution and involution. We shall use the notation 𝔄′ , 𝔄′′ , L𝜉 , R𝜂 , and so on, associated with the left Hilbert algebra 𝔄 = 𝒦 (G) (see, 2.12 and [L], 10.1–10.4). Note that 𝔄′′ = 𝔄𝜔G . By definition, for 𝜉 ∈ 𝔄 = 𝒦 (G) we have L𝜉 𝜂 = 𝜉 ∗ 𝜂 = 𝜆 (𝜉)𝜂

(𝜂 ∈ ℒ 2 (G)).

(1)

Consequently, for 𝜂 ∈ ℒ 2 (G), the operator R0𝜂 is defined by R0𝜂 𝜉 = L𝜉 𝜂 = 𝜉 ∗ 𝜂 (𝜉 ∈ 𝔄 = 𝒦 (G)). If R𝜂 is bounded, it follows that R𝜂 𝜉 = 𝜉 ∗ 𝜂

(𝜉 ∈ ℒ 2 (G)).

(2)

Indeed, if 𝜉n ∈ 𝒦 (G) and 𝜉n → 𝜉 in ℒ 2 (G), then R𝜂 𝜉n → R𝜂 𝜉 in ℒ 2 (G) and 𝜉n ∗ 𝜂 → 𝜉 ∗ 𝜂 uniformly. In particular, (2) holds whenever 𝜂 ∈ 𝔄′ . Consequently, for 𝜉 ∈ ℒ 2 (G), the operator L0𝜉 is defined by L0𝜉 𝜂 = R𝜂 𝜉 = 𝜉 ∗ 𝜂 (𝜂 ∈ 𝔄′ ). If L𝜉 is bounded, we obtain similarly L𝜉 𝜂 = 𝜉 ∗ 𝜂

(𝜂 ∈ ℒ 2 (G)).

(3)

It follows that 𝜉 ∈ ℒ 1 (G) ∩ ℒ 2 (G) ⇒ L𝜉 is bounded and L𝜉 = 𝜆 (𝜉).

(4)

Using 2.12.(2), we see that for every 𝜉, 𝜂 ∈ ℒ 1 (G) ∩ ℒ 2 (G) we have 𝜆(𝜉), 𝜆(𝜂) ∈ 𝔑𝜔G and 𝜆(𝜂)∗𝜆 (𝜉)) = (𝜉|𝜂), that is, 𝜔(𝜆 𝜆(𝜂 ♯ ∗ 𝜉)) = 𝜔G (𝜆

∫

𝜉( g)𝜂( g) dg = (𝜂 ♯ ∗ 𝜉)(e) (𝜉, 𝜂 ∈ ℒ 1 (G) ∩ ℒ 2 (G)).

(5)

Recall that the convolution of two ℒ 2 -functions is equal almost everywhere to a continuous function.

252

Crossed Products

Moreover, for every function f ∈ ℒ 1 (G), we have 𝜆( f ♯ ∗ f ) ∈ 𝔏(G)+ and 𝜔G (𝜆( f ♯ ∗ f )) =

∫

| f ( g)|2 dg = ′′( f ♯ ∗ f )(e)′′

( f ∈ ℒ 1 (G))

(6)

where the last equation is just formal. Indeed, if f ∈ ℒ 1 (G) ∩ ℒ 2 (G), then (6) follows from (5). If 𝜆( f ♯ ∗ f )) < +∞, then 𝜆 ( f ) ∈ 𝔑𝜔G and so there exists 𝜉 ∈ ℒ 2 (G) with L𝜉 = 𝜆 ( f ), that is, 𝜔G (𝜆 𝜉 ∗ 𝜂 = f ∗ 𝜂 for all 𝜂 ∈ ℒ 2 (G), which means that f = 𝜉 ∈ ℒ 2 (G). In computations involving 𝜔G , we shall often use the following: Proposition. For x ∈ 𝔏(G)+ , we have 𝜔G (x) < +∞ if and only if there exists a continuous function f ∈ ℒ 2 (G) with x = Lf ; in this case, 𝜔G (Lf ) = f (e).

(7)

𝜆( f )) < +∞ if and only if f ∈ ℒ 1 (G) ∩ ℒ 2 (G); In particular, if f ∈ ℒ 1 (G) and 𝜆 ( f ) ≥ 0, then 𝜔G (𝜆 in this case f is (equal almost everywhere to) a continuous function and 𝜆( f )) = f (e). 𝜔G (𝜆

(8)

Proof. Let f ∈ ℒ 2 (G) be a continuous function such that the operator Lf is bounded and positive. Then 𝜑 = JG f ∈ ℒ 2 (G) is also a continuous function and the operator R𝜑 = JG Lf JG (see 2.12.(4)) is bounded and positive: 0 ≤ R𝜑 ≤ 𝜆. Since, for every 𝜁 ∈ 𝒦 (G) we have (R𝜑 𝜁|𝜁 ) = (𝜁 ∗ 𝜑|𝜁) = (𝜑|𝜁 ♯ ∗ 𝜁) = ∫ 𝜑( g)(𝜁 ♯ ∗ 𝜁)( g) dg, it follows that the continuous function 𝜑 is positive definite and ∫

𝜑( g)(𝜁 ♯ ∗ 𝜁)( g) dg ≤ 𝜆‖𝜁‖22

(𝜁 ∈ 𝒦 (G)).

Let u𝜑 ∶ G → ℬ(ℋ𝜑 ) be the so-continuous cyclic unitary representation of G, with cyclic vector 𝜉𝜑 ∈ ℋ𝜑 , associated with 𝜑 (see 13.4.5). The corresponding *-representation of the involutive Banach algebra ℒ 1 (G) is just the GNS-representation defined by the positive form 𝜑 ∈ ℒ ∞ (G) = ℒ 1 (G)∗ and will also be denoted by u𝜑 . For 𝜁 ∈ 𝒦 (G), we have ‖u𝜑 (𝜁)𝜉𝜑 ‖2 = ⟨𝜁 ♯ ∗ 𝜁 , 𝜑⟩ =

∫

𝜑( g)(𝜁 ♯ ∗ 𝜁)( g) dg ≤ 𝜆‖𝜁‖22 .

Thus, there exists a bounded linear operator T ∶ ℒ 2 (G) → ℋ𝜑 , ‖T‖ ≤ 𝜆, such that T𝜁 = u𝜑 (𝜁)𝜉𝜑 (𝜁 ∈ 𝒦 (G)). The range of T is dense in ℋ𝜑 since 𝜉𝜑 ∈ 𝒦𝜑 is a cyclic vector and 𝜆( g) = u𝜑 ( g)T T𝜆

( g ∈ G ).

Hopf–von Neumann Algebras

253

If T = V|T| is the polar decomposition of T, then V ∶ ℒ 2 (G) → ℋ𝜑 is a coisometry, that is, VV ∗ = 1, and 𝜆( g) = u𝜑 ( g)V V𝜆

( g ∈ G ).

Let 𝜂 = V ∗ 𝜉𝜑 ∈ ℒ 2 (G). For every 𝜁 ∈ 𝒦 (G), we have 𝜁 ∗ 𝜂 = 𝜁 ∗ 𝜂̄ = 𝜆 (𝜁 )V ∗ 𝜉𝜑 = V ∗ u𝜑 (𝜁)𝜉𝜑 = V ∗ T𝜁 = |T|𝜁, hence the operator R𝜂 is bounded and positive. On the other hand, 𝜆( g)𝜂| 𝜑( g) = (u𝜑 ( g)𝜉𝜑 |𝜉𝜑 ) = (𝜆 ̄ 𝜂) ̄ =

∫

𝜂( ̄ g−1 t)𝜂(t) dt

(g ∈ G)

that is, 𝜑 = 𝜂 ∗ 𝜂 ♭ , where 𝜂 ♭ (s) = 𝜂(s−1 )(s ∈ G ). It follows that 𝜉 = JG 𝜂 ∈ ℒ 2 (G) has the property that the operator L𝜉 = JG R𝜂 JG is bounded and positive and f = JG 𝜑 = JG (𝜂 ∗ 𝜂 ♭ ) = 𝜉 ♯ ∗ 𝜉. Consequently, 𝜔G (Lf ) = 𝜔G (L∗𝜉 L𝜉 ) = ‖𝜉‖22 = (𝜉 ♯ ∗ 𝜉)(e) = f (e) < +∞. Consider now x ∈ 𝔏(G)+ with 𝜔G (x) < +∞. By the definition of the natural weight 𝜔G , it follows that there exists 𝜉 ∈ ℒ 2 (G) such that x1∕2 = L𝜉 and 𝜔G (x) = ‖𝜉‖2 . Then f = 𝜉 ♯ ∗ 𝜉 ∈ ℒ 2 (G) is a continuous function with x = L∗𝜉 L𝜉 = Lf and 𝜔G (x) = f (e). In particular, if x = 𝜆 (h) with h ∈ ℒ 1 (G), then 𝜆(h) = Lf , hence h = f almost everywhere. Thus, the proposition is completely proved. We have k ⋅ 𝜆 ( f ) = 𝜆 (k(⋅)f ) ( f ∈ ℒ 1 (G), k ∈ 𝒜 (G)).

(9)

̄ k⟩ = ⟨∫ f ( g)𝜆 ̄ 𝜆 ( g) dg, h 𝜆( f )), h ⊗ 𝜆( g) ⊗ Indeed, for every h ∈ 𝒜 (G), we get ⟨k ⋅ 𝜆 ( f ), h⟩ = ⟨𝛿G (𝜆 ̄ 𝜆( g), k⟩𝜆 𝜆( g) dg, h⟩ = ⟨𝜆 𝜆(k(⋅)f ), h⟩. ⊗ k⟩ = ⟨∫ f ( g)⟨𝜆 From (9) it follows that 𝛿

𝜋𝜔GG (k) = k(⋅) ∈ ℒ ∞ (G) ⊂ ℬ(ℒ 2 (G)) 𝛿

(k ∈ 𝒜 (G) = 𝔏(G)∗ ).

(10)

𝛿

𝜆(𝜉))𝜔G = (k ⋅𝜆 𝜆(𝜉))𝜔G = (𝜆 𝜆(k(⋅)𝜉))𝜔G = k(⋅)𝜉. Indeed, for every 𝜉 ∈ 𝒦 (G), we have 𝜋𝜔GG (k) = 𝜋𝜔GG (k)(𝜆 We are now able to show that the n.s.f. weight 𝜔G on 𝔏(G) is (𝛿G , jG )-invariant

(11)

by checking conditions (iv), (a)–(d), of Theorem 18.12. (a) s(𝜔G ) = 1G ∈ 𝔏(G)𝛿G . (b) The modular operator associated with 𝜔G is the multiplication operator defined by the modular function ΔG ; hence it is affiliated to ℒ ∞ (G), which is the centralizer ℬ(ℒ 2 (G))𝛿G of the action 𝛿G 𝜔 −it (18.7.(9)). Consequently, 𝜎t G = ΔitG ⋅ ΔG commutes with 𝛿G . (c) For 𝜉 ∈ 𝒦 (G) with 𝜆 (𝜉) ≥ 0 and k ∈ 𝒜 (G)+ , we have 𝜆 (k(⋅)𝜉) = k ⋅ 𝜆 (𝜉) ≥ 0 and, using (8), ̄ k⟩ = ⟨𝜆 ̄ 1G , 𝜔G ⊗ ̄ k⟩. 𝜆(𝜉)), 𝜔G ⊗ 𝜆(k(⋅)𝜉), 𝜔G ⟩ = k(e)𝜉(e) = ⟨𝜆 𝜆(𝜉) ⊗ we get ⟨𝛿G (𝜆

254

Crossed Products 𝛿

(d) This condition amounts to showing that the 𝜋𝜔GG is a *-representation, and this follows obviously from (10) as k0 (⋅) = k(⋅). We also have 𝜔G ◦ jG = 𝜔G .

(12)

𝜔

𝜆( g)) = ΔitG𝜆 ( g)Δ−it Indeed, it is easy to see that 𝜎t G (𝜆 = ΔG ( g)it𝜆 ( g)( g ∈ G, t ∈ ℝ), so G 𝜔G 𝜔 𝜔 𝜆( g))) = ΔG ( g)it jG (𝜆 𝜆( g)) = ΔG ( g−1 )−it𝜆 ( g−1 ) = 𝜎−tG (𝜆 𝜆( g−1 )) = 𝜎−tG ( jG (𝜆 𝜆( g)) ( g ∈ G, t ∈ jG (𝜎t (𝜆 ℝ), and hence the two weights appearing in (12) commute. On the other hand, for 𝜉 ∈ 𝒦 (G) with 𝜆(𝜉)) = ∫ 𝜉( g)𝜆 𝜆( g−1 ) dg = ∫ 𝜉( g−1 )ΔG ( g)−1𝜆 ( g) dg = 𝜆 (𝜂), where 𝜆 (𝜉) ≥ 0, we have 0 ≤ jG (𝜆 −1 −1 𝜆(𝜉), 𝜔G ◦ jG ⟩ = ⟨𝜆 𝜆(𝜂), 𝜔G ⟩ = 𝜂(e) = 𝜂( g) = 𝜉( g )ΔG ( g) , ( g ∈ G ), and using (8) we get ⟨𝜆 𝜆(𝜉), 𝜔G ⟩. Consequently, (12) follows using Theorem 6.2. 𝜉(e) = ⟨𝜆 ̂ = (𝔏(G), 𝛿G , jG , 𝜔G ) is a Kac algebra. Actually, G and G ̂ From (11) and (12), it follows that G are dual Kac algebras. Finally, we show that 𝜆( g)) = ΔG ( g)𝜔G ; g ∈ G. 𝜔G ◦ Ad(𝜆

(13)

𝜆( g)) commutes with Indeed, the two weights appearing in this equation commute because Ad(𝜆 𝜔 𝜆( g)))(𝜆 𝜆(𝜉)) = 𝜆 ( g)𝜆 𝜆(𝜉)𝜆 𝜆( g−1 ) = 𝜆 (𝜂), 𝜎t G (t ∈ ℝ). For 𝜉 ∈ 𝒦 (G) with 𝜆 (𝜉) ≥ 0, we have 0 ≤ (Ad(𝜆 −1 𝜆(𝜉), 𝜔 ◦ Ad(𝜆 𝜆( g))⟩ = ⟨𝜆 𝜆(𝜂), 𝜔⟩ = where 𝜂(s) = ΔG ( g)𝜉( gsg )(s ∈ G ), and using (8) we get ⟨𝜆 𝜆(𝜉), ΔG ( g)𝜔G ⟩. Thus, (13) follows by Theorem 6.2. 𝜂(e) = ΔG ( g)𝜉(e) = ⟨𝜆 18.18. The invariance property of a weight with respect to an action can be extended to a similar property of the tensor product of the weight with a normal positive form, namely: ̄ 𝒜 be an action of the coinvolutive Hopf–von Neumann algebra Proposition. Let 𝛿 ∶ ℳ → ℳ ⊗ ∗ (𝒜 , 𝛿𝒜 , j𝒜 ) on the W -algebra ℳ and let 𝜑 be a (𝛿, j𝒜 )-invariant n.s.f. weight on ℳ. For any ̄ 𝒩 )+ we have W ∗ -algebra 𝒩 and any f ∈ (𝒜 ⊗ ∗ ̄ f ) ◦ (𝛿 ⊗ ̄ 𝜄𝒩 )⟩ = ⟨X ⊗ ̄ 1𝒜 , (𝜑 ⊗ ̄ f ) ◦ (𝜄ℳ ⊗̃ ̄ ·𝒩 ,𝒜 )⟩ ⟨X, (𝜑 ⊗ + ̄ 𝒩 ) ). (X ∈ (ℳ ⊗

(1)

̄ 𝜄𝒩 ) on ℳ ⊗ ̄ 𝒩 and the normal positive ̄ f ) ⋅ (𝛿 ⊗ Proof. We consider the normal weight 𝜓 = (𝜑 ⊗ ̄ ⋅) on 𝒩 , that is, ⟨y, h⟩ = ⟨1𝒜 ⊗ ̄ y, f ⟩( y ∈ 𝒩 ). Using the 𝛿-invariance of 𝜑, we form h = f (1𝒜 ⊗ obtain, for x ∈ 𝔑𝜑 ⊂ ℳ and y ∈ 𝒩 ̄ y∗ y, 𝜓⟩ = ⟨𝛿(x∗ x) ⊗ ̄ y∗ y, 𝜑 ⊗ ̄ f ⟩ = ⟨𝛿(x∗ x), (𝜑 ⊗ ̄ f )(⋅ ⊗ ̄ y∗ y)⟩ ⟨x∗ x ⊗ ̄ ( f (⋅ ⊗ ̄ y∗ y))⟩ = ⟨x∗ x ⊗ ̄ 1𝒜 , 𝜑 ⊗ ̄ ( f (⋅ ⊗ ̄ y∗ y))⟩ = ⟨𝛿(x∗ x), 𝜑 ⊗ ̄ y∗ y, f ⟩ = ⟨x∗ x, 𝜑⟩⟨y∗ y, h⟩ = ⟨x∗ x ⊗ ̄ y∗ y, 𝜑 ⊗ ̄ h⟩. = ⟨x∗ x, 𝜑⟩⟨1𝒜 ⊗ In particular, 𝜓 is semifinite. On the other hand, for every t ∈ ℝ, 𝜎t𝜑 commutes with 𝛿 (Lemma 18.11), hence ̄ 𝜄𝒩 ) = (𝜑 ⊗ ̄ f ) ◦ ((𝛿 ◦ 𝜎t𝜑 ) ⊗ ̄ 𝜄𝒩 ) 𝜓 ◦ (𝜎t𝜑 ⊗ 𝜑 ̄ f ) ◦ (𝜎t ⊗ ̄ 𝜄𝒜 ⊗ ̄ 𝜄𝒩 ) ◦ (𝛿 ⊗ ̄ 𝜄𝒩 ) = ((𝜑 ◦ 𝜎t𝜑 ) ⊗ ̄ f ) ◦ (𝛿 ⊗ ̄ 𝜄𝒩 ) = 𝜓. = (𝜑 ⊗

Hopf–von Neumann Algebras

255

̄ h, which is equivalent If 𝒩 is a type I factor, then, using Proposition 8.10, we infer that 𝜓 = 𝜑 ⊗ to (1). In the general case, we can represent 𝒩 as a von Neumann algebra 𝒩 ⊂ ℬ(ℋ ) and extend f to a ̄ ℬ(ℋ ) so that the general case of (1) follows by restriction from the normal positive form on 𝒜 ⊗ case 𝒩 = ℬ(ℋ ), which has already been proved. Note that (1) for 𝒩 = ℂ ⋅ 1𝒩 just expresses the 𝛿-invariance of 𝜑. 18.19. We now show that the action 𝛿 of a Kac algebra on a W ∗ -algebra ℳ defines a canonical ℳ 𝛿 -valued weight on ℳ. ̄ 𝒜 be an action of the Kac algebra (𝒜 , 𝛿𝒜 , j𝒜 , 𝜔𝒜 ) on the Proposition. Let 𝛿 ∶ ℳ → ℳ ⊗ ∗ W -algebra ℳ. The formula 𝜔 ◦ j𝒜

P𝛿 (x) = Eℳ𝒜

(𝛿(x)) (x ∈ ℳ + )

(1)

defines a normal faithful ℳ 𝛿 -valued weight on ℳ. +

Proof. Let x ∈ ℳ + . (1) defines an element 𝔪 = P𝛿 (x) ∈ ℳ . We shall show that 𝔪 = (ℳ 𝛿 )+ . By the uniqueness of the spectral decomposition of 𝔪 (11.3.(2)), it is sufficient to show that 𝛿(𝔪) = ̄ 1𝒜 , that is, 𝔪⊗ ̄ 1𝒜 , f ⟩ for all f ∈ (ℳ ⊗ ̄ 𝒜 )+ . ⟨𝛿(𝔪), f ⟩ = ⟨𝔪 ⊗ ∗

(2)

̄ 𝒜 )+ . We have ⟨𝛿(𝔪), f ⟩ = ⟨𝔪, f ◦ 𝛿⟩ = ⟨𝛿(x), ( f ◦ 𝛿) ⊗ ̄ (𝜔𝒜 ◦ j𝒜 )⟩ = Let f ∈ (ℳ ⊗ ∗ ̄ (𝜔𝒜 ◦ j𝒜 )) ◦ (𝛿 ⊗ ̄ 𝜄𝒜 ) ◦ 𝛿⟩ = ⟨x, ( f ⊗ ̄ (𝜔𝒜 ◦ j𝒜 )) ◦ (𝜄ℳ ⊗ ̄ 𝛿𝒜 ) ◦ 𝛿⟩ = ⟨𝛿(x), (f ⊗ ̄ (𝜔𝒜 ◦ j𝒜 )) ⟨x, ( f ⊗ ̄ 𝛿𝒜 )⟩ and ⟨𝔪 ⊗ ̄ 1𝒜 , f ⟩ = ⟨𝔪, f (⋅ ⊗ ̄ 1𝒜 )⟩ = ⟨𝛿(x), ( f (⋅ ⊗ ̄ 1𝒜 )) ⊗ ̄ (𝜔𝒜 ◦ j𝒜 )⟩. Consequently, ◦ (𝜄ℳ ⊗ ̄ 𝒜: (2) is equivalent to the following equality of weights on ℳ ⊗ ̄ (𝜔𝒜 ◦ j𝒜 )) ◦ (𝜄ℳ ⊗ ̄ 𝛿𝒜 ) = ( f (⋅ ⊗ ̄ 1𝒜 ) ⊗ ̄ (𝜔𝒜 ◦ j𝒜 ). (f ⊗

(3)

As in the last part of the proof of Proposition 18.18, we see that it is sufficient to check (3) only when ℳ is a type I factor. In this case, there exists an involutive *-automorphism j ∶ ℳ → ℳ, j ◦ j = ̄ j𝒜 ) ◦̃·𝒜 ,ℳ ∈ (𝒜 ⊗ ̄ ℳ)+ . 𝜄ℳ . Let h = f ◦ ( j ⊗ ∗ ̄ ℳ → ̄ j) ∶ 𝒜 ⊗ By composing the weights in (3) with the *-antiautomorphism ̃·𝒜 ,ℳ ◦ ( j𝒜 ⊗ ̄ 𝒜 and using 18.2.(3), we see that (3) is equivalent to the following equality: ℳ⊗ ̄ h) ◦ (𝛿𝒜 ⊗ ̄ 𝜄ℳ ) = [(𝜔𝒜 ⊗ ̄ h) ◦ (𝜄𝒜 ⊗̃ ̄ ·ℳ,𝒜 )](⋅ ⊗ ̄ 1𝒜 ) (𝜔𝒜 ⊗ ̄ ℳ, which is an obvious consequence of Proposition 18.18. of normal weights on 𝒜 ⊗ +

Thus, P𝛿 (x) ∈ (ℳ 𝛿 ) for every x ∈ ℳ + . Taking into account the properties of the Fubini +

mappings (9.8 and 12.18), it is now easy to check that P𝛿 ∶ ℳ + → (ℳ 𝛿 ) is a normal faithful operator-valued weight. Note that if 𝒩 ⊂ ℳ is an 𝒜 -subcomodule of the 𝒜 -comodule ℳ via 𝛿, then for the action 𝛿|𝒩 of 𝒜 on 𝒩 we have P𝛿|𝒩 = P𝛿 |𝒩 ̄ 𝒜 and, clearly, E𝜔𝒜 ◦ j𝒜 = E𝜔𝒜 ◦ j𝒜 |𝒩 ⊗ ̄ 𝒜. as 𝛿(𝒩 ) ⊂ 𝒩 ⊗ ℳ 𝒩

(4)

256

Crossed Products

̄ 𝒜 is an action of 𝒜 on ℳ and 𝒩 is any other W ∗ -algebra, then for the Also, if 𝛿 ∶ ℳ → ℳ ⊗ ̄ 𝛿∶𝒩 ⊗ ̄ ℳ→𝒩 ⊗ ̄ ℳ⊗ ̄ 𝒜 of 𝒜 on 𝒩 ⊗ ̄ ℳ we have action 𝜄𝒩 ⊗ P𝜄𝒩

̄ 𝛿 ⊗

̄ P𝛿 = 𝜄𝒩 ⊗

(5)

𝜔 ◦j ̄ E𝜔𝒜 ◦ j𝒜 . as E𝒩𝒜 ⊗̄ 𝒜ℳ = 𝜄𝒩 ⊗ ℳ If 𝛿 = 𝛿𝒜 is the action of the Kac algebra 𝒜 on itself, then 𝒜 𝛿 = ℂ ⋅ 1𝒜 (18.14) and P𝜎 coincides with the weight 𝜔𝒜 ◦ j𝒜 . ̄ 𝒜 is called integrable if the operator-valued weight P𝛿 is semifinite. The action 𝛿 ∶ ℳ → ℳ ⊗

18.20. Consider, in particular, a continuous action 𝜎 ∶ G → Aut(ℳ) of the locally compact group ̄ ℒ ∞ (G) of the Kac algebra G on the W ∗ -algebra ℳ and the corresponding action 𝜋𝜎 ∶ ℳ → ℳ ⊗ ∞ 𝜎 𝜋 𝜎 G = (ℒ (G), 𝜋G , kG , 𝜇G ) on ℳ. We recall that ℳ = ℳ (18.6.(4)) and put P𝜎 = P𝜋𝜎 . We shall +

show that in this case the operator-valued weight P𝜎 ∶ ℳ + → (ℳ 𝛿 ) is given by P𝜎 (x) =

∫

𝜎g (x) dg

(x ∈ ℳ + ).

(1)

Let x ∈ ℳ + , 𝜑 ∈ ℳ∗+ and h ∈ ℒ 1 (G) = ℒ ∞ (G)∗ . Then the element h ◦ kG ∈ ℒ ∞ (G)∗ , regarded as an ℒ 1 -function, has the expression (h ◦ kG )( g) = h( g−1 )ΔG ( g−1 ) ( g ∈ G ) see (18.5.(5)). Thus, ̄ (h ◦ kG )⟩ ⟨kG (E𝜑ℒ ∞ (G) (𝜋𝜎 (x))), h⟩ = ⟨𝜋𝜎 (x), 𝜑 ⊗ =

∫

𝜑(𝜎g−1 (x))h( g−1 )ΔG ( g−1 ) dg

=

∫

𝜑(𝜎g (x))h( g) dg

hence the element kG (E𝜑ℒ ∞ (G) (𝜋𝜎 (x))) ∈ ℒ ∞ (G) is the function G ∋ g ↦ 𝜑(𝜎g (x)). Consequently ̄ (𝜇G ◦ kG )⟩ = ⟨kG (E𝜑 ∞ (𝜋𝜎 (x))), 𝜇G ⟩ ⟨P𝜎 (x), 𝜑⟩ = ⟨𝜋𝜎 (x), 𝜑 ⊗ ℒ (G) ⟨ ⟩ = 𝜑(𝜎g (x)) dg = 𝜎g (x) dg, 𝜑 , ∫ ∫ proving (1). Equation (1) justifies the notion of an integrable action (18.19). Note that if G is compact, then P𝜎 is finite, that is, P𝜎 is a normal faithful conditional expectation of ℳ onto ℳ 𝜎 . Using (1) and 18.4.(2), we see that P𝜎 (𝜎t (x)) = ΔG (t)−1 P𝜎 (x) (x ∈ ℳ + , t ∈ G ). In particular, if G is unimodular, for example, abelian or compact, then P𝜎 ◦ 𝜎t = P𝜎 (t ∈ G ).

(2)

Hopf–von Neumann Algebras

257

18.21. Consider now the continuous action ̄ ℬ(ℒ 2 (G))) ̄ Ad(𝜌𝜌) ∶ G → Aut(ℳ ⊗ 𝜎⊗ where 𝜎 ∶ G → Aut(ℳ) is any continuous action of G on ℳ and 𝜌 ∶ G → ℬ(ℒ 2 (G)) is the right regular representation of G. ̄ ℬ(ℒ 2 (G)) ⊂ We shall assume ℳ ⊂ ℬ(ℋ ) realized as a von Neumann algebra and hence ℳ ⊗ 2 ℬ(ℒ (G, ℋ )). Moreover, we may assume (2.24) that there exists an so-continuous unitary representation G ∋ g ↦ v( g) ∈ ℬ(ℋ ) such that 𝜎g = Ad(v( g))( g ∈ G ). On the other hand, ̄ 𝜆 ( g) ∈ ℳ ⊗ ̄ ℬ(ℒ 2 (G))( g ∈ G ), and recall (18.6.(5)) that 𝜋𝜎 (𝜎g (x)) = we put u( g) = 1ℳ ⊗ ∗ u( g)𝜋𝜎 (x)u( g) (x ∈ ℳ, g ∈ G ). Every compactly supported w-continuous function f ∶ G → ℳ defines an element Tf𝜎 =

∫

̄ ℬ(ℒ 2 (G)). 𝜋𝜎 ( f (t))u(t) dt ∈ ℳ ⊗

(1)

On the other hand, every compactly supported s∗ -continuous function G × G ∋ (s, r) ↦ X(s, r) ∈ ̄ ℬ(ℒ 2 (G)) uniquely determined such that ℳ defines an element X ∈ ℳ ⊗ (X𝜉|𝜂) =

∫ ∫

(X(s, r)𝜉(r)|𝜂(s))dr ds

(𝜉, 𝜂 ∈ ℒ 2 (G, ℋ )).

(2)

It is easy to check that if the operator X is positive, then X( g, g) ≥ 0 for all g ∈ G. We shall show that if the operator X just defined is positive, then P𝜎 ⊗̄ Ad(𝜌𝜌) (X) = Tf𝜎 ,

(3)

where the compactly supported w-continuous function f ∶ G → ℳ is defined by f (t) =

∫

𝜎tr (X(tr, r))ΔG (r) dr

(t ∈ G ).

Indeed, for 𝜉 ∈ ℒ 2 (G, ℋ ) and g, s ∈ G we have ̄ 𝜌( g))(X))𝜉)(s) (((𝜎g ⊗Ad(𝜌 ̄ 𝜌 ( g))X(v( g) ⊗ ̄ 𝜌 ( g))∗ 𝜉)(s) = ((v( g) ⊗ ̄ 𝜌 ( g))∗ 𝜉)(sg) = ΔG ( g)1∕2 v( g)(X(v( g) ⊗ = ΔG ( g)1∕2 v( g) = v( g)

∫

∫

̄ 𝜌 ( g−1 ))𝜉)(r) dr X(sg, r)((v( g)∗ ⊗

X(sg, r)v( g)∗ 𝜉(rg−1 ) dr

= ΔG ( g)

∫

= ΔG ( g)

∫

v( g)X(sg, rg)v( g)∗ 𝜉(r) dr 𝜎g (X(sg, rg))𝜉(r) dr

(4)

258

Crossed Products

and, using 18.20.(1), we obtain (P𝜎 ⊗̄ Ad(𝜌𝜌) (X)𝜉|𝜉) =

∫ ∫ ∫

ΔG ( g)(𝜎g (X(sg, rg))𝜉(r)|𝜉(s)) dr ds dg.

On the other hand, for the function f defined by (4) we have (Tf𝜎 𝜉|𝜉) =

∫

(𝜋𝜎 ( f (t))u(t)𝜉|𝜉) dt

=

∫ ∫

((𝜋𝜎 ( f (t))u(t)𝜉)(s)|𝜉(s)) ds dt

=

∫ ∫

(𝜎s−1 ( f (t))(u(t)𝜉)(s)|𝜉(s)) ds dt

=

∫ ∫

(𝜎s−1 ( f (t))𝜉(t−1 s)|𝜉(s)) ds dt

=

∫ ∫ ∫

ΔG (r)(𝜎s−1 tr (X(tr, r))𝜉(t−1 s)|𝜉(s)) dr ds dt

∫ ∫ ∫

ΔG (t−1 sg)(𝜎g (X(sg, t−1 sg))𝜉(t−1 s)|𝜉(s)) dg ds dt

∫ ∫ ∫

ΔG ( g)(𝜎g (X(sg, rg))𝜉(r)|𝜉(s)) dg ds dr.

(via r = t−1 sg) = (via t = sr−1 ) =

Consequently, the element P𝜎 ⊗̄ Ad(𝜌𝜌) (X) is bounded and (3) holds. ̂ = (𝔏(G), 𝛿G , jG , 𝜔G ) on ̄ 𝛿G of the Kac algebra G 18.22. Finally, we consider the action 𝛿 = 𝜄ℳ ⊗ 2 ̄ ℳ ⊗ ℬ(ℒ (G)), where ℳ ⊂ ℬ(ℋ ) is a von Neumann algebra, and compute the values of the operator-valued weight P𝛿 on two types of elements. ̄ 𝜆 ( g) ∈ ℳ ⊗ ̄ ℬ(ℒ 2 (G)) ( g ∈ G ). Recall that 𝜔G ◦ jG = 𝜔G (18.17.(12)) and u( g) = 1ℳ ⊗ Let f ∶ G → ℳ be a compactly supported w-continuous function. ̄ 𝜆 ( g)) dg ∈ ℳ ⊗ ̄ 𝔏(G) and show that if Λf ≥ 0, We first consider the operator Λf = ∫ ( f ( g) ⊗ then 𝜔

EℳG (Λf ) = f (e). Indeed, for 𝜑 ∈ ℳ∗+ and k ∈ 𝒜 (G) we have ̄ k⟩ = ⟨E𝜑𝔏(G) (Λf ), k⟩ = ⟨Λf , 𝜑 ⊗ 𝜑( f ( g))k( g) dg ∫ ⟨ ⟩ 𝜆 𝜆(𝜑 ◦ f ), k⟩ = 𝜑( f ( g))𝜆 ( g) dg, k = ⟨𝜆 ∫ hence E𝜑𝔏(G) (Λf ) = 𝜆 (𝜑 ◦ f ) ≥ 0 and, using 18.17.(8), we obtain 𝜔

𝜆(𝜑 ◦ f ), 𝜔G ⟩ = ⟨f (e), 𝜑⟩ ⟨EℳG (Λf ), 𝜑⟩ = ⟨E𝜑𝔏(G) (Λf ), 𝜔G ⟩ = ⟨𝜆 which proves (1).

(1)

Hopf–von Neumann Algebras

259

Next, we consider a continuous action 𝜎 ∶ G → Aut(ℳ), we define the operator (18.21.(1)) Tf𝜎 = ̄ ℬ(ℒ 2 (G)) and show that ∫ 𝜋𝜎 ( f ( g))u( g) dg ∈ ℳ ⊗ P𝜄ℳ ⊗𝛿 ̄ G = 𝜋𝜎 ( f (e)).

(2)

̄ ℒ ∞ (G) = (ℳ ⊗ ̄ ℬ(ℒ 2 (G))𝛿 and 𝛿(u( g)) = u( g) ⊗ ̄ 𝜆 ( g), Indeed, we have 𝜋𝜎 (ℳ) ⊂ ℳ ⊗ 𝜎 ̄ 𝜆 ( g) dg so that, using (1), we obtain P𝛿 (T 𝜎 ) = E𝜔G (𝛿(T 𝜎 )) = hence 𝛿(Tf ) = ∫ ((𝜋𝜎 ( f ( g))u( g)) ⊗ ℳ f f 𝜋𝜎 ( f (e)). Given the compactly supported w-continuous functions f, f1 , f2 ∶ G → ℳ, we define the functions f ♯ , f1 ∗ f2 ∶ G → ℳ by f ♯ ( g) = Δg ( g)−1 f ( g−1 )∗ , ( f1 ∗ f2 )( g) =

∫

f1 (t)f2 (t−1 g) dt;

( g ∈ G ).

It is easy to check that (Λf )∗ = Λf ♯

Λf1 Λf2 = Λf1 ∗f2

(3)

(Tf𝜎 )∗

Tf𝜎 Tf𝜎 1 2

(4)

=

Tf𝜎♯

=

Tf𝜎∗f 1 2

and from (1) and (2), using the polarization relation, we infer that 𝜔

EℳG (Λ∗f Λf2 ) = (f1♯ ∗ f2 )(e)

(5)

♯ 𝜎 ∗ 𝜎 P𝜄ℳ ⊗𝛿 ̄ G ((Tf ) (Tf )) = 𝜋𝜎 ((f1 ∗ f2 )(e))

(6)

1

1

2

Consider now a compactly supported w-continuous function G × G ∋ (s, r) ↦ X(s, r) ∈ ℳ, the ̄ ℬ(ℒ 2 (G)) uniquely determined by equalities 18.21.(2) and the corresponding operator X ∈ ℳ ⊗ ∞ ̄ ℒ (G) defined by the compactly supported w-continuous function operator FX ∈ ℳ ⊗ FX ( g) = ΔG ( g)X( g, g)

( g ∈ G ).

(7)

If the operator X is positive, then P𝜄ℳ ⊗𝛿 ̄ G (X) = FX .

(8)

̄ 𝛿G )(X) ∈ ℳ ⊗ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ 𝔏(G) ⊂ ℬ(ℒ 2 (G × Indeed, we first compute the element (𝜄ℳ ⊗ G, ℋ )). For 𝜁 ∈ ℒ 2 (G × G, ℋ ), we have ̄ 𝛿G )(X)𝜁|𝜁) = ((1ℳ ⊗ ̄ W ∗ )(X ⊗ ̄ 1G )(1ℳ ⊗ ̄ WG )𝜁|𝜁) ((𝜄ℳ ⊗ G ̄ 1G )(1ℳ ⊗ ̄ WG )𝜁 |(1ℳ ⊗ ̄ WG )𝜁) = ((X ⊗ =

∫ ∫ ∫

=

∫ ∫ ∫

=

∫ ∫ ∫

=

∫ ∫ ∫

̄ WG 𝜁)(r, t))|((1ℳ ⊗ ̄ WG )𝜁)(s, t)) dr ds dt (X(s, r)(((1ℳ ⊗ (X(s, r)𝜁(r, rt)|𝜁 (s, st)) dr ds dt (X(s, r)𝜁(r, rs−1 t)|𝜁(s, t)) dr ds dt (X(s, r−1 s)𝜁 (r−1 s, r−1 t)|𝜁(s, t))ΔG (r−1 s) dr ds dt.

260

Crossed Products

̄ ℬ(ℒ 2 (G))+ with 𝜉 ∈ 𝒦 (G, ℋ ) and k ∈ 𝒜 (G). There exists 𝜂 ∈ ℒ 2 (G) Let 𝜑 = 𝜔𝜉 ∈ ℳ ⊗ ∗ such that k = 𝜔𝜂 , that is, 𝜆(r), 𝜔𝜂 ⟩ = (𝜆 𝜆(r)𝜂|𝜂) = k(r) = ⟨𝜆

∫

𝜂(r−1 t)𝜂(t) dt.

̄ k = 𝜔𝜁 with 𝜁 = 𝜉 ⊗ ̄ 𝜂, that is, 𝜁 (s, t) = 𝜉(s)𝜂(t), and by the above computation it follows Then 𝜑 ⊗ that ̄ 𝛿G )(X), 𝜑 ⊗ ̄ k⟩ ⟨E𝜑𝔏(G) (𝛿(X)), k⟩ = ⟨(𝜄ℳ ⊗ ( ) 𝜆( f ), k⟩ = k(r) ΔG (r−1 s)(X(s, r−1 s)𝜉(r−1 s)|𝜉(s)) ds dr = ⟨𝜆 ∫ ∫ where f ∈ 𝒦 (G) is defined by f (r) =

∫

ΔG (r−1 s)(X(s, r−1 s)𝜉(r−1 s)|𝜉(s)) ds

(r ∈ G ).

(9)

Consequently, 0 ≤ E𝜑𝔏(G) (𝛿(X)) = 𝜆 ( f ) and, using 18.17.(8), it follows that 𝜔 ̄ 𝜔G ⟩ (P𝛿 (X)𝜉|𝜉) = ⟨EℳG ⊗̄ ℬ(ℒ 2 (G)) (𝛿(X)), 𝜑⟩ = ⟨𝛿(X), 𝜑 ⊗

𝜆( f ), 𝜔G ⟩ = f (e) = ⟨E𝜑𝔏(G) (𝛿(X)), 𝜔G ⟩ = ⟨𝜆 =

∫

ΔG (s)(X(s, s)𝜉(s)|𝜉(s)) dr = (FX 𝜉|𝜉).

Therefore, the operator P𝛿 (X) is bounded and P𝛿 (X) = FX . 18.23. Notes. The extension of the Pontryagin and Tannaka duality theories (see Hewitt & Ross, 1963, 1970) to a duality in the category of Kac algebras has been developed in a long series of papers by several different mathematicians: Segal (1953); Stinespring (1959); Kac (1963); Eymard (1964); Tatsuuma (1967); Ernest (1966); Takesaki (1972b); Vainerman and Kac (1973, 1974); Enock and Schwartz (1975); Kirchberg (1975), and so on. The actions of Hopf–von Neumann algebras on W ∗ -algebras have been considered in Strătilă, Voiculescu, and Zsidó (1976, 1977). The notions of (𝛿, j𝒜 )-invariance, Theorem 18.12, and Proposition 18.14, and also the computations given in Sections 18.21 and 18.22, are from Strătilă et al. (1976, 1977). Proposition 18.19 was proved in several special cases in Haagerup (1978); Landstad (1979); Strătilă et al. (1976, 1977) and the general case was asserted and used in Enock (1977), but the complete proof of it in the general case has only appeared in Zsidó (1978). For our exposition we have used: Haagerup (1978); Strătilă et al. (1976, 1977); Takesaki (1972b); and Zsidó (1978).

Crossed Products

261

19 Crossed Products In this section, we introduce the crossed product of a W ∗ -algebra by the continuous action of a locally compact group as a particular case of crossed products by actions of Kac algebras. The main properties of crossed products, including the duality theory, are described. ̄ 𝒜 be an action of the Kac algebra (𝒜 , 𝛿𝒜 , j𝒜 , 𝜔𝒜 ) on the W ∗ -algebra 19.1. Let 𝛿 ∶ ℳ → ℳ ⊗ ℳ. Consider the W ∗ -algebra 𝒜 realized as a von Neumann algebra 𝒜 ⊂ ℬ(ℋ𝒜 ) in the standard representation associated with the weight 𝜔𝒜 . As we have seen in Section 18.15, one can construct a dual Kac algebra (𝒜̂, 𝛿̂𝒜 , ̂j𝒜 , 𝜔̂ 𝒜 ), where 𝒜̂ ⊂ ℬ(ℋ𝒜 ) is a von Neumann algebra also acting on ℋ𝒜 . We define the crossed product of the W ∗ -algebra ℳ by the action 𝛿 of the Kac algebra 𝒜 on ℳ to be the W ∗ -algebra ̄ ℬ(ℋ𝒜 ) ℛ(ℳ, 𝛿) ⊂ ℳ ⊗ ̄ 𝒜̂. generated by 𝛿(ℳ) and 1ℳ ⊗ In particular, let G be a locally compact group and let G = (ℒ ∞ (G), 𝜋G , kG , 𝜇G ) and ̂ = (𝔏(G), 𝛿G , jG , 𝜔G ) be the two associated Kac algebras, which are dual to one another G (18.16, 18.17). If 𝜎 ∶ G → Aut(ℳ) is a continuous action of G on the W ∗ -algebra ℳ, then 𝜋𝜎 ∶ ℳ → ̄ ℒ ∞ (G) is an action of G on ℳ (18.6) and, according to the general definition above, the ℳ⊗ crossed product of the W ∗ -algebra ℳ by the continuous action 𝜎 of G on ℳ is the W ∗ -algebra ̄ ℬ(ℒ 2 (G)) ℛ(ℳ, 𝜎) ⊂ ℳ ⊗ ̄ 𝔏(G), that is, generated by 𝜋𝜎 (ℳ) and 1ℳ ⊗ ̄ 𝜆 ( g); x ∈ ℳ, g ∈ G}. ℛ(ℳ, 𝜎) = ℛ{𝜋𝜎 (x), 1ℳ ⊗

(1)

Recall (18.6.(5)) that ̄ 𝜆 ( g))𝜋𝜎 (x)(1ℳ ⊗ ̄ 𝜆 ( g))∗ 𝜋𝜎 (𝜎g (x)) = (1ℳ ⊗

(x ∈ ℳ, g ∈ G ),

(2)

which constitutes another proof of the fact (2.24) that there exists a realization ℳ ⊂ ℬ(ℋ ) of ℳ as a von Neumann algebra and an so-continuous unitary representation G ∋ g ↦ v( g) ∈ ℬ(ℋ ) with 𝜎g (x) = v( g)xv( g)∗

(x ∈ ℳ, g ∈ G ).

(3)

̂ on ℳ, then the corresponding crossed ̄ 𝔏(G) is an action of the Kac algebra G If 𝛿 ∶ ℳ ↦ ℳ ⊗ ∗ product is the W -algebra ̄ ℬ(ℒ 2 (G)) ℛ(ℳ, 𝛿) ⊂ ℳ ⊗ ̄ ℒ ∞ (G), that is, generated by 𝛿(ℳ) and 1ℳ ⊗ ̄ f; x ∈ ℳ, f ∈ ℒ ∞ (G)}. ℛ(ℳ, 𝛿) = ℳ{𝛿(x), 1ℳ ⊗

(4)

262

Crossed Products

In what follows we are interested only in crossed products by continuous actions of groups. However, the appearance of certain important duality phenomena necessitates the consideration also ̂ of crossed products by actions of Kac algebras of type G. Throughout this section, G will denote a locally compact group. 19.2 Lemma. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W ∗ -algebra ℳ. Then ̄ ℒ ∞ (G) = ℛ{𝜋𝜎 (ℳ), 1ℳ ⊗ ̄ ℬ(ℒ ∞ (G))} ℳ⊗ ̄ ℬ(ℒ 2 (G))} ̄ ℬ(ℒ 2 (G)) = ℛ{𝜋𝜎 (ℳ), 1ℳ ⊗ ℳ⊗ ̄ ℒ ∞ (G)}. = ℛ{ℛ(ℳ, 𝜎), 1ℳ ⊗ ̄

̄ ℒ ∞ (G))𝜎 ⊗ Ad(𝜌𝜌) = 𝜋𝜎 (ℳ). (ℳ ⊗

(1) (2) (3)

Proof. We may assume ℳ ⊂ ℬ(ℋ ) realized as a von Neumann algebra such that there exists an so-continuous unitary representation v ∶ G → ℬ(ℋ ) with 𝜎g = Ad(v( g))( g ∈ G ). Let V be the ̄ ℒ 2 (G) defined by the bounded so-continuous function g ↦ v( g), that is, unitary operator on ℋ ⊗ (V𝜉)( g) = v( g)𝜉( g) (𝜉 ∈ ℒ 2 (G, ℋ ), g ∈ G ). It is easy to check that the *-automorphism 𝜃 = Ad(V ∗ ) = V ∗ ⋅V leaves invariant the von Neumann ̄ ℒ ∞ (G) ⊂ ℬ(ℋ ⊗ ̄ ℒ 2 (G)); in fact algebra ℳ ⊗ ̄ 1G ) = 𝜋𝜎 (x) (x ∈ ℳ) 𝜃(x ⊗ ̄ f ) = 1ℳ ⊗ ̄ f ( f ∈ ℒ ∞ (G)). 𝜃(1ℳ ⊗

(4) (5)

̄ ℒ ∞ (G) = 𝜃(ℳ ⊗ ̄ ℒ ∞ (G)) = 𝜃(ℛ{ℳ ⊗ ̄ 1G , 1ℳ ⊗ ̄ ℒ ∞ (G)}) = Consequently, ℳ ⊗ ∞ ̄ ℒ (G)}, proving (1). Equation (2) follows from (1) because ℬ(ℒ 2 (G)) ℛ{𝜋𝜎 (ℳ), 1ℳ ⊗ ∞ ̄ 𝔏(G)}. = ℛ{𝔏(G), ℒ (G)} and, by definition, ℛ(ℳ, 𝜎) = ℛ{𝜋𝜎 (ℳ), 1ℳ ⊗ On the other hand, it is easy to check that ̄ Ad(𝜌𝜌( g))) ◦ 𝜃 = 𝜃 ◦ (𝜄ℳ ⊗ ̄ Ad(𝜌𝜌( g))) (𝜎g ⊗

(g ∈ G)

(6)

̄ ℒ ∞ (G). Consequently, for X ∈ ℳ ⊗ ̄ ℒ ∞ (G), we have X ∈ as *-automorphisms on ℳ ⊗ ̄ 𝜌 ∞ 𝜎 ⊗ Ad(𝜌 ) ̄ ℒ (G)) ̄ Ad(𝜌𝜌( g)))X = X( g ∈ G ) ⇔ (𝜄ℳ ⊗ ̄ Ad(𝜌𝜌( g)))𝜃 −1 X = 𝜃 −1 X, ( g ∈ (ℳ ⊗ ⇔ (𝜎g ⊗ ̄ 1G ⇔ X ∈ 𝜃(ℳ ⊗ ̄ 1G ) = 𝜋𝜎 (ℳ), since (ℳ ⊗ ̄ ℒ ∞ (G))𝜄⊗̄ Ad(𝜌𝜌) = ℳ ⊗ ̄ 1G . G ) ⇔ 𝜃 −1 X ∈ ℳ ⊗ This proves (3). ̄ ℒ ∞ (G) we have Similarly, one can show that for X ∈ ℳ ⊗ ̄ 𝜄G )(X) = (𝜄ℳ ⊗ ̄ 𝜋G )(X) X ∈ 𝜋𝜎 (ℳ) ⇔ (𝜋𝜎 ⊗

(7)

and that if 𝒩 ⊂ ℳ is a unital W ∗ -subalgebra, then for x ∈ ℳ we have ̄ ℒ ∞ (G) ⇒ x ∈ 𝒩 . 𝜋𝜎 (x) ∈ 𝒩 ⊗

(8)

̄ ℬ(ℒ 2 (G)) be the crossed product of the W ∗ -algebra ℳ by the 19.3. Let ℛ(ℳ, 𝜎) ⊂ ℳ ⊗ continuous action 𝜎 ∶ G → Aut(ℳ) of G on ℳ.

Crossed Products

263

̂ on ℳ ⊗ ̄ 𝛿G of G ̄ ℬ(ℒ 2 (G)). Since, by 18.7.(11), its centralizer Consider also the action 𝛿 = 𝜄ℳ ⊗ ∞ ∞ ̄ ℒ (G) and since 𝜋𝜎 (ℳ) ⊂ ℳ ⊗ ̄ ℒ (G), we have is ℳ ⊗ ̄ 1G ∈ ℛ(ℳ, 𝜎) ⊗ ̄ 𝔏(G) 𝛿(𝜋𝜎 (x)) = 𝜋𝜎 (x) ⊗

(x ∈ ℳ).

(1)

On the other hand (18.7.(2)), ̄ 𝜆 ( g)) = 1ℳ ⊗ ̄ 𝜆 ( g) ⊗ ̄ 𝜆 ( g) ∈ ℛ(ℳ, 𝜎) ⊗ ̄ 𝔏(G) 𝛿(1ℳ ⊗

( g ∈ G ).

(2)

̂ ̄ 𝔏(G) and hence ℛ(ℳ, 𝜎) is a G-subcomodule Thus (19.1.(1)), 𝛿(ℛ(ℳ, 𝜎)) ⊂ ℛ(ℳ, 𝜎) ⊗ of the ̂ ̄ ℬ(ℒ 2 (G)) via 𝛿 = 𝜄ℳ ⊗ ̄ 𝛿G . The restriction of the action 𝜄ℳ ⊗ ̄ 𝛿G to ℛ(ℳ, 𝜎) G-comodule ℳ⊗ is denoted by ̄ 𝔏(G) 𝜎̂ ∶ ℛ(ℳ, 𝜎) → ℛ(ℳ, 𝜎) ⊗ ̂ on ℛ(ℳ, 𝜎). and is called the dual action of G If G is commutative, then, according to Proposition 18.9, the dual action is determined by the continuous action ̂ → Aut(ℛ(ℳ, 𝜎)) 𝜎̂ ∶ G defined by ̂ ̄ 𝔪(𝛾))∗ X(1ℳ ⊗ ̄ 𝔪(𝛾)) (X ∈ ℛ(ℳ, 𝜎), 𝛾 ∈ G). 𝜎̂ 𝛾 (X) = (1ℳ ⊗ and equalities (1) and (2) can be reformulated as follows (see 18.9.(1)): 𝜎̂ 𝛾 (𝜋𝜎 (x)) = 𝜋𝜎 (x)

̂ (x ∈ ℳ, 𝛾 ∈ G)

̂ ̄ 𝜆 ( g)) = ⟨g, 𝛾⟩(1ℳ ⊗ ̄ 𝜆 ( g)) ( g ∈ G, 𝛾 ∈ G). 𝜎̂ 𝛾 ((1ℳ ⊗

(3) (4)

̄ ℬ(ℒ 2 (G)). ̄ Ad(𝜌𝜌) of G on ℳ ⊗ In the general case, we also consider the continuous action 𝜎 ⊗ Using 19.2.(3), 18.4.(14), and the definition 19.1.(1) of the crossed product, we see that ̄ ℬ(ℒ 2 (G))𝜎 ⊗̄ Ad(𝜌𝜌) . ℛ(ℳ, 𝜎) ⊂ (ℳ ⊗

(5)

̄ ℬ(ℒ 2 (G)) and 𝛽 = 𝜎 ⊗ ̄ Ad(𝜌𝜌); recall In the sequel, we shall write 𝒩 = ℛ(ℳ, 𝜎), 𝒫 = ℳ ⊗ ̄ also the notation 𝛿 = 𝜄ℳ ⊗ 𝛿G . ̄ 𝜄G )(𝛿G (x)) (t ∈ G ), is easily verified for x = f ∈ The identity, 𝛿G (Ad(𝜌𝜌(t))(x)) = (Ad(𝜌𝜌(t)) ⊗ ∞ 𝜆 ℒ (G) and for x = ( g) ∈ 𝔏(G), and so it remains valid for any x ∈ ℬ(ℒ 2 (G)) (see 18.4.(15)). Consequently, ̄ 𝜄G ) ◦ 𝛿 𝛿 ◦ 𝛽g = (𝛽g ⊗

(g ∈ G)

that is, the actions 𝛿 and 𝛽 on 𝒫 commute. It follows that ̄ 𝔏(G) X ∈ 𝒫 𝛿 ⇒ 𝛽g (X) ∈ 𝒫 𝛿 and X ∈ 𝒫 𝛽 ⇒ 𝛿(X) ∈ 𝒫 𝛽 ⊗

(6)

264

Crossed Products

hence, by restriction, we obtain a continuous action 𝛽 ∶ G → Aut(𝒫 𝛿 ) of G on 𝒫 𝛿 , ̂ on 𝒫 𝛽 . ̄ 𝔏(G) of G an action 𝛿 ∶ 𝒫 𝛽 → 𝒫 𝛽 ⊗

(7) (8)

Proposition. For every continuous action 𝜎 ∶ G → Aut(ℳ) of G on the W ∗ -algebra ℳ we have 𝜋𝜎 (ℳ) = ℛ(ℳ, 𝜎)𝜎̂

(9)

where 𝜎̂ is the dual action. Proof. With the above notation, we have 𝜋𝜎 (ℳ) ⊂ 𝒩 𝛿 (by (1)), 𝒩 ⊂ 𝒫 𝛽 (by (5)) and 𝒫 𝛿 = ̄ ℒ ∞ (G) (by 18.7.(11)). Since 𝛽 and 𝛿 commute, using Lemma 19.2.(3), we obtain 𝜋𝜎 (ℳ) ⊂ ℳ⊗ 𝛿 ̄ ℒ ∞ (G))𝛽 = 𝜋𝜎 (ℳ) and hence 𝜋𝜎 (ℳ) = 𝒩 𝛿 . 𝒩 ⊂ (𝒫 𝛽 )𝛿 = (𝒫 𝛿 )𝛽 = (ℳ ⊗ This proof also showed that (𝒫 𝛽 )𝛿 = 𝜋𝜎 (ℳ).

(10)

̄ f ∈ 𝒫 is a unital injective normal Finally, note that the mapping 𝜒 ∶ ℒ ∞ (G) ∋ f ↦ 1ℳ ⊗ *-homomorphism and that 𝛽g (𝜒( f )) = 𝜒(Ad(𝜌𝜌( g))( f )) ( f ∈ ℒ ∞ (G), g ∈ G ).

(11)

̄ ℬ(ℒ 2 (G)) of the W ∗ -algebra ℳ by the 19.4. Consider now the crossed product ℛ(ℳ, 𝛿) ⊂ ℳ ⊗ ̂ ̄ 𝔏(G) of G on ℳ. action 𝛿 ∶ ℳ → ℳ ⊗ ̄ Ad(𝜌𝜌) of G on ℳ ⊗ ̄ ℬ(ℒ 2 (G)). Since, by Consider also the continuous action 𝜎 = 𝜄ℳ ⊗ ̄ 𝔏(G) and since 𝛿(ℳ) ⊂ ℳ ⊗ ̄ 𝔏(G), we have 18.4.(14), the centralizer of this action is ℳ ⊗ 𝜎g (𝛿(x)) = 𝛿(x) ∈ ℛ(ℳ, 𝛿)

(x ∈ ℳ, g ∈ G ).

(1)

On the other hand, ̄ f ) = 1ℳ ⊗ ̄ (Ad(𝜌𝜌( g))( f )) ∈ ℛ(ℳ, 𝛿) ( f ∈ ℒ ∞ (G), g ∈ G ). 𝜎g (1ℳ ⊗

(2)

Thus (19.1.(4)), 𝜎g (ℛ(ℳ, 𝛿)) = ℛ(ℳ, 𝛿) ( g ∈ G ). The restriction of the continuous action ̄ Ad(𝜌𝜌) to ℛ(ℳ, 𝛿) is denoted by 𝜎 = 𝜄ℳ ⊗ 𝛿̂ ∶ G → Aut(ℛ(ℳ, 𝛿)) and is called the dual action of G on ℛ(ℳ, 𝛿). If G is commutative, then the method of defining the dual action given in that section leads to the same result as the procedure of Section 19.3. This can easily be verified using the Fourier–Plancherel isomorphism (18.8.(4); a continuous action of G can be regarded also as an action of the dual object ̂ of the group G. 19.5. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ. We then have a dual action ̂ on the crossed product ℛ(ℳ, 𝜎) (19.3) and also a second ̄ 𝔏(G) of G 𝜎̂ ∶ ℛ(ℳ, 𝜎) → ℛ(ℳ, 𝜎) ⊗

Crossed Products

265

dual action 𝜎̂̂ ∶ G → Aut(ℛ(ℛ(ℳ, 𝜎), 𝜎)) ̂ of G on the second crossed product ℛ(ℛ(ℳ, 𝜎), 𝜎) ̂ (19.4). Theorem. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact group G on the W ∗ -algebra ℳ. There exists a *-isomorphism ̄ ℬ(ℒ 2 (G)) → ℛ(ℛ(ℳ, 𝜎), 𝜎) ̂ Φ∶ℳ ⊗ such that ̄ Ad(𝜌𝜌( g))) 𝜎̂̂ g ◦ Φ = Φ ◦ (𝜎g ⊗

( g ∈ G ).

Briefly, we shall write ̂̂ ≈ (ℳ ⊗ ̄ ℬ(ℒ 2 (G)); 𝜎 ⊗ ̄ Ad(𝜌𝜌)). (ℛ(ℛ(ℳ, 𝜎), 𝜎)); ̂ 𝜎)

(1)

̄ ℬ Proof. We shall use the notation introduced in Section 19.3, that is, 𝒩 = ℛ(ℳ, 𝜎), 𝒫 = ℳ ⊗ 2 ̄ ̄ (ℒ (G)), 𝛽 = 𝜎 ⊗ Ad(𝜌𝜌) and 𝛿 = 𝜄ℳ ⊗ 𝛿G ; recall that 𝜎̂ = 𝛿|𝒩 . ̄ ℒ ∞ (G) defined by the function g ↦ For each X ∈ 𝒫 , we consider the element Ψ(X) ∈ 𝒫 ⊗ 𝛽g (X), that is, ̄ k⟩ = ⟨Ψ(X), 𝜑 ⊗

∫

𝜑(𝛽g (X))k( g) dg

(𝜑 ∈ 𝒫∗ , k ∈ ℒ 1 (G)).

(2)

̄ ℬ(ℒ 2 (G))) is a unital injective It is easy to check that Ψ ∶ 𝒫 ∋ X ↦ Ψ(X) ∈ 𝒫 ⊗ normal *-homomorphism. We define another unital injective normal *-homomorphism Φ ∶ 𝒫 → ̄ ℬ(ℒ 2 (G)) by 𝒫 ⊗ ̄ W ∗ )Ψ(X)(1ℳ ⊗ ̄ WG ) (X ∈ 𝒫 ). Φ(X) = (1ℳ ⊗ G

(3)

̄ 1G and from (3) and 18.7.(10) If X ∈ 𝒩 ⊂ 𝒫 𝛽 (19.3.(5)), then from (2) it follows that Ψ(X) = X ⊗ we obtain ̂ (X ∈ 𝒩 ). Φ(X) = 𝛿(X) = 𝜎(X)

(4)

̂

Since 𝜎(𝒩 ̂ ) ⊂ ℛ(𝒩 , 𝜎) ̂ 𝜎̂ (19.4.(1)) and 𝒩 ⊂ 𝒫 𝛽 (19.3.(5)), it follows that 𝜎̂̂ g (Φ(X)) = Φ(X) = Φ(𝛽g (X)) (X ∈ 𝒩 , g ∈ G ).

(5)

̄ f ) is the element F ∈ On the other hand, if f ∈ ℒ ∞ (G), then it is easy to check that Ψ(1ℳ ⊗ ∞ ∞ 2 ̄ ̄ ̄ ℳ ⊗ ℒ (Gs ) ⊗ ℒ (Gt ) ⊂ 𝒫 ⊗ ℬ(ℒ (Gt )) defined by the function F ∶ G × G ∋ (s, t) ↦ f (st) ⋅ 1ℳ ∈ ℳ. For every 𝜉 ∈ ℒ 2 (G × G, ℋ ) we have ̄ W ∗ )F(1ℳ ⊗ ̄ WG )𝜉)(s, t) = (F(1ℳ ⊗ ̄ WG )𝜉)(s, s−1 t) ((1ℳ ⊗ G ̄ WG )𝜉)(s, s−1 t) = f (t)𝜉(s, t) = ((1ℳ ⊗ ̄ 1G ⊗ ̄ f )𝜉)(s, t) = F(s, s−1 t)((1ℳ ⊗

266

Crossed Products

hence ̄ f ) = 1𝒫 ⊗ ̄ f Φ(1ℳ ⊗

( f ∈ ℒ ∞ (G)).

(6)

Using 19.4.(2), we obtain ̄ f )) = 1𝒫 ⊗ ̄ (Ad(𝜌𝜌( g))( f )) = Φ(𝛽g (1ℳ ⊗ ̄ f )) 𝜎̂̂ g (Φ(1ℳ ⊗ ∞ ( f ∈ ℒ (G), g ∈ G ).

(7)

Thus, using Lemma 19.2.(2), the definition of ℛ(𝒩 , 𝜎) ̂ (19.1.(4)) and (4) and (6), we get ̄ ℒ ∞ (G)}) = ℛ{𝜎(𝒩 ̄ ℒ ∞ (G)} = ℛ(𝒩 , 𝜎), Φ(P) = Φ(ℛ{𝒩 , 1ℳ ⊗ ̂ ), 1𝒫 ⊗ ̂ and from (5) and (7) it follows that 𝜎̂̂ g ◦ Φ = Φ ◦ 𝛽g . The above Theorem is usually called “the Takesaki duality theorem.” 19.6. In this section, we summarize the essential facts concerning crossed products by continuous group actions. Thus, let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W ∗ -algebra ℳ and 𝒩 = ℛ(ℳ, 𝜎). Using the *-isomorphism 𝜋𝜎 ∶ ℳ → 𝜋𝜎 (ℳ) ⊂ 𝒩 , we shall identify ℳ with a unital W ∗ -subalgebra of 𝒩 , hence ℳ ⊂ 𝒩 . There exist an s-continuous unitary representation ̄ 𝜆 ( g) ∈ 𝒩 G ∋ f ↦ u( g) = 1 ⊗

(1)

̂ on 𝒩 (19.3) and a dual action of G ̄ 𝔏(G), 𝛿 = 𝜎̂ ∶ 𝒩 → 𝒩 ⊗

(2)

such that (19.3.(2)) ̄ 𝜆 ( g) 𝛿(u( g)) = u( g) ⊗

( g ∈ G ),

(3)

which characterize the W ∗ -subalgebra ℳ of 𝒩 (Proposition 19.3) ℳ = 𝒩 𝛿,

(4)

and the continuous action 𝜎 of G on ℳ (19.1.(2)), 𝜎g (x) = u( g)xu( g)∗

(x ∈ ℳ, g ∈ G ).

(5)

In particular, 𝒩 = ℛ{𝒩 𝛿 , u(G)}.

(6)

̂ If G is commutative, then the dual action is a continuous action of the dual group G ̂ → Aut(𝒩 ) 𝜃 = 𝜎̂ ∶ G

(2’)

Crossed Products

267

such that (19.3.(4)) ̂ 𝜃𝛾 (u( g)) = ⟨g, 𝛾⟩u( g) ( g ∈ G, 𝛾 ∈ G) ℳ=𝒩

𝛿

(3’) (4’)

𝛿

𝒩 = ℛ{𝒩 , u(G)}.

(5’)

We shall show that, conversely, the existence of a unitary representation (1) and an action (2) (resp. (2’)), which satisfy the commutation relations (3) (resp. (3’)) implies the fact that 𝒩 is a crossed product ℛ(ℳ, 𝜎), where ℳ and 𝜎 are determined by (4) (resp. (4’)) and (5), in particular the generation relation (6) (resp. (6’)) holds. To this end, we need some preliminary results that are of independent interest and will also be used in other situations. 19.7 Proposition. Let 𝒩 be a W ∗ -algebra with the property that there exist an s-continuous unitary ̂ on 𝒩 such that ̄ 𝔏(G) of G representation u ∶ G → 𝒩 and an action 𝛿 ∶ 𝒩 → 𝒩 ⊗ ̄ 𝜆 ( g) 𝛿(u( g)) = u( g) ⊗

( g ∈ G ).

(1)

+

Then the faithful normal operator-valued weight P𝛿 ∶ 𝒩 + → (𝒩 𝛿 ) , 𝜔

P𝛿 (x) = E𝒩G (𝛿(x)) (x ∈ 𝒩 + )

(2)

is semifinite and P𝛿 (u( g)xu( g)∗ ) = ΔG ( g)u( g)P𝛿 (x)u( g)∗

(x ∈ 𝒩 + , g ∈ G ).

(3)

For every k ∈ 𝒦 (G) with u(k) ≥ 0 we have P𝛿 (u(k)) = k(e) ⋅ 1G .

(4)

For all h, k ∈ 𝒦 (G) we have u(h), u(k) ∈ 𝔑P𝛿 and P𝛿 (u(h)∗ u(k)) = (h∗ ∗ k)(e) ⋅ 1G .

(5)

Proof. Recall (18.17.(12)) that 𝜔G ◦ jG = 𝜔G . Thus, using the definition (2) (or 18.19.(1)) of P𝛿 , (1) and 18.17.(8), for k ∈ 𝒦 (G) with u(k) ≥ 0 and 𝜓 ∈ 𝒩∗+ we have ⟩ ⟨ ( ) ̄ 𝜔G ⟩ = 𝛿 ̄ 𝜔G ⟨P𝛿 (u(k)), 𝜓⟩ = ⟨𝛿(u(k)), 𝜓 ⊗ k( g)u( g) dg , 𝜓 ⊗ ∫ ⟨ ⟩ ⟨ ⟩ ̄ 𝜆 ( g)) dg, 𝜓 ⊗ ̄ 𝜔G = 𝜆( g) dg, 𝜔G = k( g)(u( g) ⊗ k( g)𝜓(u( g))𝜆 ∫ ∫ 𝜆(k(⋅)𝜓(u(⋅))), 𝜔G ⟩ = ⟨k(e)𝜓(u(e)) = (k(e) ⋅ 1𝒩 , 𝜓⟩, = ⟨𝜆 which proves (4).

268

Crossed Products

For every k ∈ 𝒦 (G), we have k♯ ∗ k ∈ 𝒦 (G) and u(k♯ ∗ k) = u(k)∗ u(k) ≥ 0, hence we infer from (4) that u(k) ∈ 𝔑P𝛿 and P𝛿 (u(k)∗ u(k)) = (k♯ ∗ k)(e) ⋅ 1𝒩 . Thus (5) follows on applying the polarization relation. s If {ki } is a net in 𝒦 (G) such that u(ki ) → 1𝒩 , then for every x ∈ 𝒩 we have xu(ki ) ∈ 𝔑P𝛿 and s

xu(ki ) → s. Hence P𝛿 is semifinite. Finally, using 18.17.(13), we obtain for x ∈ 𝒩 + , g ∈ G and 𝜓 ∈ 𝒩∗+ ̄ 𝜔G ⟩ ⟨P𝛿 (u( g)xu( g)∗ ), 𝜓⟩ = ⟨𝛿(u( g)xu( g)∗ ), 𝜓 ⊗ ̄ ̄ 𝜆 ( g))∗ , 𝜓 ⊗ ̄ 𝜔G ⟩ = ⟨(u( g) ⊗ 𝜆 ( g))𝛿(x)(u( g) ⊗ ̄ (𝜔G ◦ Ad(𝜆 𝜆( g))) = ⟨𝛿(x), (𝜓 ◦ Ad(u( g))) ⊗ ̄ 𝜔G ⟩ = ΔG ( g)⟨𝛿(x), 𝜓 ◦ Ad(u( g))) ⊗ = ΔG ( g)⟨P𝛿 (x), 𝜓 ◦ Ad(u( g))⟩ = ΔG ( g)⟨u( g)P𝛿 (x)u( g)∗ , 𝜓⟩, which proves (3). If G is commutative, then it is more convenient to state the above proposition as follows: Let 𝒩 be a W ∗ -algebra with the property that there exist an s-continuous unitary representation ̂ on 𝒩 such that ̂ → Aut(𝒩 ) of G u ∶ G → 𝒩 and a continuous action 𝜃 ∶ G ̂ 𝜃𝛾 (u( g)) = ⟨g, 𝛾⟩u( g) ( g ∈ G, 𝛾 ∈ G).

(1’)

Then the faithful normal operator-valued weight P𝜃 ∶ 𝒩 + → (𝒩 𝜃 )+ , P𝜃 (x) =

∫

𝜃𝛾 (x) d𝛾

(x ∈ 𝒩 + )

(2’)

is semifinite and (x ∈ 𝒩 + , g ∈ G ).

P𝜃 (u( g)xu( g)∗ ) = u( g)P𝜃 (x)u( g)∗

(3’)

Also, identities similar to (4) and (5) hold. In this case, the main part of the proof reduces to an application of the Fourier inversion theorem. Indeed, for any positive definite function k ∈ 𝒦 (G) and every 𝜓 ∈ 𝒩∗+ we have ⟨ ⟨P𝜃 (u(k)), 𝜓⟩ = =

( ∫

∫

𝜃𝛾

⟨e, 𝛾⟩

)

∫ ( ∫

k( g)u( g) dg

⟩ d𝛾, 𝜓 )

⟨g, 𝛾⟩k( g)𝜓(u( g)) dg

d𝛾 = k(e)𝜓(u(e)) = ⟨k(e) ⋅ 1𝒩 , 𝜓).

19.8. In particular, for every continuous action 𝜎 ∶ G → Aut(ℳ), the dual action 𝜎̂ ∶ ℛ(ℳ, 𝜎) → ̄ 𝔏(G) defines an n.s.f. operator-valued weight ℛ(ℳ, 𝜎) ⊗ +

P𝜎̂ ∶ ℛ(ℳ, 𝜎)+ → 𝜋𝜎 (ℳ) .

Crossed Products

269

Thus, if 𝜑 is an n.s.f. weight on ℳ, then, according to Proposition 11.6, 𝜑̂ = 𝜑 ◦ 𝜋𝜎−1 ◦ P𝜎̂ is an n.s.f. weight on ℛ(ℳ, 𝜎), called the dual weight of 𝜑. The theorem in this section will characterize the dual weight of a given 𝜑 and also the set of all dual weights on ℛ(ℳ, 𝜎). In order to obtain such characterizations, it is necessary to know the values of 𝜑̂ on a wide dass of elements of ℛ(ℳ, 𝜎). For this reason, we begin by defining and studying this ̄ 𝜆 ( g) ( g ∈ G ). class. As usual, we put u( g) = 1ℳ ⊗ Every compactly supported w-continuous function f ∶ G → ℳ defines an element Tf𝜎 =

∫

𝜋𝜎 ( f ( g))u( g) dg ∈ ℛ{𝜋𝜎 (ℳ), u(G)} = ℛ(ℳ, 𝜎).

We recall (18.21, 18.22) that the mapping f ↦ Tf𝜎 is linear and (Tf𝜎 )∗ = Tf𝜎♯ , where f ♯ ( g) = ΔG ( g)−1 f ( g−1 )∗ Tf𝜎 Tf𝜎 = Tf𝜎∗f , where (f1 ∗ f2 )( g) = 1

2

1

2

∫

(g ∈ G)

f1 (t)f2 (t−1 g) dt,

( g ∈ G ).

(1) (2)

̂ on ℛ(ℳ, 𝜎) defines a right 𝒜 (G)-module structure on We also recall (18.3) that the action 𝜎̂ of G ℛ(ℳ, 𝜎). In this connection, it is easy to check that for every k ∈ 𝒜 (G) we have k ⋅ Tf𝜎 = Tkf𝜎 , where (kf )( g) = k( g)f ( g), ( g ∈ G ).

(3)

Finally, if f ∈ 𝒦 (G) and if we put ( f x)( g) = f ( g)x, (x ∈ ℳ, g ∈ G ), then Tfx𝜎 = 𝜋𝜎 (x)u( f ).

(4)

Consequently, ℬ = {Tf𝜎 ; f ∶ G → ℋ w-continuous with compact support} is an s∗ -dense *-subalgebra of ℛ(ℳ, 𝜎). On the other hand, from the definition (19.3) of the dual action 𝜎̂ and 18.22.(2) it follows that if Tf𝜎 ≥ 0 (in particular if f is of the form f ♯ ∗ f ),then P𝜎̂ (Tf𝜎 ) = 𝜋𝜎 ( f (e)).

(5)

Theorem. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W ∗ -algebra ℳ. For any n.s.f. weight 𝜑 on ℳ, the dual weight 𝜑̂ = 𝜑 ◦ 𝜋𝜎−1 ◦ P𝜎̂ is the unique n.s.f. weight ℛ(ℳ, 𝜎) with the properties: (i) if f ∶ G → ℳ is a compactly supported w-continuous function and Tf𝜎 ≥ 0, then 𝜑(T ̂ f𝜎 ) = 𝜑( f (e));

(6)

270

Crossed Products

(ii) for all x ∈ ℳ, g ∈ G, t ∈ ℝ we have 𝜎t𝜑̂ (𝜋𝜎 (x)) = 𝜋𝜎 (𝜎t𝜑 (x)) 𝜎t𝜑̂ (1ℳ

(7)

̄ 𝜆 ( g)) = ΔG ( g)it (1ℳ ⊗ ̄ 𝜆 ( g))𝜋𝜎 ([D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t ). ⊗

(8)

Moreover, the dual weights are exactly those weights that are invariant with respect to the dual action, that is, the mapping 𝜑 ↦ 𝜑̂ establishes a bijection between the sets {𝜑; 𝜑 n.s.f. weight on ℳ} and {Ψ ∶ Ψ (𝜎, ̂ jG )-invariant n.s.f. weight on ℛ(ℳ, 𝜎) }. Proof. Let 𝜑 be an n.s.f. weight on ℳ. (6) follows obviously from (5). On the other hand, using Theorem 11.9, we obtain (7) and [D(𝜑 ◦ 𝜎g )̂ ∶ D𝜑] ̂ t = 𝜋𝜎 ([D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t )

( g ∈ G, t ∈ ℝ),

so that, by 3.10, 𝜑,(𝜑 ̂ ◦ 𝜎g )̂

𝜎t𝜑̂ (u( g)) = 𝜎t

(u( g))𝜋𝜎 ([D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t ) ( g ∈ G, t ∈ ℝ),

and (8) will follow once we show that 𝜑,(𝜑 ̂ ◦ 𝜎g )̂

𝜎t

(u( g)) = ΔG ( g)it u( g)

( g ∈ G, t ∈ ℝ),

(9)

By 3.15.(1), (9) is equivalent to ̂ g)Xu( g)∗ ) (X ∈ ℛ(ℳ, 𝜎)+ , g ∈ G ) (𝜑 ◦ 𝜎g )̂(X) = ΔG ( g)−1 𝜑(u(

(10)

and, since (𝜑 ◦ 𝜎g )̂(X) = (𝜑 ◦ 𝜋𝜎−1 )(u( g)P𝜎̂ (X)u( g)∗ ), ΔG ( g)−1 𝜑(u( ̂ g)Xu( g)∗ ) = (𝜑 ◦ 𝜋𝜎−1 )(P𝜎̂ (u( g) ∗ Xu( g) )), (10) follows using 19.7.(3). Equations (7) and (8) determine the modular automorphism group of 𝜑̂ uniquely and permit an immediate verification of the fact that the s∗ -dense *-subalgebra ℬ of ℛ(ℳ, 𝜎) is 𝜎 𝜑̂ -invariant. On the other hand, (6) determines the values 𝜑(X ̂ ∗ X) for X ∈ ℬ. Consequently, according to Theorem 6.2, conditions (i) and (ii) determine the dual weight 𝜑̂ uniquely. We now show that the dual weight 𝜑̂ is (𝜎, ̂ jG )-invariant by checking conditions (a)–(d) from Theorem 18.12. Since 𝜑̂ is faithful, condition (a) is trivially satisfied. The commutation condition ̄ 𝜄G ) ◦ 𝜎, (b), that is, 𝜎̂ ◦ 𝜎t𝜑̂ = (𝜎t𝜑̂ ⊗ ̂ follows immediately using equalities (7), (8), and 19.6.(3), ̄ k⟩ = ⟨k ⋅ T 𝜎 , 𝜑⟩ 19.6.(4). For 0 ≤ Tf𝜎 ∈ ℬ and k ∈ 𝒜 (G)+ , we have ⟨𝜎(T ̂ f𝜎 ), 𝜑̂ ⊗ ̂ = ⟨Tkf𝜎 , 𝜑⟩ ̂ = f 𝜎 𝜎 ̄ ̄ k(e)𝜑( f (e)) = ⟨1G , k⟩⟨Tf , 𝜑⟩ ̂ = ⟨Tf ⊗ 1G , 𝜑̂ ⊗ k⟩, hence condition (c) is satisfied. Consider now X = Tf𝜎 ∈ ℬ, Y = Tf𝜎 ∈ ℬ, and k ∈ 𝒜 (G). Using (3) and 18.7.(7), we see that k ⋅ X ∈ ℬ 1 2 is determined by the function f3 ( g) = k( g)f1 ( g), and k0 ⋅ Y ∈ ℬ is determined by the function f4 ( g) = k( g)f2 ( g). From (1), (2), and (6), it follows that 𝜑((T ̂ f𝜎 )∗ (Tf𝜎 )) = (f ♯ ∗ f )(e); this equality, together with the usual polarization relation, justifies the following computations: ̄ 1G )𝜎(X), ̄ k⟩ = ((k ⋅ X)𝜑̂ |Y𝜑̂ )𝜑̂ = 𝜑(Y ⟨(Y∗ ⊗ ̂ 𝜑̂ ⊗ ̂ ∗ (k ⋅ X)) = (f2♯ ∗ f3 )(e), ̄ 1G ), 𝜑̂ ⊗ ̄ k ◦ jG ⟩ = (X𝜑̂ |(k0 ⋅ Y)𝜑̂ )𝜑̂ = 𝜑((k ⟨𝜎(Y ̂ ∗ )(X ⊗ ̂ 0 ⋅ Y)∗ X) = (f4♯ ∗ f1 )(e). Since (f♯2 ∗ f3 )(e) = (f4♯ ∗ f1 )(e), condition (d) is also satisfied.

Crossed Products

271

Finally, let Ψ be a (𝜎, ̂ jG )-invariant n.s.f. weight on ℛ(ℳ, 𝜎) and 𝜑 any n.s.f. weight on ℳ. By Corollary 18.13, [DΨ ∶ D𝜑] ̂ t ∈ ℛ(ℳ, 𝜎)𝜎̂ = 𝜋𝜎 (ℳ), hence, by Theorem 5.1, there exists an n.s.f. weight 𝜓 on ℳ such that 𝜋𝜎 ([D𝜓 ∶ D𝜑]t ) = [DΨ ∶ D𝜑] ̂ t (t ∈ ℝ). Then, using Theorem 11.9, we deduce that [D𝜓̂ ∶ D𝜑] ̂ t = 𝜋𝜎 ([D𝜓 ∶ D𝜑]t ) = [DΨ ∶ D𝜑] ̂ t (t ∈ ℝ), and hence, by Corollary 3.6, Ψ = 𝜓. ̂ If 𝜓̂ = 𝜑, ̂ then 𝜋𝜎 ([D𝜓 ∶ D𝜑]t ) = [D𝜓̂ ∶ D𝜑] ̂ t = 1 (t ∈ ℝ), and hence 𝜓 = 𝜑. The proof of the theorem is complete. 19.9. The following important theorem characterizes those W ∗ -algebras that are crossed products by a continuous action of G. Theorem (Landstad). Let 𝒩 be a W ∗ -algebra with the property that there exist an s-continuous ̂ on 𝒩 such that ̄ 𝔏(G) of G unitary representation u ∶ G → 𝒩 and an action 𝛿 ∶ 𝒩 → 𝒩 ⊗ ̄ 𝜆 ( g) 𝛿(u( g)) = u( g) ⊗

( g ∈ G ).

Then 𝒩 is generated by 𝒩 𝛿 and u(G), 𝒩 = ℛ{𝒩 𝛿 , u(G)},

(1)

and u( g)𝒩 𝛿 u( g)∗ = 𝒩 𝛿 for every g ∈ G. Consider the W ∗ -algebra ℳ = 𝒩 𝛿 and the continuous action 𝜎 ∶ G → Aut(ℳ) of G on ℳ defined by 𝜎g = Ad(u( g))|ℳ, ( g ∈ G ). There exists a *-isomorphism Φ ∶ 𝒩 → ℛ(ℳ, 𝜎) such that Φ(x) = 𝜋𝜎 (x) (x ∈ ℳ) ̄ 𝜆 ( g) ( g ∈ G ) Φ(u( g)) = 1ℳ ⊗ ̄ 𝜄G ) ◦ 𝛿 = 𝜎̂ ◦ Φ. (Φ ⊗

(2) (3) (4)

(𝒩 , 𝛿) ≈ (ℛ(ℳ, 𝜎), 𝜎). ̂

(5)

Briefly,

If G is abelian, then it is more convenient to state the theorem as follows: Let 𝒩 be a W ∗ -algebra with the property that there exist an s-continuous unitary representation ̂ on 𝒩 such that ̂ → Aut(𝒩 ) of G u ∶ G → 𝒩 and a continuous action 𝜃 ∶ G ̂ 𝜃𝛾 (u( g)) = ⟨g, 𝛾⟩u( g) ( g ∈ G, 𝛾 ∈ G).

(6)

Then 𝒩 is generated by 𝒩 𝛿 and u(G), 𝒩 = ℛ{𝒩 𝛿 , u(G)}, and u( g)𝒩 𝛿 u( g)∗ = 𝒩 𝛿 for every g ∈ G.

(1’)

272

Crossed Products

Consider the W ∗ -algebra ℳ = 𝒩 𝛿 and the continuous action 𝜎 ∶ G → Aut(ℳ) of G on ℳ defined by 𝜎g = Ad(u( g))|ℳ( g ∈ G ). There exists a *-isomorphism Φ ∶ 𝒩 → ℛ(ℳ, 𝜎) such that Φ(x) = 𝜋𝜎 (x) (x ∈ ℳ) ̄ 𝜆 ( g) ( g ∈ G ) Φ(u( g)) = 1ℳ ⊗ Φ ◦ 𝜃g = 𝜎̂ g ◦ Φ ( g ∈ G ).

(2’) (3’) (4’)

(𝒩 , 𝜃) ≡ (ℛ(ℳ, 𝜎), 𝜎). ̂

(5’)

Briefly,

Of course, if G is abelian, then the above two statements are equivalent (see 18.6, 18.8, and 18.9). The proof of the theorem is contained in Sections 19.10–19.12. 19.10. The main result contained in Theorem 19.9 is the assertion that 𝒩 = ℛ{𝒩 𝛿 , u(G)}, which will be proved in the next section. In this section, we assume that 𝒩 = ℛ{𝒩 𝛿 , u(G)}; we prove the other assertions in Theorem 19.9. We shall use the notation of the statement of Theorem 19.9 and assume 𝒩 ⊂ ℬ(ℋ ) realized as a von Neumann algebra. ̂ we have Assume first that G is abelian. Then, for every x ∈ ℳ = 𝒩 𝜃 , g ∈ G and 𝛾 ∈ G, 𝜃𝛾 (u( g)xu( g)∗ ) = 𝜃𝛾 (u( g))𝜃𝛾 (x)𝜃𝛾 (u( g−1 )) = ⟨g, 𝛾⟩ ⟨g−1 , 𝛾⟩u( g)xu( g−1 ) = u( g)xu( g)∗ , hence u( g)xu( g)∗ ∈ 𝒩 𝜃 = ℳ. Consider now the general case. For every x ∈ ℳ = 𝒩 𝛿 and g ∈ G, we have 𝛿(u( g)xu( g)∗ ) = ̄ 𝜆 ( g))(x ⊗ ̄ 1G )(u( g)∗ ⊗ ̄ 𝜆 ( g)∗ ) = (u( g)xu( g)∗ ) ⊗ ̄ 1G , hence 𝛿(u( g))𝛿(x)𝛿(u( g)∗ ) = (u( g) ⊗ u( g)xu( g)∗ ∈ 𝒩 𝛿 = ℳ. ̄ ℒ ∞ (G) ⊂ ℬ(ℒ 2 (G, ℋ )) be the unitary operator defined by the bounded sLet U ∈ 𝒩 ⊗ continuous function G ∋ g ↦ u( g), that is, (U𝜉)( g) = u( g)𝜉( g)(𝜉 ∈ ℒ 2 (G, ℋ ), g ∈ G ). We define a unital injective normal *-homomorphism ̄ ℬ(ℒ 2 (G)) Φ∶𝒩 → 𝒩 ⊗ by the equality Φ( y) = U ∗ 𝛿( y)U( y ∈ 𝒩 ). ̄ 1G , hence For x ∈ ℳ = 𝒩 𝛿 , we have 𝛿(x) = x ⊗ ̄ 1G )U𝜉)( g) = u( g)∗ xu( g)𝜉( g) = 𝜎 −1 (x)𝜉( g) = (𝜋𝜎 (x)𝜉)( g); (Φ(x)𝜉)( g) = (U ∗ (x ⊗ g this proves 19.9.(2). ̄ 𝜆 (s), hence For s ∈ G, we have 𝛿(u(s)) = u(s) ⊗ ̄ 𝜆 (s))U𝜉)( g) (Φ(u(s))𝜉)( g) = (U ∗ (u(s) ⊗ ∗ ̄ 𝜆 (s))𝜉)( g); = u( g) u(s)(U𝜉)(s−1 g) = u( g)∗ u(s)u(s−1 g)𝜉(s−1 g) = ((1 ⊗ this proves 19.9.(3).

Crossed Products

273

̄ 𝔏(G)} = Thus, since 𝒩 = ℛ{𝒩 𝛿 , u(G)}, we get Φ(𝒩 ) = Φ(ℛ{ℳ, u(G)}) = ℛ{𝜋𝜎 (ℳ), 1 ⊗ ℛ(ℳ, 𝜎). Finally, 19.9.(4) is obvious when applied to elements x ∈ ℳ or u( g)( g ∈ G ), and so remains valid also when applied to an arbitrary element of 𝒩 = ℛ{ℳ, u(G)}. Note that in the commutative case this part of the proof is more complicated, as in this case 𝛿 does not appear explicitly in the statement, but has to be defined, starting from 𝜃, as the Fourier–Plancherel ̄ ℒ ∞ (G). transform of the action 𝜋𝜃 ∶ 𝒩 → 𝒩 ⊗ 19.11. Assume 𝒩 ⊂ ℬ(ℋ ) realized as a von Neumann algebra. To show that 𝒩 = ℛ{ℳ, u(G)}, ̄ 1G , 𝛿(u(G))} ⊂ ℬ(ℋ ⊗ ̄ ℒ 2 (G)) or, it is necessary and sufficient to prove that 𝛿(𝒩 ) ⊂ ℛ{ℳ ⊗ ′ ̄ equivalently, to prove the reverse inclusion for the commutants, that is, (ℳ ⊗ 1G ) ∩ 𝛿(u(G))′ ⊂ s∗

𝛿(𝒩 )′ . Since there exist nets {ki } ⊂ 𝒦 (G) with u(ki ) → 1𝒩 , it is sufficient to prove that ̄ 1G )′ ∩ 𝛿(u(G))′ ⊂ {𝛿(u(k)∗ xu(k)); k ∈ 𝒦 (G), x ∈ 𝒩 + }′ (ℳ ⊗

(1)

Before starting the proof of (1), we need some preparation. Consider the n.s.f. operator-valued weights 𝜔 ̄ 𝔏(G))+ → 𝒩̄ + , P = P𝛿 = E ◦ 𝛿 ∶ 𝒩 + → ℳ̄ + E = E𝒩G ∶ (𝒩 ⊗

̄ ℒ 2 (G) = ℒ 2 (G, ℋ ) defined by the function G ∋ g ↦ u( g), and the unitary operator U on ℋ ⊗ 2 i.e. (U𝜁 )( g) = u( g)𝜁( g)(𝜁 ∈ ℒ (G, ℋ ), g ∈ G ). ̄ 𝔏(G), h, k ∈ 𝒦 (G), 𝜉 ∈ 𝒦 , g ∈ G, We show that for X ∈ 𝒩 ⊗ ̄ k))( g). E(𝛿(u(h))X𝛿(u(k)u( g)∗ ))u( g)𝜉 = ΔG ( g)[𝛿(u(h))XU(𝜉 ⊗

(2)

Indeed, it is easy to check that 𝛿(u(𝒦 (G))) ⊂ 𝔑E (see 19.7.(5)), hence 𝛿(u(h))X𝛿(u(k)u( g)∗ ) ∈ 𝔐E and E(𝛿(u(h))X𝛿(u(k)u( g)∗ )) depends w-continuously on X (see Proposition 1.14). Consē 𝔏(G). So, we can assume quently, it is sufficient to check (2) for X in a w-dense subset of 𝒩 ⊗ ̄ 𝜆 ( f ) with x ∈ 𝒩 and f ∈ 𝒦 (G). Then that X = x ⊗ ̄ 𝜆 ( f ))𝛿(u(k)u( g)∗ ))u( g)𝜉 E(𝛿(u(h))(x ⊗ ̄ 𝜆 ( f ))𝛿(u(k))(1 ⊗ ̄ 𝜆( g)∗ ))𝜉 = E(𝛿(u(h))(x ⊗ (( )( ) ̄ 𝜆 (s)) ds ̄ 𝜆(r) dr =E h(s)(u(s) ⊗ x⊗ f (r)𝜆 ∫ ∫ ( ) ) ̄ 𝜆 (t)) dt (1 ⊗ ̄ 𝜆 ( g)∗ ) 𝜉 × k(t)(u(t) ⊗ ∫ ( ) −1 ̄ =E h(s)f (r)k(t)(u(s)xu(t) ⊗ 𝜆 (srtg )) ds dr dt 𝜉 ∫ ∫ ∫ ( = ΔG ( g)E

∫

(( ∫∫

(via t ↦ r−1 s−1 tg)

) ) ̄ h(s)f (r)u(s)xk(r s tg)u(r s tg)dsdr ⊗ 𝜆 (t) dt 𝜉 −1 −1

(using 18.22.(1))

)

−1 −1

274

Crossed Products ( = ΔG ( g)

) h(s)f (r)u(s)xk(r s g)u(r s g)𝜉 ds dr 𝜉 −1 −1

∫ ∫

= ΔG ( g)

∫ ∫

−1 −1

h(s)f (r)u(s)xk(r−1 s−1 g)u(r−1 s−1 g)𝜉 ds dr

̄ k)](r−1 s−1 g) ds dr h(s)f (r)u(s)x[U(𝜉 ⊗ ( ) ̄ 𝜆 (r))U(𝜉 ⊗ ̄ k)](s−1 g) dr ds f (r)[(1𝒩 ⊗ = ΔG ( g) h(s)u(s)x ∫ ∫ = ΔG ( g)

∫ ∫

= ΔG ( g)

∫

= ΔG ( g)

̄ 𝜆 ( f ))U(𝜉 ⊗ ̄ k)](s−1 g) ds h(s)u(s)x[(1𝒩 ⊗ ∫

̄ 𝜆 ( f ))U(𝜉 ⊗ ̄ k)](s−1 g) ds h(s)u(s)[(x ⊗

̄ 𝜆 (s))(x ⊗ ̄ 𝜆 ( f ))U(𝜉 ⊗ ̄ k)]( g) ds h(s)[(u(s) ⊗ [ ] ̄ 𝜆 ( f ))U(𝜉 ⊗ ̄ k) ds ( g) = ΔG ( g) h(s)𝛿(u(s))(x ⊗ ∫ ̄ 𝜆 ( f ))U(𝜉 ⊗ ̄ k)]( g). = ΔG ( g)[𝛿(u(h))(x ⊗

= ΔG ( g)

∫

We show that for x ∈ 𝒩 , h, k ∈ 𝒩 (G), 𝜉 ∈ ℋ , g ∈ G, we have ̄ k)]( g) P(u(h)∗ xu(k)u( g)∗ )u( g)𝜉 = ΔG ( g)[𝛿(u(h)∗ x)U(𝜉 ⊗ ̄ k)]( g−1 ) u( g)∗ P(u( g)u(h)∗ xu(k))𝜉 = [𝛿(u(h)∗ x)U(𝜉 ⊗

(3) (4)

Indeed, (3) follows from (2) replacing X by 𝛿(x) and h by h♯ , since u(h)∗ = u(h♯ ) and P = E ◦ 𝛿, while (4) follows from (3) using 19.7.(3) and replacing g by g−1 . For f ∈ 𝒦 (G), we consider the operator Rf =

∫

̄ 𝜌 ( g)) dg ∈ (𝒩 ⊗ ̄ 𝔏(G))′ . ΔG ( g)−1∕2 f ( g−1 )(1𝒩 ⊗

(5)

(Rf 𝜁)(s) =

(6)

For 𝜁 ∈ ℒ 2 (G, ℋ ), we have ∫

f ( g−1 )𝜁(sg) dg

(s ∈ G ).

It is easy to check that for f, k ∈ 𝒦 (G) an 𝜉 ∈ ℋ we have ̄ k) = 𝛿(u(k))(𝜉 ⊗ ̄ f ). Rf U(𝜉 ⊗

(7)

̄ 1G )′ ∩ 𝛿(u(G))′ and h, k ∈ 𝒦 (G), x ∈ 𝒩 . Consider We now prove inclusion (1). Let X ∈ (ℳ ⊗ also 𝜉, 𝜂 ∈ ℋ and 𝜑, 𝜓 ∈ 𝒦 (G). Using (3)–(7), we obtain ̄ 𝜑)||(𝜂 ⊗ ̄ 𝜓))) ( X𝛿(u(h)∗ xu(k))(𝜉 ⊗ ̄ 𝜑)||(𝜂 ⊗ ̄ 𝜓))) = ( X𝛿(u(h)∗ x)𝛿(u(k))(𝜉 ⊗ ∗ ̄ k)||(𝜂 ⊗ ̄ 𝜓))) = ( X𝛿(u(h) x)R𝜑 U(𝜉 ⊗

Crossed Products

275

= =

∫ ∫

∫

̄ k)](s)||[X∗ (𝜂 ⊗ ̄ 𝜓)](s))) ds ( [R𝜑 𝛿(u(h)∗ x)U(𝜉 ⊗

̄ k)](sg)||[X∗ (𝜂 ⊗ ̄ 𝜓)](s))) ds dg 𝜑( g−1 )(([𝛿(u(h)∗ x)U(𝜉 ⊗ (via g ↦ s−1 g)

= =

∫ ∫

̄ 𝜓)](s))) ds dg ΔG ( g−1 )𝜑( g−1 s)((P(u(h)∗ xu(k)u( g)∗ )u( g)𝜉||[X∗ (𝜂 ⊗

∫ ∫ =

̄ k)]( g)||[X∗ (𝜂 ⊗ ̄ 𝜓)](s))) ds dg 𝜑( g−1 s)(([𝛿(u(h)∗ x)U(𝜉 ⊗

∫ ∫

̄ 𝜆 ( g))(𝜉 ⊗ ̄ 𝜑)](s) ΔG ( g−1 )(([(P(u(h)∗ xu(k)u( g)∗ )u( g) ⊗ ̄ 𝜓)](s))) ds dg |[X∗ (𝜂 ⊗

=

̄ 𝜑)||X∗ (𝜂 ⊗ ̄ 𝜓))) dg ΔG ( g−1 )((𝛿(P(u(h)∗ xu(k)u( g)∗ )u( g))(𝜉 ⊗

∫

̄ 1G )′ ∩ 𝛿(u(G))′ ) (since X ∈ (ℳ ⊗ = =

̄ 𝜑)||𝛿(u( g)∗ P(u( g)u(k)∗ x∗ u(h)))(𝜂 ⊗ ̄ 𝜓))) dg ΔG ( g−1 )((X(𝜉 ⊗

∫

̄ 𝜑)](s)||[(u( g)∗ P(u( g)u(k)∗ x∗ u(h)) ⊗ ̄ 𝜆 ( g)∗ ) ΔG ( g−1 )(([X(𝜉 ⊗

∫ ∫

̄ 𝜓)](s))) ds dg ×(𝜂 ⊗ =

∫ ∫

̄ 𝜑)](s)||𝜓( gs)u( g)∗ P(u( g)u(k)∗ x∗ u(h))𝜂)) ds dg ΔG ( g−1 )(([X(𝜉 ⊗

=

∫ ∫

̄ 𝜑)](s)||𝜓( gs)[𝛿(u(k)∗ x∗ )U(𝜂 ⊗ ̄ h)]( g−1 ))) ds dg ΔG ( g−1 )(([X(𝜉 ⊗ (via g ↦ (sg)−1 = g−1 s−1 )

=

∫ ∫ =

∫

̄ 𝜑)](s)||𝜓( g−1 )[𝛿(u(k)∗ x∗ )U(𝜂 ⊗ ̄ h)](sg))) ds dg ( [X(𝜉 ⊗ ̄ 𝜑)](s)𝜑)](s)||[R𝜓 𝛿(u(k)∗ x∗ )U(𝜂 ⊗ ̄ h)](s))) ds ( [X(𝜉 ⊗ ̄ 𝜑)||𝛿(u(k)∗ x∗ )R𝜓 U(𝜂 ⊗ ̄ h))) = ( X(𝜉 ⊗ ∗ ∗ ̄ 𝜑)||𝛿(u(k) x )𝛿(u(h))(𝜂 ⊗ ̄ 𝜓))) = ( X(𝜉 ⊗ ∗ ̄ 𝜑)||(𝜂 ⊗ ̄ 𝜓))), = ( 𝛿(u(h) xu(k))X(𝜉 ⊗

hence X𝛿(u(h)∗ xu(k)) = 𝛿(u(h)∗ xu(k))X. This completes the proof of Theorem 19.9.

□

19.12. From the first half of the sequence of equations in the final part of Section 19.11, it follows ∗ −1 ∗ ∗ that ( for x ∈ ∗𝒩 and h, k ∈ 𝒦 ∗(G) we ) have 𝛿(u(h) xu(k)) = ∫ ΔG ( g )𝛿(P(u(h) xu(k)u( g) )u( g)) dg = 𝛿 ∫ P(u(h) xu(k)u( g))u( g) dg , hence u(h)∗ xu(k) =

∫

P(u(h)∗ xu(k)u( g))u( g−1 ) dg.

(1)

276

Crossed Products

This relation is also sufficient to conclude the proof of Theorem 19.9. Moreover, it contains the main idea of the proof, by showing that for a wide class of elements a ∈ 𝒩 we have a=

P(au( g))u( g−1 ) dg

∫

(2)

with P(au( g)) ∈ ℳ = 𝒩 𝛿 and u( g−1 ) ∈ u(G). If G is abelian, then (1) is easily reduced to the Fourier inversion theorem. Indeed, let a = ̂ → 𝒩 defined by f (𝛾) = 𝜃𝛾 (a)(𝛾 ∈ G). ̂ Recall (19.7) u(h)∗ xu(k) and consider the function f ∶ G that in this case we have P = P𝜃 = ∫ 𝜃𝛾 (⋅)d𝛾. Since a ∈ 𝔐P (see 19.7), it follows that the function f is integrable. Then ( f (𝛾) = =

∫

) ( h♯ (s)𝜃𝛾 (u(s)) ds 𝜃𝛾 (x)

∫ ∫

∫

⟨st, 𝛾⟩h♯ (s)k(t)u(s)𝜃𝛾 (x)u(t) ds dt

(via t ↦ s−1 t−1 )

( =

∫

) k(t)𝜃𝛾 (u(t)) dt

⟨t, 𝛾⟩

∫

♯

) −1 −1

−1 −1

h (s)k(s t )u(s)𝜃𝛾 (x)u(s t ) ds

dt

and ∫ ∫

‖h♯ (s)k(s−1 t−1 )u(s)𝜃𝛾 (x)u(s−1 t−1 )‖ ds dt ≤ ‖x‖

∫ ∫

|h♯ (s)k(s−1 t−1 )| ds dt < +∞

hence f is the Fourier transform of an integrable function. Consequently, we can apply the Fourier inversion theorem to get (

∫

) 𝜃𝛾 (a)𝜃𝛾 (u( g))d𝛾 u( g−1 ) dg ∫ ∫ ( ) ̂f ( g) dg = f (𝜀) = 𝜃𝜀 (a) = a = ⟨g, 𝛾⟩f (𝛾) d𝛾 u( g)u( g−1 ) dg = ∫ ∫ ∫

P(au( g))u( g−1 ) dg =

̂ denotes the neutral element of G. ̂ where 𝜀 ∈ G 19.13. As a first application of Theorem 19.9, we show that in 19.3.(5) we actually have an equality. Corollary (Takesaki & Digernes). Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W ∗ -algebra ℳ. Then ̄

̄ ℬ(ℒ 2 (G)))𝜎 ⊗ Ad(𝜌𝜌) . ℛ(ℳ, 𝜎) = (ℳ ⊗ ̄ ℬ(ℒ 2 (G)), 𝛽 = 𝜎 ⊗ ̄ Ad(𝜌𝜌), 𝛿 = Proof. We shall use the notation 𝒩 = ℛ(ℳ, 𝜎), 𝒫 = ℳ ⊗ ̄ 𝛿G as in Section 19.3 and u( g) = 1ℳ ⊗ ̄ 𝜆 ( g) ∈ 𝒩 ⊂ 𝒫 𝛽 ( g ∈ G ). As we have seen 𝜄ℳ ⊗

Crossed Products

277

in Section 19.3, g ↦ u( g) is an s-continuous unitary representation of G in 𝒫 𝛽 and 𝛿 = 𝒫 𝛽 → ̂ on 𝒫 𝛽 such that 𝛿(u( g) = u( g) ⊗ ̄ 𝔏(G) is an action of G ̄ 𝜆 ( g) ( g ∈ G ), and (𝒫 𝛽 )𝛿 = 𝜋𝜎 (ℳ). 𝒫𝛽⊗ 𝛽 By Theorem 19.9 it follows that 𝒫 = ℛ{𝜋𝜎 (ℳ), u(G)} = 𝒩 . 19.14. The previous corollary enables us to compute the commutant of the crossed product. Corollary (Takesaki & Digernes). Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the von Neumann algebra ℳ ⊂ ℬ(ℋ ) and assume that there exists an so-continuous unitary representation v ∶ G → ℬ(ℋ ) such that 𝜎g = Ad(v( g))|ℳ ( g ∈ G ). Then the commutant of the von Neumann ̄ ℒ 2 (G)) is given by algebra ℛ(ℳ, 𝜎) ⊂ ℬ(ℋ ⊗ ̄ 1G , v( g) ⊗ ̄ 𝜌 ( g); x′ ∈ ℳ ′ , g ∈ G}. ℛ(ℳ, 𝜎)′ = ℛ{x′ ⊗

(1)

Proof. By the von Neumann double commutant theorem, (1) is equivalent to ̄ 1G , v( g) ⊗ ̄ 𝜌( g); x′ ∈ ℳ ′ , g ∈ G}′ , ℛ(ℳ, 𝜎) = ℛ{x′ ⊗ □

which is an obvious consequence of Corollary 19.13.

The so-continuous unitary representation v ∶ G → ℬ(ℋ ) in the statement of the corollary also defines a continuous action 𝜎 ′ ∶ G → Aut(ℳ ′ ) where 𝜎g′ = Ad(v( g))|ℳ ′ ( g ∈ G ). We show that ℛ(ℳ, 𝜎)′ is spatially isomorphic to ℛ(ℳ ′ , 𝜎 ′ ).

(2)

To this end, recall that the regular representations 𝜆 and 𝜌 of G are unitarily equivalent. Indeed, the operator U ∈ ℬ(ℒ 2 (G)) defined by (U𝜉)(s) = ΔG (s)−1∕2 𝜉(s−1 )(𝜉 ∈ ℒ 2 (G), s ∈ G ), is unitary and for g ∈ G we have 𝜆( g)𝜉)(s) = ΔG (s)−1∕2 𝜉( g−1 s−1 ) = ΔG ( g)1∕2 ΔG (sg)−1∕2 𝜉((sg)−1 ) (U𝜆 = (𝜌𝜌( g)U𝜉)(s), that is, U ∗𝜌 ( g)U = 𝜆 ( g). ̄ ℒ 2 (G)) be the unitary operator defined by the functions On the other hand, let V ∈ ℬ(ℋ ⊗ s ↦ v(s), that is, (V𝜁)(s) = v(s)𝜁(s)(𝜁 ∈ ℒ 2 (G, ℋ ), s ∈ G ). ̄ U)V ∈ ℬ(ℋ ⊗ ̄ ℒ 2 (G)) is a unitary operator and, as easily verified, we have Then W = (1ℋ ⊗ ̄ 1G )W = 𝜋𝜎 ′ (x′ ) (x′ ∈ ℳ ′ ), W ∗ (x′ ⊗ ̄ 𝜌 ( g))W = 1ℋ ⊗ ̄ 𝜆 ( g) ( g ∈ G ), W ∗ (v( g) ⊗

(3) (4)

proving (2). ̂ on ℳ similar to 19.15. Landstad’s Theorem 19.9 enables us to prove a result for actions of G Proposition 19.3. ̂ on the W ∗ -algebra ℳ we have ̄ 𝔏(G) of G Proposition. For every action, 𝛿 ∶ ℳ → ℳ ⊗ ̂

𝛿(ℳ) = ℛ(ℳ, 𝛿)𝛿 where 𝛿̂ is the dual action.

(1)

278

Crossed Products

̄ ℬ(ℒ 2 (G)) ⊂ ℬ(𝒦 ⊗ ̄ ℒ 2 (G)) be realized as von Neumann Proof. Let ℳ ⊂ ℬ(ℋ ) and ℳ ⊗ algebras. We consider the action (18.7.(12)) ̄ 𝛿 ∗ ∶ ℬ(ℋ ) ⊗ ̄ ℬ(ℒ 2 (G)) → ℬ(ℋ ) ⊗ ̄ ℬ(ℒ 2 (G))⊗ ̄ 𝔏(G). 𝜄ℋ ⊗ G ̂ on ℬ(ℋ ⊗ ̄ ℒ 2 (G)) and show that of G. ̄ 𝛿 ∗ )(𝛿(ℳ)′ ) ⊂ 𝛿(ℳ)′ ⊗ ̄ 𝔏(G). (𝜄ℋ ⊗ G ̄ ℒ 2 (G))) and x ∈ ℳ, then Indeed, if X ∈ 𝛿(ℳ)′ ⊂ ℬ(ℋ ⊗ ̄ 𝛿 ∗ )(X))(𝛿(x) ⊗ ̄ 1G ) ((𝜄ℋ ⊗ G ̄ WG )(X ⊗ ̄ 1G )(1ℋ ⊗ ̄ W ∗ )(𝛿(x) ⊗ ̄ 1G )(1ℋ ⊗ ̄ WG )(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G G ̄ WG )(X ⊗ ̄ 1G )((𝜄ℋ ⊗ ̄ 𝛿G )(𝛿(x))(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G

̄ WG )(X ⊗ ̄ 1G )((𝛿 ⊗ ̄ 𝜄G )(𝛿(x)))(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G ̄ WG )((𝛿 ⊗ ̄ 𝜄G )(𝛿(x)))(X ⊗ ̄ 1G )(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G ̄ WG )((𝜄ℋ ⊗ ̄ 𝛿G )(𝛿(x)))(X ⊗ ̄ 1G )(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G

̄ WG )(1ℋ ⊗ ̄ W ∗ )(𝛿(x) ⊗ ̄ 1G )(1ℋ ⊗ ̄ WG )(X ⊗ ̄ 1G )(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G G ̄ 1G )((𝜄ℋ ⊗ ̄ 𝛿 ∗ )(X)). = (𝛿(x) ⊗ G

̂ on 𝛿(ℳ)′ . On the ̄ 𝛿 ∗ restricts to an action 𝛿 ′ ∶ 𝛿(ℳ)′ → 𝛿(ℳ)′ ⊗ ̄ 𝔏(G) of G Consequently, 𝜄ℋ ⊗ G ′ ′ ̄ 𝜌 ( g) ∈ other hand, there exists an s-continuous unitary representation u ∶ G ∋ g ↦ u ( g) = 1ℋ ⊗ ̄ 𝜆 ( g) ( g ∈ G ). By Theorem 19.9, it follows 𝛿(ℳ)′ and it is easy to check that 𝛿 ′ (u′ ( g)) = u′ ( g) ⊗ that 𝛿(ℳ)′ = ℛ{(𝛿(ℳ)′ )𝛿 , u′ (G)}. ′

(2)

̄ Ad(𝜌)) |ℛ(ℳ, 𝛿) and that Recall (19.4) that the dual action 𝛿̂ ∶ G → Aut(ℛ(ℳ, 𝛿)) is 𝛿̂ = (𝜄ℳ ⊗ ̂ 𝛿(ℳ) ⊂ ℬ(ℳ, 𝛿)𝛿 . To prove the reverse inclusion, we note that ̂ ̄ 𝜌) ̄ ℬ(ℒ 2 (G)))𝜄⊗Ad(𝜌 ̄ 𝔏(G) =ℳ⊗ ℛ(ℳ, 𝛿)𝛿 ⊂ (ℳ ⊗

(3)

and that for every X ∈ ℛ(ℳ, 𝛿) we have ̄ 𝛿G )(X) = (𝛿 ⊗ ̄ 𝜄G )(X). (𝜄ℳ ⊗

(4)

� on ℳ Indeed, for X = 𝛿(x)(x ∈ ℳ), (4) follows from the fact that 𝛿 is an action of G ∞ ̄ (18.2.(4)), while for X = 1ℳ ⊗ f ( f ∈ ℒ (G)), (4) is immediate (18.7.(9)). Since ℛ(ℳ, 𝛿) = ̄ ℒ ∞ (G)}, (4) is valid for every X ∈ ℛ(ℳ, 𝛿). ℛ{𝛿(ℳ), 1ℳ ⊗ ̂ ̄ 𝔏(G) and X satisfies (4). Since X ∈ ℳ ⊗ ̄ 𝔏(G), Consider now X ∈ ℛ(ℳ, 𝛿)𝛿 . Then X ∈ ℳ ⊗ ′ ′ ′ it is clear that Xu ( g) = u ( g)X for all g ∈ G. Let Y ∈ (𝛿(ℳ)′ )𝛿 , that is, Y ∈ 𝛿(ℳ)′ and

Crossed Products

279

̄ WG )(Y ⊗ ̄ 1G )(1ℋ ⊗ ̄ W∗ ) = Y ⊗ ̄ 1G , Then (1ℋ ⊗ G ̄ 1G = (X ⊗ ̄ 1G )(Y ⊗1 ̄ G) XY ⊗ ̄ WG )(1ℋ ⊗ ̄ W ∗ )(X ⊗ ̄ 1G )(1ℋ ⊗ ̄ WG )(Y ⊗ ̄ 1G )(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G G ̄ WG )((𝜄ℳ ⊗ ̄ 𝛿G )(X))(Y ⊗ ̄ 1G )(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G

̄ WG )((𝛿 ⊗ ̄ 𝜄G )(X))(Y ⊗ ̄ 1G )(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G ̄ WG )(Y ⊗ ̄ 1G )((𝛿 ⊗ ̄ 𝜄G )(X))(1ℋ ⊗ ̄ W∗ ) = (1ℋ ⊗ G ̄ WG )(Y ⊗ ̄ 1G )((𝜄ℳ ⊗ ̄ 𝛿G )(X))(1ℋ ⊗ ̄ W ∗) = (1ℋ ⊗ G

̄ 1G )(1ℋ ⊗ ̄ W ∗ )(X ⊗ ̄ 1G )(1ℋ ⊗ ̄ WG )(1ℋ ⊗ ̄ W∗ ) ̄ WG )(Y ⊗ = (1ℋ ⊗ G G ̄ 1G )(X ⊗ ̄ 1G ) = YX ⊗ ̄ 1G , = (Y ⊗ hence XY = YX. From (2) it follows that X ∈ 𝛿(ℳ)′′ = 𝛿(ℳ). ̄ 𝔏(G), we have also obtained For X ∈ ℳ ⊗ ̄ 𝛿G )(X) = (𝛿 ⊗ ̄ 𝜄G )(X). X ∈ 𝛿(ℳ) ⇔ (𝜄ℳ ⊗

(5)

19.16. We now prove a result similar to Proposition 19.7. Proposition. Let 𝜎 ∶ G → Aut(𝒩 ) be a continuous action of G on the W ∗ -algebra 𝒩 . If there exists a unital injective normal *-homomorphism 𝜒 ∶ ℒ ∞ (G) → 𝒩 such that 𝜎g (𝜒( f )) = 𝜒(Ad(𝜌𝜌( g))( f )) ( f ∈ ℒ ∞ (G), g ∈ G )

(1)

then the faithful normal operator-valued weight P𝜎 ∶ 𝒩 + → (𝒩 𝜎 )+ , P𝜎 (x) =

∫

𝜎g (x) dg

(x ∈ 𝒩 + )

(2)

) f ( g) dg ⋅ 1𝒩 .

(3)

is semifinite. For f ∈ ℒ ∞ (G)+ , we have ( P𝜎 (𝜒( f )) =

∫

For x ∈ 𝒩 + and s ∈ G, we have P𝜎 (𝜎s (x)) = ΔG (s)−1 P𝜎 (x).

(4)

Proof. For f ∈ ℒ ∞ (G)+ and k ∈ ℒ ∞ (G)+∗ ⊂ ℒ 1 (G), we have ( ∫

⟨Ad(𝜌𝜌( g))( f ), k⟩ dg =

∫ ∫

f (sg)k(s) ds dg =

) ∫

f ( g) dg ⟨1G , k⟩.

280

Crossed Products

Thus, for every 𝜓 ∈ 𝒩∗+ we obtain ∫

⟨𝜎g (𝜒( f )), 𝜓⟩ dg = =

∫ ∫

⟨𝜒(Ad(𝜌𝜌( g))( f ))𝜓⟩ dg ( ⟨Ad(𝜌𝜌( g))( f ), 𝜓 ◦ 𝜒⟩ dg =

∫

) f ( g) dg (1𝒩 , 𝜓)

which proves (3) and the semifiniteness of P𝜎 . (4) has already been proved (18.20.(2)). ̂ on ℳ, the dual action 𝛿̂ ∶ G → ̄ 𝔏(G) of G 19.17. In particular, for every action 𝛿 ∶ ℳ → ℳ ⊗ Aut(ℛ(ℳ, 𝛿)) defines an n.s.f. operator-valued weight P𝛿̂ ∶ ℛ(ℳ, 𝜎)+ → 𝛿(ℳ)+ . ̄ f In this case, the *-homomorphism 𝜒 ∶ ℒ ∞ (G) → ℛ(ℳ, 𝛿) is defined by 𝜒( f ) = 1ℳ ⊗ ∞ ( f ∈ ℒ (G)). Thus, if 𝜑 is an n.s.f. weight on ℳ, then, according to Proposition 11.6, 𝜑̂ = 𝜑 ◦ 𝛿 −1 ◦ P𝛿̂ is an n.s.f. weight on ℛ(ℳ, 𝛿), called the dual weight of 𝜑. From Theorem 11.9, it follows that 𝜎t𝜑̂ (𝛿(X)) = 𝛿(𝜎t𝜑 (x)) (x ∈ ℳ, t ∈ ℝ) [D𝜓̂ ∶ D𝜑] ̂ t = 𝛿([D𝜓 ∶ D𝜑]t ) (t ∈ ℝ)

(1) (2)

for any n.s.f. weights 𝜑 and 𝜓 on ℳ. According to 19.16.(4), the dual weights are relatively invariant with respect to the dual action, more precisely, 𝜑( ̂ 𝛿̂g (X)) = ΔG ( g)−1 𝜑(X) ̂ (X ∈ ℛ(ℳ, 𝛿)+ , g ∈ G ).

(3)

Conversely, let Ψ be an n.s.f. weight on ℛ(ℳ, 𝛿), which is relatively invariant with respect to the ̂ dual action. Then it follows that [DΨ ∶ D𝜑] ̂ t ∈ ℛ(ℳ, 𝛿)𝛿 = 𝛿(ℳ) (t ∈ ℝ), and by Theorem 5.1 there exists an n.s.f. weight 𝜓 on ℳ such that [DΨ ∶ D𝜑] ̂ t = 𝛿([D𝜓 ∶ D𝜑]t ) = [D𝜓̂ ∶ D𝜑] ̂ t (t ∈ ℝ), hence Ψ = 𝜓. ̂ Thus, the dual weights on ℛ(ℳ, 𝛿) are exactly those which are relatively invariant with respect to the dual action. If G is commutative, then the notions introduced in Sections 19.16 and 19.17 agree with the corresponding notions introduced in Sections 19.7 and 19.8. 19.18. Consider again a continuous action 𝜎 ∶ G → Aut(ℳ) of G on ℳ and an n.s.f. weight 𝜑 on ℳ. On the crossed product ℛ(ℳ, 𝜎), we have the dual weight (19.8) 𝜑̂ = 𝜑 ◦ 𝜋𝜎−1 ◦ P𝜎̂ . By Theorem 19.5, we have ̂̂ ≡ (ℳ ⊗ ̄ ℬ(ℒ 2 (G)); 𝜎 ⊗ ̄ Ad(𝜌𝜌)) (ℛ(ℛ(ℳ, 𝜎), 𝜎); ̂ 𝜎)

(1)

Crossed Products

281

̄ ℬ(ℒ 2 (G)) we obtain the dual weight 𝜑̂̂ of the weight 𝜑̂ (19.17) and hence on ℳ ⊗ 𝜑̂̂ = 𝜑̂ ◦ P𝜎 ⊗Ad(𝜌) = 𝜑 ◦ 𝜋𝜎−1 ◦ P𝜎̂ ◦ P𝜎 ⊗Ad(𝜌) . ̄ ̄

(2)

̄ ℬ(ℒ 2 (G)) we can consider the weight 𝜑 ⊗ ̄ tr where tr is the canonical On the other hand, on ℳ ⊗ 2 trace on ℬ(ℒ (G)). It is then natural to try to compute the Connes cocycle ̄ tr)]t [D𝜑̂̂ ∶ D(𝜑 ⊗

(t ∈ ℝ)

that is, to give a more explicit expression of the second dual weight 𝜑̂̂ using the *-isomorphism (1). ̄ ℒ ∞ (G) defined by the To this end, we consider for each t ∈ ℝ the unitary operator Ut ∈ ℳ ⊗ function G ∋ g ↦ Ut ( g) = [D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t ∈ ℳ and recall that the modular function Δ = ΔG can be also regarded as nonsingular positive self-adjoint operator on ℒ 2 (G), acting by multiplication. Theorem. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on the W ∗ -algebra ℳ. For every n.s.f. weight 𝜑 on ℳ, we have ̄ tr)]t = (1ℳ ⊗ ̄ Δit )Ut [D𝜑̂̂ ∶ D(𝜑 ⊗ G

(t ∈ ℝ).

The following obvious consequence is particularly useful: Corollary. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the unimodular locally compact group ̄ tr. G on the W ∗ -algebra ℳ. For every 𝜎-invariant n.s.f. weight 𝜑 on ℳ we have 𝜑̂̂ = 𝜑 ⊗ The proof of the theorem is contained in Sections 19.19–19.23. 19.19. We first reformulate Theorem 19.18. Since Δ = ΔG is a nonsingular positive self-adjoint operator on ℒ 2 (G), we can consider the n.s.f. weight trΔ on ℬ(ℒ 2 (G)) given by the Pedersen– Takesaki construction (4.4). ̄ ℒ ∞ (G) ⊂ ℳ ⊗ ̄ ℬ(ℒ 2 (G)) is a unitary cocycle with Lemma. The mapping ℝ ∋ t ↦ Ut ∈ ℳ ⊗ ̄ trΔ on ℳ ⊗ ̄ ℬ(ℒ 2 (G)). respect to the modular automorphism group of the n.s.f. weight 𝜑 ⊗ ̄ ℒ ∞ (G) is determined by Proof. Recall that the unitary operator Ut ∈ ℳ ⊗ ̄ k⟩ = ⟨Ut , 𝜓 ⊗

∫

𝜓(Ut ( g))k( g) dg

(𝜓 ∈ ℳ∗ , k ∈ ℒ 1 (G)),

where Ut ( g) = [D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t ( g ∈ G ). Since, for every g ∈ G, the function t ↦ Ut ( g) is s∗ -continuous, using the Lebesgue dominated convergence theorem we see also that the function t ↦ Ut is s∗ -continuous. ̄ trΔ . Then 𝜎 𝜃 = 𝜎t𝜑 ⊗ ̄ Ad(Δit ) and hence Let 𝜃 = 𝜑 ⊗ t ̄ ℒ ∞ (G) = 𝜎t𝜑 ⊗ ̄ 𝜄G 𝜎t𝜃 |ℳ ⊗

(t ∈ ℝ).

282

Crossed Products

̄ ℒ ∞ (G) is defined by the function g ↦ Us ( g), the element 𝜎 𝜃 (Us ) ∈ Let s, t ∈ ℝ. Since Us ∈ ℳ ⊗ t ∞ ̄ ℒ (G) is defined by the function g ↦ 𝜎t𝜑 (Us ( g)) and the element Ut 𝜎 𝜃 (Us ) ∈ ℳ ⊗ ̄ ℒ ∞ (G) ℳ⊗ t is defined by the function g ↦ Ut ( g)𝜎t𝜑 (Us ( g)) = Ut+s ( g). Hence Ut+s = Ut 𝜎t𝜃 (Us ). ̄ ℬ(ℒ 2 (G)), denoted by Thus, by Theorem 5.1, there exists a unique n.s.f. weight on ℳ ⊗ ̄ trΔ )U , such that (𝜑 ⊗ ̄ trΔ )U ) ∶ D(𝜑 ⊗ ̄ trΔ )]t = Ut [D((𝜑 ⊗

(t ∈ ℝ).

Then ̄ trΔ )U ) ∶ D(𝜑 ⊗ ̄ tr)]t = (1 ⊗ ̄ Δit )Ut [D((𝜑 ⊗

(t ∈ ℝ).

Consequently, Theorem 19.18 asserts that ̄ trΔ )U 𝜑̂̂ = (𝜑 ⊗

(1)

̄ ℬ(ℒ 2 (G)). on ℳ ⊗ The proof of (1) consists in computing the values of the two weights on elements X ∈ ̄ ℬ(ℒ 2 (G)))+ defined by compactly supported s∗ -continuous functions G × G ↦ ℳ (ℳ ⊗ (18.21.(2)) and showing that the two weights commute; then (1) will follow from Theorem 6.2. ̄ trΔ )U are not directly computable, we shall first express 19.20. Since the values of the weight (𝜑 ⊗ this weight in another form, which makes possible such computations. Without loss of generality, we may assume ℳ ⊂ ℬ(ℋ ) realized as a von Neumann algebra such that there is an so-continuous unitary representation v ∶ G → ℬ(ℋ ) with 𝜎g = Ad(v( g))|ℳ( g ∈ ̄ ℒ ∞ (G) and hence a G ). Then the function g ↦ v( g) defines a unitary operator V ∈ ℳ ⊗ ∞ ̄ ℒ (G) (compare to the proof of Lemma 19.2). It is easy *-automorphism 𝔖 = Ad(V) of ℳ ⊗ ̄ ℒ ∞ (G) is defined by a bounded w-continuous function to check that if an element X ∈ ℳ ⊗ ̄ ℒ ∞ (G) is defined by the function G ∋ g ↦ X( g) ∈ ℳ, then the element 𝔖(X) ∈ ℳ ⊗ G ∋ g ↦ 𝜎g (X( g)): [𝔖(X)]( g) = 𝜎g (X( g))

( g ∈ G ).

(1)

̄ ℒ ∞ (G) depends only on the continuous action 𝜎 ∶ G → Thus, the *-automorphism 𝔖 of ℳ ⊗ ̄ 𝜇) ◦ 𝔖 on ℳ ⊗ ̄ ℒ ∞ (G), where 𝜇 = 𝜇G . Aut(ℳ) and we can consider the n.s.f. weight (𝜑 ⊗ ̂ on ℳ ⊗ ̄ 𝛿G of G ̄ ℬ(ℒ 2 (G)) defines an n.s.f. operatorOn the other hand, the action 𝛿 = 𝜄ℳ ⊗ ̄ ℬ(ℒ 2 (G)) with values in (ℳ ⊗ ̄ ℬ(ℒ 2 (G)))𝛿 = ℳ ⊗ ̄ ℒ ∞ (G), and we valued weight P𝛿 on ℳ ⊗ ̄ 𝜇) ◦ 𝔖) ◦ P𝛿 on ℳ ⊗ ̄ ℬ(ℒ 2 (G)). can consider the n.s.f. weight ((𝜑 ⊗ Lemma. We have ̄ trΔ )U = ((𝜑 ⊗ ̄ 𝜇) ◦ 𝔖) ◦ P𝛿 . (𝜑 ⊗

(2)

̄ ℒ ∞ (G) Proof. From the proof of Lemma 19.19, it also follows that the mapping t ↦ Ut ∈ ℳ ⊗ ̄ 𝜇 on is a unitary cocycle with respect to the modular automorphism group of the n.s.f. weight 𝜑 ⊗ ̄ ℒ ∞ (G), hence there exists a unique n.s.f. weight on ℳ ⊗ ̄ ℒ ∞ (G), denoted by (𝜑 ⊗ ̄ 𝜇)U , such ℳ⊗ ̄ 𝜇)U ∶ D(𝜑 ⊗ ̄ 𝜇)]t = Ut (t ∈ ℝ). Actually, that [D(𝜑 ⊗ ̄ 𝜇)U = (𝜑 ⊗ ̄ 𝜇) ◦ 𝔖. (𝜑 ⊗

(3)

Crossed Products

283

̄ ℒ ∞ (G) defined by the bounded w-continuous function G ∋ g ↦ Indeed, for the element X ∈ ℳ ⊗ X( g) ∈ ℳ we have ̄ ̄ 𝜄G ) ◦ 𝔖)(X)]( g) = (𝜎 −1 ◦ 𝜎t𝜑 ◦ 𝜎g )(X( g)) [𝜎t(𝜑⊗𝜇) ◦ 𝔖 (X)]( g) = [(𝔖−1 ◦ (𝜎t𝜑 ⊗ g 𝜑 ◦ 𝜎g

= 𝜎t

(X( g)) = Ut ( g)𝜎t𝜑 (X( g))Ut ( g)∗ ̄

̄ 𝜄G )(X)]( g)Ut ( g)∗ = Ut 𝜎t𝜑⊗𝜇 (X)U ∗ ]( g), = Ut ( g)[(𝜎t𝜑 ⊗ ̄

̄

hence 𝜎t(𝜑⊗𝜇) ◦ 𝔖 (X) = Ut 𝜎t𝜑⊗𝜇 (X)U∗t (t ∈ ℝ). Then, we similarly verify the corresponding KMS condition and hence (3) follows from the uniqueness part of Theorem 3.1. ̄ ℒ ∞ (G))-valued weight on ℳ ⊗ ̄ ℬ(ℒ 2 (G)) and Ut ∈ Furthermore, since P𝛿 is an n.s.f. (ℳ ⊗ ∞ ̄ ℳ ⊗ ℒ (G) (t ∈ ℝ), we infer from (3) and Theorem 11.9.(3) that ̄ 𝜇)U ◦ P𝛿 = ((𝜑 ⊗ ̄ 𝜇) ◦ P𝛿 )U . (𝜑 ⊗

(4)

̂ on ℬ(ℒ 2 (G)) defines an ℒ ∞ (G)-valued n.s.f. weight On the other hand, the action 𝛿G of G 2 ̄ 𝛿G we have (18.19.(5)) P𝛿 = 𝜄ℳ ⊗ ̄ P. Consequently, P = P𝛿G on ℬ(ℒ (G)). Since 𝛿 = 𝜄ℳ ⊗ ̄ 𝜇) ◦ P𝛿 = 𝜑 ⊗ ̄ (𝜇 ◦ P). (𝜑 ⊗

(5)

Thus, (2) will follow from (3), (4), and (5) once we have proved that on ℬ(ℒ 2 (G)): trΔ = 𝜇 ◦ P.

(6)

To prove (6), we consider an operator X ∈ ℬ(ℒ 2 (G))+ defined as in 18.21.(2) by a compactly supported continuous function X ∶ G × G ∋ (s, r) ↦ X(s, r) ∈ ℂ. By 18.22.(7) and 18.22.(8), we have [P(X)]( g) = ∫ Δ( g)X( g, g) dg ( g ∈ G ), hence (𝜇 ◦ P)(X) = ∫ Δ( g)X( g, g) dg. On the other hand, the operator ΔX is defined similarly by the function (ΔX)(s, r) = Δ(s)X(s, r)(s, r ∈ G ), and using Mercer’s formula (Dunford & Schwartz, 1958, 1963, Ch. IX, §8, Ex. 49 (c)) we obtain trΔ (X) = ∫ Δ( g)X( g, g) dg. Therefore, the n.s.f. weights 𝜇 ◦ P and trΔ are equal on the linear subspace ℬ of ℬ(ℒ 2 (G)) spanned by the elements X ∈ ℬ(ℒ 2 (G)), which are defined as above by compactly supported continuous functions G × G → ℂ. It is clear that ℬ is an s∗ -dense *-subalgebra tr of ℬ(ℒ 2 (G)). Since 𝜎t Δ (X) = Δit XΔ−it (X ∈ ℬ(ℒ 2 (G)), t ∈ ℝ), it follows that ℬ is also 𝜎 trΔ invariant. Moreover, since Δit ∈ ℒ ∞ (G), P is an ℒ ∞ (G)-valued weight on ℬ(ℒ 2 (G)) and ℒ ∞ (G) is abelian, we have P(Δit XΔ−it ) = Δit P(X)Δ−it = P(X) (X ∈ ℬ(ℒ 2 (G))+ ). Consequently, the weight 𝜇 ◦ P is 𝜎 trΔ -invariant, that is the two weights appearing in (6) commute. Thus, (6) follows from Theorem 6.2. ̄ ℬ(ℒ 2 (G)))+ be defined as in 18.21.(2) by the compactly supported 19.21 Lemma. Let X ∈ (ℳ ⊗ ∗ s -continuous function G × G ∋ (s, r) ↦ X(s, r) ∈ ℳ. Then ̂̂ 𝜑(X) =

∫

̄ trΔ )U (X). 𝜑(𝜎g (X( g, g)))Δ( g) dg = (𝜑 ⊗

(1)

Proof. By 18.21.(3), we have P𝜎 ⊗Ad(𝜌) (X) = Tf𝜎 , where f (s) = ∫ 𝜎sr (X(sr, r) Δ(r) dr (s ∈ G ). Recall ̄ ̄ 𝛿G so that, by 18.19.(4) and 18.22.(2), we that the dual action 𝜎̂ is the restriction of the action 𝜄ℳ ⊗ ( ) have P𝜎̂ (Tf𝜎 ) = 𝜋𝜎 ( f (e)) = 𝜋𝜎 ∫ 𝜎r (X(r, r))Δ(r) dr . Finally, using the definition 19.18.(2) of 𝜑̂̂ and Corollary 2.10, we infer that 𝜑(X) ̂̂ = ∫ 𝜑(𝜎r (X(r, r)))Δ(r) dr.

284

Crossed Products

On the other hand, using 19.20.(2) and 18.22.(2), we obtain ̄ trΔ )U (X) = (𝜑 ⊗ ̄ 𝜇)(𝔖(P𝛿 (X))) = (𝜑 ⊗

∫

𝜑([𝔖(P𝛿 (X))]( g))dg

=

∫

𝜑(𝜎g ([P𝛿 (X)]( g))dg =

=

∫

𝜑(𝜎g (X( g, g)))Δ( g)dg.

∫

𝜑(𝜎g (Δ( g)X( g, g)))dg

̄ trΔ )U and 𝜑̂̂ commute. We shall put 19.22. Our next objective is to show that the weights (𝜑 ⊗ ̄ trΔ )U (𝜑 ⊗

̄ ℬ(ℒ 2 (G))) (t ∈ ℝ) ∈ Aut(ℳ ⊗ ̄ Ad(𝜌𝜌( g)) ∈ Aut(ℳ ⊗ ̄ ℬ(ℒ 2 (G))) ( g ∈ G ) 𝛽g = 𝜎g ⊗ 𝛼t = 𝜎t

̃ t ( g) = Δ( g)it Ut ( g) (t ∈ ℝ, g ∈ G ). U Lemma. With the above notation we have 𝛼t |ℛ(ℳ, 𝜎) = 𝜎t𝜑̂ (t ∈ ℝ) P𝛽 ◦ 𝛼t = 𝛼t ◦ P𝛽 (t ∈ ℝ).

(1) (2)

̄ trΔ )U commute. Corollary. The weiglts 𝜑̂̂ and (𝜑 ⊗ ̄ ℬ(ℒ 2 (G)) and Proof of the Corollary. Recall that P𝛽 is an ℛ(ℳ, 𝜎)-valued n.s.f. weight on ℳ ⊗ ̂𝜑̂ = 𝜑̂ ◦ P𝛽 . Using (1) and (2), we get 𝜑̂̂ ◦ 𝛼t = 𝜑̂ ◦ (P𝛽 ◦ 𝛼t ) = (𝜑̂ ◦ 𝜎 𝜑̂ ) ◦ P𝛽 = 𝜑̂ ◦ P𝛽 = 𝜑̂̂ (t ∈ ℝ), t ̄ trΔ )U commute. hence 𝜑̂̂ is 𝛼-invariant, that is, 𝜑̂̂ and (𝜑 ⊗ Proof of the Lemma. Note that ̃ t ((𝜎t𝜑 ⊗ ̃∗ ̄ 𝜄G )(X))U 𝛼t (X) = U t

̄ ℬ(ℒ 2 (G)), t ∈ ℝ). (X ∈ ℳ ⊗

(3)

Thus, for f ∈ ℒ ∞ (G) and t ∈ ℝ we have ̃ t (1ℳ ⊗ ̃ ∗ = 1ℳ ⊗ ̄ f) = U ̄ f )U ̄ f, 𝛼t (1ℳ ⊗ t

(4)

̃t ∈ ℳ ⊗ ̄ ℒ ∞ (G) and ℒ ∞ (G) is commutative. Then, for x ∈ ℳ, t ∈ ℝ and g ∈ G, we have since U ̃ t ((𝜎t𝜑 ⊗ ̃ ∗ ]( g) ̄ 𝜄G )(𝜋𝜎 (x)))U [𝛼t (𝜋𝜎 (x))]( g) = [U t ̃ t ( g)𝜎t𝜑 ([𝜋𝜎 (x)]( g))U ̃ t ( g)∗ =U = Ut ( g)𝜎t𝜑 (𝜎g−1 (x))Ut ( g)∗ 𝜑 ◦ 𝜎g

= 𝜎t

(𝜎g−1 (x))

= 𝜎g−1 (𝜎t𝜑 (x))

= [𝜋𝜎 (𝜎t𝜑 (x))]( g) = [𝜎t𝜑̂ (𝜋𝜎 (x))]( g),

Crossed Products

285

by 19.8.(7), hence 𝛼t (𝜋𝜎 ((x)) = 𝜎t𝜑̂ (𝜋𝜎 (x)),

(5)

Finally, let g ∈ G, t ∈ ℝ. We have ̃ t (1ℳ ⊗ ̃∗ ̄ 𝜆 ( g)) = U ̄ 𝜆 ( g))U 𝛼t (1ℳ ⊗ t = (1ℳ

(6)

̃ t ))U ̃ ∗. ̄ 𝜆 ( g))((Ad(1ℳ ⊗ ̄ 𝜆 ( g)∗ ))(U ⊗ t

For r ∈ G, we have ̃ t )U ̃ ∗ ](r) = U ̃ t ( gr)U ̃ t (r)∗ ̄ 𝜆 ( g)∗ ))(U [(Ad(1ℳ ⊗ t

(7)

= Δ( g) Ut ( gr)Ut (r) = Δ( g)[𝜋𝜎 ([D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t ))(r) it

∗

The last equality is justified as follows: By Corollary 3.5, we have [D(𝜑 ◦ 𝜎gr ) ∶ D(𝜑 ◦ 𝜎r )]t [D(𝜑 ◦ 𝜎r ) ∶ D𝜑]t = [D(𝜑 ◦ 𝜎gr ) ∶ D𝜑]t that is, Ut ( gr)Ut (r)∗ = [D(𝜑 ◦ 𝜎gr ) ∶ D(𝜑 ◦ 𝜎r )]t and, using Corollary 3.8, we obtain Ut ( gr)Ut (r)∗ = [D((𝜑 ◦ 𝜎g ) ◦ 𝜎r ) ∶ D(𝜑 ◦ 𝜎r )]t = 𝜎r−1 ([D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t ) = [𝜋𝜎 ([D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t )(r). From (6) and (7) we infer that ̄ 𝜆( g)) = Δ( g)it (1ℳ ⊗ ̄ 𝜆( g))𝜋𝜎 ([D(𝜑 ◦ 𝜎g ) ∶ D𝜑]t ) 𝛼t (1ℳ ⊗ and hence (19.8.(8)) ̄ 𝜆 ( g)) = 𝜎t𝜑̂ (1ℳ ⊗ ̄ 𝜆 ( g)). 𝛼t (1ℳ ⊗

(8)

̄ 𝔏(G)}, (1) follows from (5) and (8). Since ℛ(ℳ, 𝜎) ⊂ Since ℛ(ℳ, 𝜎) = ℛ{𝜋𝜎 (ℳ), 1ℳ ⊗ ̄ ℬ(ℒ 2 (G)))𝛽 , (5) and (8) also show that (ℳ ⊗ (𝛽g ◦ 𝛼t )(X) = (𝛼t ◦ 𝛽g )(X)

(9)

̄ ℒ ∞ (G). Thus, (9) is valid for all for X ∈ ℛ(ℳ, 𝜎), and using (4) we get (9) for X ∈ 1ℳ ⊗ ̄ ℬ(ℒ 2 (G)) (see Lemma 19.2.(2)), and, since P𝛽 = ∫ 𝛽g (⋅) dg (18.20), this proves (2). □ X∈ℳ⊗ 19.23. We shall construct an 𝛼-invariant s∗ -dense *-subalgebra ̄ ℬ(ℒ 2 (G)) ℬ ⊂ 𝔑(𝜑 ⊗tr ̄ Δ )U ⊂ ℳ ⊗ ̄ trΔ )U and 𝜑̂̂ are equal on ℬ. such that the weights (𝜑 ⊗

286

Crossed Products

To this end, we consider the sets ̄ ℬ(ℒ 2 (G)); X is defined (18.21.(2)) by a compactly 𝔛 = {X ∈ ℳ ⊗ supported s∗ -continuous function G × G ∋ (s, r) ↦ X(s, r) ∈ ℳ}, 𝔖 = {S ∈ ℬ(ℒ 2 (G)); S is defined (18.21.(2)) by a compactly supported continuous function G × G ∋ (s, r) ↦ S(s, r) ∈ ℂ}, 𝔚 = {W ∶ G × G → ℳ; W is an s∗ -continuous function with W(s, r) unitary for all s, r ∈ G}. For S ∈ 𝔖, W ∈ 𝔚, x ∈ 𝔑𝜑 ⊂ ℳ, we consider the operator ̄ ℬ(ℒ 2 (G)) Y = Y(S, W, x) ∈ 𝔛 ⊂ ℳ ⊗ defined by the function Y(s, r) = S(s, r)W(s, r)𝜎r−1 (x) (s, r ∈ G ).

(1)

Then Y∗ is defined by the function Y∗ (s, r) = Y(r, s)∗ = S(r, s)𝜎s−1 (x∗ )W(r, s)∗

(s, r ∈ G ),

(2)

and X = Y∗ Y is defined by the function X(s, r) =

∫

=

∫

Y∗ (s, h)Y(h, r) dh S(h, s)S(h, r)𝜎s−1 (x∗ )W(h, s)∗ W(h, r)𝜎r−1 (x) dh

(s, r ∈ G ).

In particular, ( X( g, g) =

∫

) |S(h, g)|2 dh 𝜎g−1 (x∗ x)

(g ∈ G)

and, using Lemma 19.21, we get ̄ trΔ )U (X) = (𝜑 ⊗

∫

𝜑(𝜎g (X( g, g)))Δ( g) dg = 𝜑(x∗ x)

∫ ∫

|S(h, g)|2 dh dg < +∞.

Consequently, Y(S, W, x) ∈ 𝔑(𝜑 ⊗̄ trΔ )U . Let ℬ be the *-subalgebra of ℳ generated by the elements of form Y(S, W, x) with S ∈ 𝔖, W ∈ 𝔚, and x ∈ 𝔑𝜑 . Then ℬ ⊂ 𝔑(𝜑 ⊗̄ trΔ )U .

Crossed Products

287

Also, as ℬ ⊂ 𝔛 we have, by Lemma 19.21, ̂̂ ∗ X) = (𝜑 ⊗ ̄ trΔ )U (X∗ X) 𝜑(X

(X ∈ ℬ).

(3)

If W0 ∈ 𝔚 is defined by W0 (s, r) = 1ℳ (s, r ∈ G ), then for every S ∈ 𝔖 and every x ∈ 𝔑𝜑 , we have ̄ S)𝜋𝜎 (x). Y(S, W0 , x) = (1ℳ ⊗ Using Lemma 19.2.(2), we infer that the *-subalgebra ℬ is s∗ -dense in ℳ. It remains to be shown that ℬ is 𝛼t -invariant. Let S ∈ 𝔖, W ∈ 𝔚, x ∈ 𝔑𝜑 and let the operator Y = Y(S, W, x) ∈ 𝔛 be defined by the function (1). Recall that the *-automorphism 𝛼t is determined by 19.22.(3). It follows that the operator 𝛼t (Y) ∈ 𝔛 is determined by the function ̃ t (s)𝜎t𝜑 (W(s, r))𝜎t𝜑 (𝜎 −1 (x))U ̃ t (r)∗ [𝛼t (Y)](s, r) = S(s, r)U r ̃ t (s)𝜎t𝜑 (W(s, r))U ̃ t (r)∗ U ̃ t (r)𝜎t𝜑 (𝜎 −1 (x))U ̃ t (r)∗ = S(s, r)U r

̃ t (s)𝜎t𝜑 (W(s, r))U ̃ t (r)∗ 𝜎 −1 (𝜎t𝜑 (x)). = S(s, r)U r Hence 𝛼t (Y(S, W, x)) = Y(S, Wt , 𝜎t𝜑 (x)) ̃ t (s)𝜎t𝜑 (W(s, r))U ̃ t (r)∗ (s, r ∈ G ). Consequently, ℬ is where Wt ∈ 𝔚 is the function Wt (s, r) = U indeed 𝛼t -invariant. ̄ trΔ )U commute and satisfy (2), so that they are equal by Thus, the weights 𝜑̂̂ and (𝜑 ⊗ Theorem 6.2. The proof of Theorem 19.18 is complete. □ 19.24. Notes. The crossed product construction for operator algebras appeared already in the work of Murray and von Neumann (1936, 1937, 1943, I) for actions of discrete groups on commutative W ∗ -algebras. It was precisely this construction that produced the first concrete examples of factors of types II1 , II∞ and III. Subsequently, the construction was developed, for discrete groups, in Ching (1969); Diximier (1969); Golodec (1971); Nakamura and Takeda (1958, 1960); Suzuki (1959); Turumaru (1958); Zeller-Meier (1968), and so on, and Connes (1973a) discovered its crucial importance for the structure theory of type III factors. In the general case of locally compact groups, the construction appears in Doplicher, Kastler, Robinson. and the first systematic study of it is due to Takesaki (1973b). The definition of the crossed product of a W ∗ -algebra by the action of a Kac algebra was introduced in Strătilă, Voiculescu, and Zsidó (1976). Takesaki (1973b) proved the most important special cases of the main results presented in this section (19.5, 19.8, 19.13, 19.14, 19.18), and so extended the duality principle given in Connes (1973a) to arbitrary W ∗ -algebras of type III. In the general case, for Theorem 19.5 see Landstad (1977); Strătilă et al. (1976, 1977), for Corollaries 19.13 and 19.14 see Digernes (1975), for Theorem 19.8 see Digernes (1975); Haagerup (1978); Sauvageot (1977); Strătilă et al. (1976, 1977); Zsidó (1977); Theorem 19.18 is proved in Digernes (1975) for abelian groups and in Strătilă et al. (1976, 1977); Strătilă (1977) for arbitrary locally compact groups. The characterization of crossed product W ∗ -algebras by Theorem 19.9 is due to Landstad (1979).

288

Crossed Products

The theory of crossed products of W ∗ -algebras by actions of group duals was developed in Landstad (1977); Strătilă et al. (1976, 1977). The key point in this theory is the saturation property (19.15.(5)) introduced in the general case and proved for amenable groups in Strătilă et al. (1976); for arbitrary locally compact groups, Proposition 19.15 is due to Landstad (1977) (see (Strătilă et al., 1976, 1977, II, pp. 83–85). A different approach, working for arbitrary compact groups, is due to Roberts (1976). For our exposition, we have used Haagerup (1978); Landstad (1979) and Strătilă et al. (1976, 1977). We should mention also the survey article Nakagami and Takesaki (1979) and the following related references concerning extensions of the commutation theorem for crossed products Day (1957); van Heeswijck (1979); Rousseau (1979, 1980); Rousseau and Van Daele (1979), crossed products with cocycles (Sutherland, 1977, 1980; Zeller-Meier, 1968), crossed products by actions of Kac algebras Enock (1977); Enock and Schwartz (1975) and C∗ -crossed products Doplicher et al. (1979); Landstad, Olesen, and Pedersen (1978); Olesen, Pedersen, and Størmer, (1977); Pedersen (1973–1977); Takai (1975); Pimsner and Voiculescu (1982); Rieffel (1981).

20 Comparison of cocycles In this section, we develop a comparison theory for cocycles with respect to a given continuous group action on a W ∗ -algebra, characterize the square-integrable cocycles as subcocycles of the dominant cocycle and, as an application, study the dominant actions. Moreover, this theory will give rise, via the Coones cocycle theorem (3.1 and 5.1), (§23) to a corresponding comparison theory for weights on W ∗ -algebras. 20.1. Let G be a locally compact group and 𝜎 ∶ G → Aut(ℳ) a continuous action of G on the W ∗ -algebra ℳ. Recall (5.1) that a 𝜎-cocycle (of degree 1) is an s∗ -continuous function 𝜎 ∶ G ∋ s ↦ a(s) ∈ ℳ with the properties: a(st) = a(s)𝜎s (a(t)) and a(s−1 ) = 𝜎s−1 (a(s)∗ )

(s, t ∈ G )

(1)

and that the set of all 𝜎-cocycles is denoted by Z𝜎 = Z𝜎 (G; ℳ). We denote by e ∈ G the neutral element of G. Proposition. Let a ∈ Z𝜎 (G; ℳ). Then the elements a(s) ∈ ℳ are partial isometries a(s)a(s)∗ = a(e),

a(s)∗ a(s) = 𝜎s (a(e))

(s ∈ G )

(2)

in particular a(e) is a projection and a(e)a(s) = a(s) = a(s)𝜎s (a(e))

(s ∈ G ).

(3)

The equation (a 𝜎)s (x) = a(s)𝜎s (x)a(s)∗

(x ∈ ℳa(e) , s ∈ G )

(4)

Comparison of cocycles

289

defines a continuous action a 𝜎 ∶ G → Aut(ℳa(e) ) whose centralizer is denoted by 𝜎

ℳ a = (ℳa(e) )a .

(5)

For every projection p ∈ ℳ a , the function ap ; G ∋ s ↦ pa(s) ∈ ℳ

(6)

is a 𝜎-cocycle ap ∈ Z𝜎 (G; ℳ). Proof. We have a(e) = 𝜎e−1 (a(e)∗ ) = a(e)∗ and a(e) = a(ee) = a(e)𝜎e (a(e)) = a(e)2 , hence a(e) is a projection. Since a(s)∗ = 𝜎s (a(s−1 )), we obtain a(s)a(s)∗ = a(s)𝜎s (a(s−1 )) = a(ss−1 ) = a(e) and a(s)∗ a(s) = 𝜎s (a(s−1 ))a(s) = 𝜎s (a(s−1 )𝜎s−1 (a(s))) = 𝜎s (a(s−1 s)) = 𝜎s (a(e)), thus proving (2) and (3). Using (1), (2), and (3) it is easy to see that (4) define a continuous action. For instance, for s, t ∈ G and x, y ∈ a(e)ℳa(e), we have (a 𝜎)s (xy) = a(s)𝜎s (xy)a(s)∗ = a(s)𝜎s (x)𝜎s (a(e))𝜎s ( y)a(s)∗ = a(s)𝜎s (x)a(s)∗ a(s)𝜎s ( y)a(s)∗ = (a 𝜎)s (x)(a 𝜎)s ( y) and (a 𝜎)st (x) = a(st)𝜎st (x)a(st)∗ = a(s)𝜎s (a(t)) 𝜎s (𝜎t (x))𝜎s (a(t)∗ )a(s)∗ = a(s)𝜎s (a(t)𝜎t (x)a(t)∗ )a(s)∗ = (a 𝜎)s ((a 𝜎)t (x)). Consider now a projection p ∈ ℳ a . Then p ≤ a(e) and a(r)𝜎r (p)a(r)∗ = p for all r ∈ G, hence p a (st) = pa(st) = pa(s)𝜎s (a(t)) = pa(s)𝜎s (p)a(s)∗ a(s)𝜎s (a(t)) = pa(s)𝜎s (pa(e)a(t)) = ap (s)𝜎s (ap (t)) and, since 𝜎r (p) = a(r)∗ pa(r), ap (s−1 ) = pa(s−1 ) = a(e)∗ pa(s−1 ) = (a(s−1 )𝜎s−1 (a(s)))∗ pa(s−1 ) = 𝜎s−1 (a(s)∗ )a(s−1 )∗ pa(s−1 ) = 𝜎s−1 (a(s)∗ )𝜎s−1 (p) = 𝜎s−1 ((pa(s))∗ ) = 𝜎s−1 (ap (s)∗ ). If the W ∗ -algebra ℳ a is properly infinite, then we say that the cocycle a ∈ Z𝜎 (G; ℳ) is of infinite multiplicity. A cocycle of the form ap is called a subcocycle of a. 20.2. Let ℱ2 be the type I2 factor with the system of matrix units {eij }1≤i,j≤2 . We shall identify ̄ ℱ2 with Mat2 (ℳ) in the usual way. Let 𝜄 ∶ G → Aut(ℱ2 ) be the trivial action and 𝜎 ∶ G → ℳ⊗ ̄ 𝜄 ∶ G → Aut(ℳ ⊗ ̄ ℱ2 ) is a continuous action. Aut(ℳ) a continuous action. Then 𝜎 ⊗ ̄ ℱ2 by Let a, b ∈ Z𝜎 (G; ℳ). We define a function c = c(a, b) ∶ G → ℳ ⊗ ̄ e11 + b(s) ⊗ ̄ e22 = c(s) = a(s) ⊗

(

a(s) 0

0 b(s)

) (s ∈ G )

(1)

and the set ℐ (a, b) by ℐ (a, b) = {x ∈ a(e)ℳa(e); xb(s) = a(s)𝜎s (x) for all s ∈ G}. Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action and a, b ∈ Z𝜎 (G; ℳ). Then c = c(a, b) ∈ ̄ ℱ2 ), Z𝜎 ⊗𝜄 ̄ (G; ℳ ⊗ ̄ e11 ∈ (ℳ ⊗ ̄ ℱ2 )c , b(e) ⊗ ̄ e22 ∈ (ℳ ⊗ ̄ ℱ2 )c a(e) ⊗ and the following statements are equivalent:

(2)

290

Crossed Products

̄ e11 ≺ b(e) ⊗ ̄ e22 in the W ∗ -algebra (ℳ ⊗ ̄ ℱ2 )c ; (i) a(e) ⊗ ∗ ∗ (ii) there exists u ∈ ℳ such that u u = a(e), uu ≤ b(e) and u = b(s)𝜎s (u)a(s)∗ for all s ∈ G; (iii) there exist v ∈ ℳ and a projection q ∈ ℳ b such that a(s) = v∗ bq (s)𝜎s (v) and bq (s) = va(s)𝜎s (v∗ ) for all s ∈ G. On the other hand, we have ℐ (a, a) = ℳ a

(3)

ℐ (a, b)∗ = ℐ (b, a)

(4)

ℐ (a, w)ℐ (w, b) ⊂ ℐ (a, b) for all w ∈ Z𝜎 (G; ℳ)

(5)

x ∈ ℐ (a, b) with polar decomposition x = v|x| ⇒ |x| ∈ ℐ (b, b), ) ( ℐ (a, a) ℐ (a, b) ̄ ℱ2 )c = (ℳ ⊗ ℐ (b, a) ℐ (b, b) ̄ ℱ2 )c ]a(e) ⊗̄ e = ℳ a , [(ℳ ⊗ 11

v ∈ ℐ (a, b)

̄ ℱ2 )c ]b(e) ⊗̄ e = ℳ b [(ℳ ⊗ 22

̄ e11 ) ⟂ z(b(e) ⊗ ̄ e22 ) in (ℳ ⊗ ̄ ℱ2 )c ℐ (a, b) = {0} ⇔ z(a(e) ⊗ ̄ e11 ) = z(b(e) ⊗ ̄ e22 ) = z ∈ (ℳ ⊗ ̄ ℱ2 ) ⇒ z = a(e) ⊗ ̄ e11 + b(e) ⊗ ̄ e22 . z(a(e) ⊗ c

(6) (7) (8) (9) (10)

̄ e11 ≤ c(e). For s ∈ G, we have Proof. By definition (1) it is clear that ̄ 𝜄 (G; ℳ) and a(e) ⊗ ( c ∈ Z𝜎 ⊗ ) ( )( )( ) ∗ 0 a(s) 0 𝜎 (a(e)) 0 a(s) a(s)𝜎s (a(e))a(s)∗ 0 s ∗ ̄ ̄ c(s)[(𝜎 ⊗ 𝜄)s (a(e) ⊗ e11 )]c(s) = 0 b(s) 0 0 0 b(s)∗ 0 0 ) ( a(e) 0 ̄ e11 , which proves (2). = a(e) ⊗ = 0 0 We now prove the equivalence of statements (i), (ii), and (iii). ̄ ℱ2 with x∗ x = a(e) ⊗ ̄ e11 , xx∗ ≤ b(e) ⊗ ̄ e22 . If x = [xij ] with xij ∈ ℳ, (i) ⇔ (ii). Let x ∈ ℳ ⊗ ∗ ∗ then it follows that x11 x11 + x21 x21 = a(e), x12 = x22 = 0 and x11 = 0, x21 x∗21 ≤ b(e), that is ( ) 00 ̄ ℱ2 )c(e). If we require that x= with u ∈ ℳ, u∗ u = a(e), uu∗ ≤ b(e). Hence x ∈ c(e)(ℳ ⊗ u0 ̄ ℱ2 )c , that is, c(s)[(𝜎 ⊗ ̄ 𝜄)s (x)]c(s)∗ = x, then it also follows that u = b(s)𝜎s (u)a(s)∗ (s ∈ x ∈ (ℳ ⊗ G ). Thus, (i) ⇒ (ii) and, similarly, (ii) ⇒ (i). (ii) ⇒ (iii). Let v = u and q = uu∗ ≤ b(e). We have q = uu∗ = b(s)𝜎s (u)a(s)∗ a(s)𝜎s (u∗ )b(s∗ ) = b(s)𝜎s (ua(e)u∗ )b(s)∗ = b(s)𝜎s (q)b(s)∗ , hence q ∈ ℳ b and then, successively, ua(s) = b(s)𝜎s (u)a(s)∗ a(s) = b(s)𝜎s (u)𝜎s (a(e)) = b(s)𝜎s (u), u∗ b(s)𝜎s (u) = u∗ ua(s) = a(e)a(s) = a(s), a(s) = u∗ uu∗ b(s)𝜎s (u) = v∗ qb(s)𝜎s (v) = v∗ bq (s)𝜎s (v), bq (s) = qb(s) = b(s)𝜎s (q)b(s)∗ b(s) = b(s)𝜎s (u)𝜎s (u∗ )𝜎s (b(e)) = b(s)𝜎s (u)𝜎s (u∗ ) = ua(s)𝜎s (u∗ ) = va(s)𝜎s (v∗ ). (iii) ⇒ (ii). Let u = qva(e). Then s(u∗ u) ≤ a(e), s(uu∗ ) ≤ q and a(s) = u∗ bq (s)𝜎a (u), bq (s) = ua(s)𝜎s (u∗ ) (s ∈ G ). Then a(e) = u∗ bq (e)u = u∗ ua(e)u∗ u = (u∗ u)2 , hence u∗ u = a(e). Also, q = bq (e) = ua(e)u∗ = uu∗ quu∗ = (uu∗ )2 , hence uu∗ = q ≤ b(e). Finally, we obtain bq (s)𝜎s (u) = ua(s)𝜎s (u∗ u) = ua(s) and u = ua(e) = ua(s)a(s)∗ = bq (s)𝜎s (u)a(s)∗ = qb(s)𝜎s (u)a(s)∗ = b(e)qb(s)𝜎s (u)a(s)∗ = b(s)b(s)∗ qb(s)𝜎s (u)(a(s)∗ = b(s)𝜎s (q)𝜎s (u)a(s)∗ = b(s)𝜎s (u)a(s)∗ . Assertions (3), (4), (5), (7), and (8) can be checked without any difficulty. If x ∈ ℐ (a, b), then, by (3), (4), and (5), we have |x|2 = x∗ x ∈ ℐ (b, b) = ℳ b and hence |x| ∈ ℳ b . Since x ∈ a(e)ℳb(e) and x = v|x| is the polar decomposition of x, it follows that v ∈ a(e)ℳb(e)

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291

and vb(s)𝜎s (|x|) = v|x|b(s) = xb(s) = a(s)𝜎s (x) = a(s)𝜎s (v)𝜎s (|x|). Thus, with ℳ ⊂ ℬ(ℋ ) realized as a von Neumann algebra, we have vb(s)𝜉 = a(s)𝜎s (v)𝜉 for all 𝜉 ∈ 𝜎s (|x|)ℋ . If 𝜉 ⟂ 𝜎s (|x|)ℋ and 𝜂 ∈ ℋ , then (vb(s)𝜉|𝜂) = (b(s)𝜉|v∗ 𝜂) = 0 (as v∗ ℋ = |x|ℋ ) and, for every 𝜁 ∈ ℋ , we have (b(s)𝜉||x|𝜁) = (𝜉|b(s)∗ |x|𝜁) = (𝜉|𝜎s (|x|)b(s)∗ 𝜁) = 0. Consequently, vb(s) = a(s)𝜎s (v), that is v ∈ ℐ (a, b), thus proving (6). If there exists 0 ≠ x ∈ ℐ (a, b), then (

) a(e) x ̄ ℱ2 )c and ∈ (ℳ ⊗ x∗ b(e) )( ) ( ) ( )( 00 0x a(e) 0 a(e) x = ≠0 0 b(e) 00 0 0 x∗ b(e) ̄ e11 )(ℳ ⊗ ̄ ℱ2 )c (b(e) ⊗ ̄ e22 ) ≠ 0 and consequently ([L], 4.5) the central supports of that is, (a(e) ⊗ ̄ e11 and b(e) ⊗ ̄ e22 in (ℳ ⊗ ̄ ℱ2 )c are not mutually orthogonal. Conversely, if there exists a(e) ⊗ ̄ ℱ2 )c with (a(e) ⊗ ̄ e11 )x(b(e) ⊗ ̄ e22 ) ≠ 0, then 0 ≠ a(e)x12 b(e) ∈ ℐ (a, b). We 0 ≠ x = [xij ] ∈ (ℳ ⊗ have thus proved (9). ̄ e11 and b(e) ⊗ ̄ e22 have the same central support z in (ℳ ⊗ ̄ ℱ2 )c , then z ≥ Finally, if a(e) ⊗ c ̄ e11 + b(e) ⊗ ̄ e22 = c(e). Since c(e) is the unit element in (ℳ ⊗ ̄ ℱ2 ) , it follows that z = c(e), a(e) ⊗ which proves (10). The cocycle c = c(a, b) is called the balanced cocycle associated with the cocycles a and b. If the equivalent conditions (i), (ii), and (iii) are satisfied, then we say that the cocycle a ∈ Z𝜎 is dominated by the cocycle b ∈ Z𝜎 and write a ≲ b. ̄ e11 ∼ b(e) ⊗ ̄ e22 in (ℳ ⊗ ̄ ℱ2 )c or, equivalently, if q = b(e) in (iii), then we say that a If a(e) ⊗ and b are equivalent and write a ≂ b. Note that a ≲ b if and only if a is equivalent to a cocycle of the form bq with q ∈ Proj(ℳ b ). From ([L], 4.7), it follows that a ≂ b ⇔ a ≲ b and b ≲ a.

(11)

a ≂ b ⇒ the W ∗ -algebras ℳ a and ℳ b are ∗ -isomorphic.

(12)

From (8), it follows that

If ℐ (a, b) = {0}, then we say that the cocycles a, b ∈ Z𝜎 are disjoint and write a ◦| b. ̄ e11 and b(e) ⊗ ̄ e22 have the same central support in (ℳ ⊗ ̄ ℱ2 )c , then we say that a and If a(e) ⊗ b are quasi-equivalent and write a ∼ b. Clearly, a ≂ b ⇒ a ∼ b. On the other hand, if ℳ is countably decomposable and the cocycles a, b ∈ Z𝜎 (G; ℳ) are of infinite multiplicity, then a ≂ b ⇔ a ∼ b.

(13)

̄ e11 and b(e) ⊗ ̄ e22 are properly Indeed, by assumption and assertion (8), the projections a(e) ⊗ ̄ ℱ2 )c and hence ([L], 4.13) they are infinite in the countably decomposable W ∗ -algebra (ℳ ⊗ equivalent if and only if they have same central support.

292

Crossed Products

Note that if a, b ∈ Z𝜎 (G; ℳ) and a ≂ b, then the continuous actions 𝜎 ∶ G → Aut(ℳa(e) ), b 𝜎 ∶ G → Aut(ℳb(e) ) are inner conjugate, a that is, there exists u ∈ ℳ with u∗ u = a(e), uu∗ = b(e) and (a 𝜎)s (x) = u∗ [(b 𝜎)s (uxu∗ )]u (x ∈ a(e)ℳa(e)).

(14)

Indeed, if a ≂ b, then, by the above proposition, there exists u ∈ ℳ with u∗ u = a(e), uu∗ = b(e), and a(s) = u∗ b(s)𝜎s (u) (s ∈ G ). Consequently, for every x ∈ a(e)ℳa(e), we have (a 𝜎)s (x) = a(s)𝜎s (x)a(s)∗ = u∗ b(s)𝜎s (u)𝜎s (x)𝜎s (u∗ )b(s)∗ u = u∗ [(b 𝜎)s (uxu∗ )]u. 20.3. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ. A cocycle a ∈ Z𝜎 (G; ℳ) with a(e) = 1ℳ is called a unitary cocycle. In this case all the elements a(s) (s ∈ G ), belong to the unitary group U(ℳ) of ℳ. We shall denote by Z𝜎 (G; U(ℳ)) the set of all unitary 𝜎-cocycles. The function 1 ∶ G → ℳ defined by 1(s) = 1ℳ (s ∈ G ), is a unitary cocycle, 1 ∈ Z𝜎 (G; U(ℳ)). Clearly, 1 𝜎 = 𝜎 and ℳ 1 = ℳ 𝜎 . A cocycle a ∈ Z𝜎 (G, ℳ) is called trivial if a ≂ 1. In this case, a is a unitary cocycle and there exists a unitary element v ∈ ℳ such that a(s) = v∗ 𝜎s (v), (s ∈ G ). Note that for all u ∈ Z𝜎 (G; U(ℳ)) and b ∈ Z(u 𝜎) (G; ℳ) the equation a(s) = b(s)u(s) (s ∈ G ) defines an element a ∈ Z𝜎 (G; ℳ) whose centralizer is equal to the centralizer of b ∈ Z(u 𝜎) (G; ℳ).

(1)

Indeed, for s, t ∈ G we have a(st) = b(st)u(st) = b(s)u(s)𝜎s (b(t))u(s)∗ u(s)𝜎s (u(t)) = b(s)u(s)𝜎s (b(t)u(t)) = a(s)𝜎s (a(t)) and a(s−1 ) = b(s−1 )u(s−1 ) = 𝜎s−1 (u(s)∗ b(s)∗ u(s))𝜎s−1 (u(s)∗ ) = 𝜎s−1 (u(s)∗ b(s)∗ ) = 𝜎s−1 (a(s)∗ ), hence a ∈ Z𝜎 (G; ℳ) and, clearly, u 𝜎 = b (a 𝜎). On the other hand, we now show that if ℳ is properly infinite, then there is a unitary cocycle u ∈ Z𝜎 (G; ℳ) of infinite multiplicity.

(2)

Indeed, since ℳ is properly infinite, there is a family of partial isometrics {un }n∈I ⊂ ℳ with ∑ un u∗m = 0 for n ≠ m, un u∗n = 1 and n u∗n un = 1. We define ∑ u(s) = u∗n 𝜎s (un ) (s ∈ G ). n

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Then, u(s) ∈ U(ℳ) since ∗

u(s)u(s) =

( ∑

)( u∗n 𝜎s (un )

∑

n

and u(s)∗ u(s) =

( ∑

) ∗ 𝜎s (um )um

=

∑

m

)( 𝜎s (u∗m )um

∑

m

u∗n un = 1

n

) u∗n 𝜎s (un )

( = 𝜎s

∑

n

) u∗n un

= 1.

n

Also, u ∈ Z𝜎 (G; U(ℳ)) since ( )( ) ∑ ∑ ∑ ∗ ∗ u(s)𝜎s (u(t)) = un 𝜎s (un ) 𝜎s (um )𝜎st (um ) = u∗n 𝜎st (un ) = u(st). n

m

n

Moreover, we have u∗i uj ∈ ℳ u since u(s)𝜎s (u∗i uj )u(s)∗ = u∗i 𝜎s (ui )𝜎s (u∗i uj )𝜎s (u∗j )uj = u∗i uj and hence ℳ u is properly infinite. Finally, we note that if 𝜄 ∶ G → Aut(ℳ) is the trivial action then, clearly Z𝜄 (G; U(ℳ)) = {u ∶ G → ℳ s-continuous unitary representations}.

(3)

20.4. Let 𝜎 ∶ G → Aut(ℳ) and 𝜏 ∶ G → Aut(𝒩 ) be continuous actions of G on the W ∗ -algebras ℳ and 𝒩 , respectively. If a ∈ Z𝜎 (G; ℳ) and b ∈ Z𝜏 (G; 𝒩 ), then the equation ̄ b)(s) = a(s) ⊗ ̄ b(s) (a ⊗

(s ∈ G )

̄ b ∈ Z𝜎 ⊗𝜏 ̄ 𝒩 ), called the tensor product of a and b. defines a cocycle a ⊗ ̄ (G; ℳ ⊗ Next we take ℳ countably decomposable, 𝒩 = ℱ∞ = the countably decomposable type I∞ factor and 𝜏 = 𝜄 = the trivial action. Then, for a, b ∈ Z𝜎 (G; ℳ) we have ̄ 1≂b⊗ ̄ 1. a∼b⇔a⊗

(1)

̄ ℱ2 ) is the balanced cocycle associated with a and b, then Indeed, if c ∈ Z𝜎 ⊗𝜄 ̄ (G; ℳ ⊗ ̄ ̄ ̄ ℱ2 ) is the balanced cocycle associated with a ⊗ ̄ 1 and c ⊗ 1 ∈ Z𝜎 ⊗𝜄 ̄ ⊗𝜄 ̄ (G; ℳ ⊗ ℱ∞ ⊗ ̄ c ⊗ 1 c ̄ ℱ∞ ⊗ ̄ ℱ2 ) ̄ ℱ2 ) ⊗ ̄ ℱ∞ and ̄ 1, (ℳ ⊗ b⊗ = (ℳ ⊗ ̄ 1)(e) ⊗ ̄ e11 = (a(e) ⊗ ̄ e11 ) ⊗ ̄ 1, (b ⊗ ̄ 1)(e) ⊗ ̄ e22 = (b(e) ⊗ ̄ e22 ) ⊗ ̄ 1. (a ⊗ ̄ 1) of p ⊗ ̄ 1 in In general, if p is a projection of the W ∗ -algebra 𝒫 , then the central support z(p ⊗ ̄ ̄ 𝒫 ⊗ ℱ∞ is z(p) ⊗ 1, where z(p) is the central support of p in 𝒫 . If 𝒫 is countably decomposable, ̄ 1∼q⊗ ̄ 1 ⇔ z(p ⊗ ̄ 1) = z(q ⊗ ̄ 1) ⇔ z(p) = z(q), since 𝒫 ⊗ ̄ ℱ∞ is then for p, q ∈ 𝒫 we have p ⊗ properly infinite and countably decomposable ([L], 4.13). In view of the definitions of Section 20.2, these considerations prove (1).

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Crossed Products

Now, let 𝜌 be the right regular representation of G regarded as a cocycle 𝜌 ∈ Z𝜄 (G; ℬ(ℒ 2 (G))). For every u ∈ Z𝜎 (G, U(ℳ)), we have ̄ 𝜌≂1⊗ ̄ 𝜌 in Z𝜎 ⊗̄ 𝜄 (G; ℋ ⊗ ̄ ℬ(ℒ 2 (G))). u⊗

(2)

̄ ℬ(ℒ 2 (G)) ⊂ ℬ(ℒ 2 (G; ℋ )) be realized as von Neumann Indeed, let ℳ ⊂ ℬ(ℋ ) and ℳ ⊗ ̄ ℒ ∞ (G) defined by the function s ↦ u(s−1 ), that algebras. Consider the unitary operator U ∈ ℳ ⊗ −1 2 is, (U𝜉)(s) = u(s )𝜉(s) (𝜉 ∈ ℒ (G, ℋ ), s ∈ G ). For s, t ∈ G, and 𝜉 ∈ ℒ 2 (G, ℋ ) we have ̄ 𝜌 )(t)𝜉](s) = u(s−1 )[(1 ⊗ ̄ 𝜌 (t))𝜉](s) = Δ(t)1∕2 u(s−1 )𝜉(st) U(1 ⊗ ̄ 𝜄)t U)𝜉](r) = [((𝜎t ⊗ ̄ 𝜄)U)𝜉](r) = 𝜎t (u(r−1 ))𝜉(r), and, since [((𝜎 ⊗ ̄ 𝜌 (t))((𝜎t ⊗ ̄ 𝜄)U)𝜉](s) = Δ(t)1∕2 u(t)[((𝜎t ⊗ ̄ 𝜄)U)𝜉](st) [((u(t) ⊗ 1∕2 −1 1∕2 = Δ(t) u(t)𝜎t (u((st) ))𝜉(st) = Δ(t) u(t)𝜎t (𝜎st−1 (u(st)∗ ))𝜉(st) = Δ(t)1∕2 u(t)𝜎s−1 ((u(s)𝜎s (u(t)))∗ )𝜉(st) = Δ(t)1∕2 u(t)u(t)∗ 𝜎s−1 (u(s)∗ )𝜉(st) = Δ(t)1∕2 u(s−1 )𝜉(st), ̄ 𝜌)(t) = ((u ⊗ ̄ 𝜌)(t))((𝜎s ⊗ ̄ 𝜄)U) (t ∈ G ) proving (2). we have U(1 ⊗ ̄ ℬ(ℒ 2 (G))𝜎 ⊗̄ Ad(𝜌𝜌) = (ℳ ⊗ ̄ ℬ Since, by Corollary 19.13, we have ℛ(ℳ, 𝜎) = (ℳ ⊗ ̄ 2 1 ⊗ 𝜌 𝜆 𝜌 𝜆 𝜌 (ℒ (G))) , and since the regular representations and are equivalent, that is, ≂ , it follows from (2) that for every unitary cocycle u ∈ Z𝜎 (G; U(ℳ)) we have ̄ ℬ(ℒ 2 (G)))u ⊗̄ 𝜌 ≈ (ℳ ⊗ ̄ ℬ(ℒ 2 (G)))u ⊗̄ 𝜆 . ℛ(ℳ, 𝜎) ≈ (ℳ ⊗

(3)

20.5. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of G on ℳ. A unitary cocycle a ∈ Z𝜎 (G; U(ℳ)) is called a dominant cocycle if it is of infinite multiplicity and ̄ 1≂a⊗ ̄ 𝜌 in Z𝛼 ⊗̄ 𝜄 (ℳ ⊗ ̄ ℬ(ℒ 2 (G))). a⊗

(1)

According to statement 20.4.(2), (1) is equivalent to the condition ̄ 1≂1⊗ ̄ 𝜌 in Z𝜎 ⊗̄ 𝜄 (ℳ ⊗ ̄ ℬ(ℒ 2 (G))). a⊗

(2)

̄ 𝜄 and 𝜎 ⊗ ̄ Ad(𝜌𝜌) of ℳ ⊗ ̄ ℬ(ℒ 2 (G))) are inner and this implies that the *-automorphisms (a 𝜎) ⊗ conjugate (10.2.(14)). If the locally compact group G is separable, then a ∈ Z𝜎 (G; U(ℳ)) dominant cocycle ⇒ ℳ 𝜎 ≈ ℛ(ℳ, 𝜎).

(3)

̄𝜌 ̄ ℬ(ℒ 2 (G)))a⊗𝜌 ̄ ℬ Indeed, according to (1) and 20.4.(3), we have ℛ(ℳ, 𝜎) ≈ (ℳ ⊗ ≈ (ℳ ⊗ ̄ 2 𝛼 ⊗1 a 2 ̄ (ℒ (G))) ≈ ℳ ⊗ ℬ(ℒ (G))). Since G is separable, there exists n ∈ ℕ ∪ {∞} such that ℬ(ℒ 2 (G)) ≈ ℱn = the countably decomposable type In factor and, since ℳ a is properly infinite, ̄ ℱn ≈ ℳ a . we conclude that ℳ a ⊗ Clearly, the existence of a dominant cocycle a ∈ Z𝜎 (G; ℳ) implies that ℳ is properly infinite.

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295

Theorem. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the separable locally compact group G on the properly infinite W ∗ -algebra ℳ. Then there exists a dominant cocycle a ∈ Z𝜎 (G; U(ℳ)). If, moreover, ℳ is countably decomposable, then any two dominant cocycles a, b ∈ Z𝜎 (G; ℳ) are equivalent, that is, a ≂ b. ̄ 1≂ Proof. We first prove the uniqueness assertion. If a, b ∈ Z𝜎 (G, U(ℳ)) are dominant, then a ⊗ 2 ̄ ̄ 1 ⊗ 𝜌 = b ⊗ 1. Since ℳ and ℬ(ℒ (G)) are countably decomposable, we infer, using 20.4.(1), that a ∼ b and, since a and b are of infinite multiplicity, it follows by 20.2.(13) that a ≂ b. We now prove the existence assertion. By 20.3.(2), there exists a unitary cocycle u ∈ Z𝜎 (G; U(ℳ)) of infinite multiplicity, that is, the centralizer ℳ (u 𝜎) is properly infinite. If there exists a dominant cocycle b ∈ Z(u 𝜎) (G; U(ℳ)), then by 20.3.(1), the equation a(s) = b(s)u(s) (s ∈ G ), defines a cocycle a ∈ Z𝜎 (G; ℳ) of infinite multiplicity. Since b ∈ Z(u 𝜎) (G; U(ℳ)) is dom̄ 𝜌 ≂ b ⊗ ̄ 1 in Z( 𝜎 ⊗𝜄) ̄ ℬ(ℒ 2 (G))), that is, there exists V ∈ inant, we have b ⊗ ̄ (G; ℳ ⊗ u ̄ ℬ(ℒ 2 (G))) such that V = (b(s) ⊗ ̄ 1)(u(s) ⊗ ̄ 1)((𝜎s ⊗ ̄ 𝜄)(V))(u(s)∗ ⊗ ̄ 1)(b(s)∗ ⊗ ̄ 𝜌 (s)∗ ) U(ℳ ⊗ ̄ 1)(𝜎s ⊗ ̄ 𝜄)(V))(a(s)∗ ⊗ ̄ 𝜌 (s)∗ ) (s ∈ G ), and this means that a ⊗ ̄ 𝜌 ≂ a⊗ ̄ 1 in or V = (a(s) ⊗ 2 (G))); hence a ∈ Z (G; ℳ) is dominant. ̄ Z𝜎 ⊗𝜄 (G; ℳ ⊗ ℬ(ℒ ̄ 𝜎 Thus, in order to prove the existence of a dominant cocycle a ∈ Z𝜎 (G; U(ℳ)), we may assume that the centralizer ℳ 𝜎 is properly infinite. Then, according to Corollary 9.16, we have (ℳ, 𝜎) ≈ ̄ ℱn ⊗ ̄ ℱ∞ , 𝜎 ⊗ ̄ 𝜄⊗ ̄ 𝜄) for all n ∈ ℕ ∪ {∞}. (ℳ ⊗ Since G is separable, there exists n ∈ ℕ ∪ {∞} with ℬ(ℒ 2 (G)) ≈ ℱn , so that (ℳ, 𝜎) ≈ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ ℱ∞ , 𝜎 ⊗ ̄ 𝜄 ⊗ ̄ 𝜄). Using the identification, we can define a cocycle a ∈ (ℳ ⊗ ̄ 𝜌⊗ ̄ 1. Then ℳ a ≈ ℳ 𝜎 ⊗ ̄ 𝔏(G) ℱ∞ , hence a is of infinite multiplicity. Z𝜎 (G; U(ℳ)), a ≈ 1 ⊗ ̄ 𝜌 ≂1⊗ ̄ 𝜌 , hence a ⊗ ̄ 𝜌 ≂a⊗ ̄ 1, that is, a is On the other hand, according to 20.4.(2), we have 𝜌 ⊗ a dominant cocycle. 20.6. Recall (18.19 and 18.20) that a continuous action 𝜎 ∶ G → Aut(ℳ) is called integrable if the + faithful normal operator-valued weight P𝜎 = ∫ 𝜎a (⋅) ds ∶ ℳ + → (ℳ 𝜎 ) is semifinite. A cocycle a ∈ Z𝜎 (G; ℳ) is called square-integrable, if the continuous action a 𝜎 ∶ G → Aut(ℳa(e) ) is integrable. From Section 20.2, it follows that if a, b ∈ Z𝜎 (G; ℳ), b ≲ a and a is square-integrable, then b is also square-integrable.

(1)

It follows from Proposition 19.16 that the action Ad(𝜌) ∶ G → Aut(ℬ(ℒ 2 (G))) is integrable, and hence that the cocycle 𝜌 ∈ Z𝜄 (G; ℬ(ℒ 2 (G))) is square-integrable.

(2)

It is easy to check that if 𝜎 ∶ G → Aut(ℳ) is an integrable action and 𝜏 ∶ G → Aut(𝒩 ) is an ̄ 𝜏 ∶ G → Aut(ℳ ⊗ ̄ 𝒩 ) is integrable. arbitrary continuous action, then the tensor product action 𝜎 ⊗ Conversely, if 𝜎 ∶ G → Aut(ℳ) is a continuous action, 𝜄 ∶ G → Aut(𝒩 ) is the trivial action and ̄ 𝜄 is integrable, then also the action 𝜎 is integrable. Indeed, for X ∈ (ℳ ⊗ ̄ 𝒩 )+ , 𝜓 ∈ the action 𝜎 ⊗ 𝜓 𝜓 + + ̄ 𝜓⟩ = 𝒩∗ , 𝜑 ∈ ℳ∗ , and s ∈ G, we have ⟨𝜎s (Eℳ (X)), 𝜑⟩ = ⟨Eℳ (X), 𝜑 ◦ 𝜎s ⟩ = ⟨X, (𝜑 ◦ 𝜎s ) ⊗ ̄ 𝜄)(X), 𝜑 ⊗ ̄ 𝜓⟩ = ⟨(E 𝜓 ((𝜎s ⊗ ̄ 𝜄)(X)), 𝜑⟩, hence ⟨(𝜎s ⊗ ℳ 𝜓 𝜓 P𝜎 (Eℳ (X)) = Eℳ (P𝜎 ⊗̄ 𝜄 (X)).

̄ 𝒩 )+ , ‖P𝜎 ⊗̄ 𝜄 (Xi )‖ < +∞, Xi → 1 and 𝜓j ∈ 𝒩 + , s(𝜓j ) → 1, then E𝜓j (Xi ) ∈ Thus, if Xi ∈ (ℳ ⊗ ∗ ℳ s

𝜓

𝜓

s

ℳ + , ‖P𝜎 (Eℳj (Xi ))‖ < +∞ and Eℳj (Xi ) → 1, that is, P𝜎 is semifinite.

s

296

Crossed Products

We now show that every dominant cocycle a ∈ Z𝜎 (G; U(ℳ)) is square-integrable.

(3)

̄ 1≂1⊗ ̄ 𝜌 and, by (2), 𝜌 is square-integrable, hence (3) follows Indeed, by 20.5.(2) we have a ⊗ from (1) using the above remarks. If the group G is compact, then every cocycle a ∈ Z𝜎 (G; ℳ) is square-integrable. In the general case, the following result holds: Theorem. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the separable locally compact group G on the countably decomposable properly infinite W ∗ -algebra ℳ. Let a ∈ Z𝜎 (G; U(ℳ)) be a dominant cocycle. Then a cocycle b ∈ Z𝜎 (G; ℳ) is square-integrable if and only if b ≲ a. If b ≲ a, then b is square-integrable, as follows from the previous considerations. The converse will be proved in Section 20.9 using the preliminary results given in Sections 20.7–20.8. In particular, consider the trivial action 𝜄 ∶ G → Aut(ℬ(ℋ )), where ℋ is any Hilbert space. In this case, a cocycle a ∈ Z𝜎 (G; ℬ(ℋ )) is just an so-continuous unitary representation a ∶ G → ℬ(𝒦 ) of G on a closed linear subspace 𝒦 of ℋ . The cocycle a ∈ Z𝜄 (G; ℬ(ℋ )) is squareintegrable in the sense defined above if and only if the so-continuous unitary representation a ∶ G → ℬ(𝒦 ) is square-integrable in the usual sense, that is, ∫ |(a(s)𝜉|𝜉)|2 ds < +∞ for a dense subset of vectors 𝜉 ∈ 𝒦 . If ℋ is separable and infinite dimensional, then the ̄ 1 of the right regular representation 𝜌 . It is well known dominant cocycle is a multiple 𝜌 ⊗ that an irreducible representation of G is square-integrable if and only if it is equivalent to a subrepresentation of the regular representation. Thus, the above Theorem is also an extension of this result. 20.7. With the assumptions of Theorem 20.6, the next result improves 20.3.(2): Lemma. For every cocycle b ∈ Z𝜎 (G; ℳ), there exists a unitary cocycle of infinite multiplicity ̌ If b is square-integrable, then b̌ can also be chosen squareb̌ ∈ Z𝜎 (G; U(ℳ)) such that b ≲ b. integrable. Proof. Let p = z(b(e)) be the central support of the projection b(e) ∈ ℳ. Then, for every s ∈ G we have p = z(b(s)b(s)∗ ) = z(b(s)∗ b(s)) = z(𝜎s (b(e))) = 𝜎s (p), hence p ∈ ℳ 𝜎 . Consequently, (ℳ, 𝜎) ≈ (ℳp, 𝜎) ⊕ (ℳ(1 − p), 𝜎). By Theorem 20.5, there exists a dominant cocycle in Z𝜎 (G; ℳ(1 − p)), which is square-integrable (20.6.(3)). Thus, to prove the lemma, we may assume that z(b(e)) = 1. In this case, there exists a sequence ∑ of partial isometries {un } ⊂ ℳ such that u∗n um = 0 for n ≠ m, u∗n un = b(e) and n un u∗n = 1. We define ∑ ̌ = b(s) un b(s)𝜎s (u∗n ); s ∈ G. n

∑ ∑ ∗ ∗ ̌ ̌ ̌ Then b(s) ∈ U(ℳ), since b(e) = n un b(e)un = n un un = 1, b ∈ Z𝜎 (G; U(ℳ)), since ∑ ∑ ∗ ∗ ∗ ̌b(s)𝜎s (b(t)) ̌ = n,m un b(s)𝜎s (un )𝜎s (um )𝜎s (b(t))𝜎st (um ) = n un b(s)𝜎s (b(e))𝜎s (b(t))𝜎st (un ) = ∑ ̌ ∗ ∗ ∗ ∗ ∗ b ∗ ∗ ̌ ̌ n un b(st)𝜎st (un ) = b(st), ui uj ∈ ℳ , as b(s)𝜎s (ui uj )b(s) = ui b(s)𝜎s (ui )𝜎s (ui uj )𝜎s (uj )b(s) uj = ̌ ui b(s)𝜎s (b(e))b(s)∗ u∗ = ui b(e)u∗ = ui u∗ , so that b̌ is of infinite multiplicity, qk = uk u∗ ∈ ℳ b , and j

j

j

̌ as b̌ qk (s) = qk b(s) ̌ = uk b(s)𝜎s (u∗ ). finally b ≂ b̌ qk ≂ b, k

k

Comparison of cocycles

297 s

If b is square-integrable, then there exists a net {x𝜆 } in b(e)ℳb(e), x𝜆 → b(e) with ‖P(b𝜎 ) (x𝜆 )‖ < ∗ ∗ ̌ ̌ ∗ +∞. Let x𝜆,n = un x𝜆 u∗n . Then b(s)𝜎 s (x𝜆,n )b(s) = un b(s)𝜎s (x𝜆 )b(s) un , hence ‖Pb̌ 𝜎 (x𝜆,n )‖ < +∞ and s ∑ x𝜆,n → un b(e)u∗n = un u∗n . Since n un u∗n = 1, it follows that b̌ is square-integrable. 20.8. We continue with the assumptions of Theorem 20.6. The next lemma contains the main technical part of the proof of this Theorem. Lemma. If the cocycle b ∈ Z𝜎 (G; ℳ) is square-integrable, then, with respect to the pair ̄ ℬ(ℒ 2 (G)), 𝜎 ⊗ ̄ 𝜄) we have (ℳ ⊗ ̄ 1 = ∨{s(X∗ X); X ∈ ℐ (b ⊗ ̄ 𝜌, b ⊗ ̄ 1)}. b(e) ⊗ ̄ 𝜌, b ⊗ ̄ 1)}. Since for any X ∈ ℐ (b ⊗ ̄ 𝜌, b ⊗ ̄ 1), we Proof. Indeed, let P = ∨{s(X∗ X); X ∈ ℐ (b ⊗ ̄ 1, b ⊗ ̄ 1) (see 20.2.(4) and 20.2.(5)) so that s(X∗ X) ∈ (ℳ ⊗ ̄ ℬ(ℒ 2 (G)))b ⊗̄ 1 = have X∗ X ∈ ℐ (b ⊗ ̄ ℬ(ℒ 2 (G)), it follows that P ∈ ℳ b ⊗ ̄ ℬ(ℒ 2 (G)). On the other hand, for each U ∈ ℳb ⊗ ̄ ℬ(ℒ 2 (G))) we have ℐ (b ⊗ ̄ 𝜌, b ⊗ ̄ 1)U = ℐ (b ⊗ ̄ 𝜌, b ⊗ ̄ 1) (see 20.2.(3) and 20.2.(5)) and U(ℳ b ⊗ ̄ 𝜌, b ⊗ ̄ 1) ⇒ XU ∈ ℐ (b ⊗ ̄ 𝜌, b ⊗ ̄ 1) ⇒ P ≥ s(U ∗ X∗ XU) = U ∗ s(X∗ X)U, so that hence X ∈ ℐ (b ⊗ ̄ ℬ(ℒ 2 (G))) = 𝒵 (ℳ b ) ⊗ ̄ 1. Thus, it remains to show P = U ∗ PU. Consequently, P ∈ 𝒵 (ℳ b ⊗ that ̄ 𝜌, b ⊗ ̄ 1)(q ⊗ ̄ 1) ≠ 0. 0 ≠ q ∈ 𝒵 (ℳ b ) ⇒ ℐ (b ⊗

(1)

Let 0 ≠ q ∈ 𝒵 (ℳ b ). Since b is square-integrable, there exists x ∈ ℳ such that x = xq ≠ 0 ̄ ℬ(ℒ 2 (G)) ⊂ and ‖P(b𝜎 ) (x∗ x)‖ < +∞. Let f ∈ 𝒦 (G), f ≠ 0. Consider ℳ ⊂ ℬ(ℋ ) and ℳ ⊗ ℬ(ℒ 2 (G, ℋ )) realized as von Neumann algebras. For 𝜉 ∈ ℒ 2 (G, ℋ ) and s ∈ G we define [X𝜉](s) = Δ(s)−1∕2 b 𝜎s−1 (x)

∫

f (t)𝜉(t) dt.

Then, ‖X𝜉‖22 = ≤

∫

‖Δ(s)−1∕2 b 𝜎s−1 (x)

∫

f (t)𝜉(t) dt‖2 ds

‖Δ(s)−1∕2 f (t)b 𝜎s−1 (x)𝜉(t)‖2 dt ds ( ) = | f (t)|2 Δ(s)−1 ‖b 𝜎s−1 (x)𝜉(t)‖2 ds dt ∫ ∫ ( ) 2 2 = | f (t)| ‖ 𝜎 (x)𝜉(t)‖ ds dt ∫ ∫ b s ( ) 2 ∗ = | f (t)| ( 𝜎 (x x)𝜉(t)|𝜉(t)) ds dt ∫ ∫ b s =

∫ ∫

∫

| f (t)|2 ‖P(b𝜎 ) (x∗ x)1∕2 𝜉(t)‖2 dt

≤ ‖P(b𝜎 ) (x∗ x)‖‖ f ‖∞ ‖𝜉‖2 ,

(2)

298

Crossed Products

so that (2) defines an element X ∈ ℬ(ℒ 2 (G, ℋ )). Using the von Neumann double commutant ̄ ℬ(ℒ 2 (G)). Then, theorem, it is easy to check that X ∈ ℳ ⊗ ̄ 1)𝜉](s) = Δ(s)−1∕2 b 𝜎 −1 (x) [X(q ⊗ s

= Δ(s)−1∕2 b 𝜎s−1 (x)q = Δ(s)−1∕2 b 𝜎s−1 (x)

̄ 1)𝜉](t) dt f (t)[(q ⊗

∫

f (t)𝜉(t) dt = Δ(s)−1∕2 b 𝜎s−1 (xq)

∫

∫

∫

f (t)𝜉(t) dt

f (t)𝜉(t) dt = [X𝜉](s),

̄ 1) = X ≠ 0. Finally, we show that X ∈ ℐ (b ⊗ ̄ 𝜌, b ⊗ ̄ 1). We have and X(q ⊗ ̄ 1)𝜉](s) = Δ(s)−1∕2 b 𝜎s (x) [X(b(r) ⊗

∫

f (t)b(r)𝜉(t) dt

= Δ(s)−1∕2 b 𝜎s (x)b(r)

∫

f (t)𝜉(t) dt

and ̄ 𝜌 (r))((𝜎r ⊗ ̄ 𝜄)(X))𝜉](s) = Δ(r)1∕2 b(r)[((𝜎r ⊗ ̄ 𝜄)(X))𝜉](sr) [(b(r) ⊗ = Δ(r)1∕2 b(r)Δ(sr)−1∕2 𝜎r (b 𝜎sr−1 (x))

∫

= Δ(s)−1∕2 b(r)𝜎r (b 𝜎sr−1 (x))b(r)∗ b(r)

∫

f (t)𝜉(t) dt f (t)𝜉(t) dt

= Δ(s)−1∕2 b 𝜎s−1 (b 𝜎sr−1 (x))b(r)

∫

= Δ(s)−1∕2 b 𝜎s (x)b(r)∗ b(r)

f (t)𝜉(t) dt.

∫

f (t)𝜉(t) dt

This proves (1) and also the lemma. 20.9. Proof of Theorem 20.6. Let b ∈ Z𝜎 (G; ℳ) be square-integrable. To show that b ≲ a, where a ∈ Z𝜎 (G; U(ℳ)) is the dominant cocycle, we may assume, by Lemma 20.7, that b is unitary and ̄ 𝜌, b ⊗ ̄ 1 ∈ Z𝜎 ⊗̄ 𝜄 (G; U(ℳ ⊗ ̄ ℬ(ℒ 2 (G)))) and the balanced of infinite multiplicity. Consider b ⊗ 2 ̄ 𝜌, b ⊗ ̄ 1) ∈ Z𝜎 ⊗̄ 𝜄⊗̄ 𝜄 (G; U(ℳ ⊗ ̄ ℬ(ℒ (G)) ⊗ ̄ ℱ2 )). cocycle c = c(b ⊗ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ ℱ2 )c we have ()) Indeed, By Lemma 20.8, it follows that in the W ∗ -algebra (ℳ ⊗ we have ̄ 1⊗ ̄ e11 ) ≥ z(1 ⊗ ̄ 1⊗ ̄ e22 ). z(1 ⊗

(1)

Indeed, we have ̄ ℬ(ℒ 2 (G)) ⊗ ̄ ℱ2 )c = (ℳ ⊗

(

̄ 𝜌, b ⊗ ̄ 𝜌 ) ℐ (b ⊗ ̄ 𝜌, b ⊗ ̄ 1) ℐ (b ⊗ ̄ ̄ ̄ ̄ ℐ (b ⊗ 1, b ⊗ 𝜌 ) ℐ (b ⊗ 1, b ⊗ 1)

)

Comparison of cocycles

299

and (

0 x∗

0 0

)(

1 0 0 0

)(

0 0

x 0

)

( =

0 0

0 x∗ x

) .

̄ 𝜌, b ⊗ ̄ 1) Lemma 20.8 shows that the support of x∗ x converges to 1 when x runs over ℐ (b ⊗ [Recall that for any projection f in a W ∗ -algebra 𝒩 we have z( f ) = ∨{s( y∗ fy); y ∈ 𝒩 }]. On the other hand, the projection ̄ 1⊗ ̄ e11 is properly infinite in (ℳ ⊗ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ ℱ2 )c , 1⊗

(2)

since the corresponding reduced algebra is *-isomorphic to ℳ b which, by assumption, is properly infinite. The assumptions of Theorem 20.6 also insure that all the W ∗ -algebras involved in our argument ̄ 1⊗ ̄ e22 ≺ 1 ⊗ ̄ 1⊗ ̄ e11 are countably decomposable. Therefore, we infer from (1) and (2) that 1 ⊗ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ ℱ2 )c , that is, b ⊗ ̄ 1 ≲ b⊗ ̄ 𝜌 . Thus, by 20.4.(2) and 20.5.(2), we have in (ℳ ⊗ ̄ 1 ≲ b⊗ ̄ 𝜌 ≂ 1⊗ ̄ 𝜌 ≂ a⊗ ̄ 1. Since a and b are both of infinite multiplicity, we infer using b⊗ 20.4.(1) and 20.2.(13) that b ≲ a. □ 20.10 Corollary. Let 𝜎 ∶ G → Aut(ℳ) be an integrable continuous action of the separable locally compact group G on the countably decomposable properly infinite W ∗ -algebra ℳ. Then the centralizer ℳ 𝜎 is a reduced algebra of the crossed product ℛ(ℳ, 𝜎). Proof. By assumption, the trivial cocycle, 1 ∈ Z𝜎 (G; U(ℳ)) is square-integrable and hence, by Theorem 20.6, 1 ≲ a, where a ∈ Z𝜎 (G; U(ℳ)) denotes the dominant cocycle. Thus, there exists u ∈ ℳ with u∗ u = 1, uu∗ = p ∈ ℳ 𝜎 , and u∗ a(s)𝜎s (u) = 1 for all s ∈ G. Then, x ∈ ℳ 𝜎 ⇔ 𝜎s (x) = x for all s ∈ G ⇔ a(s)𝜎s (u)𝜎s (x)𝜎s (u∗ )a(s)∗ = uxu∗ for all s ∈ G ⇔ a 𝜎s (uxu∗ ) = uxu∗ for all s ∈ G ⇔ uxu∗ ∈ ℳ a , hence ℳ 𝜎 ≈ pℳ a p. Since, by 20.5.(4), ℳ a ≈ ℛ(ℳ, 𝜎), this proves the Corollary. 20.11 Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the separable locally compact group G on the W ∗ -algebra ℳ. We assume that the centralizer ℳ 𝜎 is properly infinite. Then the following statements are equivalent: (i) the trivial cocycle 1 ∈ Z𝜎 (G; U(ℳ)) is a dominant cocycle; ̄ Ad(𝜌𝜌) and 𝜎 ⊗ ̄ 𝜄 are inner conjugate, that is, there exists U (ii) the actions 𝜎 ⊗ 2 ̄ U(ℳ ⊗ ℬ(ℒ (G))) such that

∈

̄ Ad(𝜌𝜌))s = Ad(U) ◦ (𝜎 ⊗ ̄ 𝜄)s ◦ Ad(U ∗ ) (s ∈ G ); (𝜎 ⊗ ̄ Ad(𝜌𝜌) and 𝜎 ⊗ ̄ 𝜄 are conjugate, that is, there exists Φ ∈ Aut(ℳ ⊗ ̄ ℬ(ℒ 2 (G))) (iii) the actions 𝜎 ⊗ such that ̄ Ad(𝜌𝜌))s = Φ ◦ (𝜎 ⊗ ̄ 𝜄)s ◦ Φ−1 (𝜎 ⊗

(s ∈ G ).

̄ 𝜌 ≂1⊗ ̄ 1 in Z𝜎 ⊗̄ 𝜄 (G; ℳ ⊗ ̄ ℬ(ℒ 2 (G))), that is, Proof. (i) ⇒ (ii). By assumption, we have 1 ⊗ 2 ∗ ̄ ̄ ̄ there exists U ∈ U(ℳ ⊗ ℬ(ℒ (G))) such that 1 ⊗ 𝜌 (s) = U(𝜎s ⊗ 𝜄)(U ) (s ∈ G ). Then, for every

300

Crossed Products

̄ ℬ(ℒ 2 (G)) and s ∈ G we have X∈ℳ⊗ ̄ Ad(𝜌𝜌(s))(X) = (1 ⊗ ̄ 𝜌 (s))((𝜎s ⊗ ̄ 𝜄)(X))(1 ⊗ ̄ 𝜌 (s)∗ ) (𝜎s ⊗ ̄ 𝜄)(U ∗ XU))U ∗ . = U((𝜎s ⊗ (ii) ⇒ (iii). Obvious. (iii) ⇒ (i). Since ℳ 𝜎 is properly infinite and G is separable, we have by Corollary 9.16 ̄ ℬ(ℒ 2 (G)), 𝜎 ⊗ ̄ 𝜄). Using assumption (iii), we obtain (ℳ, 𝜎) ≈ (ℳ ⊗ ̄ ℬ(ℒ 2 (G)), (ℳ, 𝜎) ≈ (ℳ ⊗ ̄ ̄ ̄ ̄ ̄ ̄ 𝜎 ⊗ Ad(𝜌𝜌)). Consequently, we have to show that 1 ⊗ 1 ⊗ 𝜌 ≂ 1 ⊗ 1 ⊗ 1 in Z𝜎 ⊗̄ Ad(𝜌𝜌) ⊗̄ 𝜄 (G; ℳ ⊗ 2 2 ̄ ℬ(ℒ (G))) or, equivalently, that 1 ⊗ ̄ 𝜌⊗ ̄ 𝜌 ≂ 1⊗ ̄ 𝜌⊗ ̄ 1 in Z𝜎 ⊗̄ 𝜄 ⊗̄ 𝜄 (G; ℳ ⊗ ̄ ℬ(ℒ (G)) ⊗ 2 (G))), which results from 20.4.(2). ̄ ℬ(ℒ 2 (G)) ⊗ℬ(ℒ The continuous action 𝜎 ∶ G → Aut(ℋ ) will be called dominant if the centralizer ℳ 𝜎 is properly ̄ Ad(𝜌𝜌) and 𝜎 ⊗ ̄ 𝜄 are inner conjugate. In this case, we have infinite and the actions 𝜎 ⊗ ̄ ℬ(ℒ 2 (G)), 𝜎 ⊗ ̄ Ad(𝜌𝜌)). (ℳ, 𝜎) ≈ (ℳ ⊗

(1)

The previous proposition gives us equivalent conditions for the action 𝜎 to be dominant when G is separable, but the implications (i) ⇒ (ii) ⇒ (iii) are valid in general, without the assumption of separability on G. 20.12. In this section, we show that the dominant actions are actually dual actions on crossed products. Although such a result is valid in general, we restrict ourselves here to the abelian case, where a more detailed analysis is possible. Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the separable locally compact abelian group G on the W ∗ -algebra ℳ. We assume that the centralizer ℳ 𝜎 is properly infinite. Then the following statements are equivalent: (i) 𝜎 is a dominant action; ̂ → U(ℳ) such that 𝜎g (u(𝛾)) = (ii) there exists an s-continuous unitary representation u ∶ G ̂ ⟨g, 𝛾⟩u(𝛾) for all g ∈ G, 𝛾 ∈ G; ̂ → U(ℳ) such that 𝜎g (u(𝛾)) = ⟨g, 𝛾⟩u(𝛾) for all g ∈ G, 𝛾 ∈ (ii’) there exists a Borel function u ∶ G ̂ G; ̂ ̂ → Aut(ℳ 𝜎 ) such that (ℳ, 𝜎) ≈ (ℛ(M𝜎 , 𝜃), 𝜃). (iii) there exists a continuous action 𝜃 ∶ G If the predual ℳ∗ is separable, then another equivalent statement is: ̂ there exists u(𝛾) ∈ U(ℳ) such that 𝜎g (u(𝛾)) = ⟨g, 𝛾⟩u(𝛾) for all g ∈ G. (ii”) for every 𝛾 ∈ G Proof. (i) ⇒ (ii). Since the action 𝜎 is dominant, we may assume (20.11.(1)) that (ℳ, 𝜎) = ̄ ℬ(ℒ 2 (G)), 𝜎 ⊗ ̄ Ad(𝜌𝜌)). In this case, denoting by m(𝛾) the multiplication operator by the (ℳ ⊗ ̂ ̂ we obtain statement (ii). ̄ m(𝛾)∗ (𝛾 ∈ G), character 𝛾 ∈ G and defining u(𝛾) = 1 ⊗ It is clear that (ii) ⇒ (ii’) ⇒ (ii”). ̂ × U(ℳ) is then a polish space and We show that (ii”) ⇒ (ii’) if ℳ∗ is separable. Indeed, G ̂ ̂ ̂ × U(ℳ); 𝜎g (u) = the projection map G × U(ℳ) → G is continuous. The set E = {(𝛾, u) ∈ G ̂ × U(ℳ) is closed and, by (ii”), its projection on G ̂ covers the whole of ⟨g, 𝛾⟩u for all g ∈ G} ⊂ G ̂ ̂ G. It follows that there exists a Borel section u ∶ G → U(ℳ) with (𝛾, u(𝛾)) ∈ E.

Comparison of cocycles

301

̂ → U(ℳ) defines a unitary element U ∈ ℳ ⊗ℒ ̄ ∞ (G), (ii’) ⇒ (i). The Borel function u ∶ G ̄ k⟩ = ∫ ⟨u(𝛾), 𝜑⟩k(𝛾) d𝛾 for all 𝜑 ∈ ℳ∗ , k ∈ ℒ 1 (G). uniquely determined, such that ⟨U, 𝜑 ⊗ ̂ is defined by the function 𝛾 ↦ 𝜎g (u(𝛾)), hence ̄ 𝜄)(U) ∈ ℳ ⊗ ̄ ℒ ∞ (G) Then the element (𝜎g ⊗ ̄ 𝜄)(U)) is defined by the function 𝛾 ↦ u(𝛾)∗ 𝜎g (u(𝛾)) = ⟨g, 𝛾⟩. Thus, considering the eleU ∗ ((𝜎g ⊗ ̂ we have U ∗ ((𝜎g ⊗ ̂ ( g ∈ G ), or, ̄ 𝜄)(U)) = 1 ⊗ ̄ m( g)∗ ∈ ℳ ⊗ ̄ ℒ ∞ (G) ment g ∈ G as a character on G, ̂ with 𝔏(G) = ℜ(G) by the Fourier–Plancherel isomorphism, U ∗ ((𝜎g ⊗ ̄ 𝜄)(U)) = identifying ℒ ∞ (G) 2 ̄ ̄ 1 ⊗ 𝜌 ( g) ∈ ℳ ⊗ ℬ(ℒ (G)) ( g ∈ G ). It follows that the cocycle 1 ∈ Z𝜎 (G; U(ℳ)) is dominant, hence 𝜎 is a dominant action. (ii) ⇒ (iii). This follows from Landstad’s Theorem (19.9). Moreover, it follows from Landstad’s theorem that the action 𝜃 is defined by the representation u as follows: 𝜃𝛾 = Ad(u(𝛾))|ℳ 𝜎

̂ ( y ∈ G).

(1)

Note that the equivalences (i) ⇔ (ii) ⇔ (iii) are valid without the separability assumption on G. Using the Takesaki duality theorem (19.5), we deduce from (iii) that if 𝜎 is a dominant action, then ̄ ℬ(ℒ 2 (G)) ≈ ℛ(ℳ, 𝜎). ℳ𝜎 ≈ ℳ𝜎 ⊗

(2)

20.13. The definition (19.1) of crossed products is invariant under *-isomorphisms. More precisely, if 𝜎 ∶ G → Aut(ℳ), 𝜏 ∶ G → Aut(𝒩 ) are continuous actions of the locally compact group G on the W ∗ -algebras ℳ, 𝒩 and there exists a *-isomorphism Φ ∶ ℳ → 𝒩 such that Φ ◦ 𝜎g = 𝜏g ◦ Φ ( g ∈ ̄ 𝜄∶ℳ ⊗ ̄ ℬ(ℒ 2 (G)) → 𝒩 ⊗ ̄ ℬ(ℒ 2 (G)) is a *-isomorphism and G ), then the mapping Ψ = Φ ⊗ ̄ 𝜆 ( g)) = 1 ⊗ ̄ 𝜆 ( g) ( g ∈ G ), it is easy to check that Ψ(𝜋𝜎 (x)) = 𝜋𝜏 (Φ(x)) (x ∈ ℳ), and that Ψ(1 ⊗ ̄ ℬ(ℒ 2 (G)) is the hence Ψ(ℛ(ℳ, 𝜎)) = ℛ(𝒩 , 𝜏). Moreover, since the restriction of Ψ to 1ℳ ⊗ ̄ 𝜄) ◦ 𝜎̂ = 𝜏̂ ◦ Ψ. We identity mapping, it follows that Ψ intertwines the dual actions, that is, (Ψ ⊗ abbreviate these remarks by writing: ̂ ≈ (ℛ(𝒩 , 𝜏), 𝜋𝜏 , 𝜆 , 𝜏) ̂ (ℳ, 𝜎) ≈ (𝒩 , 𝜏) ⇒ (ℛ(ℳ, 𝜎), 𝜋𝜎 , 𝜆 , 𝜎)

(1)

In particular, if 𝜎 ∶ G → Aut(ℳ) is a continuous action of G on ℳ, then every *-automorphism Φ ∈ Aut(ℳ) with Φ ◦ 𝜎g = 𝜎g ◦ Φ ( g ∈ G ) “extends” to a *-automorphism Ψ ∈ Aut(ℛ(ℳ, 𝜎)), uniquely determined, such that ̄ 𝜆 ( g)) = 𝜋𝜏 (Φ(x))(1 ⊗ ̄ 𝜆 ( g)) Ψ(𝜋𝜎 (x)(1 ⊗

(x ∈ ℳ, g ∈ G ).

(2)

On the other hand, suppose that 𝜎 and 𝜏 are both actions of G on the same von Neumann algebra ℳ ⊂ ℬ(ℋ ) and 𝜎 ∼ 𝜏, that is (15.11) there exists a unitary cocycle u ∈ Z𝜎 (G; U(ℳ)) such that ̄ ℒ ∞ (G) defined by the function G ∋ s ↦ u(s−1 ), 𝜏 = u 𝜎. Consider the unitary operator U ∈ ℳ ⊗ −1 2 that is, (U𝜉)(s) = u(s )𝜉(s) (𝜉 ∈ ℒ (G, ℋ ), s ∈ G ). It is easy to check that U𝜋𝜎 (x)U ∗ = 𝜋𝜏 (X) (x ∈ ℳ) ̄ 𝜆 ( g))U ∗ = 𝜋𝜏 (u( g)∗ )(1 ⊗ ̄ 𝜆 ( g)) ( g ∈ G ). U(1 ⊗

(3)

302

Crossed Products

Consequently, Ad(U) establishes a (spatial) *-isomorphism between ℛ(ℳ, 𝜎) and ℛ(ℳ, 𝜏), ̄ ℒ ∞ (G) and ℒ ∞ (G) is sending 𝜋𝜎 (x) into 𝜋𝜏 (x) for all x ∈ ℳ. Moreover, since U ∈ ℳ ⊗ commutative, Ad(U) intertwines the dual actions 𝜎̂ and 𝜏. ̂ More generally, we shall write (ℳ, 𝜎) ∼ (𝒩 , 𝜏), if there exists u ∈ Z𝜎 (G; U(ℳ)) such that (ℳ, u 𝜎) ≈ (𝒩 , 𝜏). Combining the above two remarks, we conclude that (ℳ, 𝜎) ∼ (𝒩 , 𝜏) ⇒ (ℛ(ℳ, 𝜎), 𝜋𝜎 , 𝜎) ̂ ≈ (ℛ(𝒩 , 𝜏), 𝜋𝜏 , 𝜏). ̂

(4)

Furthermore, if 𝜑, 𝜓 are n.s.f. weights on ℳ, 𝒩 respectively and if (ℳ, 𝜎, 𝜑) ≈ (𝒩 , 𝜏, 𝜓), then (ℛ(ℳ, 𝜎), 𝜑) ̂ ≈ (ℛ(𝒩 , 𝜏), 𝜓) ̂ by the same *-isomorphism Ψ as in (1). Also, if 𝜎 ∼ u 𝜎 = 𝜏 are both actions of G on ℳ and 𝜑 is an n.s.f. weight on ℳ, then the *-isomorphism Ad(U) appearing in (3) transports the dual weight 𝜑̂ on ℛ(ℳ, 𝜎) into the dual weight 𝜑̂ on ℛ(ℳ, 𝜏). Therefore, ̂ 𝜑) ̂ ≈ (ℛ(𝒩 , 𝜏), 𝜋𝜏 , 𝜏, ̂ 𝜓). ̂ (ℳ, 𝜎, 𝜑) ∼ (𝒩 , 𝜏, 𝜓) ⇒ (ℛ(ℳ, 𝜎), 𝜋𝜎 , 𝜎,

(5)

20.14. If ℳ is a properly infinite W ∗ -algebra, then for any continuous action 𝜎 ∶ G → Aut(ℳ) of a locally compact group G on ℳ we have ̄ ℱ,𝜎 ⊗ ̄ 𝜄) (ℳ, 𝜎) ∼ (ℳ ⊗

(1)

where ℱ is any countably decomposable type I factor. Indeed, by 20.3.(2), there exists a unitary cocycle u ∈ Z𝜎 (G; U(ℳ)) of infinite multiplicity. Then the continuous action 𝜏 = u 𝜎 has a properly infinite centralizer ℳ 𝜏 so that, using Corollary 9.16, ̄ ℱ,𝜏 ⊗ ̄ 𝜄) ∼ (ℳ ⊗ ̄ ℱ,𝜎 ⊗ ̄ 𝜄). we get (ℳ, 𝜎) ∼ (ℳ, 𝜏) ≈ (ℳ ⊗ 20.15. Notes. The results in this section are due to Connes and Takesaki [1977]. The s∗ -continuity requirement in the definition of a 𝜎-cocycle (20.l) can be replaced, without modifying this notion, by some measurability condition (see Connes [1973a], proof of 1.2.5 and 1.2.10). For our exposition, we have used Connes and Takesaki [1977].

21 Abelian Groups In this section, we undertake a more detailed study of crossed products by actions of abelian groups, which includes the connection between the Connes invariant and the dual action on the center of the crossed product, as well as a certain “Galois correspondence” between subgroups and invariant subalgebras of the crossed product. 21.1 Theorem (A. Connes, M. Takesaki). Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact abelian group G on the W ∗ -algebra ℳ. Then the Connes invariant Γ(𝜎) is the kernel of the restriction of the dual action 𝜎̂ to the center of the crossed product ℛ(ℳ, 𝜎): ̂ → Aut(𝒵 (ℛ(ℳ, 𝜎))). Γ(𝜎) = Ker(𝜎) ̂ ∶G

Abelian Groups

303

The proof will be given in Section 21.4, using some auxiliary results of independent interest which are presented in Sections 21.2 and 21.3. 21.2 Lemma. Let 𝜎 ∶ G → Aut(𝒵 ) be a continuous action of the locally compact group G on the commutative W ∗ -algebra 𝒵 . If g0 ∈ G and 𝜎g0 ≠ 𝜄, then there exist a neighborhood V of g0 and a projection 0 ≠ p ∈ 𝒵 such that p𝜎g (p) = 0 for all g ∈ V. Proof. Let 𝒜 = {x ∈ 𝒵 ; G ∋ g ↦ 𝜎g (x) ∈ 𝒵 is norm-continuous}. Then 𝒜 is an s-dense 𝜎-invariant C∗ -subalgebra of 𝒵 ; indeed, if {fi } ⊂ ℒ 1 (G) is an approximate unit of the Banach alges

bra ℒ 1 (G) with supp fi compact sets, then for every x ∈ 𝒵 we have 𝒜 ∋ 𝜎fi (x) → x. Consequently, 𝜎g0 |𝒜 ≠ 𝜄. Let 𝒜 ≅ 𝒞 (Ω) where Ω is the Gelfand spectrum of 𝒜 . For each g ∈ G, there exists a unique homeomorphism Tg ∶ Ω → Ω such that [𝜎g (x)](𝜔) = x(Tg 𝜔) (x ∈ 𝒜 , 𝜔 ∈ Ω). For each x ∈ 𝒜 , the function G × Ω ∋ ( g, 𝜔) ↦ [𝜎g (x)](𝜔) = x(Tg (𝜔)) ∈ ℂ is continuous in both variables, hence the function G × Ω ∋ ( g, 𝜔) ↦ Tg (𝜔) ∈ Ω is also continuous in both variables. Since Tg0 ≠ 𝜄, there exists 𝜔0 ∈ Ω with Tg0 (𝜔0 ) ≠ 𝜔0 . Therefore, there exist a neighborhood V of g0 and a neighborhood D of 𝜔0 such that Tg 𝜔 ≠ 𝜔 for all g ∈ V and 𝜔 ∈ D. Let x ∈ 𝒜 , 0 ≠ x ≥ 0 with supp x ⊂ D. Then x𝜎g (x) = 0 for g ∈ V. There exist 𝜀 > 0 and a spectral projection p of x with xp ≥ 𝜀p. Then 0 ≤ p𝜎g (p) ≤ 𝜀−2 x𝜎g (x) = 0, hence p𝜎g (p) = 0, for g ∈ V. 21.3. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact abelian group G on the W ∗ -algebra ℳ. Consider the faithful normal operator-valued weight P𝜎 ∶ ℳ + ∋ x ↦ ∫ 𝜎g (x) dg ∈ (ℳ 𝜎 )+ . ̂ we define For x ∈ 𝔐P𝜎 and 𝛾 ∈ G x̂ (𝛾) =

∫

⟨g, 𝛾⟩𝜎g (x) dg ∈ ℳ.

(1)

Then, x̂ (𝛾) ∈ ℳ(𝜎; {𝛾})

(2)

since for every s ∈ G we have 𝜎g (̂x(𝛾)) = ∫ ⟨g, 𝛾⟩𝜎sg (x) dg = ∫ ⟨s−1 g, 𝛾⟩𝜎g (x) dg = ⟨s, 𝛾⟩̂x(𝛾). According to the uniqueness theorem for the Fourier transform, it follows that for x ∈ 𝔐P𝜎 we have ̂ ⇔ x = 0. x̂ (𝛾) = 0 for all 𝛾 ∈ G

(3)

̂ Recall that for f ∈ ℒ 1 (G) we defined the Fourier transform ̂f by ̂f(𝛾) = ∫ ⟨g, 𝛾⟩f ( g) dg, (𝛾 ∈ G), ̂ and that the Haar measures on G and G are chosen so that the Fourier inversion theorem holds. Lemma. For every x ∈ 𝔐P𝜎 and every f ∈ ℒ 1 (G) we have 𝜎f (x) ∈ 𝔐P𝜎 and ̂ 𝜎f (x)̂(𝛾) = ̂f(𝛾)̂x(𝛾) (𝛾 ∈ G)

(4)

304

Crossed Products

̂ then and moreover, if ̂f ∈ ℒ 1 (G), 𝜎f (x) =

∫

⟨g, 𝛾⟩𝜎f (x)̂(𝛾) d𝛾.

(5)

Proof. We may assume x ≥ 0 and f ≥ 0. Then 𝜎f (x) ≥ 0 and, using the Fubini–Tonelli theorem, we obtain ( ) 𝜎 (𝜎 (x)) dg = 𝜎 f (s)𝜎s (x) ds dg P𝜎 (𝜎f (x)) = ∫ g f ∫ g ∫ ( ) = f (s)𝜎gs (x) ds dg = f (s) 𝜎 (x) dg ds ∫ ∫ ∫ ∫ gs ( )( ) = f (s) ds 𝜎 (x) dg = ‖ f‖1 P𝜎 (x) ∈ ℳ + , ∫ ∫ g ̂ we have hence 𝜎f (x) ∈ 𝔐P𝜎 . Then, for 𝛾 ∈ G 𝜎f (x)̂(𝛾) =

⟨g, 𝛾⟩𝜎g (𝜎f (x)) dg = ⟨g, 𝛾⟩f (s)𝜎gs (x) ds dg ∫ ∫ ( ) ⟨gs, 𝛾⟩𝜎gs (x) dg ds = ⟨s, 𝛾⟩f (s) ∫ ∫ ( )( ) = ⟨s, 𝛾⟩f (s) ds ⟨g, 𝛾⟩𝜎g (x) dg = ̂f (𝛾)̂x(𝛾). ∫ ∫ ∫

̂ (5) follows by the Fourier inversion theorem. Finally, if ̂f ∈ ℒ 1 (G), Corollary. If the action 𝜎 ∶ G → Aut(ℳ) is integrable, then ̂ ℳ = ℛ{ℳ(𝜎; {𝛾}); 𝛾 ∈ G}

(6)

̂ we have and, for every open set V ⊂ G, ℳ(𝜎; V) ≠ 0 ⇔ there exists 𝛾 ∈ V such that ℳ(𝜎; {𝛾}) ≠ {0}.

(7)

Proof. Let {fi } be a norm-bounded approximate unit of the Banach algebra ℒ 1 (G) with supp fi w

compact sets. For each x ∈ 𝔐P𝜎 we have 𝜎fi (x) → x and for each 𝜎fi (x) (5) holds. Since the action 𝜎 is integrable, that is, 𝔐P𝜎 is w-dense in ℳ, (6) follows using (2) and (5). We now prove assertion (7). The implication (⇐) is obvious. Conversely, assume that ℳ(𝜎; {𝛾}) = {0} for every 𝛾 ∈ V. By (2) it follows that x̂ (𝛾) = 0 for all x ∈ 𝔐P𝜎 and 𝛾 ∈ V. Furthermore, using (4) it follows that if x ∈ 𝔐P𝜎 and f ∈ ℒ 1 (G), supp ̂f ⊂ V, then 𝜎f (x)̂(𝛾) = 0 for ̂ hence 𝜎f (x) = 0. Since 𝔐P is w-dense in ℳ, it follows that 𝜎f (x) = 0 for all x ∈ ℳ and all 𝛾 ∈ G, 𝜎 all f ∈ ℒ 1 (G) with supp ̂f ⊂ V, hence (14.3.(7)) ℳ(𝜎; V) = {0}.

Abelian Groups

305

Note that if the action 𝜎 is integrable, then x=

∫

⟨g, 𝛾⟩̂x(𝛾) d𝛾

(8)

for x in a w-dense *-subalgebra of ℳ. ̄ ℬ(ℒ 2 (G)). ̄ 𝜄 and 𝜎 ⊗ ̄ Ad(𝜌) of G on ℳ ⊗ 21.4. Proof of Theorem 21.1. Consider the actions 𝜎 ⊗ By 16.16.(2) and 16.3, we have ̄ 𝜄) = Γ(𝜎 ⊗ ̄ Ad(𝜌𝜌)). Γ(𝜎) = Γ(𝜎 ⊗

(1)

On the other hand, putting ℬ = ℬ(ℒ 2 (G)), we have (19.13 and 19.3) ̄ ℬ)𝜎 ⊗̄ Ad 𝜌 , 𝜎̂ 𝛾 = Ad(1 ⊗ ̄ m(𝛾)∗ ), ℛ(ℳ, 𝜎) = (ℳ ⊗ ̄ ℬ, 𝜎 ⊗ ̄ Ad 𝜌 ) = (ℳ ⊗ ̄ ℬ⊗ ̄ ℬ)𝜎 ⊗̄ Ad 𝜌 ⊗̄ Ad 𝜌 , ℛ(ℳ ⊗ ̄ Ad 𝜌 )̂𝛾 = Ad(1 ⊗ ̄ 1⊗ ̄ m(𝛾)∗ ) (𝜎 ⊗ so that ̄ ℬ⊗ ̄ ℬ)𝜎 ⊗̄ Ad 𝜌 ⊗̄ 𝜄 ), 𝜎̂ 𝛾 ≈ Ad(1 ⊗ ̄ m(𝛾)∗ ⊗ ̄ 1), 𝒵 (ℛ(ℳ, 𝜎)) ≈ 𝒵 ((ℳ ⊗ ̄ ℬ, 𝜎 ⊗ ̄ Ad 𝜌 )) ≈ 𝒵 ((ℳ ⊗ ̄ ℬ⊗ ̄ ℬ))𝜎 ⊗̄ Ad 𝜌 ⊗̄ Ad 𝜌 ), 𝒵 (ℛ(ℳ ⊗ ̄ Ad 𝜌 )̂𝛾 ≈ Ad(1 ⊗ ̄ m(𝛾)∗ ⊗ ̄ 1), (𝜎 ⊗ ̄ W∈ℳ⊗ ̄ ℬ⊗ ̄ ℬ, where (W𝜉)(s, t) = 𝜉(s, ts), has the properties W ∗ (𝜌𝜌( g) ⊗ ̄ 1)W = the operator 1 ⊗ ∗ ̄ ̄ ̄ 𝜌 ( g) ⊗ 𝜌 ( g), W (m(𝛾) ⊗ 1)W = m(𝛾) ⊗ 1, and hence implements a *-isomorphism ̄ ℬ(ℒ 2 (G)), (𝜎 ⊗ ̄ Ad𝜌𝜌)̂ ). (𝒵 (ℛ(ℳ, 𝜎)), 𝜎) ̂ ≈ (𝒵 (ℛ(ℳ ⊗

(2)

From (1) and (2), it follows that in proving Theorem 21.1 we may replace (ℳ, 𝜎) by ̄ ℬ(ℒ 2 (G)), 𝜎 ⊗ ̄ Ad 𝜌 ). Similarly, we see that we may replace (ℳ, 𝜎) by (ℳ ⊗ ̄ ℬ(ℒ 2 (G)) ⊗ ̄ ℱ, (ℳ ⊗ ̄ ̄ ̄ ̄ 𝜎 ⊗ Ad 𝜌 ⊗ 𝜄), where ℱ is any infinite factor. In this case, 𝜎 ⊗ Ad 𝜌 ⊗ 𝜄, is a dominant action (20.12). Consequently, in proving the theorem we shall assume, as we may, that 𝜎 ∶ G → Aut(ℳ) is a ̂ → Aut(ℳ 𝜎 ) such dominant action. Then, by Proposition 20.12, there exists a continuous action 𝜃 ∶ G 𝜎 ̂ ̂ → U(ℳ) that (ℳ, 𝜎) ≈ (ℛ(ℳ , 𝜃), 𝜃), and there exists an s-continuous unitary representation u ∶ G such that 𝜎g (u(𝛾)) = ⟨g, 𝛾⟩u(𝛾) and 𝜃𝛾 = Ad(u(𝛾))|ℳ 𝜎

̂ ( g ∈ G, 𝛾 ∈ G).

(3)

Thus, we have the identifications ̂̂ ≈ (ℳ 𝜎 ⊗ ̂ 𝜃) ̄ ℬ(ℒ 2 (G)), 𝜃 ⊗ ̄ Ad 𝜌 ), (ℛ(ℳ, 𝜎), 𝜎) ̂ ≈ (ℛ(ℛ(ℳ 𝜎 , 𝜃), 𝜃) the last *-isomorphism being given by the Takesaki duality theorem (19.5). It follows that ̄ ℬ(ℒ 2 (G))), 𝜃 ⊗ ̄ Ad 𝜌) ≈ (𝒵 (ℳ 𝜎 ), 𝜃). (𝒵 (ℛ(ℳ, 𝜎)), 𝜎) ̂ ≈ (𝒵 (ℳ 𝜎 ⊗

(4)

306

Crossed Products

Thus, all we have to prove is that ̂ → Aut(𝒵 (ℳ 𝜎 ))). Γ(a) = Ker(𝜃 ∶ G

(5)

Let e ∈ 𝒵 (ℳ 𝜎 ) be a nonzero projection. From (3), it follows that u(𝛾) ∈ ℳ(𝜎; {𝛾}) ∩ U(ℳ), hence ℳ(𝜎; {𝛾}) = ℳ 𝜎 u(𝛾). Therefore, ℳe (𝜎 e ; {𝛾}) = ℳ(𝜎; {𝛾}) ∩ ℳe = eℳ(𝜎; {𝛾})e = eℳ 𝜎 u(𝛾)e = eℳ 𝜎 u(𝛾)eu(𝛾)∗ u(𝛾) = eℳ 𝜎 𝜃𝛾 (e)u(𝛾) = e𝜃𝛾 (e)ℳ 𝜎 u(𝛾), so that ℳe (𝜎 e ; {𝛾}) = {0} ⇔ e𝜃𝛾 (e) = 0.

(6)

̂ → Aut(𝒵 (ℳ 𝜎 )) ⇒ 𝜃𝛾 (e) = e for all e ∈ 𝒵 (ℳ 𝜎 ) ⇒ e𝜃𝛾 (e) ≠ 0 for all Thus, 𝛾 ∈ Ker(𝜃 ∶ G 𝜎 0 ≠ e ∈ 𝒵 (ℳ ) ⇒ ℳe (𝜎 e ; {𝛾}) ≠ {0} for all projections 0 ≠ e ∈ 𝒵 (ℳ 𝜎 ) ⇒ 𝛾 ∈ Sp 𝜎 e for all projections 0 ≠ e ∈ 𝒵 (ℳ 𝜎 ) ⇒ 𝛾 ∈ Γ(𝜎). Conversely, using Lemmas 21.2 and 21.3 and denoting ̂ we get 𝛾 ∉ Ker(𝜃 ∶ G ̂ → Aut(𝒵 (ℳ 𝜎 ))) ⇒ there by V(𝛾) the family of open neighborhoods of 𝛾 ∈ G, 𝜎 exist V ∈ V(𝛾) and 0 ≠ e ∈ Proj(𝒵 (ℳ )) such that e𝜃𝛽 (e) = 0 for all 𝛽 ∈ V ⇒ there exist V ∈ V(𝛾) and 0 ≠ e ∈ Proj(𝒵 (ℳ 𝜎 )) such that ℳe (𝜎 e ; {𝛽}) = {0} for all 𝛽 ∈ V ⇒ there exist V ∈ V(𝛾) and 0 ≠ e ∈ Proj(𝒵 (ℳ 𝜎 )) such that ℳ𝜎 (𝜎 e ; V) = {0} ⇒ there exists 0 = e ∈ Proj(𝒵 (ℳ 𝜎 )) such that 𝛾 ∉ Sp 𝜎 e ⇒ 𝛾 ∉ Γ(𝜎). The proof of Theorem 21.1 is complete. □ Note that in the above proof the notion of a dominant action appears only in order to simplify the exposition, so that the reference to Section 20.12 is not necessary. Indeed, all facts obtained ̄ ℬ(ℒ 2 (G)), 𝜎 ⊗ ̄ Ad(𝜌𝜌)) from the Takesaki duality using this reference follow obviously for (ℳ ⊗ theorem. Moreover, since (in contrast to Proposition 20.12) the Takesaki duality theorem does not require restrictive conditions on ℳ or G, the statement of Theorem 21.1 is also free of any such conditions. 21.5. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact abelian group G on the W ∗ -algebra ℳ and let 𝜎̂ ∶ G → Aut(ℛ(ℳ, 𝜎)) be the dual action. Recall (19.3.(9)) that ℛ(ℳ, 𝜎)𝜎̂ = 𝜋𝜎 (ℳ).

(1)

With the help of Landstad’s theorem (19.9), this result can be extended as follows: Consider a closed subgroup H of G and its “orthogonal” subgroup ̂ ⟨h, 𝛾⟩ = 1 for all h ∈ H}. H⟂ = {𝛾 ∈ G; It is well known (Hewitt & Ross, 1963, 1970; Rudin, 1962; Strătilă, 1967, 1968) that the dual group ̂ ⟂ ; namely, denoting by 𝛾̃ ∈ G∕H ̂ ⟂ the ̂ can be canonically identified with the quotient group G∕H H ̂ ̂ image of 𝛾 ∈ G, we have ⟨h, 𝛾̃ ⟩ = ⟨h, 𝛾⟩ (h ∈ H, 𝛾 ∈ G). Proposition. In the above situation, we have ⟂

̂ ̄ 𝜆 (H)} ≈ ℛ(ℳ, 𝜎|H). ℛ(ℳ, 𝜎)𝜎∕H = ℛ{𝜋𝜎 (ℳ), 1 ⊗

(2)

Abelian Groups

307 ⟂

̂ Proof. Let 𝒩 = ℛ(ℳ, 𝜎)𝜎∕H . We have an s-continuous unitary representation u ∶ H ∋ h ↦ ̂ ∋ 𝛾̃ ↦ 𝜃 𝛾̃ = 𝜎̂ 𝛾 |𝒩 ∈ Aut(𝒩 ) such that ̄ 𝜆 (h) ∈ 𝒩 and a continuous action 𝜃 ∶ H u(h) = 1 ⊗ ̂ From (1), it follows that 𝒩 𝜃 = 𝜋𝜎 (ℳ) and by 19.1.(2) 𝜃𝛾̃ (u(h)) = ⟨h, 𝛾̃ ⟩u(h) for all h ∈ H, 𝛾̃ ∈ H. ∗ we have 𝜋𝜎 (𝜎h (x)) = u(h)𝜋𝜎 (x)u(h) (x ∈ ℳ, h ∈ H). Thus, the proposition follows using Landstad’s theorem.

Clearly, (1) follows from (2) by taking as H the trivial subgroup, consisting of the neutral element of G. Recall that 𝔏(G) = ℛ(ℳ, 𝜎), where ℳ = ℂ ⋅ 1G and 𝜎 is the trivial action of G, while the dual action 𝜎̂ is implemented by the multiplication operators defined by the characters: 𝜎̂ 𝛾 (x) = ̂ Thus, from the above proposition it follows that for any closed m(𝛾)∗ xm(𝛾)(x ∈ 𝔏(G), 𝛾 ∈ G). subgroup H of G we have 𝜆(h); h ∈ H} ≈ 𝔏(H). {x ∈ 𝔏(G); xm(𝛾) = m(𝛾)x, (∀)𝛾 ∈ H⟂ } = ℛ{𝜆

(3)

21.6. An important problem concerning crossed products is to obtain information about the center of a crossed product. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact abelian group G on the W ∗ -algebra ℳ. We show that ̄ ℛ{𝜆 𝜆( g); g ∈ Γ(𝜎)⟂ }. 𝒵 (ℛ(ℳ, 𝜎) ⊂ 𝒵 (ℳ 𝜎 ) ⊗

(1)

̄ 𝜆( g) ( g ∈ G ), it follows that X ∈ Indeed, let X ∈ 𝒵 (ℛ(ℳ, 𝜎)). Since X commutes with 1 ⊗ ̄ Ad 𝜌 2 𝜎 ⊗ ̄ ̄ ℳ ⊗ 𝔏(G). Since X ∈ ℛ(ℳ, 𝜎) = (ℳ ⊗ ℬ(ℒ (G))) (see 19.13), it follows that X ∈ ̄ 𝔏(G). If x ∈ ℳ 𝜎 , then X commutes with 𝜋𝜎 (x) = x ⊗ ̄ 1, hence X ∈ 𝒵 (ℳ 𝜎 ) ⊗ ̄ 𝔏(G). By ℳ𝜎 ⊗ Theorem 21.1, for every 𝛾 ∈ Γ(𝜎) we have 𝜎̂ 𝛾 (X) = X, that is, ̄ m(𝛾))X = X(1 ⊗ ̄ m(𝛾)) (1 ⊗

(𝛾 ∈ Γ(𝜎)).

̄ 𝔏(G), it follows from the previous identity that Let 𝜑 ∈ ℳ∗ and a = E𝜑G (X). Since X ∈ ℳ ⊗ a ∈ 𝔏(G) and m(𝛾)a = am(𝛾) for all 𝛾 ∈ Γ(𝜎). Using 21.5.(3), we further deduce that E𝜑G (X) = a ∈ ̄ ℛ{𝜆 𝜆( g); g ∈ Γ(𝜎)⟂ }. Since 𝜑 ∈ ℳ∗ was arbitrary, we conclude that X ∈ 𝒵 (ℳ 𝜎 ) ⊗ 𝜆( g); g ∈ ℛ{𝜆 Γ(𝜎)⟂ }. On the other hand, we have ̄ 1G ⊂ 𝒵 (ℛ(ℳ, 𝜎)). 𝒵 (ℳ)𝜎 ⊗

(2)

̄ 1 obviously commutes with every 1 ⊗ ̄ 𝜆 ( g) ( g ∈ G ), and for Indeed, let z ∈ 𝒵 (ℳ)𝜎 . Then z ⊗ ̄ 1)𝜋𝜎 (x) = 𝜋𝜎 (z)𝜋𝜎 (x) = 𝜋𝜎 (zx) = 𝜋𝜎 (xz) = 𝜋𝜎 (x)𝜋𝜎 (z) = 𝜋𝜎 (x)(z ⊗ ̄ 1). any x ∈ ℳ we have (z ⊗ We now show that ̂ ⇔ 𝒵 (ℛ(ℳ, 𝜎)) = 𝒵 (ℳ)𝜎 ⊗ ̄ 1G . Γ(𝜎) = G

(3)

̄ m(𝛾)∗ ) (𝛾 ∈ Indeed, the implication (⇐) follows immediately from Theorem 21.1, as 𝜎̂ 𝛾 = Ad(1 ⊗ 𝜎 ̂ ̂ ̄ 1. Thus, G). Conversely, if Γ(𝜎) = G, then from (1) it follows that 𝒵 (ℛ(ℳ, 𝜎)) ⊂ 𝒵 (ℳ ) ⊗

308

Crossed Products

̄ 1 = 𝜋𝜎 (z), and for every x ∈ ℳ if X ∈ 𝒵 (ℛ(ℳ, 𝜎)), there exists z ∈ ℳ 𝜎 such that X = z ⊗ we have 𝜋𝜎 (zx) = X𝜋𝜎 (x) = 𝜋𝜎 (x)X = 𝜋𝜎 (xz), that is, z ∈ 𝒵 (ℳ) ∩ ℳ 𝜎 = 𝒵 (ℳ)𝜎 . Therefore, ̄ 1 and the reverse inclusion follows by (2). 𝒵 (ℛ(ℳ, 𝜎)) ⊂ 𝒵 (ℳ)𝜎 ⊗ From (3) and Theorem 21.1 it follows that ̂ and 𝒵 (ℳ)𝜎 = ℂ ⋅ 1ℳ . ℛ(ℳ, 𝜎) is a factor ⇔ Γ(𝜎) = G

(4)

Thus, if the action 𝜎 is ergodic on 𝒵 (ℳ), for instance if ℳ is a factor, or if ℳ 𝜎 is a factor, then ̂ ℛ(ℳ, 𝜎) is a factor ⇔ Γ(𝜎) = G.

(5)

In particular, we obtain the following: Corollary. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact abelian group G on the W ∗ -algebra ℳ. If the centralizer ℳ 𝜎 is a factor and 𝜎g ≠ 𝜄 for every g ≠ e (= the neutral element of G), then ℛ(ℳ, 𝜎) is a factor. Proof. Since ℳ 𝜎 is a factor, (5) holds and Γ(𝜎) = Sp 𝜎 (16.1.(3)). Let g ∈ Γ(𝜎)⟂ = (Sp 𝜎)⟂ . By Proposition 14.6, the spectrum of the operator 𝜎g ∈ ℬ(ℳ) is equal to {⟨g, 𝛾⟩; 𝛾 ∈ Sp 𝜎} = {1} and hence (14.12.(3)) 𝜎g = 𝜄. By hypothesis it follows that g = e. Thus, Γ(𝜎)⟂ = {e} and Γ(𝜎) = ̂ that is, ℛ(ℳ, 𝜎) is a factor. Γ(𝜎)⟂⟂ = G, As a further application, we give another proof of statement 16.5.(1). Indeed, let g ∈ Int 𝜎 and let u( g) ∈ U(ℳ 𝜎 ) be such that 𝜎g = Ad(u( g)). Then u( g) ∈ U(𝒵 (ℳ 𝜎 )) and using Corollary 19.13 it is easy to see that ̄ 𝜆 ( g) ∈ 𝒵 (ℛ(ℳ, 𝜎)). u( g) ⊗

(1)

̄ 𝜆 ( g) = 𝜎(u( ̄ 𝜆 ( g)) = Consequently, if 𝛾 ∈ Γ(𝜎), then by Theorem 21.1, we have u( g) ⊗ ̂ g) ⊗ ̄ m(𝛾)∗ )(u( g) ⊗ ̄ 𝜆 ( g))(1 ⊗ ̄ m(𝛾)) = ⟨g, 𝛾⟩(u( g) ⊗ ̄ 𝜆 ( g)), so that ⟨g, 𝛾⟩ = 1. Hence g ∈ Γ(𝜎)⟂ , (1 ⊗ thus proving the inclusion Int 𝜎 ⊂ Γ(𝜎)⟂ . Recall that if ℳ is a factor and Sp 𝜎∕Γ(𝜎) is compact, then, by Theorem 16.5, Int 𝜎 = Γ(𝜎)⟂ . 21.7. The Coones invariant Γ(𝜎) also gives information about the comparison of 𝜎-cocycles: Corollary. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the separable locally compact abelian ̂ then every squaregroup G on the W ∗ -algebra ℳ with separable predual ℳ∗ . If Γ(𝜎) = G, integrable unitary cocycle of infinite multiplicity a ∈ Z𝜎 (G; U(ℳ)) is a dominant cocycle. Proof. By Theorems 20.5 and 20.6, there exist a dominant cocycle u ∈ Z𝜎 (G; U(ℳ)) and a projection p ∈ ℳ u such that a ≂ up ; then the actions a 𝜎 and (u 𝜎)p are conjugate. Since u 𝜎 is a dominant action, proving the corollary amounts to showing that ̂ then for every if 𝜎 is a dominant action with Γ(𝜎) = G, properly infinite projection p ∈ ℳ 𝜎 with p ∼ 1 in ℳ the action 𝜎 p is also dominant.

(1)

̂ → If 𝜎 is a dominant action, then, by Proposition 20.12, there exists a continuous action 𝜃 ∶ G ̂ According to the Takesaki duality theorem (19.5), it Aut(ℳ 𝜎 ) such that (ℳ, 𝜎) ≈ (ℛ(ℳ 𝜎 , 𝜃), 𝜃).

Abelian Groups

309

̂ means ̂ = (𝒵 (ℳ 𝜎 ), 𝜃). By Theorem 21.1, the assumption Γ(𝜎) = G follows that (𝒵 (ℛ(ℳ, 𝜎)), 𝜎) that 𝜃 acts identically on 𝒵 (ℳ 𝜎 ). ̂ the properly infinite projections p and 𝜃𝛾 (p) of ℳ 𝜎 have the same central Then, for each 𝛾 ∈ G, 𝜎 𝜎 support in ℳ , and ℳ is countably decomposable, since ℳ∗ is separable. Consequently, for every ̂ we have p ∼ 𝜃𝛾 (p) in ℳ 𝜎 , that is, there exists w(𝛾) ∈ ℳ 𝜎 such that w(𝛾)∗ w(𝛾) = 𝜃𝛾 (p) and 𝛾 ∈G w(𝛾)w(𝛾)∗ = p. On the other hand, since the action 𝜎 is dominant, it also follows from Propsition 20.12 that ̂ → ℳ such that 𝜎g (u(𝛾)) = ( g, 𝛾)u(𝛾) and there exists an s-continuous unitary representation u ∶ G 𝜎 for all g ∈ G, 𝛾 ∈ G. ̂ 𝜃𝛾 = Ad(u(𝛾))|ℳ ̂ Then v(𝛾) ∈ U(pℳp) as v(𝛾)∗ v(𝛾) = pu(𝛾)∗ w(𝛾)∗ Let v(𝛾) = w(𝛾)u(𝛾)p ∈ pℳp, (𝛾 ∈ G). ∗ −1 w(𝛾)u(𝛾)p = pu(𝛾) 𝜃𝛾 (p)u(𝛾)p = p𝜃𝛾 (𝜃𝛾 (p))p = p and v(𝛾)v(𝛾)∗ = w(𝛾)u(𝛾)pu(𝛾)∗ w(𝛾)∗ p

= w(𝛾)𝜃𝛾 (p)w(𝛾)∗ = w(𝛾)w(𝛾)∗ = p. Also, 𝜎g (v(𝛾)) = 𝜎g (w(𝛾))𝜎g (u(𝛾))𝜎g (p) = w(𝛾)⟨( g, 𝛾)⟩u(𝛾)p = ⟨g, 𝛾⟩v(𝛾). Since ℳ∗ is separable, it follows from Proposition 20.12 that 𝜎 p is dominant. Thus, under the conditions of the corollary, all square-integrable unitary cocycles of infinite multiplicity are equivalent. 21.8. Proposition 21.5 allows us to establish a certain “Galois correspondence” between the closed subgroups of G and the 𝜎-invariant ̂ unital W ∗ -subalgebras of ℛ(ℳ, 𝜎), which contain 𝜋𝜎 (ℳ), in the case when ℳ is a factor. Theorem. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the locally compact abelian group G on the factor ℳ. If 𝒩 is a 𝜎-invariant ̂ unital W ∗ -subalgebra of ℛ(ℳ, 𝜎) containing 𝜋𝜎 (ℳ), then there exists a closed subgroup H of G, uniquely determined, such that ⟂

̂ ̄ 𝜆 (H)} ≈ ℛ(ℳ, 𝜎|H) 𝒩 = ℛ(ℳ, 𝜎)𝜎|H = ℛ{𝜋𝜎 (ℳ), 1 ⊗

(1)

̂ 𝜎̂ 𝛾 |𝒩 = 𝜄}⟂ = {g ∈ G; 1 ⊗ ̄ 𝜆 ( g) ∈ 𝒩 }. H = {𝛾 ∈ G;

(2)

namely,

Thus, the mappings H ↦ 𝒩 and 𝒩 → H defined by (1) and (2) are reciprocal order-preserving bijections between the closed subgroups H of G and the 𝜎-invariant ̂ unital W ∗ -subalgebras 𝒩 ⊃ 𝜋𝜎 (ℳ) of ℛ(ℳ, 𝜎). ̂ ⟂ and, by ̂ 𝜎̂ 𝛾 |𝒩 = 𝜄}⟂ . It is clear that 𝒩 ⊂ ℛ(ℳ, 𝜎)𝜎|H Proof. Let H = {𝛾 ∈ G; ⟂ ̂ ̂ ⟂ with Proposition 21.5, we have ℛ(ℳ, 𝜎)𝜎|H ≈ ℛ(ℳ, 𝜎|H) so that, identifying ℛ(ℳ, 𝜎)𝜎|H ℛ(ℳ, 𝜎|H) via this *-isomorphism, the subalgebra 𝒩 ⊂ ℛ(ℳ, 𝜎|H) satisfies the conditions ̂ and the subgroup {𝜒 ∈ H; ̂ (𝜎|H)̂ |𝒩 = 𝜄} reduces to the 𝒩 ⊃ 𝜋𝜎|H (ℳ), (𝜎|H)̂𝜒 (𝒩 ) = 𝒩 (𝜒 ∈ H), 𝜒 ̂ neutral element of H. ̂ 𝛾 ≠ 𝜀, where Thus, it is sufficient to prove the theorem in the case when 𝜎̂ 𝛾 |𝒩 ≠ 𝜄 for all 𝛾 ∈ G, 𝜀 stands for the neutral element of G. In this case, we identify ℳ with 𝜋𝜎 (ℳ) and put ℛ = ℛ(ℳ, 𝜎), so that ℳ ⊂ 𝒩 ⊂ ℛ. We have ℛ 𝜎̂ = 𝒩 𝜎̂ = ℳ which, by assumption, is a factor. From Corollary 21.6, it follows that ̃ If 𝒩̃ = ℛ(𝒩 , 𝜎) ̂ is a factor. Putting ℛ̃ = ℛ(ℛ, 𝜎) ̂ and ℳ̃ = ℛ(ℳ, 𝜎), ̂ we have ℳ̃ ⊂ 𝒩̃ ⊂ ℛ.

310

Crossed Products

̃ then it will follow that 𝒩 = 𝒩̃ 𝜎̂̂ = ℛ̃ 𝜎̂̂ = ℛ = ℛ(ℳ, 𝜎), thus proving we can show that 𝒩̃ = ℛ, the theorem. We have ℛ̃ = ℛ(ℛ(ℳ, 𝜎), 𝜎) ̂ and hence, by the Takesaki duality theorem (19.5), there exists a *̄ 𝜌, 𝜄 ⊗ ̄ ℬ(ℒ 2 (G)) such that (ℛ(ℳ, 𝜎), 𝜎) ̄ ℬ(ℒ 2 (G)))𝜎 ⊗Ad ̄ Ad(m)). isomorphism ℛ̃ ≈ ℳ ⊗ ̂ ≈ ((ℳ ⊗ ̄ ℬ(ℒ 2 (G)) The image of ℳ ≡ 𝜋𝜎 (ℳ) = ℛ(ℳ, 𝜎)𝜎̂ by this *-isomorphism is just 𝜋𝜎 (ℳ) ⊂ ℳ ⊗ ̂ =ℳ⊗ ̄ m(G)) ̄ ℒ ∞ (G). and hence the image of ℳ̃ = ℛ(ℳ, 𝜎) ̂ is (19.2.(1)) ℳ̃ ≈ ℛ{𝜋𝜎 (ℳ), 1 ⊗ ̄ ℒ ∞ (G) ⊂ 𝒩̃ ⊂ ℳ ⊗ ̄ ℬ(ℒ 2 (G)) or, passing to the commutants, Therefore, we have ℳ ⊗ ̄ 1G ⊂ 𝒩̃ ′ ⊂ ℳ ′ ⊗ ̄ ℒ ∞ (G), where 𝒩̃ ′ is a factor. By Corollary 10.18, it follows that ℳ′ ⊗ ̃ ̄ 1G , that is, 𝒩̃ = ℳ ⊗ ̄ ℬ(ℒ 2 (G)) = ℛ. 𝒩̃ ′ = ℳ ′ ⊗ ̂ 𝜎̂ 𝛾 |𝒩 = 𝜄}⟂ . It is then We have thus proved, in the general case, that (1) holds with H = {𝛾 ∈ G; ̄ 𝜆 (h) ∈ 𝒩 for every h ∈ H. Conversely, let g ∈ G be such that 1 ⊗ ̄ 𝜆 ( g) ∈ 𝒩 . immediate that 1 ⊗ ⟂ ̄ ̄ ̄ Then, for 𝛾 ∈ H we have 1 ⊗ 𝜆 ( g) = 𝜎̂ 𝛾 (1 ⊗ 𝜆 ( g)) = ⟨g, 𝛾⟩(1 ⊗ 𝜆 ( g)), hence ⟨g, 𝛾⟩ = 1. Thus, g ∈ H⟂⟂ = H, which proves (2). 21.9 Corollary. Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the commutative locally compact group G on the W ∗ -algebra ℳ. If (a) the action 𝜎 is dominant and ℳ 𝜎 is a factor, or if ̂ and ℳ is a factor, (b) the action 𝜎 is integrable, Γ(𝜎) = G then the mappings H ↦ 𝒩H = ℳ 𝜎|H and 𝒩 ↦ H𝒩 = {g ∈ G; 𝜎g |𝒩 = 𝜄} are reciprocal orderreversing bijections between the closed subgroups H of G and the 𝜎-invariant unital W ∗ -subalgebras 𝒩 ⊃ ℳ 𝜎 of ℳ. ̂ → Aut(ℳ) Proof. (a) If the action 𝜎 is dominant, then (20.12) there exists a continuous action 𝜃 ∶ G ̂ Since ℳ 𝜎 is assumed to be a factor, the corollary follows such that (ℳ, 𝜎) ≈ (ℛ(ℳ 𝜎 , 𝜃), 𝜃). immediately from Theorem 21.8. (b) Let ℱ be the countably decomposable infinite factor of type I. It is easy to see that if (ℳ, 𝜎) ̄ ℱ,𝜎 ⊗ ̄ 𝜄) also satisfies condition (b) and that if the conclusion of satisfies condition (b), then (ℳ ⊗ ̄ ℱ , 𝜎 ⊗ 𝜄), it remains true for (ℳ, 𝜎). the corollary is true for (ℳ ⊗ Consequently, we may assume that the centralizer ℳ 𝜎 is properly infinite. Then, by assumption ̄ ℬ(ℒ 2 (G)) ≈ (b) and Corollary 21.7 it follows that 𝜎 is a dominant action. Moreover, ℳ 𝜎 ≈ ℳ 𝜎 ⊗ ̂ and ℳ is a factor (see 21.6.(5)). We can thus ℛ(ℳ, 𝜎) (see 20.12.(2)) is a factor because Γ(𝜎) = G restrict attention to case (a), which has been already considered. 21.10. Let G be a locally compact abelian group and H ⊂ G a closed subgroup. For g ∈ G, we shall denote by g̃ ∈ G∕H its image under the canonical quotient mapping. It is known (Rudin, 1962; Strătilă, 1967, 1968) that the Haar measures on G, H and G∕H can be chosen so that ( ∫G

f ( g) dg =

∫G∕H ∫H

) f ( gh) dh

d̃g

( f ∈ ℒ 1 (G)).

(1)

More precisely, for every f ∈ ℒ 1 (G) there exists a negligible set Ω ⊂ G∕H such that for every g ∈ G with g̃ ∈ Ω, the function H ∋ h ↦ f ( gh) belongs to ℒ 1 (H), the function G∕H ∋ g̃ ↦ ∫H f ( gh) dh belongs to ℒ 1 (G∕H), and (1) holds.

Abelian Groups

311

If 𝜎 ∶ G → Aut(ℳ) is a continuous action of G on the W ∗ -algebra ℳ, then we can also define a continuous action 𝜎̃ ∶ G∕H → Aut(ℳ 𝜎|H ) by 𝜎̃ g̃ = 𝜎g |ℳ 𝜎|H

( g ∈ G ).

(2)

Using (1), it is easy to check that if x ∈ ℳ + and if ‖P𝜎 (x)‖ < +∞, then P𝜎|H (x) ∈ +

(ℳ 𝜎|H ) , ‖P𝜎|H (x)‖ < +∞ and P𝜎̃ (P𝜎|H (x)) = P𝜎 (x).

(3)

It follows that if the action 𝜎 is integrable, then the actions 𝜎|H and 𝜎̃ are also integrable. 21.11. As an application of Corollary 21.9, consider the factor ℳ = ℬ(ℒ 2 (G)) and the continuous ̂ → Aut(ℳ) defined by action 𝜎 ∶ G × G 𝜆( g)∗ 𝜎g,𝛾 (x) = 𝜆 ( g)m(𝛾)∗ xm(𝛾)𝜆

̂ (x ∈ ℳ, g ∈ G, 𝛾 ∈ G).

̂ Since ℬ(ℒ 2 (G)) = ℛ{𝔏(G), ℒ ∞ (G)} = ℛ{𝜆(G), m(G)}, we have ℳ 𝜎 = ℂ ⋅ 1G .

(1)

Consequently, Γ(𝜎) = Sp 𝜎, and it is easy to see that ̂ ̂=G ̂ × G. Γ(𝜎) = (G × G)

(2)

̄ 𝜂̄ ∈ ℬ(ℒ 2 (G)) (see Finally, the action 𝜎 is integrable. Indeed, for 𝜉, 𝜂 ∈ ℒ 2 (G) the operator 𝜉 ⊗ 4.23) belongs to 𝔐P𝜎 and ̄ 𝜂) ̄ = (𝜉|𝜂) ⋅ 1G . P𝜎 (𝜉 ⊗

(3)

By Corollary 21.9, it follows that there exists a decreasing bijection between the 𝜎-invariant von ̂ Neumann subalgebras of ℬ(ℒ 2 (G)) and the closed subgroups of G × G. ̂ be a closed subgroup. Then H⟂ = {(𝛾, g) ∈ G ̂ × G; ⟨h, 𝛾⟩⟨g, 𝜒⟩ = 1, for all Let H ⊂ G × G (h, 𝜒) ∈ H}. Using the commutation relations 18.9.(1), for g, h ∈ G and 𝛾, 𝜒 ∈ G we obtain 𝜆( g)m(𝛾), 𝜆( g)m(𝛾)) = ⟨h, 𝛾⟩ ⟨g, 𝜒⟩𝜆 𝜎(h,𝜒) (𝜆

(4)

𝜆 ( g)m(𝛾) ∈ ℳ 𝜎|H ⇔ (𝛾, g) ∈ H⟂ .

(5)

so that

̂ on ℳ also defines a natural action 𝜎̃ of (G × G)∕H ̂ The action 𝜎 of G × G on ℳ 𝜎|H whose centralizer is (ℳ 𝜎|H )𝜎̃ = ℳ 𝜎 = ℂ ⋅ 1G . Since the action 𝜎 is integrable, so is the action 𝜎̃ (21.10) and, by 21.3.(6), it follows that ℳ 𝜎|H = ℛ{ℳ 𝜎|H (𝜎; ̃ {(𝛾, g)});

(𝛾, g) ∈ H⟂ }.

(6)

312

Crossed Products

Let (𝛾, g) ∈ H⟂ and x ∈ ℳ 𝜎|H (𝜎; ̃ {(𝛾, g)}). We infer from (4) that 𝜆( g)m(𝛾) ∈ (ℳ 𝜎|H )𝜎̃ = ℂ ⋅ 1G . 𝜆 ( g)m(𝛾) ∈ ℳ 𝜎|H (𝜎; ̃ {(𝛾 −1 , g−1 )}), hence x𝜆 𝜆( g)m(𝛾))∗ . We have proved that Thus, there exists 𝜇 ∈ ℂ with x = 𝜇(𝜆 ℳ 𝜎|H (𝜎; ̃ {(𝛾, g)}) = ℂ ⋅ (𝜆( g)m(𝛾))∗ .

(7)

Using (5), (6), and (7), we obtain 𝜆( g)m(𝛾); (𝛾, g) ∈ H⟂ }. ℳ 𝜎|H = ℛ{𝜆

(8)

Equation (8) makes explicit the correspondence between the 𝜎-invariant von Neumann subalgebras ̂ of ℬ(ℒ 2 (G)) and the closed subgroups of G × G. On the other hand, by the definition of 𝜎 it is clear that 𝜆(h)m(𝜒); (h, 𝜒) ∈ H}′ , ℳ 𝜎|H = {𝜆

(9)

so that we obtain the following commutation result: ̂ a closed subgroup. Then Proposition. Let G be a locally compact abelian group and H ⊂ G × G 𝜆(h)m(𝜒); (h, 𝜒) ∈ H}′ = ℛ{𝜆 𝜆( g)m(𝛾); (𝛾, g) ∈ H⟂ }. ℛ{𝜆

(10)

In particular, let H1 , H2 be two closed subgroups of G and H = H1 ×H⟂2 . In this case, (10) becomes 𝜆(H1 ), m(H⟂2 )}′ = ℛ{𝜆 𝜆(H2 ), m(H⟂1 )}. ℛ{𝜆

(11)

Note that this contains in particular the commutation relations ℒ ∞ (G)′ = ℒ ∞ (G) ( f or H1 = H2 = {e}), 𝔏(G)′ = 𝔏(G)(= ℜ(G)) (for H1 = H2 = G), and ℬ(ℒ 2 (G)) = ℛ{𝔏(G), ℒ ∞ (G)} (for H1 = {e}, H2 = G). For every closed subgroup K ⊂ G, the set m(K⟂ ) ⊂ ℬ(ℒ 2 (G)) generates the von Neumann algebra consisting of operators of multiplication by those functions f ∈ ℒ ∞ (G), which are constant on the equivalence classes of G with respect to K; with an abuse of notation, we shall denote this von Neumann algebra by ℒ ∞ (G∕K). Then (11) can be written as follows: 𝜆(H1 ), ℒ ∞ (G∕H2 )}′ = ℛ{𝜆 𝜆(H2 ), ℒ ∞ (G∕H1 )}. ℛ{𝜆

(12)

In this form, the result can be extended to arbitrary locally compact groups (see Takesaki, 1969). 21.12. Finally, we mention without proof the following useful result: ∞ ̄ Proposition. Let ℳ be any W ∗ -algebra. Then every unitary cocycle u ∈ Z𝜄⊗Ad ̄ 𝜆 (ℝ; U(ℳ ⊗ ℒ ∞ ∗ ̄ ̄ 𝜆(t))](v ) (t ∈ ℝ). (ℝ))) is trivial, that is, there exists v ∈ U(ℳ ⊗ ℒ (ℝ)) such that u(t) = v[𝜄 ⊗Ad(𝜆 Note that, if ℳ = ℂ, the proposition follows easily using the Stone–von Neumann uniqueness theorem (see Loomis, 1952; Mackey, 1949) for the irreducible representation of the Heisenberg commutation relations 18.9.(1), in the case G = ℝ. Indeed, consider more generally a locally ̂ 𝜆(t), m(𝛾); t ∈ G, 𝛾 ∈ G} compact abelian group G and u ∈ ZAd 𝜆 (G; ℒ ∞ (G)). Then, both {𝜆

Discrete Groups

313

̂ are irreducible representation of the Heisenberg commutation 𝜆(t), m(𝛾); t ∈ G, 𝛾 ∈ G} and {u(t)𝜆 relations on ℒ 2 (G); hence (Loomis, 1952; Mackey, 1949) there exists a unitary operator v ∈ ̂ It follows that 𝜆(t)v∗ = u(t)𝜆 𝜆(t) (t ∈ G ), and vm(𝛾)v∗ = m(𝛾) (𝛾 ∈ G). ℬ(ℒ 2 (G)) such that v𝜆 ∞ 𝜆(t))](v∗ ) (t ∈ G ). v ∈ U(ℒ (G)) and u(t) = v[Ad(𝜆 For the general case, we refer to Moore (1976). 21.13. Notes. The main results in this section (21.1, 21.8, and 21.11) are due to Connes and Takesaki (1977). As mentioned in Section 21.6, Theorem 21.1 is an important tool for getting information about the center of the crossed product; this result replaces in the general case the previous result (Theorem 16.5) of Connes (1973a). The Galois correspondence type results originated in the works of Dye (1959, 1963); Nakamura and Takeda (1958, 1960). Theorem 21.8 completes the result of Proposition 21.5, which is due to Takesaki (1973b). Proposition 21.11 is an extension (in the commutative case) of the previous results on the same line obtained by Takesaki (1969) as a generalization of the Heisenberg commutation relation. Similar results concerning Galois correspondence and commutation relations in noncommutative settings are contained in Sections 22.5 and 22.9 and in Takesaki (1969). Further information is given in the survey article by Nakagami and Takesaki (1979). For our exposition, we have used Connes and Takesaki (1977) and Takesaki (1973b).

22 Discrete Groups In this section, we study crossed products by actions of discrete groups. The results obtained in this case are more precise, and their proofs simpler, than in the general case. Moreover, the first concrete examples of factors appeared as crossed products by discrete groups. 22.1. Let G be a discrete group. Consider the Haar measure on G, which assigns to each point of G the mass 1. Recall that all discrete groups are unimodular. For s, t ∈ G, we denote by 𝛿ts the Kronecker symbol, equal to 1 if s = t and equal to 0 if s ≠ t. The Dirac functions 𝛿t (s) = 𝛿ts (s, t ∈ G ) form an orthonormal basis {𝛿t ; t ∈ G} of the Hilbert space 𝓁 2 (G), and we have (𝜉|𝛿t ) = 𝜉(t) (𝜉 ∈ 𝓁 2 (G), t ∈ G ). ̄ ℬ(𝓁 2 (G)) ∋ X ↦ For any von Neumann algebra ℳ ⊂ ℬ(ℋ ), we obtain an identification ℳ ⊗ [Xs,t ] ↦ MatG (ℳ) via the matrix representation given by ̄ 𝛿t )](s) Xs,t 𝜉 = [X(𝜉 ⊗

(𝜉 ∈ ℋ , s, t ∈ G ).

In particular (if ℳ = ℂ), for X ∈ ℬ(𝓁 2 (G)), we have Xs,t = [X𝛿t ](s) = (X𝛿t |𝛿s )

(s, t ∈ G ).

Let 𝜎 ∶ G → Aut(ℳ) be an action of G on ℳ. Recall that, by definition, ℛ(ℳ, 𝜎) = ̄ 𝜆 (G)}. Thus, if a ∶ G ∋ g ↦ a( g) ∈ ℳ is a function such that the family ℛ{𝜋𝜎 (ℳ), 1ℳ ⊗ ̄ {𝜋𝜎 (a( g))(1 ⊗ 𝜆 ( g))}g∈G is w-summable, we obtain an element (compare with 18.21.(1)) Ta𝜎 =

∑ g∈G

̄ 𝜆 ( g)) ∈ ℛ(ℳ, 𝜎). 𝜋𝜎 (a( g))(1 ⊗

314

Crossed Products

In order to simplify the exposition, we shall in the sequel no longer mention explicitly the w-summability of the family defining Ta𝜎 ; however, this condition will always be satisfied, either by assumption or construction. ̄ ℬ(𝓁 2 (G)), we have It is easy to check that for every X ∈ ℳ ⊗ ̄ Ad(𝜌𝜌( g)))(X)]s,t = 𝜎g (Xsg,tg ) (s, t, g ∈ G ) (𝜎g ⊗ hence ̄ ℬ(𝓁 2 (G)))𝜎 ⊗̄ Ad 𝜌 = {X ∈ ℳ ⊗ ̄ ℬ(𝓁 2 (G)); 𝜎g (Xsg,tg ) = Xs,t (ℳ ⊗

(s, t, g ∈ G )}.

Theorem. Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the W ∗ -algebra ℳ. Then ̄ ℬ(𝓁 2 (G)))𝜎 ⊗̄ Ad 𝜌 . ℛ(ℳ, 𝜎) = {Ta𝜎 ; a ∶ G → ℳ} = (ℳ ⊗

(1)

̄ ℬ(𝓁 2 (G))) we have More precisely, for X ∈ ℋ ⊗ X ∈ ℛ(ℳ, 𝜎) ⇔ 𝜎g (Xsg,tg ) = Xs,t for all s, t, g ∈ G

(2)

and in this case there exists a unique function a ∶ G → ℳ such that X = Ta𝜎 , namely X = Ta𝜎 ⇔ a( g) = 𝜎g (Xg,e ) for all g ∈ G, X=

Ta𝜎

⇔ Xs,t =

𝜎s−1 (a(st−1 ))

(3)

for all s, t ∈ G.

(4)

In particular, [𝜋𝜎 (x)]s,t = 𝛿ts 𝜎s−1 (x) (x ∈ ℳ, s, t ∈ G ) ̄ 𝜆 ( g)]s,t = 𝛿 s (s, t, g ∈ G ). [1 ⊗ gt

(5) (6)

Proof. Consider ℳ ⊂ ℬ(ℋ ) realized as a von Neumann algebra. For x ∈ ℳ, g ∈ G, 𝜉 ∈ ℋ , and s, t ∈ G we have ̄ 𝛿t )](s) = 𝜎 −1 (x)[𝜉 ⊗ ̄ 𝛿t ](s) = 𝛿 s 𝜎 −1 (x)𝜉 𝜋𝜎 (x)]s,t 𝜉 = [𝜋𝜎 (x)(𝜉 ⊗ s t s ̄ 𝜆( g)]s,t 𝜉 = [(1 ⊗ ̄ 𝜆( g))(𝜉 ⊗ ̄ 𝛿t )](s) = [𝜉 ⊗ ̄ 𝛿t ]( g−1 s) = 𝛿tg s 𝜉 = 𝛿 s 𝜉, [1 ⊗ gt −1

proving (5) and (6). Then ̄ 𝜆( g))]s,t = [𝜋𝜎 (x)(1 ⊗𝜆

∑

̄ 𝜆( g)]r,t [𝜋𝜎 (x)]s,r [1 ⊗𝜆

r

=

∑

r −1 s −1 𝛿rs 𝛿gt 𝜎s (x) = 𝛿gt 𝜎s (x).

r

∑ s −1 Thus, if X = Ta for some function a ∶ G → ℳ, then Xs,t = g 𝛿gt 𝜎s (a( g)) = 𝜎s−1 (a(st−1 )) (s, t ∈ −1 (a(st−1 ))) = G ). In particular, Xg,e = 𝜎g−1 (a( g)), hence a( g) = 𝜎g (Xg,e ); and 𝜎g (Xsg,tg ) = 𝜎g (𝜎sg −1 −1 𝜎s (a(st )) = Xs,t ( g, s, t ∈ G ), so that ̄ ℬ(𝓁 2 (G)))𝜎 ⊗̄ Ad 𝜌 . {Ta𝜎 ; a ∶ G → ℳ} ⊂ ℛ(ℳ, 𝜎) ⊂ (ℳ ⊗

Discrete Groups

315

̄ ℬ(𝓁 2 (G)) be such that 𝜎g (Xsg,tg ) = Xs,t for all s, t, g ∈ G. Then we Conversely, let X ∈ ℳ ⊗ define a function a ∶ G → ℳ by putting a( g) = 𝜎g (Xg,e ) ( g ∈ G ), and we have 𝜎s−1 (a(st−1 )) = 𝜎s−1 ((𝜎st−1 (Xst−1 ,e )) = 𝜎t−1 (Xst−1 ,tt−1 ) = Xs,t (s, t ∈ G ), hence X = Ta𝜎 . The equality of the extreme terms in (1) has been proved also in the general case (19.13), but the possibility of writing every element X ∈ ℛ(ℳ, 𝜎) in the form X = Ta𝜎 occurs only in the case of discrete groups. Note that if a( g) = 𝛿ge 1ℳ ( g ∈ G ), then Ta = 1 ∈ ℛ(ℳ, 𝜎). 22.2. Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the W ∗ -algebra ℳ and let ̄ 𝔏(G) be the dual action (19.3). Note that in the discrete case the 𝜎̂ ∶ ℛ(ℳ, 𝜎) → ℛ(ℳ, 𝜎) ⊗ identity ℛ(ℳ, 𝜎)𝜎̂ = 𝜋𝜎 (ℳ) (19.3.(9)) has a simple proof based on Theorem 22.1. +

The dual action 𝜎̂ defines (19.7, 19.8) an n.s.f. operator-valued weight P𝜎̂ ∶ ℛ(ℳ, 𝜎)+ → 𝜋𝜎 (ℳ) such that for every finitely supported function a ∶ G → ℳ with Ta𝜎 ≥ 0 we have P𝜎̂ (Ta𝜎 ) = 𝜋𝜎 (a(e)).

(1)

In particular, we have P𝜎̂ (1) = 1, so that P𝜎̂ is a faithful normal conditional expectation of the crossed product ℛ(ℳ, 𝜎) onto the image 𝜋𝜎 (ℳ) of ℳ, and (1) can be extended by continuity to any function a ∶ G → ℳ defining an element of ℛ(ℳ, 𝜎). Thus using 22.1.(3), we get P𝜎̂ (X) = 𝜋𝜎 (Xe,e ) (X ∈ ℛ(ℳ, 𝜎)).

(2)

Actually, it is easy to check directly that (1) or (2) define a faithful normal conditional expectation of ℛ(ℳ, 𝜎) onto 𝜋𝜎 (ℳ) and that (19.2.(3)) ̄ 𝜆 ( g))X(1 ⊗ ̄ 𝜆 ( g))∗ ) = (1 ⊗ ̄ 𝜆 ( g))P𝜎̂ (X)(1 ⊗ ̄ 𝜆 ( g))∗ P𝜎̂ ((1 ⊗

(3)

for every X ∈ ℛ(ℳ, 𝜎) and g ∈ G. Moreover, from (1) and (2) it follows that ̄ 𝜆 ( g)) = 0 (e ≠ g ∈ G ) P𝜎̂ (1 ⊗ P𝜎̂ (𝜋𝜎 (x)) = 𝜋𝜎 (x) (x ∈ ℳ) and for every element X ∈ ℛ(ℳ, 𝜎) we have (compare with 19.12.(2)) ∑ ̄ 𝜆 ( g−1 )))(1 ⊗ ̄ 𝜆 ( g)) X= P𝜎̂ (X(1 ⊗

(4) (5)

(6)

g

=

∑

̄ 𝜆 ( g)))(1 ⊗ ̄ 𝜆 ( g−1 )). P𝜎̂ (X(1 ⊗

g

The next proposition gives us a characterization of crossed products by discrete groups, which is different from Landstad’s theorem. Proposition. Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the countably decomposable W ∗ -algebra ℳ. Let 𝒩 be a W ∗ -algebra with the property that there exist: – an injective unital normal *-homomorphism 𝜋 ∶ ℳ → 𝒩 , – a faithful normal conditional expectation P ∶ 𝒩 → 𝜋(ℳ), – a unitary representation u ∶ G → 𝒩 ,

316

Crossed Products

such that (a) 𝒩 = ℛ{𝜋(ℳ), u(G)}, (b) 𝜋(𝜎g (x)) = u( g)𝜋(x)u( g)∗ for all x ∈ ℳ and g ∈ G, (c) P(u( g)) = 0 for all e ≠ g ∈ G. Then there exists a *-isomorphism Φ ∶ 𝒩 → ℛ(ℳ, 𝜎) such that Φ(𝜋(x)) = 𝜋𝜎 (x) (x ∈ ℳ) ̄ 𝜆 ( g) ( g ∈ G ) Φ(u( g)) = 1 ⊗ Φ(P(X)) = P𝜎̂ (Φ(X)) (X ∈ 𝒩 ).

(1) (2) (3)

Proof. Since ℳ is countably decomposable, there exists a faithful normal state 𝜑 ∈ ℳ∗ . Then 𝜓 = 𝜑 ◦ 𝜋 −1 ◦ P is a faithful normal state on 𝒩 . Let 𝜋𝜓 ∶ 𝒩 → ℬ(ℋ𝜓 ) be the GNS representation associated with 𝜓, with cyclic vector 𝜉𝜓 ∈ ℋ𝜓 . For x, y ∈ ℳ and s, t ∈ ℳ, we have (𝜋𝜓 (𝜋(x)u(s))𝜉𝜓 |𝜋𝜓 (𝜋( y)u(t))𝜉𝜓 ) = (𝜋𝜓 (u(t)∗ 𝜋( y∗ x)u(s))𝜉𝜓 |𝜉𝜓 ) = (𝜋𝜓 (u(t−1 s)𝜋(𝜎s−1 ( y∗ x))𝜉𝜓 |𝜉𝜓 ) = 𝜓(u(t−1 s)𝜋(𝜎s−1 ( y∗ x))) = (𝜑 ◦ 𝜋 −1 )(P(u(t−1 s)𝜋(𝜎s−1 ( y∗ x))) = 𝛿ts 𝜑(𝜎s−1 ( y∗ x)). ̄ 𝜆 ) and Consider now (𝒩1 , 𝜋1 , P1 , u1 ) = (𝒩 , 𝜋, P, u) and (𝒩2 , 𝜋2 , P2 , u2 ) = (R(ℳ, 𝜎), 𝜋𝜎 , P𝜎̂ , 1 ⊗ the corresponding states 𝜓1 and 𝜓2 . The previous computation shows that ‖2 ‖2 ‖∑ ‖∑ ‖ n ‖ ‖ ‖ n ‖ 𝜋𝜓 (𝜋1 (xj )u1 (sj ))𝜉𝜓 ‖ = ‖ 𝜋𝜓 (𝜋2 (xj )u2 (sj ))𝜉𝜓 ‖ ‖ ‖ 1‖ 2 2‖ 1 ‖ ‖ j=1 ‖ ‖ j=1 ‖ ‖ ‖ ‖ for every x1 , … , xn ∈ ℳ and every s1 , … , sn ∈ G. Since 𝒩k = ℛ{𝜋k (ℳ), uk (G)} and 𝜋𝜓k (𝒩k )𝜉𝜓k = ℋ𝜓k , it follows that there exists a unitary operator V ∶ ℋ𝜓1 → ℋ𝜓2 such that V𝜋𝜓1 (𝜋1 (x)u1 (s))𝜉𝜓1 = 𝜋𝜓2 (𝜋2 (x)u2 (s))𝜉𝜓2

(x ∈ ℳ, s ∈ G ).

It is then easy to check (on vectors of the form 𝜋𝜓2 (𝜋2 ( y)u2 (t))𝜉𝜓2 with y ∈ ℳ, t ∈ G) that V𝜋𝜓1 (𝜋1 (x))V ∗ = 𝜋𝜓2 (𝜋2 (x))

(x ∈ ℳ)

V𝜋𝜓1 (u1 ( g))V = 𝜋𝜓2 (u2 ( g))

( g ∈ G ).

∗

Thus the mapping Φ ∶ 𝒩 = 𝒩1 ∋ X ↦ 𝜋𝜓−1 (V𝜋𝜓1 (X)V ∗ ) ∈ 𝒩2 = ℛ(ℳ, 𝜎) is the required 2 *-isomorphism. Note that if in Proposition 22.2 the action 𝜎 is assumed to be properly outer, then condition (c) can be omitted from the statement, since it is automatically satisfied. Indeed, from (b) it follows that 𝜋(𝜎g (x))u( g) = u( g)𝜋(x); hence 𝜋(𝜎g (x)) P(u( g)) = P(u( g))𝜋(x), so that either g = e, or P(u( g)) = 0, since the action 𝜎 is properly outer (see 17.6 and 17.4.(2)).

Discrete Groups

317

22.3. Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the W ∗ -algebra ℳ. By the last remark of Section 17.4, 𝜎 is properly outer if and only if a ∈ ℳ, g ∈ G, xa = a𝜎g (x) for all x ∈ ℳ ⇔ g = e or a = 0. We shall say that 𝜎 acts freely on 𝒵 (ℳ) if each 𝜎g (e ≠ g ∈ G ), acts freely on 𝒵 (ℳ) (17.5). By Proposition 17.5, 𝜎 acts freely on 𝒵 (ℳ) if and only if a ∈ ℳ, g ∈ G, za = a𝜎g (z) for all z ∈ 𝒵 (ℳ) ⇔ g = e or a = 0. Theorem (Relative Commutant Theorem). Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the W ∗ -algebra ℳ. Then 𝜎 is properly outer ⇔ 𝜋𝜎 (ℳ ′ ) ∩ ℛ(ℳ, 𝜎) = 𝜋𝜎 (𝒵 (ℳ)), 𝜎 acts freely on 𝒵 (ℳ) ⇔ 𝜋𝜎 (𝒵 (ℳ))′ ∩ ℛ(ℳ, 𝜎) = 𝜋𝜎 (ℳ). ∑ ̄ 𝜆 ( g)) ∈ ℛ(ℳ, 𝜎) and x ∈ ℳ. Then 𝜋𝜎 (x)X Proof. Let X = g 𝜋𝜎 (a( g))(1 ⊗ ∑ ∑ ̄ ̄ g 𝜋𝜎 (xa( g))(1 ⊗ 𝜆 ( g)) and X𝜋𝜎 (x) = g 𝜋𝜎 (a( g)𝜎g (x))(1 ⊗ 𝜆 ( g)) so that

(1) (2) =

X ∈ 𝜋𝜎 (ℳ)′ ⇔ xa( g) = a( g)𝜎g (x) for all x ∈ ℳ, g ∈ G,

(3)

X ∈ 𝜋𝜎 (𝒵 (ℳ)) ⇔ za( g) = a( g)𝜎g (z) for all z ∈ 𝒵 (ℳ), g ∈ G,

(4)

′

from which (1) and (2) follow immediately. Note that if 𝜎 acts freely on 𝒵 (ℳ), then both (1) and (2) hold, so that (𝜋𝜎 (ℳ)′ ∩ ℛ(ℳ, 𝜎))′ ∩ ℛ(ℳ, 𝜎) = 𝜋𝜎 (ℳ), (𝜋𝜎 (𝒵 (ℳ))′ ∩ ℛ(ℳ, 𝜎))′ ∩ ℛ(ℳ, 𝜎) = 𝜋𝜎 (𝒵 (ℳ)).

(5) (6)

22.4. Consider a W ∗ -algebra 𝒩 and a unital W ∗ -subalgebra ℳ of 𝒩 such that ℳ ′ ∩ 𝒩 ⊂ ℳ and such that there exists a normal conditional expectation P ∶ 𝒩 → ℳ. By Proposition 10.17, P is uniquely determined and faithful and for the normalizer  (P) of P we have  (P) = {v ∈ U(𝒩 ); P ◦ Ad(v) = Ad(v) ◦ P} = {v ∈ U(𝒩 ); vℳv∗ = ℳ}. Lemma. For every v ∈  (P), we have vP(v∗ ) = P(v)v∗ = P(v)P(v∗ ) = P(v∗ )P(v) = v∗ P(v) = P(v∗ )v = p(Ad(v)|ℳ) (see 17.2). Proof. Indeed, we have vP(v∗ )v∗ = P(vv∗ v∗ ) = P(v∗ ), hence vP(v∗ ) = P(v∗ )v. Similarly, P(v)v∗ = v∗ P(v). For x ∈ ℳ, we have vxv∗ ∈ ℳ, hence P(v)x = P(vx) = P(vxv∗ v) = vxv∗ P(v), so that v∗ P(v) ∈ ℳ ′ ∩ 𝒩 ⊂ ℳ. Therefore, v∗ P(v) = P(v∗ P(v)) = P(v∗ )P(v). Replacing v here by v∗ we get vP(v∗ ) = P(v)P(v∗ ). Finally, applying P to the identity P(v)v∗ = v∗ P(v), we obtain P(v)P(v∗ ) = P(v∗ )P(v). Thus, the first six terms appearing in the statement are all equal to some element p ∈ ℳ ′ ∩𝒩 = 𝒵 (ℳ), which is a projection since p∗ = (P(v)v∗ )∗ = vP(v∗ ) = p and p2 = v∗ P(v)vP(v∗ ) = P(v∗ vv)P(v∗ ) = P(v)P(v∗ ) = p.

318

Crossed Products

On the other hand, we have [Ad(v)](p) = vpv∗ = vv∗ P(v)v∗ = vP(v∗ ) = p. Since P(v)P(v)∗ = p = P(v)∗ P(v), P(v) is a unitary element in ℳp and for every x ∈ ℳp we have [Ad(v)](x) = vxv∗ = vpxpv∗ = vv∗ P(v)xP(v∗ )vv∗ = P(v)xP(v)∗ . Thus, p ≤ p(Ad(v)|ℳ). Let q = p(Ad(v)|ℳ) − p and consider a unitary element u in ℳq such that [Ad(v)](x) = uxu∗ for all x ∈ ℳq. Then v∗ u ∈ 𝒩 , u∗ vv∗ u = u∗ u = q, v∗ uu∗ v = v∗ qv = q and v∗ uxu∗ v = x for all x ∈ ℳq. Since q ∈ 𝒵 (ℳ), it follows that v∗ u ∈ ℳ ′ ∩ 𝒩 = 𝒵 (ℳ), so that there exists a unitary element z ∈ 𝒵 (ℳq) such that u = vz; we have q = uu∗ = P(u)P(u∗ ) = P(v)zz∗ P(v∗ ) = qP(v)P(v∗ ) = qp = 0. Recall (17.3) that [G] denotes the full group associated with a group G ⊂ Aut(ℳ). Proposition. Let 𝒩 be a W ∗ -algebra and ℳ ⊂ 𝒩 a unital W ∗ -subalgebra such that ℳ ′ ∩ 𝒩 ⊂ ℳ and such that there exists a normal conditional expectation P ∶ 𝒩 → ℳ. Let 𝒢 ⊂  (P) be a subgroup such that 𝒩 = ℛ{ℳ, 𝒢 }. Then {Ad(v)|ℳ; v ∈  (P)} = {Ad(w)|ℳ; w ∈ 𝒢 }. Proof. Let v ∈  (P). From the previous lemma it follows that p(Ad(w∗ v)|ℳ) = s(P(w∗ v))

(w ∈ 𝒢 ).

Thus, in order to prove that Ad(v)|ℳ ∈ [Ad(𝒢 )|ℳ], it is sufficient to show that for every nonzero projection q ∈ 𝒵 (ℳ) there exists w ∈ 𝒢 such that qP(w∗ v) ≠ 0. Let q ∈ 𝒵 (ℳ) be a nonzero projection. Then 0 ≠ q = qP(v∗ v). Since 𝒩 = ℛ{ℳ, 𝒢 }, there exist a ∈ ℳ, w ∈ 𝒢 such that 0 ≠ qP(aw∗ v) = qaP(w∗ v), hence qP(w∗ v) ≠ 0. Conversely, let 𝜎 ∈ [Ad(𝒢 )|ℳ] ⊂ Aut(ℳ). By Section 17.3, there exists a family {(qi , ui , wi )}i∈I , ∑ where qi ∈ 𝒵 (ℳ) are projections, i qi = 1, ui ∈ ℳ, u∗i ui = ui u∗i = qi , wi ∈ 𝒢 , 𝜎(qi ) = wi qi w∗i , ∑ ∑ and 𝜎(x) = i wi ui xu∗i w∗i for all x ∈ ℳ. Then v = i wi ui ∈ U(𝒩 ) and 𝜎 = Ad(v)|ℳ, hence v ∈ 𝒩 (P). Thus, in the setting of the proposition, for every v ∈  (P) there exists a family {(qi , ui , wi )}i∈I , ∑ where the qi ∈ 𝒵 (ℳ) are projections, i qi = 1, ui ∈ ℳ, u∗i ui = ui u∗i = qi , wi ∈ 𝒢 , vqi v∗ = wi qi w∗i , ∑ and vxv∗ = i wi ui xu∗i w∗i for all x ∈ ℳ. If x ∈ ℳqi , then vxv∗ = wi ui xu∗i w∗i , and v∗ wi ui ∈ (ℳqi )′ ∩ 𝒩 . It follows that v∗ wi ui ∈ 𝒵 (ℳ)qi so that, multiplying ui by some unitary element of ∑ 𝒵 (ℳ)qi if necessary, we may assume that v∗ wi ui = qi . Since i qi = 1, it follows that v=

∑ i

wi ui with wi ∈ 𝒢 , ui ∈ U(𝒵 (ℳ)qi ),

∑

qi = 1, vqi = wi ui .

(1)

i

In particular, let 𝜎 ∶ G → Aut(ℳ) be a properly outer action of the discrete group G on the W ∗ -algebra ℳ. Then the conclusion of the previous proposition is valid for 𝒩 = ℛ(ℳ, 𝜎), ℳ = ̄ 𝜆( g); g ∈ G} ⊂  (P𝜎̂ ) (see 22.2 and 22.3), and can be formulated 𝜋𝜎 (ℳ) ⊂ 𝒩 , P = P𝜎̂ , 𝒢 = {1 ⊗ as follows: [𝜎(G)] = {𝛼 ∈ Aut(ℳ); 𝛼 extends to an inner *-automorphism of ℛ(ℳ, 𝜎)}. 22.5. The next result concerns the structure of certain “intermediate subalgebras” of the crossed product (compare with Theorem 21.8). Proposition. Let 𝜎 ∶ G → Aut(ℳ) be a (properly) outer action of the discrete group G on the factor ℳ. If 𝒩 is a unital W ∗ -subalgebra of ℛ(ℳ, 𝜎) containing 𝜋𝜎 (ℳ) and there exists a normal conditional expectation P𝒩 ∶ ℛ(ℳ, 𝜎) → 𝒩 , then ̄ 𝜆( g) ∈ 𝒩 } H = {g ∈ G; 1 ⊗

(1)

Discrete Groups

319

is a subgroup of G, ̄ 𝜆 (H)} ≈ ℛ(ℳ, 𝜎|H) 𝒩 = ℛ{𝜋𝜎 (ℳ), 1 ⊗ ̄ 𝜆 ( g)) = 0 for all g ∈ G ⧵ H P𝒩 (1 ⊗

(2) (3)

̄ 𝜆 ( g) and identify ℳ = 𝜋𝜎 (ℳ) ⊂ 𝒩 . Proof. We put u( g) = 1 ⊗ We first prove (3). For g ∈ G and x ∈ ℳ we have xP𝒩 (u( g)) = P𝒩 (xu( g)) = P𝒩 (u( g)u( g)∗ xu( g)) = P𝒩 (u( g))u( g)∗ xu( g), hence P𝒩 (u( g))u( g)∗ ∈ ℳ ′ ∩ ℛ(ℳ, 𝜎). Since 𝜎 is properly outer, it follows that P𝒩 (u( g))u( g)∗ ∈ 𝒵 (ℳ) (22.3). Since ℳ is a factor, we deduce also that P𝒩 (u( g)) = 𝜆u( g) for some 𝜆 ∈ ℂ. If 𝜆 ≠ 0, then u( g) ∈ 𝒩 , that is, g ∈ H. Thus, P𝒩 (u( g)) = 0 for all g ∈ G ⧵ H. ∑ Consider now X ∈ 𝒩 ⊂ ℛ(ℳ, 𝜎), X = g a( g)u( g) (22.1). Then, using (3), we get X = P𝒩 (X) = ∑ ∑ g a( g)P𝒩 (u( g)) = g a( g)u( g) ∈ ℛ{ℳ, u(H)}, proving (2). 22.6. We now study the center of the crossed product. Theorem. Let 𝜎 ∶ G → Aut(ℳ) be an action of the discrete group G on the W ∗ -algebra ℳ and let ∑ ̄ 𝜆 ( g)) ∈ ℛ(ℳ, 𝜎) an arbitrary element of the crossed product (a( g) ∈ ℳ X = g 𝜋𝜎 (a( g))(1 ⊗ for all g ∈ G). Then we have X ∈ 𝒵 (ℛ(ℳ, 𝜎)) if and only if the following conditions are satisfied: (x ∈ ℳ, g ∈ G )

a( g)𝜎g (x) = xa( g)

(1)

𝜎s (a( g)) = a(sgs ) (s, g ∈ G ).

(2)

a(e) ∈ 𝒵 (ℳ)𝜎 .

(3)

−1

In this case

Proof. By 22.3.(3), (1) means that X ∈ 𝜋𝜎 (ℳ)′ . On the other hand, it is easy to check that (2) ̄ 𝜆 (s) (s ∈ G ). Finally, (3) follows is equivalent to the statement that X commutes with all 1 ⊗ immediately from (1) and (2), with g = e. Corollary 1. If the action 𝜎 ∶ G → Aut(ℳ) of the discrete group G on the W ∗ -algebra ℳ is properly outer and its restriction to 𝒵 (ℳ) is ergodic, then the crossed product ℛ(ℳ, 𝜎) is a factor. In particular, if ℳ is a factor, then the ergodicity condition 𝒵 (ℳ)𝜎 = ℂ ⋅ 1ℳ 1s automatically satisfied. Also, if G is commutative, we can also deduce the discrete case of Corollary 21.6 from the above Theorem. Indeed, from (2) it follows that a( g) ∈ ℳ 𝜎 for all g ∈ G and using (1) we deduce also that a( g) ∈ 𝒵 (ℳ 𝜎 ). Thus, if ℳ 𝜎 is a factor, then a( g) are all scalars. Furthermore, if a( g) ≠ 0, then it follows from (1) that 𝜎g = 𝜄. Consequently, if 𝜎g ≠ 𝜄 for all g ≠ e, then a( g) = 0 for all g ≠ e. If ℳ = ℂ and 𝜎 ∶ G → Aut(ℳ) is the trivial action, then ℛ(ℳ, 𝜎) = 𝔏(G). In this case an ∑ 𝜆( g) ∈ 𝔏(G) belongs to 𝒵 (𝔏(G)) if and only if 𝛼( g) = 𝛼(sgs−1 ) for all element X = g 𝛼( g)𝜆 s, g ∈ G. However, the function 𝛼 ∶ G → ℂ is not arbitrary, as X ∈ ℬ(𝓁 2 (G)). Thus, X𝛿e ∈ 𝓁 2 (G), that is, ∑ ∑ |𝛼( g)|2 = |(X𝛿e )( g)| < +∞. (4) g

g

320

Crossed Products

The discrete group G will be called an ICC-group if for every g ∈ G, g ≠ e, the conjugacy class {sgs−1 ; s ∈ G} is an infinite set. The above discussion shows Corollary 2. Let G be a discrete group. Then 𝔏(G) is a factor if and only if G is an ICC-group, or G = {e}. It is clear that the free group Fk with k generators (2 ≤ k ≤ ∞) is an ICC-group and hence that 𝔏(Fk ) is a factor. Also, it is easy to check that the group S(∞) of all bijections 𝛾 ∶ ℕ → ℕ with the property that 𝛾(n) = n for sufficiently large n ∈ ℕ, is also an ICC-group, so that 𝔏(S(∞)) is a factor. In order to extend the result of Corollary 2, let us now consider an arbitrary action 𝜎 ∶ G → Aut(ℳ) ∑ ̄ 𝜆 ( g)) ∈ of the discrete group G on the von Neumann algebra ℳ ⊂ ℬ(ℋ ). Let X = g 𝜋𝜎 (a( g))(1 ⊗ 2 −1 ̄ ℛ(ℳ, 𝜎) ⊂ ℬ(𝓁 (G, ℋ )) and 𝜉 ∈ ℋ . We have [X(𝜉 ⊗ 𝛿e ))( g) = Xg,e 𝜉 = 𝜎g (a( g))𝜉, hence ∑

̄ 𝛿e )‖2 < +∞. ‖𝜎g−1 (a( g))𝜉‖2 = ‖X(𝜉 ⊗

g

Furthermore, assuming that ‖𝜎t (x)𝜉‖ = ‖x𝜉‖ (x ∈ ℳ, t ∈ G ), the above inequality becomes ∑

‖a( g)𝜉‖2 < +∞,

g

and if X ∈ 𝒵 (ℛ(ℳ, 𝜎)), then (2) shows that ‖a( g)𝜉‖ = ‖a(sgs−1 )𝜉‖

(s, g ∈ G ).

If G is an ICC-group, then we conclude that X ∈ 𝒵 (ℛ(ℳ, 𝜎)) ⇒ a( g)𝜉 = 0 for all e ≠ g ∈ G. If, moreover, 𝜉 ∈ ℋ is a separating vector for ℳ ⊂ ℬ(ℋ ), this means that X ∈ 𝒵 (ℛ(ℳ, 𝜎)) ⇒ a( g) = 0 for all e ≠ g ∈ G. Finally, if 𝜎 is ergodic on 𝒵 (ℳ), then we infer from (3) that a(e) ∈ ℂ ⋅ 1ℳ . We have thus proved that if G is an ICC-group acting ergodically on 𝒵 (ℳ) and if there exists a separating vector 𝜉 ∈ ℋ for ℳ such that ‖𝜎t (x)𝜉‖ = ‖x𝜉‖ (x ∈ ℳ, t ∈ G ), then ℛ(ℳ, 𝜎) is a factor. Using the GNS-representation, this result can be formulated as follows: Corollary 3. Let 𝜎 ∶ G → Aut(ℳ) be an action of the ICC-group G on the W ∗ -algebra ℳ whose restriction to 𝒵 (ℳ) is ergodic and such that there exists a 𝜎-invariant faithful normal state on ℳ. Then ℛ(ℳ, 𝜎) is a factor. 22.7. We now study the type of the crossed product. Let 𝜎 ∶ G → Aut(ℳ) be an action of the ̄ 𝜆 ( g) ( g ∈ G ), and we identify ℳ = discrete group G on the W ∗ -algebra ℳ. We put u( g) = 1 ⊗ ∑ 𝜋𝜎 (ℳ) ⊂ ℛ(ℳ, 𝜎). Thus, an arbitrary element of ℛ(ℳ, 𝜎) is of the form X = g a( g)u( g) with a( g) ∈ ℳ ( g ∈ G ), and the conditional expectation P = P𝜎̂ of ℛ(ℳ, 𝜎) onto ℳ is defined by P(X) = a(e).

Discrete Groups

321

Theorem 1. The crossed product ℛ(ℳ, 𝜎) is a finite W ∗ -algebra if and only if there exists a separating family of 𝜎-invariant finite normal traces on ℳ. Proof. Assume that ℛ(ℋ , 𝜎) is finite. Let a ∈ ℳ + , a ≠ 0. There exists a finite normal trace 𝜇 on ℛ(ℳ, 𝜎) with 𝜇(a) ≠ 0. Then 𝜏 = 𝜇|ℳ is a finite normal trace on ℳ and 𝜏(a) ≠ 0. For x ∈ ℳ, g ∈ G, we have 𝜏(𝜎g (x)) = 𝜇(𝜎g (x)) = 𝜇(u( g)xu( g)∗ ) = 𝜇(x) = 𝜏(x), hence 𝜏 is 𝜎-invariant. Conversely, let 𝜏 be a 𝜎-invariant finite normal trace on ℳ. Then 𝜇 = 𝜏 ◦ P is a positive ∑ normal linear form on ℛ(ℳ, 𝜎) with s(𝜇) = s(𝜏). For X = g a( g)u( g) ∈ ℛ(ℳ, 𝜎), ∑ ∑ ∑ ∑ ∗ −1 (a(s)∗ a(sg)))u( g), XX∗ gs)𝜎 = a( we have X∗ X = ( 𝜎 ( g (a(s) ))u( g), so that g s s g s ∑ ∑ ∑ P(X∗ X) = s 𝜎s−1 (a(s)∗ a(s)), P(XX∗ ) = s a(s)a(s)∗ and hence 𝜇(X∗ X) = s 𝜏(𝜎s−1 (a(s)∗ (a(s)) = ∑ ∑ ∗ ∗ ∗ s 𝜏(a(s) (a(s))) = s 𝜏(a(s)a(s) ) = 𝜇(XX ), that is, 𝜇 is a trace. Thus, if G is a discrete ICC-group, then 𝔏(G) is a finite infinite-dimensional factor, that is a factor of type II1 . Theorem 2. If there exists a 𝜎-invariant n.s.f. trace on ℳ, then the crossed product ℛ(ℳ, 𝜎) is semifinite. Conversely, if the crossed product ℛ(ℳ, 𝜎) is semifinite and 𝜎 acts freely on 𝒵 (ℳ), then there exists a 𝜎-invariant n.s.f. trace on ℳ. Proof. The first assertion is proved with the same arguments as in Theorem 1. If ℛ(ℳ, 𝜎) is semifinite, then there exists an n.s.f. trace 𝜇 on ℛ(ℳ, 𝜎). As in the proof of Theorem 1 one can show that 𝜏 = 𝜇|ℳ is a 𝜎-invariant faithful normal trace on ℳ. Since 𝜇 is semifinite, there exists a w w net {xi } ⊂ ℛ(ℳ, 𝜎) such that 𝜇(x∗i xi ) < +∞ and xi → 1. Then ai = P(xi ) ∈ ℳ and ai → P(1) = 1. Thus, in order to prove that 𝜏 is semifinite, it is sufficient to show that x ∈ ℛ(ℳ, 𝜎), 𝜇(x∗ x) < +∞ ⇒ 𝜇(P(x)∗ P(x)) < +∞. w

By Proposition 10.14, there exists a ∈ co {uxu∗ ; 𝜎 ∈ U(𝒵 (ℳ))} ∩ 𝒵 (ℳ)′ . Since 𝜎 acts freely on 𝒵 (ℳ), it follows by Theorem 22.3.(2) that a ∈ ℳ, hence P(a) = a. For u ∈ U(𝒵 (ℳ)), we have P(uxu∗ ) = uP(x)u∗ = P(x), hence a = P(a) = P(x). Using Proposition 10.14 again, we conclude that 𝜇(a∗ a) < +∞, that is, 𝜇(P(x)∗ P(x)) < +∞. 22.8. In this section, we consider crossed products of abelian W ∗ -algebras by actions of countable discrete groups and some concrete examples. Let Ω be a locally compact Hausdorff topological space with a countable basis of open sets and 𝜇 a sigma-finite positive Borel measure on Ω. Consider the Hilbert space ℋ = ℒ 2 (Ω, 𝜇) and the maximal abelian von Neumann algebra ℳ = ℒ ∞ (Ω, 𝜇) ⊂ ℬ(ℋ ). Let G be a countable discrete group of homeomorphisms of Ω; for each g ∈ G we shall denote by Tg ∶ Ω → Ω the corresponding homeomorphism of Ω. We shall assume that the measure 𝜇 is G-quasi-invariant, that is, for every g ∈ G the measures 𝜇 and 𝜇 ◦ Tg are equivalent in the sense of absolute continuity. In this case, it is clear that the formula 𝜎g (x) = x ◦ Tg−1 (x ∈ ℳ, g ∈ G ) defines an action 𝜎 ∶ G → Aut(ℳ) of G on ℳ. We shall say that G acts (almost) freely on (Ω, 𝜇) if for every g ∈ G, g ≠ e, we have 𝜇({𝜔 ∈ Ω; Tg 𝜔 = 𝜔}) = 0. In this case, 𝜎 also acts freely on ℳ. Indeed, let g ∈ G, g ≠ e, and let E ⊂ Ω be a 𝜇-measurable set with 𝜇(E) > 0; we have to show that there exists a 𝜇-measurable set F ⊂ E with 𝜇(F) > 0 such that F ∩ Tg F = ∅ (see 17.5). Since G acts freely on (Ω, 𝜇), we may assume that

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Tg 𝜔 ≠ 𝜔 for all 𝜔 ∈ Ω. Then, for every 𝜔 ∈ E there exists a compact neighborhood V(𝜔) of 𝜔 such that V(𝜔) ∩ Tg V(𝜔) = ∅. We have W(𝜔) = V(𝜔) ∩ E ⊂ E and W(𝜔) ∩ Tg W(𝜔) = ∅. Since Ω has a countable basis of open sets and 𝜇(E) > 0, there exists 𝜔0 ∈ E with 𝜇(W(𝜔0 )) > 0 and we can take F = W(𝜔0 ). We shall say that G acts ergodically on (Ω, 𝜇) if the only 𝜇-measurable sets E ⊂ Ω such that 𝜇((E ∪ Tg E)∖(E ∩ Tg E)) = 0 for all g ∈ G satisfy either 𝜇(E) = 0 or 𝜇(Ω∖E) = 0. In this case it is easy to check that also 𝜎 is ergodic, that is, ℳ 𝜎 = ℂ ⋅ 1ℳ . Thus, according to Corollary 1/22.6, if G acts freely and ergodically on (Ω, 𝜇), then ℛ(ℳ, 𝜎) is a factor. In the sequel, we assume that these conditions are satisfied. We shall say that G is 𝜇-measurable, if there exists a sigma-finite positive Borel measure ν on Ω, equivalent to 𝜇 and G-invariant, that is, ν ◦ Tg = ν ( g ∈ G ). Since G acts ergodically on (Ω, 𝜇), the measure ν with the previous properties is unique up to a multiplicative scalar factor. The measure ν defines a 𝜎-invariant n.s.f. trace 𝜏 on ℳ by 𝜏(x) = ∫ x dν (x ∈ ℳ + ), and any 𝜎-invariant n.s.f. trace 𝜏 on ℳ is of this form; clearly, the trace 𝜏 is finite if and only if the measure ν is finite. Since G acts freely on (Ω, 𝜇), it follows according to Theorem 2/22.7 that the factor ℛ(ℳ, 𝜎) is semifinite if and only if G is 𝜇-measurable; moreover, according to Theorem 1/22.7, the factor ℛ(ℳ, 𝜎) is finite if and only if the measure ν is finite. Thus, if G acts freely and ergodically on (Ω, 𝜇) and is not 𝜇-measurable, then ℛ(ℳ, 𝜎) is a type III factor. In order to give a concrete example of a type III factor with a separable predual, we first notice the following simple fact: if the subgroup G0 = {g ∈ G; 𝜇 ◦ Tg = 𝜇} ⊂ G acts ergodically on (Ω, 𝜇) and G0 ≠ G, then G is not 𝜇-measurable. Indeed, assume to the contrary, that is, there exists a G-invariant sigma-finite positive Bore measure ν on Ω, equivalent to 𝜇, and consider the Radon– Nikodym derivative f = d𝜇∕dν. Since for g ∈ G0 we have f ◦ Tg = d(𝜇 ◦ Tg )∕dν = d𝜇∕dν = f and since G0 acts ergodically on (Ω, 𝜇), it follows that the function f is constant and hence the measure 𝜇 is G-invariant, contradicting G0 ≠ G. Consider now Ω = ℝ, 𝜇 = the Lebesgue measure on ℝ and the countable group G of homeomorphisms of ℝ consisting of the transformations T(𝛼, 𝛽) ∶ ℝ ∋ 𝜔 ↦ 𝛼𝜔 + 𝛽, where 𝛼, 𝛽 are rational numbers and 𝛼 > 0. Then the measure 𝜇 is G-quasi-invariant, G acts freely and G0 = {g ∈ G; 𝜇 ◦ Tg = 𝜇} = {T(𝛼, 𝛽); 𝛼 = 1} (≠ G ) acts ergodically, hence G is not 𝜇-measurable. Thus, the crossed product ℛ(ℳ, 𝜎) is in this case a concrete example of a type III factors with separable predual. This was the first example of a factor of type III due to Murray and von Neumann. It is actually a factor of type III1 . If the group G acts freely and ergodically on (Ω, 𝜇), is 𝜇-measurable, and if the G-invariant measure ν equivalent to 𝜇 is diffuse, that is, ν({𝜔}) = 0 for every 𝜔 ∈ Ω, then the factor ℛ(ℳ, 𝜎) is of type II. Indeed, in this case there exists a decreasing sequence {En }n≥1 of 𝜇-measurable sets in Ω with ν(En ) > ν(En+1 ), (n ≥ 1), and limn ν(En ) = 0. Let 𝜏 be the 𝜎-invariant n.s.f. trace on ℳ defined by ν and let 𝜏̃ = 𝜏 ◦ P be the corresponding n.s.f. trace on ℛ(ℳ, 𝜎). Then en = 𝜒En ∈ ℳ ⊂ ℛ(ℳ, 𝜎) is a decreasing sequence of non-zero projections in ℛ(ℳ, 𝜎) and 𝜏(e ̃ n ) = ν(En ) → 0 and so the factor ℛ(ℳ, 𝜎) is of type II. For instance, consider Ω = {𝜔 ∈ ℂ; |𝜔| = 1} the one-dimensional torus, let 𝜇 = the Haar measure on Ω with 𝜇(Ω) = 1 and let G be any dense countable infinite subgroup of Ω, which

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acts on Ω by translation, Tg 𝜔 = g𝜔 ( g ∈ G, 𝜔 ∈ Ω) (e.g., G = {exp(2𝜋it); t ∈ ℝ∖ℚ} or G = {exp(2𝜋int0 ); n ∈ ℤ} with t0 ∈ ℝ a fixed irrational number). Then the Haar measure 𝜇 is finite, diffuse and G-invariant, and G acts freely on (Ω, 𝜇). We shall show that G acts ergodically on (Ω, 𝜇), so that ℛ(ℳ, 𝜎) is in this case a type II1 factor. Indeed, let E ⊂ Ω be a 𝜇-measurable set such that 𝜇((E∪Tg E)∖(E∩Tg E)) = 0 for all g ∈ G. Since the functions 𝜑n (z) = zn (n ∈ ℤ) form an orthonormal ∑ basis in ℒ 2 (Ω, 𝜇), the characteristic function 𝜒E ∈ ℒ 2 (Ω, 𝜇) can be written 𝜒E (𝜔) = n∈ℤ 𝜆n 𝜔n ∑ with n∈ℤ |𝜆n |2 < +∞. By assumption, we have 𝜒E ◦ T𝜃 = 𝜒E in ℒ 2 (Ω, 𝜇) for every g ∈ G. Since ∑ (𝜒E ◦ Tg )(𝜔) = n 𝜆n gn 𝜔n , it follows that 𝜆n = 𝜆n gn for every g ∈ G, n ∈ ℤ. Since G is dense in Ω, we infer that 𝜆n = 0 for all n ≠ 0 hence either 𝜇(E) = 0, or 𝜇(Ω∖E) = 0. Similarly, if Ω = ℝ, 𝜇 = the Lebesgue measure on ℝ and G is a dense countable infinite subgroup of Ω, which acts on Ω by translation (e.g., G = ℚ), then the corresponding crossed product ℛ(ℳ, 𝜎) is a type II∞ factor. 22.9. Using Corollary 10.6 and Theorem 1/22.7, we obtain from Proposition 22.5 the following “Galois correspondence” (compare with Theorem 21.8): Corollary. Let 𝜎 ∶ G → Aut(ℳ) be a (properly) outer action of the discrete group G on the factor ℳ. We assume that there exists a 𝜎-invariant faithful normal finite trace on ℳ. Then the mappings 𝒩 → H and H → 𝒩 defined by ̄ 𝜆 ( g) ∈ 𝒩 }, H = {g ∈ G; 1 ⊗

̄ 𝜆 (H)} 𝒩 = ℛ{𝜋𝜎 (ℳ), 1 ⊗

are reciprocal increasing bijections between the subgroups H of G and the unital W ∗ -subalgebras 𝒩 ⊃ 𝜋𝜎 (ℳ) of the crossed product ℛ(ℳ, 𝜎). If ℳ is no longer assumed to be a factor, the corresponding result cannot be expressed in terms of subgroups of G, but can be expressed in terms of full subgroups of the full group [𝜎(G)]. More precisely, let 𝜎 ∶ G → Aut(ℳ) be a properly outer action of the discrete group G on the W ∗ -algebra ℳ. We assume that there exists a 𝜎-invariant faithful normal finite trace 𝜏 on ℳ. Let P𝜎̂ be the faithful normal conditional expectation of the crossed product ℛ(ℳ, 𝜎) onto 𝜋𝜎 (ℳ) = ℳ. By Proposition 22.4, Theorem 22.3.(1), and Proposition 10.17, the full group [𝜎(G)] ⊂ Aut(ℳ) consists of all *-automorphisms of the form Ad(v)|ℳ with v ∈  (P𝜎̂ ) = {v ∈ U(ℛ(ℳ, 𝜎)); vℳv∗ = ℳ}. If 𝒩 ⊃ 𝜋𝜎 (ℳ) is a unital W ∗ -subalgebra of ℛ(ℳ, 𝜎), then H𝒩 = {Ad(v)|ℳ; v ∈  (P𝜎̂ ) ∩ 𝒩 } is a full subgroup of [𝜎(G)] and 𝒩 = ℛ{v ∈ U(ℛ(ℳ, 𝜎)); Ad(v)|ℳ ∈ H𝒩 }. Conversely, if H is a full subgroup of [𝜎(G)], then 𝒩H = ℛ{v ∈ U(ℛ(ℳ, 𝜎)); Ad(v)|ℳ ∈ H} is a unital W ∗ -subalgebra of ℛ(ℳ, 𝜎) containing 𝜋𝜎 (ℳ) and H = {Ad(v)|ℳ; v ∈  (P𝜎̂ ) ∩ 𝒩H }. Hence the mappings 𝒩 ↦ H𝒩 and H ↦ 𝒩H are reciprocal increasing bijections between the full subgroups H of [𝜎(G)] and the unital W ∗ -subalgebras 𝒩 ⊃ 𝜋𝜎 (ℳ) of ℛ(ℳ, 𝜎). For the full proofs of these results, we refer to Haga and Takeda (1972); Henle (1979). 22.10. Let T and G be discrete groups and let 𝜎 ∶ T → Aut(G) be a group homomorphism. The product set G × T endowed with the operation ( g, t)( g′ , t′ ) = ( g𝜎t ( g′ ), tt′ ) ( g, g′ ∈ G, t, t′ ∈ T) is a group, denoted by G × 𝜎 T, called the semidirect product of G by the action 𝜎 of T, with neutral element (e, 1) ∈ G × 𝜎 T (where e ∈ G and 1 ∈ T are the corresponding neutral elements of G and T) and inverse given by ( g, t)−1 = (𝜎t−1 ( g−1 ), t−1 ) ( g ∈ G, t ∈ T). We have a split short exact sequence 𝛼

𝛽

{e} → G → G × 𝜎 T ⇄ T → {1} 𝛾

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where 𝛼( g) = ( g, 1), 𝛽( g, t) = t, 𝛾(t) = (e, t) ( g ∈ G, t ∈ T); 𝛼(G) is a normal subgroup of G × 𝜎 T, 𝛾(T) is a subgroup of G × 𝜎 T, G × 𝜎 T is generated by 𝛼(G) ∪ 𝛾(T) and (G × 𝜎 T)∕𝛼(G) is isomorphic to 𝛾(T). The von Neumann algebra 𝔏(G×𝜎 T) ⊂ ℬ(𝓁 2 (G×𝜎 T)) is generated by the left translation operators 𝜆 ( g, t) ∈ ℬ(𝓁 2 (G × 𝜎 T)) ( g ∈ G, t ∈ T). On the other hand, the von Neumann algebra 𝔏(G) ⊂ ℬ(𝓁2 (G)) is generated by the left translation operators 𝜆 ( g) ∈ ℬ(𝓁 2 (G)) ( g ∈ G ), and the action 𝜎 ∶ T → Aut(G) extends to an action 𝜎 ∶ T → Aut(𝔏(G)) defined by ( 𝜎t

∑ g

) 𝜆( g) 𝛼( g)𝜆

=

∑

𝜆(𝜎t ( g)) = 𝛼( g)𝜆

g

∑

𝜆( g) (t ∈ T); 𝛼(𝜎t−1 ( g))𝜆

g

so we can consider the crossed product von Neumann algebra ℛ(𝔏(G), 𝜎) ⊂ ℬ(𝓁 2 (T, 𝓁 2 (G))) ̄ 𝜆(t) (t ∈ T), acting on 𝓁 2 (T, 𝓁 2 (G)). 𝜆( g)), ( g ∈ G ) and 1 ⊗ generated by the operators 𝜋𝜎 (𝜆 It is easy to check that the equation [U𝜉]( g, t) = [𝜉(T)](𝜎t−1 ( g)) (𝜉 ∈ 𝓁 2 (T, 𝓁 2 (G)), g ∈ G, t ∈ T) defines a unitary operator U ∶ 𝓁 2 (T, 𝓁 2 (G)) → 𝓁 2 (G × 𝜎 T) such that 𝜆( g)) ( g ∈ G ), U−1𝜆( g, 1)U = 𝜋𝜎 (𝜆 ̄ 𝜆 (t)) (t ∈ T). U−1𝜆 (e, t)U = 1 ⊗

(1) (2)

Indeed, for 𝜉 ∈ 𝓁 2 (T, 𝓁 2 (G)), h ∈ G and s ∈ T we have 𝜆( g, 1)U𝜉](𝜎s (h), s) [(U−1𝜆 ( g, 1)U𝜉)(s)](h) = [𝜆 = [U𝜉](( g, 1)−1 (𝜎s (h), s)) = [U𝜉](( g−1 , 1)(𝜎s (h), s)) = [U𝜉]( g−1 𝜎s (h), s) = [𝜉(s)](𝜎s−1 ( g−1 𝜎s (h))) 𝜆(𝜎s−1 ( g))𝜉(s)](h) = [𝜉(s)](𝜎s−1 ( g)−1 h) = [𝜆 𝜆( g)𝜉)(s)](h), 𝜆( g))𝜉(s)](h) = [(𝜋𝜎 (𝜆 = [𝜎s−1 (𝜆 𝜆(e, t)U𝜉](𝜎s (h), s) [(U−1𝜆 (e, t)U𝜉)(s)](h) = [𝜆 = [U𝜉]((e, t)−1 (𝜎s (h), s)) = [U𝜉]((e, z−1 )(𝜎s (h), s)) = [U𝜉](𝜎t−1 s , t−1 s) = [𝜉(t−1 s)](𝜎t−1 −1 s (𝜎t−1 s (h))) ̄ 𝜆(t))𝜉)(s)](h). = [𝜉(t−1 s](h) = [((1 ⊗𝜆

Consequently, U−1 𝔏(G × 𝜎 T)U = ℛ(𝔏(G), 𝜎)

(3)

and hence we have proved the following Proposition. Let G, T be discrete groups and 𝜎 ∶ T → Aut(G) a group homomorphism. Then the von Neumann algebras 𝔏(G × 𝜎 T) and ℛ(𝔏(G), 𝜎) are spatially isomorphic.

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22.11. In particular, let us consider the case G = F∞ , the free group on a countable infinity of generators {xn }n∈ℤ ⊂ F∞ , T = ℤ, m ∈ ℕ and m 𝜎 ∈ Aut(F∞ ) the unique automorphism of F∞ such that m 𝜎(xn ) = xn+m (n ∈ ℤ). We obtain a group homomorphism, also denoted by m 𝜎 ∶ ℤ → Aut(F∞ ), by putting m 𝜎k = (m 𝜎)k (k ∈ ℤ). Proposition. The semidirect product F∞ × generators.

m𝜎

ℤ is isomorphic to the free group Fm+1 on m + 1

Proof. We shall put 𝜎 = m 𝜎. The neutral element of F∞ × 𝜎 ℤ is (e, 0) and for ( y, i), (z, j) ∈ F∞ × 𝜎 ℤ we have ( y, i)(z, j) = ( y𝜎 i (z), i + j), ( y, i)−1 = (𝜎 −i ( y−1 ), −i). Let y0 , y1 , … , ym be the generators of the free group Fm+1 . We define a homomorphism 𝜑 ∶ Fm+1 → F∞ × 𝜎 ℤ by putting 𝜑( yk ) = (xk , 1) for k = 0, 1, … , m−1 and 𝜑( ym ) = (e, 1). Since (e, 1)−1 = (e, −1), (xk , 1)(e, −1) = (xk , 0), and (e, 1)(xk , 0) = (xk+m , 1), it follows that 𝜑 is surjective. p −p−1 Consider also the homomorphism 𝜓1 ∶ F∞ → Fm+1 defined by 𝜓1 (xn ) = ym yk ym (n, p, k ∈ ℤ, n = mp + k, 0 ≤ k < m), and the homomorphism 𝜓2 ∶ ℤ → Fm+1 defined by 𝜓2 (i) = yim (i ∈ ℤ), and define a mapping 𝜓 ∶ F∞ × 𝜎 ℤ → Fm+1 by 𝜓( y, i) = 𝜓1 ( y)𝜓2 (i) ( y ∈ F∞ , i ∈ ℤ). We show that 𝜓 is a group homomorphism. For ( y, i) and (z, j) in F∞ × 𝜎 ℤ, we have 𝜓(( y, i)(z, j)) = 𝜓1 ( y)𝜓1 (𝜎 i (z))𝜓2 (i)𝜓2 ( j) and 𝜓( y, i)𝜓(z, j) = 𝜓1 ( y)𝜓2 (i)𝜓1 (z)𝜓2 ( j), and so it is sufficient to show that 𝜓1 (𝜎 i (z))𝜓2 (i) = 𝜓2 (i)𝜓1 (z), that is, 𝜓1 (𝜎 i (z)) = yim 𝜓1 (z)y−i m . Moreover, we may assume that z = xn . In this case writing n = mp + k with p, k ∈ ℤ, 0 ≤ k < m, we have 𝜓1 (𝜎 i (xn )) = 𝜓1 (xn+mi ) = i+p −i−p−1 p −p−1 𝜓1 (xm(i+p)+k ) = ym yk ym = yim ( ym yk ym )yim = yim 𝜓1 (xn )y−i m. Finally, we show that 𝜓 ◦ 𝜑 = the identity mapping, so that 𝜑 is the required isomorphism. For k = 0, 1, … , m − 1 we have (𝜓 ◦ 𝜑)( yk ) = 𝜓(xk , 1) = 𝜓1 (xk )𝜓2 (1) = yk y−1 m ym = yk and for k = m we have (𝜓 ◦ 𝜑)( ym ) = 𝜓(e, 1) = 𝜓1 (e)𝜓2 (1) = eym = ym . Using Proposition 22.10, we deduce that Corollary. ℛ(𝔏(F∞ ), m 𝜎) is *-isomorphic to 𝔏(Fm+1 ). One can also show (Phillips, 1976) that for every m, n ≥ 2 there exists 𝜎 ∈ Aut(𝔏(Fm(n−1)+1 )) such that the *-automorphisms 𝜎, 𝜎 2 , … 𝜎 m−1 are all outer, 𝜎 m = 𝜄 and ℛ(𝔏(Fm(n−1)+1 ), 𝜎) is *-isomorphic to 𝔏(Fn ). Let us mention that it is not yet known whether the factors 𝔏(F2 ), 𝔏(F3 ), … are *-isomorphic or not. 22.12. In this section, we give some examples of outer *-automorphisms of the W ∗ -algebras of the form 𝔏(G). Recall that every automorphism 𝜎 of the discrete group G has an extension to *-automorphism 𝜎 of 𝔏(G) such that ( 𝜎

∑ g

) 𝜆( g) 𝛼( g)𝜆

=

∑ g

𝜆(𝜎( g)) = 𝛼( g)𝜆

∑

𝜆( g), 𝛼(𝜎 −1 ( g))𝜆

(𝛼( g) ∈ ℂ, g ∈ G ).

g

Proposition. Let 𝜎 be an automorphism of the discrete group G. For the corresponding *-automorphism 𝜎 ∈ Aut(𝔏(G)) we have (i) 𝜎 is properly outer if and only if for every g ∈ G the set {𝜎(s)gs−1 ; s ∈ G} is infinite; (ii) 𝜎 is ergodic if and only if for every g ∈ G, g ≠ e, the set {𝜎 n ( g); n ∈ ℤ} is infinite.

326

Crossed Products

∑ 𝜆( g) ∈ Proof. Assume that for every g ∈ G the set {𝜎(s)gs−1 ; s ∈ G} is infinite. Let a = g 𝛼( g)𝜆 𝜆(s) = 𝜎(𝜆 𝜆(s))a = 𝜆 (𝜎(s))a, that is, a = 𝜆 (𝜎(s−1 ))a𝜆 𝜆(s) for every s ∈ G. 𝔏(G) be such that a𝜆 ∑ 𝜆(s) = g 𝛼(𝜎(s)gs−1 )𝜆 𝜆( g), it follows that 𝛼( g) = 𝛼(𝜎(s)gs−1 ) for s, g ∈ G. Since Since 𝜆 (𝜎(s−1 ))a𝜆 ∑ 2 g |𝛼( g)| < +∞ (22.6.(4)), using the assumption we conclude 𝛼( g) = 0 for all g ∈ G, that is, a = 0. Hence 𝜎 is properly outer. Conversely, assume that 𝜎 is properly outer and that there exists g ∈ G such that the set S = ∑ 𝜆(s−1 ), that is, 𝜎(𝜆 𝜆(s))a = {𝜎(s)gs−1 ; s ∈ G} is finite. Then a = s∈S 𝜆 (s) ∈ 𝔏(G) and a = 𝜆 (𝜎(s))a𝜆 𝜆(s) for each s ∈ G, hence 𝜎(x)a = ax for all x ∈ 𝔏(G), which implies that a = 0, a contradiction. a𝜆 We have proved assertion (i); (ii) can be proved similarly. Corollary 1. If G is a discrete ICC-group and 𝜎 is an outer automorphism of G, then the corresponding *-automorphism of the factor 𝔏(G) is also outer. ∑ 𝜆( g) ∈ Proof. Assume that 𝜎 ∈ Aut(𝔏(G)) is inner, that is 𝜎 = Ad(a) with a = g 𝛼( g)𝜆 ∑ U(𝔏(G)), g |𝛼( g)|2 < +∞. If g ∈ G and 𝛼( g) ≠ 0, then, as in the proof of the proposition, one shows that the set {𝜎(s)gs−1 ; s ∈ G} is finite. Since 𝜎 is an outer automorphism of G, we infer that there exist g1 , g2 ∈ G, g1 ≠ g2 , such that the sets S1 = {𝜎(s)g1 s−1 ; s ∈ G} and S2 = {𝜎(s)g2 s−1 ; s ∈ G} be finite. Then the set {sg−1 g s−1 ; s ∈ G} ⊂ S−1 S is finite, hence g1 = g2 , 1 2 1 2 since G is an ICC-group. This contradiction proves that 𝜎 is an outer *-automorphism of 𝔏(G). Let us mention that there exist countable discrete ICC-groups without outer *-automorphisms, for instance (Kallman, 1969) the semidirect product G = ℚ ×𝜎 ℚ∗ , where ℚ is the additive group of rational numbers, ℚ∗ is the multiplicative group of nonzero rational numbers and 𝜎 ∶ ℚ∗ → Aut(ℚ) is defined by 𝜎(r)(q) = rq, (r ∈ ℚ∗ , q ∈ ℚ). By the above results, we get the following examples of outer *-automorphisms: Corollary 2. Let F2 be the free group on two generators and 𝜎 the automorphism of F2 intertwining the two generators. Then 𝜎 ∈ Aut(𝔏(F2 )) is outer. Corollary 3. Let S(∞) be the group of finite permutations of ℤ, let 𝜋 be an infinite permutation of ℤ and let 𝜎 be the automorphism of S(∞) defined by 𝜎(s) = 𝜋 ◦ s ◦ 𝜋 −1 (s ∈ S(∞)). Then 𝜎 ∈ Aut(𝔏(S(∞))) is outer. With the same methods one can show that every countable discrete group can be represented as a group of outer *-automorphisms of the II1 factor 𝔏(S(∞)) (Suzuki, 1958; Kallman, 1969). Also, there exist factors of types II∞ and III enjoying the same property (Saito, 1961; Kallman, 1969). 22.13. We now describe another type of outer *-automorphisms of factors of the form 𝔏(G). Let G be a discrete ICC-group and 𝛾 a character of G, that is, a group homomorphism of G into the one-dimensional torus {𝜔 ∈ ℂ; |𝜔| = 1}; clearly, 𝛾( g) = 𝛾( g−1 ) = 𝛾( g)−1 ( g ∈ G ). 𝛾 being in 𝓁 ∞ (G) defines a multiplication operator m(𝛾) ∈ ℬ(𝓁 2 (G)); we have m(𝛾)𝔏(G)m(𝛾)∗ = 𝔏(G). Consequently, we obtain a *-automorphism 𝜎𝛾 = Ad(m(𝛾))|𝔏(G) ∈ Aut(𝔏(G)), uniquely determined, such that 𝜎𝛾 (𝜆( g)) = 𝛾( g)𝜆( g) ( g ∈ G ). Proposition. Let G be a discrete ICC-group and 𝛾 ≠ 1 a nontrivial character of G. Then the *automorphism 𝜎𝛾 ∈ Aut(𝔏(G)) is outer. ∑ 𝜆( g) ∈ U(𝔏(G)) with 𝜎𝛾 = Proof. Assume to the contrary, so that there exists v = g 𝛼( g)𝜆 ∑ ∗ 𝜆(s) = 𝜎𝛾 (𝜆 𝜆(s)) = v𝜆 𝜆(s)v = g,h 𝛼( g)𝛼(h)𝜆 𝜆( gsh−1 ) = Ad(v). For every s ∈ G, we have 𝛾(s)𝜆

Discrete Groups

327

∑ ∑ −1 𝜆(t). Let 𝜉s ∈ 𝓁 2 (G) be defined by 𝜉s ( g) = 𝛼(s−1 gs) (s, g ∈ G ). The previous t ( g 𝛼( g)𝛼(t gs))𝜆 ∑ computation shows that (𝜉e |𝜉s ) = g 𝛼( g)𝛼(s−1 gs) = |𝛾(s)| = 1 and ‖𝜉s } = ‖𝜉e ‖ = 1, so that the vectors 𝜉s and 𝜉e are proportional, in particular |𝛼(s−1 gs)| = |𝛼( g)| (s, g ∈ G ). Since 𝛾 ≠ 1, we have 𝜎𝛾 ≠ 𝜄, and there is g0 ∈ G, g0 ≠ e, with 𝛼( g0 ) ≠ 0. Since G is an ICC-group, it follows that ∑ 2 g |𝛼( g)| = +∞, a contradiction. In particular, if F∞ is the free group on a countable infinity of generators {xn }n≥1 ⊂ F∞ and G = {1, 𝛾1 , 𝛾2 , …} is an ordered countable infinite subgroup of the one-dimensional torus, then there exists a unique *-automorphism 𝜎G ∈ Aut(𝔏(F∞ )) such that 𝜎G (𝜆(xn )) = 𝛾n 𝜆(xn ) (n ≥ 1). One can prove that if G and H are two ordered countable infinite subgroups of the one-dimensional torus, then the *-automorphisms 𝜎G , 𝜎H ∈ Aut(𝔏(F∞ )) are outer conjugate if and only if G = H (Phillips, 1976). Thus Aut(𝔏(F∞ )) has a continuous infinity of outer conjugacy classes; moreover, there is no “good” classification of these classes (Phillips, 1976). On the other hand, the set of outer conjugacy classes of Aut(𝔏(S(∞)) is countable and completely described by certain simple invariants introduced by Connes (1977a, 1975e). 22.14. In this section we examine, in a slightly more general framework, some interesting consequences of the relative commutant theorem (22.3.(1)). Proposition. Let 𝜎 ∶ G → Aut(ℳ) be an integrable continuous action of the locally compact unimodular group G on the countably decomposable W ∗ -algebra ℳ. If 𝜋𝜎 (ℳ)′ ∩ ℛ(ℳ, 𝜎) = 𝜋𝜎 (𝒵 (ℳ)), then (ℳ 𝜎 )′ ∩ ℳ = 𝒵 (ℳ), in particular 𝒵 (ℳ 𝜎 ) = 𝒵 (ℳ 𝜎 ). Proof. Since ℳ is countably decomposable, we may assume ℳ ⊂ ℬ(ℋ ) realized as a von Neumann algebra with a cyclic and separating vector 𝜉0 ∈ ℋ . Since 𝜎 is integrable and G is unimodular, the formula 𝜑(x) = ∫ (𝜎g (x)𝜉0 |𝜉0 ) dg (x ∈ ℳ + ), defines a 𝜎-invariant n.s.f. weight 𝜑 on ℳ. By Corollary 2.24, there exists an so-continuous unitary representation u ∶ G → ℬ(ℋ𝜑 ) such that 𝜋𝜑 (𝜎g (x)) = u( g)𝜋𝜑 (x)u( g)∗ (x ∈ ℳ, g ∈ G ), and moreover, by Section 2.25, we have u( g)a𝜑 = (𝜎g (a))𝜑 and u( g)J𝜑 = J𝜑 u( g) ( g ∈ G, a ∈ 𝔑𝜑 ). Since ∫ ‖𝜎t−1 (a)𝜉0 ‖2 dt = 𝜑(a∗ a) = ‖a𝜑 ‖2 (a ∈ 𝔑𝜑 ), there exists an isometric linear operator U ∶ ℋ𝜑 → ℒ 2 (G, ℋ ), uniquely determined, such that [Ua𝜑 ](t) = 𝜎t−1 (a)𝜉0 for all a ∈ 𝔑𝜑 , t ∈ G. Recall that the crossed product ℛ(ℳ, 𝜎) ⊂ ℬ(ℒ 2 (G, ℋ )) is the von Neumann algebra generated by ̄ 𝜆 ( g) ( g ∈ G ). It is easy to check that U𝜋𝜑 (x) = 𝜋𝜎 (x)U, (x ∈ the operators 𝜋𝜎 (x) (x ∈ ℳ) and 1 ⊗ ̄ 𝜆 ( g))U ( g ∈ G ), so that the mapping Φ ∶ ℛ(ℳ, 𝜎) ∋ X ↦ U ∗ XU ∈ ℳ), and Uu( g) = (1 ⊗ ℛ{𝜋𝜑 (ℳ), u(G)} is a surjective normal *-homomorphism. Let Q be the unique central projection in ℛ(ℳ, 𝜎) such that Ker Φ = (1 − Q) ℛ(ℳ, 𝜎). It is clear that 𝜋𝜑 (ℳ 𝜎 ) = 𝜋𝜑 (ℳ) ∩ u(G)′ and hence that J𝜑 𝜋𝜑 ((ℳ 𝜎 )′ ∩ ℳ)J𝜑 = ℛ{𝜋𝜑 (ℳ), u(G)} ∩ 𝜋𝜑 (ℳ)′ . Thus, if z ∈ (ℳ 𝜎 )′ ∩ ℳ, there exists X = XQ ∈ ℛ(ℳ, 𝜎) such that Φ(X) = J𝜑 𝜋𝜑 (z)J𝜑 ∈ 𝜋𝜑 (ℳ)′ = Φ(𝜋𝜎 (ℳ))′ . It follows that X ∈ 𝜋𝜎 (ℳ)′ ∩ ℛ(ℳ, 𝜎) = 𝜋𝜎 (𝒵 (ℳ)) and therefore 𝜋𝜑 (z) = J𝜑 Φ(X)J𝜑 ∈ J𝜑 Φ(𝜋𝜎 (𝒵 (ℳ)))J𝜑 = J𝜑 𝜋𝜑 (𝒵 (ℳ))J𝜑 = 𝜋𝜑 (𝒵 (ℳ)), that is, z ∈ 𝒵 (ℳ).

328

Crossed Products

In particular, 𝒵 (ℳ 𝜎 ) = 𝒵 (ℳ)𝜎

(1)

for any properly outer action 𝜎 ∶ G → Aut(ℳ) of a finite group G on the countably decomposable W ∗ -algebra ℳ. This is an extension of 16.17.(3). 22.15. As an application, we prove the following extension of Corollary 16.17. Corollary. Let 𝜎 ∶ G → Aut(ℳ) be a properly outer action of the finite group G on the countably decomposable W ∗ -algebra ℳ. Then, for e, f ∈ Proj(ℳ 𝜎 ) we have e ∼ f in ℳ ⇔ e ∼ f in ℳ 𝜎

(1)

and every unitary cocycle u ∈ Z𝜎 (G; U(ℳ)) is trivial, that is, there exists v ∈ U(ℳ) such that u( g) = v𝜎g (v∗ ) ( g ∈ G ). Proof. (1) follows by the same arguments as in the proof of Corollary 16.17, using Corollary 17.24 and 22.14.(1). ̄ 𝜄, of G on the Consider now any two unitary cocycles a, b ∈ Z𝜎 (G; U(ℳ)), the action 𝜎 ⊗ ∗ ̄ ̄ W -algebra ℳ ⊗ F2 and the balanced cocycle c = c(a, b) ∈ Z𝜎 ⊗̄ 𝜄 (G; U(ℳ ⊗ F2 )) (see 20.2). ̄ 𝜄) of G on ℳ ⊗ ̄ F2 is also properly outer. On According to 17.2.(1) and 17.6.(1), the action c (𝜎 ⊗ c ̄ ̄ ̄ ̄ F2 . Hence, the other hand, the projections 1 ⊗ e11 , 1 ⊗ e22 ∈ (ℳ ⊗ F2 ) are equivalent in ℳ ⊗ c ̄ ̄ ̄ using (1) we conclude that 1 ⊗ e11 ∼ 1 ⊗ e22 in (ℳ ⊗ F2 ) , that is, a ≂ b. Thus, if 𝜏 ∶ G → Aut(𝒩 ) is another action satisfying the assumptions of the previous corollary and if (ℳ, 𝜎) ∼ (𝒩 , 𝜏), then (ℳ, 𝜎) ≈ (𝒩 , 𝜏). If, moreover, ℳ is properly infinite, there exists a projection p ∈ ℳ such that the 𝜎g (p) ( g ∈ G ) ∑ are mutually orthogonal and g∈G 𝜎a (p) = 1. Indeed, by the previous remark and 20.14.(1) we have ̄ ℬ(𝓁 2 (G)), 𝜎 ⊗ ̄ Ad(𝜌𝜌)), and the stated property is obvious for the action in this case (ℳ, 𝜎) ≈ (ℳ ⊗ 2 Ad(𝜌𝜌) ∶ G → Aut(ℬ(𝓁 (G))). 22.16. Notes. For the classical results concerning crossed products by discrete groups (22.1, 22.3, 22.6, 22.7, 22.8, 22.10) we refer to Ching (1969); Dixmier (1957, 1969); Nakagami and Takesaki (1979); Nakamura and Takeda (1958, 1960); Sakai (1971); Turumaru (1958); Zeller-Meier (1968). The results of Sections 22.2 and 22.4 appeared in Connes (1973a); Haga and Takeda (1972). The material of Section 22.12 is from Dixmier (1957, 1969) and Kallman (1969). Proposition 22.11 is due to Phillips (1976), Proposition 22.13 is due to Paschke (1976) and the Galois correspondence type results 22.5 and 22.9 are due to Haga and Takeda (1972). Extensions of the relative commutant theorem (22.3.(2)) and of the results concerning the type of the crossed products (22.7) have been obtained by Sauvageot (1977). A deep commutative nondiscrete relative commutant theorem, due to Connes and Takesaki (1977), will be given in Section 23.19. A detailed analysis of several relative commutant theorems in general situations, including Proposition 22.14, is due to Paschke (1978) (see also Nakagami & Takesaki, 1979). Corollary 22.15 is due to Connes and Takesaki (1977).

Discrete Groups

329

For our exposition, we have used Ching (1969); Connes (1973a); Connes and Takesaki (1977); Dixmier (1957, 1969); Kallman (1969); Nakagami and Takesaki (1979); Paschke (1976, 1978); Phillips (1976) and Sakai (1971). Important examples of factors arising by the “group measure space construction” (22.8) are the Pukánszky factors 𝒫𝜆 (0 < 𝜆 < 1) (Pukánszky, 1956; Sakai, 1971, p. 192; Connes, 1973a, p. 207), ∏ which we now describe. Let F2 be the free group on two generators and let Ω = g∈F2 Ωg , where Ωg = {0, 1} for each g ∈ F2 . Let p > 0, q > 0 be such that p + q = 1 and p = 𝜆q. For each g ∈ F2 , we denote by 𝜇g the measure on Ωg defined by 𝜇g ({0}) = p, 𝜇g ({1}) = q and then consider the ⨂ product measure 𝜇 = g∈F2 𝜇g on Ω. Each g0 ∈ F2 defines a shift transformation Tg0 ∶ {𝜔g }g∈F2 ↦ {𝜔g0 g }g∈F2 on Ω and also a transformation Sg0 ∶ {𝜔g }g∈F2 ↦ {𝜔′g }g∈F2 on Ω such that 𝜔′g = 1 − 𝜔g0 0 and 𝜔′g = 𝜔g for g ≠ g0 . Let G be the transformation group on Ω generated by the transformations Tg and Sg ( g ∈ F2 ). Then 𝜇 is G-quasi-invariant and G acts freely and ergodically on Ω (cf. loc. cit.), so that the corresponding crossed product W ∗ -algebra is a factor denoted by 𝒫𝜆 . Note also that the Powers factors ℛ𝜆 (A.17) can be obtained by a similar group measure space construction. In order to obtain factors by the group measure space construction, one has to assume that the action of the transformation group is free and ergodic. Krieger (1970a, 1970b, 1971a, 1971b, 1972, 1974, 1976a) showed how to modify this construction in order to obtain factors even if the action is not free (as long as it is still ergodic). Several variants, extensions and concrete examples of Krieger’s construction have appeared in Guichardet (1974); Nielsen (1974–1975); Samuelides and Sauvageot (1975); Strătilă and Voiculescu (1975, 1978). Zimmer (1977a, 1977b) proved that the factors arising by the classical group measure space construction, or the Krieger construction, are approximately finite dimensional if and only if the action of the group is amenable, as defined in Zimmer (1978) (see also Delaroche, 1979). For instance, the Pukánszky factors are not approximately finite dimensional. Recently, Connes, Feldman, and Weiss (1981) proved that amenable actions of discrete groups are necessarily singly generated. From the works of Dye (1959, 1963) and Krieger (1968, 1969, 1976b) it follows that the *-isomorphism class of a factor arising by the group measure space construction depends only on the equivalence relation defined by the orbits of the group. A detailed construction and analysis of the von Neumann algebras associated with an ergodic equivalence relation is given by Feldman and Moore (1977). A factor that is the crossed product of an abelian W ∗ -algebra by a single *-automorphism is called a Krieger factor (cf. Connes, 1975d). There are several abstract characterizations of Krieger factors due to Connes (1974d, 1975d). Two Krieger factors are *-isomorphic if and only if the corresponding transformations are weakly equivalent, that is, if and only if they define isomorphic equivalence relations (Krieger, 1976b). All Krieger factors are approximately finite dimensional. Connes (1975d) proved that any approximately finite dimensional factor of type III0 acting on a separable Hilbert space is a Krieger factor and these factors are classified by their flow of weights. On the other hand, every Araki– Woods factor (A.17) can be obtained as a Krieger factor (Krieger, 1970b; Connes, 1977b). Connes [1973a] proved that the class of Krieger factors (and hence the class of approximately finite dimensional factors) is strictly larger than the class of Araki–Woods factors; a simpler proof of this result has been obtained by Connes and Woods (1980a). Solving an old open problem, Connes (1975a, 1975b) constructed examples or factors of types II1 and III, which are not *-antiisomorphic to themselves. In particular, not every factor arises as the 𝔏(G) for some ICC-group G.

CHAPTER V

Continuous Decompositions

23 Dominant Weights and Continuous Decompositions Let ℳ be a properly infinite W ∗ -algebra and denote by Wn (ℳ) (resp. Wns (ℳ); resp. Wnsf (ℳ)) the set of all normal (resp. normal semifinite; resp. n.s.f.) weights on ℳ. In Sections 23.1–23.3, we consider a fixed weight 𝜔 ∈ Wnsf (ℳ) and, using the canonical bijection Wns (ℳ) ∋ 𝜑 ↦ [D𝜑 ∶ D𝜔] ∈ Z𝜎 𝜔 (ℝ; ℳ) given by Theorems 3.1 and 5.1, we apply the results obtained in Section 20 in order to get a comparison theory for weights. In particular, we define the notion of a dominant weight on ℳ, this yields the continuous decomposition (𝒩 , 𝜃, 𝜏) of ℳ, the properties of which are analyzed further. The main feature of the continuous decomposition is that it reduces the study of type III W ∗ -algebras to the study of actions of ℝ on semifinite W ∗ -algebras, modulo (outer) conjugation. 23.1. Recall (2.21) that if 𝜑 ∈ Wns (ℳ) and u ∈ ℳ is a partial isometry with uu∗ ∈ ℳ 𝜑 , then the formula 𝜑u (x) = 𝜑(uxu∗ ) (x ∈ ℳ + ) defines a weight 𝜑u ∈ Wns (ℳ) with s(𝜑u ) = u∗ u; moreover, if u∗ u ∈ ℳ 𝜑 , then 𝜑u = 𝜑u∗ u ⇔ u ∈ ℳ 𝜑 . Consider a fixed weight 𝜔 ∈ Wnsf (ℳ). Proposition. Let 𝜑, 𝜓 ∈ Wns (ℳ) and u ∈ ℳ with u∗ u = s(𝜓) and uu∗ ∈ ℳ 𝜑 . Consider the ̄ ℱ2 and the balanced weight 𝜃 = 𝜃(𝜑, 𝜓) ∈ Wns (𝒫 ). The following W ∗ -algebra 𝒫 = ℳ ⊗ statements are equivalent: (i) 𝜓 ; ( = 𝜑u ) 0 u (ii) ∈ 𝒫 𝜃; 0 0 (iii) [D𝜓 ∶ D𝜔]t = u∗ [D𝜑; D𝜔]t 𝜎t𝜔 (u) for all t ∈ ℝ. ( ) ( ) s(𝜑) 0 0 u Proof. (i) ⇔ (ii). We have s(𝜃) = , hence U = ∈ s(𝜃)𝒫 s(𝜃), since F = 0 s(𝜓) 0 0 ( ∗ ) ( ) uu 0 0 0 UU∗ = and E = U∗ U = ; it also follows that E, F ∈ 𝒫 𝜃 . By Proposition 0 0 0 s(𝜓) ( ) x11 x12 2.21, we have U ∈ 𝒫 𝜃 if and only if 𝜃U = 𝜃E . For X = ∈ 𝒫 + , we have UXU∗ = x21 x22 ( ) ( ) ux22 u∗ 0 0 0 and EXE = , hence 𝜃U = 𝜃E if and only if 𝜓 = 𝜑u . 0 0 0 s(𝜓)x22 s(𝜓)

331

332

Continuous Decompositions

̄ ℱ3 and the weight 𝜏 ∈ Wns (Q) defined as (ii) ⇔ (iii). Consider now the W ∗ -algebra Q = ℳ ⊗ 𝜏([xij ]) = 𝜑(x11 ) + 𝜓(x22 ) + 𝜔(x33 ), ([xij ] ∈ Q+ ). Then by Proposition 3.3, we have ⎛0 0 𝜎t𝜏 ⎜ 0 0 ⎜ ⎝0 0

s(𝜑) ⎞ ⎛ 0 0 ⎟ = ⎜0 ⎟ ⎜ 0 ⎠ ⎝0

0 0 0

[D𝜑 ∶ D𝜔]t ⎞ ⎟; 0 ⎟ 0 ⎠

⎛0 0 𝜎t𝜏 ⎜ 0 0 ⎜ ⎝0 0

0 ⎞ ⎛0 s(𝜓) ⎟ = ⎜ 0 ⎟ ⎜ 0 ⎠ ⎝0

0 0 0

0 ⎞ [D𝜓 ∶ D𝜔]t ⎟ ; ⎟ 0 ⎠

⎛0 0 𝜎t𝜏 ⎜ 0 0 ⎜ ⎝0 0 𝜎t𝜏

0⎞ ⎛0 0⎟ = ⎜0 ⎟ ⎜ u⎠ ⎝0

⎛ 0 u 0 ⎞ ⎛ 𝜎t𝜃 ⎜ 0 0 0 ⎟ = ⎜⎜ ⎜ ⎟ ⎝ 0 0 0 ⎠ ⎜⎝

0 0 0 (

0 ⎞ 0 ⎟; ⎟ 𝜔 𝜎t (u) ⎠ ) 0 u 0 0 ⎞⎟ 0 0 0 ⎟⎟ 0 0 0⎠

so that ⎛ 𝜎𝜃 ⎜ t ⎜ ⎜ ⎝

(

0 u 0 0 0

)

0

0 ⎞⎟ 0 ⎟⎟ 0⎠

⎛⎛ 0 0 s(𝜑) ⎞ ⎛ 0 0 0 ⎟⎜0 0 = 𝜎t𝜏 ⎜⎜ 0 0 ⎜⎜ ⎟⎜ 0 ⎠⎝0 0 ⎝⎝ 0 0

0⎞⎛0 0 0⎟⎜0 0 ⎟⎜ u⎠⎝0 0

⎛ 0 [D𝜑 ∶ D𝜔]t 𝜎t𝜔 (u)[D𝜓 ∶ D𝜔]∗t 0 = ⎜0 ⎜ 0 0 ⎝

0 ⎞⎞ s(𝜓) ⎟⎟ ⎟⎟ 0 ⎠⎠ 0⎞ 0⎟ ⎟ 0⎠

hence U ∈ 𝒫 𝜃 if and only if [D𝜓 ∶ D𝜔]t = u∗ [D𝜑 ∶ D𝜔]t 𝜎t𝜔 (u) for every t ∈ ℝ. ( ) a 0 ∈ Z𝜎 𝜔 ⊗̄ 𝜄 (ℝ, 𝒫 ), then we If a = [D𝜑 ∶ D𝜔], b = (D𝜓 ∶ D𝜔] ∈ Z𝜎 𝜔 (ℝ, ℳ) and c = 0 b have ℳ a = ℳ 𝜑 , ℳ b = ℳ 𝜑 , and 𝒫 c = 𝒫 𝜃 , so that the equivalence (ii) ⇔ (iii) also follows from Proposition 20.2. By the previous proposition we have that for 𝜑, 𝜓 ∈ Wns (ℳ) the following conditions are equivalent: (i) there exists u ∈ ℳ with u∗ u = s(𝜓), uu∗ ( ∈ ℳ 𝜑 and ) 𝜓 = 𝜑u ; 0 u (ii) there exists u ∈ ℳ with u∗ u = s(𝜓) and ∈ 𝒫 𝜃; 0 0 (iii) [D𝜓 ∶ D𝜔] ≲ [D𝜑 ∶ D𝜔] in Z𝜎 𝜔 (ℝ; ℳ). If these conditions are satisfied, we write 𝜓 ≲ 𝜑 and we say that the weight 𝜓 is dominated by the weight 𝜑. If in condition (i) we have also uu∗ = s(𝜑) or, equivalently, in condition (iii) we have

Dominant Weights and Continuous Decompositions

333

[D𝜓 ∶ D𝜔] ≂ [D𝜑 ∶ D𝜔], then we write 𝜓 ≂ 𝜑 and say that the weights 𝜓 and 𝜑 are equivalent. For the relations “≲” and “≂,” we can thus apply all the results obtained in Section 20. In particular, if 𝜓 ≲ 𝜑 and 𝜑 ≲ 𝜓, then 𝜓 ≂ 𝜑. 23.2 Proposition. Let 𝜑 ∈ Wns (ℳ) and u, v ∈ ℳ be partial isometries such that uu∗ , vv∗ ∈ ℳ 𝜑 . Then 𝜑u ≲ 𝜑v if and only if uu∗ ≺ vv∗ in ℳ 𝜑 . Proof. Put e = uu∗ , f = vv∗ . We have u ∈ ℳ 𝜑e , uu∗ = e = s(𝜑e ), u∗ u = s(𝜑u ) and 𝜑u = (𝜑e )u , hence 𝜑u ≂ 𝜑e and, similarly, 𝜑v ≂ 𝜑f . Thus, we have to prove that 𝜑e ≲ 𝜑f if and only if e ≺ f in ℳ 𝜑 . If e ≺ f in ℳ 𝜑 , there exists w ∈ ℳ 𝜑 with w∗ w = e and ww∗ = f; hence 𝜑e = 𝜑w ≲ 𝜑f , by Proposition 2.21. Conversely, if 𝜑e ≲ 𝜑f , then there exists w ∈ ℳ with ww∗ ∈ ℳ 𝜑f , ww∗ ≤ f, w∗ w = e and 𝜑e = (𝜑f )w = 𝜑w . By Proposition 2.21, this implies that w ∈ ℳ 𝜑 , and so e ≺ f in ℳ 𝜑 . 23.3. Recall that a weight 𝜑 ∈ Wns (ℳ) is said to be of infinite multiplicity if its centralizer ℳ 𝜑 is properly infinite. Since ℳ 𝜑 = ℳ [D𝜑∶D𝜔] , the weight 𝜑 is of infinite multiplicity if and only if the cocycle [D𝜑 ∶ D𝜔] ∈ Z𝜎 𝜔 (ℝ; ℳ) is of infinite multiplicity. Proposition. Let ℳ be a W ∗ -algebra with separable predual and 𝜑 ∈ Wnsf (ℳ) a weight of infinite multiplicity. Then the following statements are equivalent: (i) 𝜑 ≂ 𝜆𝜑 for every 𝜆 > 0; (ii) 𝜎 𝜑 ∶ ℝ → Aut(ℳ) is a dominant action; (iii) [D𝜑 ∶ D𝜔] ∈ Z𝜎 𝜔 (ℝ; U(ℳ)) is a dominant cocycle. Proof. (i) ⇔ (ii). We have 𝜑 ≂ 𝜆𝜑 for each 𝜆 > 0 if and only if for every s ∈ ℝ there exists u(s) ∈ U(ℳ) with es 𝜑 = 𝜑u(s) , that is, (3.6) if and only if for every s ∈ ℝ there exists u(s) ∈ U(ℳ) with [D(es 𝜑) ∶ D𝜑]t = [D(𝜑u(s) ) ∶ D𝜑]t (t ∈ ℝ), or (3.7) if and only if for every s ∈ ℝ there exists u(s) ∈ U(ℳ) with 𝜎t𝜑 (u(s)) = eist u(s) (t ∈ ℝ). By Proposition 20.12, this last condition means that the action 𝜎 𝜑 ∶ ℝ → Aut(ℳ) is dominant. (ii) ⇔ (iii). The action 𝜎 𝜑 is dominant if and only if the trivial cocycle 1 ∈ Z𝜎 𝜑 (ℝ; U(ℳ)) is dominant (20.11) and, since 𝜎 𝜑 = [D𝜑∶D𝜔] 𝜎 𝜔 , this is equivalent to the fact that the cocycle [D𝜑 ∶ D𝜔] ∈ Z𝜎 𝜔 (ℝ; U(ℳ)) is dominant (see the proof of Theorem 20.5). If the predual ℳ∗ is not assumed to be separable, then the previous equivalence still holds if we replace condition (i) by the condition that there exists an s-continuous function 𝜆 ↦ u(𝜆) ∈ U(ℳ) such that 𝜆𝜑 = 𝜑u(𝜆) . A weight 𝜑 ∈ Wnsf (ℳ) of infinite multiplicity is called a dominant weight if 𝜎 𝜑 ∶ ℝ → Aut(ℳ) is a dominant action. We shall see later (23.16) that if ℳ is a type III W ∗ -algebra with separable predual, then any weight 𝜑 ∈ Wnsf (ℳ) such that 𝜑 ≂ 𝜆𝜑 for every 𝜆 > 0 is automatically of infinite multiplicity and hence dominant. 23.4. A weight 𝜑 ∈ Wns (ℳ) is called integrable if the action 𝜎 𝜑 ∶ ℝ → Aut(ℳs(𝜑) ) is integrable, that is, (20.6) if the cocycle [D𝜑 ∶ D𝜔0 ] ∈ Z𝜎 𝜔0 (ℝ; ℳ) is square integrable, where 𝜔0 is any fixed n.s.f. weight on ℳ. By Theorems 20.5 and 20.6, we have the following result:

334

Continuous Decompositions

Theorem. Let ℳ be a properly infinite W ∗ -algebra. Then there exists a dominant weight 𝜑 ∈ Wnsf (ℳ). If, moreover, ℳ is countably decomposable, then all dominant weights on ℳ are equivalent and a weight 𝜓 ∈ Wns (ℳ) is integrable if and only if 𝜓 ≲ 𝜑. There exists also a simpler and more natural proof of the existence of a dominant weight, which we present in the following. By Stone’s theorem, there exists a unique nonsingular positive self-adjoint operator A on the Hilbert space ℒ 2 (ℝ) such that Ait = 𝜌 (t) (t ∈ ℝ), where we denote by 𝜌 the regular representation of ℝ on ℒ 2 (ℝ). Then the n.s.f. weight 𝜔 = trA on ℱ = ℬ(ℒ 2 (ℝ)) has the following property: 𝜆𝜔 ≂ 𝜔

(𝜆 > 0).

Indeed, let m(s) ∈ ℬ(ℒ 2 (ℝ)) be the unitary operator defined by [m(s)𝜉](t) = e−ist 𝜉(t) (𝜉 ∈ ℒ 2 (ℝ), s, t ∈ ℝ). We have [D𝜔m(s) ∶ D tr]t = [D𝜔m(s) ∶ D𝜔]t [D𝜔 ∶ D tr]t = m(s)∗ 𝜎t𝜔 (m(s))𝜌𝜌(t) = m(−s)𝜌𝜌(t)m(s)𝜌𝜌(−t)𝜌𝜌(t) = e−ist𝜌 (t) = [D(e−s 𝜔) ∶ D𝜔]t [D𝜔 ∶ D tr]t = [D(e−s 𝜔) ∶ D tr]t , hence e−s 𝜔 = 𝜔m(s) ≂ 𝜔, for all s ∈ ℝ. ̄ 𝜔 is a dominant weight Consider now any n.s.f. weight 𝜑 on ℳ of infinite multiplicity. Then 𝜑 ⊗ ̄ ℱ since ℳ 𝜑 ⊗̄ 𝜔 ⊃ ℳ 𝜑 ⊗ ̄ 1 is properly infinite and 𝜆(𝜑 ⊗ ̄ 𝜔) = 𝜑 ⊗ ̄ (𝜆𝜔) ≂ 𝜑 ⊗ ̄ 𝜔 for on ℳ ⊗ ̄ ℱ and hence we can say that 𝜑 ⊗ ̄ 𝜔 every 𝜆 > 0. Since ℳ is properly infinite, we have ℳ ≈ ℳ ⊗ is a dominant weight on ℳ. If ℳ is countably decomposable, then the uniqueness modulo equivalence of the dominant weight ̄ ℱ) shows that for any two n.s.f. weights 𝜑, 𝜓 on ℳ there exists a unitary operator U ∈ U(ℳ ⊗ ̄ 𝜔 = (𝜑 ⊗ ̄ 𝜔) ◦ Ad(U). such that 𝜓 ⊗ ̄ 𝜎𝜔 = 𝜎𝜑 ⊗ ̄ Ad(𝜌𝜌), it follows that the centralizer of the dominant Also, since 𝜎 𝜑 ⊗̄ 𝜔 = 𝜎 𝜑 ⊗ ̄ 𝜔 on ℳ ⊗ ̄ ℬ(ℒ 2 (ℝ)) is equal to the crossed product ℛ(ℳ, 𝜎 𝜑 ) (see Corollary 19.13). weight 𝜑 ⊗ Therefore, for any dominant weight 𝜑 on ℳ, the centralizer ℳ 𝜑 is *-isomorphic to the crossed product ℛ(ℳ, 𝜎 𝜑 ) (see also 20.5.(3)). On the other hand, consider also the unique n.s.f. weight 𝜔′ on ℱ = ℬ(ℒ 2 (ℝ)) such that [D𝜔′ ∶ D tr]s = m(s) (s ∈ ℝ). Since m(s)𝜌𝜌(t) = eist𝜌 (t)m(s), it follows that the modular automorphism ̄ ̄ ′ ′ groups 𝜎t𝜔 = Ad(𝜌𝜌(t)) and 𝜎s𝜔 = Ad(m(s)) commute, hence 𝜎t𝜑 ⊗ 𝜔 and 𝜎t𝜑 ⊗ 𝜔 also commute. ̄ 𝜔 and 𝜑 ⊗ ̄ 𝜔′ do not commute. Indeed, if they did commute, then However, the weights 𝜑 ⊗ ′ ′ ̄ ̄ ̄ [D(𝜑 ⊗ 𝜔 ) ∶ D(𝜑 ⊗ 𝜔)] = 1 ⊗ [D𝜔 ∶ D𝜔] would be a one-parameter group of unitary operators (see Theorem 4.10); however [D𝜔′ ∶ D𝜔]t = [D𝜔′ ∶ D tr]t [D tr ∶ D𝜔]t = m(t)𝜌𝜌(−t) is not a one-parameter group of unitary operators (see also 4.15). It follows that if 𝜑 is a dominant weight on the countably decomposable W ∗ -algebra ℳ, then there exists an n.s.f. weight 𝜓 on ℳ which anticommutes with 𝜑, that is, 𝜓 does not commute with 𝜑, but the corresponding modular automorphism groups commute, 𝜎t𝜑 𝜎s𝜓 = 𝜎s𝜓 𝜎t𝜑 (s, t ∈ ℝ). We shall see later (23.16, 23.17) that this property characterizes the dominant weights on countably decomposable type III factors. 23.5. Consider again the arguments given in the last part of Section 12.4. Let 𝜑 be any n.s.f. weight on the W ∗ -algebra ℳ. On the crossed product ℛ(ℳ, 𝜎 𝜑 ), we have a dual weight 𝜑̂ (19.8) and a dual ̄ m(s))|ℛ(ℳ, 𝜎 𝜑 ) (s ∈ action 𝛼 = (𝜎 𝜑 )∧ (19.3). Namely, the dual action 𝛼 is defined by 𝛼s = Ad(1 ⊗ ℝ), and the modular automorphism group associated with the dual weight 𝜑̂ is characterized by ̄ 𝜆 (t))](𝜋𝜎 𝜑 (x)) (x ∈ ℳ), and 𝜎t𝜑̂ (1 ⊗ ̄ 𝜆 (r)) = 1 ⊗ ̄ 𝜆 (r) = 𝜎t𝜑̂ (𝜋𝜎 𝜑 (x)) = 𝜋𝜎 𝜑 (𝜎t𝜑 (x)) = [Ad(1 ⊗ 𝜑̂ 𝜑 ̄ ̄ ̄ [Ad(1 ⊗ 𝜆 (t))](1 ⊗ 𝜆 (r)) (r ∈ ℝ), so that 𝜎t = Ad(1 ⊗ 𝜆 (t)|ℛ(ℳ, 𝜎 ) (t ∈ ℝ). Thus, if A is the unique nonsingular positive self-adjoint operator in ℒ 2 (ℝ) such that 𝜆 (t) = A−it (t ∈ ℝ), then ̄ 𝜆 (t) (t ∈ ℝ). 𝜇 = 𝜑̂ 1 ⊗̄ A is an n.s.f. trace on ℛ(ℳ, 𝜎 𝜑 ) (since 𝜎 𝜇 = 𝜄) and we have [D𝜑̂ ∶ D𝜇]t = 1 ⊗

Dominant Weights and Continuous Decompositions

335

̄ A)it ) = 𝛼s (1 ⊗ ̄ 𝜆 (−t)) = 1 ⊗ ̄ m(s)𝜆 ̄ 𝜆 (−t)) = eist (1 ⊗ ̄ A)it (s, t ∈ 𝜆(−t)m(−s) = eist (1 ⊗ Since 𝛼s ((1 ⊗ s ̄ A) = e (1 ⊗ ̄ A). Since the dual weight 𝜑̂ is invariant with respect to the ℝ), it follows that 𝛼s (1 ⊗ dual action 𝛼 (19.8), we infer that 𝜇 ◦ 𝛼s = e−s 𝜇 (s ∈ ℝ).

(1)

Thus, the crossed product ℛ(ℳ, 𝜎 𝜑 ) is a semifinite W ∗ -algebra and there exists an n.s.f. trace 𝜇 on ℛ(ℳ, 𝜎 𝜑 ) which satisfies (1), where 𝛼 = (𝜎 𝜑 )∧ is the dual action. 23.6 Theorem (Connes & Takesaki). Let ℳ be a properly infinite W ∗ -algebra and 𝜑 a dominant weight on ℳ. Then the centralizer ℳ 𝜑 is a semifinite W ∗ -algebra and there exist a continuous ̂ and action 𝜃 ∶ ℝ ↦ Aut(ℳ 𝜑 ) and an n.s.f trace 𝜏 on ℳ 𝜑 such that (ℳ, 𝜎 𝜑 ) ≈ (ℛ(ℳ 𝜑 , 𝜃), 𝜃) −s 𝜏 ◦ 𝜃s = e 𝜏 (s ∈ ℝ). Proof. Since 𝜑 is a dominant weight, we infer by Proposition 20.12 that there exists a continuous action 𝜃 ′′ ∶ ℝ → Aut(ℳ 𝜑 ) such that (ℳ, 𝜎 𝜑 ) ≈ (ℛ(ℳ 𝜑 , 𝜃 ′′ ), 𝜃̂ ′′ )

(1)

By the Takesaki duality theorem (19.5), we have ̄ ℬ(ℒ 2 (ℝ)), 𝜃 ′′ ⊗ ̄ Ad(𝜌)). (ℛ(ℳ 𝜑 , 𝜎 𝜑 ), (𝜎 𝜑 )∧ ) ≈ (ℳ 𝜑 ⊗

(2)

Since ℳ 𝜑 is properly infinite, by Corollary 9.16 and the result in Section 20.14 we deduce that ′ there exists a continuous action 𝜃 ′ ∶ ℝ → Aut(ℳ 𝜑 ), 𝜃 ′ ∼ 𝜃 ′′ , whose centralizer (ℳ 𝜑 )𝜃 is properly ̄ ℬ(ℒ 2 (ℝ)), 𝜃 ′ ⊗ ̄ 𝜄). Since infinite. Then, using Corollary 9.16 again we get (ℳ 𝜑 , 𝜃 ′ ) ≈ (ℳ 𝜑 ⊗ ̄ 𝜄 ∼ 𝜃 ′′ ⊗ ̄ 𝜄 ∼ 𝜃 ′′ ⊗ ̄ Ad(𝜌𝜌), it follows that there exists a continuous action 𝜃 ∶ ℝ → 𝜃′ ⊗ Aut(ℳ 𝜑 ), 𝜃 ∼ 𝜃 ′ ∼ 𝜃 ′′ , such that ̄ ℬ(ℒ 2 (ℝ)), 𝜃 ′′ ⊗ ̄ Ad(𝜌𝜌)). (ℳ 𝜑 , 𝜃) ≈ (ℳ 𝜑 ⊗

(3)

From (2) and (3), it follows that (ℛ(ℳ, 𝜎 𝜑 ), (𝜎 𝜑 )∧ ) ≈ (ℳ 𝜑 , 𝜃).

(4)

and, since 𝜃 ∼ 𝜃 ′′ , we infer from (1) and 20.14 that ̂ (ℳ, 𝜎 𝜑 ) ≈ (ℛ(ℳ 𝜑 , 𝜃), 𝜃).

(5)

Finally, by (4) and 23.5, there exists an n.s.f. trace 𝜏 on ℳ 𝜑 such that 𝜏 ◦ 𝜃s = e−s 𝜏

(s ∈ ℝ).

(6)

23.7. Any triple (𝒩 , 𝜃, 𝜏) consisting of a properly infinite semifinite W ∗ -algebra 𝒩 , a continuous action 𝜃 ∶ ℝ → Aut(𝒩 ) and an n.s.f. trace 𝜏 on 𝒩 such that 𝜏 ◦ 𝜃s = e−s 𝜏 (s ∈ ℝ), is called a continuous decomposition.

336

Continuous Decompositions

By Theorem 23.6, every properly infinite W ∗ -algebra ℳ has a continuous decomposition (𝒩 , 𝜃, 𝜏) such that ℳ = ℛ(𝒩 , 𝜃); in this case, (𝒩 , 𝜃, 𝜏) will be called a continuous decomposition of ℳ. Proposition. Let (𝒩 , 𝜃, 𝜏) be any continuous decomposition and let ℳ = ℛ(𝒩 , 𝜃) with the dual action 𝜃̂ ∶ ℝ → Aut(ℳ) and the dual weight 𝜑 = 𝜏̂ ∈ Wnsf (ℳ). Then 𝜑 is a dominant weight on ℳ, 𝜎 𝜑 = 𝜃̂ and ℳ 𝜑 = 𝜋𝜃 (𝒩 ). ̄ 𝜆 (s)) = Proof. Indeed, we have 𝜎t𝜑 (𝜋𝜃 (x)) = 𝜋𝜃 (𝜎t𝜏 (x)) = 𝜋𝜃 (x) (x ∈ 𝒩 ), and 𝜎t𝜑 (1 ⊗ 𝜑 −ist ̄ 𝜆 (s))𝜋𝜃 ([D(𝜏 ◦ 𝜃s ) ∶ D𝜏]t ) = e (1 ⊗ ̄ 𝜆 (s)) (s ∈ ℝ), hence 𝜎t = Ad(1 ⊗ ̄ m(t)) = 𝜃̂t (t ∈ ℝ). (1 ⊗ ̂ 𝜑 𝜃 Then ℳ = ℳ = 𝜋𝜃 (𝒩 ) (see 19.3.(9)) and the fact that 𝜑 is dominant now follows by using Proposition 20.12. Note also that if (𝒩 , 𝜃, 𝜏) is a continuous decomposition, ℳ = ℛ(𝒩 , 𝜃) and 𝜑 = 𝜏, ̂ then ℝ ∋ s ↦ ̄ 𝜆(s) ∈ ℳ is an s-continuous unitary representation and, identifying 𝒩 ≡ 𝜋𝜃 (𝒩 ) ⊂ ℳ, u(s) = 1 ⊗ we have 𝒩 = ℳ 𝜑 , ℳ = ℛ{𝒩 , u(ℝ)}, 𝜃s = Ad(u(s))|𝒩 (s ∈ ℝ); also 𝜎 𝜑 = 𝜃̂ is determined by 𝜎t𝜑 |𝒩 = 𝜄 and 𝜎t𝜑 (u(s)) = e−ist u(s) (s, t ∈ ℝ), hence 𝜑u(s) = e−s 𝜑 (s ∈ ℝ). 23.8. With regard to the uniqueness of the continuous decomposition of a W ∗ -algebra, we first prove the following result: Proposition. Let (𝒩1 , 𝜃1 , 𝜏1 ) and (𝒩2 , 𝜃2 , 𝜏2 ) be two continuous decompositions with 𝒩1 , 𝒩2 countably decomposable W ∗ -algebras. Then ℛ(𝒩1 , 𝜃1 ) ≈ ℛ(𝒩2 , 𝜃2 ) if and only if (𝒩1 , 𝜃1 ) ∼ (𝒩2 , 𝜃2 ). Proof. The “if” part is clear by statement 20.13.(4). Conversely, assume that ℛ(𝒩1 , 𝜃1 ) = ℛ(𝒩2 , 𝜃2 ) = ℳ. By assumption, ℳ is countably decomposable, hence the dominant weights (23.7) 𝜑1 = 𝜏̂1 , 𝜑2 = 𝜏̂2 on ℳ are equivalent, that is, (ℛ(𝒩1 , 𝜃1 ), 𝜑1 ) ≈ (ℛ(𝒩2 , 𝜃2 ), 𝜑2 ). Since 𝜎 𝜑1 = 𝜃̂1 , 𝜎 𝜑2 = 𝜃̂2 (23.7), we infer that (ℛ(𝒩1 , 𝜃1 ), 𝜃̂1 ) ≈ (ℛ(𝒩2 , 𝜃2 ), 𝜃̂2 ). Using the Takesaki ̄ ℬ(ℒ 2 (ℝ)), 𝜃1 ⊗ ̄ Ad(𝜌𝜌)) ≈ (𝒩2 ⊗ ̄ ℬ(ℒ 2 (ℝ)), 𝜃2 ⊗ ̄ Ad(𝜌𝜌)). duality theorem, we also obtain (𝒩1 ⊗ ′ On the other hand, for each j = 1, 2, there exists a continuous action 𝜃j ∶ ℝ → Aut(𝒩j ), 𝜃j′ ∼ 𝜃j , ̄ ℬ(ℒ 2 (ℝ)), 𝜃j ⊗ ̄ Ad(𝜌𝜌)) ≈ (𝒩j , 𝜃 ′ ). It follows that (𝒩1 , 𝜃 ′ ) ≈ (𝒩2 , 𝜃 ′ ) and hence such that (𝒩j ⊗ j 1 2 (𝒩1 , 𝜃1 ) ∼ (𝒩2 , 𝜃2 ). Actually, we shall see (23.12) that ℛ(𝒩1 , 𝜃1 ) ≈ ℛ(𝒩2 , 𝜃2 ) if and only if (𝒩1 , 𝜃1 ) ≈ (𝒩2 , 𝜃2 ). So far as the determination of the trace 𝜏 in a continuous decomposition (𝒩 , 𝜃, 𝜏) is concerned we recall (4.11) that any other n.s.f. trace on 𝒩 is of the form 𝜏A with A a nonsingular positive self-adjoint operator affiliated to 𝒵 (𝒩 ); it is easy to check that 𝜏A satisfies the same condition as 𝜏 with respect to (𝒩 , 𝜃) if and only if A is affiliated to 𝒵 (𝒩 )𝜃 . 23.9. In this section, we give some auxiliary results, which, in particular, will enable us to complete the previous uniqueness result. Proposition. Let 𝒩 be a semifinite W ∗ -algebra, 𝜃 ∈ Aut(𝒩 ) and 𝜏 an n.s.f. trace on 𝒩 such that 𝜏 ◦ 𝜃 ≤ 𝜆𝜏 for some 𝜆 such that 0 < 𝜆 < 1. Then the action ℤ ∋ n ↦ 𝜃 n ∈ Aut(𝒩 ) is integrable, there exists an s-continuous unitary representation u ∶ 𝕋 ↦ U(𝒩 ) such that 𝜃 n (u(𝜔)) = ∑ 𝜔n u(𝜔) (𝜔 ∈ 𝕋 , n ∈ ℤ), and there exists a projection e ∈ 𝒩 such that n∈ℤ 𝜃 n (e) = 1. Proof. The family {f ∈ Proj(𝒩 ); 𝜃 n ( f ) (n ∈ ℤ) are mutually orthogonal} is inductively ordered and hence, by Zorn’s lemma, it has a maximal element which we denote by e ∈ 𝒩 . Let e0 = ∑ n n∈ℤ 𝜃 (e). If f0 = 1 − e0 ≠ 0, there exists a projection p ∈ 𝒩 , 𝜃 ≠ p ≤ f0 , with 𝜏(p) < +∞.

Dominant Weights and Continuous Decompositions

337

⋁∞ ∑∞ ∑∞ Putting q = n=0 𝜃 n (p) we have 𝜏(q) ≤ n=0 𝜏(𝜃 n (p)) ≤ n=0 𝜆n 𝜏(p) = (1 − 𝜆)−1 𝜏(p) < +∞ and 𝜃(q) ≤ q, 𝜏(𝜃(q)) ≤ 𝜆𝜏(q) < 𝜏(q); hence f = q − 𝜃(q) ≠ 0. However the existence of the projection e+f contradicts the maximality of e, since (e+f )(𝜃 n (e)+𝜃 n ( f )) = e𝜃 n (e)+f𝜃 n (e)+e𝜃 n ( f )+f𝜃 n ( f ) = 0; indeed, we have e𝜃 n (e) = 0 by the choice of e, then f𝜃 n (e) = 0 = e𝜃 n ( f ) since f ≤ 1 − e0 and finally f𝜃 n ( f ) = 0 as f = q − 𝜃(q). Hence e0 = 1. ∑ ∑ We have n∈ℤ 𝜃 n (a) ∈ 𝒩 + for any finite partial sum a of the series n∈ℤ 𝜃 n (e). Since these finite partial sums converge to e0 = 1 with respect to the s-topology, it follows that the action n ↦ 𝜃 n is indeed integrable. ∑ Recall that 𝕋 = {𝜔 ∈ ℂ; |𝜔| = 1} is the dual group of ℤ. By defining u(𝜔) = n∈ℤ 𝜔−m 𝜃 m (e) ∈ U(𝒩 ) (𝜔 ∈ 𝕋 ), we obtain an s-continuous unitary representation u ∶ 𝕋 → U(𝒩 ) and it is easy to check that 𝜃 n (u(𝜔)) = 𝜔n u(𝜔) (𝜔 ∈ 𝕋 , n ∈ ℤ). Using Landstad’s theorem (19.9) and Proposition 20.12, we deduce Corollary. Let 𝒩 be a semifinite W ∗ -algebra, 𝜃 ∈ Aut(𝒩 ) and 𝜏 an n.s.f. trace on 𝒩 such that 𝜏 ◦ 𝜃 ≤ 𝜆t for some 𝜆 such that 0 < 𝜆 < 1. Then there exists a continuous action u ∶ 𝕋 → Aut(𝒩 𝜃 ) such that (𝒩 , 𝜃) ≈ (ℛ(𝒩 𝜃 , 𝜎), 𝜎). ̂ In particular, if 𝒩 𝜃 is properly infinite, then 𝜃 ∶ ℤ ∋ n ↦ 𝜃 n ∈ Aut(𝒩 ) is a dominant action. 23.10. For continuous decompositions, we obtain the following similar result: Proposition. Let 𝒩 be a semifinite W ∗ -algebra, 𝜃 ∶ ℝ → Aut(𝒩 ) a continuous action and 𝜏 an n.s.f. trace on 𝒩 such that 𝜏 ◦ 𝜃s = e−s 𝜏 (s ∈ ℝ). Then 𝜃 is an integrable action, there exists a 𝜃-invariant n.s.f. weight 𝜑 on 𝒩 and there exists an s-continuous unitary representation v ∶ ℝ → U(𝒩 ) such that 𝜃s (v(t)) = eist v(t) (s, t ∈ ℝ). Proof. The *-automorphism 𝜃1 ∈ Aut(𝒩 ) satisfies 𝜏 ◦ 𝜃1 = e−1 𝜏. By Proposition 23.9, it follows ∑ that n ↦ 𝜃1n = 𝜃n is an integrable action of ℤ on 𝒩 . If a ∈ 𝒩 + is 𝜃1 -integrable, that is, n∈ℤ 𝜃n (a) ∈ 𝒩 + , then +∞

∫−∞

𝜃t (a) dt =

∑ n

n+1

∫n 1

=

∫0

𝜃t

𝜃t (a) dt =

( ∑

) 𝜃n (a)

∑ n

( 𝜃n

1

∫0

) 𝜃t (a) dt

dt ∈ 𝒩 + .

n

Hence 𝜃 is an integrable action.

+

Thus, there exists a 𝜃-invariant n.s.f. operator-valued weight P𝜃 ∶ 𝒩 + → (𝒩 𝜃 ) (see 18.19, 20.6). If 𝜓 is any n.s.f. weight on 𝒩 𝜃 , then 𝜑 = 𝜓 ◦ P𝜃 is a 𝜃-invariant n.s.f. weight on 𝒩 . Let A be the unique nonsingular positive self-adjoint operator affiliated to 𝒩 such that 𝜑 = 𝜏A . Since 𝜏 ◦ 𝜃s = e−s 𝜏 and 𝜑 ◦ 𝜃s = 𝜑, it follows that 𝜃s (A) = e−s A (s ∈ ℝ). Then ℝ ∋ t ↦ v(t) = A−it ∈ 𝒩 is an s-continuous unitary representation and 𝜃s (v(t)) = eist v(t) (s, t ∈ ℝ). Using Landstad’s theorem (19.9) and Proposition 20.12, we deduce Corollary. Let 𝒩 be a semifinite W ∗ -algebra, 𝜃 ∶ ℝ → Aut(𝒩 ) a continuous action and 𝜏 an n.s.f. trace on ℳ such that 𝜏 ◦ 𝜃s = e−s 𝜏 (s ∈ ℝ). Then there exists a continuous action 𝜎 ∶ ℝ → Aut(𝒩 𝜃 ) such that (𝒩 , 𝜃) ≈ (ℛ(𝒩 𝜃 , 𝜎), 𝜎). ̂ In particular, if 𝒩 𝜃 is properly infinite, then 𝜃 ∶ ℝ → Aut(𝒩 ) is a dominant action.

338

Continuous Decompositions

23.11. Let (𝒩 , 𝜃, 𝜏) be a continuous decomposition. In this section, we study the type of the crossed product W ∗ -algebra ℛ(𝒩 , 𝜃). We shall use the notation, conventions and results of Section 23.7. Proposition 1. We have 𝒵 (ℛ(𝒩 , 𝜃)) = 𝒵 (𝒩 )𝜃 . In particular, ℛ(𝒩 , 𝜃) is a factor if and only if 𝜃 acts ergodically on 𝒵 (𝒩 ). Proof. The inclusion 𝒵 (𝒩 )𝜃 ⊂ 𝒵 (ℛ(𝒩 , 𝜃)) is obvious (21.6.(2)). Conversely, if z ∈ 𝒵 (ℳ), then z ∈ ℳ 𝜑 = 𝒩 , z commutes with each element of 𝒩 , that is, z ∈ 𝒵 (𝒩 ), and z commutes with u(s) (s ∈ ℝ), hence z ∈ 𝒵 (𝒩 )𝜃 . Proposition 2. The following statements are equivalent: (i) (ii) (iii) (iv)

ℛ(𝒩 , 𝜃) is semifinite; 𝒩 𝜃 is semifinite; ̄ ℒ ∞ (ℝ), 𝜄 ⊗ ̄ Ad(𝜆 𝜆)); (𝒩 , 𝜃) ≈ (𝒩 𝜃 ⊗ there exists an s-continuous unitary representation ℝ ∋ t ↦ v(t) ∈ 𝒵 (𝒩 ) such that 𝜃s (v(t)) = eist v(t) (s, t ∈ ℝ).

Proof. (i) ⇔ (ii). Indeed, using Corollary 23.10 and the Takesaki duality theorem (19.5), we get ̄ ℬ(ℒ 2 (ℝ)). ℛ(𝒩 , 𝜃) ≈ 𝒩 𝜃 ⊗ (iii) ⇔ (iv). This follows immediately using Landstad’s theorem (19.9). (i) ⇒ (iv). If ℳ is semifinite, then the modular automorphism group 𝜎 𝜑 is inner, that is, there exists an s-continuous unitary representation ℝ ∋ t ↦ v(t) ∈ ℳ, such that 𝜎t𝜑 = Ad(v(t)) (t ∈ ℝ). Then v(t) ∈ ℳ 𝜑 = 𝒩 and for x ∈ 𝒩 = ℳ 𝜑 we have v(t)xv(t)∗ = 𝜎t𝜑 (x) = x, hence v(t) ∈ 𝒵 (𝒩 ) (t ∈ ℝ). Then e−ist u(s) = 𝜎t𝜑 (u(s)) = v(t)u(s)v(t)∗ , hence 𝜃s (v(t)) = u(s)v(t)u(s)∗ = eist v(t) (s, t ∈ ℝ). (iv) ⇒ (i). If condition (iv) is satisfied, then for x ∈ 𝒩 = ℳ 𝜑 we have 𝜎t𝜑 (x) = x = v(t)xv(t)∗ , and for s ∈ ℝ we have 𝜎t𝜑 (u(s)) = e−ist u(s) = v(t)u(s)v(t)∗ , hence 𝜎t𝜑 = Ad(v(t)) (t ∈ ℝ), since ℳ = ℛ{𝒩 , u(ℝ)}. Thus, 𝜎 𝜑 is inner and ℳ is semifinite. Actually, it is also clear that (iii) ⇒ (i). Since for every projection p ∈ 𝒵 (ℳ) = 𝒵 (𝒩 )𝜃 , the triple (𝒩 p, 𝜃|𝒩 p, 𝜏|𝒩 p) is also a continuous decomposition and ℳp = ℛ(𝒩 p, 𝜃|𝒩 p), we get the following Corollary. The following statements are equivalent: (i) ℛ(𝒩 , 𝜃) is of type III; (ii) for every nonzero projection p ∈ 𝒵 (𝒩 )𝜃 , 𝒩 𝜃 p is not semifinite; (iii) for every nonzero projection p ∈ 𝒵 (𝒩 )𝜃 , ̄ ∞ (ℝ), 𝜄 ⊗ ̄ Ad(𝜆 𝜆)); (𝒩 p, 𝜃|𝒩 p) ≉ (𝒩 𝜃 p ⊗ℒ (iv) for every nonzero projection p ∈ 𝒵 (𝒩 )𝜃 , there is no s-continuous unitary representation v ∶ ℝ → 𝒵 (𝒩 )p such that 𝜃s (v(t)) = eist v(t) (s, t ∈ ℝ). Proposition 3. If ℛ(𝒩 , 𝜃) is of type III, then 𝒩 is of type II∞ . Proof. Assume that ℛ(𝒩 , 𝜃) is of type III, but 𝒩 is not of type II∞ . In accordance with ̄ F, where Proposition 1, we can then assume that 𝒩 is homogeneous of type I, hence 𝒩 = 𝒵 ⊗ 𝒵 ≈ 𝒵 (𝒩 ) and ℱ is an infinite dimensional factor of type I.

Dominant Weights and Continuous Decompositions

339

̄ tr)A with 𝜇 an n.s.f. trace on 𝒵 , tr the canonical Any n.s.f. trace on 𝒩 is then of the form (𝜇 ⊗ ̄ 1F . Hence trace on ℱ and A a nonsingular positive self-adjoint operator affiliated to 𝒵 (𝒩 ) = 𝒵 ⊗ ̄ tr for some n.s.f. trace 𝜇 on 𝒵 . 𝜏=𝜇⊗ ̄ 1𝒜 and consider 𝛼𝜏 = (𝜃t |𝒵 ) ⊗ ̄ 𝜄 ∈ Aut(𝒩 ) (t ∈ ℝ). We identify 𝒵 with 𝒵 (𝒩 ) = 𝒵 ⊗ Then 𝛼−t ◦ 𝜃t ∈ Aut(𝒩 ) acts identically on 𝒵 (𝒩 ) and hence ([L], 8.11) is inner, that is, there exists w(t) ∈ U(𝒩 ) such that 𝛼−t (𝜃t (x)) = w(t)xw(t)∗ for every x ∈ 𝒩 . Consequently, we have 𝜃t (x) = 𝛼t (w(t))𝛼t (x)𝛼t (w(t))∗

(x ∈ 𝒩 , t ∈ ℝ).

Consider now a ∈ ℱ ⊂ 𝒩 , a ≥ 0, with tr(a) < +∞. For z ∈ 𝒵 ⊂ 𝒩 and t ∈ ℝ, we have e−t 𝜇(z)tr(b) = e−t 𝜏(zb) = (𝜏 ◦ 𝜃t )(zb) = 𝜏(𝛼t (w(t))𝛼t (zb)𝛼t (w(t))∗ ) = 𝜏(𝛼t (zb)) = 𝜏(𝜃t (z)b) = 𝜇(𝜃t (z))tr(b). This shows that 𝜇 ◦ 𝜃t = e−t 𝜇. Using the equivalence (i) ⇔ (iv) in Proposition 2 and the same arguments as in the proof of Proposition 23.10, we can now conclude that ℛ(𝒩 , 𝜃) is semifinite, a contradiction. 23.12 Theorem. Let (𝒩 , 𝜃, 𝜏) be a continuous decomposition with 𝒩 a countably decomposable W ∗ -algebra. Then every unitary cocycle u ∈ Z𝜃 (ℝ; U(𝒩 )) is trivial, that is, there exists v ∈ U(𝒩 ) such that u(t) = v∗ 𝜃t (v) (t ∈ ℝ). Proof. By Corollary 23.10, there exists a continuous action 𝜎 ∶ ℝ → Aut(𝒩 𝜃 ) such that (𝒩 , 𝜃) ≈ (ℛ(𝒩 𝜃 , 𝜎), 𝜎). ̂ We may assume that (𝒩 , 𝜃) = ℛ(𝒩 𝜃 , 𝜎), 𝜎). ̂ Let p ∈ 𝒵 (𝒩 𝜃 ) be the unique projection such that 𝒩 𝜃 p is semifinite and 𝒩 𝜃 (1 − p) is purely infinite. Then 𝜎t (p) = p (t ∈ ℝ), hence p ∈ 𝒵 (ℛ(𝒩 𝜃 , 𝜎)) = 𝒵 (𝒩 ). It is now easy to see that we can divide the proof into two cases, assuming first that 𝒩 𝜃 is semifinite and then that 𝒩 𝜃 is purely infinite. ̄ ℒ ∞ (ℝ), 𝜄 ⊗ ̄ Ad(𝜆 𝜆)) If 𝒩 𝜃 is semifinite, then, by Proposition 2/23.11, we have (𝒩 , 𝜃) ≈ (𝒩 ⊗ and the required conclusion follows from Proposition 21.12. If 𝒩 𝜃 is purely infinite, then, by Corollary 23.10, 𝜃 is a dominant action, hence the trivial cocycle 1 ∈ Z𝜃 (ℝ; U(𝒩 )) is a dominant cocycle. Let u ∈ Z𝜃 (ℝ; U(𝒩 )) and 𝜃 ′ = u 𝜃. Then for x ∈ 𝒩 + and s ∈ ℝ, we have 𝜏(𝜃s′ (x)) = 𝜏(u(s)𝜃s (x)u(s)∗ ) = 𝜏(𝜃s (x)) = e−s 𝜏(x), hence also (𝒩 , 𝜃 ′ , 𝜏) is a continuous decomposition. Since 𝜃 ′ ∼ 𝜃, using Corollary 23.10 and the Takesaki duality theorem ′ ̄ ℬ(ℒ 2 (ℝ)), so that 𝒩 𝜃′ is ̄ ℬ(ℒ 2 (ℝ)) ≈ ℛ(𝒩 , 𝜃 ′ ) ≈ ℛ(𝒩 , 𝜃) ≈ 𝒩 𝜃 ⊗ (19.5) we get 𝒩 𝜃 ⊗ also purely infinite. It follows that the cocycle 1 ∈ Z𝜃′ (ℝ; U(𝒩 )) is dominant, hence the cocycle u ∈ Z𝜃 (ℝ; U(𝒩 )) is dominant. By Theorem 20.5, we conclude that u ≂ 1, that is, there exists v ∈ U(𝒩 ) such that u(t) = v∗ 𝜃t (v) (𝜏 ∈ ℝ). Using the previous Theorem and Proposition 23.8 we obtain: Corollary. Let (𝒩1 , 𝜃1 , 𝜏1 ) and (𝒩2 , 𝜃2 , 𝜏2 ) be two continuous decompositions with 𝒩1 , 𝒩2 countably decomposable W ∗ -algebras. Then ℛ(𝒩1 , 𝜃1 ) ≈ ℛ(𝒩2 , 𝜃2 ) if and only if (𝒩1 , 𝜃1 ) ≈ (𝒩2 , 𝜃2 ). 23.13. In this section, we give a discrete variant of Theorem 23.12. For its proof, we require the following result (compare with 23.9).

340

Continuous Decompositions

Lemma. Let 𝜃 be a *-automorphism of the abelian W ∗ -algebra 𝒵 such that there exists a nonsingular positive self-adjoint operator A affiliated to 𝒵 with 𝜃(A) ≤ 𝜆A for some 0 < 𝜆 < 1. ∑ Then there exists a projection e ∈ 𝒵 such that n∈ℤ 𝜃 n (e) = 1 and there exists an s-continuous unitary representation u ∶ 𝕋 → U(𝒵 ) such that 𝜃 n (u(𝜔)) = 𝜔n u(𝜔) (𝜔 ∈ 𝕋 , n ∈ ℤ). Proof. In order to prove the first assertion, it is sufficient to show that every 𝜃-invariant nonzero projection p ∈ 𝒵 majorizes a nonzero projection e ∈ 𝒵 with the property that 𝜃 i (e)𝜃 j (e) = 0 for all i, j ∈ ℤ, i ≠ j. Without loss of generality, we may assume that p = 1. There exists n ∈ ℤ such that the spectral projection e of A corresponding to the interval determined by 𝜆n and 𝜆n+1 is nonzero. If n ≥ 0, then e = 𝜒(𝜆n+1 ,𝜆n ] (A) ≠ 0 and for every k ≥ 1 we have 𝜃 −k (A) ≥ 𝜆−k A, so that 𝜃 −k (e) = 𝜒(𝜆n+1 ,𝜆n ] (𝜃 −k (A)) ≤ 𝜒(0,𝜆n ] (𝜃 −k (A)) ≤ 𝜒(0,𝜆n ] (𝜆−k A) = 𝜒(0,𝜆n+k ] (A), and hence e𝜃 −k (e) = 0, as 𝜆n+k ≤ 𝜆n+1 ; therefore 𝜃 i (e)𝜃 j (e) = 0 for every i, j ∈ ℤ, i ≠ j. If n < 0, a similar argument leads to the same conclusion. The other assertion in the statement follows now as in the proof of Proposition 23.9. Theorem. Let 𝒩 be a countably decomposable semifinite W ∗ -algebra, 𝜃 ∈ Aut(𝒩 ) and 𝜏 an n.s.f. trace on 𝒩 such that 𝜏 ◦ 𝜃 ≤ 𝜆𝜏 for some 0 < 𝜆 < 1. Then for every u ∈ U(𝒩 ) there exists v ∈ U(𝒩 ) such that u = v∗ 𝜃(v). Proof. By Corollary 23.9, there exists a continuous action 𝜎 ∶ 𝕋 → Aut(𝒩 𝜃 ) such that (𝒩 , 𝜃) ≈ (ℛ(𝒩 𝜃 , 𝜎), 𝜎). ̂ As in the proof of Theorem 23.12, we can consider separately the cases when 𝒩 𝜃 is semifinite and purely infinite. Assume that 𝒩 𝜃 is semifinite and let 𝜇0 be an n.s.f. trace on 𝒩 𝜃 . Since 𝕋 is a compact group, +

the faithful normal operator-valued weight P𝜎 ∶ (𝒩 𝜃 )+ → ((𝒩 𝜃 )𝜎 ) is finite and 𝜎-invariant, hence 𝜇 = 𝜇0 ◦ P𝜎 is a 𝜎-invariant n.s.f. trace on 𝒩 𝜃 . The dual weight 𝜑 = 𝜇̂ on 𝒩 = ℛ(𝒩 𝜃 , 𝜎) is invariant under the dual action 𝜃 = 𝜎. ̂ Since 𝜇 is a 𝜎-invariant trace, we see using Theorem 19.8 that 𝜑 = 𝜇̂ is a 𝜃-invariant n.s.f. trace on 𝒩 . Therefore, there exists a nonsingular positive selfadjoint operator A affiliated to 𝒵 (𝒩 ) such that 𝜑 = 𝜏A . Since 𝜑 ◦ 𝜃 = 𝜑 and 𝜏 ◦ 𝜃 ≤ 𝜆𝜃, it follows that 𝜃(A) ≤ 𝜆A. By the above lemma there exists an s-continuous unitary representation u ∶ 𝕋 → 𝒵 (𝒩 ) such that 𝜃 n (u(𝜔)) = 𝜔n u(𝜔) for all 𝜔 ∈ 𝕋 , n ∈ ℤ. Using Landstad’s theorem (19.9), we ̄ 𝓁 ∞ (ℤ), 𝜄 ⊗ ̄ Ad(𝜆 𝜆)). Consider now a unitary element u ∈ 𝒩 ; since infer that (𝒩 , 𝜃) ≈ (𝒩 𝜃 ⊗ ̄ 𝓁 ∞ (ℤ) ≡ 𝓁 ∞ (ℤ, 𝒩 𝜃 ), u is a sequence {un }n∈ℤ of unitary elements in 𝒩 𝜃 . We define 𝒩 = 𝒩𝜃 ⊗ a unitary element v = {vn }n∈ℤ in 𝒩 by putting v0 = 1, vn+1 = vn un for n ≥ 1 and vn = vn+1 u∗n for n < 0. Then 𝜃(v) = {wn }n∈ℤ where wn = vn+1 (n ∈ ℤ), hence v∗ 𝜃(v) = {v∗n vn+1 }n∈ℤ = {un }n∈ℤ = u. If 𝒩 𝜃 is purely infinite, then, using Corollary 23.9 and the correspondence between 𝜃-cocycles and unitary elements u ∈ U(𝒩 ) (see 16.15.(2)), the proof is similar to the corresponding part of the proof of Theorem 23.12. The above proof also shows that 𝒩 𝜃 is finite if and only if 𝒩 is finite. 23.14. A continuous action 𝜃 ∶ G → Aut(𝒩 ) of a locally compact group G on the W ∗ -algebra 𝒩 will be called stable if every unitary cocycle u ∈ Z𝜃 (G; U(ℳ)) is trivial, that is, if there exists v ∈ U(𝒩 ) such that u(t) = v∗ 𝜃t (v) (t ∈ G); in this case, Ad(v) establishes a *-isomorphism (𝒩 , 𝜃) ≈ (𝒩 , u 𝜃) so that the actions 𝜃 and u 𝜃 are inner conjugate. In particular, a *-automorphism 𝜃 ∈ Aut(𝒩 ) will be called stable if the action 𝜃 ∶ ℤ ∈ n ↦ 𝜃 n ∈ Aut(𝒩 ) is stable. This means that for every u ∈ U(𝒩 ) there exists v ∈ U(𝒩 ) with v∗ 𝜃(v) = u; in this case, Ad(v) establishes a *-isomorphism (𝒩 , 𝜃) ≈ (𝒩 , Ad(u) ◦ 𝜃), hence the *-automorphisms 𝜃 and Ad(u) ◦ 𝜃 are inner conjugate.

Dominant Weights and Continuous Decompositions

341

Notable examples of stable actions are given in Theorems 23.12, 23.13, Proposition 21.12, and Corollary 22.14. Another important example is given in the next corollary. Corollary. Let 𝒩 be a type II∞ factor, 𝜏 an n.s.f. trace on 𝒩 and 𝜃 ∈ Aut(𝒩 ). Then 𝜃 is stable if and only if 𝜏 ◦ 𝜃 ≠ 𝜏. Proof. If 𝜏 ◦ 𝜃 ≠ 𝜏, then 𝜏 ◦ 𝜃 = 𝜆𝜏 for some 𝜆 ≠ 1 and, replacing 𝜃 by 𝜃 −1 if necessary, we may assume that 0 < 𝜆 < 1. By Theorem 23.13, it follows that 𝜃 is stable. Assume now that 𝜏 ◦ 𝜃 = 𝜏. There exists a projection e ∈ 𝒩 with 0 ≠ 𝜏(e) < +∞. Then e is finite and, since 𝜏(𝜃(e)) = 𝜏(e), we have 𝜃(e) ∼ e. Thus, there exists a unitary operator u ∈ U(𝒩 ) such that 𝜃(e) = ueu∗ . Then (Ad(u) ◦ 𝜃)(e) = e and hence the positive normal form 𝜏(e⋅) on 𝒩 is (Ad(u) ◦ 𝜃)-invariant. It follows that Ad(u) ◦ 𝜃 is not integrable. On the other hand, the existence of a dominant cocycle shows that there exists u′ ∈ U(𝒩 ) such that Ad(u′ ) ◦ 𝜃 is integrable. It is then obvious that Ad(u) ◦ 𝜃 and Ad(u′ ) ◦ 𝜃 are not conjugate, hence 𝜃 is not stable. 23.15. Let ℳ be a properly infinite W ∗ -algebra and 𝜑 a normal semifinite weight on ℳ. There ∑ exists a sequence {wn } ⊂ ℳ of isometries such that n wn w∗n = 1. Then un = (wn s(𝜑))∗ is a partial isometry with un u∗n = s(𝜑) and the projections u∗n un are mutually orthogonal. We define 𝜑n = 𝜑un . Then 𝜑n ≂ 𝜑 and 𝜑n have mutually orthogonal supports. Thus, we can consider the normal semifinite ∑ weight n 𝜑n on ℳ; we have ( ℳ,

∑

) 𝜑n

̄ ℱ∞ , 𝜑 ⊗ ̄ tr). ≈ (ℳ ⊗

n

∑ ̄ eij (x ∈ ℳ), Indeed, if {eij } is a system of matrix units for ℱ∞ , then the equation 𝜋(x) = ij w∗i xwj ⊗ ∑ ̄ ̄ defines a *-isomorphism 𝜋 ∶ (ℳ, n 𝜑n ) → (ℳ ⊗ ℱ∞ , 𝜑 ⊗ tr). ∑ The weight 𝜑̌ = ̌ and (ℳ, 𝜑) ̌ ≈ n 𝜑n ∈ Wns (ℳ) is of infinite multiplicity, 𝜑 ≲ 𝜑 ̄ ℱ∞ , 𝜑 ⊗ ̄ tr). If 𝜑 is of infinite multiplicity, then (ℳ, 𝜑) (ℳ ⊗ ̌ ≈ (ℳ, 𝜑) by Corollary 9.18. By construction, if 𝜑, 𝜓 ∈ Wns (ℳ) and 𝜑 ≂ 𝜓n then 𝜑̌ ≂ 𝜓. ̌ Similarly, for every sequence {𝜓n } ⊂ Wns (ℳ), we can construct a sequence {𝜑n } ⊂ Wns (ℳ) with ∑ s(𝜑n ) mutually orthogonal and 𝜑n ≂ 𝜓n such that the equivalence class of the weight n 𝜑n depends only on the equivalence classes of the weights 𝜓n . This equivalence class, or some representative of ∑⊕ it, will be denoted by n 𝜓n . 23.16. In the next Theorem, we consider some properties related to the dominance property of a weight. Let 𝜔 be the unique n.s.f. weight on ℱ∞ = ℬ(ℒ 2 (ℝ)) such that [D𝜔 ∶ D tr]t = 𝜌(t) (t ∈ ℝ). Theorem. Let ℳ be a properly infinite W ∗ -algebra with separable predual and 𝜑 an n.s.f. weight on ℳ. Consider the statements: (i) (ii) (iii) (iv) (v)

𝜑 is a dominant weight; ̄ ℱ∞ , 𝜓 ⊗ ̄ 𝜔); there exists an n.s.f. weight 𝜓 on ℳ of infinite multiplicity such that (ℳ, 𝜑) ≈ (ℳ ⊗ 𝜑 is the dual weight of the trace appearing in some continuous decomposition of ℳ; there exists an n.s.f. weight 𝜓 on ℳ which anticommutes with 𝜑; 𝜑 ≂ 𝜆𝜑 for every 𝜆 > 0.

Then, (v) ⇐ (i) ⇔ (ii) ⇔ (iii) ⇒ (iv). If ℳ is a factor, then (iv) ⇒ (v). If ℳ is of type III, then (i) ⇔ (v). Therefore, for factors of type III with separable preduals, (i)–(v) are equivalent.

342

Continuous Decompositions

Proof. The equivalence (i) ⇔ (ii) follows from Section 23.4 and the equivalence (i) ⇔ (iii) from Sections 23.6 and 23.7; the implication (i) ⇒ (v) is obvious and the implication (i) ⇒ (iv) has been proved in Section 23.4. Assume now that ℳ is a factor and that there exists an n.s.f. weight 𝜓 on ℳ which anticommutes 𝜑 with 𝜑. Let u(t) = [D𝜓 ∶ D𝜑]t (t ∈ ℝ). Then, for x ∈ ℳ and s, t ∈ ℝ, we have u(t)𝜎s+t (x)u(t)∗ = 𝜑 𝜑 𝜓 𝜑 𝜑 𝜓 𝜑 𝜑 𝜑 𝜑 𝜑 ∗ ∗ u(t)𝜎t (𝜎s (x))u(t) = 𝜎t (𝜎s (x)) = 𝜎s (𝜎t (x)) = 𝜎s (u(t)𝜎t (x)u(t) ) = 𝜎t (u(t))𝜎s+t (x)𝜎t (u(t)∗ , that is, 𝜎s𝜑 (u(t))u(t)∗ ∈ 𝒵 (ℳ) = ℂ⋅1ℳ . Thus, there exists a continuous function f ∶ ℝ×ℝ → 𝕋 such that 𝜎s𝜑 (u(t)) = f (s, t)u(t) (s, t ∈ ℝ). Since 𝜎s𝜑1 +s2 = 𝜎s𝜑1 𝜎s𝜑2 , we have f (s1 + s2 , t) = f (s1 , t)f (s2 , t) and, since u(t1 +t2 ) = u(t1 )𝜎t𝜑1 (u(t2 )), we have f (s, t1 +t2 ) = f (s, t1 )f (s, t2 ). It follows that there exists r ∈ ℝ such that f (s, t) = eirst (s, t ∈ ℝ). Since 𝜑 does not commute with 𝜓, there exists t ∈ ℝ with u(t) ∉ ℳ 𝜑 , hence r ≠ 0. We have 𝜎s𝜑 (u(t)) = (ert )is u(t) (s, t ∈ ℝ), and {ert ; t ∈ ℝ} = {𝜆 ∈ ℝ; 𝜆 > 0}. Hence for every 𝜆 > 0, there exists v(𝜆) ∈ U(ℳ) such that 𝜎s𝜑 (u(𝜆)) = 𝜆is u(𝜆) (s ∈ ℝ), so that 𝜑 ≂ 𝜑v(𝜆) = 𝜆𝜑. Thus, (iv) ⇒ (v) whenever ℳ is a factor. Finally, assume that ℳ is a type III W ∗ -algebra and 𝜑 ≂ 𝜆𝜑 for every 𝜆 > 0. Then the weight 𝜑̌ (23.15) is of infinite multiplicity and 𝜑̌ ≂ 𝜆𝜑̌ for every 𝜆 > 0, hence 𝜑̌ is dominant (23.3). Since 𝜑 ≲ 𝜑, ̌ 𝜑 is integrable. Let 𝜓 be any dominant weight on ℳ and (ℳ 𝜓 , 𝜃, 𝜏) the corresponding continuous decomposition of ℳ (23.6). Since 𝜑 is integrable we have 𝜑 ≲ 𝜓, so that we may assume 𝜑 = 𝜓f for some projection f ∈ ℳ 𝜓 . We shall show that the projection f is properly infinite in ℳ 𝜓 , that is, 𝜏( fp) = +∞ for every nonzero projection p ∈ 𝒵 (ℳ 𝜓 ). Recall (23.7) that there exists an s-continuous unitary representation u ∶ ℝ → U(ℳ) such that 𝜃s = Ad(u(s))|ℳ 𝜓 and 𝜓u(s) = e−s 𝜓 (s ∈ ℝ). Thus, e−s 𝜑 = e−s 𝜓f = (e−s 𝜓)f = (𝜓u(s) )f = 𝜓u(s)f (s ∈ ℝ). Since 𝜑 ≂ e−s 𝜑, it follows that 𝜓f = 𝜓u(s)f , hence f ∼ u(s)fu(s)∗ in ℳ 𝜓 (23.2), that is, 𝜃s ( f ) ∼ f in ℳ 𝜓 (s ∈ ℝ). Thus, for every projection q ∈ 𝒵 (ℳ 𝜓 ), we have 𝜏( f𝜃s (q)) = 𝜏(𝜃s ( fq)) = e−s 𝜏( fq) and therefore 𝜏( f𝜃s (z)) = e−s 𝜏( fz) for all z ∈ 𝒵 (ℳ 𝜓 )+ and all s ∈ ℝ. Let p be the least upper bound of the projections q ∈ 𝒵 (ℳ 𝜓 ) with 𝜏( fq) < +∞. The above arguments show that 𝜃s (p) = p for all s ∈ ℝ and that 𝜏( fp⋅) is an n.s.f. trace on 𝒵 (ℳ 𝜓 )p. Using Proposition 1/23.11, we obtain p ∈ 𝒵 (ℳ 𝜓 )𝜃 = 𝒵 (ℳ). On the other hand, using the results of Section 23.10 applied to the triple (𝒵 (ℳ 𝜓 )p, 𝜃|𝒵 (ℳ 𝜓 )p, 𝜏( fp⋅)) and the equivalence (i) ⇔ (iv) in Proposition 23.11, we conclude that ℳp is semifinite. Since ℳ is of type III, it follows that p = 0. 23.17. During the proof of Theorem 23.16, we have also shown that if 𝜑 and 𝜓 are two anticommuting n.s.f. weights on the factor ℳ, then there exists a unique real number r = r(𝜑, 𝜓) ∈ ℝ, r ≠ 0, such that 𝜎s𝜑 [D𝜓 ∶ D𝜑]t ) = eirst [D𝜓 ∶ D𝜑]t

(s, t ∈ ℝ).

(1)

If the above identity holds with r = 0, then 𝜑 and 𝜓 commute. The next Theorem shows that the number r(𝜑, 𝜓) completely determines the equivalence class of the pair (𝜑, 𝜓). Theorem. Let ℳ be a factor with separable predual. Let (𝜑, 𝜓) and (𝜑′ , 𝜓 ′ ) be two anticommuting pairs of dominant weights on ℳ. Then r(𝜑, 𝜓) = r(𝜑′ , 𝜓 ′ ) if and only if there exists u ∈ U(ℳ) such that 𝜑′ = 𝜑u and 𝜓 ′ = 𝜓u .

Dominant Weights and Continuous Decompositions

343

Proof. Assume that 𝜑′ = 𝜑u and 𝜓 ′ = 𝜓u with u ∈ U(ℳ). Then [D𝜑′ ∶ D𝜑]t = u∗ 𝜎t𝜑 (u) and 𝜑 [D𝜓 ′ ∶ D𝜓]t = u∗ 𝜎t𝜓 (u), hence [D𝜓 ′ ∶ D𝜑′ ]t = u∗ [D𝜓 ∶ D𝜑]t u and 𝜎t u (x) = u∗ 𝜎t𝜑 (uxu∗ )u (x ∈ ′ ′ ℳ, t ∈ ℝ). It follows that r(𝜑, 𝜓) = r(𝜑 , 𝜓 ). Conversely, assume that r(𝜑, 𝜓) = r(𝜑′ , 𝜓 ′ ). By the uniqueness modulo equivalence of the dominant weight, it follows that there exists u ∈ U(ℳ) with 𝜑′ = 𝜑u . We have r(𝜑u , 𝜓 ′ ) = r(𝜑, 𝜓) = r(𝜑u , 𝜓u ). Thus, we are led to showing that if 𝜑, 𝜓, 𝜓 ′ are three dominant weights on ℳ such that (𝜑, 𝜓) and (𝜑, 𝜓 ′ ) both anticommute and r(𝜑, 𝜓) = r(𝜑, 𝜓 ′ ) = r, then there exists u ∈ U(ℳ 𝜑 ) such that 𝜓 ′ = 𝜓u . Let (𝒩 , 𝜃, 𝜏) be a continuous decomposition of ℳ and let u ∶ ℝ → U(ℳ) be the s-continuous unitary representation of ℝ on ℳ such that 𝜃s = Ad(u(s))|𝒩 . We may assume that 𝜑 = 𝜏̂ (see 23.6, 23.7). Put v(t) = [D𝜓 ∶ D𝜑]t and v′ (t) = [D𝜓 ′ ∶ D𝜑]t (t ∈ ℝ). We have 𝜎s𝜑 (v(t)) = eirst v(t) and 𝜎s𝜑 (v′ (t)) = eirst v′ (t), hence v(s + t) = eirst v(s)v(t) and v′ (s + t) = eirst v′ (s)v′ (t) (s, t ∈ ℝ). Continuous 2 functions a ∶ ℝ → U(ℳ), a′ ∶ ℝ → U(ℳ) are defined by a(s) = e−irs ∕2 v(s)u(rs)∗ , a′ (s) = 2 e−irs ∕2 v′ (s)u(rs)∗ (s ∈ ℝ); we have a(s + t) = a(s)𝜃rs (a(t)), a′ (s + t) = a′ (s)𝜃rs (a′ (t)) (s, t ∈ ℝ). Using Theorem 23.12, we can find an element u ∈ U(𝒩 ) such that a(s) = ua′ (s)𝜃rs (u∗ ) (s ∈ 2 2 ℝ). It follows that for every s ∈ ℝ we have v(s) = eirs ∕2 a(s)u(rs) = eirs ∕2 ua′ (s)𝜃rs (u∗ )u(rs) = 2 eirs ∕2 ua′ (s)u(rs)u∗ u(rs)∗ u(rs) = uv′ (s)u∗ , that is, 𝜓 ′ = 𝜓u with u ∈ 𝒩 = ℳ 𝜑 . Note that if ℳ is a type III factor, then the dominance condition imposed on the weights in the statement of the theorem follows automatically from the anticommutation condition, by Theorem 23.16. 23.18. The next result asserts the possibility of approximation by integrable weights in the sense of the metric d introduced in Section 6.10. Theorem. Let 𝜑 be an n.s.f. weight of infinite multiplicity on the W ∗ -algebra ℳ. For every 𝜀 > 0, there exists an integrable n.s.f. weight 𝜓 of infinite multiplicity, commuting with 𝜑, such that d(𝜑, 𝜓) ≤ 𝜀. Proof. By assumption, ℳ 𝜑 is properly infinite, so that there exists an infinite type I W ∗ -subfactor ℱ ⊂ ℬ(ℒ 2 (ℝ)) of ℳ 𝜑 . On the other hand, there exists an operator a ∈ ℱ with absolutely continuous spectrum such that 1 − 𝜀 ≤ a ≤ 1 + 𝜀 and {a}′ ∩ ℱ is properly infinite. Since a has absolutely continuous spectrum and since the quasi-equivalence class of an so-continuous unitary representation of ℝ is completely determined by the equivalence class of its associated spectral measure, it follows that the unitary representation ℝ ∋ t ↦ ait ∈ ℬ(ℒ 2 (ℝ)) is quasi-equivalent to a subrepresentation of the regular representation 𝜌 of ℝ, so that the weight tra is quasi-equivalent to a subweight of the dominant weight on ℬ(ℒ 2 (ℝ)) and hence it is integrable. Thus, there exists an increasing sequence {en } ⊂ ℱ ⊂ ℳ 𝜑 ⊂ ℳ of projections such that en ↑ 1 and ∫ ait en a−it dt < +∞. Then 𝜓 = 𝜑a is an n.s.f. weight on ℳ with [D𝜓 ∶ D𝜑]t = ait and 𝜎t𝜓 = Ad(ait ) ◦ 𝜎t𝜑 (t ∈ ℝ), so that ∫ 𝜎t𝜓 (en ) dt = ∫ ait en a−it dt < +∞ and 𝜓 is integrable. Moreover, 𝜓 is of infinite multiplicity, since {a}′ ∩ ℱ ⊂ ℳ 𝜓 . Finally, as 1 − 𝜀 ≤ a ≤ 1 + 𝜀, it follows that d(𝜑, 𝜓) ≤ 𝜀. 23.19 Theorem (Relative Commutant Theorem). Let ℳ be a properly infinite W ∗ -algebra with separable predual. For every n.s.f. weight 𝜑 on ℳ, we have ̄ ℬ(ℒ 2 (ℝ))) = 𝒵 (ℛ(ℳ, 𝜎 𝜑 )). ℛ(ℳ, 𝜎 𝜑 )′ ∩ (ℳ ⊗

(1)

344

Continuous Decompositions

For every integrable n.s.f. weight 𝜑 on ℳ, we have (ℳ 𝜑 )′ ∩ ℳ = 𝒵 (ℳ 𝜑 ).

(2)

Proof. Since any integrable n.s.f. weight is equivalent to a subweight of the dominant weight (23.4), in proving (2) we may assume that 𝜑 is a dominant weight. Let 𝜔 be the unique n.s.f. weight on ℬ(ℒ 2 (ℝ)) such that [D𝜔 ∶ D tr]t = 𝜌 (t) (t ∈ ℝ). Then, for ̄ 𝜔 is a dominant weight on ℳ ⊗ ̄ ℬ(ℒ 2 (ℝ)) ≈ ℳ, and any n.s.f. weight 𝜑 on ℳ, 𝜑 ⊗ ̄ 2 ̄ ℬ(ℒ 2 (ℝ)))𝜎 𝜑 ⊗̄ Ad(𝜌𝜌) = ℛ(ℳ, 𝜎 𝜑 ) ̄ (ℝ)))𝜑 ⊗ 𝜔 = (ℳ ⊗ (ℳ ⊗ℬ(ℒ

by Corollary 19.13. Since any two dominant weights are equivalent (23.4), we see that, in order to prove the entire theorem, it is sufficient to prove (1) just for a particular choice of 𝜑. Thus, let 𝜑 be a faithful normal state on ℳ and 𝜎 = 𝜎 𝜑 be its modular automorphism group. We may assume ℳ ⊂ ℬ(ℋ ) realized as a von Neumann algebra with a cyclic and separating vector 𝜉0 ∈ ℋ such that 𝜑 = 𝜔𝜉0 |ℳ. ̄ 𝔏(ℝ)}, we have ℛ(ℳ, 𝜎)′ ∩ (ℳ ⊗ ̄ ℬ(ℒ 2 (ℝ)) = 𝜋𝜎 (ℳ)′ ∩ Since ℛ(ℳ, 𝜎) = ℛ{𝜋𝜎 (ℳ), 1ℳ ⊗ ′ ′ ̄ 𝔏(ℝ) ) = 𝜋𝜎 (ℳ) ∩ (ℳ ⊗ ̄ 𝔏(ℝ)) (see 18.4.(14)). Also (see 21.6.(1))) 𝒵 (ℛ(ℳ, 𝜎)) ⊂ (ℳ ⊗ ̄ 𝔏(ℝ) ⊂ ℛ(ℳ, 𝜎). Therefore, in order to prove (1), we have to show that ℳ𝜑 ⊗ ̄ 𝔏(ℝ)) ⊂ ℳ 𝜑 ⊗ ̄ 𝔏(ℝ). 𝜋𝜎 (ℳ)′ ∩ (ℳ ⊗

(3)

Let F ∶ ℒ 2 (ℝ) → ℒ 2 (ℝ) be the unitary operator defined by the Fourier–Plancherel transform (18.8), that is, [F𝜉](t) = [F∗ 𝜉](s) =

∫ ∫

e−ist 𝜉(s) ds eist 𝜉(t) dt

(𝜉 ∈ ℒ 2 (ℝ), t ∈ ℝ), (𝜉 ∈ ℒ 2 (ℝ), s ∈ ℝ),

̂ ≡ ℝ are chosen so that the Fourier inversion formula where the Haar measures ds on ℝ and dt on ℝ holds. Also, denote by Φ the *-automorphism of ℬ(ℒ 2 (ℝ)) implemented by F, that is, Φ(x) = ̃ = 𝜄ℳ ⊗ ̄ F, Φ ̄ Φ. Then (3) becomes (see Proposition 18.8): FxF∗ (x ∈ ℬ(ℒ 2 (ℝ))), and put F̃ = 1ℳ ⊗ ̃ −1 (ℳ ⊗ ̃ −1 (ℳ 𝜑 ⊗ ̄ ℒ ∞ (ℝ)) = Φ ̄ ℒ ∞ (ℝ)). 𝜋𝜎 (ℳ)′ ∩ Φ

(4)

̄ ℒ ∞ (ℝ) can be identified with the W ∗ -algebra Since the predual of ℳ is separable, ℳ ⊗ of all essentially norm-bounded w-measurable functions a(⋅) ∶ ℝ → ℳ; namely, the ̄ ℒ ∞ (ℝ) corresponding to the function a(⋅) ∈ ℒ ∞ (ℝ, ℳ) acts on ℋ ⊗ ̄ ℒ 2 (ℝ) = element a ∈ ℳ ⊗ 2 ℒ (ℝ, ℋ ) as follows (see Sakai, 1971, 3.2.2): ℒ ∞ (ℝ, ℳ)

[a𝜉](t) = a(t)𝜉(t) (𝜉 ∈ ℒ 2 (ℝ, ℋ ), t ∈ ℝ). Let 𝒞 (ℝ, ℳ) be the *-subalgebra of ℒ ∞ (ℝ, ℳ) consisting of all norm-bounded w-continuous ̃ −1 (𝒞 (ℝ, ℳ)) is w-dense in 𝜋𝜎 (ℳ)′ ∩ Φ ̃ −1 (ℒ ∞ (ℝ, ℳ)). functions ℝ → ℳ. We show that 𝜋𝜎 (ℳ)′ ∩ Φ 2 ̄ To this end, consider the continuous action 𝜃 ∶ ℝ → Aut(ℳ ⊗ ℬ(ℒ (ℝ))) defined by 𝜃s = ̄ m(s)∗ ), where [m(s)𝜉](t) = e−ist 𝜉(t) (s, t ∈ ℝ, 𝜉 ∈ ℒ 2 (ℝ)). Recall (19.3) that the dual Ad(1ℳ ⊗

Dominant Weights and Continuous Decompositions

345

action 𝜎̂ ∶ ℝ → Aut(ℛ(ℳ, 𝜎)) is defined by 𝜎̂ s = 𝜃s |ℛ(ℳ, 𝜎) so that (19.3.(3)) 𝜃s (𝜋𝜎 (ℳ)) = ̃ −1 (a) ∈ 𝜋𝜎 (ℳ) and hence 𝜃s (𝜋𝜎 (ℳ)′ ) = 𝜋𝜎 (ℳ)′ (s ∈ ℝ). Now let a ∈ ℒ ∞ (ℝ, ℳ) be such that Φ ′ ′ −1 −1 ̃ ̃ 𝜋𝜎 (ℳ) . A simple computation shows that 𝜋𝜎 (ℳ) ∋ 𝜃s (Φ (a)) = Φ (as ), where as (t) = a(t + s) (s, t ∈ ℝ). Then, for any continuous function f ∶ ℝ → ℂ with compact support, it follows that ̃ −1 (a)) = Φ ̃ −1 (af ), where af (t) = ∫ f (s)a(t + s) ds = ∫ f (s − t)a(s) ds (t ∈ ℝ). It is 𝜋𝜎 (ℳ)′ ∋ 𝜃f (Φ ̃ −1 (af ) ∈ 𝜋𝜎 (ℳ)′ ∩ Φ ̃ −1 (𝒞 (ℝ, ℳ)). On the other hand, if {fi } easy to see that af ∈ 𝒞 (ℝ, ℳ), hence Φ w ̃ −1 (af ) = 𝜃f (Φ ̃ −1 (a)) → Φ−1 (a). is a norm-bounded approximate unit of ℒ 1 (ℝ), then (15.1.(2)) Φ i i Consequently, in order to prove (4), it is sufficient to show that ̃ −1 (𝒞 (ℝ, ℳ)) ⊂ Φ ̃ −1 (𝒞 (ℝ, ℳ 𝜑 )) 𝜋𝜎 (ℳ)′ ∩ Φ or, equivalently, that ̃ 𝜎 (ℳ)′ ) ∩ 𝒞 (ℝ, ℳ) ⊂ 𝒞 (ℝ, ℳ 𝜑 ). Φ(𝜋

(5)

Let a ∈ 𝒞 (ℝ, ℳ), x ∈ ℳ, and 𝜂, 𝜁 ∈ 𝒟 (ℝ), where 𝒟 (ℝ) stands for the space of all complex valued C∞ -functions with compact support on ℝ. Then, as easily verified, ̃ 𝜎 (x)F̃ ∗ (𝜉0 ⊗ ̄ 𝜂)|a∗ (𝜉0 ⊗ ̄ 𝜁)) (F𝜋 ( ) it(r−s) = e 𝜂(r)𝜁(s)𝜑(a(s)𝜎−t (x)) dr ds dt ∫ ∫ ∫ ̃ 𝜎 (x)F̃ ∗ , we where for r and s the order of integration is irrelevant. Therefore, if a commutes with F𝜋 obtain ( ) eit(r−s) 𝜂(r)𝜁(s)𝜑(a(s)𝜎−t (x)) dr ds dt ∫ ∫ ∫ ̄ = (F𝜋 ̄ ̃ 𝜎 (x)F̃ ∗ (𝜁0 ⊗ 𝜂)|a∗ (𝜉0 ⊗ 𝜁)) ̃ 𝜎 (x∗ )F̃ ∗ (𝜉0 ⊗ 𝜁)|a(𝜉 = (F𝜋 0 ⊗ 𝜂)) ( ) = eit(r−s) 𝜂(r)𝜁 (s)𝜑(a∗ (r)𝜎−t (x∗ )) dr ds dt ∫ ∫ ∫ ( ) eit(r−s) 𝜂(r)𝜁 (s)𝜑(𝜎−t (x)a(r)) dr ds dt. = ∫ ∫ ∫ Since 𝜎 = 𝜎 𝜑 satisfies the KMS-condition (2.12), for each r ∈ ℝ there exists a function G(⋅, r) defined, continuous and bounded on the strip {𝛼 ∈ ℂ; 0 ≤ Im 𝛼 ≤ 1}, analytic in the interior of the strip and such that G(t, r) = 𝜑(𝜎−t (x)a(r)), G(t + i, r) = 𝜑(a(r)𝜎−t (x)) for all t ∈ ℝ. Then, ( ) eit(r−s) 𝜂(r)𝜁(s)𝜑(𝜎−t (x)a(r)) dr ds dt ∫ ∫ ∫ ( ) it(r−s) = e 𝜂(r)𝜁(s)G(t, r)) dr ds dt ∫ ∫ ∫ ( ) itr = e 𝜂(r)[F𝜁](t)G(t, r)) dr dt ∫ ∫ ( ) itr = e [F𝜁](t)G(t, r)) dt 𝜂(r) dr = ⋅∕. ∫ ∫

346

Continuous Decompositions

by Fubini’s theorem. Since the function 𝛼 ↦ ei𝛼r [F𝜁](𝛼)G(𝛼, r)) is analytic on the strip {𝛼 ∈ ℂ; 0 < Im 𝛼 < 1} and decays exponentially along horizontal lines, using the theorems of Cauchy and Fubini we can continue the earlier computation as follows: ( ) i(t+i)r e [F𝜁 ](t + i)G(t + i, r) dt 𝜂(r) dr ⋅∕⋅ = ∫ ∫ ( ) = ei(t+i)r 𝜂(r)[F𝜁](t + i)G(t + i, r) dr dt ∫ ∫ ( ) = ei(t+i)(r−s) 𝜂(r)𝜁(s)G(t + i, r) dr ds dt ∫ ∫ ∫ ) ( = ei(t+i)(r−s) 𝜂(r)𝜁(s)𝜑(a(r)𝜎−t (x)) dr ds dt. ∫ ∫ ∫ ̃ 𝜎 (x)F̃ ∗ , then Thus, if a commutes with F𝜋 H ∶ ℝ × ℝ ∋ (s, t) ↦ 𝜑(a(s)𝜎−t (x)) ∈ ℂ is a bounded continuous function such that for every 𝜂, 𝜁 ∈ 𝒟 (ℝ) we have ( ) it(r−s) e 𝜂(r)𝜁(s)H(s, t) dr ds dt ∫ ∫ ∫ ( ) i(t+i)(r−s) = e 𝜂(r)𝜁(s)H(r, t) dr ds dt. ∫ ∫ ∫

(6)

With these conditions, we shall show in the last part of the proof that H(s, t) = H(s, 0)

(s, t ∈ ℝ).

(7)

In our case, this means that 𝜑(a(s)𝜎−t (x)) = 𝜑(a(s)x) and hence that 𝜑(𝜎t (a(s))x) = 𝜑(a(s)x) (s, t ∈ ℝ). ̃ 𝜎 (ℳ)′ )∩𝒞 (ℝ, ℳ), then it follows that 𝜎t (a(s)) = a(s) (s, t ∈ ℝ), that is, a ∈ 𝒞 (ℝ, ℳ 𝜑 ), If a ∈ Φ(𝜋 which proves the theorem. Suppose now that a bounded continuous function H ∶ ℝ2 → ℂ satisfies (6) for all 𝜂, 𝜁 ∈ 𝒟 (ℝ). Then ( ) eit(r−s) 𝜉(r, s)H(s, t) dr ds dt ∫ ∫ ∫ ( ) = ei(t+i)(r−s) 𝜉(r, s)H(r, t) dr ds dt. ∫ ∫ ∫ for all 𝜉 ∈ 𝒟 (ℝ2 ), since both sides of this equation define distributions on 𝒟 (ℝ2 ) which, by (6), ̄ 𝒟 (ℝ). Equivalently, this means that agree on 𝒟 (ℝ) ⊗ ( ) itr e 𝜉(s + r, s)H(s, t) dr ds dt ∫ ∫ ∫ ( ) = eitr e−r 𝜉(s + r, s)H(s + r, t) dr ds dt ∫ ∫ ∫

Dominant Weights and Continuous Decompositions

347

or, since the functions (r, s) ↦ 𝜉(s + r, s) exhaust all of 𝒟 (ℝ2 ), (

)

e 𝜉(r, s)H(s, t) dr ds dt ( ) = eitr e−r 𝜉(r, s − r)H(s, t) dr ds dt ∫ ∫ ∫

∫

∫ ∫

itr

(8)

for all 𝜉 ∈ 𝒟 (ℝ2 ). The expression ̃ = ⟨𝜉, H⟩

( ∫

) ∫ ∫

e 𝜉(r, s)H(s, t) dr ds itr

dt

(𝜉 ∈ 𝒟 (ℝ2 ))

defines a distribution on 𝒟 (ℝ2 ). Being the partial Fourier transform of the bounded continuous ̃ is tempered and hence extends by continuity to the Schwartz space function H, the distribution H 2 𝒮 (ℝ ). On the other hand, consider the linear operator T defined on 𝒟 (ℝ2 ) by [T𝜉](r, s) = e−r 𝜉(r, s − r) (𝜉 ∈ 𝒟 (ℝ2 )). Then (8) means that ̃ =0 ⟨(I − T)𝜉, H)

(𝜉 ∈ 𝒟 (ℝ2 )).

(9)

For a given 𝜉 ∈ 𝒟 (ℝ2 ) with supp 𝜉 ∩ ({0} × ℝ) = ∅, we define a sequence {𝜉n } ⊂ 𝒟 (ℝ2 ) by n ⎧∑ e−kr 𝜉(r, s − kr) if r ≥ 0 ⎪ ⎪ k=0 𝜉n (r, s) = ⎨ n ⎪ − ∑ e(k+1)r 𝜉(r, s + (k + 1)r) if r ≤ 0. ⎪ ⎩ k=0

Then [(I − T)𝜉n ](r, s) = 𝜉(r, s) − e−(n+1)r 𝜉(r, s − (n + 1)r) [(I − T)𝜉n ](r, s) = 𝜉(r, s) − e−(n+1)r 𝜉(r, s + (n + 1)r)

(r ≥ 0) (r ≤ 0)

̃ = and it follows that limn (I − T)𝜉n = 𝜉 in the topology of the Schwartz space 𝒮 (ℝ2 ), so that ⟨𝜉, H⟩ ̃ = 0. This means that supp H ̃ ⊂ {0} × ℝ. limn ⟨(I − T)𝜉n , H⟩ Consequently. for every 𝜂 ∈ 𝒟 (ℝ) with supp 𝜂 ∉ 0 we get ( 0=

∫

) ∫

eitr 𝜂(r)H(s, t) dr

dt = [F𝜂](−t)H(s, t) dt.

Thus, for each fixed s ∈ ℝ, the partial Fourier transform of H is a distribution with support equal to {0} and is therefore a finite linear combination of derivatives of the Dirac delta function, which in turn implies that H(s, ⋅) is a polynomial. Since H is bounded, it follows that H(s, ⋅) is a constant function, which proves (7).

348

Continuous Decompositions

23.20. Notes. The material of this section is due to Connes (1973a); Connes and Takesaki (1973a), and Takesaki (1973b). They give (Connes & Takesaki, 1977) an explicit construction of the continuous decomposition of a factor of type III arising from the group measure space construction. Also, there are concrete descriptions of the integrable and dominant weights and the continuous decomposition of von Neumann algebras associated with foliations (Connes, 1979a, b). For our exposition we have used Connes (1973a); Connes and Takesaki (1977); Takesaki (1973b, 1976).

24 The Flow of Weights In this section, we introduce the (smooth) flow of weights on countably decomposable properly infinite W ∗ -algebras and we prove a “spectral multiplicity theorem” for integrable weights. 24.1. Let ℳ be a countably decomposable properly infinite W ∗ -algebras, 𝜔 ∈ Wnsf (ℳ) a dominant weight on ℳ and (𝒩 , 𝜃, 𝜏) the corresponding continuous decomposition of ℳ, that is, 𝒩 = ℳ 𝜔 , 𝜃 ∶ ℝ → Aut(ℳ 𝜔 ) is a continuous action and 𝜏 is an n.s.f. trace on ℳ 𝜔 with 𝜏 ◦ 𝜃s = e−s 𝜏, (s ∈ ℝ), such that ̂ 𝜏). ̂ (ℳ, 𝜎 𝜔 , 𝜔) ≈ (ℛ(ℳ 𝜔 , 𝜃), 𝜃,

(1)

Recall (23.7) that there exists an s-continuous unitary representation u ∶ ℝ → U(ℳ) such that 𝜃s = Ad(u(s))|ℳ 𝜔 and 𝜔u(s) = e−s 𝜔, that is, 𝜎t𝜔 (u(s)) = e−ist u(s) (s, t ∈ ℝ). By Corollary 23.12, the pair (𝒵 (ℳ 𝜔 ), 𝜃|𝒵 (ℳ 𝜔 )) is uniquely determined modulo *-isomorphisms and will be called the smooth flow of weights on ℳ. Sometimes it is more convenient to regard the continuous action of the additive group ℝ on 𝒵 (ℳ 𝜔 ) as a continuous action F = F ℳ of the multiplicative group ℝ+∗ = {𝜆 ∈ ℝ; 𝜆 > 0} defined by 𝜔 + F𝜆 = Fℳ 𝜆 = 𝜃−ln(𝜆) |𝒵 (ℳ ) (𝜆 ∈ ℝ∗ ).

(2)

Thus, the pair (𝒵 (ℳ 𝜔 ), F ∶ ℝ+∗ → Aut(𝒵 (ℳ 𝜔 )) is the smooth flow of weights on ℳ. The justification of this terminology will follow from the contents of this section, which we briefly present in the following: Let Wint (ℳ) be the set of all integrable normal semifinite weights on ℳ and W∞ (ℳ) the subset int of Wint (ℳ) consisting of weights of infinite multiplicity. For every 𝜑 ∈ Wns (ℳ) one defines a canonical surjective mapping c𝜑 ∶ Wns (ℳ) ∋ 𝜓 ↦ c𝜑 (𝜓) ∈ Proj(𝒵 (ℳ 𝜑 )) whose properties are studied in Sections 24.2 and 24.3. In particular, the mapping c𝜔 ∶ Wint (ℳ) ∋ 𝜑 ↦ c𝜔 (𝜑) ∈ Proj(𝒵 (ℳ 𝜔 ))

The Flow of Weights

349

has the properties c𝜔 (𝜆𝜑) = F𝜆 (c𝜔 (𝜑)), 𝜓 ≲ 𝜑 ⇒ c𝜔 (𝜓) ≤ c𝜔 (𝜑) ⇒ 𝜓̌ ≲ 𝜑, ̌ c𝜔 ∨n c𝜔 (𝜑n ) for all 𝜑, 𝜓, 𝜑n ∈ Wint (ℳ), 𝜆 > 0. Thus,

(∑⊕ n

𝜑n

)

=

W∞ int (ℳ)∕ ≂ can be identified with Proj(𝒵 (ℳ 𝜔 )) in such a way that the smooth flow of weights F𝜆 corresponds to the mapping 𝜑 ↦ 𝜆𝜑 (𝜆 > 0). For each 𝜑 ∈ Wint (ℳ), there exists a *-isomorphism p𝜑 ∶ 𝒵 (ℳ 𝜑 ) → 𝒵 (ℳ 𝜔 )c𝜔 (𝜑), uniquely determined, such that p𝜑 (c𝜑 (𝜓)) = c𝜔 (𝜓)c𝜔 (𝜑) for all 𝜓 ∈ Wns (ℳ). Also, there exist an +

+ n.s.f. operator-valued weight P𝜑 ∶ ℳs(𝜑) → (ℳ 𝜑 ) and a unique n.s.f. trace 𝜏𝜑 on ℳ 𝜑 such that + 𝜑|ℳs(𝜑) = 𝜏𝜑 ◦ P𝜑 . In particular, 𝜏𝜆𝜑 = 𝜆𝜏𝜑 (𝜆 > 0). Finally, there exists a unique mapping

ν ∶ Wint (ℳ) ∋ 𝜑 ↦ ν𝜑 ∈ Wn (𝒵 (ℳ 𝜔 ))

such that ν𝜑 (c𝜔 (𝜓)) = 𝜏𝜑 (c𝜑 (𝜓)) (𝜓 ∈ Wint (ℳ)). For all 𝜑, 𝜓, 𝜑1 , … , 𝜑m ∈ Wint (ℳ) and 𝜆 > 0, ∑ we have ν𝜆𝜑 = 𝜆ν𝜑 ◦ F−1 , 𝜓 ≤ 𝜑 ⇔ ν𝜓 ≤ ν𝜑 , ν∑⊕ 𝜑n = n ν𝜑n and, if ℳ is of type III, then the 𝜆 n mapping ν is also surjective. One thus obtains a “spectral multiplicity” theory for integrable weights. ̄ ℱ2 with s(𝜃) = 24.2. Let 𝜑, 𝜓)∈ Wns (ℳ) and 𝜃 = 𝜃(𝜑, 𝜓) be the balanced weight on 𝒫 = ℳ ⊗ ( s(𝜑) 0 . We define a projection c𝜑 (𝜓) ∈ 𝒵 (ℳ 𝜑 ) by 0 s(𝜓) (( z𝒫 𝜃

0 0 0 s(𝜓)

)) (

0 0

s(𝜑) 0

)

( =

c𝜑 (𝜓) 0 0 0

) .

It is easy to see that the left-hand side necessarily has the form of the right-hand side and that c𝜑 (𝜓) ∈ 𝒵 (ℳ 𝜑 ). The next proposition contains some computation rules for c𝜑 (𝜓). Proposition. Let 𝜑, 𝜑n , 𝜓, 𝜓n , 𝜓 ′ ∈ Wns (ℳ), 𝜆 > 0, 𝜎 ∈ Aut(ℳ), and let w ∈ ℳ be a partial isometry with ww∗ ∈ ℳ 𝜑 . Then c𝜑 (𝜑w ) = zℳ 𝜑 (ww∗ ) ∗

(1)

c𝜑w (𝜓) = w c𝜑 (𝜓)w

(2)

c𝜆𝜑 (𝜓) = c𝜑 (𝜆−1 𝜓)

(3)

c𝜑 ◦ 𝜎 (𝜓 ◦ 𝜎) = 𝜎 (c𝜑 (𝜓))

(4)

𝜓 ≲ 𝜓 ⇒ c𝜑 (𝜓 ) ≤ c𝜑 (𝜓) ∑ c∑⊕ 𝜑n (𝜓) = c𝜑n (𝜓)

(5)

−1

′

′

n

c𝜑

(⊕ ∑ n

) 𝜓n

(6)

n

=

⋁

c𝜑 (𝜓n )

(7)

n

c𝜑 (𝜓) ̌ = c𝜑 (𝜓).

(8)

350

Continuous Decompositions (

0 w 0 0

)

Proof. (1) Let 𝜃 = 𝜃(𝜑, 𝜑w ). We have s(𝜑w ) = and ∈ 𝒫 𝜃 by Proposition 23.1, ( ) ( ∗ ) 0 0 ww 0 hence the projections and are equivalent in 𝒫 𝜃 and therefore have the 0 s(𝜑w ) 0 0 same central support in 𝒫 𝜃 . Since s(𝜑) is the unit element in ℳ 𝜑 , (1) follows from the definition of c𝜑 (𝜑w ). ( ) w 0 ′ (2) Let 𝜃 = 𝜃(𝜑, 𝜓), 𝜃 = (𝜑w , 𝜓) and W = ∈ 𝒫 . Then WW∗ ∈ 𝒫 𝜃 and 𝜃 ′ = 𝜃W , 0 1 ′ hence the mapping 𝒫 𝜃 ∋ X ↦ WXW∗ ∈ 𝒫 𝜃WW∗ ⊂ 𝒫 𝜃 is a *-isomorphism. Since s(𝜑w ) = w∗ w and s(𝜑)w = w, we have ) (( )) ( ) ( c𝜑w (𝜓) 0 0 0 s(𝜑w ) 0 = z𝒫 𝜃′ 0 s(𝜓) 0 0 0 0 ) ) ( ∗ ) ( ( w w 0 0 0 = W∗ z𝒫 𝜃 W W∗ W 0 0 0 s(𝜓) )) ( ) ( ∗ ) (( w 0 w 0 0 0 = z𝒫 𝜃 0 0 0 1 0 s(𝜓) ( ∗ ) (( )) ( )( ) w 0 0 0 s(𝜑) 0 w 0 = z𝒫 𝜃 0 1 0 s(𝜓) 0 0 0 0 ( ∗ )( )( ) ( ∗ ) c𝜑 (𝜓) 0 w c𝜑 (𝜓)w 0 w 0 w 0 = = , 0 1 0 0 0 0 0 0 w∗ w

which proves (2). (3) Let 𝜃 = 𝜃(𝜆𝜑, 𝜓) and 𝜃 ′ = 𝜃(𝜑, 𝜆−1 𝜓). We have s(𝜆𝜑) = s(𝜑), s(𝜆−1 𝜓) = s(𝜓) and 𝜃 = 𝜆𝜃 ′ , ′ hence 𝒫 𝜃 = 𝒫 𝜃 and (3) now follows easily using the definition of c𝜑 (𝜓). (4) Let 𝜃 = 𝜃(𝜑, 𝜓), 𝜃 ′ = 𝜃(𝜑 ◦ 𝜎, 𝜓 ◦ 𝜎). We have s(𝜑 ◦ 𝜎) = 𝜎 −1 (s(𝜑)), s(𝜓 ◦ 𝜎) = 𝜎 −1 (s(𝜓) ̄ 𝜄), hence 𝒫 𝜃′ = (𝜎 ⊗ ̄ 𝜄)−1 (𝒫 𝜃 ) and (4) follows again by definition. and 𝜃 ′ = 𝜃 ◦ (𝜎 ⊗ ′ (5)(Let 𝜓 )= 𝜓v with a partial isometry v ∈ ℳ, vv∗ ∈ ℳ 𝜓 . Let 𝜃 = 𝜃(𝜑, 𝜓), 𝜃 ′ = 𝜃(𝜑, 𝜓 ′ ). For 1 0 ′ V= ∈ 𝒫 we have VV ∗ ∈ 𝒫 𝜃 and 𝜃 ′ = 𝜃V , hence the mapping 𝒫 𝜃 ∋ X ↦ VXV ∗ ∈ 0 v 𝒫 𝜃VV ∗ ⊂ 𝒫 𝜃 is a *-isomorphism. Since s(𝜓 ′ ) = v∗ v and vv∗ ≤ s(𝜓), we have ( ) (( )) ( ) c𝜑 (𝜓 ′ ) 0 0 0 s(𝜑) 0 ′ = z𝒫 𝜃 0 0 0 v∗ v 0 0 ( ( ) ) ( ) 0 0 s(𝜑) 0 ∗ ∗ = V z𝒫 𝜃 V V V 0 v∗ v 0 0 ( ) (( )) ( )( ) 1 0 0 0 s(𝜑) 0 1 0 = z𝒫 𝜃 0 v∗ 0 vv∗ 0 0 0 v ( ) (( )) ( )( ) 1 0 0 0 s(𝜑) 0 1 0 ≤ z𝒫 𝜃 0 v∗ 0 s(𝜓) 0 0 0 v ( )( )( ) ( ) c𝜑 (𝜓) 0 c𝜑 (𝜓) 0 1 0 1 0 = = , 0 v∗ 0 v 0 0 0 0 proving (5).

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∑ (6) Assume that the projections en = s(𝜑n ) (n ∈ ℕ) are mutually orthogonal and 𝜑 = n 𝜑n . ∑ ∑ ∑ Then en ∈ ℳ 𝜑 , 𝜑n = 𝜑en and n en = s(𝜑). Using (2), we obtain n c𝜑n (𝜓) = n en c𝜑 (𝜓)en = ∑ ( n en )c𝜑 (𝜓) = c𝜑 (𝜓). ∑ (7) Assume that the projections en = s(𝜓n ) (n ∈ ℕ) are mutually orthogonal and that 𝜓 = n 𝜓n , ∑ so that en ∈ ℳ 𝜓 , 𝜓n = 𝜓en and n en = s(𝜓). Let 𝜃n = 𝜃(𝜑, 𝜓n ), 𝜃 = 𝜃(𝜑, 𝜓). Then 𝜃n = 𝜃En , where ( ) ( ) ∑ 1 0 0 0 En = = s(𝜃n ), and 𝜃 = n 𝜃n . It is clear that has the same central projection 0 en 0 en in 𝒫 𝜃n as in 𝒫 𝜃 . It follows that ( ) (( )) ( ) c𝜑 (𝜓) 0 0 0 s(𝜑) 0 = z𝒫 𝜃 0 0 0 s(𝜓) 0 0 (( )) ( ) ⋁ 0 0 s(𝜓) 0 = z𝒫 𝜃 0 en 0 0 n ) (⋁ ) ( ⋁ c (𝜓 ) 0 𝜑 n n c𝜑 (𝜓n ) 0 = . = 0 0 0 0 n proving (7). (8) This follows easily using (5), (7), and the definition (23.15) of the weight 𝜑. ̌ 24.3 Proposition. For every 𝜑 ∈ Wns (ℳ), the mapping 𝜓 ↦ c𝜑 (𝜓) defines an order isomorphism between the equivalence classes of weights 𝜓 ∈ Wns (ℳ), 𝜓 ≲ 𝜑̌ of infinite multiplicity, and the projections in 𝒵 (ℳ 𝜑 ). ∑ Proof. Let 𝜑̌ = n 𝜑n with 𝜑n = 𝜑un ∈ Wns (ℳ), where un = (wn s(𝜑))∗ and {wn } is a sequence ∑ of isometries in ℳ such that n wn w∗n = 1 (see 23.15). Using Proposition 24.2, for 𝜓 ∈ Wns (ℳ) ∑ ∑ ∑ we obtain c𝜑̌ (𝜓) = n c𝜑n (𝜓) = n u∗n c𝜑 (𝜓)un = n wn c𝜑 (𝜓)w∗n . Recall (23.15) that the mapping ∑ ∗ ̄ eij ∈ ℳ ⊗ ̄ ℱ , where {eij } is a system of matrix units of the countably 𝜋 ∶ ℳ ∋ x ↦ ij wi xwj ⊗ ̄ ℱ,𝜑⊗ ̄ tr). decomposable infinite type I factor ℱ , establishes a *-isomorphism 𝜋 ∶ (ℳ, 𝜑) ̌ ≈ (ℳ ⊗ ̄ ℱ )𝜑 ⊗̄ tr ) = 𝒵 (ℳ 𝜑 ) ⊗ ̄ 1, and the previous equation shows We have 𝜋(𝒵 (ℳ 𝜑̌ )) = 𝒵 ((ℳ ⊗ ̄ 1. Thus, we may assume that 𝜑 is of infinite multiplicity, that is, that 𝜋(c𝜑̌ (𝜓)) = c𝜑 (𝜓) ⊗ 𝜑 ≂ 𝜑. ̌ If e ∈ 𝒵 (ℳ 𝜑 ) is a nonzero projection, then 𝜑e ≲ 𝜑 is of infinite multiplicity, since ℳ 𝜑e = ℳ 𝜑 e and ℳ 𝜑 is properly infinite. Since c𝜑 (𝜑e ) = ec𝜑 (𝜑)e = es(𝜑)e, it follows that the mapping c𝜑 is surjective. Let 𝜓 ≲ 𝜑, 𝜓 ′ ≲ 𝜑. There exist partial isometries w, w′ ∈ ℳ with e = ww∗ ∈ ℳ 𝜑 , e′ = w′ w′∗ ∈ ℳ 𝜑 , such that 𝜓 = 𝜑w , 𝜓 ′ = 𝜑w′ and then c𝜑 (𝜓) = zℳ 𝜑 (e), c𝜑 (𝜓 ′ ) = zℳ 𝜑 (e′ ). If 𝜓 and 𝜓 ′ are both of infinite multiplicity and c𝜑 (𝜓 ′ ) ≤ c𝜑 (𝜓), then the projections e, e′ ∈ ℳ 𝜑 are properly infinite and zℳ 𝜑 (e′ ) ≤ zℳ 𝜑 (e); hence e′ ≺ e in ℳ 𝜑 , so that 𝜓 ′ ≲ 𝜓 by Proposition 23.2. 24.4 Corollary. The mapping 𝜑 ↦ c𝜔 (𝜑) establishes an order isomorphism between the equivalence classes of integrable weights of infinite multiplicity on ℳ and the projections in 𝒵 (ℳ 𝜔 ); we have c𝜔 (𝜆𝜑) = F𝜆 (c𝜔 (𝜑)); 𝜑 ∈ Wint (ℳ), 𝜆 > 0.

(1)

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Continuous Decompositions

Proof. We use the notation of Section 24.1. The first part of the corollary follows clearly from Proposition 24.3 and Theorem 23.4. Since c𝜔 (𝜑) = c𝜔 (𝜑), ̌ in order to prove (1) we may assume that 𝜔 ) with 𝜑 ≂ 𝜔 . For s ∈ ℝ, we have 𝜑 ∈ W∞ (ℳ). Then there exists a projection q ∈ 𝒵 (ℳ q int e−s 𝜑 ≂ e−s 𝜔q = (e−s 𝜔)q = 𝜔u(s)q = 𝜔u(s)qu(s)∗ u(s) = 𝜔𝜃s (q)u(s) ≂ 𝜔𝜃s (q)

(2)

hence c𝜔 (𝜑) = q and c𝜔 (e−s 𝜑) = 𝜃s (q), that is, c𝜔 (e−s 𝜑) = 𝜃s (c𝜔 (𝜑)),

(3)

proving (1), as F𝜆 = 𝜃−ln(𝜆) (𝜆 > 0). 24.5 Proposition. For each 𝜑 ∈ Wint (ℳ), there exists a unique *-isomorphism p𝜑 ∶ 𝒵 (ℳ 𝜑 ) → 𝒵 (ℳ 𝜔 )c𝜔 (𝜑) such that p𝜑 (c𝜑 (𝜓)) = c𝜔 (𝜓)c𝜔 (𝜑) (𝜓 ∈ Wns (ℳ)).

(1)

Proof. Since 𝜑 ∈ Wint (ℳ), we have 𝜑 ≲ 𝜔, hence there exists a partial isometry w ∈ ℳ with ww∗ ∈ ℳ 𝜔 such that 𝜑 = 𝜔w . We thus obtain a *-isomorphism ℳ 𝜑 ∋ x ↦ wxw∗ ∈ (ℳ 𝜔 )ww∗ and, by restriction, a *-isomorphism 𝒵 (ℳ 𝜑 ) ∋ x ↦ wxw∗ ∈ 𝒵 (ℳ 𝜔 )ww∗ . On the other hand, we have c𝜔 (𝜑) = zℳ 𝜔 (ww∗ ), hence the mapping 𝒵 (ℳ 𝜔 )c𝜔 (𝜑) ∋ z ↦ zww∗ ∈ 𝒵 (ℳ 𝜔 )ww∗ is a *-isomorphism. By composing these two mappings, we obtain a *-isomorphism p𝜑 ∶ 𝒵 (ℳ 𝜑 ) → 𝒵 (ℳ 𝜔 )c𝜔 (𝜑). If e ∈ 𝒵 (ℳ 𝜑 ) is a projection, then 𝜑e = 𝜔we , hence c𝜔 (𝜑e ) = zℳ 𝜔 (wew∗ ) ∈ 𝒵 (ℳ 𝜔 ) and, clearly, c𝜔 (𝜑e ) ≤ c𝜔 (𝜑). We have c𝜔 (𝜑e )ww∗ = zℳ 𝜔 (wew∗ )ww∗ = wew∗ (since e ∈ 𝒵 (ℳ 𝜑 ), hence wew∗ ∈ 𝒵 (ℳ 𝜔 )ww∗ ) and therefore c𝜔 (𝜑e ) = p𝜑 (e). Assume now that e = c𝜑 (𝜓) with 𝜓 ∈ Wns (ℳ). Since 𝜑 = 𝜔w , we have e = c𝜔w (𝜓) = w∗ c𝜔 (𝜓)w, hence p𝜑 (e)ww∗ = ww∗ c𝜔 (𝜑)ww∗ = c𝜔 (𝜓)ww∗ = c𝜔 (𝜓)zℳ 𝜔 (ww∗ )ww∗ = c𝜔 (𝜓)c𝜔 (𝜑)ww∗ , and therefore p𝜑 (c𝜑 (𝜓)) = c𝜔 (𝜓)c𝜔 (𝜑). The uniqueness of p𝜑 follows obviously from the surjectivity of c𝜑 . 24.6. Let 𝜑 ∈ Wint (ℳ). Then the continuous action 𝜎 𝜑 ∶ ℝ → Aut(ℳs(𝜑) ) is integrable, so that it + defines a 𝜎 𝜑 -invariant n.s.f. operator-valued weight P𝜑 ∶ ℳs(𝜑) → (ℳ 𝜑 )+ . Proposition. For each 𝜑 ∈ Wint (ℳ), there exists a unique n.s.f. trace 𝜏𝜑 on ℳ 𝜑 such that + 𝜑|ℳs(𝜑) = 𝜏𝜑 ◦ P𝜑 .

(1)

Proof. If 𝜑 = 𝜔, we can take 𝜏𝜔 = 𝜏 from the continuous decomposition (ℳ 𝜔 , 𝜃, 𝜏) of ℳ constructed in Theorem 23.6 (see also Theorem 23.16). If 𝜑 is just integrable, then there exists a projection e ∈ ℳ 𝜔 such that 𝜑 ≂ 𝜔e and we may assume that 𝜑 = 𝜔e . Then s(𝜑) = e, ℳ 𝜑 = eℳ 𝜔 e, 𝜎t𝜑 = 𝜎t𝜔 |eℳe, c𝜔 (𝜑) = zℳ 𝜔 (e) and the *-isomorphism p𝜑 ∶ 𝒵 (eℳ 𝜔 e) → 𝒵 (ℳ 𝜔 )zℳ 𝜔 (e)

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is the inverse of the mapping z ↦ ze. The operator-valued weight P𝜑 ∶ (eℳe)+ → (eℳ 𝜔 e)+ ⊂ ( ) (ℳ 𝜔 )+ is defined by P𝜑 (exe) = ∫ 𝜎t𝜑 (exe) dt = e ∫ 𝜎t𝜔 (x) dt e = eP𝜔 (x)e = P𝜔 (exe) (x ∈ ℳ + ), hence P𝜑 = P𝜔 |eℳ + e. Then 𝜏𝜑 = 𝜏𝜔 |eℳ 𝜔 e

(2)

is an n.s.f. trace on ℳ 𝜑 = eℳ 𝜔 e and for every x ∈ eℳ + e we have (𝜏𝜑 ◦ P𝜑 )(x) = 𝜏𝜔 (P𝜔 (x)) = 𝜔(x) = 𝜑(x). The uniqueness of 𝜏𝜑 is obvious. Note that 𝜏𝜆𝜑 = 𝜆𝜏𝜑

(𝜆 > 0).

(3)

24.7. We now prove the “spectral multiplicity theorem” for integrable weights. Theorem. For every 𝜑 ∈ Wint (ℳ), there exists a unique normal weight ν𝜑 on 𝒵 (ℳ 𝜔 ) such that ν𝜑 (c𝜔 (𝜓)) = 𝜏𝜑 (c𝜑 (𝜓))

(𝜓 ∈ Wint (ℳ)).

(1)

The mapping 𝜑 ↦ ν𝜑 establishes a bijective correspondence between the equivalence classes of weights 𝜑 ∈ Wint (ℳ) and a certain set of normal weights ν𝜑 on 𝒵 (ℳ 𝜔 ); for 𝜑, 𝜓, 𝜑1 , … , 𝜑m ∈ Wint (ℳ) and 𝜆 > 0 we have ν𝜆𝜑 = 𝜆ν𝜑 ◦ F−1 𝜆

(2)

𝜓 ≲ 𝜑 ⇔ ν𝜓 ≤ ν𝜑 ∑ ν∑⊕ 𝜑n = ν𝜑n .

(3)

n

(4)

n

If ℳ is of type III, then every normal weight on 𝒵 (ℳ 𝜔 ) is of the form ν𝜑 for some 𝜑 ∈ Wint (ℳ), hence the mapping 𝜑 ↦ ν𝜑 establishes an order isomorphism ν ∶ Wint (ℳ)∕ ≂→ Wn (𝒵 (ℳ 𝜔 )). Proof. Using Propositions 24.5, 24.6, we define ν𝜑 by ν𝜑 (z) = 𝜏𝜑 (p−1 𝜑 (zc𝜔 (𝜑)))

(z ∈ 𝒵 (ℳ 𝜔 )+ )

(5)

−1 and then, for 𝜓 ∈ Wint (ℳ) we have ν𝜑 (c𝜔 (𝜓)) = 𝜏𝜑 (p−1 𝜑 (c𝜔 (𝜓)c𝜔 (𝜑))) = 𝜏𝜑 (p𝜑 (p𝜑 (c𝜔 (𝜓))) = 𝜏𝜑 (c𝜔 (𝜓)), which proves the existence of ν𝜑 . The uniqueness of ν𝜑 is obvious. ′ If 𝜑 ≂ 𝜑′ , then (ℳ 𝜑 , c𝜑 , 𝜏𝜑 ) ≈ (ℳ 𝜑 , c𝜑′ , 𝜏𝜑′ ), hence ν𝜑 = ν𝜑′ . Thus, in proving (3) and (4), we can consider just weights of the form 𝜑 = 𝜔e with e ∈ Proj(ℳ 𝜔 ). In this case, it follows from (5) and 24.6.(2) that

ν𝜑 (z) = 𝜏𝜔 (ze)

(z ∈ 𝒵 (ℳ 𝜔 )+ ).

(6)

354

Continuous Decompositions

Note that ν𝜔 = 𝜏𝜔 |𝒵 (ℳ 𝜔 )+ = +∞, as ℳ 𝜔 is properly infinite, that is, it has no nonzero finite central projection. Let e, f ∈ Proj(ℳ 𝜔 ) and 𝜑 = 𝜔e , 𝜓 = 𝜔f . If ν𝜓 ≤ ν𝜑 , then, using (6), we obtain 𝜏𝜔 (zf ) ≤ 𝜏𝜔 (ze) for all z ∈ 𝒵 (ℳ 𝜔 )+ . By ((L], E.7.13) we infer that f ≺ e in ℳ 𝜔 and, by Proposition 23.2, we deduce 𝜓 = 𝜔f ≲ 𝜔e = 𝜑. Conversely, if 𝜓 ≲ 𝜑, then f ≺ e in ℳ 𝜔 , hence 𝜏𝜔 (zf ) ≤ 𝜏𝜔 (ze) for every z ∈ 𝒵 (ℳ 𝜔 )+ , so that ν𝜓 ≤ ν𝜑 . We have thus proved (3). If e and f are orthogonal, then 𝜑 + 𝜓 = 𝜔e+f , hence ν𝜑+𝜓 = 𝜏𝜔 (z(e + f )) = 𝜏𝜔 (ze) + 𝜏𝜔 (zf ) = ν𝜑 (z) + ν𝜓 (z) for z ∈ 𝒵 (ℳ 𝜔 )+ , and ν𝜑+𝜓 = ν𝜑 + ν𝜓 which proves (4). Using 24.2.(3) and 24.6.(3), we obtain ν𝜆𝜑 (c𝜑 (𝜓)) = 𝜏𝜆𝜑 (c𝜆𝜑 (𝜓)) = 𝜆𝜏𝜑 (c𝜑 (𝜆−1 𝜓)) = 𝜆ν𝜑 (c𝜔 (𝜆−1 𝜓)) = 𝜆ν𝜑 (F−1 (c𝜔 (𝜓))) and this proves equality (2). 𝜆 If ℳ is of type III, then, by Proposition 3/23.11, ℳ 𝜔 is of type II∞ and so, by Corollary 12.14, every normal weight on 𝒵 (ℳ 𝜔 ) is of the form 𝜏𝜔 (e⋅) = ν𝜔e for some projection e ∈ ℳ 𝜔 ; this proves the last assertion of the theorem. 24.8. If the W ∗ -algebra ℳ is semifinite, then, by Proposition 2/23.11, the continuous decomposition ̄ ℒ ∞ (ℝ), 𝜄 ⊗ ̄ Ad(𝜆 𝜆)); hence the smooth flow of weights on (𝒩 , 𝜃) of ℳ is *-isomorphic to (𝒩 𝜃 ⊗ 𝜃 ∞ ̄ ̄ ̄ ℒ ∞ (ℝ+ ), 𝜄 ⊗ ̄ Ad(𝜆 𝜆)), or (𝒵 (𝒩 𝜃 ) ⊗ 𝜆)). If we represent ℳ is the pair (𝒵 (𝒩 ) ⊗ ℒ (ℝ), 𝜄 ⊗ Ad(𝜆 ∗ ∗ 𝜃 ∞ the abelian W -algebra 𝒵 (𝒩 ) in the form ℒ (Ω) (see Dixmier, 1957, 1969; Sakai, 1970; Strătilă & Zsidó, 1977, 2005, 9.37), then the smooth flow of weights on ℳ becomes a continuous action F ∶ ℝ+∗ → Aut(ℒ ∞ (Ω × ℝ+∗ )), defined by the point transformations F𝜆 (𝜉, 𝜇) = (𝜉, 𝜆−1 𝜇) (𝜉 ∈ Ω; 𝜆, 𝜇 ∈ ℝ+∗ ). In particular, if ℳ is a semifinite factor, then 𝒩 𝜃 is a factor (by Proposition 1/23.11), hence the smooth flow of weights is the continuous action F ∶ ℝ+∗ → Aut(ℒ ∞ (ℝ+∗ )) defined by F𝜆 (𝜇) = 𝜆−1 𝜇, (𝜆, 𝜇 ∈ ℝ+∗ ). By Proposition 1/23.11, ℳ is a factor if and only if the smooth flow of weights on ℳ is ergodic. For type III factors, there are descriptions of the smooth flow of weights based on their “discrete decomposition” (see 30.8, 30.10). 24.9. The smooth flow of weights is just the continuous part of a more general, but rather artificial, object called the global flow of weights on ℳ, which we describe in the following. There exists a pair (Qℳ , qℳ ) consisting of an abelian W ∗ -algebra Qℳ and a mapping qℳ of Wns (ℳ) onto the set of countably decomposable projections of Qℳ , such that for all 𝜑, 𝜓, 𝜑n ∈ Wns (ℳ) we hare qℳ (𝜑) = qℳ (𝜑) ̌

(1)

qℳ (𝜓) ≤ qℳ (𝜑) ⇔ 𝜓̌ ≲ 𝜑̌ (⊕ ) ∑ ⋁ qℳ 𝜑n = (qℳ (𝜑n ));

(2)

n

(3)

n

and there exist *-isomorphisms q𝜑 ∶ 𝒵 (ℳ 𝜑 ) → Qℳ qℳ (𝜑)

(𝜑 ∈ Wns (ℳ)),

(4)

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355

uniquely determined, such that q𝜑 (e) = qℳ (𝜑e ) (e ∈ Proj(𝒵 (ℳ 𝜑 ))).

(5)

Indeed, let {e𝜑,𝜓 }𝜑,𝜓∈Wns (ℳ) be a system of matrix units in ℬ(𝓁 2 (Wns (ℳ))) put ℛ = ∑ ̄ ℬ(𝓁 2 (Wns (ℳ))) and consider the n.s.f. weight Φ on ℛ defined by Φ( x𝜑,𝜓 ⊗ ̄ e𝜑,𝜓 ) = ℳ ⊗ ∑ 𝜑(x ). Then the conditions of the earlier statement are satisfied with Q = 𝒵 (ℛ Φ ) and 𝜑,𝜑 ℳ 𝜑 ∑ ̄ e𝜑,𝜑 ) = 𝜓 c𝜓 (𝜑) ⊗ ̄ e𝜓,𝜓 (𝜑 ∈ Wns (ℳ)). The verification is similar to that qℳ (𝜑) = zℛ Φ (s(𝜑) ⊗ in the case of the smooth flow of weights. The global flow of weights on ℳ is an action 𝔉ℳ ∶ ℝ+∗ → Aut(Qℳ ), uniquely determined, such that 𝔉ℳ (qℳ (𝜑)) = qℳ (𝜆𝜑) (𝜑 ∈ Wns (ℳ), 𝜆 ∈ ℝ+∗ ). Proposition. The weight 𝜑 ∈ Wns (ℳ) is integrable if and only if the mapping ℝ+∗ ∋ 𝜆 → 𝔉ℳ 𝜆 (qℳ (𝜑)) ∈ Qℳ

(6)

is s-continuous. Proof. Assume that 𝜑 is integrable. Since 𝜑̌ is integrable and qℳ (𝜑) ̌ = qℳ (𝜑), we may assume that 𝜑 is of infinite multiplicity. In this case, there exists by Corollary 24.4, a unique projection e ∈ 𝒵 (ℳ 𝜔 ) such that 𝜑 ≂ 𝜔e . Using (5) and 24.4.(2), we get + 𝔉ℳ 𝜆 (qℳ (𝜑)) = q𝜔 (F𝜆 (e)) (𝜆 ∈ ℝ∗ ),

(7)

so that the mapping (6) is s-continuous. Conversely, assume that the mapping (6) is s-continuous. Let e = qℳ (𝜑) ∈ Qℳ and f = ⋁ ℳ 𝜆>0 𝔉𝜆 (e) ∈ Qℳ . By assumption, if {𝜆n } is a sequence of nonzero positive rational numbers s

converging to 𝜆 > 0, then 𝔉ℳ (e) → 𝔉ℳ (e). It follows that f is a countably decomposable projection. 𝜆n 𝜆 ℳ Since f is clearly 𝔉 -invariant, we infer that f = qℳ (𝜔), where 𝜔 is the fixed dominant weight on ℳ. Finally, since qℳ (𝜑) = e ≤ f = qℳ (𝜔), it follows that 𝜑 ≲ 𝜑̌ ≲ 𝜔, and so 𝜑 is indeed integrable. Corollary. The projection qℳ (𝜔) ∈ Qℳ is the largest countably decomposable projection q ∈ Qℳ such that the mapping ℝ+∗ ∋ 𝜆 ↦ 𝔉ℳ (q) ∈ Qℳ is s-continuous and (Qℳ qℳ (𝜔), 𝔉ℳ |Qℳ qℳ (𝜔)) ≈ 𝜆 𝜔 ℳ (𝒵 (ℳ ), F ). More precisely, by (7) we have ℳ 𝔉ℳ 𝜆 (qℳ (𝜑)) = q𝜔 (F𝜆 (c𝜔 (𝜑)))

(𝜑 ∈ Wint (ℳ), 𝜆 ∈ ℝ+∗ )

(8)

so that q𝜔 ∶ (𝒵 (ℳ 𝜔 ), F ℳ ) ≈ (Qℳ qℳ (𝜔), 𝔉ℳ |Qℳ qℳ (𝜔)).

(9)

In what follows, we shall always understand by the flow of weights on ℳ the smooth flow of weights on ℳ.

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Continuous Decompositions

24.10. Notes. The material of this section is due to Connes and Takesaki (1977). For type III factors arising from the group measure space construction, they computed (Connes & Takesaki, 1977) the smooth flow of weights using Mackey’s procedure based on the theory of virtual groups (Mackey, 1966; Ramsay, 1971). For further results, and connections with the theory of measure groupoids, we refer to Connes (1978b); Hahn (1978a, 1978b); Ramsay (1971); Samuelides (1978); Sauvageot (1978). The flow of weights has a natural interpretation for von Neumann algebras associated with foliations (Connes, 1979a, 1979b). Connes and Woods (1980b, 1985) obtained a characterization of Araki–Woods factors in terms of the flow of weights. Previously, another characterization of Araki–Woods factors had been obtained by Størmer (1971, 1972, 1975). For our exposition, we have used (Connes & Takesaki, 1977).

25 The Fundamental Homomorphism In this section, we introduce the fundamental homomorphism mod ∶ Aut(ℳ) → Aut(F ℳ ), prove its continuity and relate it to a classical invariant, called the fundamental group. 25.1. Let ℳ be a countably decomposable properly infinite W ∗ -algebra with (smooth) flow of weights (𝒫ℳ , Fℳ ). Abstractly, 𝒫ℳ represents the equivalence classes of integrable normal semifinite weights of infinite multiplicity on ℳ and the action of F𝜆ℳ is multiplication by 𝜆 > 0, that is, 𝒫ℳ = W∞ (ℳ)∕ ≂ , F𝜆ℳ 𝜑 = 𝜆𝜑. Concretely, one considers a dominant weight 𝜔 on ℳ int and the corresponding continuous decomposition (ℳ 𝜔 , 𝜃, 𝜏), and then (24.1) 𝒫ℳ = 𝒵 (ℳ 𝜔 ), F𝜆ℳ = 𝜃− ln(𝜆) |𝒵 (ℳ 𝜔 ) (𝜆 > 0). = Fℳ ◦ 𝔊 for all 𝜆 > 0} ⊂ Consider the groups Aut(ℳ) and Aut(F ℳ ) = {𝔊 ∈ Aut(𝒫ℳ ); 𝔊 ◦ Fℳ 𝜆 𝜆 Aut(𝒫ℳ ) endowed with the u-topology (2.23). Let 𝜎 ∈ Aut(ℳ). Then 𝜔 ◦ 𝜎 −1 is also a dominant weight on ℳ. By Proposition 24.5, there exists a *-isomorphism p𝜎 ∶ 𝒵 (ℳ 𝜔 ◦ 𝜎 ) → 𝒵 (ℳ 𝜔 ),

(1)

p𝜎 (c𝜔 ◦ 𝜎 −1 (𝜑)) = c𝜔 (𝜑) (𝜑 ∈ Wint (ℳ)).

(2)

−1

uniquely determined, such that

On the other hand, we have ℳ 𝜔 ◦ 𝜎 = 𝜎(ℳ 𝜔 ), and we obtain a *-isomorphism 𝜎 ∶ 𝒵 (ℳ 𝜔 ) → −1 𝒵 (ℳ 𝜔 ◦ 𝜎 ). We thus obtain a *-automorphism −1

mod(𝜎) = p𝜎 ◦ 𝜎 ∶ 𝒵 (ℳ 𝜔 ) → 𝒵 (ℳ 𝜔 ),

(3)

uniquely determined, such that (see 24.2.(4)) (mod(𝜎))(c𝜔 (𝜑)) = c𝜔 (𝜑 ◦ 𝜎 −1 )

(𝜑 ∈ Wint (ℳ)).

(4)

Since for every 𝜑 ∈ Wint (ℳ) and 𝜆 > 0, we have [mod(𝜎)](F𝜆ℳ (c𝜔 (𝜑))) = [mod(𝜎)](c𝜔 (𝜆𝜑)) = c𝜔 (𝜆𝜑 ◦ 𝜎 −1 ) = Fℳ (c𝜔 (𝜑 ◦ 𝜎 −1 )) = F𝜆ℳ ([mod(𝜎)](c𝜔 (𝜑))), it follows that mod(𝜎) ∈ Aut(F ℳ ). 𝜆

The Fundamental Homomorphism

357

We thus obtain a group homomorphism mod ∶ Aut(ℳ) → Aut(F ℳ ), called the fundamental homomorphism associated with ℳ. Note that for every 𝜎 ∈ Aut(ℳ) the weight 𝜔 ◦ 𝜎 −1 is also dominant, hence 𝜔 ◦ 𝜎 −1 ≂ 𝜔, and we have u ∈ U(ℳ), 𝜔 ◦ 𝜎 −1 = 𝜔 ◦ Ad(u) ⇒ mod(𝜎) = (Ad(u) ◦ 𝜎)|𝒵 (ℳ 𝜔 ). Indeed, using 23.2.(2) and 23.2.(4), for 𝜑 ∈ uc𝜔 ◦ 𝜎 −1 (𝜑 ◦ 𝜎 −1 )u∗ = c𝜔 (𝜑 ◦ 𝜎 −1 ). In particular,

Wint (ℳ) we obtain (Ad(u) ◦ 𝜎)(c𝜔 (𝜑))

𝜔 ◦ 𝜎 = 𝜔 ⇔ mod(𝜎) = 𝜎|𝒵 (ℳ 𝜔 ), [ there exists u ∈ U(ℳ) such that mod(𝜎) = 𝜄 ⇔ 𝜔 ◦ Ad(u) ◦ 𝜎 = 𝜔, (Ad(u) ◦ 𝜎)|𝒵 (ℳ 𝜔 ) = 𝜄.

(5) =

(6) (7)

Thus, mod(𝜎) = 𝜄 for all 𝜎 ∈ Int(ℳ). Since Int(ℳ) is a normal subgroup of Aut(ℳ), it follows that the fundamental homomorphism can be factored to a homomorphism mod ∶ Out(ℳ) = Aut(ℳ)∕Int(ℳ) → Aut(F ℳ ). Note also that from (4), 24.2.(4), and 24.5.(1), we obtain p𝜑 ◦ 𝜎 = mod(𝜎 −1 ) ◦ p𝜑 ◦ 𝜎

(𝜑 ∈ Wint (ℳ), 𝜎 ∈ Aut(ℳ)).

(8)

25.2 Theorem. For every properly infinite W ∗ -algebra ℳ with a separable predual, the fundamental homomorphism mod ∶ Aut(ℳ) → Aut(F ℳ ) is u-continuous. The proof will be given in Section 25.5, after some preparation. Since mod(𝜎) = 𝜄 for every 𝜎 ∈ Int(ℳ), we infer that Corollary. If ℳ is a properly infinite W ∗ -algebra with separable predual, then mod(𝜎) = 𝜄 for every 𝜎 ∈ Int(ℳ) ⊂ Aut(ℳ). 25.3. Let ℳ be a W ∗ -algebra with separable predual. Then (2.23) U(ℳ) is a polish group, that is, there exists a bounded complete metric d which defines the two-sided uniform structure associated with the topology on U(ℳ). Let 𝜎 ∶ G → Aut(ℳ) be a continuous action of the separable locally compact⋃group G on ℳ. There exists an increasing sequence {Kn } of compact subsets of G with G = n K̊ n . Then the set 𝒞 (G, U(ℳ)) of all continuous functions G → U(ℳ), endowed with the topology of uniform convergence on compact subsets, is also a polish topological space, since the equation ∑ 𝛿(u, v) = 2−n sup d(u(g), v(g)) (u, v ∈ 𝒞 (G, U(ℳ)) n

g∈Kn

defines a complete metric 𝛿 on 𝒞 (G, U(ℳ)) compatible with the topology. Since the set of all unitary 𝜎-cocycles Z𝜎 (G; U(ℳ)) is closed in 𝒞 (G, U(ℳ)) it follows that Z𝜎 (G; U(ℳ)) endowed with the topology of uniform convergence on compact subsets is a polish topological space.

358

Continuous Decompositions

We show that the mapping 𝜕 = 𝜕𝜎 ∶ U(ℳ) → Z𝜎 (G; U(ℳ)) defined by (𝜕u)(g) = u∗ 𝜎g (u) (u ∈ U(ℳ), g ∈ G) is continuous. Indeed, let un → u in U(ℳ), let K ⊂ G be a compact set and ℒ ⊂ ℳ∗ a 𝜎(ℳ∗ , ℳ)-compact set. Then (13.5) the set Kℒ = {𝜑 ◦ 𝜎g ; 𝜑 ∈ ℒ , g ∈ K} ⊂ ℳ∗ is 𝜎(ℳ∗ , ℳ)compact. Since un → u with respect to the Mackey topology 𝜏𝜔 , we have 𝜑(𝜎g (un )) → 𝜑(𝜎g (u)) uniformly with respect to g ∈ K and 𝜑 ∈ ℒ . Since ℒ was arbitrary, it follows that 𝜎g (un ) → 𝜎g (u) in U(ℳ), uniformly for g ∈ K, hence (𝜕un )(g) = u∗n 𝜎g (un ) → u∗ 𝜎g (u) = (𝜕u)(g) uniformly for g ∈ K. Consequently, 𝜕un → 𝜕u in Z𝜎 (G; U(ℳ)). Since for u, v ∈ U(ℳ) we have 𝜕u = 𝜕v ⇔ uv∗ ∈ U(ℳ 𝜎 ) and since U(ℳ 𝜎 ) is a closed subgroup of U(ℳ), it follows that the homogeneous space U(ℳ)∕U(ℳ 𝜎 ) with the quotient topology is a polish space and that 𝜕 can be factored to an injective mapping 𝜕 𝜎 ∶ U(ℳ)∕U(ℳ 𝜎 ) → Z𝜎 (G; U(ℳ)). According to classical results in the theory of Borel spaces, it follows that 𝜕𝜎 (U(ℳ)) is a Borel set in Z𝜎 (G; U(ℳ)) and 𝜕 𝜎 ∶ U(ℳ)∕U(ℳ 𝜎 ) → 𝜕𝜎 (U(ℳ)) is a Borel isomorphism. 25.4. Let ℳ be a W ∗ -algebra with separable predual and 𝜔 a fixed n.s.f. weight on ℳ. By Theorem 5.1, the mapping D𝜔 ∶ Wnsf (ℳ) ∋ 𝜑 ↦ [D𝜑 ∶ D𝜔] ∈ Z𝜎 𝜔 (ℝ; U(ℳ))

(1)

is a bijection. We shall consider on Wnsf (ℳ) the unique topology such that D𝜔 be a homeomorphism. The topology introduced on Wnsf (ℳ) does not depend on the fixed n.s.f. weight 𝜔 on ℳ, as can be easily seen using Corollary 3.5. We show that the mapping Aut(ℳ) ∋ 𝜎 ↦ 𝜔 ◦ 𝜎 −1 ∈ Wnsf (ℳ)

(2)

is continuous. Indeed, let f be a faithful normal state on ℳ and 𝜎n → 𝜎 in Aut(ℳ). Since ‖f ◦ 𝜎n−1 − f ◦ 𝜎 −1 ‖ → 0, by Proposition 7.18 it follows that [D( f ◦ 𝜎n−1 ) ∶ D( f ◦ 𝜎 −1 )]t → 1 uniformly on compact subsets. Also, 𝜎n ([D𝜔 ∶ Df]t ) → 𝜎([D𝜔 ∶ Df]t ) uniformly on compact subsets (see the last remark in Section 2.23). Therefore, [D(𝜔 ◦ 𝜎n−1 ) ∶ D(𝜔 ◦ 𝜎 −1 )]t = 𝜎n ([D𝜔 ∶ Df]t )[D( f ◦ 𝜎n−1 ) ∶ D( f ◦ 𝜎 −1 )]t 𝜎([Df ∶ D𝜔]t ) converges to 1 uniformly on compact subsets, that is, 𝜔 ◦ 𝜎n−1 → 𝜔 ◦ 𝜎 −1 in Wnsf (ℳ). Clearly, Wnsf (ℳ) is a polish space. Consider now W(𝜔) = {𝜑 ∈ Wnsf (ℳ); 𝜑 ≂ 𝜔}.

(3)

We show that W(𝜔) is a Borel set in Wnsf (ℳ) and that there exists a Borel mapping u ∶ W(𝜔) ∋ 𝜑 ↦ u(𝜑) ∈ U(ℳ)

(4)

𝜑 = 𝜔 ◦ Ad(u(𝜑)) (𝜑 ∈ W(𝜔)).

(5)

such that

Indeed, if we identify Wnsf (ℳ) with Z𝜎 𝜔 (ℝ; U(ℳ)) by the mapping D𝜔 , then the range of the mapping 𝜕 considered in Section 25.3 is just the set W(𝜔) which is, therefore, a Borel set. Moreover,

The Fundamental Homomorphism

359

from Section 25.3 it follows that the mapping 𝜕 ∶ U(ℳ)∕U(ℳ 𝜔 ) → W(𝜔) is a Borel isomorphism, so that the existence of the mapping u is a consequence of the classical result concerning the existence of a Borel section for the canonical mapping U(ℳ) → U(ℳ)∕U(ℳ 𝜔 ). 25.5. Proof of Theorem 25.2. Let 𝜔 be a dominant weight on ℳ. The mapping 𝔪 ∶ Aut(ℳ) ∋ 𝜎 ↦ Ad(u(𝜔 ◦ 𝜎 −1 )) ◦ 𝜎 ∈ Aut(ℳ)

(1)

is a Borel mapping since the mapping Aut(ℳ) ∋ 𝜎 ↦ 𝜔 ◦ 𝜎 −1 ∈ W(𝜔) is continuous (25.4.(2)), the mapping W(𝜔) ∋ 𝜑 ↦ u(𝜑) ∈ U(ℳ) is a Borel mapping (25.4.(4)), and the mapping U(ℳ) ∋ u ↦ Ad(u) ∈ Aut(ℳ) is continuous (2.23). Also, by 25.1.(5) we have mod(𝜎) = 𝔪(𝜎)|𝒵 (ℳ 𝜎 ) (𝜎 ∈ Aut(ℳ)).

(2)

It follows that mod is a Borel homomorphism between the polish groups Aut(ℳ) and Aut(F ℳ ), and so is continuous. 25.6. Let ℳ be an infinite semifinite factor with separable predual. We recall (24.8) that 𝜆)). It is then easy to check that the mapping ℝ+∗ ∋ 𝜆 ↦ Fℳ ∈ Aut(F ℳ ) (𝒫ℳ , F ℳ ) ≈ (ℒ ∞ (ℝ+∗ ), Ad(𝜆 𝜆 is a surjective group isomorphism. Let 𝜏 be an n.s.f. trace on ℳ, 𝜎 ∈ Aut(ℳ) and 𝜆 ∈ ℝ+∗ . Then mod(𝜎) = F𝜆ℳ ⇔ 𝜏 ◦ 𝜎 −1 = 𝜆𝜏.

(1)

+ −1 ≂ 𝜇𝜑 for Indeed, let 𝜏 ◦ 𝜎 −1 = 𝜆𝜏 and mod(𝜎) = Fℳ 𝜇 for some 𝜆, 𝜇 ∈ ℝ∗ . Then we have 𝜑 ◦ 𝜎 ∞ + every 𝜑 ∈ Wint (ℳ). Let 𝜀 > 0. By Section 23.18, there exists a ∈ ℳ with 1 − 𝜀 ≤ a ≤ 1 + 𝜀 such that 𝜑 = 𝜏a ∈ W∞ (ℳ). Then, there exists u ∈ U(ℳ) such that 𝜇𝜑 = 𝜑 ◦ 𝜎 −1 ◦ Ad(u). int Thus, for x ∈ ℳ + we obtain 𝜇𝜏a (x) = 𝜇𝜑(x) = 𝜑(𝜎 −1 (uxu∗ )) = (𝜏 ◦ 𝜎 −1 )(𝜎(a)1∕2 uxu∗ 𝜎(a)1∕2 ) = 𝜆𝜏(uu∗ 𝜎(a)1∕2 uxu∗ 𝜎(a)1∕2 uu∗ ) = 𝜆𝜏u∗ 𝜎(a)u (x), hence 𝜇a = 𝜆u∗ 𝜎(a)u. It follows that (1 − 𝜀)𝜆 ≤ (1 + 𝜀)𝜇 and (1 − 𝜀)𝜇 ≤ (1 + 𝜀)𝜆. Since 𝜀 > 0 was arbitrary, we conclude that 𝜇 = 𝜆. Clearly, the earlier result is interesting only for type II∞ , factors since if ℳ is a type I∞ factor, Aut(ℳ) = Int(ℳ), and mod(𝜎) = 𝜄, 𝜏 ◦ 𝜎 = 𝜏 for all 𝜎 ∈ Aut(ℳ).

25.7. Consider now a type II1 factor 𝒩 and let ν be the unique faithful finite normal trace on 𝒩 ̄ ℱ∞ and the n.s.f. trace 𝜏 = ν ⊗ ̄ tr on with ν(1) = 1. Consider also the type II∞ factor ℳ = 𝒩 ⊗ ℳ, where tr is the canonical trace on the countably decomposable type I∞ factor ℱ∞ . Let 0 < 𝜆 ≤ 1 be fixed. If e and e1 are projections in 𝒩 with ν(e) = ν(e1 ) = 𝜆, then e ∼ e1 in 𝒩 and so the reduced algebras e𝒩 e and e1 𝒩 e1 are *-isomorphic. Thus, the isomorphism class of e𝒩 e does not depend on the projection e ∈ 𝒩 with ν(e) = 𝜆 and will be denoted by 𝒩 𝜆 . We show that 𝒩 𝜆 ≈ 𝒩 ⇔ there exists 𝜎 ∈ Aut(ℳ) with 𝜏 ◦ 𝜎 = 𝜆𝜎.

(1)

Indeed, let e ∈ 𝒩 be a projection with ν(e) = 𝜆 and let 𝜋 ∶ 𝒩 → e𝒩 e ⊂ 𝒩 be a *-isomorphism. ̄ 1ℱ and 1𝒩 ⊗ ̄ 1ℱ = 1ℳ are properly infinite projections in the countably decomposable Since e ⊗ ̄ 1ℱ ∼ 1ℳ , hence there exists w ∈ ℳ such that w∗ w = e ⊗ ̄ 1ℱ and factor ℳ, we have e ⊗ ∗ ̄ ℱ∞ → 𝒩 ⊗ ̄ ℱ∞ . ww = 1ℳ . Then the mapping x ↦ wxw∗ defines a *-isomorphism 𝜌 ∶ e𝒩 e ⊗

360

Continuous Decompositions

̄ 𝜄) ∈ Aut(ℳ). If f ∈ ℱ∞ is any minimal projection, We thus obtain a *-automorphism 𝜎 = 𝜌 ◦ (𝜋 ⊗ ̄ f ) = 1 and (𝜏 ◦ 𝜎)(1𝒩 ⊗ ̄ f ) = 𝜏(w(e ⊗ ̄ f )w∗ ) = 𝜏(e ⊗ ̄ f ) = ν(e) = 𝜆, hence 𝜏 ◦ 𝜎 = 𝜆𝜏. then 𝜏(1𝒩 ⊗ Conversely, let 𝜎 ∈ Aut(ℳ) with 𝜏 ◦ 𝜎 = 𝜆𝜏, let e ∈ 𝒩 be a projection with ν(e) = 𝜆 ̄ f )) = 𝜆𝜏(1𝒩 ⊗ ̄ f ) = 𝜏(e ⊗ ̄ f ), and let f ∈ ℱ∞ be any minimal projection. Then 𝜏(𝜎(1𝒩 ⊗ ̄ f ∼ 𝜎(1𝒩 ⊗ ̄ f ) in ℳ. It follows that e𝒩 e ≈ (e ⊗ ̄ f )(𝒩 ⊗ ̄ ℱ∞ )(e ⊗ ̄ f) ≂ hence e ⊗ ̄ f )(𝒩 ⊗ ̄ ℱ∞ )𝜎(1𝒩 ⊗ ̄ f ) ≈ (1𝒩 ⊗ ̄ f )(𝒩 ⊗ ̄ ℱ∞ )(1𝒩 ⊗ ̄ f ) ≈ 𝒩 , and so 𝒩 𝜆 ≈ 𝒩 . 𝜎(1𝒩 ⊗ 𝜆 1∕𝜆 For 𝜆 ≥ 1, we shall write 𝒩 ≈ 𝒩 if 𝒩 ≈ 𝒩 . The assertion (1) extends obviously to every 𝜆 > 0. It follows that the set G(𝒩 ) = {𝜆 > 0; (𝒩 𝜆 ≈ 𝒩 )} is a subgroup of ℝ+∗ , called the fundamental group of the type II1 factor 𝒩 . For the type II∞ factor ℳ, we shall define the fundamental group G(ℳ) by G(ℳ) = {𝜆 > 0; there exists 𝜎 ∈ Aut(ℳ) with 𝜏 ◦ 𝜎 = 𝜆𝜏} where 𝜏 is any n.s.f. trace on ℳ. From the considerations of this section and Section 25.6, it follows that the fundamental homomorphism, more precisely its range, is an extension of the fundamental group to properly infinite factors. For type III factors, the fundamental homomorphism will be studied in more detail later (see 30.9, 30.10). 25.8. Notes. The material or this section is due to Connes and Takesaki (1977). The fundamental group of a type II1 factor is a classical invariant introduced by Murray and von Neumann (1936, 1937, 1943, IV). For our exposition, we have used Connes and Takesaki (1977); Murray and von Neumann (1936, 1937, 1943, IV).

26 The Extension of the Modular Automorphism Group In this section, we extend the modular automorphism groups {𝜎t𝜑 }t∈ℝ to families {𝜎c𝜑 }c indexed by unitary F ℳ -cocycles; similar extensions are given for Connes cocycles. 26.1. Let ℳ be a countably decomposable properly infinite W ∗ -algebra and (𝒫ℳ , F ℳ ) the flow of weights on ℳ. Consider the set Z(F ℳ ) = ZF ℳ (ℝ+∗ ; U(𝒫ℳ )) of all unitary F ℳ -cocycles, the mapping 𝜕 ∶ U(𝒫ℳ ) → Z(F ℳ ) defined by (𝜕v)(𝜆) = v∗ Fℳ (v) (v ∈ U(𝒫ℳ ), 𝜆 ∈ ℝ+∗ ), and put B(F ℳ ) = 𝜆 𝜕(U(𝒫ℳ )). It is easy to check that, with respect to pointwise multiplications, Z(F ℳ ) is an abelian group and B(F ℳ ) is a subgroup of Z(F ℳ ), so that we can consider the quotient group H(F ℳ ) = Z(F ℳ )∕B(F ℳ ). We define an injective group homomorphism ℝ ∋ t ↦ ̄t ∈ Z(F ℳ ) by ̄t(𝜆) = 𝜆−it , (t ∈ ℝ, 𝜆 ∈ ℝ+∗ ). Let 𝜔 be a dominant weight on ℳ and (𝒩 , 𝜃, 𝜏) a continuous decomposition of ℳ with 𝒩 = ℳ 𝜔 ̂ 𝜏). and (ℳ, 𝜎 𝜔 , 𝜔) ≈ (ℛ(𝒩 , 𝜃), 𝜃, ̂ The *-isomorphism ℳ ≈ ℛ(ℳ 𝜔 , 𝜃) shows that there exists an s-continuous unitary representation u ∶ ℝ → U(ℳ) such that ℳ = ℛ{ℳ 𝜔 , u(ℝ)}, 𝜃s = Ad(u(s))|ℳ 𝜔 and 𝜔 ◦ Ad(u(s)) = e−s 𝜔, that is, 𝜎t𝜔 (u(s)) = e−ist u(s) (s, t ∈ ℝ) (see 23.7).

The Extension of the Modular Automorphism Group

361

We shall identify ℝ with ℝ+∗ via the mapping s ↦ e−s . Then the flow of weights is the pair (𝒵 (ℳ 𝜔 ), 𝜃|𝒵 (ℳ 𝜔 )), Z(F ℳ ) consists of all continuous functions c ∶ ℝ → U(𝒵 (ℳ 𝜔 )) with the property c(s + t) = c(s)𝜃s (c(t)) (s, t ∈ ℝ), for v ∈ U(𝒵 (ℳ 𝜔 )) we have (𝜕v)(s) = v∗ 𝜃s (v) (s ∈ ℝ), and for t ∈ ℝ we have ̄t(s) = e−ist (s ∈ ℝ). 26.2. Assume that the W ∗ -algebra ℳ is realized as a von Neumann algebra ℳ ⊂ ℬ(ℋ ). Note that ℳ ≈ ℛ(ℳ 𝜔 , 𝜃) ⊂ ℬ(ℒ 2 (ℝ, ℋ )) is also a realization of ℳ as a von Neumann algebra. ̄ ℒ ∞ (ℝ) ⊂ ℬ(ℒ 2 (ℝ, ℋ )) by Let c ∈ Z(F ℳ ). We define a unitary operator Uc ∈ 𝒵 (ℳ 𝜔 ) ⊗ (Uc 𝜉)(s) = 𝜃s−1 (c(s))𝜉(s) = c(−s)−1 𝜉(s) (𝜉 ∈ ℒ 2 (ℝ, ℋ ), s ∈ ℝ). ̄ ℒ ∞ (ℝ), hence Uc 𝜋𝜃 (a)U∗ = 𝜋𝜃 (a). For u(s) = 1 ⊗ ̄ 𝜆 (s) (s ∈ For a ∈ ℳ 𝜔 , we have 𝜋𝜃 (a) ∈ ℳ 𝜔 ⊗ c ℝ), we have ̄ 𝜆 (s))U∗ 𝜉)(r) = c(−r)−1 ((1 ⊗ ̄ 𝜆 (s))U∗ 𝜉)(r) = c(−r)−1 (U∗ 𝜉)(r − s) (Uc (1 ⊗ c c c = c(−r)−1 c(s − r)𝜉(r − s) = c(−r)−1 c(−r)𝜃r−1 (c(s))𝜉(r − s) ̄ 𝜆 (s))𝜉)(r) = (𝜋𝜃 (c(s))(1 ⊗ ̄ 𝜆 (s))𝜉)(r) (𝜉 ∈ ℒ 2 (ℝ, ℋ ), r ∈ ℝ). = 𝜃 −1 (c(s))((1 ⊗ r

It follows that Uc determines a unique *-automorphism 𝜎c𝜔 = Ad(Uc )|ℳ ∈ Aut(ℳ)

(1)

𝜎c𝜔 (a) = a (a ∈ ℳ 𝜔 ) 𝜎c𝜔 (u(s)) = c(s)u(s) (s ∈ ℝ).

(2) (3)

unique with the properties

̄ ℒ ∞ (ℝ) commute and If c1 , c2 ∈ Z(F ℳ ), then the unitary operators Uc1 , Uc2 ∈ 𝒵 (ℳ 𝜔 ) ⊗ Uc1 Uc2 = Uc1 c2 , so that 𝜎c𝜔 c = 𝜎c𝜔 𝜎c𝜔 . 1 2

1

2

(4)

Hence the mapping Z(F ℳ ) ∋ c ↦ 𝜎c𝜔 ∈ Aut(ℳ) is a group homomorphism. For t ∈ ℝ, the *-automorphism 𝜎̄t𝜔 ∈ Aut(ℳ) is uniquely determined by 𝜎̄t𝜔 (a) = a (a ∈ ℳ 𝜔 ) and 𝜎̄t𝜔 (u(s)) = e−ist u(s) (s ∈ ℝ), hence 𝜎̄t𝜔 = 𝜎t𝜔

(t ∈ ℝ).

(5)

Since 𝜔 is the dual weight corresponding to the trace 𝜏, we have 𝜔 = 𝜏 ◦ P𝜔 where P𝜔 ∶ ℳ + → (ℳ 𝜔 )+ is the n.s.f. operator-valued weight defined by P𝜔 (x) = ∫ 𝜎t𝜔 (x) dt (x ∈ ℳ + ). Since 𝜎c𝜔 commutes with 𝜎t𝜔 (t ∈ ℝ), and 𝜎c𝜔 |ℳ 𝜔 = 𝜄, it follows that P𝜔 ◦ 𝜎c𝜔 = 𝜎c𝜔 ◦ P𝜔 = P𝜔 and hence 𝜔 ◦ 𝜎c𝜔 = 𝜔

(c ∈ Z(F ℳ )).

(6)

362

Continuous Decompositions

If x ∈ ℳ(𝜎 𝜔 , {s}), that is, x ∈ ℳ and 𝜎t𝜔 (x) = e−ist x (t ∈ ℝ), then u(s)∗ x ∈ ℳ 𝜔 , so that u(s)∗ x = 𝜎c𝜔 (u(s)∗ x) = u(s)∗ c(s)∗ x. Therefore, x ∈ ℳ(𝜎 𝜔 , {s}) ⇒ 𝜎c𝜔 (x) = c(s)x for all c ∈ Z(F ℳ ).

(7)

This condition also determines uniquely the *-automorphism 𝜎c𝜔 . Let 𝛼 ∈ Aut(ℳ) be such that 𝜔 ◦ 𝛼 = 𝜔 and let c ∈ Z(F ℳ ). Then 𝛼(𝒵 (ℳ 𝜔 )) = 𝒵 (ℳ 𝜔 ) and the function 𝛼(c) ∶ ℝ ∋ s ↦ 𝛼(c(s)) ∈ 𝒵 (ℳ 𝜔 ) defines a unique element 𝛼(c) ∈ Z(F ℳ ). If 𝜔 x ∈ ℳ(𝜎 𝜔 , {s}), then 𝛼(x) ∈ ℳ(𝜎 𝜔 , {s}) and so 𝜎𝛼(c) (𝛼(x)) = 𝛼(c(s))𝛼(x) = 𝛼(c(s)x) = 𝛼(𝜎c𝜔 (x)). Consequently, 𝜔 𝛼 ∈ Aut(ℳ), 𝜔 ◦ 𝛼 = 𝜔 ⇒ 𝛼 ◦ 𝜎c𝜔 = 𝜎u(c) ◦ 𝛼 for all c ∈ Z(F ℳ ).

(8)

We show that for c ∈ Z(F ℳ ) we have 𝜎c𝜔 ∈ Int(ℳ) ⇔ c ∈ B(F ℳ ).

(9)

Indeed, if c ∈ B(F ℳ ), then there exists a unitary element v ∈ 𝒵 (ℳ 𝜔 ) such that c(s) = v∗ 𝜃s (v) (s ∈ ℝ). We have [Ad(v∗ )](a) = a for a ∈ ℳ 𝜔 and [Ad(v∗ )](u(s)) = v∗ u(s)vu(s)∗ u(s) = v∗ 𝜃s (v)u(s) = c(s)u(s) for s ∈ ℝ, hence 𝜎c𝜔 = Ad(v∗ ) ∈ Int(ℳ). Conversely, let 𝜎c𝜔 = Ad(v∗ ) for some v ∈ U(ℳ). Using (2), we infer that v ∈ (ℳ 𝜔 )′ ∩ ℳ; hence v ∈ 𝒵 (ℳ 𝜔 ) by the relative commutant theorem (23.19) and from (3) it follows that c = 𝜕v ∈ B(F ℳ ). Thus, the mapping c ↦ 𝜎c𝜔 can be factored to a group homomorphism 𝛿𝜔 ∶ H(F ℳ ) → Out(ℳ),

(10)

where 𝛿𝜔 (c∕B(F ℳ )) = 𝜎c𝜔 ∕Int(ℳ) (c ∈ Z(F ℳ )). We shall see later that the mapping 𝛿𝜔 does not depend on the choice of the dominant weight 𝜔 on ℳ. Finally, we notice the following important result {𝜎c𝜔 ; c ∈ Z(F ℳ )} = {𝜎 ∈ Aut(ℳ); 𝜎|ℳ 𝜔 = 𝜄}.

(11)

Indeed, the inclusion “⊂” follows from (2). Conversely, let 𝜎 ∈ Aut(ℳ) be such that 𝜎|ℳ 𝜔 = 𝜄, and define c(s) = 𝜎(u(s))u(s)∗ ∈ U(ℳ) (s ∈ ℝ). For x ∈ ℳ 𝜔 and s ∈ ℝ, we have c(s)xc(s)∗ = 𝜎(u(s))u(s)∗ xu(s)𝜎(u(s)∗ ) = 𝜎(u(s))𝜃s−1 (x)𝜎(u(s)∗ ) = 𝜎(u(s)𝜃s−1 (x)u(s)∗ ) = 𝜎(𝜃s (𝜃s−1 (x))) = 𝜎(x) = x, hence c(s) ∈ (ℳ 𝜔 )′ ∩ ℳ = 𝒵 (ℳ 𝜔 ) by the relative commutant theorem (23.19). For s, t ∈ ℝ, we have c(s + t) = 𝜎(u(s + t))u(s + t)∗ = 𝜎(u(s))𝜎(u(t))u(t)∗ u(s)∗ = 𝜎(u(s))u(s)∗ u(s)𝜎(u(t))u(t)∗ u(s)∗ = 𝜎(u(s))u(s)∗ 𝜃s (𝜎(u(t))u(t)∗ ) = c(s)𝜃s (c(t)), hence c ∈ Z(F ℳ ). Since 𝜎(u(s)) = c(s)u(s)∗ (s ∈ ℝ), and 𝜎(a) = a (a ∈ ℳ 𝜔 ), it follows that 𝜎 = 𝜎c𝜔 . 26.3. We now extend the previous construction to integrable n.s.f. weights on ℳ. In formulating the uniqueness condition, we shall use the notation c𝜔 (𝜑) and p𝜑 of Sections 24.2, 24.6. Theorem. Let ℳ be a properly infinite W ∗ -algebra with separable predual and 𝜔 a dominant weight on ℳ.

The Extension of the Modular Automorphism Group

363

Let 𝜑 be an integrable n.s.f. weight on ℳ. For every c ∈ Z(F ℳ ), there exists a unique *-automorphism 𝜎c𝜑 ∈ Aut(ℳ) such that x ∈ ℳ(𝜎 𝜑 ; {s}) ⇒ 𝜎c𝜑 (x) = p−1 𝜑 (c(s)c𝜔 (𝜑))x

(s ∈ ℝ).

(1)

The mapping Z(F ℳ ) ∋ c ↦ 𝜎c𝜑 ∈ Aut(ℳ) is a group homomorphism and 𝜑 ◦ 𝜎c𝜑 = 𝜑 𝜎̄t𝜑 = 𝜎t𝜑 .

(2) (3)

Let 𝜑, 𝜓 be integrable n.s.f. weights on ℳ and 𝜃(𝜑, 𝜓) the corresponding balanced weight on Mat2 (ℳ). For every c ∈ Z(F ℳ ), there exists a unique unitary element [D𝜓 ∶ D𝜑]c ∈ ℳ such that 𝜎c𝜃(𝜑,𝜓)

((

0 1

0 0

)) (

0 [D𝜓 ∶ D𝜑]c

0 0

) (4)

and we have 𝜎c𝜓 = Ad([D𝜓 ∶ D𝜑]c ) ◦ 𝜎c𝜑 [D𝜓 ∶ D𝜑]c1 c2 = [D𝜓 ∶ D𝜑]c1 𝜎c𝜑 ([D𝜓 ∶ D𝜑]c2 )

(5) (6)

[D𝜓, ∶ D𝜑]̄t = [D𝜓 ∶ D𝜑]t .

(7)

1

Proof. Since 𝜑 is integrable, we have 𝜑 ≲ 𝜔 and, as 𝜑 is faithful, there exists v ∈ ℳ such that v∗ v = s(𝜑) = 1, e = vv∗ ∈ ℳ 𝜔 , and 𝜑 = 𝜔v . The mapping Φv ∶ ℳ ∋ x ↦ vxv∗ ∈ eℳe is a *-isomorphism such that 𝜔e ◦ Φv = 𝜑. It follows that (𝜎t𝜔 |eℳe) ◦ Φv = Φv ◦ 𝜎t𝜑 (t ∈ ℝ), so that Φv (ℳ 𝜑 ) = eℳ 𝜔 e and Φv (𝒵 (ℳ 𝜑 )) = 𝒵 (ℳ 𝜔 )e. Also, we have c𝜔 (𝜑) = zℳ 𝜔 (e) = c𝜔 (𝜔e ) (see −1 𝜔 𝜔 24.2) and Φv ◦ p−1 𝜑 = p𝜔e ∶ 𝒵 (ℳ )c𝜔 (𝜑) → 𝒵 (ℳ )e is just the mapping z → ze. We define 𝜎c𝜑 = Φv−1 ◦ (𝜎c𝜔 |eℳe) ◦ Φv , that is, 𝜎c𝜑 (x) = v∗ 𝜎c𝜔 (vxv∗ )v (x ∈ ℳ). If x ∈ ℳ(𝜎 𝜑 , {s}), then vxv∗ = Φv (x) ∈ eℳ(𝜎 𝜔 , {s})e and 𝜎c𝜑 (x) = v∗ 𝜎c𝜔 (vxv∗ )v = v∗ c(s)vxv∗ v = ∗ −1 −1 v c(s)c𝜔 (𝜑)evx = [Φ−1 (1). Since 𝜑 is v (Φv ◦ p𝜑 )(c(s)c𝜔 (𝜑))]x = p𝜑 (c(s)c𝜔 (𝜑))x, thus proving ⋃ integrable, it follows by Corollary 21.3 that ℳ is generated by the union s ℳ(𝜎 𝜑 ; {s}), hence condition (1) completely determines the *-automorphism 𝜎c𝜑 . The fact that the mapping c → 𝜎c𝜑 is a group homomorphism, and equations (2) and (3), now follow easily from the previous considerations and from the similar properties of 𝜔, already proved in Section 26.2. If the weights 𝜑 and 𝜓 are integrable, then, as easily verified, also the weight 𝜃(𝜑, 𝜓) on ℳ̄ = ̄ ℳ𝜔 = ℳ𝜔 ⊗ ̄ ℱ2 is also integrable. Furthermore, 𝜔̄ = 𝜔 ⊗ ̄ tr is a dominant weight on ℳ, ̄ ℱ2 ℳ⊗ 𝜔 ̄ ̄ ̄ ̄ and (ℳ ⊗ ℱ2 , 𝜃 ⊗ 𝜄, 𝜏 ⊗ tr) is a continuous decomposition of ℳ. It follows that the flow of weights on ℳ̄ is isomorphic to the flow of weights on ℳ. Using the uniqueness assertion based on (1), for x ∈ ℳ we obtain (( )) ( 𝜑 ) (( )) ( ) x 0 𝜎c (x) 0 0 0 0 0 𝜃(𝜑,𝜓) 𝜃(𝜑,𝜓) . 𝜎c = , 𝜎c = 0 0 0 0 0 x 0 𝜎c𝜓 (x) Equations (4), (5), and (6) can now be proved with the same arguments as in the proof of Theorem 3.1, while (7) follows immediately from (3) and (4).

364

Continuous Decompositions

Besides the previous basic properties, we also have the following computation rules for the extensions 𝜎c𝜑 and [D𝜓 ∶ D𝜑]c : 𝜑 𝜎c𝜓 ◦ 𝛼 = 𝛼 −1 ◦ 𝜎[mod(𝛼)](c) ◦𝛼

(8)

[D(𝜓 ◦ 𝛼) ∶ D(𝜑 ◦ 𝛼)]c = 𝛼 ([D𝜓 ∶ D𝜑][mod(𝛼)](c)

(9)

[D𝜓u ∶ D𝜑]c = u [D𝜓 ∶ D𝜑]c 𝜎c𝜓 (u) [D𝜓 ∶ D𝜑]c = [D𝜑 ∶ D𝜓]∗c

(10) (11) (12)

−1

∗

[D𝜒 ∶ D𝜑]c = [D𝜒 ∶ D𝜓]c [D𝜓 ∶ D𝜑]c

where 𝜑, 𝜓, 𝜒 are integrable n.s.f. weights on ℳ, 𝛼 ∈ Aut(ℳ), u ∈ U(ℳ) and c ∈ Z(F ℳ ). (8) follows using (1) and 25.1.(8), (9) follow from (4) and (8), and (10), (11), (12) are easily checked using the definitions and the special case c = ̄t considered in Section 3. It is possible (see Connes & Takesaki, 1977, IV.2.2) to show that 26.2.(11) can also be extended to integrable n.s.f. weights 𝜑 on ℳ: {𝜎c𝜑 ; c ∈ Z(F ℳ )} = {𝜎 ∈ Aut(ℳ); 𝜎|ℳ 𝜑 = 𝜄}.

(13)

Note that the condition s(𝜑) = 1 was necessary only in order to obtain 𝜎c𝜑 as a *-automorphism of ℳ itself. Without assuming this condition we can still define, by the same method, 𝜎c𝜑 ∈ Aut(ℳs(𝜑) ). 26.4. From 26.3.(5) and 26.2.(9), it follows that for every integrable n.s.f. weight 𝜑 on ℳ and c ∈ Z(F ℳ ) we have 𝜎c𝜑 ∈ Int(ℳ) ⇒ c ∈ B(F ℳ ).

(1)

Thus, the mapping c∕B(F ℳ ) → 𝜎c𝜑 ∕Int(ℳ) (c ∈ Z(F ℳ )) defines an injective group homomorphism 𝛿ℳ ∶ H(F ℳ ) → Out(ℳ) which, by 26.3.(5), does not depend on the integrable n.s.f. weight 𝜑 on ℳ. In particular, 𝛿ℳ = 𝛿𝜔 (26.2.(10)). Since 𝜔 ◦ 𝜎c𝜔 = 𝜔 and hence mod(𝜎c𝜔 ) = 𝜎c𝜔 |𝒵 (ℳ 𝜔 ) (see 25.1.(6)), it follows that the composition of 𝛿ℳ with mod ∶ Out(ℳ) → Aut(F ℳ ) is the trivial homomorphism: mod(𝛿ℳ (c)) = 𝜄

(c ∈ H(F ℳ )).

(2)

Note that 𝛿ℳ (H(F ℳ )) is a normal subgroup of Out(ℳ), in fact for every c ∈ H(F ℳ ) and 𝛼 ∈ Aut(ℳ) we have 𝛼𝛿ℳ (c)𝛼 −1 = 𝛿ℳ ([mod(𝛼)](c)).

(3)

Indeed, 𝜔 ◦ 𝛼 is still a dominant weight, so that there exists v ∈ U(ℳ) with 𝜔 ◦ 𝛼 ◦ Ad(v) = 𝜔 and we may assume that 𝜔 ◦ 𝛼 = 𝜔; in this case, using 26.2.(8) and 25.1.(6), we obtain 𝛼 ◦ 𝜎c𝜔 ◦ 𝜎 −1 = 𝜔 𝜎[mod(v)](c) .

The Extension of the Modular Automorphism Group

365

As an application, in the next theorem we compute the group Out(ℳ) in terms of the continuous decomposition (𝒩 , 𝜃, 𝜏) of ℳ. To this end, we consider the subgroups Aut𝜃,𝜏 (𝒩 ) = {𝛽 ∈ Aut(𝒩 ); 𝜏 ◦ 𝛽 = 𝜏, 𝛽 ◦ 𝜃s = 𝜃s ◦ 𝛽 for all s ∈ ℝ} ⊂ Aut(𝒩 ) and Out𝜃,𝜏 (𝒩 ) = {𝛽∕Int(𝒩 ); 𝛽 ∈ Aut𝜃,𝜏 (𝒩 )} ⊂ Out(𝒩 ). Theorem. Let ℳ be a countably decomposable properly infinite W ∗ -algebra and (𝒩 , 𝜃, 𝜏) a continuous decomposition of ℳ. There exists a homomorphism 𝛾̄ of Out(ℳ) onto Out𝜃,𝜏 (𝒩 ) such that the following sequence is exact: 𝛿ℳ

𝛾̄

{1} → H(F ℳ ) → Out(ℳ) → Out𝜃,𝜏 (𝒩 ) → {1}.

(4)

Proof. We shall use the notation introduced in Section 26.1, in particular 𝒩 = ℳ 𝜔 . Let Aut𝜔 (ℳ) = {𝜎 ∈ Aut(ℳ); 𝜔 ◦ 𝜎 = 𝜔}. For every 𝜎 ∈ Aut(ℳ), 𝜔 ◦ 𝜎 is also a dominant weight and so there exists u ∈ U(ℳ) such that 𝜔 ◦ 𝜎 ◦ Ad(u) = 𝜔. It follows that Out(ℳ) = Aut𝜔 (ℳ)∕Int(ℳ). If 𝜎 ∈ Aut𝜔 (ℳ), then 𝜎(ℳ 𝜔 ) = ℳ 𝜔 , so that we can define a homomorphism 𝛾 ∶ Aut𝜔 (ℳ) → Out(𝒩 ) by putting 𝛾(𝜎) = (𝜎|𝒩 )∕Int(𝒩 ) (𝜎 ∈ Aut𝜔 (ℳ)).

(5)

Since Aut𝜔 (ℳ) ∩ Int(ℳ) = {Ad(u); u ∈ U(ℳ 𝜔 )}, it follows that 𝛾 can be factored to a homomorphism 𝛾̄ ∶ Out(ℳ) → Out(𝒩 ). If 𝜎 = 𝜎c𝜔 with c ∈ Z(F ℳ ), then 𝜎|𝒩 = 𝜄 (26.2.(2)), and 𝛾(𝜎) = 𝜄. Conversely, if 𝜎 ∈ Aut𝜔 (ℳ) and 𝛾(𝜎) = 𝜄, that is, 𝜎|𝒩 = Ad(v∗ ) with v ∈ U(𝒩 ), then (𝜎 ◦ Ad(v))|𝒩 = 𝜄 and so there exists c ∈ Z(F ℳ ) such that 𝜎 ◦ Ad(v) = 𝜎c𝜔 (26.2.(11)). Thus, Ker(̄𝛾 ) = Range(𝛿ℳ ). Let 𝜎 ∈ Aut𝜔 (ℳ) and 𝛽 = 𝜎|𝒩 ∈ Aut(𝒩 ). For s ∈ ℝ, we have 𝜎t𝜔 (𝜎(u(s))) = 𝜎(𝜎t𝜔 (u(s))) = −ist e 𝜎(u(s)), hence a(s) = 𝜎(u(s))u(s)∗ ∈ U(𝒩 ) (s ∈ ℝ). As in the proof of 26.2.(11) one sees that the function s → a(s) defines a cocycle a ∈ Z𝜃 (ℝ; U(𝒩 )). By Theorem 23.12, there exists v ∈ U(𝒩 ) such that a(s) = v∗ 𝜃s (v) (s ∈ ℝ). It follows that 𝜎(u(s)) = v∗ 𝜃s (v)u(s), or (𝜎 ◦ Ad(v))(u(s)) = u(s), that is, 𝛽 ◦ Ad(v) = (𝜎 ◦ Ad(v))|𝒩 commutes with 𝜃s (s ∈ ℝ). Let P𝜔 ∶ ℳ + → 𝒩̄ + be the n.s.f. operator-valued weight associated with 𝜔 (24.6). Then 𝜔 = 𝜏 ◦ P𝜔 and, as 𝜔 ◦ 𝜎 = 𝜔, we get (𝜏 ◦ 𝛽) ◦ P𝜔 = 𝜏 ◦ (𝜎|𝒩 ) ◦ P𝜔 = 𝜏 ◦ P𝜔 ◦ 𝜎 = 𝜔 ◦ 𝜎 = 𝜔 = 𝜏 ◦ P𝜔 , hence 𝜏 ◦ 𝛽 = 𝜏 and 𝜏 ◦ (𝛽 ◦ Ad(v)) = 𝜏. Thus, Range (̄𝛾 ) ⊂ Out𝜃,𝜏 (𝒩 ). ̂ 𝜏), Finally, let 𝛽 ∈ Aut𝜃,𝜏 (𝒩 ). Since (ℳ, 𝜎 𝜔 , 𝜔) ≈ (ℛ(𝒩 , 𝜃), 𝜃, ̂ it follows that there exists 𝜎 ∈ Aut𝜔 (ℳ) with 𝜎|𝒩 = 𝛽 (20.13.(2)). Thus, Out𝜃,𝜏 (𝒩 ) ⊂ Range(̄𝛾 ). If 𝒩 has separable predual, then the subgroup Out𝜃,𝜏 (𝒩 ) also has the following description: Out𝜃,𝜏 (𝒩 ) = {𝛽 ∈ Out(𝒩 ); 𝜏 ◦ 𝛽 = 𝜏, 𝔬𝒩 (𝜃s )𝛽 = 𝛽 ◦ 𝔬𝒩 (𝜃s ) (s ∈ ℝ)}

(6)

where 𝔬𝒩 ∶ Aut(𝒩 ) → Out(𝒩 ) is the canonical quotient mapping. Instead of a complete proof, which involves some nontrivial technical complications (see Connes & Takesaki, 1977, IV.3.2), we give just a sketch of the proof here. By familiar arguments it follows from the theory of standard Borel spaces that there exists a Borel mapping v ∶ Int(𝒩 ) → U(𝒩 ) such that 𝛼 = Ad(v(𝛼)) for every 𝛼 ∈ Int(𝒩 ). If 𝛽 ∈ Aut(𝒩 ) and 𝔬𝒩 (𝛽) commutes with 𝔬𝒩 (𝜃s ), then 𝛽 ◦ 𝜃s ◦ 𝛽 −1 ◦ 𝜃s−1 ∈ Int(𝒩 ) and b(s) = v(𝛽 ◦ 𝜃s ◦ 𝛽 −1 ◦ 𝜃s−1 ) ∈ U(𝒩 ) (s ∈ ℝ). Then Ad(b(s)) ◦ 𝜃s = 𝛽 ◦ 𝜃s ◦ 𝛽 −1 , so that Ad(b(s)𝜃s (b(t))) = Ad(b(s + t)) and so c(s, t) = b(s)∗ b(s + t)𝜃s (b(t)∗ ) ∈ 𝒵 (𝒩 ) (s, t ∈ ℝ). By direct computation, one

366

Continuous Decompositions

checks that c(r, s)c(r + s, t) = 𝜃s (c(s, t))c(r, s + t), (r, s, t ∈ ℝ), and by cohomological arguments, it follows that there exists a Borel function d ∶ ℝ → U(𝒵 (𝒩 )) such that c(s, t) = d(s)∗ d(s + t)𝜃s (d(t)∗ ) for almost all s, t ∈ ℝ. Then bd ∶ ℝ → U(𝒩 ) is a Borel function which, modified on a negligible set, gives rise to a cocycle a ∈ Z𝜃 (ℝ; U(𝒩 )) such that Ad(a(s)) ◦ 𝜃s = 𝛽 ◦ 𝜃s ◦ 𝛽 −1 (s ∈ ℝ). Using Theorem 23.12, we infer that a(s) = w∗ 𝜃s (w) (s ∈ ℝ) for some w ∈ U(𝒩 ). Then Ad(w) ◦ 𝛽 commutes with 𝜃s (s ∈ ℝ). We thus obtain the inclusion “⊃” in (6), the other inclusion being obvious. 26.5. In order to explain the nature of the extensions 𝜎c𝜑 and [D𝜓 ∶ D𝜑]c defined in Section 26.3, we shall consider the case of an infinite semifinite factor ℳ with separable predual. Let 𝜇 be an n.s.f. trace on ℳ. By Theorem 4.10, every n.s.f. weight 𝜑 on ℳ is of the form 𝜑 = 𝜇A for some nonsingular positive self-adjoint operator A affiliated to ℳ. In particular, for the dominant weight 𝜔 on ℳ, there exists a nonsingular positive self-adjoint operator D affiliated to ℳ such that 𝜔 = 𝜇D . The weight 𝜑 = 𝜇A is integrable if and only if A is unitarily equivalent to the restriction of D to some invariant subspace (see 23.4). The structure of the operator D can be better analyzed if we consider the *-isomorphism (ℳ, 𝜇) ≈ ̄ ℱ,𝜇 ⊗ ̄ tr) (see 9.18) where ℱ = ℬ(ℒ 2 (ℝ+ )). Let D0 be the operator defined in ℒ 2 (ℝ+ ) (ℳ ⊗ ∗ ∗ by (D0 𝜉)(𝜆) = 𝜆𝜉(𝜆) (𝜉 ∈ Dom(D0 ) ⊂ ℒ 2 (ℝ+∗ ), 𝜆 ∈ ℝ+∗ ). As in Section 23.4, it is easy to see that ̄ trD is a dominant weight, so that the operator D = 1 ⊗ ̄ D0 has absolutely continuous spectrum. 𝜇⊗ 0 Thus, 𝜑 = 𝜇A is integrable if and only if A has absolutely continuous spectrum. In this case, the functional calculus associated with A can be extended to functions f ∈ ℒ ∞ (ℝ+∗ ). Indeed, this fact is obvious for D0 ∶ ( f(D0 )𝜉)(𝜆) = f (𝜆)𝜉(𝜆) (𝜉 ∈ Dom( f (D0 )) ⊂ ℒ 2 (ℝ+∗ ), 𝜆 ∈ ℝ+∗ ). Note that the commutant of D0 in ℱ is just the set {f (D0 ); f ∈ ℒ ∞ (ℝ+∗ )}. The centralizer ℳ 𝜑 of 𝜑 = 𝜇A is the commutant of A in ℳ. Consequently, 𝒵 (ℳ 𝜑 ) = {f (A); f ∈ ℒ ∞ (ℝ+∗ )}.

(1)

By the construction (24.5) of the *-isomorphism p𝜑 ∶ 𝒵 (ℳ 𝜑 ) → 𝒵 (ℳ 𝜔 )c𝜔 (𝜑) it follows that p𝜑 ( f (A)) = f (D)c𝜔 (𝜑)

( f ∈ ℒ ∞ (ℝ+∗ )).

(2)

Recall (24.8) that the flow of weights on ℳ is the pair consisting of 𝒵 (ℳ 𝜔 ), identified with via the mapping

ℒ ∞ (ℝ∗+ )

ℒ ∞ (ℝ+∗ ) ∋ f ↦ f (D) ∈ 𝒵 (ℳ 𝜔 ),

(3)

and the continuous action F ℳ of ℝ+∗ on ℒ ∞ (ℝ+∗ ) defined by (F𝜆ℳ f )(𝛼) = f (𝜆−1 𝛼) ( f ∈ ℒ ∞ (ℝ+∗ , 𝜆, 𝛼 ∈ ℝ+∗ ).

(4)

As mentioned in Section 21.12, every unitary cocycle c ∈ Z(F ℳ ) is trivial, that is, there exists a unitary element f ∈ ℒ ∞ (ℝ+∗ ), uniquely determined up to a scalar multiple, such that c(𝜆) = fFℳ ( f ∗ ) (𝜆 ∈ ℝ+∗ ). 𝜆

The Extension of the Modular Automorphism Group

367

If 𝜑 = 𝜇A , 𝜓 = 𝜇B are integrable n.s.f. weights on ℳ and c ∈ Z(F ℳ ) is defined by c(𝜆) = (𝜆 ∈ ℝ+∗ ), then

fFℳ ( f ∗) 𝜆

𝜎c𝜑 = Ad( f (A)) [D𝜓 ∶ D𝜑]c = f (B)f (A)∗ .

(5) (6)

Indeed, let 𝜆 > 0 and let w ∈ ℳ (𝜎 𝜑 , {− ln(𝜆)}) be a partial isometry. Then w∗ w and ww∗ belong to ℳ 𝜑 so that they commute with A. We have Ait wA−it = 𝜎t𝜑 (w) = 𝜆it w, hence w∗ Ait w = w∗ w(𝜆A)it = w∗ w(𝜆A)it w∗ w (t ∈ ℝ). For h ∈ ℒ ∞ (ℝ+∗ ), it follows that w∗ h(A)w = w∗ wh(𝜆A)w∗ w = w∗ wh(𝜆A), whence h(A)w = wh(𝜆A).

(7)

Using (7), (4), (2), (3), and 26.3.(1), we get [Ad( f (A))](w) = f (A)wf (A)∗ = wf (𝜆A)f (A)∗ = 𝜑 −1 f (A)f (𝜆−1 A))∗ w = [c(𝜆)](A)w = p−1 𝜑 ([c(𝜆)](D)c𝜔 (𝜑))w = p𝜑 (c(𝜆)c𝜔 (𝜑))w = 𝜎c (w). Thus, the 𝜑 *-automorphisms 𝜎c and Ad( f (A)) coincide on every partial isometry w ∈ ℳ(𝜎 𝜑 , {− ln(𝜆)}). Since both these *-automorphisms act identically on ℳ 𝜑 , it follows that they are equal, which proves (5). weight 𝜃(𝜑, 𝜓) is obtained from the trace 𝜃(𝜇, 𝜇) with the help of the operator ( The balanced ) A 0 so that, using (5) and 26.3.(4), we obtain 0 B (

0 [D𝜓 ∶ D𝜑]c

0 0

)

= 𝜎c𝜃(𝜑,𝜓) ( =

f (A) 0

((

0 0 1 0

0 f (B)

))

)(

0 0 1 0

)(

f (A)∗ 0

0 f (B)∗

)

( =

0 f (B)f (A)∗

0 0

)

so that (6) is also proved. Thus, the extension of the modular automorphism group to a family of *-automorphism indexed by Z(F ℳ ) is actually a generalization of the usual functional calculus. 26.6 Proposition. Let ℳ be a properly infinite W ∗ -algebra with separable predual and let c ∈ Z(F ℳ ). Let 𝜑 be an integrable n.s.f. weight on ℳ and A be a nonsingular positive self-adjoint operator affiliated to ℳ 𝜑 such that the weight 𝜑A is integrable. Then c(A) = [D𝜑A ∶ D𝜑]c belongs to the center of {A}′ ∩ ℳ 𝜑 . ( ) 0 0 ∈ 𝒫. Proof. Let 𝜓 = 𝜃(𝜑, 𝜑A ) be the balanced weight on 𝒫 = Mat2 (ℳ) and let u = 1 0 ( ) 0 0 Then (26.3.(4)) = 𝜎c𝜓 (u). We have 𝜎t𝜓 (u∗ )u ∈ 𝒫 𝜓 , so that 𝜎t𝜓 (u∗ )u = 𝜎c𝜓 (𝜎t𝜓 (u∗ )u) = c(A) 0 ( ) c(A) 0 𝜓 𝜓 ∗ 𝜓 𝜎t (𝜎c (u ))𝜎c (u) and hence = u∗ 𝜎c𝜓 (u) ∈ 𝒫 𝜓 , that is, c(A) ∈ ℳ 𝜑 . For every x ∈ 0 0 𝜑 {A}′ ∩ ℳ 𝜑 ⊂ ℳ 𝜑 ∩ ℳ 𝜑A , we have x = 𝜎c A (x) = c(A)𝜎c𝜑 (x)c(A)∗ = c(A)xc(A)∗ and therefore c(A) ∈ ({A}′ ∩ ℳ 𝜑 )′ ∩ ℳ 𝜑 = 𝒵 ({A}′ ∩ ℳ 𝜑 ).

368

Continuous Decompositions

Using Theorem 26.3.(6), we infer that (c1 c2 )(A) = c1 (A)c2 (A)

(c1 , c2 ∈ Z(F ℳ )).

(1)

26.7. Notes. The material of this section is due to Connes and Takesaki (1977). In our exposition, we have defined the extended modular automorphisms and the extended Connes cocycles only for integrable weights. In fact, it is possible to define these objects for arbitrary n.s.f. weigh by restricting the class of cocycles c ∈ Z(F ℳ ) to those which are twice continuously differentiable in norm (Connes & Takesaki, 1977, IV.2.6). For factors arising by the group measure space construction, the extended modular automorphisms are explicitly computed in (Connes & Takesaki, 1977).

CHAPTER VI

Discrete Decompositions

27 The Connes Invariant T(𝓜) In this section, we introduce the invariant T (ℳ) as the group of inner periods of modular automorphism groups on ℳ. 27.1. Let ℳ be a W ∗ -algebra and 𝔬ℳ ∶ Aut(ℳ) → Out(ℳ) the canonical quotient mapping. Recall (3.2) that the modular homomorphism 𝛿ℳ ∶ ℝ → Out(ℳ) defined by 𝛿ℳ (t) = 𝔬ℳ (𝜎t𝜑 ) (t ∈ ℝ) does not depend on the choice of the n.s.f. weight 𝜑 on ℳ and that 𝛿ℳ (ℝ) is contained in the center of the group Out(ℳ). Therefore, the kernel of the modular homomorphism is a subgroup of the additive group ℝ and an algebraic invariant of ℳ, which we denote by T (ℳ) = {t ∈ ℝ; 𝛿ℳ (t) = 𝜄}. Theorem. Let ℳ be a W ∗ -algebra and t ∈ ℝ. The following statements are equivalent: (i) (ii) (iii) (iv)

t ∈ T (ℳ); for every 𝜑 ∈ Wnsf (ℳ) we have 𝜎t𝜑 ∈ Int(ℳ); for every 𝜑 ∈ Wnsf (ℳ) there exists a unitary element u ∈ 𝒵 (ℳ 𝜑 ) such that 𝜎t𝜑 = Ad(u); for every 𝜑 ∈ Wnsf (ℳ) there exists a (strictly semifinite) 𝜓 ∈ Wnsf (ℳ) commuting with 𝜑 such that 𝜎t𝜓 = 𝜄; (v) there exists 𝜑 ∈ Wnsf (ℳ) such that 𝜎t𝜑 ∈ Int(ℳ).

If ℳ is countably decomposable and t ∈ T (ℳ), then (vi) there exists a faithful normal state 𝜑 on ℳ with 𝜎t𝜑 = 𝜄. Proof. It is clear that (vi) ⇒ (iv) ⇒ (v) ⇒ (i) ⇒ (ii). (ii) ⇒ (iii). If 𝜎t𝜑 = Ad(u) with u ∈ U(ℳ), then 𝜑(uxu∗ ) = 𝜑(𝜎t𝜑 (x)) = 𝜑(x) for x ∈ ℳ + , whence u ∈ ℳ 𝜑 . For x ∈ ℳ 𝜑 , we have uxu∗ = 𝜎t𝜑 (x) = x, so u ∈ 𝒵 (ℳ 𝜑 ). (iii) ⇒ (iv). By assumption, we have 𝜎t𝜑 = Ad(u) for some unitary u ∈ 𝒵 (ℳ 𝜑 ). There exists a nonsingular positive self-adjoint operator A affiliated to 𝒵 (ℳ 𝜑 ) such that A−it = u. Then the n.s.f. weight 𝜓 = 𝜑A commutes with 𝜑 and 𝜎t𝜓 = Ad(Ait ) ◦ 𝜎t𝜑 = 𝜄. On the other hand, we recall (10.9) that if 𝜎t𝜓 = 𝜄, then the weight 𝜓 is strictly semifinite. (iii) ⇒ (vi). Assume that ℳ is countably decomposable. By (iii) there exists an n.ss.f. weight 𝜓 on ℳ such that 𝜎t𝜓 = 𝜄. Since 𝜓 is strictly semifinite and ℳ is countably decomposable, it follows by Theorem 10.9 that there exists a sequence {𝜓n }n≥1 ⊂ ℳ∗+ with supports s(𝜓n ) ∈ ℳ 𝜓 ,

369

370

Discrete Decompositions

mutually orthogonal, such that 𝜓 = with 𝜎t𝜑 = 𝜄.

∑ n

𝜓n . Then 𝜑 =

∑

n ‖𝜓

n (2

n ‖)

−1 𝜓

n

is a faithful normal state

27.2 Proposition. Let ℳ be a W ∗ -algebra. If ℳ is semifinite, then T (ℳ) = ℝ. If T (ℳ) = ℝ and ℳ has separable predual, then ℳ is semifinite. Proof. If 𝜇 is an n.s.f. trace on ℳ, 𝜎t𝜇 = 𝜄, for all t ∈ ℝ, and T (ℳ) = ℝ. Conversely, if T (ℳ) = ℝ and 𝜑 is any n.s.f. weight on ℳ, then 𝜎t𝜑 ∈ Int(ℳ) for all t ∈ ℝ. If ℳ has separable predual, then, by Theorem 15.16, there exists an s-continuous unitary representation ℝ ∋ t → u(t) ∈ ℳ such that 𝜎t𝜑 = Ad(u(t)) (t ∈ ℝ). Also, there exists a nonsingular positive self-adjoint operator A affiliated to 𝜑 ℳ 𝜑 such that u(t) = A−it and hence 𝜎t A = 𝜄, for all t ∈ ℝ, that is, 𝜇 = 𝜑A is an n.s.f. trace on ℳ. Thus, for any nonsemifinite W ∗ -algebra ℳ with separable predual the center of the group Out(ℳ) is nontrivial. 27.3 Proposition. For any two W ∗ -algebras ℳ, 𝒩 we have ̄ 𝒩 ) = T (ℳ) ∩ T (𝒩 ), T (ℳ ⊗ T (ℳ ⊕ 𝒩 ) = T (ℳ) ∩ T (𝒩 ),

(1) (2)

if e ∈ ℳ is a projection with central support equal to 1, T (ℳe ) = T (ℳ),

(3)

and if ℳ ⊂ ℬ(ℋ ) is realized as a von Neumann algebra T (ℳ ′ ) = T (ℳ).

(4)

̄ 𝜓 is an n.s.f. weight Proof. Let 𝜑 and 𝜓 be n.s.f. weights on ℳ and 𝒩 , respectively. Then 𝜑 ⊗ ̄ 𝜓 𝜑 ⊗ 𝜑 𝜓 ̄ 𝒩 and, by Proposition 17.6.(3), 𝜎t ̄ 𝜎t ∈ Int(ℳ ⊗ ̄ 𝒩 ) if and only if 𝜎t𝜑 ∈ on ℳ ⊗ = 𝜎t ⊗ 𝜓 Int(ℳ) and 𝜎t ∈ Int(𝒩 ), which proves (1). Also, 𝜑 ⊕ 𝜓 is an n.s.f. weight on ℳ ⊕ 𝒩 and 𝜎t𝜑 ⊕ 𝜓 = 𝜎t𝜑 ⊕ 𝜎t𝜓 ∈ Int(ℳ ⊕ 𝒩 ) if and only if 𝜎t𝜑 ∈ Int(ℳ) and 𝜎t𝜓 ∈ Int(𝒩 ), from which (2) follows. In view of (2), to prove (3) we may consider separately the cases e finite and e properly infinite. If e is finite, then ℳe and ℳ are both semifinite, whence T (ℳe ) = ℝ = T (ℳ). If e is properly infinite and ℳ is countably decomposable, then e ∼ 1 in ℳ, whence ℳe ≈ ℳ and T (ℳe ) = T (ℳ). If ℳ is not countably decomposable, then we can decompose ℳ into a direct sum of uniform W ∗ -algebras (see [L], 8.4) and, using (2), (1), Proposition 27.2, and the previous remarks, we conclude again that T (ℳe ) = T (ℳ). Equation (4) is obvious if ℳ ⊂ ℬ(ℋ ) is in standard form, since then ℳ ′ is *-antiisomorphic to ℳ. In general, ℳ is *-isomorphic to some standard form of itself via an amplification and a faithful ̄ ℬ(𝒦 ) (for induction ([L], E.8.8). These operations change the commutant ℳ ′ either into ℳ ′ ⊗ some Hilbert space 𝒦 ) or into ℳe′′ (for some projection e′ ∈ ℳ ′ with central support equal to 1) without modifying T (ℳ ′ ). 27.4. In this section, we compute the invariant T (ℳ) for crossed products by properly outer actions of discrete groups. More generally, let ℳ be a W ∗ -algebra, 𝒩 ⊂ ℳ a semifinite unital

The Connes Invariant T(ℳ)

371

W ∗ -subalgebra with 𝒵 (𝒩 ) = 𝒩 ′ ∩ ℳ, P ∶ ℳ → 𝒩 a faithful normal conditional expectation of ℳ onto 𝒩 and 𝒢 ⊂  (P) a subgroup of the normalizer  (P) of P such that ℳ = ℛ{𝒩 , 𝒢 } (see 10.17). By Theorem 4.10, for every n.s.f. trace 𝜏 on ℳ and u ∈  (P) there exists a unique nonsingular positive self-adjoint operator A𝜏,u affiliated to 𝒵 (𝒩 ) such that 𝜏(u ⋅ u∗ ) = 𝜏A𝜏,u .

(1)

Note that 𝜎t𝜏 ◦ P = uAit𝜏,u

(u ∈  (P), t ∈ ℝ).

(2)

Indeed, using Corollary 3.7, Theorem 11.9, and Corollary 4.8, we get u∗ 𝜎t𝜏 ◦ P (u) = [D(𝜏 ◦ P)u ∶ D(𝜏 ◦ P)]t = [D(𝜏(u ⋅ u∗ ) ◦ P) ∶ D(𝜏 ◦ P)]t = [D(𝜏(u ⋅ u∗ )) ∶ D𝜏]t = [D(𝜏A𝜏,u ) ∶ D𝜏]t = Ait𝜏,u . Theorem. In the previous setting, the following statements concerning t ∈ ℝ are equivalent: (i) t ∈ T (ℳ); (ii) for every n.s.f. trace 𝜏 on 𝒩 there exists a unitary element v ∈ 𝒵 (𝒩 ) such that Ait𝜏,u = u∗ vuv∗ for all u ∈ 𝒢 ; (iii) there exists an n.s.f. trace 𝜏 on 𝒩 such that Ait𝜏,u = 1 for all u ∈ 𝒢 . Proof. (i) ⇒ (ii). Since t ∈ T (ℳ), given the weight 𝜑 = 𝜏 ◦ P on ℳ there exists v ∈ U(ℳ) such that 𝜎t𝜑 = Ad(v). For y ∈ 𝒩 , we have (11.9) vyv∗ = 𝜎t𝜑 (y) = 𝜎t𝜏 (y) = y, hence v ∈ 𝒩 ′ ∩ ℳ = 𝒵 (𝒩 ). By (2), for u ∈ 𝒢 we have vuv∗ = 𝜎t𝜑 (u) = uAit𝜏,u , hence Ait𝜏,u = v∗ vuv∗ . (ii) ⇒ (iii). Let 𝜇 be any n.s.f. trace on 𝒩 and let v ∈ 𝒵 (𝒩 ) be such that u∗ vuv∗ = Ait𝜏,u = [D(𝜇(u ◦ u∗ )) ∶ D𝜇]t for all u ∈ 𝒢 . There exists an invertible element b ∈ 𝒵 (𝒩 )+ such that bit = v∗ . Then 𝜏 = 𝜇b is an n.s.f. trace on 𝒩 with [D𝜏 ∶ D𝜇]t = bit = v∗ , so that for u ∈ 𝒢 we obtain (3.8) [D(𝜏(u ⋅ u∗ )) ∶ D(𝜇(u ⋅ u∗ ))]t = u∗ [D𝜏 ∶ D𝜇]t u = u∗ v∗ u and Ait𝜏,u = [D(𝜏(u ⋅ u∗ )) ∶ D𝜏]t = (u∗ v∗ u)(u∗ vuv∗ )v = 1. (iii) ⇒ (i). Let 𝜑 = 𝜏 ◦ P with 𝜏 as in (iii). For y ∈ 𝒩 we have 𝜎t𝜑 (y) = 𝜎t𝜏 (y) = y and for u ∈ 𝒢 we have 𝜎t𝜑 (u) = uAit𝜏,u = u, whence 𝜎t𝜑 = 𝜄 since ℳ = ℛ{𝒩 , 𝒢 }. Proposition. In the above setting, consider the following statements concerning t ∈ ℝ and 𝜓 ∈ Wnsf (𝒩 ): (i) t ∈ T (ℳ); (ii) 𝜎t𝜓 ∈ [{Ad(w)|𝒩 ; w ∈ 𝒢 }]; (iii) there exists u ∈ 𝒢 such that 𝜎t𝜓 = Ad(u)|𝒩 . Then (i) ⇒ (ii) and, if 𝒢 is abelian and 𝜓 ◦ Ad(w) = 𝜓 for all w ∈ 𝒢 , (iii) ⇒ (i). Proof. Assume that t ∈ T (ℳ) and consider the n.s.f. weight 𝜑 = 𝜓 ◦ P on ℳ. Then there exists u ∈ U(ℳ) such that 𝜎t𝜑 = Ad(u). For y ∈ 𝒩 , we have uyu∗ = 𝜎t𝜑 (y) = 𝜎t𝜓 (y) ∈ 𝒩 , hence y ∈  (P) and 𝜎t𝜓 = 𝜎t𝜑 |𝒩 = Ad(u)|𝒩 ∈ [{Ad(w)|𝒩 ; w ∈ 𝒢 }], by Proposition 22.4.

372

Discrete Decompositions

Assume now that 𝒢 is abelian, that 𝜓 ◦ Ad(w) = 𝜓 for all w ∈ 𝒢 and 𝜎t𝜓 = Ad(u)|𝒩 for some u ∈ 𝒢 . Let 𝜑 = 𝜓 ◦ P ∈ Wnsf (ℳ). For every w ∈ 𝒢 , we have (3.7, 11.9) w∗ 𝜎t𝜑 (w) = [D(𝜑 ◦ Ad(w)) ∶ D𝜑]t = [D(𝜓 ◦ Ad(w)) ∶ D𝜓]t = 1, hence 𝜎t𝜑 (w) = w = uwu∗ = [Ad(u)](w) since 𝒢 is abelian. For y ∈ 𝒩 , we have 𝜎t𝜑 (y) = 𝜎t𝜓 (y) = [Ad(u)](y). Since ℳ = ℛ{𝒩 , 𝒢 }, it follows that 𝜎t𝜑 = Ad(u), so that t ∈ T (ℳ). 27.5. The previous results are valid if, for instance, ℳ = ℛ(𝒩 , 𝜎) is the crossed product of the semifinite W ∗ -algebra 𝒩 by a properly outer action 𝜎 ∶ G → Aut(𝒩 ) of the discrete group G, since ̄ 𝜆 (g); g ∈ G}, 𝒩 ′ ∩ ℳ = 𝒵 (𝒩 ) (see then ℳ is generated by 𝒩 ≡ 𝜋𝜎 (𝒩 ) ⊂ ℳ and 𝒢 = {1 ⊗ 22.3), and P𝜎̂ ∶ ℳ → 𝒩 is a faithful normal conditional expectation such that 𝒢 ⊂  (P𝜎̂ ) (see 22.2.(3)). In particular, let G ∋ g ↦ Tg be an action of the discrete countable group G as homeomorphisms of the locally compact Hausdorff topological space Ω with a countable basis of open sets and let 𝜇 be a G-quasi-invariant sigma-finite positive Borel measure on Ω; we assume that G acts freely on (Ω, 𝜇) (see 22.8). We then obtain a free action 𝜎 ∶ G → Aut(𝒩 ) of G on the abelian W ∗ -algebra 𝒩 = ℒ ∞ (Ω, 𝜇) and a corresponding crossed product W ∗ -algebra ℳ = ℛ(𝒩 , 𝜎) (see 22.8). For any n.s.f. trace 𝜏 on 𝒩 , there is a Borel measure ν on Ω, equivalent to 𝜇, such that 𝜏(x) = ∫ x dν, (x ∈ 𝒩 + ). For every g ∈ G, there exists a unique nonsingular positive self-adjoint operator A𝜏⋅g in ℋ = ℒ 2 (Ω, 𝜇), affiliated to 𝒩 = ℒ ∞ (Ω, 𝜇), such that 𝜏 ◦ 𝜎g = 𝜏A𝜏,g . Since for x ∈ ℒ ∞ (Ω, 𝜇)+ and g ∈ G, we have 𝜏(𝜎g (x)) = ∫ x(Tg−1 (𝜔)) dν(𝜔) = ∫ x(𝜔) d(ν ◦ Tg )(𝜔) = ∫ x(𝜔)[d(ν ◦ Tg )∕dν))(𝜔)dν(𝜔), it follows that A𝜏,g is the multiplication operator defined by the Radon–Nikodym derivative A𝜏,g = d(ν ◦ Tg )∕dν. Note that a function f ∶ Ω → ℝ+ has the property fit = 1 if and only if f (𝜔) ∈ {e2n𝜋∕t ; n ∈ ℤ}. We thus infer from Theorem 27.4 the following Corollary. In the previous setting, a real number t belongs to T (ℳ) if and only if there exists a positive measure ν on Ω, equivalent to 𝜇, such that [d(ν ◦ Tg )∕dν](𝜔) ∈ {e2n𝜋∕t ; n ∈ ℤ} for all g ∈ G, 𝜔 ∈ Ω. 27.6 Proposition. Let ℳ be a countably decomposable W ∗ -algebra, 𝜑 a faithful normal state on ℳ and t ∈ ℝ. Then t ∈ T (ℳ) if and only if there exists a unitary element u ∈ ℳ such that Δit𝜑 = 𝜋𝜑 (u)J𝜑 𝜋𝜑 (u)J𝜑 .

(1)

Proof. If t ∈ T (ℳ), there exists u ∈ U(ℳ 𝜑 ) such that 𝜎t𝜑 = Ad(u) and for x ∈ ℳ we have 𝜑 ∗ Δit𝜑 𝜋𝜑 (x)𝜉𝜑 = Δit𝜑 𝜋𝜑 (x)Δ−it 𝜑 𝜉𝜑 = 𝜋𝜑 (𝜎t (x))𝜉𝜑 = 𝜋𝜑 (u)𝜋𝜑 (x)𝜋𝜑 (u) 𝜉𝜑 = 𝜋𝜑 (u)𝜋𝜑 (x)S𝜑 𝜋𝜑 (u)𝜉𝜑 = 1∕2

𝜋𝜑 (u)𝜋𝜑 (x)J𝜑 Δ𝜑 𝜋𝜑 (u)𝜉𝜑 = 𝜋𝜑 (u)𝜋𝜑 (x)J𝜑 𝜋𝜑 (u)𝜉𝜑 = 𝜋𝜑 (u)𝜋𝜑 (x)J𝜑 𝜋𝜑 (u)J𝜑 𝜉𝜑 = 𝜋𝜑 (u)J𝜑 𝜋𝜑 (u) J𝜑 𝜋𝜑 (x)𝜉𝜑 , since u ∈ ℳ 𝜑 and J𝜑 𝜋𝜑 (u)J𝜑 ∈ 𝜋𝜑 (ℳ)′ ; this proves (1). Conversely, if there exists u ∈ U(ℳ 𝜑 ) satisfying (1), then for every x ∈ ℳ we have 𝜋𝜑 (𝜎t𝜑 (x))𝜉𝜑 = Δit𝜑 𝜋𝜑 (x)𝜉𝜑 = 𝜋𝜑 (u)J𝜑 𝜋𝜑 (u)J𝜑 𝜋𝜑 (x)𝜉𝜑 = … = 𝜋𝜑 (uxu∗ )𝜉𝜑 , hence 𝜎t𝜑 = Ad(u) and t ∈ T (ℳ). If ℳ has separable predual, the U(ℳ 𝜑 ) and U(ℬ(ℋ𝜑 )) are polish topological groups and the mappings t ↦ Δit𝜑 and u ↦ 𝜋𝜑 (u)J𝜑 𝜋𝜑 (u)J𝜑 are continuous. Consequently, if ℳ has separable predual, then T (ℳ) is an analytic subgroup of ℝ.

(2)

The Connes Invariant T(ℳ)

373

In particular, T (ℳ) is in this case Lebesgue measurable. If T (ℳ) has positive measure, then T (ℳ) is open and hence T (ℳ) = ℝ. Therefore, if ℳ is a type III W∗ -algebra with separable predual, then T (ℳ) has zero Lebesgue measure.

(3)

On the other hand, we mention that every subgroup of ℝ is of the form T (ℳ) for some countably decomposable factor ℳ (Connes, 1973a, 1.5.8). 27.7. A W ∗ -algebra ℳ is called normal if for every unital W ∗ -subalgebra 𝒩 ⊂ ℳ the condition 𝒩 = (𝒩 ′ ∩ ℳ)′ ∩ ℳ holds. By the von Neumann double commutant theorem ([L], 3.2), any type I factor is normal. As an application of the invariant T (ℳ), we prove the following result. Proposition. No type III factor with separable predual is normal. Proof. Let ℳ ⊂ ℬ(ℋ ) be a type III factor acting on the separable Hilbert space ℋ . Then 𝒩 = ̄ ℱ , where ℱ is the ℳ ′ ∩ ℬ(ℋ ) is a type III factor in standard form (see [L]. 8.13) and ℳ ≈ 𝒩 ′ ⊗ countably decomposable infinite type I factor. Since T (𝒩 ) has zero Lebesgue measure (27.6.(3)), there is a t ∈ ℝ such that nt ∉ T (𝒩 ) for every n ∈ ℤ. Let 𝜓 be an n.s.f. weight on 𝒩 and 𝜎 = 𝜎t𝜓 ∈ Aut(𝒩 ). Then, for every n ∈ ℤ, 𝜎 n is an outer *-automorphism of 𝒩 . On the other hand, as ̄ 1, v ⊗ ̄ 𝜌 (1)} ⊂ 𝒩 is in standard form, there is v ∈ U(ℬ(ℋ )) such that 𝜎 = Ad(v). Let 𝒫 = ℛ{𝒩 ⊗ ̄ 𝓁 2 (ℤ)), where 𝜌 is the right regular representation of ℤ, and Q = 𝒫 ′ ⊂ (𝒩 ⊗ ̄ 1)′ = ℬ(ℋ ⊗ ̄ ℱ = ℳ. We have 𝒩 ⊗ ̄ 1 ≠ 𝒫 since v ⊗ ̄ 𝜌 (1) ∉ 𝒩 ⊗ ̄ 1, hence Q ≠ ℳ. On the other hand, 𝒩 ′⊗ ̄ 1)′ = 𝒫 ∩ (𝒩 ⊗ ̄ 1)′ = ℂ ⋅ 1; the last equality follows as in the we have Q′ ∩ ℳ = 𝒫 ′′ ∩ (𝒩 ⊗ relative commutant theorem (22.3.(1)), since the action n ↦ 𝜎 n of ℤ on 𝒩 is properly outer (see also 19.14.(1)). Thus, (Q′ ∩ ℳ)′ ∩ ℳ = ℳ ≠ Q. It is a classical result (Fuglede & Kadison, 1951) that no type II factor is normal. Thus the only normal factors with separable predual are those of type I. 27.8. Notes. The invariant T (ℳ) was introduced by Connes (1973a), who obtained the results presented in this section. Moreover, Connes (Connes, 1973a, 1.3.7) computed the invariant T (ℳ) for infinite tensor product factors, and obtained an explicit expression of T (ℳ) in terms of the eigenvalue list for Araki–Woods factors (Connes, 1973a, 1.3.9). In particular, for the Powers factors ℛ𝜆 (0 < 𝜆 < 1), we have T (ℛ𝜆 ) = {nt; n ∈ ℤ} where t = −2𝜋∕ ln(𝜆), from which it follows that {ℛ𝜆 }0 0, 0 ≠ e ∈ Proj(𝒵 (𝒩 )) and F = 𝜆 − 2𝜀 , 𝜆 + 2𝜀 . From (2), ⋁ we infer that 𝜆 ∈ Sp 𝜎 𝜑e , hence {r(x); x ∈ ℳ(𝜎 𝜑 ; F) ∩ eℳe} ≠ 0 and from (1), it follows that there exists u ∈ 𝒢 such that p = e(u∗ eu)𝜒F (Au ) ≠ 0. Then p ∈ 𝒵 (𝒩 ), p ≤ e, upu∗ ≤ e, and Sp(Au p|𝒩 p) ⊂ F ⊂ (𝜆 − 𝜀, 𝜆 + 𝜀). Conversely, let 0 ≠ e ∈ Proj(𝒵 (𝒩 )), 𝜀 > 0 and F = [𝜆−𝜀, 𝜆+𝜀]. By assumption, we can find 0 ≠ ⋁ p ∈ Proj(𝒵 (𝒩 )), p ≤ e, and u ∈ 𝒢 with upu∗ ≤ e and Sp(Au |𝒩 p) ⊂ F; thus {e(u∗ eu)𝜒F (Au ); u ∈ 𝒢 } ≠ 0. From (1), it follows that ℳ(𝜎 𝜑 ; F) ∩ eℳe ≠ 0, hence 𝜆 ∈ Sp 𝜎 𝜑e . Using (2), we conclude that 𝜆 ∈ S(ℳ). 28.6. For instance, Theorem 28.5 holds if ℳ = ℛ(𝒩 , 𝜎) is the crossed product of a semifinite W ∗ -algebra 𝒩 by a properly outer action 𝜎 ∶ G → Aut(𝒩 ) of a discrete group G which is ergodic on 𝒵 (𝒩 ) (see 27.5 and Corollary 1/22.6). In particular, let G ∋ g → Tg be a free ergodic action of G on the measure space (Ω, 𝜇) with 𝜇 a G-quasi-invariant measure (see 22.8, 27.5). We then obtain a free ergodic action 𝜎 ∶ G → Aut(𝒩 ) of G on the abelian W ∗ -algebra 𝒩 = ℒ ∞ (Ω, 𝜇) and the crossed product ℳ = ℛ(𝒩 , 𝜎) is a factor. For the dynamical system (Ω, 𝜇, G), one defines (cf. Araki & Woods, 1968; Krieger, 1970b) an invariant called “ratio set” as being the set r(G) of all 𝜆 ≥ 0 with the following property: for every 𝜀 > 0 and every 𝜇-measurable set A ⊂ Ω with 𝜇(A) > 0, there exist a 𝜇-measurable set B ⊂ A with 𝜇(B) > 0 and g ∈ G such that Tg (B) ⊂ A and |[d(𝜇 ◦ Tg )∕d𝜇](𝜔) − 𝜆| < 𝜀 for all 𝜔 ∈ B. It is easy to see that 0 ∉ r(G) if and only if there exists a G-invariant sigma-finite positive measure v on Ω which is equivalent to 𝜇 (see Krieger, 1970b), that is, if and only if the crossed product ℳ = ℛ(𝒩 , 𝜎) is semifinite (22.8). If 𝜏 is the n.s.f. trace on 𝒩 defined by the measure 𝜇, then the unique positive nonsingular selfadjoint operator Ag affiliated to 𝒩 such that 𝜏 ◦ 𝜎g = 𝜏Ag is the multiplication operator defined by the function Ag = d(𝜇 ◦ Tg )∕d𝜇 (g ∈ G) (see 27.5). Thus, we infer from Theorem 28.5 the following: Corollary. In the previous situation, we have S(ℳ) = r(G). 28.7. In this section, we give a spatial characterization of the invariant S(ℳ). Let Δ be a positive self-adjoint operator in the Hilbert space ℋ and D ⊂ Dom(Δ) a vector subspace such that Δ = Δ|D. For 𝜆 ∈ ℂ, it is easy to check that 𝜆 ∈ Sp(Δ) ⇔ there exist 𝜉n ∈ D, ‖𝜉n ‖ = 1, ‖(𝜆 − D)𝜉n ‖ → 0,

(1)

𝜆 ∉ Sp(Δ) ⇒ ‖(𝜆 − Δ) ‖ = dist(𝜆, Sp(Δ)) ;

(2)

−1

−1

for 𝜀 > 0, it follows that dist(𝜆, Sp(Δ)) < 𝜀 ⇔ there exists 𝜉 ∈ D, ‖𝜉‖ > 1, ‖(𝜆 − Δ)𝜉‖ < 𝜀.

(3)

The Connes Invariant S(ℳ)

379

Lemma. Let ℳ ⊂ ℬ(ℋ ) be a von Neumann algebra with a cyclic and separating vector 𝜉 ∈ ℋ , 𝜆 ≥ 0, and 𝜀 > 0. Consider the statements: 1∕2

dist(𝜆1∕2 , Sp(Δ𝜉 )) < 𝜀 there exist x ∈ ℳ and ∈ such that ′ ‖𝜆1∕2 x𝜉 − x′ 𝜉‖ < 𝜀, ‖x∗ 𝜉 − 𝜆1∕2 x ∗ 𝜉‖ < 𝜀, ‖x𝜉‖ > 1. x′

ℳ′

(A(𝜀)) (B(𝜀))

The A(𝜀) ⇒ B(𝜀) ⇒ A(2𝜀). Proof. Put Δ = Δ𝜉 , J = J𝜉 . Since Δ1∕2 = Δ1∕2 |ℳ𝜉, from statement A(𝜀) it follows that there is an x ∈ ℳ such that ‖x𝜉‖ > 1 and ‖(𝜆1∕2 − Δ1∕2 )x𝜉‖ < 𝜀. Then x′ = Jx∗ J ∈ ℳ ′ and ‖𝜆1∕2 x𝜉 − x′ 𝜉‖ = ‖𝜆1∕2 x𝜉 − Jx∗ 𝜉‖ = ‖𝜆1∕2 x𝜉 − Δ1∕2 x𝜉‖ < 𝜀, ‖x∗ 𝜉 − 𝜆1∕2 x′∗ 𝜉‖ = ‖JΔ1∕2 x𝜉 − 𝜆1∕2 Jx𝜉‖ = ‖Δ1∕2 x𝜉 − 𝜆1∕2 x𝜉‖ < 𝜀. Hence A(𝜀) ⇒ B(𝜀). Conversely, assume that B(𝜀) holds and consider 𝜂 = x𝜉, 𝜁 = Jx′∗ 𝜉 = Δ−1∕2 x′ 𝜉. Then 𝜂, 𝜁 ∈ Dom(Δ1∕2 ), ‖𝜂‖ > 1 and ‖𝜆1∕2 𝜂 − Δ1∕2 𝜁 ‖ < 𝜀, ‖Δ1∕2 𝜂 − 𝜆1∕2 𝜁‖ < 𝜀. Since ‖Δ1∕2 (𝜆1∕2 + Δ1∕2 )−1 ‖ ≤ 1, ‖𝜆1∕2 (𝜆1∕2 + Δ1∕2 )−1 ‖ ≤ 1, it follows that ‖𝜆(𝜆1∕2 + Δ1∕2 )−1 𝜂 − 𝜆1∕2 Δ1∕2 (𝜆1∕2 + Δ1∕2 )−1 𝜁‖ < 𝜀, ‖Δ(𝜆1∕2 +Δ1∕2 )−1 𝜂 −𝜆1∕2 Δ1∕2 (𝜆1∕2 +Δ1∕2 )−1 𝜁‖ < 𝜀; hence ‖(𝜆−Δ)(𝜆1∕2 +Δ1∕2 )−1 𝜂‖ < 2𝜀. Thus ‖𝜂‖ > 1 and ‖(𝜆1∕2 − Δ1∕2 )𝜂‖ < 2𝜀, that is, dist(𝜆1∕2 , Sp(Δ1∕2 )) < 2𝜀. Therefore, B(𝜀) ⇒ A(2𝜀). Theorem. Let ℳ ⊂ ℬ(ℋ ) be a factor and let 𝜆 ≥ 0. Then 𝜆 ∈ S(ℳ) if and only if for every 0 ≠ 𝜉 ∈ ℋ and 𝜀 > 0 there exist x ∈ ℳ and x′ ∈ ℳ ′ such that ‖x𝜉‖ > 1, ‖x𝜉 − x′ 𝜉‖ < 𝜀, ‖x∗ 𝜉 − 𝜆x′∗ 𝜉‖ < 𝜀.

(4)

Proof. Assume that 𝜆 ∈ S(ℳ). Let 𝜀 > 0, 0 ≠ 𝜉 ∈ ℋ and e = p𝜉 ∈ ℳ, e′ = p′𝜉 ∈ ℳ ′ be the cyclic projections onto ℳ ′ 𝜉, ℳ𝜉, respectively. Then e′e = ee′ is a projection in ℳe′ ⊂ ℬ(eℋ ) with central support equal to the identity of ℳe′ , so that the canonical induction I ∶ .ℳe → 𝒩 = ℳee′ ⊂ ℬ(ee′ ℋ ) is a *-isomorphism and 𝜉 ∈ ee′ ℋ is a cyclic and separating vector for 𝒩 . Let Δ be the modular operator associated with the factor 𝒩 ⊂ ℬ(ee′ ℋ ) and the vector 𝜉 ∈ ee′ ℋ . By 28.4.(2), we have 𝜆 ∈ S(ℳ) = S(ℳe ) = S(𝒩 ), hence 𝜆 ∈ Sp(Δ). If 𝜆 = 0, the previous lemma shows that there exists y ∈ 𝒩 such that ‖y𝜉‖ > 1 and ‖y∗ 𝜉| < 𝜀. Since y𝜉 ∈ ee′ ℋ ⊂ ℳ ′ 𝜉, there exists x′ ∈ ℳ ′ such that ‖y𝜉 − x′ 𝜉‖ < 𝜀. Then, with x = I−1 (y) we have x𝜉 = y𝜉, x∗ 𝜉 = y∗ 𝜉. The pair (x, x′ ) satisfies (4). If 𝜆 ≠ 0, the previous lemma shows that there exist y ∈ 𝒩 , y′ ∈ 𝒩 ′ such that ‖y𝜉‖ > 1, ‖y𝜉 − 𝜆−1∕2 y′ 𝜉‖ < 𝜀, and ‖y∗ 𝜉 − 𝜆1∕2 x′∗ 𝜉‖ < 𝜀. Then there exist x ∈ ℳ, x′ ∈ ℳ ′ such that x𝜉 = y𝜉, x∗ 𝜉 = y∗ 𝜉, 𝜆1∕2 x′ 𝜉 = y′ 𝜉, 𝜆1∕2 x′∗ 𝜉 = y′∗ 𝜉, and the pair (x, x′ ) satisfies (4). Conversely, assume that the condition in the statement is satisfied. If ℳ contains a finite projection, then it also contains a cyclic finite projection. Otherwise, every projection is purely infinite. Consequently, there exists a cyclic projection 0 ≠ e ∈ ℳ which is not simultaneously infinite and semifinite. Thus, it follows from 28.4.(2) and 28.3.(6) that we have to prove that 𝜆 ∈ Sp(Δ𝜑 ) for every faithful normal state 𝜑 on ℳe .

380

Discrete Decompositions

Let 𝜑 be any faithful normal state on ℳe . Since e is cyclic, there exists 𝜉 ∈ ℋ such that p𝜉 = e ′ ′ and 𝜑 = 𝜔𝜉 |eℳe. Put e′ = p′𝜉 ∈ ℳ ′ , 𝒩 = ℳee′ , 𝒩 ′ = ℳee ′ and denote by I ∶ ℳe → 𝒩 , I ∶ ′ ′ ′ ℳe′ → 𝒩 the canonical inductions. Note that 𝜉 ∈ ee ℋ is a cyclic and separating vector for 𝒩 . By assumption, there exist x ∈ ℳ, x′ ∈ ℳ ′ satisfying (4). Let x1 = exe ∈ ℳe , x′1 = e′ x′ e′ ∈ ℳe′′ . We have ‖x1 𝜉‖ ≥ ‖ex′ 𝜉‖ − ‖ex𝜉 − ex′ 𝜉‖ ≥ ‖x′ 𝜉‖ − ‖x𝜉 − x′ 𝜉‖ ≥ ‖x𝜉‖ − 2‖x𝜉 − x′ 𝜉‖ ≥ 1 − 2𝜀, ‖x1 𝜉 − x′1 𝜉‖ = ‖ee′ (x𝜉 − x′ 𝜉)‖ < 𝜀, ‖x∗1 𝜉 − 𝜆x′∗ 𝜉‖ = ‖ee′ (x∗ 𝜉 − 𝜆x′∗ 𝜉)‖ < 𝜀. It follows 1 −1 I(x ) ∈ 𝒩 , y′ = 𝜆1∕2 (1 − 2𝜀)−1 I′ (x′ ) ∈ that y = (1 − 2𝜀) 𝒩 ′ and ‖y𝜉‖ > 1, ‖𝜆1∕2 y𝜉 − y′ 𝜉‖ < 1 1 𝜆1∕2 (1 − 2𝜀)−1 𝜀, ‖y∗ 𝜉 − 𝜆1∕2 y′∗ 𝜉‖ < (1 − 2𝜀)−1 𝜀. As 𝜀 > 0 was arbitrary, using the previous lemma we conclude that 𝜆 ∈ Sp(Δ𝜑 ). 28.8. We shall say that the W ∗ -algebra ℳ has property L𝜆 (0 ≤ 𝜆 ≤ 1∕2) if for every faithful normal state 𝜑 on ℳ and 𝜀 > 0 there exists a partial isometry u ∈ ℳ such that u2 = 0, uu∗ + u∗ u = 1, and ‖𝜆𝜑(u⋅) − (1 − 𝜆)𝜑(⋅u)‖ ≤ 𝜀. Clearly, property L𝜆 is an algebraic invariant of ℳ. As an application of the spatial characterization of S(ℳ), we shall prove the following result. Proposition. If ℳ is a factor with property L𝜆 , then 𝜆∕(1 − 𝜆) ∈ S(ℳ) (0 < 𝜆 < 1∕2). For the proof, we need the following: Lemma. Let ℳ ⊂ ℬ(ℋ ) be a factor in standard form, let 𝜀 > 0 and let 𝜉1 , 𝜉2 ∈ ℋ with ‖(𝜔𝜉1 − 𝜔𝜉2 )|ℳ‖ < 𝜀. Then there exists a unitary element u′ ∈ ℳ ′ such that ‖u′ 𝜉1 − 𝜉2 ‖2 < 4𝜀. Proof. Assume first that 𝜔𝜉1 = 𝜔𝜉2 on ℳ. Then there exists a partial isometry w′ ∈ ℳ ′ such that w′ x𝜉1 = x𝜉2 (x ∈ ℳ) and w′ ([ℳ𝜉1 ]⟂ ) = 0, whence w′∗ w′ = p′𝜉 , w′ w′∗ = p′𝜉 . Using the comparison 1 2 theorem in ℳ ′ ([L], 4.6), it follows, for instance, that 1 − p′𝜉 < 1 − p′𝜉 , and there exists a partial 1 2 isometry v′ ∈ ℳ ′ such that v′∗ v′ = 1 − p′𝜉 , v′ v′∗ ≤ 1 − p′𝜉 . Then w′ + v′ ∈ ℳ ′ is an isometry with 1 2 (w′ + v′ )𝜉1 = 𝜉2 . Since every isometry is the s-limit of a sequence of unitary elements (see Dixmier & Marechal, 1971, Lemma 2), it follows that there exists u′ ∈ U(ℳ ′ ) with ‖u′ 𝜉1 − 𝜉2 ‖2 < 𝜀. In the general case, consider 𝜑1 = 𝜔𝜉1 , 𝜑2 = 𝜔𝜉2 with ‖𝜑1 − 𝜑2 ‖ < 𝜀 and 𝜓 = 𝜑1 + (𝜑1 − 𝜑2 )− = 𝜑2 + (𝜑1 − 𝜑2 )+ . Since ℳ is in standard form, there exists 𝜂 ∈ ℋ such that 𝜓 = 𝜔𝜂 ((L], 10.25) and since 𝜑1 ≤ 𝜓, there exists a′ ∈ ℳ ′ , 0 ≤ a′ ≤ 1 such that 𝜑1 = 𝜔a′ 𝜂 . Then ‖a′ 𝜂 − 𝜂‖2 = ((1 − a′ )2 𝜂|𝜂) ≤ ((1 − a′ )(1 + a′ )𝜂|𝜂) = ‖𝜂‖2 − ‖a′ 𝜂‖2 = (𝜓 − 𝜑1 )(1) = ‖𝜓 − 𝜑1 ‖ ≤ ‖𝜑1 − 𝜑2 ‖ < 𝜀, and there exists 𝜂1 ∈ ℋ with 𝜑1 = 𝜔𝜂1 and ‖𝜂1 −𝜂‖2 < 𝜀. By the first part of the proof it follows that there is a u′1 ∈ U(ℳ ′ ) with ‖u′1 𝜉1 −𝜂‖2 < 2𝜀. Similarly, there is a u′2 ∈ U(ℳ ′ ) with ‖u′2 𝜉2 −𝜂‖2 < 2𝜀. Hence u′ = (u′2 )−1 u′1 ∈ U(ℳ ′ ) and ‖u′ 𝜉1 − 𝜉2 ‖2 < 4𝜀. Proof of the Proposition. We may assume that ℳ ⊂ ℬ(ℋ ) is in standard form. Let ℋ1 be a fourdimensional Hilbert space with orthonormal basis {𝜀ij }1≤i,j≤2 , let ℱ be the type I2 factor generated in ℬ(ℋ1 ) by the system of matrix units {eij }1≤i,j≤2 , where eij 𝜀kl = 𝛿jk 𝜀il , and let ℱ ′ be the type I2 factor generated in ℬ(ℋ1 ) by the system of matrix units {e′ij }1≤i,j≤2 , where e′ij 𝜀kl = 𝛿il 𝜀kj and 𝜂1 = 𝜆1∕2 𝜀11 + (1 − 𝜆)1∕2 𝜀22 ∈ ℋ1 . Then ℱ ′ is the commutant of ℱ and 𝜂1 is a cyclic and separating vector for ℱ . Consider now 𝜉 ∈ ℋ , ‖𝜉‖ = 1, 𝜑 = 𝜔𝜉 |ℳ, and 0 < 𝜀 < 𝜆∕2. By assumption, there is a partial isometry u ∈ ℳ such that u2 = 0, uu∗ + u∗ u = 1 and |𝜆𝜑(ux) − (1 − 𝜆)𝜑(xu)| ≤ 𝜀‖x‖ (x ∈ ℳ) |𝜆𝜑(xu∗ ) − (1 − 𝜆)𝜑(u∗ x)| ≤ 𝜀‖x‖ (x ∈ ℳ).

(1) (2)

The Connes Invariant S(ℳ)

381

Let e = u∗ u ∈ ℳ. Then 1 − e = uu∗ and from (1) with x = u∗ we deduce that |𝜆𝜑(1 − e) − (1 − 𝜆)𝜑(e)| ≤ 𝜀, |𝜆 − 𝜑(e)| ≤ 𝜀, and, as 𝜀 < 𝜆∕2, we get 𝜑(e) < 𝜆∕2.

(3)

On the other hand, since (u∗ )2 = 0, from (2) with x = uyuu∗ we deduce that |(1−𝜆)𝜑(ey(1−e))| ≤ 𝜀‖y‖ and, as 𝜆 < 1∕2, 1 − 𝜆 > 1∕2, we obtain |𝜑(ey(1 − e))| ≤ 2𝜀‖y‖, |𝜑((1 − e)ye)| ≤ 2𝜀‖y‖, so that |𝜑(y) − 𝜑(eye) − 𝜑((1 − e)y(1 − e))| ≤ 4𝜀‖y‖ (y ∈ ℳ). Using this inequality and also (1) with x = eyu∗ and with x = u∗ y(1 − e), we conclude that ‖𝜑(⋅) − 𝜆𝜑(e ⋅ e) − 𝜆𝜑(u ⋅ u∗ ) − (1 − 𝜆)𝜑((1 − e) ⋅ (1 − e)) − (1 − 𝜆)𝜑(u∗ ⋅ u)‖ ≤ 6𝜀.

(4)

Let 𝒩 be the subalgebra of ℳ generated by u and u∗ . Then {E11 = e = u∗ u, E12 = u, E21 = ∗ 22 = 1 − e = uu } is a system of matrix units in 𝒩 , so that 𝒩 is a type I2 factor. Every element x ∈ ℳ can be uniquely written in the form (9.15.(3)) u∗ , E

x = x11 e + x12 u + x21 u∗ + x22 (1 − e)

(5)

with xij ∈ 𝒩 ′ ∩ ℳ. Since ℳ is a factor, it follows that 𝒩 ′ ∩ ℳ is also a factor. Let 𝒩1 = 𝒩 and 𝜋1 ∶ 𝒩1 → ℱ be the *-isomorphism defined by 𝜋1 (Eij ) = eij (1 ≤ i, j ≤ 2). Let 𝒩2 = 𝒩 ′ ∩ ℳ and 𝜋2 ∶ 𝒩2 → ℬ(ℋ2 ) be the faithful standard representation of 𝒩2 associated with 𝜑|𝒩2 = 𝜔𝜉 |𝒩2 with the cyclic and separating vector 𝜂2 ∈ ℋ2 , that is, 𝜑(x) = (𝜋2 (x)𝜂2 |𝜂2 ) (x ∈ 𝒩2 ). ̄ 𝜋2 ∶ ℳ → ℱ ⊗ ̄ 𝜋2 (𝒩2 ) ⊂ Using (5), we define a faithful standard representation 𝜋 = 𝜋1 ⊗ ∑ ̄ ℋ2 ), that is, 𝜋(x) = ij eij ⊗ ̄ 𝜋2 (xij ) (x ∈ ℳ). Thus, for x ∈ ℳ we have ℬ(ℋ1 ⊗ ̄ 𝜂2 )|𝜂1 ⊗ ̄ 𝜂2 ) = (𝜋(x)(𝜂1 ⊗

∑

(eij 𝜂1 |𝜂1 )𝜑(xij ) = 𝜆𝜑(x11 ) + (1 − 𝜆)𝜑(x22 )

(6)

ij

Since x11 = exe + uxu∗ , x22 = (1 − e)x(1 − e) + u∗ xu, from (6) and (4) it follows that ‖(𝜋(⋅)(𝜂1 ⊗ 𝜂2 )|𝜂1 ⊗ 𝜂2 ) − 𝜑(⋅)‖ ≤ 6𝜀.

(7)

Since any two standard forms are spatially isomorphic, from the previous lemma and (7) it ̄ ℋ2 such that 𝜋(x) = VxV ∗ (x ∈ ℳ) follows that there exists a unitary operator V ∶ ℋ → ℋ1 ⊗ and ‖V ∗ (𝜂1 ⊗ 𝜂2 ) − 𝜉‖2 ≤ 25𝜀.

(8)

̄ 1)V ∈ ℳ ′ . By (3), we have ‖u𝜉‖2 = 𝜑(u∗ u) = Let t = 𝜆1∕2 (1 − 𝜆)1∕2 and u′ = t−1 V ∗ (e′12 ⊗ 𝜑(e) ≥ 𝜆∕2, hence ‖u𝜉‖ ≥ 𝜆1∕2 ∕2.

(9)

From (8), we obtain ‖u𝜉 − u′ 𝜉‖ ≤ ‖uV ∗ (𝜂1 ⊗ 𝜂2 ) − u′ V ∗ (𝜂1 ⊗ 𝜂2 )‖ + 5𝜀1∕2 (‖u‖ + ‖u′ ‖), ‖u∗ 𝜉 − t2 u′∗ 𝜉‖ ≤ ‖u∗ V ∗ (𝜂1 ⊗ 𝜂2 ) − t2 u′∗ V ∗ (𝜂1 ⊗ 𝜂2 )‖ + 5𝜀1∕2 (‖u‖ + t2 ‖u′ ‖). Since e12 𝜂1 = t−1 e′12 𝜂1 ,

382

Discrete Decompositions

̄ 1, Vu′ V ∗ = t−1 (e′ ⊗ ̄ 1), it follows that e21 𝜂1 = te′21 𝜂1 , and VuV ∗ = 𝜋(u) = e12 ⊗ 12 ‖u𝜉 − u′ 𝜉‖ ≤ 5𝜀1∕2 (1 + t−1 ), ‖u∗ 𝜉 − t2 u′∗ 𝜉‖ ≤ 5𝜀1∕2 (1 + t). From Theorem 28.7, (9), and (10), we conclude that 𝜆∕(1 − 𝜆) = t2 ∈ S(ℳ).

(10) □

̂ is a closed subgroup, then its annihilator in ℝ is the set T = {t ∈ 28.9. Recall that if S ⊂ ℝ+∗ ≡ ℝ it ℝ; 𝜆 = 1 for all 𝜆 ∈ S}; T is a closed subgroup of ℝ and S = {𝜆 ∈ ℝ+∗ ; 𝜆it = 1 for all t ∈ T}. The invariants T (ℳ) and S(ℳ) are connected by the following partial duality result. Theorem. If ℳ is a factor and S(ℳ) ≠ {0, 1}, then T (ℳ) is the annihilator of S(ℳ) ∩ ℝ+∗ in ℝ. Proof. If S(ℳ) = {1}, then ℳ is semifinite (28.2) and hence T (ℳ) = ℝ (27.2.). Let 𝜑 be an n.s.f. weight on ℳ. If (28.3.(1)) Γ(𝜎 𝜑 ) = S(ℳ) ∩ ℝ+∗ ≠ {1}, then the quotient 𝜑 ) is compact and so, by Theorem 16.5, the annihilator of S(ℳ) ∩ ℝ+ = Γ(𝜎 𝜑 ) is ̂ group ℝ∕Γ(𝜎 ∗ 𝜑 Int(𝜎 ) = T (ℳ). We mention that for every t ∈ ℝ there exists a factor ℳ with separable predual such that T (ℳ) = {nt; n ∈ ℤ} and S(ℳ) = {0, 1} (see Connes, 1973a, 3.4.4). 28.10. A general duality result can be obtained by considering the Bohr compactification 𝔹 of ℝ+∗ (Rudin (1962)), that is, 𝔹 is the dual group of the discrete additive group ℝ which we shall denote by ℝd . The identity mapping t ∶ ℝd → ℝ is a continuous group isomorphism and the dual mapping 𝛽 ∶ ℝ∗+ → 𝔹 is an injective continuous group homomorphism with dense range. Proposition. For any factor ℳ, the invariant T (ℳ) ⊂ ℝd is the annihilator of the set ⋂ {𝛽(Sp(Δ𝜑 ) ∩ ℝ+∗ ); 𝜑 ∈ Wnsf (ℳ)} ⊂ 𝔹. Proof. Using Propositions 28.1 and 14.9, for every 𝜑 ∈ Wnsf (ℳ) we have 𝛽(Sp(Δ𝜑 ) ∩ ℝ+∗ ) = Sp(𝜎 𝜑 ◦ 𝜄). Let 𝜔 ∈ Wnsf (ℳ) be fixed. By Theorem 16.5, T (ℳ) ⊂ ℝd is the annihilator of Γ(𝜎 𝜔 ◦ 𝜄) ⊆ 𝔹. It is therefore sufficient to show that ⋂ Γ(𝜎 𝜔 ◦ 𝜄) = {Sp(𝜎 𝜑 ◦ 𝜄); 𝜑 ∈ Wnsf (ℳ)}. (1) For every 𝜑 ∈ Wnsf (ℳ) we have 𝜎 𝜑 ∼ 𝜎 𝜔 (3.1), hence 𝜎 𝜑 ◦ 𝜄 ∼ 𝜎 𝜔 ◦ 𝜄 and therefore Γ(𝜎 𝜔 ◦ 𝜄) = Γ(𝜎 𝜑 ◦ 𝜄) ⊂ Sp(𝜎 𝜑 ◦ 𝜄) (16.3). We have thus proved the inclusion “⊂” in (1). By the definition of the invariant Γ (see 16.1), we have ⋂ Γ(𝜎 𝜔 ◦ 𝜄) = {Sp(𝜎 𝜔e ◦ 𝜄); 0 ≠ e ∈ Proj(ℳ 𝜔 )}. Let 0 ≠ e ∈ Proj(ℳ 𝜔 ). By Theorems 16.6 and 5.1, we can find an n.s.f. weight 𝜑 on ℳ such that Sp 𝜎 𝜑 ⊂ Sp 𝜎 𝜔e and by Proposition 14.9 it follows that Sp(𝜎 𝜑 ◦ 𝜄) ⊂ Sp(𝜎 𝜔e ◦ 𝜄). This proves the inclusion “⊃” in (1).

Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

383

28.11. Let ℳ be a type III factor. Since S(ℳ) ∩ ℝ+∗ is a closed subgroup of ℝ+∗ and S(ℳ) is a closed subset of [0, +∞), we have either S(ℳ) = {0, 1}, or S(ℳ) = [0, +∞), or S(ℳ) = {0} ∪ {𝜆n ; n ∈ ℤ} for some 0 < 𝜆 < 1. If S(ℳ) = {0, 1}, we say that ℳ is of type III0 . If S(ℳ) = [0, +∞), we say that ℳ is of type III1 . In this case, we have T (ℳ) = {0}, by Theorem 28.9. If S(ℳ) = {0} ∪ {𝜆n ; n ∈ ℤ}, we say that ℳ is of type III𝜆 , (0 < 𝜆 < 1). In this case, by Theorem 28.9, we have T (ℳ) = {2𝜋k∕ ln(𝜆); k ∈ ℤ}; note that −2𝜋∕ ln(𝜆) is the smallest positive member of T (ℳ). Recall (28.2) that ℳ is semifinite if and only if 0 ∉ S(ℳ) and that in this case S(ℳ) = {1}. In the next sections, we shall study in more detail the type III𝜆 factors for 0 ≤ 𝜆 < 1. For these factors there exists, besides the continuous decomposition (§23), a discrete decomposition. 28.12. Notes. The invariant S(ℳ) was introduced by Connes (1973a). Previously, similar (but different) invariants had been considered by Araki and Woods (1968), Golodec (1972), and Krieger (1970b). Actually, the starting point for Connes’ work (Connes, 1973a) was the confrontation between the classification by Araki and Woods of ITPFI-factors and the Tomita–Takesaki modular theory of W ∗ -algebras. The group property of the invariant S(ℳ) and 28.3.(4) was first proved by Connes and van Daele (1973) by a different method. The other results presented in this section are due to Connes (1973a). For an arbitrary factor ℳ, Araki and Woods (1968) introduced the invariant r∞ (ℳ) as the set of ̄ ℛ𝜆 ≈ ℳ and Connes (1973a, 3.6.1) proved that 𝜆 ∈ r∞ (ℳ) ⇒ ℳ has all 𝜆 ≥ 0 such that ℳ ⊗ property L1∕(1+𝜆) ⇒ 𝜆 ∈ S(ℳ). If ℳ is an Araki–Woods factor, then S(ℳ) = r∞ (ℳ) coincides with the “asymptotic ratio set” (Araki & Woods, 1968) which is expressed only in terms of the eigenvalue list of ℳ (Araki & Woods, 1968; Connes, 1973a, 3.6.1). Property L𝜆 (0 < 𝜆 < 1∕2) was introduced by Powers (1967, 1970). Then Araki (1971) considered a stronger property, called L′𝜆 , and showed that a W ∗ -algebra ℳ has this property if and only if 𝜆∕(1 − 𝜆) ∈ r∞ (ℳ). Connes (1973a, 3.7.9) proved that the Pukánszky factor P𝜆∕(1−𝜆) has property L𝜆 without having property L′𝜆 . For some classes of factors, including those with {0, 1} ≠ S(ℳ) ≠ [0, +∞), Connes (1973a, 3.7.2. 3.7.7) proved that property L𝜆 is equivalent with 𝜆∕(1 − 𝜆) ∈ S(ℳ) (0 < 𝜆 < 1∕2). Using Corollary 28.6 and the results of Krieger (1971a, 1971b), Connes (1973a, 3.3.5) proved that for every subgroup 𝔾 ≠ {1} of ℝ+∗ there exists a factor ℳ with separable predual such that ̄ ℳ) S(ℳ) = {0, 1} and S(ℳ ⊗ ℳ) = {0} ∪ 𝔾. In particular, using the duality between S(ℳ ⊗ ̄ ℳ) = T (ℳ) (28.9), it follows that for every t ∈ ℝ there exists a factor ℳ with and T (ℳ ⊗ separable predual such that S(ℳ) = {0, 1} and T (ℳ) = {nt; n ∈ ℤ} (Connes, 1973a, 3.4.4). This shows on the one hand the wealth of the class of factors ℳ with S(ℳ) = {0, 1} (see also Araki & Woods, 1968) and on the other that the assumptions of Theorems 16.5 and 28.9 are essential. For our exposition we have used Connes (1973a).

29 Factors of Type III𝝀 (0 ≤ 𝝀 < 1) In this section, we give canonical constructions for factors of type III𝜆 (0 ≤ 𝜆 < 1) and study some important classes of weights on these factors. All W ∗ -algebras that appear will be countably decomposable, either by assumption or by construction.

384

Discrete Decompositions

29.1. We first give a construction which leads to factors of type III𝜆 with 0 < 𝜆 < 1. Let 0 < 𝜆 < 1 be fixed and let t = −2𝜋∕ln(𝜆). An n.s.f. weight 𝜑 on a factor of type III𝜆 such that 𝜑(1) = +∞ and 𝜎t𝜑 = 𝜄 will be called a 𝜆-trace. Note that every 𝜆-trace is strictly semifinite. Proposition. Let 𝒩 be a factor of type II∞ , 𝜏 an n.s.f. trace on 𝒩 and let 𝜃 ∈ Aut(𝒩 ) be such that 𝜏 ◦ 𝜃 = 𝜆𝜏. Then the action 𝜃 ∶ ℤ ∋ n ↦ 𝜃 n ∈ Aut(𝒩 ) is properly outer, the crossed product ℳ = ℛ(𝒩 , 𝜃) is a factor of type III𝜆 , and the dual weight 𝜑 = 𝜏̂ is a 𝜆-trace on ℳ. Moreover, ̄ 𝜆 (1) ∈ ℳ, we have identifying 𝒩 ≡ 𝜋𝜃 (𝒩 ) ⊆ ℳ and putting u = 1 ⊗ 𝒩 = ℳ 𝜑 , 𝜑|𝒩 = 𝜏 u ∈ ℳ(𝜎 𝜑 , {𝜆}), 𝜑 ◦ Ad(u) = 𝜆𝜑.

(1) (2)

Proof. Since 𝜏 ◦ 𝜃 n = 𝜆n 𝜏 ≠ 𝜏, the *-automorphism 𝜃 n (n ≠ 0) is properly outer, hence the action 𝜃 ∶ ℤ → Aut(𝒩 ) is properly outer. By Corollary 1/22.6 and Theorem 22.3, ℳ is a factor and 𝒩 ′ ∩ ℳ = 𝒵 (𝒩 ). Let P ∶ ℳ → 𝒩 be the faithful normal conditional expectation associated with the crossed ̄ 𝜆 (n); n ∈ ℤ} = {un ; n ∈ ℤ} ⊂  (P). Since 𝜏(un ⋅ u−n ) = 𝜏 ◦ 𝜃 n = product (22.2) and let 𝒢 = {1 ⊗ n 𝜆 𝜏, we infer from Theorem 28.5 that a positive number 𝜇 > 0 belongs to the Connes invariant S(ℳ) if and only if for each 𝜀 > 0 there exists n ∈ ℤ such that Sp(𝜆−n ⋅ 1) ⊂ (𝜇 − 𝜀, 𝜇 + 𝜀), hence S(ℳ) = {0} ∪ {𝜆n ; n ∈ ℤ}, that is, ℳ is a factor of type III𝜆 . The dual weight 𝜑 = 𝜏̂ = 𝜏 ◦ P is an n.s.f. weight on ℳ and for every x ∈ ℳ + we have (𝜑 ◦ Ad(u))(x) = 𝜑(uxu∗ ) = 𝜏(P(uxu∗ )) = 𝜏(uP(x)u∗ ) = (𝜏 ◦ 𝜃)(P(x)) = 𝜆𝜏(P(x)) = 𝜆𝜑(x), whence 𝜑 ◦ Ad(u) = 𝜆𝜑. Then, for s ∈ ℝ we have 𝜆is = [D(𝜆𝜑) ∶ D𝜑]s = [D(𝜑u ) ∶ D𝜑]s = u∗ 𝜎s𝜑 (u), that is, 𝜎s𝜑 (u) = 𝜆is u = ⟨s, 𝜆⟩u, so that u ∈ ℳ(𝜎 𝜑 , {𝜆}). On the other hand, we have 𝜎t𝜑 |𝒩 = 𝜄 and 𝜎t𝜑 (u) = 𝜆it u = u, hence 𝜎t𝜑 = 𝜄, since ℳ = ℛ{𝒩 , u}. Also, 𝜑|𝒩 = (𝜏 ◦ P)|𝒩 = 𝜏, in particular 𝜑(1) = 𝜏(1) = +∞, so that 𝜑 is a 𝜆-trace. Finally, since 𝜑 = 𝜏 ◦ P, from Theorem 11.9 we infer that 𝒩 ⊂ ℳ 𝜑 . The mapping E ∶ ℳ → ℳ 𝜑 t defined by E(x) = t−1 ∫0 𝜎s𝜑 (x)ds (x ∈ ℳ) is a faithful normal conditional expectation of ℳ onto t ℳ 𝜑 , for x ∈ 𝒩 ⊂ ℳ 𝜑 we have E(x) = x and E(un ) = (t−1 ∫0 𝜆ins dt)un = 0 for 0 ≠ n ∈ ℤ. Since ∑ every element of ℳ is of the form n an un with an ∈ 𝒩 (see 22.1), it follows that E = P and hence ℳ𝜑 = 𝒩 . ̂ → Aut(ℳ). Consider Recall (19.3) that on ℳ we also have the dual action 𝜃̂ ∶ 𝕋 = ℤ the surjective continuous group homomorphism 𝛼 ∶ ℝ ∋ s ↦ e2𝜋is∕t = 𝜆−is ∈ 𝕋 . Since ℳ = ℛ{𝒩 , u}, 𝜃̂𝛼(s) |𝒩 = 𝜄 = 𝜎s𝜑 |𝒩 , 𝜎s𝜑 (u) = 𝜆is u, and (19.3.(4)) 𝜃̂𝛼(s) (u) = ⟨1, 𝛼(s)⟩u = 𝜆is u, it follows that 𝜃̂𝛼(s) = 𝜎s𝜑

(s ∈ ℝ).

(3)

Thus, the identity E = P established in the above proof is nothing but the equality of P𝜃̂ and P. that is, just the definition of P (22.2). Also, some other parts of the previous proof are consequences of (3) and of general properties of the dual action. A triple (𝒩 , 𝜃, 𝜏) consisting of a type II∞ factor 𝒩 , a *-automorphism 𝜃 ∈ Aut(𝒩 ), and an n.s.f. trace 𝜏 on 𝒩 such that 𝜏 ◦ 𝜃 = 𝜆𝜏 will be called a discrete decomposition of type III𝜆 (0 < 𝜆 < 1).

Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

385

29.2. Before giving a similar construction for factors of type III0 we shall study some properties of diffuse abelian W ∗ -algebras. An abelian W ∗ -algebra is called diffuse if it has no minimal projections. Lemma. Let 𝒵 be a diffuse abelian W ∗ -algebra and 𝜃 ∈ Aut(𝒵 ) an ergodic *-automorphism. For each projection 0 ≠ e ∈ 𝒵 , there exists a sequence of projections {en }n≥1 ⊂ 𝒵 , uniquely determined, such that ∑ (1) e= en , 𝜃 n (en ) ≤ e and e𝜃 k (en ) = 0 for k = 1, … , n − 1. n≥1

Moreover, we have e=

∑

𝜃 n (en ).

(2)

n≥1

Proof. Since 𝜃 is ergodic and 𝒵 is diffuse, it follows that 𝜃 is conservative, that is, p ∈ Proj(𝒵 ), p𝜃 n (p) = 0 for all n ≥ 1 ⇒ p = 0.

(3)

Indeed, if p𝜃 n (p) = 0 for all n ≥ 1, then 𝜃 i (q)𝜃 j (q) = 0 for all i ≠ j in ℤ and for every projection ∑ 0 ≠ q ≤ p in 𝒵 , hence k∈ℤ 𝜃 k (q) = 1 by the ergodicity of 𝜃. It follows that 0 ≠ q ≤ p ⇒ q = p, that is, p is a minimal projection, contradicting the fact that 𝒵 is diffuse. Let e1 be the largest projection of 𝒵 such that e1 ≤ e and 𝜃(e1 ) ≤ e. Then, if e1 , … , en−1 have been already chosen, we define en inductively to be the largest projections of 𝒵 such that en ≤ e − (e1 + … + en−1 ), 𝜃 n (en ) ≤ e. By this definition, we have 𝜃 n (en ) ≤ e. Let 1 ≤ k < n, p = 𝜃 k (en )e, and q = 𝜃 −k (p) = en 𝜃 −k (e). We have qek = 0, q ≤ e − (e1 + … + ek−1 ) and 𝜃 n (q) = p ≤ e, hence q = 0 by the definition of ek . Thus, 𝜃 k (en )e = 𝜃 k (q) = 0. Let p ≤ e be such that pen = 0 for every n ≥ 1 and qn = p𝜃 −n (e). We have qn ≤ p ≤ e − (e1 + … + en−1 ) and 𝜃 k (qn ) = 𝜃 n (p)e ≤ e, whence qn ≤ en . Since qn ≤ p, we have qn ≤ pen = 0, so that ∑ 𝜃 n (p)e = 0. In particular, p𝜃 n (p) = 0 for every n ≥ 1, hence p = 0, by (3). Thus, e = n≥1 en . ∑ To prove the uniqueness assertion, let e = n≥1 fn with 𝜃 n (fn ) ≤ e and e𝜃 k (fn ) = 0 for 1 ≤ k < n. Let n ≥ 1 be fixed. For k < n we have 𝜃 k (ek fn ) ≤ e, hence ek fn = 0. Thus, fn ≤ e − (e1 + … + en−1 ) ∑ ∑ and 𝜃 n (fn ) ≤ e, so fn ≤ en . Since n≥1 fn = e = n≥1 en , it follows that fn = en . It remains to prove (2). If n < m, then 𝜃 n (en )𝜃 m (em ) = 𝜃 n (en 𝜃 m−n (em )) ≤ 𝜃 n (e𝜃 m−n (em )) = 0, ∑ hence n≥1 𝜃 n (en ) is a projection ≤ e. Consider now a projection p ≤ e with p𝜃 k (ek ) = 0 for every k ≥ 1. Since p ≤ e and 𝜃 n (ek )e = 0 for k > n it follows that p𝜃 n (ek ) = 0 for every k ≥ n, hence p𝜃 n (e − (e1 + … + en−1 ) = 0.

(4)

p𝜃 n (e) = 0 for every n ≥ 1.

(5)

We show by induction that

For n = 1 this is just (4). We assume that (5) has already been proved for 1, … , n − 1 and prove it for n. Let 1 ≤ j < n. By the induction hypothesis, we have p𝜃 n−j (e) = 0, hence 𝜃 −n (p)𝜃 −j (e) = 0. As

386

Discrete Decompositions

𝜃 j (ej ) ≤ e, we have ej ≤ 𝜃 −j (e). Thus, 𝜃 −n (p)ej = 0 and hence p𝜃 n (ej ) = 0 for every j = 1, … , n − 1. Using (4) we now obtain (5), and from (5) it follows, in particular, that p𝜃 n (p) = 0 for every n ≥ 1, that is, p = 0 by (3). □ In particular, for every n ≥ 1 and every nonzero projection e ∈ 𝒵 , there exists a nonzero projection p ∈ 𝒵 , p ≤ e, such that p𝜃 n (p) = 0. Thus, if 𝜃 is an ergodic *-automorphism of a diffuse abelian W ∗ -algebra 𝒵 , then the action k ↦ 𝜃 k of ℤ on 𝒵 is free. 29.3. We now give the construction that leads to factors of type III0 . An n.s.f. weight 𝜑 on a W ∗ -algebra such that 1 is an isolated point in Sp(Δ𝜑 ), that is, (28.1.(4)) in Sp 𝜎 𝜑 will be called a lacunary weight. It is easy to see that every lacunary weight is strictly semifinite. Proposition. Let 𝒩 be a type II∞ W ∗ -algebra with diffuse center, 𝜏 an n.s.f. trace on 𝒩 , 𝜃 ∈ Aut(𝒩 ) a *-automorphism acting ergodically on 𝒵 (𝒩 ) and 0 < 𝜆 < 1 such that 𝜏 ◦ 𝜃 ≤ 𝜆𝜏. Then the action 𝜃 ∶ ℤ ∋ n ↦ 𝜃 n ∈ Aut(𝒩 ) is free on the center of 𝒩 , the crossed product ℳ = ℛ(𝒩 , 𝜃) is a factor of type III0 and the dual weight 𝜑 = 𝜏̂ is a lacunary weight of infinite multiplicity on ℳ. ̄ 𝜆 (1) ∈ ℳ and considering the faithful Moreover, identifying 𝒩 and 𝜋𝜃 (𝒩 ) ⊂ ℳ, putting u = 1 ⊗ normal conditional expectation P ∶ ℳ → 𝒩 associated with the crossed product, we have 𝒩 = ℳ 𝜑 , 𝜑|𝒩 = 𝜏 u ∈ ℳ(𝜎 𝜑 ; (0, 1)) v ∈  (P) ∩ ℳ(𝜎 𝜑 ; (0, 1)), ℳ = ℛ{𝒩 , v} ⇒ u∗ v ∈ ℳ 𝜑

(1) (2) (3)

x − P(x) ∈ ℳ(𝜎 𝜑 ; ℝ+∗ ∖(𝜆, 𝜆−1 )) (x ∈ ℳ).

(4)

Proof. Since 𝒵 (𝒩 ) is diffuse and 𝜃 is ergodic on 𝒵 (𝒩 ), the action 𝜃 ∶ ℤ → Aut(𝒩 ) is ergodic and free on 𝒵 (𝒩 ) (29.2) and hence (17.5) properly outer on 𝒩 . Using 22.6 and 22.3, we conclude that ℳ is a factor and 𝒵 (𝒩 )′ ∩ ℳ = 𝒩 , 𝒩 ′ ∩ ℳ = 𝒵 (𝒩 ). We now compute the invariant S(ℳ) using Theorem 28.5. Let An be the unique nonsingular positive self-adjoint operator affiliated to 𝒵 (𝒩 ) such that 𝜏 ◦ 𝜃 n = 𝜏An (n ∈ ℤ). Since 𝜏 ◦ 𝜃 ≤ 𝜆𝜏, it follows that n > 0 ⇒ An ≤ 𝜆n < 1, n < 0 ⇒ An ≥ 𝜆n > 1.

(5) (6)

Let 𝜇 ∈ (0, 1). There exists m ≥ 1 such that 𝜆m < 𝜇 < 1. Since 𝒵 (𝒩 ) is diffuse and 𝜃 is ergodic on 𝒵 (𝒩 ), it follows from Lemma 29.2 that there exists a nonzero projection e ∈ 𝒵 (𝒩 ) such that e𝜃 n (e) = 0 for all n = 1, … , m. Let 𝜀 > 0 be such that 𝜆m < 𝜇 − 𝜀 < 𝜇 + 𝜀 < 1. Assume that 𝜇 ∈ S(ℳ). Then (28.5) there exist n ∈ ℤ and a nonzero projection p ∈ 𝒵 (𝒩 ) such that p ≤ e, 𝜃 n (p) ≤ e and Sp(An p|𝒩 p) ⊂ (𝜇 − 𝜀, 𝜇 + 𝜀). If n > 0, then An ≤ 𝜆n , that is, Sp(An p|𝒩 p) ⊂ [0, 𝜆n ], so that n ≤ m; it follows that 𝜃 n (p) ≤ e𝜃 n (e) = 0, that is, p = 0, a contradiction. If n ≤ 0, then An ≥ 1, hence Sp(An p|𝒩 p) ⊂ [1, +∞), which is again a contradiction, since 𝜇 + 𝜀 < 1. As S(ℳ) ∩ ℝ+∗ is a closed subgroup of ℝ+∗ , it follows that S(ℳ) ⊂ {0, 1}. Consequently, in order to show that S(ℳ) = {0, 1}, it is enough to show that ℳ is not semifinite (see 28.2). By Theorem 2/22.7, this amounts to showing that there exist no 𝜃-invariant n.s.f. traces on 𝒩 .

Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

387

Otherwise, let 𝜇 be a 𝜃-invariant n.s.f. trace on 𝒩 and let A be the unique positive nonsingular self-adjoint operator affiliated to 𝒵 (𝒩 ) such that 𝜏 = 𝜇A . Then 𝜇𝜃−1 (A) = 𝜏 ◦ 𝜃 ≤ 𝜆𝜏 = 𝜇𝜆A , hence 𝜃 −1 (A) ≤ 𝜆A and 𝜃 −k (A) ≤ 𝜆k A, 𝜃 k (A) ≥ 𝜆−k A, for all k > 0. There exists n ∈ ℤ such that e = 𝜒(𝜆n+1 ,𝜆n ] (A) ≠ 0 and for every k > 0 we have 𝜃 k (e) = 𝜒(𝜆n+1 ,𝜆n ] (𝜃 k (A)) ≤ 𝜒(0,𝜆n ] (𝜃 k (A)) ≤ 𝜒(0,𝜆n ] (𝜆−k A) = 𝜒(0,𝜆n+k ] (A), hence e𝜃 k (e) = 0. As 𝒵 (𝒩 ) is diffuse and 𝜃 is ergodic on 𝒵 (𝒩 ), it follows that e = 0 (29.2.(3)), a contradiction. Hence ℳ is a factor of type III0 . By the general theory of crossed products, we know that u ∈  (P) and 𝜃 n = Ad(un )|𝒩 (n ∈ ℤ). Since 𝜑 = 𝜏̂ = 𝜏 ◦ P, it follows (see 27.4.(2)) that 𝜎t𝜑 (un ) = un Aitn (n ∈ ℤ, t ∈ ℝ). Thus, for ( ) f ∈ ℒ 1 (ℝ), we obtain 𝜎f𝜑 (un ) = ∫ f (t)𝜎t𝜑 (un ) dt = un ∫ f (t)Aitn dt = un ̂f (An ) and hence Sp𝜎 𝜑 (un ) = Sp(An ) ∩ ℝ+∗ . Using (5) and (6), we infer that n > 0 ⇒ Sp𝜎 𝜑 (un ) ⊂ (0, 𝜆n ] ⊂ (0, 𝜆] ⊂ (0, 1) n < 0 ⇒ Sp𝜎 𝜑 (un ) ⊂ [𝜆n , +∞) ⊂ [𝜆−1 , +∞) ⊂ [1, +∞). In particular, this proves (2). Since 𝜑 = 𝜏 ◦ P, we have 𝜑|𝒩 = 𝜏 and 𝒩 ⊂ ℳ 𝜑 = ℳ(𝜎 𝜑 ; {1}) (11.9). Let a ∈ 𝒩 and 0 ≠ n ∈ ℤ. Using the previous remarks and 15.3.(1), it follows that aun ∈ ℳ(𝜎 𝜑 ; ℝ+∗ ∖(𝜆, 𝜆−1 )). Since P(aun ) = aP(un ) = 0 (22.2.(4)), we also have aun − P(aun ) ∈ ℳ(𝜎 𝜑 ; ℝ+∗ ∖(𝜆, 𝜆−1 )). Since the set {aun ; a ∈ 𝒩 , n ∈ ℤ} is w-total in ℳ (22.1), it follows (14.5.1) that (Sp 𝜎 𝜑 ) ∩ (𝜆, 𝜆−1 ) = {1} and x − P(x) ∈ ℳ(𝜎 𝜑 ; ℝ+∗ ∖(𝜆, 𝜆−1 )) for all x ∈ ℳ. If x ∈ ℳ 𝜑 , then x − P(x) ∈ ℳ 𝜑 ∩ ℳ(𝜎 𝜑 ; ℝ+∗ ∖(𝜆, 𝜆−1 )) = ℳ(𝜎 𝜑 ; ∅) = {0}, hence x = P(x) ∈ 𝒩 . Thus, ℳ 𝜑 = 𝒩 and 𝜑|ℳ 𝜑 = 𝜏 is semifinite. Therefore, 𝜑 is a lacunary weight of infinite multiplicity on ℳ. Also, we have proved assertions (1) and (4). To prove (3) we consider u ∈  (P) and 𝒢 = {vn ; n ∈ ℤ} ⊂  (P). By assumption ℳ = ℛ{𝒩 , 𝒢 }, and so (22.4.(1)) there exist a family of mutually orthogonal projections {qk }k∈ℤ ⊂ ∑ 𝒵 (𝒩 ) with k qk = 1 and a family {wk }k∈ℤ ⊂ 𝒩 with w∗k wk = wk w∗k = qk (k ∈ ℤ) such that ∑ u = k vk wk and uqk = vk wk (k ∈ ℤ). Since v ∈ ℳ(𝜎 𝜑 ; (0, 1)), it follows that vk ∈ ℳ(𝜎 𝜑 ; [1, +∞)) for all k ≤ 0 and, as u ∈ ℳ(𝜎 𝜑 ; (0, 1)), qk ∈ ℳ 𝜑 , wk ∈ ℳ 𝜑 , we obtain ℳ(𝜎 𝜑 ; (0, 1)) ∋ uqk = vk wk ∈ ℳ(𝜎 𝜑 ; [1, +∞)), so that uqk = vk wk = 0 and wk = qk = 0 for all k ≤ 0. Thus, there ∑ exist a family of mutually orthogonal projections {qn }n≥1 ⊂ 𝒵 (𝒩 ) with n≥1 qn = 1 and a family {wn }n≥1 ⊂ 𝒩 with w∗n wn = wn w∗n = qn such that u=

∑

vn wn and uqn = vn wn for all n ≥ 1.

(7)

n≥1

Similarly, there exist a family of mutually orthogonal projections {q′n }n≥1 ⊂ 𝒵 (𝒩 ) with ∑ ′ ′ ′∗ ′ ′ ′∗ ′ n≥1 qn = 1 and a family {wn }n≥1 ⊂ 𝒩 with wn wn = wn wn = qn such that v=

∑ n≥1

un w′n and vq′n = un w′n for all n ≥ 1.

(8)

388

Discrete Decompositions

Let m > 1 be fixed. We show that P(u∗ vn q′m ) = 0 for all n ≥ 1.

(9)

Indeed, vq′m = um w′m , u∗ vq′m = um−1 w′m and hence P(u∗ vq′m ) = P(um−1 w′m ) = P(um−1 )w′m = 0, which proves equality (9) for n = 1. If n > 1, then, by the same argument as above, we can find a family ∑ ∑ {w′′j }j≥1 ⊂ 𝒩 such that vn−1 = j≥1 uj w′′j and we have u∗ vq′m = (u∗ vn−1 )vq′m = j≥1 u∗ uj w′′j um w′m = ∑ ∑ i j−1 um (u−m w′′ um )w′ = i≥1 u ai with ai ∈ 𝒩 , hence j≥1 u m j P(u∗ vn q′m ) =

∑

P(ui )ai = 0.

i≥m

∑ ∑ From (7) and (9), it follows that q′m = P(u∗ uq′m ) = P(u∗ ( n≥1 vn wn )q′m ) = n≥1 P(u∗ vn q′m )wn = 0. Since m > 1 was arbitrary, we conclude from (8) that v = uw′1 , hence u∗ v = w′1 ∈ 𝒩 = ℳ 𝜑 . A triple (𝒩 , 𝜃, 𝜏) consisting of a type II∞ W ∗ -algebra with a diffuse center, a *-automorphism 𝜃 ∈ Aut(𝒩 ) which acts ergodically on 𝒵 (𝒩 ), and an n.s.f. trace 𝜏 on 𝒩 such that 𝜏 ◦ 𝜃 ≤ 𝜆𝜏 for some 0 < 𝜆 < 1 will be called a discrete decomposition of type III0 . 29.4. Let (𝒩 , 𝜃, 𝜏) be a discrete decomposition of type III0 and e ∈ 𝒵 (𝒩 ) a nonzero projection. By Lemma 29.2, there exists a sequence of projections {en }n≥1 ⊂ 𝒵 (𝒩 ), uniquely determined, such ∑ ∑ that e = n en , 𝜃 n (en ) ≤ e and e𝜃 k (en ) = 0 for k = 1, … , n − 1; moreover, we have e = n 𝜃 n (en ). Then a *-automorphism 𝜃e ∈ Aut(𝒩 e) is defined by 𝜃e (x) =

∑

𝜃 n (xen ) (x ∈ 𝒩 e).

n

Also, 𝜏e = 𝜏|𝒩 e is an n.s.f. trace on 𝒩 e. The triple (𝒩 , 𝜃, 𝜏) determines a crossed product W ∗ -algebra ℳ = ℛ(𝒩 , 𝜃), a canonical embedding 𝜋 = 𝜋𝜃 ∶ 𝒩 → ℳ, a conditional expectation P ∶ ℳ → 𝜋(𝒩 ), a unitary element ̄ 𝜆 (1) ∈ ℳ, and a dual weight 𝜑 = 𝜏̂ on ℳ. Similarly, the triple (𝒩 e, 𝜃e , 𝜏e ) determines the u=1⊗ objects ℳ0 = ℛ(𝒩 e, 𝜃e ), 𝜋0 ∶ 𝒩 e → ℳ0 , P0 ∶ ℳ0 → 𝜋0 (𝒩 e), u0 ∈ ℳ0 , and 𝜑0 = 𝜏̂e . Proposition. Let (𝒩 , 𝜃, 𝜏) be a discrete decomposition of type III0 and e ∈ 𝒵 (𝒩 ) a nonzero projection. Then (𝒩 e, 𝜃e , 𝜏e ) is also a discrete decomposition of type III0 and there exists a *-isomorphism Φ ∶ (ℳ0 , 𝜋0 , P0 , 𝜑e ) → (𝜋(e)ℳ𝜋(e), 𝜋|𝒩 e, P|𝜋(e)ℳ𝜋(e), 𝜑𝜋(e) )

(1)

such that Φ(u0 ) =

∑

un 𝜋(en ).

(2)

n≥1

Proof. Indeed, 𝒩 e is a type II∞ W ∗ -algebra with diffuse center, 𝜃e ∈ Aut(𝒩 e), 𝜏e is an n.s.f. trace on 𝒩 e and, if 𝜏 ◦ 𝜃 ≤ 𝜆𝜏(0 < 𝜆 < 1), then 𝜏e ◦ 𝜃e ≤ 𝜆𝜏e since for x ∈ (𝒩 e)+ we have 𝜏e (𝜃e (x)) = ∑ ∑ 𝜏( n≥1 𝜃 n (xen )) ≤ n≥1 𝜆n 𝜏(xen ) ≤ 𝜆𝜏e (x). Also, we show that 𝜃e acts ergodically on 𝒵 (𝒩 e) = ∑ ∑ 𝒵 (𝒩 )e. Let p ∈ 𝒵 (𝒩 )e be a projection such that 𝜃e (p) = p, that is, n 𝜃 n (pen ) = n pen . For

Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

389

j < m, k < n and (j, m) ≠ (k, n), we have 𝜃 j (em )𝜃 k (en ) = 0, since, if j < k then e𝜃 k−j (en ) = 0, hence 𝜃 j (em )𝜃 k (en ) = 𝜃 j (em 𝜃 k−j (en )) ≤ 𝜃 j (e𝜃 k−j (en )) = 0 and if j = k then m ≠ n, hence 𝜃 j (em )𝜃 k (en ) = 𝜃 k (em en ) = 0. Consequently, we obtain a projection q=

∞ n−1 ∑ ∑

𝜃 k (pen ) ∈ 𝒵 (𝒩 ).

n=1 k=0

∑∞ Since en ≤ e and e𝜃 k (en ) = 0 for 1 ≤ k ≤ n − 1, it follows that eq = n=1 pen = p. On the other ∑∞ ∑n ∑ ∞ hand, 𝜃(q) = n=1 k=1 𝜃 k (pen ), hence 𝜃(q) − q = n=1 (𝜃 n (pen ) − pen ) = 0. Since 𝜃 is ergodic on 𝒵 (𝒩 ), it follows that either q = 0, that is, p = 0, or q = 1, that is, p = e. Hence (𝒩 e, 𝜃e , 𝜏e ) is a discrete decomposition of type III0 . Consider the W ∗ -algebra 𝒩 e and 𝜃e ∈ Aut(𝒩 e). In order to establish the desired *-isomorphism, we shall use Proposition 22.2. We have a W ∗ -algebra ℳ1 = 𝜋(e)ℳ𝜋(e), an injective unital normal *-homomorphism 𝜋1 = 𝜋|𝒩 e ∶ 𝒩 e → ℳ1 , a faithful normal conditional expectation P1 = P|ℳ1 ∶ ℳ1 → 𝜋1 (𝒩 e) and a unitary element ∑ u1 = n≥1 un 𝜋(en ) ∈ ℳ1 such that 𝜋1 (𝜃e (x)) = u1 𝜋1 (x)u∗1 (x ∈ 𝒩 e) ℳ1 = ℛ{𝜋1 (𝒩 e), u1 }.

(3) (4)

Indeed, u1 is unitary: u1 u∗1 =

∑

un 𝜋(en )𝜋(em )u−m =

n,m

(

∑

=𝜋 u∗1 u1 =

∑

𝜋(en )u−n um 𝜋(em ) =

=

u u 𝜋(𝜃

𝜋(𝜃 n (en ))

n

∑ n,m

−n m

∑

= 𝜋(e)

n

n,m

∑

un 𝜋(en )u−n =

n

) 𝜃 n (en )

∑

n−m

(en )em ) =

n,m

u−n 𝜋(𝜃 n (en ))um 𝜋(em ) ∑

𝜋(en ) = 𝜋(e),

n

since 𝜃 n−m (en )em = 𝜃 −m (𝜃 n (en )𝜃 m (em )) = 0 for n ≠ m. For x ∈ 𝒩 e, we have u1 𝜋1 (x)u∗1 =

∑

un 𝜋(en )𝜋(x)𝜋(em )u−m =

n,m

=

∑ n

u 𝜋(xen )u

−n

=𝜋

un 𝜋(xen em )u−m

n,m

( n

∑

∑ n

𝜃 (xen ) n

) = 𝜋1 (𝜃e (x)).

390

Discrete Decompositions

This proves (3). Let ℳ2 = ℛ{𝜋1 (𝒩 e), u1 }. It is clear that ℳ2 ⊂ ℳ1 and, since ℳ = ℛ{𝜋(𝒩 ), u} and ℳ1 = 𝜋(e)ℳ𝜋(e), we have ℳ1 = ℛ{𝜋(e)𝜋(𝒩 )𝜋(e), 𝜋(e)un 𝜋(e)} (n ∈ ℤ). As 𝜋(e)𝜋(𝒩 )𝜋(e) = 𝜋1 (𝒩 e) and 𝜋 ◦ 𝜃 = Ad(u) ◦ 𝜋, to prove (4) (i.e., ℳ1 = ℳ2 ) it is sufficient to show that for every projection p ∈ 𝒵 (𝒩 ) we have p ≤ e, 𝜃 n (p) ≤ e ⇒ un 𝜋(p) ∈ ℳ2

(n ≥ 1).

Since e𝜃 n (pek ) ≤ e𝜃 n (ek ) = 0 for k > n, it follows that pek = 0 for k > n, and we may assume that p ≤ ek1 , with k1 ≤ n; then 𝜃 k1 (p) ≤ e, 𝜃 n−k1 (𝜃 k1 (p)) ≤ e and, by the same argument, we may assume also that 𝜃 k1 (p) ≤ ek2 , with k1 + k2 ≤ n; by continuing this procedure inductively, we may assume that 𝜃 k1 +k2 (p) ≤ ek3 with k1 + k2 + k3 ≤ n …………… 𝜃

k1 +k2 +…+kj−1

(p) ≤ ekj with k1 + k2 + … + kj = n.

In this case, we have u1 𝜋1 (p) =

∑

k

un 𝜋(en )𝜋(p) = u11 𝜋(p) = 𝜋(𝜃 k1 (p))uk1 ,

n

u21 𝜋1 (p) = u1 uk1 𝜋(ek1 ) = = 𝜋(𝜃

k1 +k2

∑

(p))u

n k1 +k2

un 𝜋(en )𝜋(𝜃 k1 (p))uk1 = uk2 𝜋(𝜃 k1 (p))uk1

,

and, by induction, j

u1 𝜋1 (p) = 𝜋(𝜃 n (p))un = un 𝜋(p), j

so that un 𝜋(p) = u1 𝜋1 (p) ∈ ℳ2 . Thus, (4) is proved. Since (𝒩 e, 𝜃e , 𝜏e ) is a discrete decomposition of type III0 , the action 𝜃e ∶ ℤ → Aut(𝒩 e) is properly outer (29.3). Using Proposition 22.2 and the last remark in Section 22.2, we deduce that there exists a *-isomorphism Φ ∶ (ℳ0 , 𝜋0 , P0 , u0 ) → (ℳ1 , 𝜋1 , P1 , u1 ). Consider now x ∈ ℳ0+ and y ∈ (𝒩 e)+ with P0 (x) = 𝜋0 (y). Then 𝜑0 (x) = 𝜏(y). On the other hand, P1 (Φ(x)) = Φ(P0 (x)) = Φ(𝜋0 (y)) = 𝜋1 (y), so that P(Φ(x)) = 𝜋(y), and 𝜑(Φ(x)) = 𝜏(y). Thus, 𝜑0 = 𝜑n(e) ◦ Φ.

Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

391

Since 𝜑0 = 𝜑n(e) ◦ Φ, we also have 𝜑

Φ ◦ 𝜎s 0 = 𝜎s𝜑 ◦ Φ

(s ∈ ℝ).

(5)

Note that, as the factor ℳ is of type III (and countably decomposable), we have, for every nonzero projection e ∈ 𝒵 (𝒩 ), ℳ ≈ ℳ𝜋(e) . Thus, the discrete type III0 decompositions (𝒩 , 𝜃, 𝜏) and (𝒩 e, 𝜃e , 𝜏e ) give rise to *-isomorphic crossed products which are factors of type III0 . In the constructions of factors described in Sections 29.1 and 29.3, the weights dual to the traces appearing in the corresponding discrete decompositions are lacunary. Thus, in order to obtain a discrete decomposition corresponding to any given factor of type III𝜆 (0 ≤ 𝜆 < 1), it is first necessary to study the lacunary weights on such factors. 29.5. Consider first a factor ℳ of type III𝜆 , for 0 < 𝜆 < 1, and put t = −2𝜋∕ ln(𝜆). Then t ∈ T (ℳ) (see 28.11) and so (27.1.(vi)) there exists a faithful normal state 𝜓 on ℳ such that 𝜎t𝜓 = 𝜄. Also ̄ tr is an n.s.f. weight on ℳ ≈ ℳ ⊗ ̄ ℱ∞ with 𝜑(1) = +∞ and 𝜎t𝜑 = 𝜄, that is, a 𝜆-trace. 𝜑=𝜓⊗ On the other hand, since t is the smallest positive number in T (ℳ) (28.11), for every n.s.f. weight 𝜑 on ℳ with 𝜎t = 𝜄 we have 𝜎s𝜑 = 𝜄 if and only if there exists k ∈ ℤ with s = kt; using 16.4.(2), 28.1.(4), and 28.3.(1), we infer that Sp 𝜎 𝜑 = {𝜆n ; n ∈ ℤ} = Γ(𝜎 𝜑 ) and Sp(Δ𝜑 ) = S(ℳ), in particular Sp(Δ𝜑 ) ∩ (𝜆, 𝜆−1 ) = {1}. Hence every n.s.f. weight 𝜑 on ℳ with 𝜎t𝜑 = 𝜄 is a lacunary weight and ℳ 𝜑 is a factor (see 16.4.(5)). Consider now two 𝜆-traces 𝜑 and 𝜓 on ℳ. Since 𝜎t𝜑 = 𝜎t𝜓 (= 𝜄) and ℳ is a factor, it follows that ̄ ℱ2 is a factor of type III𝜆 (28.4) [D𝜓 ∶ D𝜑]t = 𝛼 ⋅ 1ℳ . Let 𝜇 > 0 be such that 𝜇 it = 𝛼. Then ℳ ⊗ ̄ and the balanced weight ν = 𝜃(𝜇𝜑, 𝜓) is a 𝜆-trace on ℳ ⊗ ℱ2 , since ( ) ( 𝜑 ) ( ) x 0 𝜎t (x) 0 x 0 ν 𝜎t = = (x, y ∈ ℳ) 0 y 0 𝜎t𝜓 (y) 0 y ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 ν 𝜎t = = = ; 1 0 [D𝜓 ∶ D𝜇𝜑]t 0 𝛼𝜇 −it 0 1 0 ̄ ℱ2 )ν is a factor and that the restriction of v to hence 𝜎tν = 𝜄 and ν(1) = +∞. It follows that (ℳ ⊗ ( ) ( ) ̄ ℱ2 is an n.s.f. trace on ℳ ⊗ ̄ ℱ2 (see 10.9). Since ν 1 0 = ν 0 0 = +∞, it follows ℳ⊗ 0 0 0 1 ( ) ( ) 1 0 0 0 ̄ ℱ2 )ν . Using Proposition 23.1, that the projections and are equivalent in (ℳ ⊗ 0 0 0 1 we conclude that there exists a unitary element v ∈ ℳ such that 𝜓 = 𝜇(𝜑 ◦ Ad(v)). We have thus proved the following. Proposition. Let ℳ be a factor of type III𝜆 (0 < 𝜆 < 1) and t = −2𝜋∕ ln(𝜆). There exists a 𝜆-trace on ℳ. For any two 𝜆-traces 𝜑, 𝜓 on ℳ, there exist a unitary element v ∈ ℳ and 𝜇 > 0 such that 𝜓 = 𝜇(𝜑 ◦ Ad(v)); in this case 𝜇 it = [D𝜓 ∶ D𝜑]t . □ In particular, for every 𝜆-trace 𝜑 on ℳ, there exists a unitary element u ∈ ℳ such that 𝜆𝜑 = 𝜑 ◦ Ad(u). 29.6. For type III0 factors, we first prove the existence of lacunary weights of infinite multiplicity. Proposition. Let ℳ be a factor of type III0 , 𝜑 a faithful normal strictly semifinite weight on ℳ and p ∈ ℳ 𝜑 a nonzero projection. There exist a nonzero projection e ∈ ℳ 𝜑 , e ≤ p, an element

392

Discrete Decompositions

a ∈ ℳ 𝜑 ∩ eℳe, a ≥ 0, which is invertible in eℳe, and a faithful normal lacunary state 𝜓 on eℳe such that a ∈ (eℳe)𝜓 and 𝜑e = 𝜓(a⋅). Proof. Since 𝜑 is strictly semifinite, 𝜑|ℳ 𝜑 is an n.s.f. trace on ℳ 𝜑 and we may hence assume that 𝜑(p) < +∞. We have Γ(𝜎 𝜑 p) = Γ(𝜎 𝜑 ) = S(ℳ) ∩ ℝ+∗ so that, by Corollary 2/16.2, there exist a nonzero projection e ∈ ℳ 𝜑 p = ℳ 𝜑 ∩ pℳp and 𝜀 > 0 such that (Sp 𝜎 𝜑e ) ∩ exp([−2𝜀, −𝜀] ∪ [𝜀, 2𝜀]) = ∅. Using Proposition 15.12, we obtain an element h ∈ ℳ 𝜑 ∩ eℳe, −𝜀∕2 ≤ h ≤ 𝜖∕2, such that 𝜑 for the action 𝜎 ∶ ℝ → Aut(eℳe) defined by 𝜎s (x) = e−ish 𝜎s e (x)eish (x ∈ eℳe, s ∈ ℝ) we have (Sp 𝜎) ∩ exp(−𝜀, 𝜀) = {1}. Then the element a = eh ∈ ℳ 𝜑 ∩ eℳe is positive and invertible in eℳe and 𝜓 = 𝜑e (a−1 ⋅) is a faithful normal state on eℳe with 𝜎 𝜓 = 𝜎, so that 𝜓 is lacunary. It is clear that a ∈ (eℳe)𝜑 and 𝜑e = 𝜓(a⋅). Corollary. On every factor ℳ of type III0 , there exists a lacunary weight of infinite multiplicity. Proof. Since every nonzero projection in ℳ is equivalent to 1, the previous proposition shows that ̄ tr is a lacunary weight of there exists a faithful normal lacunary weight 𝜓 on ℳ. Then 𝜑 = 𝜓 ⊗ ̄ ℱ∞ ≈ ℳ. infinite multiplicity on ℳ ⊗ 29.7 Proposition. Let 𝜑1 , 𝜑2 be lacunary weights of infinite multiplicity on the type III0 factor ℳ. Given 𝜀 > 0 there exist nonzero projections e1 ∈ 𝒵 (ℳ 𝜑1 ), e2 ∈ 𝒵 (ℳ 𝜑2 ) and a partial isometry v ∈ ℳ with v∗ v = e1 , vv∗ = e2 , such that the mapping x ↦ vxv∗ defines a *-isomorphism Φ ∶ (e1 ℳe1 , ℳ 𝜑1 ∩ e1 ℳe1 ) → (e2 ℳe2 , ℳ 𝜑2 ∩ e2 ℳe2 ) with the property that for every x ∈ e1 ℳe1 we have Sp𝜎 𝜑2 (Φ(x)) ⊂ Sp𝜎 𝜑1 (x) ⋅ exp[−𝜀, 𝜀], Sp𝜎 𝜑1 (x) ⊂ Sp𝜎 𝜑2 (Φ(x)) ⋅ exp[−𝜀, 𝜀]. Proof. Put 𝜎 𝜑1 = 𝜎1 , 𝜎 𝜑2 = 𝜎2 and consider the balanced weight 𝜑 = 𝜃(𝜑1 , 𝜑2 ) ( on 𝒫 x and the *-isomorphisms I1 ∶ ℳ → 𝒫 , I2 ∶ ℳ → 𝒫 defined by I1 (x) = 0 ( ) 0 0 , (x ∈ ℳ). 0 x With 𝜎 = 𝜎 𝜑 , we have 𝜎 ◦ Ik = Ik ◦ 𝜎k

(k = 1, 2).

=)Mat2 (ℳ) 0 , I2 (x) = 0

Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

393 (

Since 𝜑1 and 𝜑2 are of infinite multiplicity, the projections in 𝒫

𝜎,

𝜎

in particular 𝒫 is properly infinite. Let (( p1 = z𝒫 𝜎

1 0 0 0

))

1 0

0 0

((

𝜎

∈ 𝒵 (𝒫 ), p2 = z𝒫 𝜎

) ( 0 , 0

0 0

0 1

))

0 1

) are properly infinite

∈ 𝒵 (𝒫 𝜎 ).

We have Sp 𝜎 pk = Sp 𝜎k

(k = 1, 2),

and 1 is an isolated point in Sp 𝜎 pk (k = 1, 2). By Corollary 15.17, there exists a nonzero partial isometry w ∈ 𝒫 such that p1 ≥ q1 = w∗ w ∈ 𝒵 (𝒫 𝜎 ), p2 ≥ q2 = ww∗ ∈ 𝒵 (𝒫 𝜎 ) and (Sp𝜎 (w))(Sp𝜎 (w))−1 ⊂ exp[−𝜀, 𝜀]. We define ( r1 =

1 0

0 0

)

( q1 ,

r2 =

0 0

0 1

) q2 .

Each rk is a properly infinite projection in 𝒫 𝜎 and z𝒫 𝜎 (rk ) = qk pk = qk ; hence rk ∼ qk in 𝒫 𝜎 , so that there exists a partial isometry wk ∈ 𝒫 𝜎 with w∗k wk = rk and wk w∗k = qk (k = 1, 2). Then u = w∗2 ww1 ∈ 𝒫 , u∗ u = r1 , uu∗ = r2 and, since w1 , w2 ∈ 𝒫 𝜎 , (Sp𝜎 (u))(Sp𝜎 (u))−1 ⊂ exp[−𝜀, 𝜀]. On the other hand, it is easy to see that rk ∈ Ik (𝒵 (ℳ 𝜎k )), so that there exists ek ∈ 𝒵 (ℳ 𝜎k ) such that = Ik (ek ) (k = 1, 2). Since u∗ u = r1 , uu∗ = r2 , it follows that u is of the form ( rk ) 0 0 u = with v ∈ ℳ and v∗ v = e1 , vv∗ = e2 . For x ∈ e1 ℳe1 , we have Sp𝜎1 (x) = 0 v (( )) (( )( ) 0 0 0 0 x 0 ∗ ∗ Sp𝜎 (I1 (x)) and Sp𝜎2 (vxv ) = Sp𝜎 (I2 (vxv )) = Sp𝜎 = Sp𝜎 0 vxv∗ v 0 0 0 ( )) ∗ 0 v × = Sp𝜎 (uI1 (x)u∗ ). 0 0 The assertions of the proposition are now easily verified. 29.8 Proposition. Let 𝜑 be a faithful normal lacunary weight on the W ∗ -algebra ℳ. Then every maximal abelian *-subalgebra 𝒜 of ℳ 𝜑 is maximal abelian in ℳ. Proof. Put 𝜎 = 𝜎 𝜑 . By assumption, there exists 𝜀 > 0 such that (Sp 𝜎) ∩ exp[−𝜀, 𝜀] = {1}. Since 𝒜 ⊂ ℳ 𝜎 , 𝒜 ′ ∩ ℳ is a 𝜎-invariant vector subspace of ℳ. Assume that there exists x ∈ 𝒜 ′ ∩ ℳ, x ∉ ℳ 𝜑 . In view of Lemma 15.1, we may further assume that (Sp𝜎 (x))(Sp𝜎 (x))−1 ⊂ exp[−𝜀, 𝜀].

394

Discrete Decompositions

If 1 ∈ Sp𝜎 (x), then Sp𝜎 (x) ⊂ (Sp 𝜎) ∩ exp[−𝜀, 𝜀] = {1}, and x ∈ ℳ 𝜑 , which is not possible. Consequently, 1 ∉ Sp𝜎 (x). Since Sp𝜎 (x) ∩ exp[−𝜀, 𝜀) ⊂ (Sp𝜎) ∩ exp[−𝜀, 𝜀] = {1}, it follows that we have either Sp𝜎 (x) ⊂ exp(𝜀, +∞), or Sp𝜎 (x) ⊂ exp(−∞, −𝜀). By replacing if necessary x by x∗ , we may assume that Sp𝜎 (x) ⊂ exp(−∞, −𝜀). Let x = u|x| be the polar decomposition of x, with u∗ u = r(x), uu∗ = l(x). Since (Sp𝜎 (x))(Sp𝜎 (x))−1 ∩ (Sp 𝜎) = {1}, using Proposition 15.17.(2) we get Sp𝜎 (u) = Sp𝜎 (x) ⊂ exp(−∞, −𝜀), u ∈ 𝒜 ′ ∩ ℳ, and u∗ u, uu∗ ∈ 𝒜 ′ ∩ ℳ 𝜑 = 𝒜 . It follows that (u∗ u)u = u(u∗ u) = u and u(uu∗ ) = (uu∗ )u = u, so that u∗ u = u∗ uuu∗ = uu∗ . Since Sp𝜎 (u) is a compact subset of exp(−∞, −𝜀), there is a function f ∈ ℒ 1 (ℝ) such that u = 𝜎f (u), 0 ≤ ̂f ≤ 1, and supp ̂f ⊂ exp(−∞, −𝜀∕2). Then sup{𝛾 1∕2 ̂f (y); 𝛾 ∈ ℝ+ } ≤ exp(−𝜀∕4) and ∗ 1∕2 1∕2 1∕2 1∕2 (28.1.(3)) Δ𝜑 ̂f (Δ𝜑 )u𝜑 = Δ𝜑 (𝜎f (u))𝜑 = Δ𝜑 u𝜑 , whence 𝜑(uu∗ ) = ‖(u∗ )𝜑 ‖𝜑 = ‖Δ𝜑 u𝜑 ‖𝜑 ∗ ∗ ∗ ≤ exp(−𝜀∕4)‖u𝜑 ‖𝜑 = exp(−𝜀∕4)𝜑(u u), contradicting u u = uu . ∑ Note that if {ei } is a family of projections of the W ∗ -algebra ℳ with i ei = 1 and if, for each i, 𝒜i ∑ is a maximal abelian *-subalgebra in ei ℳei , then 𝒜 = ( i 𝒜i )w is a maximal abelian *-subalgebra ∑ in ℳ. Indeed, if x ∈ 𝒜 ′ ∩ ℳ, then x commutes with each ei , so that x = i ei xei and ei xei ∈ 𝒜i′ ∩ ei ℳei = 𝒜i . 29.9 Theorem. Let 𝜑 be a faithful normal stricly semifinite weight on the factor ℳ of type III𝜆 (0 ≤ 𝜆 < 1). There exists a maximal abelian *-subalgebra of ℳ contained in ℳ 𝜑 . In particular, (ℳ 𝜑 )′ ∩ ℳ = 𝒵 (ℳ 𝜑 ). Proof. Consider first the case 𝜆 > 0 and put t = −2𝜋∕ ln(𝜆). Since 𝜑 is strictly semifinite, there exists a family of mutually orthogonal nonzero projections {ei } ⊂ ℳ 𝜑 , with 𝜑(ei ) < +∞ and ∑ ∗ i ei = 1. For each i, 𝜑i = 𝜑ei is a faithful normal positive form on the W -algebra ℳi = eℳei and ei ∼ 1 in ℳ, so that ℳi ≈ ℳ and T (ℳi ) = T(ℳ) ∋ t. By Theorem 27.1, we see that there 𝜑 exists a positive element ai ∈ ℳi i = ℳ 𝜑 ∩ ℳi , invertible in ℳi , such that for 𝜓i = 𝜑i (ai ⋅) we have 𝜓i 𝜎t = 𝜄. Thus, 𝜓i is a lacunary faithful normal positive form on ℳi . Let 𝒜i be any maximal abelian 𝜑 *-subalgebra of ℳi i containing ai . Then 𝒜i is maximal abelian in ℳi , by Proposition 29.8, and for 𝜓 𝜑 every x ∈ 𝒜i we have xaiti = aiti x and 𝜎t i (x) = x, hence x ∈ ℳi i ⊂ ℳ 𝜑 . Using the last remark in ∑ Section 29.8, we conclude that 𝒜 = ( i 𝒜i )w ⊂ ℳ 𝜑 is a maximal abelian *-subalgebra of ℳ. If 𝜆 = 0, the desired conclusion can be obtained by the same arguments, using Proposition 29.6. Finally, if 𝒜 is a maximal abelian *-subalgebra of ℳ contained in ℳ 𝜑 and x ∈ (ℳ 𝜑 )′ ∩ ℳ, then x ∈ 𝒜 ′ ∩ ℳ = 𝒜 ⊂ ℳ 𝜑 , hence (ℳ 𝜑 )′ ∩ ℳ = 𝒵 (ℳ 𝜑 ). 29.10 Corollary. For every faithful normal strictly semifinite weight 𝜑 on a factor ℳ of type III𝜆 (0 ≤ 𝜆 < 1), there exists a unique faithful normal conditional expectation P𝜑 ∶ ℳ → ℳ 𝜑 . Proof. This follows from Theorem 29.9, Corollary 10.9, and Proposition 10.17. 29.11 Corollary. Let 𝜑, 𝜓 be n.s.f. weights on the factor ℳ of type III𝜆 (0 ≤ 𝜆 < 1). If 𝜑 is strictly semifinite, then we have ℳ 𝜑 ⊂ ℳ 𝜓 if and only if there exists a nonsingular positive self-adjoint operator A affiliated to 𝒵 (ℳ 𝜑 ), such that 𝜓 = 𝜑A .

Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

395

Proof. Let us = [D𝜓 ∶ D𝜑]s (s ∈ ℝ). If ℳ 𝜑 ⊂ ℳ 𝜓 , then x ∈ ℳ 𝜑 ⇒ x ∈ ℳ 𝜓 ⇒ x = 𝜎s𝜓 (x) = us 𝜎s𝜑 (x)u∗s = us xu∗s , and us ∈ (ℳ 𝜑 )′ ∩ ℳ = 𝒵 (ℳ 𝜑 ) (s ∈ ℝ), by Theorem 29.9, that is, 𝜓 = 𝜑A with A affiliated to 𝒵 (ℳ 𝜑 ). Conversely, if this condition holds, then us ∈ 𝒵 (ℳ 𝜑 ) (s ∈ ℝ), and hence x ∈ ℳ 𝜑 ⇒ 𝜎s𝜓 (x) = us 𝜎s𝜑 (x)u∗s = us xus∗ = x (s ∈ ℝ) ⇒ x ∈ ℳ 𝜓 . 29.12 Corollary. Let 𝜑 be a faithful normal strictly semifinite weight on a factor ℳ of type III𝜆 (0 < 𝜆 < 1), and let t = −2𝜋∕ ln(𝜆). The following statements are equivalent: (i) (ii) (iii) (iv) (v) (vi)

𝜎t𝜑 = 𝜄; Sp(Δ𝜑 ) = S(ℳ), that is, Sp 𝜎 𝜑 = Γ(𝜎 𝜑 ); ℳ 𝜑 is a factor; (ℳ 𝜑 )′ ∩ ℳ = ℂ ⋅ 1ℳ ; every n.s.f. weight 𝜓 on ℳ with ℳ 𝜑 ⊂ ℳ 𝜓 is proportional to 𝜑; ℳ 𝜑 is maximal among the semiftnite unital W ∗ -subalgebras 𝒩 ⊂ ℳ with the property that there exists a faithful normal conditional expectation P ∶ ℳ → 𝒩 .

Proof. The implication (i) ⇒ (ii) has already been proved in Section 29.5, the equivalence (ii) ⇔ (iii) follows from 16.4.(5), the equivalence (iii) ⇔ (iv) follows from Theorem 29.9, and the implication (iv) ⇒ (v) follows easily from Theorem 29.9 and Corollary 29.11. (v) ⇒ (vi). Let ℳ 𝜑 ⊂ 𝒩 with 𝒩 ⊂ ℳ a semifinite unital W ∗ -subalgebra, let P ∶ ℳ → 𝒩 be a faithful normal conditional expectation and let 𝜏 be an n.s.f. trace on 𝒩 . Then 𝜓 = 𝜏 ◦ P is an n.s.f. weight on ℳ with ℳ 𝜑 ⊂ 𝒩 ⊂ ℳ 𝜓 , so that 𝜓 is proportional to 𝜑 and therefore ℳ 𝜑 = 𝒩 = ℳ 𝜓 . (vi) ⇒ (i). Since t ∈ T (ℳ), there exists a nonsingular positive self-adjoint operator A affiliated to 𝒵 (ℳ 𝜑 ) such that 𝜓 = 𝜑A satisfies 𝜎i𝜓 = 𝜄. Then the weight 𝜑 is strictly semifinite, that is, there exists a faithful normal conditional expectation P ∶ ℳ → ℳ 𝜓 and ℳ 𝜓 is semifinite (10.9). By Corollary 29.11, we know that ℳ 𝜑 ⊂ ℳ 𝜓 ; assumption (vi) implies that ℳ 𝜑 = ℳ 𝜓 . Since the weight 𝜓 satisfies condition (i), it also satisfies condition (v), so that 𝜑 is proportional to 𝜓 and 𝜎t𝜑 = 𝜎t𝜓 = 𝜄. 29.13. Notes. The constructions and properties of factors of types III𝜆 (0 ≤ 𝜆 < 1) given in this section are due to Connes (1972, 1973a) (see also Araki, 1973; Takesaki, 1973, (𝜆 > 0)). An important feature of factors concerns the existence of almost periodic states. A faithful normal state 𝜑 on a W ∗ -algebra ℳ is called almost periodic if the modular operator Δ𝜑 is diagonalizable, that is, if the set of eigenvalues of Δ𝜑 is total in ℋ𝜑 (Connes, 1973a, 3.7.1; see Connes (1972, 1974d) for equivalent definitions). Clearly, any 𝜆-trace on a type III𝜆 factor is (almost) periodic. Using results of Krieger (1970b), Connes (1973a, 5.3.8; 1974d, 1.5) proved the existence of almost periodic faithful normal states on every factor of type III0 . Connes (1974d) also introduced the notion of a full factor (ℳ is a full W ∗ -algebra if Int(ℳ) is closed in Aut(ℳ)) and showed that there exist full factors of type III1 with faithful normal almost periodic states as well as factors of type III1 with no almost periodic state or weight. Moreover, Connes (1974d) exhibited a nonsmooth uncountable family of type III1 factors with separable preduals. While every factor of type III𝜆 with 0 ≤ 𝜆 < 1 arises from an essentially unique discrete decomposition (see §30), Connes (1974d) showed that there exist factors of type III1 with separable preduals which are isomorphic to no crossed product of a semifinite von Neumann algebra by a discrete abelian group. Connes (1979b) has shown that every type of factor can be obtained as von Neumann algebras associated with a foliation. For our exposition, we have used Connes (1973a).

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Discrete Decompositions

30 The Discrete Decomposition of Factors of Type III𝝀 (0 ≤ 𝝀 < 1) In this section, we show that every factor of type III𝜆 (0 ≤ 𝜆 < 1) has a certain type of discrete decomposition which is essentially unique; we relate the discrete to the continuous decomposition. As in the preceding section, all W ∗ -algebras are countably decomposable, either by assumption or by construction. 30.1 Theorem. Let 0 < 𝜆 < 1 and ℳ be factors of type III𝜆 . There exists a discrete decomposition (𝒩 , 𝜃, 𝜏) of type III𝜆 such that ℳ ≈ ℛ(𝒩 , 𝜃). If (𝒩1 , 𝜃1 , 𝜏1 ) and (𝒩2 , 𝜃2 , 𝜏2 ) are two discrete decompositions of type III𝜆 such that ℛ(𝒩1 , 𝜃1 ) ≈ ℛ(𝒩2 , 𝜃2 ), then (𝒩1 , 𝜃1 ) ≈ (𝒩2 , 𝜃2 ). Proof. Let t = −2𝜋∕ ln(𝜆). By Proposition 29.5, there exist a 𝜆-trace 𝜑 on ℳ and a unitary element u ∈ ℳ such that 𝜆𝜑 = 𝜑 ◦ Ad(u). Let 𝒩 = ℳ 𝜑 and let P = P𝜑 ∶ ℳ → 𝒩 be the faithful normal t conditional expectation defined by 𝜑 ∶ P𝜑 (x) = t−1 ∫0 𝜎s𝜑 (x) ds (x ∈ ℳ). 𝜑 ∗ Since 𝜆𝜑 = 𝜑 ◦ Ad(u), we have (3.7) u 𝜎s (u) = [D(𝜑 ◦ Ad(u)) ∶ D𝜑]s = 𝜆is , whence 𝜎s𝜑 (u) = is 𝜆 u (s ∈ ℝ), that is, u ∈ ℳ(𝜎 𝜑 ; {𝜆}) and un ∈ ℳ(𝜎 𝜑 ; {𝜆n }) for all n ∈ ℤ. Since 𝜎t𝜑 = 𝜄, it follows by Corollary 29.12 that Sp 𝜎 𝜑 = {𝜆n ; n ∈ ℤ}, so that there exists an open covering {Vn }n∈ℤ of ℝ+∗ such⋃that Vn ∩ (Sp 𝜎 𝜑 ) = {𝜆n } for each n ∈ ℤ. Using Proposition 14.3.(4), we infer that ℳ = ℛ{ n∈ℤ ℳ(𝜎 𝜑 ; {𝜆n })}. Let x ∈ ℳ(𝜎 𝜑 ; {𝜆n }). As u−n ∈ ℳ(𝜎 𝜑 ; {𝜆−n }), it follows that 𝜑 n n xu−n ∈ ℳ 𝜑 = 𝒩 ( (15.3.(2)). Thus ) ℳ(𝜎 ; {𝜆 }) = u 𝒩 and ℳ = ℛ{𝒩 , u}. Also, for every n ≠ 0 t we have P(un ) = t−1 ∫0 𝜆ins ds un = 0. Finally, since u ∈ ℳ(𝜎 𝜑 ; {𝜆}) and u∗ ∈ ℳ(𝜎 𝜑 ; {𝜆−1 }), for x ∈ 𝒩 = ℳ 𝜑 , we have uxu∗ ∈ ℳ 𝜑 = 𝒩 (15.3.(2)). We thus obtain a *-automorphism 𝜃 = Ad(u)|𝒩 ∈ Aut(𝒩 ) and conclude, using Proposition 22.2, that ℳ ≈ ℛ(𝒩 , 𝜃). By Corollary 29.12, 𝒩 = ℳ 𝜑 is a factor. Since 𝜑 is strictly semifinite, its restriction 𝜏 = 𝜑|𝒩 is an n.s.f. trace on 𝒩 with 𝜏(1) = 𝜑(1) = +∞ and 𝜏 ◦ 𝜃 = (𝜑 ◦ Ad(u))|𝒩 = 𝜆𝜑|𝒩 = 𝜆𝜏. Thus, 𝒩 is a factor of type II∞ and (𝒩 , 𝜃, 𝜏) is a discrete decomposition of type III𝜆 . Consider now two discrete decompositions (𝒩1 , 𝜃1 , 𝜏1 ), (𝒩2 , 𝜃2 , 𝜏2 ) of type III𝜆 with ℛ(𝒩1 , 𝜃1 ) = ̄ 𝜆 (1) in the two crossed ℛ(𝒩2 , 𝜃2 ) = ℳ; let u1 , u2 be the unitary elements corresponding to 1 ⊗ ∗ products and, identifying 𝒩1 and 𝒩2 with the corresponding W -subalgebras of ℳ, denote by P1 ∶ ℳ → 𝒩1 , P2 ∶ ℳ → 𝒩2 the canonical faithful normal conditional expectations. We know (29.1) that the dual weights 𝜑1 = 𝜏1 ◦ P1 , 𝜑2 = 𝜏2 ◦ P2 are 𝜆-traces on ℳ and 𝒩1 = ℳ 𝜑1 , 𝒩2 = ℳ 𝜑2 . Using Proposition 29.5 and modifying, if necessary, the trace 𝜏2 , we obtain a unitary element u ∈ ℳ such that 𝜑2 = 𝜑1 ◦ Ad(u) and, in particular, 𝒩2 = [Ad(u)](𝒩1 ). Since 𝜆𝜑1 = 𝜑1 ◦ Ad(u1 ), 𝜆𝜑2 = 𝜑2 ◦ Ad(u2 ) (see 29.5), it follows that 𝜆𝜑2 = 𝜑2 ◦ Ad(u∗ u1 u). Thus, u2 and u∗ u1 u belong to the spectral subspace ℳ(𝜎 𝜑1 ; {𝜆}) and hence v = uu∗1 u∗ u2 ∈ ℳ 𝜑1 = 𝒩2 . Since 𝜃1 = Ad(u1 )|𝒩1 and 𝜃2 = Ad(u2 )|𝒩2 , it follows that [(Ad(u))𝜃1−1 (Ad(u))−1 ]𝜃2 = Ad(v) ∈ Int(𝒩2 ). Finally, using Theorem 23.13, we conclude that (𝒩1 , 𝜃1 ) ≈ (𝒩2 , 𝜃2 ). The discrete decomposition (𝒩 , 𝜃, 𝜏) of type III𝜆 such that ℛ(𝒩 , 𝜃) ≈ ℳ will be called the discrete decomposition of the type III𝜆 factor ℳ (0 < 𝜆 < 1). The previous proof and Proposition 29.1 show that the 𝜆-traces on ℳ are the weights dual to the traces appearing in the discrete decomposition of ℳ. The previous theorem can also be proved, using Landstad’s theorem and the Takesaki duality theorem as in the proof of Theorem 23.6, by factorizing the action 𝜎 𝜑 ∶ ℝ → Aut(ℳ) with 𝜎t𝜑 = 𝜄 to get an action 𝜎 ∶ 𝕋 → Aut(ℳ).

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

397

30.2. For a factor ℳ of type III0 , it is natural to start with a lacunary weight 𝜑 of infinite multiplicity on ℳ and to take 𝒩 = ℳ 𝜑 , 𝜏 = 𝜑|𝒩 , but the procedure for finding a unitary element u ∈ ℳ which implements a convenient *-automorphism 𝜃 = Ad(u)|𝒩 is considerably more complicated. We consider separately this part of the proof in the following. Lemma. Let 𝜎 ∶ ℝ → Aut(ℳ) be a continuous action of ℝ on the factor ℳ. Assume that ℳ 𝜎 is properly infinite, that 1 is an isolated point in Sp 𝜎 ⊂ ℝ+∗ and that 𝜎 is not outer conjugate with the trivial action. Then there exists a unitary element u ∈ ℳ such that ℳ = ℛ{ℳ 𝜎 , u}, uℳ 𝜎 u∗ = ℳ 𝜎 , and Sp𝜎 (u) ⊂ (1, +∞). ̂ of ℝ Proof. For the proof it is more convenient to use the natural identification of the dual group ℝ with ℝ itself, rather than ℝ+∗ , as in the statement. Thus, by assumption, there exists 𝜀 > 0 such that (Sp 𝜎) ∩ [−𝜀, 𝜀] = {0}. Let 𝒱 be the set of all partial isometries v ∈ ℳ with the properties v∗ v ∈ 𝒵 (ℳ 𝜎 ), vv∗ ∈ 𝒵 (ℳ 𝜎 ), vℳ 𝜎 v∗ ⊂ ℳ 𝜎 , v∗ ℳ 𝜎 v ⊂ ℳ 𝜎 . It is easy to check that

v1 , v2 ∈ 𝒱 ,

v1 , v2 ∈ 𝒱 ∗ v1 v1 ⟂ v∗2 v2 ,

⇒ v1 v2 ∈ 𝒱 , v1 v∗1 ⟂ v∗2 v2 ⇒ v1 + v2 ∈ 𝒱 .

Putting e = v∗ v, e′ = vv∗ , e1 = v∗1 v1 , e′1 = v1 v∗1 , e2 = v∗2 v2 , e′2 = v2 v∗2 , we have v = v1 v2 ⇒ e = v∗2 e1 v2 , e′ = v1 e′2 v∗1 v = v1 + v2 ⇒ e = e1 + e2 , e′ = e′1 ≠ e′2 , and we shall write v1 ≺ v2 ⇔ e1 ≤ e2

and

v1 = v2 e1 ;

in this case we also have v1 = e′1 v2 . Note that v1 ≺ v2 ⇔ v∗1 ≺ v∗2 ,

vv1 ≺ vv2 ,

v1 v ≺ v2 v.

We shall consider the sets 𝒱0 = {v ∈ 𝒱 ; Sp𝜎 (v) ⊂ (𝜀, +∞), Sp𝜎 (v) − Sp𝜎 (v) ⊂ [−𝜀∕2, 𝜀∕2]} 𝒱1 = {v ∈ 𝒱0 ; v1 , v2 ∈ 𝒱0 , v1 v2 ≺ v ⇒ v1 v2 = 0} and establish five properties for them. (I) For each 0 ≠ v ∈ 𝒱0 there exist v1 , … , vp ∈ 𝒱1 such that 0 ≠ v1 … vp ≺ v. We prove this property by induction over n, assuming at step n that Sp𝜎 (v) ⊂ (0, n𝜀). For n = 1, we have {v ∈ 𝒱0 ; Sp𝜎 (v) ⊂ (0, 𝜀)} = ∅. Suppose that the property holds for all elements of 𝒱0 with spectrum contained in (0, n𝜀). Let 0 ≠ v ∈ 𝒱0 be such that Sp𝜎 (v) ⊂ (0, (n + 1)𝜀). If v ∈ 𝒱1 , then the property is satisfied with p = 1 and v1 = v, so that we may assume that v ∉ 𝒱1 . Then there exist w1 , w2 ∈ 𝒱0 with w1 w2 ≺ v but w1 w2 ≠ 0. Let e = w∗1 w1 and w3 = ew2 . We have w3 ∈ 𝒱0

398

Discrete Decompositions

since e ∈ ℳ 𝜎 and w1 w2 = w1 ew2 = w1 w3 , w3 w∗3 = ew2 w∗2 e ≤ e = w∗1 w1 , w3 = w∗1 w1 w2 = w∗1 fv, where f = (w1 w2 )(w1 w2 )∗ . Since f ∈ ℳ 𝜎 and w1 ∈ 𝒱0 , we have Sp𝜎 (w3 ) ⊂ Sp𝜎 (v) − Sp𝜎 (w1 ) ⊂ (0, (n + 1)𝜀) − (𝜀, +∞) ⊂ (0, n𝜀). Since w3 ∈ 𝒱0 and Sp𝜎 (w3 ) ⊂ (0, n𝜀), by the induction hypothesis there exist v1 , … , vp ∈ 𝒱1 such that 0 ≠ v1 … vp ≺ w3 . We have 𝒵 (ℳ 𝜎 ) ∋ f ′ = v1 … vp v∗p … v∗1 ≤ e and 0 ≠ w1 (v1 … vp ) ≺ w1 w3 = w1 w2 ≺ v, that is, w1 f ′ v1 … vp = f ′′ v with f ′′ ∈ 𝒵 (ℳ 𝜎 ), f ′′ ≤ vv∗ , hence w4 = w1 f ′ = f ′′ v(v1 … vp )∗ . Since f ′′ ∈ ℳ 𝜎 and v1 , … , vp ∈ 𝒱0 , we have Sp𝜎 (w4 ) ⊂ ∑p Sp𝜎 (v) − j=1 Sp𝜎 (vj ) ⊂ (0, n𝜀) and since f ′ ∈ ℳ 𝜎 and w1 ∈ 𝒱0 , we have w4 ∈ 𝒱0 . Again by the induction hypothesis, we can find vp+1 , … , vp+q ∈ 𝒱1 such that 0 ≠ vp+1 … vp+q ≺ w4 = w1 f ′ . Since r(vp+1 … vp+q ) ≤ f ′ = l(v1 … vp ), we have 0 ≠ (vp+1 … vp+q )(v1 … vp ) ≺ w1 w3 = w1 w2 ≺ v. (II) If u ∈ 𝒱 , Sp𝜎 (u) ⊂ (𝜀, +∞), and Sp𝜎 (u) − Sp𝜎 (u) ⊂ [−𝜀, 𝜀], then there exists 0 ≠ w ∈ 𝒱0 such that w ≺ u. By Lemma 15.1, there exists h ∈ ℒ 1 (ℝ) with supp ĥ − supp ĥ ⊂ [−𝜀∕2, 𝜀∕2] such that (𝜎h (u))∗ u ≠ 0. By Proposition 15.17, there exist v ∈ 𝒱 and a ∈ ℳ 𝜎 such that va = 𝜎h (u) and ̂ ∩ Sp𝜎 (v) ⊂ Sp𝜎 (𝜎h (u)). Since a∗ v∗ u = (𝜎h (u))∗ u ≠ 0, we have v∗ u ≠ 0 and, since Sp𝜎 (v) ⊂ (supp h) ∗ Sp𝜎 (u), it follows that v ∈ 𝒱0 . Then, Sp𝜎 (v u) ⊂ (Sp𝜎 (u) − Sp𝜎 (u)) ∩ (Sp 𝜎) ⊂ [−𝜀, 𝜀] ∩ (Sp 𝜎) = {0}, hence v∗ u ∈ ℳ 𝜎 . Let w = vv∗ u. We have w ≠ 0 since v∗ w = v∗ u ≠ 0, then w ∈ 𝒱0 since v ∈ 𝒱0 , v∗ u ∈ ℳ 𝜎 and u, v ∈ 𝒱 , and finally w ≺ u since vv∗ ∈ 𝒵 (ℳ 𝜎 ). (III) If v1 , v2 ∈ 𝒱1 , then v∗1 v2 , v1 v∗2 ∈ ℳ 𝜎 . We show that v∗1 v2 ∈ ℳ 𝜎 . We have Sp𝜎 (v∗1 v2 ) − Sp𝜎 (v∗1 v2 ) ⊂ Sp𝜎 (v2 ) − Sp𝜎 (v1 ) + Sp𝜎 (v1 ) − Sp𝜎 (v2 ) ⊂ [−𝜀∕2, 𝜀∕2] + [−𝜀∕2, 𝜀∕2] = [−𝜀, 𝜀]. Since (Sp 𝜎) ∩ [−𝜀, 𝜀] = {0}, it follows that either Sp𝜎 (v∗1 v2 ) = {0}, that is, v∗1 v2 ∈ ℳ 𝜎 , or Sp𝜎 (v∗1 v2 ) ⊂ (𝜀, +∞), or Sp(v∗2 v1 ) ⊂ (𝜀, +∞). Thus, we have to show that the inclusion Sp(v∗1 v2 ) ⊂ (𝜀, +∞) leads to a contradiction. Consider u = v∗1 v2 ∈ 𝒱 with Sp𝜎 (u) ⊂ (𝜀, +∞) and Sp𝜎 (u) − Sp𝜎 (u) ⊂ [−𝜀, 𝜀]. By property (II), there exists 0 ≠ w ∈ 𝒱0 with w ≺ u. We have v1 w ≺ v1 u = v1 v∗1 v2 ≺ v2 and ww∗ ≤ uu∗ ≤ v∗1 v1 , hence v1 w ≠ 0, contradicting v2 ∈ 𝒱1 . Similarly, in order to show that u = v1 v∗2 ∈ ℳ 𝜎 , it is sufficient to show that the inclusion Sp𝜎 (u) ⊂ (𝜀, +∞) leads to a contradiction. As above, we can find using property (II) 0 ≠ w ∈ 𝒱0 , w ≺ u, and we have wv2 ≺ uv2 = v1 v∗2 v2 ≺ v1 and w∗ w ≤ u∗ u ≤ v2 v∗2 , so that wv2 ≠ 0, contradicting v1 ∈ 𝒱1 . (IV) ℳ = ℛ{ℳ 𝜎 , 𝒱1 }. Let x ∈ ℳ. In order to show that x ∈ ℛ{ℳ 𝜎 , 𝒱1 }, we may assume that Sp𝜎 (x) − Sp𝜎 (x) ⊂ [−𝜀∕2, 𝜀∕2] (see Lemma 15.1). Since, by assumption, (Sp 𝜎) ∩ [−𝜀, 𝜀] = {0}, we may assume further that Sp𝜎 (x) ⊂ (𝜀, +∞). Using Proposition 15.17, we can write x = va with v ∈ 𝒱0 , a ∈ ℳ 𝜎 . Thus it remains to be shown that 𝒱0 ⊂ ℛ{ℳ 𝜎 , V1 }. Let v ∈ 𝒱0 and e = v∗ v. From property (I), it follows that there exists a projection f ∈ 𝒵 (ℳ 𝜎 ), 0 ≠ f ≤ e with vf ∈ ℛ{ℳ 𝜎 , 𝒱1 }. By a standard maximality argument, we infer that v = ve ∈ ℛ{ℳ 𝜎 , 𝒱1 }. ∑ (V) If {vj } ⊂ 𝒱1 is a maximal family with the property that vj v∗k = v∗j vk = 0 (k ≠ j), then j vj is unitary. ∑ ∑ Let ej = v∗j vj , e′j = vj v∗j , e = j ej , e′ = j e′j ; note that ej , e′j , e, e′ ∈ 𝒵 (ℳ 𝜎 ). Assume that 1 − e ≠ 0. Using the assumption and Theorem 16.6, we see that the action 𝜎 1−e is not trivial, so that there exists y ∈ (1 − e)ℳ(1 − e) with Sp𝜎 (y) ≠ {0}. Since (Sp 𝜎) ∩ [−𝜀, 𝜀] = {0}, we infer using Lemma 15.1 that there exists x ∈ (1 − e)ℳ(1 − e) with Sp𝜎 (x) ⊂ (𝜀, +∞) and Sp𝜎 (x) − Sp𝜎 (x) ⊂ [−𝜀∕2, 𝜀∕2]. Furthermore, using Proposition 15.17, we deduce that there exists 0 ≠ v ∈ 𝒱0 with v∗ v ≤ 1 − e, vv∗ ≤ 1 − e and, finally, using property (I), we find an element 0 ≠ w ∈ 𝒱1 with w∗ w ≤ 1 − e. It is clear that vj w∗ = 0 for all j. By property (III), it follows that w∗ vj ∈ ℳ 𝜎 ; then r(w∗ vj ) ≤ ej , l(w∗ vj ) ≤ 1 − e ≤ 1 − ej , and, since ej ∈ 𝒵 (ℳ 𝜎 ), we deduce that w∗ vj = 0. The existence of w contradicts the maximality of the family {vj }. Hence e = 1.

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

399

Assuming that 1 − e′ ≠ 0, we obtain similarly an element 0 ≠ w ∈ 𝒱1 with ww∗ ≤ 1 − e′ ; we deduce that v∗j w = 0 = wv∗j for all j, which contradicts the maximality of the family {vj }. Hence e′ = 1. We are now in a position to finish the proof of the lemma. Let {vj } ⊂ 𝒱1 be a maximal family with ∑ the property that vj v∗k = v∗j vk = 0, (j ≠ k), and let u = j vj . By (V) we know that u ∈ ℳ is unitary. Since vj ∈ 𝒱 , we have uℳ 𝜎 u∗ = ℳ 𝜎 . Since Sp𝜎 (vj ) ⊂ [𝜀, +∞), it follows that Sp𝜎 (u) ⊂ [𝜀, +∞) ⊂ (0, +∞). By (III), for every v ∈ 𝒱1 we have v∗ vj ∈ ℳ 𝜎 , hence v∗ u ∈ ℳ 𝜎 , that is, v ∈ uℳ 𝜎 and by (IV) this implies that ℳ = ℛ{ℳ 𝜎 , u}. Note that u∗ ∈ ℳ is unitary, ℳ = ℛ{ℳ 𝜎 , u}, u∗ ℳ 𝜎 u = ℳ 𝜎 , and Sp𝜎 (u∗ ) ⊂ (0, 1). 30.3 Theorem. Let ℳ be a factor of type III0 . There exists a discrete decomposition (𝒩 , 𝜃, 𝜏) of type III0 such that ℳ ≈ ℛ(𝒩 , 𝜃). If (𝒩1 , 𝜃1 , 𝜏1 ) and (𝒩2 , 𝜃2 , 𝜏2 ) are two discrete decompositions of type III0 such that ℛ(𝒩1 , 𝜃1 ) ≈ ℛ(𝒩2 , 𝜃2 ), there exist nonzero projections e1 ∈ 𝒵 (𝒩1 ) and e2 ∈ 𝒵 (𝒩2 ) such (𝒩1 e1 , 𝜃1e1 ) ≈ (𝒩2 e2 , 𝜃2e2 ). Proof. By Corollary 29.6, there exists a lacunary weight 𝜑 on ℳ of infinite multiplicity. Since ℳ is of type III, the action 𝜎 = 𝜎 𝜑 ∶ ℝ → Aut(ℳ) is not outer conjugate to the trivial action. By Lemma 30.2, there exists a unitary element u ∈ ℳ such that uℳ 𝜑 u∗ = ℳ 𝜑 , ℳ = ℛ{ℳ 𝜑 , u}, and Sp𝜎 (u) ⊂ (0, 1). We define 𝒩 = ℳ 𝜑 , 𝜃 = Ad(u)|𝒩 , and 𝜏 = 𝜑|𝒩 . Since 𝜑 is strictly semifinite, there exists a faithful normal conditional expectation P ∶ ℳ → 𝒩 , 𝜏 is an n.s.f. trace on 𝒩 and 𝜑 = 𝜏 ◦ P. Let A be the unique nonsingular positive self-adjoint operator affiliated to 𝒵 (𝒩 ) such that 𝜏 ◦ 𝜃 = 𝜏A . Then, by 27.4.(2), we have 𝜎s𝜑 (u) = uAis (s ∈ ℝ), and for every f ∈ ℒ 1 (ℝ) we obtain (27.1.(1)) 𝜎f𝜑 (u) = ûf (A). It follows that Sp(A) = Sp𝜎 (u) ⊂ (0, 1), so that there is 0 < 𝜆 < 1, such that A ≤ 𝜆, and 𝜏 ◦ 𝜃 ≤ 𝜆𝜏. It follows that the action 𝜃 ∶ ℤ → Aut(𝒩 ) is properly outer, that is, p(𝜃 n ) = 0 for every n ≠ 0. Indeed, let n > 0 and e ∈ 𝒵 (𝒩 ) be a projection such that 𝜃 n (e) = e and 𝜃 n |e𝒩 e ∈ Int(e𝒩 e). Then, for every x ∈ (e𝒩 e)+ , we have 𝜏(x) = 𝜏(𝜃 n (x)) ≤ 𝜆n 𝜏(x), which is not possible, since 𝜆 < 1 and 𝜏|e𝒩 e is an n.s.f. trace on e𝒩 e. Using Proposition 22.2 and the last remark in Section 22.2, we conclude that ℳ ≈ ℛ(𝒩 , 𝜃). The W ∗ -algebra 𝒩 = ℳ 𝜑 is semifinite and properly infinite, since 𝜑 is strictly semifinite and of infinite multiplicity. The *-automorphism 𝜃 acts ergodically on 𝒵 (𝒩 ) since, if z ∈ 𝒵 (𝒩 ) and uz = zu, then z ∈ 𝒵 (ℛ{𝒩 , u}) = 𝒵 (ℳ), and ℳ is a factor. The center 𝒵 (𝒩 ) of 𝒩 is diffuse since otherwise, by Proposition 16.10 and Theorem 5.1, we could find an n.s.f. weight 𝜓 on ℳ with Sp 𝜎 𝜓 = Γ(𝜎 𝜓 ) = Γ(𝜎 𝜑 ) = {1} (as ℳ is of type III0 ), that is, 𝜎 𝜓 would be the trivial action, which contradicts the fact that ℳ is of type III. We now prove that 𝒩 is of type II∞ . Since the splitting of 𝒩 into types I∞ and II∞ is given by a 𝜃-invariant projection in 𝒵 (𝒩 ) and since 𝜃 acts ergodically on 𝒵 (𝒩 ), it is sufficient to assume ̄ ℱ∞ where, that 𝒩 is of type I∞ and reach a contradiction. If 𝒩 is of type I∞ , then 𝒩 = 𝒵 (𝒩 ) ⊗ as usually, ℱ∞ stands for the countably decomposable type I∞ factor. Consider the *-automorphism ̄ 𝜄. Then 𝜃 −1 𝜃 ′ acts identically on 𝒵 (𝒩 ) and hence 𝜃 ′ ∈ Aut(𝒩 ) defined by 𝜃 ′ = (𝜃|𝒵 (𝒩 )) ⊗ ′ is inner ([L], 8.11), that is, 𝜃 = 𝜃 ◦ Ad(v) for some v ∈ U(𝒩 ). It follows that 𝜏 ◦ 𝜃 ′ ≤ 𝜆𝜏 ′ and ℳ ≈ ℛ(𝒩 , 𝜃 ′ ). By Corollary 23.9, there exists a continuous action 𝛼 ∶ 𝕋 → Aut(𝒩 𝜃 ) ′ ′ 𝜃 such that (𝒩 , 𝜃 ) ≈ (ℛ(𝒩 , 𝛼), 𝛼) ̂ and, by the Takesaki duality theorem (19.5) we infer that ′ ̄ ℱ∞ = 𝒵 (𝒩 )𝜃 ⊗ ̄ ℱ∞ ⊗ ̄ ℱ∞ ≈ ℱ∞ , which is impossible, since ℳ ℳ ≈ ℛ(𝒩 , 𝜃 ′ ) ≈ 𝒩 𝜃 ⊗ is of type III.

400

Discrete Decompositions

Thus, (𝒩 , 𝜃, 𝜏) is a discrete decomposition of type III0 . Consider now two discrete decompositions of type III0 (𝒩1 , 𝜃1 , 𝜏1 ) and (𝒩2 , 𝜃2 , 𝜏2 ) with ℛ(𝒩1 , 𝜃1 ) = ℛ(𝒩2 , 𝜃2 ) = ℳ and, identifying 𝒩1 and 𝒩2 with the corresponding W ∗ -subalgebras of ℳ, let P1 ∶ ℳ → 𝒩1 and P2 ∶ ℳ → 𝒩2 be the canonical faithful normal conditional expectations. By Proposition 29.3, the dual weights 𝜑1 = 𝜏1 ◦ P1 and 𝜑2 = 𝜏2 ◦ P2 are lacunary, of infinite multiplicity on ℳ, and 𝒩1 = ℳ 𝜑1 , 𝒩2 = ℳ 𝜑2 . If 𝜏1 ◦ 𝜃1 ≤ 𝜆1 𝜏1 and 𝜏2 ◦ 𝜃2 ≤ 𝜆2 𝜏2 with 𝜆1 , 𝜆2 ∈ (0, 1), then we can find 𝜀 > 0 such that 𝜆1 e𝜀 < 1 and 𝜆2 e𝜀 < 1. By Proposition 29.7, there exist nonzero projections e1 ∈ 𝒵 (𝒩1 ), e2 ∈ 𝒵 (𝒩2 ), and a *-isomorphism Φ ∶ (e1 ℳe1 , 𝒩1 e1 ) → (e2 ℳe2 , 𝒩2 e2 )

(1)

such that, putting 𝜎1 = 𝜎 𝜑1 , 𝜎2 = 𝜎 𝜑2 , we have Sp𝜎2 (Φ(x)) ⊂ Sp𝜎1 (x) ⋅ exp[−𝜀, 𝜀] (x ∈ e1 ℳe1 ).

(2)

By Proposition 29.4, we have e1 ℳe1 ≈ ℛ(𝒩1 e1 , 𝜃1e1 ), the canonical image of 𝒩1 e1 in the crossed product corresponding to the W ∗ -subalgebra 𝒩1 e1 of e1 ℳe1 , and there is a unitary element u1 ∈ e1 ℳe1 such that e1 ℳe1 = ℛ{𝒩1 e1 , u1 }, 𝜃1e1 = Ad(u1 )|𝒩1 u1 ,

(3) (4)

Sp𝜎1 (u1 ) ⊂ (0, 𝜆1 ).

(5)

Then v2 = Φ(u1 ) ∈ e2 ℳe2 is a unitary element and from (3) and (1) it follows that e2 ℳe2 = ℛ{𝒩2 e2 , v2 },

(6)

v2 (𝒩2 e2 )v∗2 = 𝒩2 e2 ,

(7)

Using (4) and (1), we obtain

and from (5), (2) and the choice of 𝜀, it follows that Sp𝜎2 (v2 ) ⊂ (0, 1).

(8)

Since, by Proposition 29.4, (𝒩2 e2 , 𝜃2e2 , 𝜏2e2 ) is a discrete decomposition of type III0 and e2 ℳe2 ≈ ℛ(𝒩2 e2 , 𝜃2e2 ), using statement 29.3.(3), we infer from (6), (7), and (8) the existence of a unitary element w2 ∈ 𝒩2 e2 such that the unitary element u2 = w2 v2 ∈ e2 ℳe2 satisfies the equality 𝜃2e2 = Ad(u2 )|𝒩2 e2 . We have thus obtained a *-isomorphism Φ ∶ (𝒩1 e1 , 𝜃1e1 ) → (𝒩2 e2 , 𝜃2′ ), where 𝜃2′ = Ad(v2 )|𝒩2 e2 and (𝜃2′ )−1 𝜃2e2 = Ad(w) ∈ Int(𝒩2 e2 ). Using Theorem 23.13, we conclude that (𝒩1 e1 , 𝜃1e1 ) ≈ (𝒩2 e2 , 𝜃2e2 ). A discrete decomposition (𝒩 , 𝜃, 𝜏) of type III0 such that ℛ(𝒩 , 𝜃) ≈ ℳ will be called a discrete decomposition of the type III0 factor ℳ. For every nonzero projection e ∈ 𝒵 (𝒩 ), (𝒩 e, 𝜃e , 𝜏e ) is also a discrete decomposition of ℳ. The previous proof and Proposition 29.3 show that the lacunary weights of infinite multiplicity on ℳ are those dual to the traces appearing in some discrete decomposition of ℳ.

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

401

30.4. In this section, we recapitulate the construction which leads to the discrete decomposition of type III factors (0 ≤ 𝜆 < 1) and establish the notation for the following sections, where we shall relate the continuous decomposition to the discrete decomposition. The results that follow have already been proved for type III0 and can be easily checked, with similar arguments, also for type III𝜆 with 𝜆 ≠ 0. Let ℳ be a factor of type III𝜆 (0 ≤ 𝜆 < 1). We choose a lacunary weight 𝜑 of infinite multiplicity on ℳ (29.5, 29.6). Then 𝒩 = ℳ𝜑

(1)

is a type II∞ W ∗ -algebra (30.3). If 𝜆 = 0, then the center 𝒵 (𝒩 ) of 𝒩 is diffuse (30.3). If 𝜆 > 0, then we can choose 𝜑 so that 𝒩 is a factor (30.1). In both cases, we have (29.9) 𝒩 ′ ∩ ℳ = 𝒵 (𝒩 ).

(2)

There exists unique faithful normal conditional expectation (10.9, 10.17) P∶ℳ →𝒩. Also (10.9) 𝜏 = 𝜑|𝒩 is an n.s.f. trace on 𝒩 such that 𝜑 = 𝜏 ◦ P.

(3)

There exists a unitary element u ∈ ℳ such that (30.2) ℳ = ℛ{𝒩 , u}, u𝒩 u∗ = 𝒩 , Sp𝜎 𝜑 (u) ⊂ (0, 1).

(4) (5) (6)

𝜃 = Ad(u)|𝒩 ∈ Aut(𝒩 ), 𝜏 ◦ 𝜃 ≤ 𝜆0 𝜏.

(7) (8)

Let 𝜆0 = sup Sp𝜎 𝜑 (u) < 1. Then (30.3)

For each n ∈ ℤ, there exists a unique nonsingular positive self-adjoint operator An , affiliated to 𝒵 (𝒩 ), such that 𝜏 ◦ 𝜃 n = 𝜏An .

(9)

Am+n = Am 𝜃 −m (An ), n > 0 ⇒ An ≤ 𝜆n0 < 1,

(10) (11)

Put A = A1 . It is easy to check that

402

Discrete Decompositions n < 0 ⇒ An ≥ 𝜆n0 > 1, 𝜑 ◦ (Ad(u))n = 𝜑An ,

(12) (13)

𝜎s𝜑 (un ) = un Aisn , Sp(A) = Sp𝜎 𝜑 (u).

(14) (15)

If v ∈ ℳ is another unitary element satisfying conditions (4), (5), (6), and 𝜃 ′ = Ad(v)|𝒩 , then (29.3.(3)) u∗ v ∈ 𝒩 , hence 𝜃 −1 𝜃 ′ ∈ Int(𝒩 ) and 𝜏 ◦ 𝜃 n = 𝜏 ◦ 𝜃 ′n ; thus, the corresponding operators An remain unchanged. Recall that if H and K are positive self-adjoint operators, then we write H < K if H ≤ K and s((K − H)s(K )) = s(K ). 30.5. Let ℳ be a factor of type III𝜆 (0 ≤ 𝜆 < 1) and let 𝜑 be a lacunary weight of infinite multiplicity on ℳ. We use the notation introduced in Section 30.4. Theorem. Let a ∈ 𝒩 + with As(a) ≤ a < 1 and 𝜓 = 𝜑a = 𝜏a ◦ P, a′ ∈ 𝒩 + with As(a′ ) ≤ a′ < 1 and 𝜓 ′ = 𝜑a′ = 𝜏a′ ◦ P. If v ∈ ℳ, vv∗ = s(𝜓), v∗ v = s(𝜓 ′ ), and 𝜓 ′ = 𝜓v , then v ∈ 𝒩 . In particular, if 𝜑a ≂ 𝜑a′ in ℳ, then 𝜏a ≂ 𝜏a′ in 𝒩 . Proof. Let B = a + A(1 − s(a)), B ′ = a′ + A(1 − s(a′ )). Then B and B ′ are nonsingular positive self-adjoint operators affiliated to 𝒩 and A ≤ B < 1, A ≤ B ′ < 1. Since 𝜓 ′ = 𝜓v , we have (23.1) [D𝜓 ′ ∶ D𝜑]t = v∗ [D𝜓 ∶ D𝜑]t 𝜎t𝜑 (v) (t ∈ ℝ). Since [D𝜓 ′ ∶ D𝜑]t = a′it , [D𝜓 ∶ D𝜑]t = ait , and s(a′ ) = s(𝜓 ′ ) = v∗ v, s(a) = s(𝜓) = vv∗ , it follows that a′it = v∗ ait 𝜎t𝜑 (v) and vB ′it = vs(a′ )B ′it = va′it = vv∗ ait 𝜎t𝜑 (v) = s(a)ait 𝜎t𝜑 (v) = Bit s(a)𝜎t𝜑 (v) = Bit 𝜎t𝜑 (s(a)v) = Bit 𝜎t𝜑 (v), that is, vB ′it = Bit 𝜎t𝜑 (v) (t ∈ ℝ).

(1)

∑ k Since ℳ = ℛ(𝒩 , 𝜃), we can write v = 𝒩 ; we have Bit 𝜎t𝜑 (v) = k xk u with xk ∈ ∑ ∑ it k ∑ ∑ it it k it k ′it = B ( k xk u Ak ) = k B xk 𝜃 (Ak ) u and vB = k xk uk B ′it = xk 𝜃 k (B ′ )it uk . Consequently, it follows from (1) that Bit xk 𝜃 k (Ak )it = xk 𝜃 k (B ′ )it , that is, ∑ Bit ( k xk 𝜎t𝜑 (uk ))

it Bit xk = xk 𝜃 k (B ′ A−1 k )

(k ∈ ℤ, t ∈ ℝ).

(2)

We choose and fix k ≠ 0 and show that xk = 0. Note that in general if u1 and u2 are unitary elements and x = w|x| is the polar decomposition of some operator x such that u1 x = xu2 , then u1 w = wu2 , u2 (w∗ w) = (w∗ w)u2 and u1 (ww∗ ) = (ww∗ )u1 . Consequently, we can assume that the operator xk appearing in (2) is a partial isometry such that x∗k xk commutes with 𝜃 k (B ′ A−1 )it and xk x∗k k it commutes with B . If k > 0, then it Bit xk x∗k = xk 𝜃 k (B ′ A−1 k ) xk

(t ∈ ℝ)

(3)

and if k < 0, then x∗k xk 𝜃 k (B ′ )it = x∗k (B𝜃 k (Ak ))it

(t ∈ ℝ)

(4)

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

403

since Bit 𝜃 k (Ak )it xk = xk 𝜃 k (B ′ )it . On the other hand, if k > 0, then 0 ≤ B < 1 and Ak ≤ A ≤ B ′ , so that 0 ≤ B ≤ 1 and 𝜃 k (B ′ A−1 k ) ≥ 1;

(5)

, 𝜃 k (Ak )−1 = A−k ≤ if k < 0, then 0 ≤ B ′ ≤ 1 and 1 = A0 = A−k+k = A−k 𝜃 k (Ak ), i.e. 𝜃 k (Ak ) = A−1 −k A ≤ B, and hence 0 ≤ 𝜃 k (B ′ ) ≤ 1 and B𝜃 k (Ak ) ≥ 1.

(6)

Note that, in general, if H and K are nonsingular positive self-adjoint operators such that 0 ≤ H ≤ 1 and K ≥ 1, then the functions {z ∈ ℂ; Im(z) ≤ 0} ∋ z ↦ Hiz {z ∈ ℂ; Im(z) ≥ 0} ∋ z ↦ Kiz are analytic and bounded, hence an identity of the form xHit = yKit (t ∈ ℝ) leads to a bounded entire analytic function which is necessarily constant (by Liouville’s theorem), so that xHit = 0 and x = 0. Therefore, it follows from (3) and (5) that xk = 0 for all k > 0, and it follows from (4) and (6) that xk = 0 for all k < 0, so that v = x0 ∈ 𝒩 . Corollary. Let a ∈ 𝒩 with As(a) ≤ a < 1. Then ℳ 𝜑a = 𝒩 𝜏a = {x ∈ 𝒩 ; xa = ax}. In particular, every subweight of 𝜑a is of the form 𝜑b for some b ∈ 𝒩 with As(b) ≤ b < 1. Proof. Put 𝜓 = 𝜑a , 𝜇 = 𝜏a . Let v ∈ ℳ 𝜓 be unitary. We have vv∗ = v∗ v = s(𝜓) and 𝜓 = 𝜓v by Proposition 2.21. According to the previous theorem, it follows that v ∈ 𝒩 . Hence ℳ 𝜓 ⊂ 𝒩 . Since 𝜓 = 𝜇 ◦ P, for x ∈ 𝒩 we have (11.9) 𝜎t𝜓 (x) = 𝜎t𝜇 (x) = ait xa−it (t ∈ ℝ). Thus, ℳ 𝜓 = 𝒩 𝜇 = {x ∈ 𝒩 ; xa = ax}. A subweight of 𝜓 is of the form 𝜓e for some projection e ∈ ℳ and is hence of the form 𝜑b with b = ae ∈ 𝒩 , As(b) ≤ b < 1. 30.6. Recall that the operators An are affiliated to 𝒵 (𝒩 ) and that we have An ≤ 𝜆0 An−1 . We now show that for every positive self-adjoint operator D affiliated to 𝒩 there exists a sequence of mutually orthogonal projections {en }n∈ℤ ⊂ {D}′ ∩ 𝒩 , uniquely determined, such that ∑

en = s(D),

(1)

n

en An ≤ en D < en An−1 .

(2)

Indeed, let en = s((An−1 − D)+ ) − s((An − D)+ ), (n ∈ ℤ). Then en are clearly mutually orthogonal projections and ∑ en = lim s((An − D)+ ) − lim s((An − D)+ ). n

n→−∞

n→+∞

404

Discrete Decompositions

If e ∈ 𝒩 is a projection such that e ≤ s((An − D)+ ) for all n > 0, then e(An − D) ≥ 0 and eD ≤ eAn ≤ 𝜆n0 e → 0, hence e ≤ 1 − s(D). Consequently, limn→+∞ s((An − D)+ ) = 1 − s(D). On the other hand, if e ≥ s((An − D)+ ) for every n < 0, then 1 − e ≤ 1 − s((An − D)+ ) = 1 − s(An − D) + s((An − D)− ), and (1 − e)(An − D) ≤ 0, or (1 − e)D ≥ (1 − e)An ≥ 𝜆n0 (1 − e) → +∞, and hence 1 − e = 0, e = 1. Consequently, limn→−∞ s((An − D+ ) = 1. Hence ∑

en = s(D).

n

Since en ≤ s((An−1 − D)+ ), we have en (An−1 − D) ≥ 0, that is, en D ≤ en An−1 . If e ≤ en and e(An−1 − D) = 0, then es(An−1 − D) = 0, so that es((An−1 − D)+ ) = 0 and e = 0. Therefore, en D < en An−1 . Since en ≤ 1 − s((An − D)+ ), we have en (An − D) ≤ 0, hence en An ≤ en D. Thus, if D ≠ 0, then there exist n ∈ ℤ and a nonzero projection e ∈ {D}′ ∩ 𝒩 , such that eAn ≤ eD < eAn−1 .

(3)

Consider now a normal semifinite weight 𝜇 on 𝒩 . There exists a unique positive self-adjoint operator D𝜇 affiliated to 𝒩 , such that 𝜇 = 𝜏D𝜇 .

(4)

For each k ∈ ℤ, we have a normal semifinite weight 𝜇k = 𝜇 ◦ 𝜃 k on 𝒩 and a normal semifinite weight 𝜓k = 𝜇k ◦ P on ℳ. Note that 𝜇 = 𝜇0 and write 𝜓 = 𝜓0 = 𝜇 ◦ P. Then 𝜓 = 𝜑D𝜇 .

(5)

For every k ∈ ℤ, we have 𝜓k ≂ 𝜓, more precisely 𝜓k = 𝜓 ◦ Ad(uk ) (k ∈ ℤ).

(6)

Indeed, for every x ∈ ℳ + , we have 𝜓k (x) = 𝜇k (P(x)) = 𝜇(𝜃 k (P(x))) = 𝜇(uk P(x)u−k ) = 𝜇(P(uk xu−k )) = (𝜓 ◦ Ad(uk ))(x). Also, D𝜇k = Ak 𝜃 −k (D𝜇 ) (k ∈ ℤ). Indeed, for every x ∈ 𝒩 + , we have 𝜇k (x) = 𝜇(𝜃 k (x)) = 𝜏(D𝜇 𝜃 k (x)) = 𝜏(𝜃 k (𝜃 −k (D𝜇 )x)) = 𝜏(Ak 𝜃 −k (D𝜇 )x).

(7)

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

405

Finally, for every k ∈ ℤ, we have s(D𝜇 )An ≤ D𝜇 < s(D𝜇 )An−1 ⇒ s(D𝜇k )Ak+n ≤ D𝜇k < s(D𝜇k )Ak+n−1 .

(8)

Indeed, the desired conclusion can be obtained by applying 𝜃 −k to the left hand side, multiplying by Ak and using the fact that Ak 𝜃 −k (Am ) = Ak+m . By (3), we can show inductively that the inequalities in (8) are satisfied, starting with an arbitrary nonzero normal semifinite weight 𝜇 and then considering a convenient subweight of 𝜇. Also, the previous arguments show that if 𝜇 is a normal semifinite weight on 𝒩 such that s(D𝜇 )An ≤ D𝜇 < s(D𝜇 )An−1 for some n ∈ ℤ, then, for any m ∈ ℤ, v = 𝜇m−n is a normal semifinite weight on 𝒩 , v ◦ P ≂ 𝜇 ◦ P on ℳ and s(Dν )Am ≤ Dν < s(Dν )Am−1 . 30.7. Let ℳ be a factor of type III𝜆 (0 ≤ 𝜆 < 1) and 𝜑 a lacunary weight of infinite multiplicity on ℳ. We use the notation introduced in Section 30.4. Theorem. For every normal semifinite weight 𝜓 on ℳ, there exists a ∈ 𝒩 with As(a) ≤ a < 1 such that 𝜓 ≂ 𝜑a = 𝜏a ◦ P. Proof. (I) We assume first that the weight 𝜓 is of infinite multiplicity and show that there exists a nonzero positive self-adjoint operator D, affiliated to 𝒩 , such that 𝜑D ≲ 𝜓. Since 1 is an isolated point in S(ℳ) ∩ ℝ+∗ , we can assume using Theorem 28.3.(2) that (Sp 𝜎 𝜓 ) ∩ exp((−2𝜀, −𝜀) ∪ (𝜀, 2𝜀)) = ∅ for some 𝜀 > 0. By Proposition 15.12, it follows that there exists an invertible positive element c ∈ 𝒵 (ℳ 𝜓 ) such that the weight 𝜓c is lacunary. Then ℳ 𝜓 ⊂ ℳ 𝜓c , and so 𝜓c is also of infinite multiplicity. By Proposition 29.7, there exist nonzero projections e ∈ 𝒵 (ℳ 𝜑 ), f ∈ 𝒵 (ℳ 𝜓c ) and v ∈ ℳ such that e = v∗ v, f = vv∗ and v(ℳ 𝜑e )v∗ = ℳ (𝜓c )f = ℳ 𝜓cf . We have c−1 ∈ ℳ 𝜓c and 𝜓 = (𝜓c )c−1 ; as f ∈ 𝒵 (ℳ 𝜓c ), f commutes with c−1 and so, 𝜎t𝜓 (f) = 𝜓 c−it 𝜎t c (f)cit = f (t ∈ ℝ), that is, f ∈ ℳ 𝜓 . It is therefore meaningful to consider the weight 𝜓f and we have 𝜓f = ((𝜓c )c−1 )f = (𝜓c )c−1 f , that is, 𝜓f = (𝜓c )b , where b = c−1 f. Since vv∗ = f ∈ ℳ 𝜓f , we may also consider the weight (𝜓f )v = 𝜓v ≲ 𝜓,

406

Discrete Decompositions

and we have 𝜓fv = ((𝜓c )b )v = (𝜓c )bv = ((𝜓c )v )v∗ bv . Since ℳ (𝜓c )v = v∗ (ℳ 𝜓cf )v = ℳ 𝜑e , there exists by Corollary 29.11 a positive self-adjoint operator H, affiliated to 𝒵 (ℳ 𝜑e ), such that (𝜑c )v = (𝜑e )H . It follows that the positive self-adjoint operator D = eHv∗ bv is affiliated to 𝒩 = ℳ 𝜑 and 𝜑D = 𝜓v ≲ 𝜓. (II) Using the results of Section 30.6, it follows from (I) that if 𝜓 is of infinite multiplicity, then there is an a ∈ 𝒩 with As(a) ≤ a < 1 such that 𝜑a ≲ 𝜓. (III) We now prove the same result, but without assuming that 𝜓 is of infinite multiplicity. We have 𝜓 ≲ 𝜓̌ with 𝜓̌ of infinite multiplicity (23.15). Thus, there exists a projection f ∈ ℳ 𝜓̌ such that 𝜓 ≂ (𝜓) ̌ f and, by (II), there exists a ∈ 𝒩 with As(a) ≤ a < 1 such that 𝜑a ≲ 𝜓; ̌ 𝜓 ̌ consequently, 𝜑a ≂ (𝜓) ̌ e for some projection e ∈ ℳ . By the comparison theorem ([L], 4.6), there exists a projection p ∈ 𝒵 (ℳ 𝜓̌ ) such that pe ≺ pf and (1 − p)f ≺ (1 − p)e in ℳ 𝜓̌ . If p ≠ 0, there exists 0 ≠ v ∈ ℳ 𝜓̌ with vv∗ = pe and v∗ v ≤ pf. Using the results of Section 2.21, we obtain (𝜓̌ pe )v = (𝜓̌ pf )v∗ v ≲ 𝜓̌ pf ≲ 𝜓 and (𝜓̌ pe )v ≂ 𝜓ep ≲ 𝜑a so that, by Corollary 30.5, we have (𝜓̌ pe )v ≂ 𝜑b for some b ∈ 𝒩 with As(b) ≤ b < 1, and thus 𝜑b ≲ 𝜓. If p = 0, then f ≺ e in ℳ 𝜓̌ , and there exists v ∈ ℳ 𝜓̌ such that vv∗ = f and v∗ v ≤ e. By the results of Section 2.21, we have 𝜓 ≂ (𝜓) ̌ f ≂ ((𝜓) ̌ f )v = ((𝜓) ̌ e )v∗ v ≲ 𝜑e and, again by Corollary 30.5, 𝜓 ≂ 𝜑b for some b ∈ 𝒩 with As(b) ≤ b < 1. (IV) The above part of the proof shows that if {en } is a maximal family of mutually orthogonal nonzero projections in ℳ such that for each n there exists an ∈ 𝒩 with As(an ) ≤ an < 1 and ∑ 𝜓en = 𝜑an , then n en = s(𝜓). The supports s(an ) = s(𝜑an ) = s(𝜓en ) = en are mutually orthogonal, ∑ hence a = n an ∈ 𝒩 , As(a) ≤ a < 1, and 𝜓 = 𝜑a . In view of the results presented in Section 30.6, Theorems 30.5 and 30.7 remain still valid when the condition As(a) ≤ a < 1 is replaced by the condition An s(a) ≤ a < An−1 s(a) for some n ∈ ℤ. Corollary. For every n.s.f. weight 𝜓 on a factor ℳ of type III0 , we have 𝒵 (ℳ 𝜓 )′ ∩ ℳ = ℳ 𝜓 . Proof. Since ℳ is of type III0 , 𝒵 (𝒩 ) is diffuse and 𝜃 acts ergodically on 𝒵 (𝒩 ) (30.3), hence the action 𝜃 ∶ ℤ → Aut(𝒵 (𝒩 )) is free (29.2), and 𝒵 (𝒩 )′ ∩ ℳ = 𝒩 . By the previous theorem, we may assume that 𝜓 = 𝜑a for some a ∈ 𝒩 with s(a) = 1 and A ≤ a < 1. In this case, by Theorem 30.5 we have ℳ 𝜓 = {y ∈ 𝒩 ; ya = ay}, whence 𝒵 (𝒩 ) ⊂ 𝒵 (ℳ 𝜓 ). Let x ∈ 𝒵 (ℳ 𝜓 )′ ∩ ℳ. Then x ∈ 𝒵 (𝒩 )′ ∩ ℳ = 𝒩 and, as a ∈ 𝒵 (ℳ 𝜓 ), we have xa = ax, so that x ∈ ℳ𝜓 . 30.8. Consider now a fixed dominant weight ν on 𝒩 which defines the (smooth) flow of weights (𝒵 (𝒩 v ), F𝒩 ) on 𝒩 (24.1). Recall that for every normal semifinite weight 𝜇 on 𝒩 we denote by D𝜇 the unique positive self-adjoint operator affiliated to 𝒩 , such that 𝜇 = 𝜏D𝜇 .

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

407

Let H and K be positive self-adjoint operators affiliated to 𝒵 (𝒩 ) such that H < K. Put W(H, K ) = {𝜇 ∈ W∞ int (𝒩 ); s(D𝜇 )H ≤ D𝜇 < s(D𝜇 )K}, ⋁ [H, K) = {cν (𝜇); 𝜇 ∈ W(H, K )} ∈ Proj(𝒵 (𝒩 ν )). It is easy to check that ′ ′ 𝜇 ′ ∈ W∞ int (𝒩 ), 𝜇 ≲ 𝜇 ∈ W(H, K ) ⇒ 𝜇 ∈ W(H, K ), ⊕ ∑

{𝜇n } ⊂ W(H, K ), 𝜇 =

𝜇n ⇒ 𝜇 ∈ W(H, K )

n

and, using these remarks, it follows for 𝜇 ∈ W∞ (𝒩 ) that int cv (𝜇) ≤ [H, K ) ⇔ s(D𝜇 )H ≤ D𝜇 < s(D𝜇 )K.

(1)

In particular, we show that, for the projections [An , An−1 ) ∈ 𝒵 (𝒩 ν ), ∑

[An , An−1 ) = 1.

(2)

n∈ℤ

Indeed, let n < m. If 𝜇 ∈ W∞ (𝒩 ) and cν (𝜇) ≤ [An , An−1 ), cν (𝜇) ≤ [Am , Am−1 ), then, by (1), int An s(D𝜇 ) ≤ D𝜇 < Am−1 s(D𝜇 ), while Am−1 < An since n < m. It follows that D𝜇 = 0, 𝜇 = 0 and cν (𝜇) = 0. Hence [An , An−1 )[Am , Am−1 ) = 0. On the other hand, using the first result in Section 30.6, we see that every 𝜇 ∈ W∞ (𝒩 ν ) can be int ∑⊕ ∑ written in the form 𝜇 = n∈ℤ 𝜇n with cν (𝜇n ) ≤ [An , An−1 ), so that cν (𝜇) ≤ n∈ℤ [An , An−1 ). This proves (2). The *-automorphism 𝜃 ∈ Aut(𝒩 ) defines a *-automorphism (25.1) mod(𝜃) ∈ Aut(F𝒩 ) ⊂ Aut(𝒵 (𝒩 ν )), uniquely determined, such that [mod(𝜃)](cν (𝜇)) = cν (𝜇 ◦ 𝜃 −1 ) for every 𝜇 ∈ Wns (𝒩 ). Using the definitions of mod(𝜃) and [An , An−1 ), and assertion 30.6.(8) with k = −1, we get [mod(𝜃)]([An , An−1 )) = [An−1 , An−2 ).

(3)

Furthermore, by (2) and (3), it follows that the mappings Φn ∶ 𝒵 (𝒩 ν )mod(𝜃) ∋ z ↦ z[An , An−1 ) ∈ 𝒵 (𝒩 ν )[An , An−1 ) ∑ Ψn ∶ 𝒵 (𝒩 ν )[An , An−1 ) ∋ z ↦ [mod(𝜃)]n (z) ∈ 𝒵 (𝒩 ν )mod(𝜃) n∈ℤ

are *-isomorphisms, inverse to one another. Thus, 𝒵 (𝒩 ν )mod(𝜃) ≈ 𝒵 (𝒩 ν )[An , An−1 ).

(4)

408

Discrete Decompositions

Note that for 𝜇 ∈ Wns (𝒩 ) and 𝜓 = 𝜇 ◦ P ∈ Wns (ℳ) we have 𝜓 ∈ Wint (ℳ) ⇔ 𝜇 ∈ Wint (𝒩 ).

(5) w

Indeed, if 𝜇 ∈ Wint (𝒩 ), there exists a net {yi } ⊂ 𝒩 + with ∫ 𝜎t𝜇 (yt )dt < +∞, such that yi → 1. Since 𝜎t𝜓 |𝒩 = 𝜎t𝜇 (t ∈ ℝ), the integrability of 𝜓 follows using the net. Conversely, if 𝜓 ∈ Wint (ℳ), w

then there exists a net {xi } ⊂ ℳ + with ∫ 𝜎t𝜓 (xt )dt < +∞, such that xi → 1. Then yi = P(xi ) ∈ ( ) w 𝒩 + , yi → 1, and (10.5) ∫ 𝜎t𝜇 (yi )dt = ∫ 𝜎t𝜇 (P(xi ))dt = ∫ P(𝜎t𝜓 (xi )dt = P ∫ 𝜎t𝜓 (xi )dt < +∞, whence 𝜇 ∈ Wint (𝒩 ). The previous remark and Theorems 30.7, 30.5, show that, given an arbitrary n ∈ ℤ, every weight 𝜓 ∈ W∞ (ℳ) is equivalent to a weight of the form 𝜏a ◦ P with a ∈ 𝒩 , An s(a) ≤ a < An−1 s(a), and int 𝜏a ∈ W∞ (𝒩 ); moreover the equivalence class of the weight 𝜏a with these properties depends only int on the equivalence class of the weight 𝜓. We now consider a dominant weight 𝜔 on ℳ which defines the (smooth) flow of weights (𝒵 (ℳ 𝜔 ), Fℳ ) on ℳ (24.1). Define a mapping I ∶ Proj(𝒵 (𝒩 ν )) → Proj(𝒵 (ℳ 𝜔 ))

by I(cν (𝜇)) = c𝜔 (𝜇 ◦ P) (𝜇 ∈ W∞ (𝒩 )). int This mapping is well defined and increasing, since if 𝜇, 𝜇 ′ ∈ W∞ (𝒩 ) and cν (𝜇) ≤ cν (𝜇′ ), then int ′ ′ (24.4) 𝜇 ≲ 𝜇 in 𝒩 , so that 𝜇 ◦ P ≲ 𝜇 ◦ P in ℳ and hence c𝜔 (𝜇 ◦ P) ≤ c𝜔 (𝜇′ ◦ P). Similarly, one shows that this mapping is completely additive. For n ∈ ℤ, the restriction of the mapping I to [An , An−1 ) is bijective, so that it defines a *-isomorphism I ∶ 𝒵 (𝒩 ν )[An , An−1 ) → 𝒵 (ℳ 𝜔 ). Indeed, this follows from the fact that every weight 𝜓 ∈ W∞ (ℳ) can be written in the form 𝜓 = int 𝜇 ◦ P for some 𝜇 ∈ Wint (ℳ) with cν (𝜇) ≤ [An , An−1 ); in this case we have (30.5) ℳ 𝜇 = ℳ 𝜓 , hence 𝜇 ∈ W∞ (ℳ), and the equivalence class of 𝜇 depends only on that of 𝜓. int Also, we have I ◦ mod(𝜃) = I. Indeed, for 𝜇 ∈ W∞ (𝒩 ) we have (30.6.(6)) I([mod(𝜃)](cν (𝜇))) = I(cν (𝜇 ◦ 𝜃 −1 )) = c𝜔 (𝜇 ◦ 𝜃 −1 ◦ P) = int −1 c𝜔 (𝜇 ◦ P ◦ Ad(u )) = c𝜔 (𝜇 ◦ P) = I(cν (𝜇)). It follows that J(z) = I(Φu (z)) (z ∈ 𝒵 (𝒩 ν )mod(𝜃) ) defines a *-isomorphism J ∶ 𝒵 (𝒩 ν )mod(𝜃) → 𝒵 (ℳ 𝜔 ), which is independent of n ∈ ℤ. Moreover, ℳ JF𝒩 𝜆 = F𝜆 J

(𝜆 ∈ ℝ+∗ ),

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

409

since for 𝜇 ∈ W∞ (ℳ) and 𝜆 ∈ ℝ+∗ we have I(F𝒩 (cν (𝜇))) = I(cν (𝜆𝜇)) = c𝜔 ((𝜆𝜇) ◦ P) = int 𝜆 ℳ c𝜔 (𝜆(𝜇 ◦ P)) = F𝜆 c𝜔 (𝜇 ◦ P) = Fℳ I(c (𝜇)). ν 𝜆 Thus, we obtain the following result for the smooth flows of weights (𝒫ℳ , Fℳ ) and (𝒫𝒩 , F𝒩 ). Theorem. (𝒫ℳ , Fℳ ) ≈ ((𝒫𝒩 )mod(𝜃) , F𝒩 ). A similar result can be obtained for the global flow of weights (24.9). 30.9. We mention the following result concerning the fundamental homomorphism mod ∶ Out(ℳ) → Aut(Fℳ ). Proposition. Let ℳ be a factor of type III𝜆 (0 ≤ 𝜆 < 1) with separable predual and let 𝜎 ∈ Aut(ℳ). The following statements are equivalent: (i) mod(𝜎) = 𝜄; (ii) there exist 𝜎 ′ ∈ Aut(ℳ) with 𝜎 ′ ∕Int(ℳ) = 𝜎∕Int(ℳ) and an n.s.f. weight 𝜓 on ℳ such that 𝜓 ◦ 𝜎 ′ = 𝜓 and 𝜎 ′ |𝒵 (ℳ 𝜓 ) = 𝜄; (iii) for each 𝜀 > 1 with S(ℳ) ∩ [𝜀−1 , 𝜀] = {1} there exist 𝜎 ′ ∈ Aut(ℳ) with 𝜎 ′ ∕Int(ℳ) = 𝜎∕Int(ℳ) and a faithful normal state 𝜓 on ℳ such that 𝜓 ◦ 𝜎 ′ = 𝜓, 𝜎 ′ |𝒵 (ℳ 𝜓 ) = 𝜄 and Sp(Δ𝜓 ) ∩ [𝜀−1 , 𝜀] = {1}. Proof. (iii) ⇒ (ii). Obvious. (ii) ⇒ (i). Assume that there exists an n.s.f. weight 𝜓 on ℳ such that 𝜓 ◦ 𝜎 = 𝜓 and 𝜎|𝒵 (ℳ 𝜓 ) = 𝜄. Since ℳ is properly infinite, modifying 𝜎 if necessary by an inner *-automorphism, we may ̄ ℱ∞ , 𝜎 ⊗ ̄ 𝜄) (20.14). Let 𝜔0 be the n.s.f. weight on ℱ∞ = ℬ(ℒ 2 (ℝ)) assume that (ℳ, 𝜎) ≈ (ℳ ⊗ with [D𝜔0 ∶ D tr]t = 𝜌 (t) (t ∈ ℝ), where, as usual, 𝜌 stands for the right regular representation of ̄ ℱ∞ ⊗ ̄ ℱ∞ , 𝜎̃ = 𝜎 ⊗ ̄ 𝜄⊗ ̄ 𝜄, 𝜔̃ = 𝜓 ⊗ ̄ 𝜔0 ⊗ ̄ tr. Then 𝜔̃ is a dominant weight on ℝ. Put ℳ̃ = ℳ ⊗ ̃ ℳ and we have 𝜔̃ ◦ 𝜎̃ = 𝜔. ̃

(1)

On the other hand, ̄ ℱ∞ )𝜓 ℳ̃ 𝜔̃ = (ℳ ⊗

̄ 𝜔0 ⊗

̄ ℱ∞ = (ℳ ⊗ ̄ ℱ∞ )𝜎𝜓 ⊗

̄ Ad(𝜌𝜌) ⊗

̄ ℱ∞ ; ⊗

hence (19.13, 21.6.(1)) ̄ ℱ∞ ℳ̃ 𝜔̃ = ℛ(ℳ, 𝜎 𝜓 ) ⊗ and ̄ ℂ ⊂ 𝒵 (ℳ 𝜓 ) ⊗ ̄ 𝔏(ℝ) ⊗ ̄ ℂ, 𝒵 (ℳ̃ 𝜔̃ ) = 𝒵 (ℛ(ℳ, 𝜎 𝜓 )) ⊗ and therefore 𝜎|𝒵 ̃ (ℳ̃ 𝜔̃ ) = 𝜄. From (1), (2), and 25.1.(7), it follows that mod(𝜎) = mod(𝜎) ̃ = 𝜄.

(2)

410

Discrete Decompositions

(i) ⇒ (iii). Let 𝜔 be a dominant weight on ℳ. Using 25.1.(7) and modifying 𝜎 if necessary by an inner *-automorphism, we can rephrase the condition mod(𝜎) = 𝜄 as follows: 𝜔 ◦ 𝜎 = 𝜔, 𝜎|𝒵 (ℳ 𝜔 ) = 𝜄.

(3)

As in the first part of the proof of Theorem 30.7, we can find an element a ∈ 𝒵 (ℳ 𝜔 )+ such that 𝜓 ′ = 𝜔a is a lacunary weight, namely (Sp 𝜎 𝜓 ) ∩ [𝜀−1 , 𝜀] = {1}. ′

(4)

Since 𝜓 ′ = 𝜔a with a ∈ 𝒵 (ℳ 𝜔 ), we infer from (3) that 𝜓 ′ ◦ 𝜎 = 𝜓 ′.

(5)

Let e = s(𝜓 ′ ) = s(a) ∈ 𝒵 (ℳ 𝜔 ). For x ∈ eℳ 𝜔 e, we have 𝜎t𝜓 (x) = ait 𝜎t𝜔 (x)a−it = ait xa−it = ′ ′ x (t ∈ ℝ), whence eℳ 𝜔 e ⊂ ℳ 𝜓 . Using the relative commutant theorem (23.19), we get 𝒵 (ℳ 𝜓 ) ⊂ ′ (ℳ 𝜓 )′ ∩ ℳ ⊂ (eℳ 𝜔 e)′ ∩ ℳ ⊂ 𝒵 (ℳ 𝜔 ) and so from (3) it follows that ′

𝜎|𝒵 (ℳ 𝜓 ) = 𝜄. ′

(6)

Being lacunary, the weight 𝜓 ′ is strictly semifinite and hence 𝜇 = 𝜓 ′ |ℳ 𝜓 is an n.s.f. trace on ′ ′ ′ ℳ 𝜓 . Since 𝜇 ◦ 𝜎 = 𝜇 and 𝜎|𝒵 (ℳ 𝜓 ) = 𝜄, it follows that every projection p ∈ ℳ 𝜓 is equivalent in ′ ′ ℳ 𝜓 to 𝜎(p). Let p ∈ ℳ 𝜓 be a projection such that 0 ≠ 𝜓(p) = 𝜇(p) < +∞. Since p ∼ 𝜎(p), e−p ∼ 𝜎(e − p) = e − 𝜎(p), there exists v ∈ ℳ 𝜓 such that v∗ v = vv∗ = e and 𝜎(p) = v∗ pv. Composing with the inner *-automorphism defined by the unitary operator u = v + (1 − e), we may assume that 𝜎(p) = p. Then 𝜓 ′′ = 𝜓 ′ (p ⋅ p)∕𝜓 ′ (p) is a faithful normal state on pℳp which satisfies conditions similar to (4), (5), and (6). Since ℳ is a countably decomposable type III factor, the projection p is equivalent to 1 in ℳ and so there exists w ∈ ℳ such that w∗ w = 1 and ww∗ = p. We then define 𝜓 = 𝜓w′′ and 𝜎 ′ ∈ Aut(ℳ) by 𝜎 ′ (x) = w∗ 𝜎(wxw∗ )w (x ∈ ℳ), that is, 𝜎 ′ = Ad(w∗ 𝜎(w)) ◦ 𝜎. Then 𝜓 and 𝜎 ′ satisfy the requirements of statement (iii). ′

30.10. Consider now a factor ℳ of type III𝜆 (0 < 𝜆 < 1) with separable predual and let (𝒩 , 𝜃, 𝜏) be the discrete decomposition of ℳ (30.1). By Theorem 30.8, we know that (𝒫ℳ , Fℳ ) ≈ ((𝒫𝒩 )mod(𝜃) , F𝒩 ). On the other hand, by the results of Sections 24.8 and 25.6, we have (𝒫𝒩 , F𝒩 ) ≈ 𝜆)), and mod(𝜃) = F𝒩 𝜆(𝜆−1 )), since 𝜏 ◦ 𝜃 = 𝜆𝜏. As Γ(ℳ) = S(ℳ) ∩ ℝ+∗ = (ℒ ∞ (ℝ+∗ ), Ad(𝜆 = Ad(𝜆 𝜆−1 {𝜆n ; n ∈ ℤ}, it follows that 𝜆)). (𝒫ℳ , Fℳ ) ≈ (ℒ ∞ (ℝ+∗ ∕Γ(ℳ)), Ad(𝜆

(1)

It is now easy to check that the mapping ℝ+∗ ∕Γ(ℳ) ∋ ν∕Γ(ℳ) ↦ Fℳ ∈ Aut(Fℳ ) ν

(2)

is a continuous surjective group isomorphism. Since ℝ+∗ ∕Γ(ℳ) is compact, this mapping is also a topological isomorphism.

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

411

Let 𝜎 ∈ Aut(ℳ). Then mod(𝜎) ∈ Aut(Fℳ ) and there exists ν ∈ ℝ+ such that mod(𝜎) = Fℳ , that ν is, 𝜓 ◦ 𝜎 −1 ≂ ν𝜓, for every 𝜓 ∈ W∞ (ℳ). On the other hand, by Proposition 29.5, there exists a int 𝜆-trace 𝜑 on ℳ and 𝜇 ∈ ℝ+ such that 𝜑 ◦ 𝜎 −1 ≂ 𝜇𝜑. We shall show that 𝜇 ν−1 ∈ Γ(ℳ): , mod(𝜎) = Fℳ ν

𝜑 ◦ 𝜎 −1 ≂ 𝜇𝜑 ⇒ 𝜇v−1 ∈ Γ(ℳ).

(3)

Indeed, let 𝜀 > 0. As in Section 23.18, there exists a𝜀 ∈ ℳ 𝜑 such that 1 − 𝜀 ≤ a𝜀 ≤ 1 + 𝜀 and 𝜓 = 𝜑a𝜀 ∈ W∞ (ℳ). Also, there exist u, v ∈ U(ℳ) such that 𝜑 ◦ 𝜎 −1 = 𝜇𝜑 ◦ Ad(u) and int −1 𝜓 ◦ 𝜎 = ν𝜓 ◦ Ad(v). Then, for x ∈ ℳ + we have ∗ ∗ 1∕2 ) = ν(𝜓 ◦ Ad(v))(x) = 𝜓(𝜎 −1 (x)) ν𝜑(a1∕2 𝜀 vxv a𝜀 ) = ν𝜑a (vxv −1 1∕2 −1 1∕2 1∕2 = 𝜑(a1∕2 𝜀 𝜎 (x)a𝜀 ) = (𝜑 ◦ 𝜎 )(𝜎(a𝜀 )x𝜎(a𝜀 )) 1∕2 = 𝜇(𝜑 ◦ Ad(u))(𝜎(a1∕2 𝜀 )x𝜎(a𝜀 )) 1∕2

1∕2

or ν𝜑(vv∗ a𝜀 vxv∗ a𝜀 vv∗ ) = 𝜇(𝜑 ◦ Ad(u))(𝜎(a𝜀 )1∕2 x𝜎(a𝜀 )1∕2 ), that is, ν(𝜑 ◦ Ad(v))v∗ a𝜀 v = 𝜇(𝜑 ◦ Ad(u))𝜎(a𝜀 ) .

It follows that [D(𝜑 ◦ Ad(v))v∗ a𝜀 v ∶ D(𝜑 ◦ Ad(u))𝜎(a𝜀 ) ]t = 𝜇it v−it for every t ∈ ℝ. Using the “chain rule,” we deduce that it −it v∗ ait𝜀 v[D(𝜑 ◦ Ad(v)) ∶ D(𝜑 ◦ Ad(u))]t 𝜎(a−it 𝜀 )=𝜇 v

for all t ∈ ℝ. For t = −2𝜋∕ln(𝜆), we have D(𝜑 ◦ Ad(v)) ∶ D(𝜑 ◦ Ad(u))]t = v∗ 𝜎t𝜑 (v)𝜎t𝜑 (u∗ )u = 1, so that it −it v∗ ait𝜀 v𝜎(a−it 𝜀 )=𝜇 ν

and therefore, letting 𝜀 tend to 0, (𝜇 ν−1 )it = 1; this means that 𝜇 ν−1 ∈ Γ(ℳ). 30.11. For factors of type III𝜆 with 0 < 𝜆 < 1, it is possible also to compute the group Out(ℳ) in terms of the discrete decomposition (compare with Theorem 26.4). Theorem. Let ℳ be a factor of type III𝜆 (0 < 𝜆 < 1) with discrete decomposition (𝒩 , 𝜃, 𝜏). Let t = −2𝜋∕ln(𝜆) and denote by G the quotient of the commutant of 𝔬(𝜃) in Out(𝒩 ) by {𝔬𝒩 (𝜃)k ; k ∈ ℤ}. There exists a group homomorphism 𝛾 ∶ Out(ℳ) → G such that the following sequence is exact: n→nt

𝛿ℳ

{0} → ℤ ⟶ ℝ ⟶ Out(ℳ) ⟶ G ⟶ {𝜄}.

(1)

412

Discrete Decompositions

Proof. Recall that ℳ = ℛ(𝒩 , 𝜃) and let 𝜋 ∶ 𝒩 → 𝜋(𝒩 ) ⊂ ℳ denote the canonical embedding, P ∶ ℳ → 𝜋(𝒩 ) the faithful normal conditional expectation, 𝜑 = 𝜏 ◦ 𝜋 −1 ◦ P the corresponding ̄ 𝜆(1) the unitary element of ℳ such that 𝜋(𝜃(x)) = u𝜋(x)u∗ (x ∈ 𝒩 ). 𝜆-trace on ℳ and u = 1 ⊗𝜆 Let 𝜉 ∈ Out(ℳ) and 𝜎1 ∈ Aut(ℳ) be such that 𝔬ℳ (𝜎1 ) = 𝜉. Then 𝜑 ◦ 𝜎1 is also a 𝜆-trace on ℳ so that, by Proposition 29.5, there exists 𝜎 ∈ Aut(ℳ) such that 𝔬ℳ (𝜎) = 𝜉 and 𝜑 ◦ 𝜎 is proportional to 𝜑. Since 𝜋(𝒩 ) = ℳ 𝜑 = ℳ 𝜑 ◦ 𝜎 , it follows that 𝜎(𝜋(𝒩 )) = 𝜋(𝒩 ) and so there exists ν ∈ Aut(𝒩 ) such that 𝜎 ◦ 𝜋 = 𝜋 ◦ ν. For x ∈ 𝒩 , we have 𝜋(ν𝜃 ν−1 𝜃 −1 (x)) = 𝜎(u)u∗ 𝜋(x)u𝜎(u)∗ −1 and, since 𝜎s𝜑 (u) = 𝜆it u and 𝜎s𝜑 (𝜎(u)) = 𝜎s𝜑 ◦ 𝜎 (𝜎(u)) = 𝜎(𝜎s𝜑 (u)) = 𝜆is 𝜎(u) (s ∈ ℝ), we have 𝜎(u)u∗ ∈ ℳ 𝜑 = 𝜋(𝒩 ). Consequently, ν𝜃 ν−1 𝜃 −1 ∈ Int(𝒩 ) and so 𝔬𝒩 (ν) belongs to the commutant of 𝔬𝒩 (𝜃) in Out(𝒩 ). In order to define 𝛾(𝜉) = the image of 𝔬𝒩 (ν) in G,

(2)

we must show that 𝛾(𝜉) does not depend on 𝜎 ∈ Aut(ℳ) with 𝔬ℳ (𝜎) = 𝜉 and that 𝜑 ◦ 𝜎 is proportional to 𝜑. Accordingly, consider 𝜎, 𝜎 ′ ∈ Aut(ℳ) with 𝔬ℳ (𝜎) = 𝔬ℳ (𝜎 ′ ) = 𝜉 and 𝜑 ◦ 𝜎, 𝜑 ◦ 𝜎 ′ both proportional to 𝜑. By the above arguments, there exist ν, ν′ ∈ Aut(𝒩 ) such that 𝜎 ◦ 𝜋 = 𝜋 ◦ ν, 𝜎 ′ ◦ 𝜋 = 𝜋 ◦ ν′ , and 𝔬𝒩 (ν), 𝔬𝒩 (ν′ ) belong to the commutant of 𝔬𝒩 (𝜃) in Out(𝒩 ). We have 𝜎 −1 𝜎 ′ ∈ Int(ℳ) and 𝜎 −1 𝜎 ′ (𝜋(𝒩 )) = 𝜋(𝒩 ), so that 𝜎 −1 𝜎 ′ = Ad(v) for some v ∈ U(ℳ) with v𝜋(𝒩 )v∗ = 𝜋(𝒩 ). Then v belongs to the normalizer (10.17)  (P) of P. Since u ∈  (P) and ℳ = ℛ{𝜋(𝒩 ), u}, we infer by Proposition 22.4 that ν−1 ν′ belongs to the full group [Ad(un )|𝒩 ; n ∈ ℤ] ⊂ Aut(𝒩 ); since 𝒩 is a factor it follows that there exists k ∈ ℤ such that ν−1 ν′ 𝜃 −k ∈ Int(𝒩 ) and hence 𝔬𝒩 (ν)−1 𝔬𝒩 (ν′ ) = 𝔬𝒩 (𝜃)k . The mapping 𝛾 ∶ Out(ℳ) → G is thus well defined by (2). Also, by construction, it is a group homomorphism. We now compute the kernel of 𝛾. Let 𝜉 ∈ Out(ℳ) be such that 𝛾(𝜉) = 𝜄. Then there is a 𝜎 ∈ Aut(ℳ) with 𝔬ℳ (𝜎) = 𝜉 and 𝜑 ◦ 𝜎 proportional to 𝜑 and a ν ∈ Aut(𝒩 ) with 𝜎 ◦ 𝜋 = 𝜋 ◦ ν such that 𝔬𝒩 (v) ∈ {𝔬𝒩 (𝜃)k ; k ∈ ℤ}. Thus there exist k ∈ ℤ and a ∈ U(𝒩 ) such that ν ◦ 𝜃 k ◦ Ad(a) = 𝜄. Replacing 𝜎 by 𝜎 ◦ Ad(uk a) we may assume that ν = 𝜄, that is, 𝜎 ◦ 𝜋 = 𝜋. Then for x ∈ 𝒩 we get 𝜎(u)𝜋(x)𝜎(u∗ ) = 𝜎(u𝜋(x)u∗ ) = u𝜋(x)u∗ , hence u∗ 𝜎(u) ∈ 𝜋(𝒩 )′ ∩ ℳ = ℂ ⋅ 1ℳ and so u∗ 𝜎(u) = 𝜆is for some s ∈ ℝ. Since also 𝜎s𝜑 ◦ 𝜋 = 𝜋 and 𝜎s𝜑 (u) = 𝜆is u, it follows that 𝜎 = 𝜎s𝜑 , and so 𝜉 = 𝛿ℳ (s). Thus, the sequence (1) is exact at Out(ℳ). Finally, we compute the range of 𝛾. Let g ∈ G. There exists ν ∈ Aut(𝒩 ) with 𝔬𝒩 (ν)𝔬𝒩 (𝜃) = 𝔬𝒩 (𝜃)𝔬𝒩 (ν) such that g is the image of 𝔬𝒩 (ν) in G. Then ν ◦ 𝜃 = Ad(a) ◦ 𝜃 ◦ ν for some a ∈ U(𝒩 ). Let 𝜋1 = 𝜋 ◦ ν ∶ 𝒩 → ℳ. For x ∈ 𝒩 we have 𝜋1 (𝜃(x)) = 𝜋(ν(𝜃(x))) = 𝜋(a)𝜋(𝜃(ν(x)))𝜋(a)∗ = 𝜋(a)u𝜋(ν(x))u∗ 𝜋(a)∗ = (𝜋(a)u)𝜋1 (x)(𝜋(a)u)∗ . Also, 𝜋t (𝒩 ) = 𝜋(𝒩 ), ℳ = ℛ{𝜋1 (𝒩 ), 𝜋(a)u}, and there exists a faithful normal conditional expectation P ∶ ℳ → 𝜋1 (𝒩 ) = 𝜋(𝒩 ) such that P(𝜋(a)u) = 𝜋(a)P(u) = 0. By Proposition 22.2 we infer that there exists 𝜎 ∈ Aut(ℳ) such that 𝜎 ◦ 𝜋 = 𝜋1 and 𝜎(u) = 𝜋(a)u. It follows that 𝜑 ◦ 𝜎 and 𝜑 are proportional and, as 𝜎 ◦ 𝜋 = 𝜋 ◦ ν, we get 𝛾(𝔬ℳ (𝜎)) = g. Thus, the sequence (1) is also exact at G. The exactness of (1) at ℤ and ℝ is obvious. 30.12. Finally, we state without proof two important results: Theorem 1. (U. Haagerup). Let ℳ be a factor of type III𝜆 (0 < 𝜆 < 1) and let 𝜑, 𝜓 be n.s.f. weights on ℳ. If either 𝜑(1) = 𝜓(1) = 1 or 𝜑(1) = 𝜓(1) = +∞, then there exists a unitary element u ∈ ℳ such 𝜆𝜓 ≤ 𝜑 ◦ Ad(u) ≤ 𝜆−1 𝜓.

The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1)

413

This result might suggest that any two faithful normal states on a factor of type III1 are unitarily equivalent. However, if this is the case then ℳ = ℂ ⋅ 1ℳ (Haagerup, U. [1979]). Instead, the following holds. Theorem 2. (A. Connes, E. Størmer). Let ℳ be a factor of type III1 . For any two faithful normal states 𝜑, 𝜓 on ℳ and 𝜀 > 0 there exists a unitary element u ∈ ℳ such that ‖𝜓 − 𝜑 ◦ Ad(u)‖ ≤ 𝜀. Note that if ℳ is a factor of type III1 , then the flow of weights Fℳ is trivial Fℳ = 𝜄 𝜆 (𝜆 ∈ ℝ+∗ ). It follows that any integrable n.s.f. weight of infinite multiplicity on ℳ is dominant. Using Theorem 23.18, it is easy to see that for any two n.s.f. weights of infinite multiplicity 𝜑, 𝜓 on ℳ and for any 𝜀 > 0 there exists a unitary element u ∈ ℳ such that d(𝜓, 𝜑 ◦ Ad(u)) < 𝜀 and 𝔐𝜓 = u𝔐𝜑 u∗ (see Connes & Takesaki, 1977; II.4.8–II.4.10). 30.13. We record here some results of Florin Rădulescu (1992a, 1992b, 1995), concerning some examples of nonhyperfinite factors of type III1 and of the type III𝜆 , (𝜆 ∈ (0, 1)), based on D.-V. Voiculescu’s theory of free probability. Theorem ( ) 1. There exists a type III1 factor with core isomorphic to the tensor product 𝔏 F∞ ⊗ ℬ (ℋ ), where F∞ is the free group with a countable infinity of generators and ℋ is a separable infinite dimensional Hilbert space. ( ) Corollary. There exists a one-parameter group 𝛼t t>0 of automorphisms of the tensor product ( ) 𝔏 F∞ ⊗ ℬ (ℋ ) such that 𝛼t scales the canonical semifinite trace on the tensor product ( ) 𝔏 F∞ ⊗ ℬ (ℋ ) by t, for all t > 0. ( ) In particular, this implies that the fundamental group 𝔏 F∞ is ℝ+ ∖{0}. Theorem 2. Let 𝜆 be any number in (0, 1). Consider the state 𝜑𝜆 on Mat2 (ℂ) having the matrix unit { } eij i,j=1,2 , defined by the conditions ( ) ( ) 𝜑𝜆 e11 ∕𝜑𝜆 e22 = 𝜆 and

( ) 𝜑𝜆 eij = 0,

if

i ≠ j.

Let 𝜓 be the state on ℒ ∞ ([0, 1]) induced by the Lebesgue measure on [0, 1]. Consider the free product state 𝜑 ∗ 𝜓 on the algebraic free product Mat2 (ℂ) ∗ ∞ ([0, 1]) and the free product von Neumann algebra 𝒜 = Mat2 (ℂ) ∗ ℒ ∞ ([0, 1]) obtained by the GNS construction with respect to 𝜑𝜆 ∗ 𝜓. ( ) Then 𝒜 is a type III𝜆 factor with core isomorphic to 𝔏 F∞ ⊗ ℬ (ℋ ). Later, this was generalized by Dykema (1994a). Further, Shlyakhtenko (1997) introduced a theory of quasi-free factors that contained the above examples. Invariants for such factors were widely investigated. Related references are Houdayer and Vaes (2013) and Houdayer, Shlyakhtenko, and Vaes (2019). 30.14. Notes. The existence and (essential) uniqueness of the discrete decomposition for factors of types III𝜆 (0 ≤ 𝜆 < 1), as well as the computation of Out(ℳ) in terms of the discrete decomposition, are due to Connes (1973a). The connection between the discrete and the continuous decompositions was established by Connes and Takesaki (1977). The results mentioned in Section 30.12 are due to Haagerup (1979b); Connes and Størmer (1978).

414

Discrete Decompositions

The structure theory developed by Connes (1973a, 1974b) for factors of types III𝜆 (0 ≤ 𝜆 < 1), made necessary the study of outer conjugacy classes of *-automorphisms of W ∗ -algebras. This work was done by Connes (1977a, 1975e); see also Connes (1976b, 1978b) for the approximately finite dimensional factors of types II1 and II∞ . A similar classification for measure space automorphisms was obtained by Connes and Krieger (1977). From the work of Araki and Woods (1968) (see also Connes, 1973a, 3.6.3) it is known that the Powers factor ℛ𝜆 is the only Araki–Woods factor of type III𝜆 (0 < 𝜆 < 1). As a consequence of the work of Connes (1977a, 1975e, 1976a), we now know that ℛ𝜆 is in fact the only injective factor of type III𝜆 (0 < 𝜆 < 1). Nevertheless, for each fixed 0 < 𝜆 < 1, there exists an uncountable infinity of nonisomorphic (noninjective) factors of type III𝜆 (Connes, 1973a, 4.4.5). Connes (1976a, 1975e) proved that any injective factor of type III0 is a Krieger factor so that, according to the classification theorem of Krieger [1976b], it follows that two injective factors of type III0 are *-isomorphic if and only if their flows of weights are isomorphic; moreover, any ergodic nontransitive flow appears in this way. A direct proof of this result has been proposed by Connes (1978b, p. 476). There are also uncountable many noninjective nonisomorphic factors of type III0 . There are factors of type III1 for which there is nothing resembling a discrete decomposition (Connes, 1974d, 5.5); this fact makes the classification of type III1 factors more difficult. Araki and Woods (1968) proved the existence of a unique Araki–Woods factor of type III1 . A. Connes conjectured and U. Haagerup (1987) proved the uniqueness of the approximate finite dimensional factor of type III1 (see also Connes (1985)). There are uncountably many nonisomorphic factors of type III1 (Connes, 1974d, 4.5). For details concerning the classification of injective factors we refer to the fundamental contributions of Connes (1977a, 1975e, 1976a) and the survey articles of Connes (1974b, 1976b, 1978b, 1977b). The classification of noninjective factors is an open problem. A definite negative result in this direction says that the Borel space of isomorphism classes of noninjective type III factors acting on a separable Hilbert space is not standard and not even countably separated (Woods, 1973; Connes, 1973a, 3.6.4; Connes, 1974d, 4.5). So far as type II1 factors are concerned, besides the uncountable infinity pointed out by D. McDuff and S. Sakai (1971), we have the examples given by Connes (1975b). For our exposition, we have used Connes (1973a, 1978b); Connes and Størmer (1978); Connes and Takesaki (1977); and Haagerup (1979b).

Appendix

In this section, we review some facts used in the main text concerning positive self-adjoint operators in Hilbert spaces, W∗ -algebras and infinite tensor products. Throughout this section, ℋ will denote a complex Hilbert space. A.1. Let A be a positive self-adjoint linear operator on ℋ with domain D(A) ⊂ ℋ . For each n ∈ ℕ, we put en (A) = 𝜒[0,n] (A) = 𝜒[1∕(n+1),1] ((1 + A)−1 ) (see [L], 9.9) and for each 𝜀 > 0, we define A𝜀 = A(1 + 𝜀A)−1 ; note that 𝜀A𝜀 + (1 + 𝜀A)−1 = 1. From ([L], 9.9–9.11) we see, using Lebesgue’s dominated convergence theorem, that a vector 𝜉 ∈ ℋ belongs to D(A1∕2 ) if and only if ‖A1∕2 𝜉‖2 = lim (Aen (A)𝜉|𝜉) = lim(A𝜀 𝜉|𝜉) < +∞; 𝜀→0

n→∞

if 𝜉 ∉ D(A1∕2 ), then we put ‖A1∕2 𝜉‖ = +∞, so that the above limit gives ‖A1∕2 𝜉‖ for any vector 𝜉 ∈ ℋ. Recall ([L], 9.20) that the restriction of A to s(A)ℋ is the analytic generator of the so-continuous unitary representation ℝ ∋ t ↦ Ait ∈ ℬ(s(A)ℋ ); the operators Ait can be also regarded as partial isometries acting on the whole of ℋ . A.2. Let Ω a be a subset of ℂ. A continuous function f ∶ Ω → ℂ is said to be operator continuous if the mapping {x ∈ ℬ(ℋ ); x normal, Sp(x) ⊂ Ω} ∋ x ↦ f(x) ∈ B(H) is continuous with respect to the so-topology and the s-topology. Theorem (I. Kaplansky, R. V. Kadison). Let Ω ⊂ ℂ be such that (Ω∖Ω) ∩ Ω = ∅ and let f ∶ Ω → ℂ be a continuous function such that sup{|f(𝜆)|(1 + |𝜆|)−1 ; 𝜆 ∈ Ω} < +∞. Then f is operator continuous. Proof. We first prove the theorem for Ω = ℂ. Note that the functions 𝜆 ↦ 𝜆, 𝜆 ↦ 𝜆̄ and 𝜆 ↦ (1 + |𝜆|2 )−1 are operator continuous on ℂ. Indeed, if xi , x ∈ ℬ(ℋ ) are normal operators such that so

xi → x and 𝜉 ∈ ℋ , then ‖x∗i 𝜉 − x∗ 𝜉‖2 = ‖x∗i 𝜉‖2 − (x∗i 𝜉|x∗ 𝜉) − (x∗ 𝜉|x∗i 𝜉) + ‖x∗ 𝜉‖2 = ‖x∗i 𝜉‖2 − (𝜉|xi x∗ 𝜉) − (xi x∗ 𝜉|𝜉) + ‖x𝜉‖2 → 0

415

416

Appendix

and ‖(1 + x∗i xi )−1 𝜉 − (1 + x∗ x)−1 𝜉‖ = ‖(1 + x∗i xi )−1 [x∗i (xi − x) + (x∗i − x∗ )x](1 + x∗ x)−1 𝜉‖ ≤ ‖(xi − x)(1 + x∗ x)−1 𝜉‖ + ‖(x∗i − x∗ )x(1 + x∗ x)−1 𝜉‖ → 0. A similar argument holds for the s-topology. Denote by 𝒞 the set of all operator continuous functions on ℂ and by 𝒞b the subset of bounded functions in 𝒞 . Then 𝒞 is a uniformly closed self-adjoint vector subspace of the *-algebra of all continuous complex functions on ℂ and 𝒞b 𝒞 ⊂ 𝒞 . Hence 𝒞b is a uniformly closed *-subalgebra containing the functions 𝜆 → (1+|𝜆|2 )−1 and 𝜆 ↦ 𝜆(1+|𝜆|2 )−1 . By the Stone–Weierstrass theorem, it follows that 𝒞b contains the *-subalgebra 𝒞0 (ℂ) of continuous complex functions vanishing at infinity on ℂ. Let f ∶ ℂ → ℂ be a continuous function such that sup{|f(𝜆)|(1+|𝜆|)−1 ; 𝜆 ∈ ℂ} < +∞. Then 𝜆 → (1 + |𝜆|2 )−1 f(𝜆) belongs to 𝒞0 (ℂ) ⊂ 𝒞b and since 𝒞b 𝒞 ⊂ 𝒞 it follows that 𝜆 → ̄ + |𝜆|2 )−1 f(𝜆) belongs to 𝒞 (actually to 𝒞b ), so that 𝜆 ↦ 𝜆𝜆(1 ̄ + |𝜆|2 )−1 f(𝜆) belongs to 𝒞 and we 𝜆(1 conclude that f, as a sum of two functions in 𝒞 , also belongs to 𝒞 . Consider now the general case. Let xi , x ∈ ℬ(ℋ ) be normal operators with Sp(xi ) and Sp(x) so

contained in Ω and xi → x. Since (Ω∖Ω) ∩ Ω = ∅, there exists a compact neighborhood N of Sp(x) such that (Ω∖Ω) ∩ N = ∅. Then Ω ∩ N = Ω ∩ N, hence Ω is closed in Ω ∪ N and, by the Tietze– Urysohn extension theorem, f can be extended to a continuous function g on Ω ∪ N. Let h ∶ ℂ → ℂ be a continuous function such that supp h ⊂ N and h(𝜆) = 1 for 𝜆 ∈ Sp(x). We obtain a continuous function with compact support by putting k(𝜆) = g(𝜆)h(𝜆) for 𝜆 ∈ Ω ∪ N and k(𝜆) = 0 for 𝜆 ∉ Ω ∪ N. Consider also the functions l ∶ ℂ → ℂ, F ∶ Ω → ℂ defined by l(𝜆) = (1 + |𝜆|)(1 − h(𝜆)) and F(𝜆) = f(𝜆)(1 + |𝜆|)−1 . By construction, l(x) = 0 and, by assumption, there exists a 𝜇0 ∈ (0, +∞) such that |F(𝜆)| ≤ 𝜇0 for all 𝜆 ∈ Ω. Since f(𝜆) = f(𝜆)h(𝜆) + f(𝜆)(1 − h(𝜆)) = k(𝜆) + F(𝜆)l(𝜆) (𝜆 ∈ Ω), it follows that f(x) − f(xi ) = [k(x) − k(xi )] + F(xi )[l(x) − l(xi )]. so

so

By the first part of the proof, we have k(xi ) → k(x) and l(xi ) → l(x) = 0, while so ‖F(xi )‖ remains so

bounded by 𝜇0 , so that f(xi ) → f(x). The same argument applies for the s-topology. Hence f is operator continuous. Note that this result holds in particular for Ω = [0, +∞). so

so

A.3. Let Ak , A be positive self-adjoint operators on ℋ . We shall write Ak → A if (1 + Ak )−1 → so

so

(1 + A)−1 . If Ak → A, then Aitk → Ait for all t ∈ ℝ. Actually, we have (see Reed & Simon, 1972, VIII.21 and Exercise VIII.21; Araki & Woods, 1973; Kallman, 1971): so

so

so

so

Ak → A ⇔ Aitk → Ait for all t ∈ ℝ

(1)

Ak → A ⇔ Aitk → Ait uniformly for |t| ≤ t0 .

(2)

It follows that so

so

so

Ak → A ⇔ (1 + 𝜀Ak )−1 → (1 + 𝜀A)−1 ⇔ (Ak )𝜀 → A𝜀

(𝜀 > 0)

(3)

Appendix

417

Proposition. Let Ak , A be positive self-adjoint operators on ℋ . If there exists a vector subspace so ⋂ D ⊂ D(A) ∩ k D(Ak ) such that A|D = A and Ak 𝜉 → A𝜉 for all 𝜉 ∈ D, then Ak → A. Proof. Indeed, let 𝜉 ∈ D and 𝜂 = (1 + A)𝜉. Then ‖(1 + A)−1 𝜂 − (1 + Ak )−1 𝜂‖ = ‖(1 + Ak )−1 (Ak 𝜉 − A𝜉)‖ ≤ ‖Ak 𝜉 − A𝜉‖ → 0. so

Since A|D = A, by ([L], E.9.1), (1 + A)D is dense in ℋ and (1 + Ak )−1 → (1 + A)−1 . A.4. Let A, B be positive self-adjoint operators on ℋ . We shall write A ≤ B if (1 + A)−1 ≥ (1 + B)−1 . This condition means that ([L], E.2.6) (1+B)−1∕2 x = (1+A)−1∕2 x for some x ∈ ℬ(ℋ ), ‖x‖ ≤ 1, that is, D((1 + B)1∕2 ) ⊂ D((1 + A)1∕2 ) and ‖(1 + A)1∕2 𝜉‖2 ≤ ‖(1 + B)1∕2 𝜉‖2 for every 𝜉 ∈ D((1 + B)1∕2 ). Using ([L], E.9.29) we infer that A ≤ B ⇔ D(B1∕2 ) ⊂ D(A1∕2 ) and ‖A1∕2 𝜉‖ ≤ ‖B1∕2 𝜉‖, (𝜉 ∈ D(B1∕2 )).

(1)

A ≤ B ⇔ (1 + 𝜀A)−1 ≥ (1 + 𝜀B)−1 ⇔ A𝜀 ≤ B𝜀 ; (𝜀 > 0) A ≤ B and B ≤ A ⇔ A = B.

(2) (3)

It follows that

Also, if A ≤ B, then A1∕2 ≤ B1∕2 and x + A ≤ x + B for every x ∈ ℬ(ℋ )+ . Note that there exist characterizations of the relation A ≤ B in terms of the so-continuous unitary representations associated with A and B, which are in fact particular cases of Corollary 3.13 and Proposition 4.5 in the main text. A.5. Let Ak , A be positive self-adjoint operators on ℋ . We shall write Ak ↑ A if (1+Ak )−1 ↓ (1+A)−1 ; so

this means that {(1 + Ak )−1 } is a decreasing net and (1 + Ak )−1 → (1 + A)−1 . Using (A.2, A.4) we obtain Ak ↑ A ⇔ (1 + 𝜀Ak )−1 ↓ (1 + 𝜀A)−1 ⇔ (Ak )𝜀 ↑ A𝜀

(𝜀 > 0).

(1)

Note that A𝜀 ↑ A for 𝜀 ↓ 0

(2)

since 0 < 𝜀′ ≤ 𝜀 ⇒ A𝜀 ≤ A𝜀′ ≤ A and, with a = (1 + A)−1 , we have so

(1 + A𝜀 )−1 = (a + 𝜀(1 − a))(1 + 𝜀(1 − a))−1 → a = (1 + A)−1 by ([L], 2.20). Also, Aen (A) ↑ A for n ↑ ∞

(3)

since m ≤ n ⇒ Aem (A) ≤ Aen (A) and so

(1 + Aen (A))−1 = (1 − en (A)) + (1 + A)−1 en (A) → (1 + A)−1 .

418

Appendix

Let Bk , B to other positive self-adjoint operators on ℋ . Using (1) and (A.4) it is easy to check that Ak ↑ A, Bk ↑ B, Ak ≤ Bk for all k ⇒ A ≤ B.

(4)

Proposition. Let {Ak } be an increasing net of positive self-adjoint operators on ℋ . There exists a positive self-adjoint operator A on ℋ such that Ak ↑ A if and only if the vector subspace D = {𝜉 ∈ 1∕2 ℋ ; limk ‖Ak 𝜉‖ < +∞} is dense in ℋ . In this case, D(A1∕2 ) = D. 1∕2

Proof. Assume that Ak ↑ A. Since (Ak )𝜀 ↑ A𝜀 , (Ak )𝜀 ↑𝜀 Ak and ‖Ak 𝜉‖2 = lim𝜀 (A𝜀 𝜉|𝜉) for every 1∕2 𝜉 ∈ ℋ , we have 𝜉 ∈ D(A1∕2 ) ⇒ ((Ak )𝜀 𝜉|𝜉) ≤ (A𝜀 𝜉|𝜉) ≤ ‖Ak 𝜉‖2 < +∞ for all k and 𝜀 ⇒ 1∕2 (Ak 𝜉|𝜉) ≤ ‖Ak 𝜉‖2 for all k ⇒ 𝜉 ∈ D and 𝜉 ∈ D ⇒ ((Ak )𝜀 𝜉|𝜉) ≤ (Ak 𝜉|𝜉) ≤ 𝜇 < +∞ for all k and 𝜀 ⇒ (A𝜀 𝜉|𝜉) ≤ 𝜇 < +∞ ⇒ ‖A1∕2 𝜉‖2 ≤ 𝜇 < +∞ ⇒ 𝜉 ∈ D(A1∕2 ). Therefore D = D(A1∕2 ) is dense in ℋ . Conversely, assume that D is dense in ℋ . Since the net {Ak } is increasing, we infer from ([L], 2.16) that (1 + Ak )−1 ↓ a for some a ∈ ℬ(ℋ ), 0 ≤ a ≤ 1. If 𝜂 ∈ ℋ and a𝜂 = 0, then for every 𝜉 ∈ D we have (|𝜉|𝜂)|2 = |(𝜉|(1 + Ak )1∕2 (1 + Ak )−1∕2 𝜂)|2 = |((1 + Ak )1∕2 𝜉|(1 + Ak )−1∕2 𝜂)|2 ≤ ((1 + Ak )𝜉|𝜉)((1 + Ak )−1 𝜂|𝜂)| → 0, hence 𝜂 ⟂ D and 𝜂 = 0 since D is dense in ℋ . Thus, a is injective, so that A = a−1 − 1 is a positive self-adjoint operator on ℋ . Moreover, (1 + Ak )𝜀 = (𝜀 + (1 + Ak )−1 )−1 ↑ (𝜀 + a)−1 = (1 + A)𝜀 , so that 1 + Ak ↑ 1 + A and Ak ↑ A. A.6. Let A, B be positive self-adjoint operators on ℋ and assume that A and B commute ([L], E.9.24). Then ⋃ D= en (A)em (B)ℋ n,m

is a dense vector subspace of ℋ and the linear operator AB defined by (AB)𝜉 = A(B𝜉) (𝜉 ∈ D) is positive and AB ⊂ (AB)∗ , so that AB is preclosed, AB is positive and AB ⊂ (AB)∗ . Conversely, let 𝜂 ∈ D((AB)∗ ). There exists 𝜂 ∗ = (AB)∗ 𝜂 ∈ ℋ such that (AB𝜉|𝜂) = (𝜉|𝜂 ∗ ) for all 𝜉 ∈ D. With pnm = en (A)em (B) we have D ∋ pnm 𝜂 → 𝜂 and, for every 𝜉 ∈ D, m,n

(𝜉|ABpnm 𝜂) = (𝜉|(AB)∗ pnm 𝜂) = (AB𝜉|pnm 𝜂) = (ABpnm 𝜉|𝜂) = (pnm 𝜉|𝜂 ∗ ) = (𝜉|pnm 𝜂 ∗ ), whence ABpnm 𝜂 = pnm 𝜂 ∗ → 𝜂 ∗ = (AB)∗ 𝜂. Thus, 𝜂 ∈ D(AB) and (AB)𝜂 = (AB)∗ 𝜂. We conclude n,m

that AB is a positive self-adjoint operator on ℋ . Since Aen (A)Bem (B)𝜉 → AB𝜉 for all 𝜉 ∈ D, by n,m

Appendix

419

Proposition A.3 we obtain Aen (A)Bem (B) ↑ AB.

(1)

Also, for fixed n, m ∈ ℕ, 𝜉 ∈ ℋ and 𝜀 ↓ 0, 𝛿 ↓ 0 we have A𝜀 B𝛿 en (A)em (B)𝜉 = (Aen (A))𝜀 (Bem (B))𝛿 𝜉 → Aen (A)Bem (B)𝜉 = ABen (A)em (B)𝜉 whence A𝜀 B𝛿 ↑ AB.

(2)

In particular, AB = BA. Using ([L], 9.20) it is easy to check that (AB)it = Ait Bit

(t ∈ ℝ).

(3)

A.7. A positive quadratic form on ℋ is a mapping q ∶ D(q) → [0, +∞) such that D(q) is a dense vector subspace ofℋ

(1)

q(𝜆𝜉) = |𝜆| q(𝜉) for all 𝜉 ∈ D(q) and all 𝜆 ∈ ℂ q(𝜉 + 𝜂) + q(𝜉 − 𝜂) = 2q(𝜉) + 2q(𝜂) for all 𝜉, 𝜂 ∈ D(q).

(2) (3)

2

Then q(𝜉, 𝜂) =

1 (q(𝜉 + 𝜂) − q(𝜉 − 𝜂) + iq(𝜉 + i𝜂) − iq(𝜉 − i𝜂)) 4

(𝜉, 𝜂 ∈ D(q))

defines a positive sesquilinear form q ∶ D(q) × D(q) → ℂ, uniquely determined, such that q(𝜉, 𝜉) = q(𝜉)

(𝜉 ∈ D(q)).

The positive quadratic form q is said to be closed if D(q) is complete with respect to the scalar product (𝜉|𝜂)q = (𝜉|𝜂) + q(𝜉, 𝜂), (𝜉, 𝜂 ∈ D(q)), that is, if the conditions {𝜉n } ⊂ D(q), 𝜉 ∈ ℋ , ‖𝜉n − 𝜉‖ → 0, q(𝜉n − 𝜉m ) → 0 imply 𝜉 ∈ D(q) and q(𝜉n − 𝜉) → 0. For every positive self-adjoint operator A on ℋ , we obtain a closed positive quadratic form qA on ℋ by putting qA (𝜉) = ‖A1∕2 𝜉‖2

(𝜉 ∈ D(qA ) = D(A1∕2 )).

Conversely, Proposition. Let q be a closed positive quadratic form on ℋ . There exists a unique positive selfadjoint operator A on ℋ such that qA = q.

420

Appendix

Proof. For any fixed 𝜉 ∈ ℋ , the mapping D(q) ∋ 𝜂 ↦ (𝜉|𝜂) is a bounded linear functional on the Hilbert space D(q) with norm ≤ ‖𝜉‖ (since |(𝜉|𝜂)| ≤ ‖𝜉‖‖𝜂‖ ≤ ‖𝜉‖‖𝜂‖q for 𝜂 ∈ D(q)) and therefore there is a unique vector T𝜉 ∈ D(q), ‖T𝜉‖q ≤ ‖𝜉‖, such that (𝜉|𝜂) = (T𝜉|𝜂)q = (T𝜉|𝜂) + q(T𝜉, 𝜂)

(𝜂 ∈ D(q)).

(4)

We thus obtain a bounded linear operator T ∶ ℋ → D(q) with norm ≤ 1 such that q(T𝜉, 𝜂) = (𝜉 − T𝜉|𝜂)

(𝜉 ∈ ℋ , 𝜂 ∈ D(q)).

(5)

From (4) it follows that T is injective and T(ℋ ) is dense in the Hilbert space D(q), so that T(ℋ ) is dense in ℋ . Therefore, we obtain a densely defined linear operator A, on ℋ by putting A𝜁 = T−1 𝜁 − 𝜁

(𝜁 ∈ D(A) = T(ℋ ) ⊂ D(q))

and we have q(𝜁, 𝜂) = (A𝜁 |𝜂) (𝜁 ∈ D(A), 𝜂 ∈ D(q)).

(6)

It follows that A is positive and, as (1 + A)D(A) = T−1 (T(ℋ )) = ℋ , A is self-adjoint ([L], 9.5). For 𝜁 ∈ D(A) ⊂ D(A1∕2 ), we have ‖A1∕2 𝜁‖2 = (A𝜁|𝜁) = q(𝜁). Let 𝜉 ∈ D(q). Since D(A) = T(ℋ ) is dense in the Hilbert space D(q), there exists a sequence {𝜉n } ⊂ D(A) such that ‖𝜉n − 𝜉‖ → 0 and ‖A1∕2 𝜉n − A1∕2 𝜉m ‖2 = q(𝜉n − 𝜉m ) → 0. Since A1∕2 is closed, it follows that 𝜉 ∈ D(A1∕2 ) and ‖A1∕2 𝜉n − A1∕2 𝜉‖ → 0, so that ‖A1∕2 𝜉‖2 = limn ‖A1∕2 𝜉n ‖2 = limn q(𝜉n ) = q(𝜉). Conversely, using the fact that A1∕2 = A1∕2 |D(A) and the closedness of the form q, we see that D(A1∕2 ) ⊂ D(q). Hence qA = q. The uniqueness of A, subject to this condition, follows from (6). A.8. Let p, q be two positive quadratic forms on ℋ . We say that p is an extension of q if D(q) ⊂ D(p) and p(𝜉) = q(𝜉) for every 𝜉 ∈ D(q). The form q is said to be preclosed (or closable) if it has a closed extension. Proposition. A positive quadratic form q on ℋ is preclosed if and only if {𝜉}n ⊂ D(q), ‖𝜉n ‖ → 0, q(𝜉n − 𝜉m ) → 0 ⇒ q(𝜉n ) → 0.

(1)

In this case, there exists a closed extension q̄ of q such that any closed extension of q is also an extension of q̄ . Proof. Assume that q is preclosed and let p be a closed extension of q. Then from the assumptions in (1) it follows that q(𝜉n ) = p(𝜉n − 0) → 0. Conversely, assume that (1) holds. Let D(̄q) be the set of all 𝜉 ∈ ℋ such that there exists a sequence {𝜉n } ⊂ D(q) with the properties ‖𝜉n − 𝜉‖ → 0 and q(𝜉n − 𝜉m ) → 0. From (1) it follows that for 𝜉 ∈ D(̄q) the number q̄ (𝜉) = limn q(𝜉n ) does not depend on the sequence {𝜉n } ⊂ D(q) with ‖𝜉n − 𝜉‖ → 0 and q(𝜉n − 𝜉m ) → 0. It is then easy to check that q̄ ∶ D(̄q) → [0, +∞) is a closed extension of q and that any closed extension of q is also an extension of q̄ .

Appendix

421

The form q̄ is called the closure of q. A.9. Consider again a positive quadratic form q ∶ D(q) → [0, +∞) on ℋ . We extend the definition of q to the whole of ℋ by putting q(𝜉) = +∞ for 𝜉 ∈ ℋ ∖D(q). Proposition. Let q be a positive quadratic form on ℋ . Then q is closed ⇔ q is lower semicontinuous on ℋ ; q is preclosed ⇔ q is lower semicontinuous on D(q).

(1) (2)

Proof. Assume that q is closed and let A be the unique positive self-adjoint operator on ℋ such that q = qA (A.7). Then q(𝜉) = sup{|(A𝜁|𝜉)|; 𝜁 ∈ D(A), (A𝜁|𝜁) = 1}

(𝜉 ∈ ℋ ).

(3)

Indeed, for every 𝜁 ∈ D(A) ⊂ D(A1∕2 ) and 𝜉 ∈ D(q) we have |(A𝜁|𝜉)|2 = |(A1∕2 𝜁|A1∕2 𝜉)|2 ≤ ‖A1∕2 𝜁‖2 ‖A1∕2 𝜉‖2 = q(𝜁)q(𝜉), hence |(A𝜁|𝜉)|2 ≤ q(𝜁 )q(𝜉)

(𝜁 ∈ D(A), 𝜉 ∈ ℋ )

(4)

and (3) follows replacing 𝜁 here by 𝜁n = (Aen (A)𝜉|𝜉)−1∕2 en (A)𝜉 and taking the limit as n → +∞. Now from (3) it is clear that q is (even weakly) lower semicontinuous on ℋ . If q is preclosed, then its closure q̄ is lower semicontinuous on ℋ , and q is lower semicontinuous on D(q). Conversely, assume that q is lower semicontinuous on D(q) and consider {𝜉n } ⊂ D(q), ‖𝜉n ‖ → 0, q(𝜉n − 𝜉m ) → 0. Let 𝜀 > 0 and choose n𝜀 ∈ ℕ such that q(𝜉n − 𝜉m ) ≤ 𝜀 for n, m ≥ n𝜀 . Since 𝜉n → 0 and q is lower semicontinuous on D(q), it follows that q(𝜉n ) ≤ limm inf q(𝜉n − 𝜉m ) ≤ 𝜀 for all n ≥ n𝜀 . Hence q(𝜉n ) → 0. By Proposition A.8, it follows that q is preclosed. Finally, assume that q is preclosed but is not closed. Then there exists 𝜉 ∈ D(̄q), 𝜉 ∉ D(q). By the construction of q̄ (A.8) there exists a sequence {𝜉n } ⊂ D(q), ‖𝜉n − 𝜉‖ → 0, such that limn q(𝜉n ) = q̄ (𝜉) < +∞ = q(𝜉). Thus q is not lower n semicontinuous on ℋ . A.10. Let q ∶ D(q) → [0, +∞) be a lower semicontinuous positive quadratic form on ℋ . By Propositions A.7–A.9, there exists a unique positive self-adjoint operator A on ℋ such that qA = q̄ , that is, ‖A1∕2 𝜉‖2 = q̄ (𝜉) (𝜉 ∈ D(A1∕2 ) = D(̄q)).

(1)

D(A1∕2 ) ⊃ D(q) and ‖A1∕2 𝜉‖2 = q(𝜉) for all 𝜉 ∈ D(q)

(2)

In particular,

and, as is easily verified, A1∕2 = A1∕2 |D(q).

(3)

422

Appendix

Moreover, A is the greatest positive self-adjoint operator on ℋ satisfying (2). Indeed, let B be another positive self-adjoint operator in ℋ such that D(B1∕2 ) ⊃ D(q) and ‖B1∕2 𝜉‖2 = q(𝜉) for all 𝜉 ∈ D(q). Let 𝜉 ∈ D(A1∕2 ) = D(̄q). There exists a sequence {𝜉n } ⊂ D(q) such that ‖𝜉n − 𝜉‖ → 0 and q(𝜉n ) → q̄ (𝜉). Then ‖B1∕2 ‖2 ≤ lim ‖B1∕2 𝜉n ‖2 = lim q(𝜉n ) = q̄ (𝜉) = ‖A1∕2 𝜉‖2 . n

n

Consequently, B ≤ A. Recall (A.9) that if q is lower semicontinuous on the whole ℋ , then q = q̄ is closed, so that in this case the best characterization of A is given by (1). A.11. Let A, B be positive self-adjoint operators in ℋ . Assume that D = D(A1∕2 ) ∩ D(B1∕2 ) is dense in ℋ . Then q ∶ ℋ ∋ 𝜉 → qA (𝜉) + qB (𝜉) ∈ [0, +∞] is a lower semicontinuous positive quadratic form on ℋ with D(q) = D. By (A.9) it follows that there exists a unique positive self-adjoint operator ̂ on ℋ such that A+B ̂ 1∕2 𝜉‖2 = ‖A1∕2 𝜉‖2 + ‖B1∕2 𝜉‖2 ‖(A+B)

̂ 1∕2 ) = D). (𝜉 ∈ D((A+B)

(1)

̂ is called the weak sum (or form sum) of A and B. The operator A+B A.12. Let 𝜑 ∶ ℝ∖{0} → ℝ be the function defined by 𝜑(t) = 2(e𝜋t − e−𝜋t )−1

(t ∈ ℝ∖{0}).

For any bounded continuous function f ∶ ℝ → ℂ, we shall write [

+∞

PV

∫−∞

𝜑(t)f(t) dt = lim

𝜀→0

−𝜀

∫−∞

]

+∞

𝜑(t)f(t) dt +

𝜑(t)f(t) dt

∫+𝜀

+∞

= lim

𝜀→0 ∫+𝜀

𝜑(t)(f(t) − f(−t)) dt

whenever the limit exists. Similar notation will be used for functions f with values in ℋ or ℬ(ℋ ). Proposition. For every nonsingular positive self-adjoint operator A on ℋ , we have +∞

PV

∫−∞

𝜑(t)Ait 𝜉 dt = i(A − 1)(A + 1)−1 𝜉

(𝜉 ∈ ℋ ).

(1)

Proof. Since (1 + A1∕2 )−1 + (1 + A−1∕2 )−1 = 1, we have D(A1∕2 ) + D(A−1∕2 ) = ℋ ; thus it is sufficient to prove (1) only for 𝜉 ∈ D(A1∕2 ) and for 𝜉 ∈ D(A−1∕2 ). Let 𝜉 ∈ D(A1∕2 ). Then ([L], 9.15) the function 𝛼 ↦ A𝛼 A1∕2 𝜉 is defined, bounded, continuous on the strip {𝛼 ∈ ℂ; −1∕2 ≤ Re 𝛼 ≤ 0} and analytic in the interior of the strip. On the other hand, the function 𝛼 ↦ 2(e−𝜋i𝛼 + e𝜋i𝛼 )−1 has the same properties, except at the point 𝛼 = −1∕2 where it has a pole with residue (𝜋i)−1 ; moreover, this function tends to zero when z → ∞ on this strip. Therefore,

Appendix

423

by integrating along the boundary of the strip the function 𝛼 ↦ 2(e−𝜋it + e𝜋it )−1 A𝛼 A1∕2 , we obtain +∞

∫−∞

2(e−𝜋t + e𝜋t )−1 Ait A1∕2 𝜉 dt = 𝜉 + PV

+∞

∫−∞

−i𝜑(t)Ait 𝜉 dt.

By ([L], Corollary 9.23), the left hand side is equal to 2A(A + 1)−1 𝜉 and (1) follows. For 𝜉 ∈ D(A−1∕2 ), note that (A − 1)(A + 1)−1 = −(A−1 − 1)(A−1 + 1)−1 and apply the first part of the proof. A.13. Let b ∈ ℬ(ℋ ), 0 ≤ b ≤ 1, and let A be a nonsingular positive self-adjoint on ℋ . Put +∞

a=

∫−∞

2(e−𝜋t + e𝜋t )−1 Ait bA−it dt ∈ ℬ(ℋ )

(1)

and note that 0 ≤ a ≤ 1. With the same arguments as in Section A.12, applied to the function 𝛼 ↦ 2(e−𝜋i𝛼 + e𝜋i𝛼 )−1 (bA−𝛼 A−1∕2 𝜉|A𝛼̄ A1∕2 𝜂) on the strip {𝛼 ∈ ℂ; −1∕2 ≤ Re 𝛼 ≤ 0}, we obtain +∞

PV

∫−∞

𝜑(t)(Ait bA−it 𝜉|𝜂) dt = i(aA−1∕2 𝜉|A1∕2 𝜂) − i(b𝜉|𝜂)

(2)

whenever 𝜉 ∈ D(A−1∕2 ) and 𝜂 ∈ D(A1∕2 ). +∞

Proposition. If 𝜉 ∈ D(ln(A)), then PV ∫−∞ 𝜑(t)Ait bA−it 𝜉 dt exists. If, moreover, 𝜉 ∈ D(A−1∕2 ), then aA−1∕2 𝜉 ∈ D(A1∕2 ) and +∞

A1∕2 aA−1∕2 𝜉 = b𝜉 − iPV

∫−∞

𝜑(t)Ait bA−it 𝜉 dt.

Proof. Note that Ait aA−it 𝜉 = Ait a𝜉 + Ait a(A−it − 1) and that if 𝜉 ∈ D(ln(A)), then (see Reed & Simon, 1972, Theorem VIII.7) the function t ↦ 𝜑(t)A a(A it

−it

[ −it ] 2t A −1 − 1)𝜉 = 𝜋t a 𝜉 e − e−𝜋t t

is continuous at t = 0 and hence everywhere; as it tends to zero rapidly at infinity it is integrable. This remark, combined with Proposition A.12 proves the first statement. The second statement is now an obvious consequence of (2) since A1∕2 is self-adjoint. A.14. Let ℋ be the Hilbert space ℒ 2 (ℝ). Consider the operator b ∈ ℬ(ℋ ), 0 ≤ b ≤ 1, defined as multiplication by the function b(s) = 1 when s < 0 and b(s) = 0 when s ≥ 0, the unique nonsingular positive self-adjoint operator A on ℋ such that ([L], 9.20) [Ait 𝜉](s) = 𝜉(s − t) (𝜉 ∈ ℒ 2 (ℝ), s, t ∈ ℝ)

424

Appendix

and the corresponding operator a ∈ ℬ(ℋ ), 0 ≤ a ≤ 1, defined by A.13.(1). Note that if we replace A by A−1 , the operator a remains unchanged. We shall show, by an indirect argument, that either there exists 𝜁 ∈ D(A−1∕2 ) such that aA−1∕2 𝜁 ∉ D(A1∕2 ) or there exists 𝜁 ∈ D(A

1∕2

) such that aA

1∕2

𝜁 ∉ D(A

−1∕2

(1)

).

So assume that 𝜁 ∈ D(A−1∕2 ) ⇒ aA−1∕2 𝜁 ∈ D(A1∕2 ) and 𝜁 ∈ D(A1∕2 ) ⇒ aA1∕2 𝜁 ∈ D(A−1∕2 ). Then we can define the linear operators T1 , T2 on ℋ by T1 𝜁 = A1∕2 aA−1∕2 𝜁 − b𝜁 (𝜁 ∈ D(A−1∕2 )) T2 𝜁 = −A−1∕2 aA1∕2 𝜁 + b𝜁 (𝜁 ∈ D(A1∕2 )). For 𝜉 ∈ D(A−1∕2 ) ∩ D(A1∕2 ), we have 𝜉 ∈ D(ln(A)) (see Reed & Simon, 1972, Theorem VIII.7 and [L], Corollary 9.21) and hence, by Proposition A.13, +∞

T1 𝜉 = −iPV

∫−∞

𝜑(t)Ait bA−it 𝜉 dt = T2 𝜉.

As ℋ = D(A−1∕2 ) + D(A1∕2 ), we can define a linear operator T on ℋ by putting T(𝜉1 + 𝜉2 ) = T1 𝜉1 + T2 𝜉2

(𝜉1 ∈ D(A−1∕2 ), 𝜉2 ∈ D(A1∕2 )).

For 𝜉 ∈ D(A−1∕2 ) ∩ D(A1∕2 ) and 𝜂 ∈ D(A−1∕2 ), we have (−T𝜉|𝜂) = (−T2 𝜉|𝜂) = (A−1∕2 aA1∕2 𝜉|𝜂) − (b𝜉|𝜂) = (𝜉|A1∕2 aA−1∕2 𝜂) − (𝜉|b𝜂) = (𝜉|T1 𝜂) = (𝜉|T𝜂). Hence −T ⊂ T∗ , so that T is closed and hence bounded. Furthermore, for 𝜁 ∈ D(A−1∕2 ) ∩ D(ln(A)) and 𝜁 ∈ D(A1∕2 ) ∩ D(ln(A)) we have +∞

T𝜁 = −iPV

𝜑(t)Ait bA−it 𝜁 dt.

∫−∞

Since D(ln(A)) = D(ln(A)) ∩ D(A−1∕2 ) + D(ln(A)) ∩ D(A1∕2 ), this identity holds for any 𝜁 ∈ D(ln(A)). As T is bounded, it follows that the mapping +∞

D(ln(A)) ∋ 𝜁 ↦ PV

∫−∞

𝜑(t)Ait bA−it 𝜁 dt ∈ ℋ

(2)

is continuous. However, using again Proposition A.13 and Fubini’s theorem, for 𝜁 ∈ D(ln(A)) and 𝜂 ∈ ℋ we get +∞

PV

∫−∞

+∞

𝜑(t)(Ait bA−it 𝜁|𝜂) dt =

∫−∞

𝜓(s)𝜁(s)𝜂(s) ds

Appendix

425 +∞

with 𝜓(s) = ∫|s| 𝜑(t) dt (s ∈ ℝ). Since 𝜑(t) ∼ t−1 for small |t|, 𝜑 is not integrable and therefore 𝜓 is not bounded. In this case, it is easy to see that the mapping (2) cannot be continuous. Indeed, consider the functions 𝜁n (t) = n1∕2 exp(−n2 t2 ), t ∈ ℝ. Then 𝜁n ∈ D(ln(A)) (since the 𝜁n are “entire analytic +∞ vectors” for t ↦ Ait ), ‖𝜁n ‖ is independent of n, while ∫−∞ 𝜓(s)𝜁n (s)𝜂(s) ds → +∞ whenever 𝜂 is not zero in a neighborhood of s = 0. This contradiction proves (1). In the next sections, we review some basic facts concerning W∗ -algebras. A.15. Let 𝒜 be a C∗ -algebra, let ℋ𝒜 = ⊕{ℋ𝜑 ; 𝜑 ∈ S(𝒜 )} and let 𝜋𝒜 =

⨂ 𝜑∈S(𝒜 )

𝜋𝜑 ∶ 𝒜 → ℬ(ℋ𝒜 )

where S(𝒜 ) denotes the set of all positive linear forms 𝜑 on 𝒜 with ‖𝜑‖ = 1 and 𝜋𝜑 ∶ 𝒜 → ℬ(ℋ𝒜 ) is the corresponding Gelfand–Naimark–Segal representation ([L], 5.18; Sakai, 1971; Strătilă & Zsidó, 1977, 2005). Then 𝜋𝒜 is an isometric nondegenerate *-representation of 𝒜 , called the universal *-representation of 𝒜 , the w-closure 𝒩𝒜 of 𝜋𝒜 (𝒜 ) is a von Neumann subalgebra of ℬ(ℋ𝒜 ) called the enveloping von Neumann algebra of 𝒜 and the following statements hold: For every 𝜓 ∈ 𝒜 ∗ , there exists a unique wo-continuous form F𝒜 (𝜓) on 𝒩𝒜 such that 𝜓 = F𝒜 (𝜓)◦𝜋𝒜 and the map F𝒜 is a linear isometry of A∗ onto (𝒩𝒜 )∗ . The map Φ𝒜 ∶ 𝒩𝒜 → 𝒜 ∗∗ defined by [Φ𝒜 (y)](𝜓) = [F𝒜 (𝜓)](y) (𝜓 ∈ A∗ , y ∈ 𝒩𝒜 ) is a surjective linear isometry and a (w, 𝜎(𝒜 ∗∗ , 𝒜∗ ))-homeomorphism; Φ𝒜 ◦𝜋𝒜 is the canonical embedding of 𝒜 into 𝒜 ∗∗ . A.16. A W∗ -algebra is a C∗ -algebra ℳ which is isometrically isomorphic to the dual space of some Banach space. In this case, there is a unique norm-closed vector subspace ℱ of ℳ ∗ such that the map Φ ∶ ℳ → ℱ ∗ defined by [Φ(x)](𝜓) = 𝜓(x) (x ∈ ℳ, 𝜓 ∈ ℱ ), is a surjective linear isometry. We write ℳ∗ = ℱ and call ℳ∗ the predual of ℳ. Let 𝜋ℳ ∶ ℳ → 𝒩ℳ ⊂ ℬ(ℋℳ ) be the universal *-representation of ℳ. There exists a unique central projection pℳ in 𝒩ℳ such that the map x ↦ 𝜋ℳ (x)pℳ is a *-isomorphism of ℳ onto the von Neumann algebra 𝒩ℳ pℳ ⊂ ℬ(pℳ ℋℳ ). Moreover, this map is a (𝜎(ℳ, ℳ∗ ), w) homeomorphism and ℳ∗ = pℳ ℳ ∗ . ∑ ∑ A linear form 𝜓 ∈ ℳ ∗ satisfies 𝜓 = pℳ ⋅𝜓, that is, 𝜓 ∈ ℳ∗ , if and only if 𝜓( i∈I ei ) = i∈I 𝜓(ei ) for every family {ei }i∈I of mutually orthogonal projections in ℳ. In this case, 𝜓 is called a normal linear form on ℳ. A linear functional 𝜓 ∈ ℳ ∗ satisfies pℳ ⋅ 𝜓 = 0 if and only if for every nonzero projection e ∈ ℳ there exists a nonzero projection f ∈ ℳ, f ≤ e, with 𝜓(f) = 0; moreover, f can be chosen such that f ⋅ 𝜓 ⋅ f = 0. In this case, 𝜓 is called a singular linear form on ℳ. Thus, W∗ -algebras are essentially the same thing as von Neumann algebras. The 𝜎(ℳ, ℳ∗ )topology on ℳ is also called the w-topology on ℳ. The s-topology (resp. the s∗ -topology) on ℳ is defined by the seminorms x ↦ 𝜓(x∗ x)1∕2 (resp. x ↦ 𝜓(x∗ x + xx∗ )1∕2 where 𝜓 ranges over all normal positive forms on ℳ. The uniqueness of the predual of a W∗ -algebra shows that every *-isomorphism between ∗ W -algebras is w-continuous, and also s-continuous and s∗ -continuous. Every *-isomorphism of a W∗ -algebra ℳ onto a von Neumann algebra will be called a realization of ℳ as a von Neumann algebra. A W∗ -algebra ℳ with separable predual has a realization as a von Neumann algebra acting on a separable Hilbert space.

426

Appendix

A W∗ -subalgebra of the W∗ -algebra ℳ is a w-closed *-subalgebra of ℳ. Note that, while every ℳ has a unit element 1ℳ , the unit element of a W∗ -subalgebra 𝒩 of ℳ can be a proper projection e𝒩 in ℳ. If e𝒩 = 1ℳ , then 𝒩 is called a unital W∗ -subalgebra of ℳ. If 𝒜 is any C∗ -algebra, there exists a unique C∗ -algebra structure on the Banach space 𝒜 ∗∗ such that the canonical image of 𝒜 in 𝒜 ∗∗ is a C∗ -subalgebra of 𝒜 . Moreover, the C∗ -algebra 𝒜 ∗∗ is a W∗ -algebra, *-isomorphic to the enveloping von Neumann algebra of 𝒜 . Let Φ be a *-isomorphism of the von Neumann algebra ℳ ⊂ ℬ(ℋ ) onto the von Neumann algebra 𝒩 ⊂ ℬ(ℋ ). By ([L], 9.25), 𝜙 extends to a one-to-one correspondence A ↦ B = Φ(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 ; Φ is unique subject to the condition Φ((1 + A)−1 ) = (1 + B)−1 . It is therefore meaningful to speak about positive self-adjoint operators A affiliated to a W∗ -algebra ℳ without explicitly mentioning the specific realization of ℳ as a von Neumann algebra. For the proof of the preceding results, we refer to Sakai (1971); Strătilă and Zsidó (1977, 2005); Takesaki (1969–1970). W∗ -algebra

A.17. For each k ∈ ℕ, consider a Hilbert space ℋk and a vector 𝜉k ∈ ℋk , ‖𝜉k ‖ = 1. Then the mappings ̄ … ⊗ℋ ̄ k ⊗ ℋk+1 ̄ … ⊗ℋ ̄ k ∋ 𝜁 ↦ 𝜁 ⊗ 𝜉k+1 ∈ ℋ1 ⊗ Ik ∶ ℋ1 ⊗ ̄ … ⊗ ̄ ℋk are linear and isometric. The completion of the direct limit of the Hilbert spaces ℋ1 ⊗ along the mappings Ik is again a Hilbert space, denoted by ⊗k (ℋk , 𝜉k ) and called the (incomplete) infinite tensor product of the Hilbert spaces ℋk along the vectors 𝜉k ∈ ℋk . For any sequence of vectors 𝜂k ∈ ℋk such that ∑

|1 − ‖𝜂k ‖| < +∞ and

∑

k

|1 − (𝜂k |𝜉k )| < +∞

(1)

k

there corresponds a “decomposable vector” ⊗k 𝜂k = lim 𝜂1 ⊗ … ⊗ 𝜂k ⊗ 𝜉k+1 ⊗ 𝜉k+2 ⊗ … ∈ ⊗k (ℋk , 𝜉k ) k

(2)

which depends linearly on each 𝜂k . If ⊗k 𝜁k ∈ ⊗k (ℋk , 𝜉k ) is another decomposable vector, then (⊗k 𝜂k | ⊗k 𝜁k ) =

∏

(𝜂k |𝜁k )

(3)

k

where the infinite product is absolutely convergent. The set of decomposable vectors is total in ⊗k (ℋk , 𝜉k ). Let xk ∈ ℬ(ℋk ) (k ∈ ℕ). For each fixed k0 ∈ ℕ, there exists a unique bounded linear operator x1 ⊗ … ⊗ xk0 ⊗ 1 on ⊗k (ℋk , 𝜉k ) such that (x1 ⊗ … ⊗ xk0 ⊗ 1)(⊗k 𝜂k ) = x1 𝜂1 ⊗ … ⊗ xk0 𝜂k0 ⊗ 𝜂k0 +1 ⊗ 𝜂k0 +2 ⊗ …

(4)

for each decomposable vector ⊗k 𝜂k . In certain special situations, the sequence {x1 ⊗…⊗xk ⊗1}k∈ℕ is so-convergent to a bounded linear operator on ⊗k (ℋk , 𝜉k ) which is then denoted by ⊗k xk .

Appendix

427

For each k ∈ ℕ, consider also a von Neumann algebra ℳk ⊂ ℬ(ℋk ). Then ⊗k (ℳk , 𝜉k ) = {x1 ⊗ … ⊗ xk ⊗ 1; x1 ∈ ℳ1 , … , xk ∈ ℳk , k ∈ ℕ}′′ ⊂ ℬ(⊗k (ℋk , 𝜉k ))

(5)

is a von Neumann algebra, called the infinite tensor product of the von Neumann algebras ℳk ⊂ ℬ(ℋk ) along the vectors 𝜉k ∈ ℋk . On the other hand, for each k ∈ ℕ, let ℳk be a W∗ -algebra and let 𝜑k be a normal state on ℳk . Let 𝜋k ∶ ℳk → 𝜋k (ℳk ) ⊂ ℬ(ℋk ) be the normal cyclic *-representation of ℳk associated with 𝜑k , with the cyclic vector 𝜉k ∈ ℋk , ‖𝜉k ‖ = 1, that is, 𝜑k (x) = (𝜋k (x)𝜉k |𝜉k ) for every x ∈ ℳk (k ∈ ℕ). Then the von Neumann algebra ⊗k (ℳk , 𝜑k ) = ⊗k (𝜋k (ℳk ), 𝜉k )

(6)

is called the infinite tensor product of the W∗ -algebras ℳk along the normal states 𝜑k on ℳk . An alternative equivalent definition is given in Sakai (1971), 4.4. If all the ℳk are factors, then ⊗k (ℳk , 𝜉k ) and ⊗k (ℳk , 𝜑k ) are also factors. Assume that each ℳk is a finite discrete factor, say of type Ink (1 < nk < +∞) and denote by trk the canonical trace on ℳk (trk (1) = nk ). For any (normal) state 𝜑k on ℳk , there exists a unique positive element ak ∈ ℳk such that 𝜑k = trk (ak ⋅). We shall denote by Sp(𝜑k ∕ℳk ) = {𝜆k,1 ≥ 𝜆k,2 ≥ … 𝜆k,nk }

(7)

the set of eigenvalues of ak , each repeated according to its multiplicity; note that 𝜆k,1 +…+𝜆k,nk = 1. In this case, the factor ⊗k (ℳk , 𝜑k ) is called an Araki–Woods factor, or an ITPFI-factor (ITPFI = infinite tensor product of factors of type I), and the set {𝜆k,j ; 1 ≤ j ≤ nk , k ∈ ℕ} is called the eigenvalue list of ⊗k (ℳk , 𝜑k ). The *-isomorphy class of an ITPFI-factor depends only on its eigenvalue list. Actually, Araki and Woods (1968) (Definition 3.2) introduced an invariant r∞ (ℳ, Ω) for an ITPFI factor ℳ based on its eigenvalue list Ω, called asymptotic ratio set, and Connes (1973a) proved that r∞ (ℳ, Ω) = S (ℳ) and also that this set is equal to the set } { r∞ (ℳ) = 𝜆 > 0; ℳ is isomorphic to ℳ⊗ ℛ𝜆 , where the Powers factor ℛ𝜆 is defined below. In particular, assume that all ℳk are factors of type I2 . Then Sp(𝜑k ∕ℳk ) = {qk , pk } with qk ≥ pk ≥ 0 and pk + qk = 1, and the *-isomorphism class of ⊗k (ℳk , 𝜑k ) depends only on 𝜆k = pk ∕qk ∈ [0, 1] (k ∈ ℕ). We shall denote this factor by ℛ({𝜆k }k∈ℕ = ⊗k (ℳk , 𝜑k ). If 𝜆k = 𝜆 ∈ (0, 1] for all k ∈ ℕ, then ℛ𝜆 = ⊗k (ℳk , 𝜑k )

428

Appendix

is called the Powers factor corresponding to 𝜆 ∈ (0, 1]. For 𝜆 = 1, ℛ1 is nothing but the hyperfinite type II1 factor ℛ ≈ 𝔏(S(∞)) (see 10.31, 22.6, 6, 22.16). There are approximately finite dimensional non-ITPFI factors (Connes & Woods, 1980a). There is a characterization of ITPFI factors by their flow of weights (Connes & Woods, 1980b, 1985). For details concerning the construction of infinite tensor products, we refer to Araki and Woods (1968); Bures (1963, 1968); Connes (1973a); Guichardet (1969); Neumann (1938); Sakai (1971); Størmer (1971b). A.18. Notes. The material contained in Sections A.12–A.14 is due to van Daele (1976). Proposition A.9 is a result due to T. Kato. The other results in this Appendix are classical. For our exposition, we have used Araki and Woods (1968, 1973); Bures (1963, 1968); Van Daele (1976); Faris (1975); Guichardet (1969); Kadison (1968); Kallman (1971); Kaplansky (1951); Kato (1966); Neumann (1938); Pedersen and Takesaki (1973); Reed and Simon (1972); Sakai (1971); Størmer (1971b); Strătilă and Zsidó (1977, 2005); Takesaki (1969–1970).

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Notation Index

General notation: ℕ, ℤ, ℚ, ℝ, ℂ, 𝕋 : the usual number sets (natural, integral, rational, real,. complex, complex unimodular) 1: the unit element of an algebra; whenever it appears in a tensor product of algebras, it stands for the scalar algebra ℂ ⋅ 1 𝜄: the identity mapping 𝒜 (G) (18.7); 𝔘𝜑 (2.12); 𝔘′𝜑 (2.12); A𝜏,u (27.4); Ad(u) (2.23); Ad(𝜎) (15.15); ad(𝛿) (15.15); Aut(ℳ) (2.23); Aut𝜑 (ℳ) (2.25); Aut(Fℳ ) (25.1) 𝔹 (28.10); ℬw (ℳ) (2.23); ℬ(𝒳 , 𝒴 ) (13.1); ℬw (𝒳 , 𝒴 ) (13.1); ℬw (𝒳 , 𝒴 )∗ (13.1); B(Fℳ ) (26.1) c(a, b) (20.2); c(e, f) (17.15); c𝜑 (𝜓) (24.2); co w (convex hull); co (w-closed convex hull) D(A), Dom(A) (domain of an operator, A.1); D(ℋ , 𝜓) (7.1); d(𝜑1 , 𝜑2 ) (6.10); dg, d𝛾 (18.8); d𝜇∕dν (Radon–Nikodym derivative); [DF ∶ DE]t (11.15); [D𝜓 ∶ D𝜑]t (3.1); [D𝜓 ∶ D𝜑]c (26.3) E𝜑𝒩 (9.8, 12.18); E𝒩 𝜑 (10.4); en (4.4); en (A) (A.1); 𝜎 {eij }1≤i,j≤n (system of matrix units, 3.2); ℳexp (11.10) F (18.8); 𝔉(𝜎) (16.2); ℱn (the type In factor); Fk (22.6); F𝜆 = Fℳ (24.1); 𝔉ℳ (24.9); 𝔉E (11.5); 𝜆 𝜆 𝔉𝜑 (1.1); f𝛼 (1.5) ̂ (l8.7); G(𝒩 ) (25.7) G (18.5); G ℋ𝜑 (1.2); H(Fℳ ) (26.1) ℐ (a, b) (20.2); ℐxU (14.2); Int(ℳ) (3.2); Int 𝜎 (16.4) J𝜑 (2.12); j𝜑 (2.12); J𝜓,𝜑 (3.11); 𝒥 (ℋ , 𝜓) (7.1); JG (18.4); jG (18.7); 𝒥0 (F) (14.1); 𝒥00 (F) (14.1) 𝒦 (G) (18.4); KG (18.4); kG (18.5); 𝒦 (F) (14.1); Ker 𝜎 (16.4) 𝔏(G) (18.4); ℒ ∞ (G) (18.4); L0𝜉 , L𝜉 (2.12); ℒ p (ℋ , 𝜓) (7.16); l(𝜑) (4.19); l(x) (left support of an element, [L]) ℳ(G) (13.2); 𝔐𝜑 (1.1); 𝔐E (11.5); m(g) (18.8); m(𝛾) (18.9); mod(𝜎) (25.1); Mat2 (ℳ) (matrix algebra over ℳ, 3.1, 3.3)  (E) (10.17); 𝔑𝜑 (1.1); 𝔑E (11.5) 𝔬ℳ (3.2); Out(ℳ) (3.2)

P𝛿 (18.19, 19.7); P𝜎 (18.20), 19.16); P𝜎̂ (19.8, 22.2); P𝜑 (24.6); (𝒫ℳ , Fℳ ) (25.1); 𝒫𝜆 (22.16); 𝔓𝜑 (2.23); pu (⋅) (14.10); p(𝜎) (17.2); p𝜑 (24.5); p𝜎 +∞ (25.1); PV ∫−∞ (A.12); Proj(ℳ) (the projection lattice of ℳ; p𝜉 (cyclic projection); P(ℳ, 𝒩 ) (11.5) 𝒬(𝜎; E) (15.5); q(𝜎; E) (15.5); q𝜎t (15.6); q𝜎∞ (15.6) ℜ(G) (18.4); R0𝜂 , R𝜂 (2.12); ℛ(ℳ, 𝛿) (19.1); ℛ(ℳ, 𝜎) (19.1, 22.1); R𝜓𝜂 (7.1); ℝd (28.10); ℛ𝜆 (A.17); ℛ({𝜆k }k∈ℕ (A.17); r(𝜑, 𝜓) (23.17); 𝔯(G) (28.6); r(𝜑) (4.19); r(x) (right support of an element, [L]); ℛ{𝒳 } (von Neumann algebra generated by 𝒳 ) S(ℳ) (18.2); 𝔖𝜑 (6.5); S0𝜑 , S𝜑 (2.12); S0𝜓,𝜑 , S𝜓,𝜑 (3.11); S𝜑t (7.6); Sp U (14.5); SpU (x) (14.2); Sp(x), Sp𝒜 (x) (usual spectrum, 14.6); Sp(𝜑k ∕ℳk ) (A.17); S(∞) (22.6); s(e, f) (17.15); s(𝜑) (2.1,4.19); s(E) (11.5); s(x) (support of an element); 𝒮 (A) (𝒮A ) [L, 9.9)]) T(ℳ) (27.1); 𝔗𝜑 (2.12); 𝒯𝜑′ (2.4); T𝜎f (18.21); T𝜎a (22.1); tr (the canonical trace on ℬ(ℋ )) U(ℋ ) (2.23); U(ℳ) (3.2); u𝜎 (2.23) VG (18.4); V𝜓,𝜑 (3.16) WG (18.4); Wn (ℳ) (§23); Wns (ℳ) (§23); Wnsf (ℳ) (6.9), §23); Wint (ℳ) (24.1); W∞ (ℳ) (24.1); int W(𝜔) (25.4); W(H, K) (30.8) Z𝜎 (G; ℳ) (5.1, 20.1); Z𝜎 (G; U(ℳ)) (20.3); Z(Fℳ ) (26.1); Z(𝒥 ) (14.1); 𝒵 (ℳ) (centre of ℳ). Γ(𝜎) (16.1); Γ(ℳ) (30.10) ΔG (18.4); Γ𝜑 (2.12); Δ𝜓𝜑 (3.11); Δ(𝜑∕𝜓) (7.3) 𝛿G (l8.7); 𝛿G∗ (18.7); 𝛿ℳ (3.2, 26.4); 𝛿x , 𝛿a (10.29, 15.13); 𝛿ts (Kronecker symbol, 22.1) 𝜕 = 𝜕𝜎 (25.3); 𝜕 ∶ U(𝒫ℳ ) → Z(Fℳ ) (26.1) ∇G (18.4) 𝜒I (characteristic function of I) Λf (18.22) 𝜆 (18.4) 𝜇G (18.4) ν𝜑 (24.7)

444 𝜋G (18.5); 𝜋𝜑 (1.2); 𝜋𝜎 (18.6); 𝜋𝜑𝛿 (18.10) 𝜌 (18.4) ∑⊗ n 𝜓n (23.15) 𝜎t𝜑 (2.12); 𝜎c𝜑 (26.3); 𝜎𝛼𝜑 (2.14); 𝜎tE (11.15); 𝜎t𝜓,𝜑 (3.10); 𝜎𝛼𝜓,𝜑 (3.12); 𝜎(𝒳 , 𝒴 ) (weak topology defined by 𝒴 on 𝒳 ) 𝜃(𝜑, 𝜓) (3.1) 𝜏𝜑 (24.6) 𝜔G (18.4) Other symbols: ≈ : isomorphism ∼∶ e ∼ f (equivalence of projections); 𝜎 ∼ 𝜏 (15.11); (ℳ, 𝜎) ∼ (𝒩 , 𝜏) (20.13); a ∼ b (20.2); ̄ 𝒦 →𝒦 ⊗ ̄ ℋ (18.1); ∼∶ ℋ ⊗ ̄ 𝒩 →𝒩 ⊗ ̄ ℳ (18.1) ̃⋅ ∶ ℳ ⊗ ≂: a ≂ b (20.2); 𝜓 ≂ 𝜑 (23.1) ≲: a ≲ b (20.2); 𝜓 ≲ 𝜑 (23.1) ≤∶ A ≤ B (A.4); 𝜑2 ≤ 𝜑1 (𝜆) (6.9)