Noncommutative Integration and Operator Theory 9783031496530, 9783031496547

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Progress in Mathematics 349

Peter G. Dodds Ben de Pagter Fedor A. Sukochev

Noncommutative Integration and Operator Theory

Progress in Mathematics Volume 349

Series Editors Antoine Chambert-Loir

, Université Paris-Diderot, Paris, France

Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Ghent University, Belgium Queen Mary University of London, London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

Progress in Mathematics is a series of books intended for professional mathematicians and scientists, encompassing all areas of pure mathematics. This distinguished series, which began in 1979, includes research level monographs, polished notes arising from seminars or lecture series, graduate level textbooks, and proceedings of focused and refereed conferences. It is designed as a vehicle for reporting ongoing research as well as expositions of particular subject areas.

Peter G. Dodds • Ben de Pagter • Fedor A. Sukochev

Noncommutative Integration and Operator Theory

Peter G. Dodds College of Science and Engineering Flinders University at Tonsley Tonsley, SA, Australia

Ben de Pagter Delft Institute of Applied Mathematics Delft University of Technology Delft, The Netherlands

Fedor A. Sukochev School of Mathematics & Statistics University of New South Wales Sydney, NSW, Australia

ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-031-49653-0 ISBN 978-3-031-49654-7 (eBook) https://doi.org/10.1007/978-3-031-49654-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The purpose of this monograph is to provide a systematic account of the theory of noncommutative integration in semi-finite von Neumann algebras. It is designed to serve as an introductory graduate-level text as well as a basic reference for more established mathematicians with interests in the continually expanding areas of noncommutative analysis and probability. Its origins lie in two apparently distinct areas of mathematical analysis: the theory of operator ideals going back to von Neumann and Schatten and the general theory of rearrangement invariant Banach lattices of measurable functions which has its roots in many areas of classical analysis related to the well-known .Lp -spaces. A principal aim, therefore, is to present a general theory which contains each of these motivating areas as special cases. In some sense, this book then serves as a rewriting of the two classic books, Classical Banach Spaces by Lindenstrauss and Tzafriri and Interpolation of Linear Operators By Krein, Petunin, and Semenov, wherein these ideas are now reworked through the setting of von Neumann algebras, and the dual lenses of functional analysis and operator theory. The heart of this theory is the development of noncommutative integration created by Dixmier and Segal in the 1950s, which is in turn based on the fundamental work of von Neumann and Murray on algebras of operators in some Hilbert space. In this theory, the notion of a classical measure space is replaced by a von Neumann algebra playing the role of .L∞ (X) and a trace (satisfying some continuity conditions) standing in for the integral. Measurable functions are then replaced by “measurable operators,” i.e., those operators well behaved with respect to the trace. The role of the measure algebra is played by the lattice of projections, with the measure being the restriction of the trace to the lattice of projections. The introduction of noncommutative measure theory permits the detailed study of Banach spaces whose elements are measurable operators. This study parallels the classical theory of Banach function spaces and draws much of its inspiration from this theory. From this general study, the theory of Schatten ideals as well as much of

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the classical theory of rearrangement invariant Banach function spaces now emerge as special cases. In turn, this unification yields considerable new insights. These parallels with the classical theory of function spaces suggest very much the overall directions and even the general results that might be expected. That said, the lack of commutativity presents substantial technical barriers. At one end of the spectrum are the (commutative) function spaces in which techniques from the theory of vector lattices may be employed. On the other hand, the usual triangle inequality for the absolute value fails in the simplest noncommutative von Neuman algebra of all two-by-two matrices which is an anti-lattice with respect to the usual quadratic form ordering. These facts illustrate the intrinsic difficulty in passing from the commutative to the noncommutative situation. The idea to write this monograph arose a number of years ago, perhaps more than the authors are willing to admit, during collaborative visits of the authors to their respective institutions. However, it is only relatively recently that some sections of the material could be considered to be in final form. As is often the case, the possible material that might be included has also expanded rapidly, and so the choice was made to present in this volume what the authors see as the indispensable core of the subject, which can then form the framework for the presentation of further, though no less interesting and important subject areas, in later volume(s). The monograph itself is no mere synthesis of existing results from the literature; indeed, many of the main results have been carefully reworked, often leading to newer and more transparent proofs and yielding a deeper insight through the development of a coherent structure. Tonsley, SA, Australia Delft, The Netherlands Sydney, NSW, Australia

Peter G. Dodds Ben de Pagter Fedor A. Sukochev

Acknowledgments

This book cannot exist and could not have been written without the aid of many others. Too many to name in fact. Of course, there would be no mathematics to discuss if it were not for the school of research in noncommutative analysis and Banach space geometry based in Kazan, Tashkent, Delft, Adelaide, and Sydney. From the bottom of our hearts, we would like to thank the members of these schools for their many collaborations and contributions to the theory, to which the authors have dedicated their professional lives. Of particular and personal standing, we must acknowledge the contributions of Ovchinnikov (Voronezh); Bikchentaev and Tikhonov (Kazan); Ayupov, Ber, and Chilin (Tashkent); Lust-Piquard, Pisier, and Xu (France); Junge (UrbanaChampaign); Hiai and Kosaki (Japan); and Pietsch (Germany). Special mention must also be made for those who not only paved the way for our field, but have since unfortunately passed. Their work will always act as a guiding light and inspiration for the authors and so many other mathematicians. We pay tribute to those contributions due to Uffe Haagerup, Nigel Kalton, Wim Luxemburg, Aleksander Pełczy´nski, Haskell Rosenthal, and Nicole Tomczak-Jaegermann. It is only through the dedicated and diligent efforts of Eva-Maria Hekkelman, Jinghao Huang, and Thomas Scheckter in Sydney, as well as Sjoerd Dirksen and Anna Kaminska, that the book has been able to be edited and reviewed, and is finally in a publishable state. The authors, and in particular Peter, would like to thank Theresa Dodds for her enduring collaboration and support during this project.

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Contents

1

A Review of Relevant Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Bounded Hilbert Space Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Topologies on the Space B (H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Lattice of Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Closed Linear Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Algebras with an Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Singular Functionals on von Neumann Algebras . . . . . . . . . . . . . . . . . . . . 1.13 Direct Products of von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Comparison of Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5 9 15 18 19 24 26 30 34 37 39 44

2

Measurable Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.1 Affiliated Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 The Algebra of Measurable Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3 The Algebra of τ -Measurable Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.4 The Algebra of τ -Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.5 The Measure Topology in S (τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.6 Measure Topology and Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.7 The Local Measure Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.8 Continuity of Operator Functions with Respect to the Measure Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.9 Trace Preserving ∗-Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3

Singular Value Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.1 The Right-Continuous Inverse of a Decreasing Function. . . . . . . . . . . . 121 3.2 The Singular Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 ix

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3.3 Extension of the Trace to S (τ )+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Space L1 (τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Eigenvalue Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Properties of the Eigenvalue Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Some Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Submajorization for the Eigenvalue Function . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Submajorization for the Singular Value Function . . . . . . . . . . . . . . . . . . . . 3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ )) . . . . . . . . . . . . . . . . . . . . . . .

146 154 172 179 185 194 205 217

4

Symmetric Spaces of .τ-Measurable Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Normed M-Bimodules of τ -Measurable Operators . . . . . . . . . . . . . . . . . 4.2 The Dual of a Normed M-Bimodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Köthe Dual of a Normed M-Bimodule . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Symmetric Spaces of τ -Measurable Operators. . . . . . . . . . . . . . . . . . . . . . . 4.5 The Köthe Dual of a Symmetric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 239 252 261 271 290

5

Strongly Symmetric Spaces of .τ-Measurable Operators . . . . . . . . . . . . . . . . 5.1 Strongly Symmetric Spaces and Their Köthe Duals . . . . . . . . . . . . . . . . . 5.2 Normal and Singular Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Order Continuity of the Norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Elements of Order Continuous Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Elements of Absolutely Continuous Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Further Characterizations of Order Continuity . . . . . . . . . . . . . . . . . . . . . . . 5.7 Sets of Uniformly Absolutely Continuous Norm . . . . . . . . . . . . . . . . . . . . 5.8 Weakly Compact Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Embedding Copies of c0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297 297 306 322 326 330 341 355 373 380

6

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Construction of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Lp -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 p-Convexification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Lorentz Lp,q -Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391 391 399 400 405 412 430 435 443

7

Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Interpolation Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Interpolation Spaces for the Couple (L1 (τ ), M). . . . . . . . . . . . . . . . . . . . . 7.3 Principal Theorems on Noncommutative Interpolation . . . . . . . . . . . . . . 7.4 The K-Functional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Real Interpolation and the Noncommutative Marcinkiewicz Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

455 455 456 458 459 460 464 466

Contents

7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14

7.15 7.16 7.17

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Holmstedt Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operators of Weak Type and the Calderón Operator. . . . . . . . . . . . . . . . . Noncommutative Marcinkiewicz Theorem for Sublinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Boyd Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak Type (p, p) and Strong Type (∞, ∞) Interpolation . . . . . . . . . . Strong Type (1, 1) and Weak Type (q, q) Interpolation, 1 < q < ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpolation for the Pair (Lp (τ ), Lq (τ )), 1 ≤ p < q ≤ ∞ . . . . . . . . The Complex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of Complex Method to the Geometry of Noncommutative Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.1 Clarkson Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.2 Uniform Convexity and Uniform Smoothness . . . . . . . . . . . . . 7.14.3 Type and Cotype of the Spaces Lp (τ ), 1 ≤ p < ∞ . . . . . . . The Calderón Family for a Banach Couple of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Köthe Duality and the Calderón–Lozanovskii Construction . . . . . . . . Schmidt Decomposition and Lozanovskii Factorization. . . . . . . . . . . . .

469 471 476 478 483 484 490 509 511 511 517 524 531 534 539

Brief Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Chapter 1

A Review of Relevant Operator Theory

Abstract This chapter reviews some of the basic elements of Hilbert space operator theory and von Neumann algebras which will be used throughout this book. Most of these results are presented without proofs, which can be readily found in the relevant standard literature.

In this chapter we collect some notation and terminology concerning Hilbert space operators and von Neumann algebras which will be used throughout this book. Most of the results will be stated without proofs. For further details and proofs, we refer the reader to [13, 23, 35, 70, 71, 74, 115, 125].

1.1 Bounded Hilbert Space Operators Let H be a complex Hilbert space equipped with the inner product .〈·, ·〉 and corresponding norm .‖·‖H . The inner product is linear in the first and conjugate linear in the second variable. The elements of H will usually be denoted by small Greek letters .ξ, η, ζ, . . .. The space of all bounded linear operators in H is denoted by .B (H ). The elements of .B (H ) will be denoted by small Latin letters .x, y, u, v, . . .. The identity operator on .H is denoted by .1 = 1H (which is the unit element in the algebra .B (H )). Equipped with the operator norm .‖·‖B(H ) , given by .‖x‖B(H ) = sup‖ξ ‖ ≤1 ‖xξ ‖H , the space .B (H ) is a Banach algebra. The closed unit H ball in .B (H ) is denoted by .BB(H ) . For any .x ∈ B (H ) its adjoint is denoted by .x ∗ , so .〈xξ, η〉 = 〈ξ, x ∗ η〉 for all .ξ, η ∈ H . The mapping .x │−→ x ∗ is a conjugate linear involution in .B (H ) satisfying .‖x ∗ ‖B(H ) = ‖x‖B(H ) and .‖x‖2B(H ) = ‖x ∗ x‖B(H ) for all .x ∈ B (H ) (so, .B (H ) is an example of a .C ∗ -algebra, which we will discuss in Sect. 1.10). An operator .x ∈ B (H ) satisfying .x ∗ = x is called self-adjoint (or, Hermitian). The collection of all self-adjoint operators is denoted by .Bh (H ), which is a real linear subspace of .B (H ). An operator .x ∈ B (H ) is called normal if .xx ∗ = x ∗ x. Furthermore, if .u ∈ B (H ) satisfies .u∗ u = uu∗ = 1 (equivalently, .u−1 = u∗ ), then © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. G. Dodds et al., Noncommutative Integration and Operator Theory, Progress in Mathematics 349, https://doi.org/10.1007/978-3-031-49654-7_1

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1 A Review of Relevant Operator Theory

u is called unitary. An operator .p ∈ B (H ) is said to be an (orthogonal) projection if .p2 = p = p∗ and the set of all projections is denoted by .P (B (H )). If .p ∈ P (B (H )), then it is clear that .1 − p ∈ P (B (H )), which is called the complement of p and this projection will also be denoted by .p⊥ . A self-adjoint operator .a ∈ Bh (H ) is called positive if .〈aξ, ξ 〉 ≥ 0 for all .ξ ∈ H . The collection of all positive elements of .Bh (H ) is denoted by .Bh (H )+ . This set is a proper closed cone in .Bh (H ) and it induces a partial ordering in .Bh (H ) by defining .a ≤ b whenever .b − a ∈ Bh (H )+ , which turns .Bh (H ) into a partially ordered vector space.

1.2 Topologies on the Space B (H ) In addition to the norm topology, generated by the operator norm .‖·‖B(H ) , there are a number of other important topologies on .B (H ). For every .ξ ∈ H , we define convex the semi-norm .ρξ on .B (H ) by .ρξ (x) = ‖xξ ‖H , .x ∈ B (H ). The locally  Hausdorff topology on .B (H ) generated by the family of semi-norms . ρξ : ξ ∈ H is called the strong operator topology (briefly, so-topology). A net .{xα } in .B (H ) so so-converges to an operator .x ∈ B (H ), denoted by .xα → x, if and only if .‖xα ξ − xξ ‖H → 0 for all .ξ ∈ H . Clearly, the so-topology is weaker than the topology generated by the norm .‖·‖B(H ) . Multiplication in .B (H ) is continuous with respect to the so-topology in each factor separately, but in general not jointly so-continuous in both factors (however, multiplication is jointly so-continuous when restricted to norm bounded sets). The mapping .x │−→ x ∗ is not so-continuous (unless H is finite dimensional). For .ξ, η ∈ H , we define the semi-norm .ρξ,η by .ρξ,η (x) = |〈xξ, η〉|, .x ∈ B (H ). The locally convex Hausdorff topology on .B (H ) generated by the family of semi  norms . ρξ,η : ξ, η ∈ H is called the weak operator topology (briefly, wo-topology). wo

A net .{xα } in .B (H ) wo-converges to an operator .x ∈ B (H ), denoted by .xα → x, if and only if .〈xα ξ, η〉 → 〈xξ, η〉 for all .ξ, η ∈ H . Obviously, the wo-topology is weaker than the so-topology and coincides with the latter only if H is finite dimensional. Multiplication is wo-continuous in each factor separately but is not jointly wo-continuous (unless H is finite dimensional). The mapping .x │−→ x ∗ is evidently wo-continuous. For .ξ, η ∈ H the linear functional .ωξ,η on .B (H ) is defined by .ωξ,η (x) = 〈xξ, η〉 for all .x ∈ B (H ). Evidently, .ωξ,η belongs to the Banach dual space .B (H )∗ of  ∗ .B (H ). Let .ΩH be the linear subspace of .B (H ) generated by . ωξ,η : ξ, η ∈ H . It is clear that the wo-topology on .B (H ) coincides with .σ (B (H ) , ΩH ), the weak topology in .B (H ) generated by .ΩH . From this it follows that the wo-dual space ' .B (H )wo of .B (H ) is precisely .ΩH . Denoting the so-dual space of .B (H ) by ' ' ' .B (H )so , it is evident that .B (H )wo ⊆ B (H )so . It turns out that the reverse inclusion also holds.

1.3 The Lattice of Projections

3

Theorem 1.2.1 A linear functional on .B (H ) is wo-continuous if and only if it is so-continuous, that is, .B (H )'so = B (H )'wo . Consequently, convex subsets of .B (H ) have the same closures with respect to the so- and wo-topology and for ' every .ϕ ∈ B (H )so there exist .ξ1 , . . . , ξn ∈ H and .η1 , . . . , ηn ∈ H such that n .ϕ = j =1 ωξj ,ηj . Another useful property of the wo-topology is given in the next theorem. Theorem 1.2.2 The closed unit ball .BB(H ) of .B (H ) is wo-compact. Next, we consider the locally convex Hausdorff topology on .B (H   ) generated∞by  〈xξ 〉 the family of semi-norms given by .ρ{ξi },{ηi } (x) =  ∞ , η i i , where .{ξi }i=1 i=1 ∞  ∞ 2 2 and .{ηi }i=1 are sequences in H satisfying . i=1 ‖ξi ‖H < ∞ and . ∞ i=1 ‖ηi ‖H < ∞. This topology is called the ultra-weak operator topology (briefly, uwo-topology). The ultra-weak operator topology is stronger than the wo-topology. On norm bounded subsets of .B (H ) the uwo- and wo-topology coincide. In particularly, .BB(H ) is uwo-compact. Convergence of a net .{xα } to an element x in .B (H ) with uwo respect to the uwo-topology is denoted by .xα →  x. ∞ ‖ξi ‖2H < ∞, the semi-norm Given a sequence .{ξi }i=1 in H satisfying . ∞  i=1 ∞ 2 1/2 .ρ{ξi } on .B (H ) is defined by .ρ{ξi } (x) = . The Hausdorff i=1 ‖xξi ‖H locally convex topology on .B (H ) generated by the family of these semi-norms .ρ{ξi } is called the ultra-strong operator topology (briefly, uso-topology). The usotopology is stronger than the so- and uwo-topologies and is weaker than the norm topology. On norm bounded subsets of .B (H ), the uso- and so-topology coincide. Convergence of a net .{xα } to an element x in .B (H ) with respect to the uso-topology uso is denoted by .xα → x. The uwo-dual and uso-dual spaces of .B (H ) are denoted by .B (H )'uwo and ' .B (H )uso , respectively. The following result is similar to Theorem 1.2.1. Theorem 1.2.3 A linear functional on .B (H ) is uwo-continuous if and only if it is uso-continuous, that is, .B (H )'uso = B (H )'uwo . Consequently, convex subsets of .B (H ) have the same closures with respect to the uso- and uwo-topology and for  2  ∞  ∞    every .ϕ ∈ B (H )'uso there exist . ξj j =1 and . ηj j =1 in H such that . ∞ j =1 ξj H < 2 ∞   ηj  < ∞ and .ϕ = ∞ ωξ ,η as a convergent series in .B (H )∗ . ∞, . j =1

H

j =1

j

j

1.3 The Lattice of Projections In the sequel we shall frequently use the following notation concerning partial ordering. Let .(X, ≤) be a partially ordered set. If D is a non-empty subset of X for which the least upper bound (or, supremum) exists,

then this least upper bound is denoted by .sup D or . D. Similarly, .inf D or . D denotes the greatest lower bound (or, infimum) of D whenever it exists. In the case that .D = {x, y}, we also write .sup D = x ∨ y and .inf D = x ∧ y. A net .{xα }α∈∆ in X is called increasing

4

1 A Review of Relevant Operator Theory

(or, upwards directed) if .xα ≤ xβ whenever .α ≤ β in .∆. This is sometimes written as .xα ↑. If .{xα }α∈∆ is increasing and .x = supα xα exists, then we write .xα ↑ x. Decreasing nets are defined analogously and .xα ↓ x means that the decreasing net .{xα } has infimum x. A bijection .ϕ : X → X is called an order isomorphism if .x ≤ y in X if, and only if, .ϕ (x) ≤ ϕ (y). The following simple observation is included for later reference. Lemma 1.3.1 Suppose that D is a non-empty subset of the partially ordered set (X, ≤) such that .x = sup D exists. If .Ф is a collection of order isomorphisms in X such that .ϕ (D) = D for all .ϕ ∈ Ф, then .ϕ (x) = x for all .ϕ ∈ Ф.

.

A partially ordered set .(X, ≤) in which the supremum and infimum exist for any pair of elements is called a lattice. If the supremum and infimum exist for any non-empty subset of X, then X is called a complete lattice. Given a Hilbert space H , we denote by .Lat (H ) the collection of all closed linear subspaces of H . The set .Lat (H ) is partially ordered by inclusion, that is, if .L1 , L2 ∈ Lat (H ) then .L1 ≤ L2 whenever .L1 ⊆ L2 . With respect to this partial ordering, .Lat (H ) is a complete lattice. Indeed, given any non-empty subset

 {L D = : L ∈ D} and the supremum .D ⊆ Lat (H ), the infimum is given by . is given by . D = span {L : L ∈ D}. The smallest and largest element of .Lat (H ) are .{0} and H , respectively. We point out that the lattice .Lat (H ) is not distributive. If .{Li : i ∈ I } is a collection of pairwise of H (that is, orthogonal closed subspaces  .Li ⊥Lj whenever .i /= j in I ), then . L is also denoted by . L i i∈I i∈I i . The set .P (B (H )) of all projections in H is a subset of .Bh (H ), so we may equip .P (B (H )) with the partial ordering inherited from .Bh (H ), that is, if .p, q ∈ P (B (H )), then .p ≤ q if and only if .〈pξ, ξ 〉 ≤ 〈qξ, ξ 〉 for all .ξ ∈ H . As is easily verified, .p ≤ q in .P (B (H )) is equivalent to .p (H ) ≤ q (H ) in .Lat (H ). Hence, the mapping .p │−→ p (H ) is an order isomorphism from .P (B (H )) onto .Lat (H ). Consequently, .P (B (H )) is a complete lattice with smallest element 0 and largest element .1. Note that .p ≤ q in .P (B (H )) is also equivalent to saying that ⊥ = 1 − p is a .p = pq (or, .p = qp). For every .p ∈ P (B (H )), the projection .p ⊥ ⊥ complement of p, that is, .p ∨ p = 1 and .p ∧ p = 0. The mapping .p │−→ p⊥ reverses the ordering in .P (B (H )), i.e., .p ≤ q if and only if .q ⊥ ≤ p⊥ , and so, ⊥ .(p ∨ q) = p⊥ ∧ q ⊥ and .(p ∧ q)⊥ = p⊥ ∨ q ⊥ for all .p, q ∈ P (B (H )). If .p, q ∈ P (B (H )) commute, then .p ∧ q = pq and .p ∨ q = p + q − pq. The supremum of an upwards directed system in .P (B (H )) can be characterized as follows. Proposition 1.3.2 If .{pα } is an increasing net in .P (B (H )) and .p ∈ P (B (H )), so then .pα ↑ p in .P (B (H )) if and only if .pα → p. Two projections .p, q ∈ P (B (H )) are called mutually orthogonal if .pq = 0 (equivalently, .p (H ) and .q (H ) are mutually orthogonal subspaces). Suppose that .{pi : i ∈ I } is a pairwise orthogonal collection in .P (B (H )) (i.e., .pi pj = 0 whenever .i /= j in I ). For each finite subset F of I we may define the projection .pF = i∈F pi . It is clear that .{pF } is an increasing net (with respect to the

1.4 Closed Linear Operators

5

inclusion ordering of the finite subsets of I ). Hence, there  exists .p ∈ P (B (H )) such that .pF ↑ p. This projection p is denoted by . i∈I pi . It follows from Proposition 1.3.2 that this series is actually so-convergent in .B (H ). The set .P (B (H )) is not a sublattice of .Bh (H ). Actually, if .a, b ∈ Bh (H ), then .a ∨ b exists in .Bh (H ) if and only if .a ≤ b or .b ≤ a, by the anti-lattice theorem of R.V. Kadison. However, for increasing nets we have the following important result (an analogous statement holds for decreasing nets). Theorem 1.3.3 If .{aα } is an increasing net in .Bh (H ) which is bounded above, then so there exists .a ∈ Bh (H ) such that .aα → a and .aα ↑ a in .Bh (H ). It follows, in particular, from Proposition 1.3.2 and Theorem 1.3.3 that, if .{pα } is an increasing net in .P (B (H )) such that .pα ↑ p in .P (B (H )), then .pα ↑ p in .Bh (H ). Given a unitary operator .u ∈ B (H ), the mapping .a │−→ uau∗ , .a ∈ Bh (H ), is an order isomorphism in .Bh (H ). Similarly, the mapping .p − │ → upu∗ , .p ∈ P (B (H )), is an order isomorphism in .P (B (H )). Hence, the following observation is an immediate consequence of Lemma 1.3.1. Lemma 1.3.4 Suppose that .U0 is a collection of unitary operators on H . (i) Let D be a non-empty subset of .Bh (H ) for which .a0 = D exists in .Bh (H ). If .uDu∗ = D for all .u ∈ U0 (in particular, if .uau∗ = a for all .u ∈ U0 ), then ∗ .ua0 u = a0 for all .u ∈ U0 . (ii) Let D be a non-empty subset of .P (B (H )) and .p0 = D in .P (B (H )). If ∗ ∗ .uDu = D for all .u ∈ U0 (in particular, if .upu = p for all .u ∈ U0 ), then ∗ .up0 u = p0 for all .u ∈ U0 . Note that, via complementation, similar statements hold for infima in .Bh (H ) and P (B (H )).

.

1.4 Closed Linear Operators Many of the linear operators we shall encounter will not be bounded and are only defined on a (dense) subspace of the Hilbert space H . Here we shall introduce the necessary notions to deal with such operators. A linear operator x in H is a linear mapping from its domain .D (x), which is a linear subspace of H , into the space H . Given two such linear operators x and y in H , the operator y is said to be an extension of x (or, x is a restriction of y), if .D (x) ⊆ D (y) and .xξ = yξ for all .ξ ∈ D (x). This is denoted as .x ⊆ y. If .x ⊆ y as well as .y ⊆ x, then we say that .x = y. The range and kernel of a linear operator x are denoted by .Ran (x) and .Ker (x), respectively. In the collection of linear operators we may introduce the algebraic operations of scalar multiplication, addition, and multiplication as follows. Given linear operators .x, y in H and .λ ∈ C we define:

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1 A Review of Relevant Operator Theory

• .λx by setting .D (λx) = D (x) and .(λx) ξ = λ (xξ ) for all .ξ ∈ D (λx). • .x + y by setting .D (x + y) = D (x) ∩ D (y) and .(x + y) ξ = xξ + yξ for all .ξ ∈ D (x + y). • xy by setting .D (xy) = {ξ ∈ D (y) : yξ ∈ D (x)} and .(xy) ξ = x (yξ ) for all .ξ ∈ D (xy).   • The inverse operator .x −1 whenever x is injective, by setting .D x −1 = .Ran (x) and .x −1 ξ = η whenever .ξ = xη for some .η ∈ D (x). We note that in general it may happen that .D (x + y) = {0} or .D (xy) = {0}. Furthermore, with respect to these algebraic operations the set of all linear operators is not a vector space. However, we do have the following relations for arbitrary linear operators .x, y, and z: (a) (b) (c) (d)

(x + y) + z = x + (y + z). (xy) z = x (yz). .(x + y) z = xz + yz. .zx + zy ⊆ z (x + y). . .

It follows from (a) and (b) above that we can form, without ambiguity, sums and products of an arbitrary number of linear operators. In particular, polynomials in linear operators are well defined. For a linear operator x in H the graph .𝚪 (x) is defined to be the linear subspace of .H × H given by .𝚪 (x) = {(ξ, xξ ) : ξ ∈ D (x)}. Note that .x ⊆ y is equivalent to .𝚪 (x) ⊆ 𝚪 (y). A linear operator x is called closed if .𝚪 (x) is a closed subspace of .H × H (equipped with the product topology). In other words, x is closed if and only if it follows from .{ξn }∞ n=1 ⊆ D (x), .ξ, η ∈ H , .ξn → ξ and .xξn → η as .n → ∞, that .ξ ∈ D (x) and .xξ = η. If .x is closed, then .Ker (x) is a closed subspace of .H . It is clear that any bounded linear operator in H is closed. Conversely, if x is a closed linear operator and if the domain .D (x) is a closed subspace of H , then it follows from the Closed Graph Theorem that x is bounded on its domain .D (x). This applies, in particular, if .D (x) = H . Furthermore, if x is a closed injective −1 is also closed. Consequently, if x is in linear operator in H , then its inverse .x  −1 = Ran (x) = H and so, .x −1 ∈ B (H ). addition surjective, then .D x The linear operator x in H is called pre-closed (or, closable) if the closure .𝚪 (x) of the graph .𝚪 (x) in .H × H is the graph of some linear operator in H . Note that .𝚪 (x) is the graph of a linear operator if, and only if, .(0, η) ∈ 𝚪 (x) implies that ∞ .η = 0. In other words, x is pre-closed if, and only if, it follows from .{ξn } n=1 ⊆ D (x), .η ∈ H , .ξn → 0 and .xξn → η as .n → ∞, that .η = 0. If a linear operator x is pre-closed, then the linear operator whose graph is .𝚪 (x) is denoted by .x¯ and is called the closure of x. Clearly, .x ⊆ x. ¯ Suppose that x is a linear operator in H and that .D is a linear subspace of .D (x).   Then .D is called a core of x if .𝚪 (x) ⊆ 𝚪 x|D , where .x|D denotes the restriction of x to .D. In other words, .D is a core of x if and only if for every .ξ ∈ D (x) there exists a sequence .{ξn }∞ n=1 ⊆ D such that .ξn → ξ and .xξn → xξ as .n → ∞. It is easily verified that two pre-closed operators which coincide on a common core have identical closures.

1.4 Closed Linear Operators

7

A linear operator x in H is called densely defined if .D (x) is a dense subspace of H . Note that if x is a closed and densely defined operator, then x is bounded if, and only if, .D (x) = H . Now suppose that x is a densely defined operator in .H and consider the linear subspace .D of .H given by D = {η ∈ H : ∃ζ ∈ H such that 〈xξ, η〉 = 〈ξ, ζ 〉 ∀ξ ∈ D (x)} .

.

If .η ∈ D, then the element .ζ ∈ H satisfying .〈xξ, η〉 = 〈ξ, ζ 〉 for all .ξ ∈ D (x) is uniquely determined by .η, as .D (x) is dense in H . Therefore, we may define the │ → ζ from .D into H . It is readily verified that .x ∗ is linear. Hence, mapping .x ∗ : η − ∗ ∗ ∗ .x is a linear operator in H with domain .D (x ) = D. The operator .x is called the adjoint of x. Note that, by definition, we have .



  〈xξ, η〉 = ξ, x ∗ η , ξ ∈ D (x) , η ∈ D x ∗ .

(1.1)

From (1.1) and the definition of .x ∗ it is immediately clear that .x ∗ is closed. If x is densely defined and pre-closed, then .x¯ ∗ = x ∗ . The following theorem lists some elementary properties of adjoint operators. Theorem 1.4.1 Let x and y be densely defined linear operators in H . If .x ⊆ y, then .y ∗ ⊆ x ∗ . ∗ ¯ x ∗ for all .λ ∈ C, .λ /= 0. .(λx) = λ If .x + y is densely defined, then .x ∗ + y ∗ ⊆ (x + y)∗ . If xy is densely defined, then .y ∗ x ∗ ⊆ (xy)∗ . If .y ∈ B (H ), then .(x + y)∗ = x ∗ + y ∗ and .(yx)∗ = x ∗ y ∗ . If .u ∈ B (H ) is unitary, then .(ux)∗ = x ∗ u∗ , .(xu)∗ = u∗ x ∗ and .(uxu∗ )∗ = ux ∗ u∗ .  ∗ (vii) If x is injective and .Ran (x) is dense, then . x −1 = (x ∗ )−1 . (i) (ii) (iii) (iv) (v) (vi)

Given the densely defined linear operator x in H , the adjoint .x ∗ : D (x ∗ ) → H is closed. If .x ∗ is densely defined, then we may consider .(x ∗ )∗ = x ∗∗ and it is easy to see that .x ⊆ x ∗∗ . Hence, x is pre-closed. Conversely, if x is pre-closed, then it can be shown that .x ∗ is densely defined. This is contained in the next theorem. Theorem 1.4.2 If x is a densely defined linear operator in H , then .x ∗ is densely defined if, and only if, x is pre-closed. Moreover, if x is pre-closed, then .x¯ = x ∗∗ . In particular, for every densely defined closed operator x we have .x ∗∗ = x. A densely defined linear operator a in H is called self-adjoint if .a = a ∗ . A selfadjoint operator a in H is called positive if .〈aξ, ξ 〉 ≥ 0 for all .ξ ∈ D (a). This is denoted by .a ≥ 0. Furthermore, as in the case of bounded linear operators on H , a closed and densely defined linear operator x in H is called normal when .x ∗ x = xx ∗ . There are a number of important projections associated with a closed, densely defined operator in H . First we recall the following simple result.

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1 A Review of Relevant Operator Theory

Proposition 1.4.3 If x is a densely defined closed linear operator in H , then Ran (x)⊥ = Ker (x ∗ ) and .Ker (x)⊥ = Ran (x ∗ ). In particular, if a is a self-adjoint operator in H , then .Ran (a)⊥ = Ker (a) and .Ker (a)⊥ = Ran (a).

.

Definition 1.4.4 Given a closed, densely defined linear operator x in H we define: (i) The projection onto .Ker (x) is called the null projection of x, denoted by .n (x). (ii) The projection onto .Ran (x) is called the range projection of x, denoted by .r (x). (iii) The projection .1 − n (x), which is the projection onto .Ran (x ∗ ), is called the support projection, denoted by .s (x). Observe that .n (x ∗ ) = 1 − r (x), .r (x ∗ ) = s (x) and .s (x ∗ ) = r (x). It is easy to see that .x = r (x) x = xs (x). In particular, if a is a self-adjoint operator in H , then .s (a) = r (a) and .a = s (a) a = as (a). The following theorem presents alternative descriptions of .s (x) and .r (x). Theorem 1.4.5 Let x be a closed and densely defined linear operator in .H . (i) .s (x) is the smallest projection .p ∈ P (B (H )) satisfying .x = xp. (ii) .r (x) is the smallest projection .p ∈ P (B (H )) satisfying .x = px. In view of this characterization of .s (x) and .r (x), these projections are sometimes called the right and left support projections of x, respectively. Next we recall the definition of the spectrum of a linear operator. Suppose that x is a densely defined closed linear operator. A complex number .λ belongs to the resolvent set .ρ (x) of .x if the operator .λ1−x : D (x) → H is bijective. If .λ ∈ ρ (x), then the inverse .r (λ, x) = (λ1 − x)−1 , which is called the resolvent of x at the point .λ, is a bounded linear operator on H , as follows from the Closed Graph Theorem. The spectrum .σ (x) of x is defined by .σ (x) = C \ ρ (x). In contrast to the case of bounded operators, it may happen that .σ (x) = ∅ or .σ (x) = C. In the next theorem we list some properties of the resolvent and the spectrum. Theorem 1.4.6 Let x be a closed and densely defined linear operator in H . Then: (i) The resolvent set .ρ (x) is an open subset of .C and so, the spectrum is closed. (ii) The mapping .λ │−→ r (λ, x) is an analytic function from .ρ (x) into .B (H ). Next we discuss the construction of closed, densely defined linear operators in H ∞ as direct sums of bounded linear operators. ∞ Let .{pn }n=1 be a sequence of pairwise orthogonal projections in H satisfying . n=1 pn = 1 and suppose that .{xn }∞ n=1 is a sequence in .B (H ) such that .xn = pn xn = xn pn for all n. Define  D (x) = ξ ∈ H :

∞ 

.

n=1

 ‖xn ξ ‖ < ∞ 2

1.5 The Spectral Theorem

9

∞ and .xξ = n=1 xn ξ for all .ξ ∈ D (x). The linear operator .x : D (x) → H is ∞ denoted by .⊕∞ n=1 xn and is called the direct sum of .{xn }n=1 . We list some of its properties. Proposition 1.4.7 Given the sequence .{xn }∞ n=1 in .B (H ) as above, the following statements hold: ∞ (i) The operator .⊕∞ n=1 xn is closed and densely defined; .⊕n=1 xn is bounded if and only if .supn ‖xn ‖B(H ) < ∞. ∞ ∗ (ii) The adjoint of .⊕∞ n=1 xn is equal to .⊕n=1 xn . (iii) If the operators .xn (.n = 1, 2, . . .) are normal, then .⊕∞ n=1 xn is normal. (iv) If the operators .xn (.n = 1, 2, . . .) are self-adjoint, then .⊕∞ n=1 xn is self-adjoint.

1.5 The Spectral Theorem As before we assume that .(H, 〈·, ·〉) is a complex Hilbert space. Suppose that .Ω is a non-empty set and that .A is a .σ -algebra of subsets of .Ω, so .(Ω, A) is a measurable space. Definition 1.5.1 A spectral measure on .(Ω, A) is a mapping .e : A → B (H ) such that: (i) (ii) (iii) (iv)

e (δ) is a projection in H for each .δ ∈ A. e (∅) = 0 and .e (Ω) = 1. .e (δ1 ∩ δ2 ) = e (δ1 ) e (δ2 ) for all .δ1 , δ2 ∈ A.   = If .δj ∈ A (.j = 1, 2, . . .) are pairwise disjoint, then .e ∪∞ j =1 δj ∞   j =1 e δj , where the series is strongly convergent in .B (H ). . .

Observe that condition (iii) in particular implies that .e (δ1 ) and .e (δ2 ) commute for any two .δ1 , δ2 ∈ A. If condition (iv) is satisfied only for finite disjoint collections .δ1 , . . . , δn (.n ∈ N) in .A, then e is referred to as a finitely additive spectral measure. A set .δ ∈ A is called an e-null set if .e (δ) = 0. The collection of all e-null sets, denoted by .Ne , is a .σ -ideal in .A. Given a spectral measure e on .(Ω, A) and .ξ, η ∈ H , we define the .σ -additive   measure .eξ,η : A → C by .eξ,η (δ) = 〈e (δ) ξ, η〉 for all .δ ∈ A. The variation .eξ,η  of .eξ,η satisfies .

  eξ,η  (δ) ≤ ‖e (δ) ξ ‖H ‖e (δ) η‖H

  for all .δ ∈ A.In particular, the total variation of .eξ,η , which is denoted by .eξ,η , satisfies .eξ,η  ≤ ‖ξ ‖H ‖η‖H .

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1 A Review of Relevant Operator Theory

The space of all .A-simple complex valued functions on .Ω is denoted by sim (Ω, A), i.e., ⎧ ⎫ n ⎨ ⎬ .sim (Ω, A) = αj χδj : αj ∈ C, δj ∈ A, j = 1, . . . , n; n ∈ N . ⎩ ⎭

.

j =1

For .s ∈ sim (Ω, A) we denote .‖s‖∞ = maxω∈Ω |s (ω)|. Clearly, .sim (Ω, A) is an algebra with respect  to the pointwise operations. Given a spectral measure .e : A → B (H ) and .s = nj=1 αj χδj in .sim (Ω, A), we define  sde =

.

Ω

n 

  αj e δj .

j =1

 The integration mapping .s │−→ Ω sde is an algebra homomorphism from .sim (Ω, A) into .B (H ) satisfying in addition for all .s ∈ sim (Ω, A) and all .ξ, η ∈ H :   ∗ (a) .Ω  Ω sde .  s¯ de = (b) . Ω sdeB(H ) ≤ ‖s‖∞ . 

  (c) . Ω sde ξ, η = Ω sdeξ,η . Let .Bb (Ω, A) be the algebra of all bounded complex valued .A-measurable functions on .Ω. For .f ∈ Bb (Ω, A) let .‖f ‖∞ = supω∈Ω |f (ω)|. With respect to the norm .‖·‖∞ , the space .Bb (Ω, A) is a Banach algebra and .sim (Ω, A) is a subalgebra. Now we shall extend the definition of the integral with respect to a spectral measure e on .(Ω, A) from .sim (Ω, A) to .Bb (Ω, A). For any .f ∈ − sn ‖∞ → 0 Bb (Ω, A) there exists a sequence .{sn }∞ n=1 in .sim (Ω, A) such that .‖f  ∞ as .n → ∞. From the estimate given in (b) above it follows that . Ω sn de n=1 is a Cauchy sequence in .B (H ). Consequently, this sequence is convergent in .B (H ) and it is readily verified that its limit depends only on the function f and not on the choice of the particular sequence .{sn }∞ n=1 . This justifies the following definition. Definition 1.5.2 Let .e : A → B (H ) be a spectral measure on the measurable by space .(Ω, A). For .f ∈ Bb (Ω, A) the integral with respect to e is defined ∞ {s } . f de = lim s de (norm convergence in . B where . is any (H )), n→∞ Ω n n n=1 Ω sequence in .sim (Ω, A) such that .‖f − sn ‖∞ → 0 as .n → ∞. In the next theorem we collect the basic properties of the integration mapping. Theorem 1.5.3 If .e : A → B (H ) is a spectral measure on the measurable space (Ω, A), then the following statements hold.  (i) The integration mapping .f │−→ Ω f de is an algebra homomorphism from .Bb (Ω, A) into .B (H ).  ∗  (ii) . Ω f¯de = Ω f de for all .f ∈ Bb (Ω, A).

.

1.5 The Spectral Theorem

11

  (iii) . Ω f deB(H ) ≤ ‖f ‖∞ for all .f ∈ Bb (Ω, A).  (iv) For every .f ∈ Bb (Ω, A) the operator . Ω f de is normal and if f is real valued, then  is self-adjoint.   . Ω f de (v) . Ω f de ξ, η = Ω f deξ,η for all .ξ, η ∈ H and all .f ∈ Bb (Ω, A).    2 (vi) . Ω f de ξ H = Ω |f |2 deξ,ξ for all .ξ ∈ H and all .f ∈ Bb (Ω, A). (vii) If .f ≥ 0 in .Bb (Ω, A), then . Ω f de ≥ 0 in .B (H ).    (viii) If .f, g ∈ Bb (Ω, A) are such that .|f | ≤ |g|, then . f de ξ H ≤ Ω             Ω gde ξ H for all .ξ ∈ H . In particular, . Ω f de B(H ) ≤ Ω gde B(H ) . ∞ (ix) If .{fn }n=1 is a uniformly bounded sequence in .Bb (Ω, A) and if .f ∈ Bb (Ω, A) such  that .fn (ω) → f (ω) as .n → ∞ for all .ω ∈ Ω, then . f de → Ω n Ω f de strongly as .n → ∞. The integral of a function over a subset of .Ω is defined in the usual manner.  Definition 1.5.4 Given .f ∈ Bb (Ω, A) and .∆ ∈ A, we define . ∆ f de =  Ω f χ∆ de. From the multiplicativity of the integration mapping it is obvious that 



 f de = e (∆)

.

∆

f de = Ω

 f de e (∆) .

Ω

Our next objective is to extend the integration mapping to all measurable functions on .(Ω, A). We assume that .e : A → B (H ) is a spectral measure. The space of all complex valued .A-measurable functions on .Ω is denoted by .B (Ω, A).  then . f χ If .f ∈ B (Ω, A) and .∆ ∈ A such that .f χ∆ ∈ Bb (Ω, A), Ω  ∆ de is defined  in Definition 1.5.2. In this situation we will also write . ∆ f de = Ω f χ∆ de. Note that this 1.5.4 in case f is bounded. Moreover, . ∆ f de =  agrees with  Definition  e (∆) ∆ f de = ∆ f de e (∆). Definition 1.5.5 Given .f ∈ B (Ω, A), a countable collection .{∆n }∞ n=1 in .A is called admissible for the function f if: ∞ (i) .{∆n }∞ n=1 consists of mutually disjoint sets and .∪n=1 ∆n = Ω; (ii) .f χ∆n ∈ Bb (Ω, A) for all .n = 1, 2, . . ..  Note that if follows from (i) that . ∞ n=1 e (∆n ) = 1 strongly in .B (H ). Let ∞ .f ∈ B (Ω, A) be given and let .{∆n } n=1 be any admissible sequence for f (evidently, such a sequence always exists). For each .n = 1, 2, . . . define .xn =  f de. Then .xn ∈ B (H ) is a normal operator satisfying .xn = e (∆n ) xn = ∆n xn e (∆n ). Consequently, we may define .x = ⊕∞ n=1 xn as in the discussion preceding Proposition 1.4.7, that is,

 D (x) = ξ ∈ H :

∞ 

.

n=1

 ‖xn ξ ‖2H

0, there exists .N ∈ N such that .‖ξn − ξm ‖q ≤ ε for all .n, m ≥ N and hence, .‖ξn − ξm ‖qα ≤ ε for all .n, m ≥ N and all .α. Since .‖ξm − ξ ‖q → 0 for each .α, it follows that .‖ξn − ξ ‖q ≤ ε for all .n ≥ N and all α α .α and so, .supα ‖ξn − ξ ‖q ≤ ε for all .n ≥ N . This implies that for .n ≥ N we have α .ξn − ξ ∈ D (q), hence .ξ ∈ D (q), and .‖ξn − ξ ‖q ≤ ε for all .n ≥ N . Consequently, .ξn → ξ in .D (q) with respect to .‖·‖q . We may conclude that .D (q) is complete with respect to .‖·‖q and so, .q is closed. ⨆ ⨅ .

If the domain .D (q) is dense in H , then we say that the form .q is densely defined. If two positive forms .q1 and .q2 satisfy .q1 ≤ q2 and if .q2 is densely defined, then .q1 is also densely defined, as .D (q2 ) ⊆ D (q1 ). The set of all closed positive densely defined forms on H is denoted by .Q+ . We equip .Q+ with the partial ordering inherited from the set of all positive forms. Corollary 1.8.6 If .{qα } is an increasing net in .Q+ and if .q0 ∈ Q+ is such that + + .qα ≤ q0 for all .α, then .sup qα ∈ Q (and .sup qα is the supremum of .{qα } in .Q ). Let .a : D (a) → H be operator in H . Define the form  a positive self-adjoint qa by .D (qa ) = D a 1/2 and .qa (ξ, η) = a 1/2 ξ, a 1/2 η for all .ξ, η ∈ D (qa ).

.

22

1 A Review of Relevant Operator Theory

2  Clearly, .qa is a positive densely defined form on H and .qa (ξ ) = a 1/2 ξ H for all 1/2 is a closed operator, it follows that .q is also closed. .ξ ∈ D (qa ). Using that .a a The form .qa is called the quadratic form corresponding to the positive selfadjoint operator a. It is easily verified that the operator a is uniquely determined by its quadratic form in the following sense: if .a and b are two positive self-adjoint operators in H such that .qa = qb , then .a = b. Furthermore, we observe that a linear subspace D of .D (qa ) is a core of .qa if and only if D is a core of .a 1/2 . In particular, .D (a) is a core of .D (qa ). Let us denote the set of all positive self-adjoint operators in H by .H+ . By the │ → qa from .H+ into .Q+ is injective. The next above observation, the map .a − proposition shows that this map is also surjective (and hence, it is a bijection). Proposition 1.8.7 If .q is a closed densely defined positive form on H , then there exists a unique self-adjoint positive operator a in H such that .D (q) = D a 1/2  2 and .q (ξ ) = a 1/2 ξ  for all .ξ ∈ D (q) (in other words, .q = qa ). H

Proof Since .q is closed, .D (q) is a Hilbert space with respect to .〈·, ·〉q . For .ξ ∈ H , define the linear functional .ϕξ on .D (q) by .ϕξ (η) = 〈η, ξ 〉. Since .

  ϕξ (η) = |〈η, ξ 〉| ≤ ‖ξ ‖H ‖η‖H ≤ ‖ξ ‖H ‖η‖q

  for all .η ∈ D (q), it follows that .ϕξ is bounded with respect to .‖·‖q and .ϕξ  ≤ ‖ξ ‖H . Consequently, there exists a unique .bξ ∈ D (q) such that .ϕξ (η) =  element  〈η, bξ 〉q for all .η ∈ D (q) and .‖bξ ‖q = ϕξ  ≤ ‖ξ ‖H . The map .ξ │−→ bξ is linear from H into .D (q). Composing this map with the embedding of .D (q) into H , we obtain a linear operator .b : H → H , which is bounded with .‖b‖B(H ) ≤ 1. From the definition of .bξ it follows that .

〈η, ξ 〉 = 〈η, bξ 〉q = q (η, bξ ) + 〈η, bξ 〉 , η ∈ D (q) , ξ ∈ H,

(1.4)

and so, q (η, bξ ) = 〈η, ξ − bξ 〉 , η ∈ D (q) , ξ ∈ H.

.

(1.5)

Observe the following facts: (i) b is self-adjoint. Indeed, if .ξ, ζ ∈ H , then .bζ ∈ D (q) and so, it follows from (1.4) that .

〈bζ, ξ 〉 = 〈bζ, bξ 〉q = 〈bξ, bζ 〉q = 〈bξ, ζ 〉 = 〈ζ, bξ 〉 .

(ii) b is injective. Indeed, if .ξ ∈ H satisfies .bξ = 0, then .〈η, ξ 〉 = 〈η, bξ 〉q = 0 for all .η ∈ D (q). Since .D (q) is dense in H , this implies that .ξ = 0.

1.8 Quadratic Forms

23

(iii) .Ran (b) is dense in .D (q) with respect to .‖·‖q (and hence, .Ran (b) is dense in H ). Indeed, if .η ∈ D (q) satisfies .〈η, bξ 〉q = 0 for all .ξ ∈ H , then .〈η, ξ 〉 = 〈η, bξ 〉q = 0 for all .ξ ∈ H and hence, .η = 0.     Consider the inverse .b−1 : D b−1 → H , where .D b−1 = Ran (b). From the above observations that .b−1 is closed, densely defined and self-adjoint  −1 ∗ it follows −1 ∗ (recall that . b = (b ) = b−1 ). Equation (1.5) can then be written as       q (η, ζ ) = η, b−1 ζ − ζ = η, b−1 − 1 ζ , η ∈ D (q) , ζ ∈ D b−1 .

.

−1 − 1 is closed, densely defined and self-adjoint with .D (a) = The .a = b   operator −1 = Ran (b) and satisfies D b

q (η, ζ ) = 〈η, aζ 〉 , η ∈ D (q) , ζ ∈ D (a) .

.

In particular, if .ξ, η ∈ D (a), then q (ξ, η) = 〈ξ, aη〉 = 〈aξ, η〉

.

and .〈aξ, ξ 〉 = q (ξ, ξ ) = q (ξ ) ≥ 0 for all .ξ ∈ D (a), which shows that a is positive.  2   Finally we show that .D (q) = D a 1/2 and .q (ξ ) = a 1/2 ξ H for all .ξ ∈ D (q).   Let .qa be the closed positive form corresponding to a, that is, .D (qa ) = D a 1/2 2  and .qa (ξ ) = a 1/2 ξ H for all .ξ ∈ D (qa ). Since .D (a) is a core of the operator .a 1/2 , it follows that .D (a) is a core for .qa . As has been observed above, .D (a) = Ran (b) is dense in .D (q) with respect to .‖·‖q , which implies that .D (a) is a core for .q. Furthermore,  2    q (ξ ) = 〈aξ, ξ 〉 = a 1/2 ξ, a 1/2 ξ = a 1/2 ξ  = qa (ξ ) ,

.

H

This implies that .q = qa and the proof is complete.

ξ ∈ D (a) . ⨆ ⨅

It follows from the above results that the map .a │−→ qa is a bijection from .H+ onto .Q+ . Via this map we transfer the partial ordering in .Q+ to .H+ : if a and b are two self-adjoint positive operators, then we write .a ≤ b whenever .qa ≤ qb . This partial ordering in .H+ is called the quadratic form ordering in .H+ . By definition, the map .a − │ → qa is an order isomorphism from .H+ onto .Q+ . Therefore, the following result is now an immediate consequence of Corollary 1.8.6 and Lemma 1.8.4.   Proposition 1.8.8 If . aβ is an increasing net in .H+ and .b ∈ H+ is such that + .aβ ≤ b for all .β, then .a = supβ aβ exists in .H . Moreover, this supremum a is

24

1 A Review of Relevant Operator Theory

characterized by: ⎧ ⎫   ⎬  ⎨    1/2    1/2 1/2 = ξ∈ : sup aβ ξ  < ∞ , .D a D aβ ⎩ ⎭ H β    1/2  a ξ 

β

H

   1/2  = sup aβ ξ  , H

β

  ξ ∈ D a 1/2 .

If .a ∈ H+ and .u ∈ B (H ) is unitary, then .u∗ au ∈ H+ and so, we may define the map .ϕu : H+ → H+ by .ϕu (a) = u∗ au. It is straightforward to show that .ϕu : H+ → H+ is an order isomorphism. Therefore, the next result follows immediately from Lemma 1.3.1. Corollary 1.8.9  Let .U0 be a non-empty collection of unitary operators on H . Suppose that . aβ is an increasing net in .H+ and .a ∈ H+ such that .a = supβ aβ . If ∗ ∗ .u aβ u = aβ for all .u ∈ U0 and all .β, then .u au = a for all .u ∈ U0 . Remark 1.8.10 We wish to point out another useful consequence of the above considerations. Suppose that a is a positive self-adjoint operator in the Hilbert space H . If .f : σ (a) → [0, ∞) is a Borel function, then .f (a) ∈ H+ and it follows from Theorem 1.5.7 (i) that the corresponding quadratic form .qf (a) is given by !    .D qf (a) = ξ ∈ H :

f (λ) deξ,ξ

" (λ) < ∞

σ (a)

and  qf (a) (ξ ) =

f (λ) deξ,ξ (λ) ,

.

  ξ ∈ D qf (a) .

σ (a)

A combination of Proposition 1.8.8 and the monotone convergence theorem immediately yields the following result: if .f, fn : σ (a) → [0, ∞) (.n = 1, 2, . . .) are Borel functions such that .0 ≤ fn (λ) ↑ f (λ) for all .λ ∈ σ (a), then .fn (a) ↑ f (a) in .H+ .

1.9 Algebras with an Involution In this and the next section, we discuss some of the aspects of the theory of abstract operator algebras, which will be convenient to have available. We start with the general concept of a .∗-algebra.

1.9 Algebras with an Involution

25

Let .A be an algebra over the complex numbers. The mapping .x │−→ x ∗ from .A into itself is said to be an involution if (i) (ii) (iii) (iv)

(x + y)∗ = x ∗ + y ∗ ; ∗ ¯ x∗; .(λx) = λ ∗ ∗ ∗ .(xy) = y x ; ∗ ∗ .(x ) = x, .

whenever .x, y ∈ A and .λ ∈ C. An algebra equipped with an involution is called a .∗-algebra. If .A is a .∗-algebra, then an element .x ∈ A is called self-adjoint (or, Hermitian) if .x ∗ = x. The set of all self-adjoint elements in .A is denoted by .Ah , which is clearly a real linear subspace of .A. If .x, y ∈ Ah , then .xy ∈ Ah if and only if .xy = yx. Note furthermore that .x ∗ x and .xx ∗ belong to .Ah for every .x ∈ A. For .x ∈ A we set Re (x) =

.

  1  1 x + x ∗ , Im (x) = x − x∗ . 2 2i

Clearly, .Re (x) , Im (x) ∈ Ah and .x = Re (x) + iIm (x) for all .x ∈ A. Conversely, if for a given .x ∈ A we have .x = x1 + ix2 with .x1 , x2 ∈ Ah , then necessarily .x1 = Re (x) and .x2 = Im (x). The .∗-algebra .A is called unital if it possesses a multiplicative identity, a unit element, denoted by .1 = 1A . Note that .1∗ = 1. An element x in the unital algebra .A is said to be invertible if there exists .y ∈ A such that .xy = yx = 1; in this case, the element y is unique and denoted by .x −1 , the inverse of x. It is easy  to see that ∗ −1 ∗ = (x ∗ )−1 . .x ∈ A is invertible if and only if .x is invertible and, in this case, . x An element .x ∈ A is called normal if .x ∗ x = xx ∗ . Furthermore, .u ∈ A is said to be unitary if .u∗ u = uu∗ = 1 (equivalently, u is invertible and .u−1 = u∗ ). All unitary elements in .A form a (multiplicative) group, which we shall denote by 2 ∗ .U (A). An element .p ∈ A is said to be a projection if .p = p and .p = p. The set of all projections in .A is denoted by .P (A). Let .A and .B be two .∗-algebras. The mapping .φ : A → B is called a .∗homomorphism whenever .φ is an algebra homomorphism (that is, .φ is linear and ∗ ∗ .φ (xy) = φ (x) φ (y) for all .x, y ∈ A) satisfying .φ (x ) = φ (x) for all .x ∈ A. If, in addition, .A and .B are unital and .φ (1A ) = 1B , then .φ is called a unital .∗-homomorphism. If .A and .B are .∗-algebras and .φ : A → B is an injective .∗homomorphism, then .φ is said to be a .∗-isomorphism. The algebras .A and .B are said to be .∗-isomorphic if there exists a .∗-isomorphism from .A onto .B. If .φ : A → B is a .∗-isomorphism onto and .A is unital, then .B is also unital and .φ is a unital .∗isomorphism. A .∗-isomorphism from .A onto itself is called a .∗-automorphism of .A. A subset .S of a .∗-algebra .A is called self-adjoint if .x ∗ ∈ S whenever .x ∈ S. A self-adjoint subalgebra .S of .A is said to be a .∗-subalgebra and, in this case, .S itself is a .∗-algebra with respect to the algebraic operations and involution inherited from .A.

26

1 A Review of Relevant Operator Theory

1.10 C ∗ -Algebras An algebra .A equipped with a norm .‖·‖A such that .A is a Banach space and (i) .‖xy‖A ≤ ‖x‖A ‖y‖A for all .x, y ∈ A, is called a Banach algebra. If .A has a unit element .1, then we will assume that ‖1‖A = 1. Furthermore, if .A is a .∗-algebra and the norm satisfies in addition

.

(ii) .‖x ∗ ‖A = ‖x‖A for all .x ∈ A, then .A is called a Banach .∗-algebra. A .C ∗ -algebra is a .∗-algebra .A equipped with a norm .‖·‖A such that .A is a Banach algebra and (iii) .‖x ∗ x‖A = ‖x‖2A for all .x ∈ A. If .A is a .C ∗ -algebra, then it is easy to see that the norm also satisfies condition (ii). So, any .C ∗ -algebra is a Banach .∗-algebra. Moreover, if the .C ∗ algebra .A has a unit element .1, then the equality .‖1‖A = 1 is automatically satisfied. If .A is a .C ∗ -algebra and .A1 is a closed .∗-subalgebra of .A, then .A1 is a .C ∗ algebra with respect to the structure inherited from .A and we say that .A1 is a .C ∗ subalgebra of .A. Suppose that .A and .B are two .C ∗ -algebras and that .φ : A → B is a .∗homomorphism. Then, .‖φ (x)‖B ≤ ‖x‖A for all .x ∈ A and so, in particular, .φ is continuous. Moreover, if .φ is a .∗-isomorphism, then .‖φ (x)‖B = ‖x‖A for all ∗ .x ∈ A. Consequently, two .∗-isomorphic .C -algebras are isometrically isomorphic as well. We list a few simple examples of .C ∗ -algebras. Example 1.10.1 1. Let X be an arbitrary non-empty set. We denote by .𝓁∞ (X) the .∗-algebra of all bounded complex-valued functions .f : X → C, with pointwise algebraic operations and involution given by pointwise complex conjugation. Equipped with the norm given by .‖f ‖∞ = supt∈X |f (t)| for all .f ∈ 𝓁∞ (X), the algebra ∗ .𝓁∞ (X) is a commutative unital .C -algebra. 2. Let X be a locally compact Hausdorff space. A continuous function .f : X → C is said to vanish at infinity if, for every .ε > 0 the set .{t ∈ X : |f (t)| ≥ ε} is compact. We denote by .C0 (X) the .∗-algebra of all continuous complexvalued functions on X vanishing at infinity (which is a .∗-subalgebra of .𝓁∞ (X)), equipped with the .sup-norm .‖·‖∞ . Clearly, .C0 (X) is a commutative .C ∗ -algebra, which has a unit element if and only if X is compact. By the Gelfand–Naimark theorem, every commutative .C ∗ -algebra .A is .∗isomorphic to an algebra .C0 (X) for some suitable locally compact Hausdorff space X (which is compact if, and only if, .A is unital). 3. Let .(Ω, ∑, μ) be a measure space and .L∞ (Ω, ∑, μ) = L∞ (μ) be the space of all .μ-essentially bounded complex-valued .∑-measurable functions on .Ω,

1.10 C ∗ -Algebras

27

with identification of functions which coincide .μ-almost everywhere. The space L∞ (μ), equipped with the norm given by

.

.

‖f ‖∞ = esssupt∈Ω |f (t)|

for all .f ∈ L∞ (μ), is a commutative .C ∗ -algebra with respect to the pointwise algebraic operations and the involution given by complex conjugation. All measure spaces we shall consider will be assumed to be so-called Maharam measure spaces. A measure space .(Ω, ∑, μ) is said to be a Maharam measure space if it is semi-finite (that is, for every .E ∈ ∑ with .μ (E) > 0 there exists .F ∈ ∑ such that .F ⊆ E and .0 < μ (F ) < ∞) and its measure algebra (that is, the quotient Boolean algebra of .∑ by the ideal of .μ-null sets) is complete. Clearly, all .σ -finite measure spaces are Maharam. We note that a measure space .(Ω, ∑, μ) is Maharam if and only if the space .LR ∞ (μ) of all real-valued functions in .L∞ (μ), is a Dedekind complete vector lattice. Another equivalent condition is that .L∞ (μ) is (isometrically isomorphic to) the Banach dual space of .L1 (μ) (equivalently, the Radon–Nikodym theorem is valid for the measure space .(Ω, ∑, μ)). 4. If H is a Hilbert space, then the algebra .B (H ) of all bounded linear operators on H is a .C ∗ -algebra, as we have observed already in Sect. 1.1. This .C ∗ algebra is noncommutative whenever .dim H > 1. Any closed self-adjoint subalgebra of ∗ .B (H ) is also a .C -algebra. The Gelfand–Naimark–Segal theorem asserts that an arbitrary .C ∗ -algebra .A is .∗-isomorphic to a .C ∗ -subalgebra of .B (H ) for some Hilbert space H . For the remaining part of this section we will assume that .A is an arbitrary unital C ∗ -algebra, although most of the results mentioned can be adjusted to the non-unital situation via adjunction of a unit element. As in any unital Banach algebra, for .x ∈ A the set .σ (x) = σA (x) of all .λ ∈ C for which .λ1 − x is not invertible in .A, is called the spectrum of x. The spectrum .σ (x) is a non-empty compact subset of .C. If .u ∈ U (A), then .σ (u) ⊆ {λ ∈ C : |λ| = 1} and if .a ∈ Ah , then .σ (a) ⊆ R. In general, if .B is a Banach subalgebra of the Banach algebra .A and .x ∈ B, then .σA (x) ⊊ σB (x). However, for .C ∗ -algebras we have the following important observation: if .B is a ∗ ∗ .C -subalgebra of the .C -algebra .A, then .σA (x) = σB (x) for all .x ∈ B. The following theorem, which is a consequence of the Gelfand–Naimark theorem for commutative .C ∗ -algebras, gives the functional calculus for normal elements. .

Theorem 1.10.2 Let .A be a unital .C ∗ -algebra and .x be a normal element of .A. There exists a unique unital .∗-isomorphism .φ : C (σ (x)) → A satisfying .φ (ι) = x, where .ι (λ) = λ for all .λ ∈ C. Actually, .φ is a .∗-isomorphism from .C (σ (x)) onto the .C ∗ -subalgebra of .A generated by x and .1. In the situation of the above theorem, for any .f ∈ C (σ (x)) the corresponding │ → f (x) is called the element .φ (f ) is denoted by .f (x) and the mapping .f − functional calculus of the normal element .x. The spectral mapping theorem holds

28

1 A Review of Relevant Operator Theory

for this functional calculus, that is, for every .f ∈ C (σ (x)) we have σ (f (x)) = f (σ (x)) = {f (λ) : λ ∈ σ (x)} .

.

(1.6)

Moreover, if .f ∈ C (σ (x)) and .g ∈ C (σ (f (x))), then .(g ◦ f ) (x) = g (f (x)). Remark 1.10.3 Suppose that H is a Hilbert space and .x ∈ B (H ) is a normal operator. Let .φ : C (σ (x)) → B (H ) be the .∗-isomorphism which exists according to Theorem 1.10.2. On the other hand, it follows from Theorems 1.5.10 and 1.5.13, that there exists a unique spectral measure .ex : B (σ (x)) → B (H ) for the operator x. Let .Bb (σ (x)) denote the .C ∗ -algebra of all bounded Borel functions on .σ (x). For .f ∈ Bb (σ (x)) define .Ф (f ) ∈ B (H ) by  Ф (f ) =

f (λ) dex (λ) .

.

σ (x)

It follows from Theorem 1.5.3 that .Ф : Bb (σ (x)) → B (H ) is a unital .∗homomorphism satisfying .Ф (ι) = x (by the definition of the spectral measure .ex ). Consequently, if p is a polynomial in .λ and .λ¯ , then .φ (p) = p (x, x ∗ ) = Ф (p). Since the subalgebra of .C (σ (x)) consisting of all such polynomials is dense in .C (σ (x)) with respect to .‖·‖∞ , the continuity of .φ and .Ф implies that .φ (f ) = Ф (f ) for all .f ∈ C (σ (x)). Hence,  f (x) = φ (f ) =

f (λ) dex (λ)

.

σ (x)

for all .f ∈ C (σ (x)). So, we may use the notation  f (x) =

f (λ) dex (λ)

.

σ (x)

for all .f ∈ Bb (σ (x)) (and also for .f ∈ B (σ (x)); see Definition 1.5.11) without any danger of confusion. Actually, one may prove the existence of the spectral measure of a normal operator .x ∈ B (H ) via the result of Theorem 1.10.2 (a method which is analogous to the Riesz representation theorem for positive functionals on a .C (K)-space). Let .A be a (unital) .C ∗ -algebra. An element .x ∈ A is said to be positive if .x ∗ = x and .σ (x) ⊆ [0, ∞). This is denoted by .x ≥ 0. The set of all positive elements in .A is denoted by .A+ . If .x ∈ A+ , then it follows from Theorem 1.10.2 that there exists an element .y ∈ A+ such that .y 2 = x (indeed, take .y = f (x), where .f (λ) = λ1/2 for .λ ∈ σ (x) ⊆ [0, ∞) and use (1.6) that .y ≥ 0). The element .y ∈ A+ with √ to see 1/2 2 .y = x is unique and denoted by . x or .x , the positive square root of x. The following proposition describes the basic properties of positive elements.

1.10 C ∗ -Algebras

29

Proposition 1.10.4 + (i) .A+ is a cone in .A (that is, .λx + μy ∈ A+ whenever .x, y ∈ A and .λ, μ ∈  + + R+ ) which is closed and proper (that is, .A ∩ −A = {0});   (ii) .A+ = {x ∗ x : x ∈ A} = a 2 : a ∈ Ah ; (iii) if .A is a .C ∗ -subalgebra of .B (H ), for some Hilbert space H , and if .x ∈ A, then .x ∈ A+ if and only if .〈xξ, ξ 〉 ≥ 0 for all .ξ ∈ H (that is, x is a positive self-adjoint operator on H ).

In the real vector space .Ah we define a partial ordering by setting .a ≤ b whenever .a, b ∈ Ah satisfy .b − a ∈ A+ . In the next proposition we collect some of the properties of this partial ordering. Proposition 1.10.5 (i) .Ah is a partially ordered vector space (that is, .a + c ≤ b + c and .λa ≤ λb whenever .a ≤ b in .Ah , .c ∈ Ah and .λ ∈ R+ ); (ii) if .a ≤ b in .Ah , then .x ∗ ax ≤ x ∗ bx for all .x ∈ A; (iii) if .a, b ∈ A+ and .ab = ba, then .ab ∈ A+ ; (iv) if .0 ≤ a ≤ b in .Ah , then .‖a‖A ≤ ‖b‖A and .a 1/2 ≤ b1/2 ; moreover, if in addition a is invertible, then b is also invertible and .0 ≤ b−1 ≤ a −1 ; (v) if .x ∈ A is normal and .f ∈ C (σ (x)) satisfies .f (λ) ≥ 0 for all .λ ∈ σ (x), then .f (x) ≥ 0 in .Ah . Define the functions .f0 , f1 and .f2 on .R by .f0 (λ) = |λ|, .f1 (λ) = λ+ and − respectively, for all .λ ∈ R. For .a ∈ A we set .|a| = f (a), the .f2 (λ) = λ h 0 absolute value of a, .a + = f1 (a), the positive part of a, and .a − = f2 (a), the negative part of a. From the properties of the functional calculus it is clear that + − ∈ A+ , .a = a + − a − and .|a| = a + + a − . This shows in particular that .|a| , a , a the positive cone .A+ is generating in .Ah , that is, .Ah = A+ − A+ . Note that the  1/2 . absolute value of .a ∈ Ah is also given by .|a| = a 2 It is worth noting, that for every .a ∈ Ah with .‖a‖A ≤ 1, the element .1 − a 2 is 1/2 1/2   and .u2 = a − i 1 − a 2 are positive and the elements .u1 = a + i 1 − a 2 unitary. Moreover, .a = 12 (u1 + u2 ). This implies, in particular, that .A is the linear span of its unitary elements. A linear functional f on a .C ∗ -algebra .A is said to be positive if .f (a) ≥ 0 for all .a ∈ A+ . If f is a positive functional on .A, then f is necessarily continuous, 2 ∗ ∗ .f (x ) = f (x) and .|f (x)| ≤ ‖f ‖A∗ f (x x) for all .x ∈ A. Moreover, if .A is unital, then  .‖f ‖A∗ = f (1). Let . Aj j ∈J be a family of .C ∗ -algebras, where J is an arbitrary set of indices.     We denote by .A the set of all families . xj = xj j ∈J , with .xj ∈ Aj for all .j ∈ J   and .supj xj A < ∞. Defining the algebraic operations, involution and norm in j .A by                xj yj = xj yj , λ xj = λxj , . xj + yj = xj + yj ,    ∗  ∗    xj = xj ,  xj A = sup xj A , j

j

30

1 A Review of Relevant Operator Theory

∗ respectively, it follows which is called the .C ∗ -algebra  that  .A is a .C -algebra, # ∗ product of the family . Aj j ∈J . This product .C -algebra is denoted by .C ∗ − j Aj , # or simply by . j Aj , if no confusion can arise.

1.11 Von Neumann Algebras Given a non-empty subset .S of .B (H ), the commutant .S' of .S is defined by S' = {x ∈ B (H ) : xy = yx ∀y ∈ S} ,

.

which is a wo-closed unital subalgebra of .B (H ). If .S is self-adjoint, then .S' is a '' ∗ wo-closed .B (H ). Defining the bi-commutant .S of .S by  ' 'unital .C -subalgebra of '' '' ' ''' .S = S , it is clear that .S ⊆ S and .S = S . Definition 1.11.1 A .∗-subalgebra .M of .B (H ) is said to be a von Neumann algebra if .M = M'' . In order to indicate that .M is contained in the algebra .B (H ), we say that .M acts on the Hilbert space H (briefly, .M is a von Neumann algebra on H ). If .M is a von Neumann algebra, then .M is a wo-closed unital .C ∗ -subalgebra of .B (H ). The simplest examples of von Neumann algebras are given by the algebra .B (H ) itself and the subalgebra .C1 = CH = {λ1 : λ ∈ C}. For any non-empty subset ' '' .S of .B (H ), the commutant .S is a von Neumann algebra. Similarly, .S is a von '' Neumann algebra. Actually, .S is the von Neumann algebra generated by .S, that is, the smallest von Neumann algebra on H containing .S. Suppose that .M is a von Neumann algebra. By the observations at the end of Sect. 1.10, every element of .M' is a linear combination of unitary elements of .M' . Consequently, if .x ∈ B (H  ), then .x ∈ M if and only if .xu = ux (equivalently, ' ∗ .x = uxu ) for all .U M . Furthermore, if .x ∈ B (H ) is normal, it follows from Theorem 1.5.12 that the following statements are equivalent: 1. .x ∈ M; 2. .ex (δ) ∈ M for all Borel sets .δ ⊆ C; 3. .f (x) ∈ M for all bounded Borel functions .f : σ (x) → C. In particular, if .a ∈ M+ , then .a 1/2 ∈ M+ . Consequently, if .x ∈ M, then ∈ M+ and so, .|x| = (x ∗ x)1/2 ∈ M+ . Furthermore, if .x ∈ M, with polar ∗ decomposition  ' .x = v |x|, then it follows from Proposition 1.7.4 that .v = uvu for all .u ∈ U M and hence, .v ∈ M. This also implies that the support projection ∗ ∗ .s (x) = v v and the range projection .r (x) = vv both belong to .M. The following result is an immediate consequence of Lemma 1.3.4 and Theorem 1.3.3. ∗ .x x

1.11 Von Neumann Algebras

31

  Theorem 1.11.2 Suppose that . aβ is an increasing net in .Mh . (i) If .a ∈ Bh (H ) is such that .aβ ↑ a in .Bh (H ), then .a ∈ Mh (and hence, .aβ ↑ a in .Mh). (ii) If . aβ is bounded from above, then there exists .a ∈ Mh such that .aβ ↑ a in .Mh (and .aβ ↑ a in .Bh (H )). The following observation is easily checked.   Lemma 1.11.3 If . aβ is an increasing net in .Mh and .a ∈ Mh , then .aβ ↑ a in uwo

Mh if and only if .aβ → a.

.

Suppose that .{Mi : i ∈ I } is a collection of von Neumann algebras acting on the '  Hilbert space H and define .M = i∈I Mi . Since .x ∈ M if and only if .x ∈ M'i for all .i ∈ I , it is clear that M=

$ %

.

&' M'i

,

i∈I

which shows, in particular, that .M is a von Neumann algebra. Moreover, '

M =

.

$ %

&'' M'i

,

i∈I

  so .M' is the von Neumann algebra generated by . M'i : i ∈ I . The center .Z (M) of a von Neumann algebra .M is defined by Z (M) = {x ∈ M : xy = yx ∀y ∈ M} .

.

Since .Z (M) = M ∩ M' , it follows from the above observationsthat.Z (M) is a von Neumann algebra, which is clearly commutative. Note that .Z M' = Z (M). If .Z (M) = CH , then the von Neumann algebra .M is said to be a factor. Since ' .B (H ) = CH , it is evident that .B (H ) is a factor. As we have observed before, if .M is a von Neumann algebra, then .M is a wo-closed subalgebra of .B (H ) (and hence, .M is also so-, uwo-, and usoclosed, as these topologies are stronger than the wo-topology). Moreover, by Theorem 1.2.2, the norm closed unit ball .BB(H ) is wo-compact and so, .BM is wo-closed (and hence, .BM is also so-, uwo-, and uso-closed). These topological properties actually characterize von Neumann algebras. This is the famous Double Commutant Theorem of J. von Neumann.

32

1 A Review of Relevant Operator Theory

Theorem 1.11.4 (Double Commutant Theorem) Let .M be a unital .∗-subalgebra of .B (H ) and let .BM be its unit ball (with respect to the operator norm). The following statements are equivalent. (i) .M is a von Neumann algebra, that is, .M = M'' . (ii) .M is wo-closed (or, equivalently, so-, uwo-, uso-closed). (iii) .BM is wo-closed (or, equivalently, so-, uwo-, uso-closed). Note that the only non-trivial implication in this theorem is that .M is a von Neumann algebra whenever .BM is uso-closed. If .M is a von Neumann algebra, then .BM = M ∩ BB(H ) is wo-compact and hence, .BM is uwo-compact, as the wo- and uwotopology coincide on norm bounded subsets of .B (H ). The dual space of .M with respect to the norm topology is denoted by .M∗ . For .ξ, η ∈ H we denote (with slight abuse of notation) by .ωξ,η the linear functional on .M given by .ωξ,η (x) = 〈xξ, η〉 for  all .x ∈ M (see  Sect. 1.2). Let .M∼ be the linear subspace of .M∗ generated by . ωξ,η : ξ, η ∈ H . Evidently, the wo-topology is equal to .σ (M, M∼ ). The norm closure of .M∼ is denoted by .M∗ . Furthermore, we denote by .M'wo , .M'so , .M'uwo and .M'uso the dual spaces of .M with respect to the wo-, so-, uwo- and uso-topology, respectively. In the following theorem we collect the main features of the duality theory of von Neumann algebras. Theorem 1.11.5 Let .M be a von Neumann algebra on the Hilbert space H .  (i) .M'so = M'wo and .ϕ ∈ M'wo if and only if .ϕ is given by .ϕ = nj=1 ωξj ,ηj with .ξ1 , . . . , ξn ∈ H and .η1 , . . . , ηn ∈ H . ' ' ' (ii) .M is given by .ϕ uso = Muwo = M∗ and .ϕ ∈ Muwo if and only if  .ϕ∞ ∞= ∞ ∗ ω , as a norm convergent series in . M , with . ξ and . η ξ ,η j j j j j =1 j =1 j =1  2    ξj  < ∞ and .∞ ηj 2 < ∞. in H satisfying . ∞ j =1 j =1 H H (iii) If .ϕ is a linear functional on .M, then .ϕ belongs to .M∗ if and only if the restriction of .ϕ to the unit ball .BM is wo-continuous (equivalently, is so-, usoor uwo-continuous). (iv) The uwo-topology in .M is equal to .σ (M, M∗ ). (v) Every .x ∈ M defines a bounded linear functional .xˆ ∈ (M∗ )∗ by defining .x ˆ (ϕ) = ϕ (x) for all .ϕ ∈ M∗ . The mapping .x │−→ xˆ is a isometrically isomorphism from .M onto .(M∗ )∗ . By (v) of the above theorem, .M may be identified with the Banach dual space of M∗ via the mapping .x − │ → x. ˆ Therefore, .M∗ is called the pre-dual of .M. For positive linear functionals on .M there is another important characterization of uwo-continuity. Recall that a linear functional .ϕ on .M is called positive if .ϕ (a) ≥ 0 whenever .a ∈ M+ . This is denoted by .ϕ ≥ 0. If .ϕ, ψ ∈ M∗ are such that ∗ .ϕ − ψ ≥ 0, then we write .ψ ≤ ϕ. Such positive functionals belong to .M and ∗ it can be shown that every element of .M can be written as a linear combination positive linear functionals. Examples of positive functional on .M are given by the functionals of the form .ωξ = ωξ,ξ , with .ξ ∈ H . Note that it follows immediately from (ii) of Theorem 1.11.5, via polarization, that every .ϕ ∈ M∗ can be written as a linear combination of positive functionals in .M∗ . .

1.11 Von Neumann Algebras

33

Definition 1.11.6 Let .ϕ be a positive linear functional on the von Neumann algebra M.   (i) .ϕ is said to be normal if .aβ ↑ a in .M+implies  that.ϕ aβ ↑ ϕ (a). (ii) .ϕ is called completely additive if .ϕ α pα = α ϕ (pα ) for every system .{pα } of pairwise orthogonal projections in .M.

.

Theorem 1.11.7 For a positive functional .ϕ on .M the following statements are equivalent: (i) (ii) (iii) (iv)

ϕ is normal; ϕ is completely additive; .ϕ is uwo-continuous (equivalently, .ϕ ∈ M∗ );  ∞ ∞ ∗ .ϕ is given by .ϕ = j =1 ωξj , as a norm convergent series in .M , with . ξj j =1  2  in H satisfying . ∞ ξj  < ∞. . .

j =1

H

For normal positive functionals the following noncommutative version of the Radon–Nikodym theorem holds, which is due to S. Sakai. Theorem 1.11.8 If .ϕ ∈ M∗ and .ψ ∈ M∗ are such that .0 ≤ ψ ≤ ϕ, then .ψ ∈ M∗ and there exists .a ∈ Mh such that .0 ≤ a ≤ 1 and .ψ (x) = ϕ (axa) for all .x ∈ M. The notions of positivity and normality can also be introduced for linear mappings between von Neumann algebras. Definition 1.11.9 Let .M and .N be von Neumann algebras, acting on the Hilbert spaces H and K respectively, and .π be a linear mapping from .M into .N. (i) .π is called positive if .π (a) ≥ 0 in .N whenever .a ≥ 0 in .M. (ii) If .π is positive, then .π is said to be normal if .π aβ ↑ π (a) in .N whenever .aβ ↑ a in .M. Note that any .∗-homomorphism .π : M → N is positive. Furthermore, a positive linear mapping .π : M → N is normal if and only if it is continuous with respect to the ultra-weak operator topologies. Theorem 1.11.10 Let .M and .N be von Neumann algebras (acting on the Hilbert spaces H and K respectively). If .π : M → N is a normal unital .∗-homomorphism, then .π (M) is a von Neumann algebra. Next we discuss some important constructions concerning von Neumann algebras. First we introduce the necessary notation. Let .e ∈ P (B (H )) a projection onto .K = e (H ). For .x ∈ B (H ) we define .xe ∈ B (K) by .xe (ξ ) = exξ for all .ξ ∈ K, that is, .xe = (ex)|K . Note that .xe = (exe)e for all .x ∈ B (H ). For any non-empty subset .D ⊆ B (H ) we denote .De = {xe : x ∈ D}. We consider, in particular, the following two situations: 1. Let .M ⊆ B (H ) be a von Neumann algebra and .e ∈ P (M). Define eMe = {exe : x ∈ M} ,

.

34

1 A Review of Relevant Operator Theory

which is a .∗-subalgebra of .M with unit element e.  that an element .x ∈ M  Note belongs to .eMe if and only if .x (K) ⊆ K and .x K ⊥ = {0}. If .x ∈ eMe, then .xe = x|K . In this situation, .Me is a unital .∗-subalgebra of .B (K) and the mapping .φe : eMe → Me , defined by .φe (x) = xe for all .x ∈ eMe, is a surjective unital .∗-isomorphism.   2. Let .M ⊆ B (H ) be a von Neumann algebra and .e ∈ P M' . If .x ∈ M, then  ⊥ .xe = ex, that is, .x (K) ⊆ K and .x K ⊆ K ⊥ . Moreover, .xe = x|K for all .x ∈ M. The set .Me is a unital .∗-subalgebra of .B (K) and the mapping .ψe : M → Me , defined by .ψe (x) = xe , is a surjective .∗-homomorphism. The kernel of .ψe is given by Ker (ψe ) = {x (1 − z (e)) : x ∈ M} ,

.

where .z(e) denotes the central support of e; see Definition 1.14.2 below. In particular, .ψe is an isomorphism if and only if .z (e) = 1. Theorem 1.11.11 Let .M be a von Neumann algebra on the Hilbert space H . Suppose that .e ∈ P (M) and put .K = e (H ).   (i) .Me and . M' e are von Neumann algebras on the Hilbert space K and   ' ' .(Me ) = M . e (ii) The center of .Me is equal to .(Z  (M))  e. (iii) If .M is a factor, then .Me and . M' e are also factors.   By (i) of the above theorem, there is no danger of confusion to denote . M' e simply by .M'e whenever .e ∈ P (M). The von Neumann algebra .Me is called the reduced von Neumann algebra of .M with respect to .e ∈ P (M). The von Neumann algebra .M'e is called the induced von Neumann algebra by .M' on .K = e (H ). Frequently, we shall identify the .∗-algebra .eMe with the von Neumann algebra .Me on K.

1.12 Singular Functionals on von Neumann Algebras Let .M be a von Neumann algebra on a Hilbert space H and recall that .M∗ denotes the predual of .M, that is, .M∗ ⊆ M∗ is the space of all uwo-continuous linear functionals on .M. For .ϕ ∈ M∗ and .y ∈ M the linear functionals .ϕy, yϕ ∈ M∗ are defined by .

(ϕy) (x) = ϕ (yx) , (yϕ) (x) = ϕ (xy) , x ∈ M,

respectively. For a discussion of singular functionals on .M it will be convenient to introduce the universal enveloping von Neumann algebra .Mu of .M. For the definition and properties of .Mu the reader is referred to [71], Section 10.1 or [125], Section III.2.

1.12 Singular Functionals on von Neumann Algebras

35

In particular, .Mu is a von Neumann algebra on some Hilbert space K and there is a natural embedding of .M into .Mu . Furthermore, .Mu can be identified with the bidual space .M∗∗ of .M, such that the embedding of .M into .Mu ∼ = M∗∗ coincides ∗∗ with the canonical embedding of .M into .M . This implies that .M∗ is the predual of .Mu , that is, .M∗ = (Mu )∗ . It follows that there exists a central projection .z0 in ∗ .Mu such that .M∗ = M z0 , that is,   M∗ = ϕz0 : ϕ ∈ M∗

.

(cf. [125], p. 126 or [71], Proposition 10.1.14). Definition 1.12.1 The subspace .M∗s ⊆ M∗ of all singular functionals on .M is defined by setting   M∗s = M∗ (1 − z0 ) = ϕ (1 − z0 ) : ϕ ∈ M∗ .

.

From this definition it is clear that M∗ = M∗ ⊕ M∗s .

.

(1.7)

For further properties of this decomposition the reader is referred to, for instance, Theorem III.2.14 of [125]. Any .ϕ ∈ M∗ can be written as ϕ = (ϕ1 − ϕ2 ) + i (ϕ3 − ϕ4 ) ,

.

+  where .ϕj ∈ M∗ , for .j = 1, . . . , 4. Since .z0 is a central projection of .Mu , this implies the following result. Proposition 1.12.2 (i) Any .ϕ ∈ M∗s can be written as ϕ = (ϕ1 − ϕ2 ) + i (ϕ3 − ϕ4 ) ,

.

+  where .ϕj ∈ M∗s , for .j = 1, . . . , 4. (ii) If .ϕ ∈ M∗s , then also .ϕy, yϕ ∈ M∗s for all .y ∈ M. Another relevant observation is the following proposition (see [125], Theorem 3.8). Here .P (M) denotes the collection of all (orthogonal) projections belonging to .M. +  Proposition 1.12.3 Let .ϕ ∈ M∗ . The following two statements are equivalent. (i) .ϕ ∈ M∗s . (ii) For every .0 /= p ∈ P (M) there exists .q ∈ P (M) with .0 < q ≤ p satisfying .ϕ (q) = 0.

36

1 A Review of Relevant Operator Theory

To exhibit an alternative characterization of singular functionals it is convenient to introduce some further terminology. A linear subspace .J ⊆ M is called an order ideal in .M if J is .∗-closed, .x ∈ J implies .|x| ∈ J and if .x ∈ M and .y ∈ J satisfy .0 ≤ x ≤ y, then .x ∈ J . Furthermore, an order ideal .J ⊆ M is said to be order dense in .M if for every .0 < x ∈ M+ there exists .y ∈ J + such that .0 < y ≤ x. The following characterization of order denseness is readily verified. Lemma 1.12.4 If .J ⊆ M is an order ideal, then the following three statements are equivalent: (i) J is order dense in .M; (ii) for every .0 < x ∈ M+ there exists and upwards directed system .{xα } in .J + such that .0 ≤ xα ↑α x; (iii) for every .0 < p ∈ P (M) there exists .q ∈ P (M) ∩ J such that .0 < q ≤ p. +  For any .φ ∈ M∗ we denote by .N (φ) its absolute kernel that is,     N (φ) = x ∈ M : φ (|x|) = φ x ∗  = 0 ,

.

which is an order ideal in .M. Furthermore, .φ vanishes on .N (φ). Using the terminology introduced above, singularity of a functional on .M may also be characterized as follows. Lemma 1.12.5 (i) A positive functional .0 ≤ φ ∈ M∗ is singular if and only if .N (φ) is order dense. (ii) A functional .φ ∈ M∗ is singular if and only if .φ vanishes on some order dense order ideal in .M. Proof

+  (i) From Proposition 1.12.3 it is clear that a functional .φ ∈ M∗ is singular if and only if for every .0 /= p ∈ P (M) there exists .q ∈ P (M) ∩ N (φ) such that .0 < q ≤ p. Consequently, assertion (i) follows immediately from the equivalence of statements (i) and (iii) in Lemma 1.12.4. (ii) Let .φ ∈ M∗s . By Proposition 1.12.2 (i) there exist positive singular functional ∗ .0 ≤ φj ∈ Ms , .1 ≤ j ≤ 4, such that φ = φ1 − φ2 + i (φ3 − φ4 ) .

.

  As observed in part (i), each .N φj is an order dense order ideal in .M and so,   4 .J = j =1 N φj is an order dense order ideal on which .φ vanishes. Conversely, suppose that .φ ∈ M∗ vanishes on some order ideal J which is order dense in .M. Writing .φ = φ1 + φ2 , with .φ1 ∈ M∗ and .φ2 ∈ M∗s , it follows from what just has been proved that .φ2 vanishes on some order dense order ideal .J2 ⊆ M. Hence, .φ1 vanishes on the order dense order ideal .J ∩ J2 . If .0 ≤ x ∈ M, then it follows from Lemma 1.12.4 (ii) that there exists an

1.13 Direct Products of von Neumann Algebras

37

upwards directed system .{xα } in .J ∩ J2 such that .0 ≤ xα ↑α x. Since .φ1 is uwo-continuous, this implies that .φ1 (xα ) →α φ1 (x). On the other hand, .φ1 (xα ) = 0 for all .α and so, .φ1 (x) = 0. This shows that .φ1 = 0 and hence, .φ = φ2 , that is, .φ is singular. This completes the proof of assertion (ii). ⨆ ⨅  ∗ + Recall from Definition 1.11.6 that a positive functional .φ ∈ M is normal whenever .xα ↓α 0 in .M implies that .φ (xα ) ↓α 0. Furthermore, it is clear that any .φ ∈ M∗ has the property that .φ (xα ) → 0 whenever .xα ↓α 0 in .M (as .xα → 0 with respect to the uwo-topology). This motivates the following extension of Definition 1.11.6. Definition 1.12.6 A linear functional .φ ∈ M∗ is called normal whenever .xα ↓α 0 in .M implies that .φ (xα ) → 0. The collection of all normal functionals on .M is denoted by .M∗n . By what just has been observed, it follows that .M∗ ⊆ M∗n . Next it will be shown that actually ∗ .M∗ = Mn , extending Theorem 1.11.7. Proposition 1.12.7 A linear functional .φ ∈ M∗ is normal if and only if .φ ∈ M∗ . Proof Let .φ ∈ M∗n be given and write .φ = φ1 + φ2 with .φ1 ∈ M∗ and .φ2 ∈ M∗s . By Lemma 1.12.5 (ii) there exists an order dense order ideal .J ⊆ M on which .φ2 vanishes. Let .x ∈ M+ . By Lemma 1.12.4 (ii) there exists and upwards directed system .{xα } in .J + such that .0 ≤ xα ↑α x. Since .φ2 (xα ) = 0 it is clear that .φ (xα ) = φ1 (xα ) for all .α. Both .φ and .φ1 are normal and so .φ (xα ) → φ (x) and .φ1 (xα ) → φ1 (x). Hence, .φ (x) = φ1 (x), that is, .φ2 (x) = 0. This holds for all + .x ∈ M , which implies that .φ2 = 0, that is, .φ = φ1 ∈ M∗ . This suffices for the proof. ⨆ ⨅ In view of this observation, decomposition (1.7) can also be written as M∗ = M∗n ⊕ M∗s

.

and any .φ ∈ M∗ can be uniquely decomposed as .φ = φn + φs with .φn ∈ M∗n and ∗ .φs ∈ Ms . This is frequently referred to as the decomposition of .φ in its normal and singular parts.

1.13 Direct Products of von Neumann Algebras   Suppose that . Hj : j ∈ J is a collection of Hilbert spaces. The inner product on each .Hj will be denoted simply by .〈·, ·〉. We denote by . j ∈J Hj the set of all  2      families . ξj j ∈J = ξj , with .ξj ∈ Hj for all .j ∈ J , satisfying . j ∈J ξj H < ∞. j

38

1 A Review of Relevant Operator Theory

With addition and scalar multiplication given by .

          ξj + ηj = ξj + ηj , λ ξj = λξj

      for all . ξj , ηj ∈ j ∈J Hj and all .λ ∈ C, the set . j ∈J Hj is a vector space.      Defining the inner product of two vectors . ξj , ηj ∈ j ∈J Hj by .

    

ξj , ηj = ξj , ηj , j

    it follows that . j ∈J Hj is a Hilbert space. Note that the norm of . ξj ∈ j ∈J Hj    1/2    2 is given by . . The Hilbert space . j ∈J Hj , also denoted by . j Hj , j ξ j Hj   is called the direct sum of the Hilbert spaces . Hj j ∈J . If .J = {1, . . . , n}, then  ' . j Hj is also  written as .H1 ⊕ · · · ⊕ Hn . Note that the algebraic direct sum . j Hj is dense in . j Hj . For any .k ∈ J , the mapping .Uk , which assigns to every .ξ ∈ Hk    the vector . ξj ∈ j Hj given by .ξk = ξ and .ξj = 0 whenever .j /= k, is an  isometric isomorphism from .Hk onto the closed subspace .H˜ k of . j Hj . Usually, ˜ we will identify   .Hk as a closed subspace of the  .Hk with .Hk and so, we consider direct sum . j Hj . Note that the subspaces . Hj j ∈J are pairwise orthogonal and   that . j Hj = j Hj . For .k ∈ J we denote by .pk the projection in . j Hj onto the subspace .Hk .  For the sake of convenience, we denote the direct sum Hilbert space . j Hj   for the moment simply  by  H . Suppose that operators .xj ∈ B Hj , .j ∈ J , are given such that .supj xj B (H ) < ∞. Then, we may define the linear operator    j     .⊕j xj on H by . ⊕j xj .⊕j xj is   ξj = xjξj for all . ξj ∈ H . The operator  bounded and .⊕j xj B(H ) = supj xj B (H ) . The mapping . xj │−→ ⊕j xj is j   # a .∗-isomorphism from the .C ∗ -algebra product. j B Hj into .B (H ), the range of    which will be denoted by . j B Hj . It is easily verified that an operator .x ∈ B (H )    belongs to the .C ∗ -subalgebra . j B Hj if and only if .xpj = pj x for all .j ∈ J .         Hence, . j B Hj is the commutant of the set . pj : j ∈ J . Therefore, . j B Hj is a von Neumann algebra on the Hilbert space H . Now suppose that .Mj is a von Neumann algebra on each of the Hilbert spaces .Hj , .j ∈ J . Define  .

j



     Mj = ⊕j xj : xj ∈ Mj ∀j ∈ J, sup xj B (H ) < ∞ , j j

    which is a .C ∗ -subalgebra of . j B Hj and hence, of .B (H ),  where .H = j Hj .  Note that .pk ∈ j Mj for all .k ∈ J . The commutant of . j Mj is given by

1.14 Comparison of Projections

39



'  ''    ' M = M and so, . M = j Mj , which shows that . j Mj j j j j j j  is a von Neumann algebra on H . The algebra . j Mj is called the product von   Neumann algebra of the family . Mj j ∈J . The projections .pk , .k ∈ J , belong to the     center of . j Mj , satisfy . k pk = 1 and . M = Mk . If the index set J is j j pk  finite, say .J = {1, . . . , n}, then . j Mj is also denoted by .M1 ⊕ · · · ⊕ Mn . As a converse to the above construction, we observe the following. Suppose that .M is a von Neumann algebra on a Hilbert space H and assume that .{pα } is a collection of pairwise orthogonal projections in the center .Z (M) of .M such that  . , the reduced von  Neumann algebra on the α pα = 1. If we define .Mα = Mpα  Hilbert space .Hα = pα (H ), then .H = α Hα and .M = α Mα . .

1.14 Comparison of Projections In this section, .M is a von Neumann algebra on the Hilbert space H . We denote by P (M) the collection of all (orthogonal) projections belonging to .M, that is,

.

( ) P (M) = p ∈ M : p 2 = p, p∗ = p .

.

Evidently, .P (M) ⊆ P (B (H )). As we have seen in Sect. 1.3, .P (B (H )) is a complete lattice. The following fundamental observation follows immediately from Lemma 1.3.4. Proposition 1.14.1 For any non-empty subset .D ⊆ P (M) the supremum . D of D in .P (B (H )) belongs to .P (M). Consequently, .P (M) is a complete sublattice of .P (B (H )) (that is, .P (M) is a complete lattice and the lattice operations, finite and infinite, in .P (M) coincide with the lattice operations in .P (B (H ))). If the von Neumann algebra .M is commutative, then .p ∧ q = pq and .p ∨ q = p + q − pq for all .p, q ∈ P (M). In this case .P (M) is a complete Boolean algebra, where the complement of each .p ∈ P (M) is given by .p⊥ = 1 − p. In particular, .P (Z (M)) is a complete Boolean algebra and a complete sublattice of .P (M), for any von Neumann algebra .M. For any .x ∈ M, the support projection .s (x) and the range projection .r (x) both belong to .P (M) (see the observations following Definition 1.11.1). Therefore, it follows from Theorem 1.4.5 that .s (x) (respectively, .r (x)) is the smallest of all projections .p ∈ P (M) satisfying .x = xp (respectively, .x = px). Projections belonging to the center .Z (M) of .M are called central projections. Definition 1.14.2 For .x ∈ M the central support .z (x) ∈ P (Z (M)) is defined by z (x) = inf {p ∈ P (Z (M)) : x = xp} .

.

40

1 A Review of Relevant Operator Theory

Note that .x = xz (x), so the above infimum is actually a minimum. For .p ∈ P (Z (M)), the conditions .x = xp and .x ∗ = x ∗ p are equivalent. Hence, .z (x) = z (x ∗ ) for all .x ∈ M. From the discussion preceding Definition 1.14.2, it is clear that .s (x) , r (x) ≤ z (x) and that .z (x) is also given by z (x) = inf {p ∈ P (Z (M)) : s (x) ≤ p} = inf {p ∈ P (Z (M)) : r (x) ≤ p} .

.

Note furthermore that, for any .q ∈ P (M), we have z (q) = inf {p ∈ P (Z (M)) : q ≤ p} .

.

In particular, .z (x) = z (s (x)) = z (r (x)) for all .x ∈ M. An alternative description of the central support is given in the next proposition. Proposition 1.14.3 The central support .z (x) of an element .x ∈ M is the projection onto the closed subspace given by

span

.

⎧ n ⎨ ⎩

j =1

⎫ ⎬ yj xξj : yj ∈ M, ξj ∈ H, j = 1, . . . , n; n ∈ N . ⎭

It is easily verified that z

$ *

.

α

& pα

=

*

z (pα )

(1.8)

α

for any non-empty collection .{pα } in .P (M). Definition 1.14.4 Let .e, f ∈ P (M) be given. (i) The projections e and f are said to be equivalent (relative to the von Neumann algebra .M) if there exists a partial isometry .v ∈ M with initial projection e and final projection f (that is, .e = v ∗ v and .f = vv ∗ ). This is denoted by .e ∼ f (or M

by .e ∼ f , if it is necessary to emphasize the von Neumann algebra relative to which the projections are equivalent). (ii) The projection e is said to be majorized by f (relative to .M) if there exists a projection .f1 ∈ P (M) such that .f1 ≤ f and .e ∼ f1 . This is denoted by .e  f (or, .e M f ). If .x ∈ M with polar decomposition .x = v |x|, then .v ∈ M and .v ∗ v = s (x) and = r (x). Therefore, .s (x) ∼ r (x). Evidently, if .M is an abelian von Neumann algebra and .e, f ∈ P (M), then .e ∼ f if and only if .e = f and .e  f if and only if .e ≤ f . In the next proposition we list some of the properties of the relation .∼.

∗ .vv

1.14 Comparison of Projections

41

Proposition 1.14.5 (i) The relation .∼ is an equivalence relation on the set .P (M). (ii) If .x ∈ M, then .s (x) ∼ r (x). (iii) If .e, f ∈ P (M), then e ∨ f − f ∼ e − e ∧ f.

.

(iv) (v)

(vi) (vii)

In particular, if .e ∧ f = 0, then .e  f ⊥ . If .e, f ∈ P (M) and .e ∼ f , then .z (e) = z (f ). Given .e, f ∈ P (M), there exist .e1 , f1 ∈ P (M) such that .e1 ≤ e, .f1 ≤ f and .e1 ∼ f1 if and only if .z (e) z (f ) /= 0 (equivalently, there exists .x ∈ M such that .exf /= 0). If .e, f ∈ P (M) such that .e ∼ f , then .ep ∼ fp for all .p ∈ P (Z (M)). Suppose that .{fi }i∈I and .{gi }i∈I are two families of pairwise  orthogonal projections in .P (M). If .ei ∼ fi for all .i ∈ I , then . i∈I ei ∼ i∈I fi .

Some properties of the relation . are collected in the following proposition. Proposition 1.14.6 (i) If .e, f, g ∈ P (M) are such that .e  f and .f  g, then .e  g. (ii) If .e, f ∈ P (M) such that .e  f and .f  e, then .e ∼ f . (iii) Suppose that .{fi }i∈I and .{gi }i∈I are two families  of pairwise  orthogonal projections in .P (M). If .ei  fi for all .i ∈ I , then . i∈I ei  i∈I fi . (iv) If .e, f ∈ P (M), then there exists a central projection .p ∈ P (Z (M)) such that .pe  pf and .p⊥ f  p⊥ e. (v) Suppose that .M is a factor. For .e, f ∈ P (M) we have either .e  f or .f  e. Definition 1.14.7 let .M be a von Neumann algebra on the Hilbert space H . (i) A projection .e ∈ P (M) is said to be finite (relative to .M) if it follows from .f ∈ P (M), .e ∼ f and .f ≤ e that .f = e. If e is not finite, then we say that e is infinite. (ii) A projection .e ∈ P (M) is said to be properly infinite (relative to .M) if .e /= 0 and for every .p ∈ P (Z (M)), either .pe = 0 or pe is infinite. Before we collect some properties of finite and infinite projections in the next proposition, we recall that a projection .p ∈ P (M) is said to be countably decomposable (also called .σ -finite or, of countable type) if every system of .{eα } of non-zero pairwise orthogonal projections in .P (M), satisfying .eα ≤ e for all .α, is at most countable. On a separable Hilbert space, every .p ∈ P (M) is clearly countably decomposable. Proposition 1.14.8 (i) If .e ∈ P (M) is finite and if .f ∈ P (M) such that .f  e, then f is finite. (ii) If .e, f ∈ P (M) are finite, then .e ∨ f is a finite projection.

42

1 A Review of Relevant Operator Theory

(iii) If .e ∈ P (M) is properly infinite, then there exists .f ∈ P (M) such that .f ≤ e, .f ∼ e and .f ∼ e − f . (iv) Let .{pα } be a collection of projections in .P (Z (M)) and .p = α pα . If .e ∈ P (M) is such that .pα e is finite for all .α, then pe is finite. (v) If .e ∈ P (M) is infinite, then there exists a unique .p ∈ P (Z (M)), satisfying .p ≤ z (e), such that .(1 − p) e is finite and pe is properly infinite. (vi) If .e ∈ P (M) is properly infinite and if .f ∈ P (M) such that .f ∼ e, then f is properly infinite. (vii) If .e ∈ P (M) is properly infinite and if .f ∈ P (M) is countably decomposable and .z (f ) ≤ z (e), then .f  e. Consequently, if .e, f ∈ P (M) are properly infinite and countably decomposable, then .e ∼ f if and only if .z (e) = z (f ). In particular, if .M is a factor on a separable Hilbert space, then any two properly infinite projections in .P (M) are equivalent. Definition 1.14.9 A projection .e ∈ P (M) is said to be abelian if the reduced von Neumann algebra .Me is abelian (equivalently, .eMe is a commutative .∗-subalgebra of .M). We list some of the properties of abelian projections. Proposition 1.14.10 (i) If .e ∈ P (M) is abelian, then e is finite. (ii) A projection .e ∈ P (M) is abelian if and only if e is a minimal element of the set .{p ∈ P (M) : z (p) = z (e)}. In particular, if .M is a factor, a non-zero projection .e ∈ P (M) is abelian if and only if e is a minimal projection of .M. (iii) If .{eα } is a collection of abelian projections in .P (M) such that the central supports .{z (eα )} are pairwise orthogonal, then . α eα is an abelian projection. (iv) If .e ∈ P (M) is abelian and if .f ∈ P (M) such that .f  e, then f is also abelian. (v) If .e ∈ P (M) is abelian and if .f ∈ P (M) such that .z (e) ≤ z (f ), then .e  f . (vi) If .e, f ∈ P (M) are abelian and .z (e) = z (f ), then .e ∼ f . (vii) Suppose that .e ∈ P (M) is abelian. If .f ∈ P (M) satisfies .f ≤ z (e), then f is a sum of pairwise orthogonal abelian projections. Now, we discuss the type decomposition of von Neumann algebras. Definition 1.14.11 let .M be a von Neumann algebra on the Hilbert space H . (i) .M is of type I if there exists an abelian projection .e ∈ P (M) with .z (e) = 1. (ii) .M is of type II if .M does not contain any non-zero abelian projections and there exists a finite projection .e ∈ P (M) such that .z (e) = 1. (iii) .M is of type III if .M does not contain any non-zero finite projection. (iv) .M is of type I.n , where n is a cardinal number satisfying .1 ≤ n ≤ dim H , if .1 is the sum of n mutually equivalent abelian projections in .P (M). (v) If .M is of type II, then .M is said to be of type II.1 (respectively, type II.∞ ), if .1 is a finite projection (respectively, .1 is a properly infinite projection).

1.14 Comparison of Projections

43

Type I von Neumann algebras are also called discrete and type III von Neumann algebras are also known as purely infinite von Neumann algebras. Observe that any von Neumann algebra .M of type I.n , for some n, is also of type I. Indeed, suppose that .{eα }α∈A is a collection of pairwise disjoint, mutually equivalent abelian projections in .P (M) such that . α eα = 1, where the cardinality of A satisfies   = z eα2 for all .|A| = n. It follows from Proposition 1.14.5 (iv) that .z eα1 .α1 , α2 ∈ A and so, (1.8) implies that .z (eα ) = 1 for all .α ∈ A. Hence, .M is of type I. Theorem 1.14.12 (Type Decomposition) Suppose that .M is a von Neumann algebra on H . (i) There exist unique, pairwise orthogonal, central projections .pI , .pI I , .pI I I ∈ P (Z (M)), satisfying .pI + pI I + pI I I = 1, such that .MpI is of type I or .pI = 0, .MpI I is of type II or .pI I = 0, and .MpI I I is of type III or .pI I I = 0. (ii) Suppose that .M is of type I. There exists a unique system .{pn : 1 ≤ n ≤ dim H } of pairwise orthogonal projections in .P (Z (M)), satisfying . n pn = 1, such that .Mpn is of type I.n or .pn = 0, for each n. (iii) Suppose that .M is of type II. There exist unique, mutually orthogonal projections .q1 , .q∞ ∈ P (Z (M)), with .q1 + q∞ = 1, such that .Mq1 is of type I.1 or .q1 = 0, and .Mq∞ is of type II.∞ or .q∞ = 0. Clearly, it is possible to apply (ii) and (iii) in the above theorem to .MpI and MpI I of (i) to obtain a further decomposition of an arbitrary von Neumann algebra .M. .

Corollary 1.14.13 A factor is either of type I.n (for a unique cardinal n satisfying 1 ≤ n ≤ dim H ), or type II.1 , or type II.∞ , or type III.

.

We introduce some further terminology. Definition 1.14.14 We use the notation introduced in Theorem 1.14.12. If .pI I I = 0, then .M is said to be a semi-finite. If .pI = 0, then .M is called a continuous von Neumann algebra. If .1 is a finite projection, then .M is called a finite von Neumann algebra. If .1 is a properly infinite projection, then .M is called a properly infinite von Neumann algebra. (v) .M is said to be of type I.f in if .M is of type I and .M is finite. (vi) .M is said to be of type I.∞ if .M is of type I and .M is properly infinite. (vii) .1 is a countably decomposable projection, then .M is said to be countably decomposable (or, .σ -finite). (i) (ii) (iii) (iv)

In the next proposition we present some alternative characterizations of the types of von Neumann algebras we have introduced above.

44

1 A Review of Relevant Operator Theory

Proposition 1.14.15 (i) .M is of type I if and only if for every non-zero central projection .p ∈ P (Z (M)) there exists a non-zero abelian projection .e ∈ P (M) such that .e ≤ p. (ii) .M is semi-finite if and only if for every non-zero central projection .p ∈ P (Z (M)) there exists a non-zero finite projection .e ∈ P (M) such that .e ≤ p. (iii) .M is of type I.f in (respectively, type I.∞ ) if and only if .M is of type I and .pn = 0 for all infinite (respectively, finite) cardinals n (here, the central projections .pn are as introduced in (ii) of Theorem 1.14.12). (iv) If .M is of type I, then there exist unique projections .ef in , .e∞ ∈ P (Z (M)), with .ef in + e∞ = 1, such  that .Mef in is of type I.f in  and .Me∞ is of type I.∞ or .e∞ = 0. Actually, .ef in = p and . e = ∞ {n:n finite} n {n:n infinite} pn .

1.15 Traces First we discuss the notion of a center-valued trace on von Neumann algebras. Let M be a von Neumann algebra, with center .Z (M), on the Hilbert space H .

.

Definition 1.15.1 A center-valued trace on .M is a linear mapping .T : M → Z (M) satisfying: (i) .T (a) > 0 whenever .0 < a ∈ M+ ; (ii) .T (xy) = T (yx) for all .x, y ∈ M; (iii) .T (z) = z whenever .z ∈ Z (M). Suppose that .T : M → Z (M) is a center-valued trace on the von Neumann algebra .M. If .e, f ∈ P (M) are equivalent projections, then there exists a partial isometry .v ∈ M such that .e = v ∗ v and .f = vv ∗ . Hence, it follows from (ii) above that .T (e) = T (v ∗ v) = T (vv ∗ ) = T (f ). In particular, if .e ∈ P (M) is such that .e ∼ 1, then .T (e) = T (1), that is, .T (1 − e) = 0. Hence, by (i) in the above definition, it follows that .e = 1. This shows that any von Neumann algebra on which a center-valued trace exists, is necessarily finite. Theorem 1.15.2 If .M is a finite von Neumann algebra, then there exists a unique center-valued trace .T : M → Z (M). Moreover, T has the following properties: (i) .T (zx) = zT (x) whenever .x ∈ M and .z ∈ Z (M); (ii) .‖T (x)‖B(H ) ≤ ‖x‖B(H ) for all .x ∈ M; (iii) T is uwo-continuous. Observe that it follows from (iii) of the above theorem, in combination with Lemma 1.11.3, that the center-valued trace T is normal, that is, .T aβ ↓ 0 in .Z (M) whenever .aβ ↓ 0 in .Mh . Next we consider numerical traces on von Neumann algebras. Let .M be a von Neumann algebra with positive cone .M+ .

1.15 Traces

45

Definition 1.15.3 A function .τ : M+ → [0, ∞] is said to be a weight on .M+ if: (i) .τ (a + b) = τ (a) + τ (b) for all .a, b ∈ M+ ; (ii) .τ (λa) = λτ (a) for all .a ∈ M+ and .0 ≤ λ ∈ R (with the convention that .0 · ∞ = 0). If .τ has the additional property that (iii) .τ (u∗ au) = τ (a) whenever .a ∈ M+ and .u ∈ U (M), then .τ is called a trace (or, tracial weight) on .M+ . If .τ : M+ → [0, ∞] is a weight, then it follows immediately from (i) in the above definition that .τ (a) ≤ τ (b) whenever .a ≤ b in .M+ . Furthermore, observe that a weight .τ is a trace if and only if .τ (x ∗ x) = τ (xx ∗ ) for all .x ∈ M. Definition 1.15.4 A trace .τ : M+ → [0, ∞] is called: (i) finite if .τ (1) < ∞; (ii) semi-finite if   τ (a) = sup τ (b) : b ∈ M+ , b ≤ a, τ (b) < ∞

.

for all .a ∈ M+ ; (iii) faithful if .a ∈ M+ and .τ (a) = 0 imply that  .a = 0; (iv) normal if .aβ ↑ a in .M+ implies that .τ aβ ↑ τ (a). Remark 1.15.5 Suppose that .τ : M+ → [0, ∞] is a normal trace. Then, .τ is semifinite if and only if for every .0 < a ∈ M+ , there exists .b ∈ M+ such that .0 < b ≤ a and .τ (b) < ∞. The following simple observation will also be used. Lemma 1.15.6 Let .τ : M+ → [0, ∞] be a semi-finite faithful trace. (i) If .0 /= p ∈ P (M), then there exists .q ∈ P (M) such that .0 < q ≤ p and .τ (q) < ∞. (ii) If .0 /= p ∈ P (M), then there exists a mutually orthogonalsystem .{qα } in .P (M) satisfying .τ (qα ) < ∞ and .qα ≤ p for all .α, such that . α qα = p. Proof (i) Since .τ is semi-finite, it follows that there exists a .b ∈ M such that .0 < b ≤ p such that .τ (b) < ∞. Let .0 < ε ∈ R be such that .eb (ε, ∞) /= 0. Since b b b .εe (ε, ∞) ≤ b, it is clear that .τ e (ε, ∞) < ∞. Furthermore, .e (ε, ∞) ≤ b b e (0, ∞) = s (b) ≤ p and so, .q = e (ε, ∞) has the desired properties. (ii) Let .{qα } be a maximal disjoint system in .P (M) satisfying .τ (qα ) <  ∞ and .qα ≤ p for all .α (such a system exists by Zorn’s lemma) and put .q = α qα . If .q < p, then .0 < p − q ∈ P (M). By (i), there exists .e ∈ P (M) such that .0 < e ≤ p − q and .τ (e) < ∞, which contradicts the maximality of the system  .{qα }. Therefore, .p = α qα . ⨆ ⨅

46

1 A Review of Relevant Operator Theory

The next theorem characterizes finite and semi-finite von Neumann algebras in terms of traces. Theorem 1.15.7 Let .M be a von Neumann algebra. (i) .M is finite if and only if for every non-zero .a ∈ M+ there exists a finite trace .τ on .M+ such that .τ (a) > 0. (ii) .M is semi-finite if and only if there exists a faithful normal semi-finite trace .τ on .M+ . Suppose that .M is a semi-finite von Neumann algebra and that .τ : M+ → [0, ∞] is a trace. Define     Nτ = x ∈ M : τ x ∗ x < ∞ .

.

It can be shown that .Nτ is an (two-sided) ideal in .M. We denote by .Mτ the ideal N2τ , that is,

.

Mτ =

.

⎧ n ⎨ ⎩

j =1

⎫ ⎬ xj yj : xj , yj ∈ Nτ , j = 1, . . . , n; n ∈ N . ⎭

Evidently, .Mτ ⊆ Nτ . Since .Mτ is an ideal in .M, it follows, in particular, that .Mτ is self-adjoint and that .Mτ is the linear span of its positive cone .(Mτ )+ = Mτ ∩ M+ . Moreover, if .x ∈ M, then .x ∈ Mτ if and only if .|x| ∈ (Mτ )+ . Theorem 1.15.8 Suppose that .τ : M+ → [0, ∞] is a trace.   (i) .(Mτ )+ = a ∈ M+ : τ (a) < ∞ and .Mτ = {x ∈ M : τ (|x|) < ∞}. (ii) The trace .τ extends uniquely from .(Mτ )+ to a positive linear functional .τ˙ : Mτ → C, satisfying the following conditions: a. .τ˙ (x ∗ ) = τ˙ (x) for all .x ∈ Mτ ; b. .τ˙ (xy) = τ˙ (yx) for all .x ∈ Mτ and all .y ∈ M; c. .τ˙ (xy) = τ˙ (yx) for all .x, y ∈ Nτ . (iii) Given .y ∈ Mτ , the linear functional .ψy : M → C is defined by .ψy (x) = τ˙ (xy) for all .x ∈ M. If .y ∈ (Mτ )+ , then .ψy is a positive functional on .M. If we assume that .τ is a normal trace, then .ψy is uwo-continuous for every .y ∈ Mτ (equivalently, .ψy is a normal positive functional for every .y ∈ (Mτ )+ ). The ideal .Mτ is called the ideal of definition of the trace .τ . We end this section with some facts that will frequently be used in the sequel. Proposition 1.15.9 Let .M be a von Neumann algebra equipped with a normal semi-finite faithful trace .τ : M+ → [0, ∞]. The following statements hold. (i) If .p, q ∈ P (M) and .p ∼ q, then .τ (p) = τ (q). (ii) If .p, q ∈ P (M) and .p  q, then .τ (p) ≤ τ (q).

1.15 Traces

47

  ≤ nj=1 τ (pj ). p j j =1    ∞ ∞ (iv) If .{pj }∞ j =1 pj ≤ j =1 τ (pj ). j =1 is a sequence in .P (M), then .τ

(iii) Let .p1 , . . . , pn ∈ P (M). Then .τ

 n

Proof (i) If .p ∼ q, then there exists a partial isometry .v ∈ M such that .p = v ∗ v and ∗ ∗ ∗ .q = vv (see Definition 1.14.4 (i)). Hence, .τ (p) = τ (v v) = τ (vv ) = τ (q). (ii) This follows immediately from the definition of the relation . in combination with part (i) and the monotonicity of the trace. (iii) It suffices to prove this for .n = 2 (the general case then follows via induction). Let .p, q ∈ P (M) be given. Recall from Proposition 1.14.5 (iii) that .p∨q−p ∼ q − p ∧ q and so it follows from (i) that τ (p ∨ q − p) = τ (q − p ∧ q).

.

Writing .p ∨ q = p + (p ∨ q − p), it follows that τ (p ∨ q) = τ (p) + τ (p ∨ q − p)

.

= τ (p) + τ (q − p ∧ q) ≤ τ (p) + τ (q).   n n ≤ (iv) For each .n ∈ N, (iii) implies that .τ j =1 pj j =1 τ (pj ). Since n ∞ . 1.14.1 j =1 pj ↑n j =1 pj in .P (M) (and hence in .M; cf. Proposition   n ↑n and Theorem 1.11.2), the normality of the trace implies that .τ p j j =1   ∞ τ j =1 pj . This suffices for the proof of (iv). ⨆ ⨅

Chapter 2

Measurable Operators

Abstract This chapter presents the basic theory of measurable and .τ -measurable operators affiliated with a semi-finite von Neumann algebra. Particular attention is given to the properties of the measure topology and the order structure in spaces of .τ -measurable operators. Properties of operator functions on these spaces are studied in some detail.

2.1 Affiliated Operators In this section the notion of an affiliated operator is introduced and its basic properties are discussed. Throughout this section, H is a Hilbert space and .M denotes a von Neumann subalgebra of .B (H ). The lattice of all (orthogonal) projections in .M is denoted by .P (M) and .U (M) is the unitary group of .M. The commutant of .M is denoted by .M' . Definition 2.1.1 A linear subspace D of H is said ( to )be affiliated with .M, denoted DηM, if and only if .u (D) ⊆ D for every .u ∈ U M' . ( ) ( ) Since .u−1 = u∗ ∈ U M' whenever .u ∈ U M'( , it)is evident that .DηM is equivalent to requiring that .u (D) = D for all .u ∈ U M' . Furthermore, if D is a closed subspace of H and p is(the )orthogonal projection onto D, then .DηM if and only if .up = pu for all .u ∈ U M' , or equivalently, .p ∈ P (M).

.

Definition 2.1.2 A linear operator x in H , with domain .D (x), is said to be affiliated with ( a )von Neumann algebra .M on H , denoted by .xηM, if .ux ⊆ xu for all .u ∈ U M' . It should be observed that (.xηM ) if and only if .D (x) ηM and .uxξ = xuξ for ' all .ξ ∈ D (x) and all .u ∈ U M . Also note that .xηM if and only if .ux = xu ( ) for( all ).u ∈ U M' . This is equivalent to requiring that .x = uxu∗ for all .u ∈ ' U M . Furthermore, since every operator in .M' is a linear combination of at most four unitary operators in .M' , the condition .xηM is also equivalent to requiring that ' .yx ⊆ xy for all .y ∈ M . In particular, if .x ∈ B (H ), then .xηM if and only if .x ∈ M. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. G. Dodds et al., Noncommutative Integration and Operator Theory, Progress in Mathematics 349, https://doi.org/10.1007/978-3-031-49654-7_2

49

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2 Measurable Operators

Remark 2.1.3 Recall that the graph .r (x) of any linear operator x in H is the linear subspace of .H ⊕ H given by r (x) = {(ξ, xξ ) : ξ ∈ D (x)} .

.

Consider the von Neumann algebra .M2 (M) = M ⊗ M2 (C) in .H ⊕ H . It is not difficult to show that the operator x is affiliated with .M if and only if the subspace .r (x) is affiliated with .M2 (M). The details are left to the reader. The next proposition gathers some properties of operators affiliated with a von Neumann algebra .M. Proposition 2.1.4 Let x and y be linear operators in the Hilbert space H . If .xηM and .yηM, then .(x + y) ηM, .(xy) ηM and .(λx) ηM for all .λ ∈ C. If x is pre-closed and .xηM, then .xηM. ¯ If x is densely defined and .xηM, then .x ∗ ηM. If x is normal, then .xηM if and only if .ex (B) ∈ M for all Borel sets .B ⊆ C. If x is normal and .xηM, then .f (x) ηM for any Borel function .f : C → C . If x is closed and densely defined and has polar decomposition .x = v |x|, then .xηM if and only if .v ∈ M and .|x| ηM. Moreover, if .xηM, then the range projection .r (x) and the support projection .s (x) both belong to .M and .s (x) ∼ r (x) (with respect to .M). (vii) If x is a closed densely defined operator with non-empty resolvent set .ρ (x), then .xηM if and only if .r (λ, x) ∈ M for all (equivalently, for some) .λ ∈ ρ (x). (i) (ii) (iii) (iv) (v) (vi)

Proof Assertions (i) and (ii) are easy consequences of the definitions of sum, product, and closure. ∗ ∗ ∗ ∗ (iii) As observed in Theorem 1.4.1 (vi), .(uxu operator ( ' ) ) = ux ∗u ∗for any unitary u on H . Hence, if .xηM and .u ∈ U M , then .ux u = (uxu∗ )∗ = x ∗ . This shows that .x ∗ ηM. (iv) If .y ∈ B (H ), then it follows from Theorem 1.5.12 that .yx ⊆ xy if and only if .yex (B) = ex (B) y for all .B ∈ B (C). If .xηM and ( .B' ) ∈ B (C), then x x .ux ⊆ xu and hence, .ue (B) = e (B) u for all .u ∈ U M . This implies that .ex (B) suppose that .ex (B) ∈ M for all .B ∈ B (C) . ( ∈' )M. Conversely, x If .u ∈ U M , then .ue (B) = ex (B) u for all .B ∈ B (C) and so, .ux ⊆ xu, which shows that .xηM. (v) By Theorem 1.5.14, the spectral measure of .f (x) is given by

( ) ef (x) (B) = ex f −1 (B)

.

for all .B ∈ B (C). Therefore, (v) is an immediate consequence of (iv). (vi) Let x be closed and densely defined with polar decomposition .x = v |x|. If .v ∈ M and .|x| ηM, then it is clear that .xηM (see (i)). Now assume that .xηM. It follows from Proposition 1.7.4 that, given any unitary operator .u ∈ B (H ),

2.1 Affiliated Operators

51

the polar decomposition of .uxu∗ is given by )( ) ( uxu∗ = uvu∗ u |x| u∗ .

.

( ) Consequently, if .u ∈ U M' , then .x = uxu∗ and so, by the uniqueness of the polar decomposition, it follows that .v = uvu∗ and .|x| = u |x| u∗ . Hence, .v ∈ M and .|x| ηM. Since v is the partial isometry with initial projection .s (x) and final projection .r (x), it follows that .s (x) = v ∗ v ∈ M and .r (x) = vv ∗ ∈ M. This also shows that .s (x) ∼ r (x) with respect to .M. (vii) Let .λ ∈ ρ (x) be fixed. First suppose ( )that .xηM. Given .ξ ∈ H define .ζ ∈ D (x) by .ζ = r (λ, x) ξ . If .u ∈ U M' , then .uζ ∈ D (x) and .(λ1 − x) uζ = u (λ1 − x) ζ = uξ . Hence, .uζ = r (λ, x) uξ , that is, .ur (λ, x) ) r (λ, x) uξ . (ξ= This shows that .ur (λ, x) = r (λ, x) u for all .u ∈ U M' and hence, .r (λ, x) ∈ M. ( ) Now assume that .r (λ, x) ∈ M. If .u ∈ U M' and .ξ ∈ D (x), then .ξ = r (λ, x) ζ for some .ζ ∈ H and so, it follows from .ur (λ, x) = r (λ, x) u that uξ = ur (λ, x) ζ = r (λ, x) uζ ∈ D (x)

.

and .(λ1 − x) uξ = uζ( = )u (λ1 − x) ξ , that is, .xuξ = uxξ . Consequently, ux ⊆ xu for all .u ∈ U M' , that is, .xηM. u n

.

Example 2.1.5 The notion of an affiliated operator is illustrated by two simple examples. (a) Let H be a Hilbert space and consider the von Neumann algebra .M = B (H ). Since .M' = C1, it is evident that all linear operators in H are affiliated with .M. (b) Let .(X, Σ, μ) be a measure space. So, X is a non-empty set, .Σ is a .σ -algebra of subsets of X, and .μ : Σ → [0, ∞] is a (countably additive) measure. It is assumed that .(X, Σ, μ) is a Maharam measure space, that is, .μ is locally finite (that is, for every set .E ∈ Σ with .μ (E) > 0 there exists a set .F ∈ Σ such that .F ⊆ E and .0 < μ (F ) < ∞) and localizable (that is, the corresponding measure algebra is a complete Boolean algebra). Note that, in particular, if .μ is a .σ -finite measure, then .(X, Σ, μ) is a Maharam measure space. Let .H = L2 (μ) be the Hilbert space of all (equivalence classes of) .μ-square integrable functions on X, let .L∞ (μ) be the commutative .C ∗ -algebra of all (equivalence classes of) .μ-essentially bounded functions on X, and let .L0 (μ) be the .∗-algebra of all (equivalence classes of) complex-valued .μ-measurable functions on X. For .f ∈ L∞ (μ), the bounded linear operator on .L2 (μ) given by multiplication with the function f is denoted by .Mf , that is, .Mf ξ = f ξ , .ξ ∈ L2 (μ). The map .f |−→ Mf is an isometric .∗-homomorphism from .L∞ (μ) into .B (L2 (μ)). Define the .C ∗ -subalgebra .M of .B (L2 (μ)) by { } M = Mf : f ∈ L∞ (μ) .

.

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Since .(X, Σ, μ) is a Maharam measure space, .M is a maximal abelian von ' Neumann algebra on .L2 (μ), that is, .M = M. The projections in .M are given by { } P (M) = MχA : A ∈ Σ

.

and the unitary group of .M is given by { } U (M) = Mf : f ∈ L∞ (μ) , |f | = χX .

.

For .f ∈ L0 (μ) the operator .Mf in .L2 (μ) is defined by setting ( ) D Mf = {ξ ∈ L2 (μ) : f ξ ∈ L2 (μ)}

.

and Mf ξ = f ξ,

.

( ) ξ ∈ D Mf .

( ) The domain .D Mf is a dense order ideal (that is, an absolutely solid linear subspace) in .L2 (μ) ) .Mf is closed. The operator .Mf is affiliated with .M. ( and Indeed, if .u ∈ U M(' =) U (M), then .u = Mg for some .g ∈ L∞ (μ) with .|g| = χX . If .ξ ∈ D Mf , then .Mf u (ξ ) = fgξ ∈ L2 (μ) and so, .u (ξ ) ∈ ( ) ( ) D Mf . Moreover, .Mf uξ = f gξ = (gf ξ )= uMf ξ for all .ξ ∈ D Mf . This shows that .uMf ⊆ Mf u for all .u ∈ U M' and hence, .Mf ηM. It will be shown that every closed and densely defined operator x on .L2 (μ), which is affiliated with .M, is of the form .Mf for a uniquely determined .f ∈ L0 (μ). First, suppose that a is a positive self-adjoint operator on .L2 (μ), affiliated with .M. Denoting by .ea the spectral measure of a, it follows from Proposition 2.1.4 (iv) that .ea (B)f ∈ M for all Borel sets .B ⊆ R. Define a a .pn = e ([n, n + 1)) and .an = [n,n+1) λde (λ) for all .n = 0, 1, . . .. Since + .an ∈ M , there exists .0 ≤ gn ∈ L∞ (μ) such that .an = Mgn and since .pn ∈ P (M), there exists .An ∈ Σ such that .pn = MχA , for each .n ≥ 0. n Furthermore, .pn pm = 0 whenever .n /= m and .an = an pn for all n. This implies that .χAn χAm = 0, .n /= m, and .gn = gn χAn for all n. Therefore, it may be assumed that the sets .{An }∞ that the function .gn is n=0 are pairwise disjoint and Σ supported on .An . If .g ∈ L0 (μ) is defined by setting .g = ∞ n=0 gn , with Σthe sum taken pointwise on X, then .Mg = a. Indeed, if .ξ ∈ D (a), then .aξ = ∞ n=0 an ξ as a norm convergent series in .L2 (μ) and so, aξ =

∞ Σ

.

n=0

Mgn ξ =

∞ Σ n=0

gn ξ =

(∞ Σ

) gn ξ.

n=0

( ) This shows that .ξ ∈ D Mg and .aξ = Mg ξ(. Hence, .a ⊆ Mg . For the proof of ) Σ the converse inclusion, suppose that .ξ ∈ D Mg , that is, . ∞ n=0 gn ξ ∈ L2 (μ).

2.2 The Algebra of Measurable Operators

53

Σ It follows from the dominated convergence theorem that the series . ∞ n=0 gn ξ is convergent in .L2 (μ) and so, ∞ Σ .

n=0

an ξ =

∞ Σ

gn ξ = Mg ξ,

n=0

as a norm convergent series. This implies that .ξ ∈ D (a) and .aξ = Mg ξ . Consequently, .Mg ⊆ a and so, .a = Mg . Now, let .x : D (x) → L2 (μ) be a closed densely defined operator which is affiliated with .M and .x = v |x| be its polar decomposition. By Proposition 2.1.4 (vi), .v ∈ M and .|x| ηM. So, .v = Mh for some .h ∈ L∞ (μ) and .|x| = Mg for some .0 ≤ g ∈ L0 (μ), by the special case treated above. Consequently, .x = Mf with .f = hg. Finally, if .f1 , f2 ∈ L0 (μ) are such that .Mf1 = Mf2 , then .f1 χF = f2 χF for all sets F of finite measure, which implies that .f1 = f2 in .L0 (μ) (since in a Maharam measure space, local .μ-null sets in X are .μ-null sets). References: [92].

2.2 The Algebra of Measurable Operators In general, the set of all closed, densely defined linear operators affiliated with a von Neumann algebra .M, has the structure neither of an algebra nor of a vector space. Indeed, if .M = B (H ), then this set is equal to .C (H ) , the collection of all closed, densely defined operators in H (see Example 2.1.5 (a)), which does not have a natural structure of a vector space if H is infinite dimensional. In this section, a subset of all closed, densely defined operators affiliated with .M will be singled out, which may be equipped with a natural structure of a .∗ -algebra. This is the algebra .S (M) of all measurable operators with respect to an arbitrary von Neumann algebra .M on a Hilbert space H . Some further terminology is necessary. Definition 2.2.1 A linear subspace D of H is said to be strongly dense in H with respect to .M, if there exists a sequence .{pn }∞ n=1 in .P (M) such that .pn ↑ 1, ⊥ .pn (H ) ⊆ D and .pn is a finite projection for every .n = 1, 2, . . .. The sequence ∞ .{pn } n=1 is called a determining sequence for D. From the assumption .pn ↑ 1 in Definition 2.2.1, it may immediately be inferred that every strongly dense subspace of H is dense in H . The next objective is to show that the intersection of two strongly dense subspaces is also strongly dense (see Proposition 2.2.3). The following lemma will be needed. For later purposes, this result will be formulated in a slightly more general form than is needed at present. ∞ Lemma 2.2.2 Suppose that .{en }∞ n=1 and .{fn }n=1 are sequences of finite projections ∞ in .P (M) such that .en ↓ 0 and .fn ↓ 0. If .{pn }∞ n=1 and A .{qn }n=1 are sequences in .P (M) such that .pn < en and .qn < fn for all n, then . n (pn ∨ qn ) = 0.

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2 Measurable Operators

Proof Consider first the following special case. Assume that .{en }∞ n=1 is a sequence of finite projections in .M such that .en ↓ 0 and let .p ∈ P (M) be such that .p < en for all n. This implies that .p = 0 . Indeed, since .e1 is finite and .p < e1 , it follows that p is finite and so, the projection .e = e1 ∨ p is also finite. The reduced von Neumann algebra .eMe is finite, .en ↓ 0 in .eMe and .p < en in .eMe for all n. Let .T : eMe → Z (eMe) be the center-valued trace (see Definition 1.15.1). It follows from .en ↓ 0 that .T (en ) ↓ 0, and .p < en implies that .T (p) ≤ T (en ) for all n. Hence, .T (p) = 0 and so .p = 0, which proves the assertion. To return to the general setting of the lemma, consider the von Neumann algebra A .M2 (M) = M2 (C) ⊗ M in the Hilbert space .H ⊕ H . Defining .p = ∨ qn ) (p n n in .P (M), it is easily verified that [ .

] [ ] [ ] ] [ p0 pn 0 qn 0 pn ∨ qn 0 = ∨ ≤ 0 0 0 0 0 0 00

(2.1)

in .P (M2 (M)). Furthermore, it follows from .pn < en and .qn < fn that [ .

] [ ] [ ] [ ] pn 0 en 0 qn 0 fn 0 < and < 0 0 0 0 0 0 0 0

(2.2)

for all n. Since [ .

] ] [ fn 0 0 0 ∼ 0 0 0 fn

relative to .M2 (M) and the projections [ .

] ] [ en 0 0 0 , 0 0 0 fn

are mutually orthogonal, it follows from (2.1) and (2.2) that [ .

] ] [ p0 e 0 < n 0 fn 00

for all n. The projections [ .

en 0 0 fn

]

are all finite and are monotonically decreasing to zero in .P (M2 (M)). Therefore, it follows from the special case, treated at the beginning of the present proof, that .p = 0. u n

2.2 The Algebra of Measurable Operators

55

Proposition 2.2.3 If .D1 and .D2 are strongly dense subspaces of H , then the subspace .D1 ∩ D2 is also strongly dense in H . ∞ Proof Let .{pn }∞ n=1 and .{qn }n=1 be determining sequences in .P (M) for .D1 and .D2 , respectively. Defining .en = pn ∧ qn in .P (M) for .n = 1, 2, . . ., it is clear that ⊥ ⊥ ⊥ ⊥ ⊥ .en (H ) ⊆ D1 ∩ D2 . Since .pn and .qn are finite projections, .en = pn ∨ qn is also ⊥ ⊥ finite for all n. Moreover, .pn ↓ 0 and .qn ↓ 0 and so, it follows from Lemma 2.2.2 that .en⊥ ↓ 0, equivalently, .en ↑ 1. This shows that .D1 ∩ D2 is strongly dense in H with determining sequence .{en }∞ u n n=1 .

A further consequence of Lemma 2.2.2 now follows. Lemma 2.2.4 If D is a strongly dense subspace of H and L is a closed subspace of H such that .LηM and .L ∩ D = {0}, then .L = {0}. Proof Let .{pn }∞ n=1 be a determining sequence in .P (M) for D and p be the projection in H onto .L. As observed in the remarks following Definition 2.1.1, ⊥ .LηM implies that .p ∈ P (M). Moreover, .p∧pn = 0 and so, .p ∼ p∨pn −pn ≤ pn , ⊥ ⊥ ⊥ that is, .p < pn . Since the projections .pn are finite and .pn ↓ 0, it follows from u n Lemma 2.2.2 that .p = 0. Hence, .L = {0}. In the next definition, measurable operators are introduced. Definition 2.2.5 A linear operator x in the Hilbert space H is called premeasurable with respect to the von Neumann algebra .M, if: (i) .xηM. (ii) The domain .D (x) of x is strongly dense in H . (iii) The operator x is pre-closed. If, in addition, the operator x is closed, then x is called measurable with respect to .M. The collection of all operators which are measurable with respect to .M is denoted by .S (M). When the von Neumann algebra .M is fixed and no confusion may arise, the additional “with respect to .M” is usually omitted. Evidently, a bounded operator .x ∈ B (H ) is measurable if and only if .x ∈ M. It is also clear that the closure of a pre-measurable operator is measurable. If x is pre-measurable and .{pn }∞ n=1 is a determining sequence in .P (M) for the domain .D (x), then .{pn }∞ is called a n=1 determining sequence for the operator x. Suppose that the operator x is pre-measurable with respect to .M and that .p ∈ P (M) is such that .p (H ) ⊆ D (x). The linear operator xp is closed with domain H (and )so, the Closed Graph Theorem implies that .xp ∈ B (H ). Moreover, if .u ∈ U M' , then .u (xp) = (xp) u and hence, .xp ∈ M. In particular, .xpn ∈ M for any determining sequence .{pn }∞ n=1 for the operator x. Proposition 2.2.6 If x is a closed, densely defined operator in H with polar decomposition .x = v |x|, then x is measurable with respect to the von Neumann algebra .M if and only if .v ∈ M and .|x| is measurable.

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Proof The result follows by observing that .D (x) = D (|x|) and, by (vi) of Proposition 2.1.4, .xηM if and only if .v ∈ M and .|x| ηM. n u The next proposition presents a criterion for the measurability of a closed, densely defined operator x affiliated with .M in terms of the spectral measure .e|x| of the operator .|x|. The following lemmas are needed for the proof. Lemma 2.2.7 Let x be a closed densely ( ) defined operator in the Hilbert space H and .λ ≥ 0. If .0 /= ξ ∈ Ran e|x| (λ, ∞) ∩ D (x), then .||xξ ||H > λ ||ξ ||H . < > |x| |x| Proof Recall that the measure .eξ,ξ is defined by .eξ,ξ (B) = e|x| (B) ξ, ξ for all |x|

Borel sets B in .R. It follows from .ξ = e|x| (λ, ∞) ξ that .eξ,ξ (−∞, λ] = 0 and so, |x|

|x|

eξ,ξ (λ, ∞) = eξ,ξ (R) = ||ξ ||2H > 0.

.

|x|

Hence, there exists .γ > λ such that .eξ,ξ (γ , ∞) > 0. Since .ξ ∈ D (x) = D (|x|) and .||xξ ||H = |||x| ξ ||H , it follows that f .

||xξ ||2H =

R f

=

2

μ (λ,γ ]

≥

f

|x|

μ2 deξ,ξ (μ) = |x| deξ,ξ

|x|

(λ,∞)

μ2 deξ,ξ (μ)

f

|x|

(μ) + (γ ,∞)

|x| λ2 eξ,ξ

|x| (λ, γ ] + γ 2 eξ,ξ

|x|

|x|

μ2 deξ,ξ (μ)

(γ , ∞) |x|

> λ2 eξ,ξ (λ, γ ] + λ2 eξ,ξ (γ , ∞) = λ2 eξ,ξ (λ, ∞) = λ2 ||ξ ||2H . The proof is finished.

u n

Remark 2.2.8 By the same method of proof, it follows that if .x = x ∗ and if .λ ∈ R, then . > λ for all .0 /= ξ ∈ Ran (ex (λ, ∞)) ∩ D (x). Lemma 2.2.9 Let x be a closed, densely defined operator, which is affiliated with the von Neumann algebra .M. If .p ∈ P (M) such that .p (H ) ⊆ D (x), then |x| (λ, ∞) < p ⊥ whenever .λ ≥ ||xp|| .e B(H ) . Proof Fix .λ ≥ ||xp||B(H ) . Put .q = p ∧ e|x| (λ, ∞) and suppose that .q /= 0. ( ) If .0 /= ξ ∈ q (H ), then .ξ ∈ Ran e|x| (λ, ∞) ∩ D (x) and so, it follows from Lemma 2.2.7 that .||xpξ ||H = ||xξ ||H > λ ||ξ ||H . This contradicts the assumption that .||xp||B(H ) ≤ λ. Therefore, .q = 0 and so, e|x| (λ, ∞) ∼ p ∨ e|x| (λ, ∞) − p ≤ p⊥ ,

.

in other words, .e|x| (λ, ∞) < p⊥ .

u n

2.2 The Algebra of Measurable Operators

57

Proposition 2.2.10 For any closed and densely defined linear operator x affiliated with the von Neumann algebra .M, the following three statements are equivalent: (i) x is measurable with respect to .M. (ii) There exists a projection .p ∈ P (M) such that .p (H ) ⊆ D (x) and .p⊥ is finite. (iii) .e|x| (λ, ∞) is a finite projection for some .λ > 0. Proof It is evident that (i) implies (ii). Suppose that condition (ii) is satisfied and let p ∈ P (M) be such that .p (H ) ⊆ D (x) and .p⊥ is finite. Note that .xp ∈ B (H ) (see the comments following Definition 2.2.5). If .λ ≥ ||xp||B(H ) , then it follows from Lemma 2.2.9 that .e|x| (λ, ∞) < p⊥ and therefore, .e|x| (λ, ∞) is finite. Now assume that condition (iii) holds. Let .{λn }∞ n=1 be any sequence in .R such that .λ ≤ λn ↑ ∞ and define .pn = e|x| [0, λn ] for .n = 1, 2, . . .. It follows from Proposition 2.1.4 (iv), (vi) that .|x| ηM and .pn ∈ P (M) for all n. Furthermore, |x| (λ, ∞) and ⊥ .pn ↑ 1 and .pn (H ) ⊆ D (|x|) = D (x) for all n. Moreover, .pn ≤ e ⊥ so, .pn is finite for all n. Hence, .D (x) is strongly dense and this suffices to show that x is measurable with respect to .M. u n .

If the von Neumann algebra .M is finite (in particular, if .M is abelian), then condition (iii) of Proposition 2.2.10 is fulfilled for every closed, densely defined operator x affiliated with .M. Therefore, the following result is an immediate consequence. Corollary 2.2.11 If .M is a finite von Neumann algebra, then every closed and densely defined operator affiliated with .M is measurable with respect to .M. It will be shown next that the set .S (M) actually has the structure of a .∗-algebra (see Theorem 2.2.16). For this purpose, some properties of measurable operators need to be studied in more detail. This will be carried out in the propositions which follow. Proposition 2.2.12 If the linear operator x is pre-measurable with respect to .M and x is symmetric, then its closure .x¯ is self-adjoint. Proof The closure .x¯ of x is measurable and symmetric. Therefore, without loss of generality, it may be assumed that x is measurable and symmetric. To show that x is self-adjoint, it is sufficient to prove that .Ran (x ± i1)⊥ = {0}. Let ⊥ .L = Ran (x − i1) and p be the projection onto L. Note that .p = 1 − r (x − i1) and hence, by Proposition 2.1.4 (vi), .p ∈ P (M) , that is, .LηM. It follows that .D (x) ∩ L = {0}. Indeed, if .ξ ∈ D (x) ∩ L, then . = 0 and so, . = i . Since x is symmetric, it follows that . ∈ R and hence .ξ = 0. This proves the claim. The domain .D (x) is strongly dense in H , so Lemma 2.2.4 implies that .L = 0. The proof of the equality .Ran (x + i1)⊥ = {0} is similar. u n The result of the following lemma plays an important role in the sequel.

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Lemma 2.2.13 If x is pre-measurable with respect to the von Neumann algebra .M and if D is a strongly dense subspace of H , then .x −1 (D) is also a strongly dense subspace. ∞ Proof Let .{pn }∞ n=1 and .{qn }n=1 be determining sequences in .P (M) for .D (x) and D, respectively. For each n, the operator .x(n = xp to .M and so, .qn⊥ xn ∈ ) n belongs ⊥ ⊥ M. Therefore, the null projection .fn = n qn xn of .qn xn belongs to .P (M) (see Definition 1.4.4). Note that

xn fn = qn xn fn + qn⊥ xn fn = qn xn fn

.

(2.3)

for all n. Furthermore, ) ( ) ( fn⊥ = s qn⊥ xn ∼ r qn⊥ xn ≤ qn⊥

.

and so, .fn⊥ < qn⊥ for all n. This shows, in particular, that .fn⊥ is a finite projection. The sequence .{en }∞ n=1 in .P (M), defined by .en = fn ∧ pn for all n, is a determining sequence for .x −1 (D). Indeed, it will be shown first that .{en }∞ n=1 is increasing. Using ⊥ q = 0, it follows from (2.3) that the facts that .en = fn en = pn en and .qn+1 n ⊥ ⊥ ⊥ qn+1 xn+1 en = qn+1 xn+1 pn en = qn+1 xn en

.

⊥ ⊥ = qn+1 xn fn en = qn+1 qn xn fn en = 0.

) ( ⊥ Hence, .en ≤ n qn+1 xn+1 = fn+1 . Moreover, .en ≤ pn ≤ pn+1 and so, .en ≤ fn+1 ∧ pn+1 = en+1 . Furthermore, since .en⊥ = fn⊥ ∨ pn⊥ and the projections .fn⊥ and .pn⊥ are finite, it follows that .en⊥ is also finite. It is clear that .en (H ) ⊆ pn (H ) ⊆ D (x). The equalities in (2.3) imply that xen ξ = xpn en ξ = xn en ξ = xn fn en ξ = qn xn fn en ξ ∈ qn (H ) ⊆ D,

.

for any .ξ ∈ H and this shows that .en (H ) ⊆ x −1 (D) for all n. It remains to be shown that .en ↑ 1. Since .en⊥ = fn⊥ ∨ pn⊥ , .fn⊥ < qn⊥ for all n { }∞ { }∞ and . pn⊥ n=1 , . qn⊥ n=1 are two sequences of finite projections satisfying .pn⊥ ↓ 0, ⊥ ⊥ ⊥ ⊥ .qn ↓ 0, it follows from Lemma 2.2.2 that .en = fn ∨ pn ↓ 0 and, therefore, .en ↑ 1. u n Proposition 2.2.14 Let x and y be pre-measurable operators with respect to the von Neumann algebra .M. (i) (ii) (iii) (iv)

The operator .x ∗ is measurable. If .x ⊆ y, then .x¯ = y. ¯ If D is a strongly dense subspace of .D (x), then D is a core of .x. ¯ If D is a strongly dense subspace of .D (x) ∩ D (y) and .x|D = y|D , then .x¯ = y. ¯

2.2 The Algebra of Measurable Operators

59

Proof (i) Since .x ∗ = x¯ ∗ and .x¯ is measurable by assumption, it may be assumed, without loss of generality, that x is measurable. Let .x = v |x| be the polar decomposition of .x. As observed in Proposition 2.2.6, this implies that .v ∈ M and .|x| is measurable. Since .x ∗ = |x| v ∗ , it follows that .D (x ∗ ) = (v ∗ )−1 (D (x)), which is strongly dense by Lemma 2.2.13. Proposition 2.1.4 (iii) implies that .x ∗ ηM and so, .x ∗ is measurable. (ii) Since .x ⊆ y implies that .x¯ ⊆ y¯ and .x¯ and .y¯ are measurable, it may be assumed that both x and y are measurable. By (i), .D (y ∗ ) is strongly dense and further .y ∗ ⊆ x ∗ . Defining .D0 = D (x) ∩ y −1 (D (y ∗ )), it follows from Lemma 2.2.13 in combination with Proposition 2.2.3 that .D0 is strongly dense in H . If .ξ ∈ D0 , then .xξ = yξ ∈ D (y ∗ ) and .y ∗ yξ = x ∗ yξ = x ∗ xξ . Defining ∗ .z = (x x)|D , it is easy to see that the operator z is symmetric and pre0 measurable. Hence, by Proposition 2.2.12, the closure .z¯ is self-adjoint. Since ∗ .z ⊆ x x, it is also clear that .z ¯ = x ∗ x, as self-adjoint operators are maximal symmetric. Similarly, it follows that .z¯ = y ∗ y and hence, .x ∗ x = y ∗ y. This implies that .|x| = |y| and so, it may be concluded that .D (x) = D (|x|) = D (|y|) = D (y). Therefore, .x = y. (iii) Suppose that x is pre-measurable and that .D ⊆ D (x) is strongly dense and put .y =Ux|D . Let .{pn }∞ n=1 be a determining sequence in .P (M) for D and let ∞ .D0 = p Defining .y0 = x|D0 , it is readily verified that .y0 is pre(H ). n=1 n measurable and it is evident that .y0 ⊆ y ⊆ x. Hence, it follows from (ii) that .y ¯0 = x¯ and so, .x¯ = y. ¯ This shows that D is a core for .x. ¯ (iv) If two pre-closed operators coincide on a common core, then their closures are equal. u n Proposition 2.2.15 If x and y are pre-measurable operators with respect to a von Neumann algebra .M, then .x + y and xy are also pre-measurable. Proof The operators .x + y and xy are affiliated with .M, as observed in (i) of Proposition 2.1.4. It follows from Proposition 2.2.3 that .D (x + y) = D (x) ∩ D (y) is strongly dense. In particular, .x + y is densely defined and so, .x ∗ + y ∗ ⊆ (x + y)∗ . By Proposition 2.2.14 (i), .x ∗ and .y ∗ are measurable and hence, .D (x ∗ + y ∗ ) = D (x ∗ )∩D (y ∗ ) is strongly dense in H . This implies that .(x + y)∗ is densely defined and so, by Theorem 1.4.2, .x + y is pre-closed. Therefore, .x + y is pre-measurable. From Lemma 2.2.13 in combination with Proposition 2.2.3, it follows that −1 (D (x)) is strongly dense in H . Furthermore, .y ∗ x ∗ ⊆ (xy)∗ .D (xy) = D (y) ∩ y ∗ ∗ and .x and .y are measurable. Hence, ) ( ) ( )−1 ( ( ∗ )) ( D y D y∗x∗ = D x∗ ∩ x∗

.

is strongly dense. This implies, in particular, that .(xy)∗ is densely defined and hence, xy is pre-closed. Therefore, xy is pre-measurable. u n

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It is now possible to introduce the algebraic structure in the set .S (M) of all operators which are measurable with respect to the von Neumann algebra .M. Given .x, y ∈ S (M), it follows from Proposition 2.2.15 that the operators .x + y and xy are ˆ and strong product .xˆ·y, defined pre-measurable and hence, their strong sum .x +y as the closure of .x + y and xy, respectively, both exist and are measurable, that is, ˆ xˆ·y ∈ S (M). If .x ∈ S (M) and .λ ∈ C, .λ /= 0, then .λx ∈ S (M). If .λ = 0, .x +y, then 0x need not be closed, but it is pre-closed with closure the zero operator 0 on H . Therefore, set .0x = 0 for all .x ∈ S (M). With these operations of scalar multiplication, addition, and multiplication, the set .S (M) has the structure of an algebra. In order to verify the necessary algebraic identities, it suffices to verify these equalities on some strongly dense subspace, Proposition 2.2.14 (iv). ( )as follows from ( ) ˆ +z ˆ = x+ ˆ y +z ˆ holds for all .x, y, z ∈ To illustrate this, it will be shown that . x +y S (M). By Proposition 2.2.3, the .D = D (x) ∩ D (y) ∩ D (z) is strongly ( subspace ) ( ) ˆ +z ˆ and .x + ˆ y +z ˆ coincide with the algebraic dense and on D, the operators . x +y sum .x + y + z. Since D is a core for both operators (see Proposition 2.2.14 (iii)), it follows that these operators are equal. ( ) ˆ = zˆ·x +zˆ ˆ ·y for all .x, y, z ∈ S (M). A further illustration is the equality .zˆ· x +y subspace Setting .D = x −1 (D (z)) ∩ y −1 (D (z)), observe that D is a strongly ( dense ) ˆ and .zˆ·x +zˆ ˆ ·y of H (by Lemma 2.2.13 and Proposition 2.2.3). The operators .zˆ· x +y coincide on D with the operator .zx + zy. Since D is a core for both operators, it follows that these operators are equal. Evidently, .M = S (M) ∩ B (H ) and .M is a subalgebra of .S (M). Moreover, if .x ∈ S (M) and .y ∈ M, then it is easy to see that .x + y is closed and hence, measurable. Similarly, the operator xy is closed in this case and so, xy is measurable. If .x ∈ S (M), then it follows from Proposition 2.2.14 (i) that .x ∗ ∈ S (M). The mapping .x |−→ x ∗ is an involution in .S (M). Indeed, it is clear that .(x ∗ )∗ = x ¯ ∗ for all .x ∈ S (M) and .λ ∈ C. Given .x, y ∈ S (M), the operator and .(λx)∗ = λx ( ) ∗ ∗ ∗ ˆ ∗ . Hence, the two .x + y is densely defined and so, .x + y ⊆ (x + y) = x +y ( ) ˆ ∗ coincide on the strongly dense subspace ˆ ∗ and . x +y measurable operators .x ∗ +y ( ) ∗ ∗ ˆ ∗ = x ∗ +y ˆ ∗ . Via a similar argument, it is shown that .D (x ) ∩ D (y ) and so, . x +y ( )∗ ·y = y ∗ˆ·x ∗ for all .x, y ∈ S (M). . xˆ The results of the preceding discussion are now gathered in the next theorem. Theorem 2.2.16 The set .S (M) is a complex .∗ -algebra with unit element .1, with respect to the operations of strong sum and strong product and the .∗-operation of taking adjoints. The von Neumann algebra .M is a .∗-subalgebra of .S (M). ˆ and the strong product .xˆ·y of two elements From now on, the strong sum .x +y x, y ∈ S (M) will be denoted simply by .x + y and xy, respectively, unless stated otherwise. The set of all self-adjoint elements in .S (M) is denoted by .Sh (M), which is a real linear subspace of .S (M). For every .x ∈ S (M) the real and imaginary part are

.

2.2 The Algebra of Measurable Operators

61

defined by setting Re (x) =

.

) ) 1 ( 1( x − x∗ , x + x ∗ and Im (x) = 2i 2

respectively. Evidently, .Re (x) , Im (x) ∈ Sh (M) and .x = Re (x) + iIm (x). This shows that .S (M) = Sh (M) ⊕ iSh (M), as a direct sum of real linear subspaces. An operator .a ∈ Sh (M) is called positive, denoted by .a ≥ 0, whenever . ≥ 0 for all .ξ ∈ D (a). If .a ∈ Sh (M) and .D ⊆ D (a) is a core for a, then it is easily verified that .a ≥ 0 if and only if . ≥ 0 for all .ξ ∈ D. Note that .x ∗ x ≥ 0 for all .x ∈ S (M) and, in particular, .a 2 ≥ 0 for all .a ∈ Sh (M). The set of all positive elements in .Sh (M) is denoted by .S (M)+ . If .a ∈ Sh (M) satisfies .a ≥ 0 as well as .−a ≥ 0, then . = 0 for all .ξ ∈ D (a) and so, via polarization, it follows that ( .) = 0 for all .ξ, η ∈ D (a). Hence, .a = 0. This shows that + .S (M) ∩ −S (M)+ = {0}. Furthermore, .a + b ∈ S (M)+ and .λa ∈ S (M)+ whenever .a, b ∈ S (M)+ and .0 ≤ λ ∈ R. Consequently, .S (M)+ is a proper cone in .Sh (M) and a partial ordering in .Sh (M) may be introduced by defining .a ≤ b whenever .b − a ∈ S (M)+ . With respect to this ordering, .Sh (M) is a partially ordered vector space. Clearly, this partial ordering extends the ordering in the space .Mh . Moreover, if .a ∈ Sh (M), then the positive self-adjoint operators .a + and .a − are defined by a+ =

.

f R

λ+ dea (λ) = aea [0, ∞) , a − =

f R

λ− dea (λ) = −aea (−∞, 0] .

+ − (see Sect. 1.6). From (v) of Proposition ( 2.1.4, ) it follows that .a and( .a+ )are affiliated ( +) ( ) − with .M. Since .D (a) = D a ∩ D a , it is also clear that .D a and .D a − are strongly dense in .H . Consequently, .a + , a − ∈ S (M)+ and .a = a + − a − in + .Sh (M). This shows, in particular, that the positive cone .S (M) is generating in 1 + .Sh (M). Note, furthermore, that the equalities .a = 2 (|a| + a), .a − = 12 (|a| − a) and .a + a − = 0 hold in .S (M). The decomposition .a = a + − a − of an element .a ∈ Sh (M) is unique in the following sense.

Proposition 2.2.17 If .a ∈ Sh (M) and .b, c ∈ S (M)+ are such that .a = b − c and + and .c = a − . .bc = 0, then .b = a Proof Since .bc = 0 implies that also .cb = (bc)∗ = 0, it follows that .a 2 = (b − c)2 = b2 + c2 = (b + c)2 . By the uniqueness of the positive square root, ( )1/2 = b + c and so, .b = 12 (|a| + a) = a + and this implies that .|a| = a 2 1 − .c = u n 2 (|a| − a) = a . Remark 2.2.18 In connection with the order structure of .Sh (M), note that .M is an absolutely solid subspace of .S (M) in the following sense. If .x ∈ S (M) and .y ∈ M such that .|x| ≤ |y| in .Sh (M), then .x ∈ M. Indeed, it follows from .|x| ≤ |y| that || || 1/2 1/2 2 || || . ≤ ≤ ||y||B(H ) ||ξ || ξ H ≤ ||y||B(H ) ||ξ ||H for H and so, . |x|

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all .ξ ∈ D (|x|). This implies that .|x| is bounded, so .|x| ∈ M and hence, .x ∈ M. In the sequel, many other examples of absolutely solid (unitarily invariant) subspaces of .S (M) will be considered. Example 2.2.19 The notion of measurable operator may be illustrated by two simple examples. (a) Let .M = B (H ), where H is any Hilbert space. As follows from Example 2.1.5 (a), all closed and densely defined operators in H are affiliated with .M. Note that a projection .p ∈ P (M) is finite if and only if its range .Ran (p) is finite dimensional. Suppose that the linear subspace D of H is strongly dense. By definition, there exists a sequence .{pn }∞ n=1 in .P (M) such that .pn ↑ 1, ⊥ ⊥ ⊥ .pn (H ) ⊆ D and .pn is finite for all n. Since .pn ↓ 0 and each .pn has finite⊥ dimensional range, there must be an .n0 such that .pn0 = 0 and hence, .pn0 = 1. This shows that .D = H . Consequently, .S (M) = M = B (H ). (b) Let { .(X, Σ, μ) be }a Maharam measure space, .H = L2 (μ) and .M = of all closed, }densely Mf : f ∈ L∞ (μ) . As seen in 2.1.5 (b), the collection { defined linear operators affiliated with .M is given by . Mf : f ∈ L0 (μ) . Since .M is abelian (and hence, finite), it follows from Corollary 2.2.11 that every closed densely defined operator { } affiliated with .M is measurable. Consequently, .S (M) = Mf : f ∈ L0 (μ) . The bijective map .M : L0 (μ) → S (M), given by .M (f ) = Mf , .f ∈ L0 (μ), is a .∗-isomorphism. The proof of this follows easily from the following observation. If ( .f )∈ L0 (μ) and if D is a norm dense order ideal in .(L2 (μ) such that .D ⊆ D Mf , then D is a core for .Mf . Indeed, ) given .ξ ∈ D Mf , there exists a sequence .{ξn }∞ n=1 in D such that .ξn → ξ as .n → ∞ in .L2 (μ) . By passing to a subsequence, it may be assumed that .ξn → ξ .μ-a.e. as .n → ∞. Since D is an order ideal, it may be assumed further that .|ξn | ≤ |ξ | for all n. Using that .f ξ ∈ L2 (μ) , it follows from the Dominated Convergence Theorem that .f ξn → f ξ in .L2 (μ) as .n → ∞. Hence, D is a core for .Mf . The remaining details are left to the reader. The functional calculus of an element .a ∈ Sh (M) will now be considered. Recall that .B (σ (a)) denotes the space of all complex-valued Borel functions on the spectrum .σ (a) of a. For any .f ∈ B (σ (a)), the normal operator .f (a) is defined by f f (a) =

f (λ) dea (λ) .

.

σ (a)

The subspace consisting of all bounded functions in .B (σ (a)) is denoted by Bb (σ (a)). Furthermore, .Bbc (σ (a)) is defined to be the subspace of .B (σ (a)) consisting of all Borel functions which are bounded on compact subsets of .σ (a). Clearly, .Bbc (σ (a)) is a .∗-algebra with respect to pointwise addition, multiplication, and complex conjugation.

.

Proposition 2.2.20 If .a ∈ Sh (M), then .f (a) ∈ S (M) for all .f ∈ Bbc (σ (a)). Moreover, the mapping .f |−→ f (a) is a .∗-homomorphism from .Bbc (σ (a)) into .S (M).

2.2 The Algebra of Measurable Operators

63

Proof Given .f ∈ Bbc (σ (a)), it follows from Proposition 2.1.4 (v) that .f (a) ηM . Furthermore, .f (a) is closed and densely defined. Since a is measurable, it follows from Proposition 2.2.10 (iii) that there exists .λ > 0 such that .e|a| (λ, ∞) is a finite projection. Since f is bounded on .σ (a) ∩ [−λ, λ], there exists .0 < ρ ∈ R such that .|μ| > λ whenever .μ ∈ σ (a) and .|f (μ)| > ρ. Hence, e|f (a)| (ρ, ∞) = e|f |(a) (ρ, ∞) = ea {μ ∈ σ (a) : |f (μ)| > ρ}

.

≤ e {μ ∈ σ (a) : |μ| > λ} = e a

|a|

(2.4)

(λ, ∞) .

This shows that .e|f (a)| (ρ, ∞) is a finite projection and so, it follows from Proposition 2.2.10 that .f (a) is measurable. From the properties of the functional calculus, it is clear that the mapping .f |−→ f (a) is a .∗-homomorphism from .Bbc (σ (a)) into .S (M). u n Remark 2.2.21 Suppose that .M is a finite von Neumann algebra (that is, .1 is a finite projection). As observed in Corollary 2.2.11, in this case .S (M) coincides with the collection of all closed, densely defined operators affiliated with .M. Therefore, it follows from Proposition 2.1.4 (v) that .f (a) ∈ S (M) for all .a ∈ Sh (M) and all .f ∈ B (σ (a)). If the von Neumann algebra .M is not finite, then the latter statement does not hold in general, as can be easily seen by considering the von Neumann algebra .B (H ), whenever H is infinite dimensional. Given .a ∈ Sh (M), the above proposition implies, in particular, that .f (a) and g (a) are commuting elements of .S (M), for any two functions .f, g ∈ Bbc (σ (a)). An immediate consequence of Proposition 2.2.20 is that every .a ∈ S (M)+ has a unique positive square root .a 1/2 ∈ S (M)+ . If .f, g ∈ Bbc (σ (a)) are real-valued and .f (λ) ≤ g (λ) for all .λ ∈ σ (a), then .f (a) ≤ g (a) in .Sh (M) . Indeed,

.

g (a) − f (a) = (f − g) (a) ≥ 0,

.

where the last inequality follows from the properties of the functional calculus. If .a ∈ S (M)+ such that .a ≥ ε1 for some .ε > 0, then .σ (a) ⊆ [ε, ∞) and so, the function .f : σ (a) → R+ , given by .f (λ) = λ−1 , belongs to .Bb (σ (a)) and ≤ ε−1 . Hence, .f (a) = a −1 exists and satisfies .a −1 ∈ M+ and satisfies || || .||f ||∞ −1 −1 || .||a ≤ε . B(H ) Proposition 2.2.22 For .a, b ∈ Sh (M), the following statements are equivalent: (i) a and b commute in .S (M); (ii) .ea (δ1 ) eb (δ2 ) = eb (δ2 ) ea (δ1 ) for all Borel sets .δ1 , δ2 in .R; (iii) .f (a) g (b) = g (b) f (a) for all .f, g ∈ Bbc (R). Proof If .a, b ∈ Sh (M) are such that .ab = ba, then .(λ1 − a) (μ1 − b) = (μ1 − b) (λ1 − a) and hence, .r (λ, a) r (μ, b) = r (μ, b) r (λ, a) for all .λ, μ ∈ C \ R, where .r (λ, a) = (λ1 − a)−1 and .r (μ, b) = (μ1 − b)−1 . Now, it follows from Proposition 1.6.6 that condition (ii) is satisfied. Therefore, (i) implies (ii).

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It will be shown next that (ii) implies (i). If condition (ii) is satisfied, then it follows from Proposition 1.6.6 that the operators a and b are resolvent commuting, that is, r (λ, a) r (μ, b) = r (μ, b) r (λ, a) ,

.

λ, μ ∈ C \ R.

Fixing any two .λ, μ ∈ C \ R, the elements .r (λ, a) and .r (μ, b) commute in the algebra .S (M) and so, their inverses .λ1 − a and .λ1 − b commute as well. This implies that .a and b commute in .S (M). Hence, condition (i) is satisfied. Evidently, (i) follows from (iii) and therefore, it remains to prove that (ii) implies (iii). Let .f, g ∈ Bbc (R) be given. Considering the real and imaginary parts of these functions separately, it may be assumed, without loss of generality, that f and g are real( valued.)Recall that the spectral measure .ef (a) of .f (a) is given by .ef (a) (δ) = ea f −1 (δ) , .δ ⊆ R, and similarly for .eg(b) . Consequently, condition (ii) is satisfied for the operators .f (a) and .g (b). Since (ii) implies (i), it follows that .f (a) and .g (b) commute in .S (M). u n Note that it follows in particular from the above proposition that the self-adjoint operators .a, b ∈ S (M) commute in .S (M) if and only if .aeb (δ) = eb (δ) a in .S (M) for all Borel sets .δ ⊆ R. Corollary 2.2.23 If .M is an abelian von Neumann algebra, then the algebra .S (M) is also abelian (and .S (M) consists of all closed densely defined operators affiliated with .M). Proof First take .a, b ∈ Sh (M). It follows from Proposition 2.1.4 (iv) that ea (δ) , eb (δ) ∈ M for all Borel sets .δ ⊆ R. Since .M is abelian, it is clear that condition (ii) of Proposition 2.2.22 is satisfied and so, .ab = ba in .S (M). Writing .x ∈ S (M) as .x = Re (x) + iIm (x), with .Re (x) , Im (x) ∈ Sh (M), it is now evident that .S (M) is abelian. The final statement has already been observed in Corollary 2.2.11, so the proof is complete. u n .

In the remaining part of this section a number of results concerning the order structure of the space .Sh (M) will be discussed. Proposition 2.2.24

) ( ) ( + (i) If b ||∈ S (M) , ||then .a ≤ b if and only if .D b1/2 ⊆ D a 1/2 and || || .a, ) ( 1/2 ξ || ≤ ||b1/2 ξ || for all .ξ ∈ D b1/2 . .||a H H (ii) If .a, b ∈ S (M)+ , then .a ≤ b if and only if there exists .x ∈ M such that 1/2 = xb1/2 and .||x|| .a B(H ) ≤ 1. + (iii) If .a, b ∈ S (M) and .ab = ba, then .ab ∈ S (M)+ . (iv) If .a, b ∈ Sh (M) such that .a ≤ b , then .x ∗ ax ≤ x ∗ bx for every .x ∈ S (M). (v) If .a ∈ S (M)+ is invertible in .S (M), then .a −1 ≥ 0. (vi) If .0 ≤ a ≤ b in .Sh (M) and a is invertible in .S (M), then b is invertible in −1 ≤ a −1 . .S (M) and .0 ≤ b (vii) If .0 ≤ a ≤ b in .Sh (M) then .ea (λ, ∞) < eb (λ, ∞) for all .λ ≥ 0.

2.2 The Algebra of Measurable Operators

65

Proof (i) First assume that .0 ≤ a ≤ b in .Sh (M).||This implies || ||that .0||≤ ≤ for all .ξ ∈ D (a) ∩ D (b). Hence, .||a 1/2 ξ ||H ≤ ||b1/2 ξ ||H for all .ξ ∈ ) ( D (a) ∩ D (b). Furthermore, since .D (a) ∩ D (b) ⊆ D b1/2 and .D (a) ∩ D (b) is strongly dense, it follows ( from ) Proposition 2.2.14 (iii) that .D (a)∩D (b) is a core for .b1/2 . Take .ξ ∈ D b1/2 and let .{ξn }∞ n=1 be a sequence in .D (a)∩D (b) such that .ξn → ξ and .b1/2 ξn → b1/2 ξ as .n → ∞. Since || || || || || 1/2 || || || . ||a ξn − a 1/2 ξm || ≤ ||b1/2 ξn − b1/2 ξm || , n, m ∈ N, H

H

{ }∞ the sequence . a 1/2 ξn n=1 is Cauchy in H . Hence, there exists .η ∈ H such ) ( 1/2 is closed, it follows that .ξ ∈ D a 1/2 that .a 1/2 ξn → η as .n → ∞. Since .a || || || || and .η = a 1/2 ξ . Moreover, since .||a 1/2 ξn ||H ≤ ||b1/2 ξn ||H for all n, it is also || || || || clear that .||a 1/2 ξ ||H ≤ ||b1/2 ξ ||H . ( 1/2 ) ( 1/2 ) ⊆ D a .D b For the proof of the reverse implication, suppose that || 1/2 || || 1/2 || ( 1/2 ) || || || || and . a ξ H ≤ b ξ H for all .ξ ∈ D b . If .ξ ∈ D (a) ∩ D (b) ⊆ ) ( D b1/2 , then .

|| ||2 || ||2 || || || || = ||a 1/2 ξ || ≤ ||b1/2 ξ || = H

H

and so, . ≥ 0 for all .ξ ∈ D (a) ∩ D (b). Consequently, .b − a ≥ 0 in .Sh (M), since .D (a) ∩ D (b) is a core for .b − a (by Propositions 2.2.3 and 2.2.14 (iii)). (ii) If .0 ( ≤ )a ≤ b, then it follows from (i) (that the ) linear operator( .x0 ): Ran b1/2 → H may be defined by setting .x0 b1/2 ξ = a 1/2 ξ , .ξ ∈ D b1/2 . ) ( Since, by (i), .||x0 η||H ≤ ||η||H for all .η ∈ Ran b1/2 , it follows that .x0 extends uniquely to a linear operator ) ( x¯0 : Ran b1/2 → H

.

) ( satisfying .||x¯0 || ≤ 1. Defining .xξ = x¯0 r b1/2 ξ for all .ξ ∈ H , it is clear that ) ( 1/2 ξ = xb1/2 ξ for all .ξ ∈ D b1/2 . .x ∈ B (H ) with .||x||B(H ) ≤ 1 and that .a ( ') It is easy to see that .ux = xu for all .u ∈ U M , so .x ∈ M. Moreover, since ( 1/2 ) .D b is strongly dense in H , it follows that .a 1/2 = xb1/2 in .S (M). The proof of the converse implication is easy. (iii) Assume that .a, b ∈ S (M)+ are such that .ab( = ba. By )∗ (Proposition ) 2.2.22, this implies that .ab1/2 = b1/2 a and so, .ab = a 1/2 b1/2 a 1/2 b1/2 ≥ 0. (iv) Since .x ∗ bx −x ∗ ax = x ∗ (b − a) x, it is clearly enough to prove that .x ∗ ax ≥ 0 whenever .a ≥ 0 in .Sh (M). Suppose that .0 ≤ a ∈ Sh (M) and .x ∈ S (M) and

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( ) put .D = x −1 a −1 (D (x ∗ )) . Then .

< ∗ > x axξ, ξ = ≥ 0

for all .ξ ∈ D. By Lemma 2.2.13, D is strongly dense and so, by Proposition 2.2.14 (iii),{ it is a core operator .x ∗}ax and consequently, .x ∗ ax ≥ 0. ) the ( −1for −1 : a ξ ∈ D (a) , it is clear that (v) Defining .D = ξ ∈ D a / \ / ( )\ a −1 ξ, ξ = a −1 ξ, a a −1 ξ ≥ 0,

.

ξ ∈ D.

By Lemma 2.2.13, D is strongly dense in H and so, it is a core for .a −1 . Hence −1 ≥ 0. .a (vi) That .a −1 ≥ 0 follows from (v) and so, .a −1/2 ∈ Sh (M)+ . It follows from (iv) that .1 ≤ a −1/2 ba −1/2 and hence, by (ii), there exists .x ∈ M such that )1/2 ( )1/2 ( 1 = x a −1/2 ba −1/2 = a −1/2 ba −1/2 x∗.

.

( )1/2 This shows that . a −1/2 ba −1/2 is invertible in .S (M) and so, b is invertible in .S (M) with .b−1 ≥ 0. Using (iv), it follows from .0 ≤ a ≤ b that .0 ≤ b−1/2 ab−1/2 ≤ 1. Writing )∗ ( ) ( a 1/2 b−1/2 , b−1/2 ab−1/2 = a 1/2 b−1/2

.

|| || this implies that .a 1/2 b−1/2 ∈ M and .||a 1/2 b−1/2 ||B(H ) ≤ 1. Therefore, ||( || || 1/2 −1/2 )∗ || .|| a b ≤ 1 and hence, || B(H )

)( )∗ ( 0 ≤ a 1/2 b−1 a 1/2 = a 1/2 b−1/2 a 1/2 b−1/2 ≤ 1.

.

Using (iv) once again (with .x = a −1/2 ) yields .0 ≤ b−1 ≤ a −1 . (vii) Let .λ ≥ 0 be fixed and set .p = ea (λ, ∞) ∧ eb [0, λ]. If .0 /= ξ ∈ H satisfies .pξ = ξ , then ξ = eb [0, λ] ξ = eb

.

1/2

[

] ( ) 0, λ1/2 ξ ∈ D b1/2

|| || ) ( and .||b1/2 ξ ||H ≤ λ1/2 ||ξ ||H . It follows from (i) that .ξ ∈ D a 1/2 and ( || || || 1/2 || )) 1/2 ( 1/2 λ ,∞ .||a ξ ||H ≤ ||b1/2 ξ ||H . On other hand, since .0 /= ξ ∈ Ran ea || || ) ( ∩ D a 1/2 , it follows from Lemma 2.2.7 that .||a 1/2 ξ ||H > λ1/2 .||ξ ||H . This is

2.2 The Algebra of Measurable Operators

67

a contradiction and hence, it may be concluded that .p = 0. Consequently, ea (λ, ∞) ∼ ea (λ, ∞) ∨ eb [0, λ] − eb [0, λ] ≤ eb (λ, ∞)

.

u n

and so, .ea (λ, ∞) < eb (λ, ∞).

Statement (i) of the above proposition shows that the partial ordering in .Sh (M) coincides on .S (M)+ with the quadratic form ordering in the set .H+ of all positive self-adjoint operators in H (see Sect. 1.8). This fact is exploited in the proof of the next proposition. Proposition 2.2.25 { } (i) Suppose that . aβ is an increasing net in .Sh (M) and there exists .b ∈ Sh (M) such that .aβ ≤ b for all .β. Then .supβ aβ exists in .Sh (M). { } (ii) If . aβ is an increasing net in .S (M)+ and .a ∈ S (M)+ , then .aβ ↑ a if and only if )

(

D a 1/2 =

.

|| || || 1/2 || ||a ξ ||

H

⎧ ⎨ ⎩

ξ∈

n

|| || ) ( || 1/2 || 1/2 : sup ||aβ ξ || D aβ β

β

|| || || 1/2 || = sup ||aβ ξ || , β

H

H

} ⎬ 0 such that .0 ≤ f (λ) ≤ C max (1, λ) for all .λ ≥ 0, it follows from Proposition 2.2.20 that .f (a) , f (b) ∈ S (M)+ . Furthermore, note that it may be assumed, without loss of generality, that .ν ({0}) = 0. If .0 ≤ a ≤ b in .Sh (M), then .t1 ≤ a + t1 ≤ b + t1 for all .t ∈ (0, ∞) and so, by Proposition 2.2.24 (vi), .0 ≤ (b + t1)−1 ≤ (a + t1)−1 . This implies that a (a + t1)−1 = 1 − t (a + t1)−1 ≤ 1 − t (b + t1)−1 = b (b + t1)−1

(2.8)

.

for all .t > 0. Note that the elements .a (a + t1)−1 and .b (b + t1)−1 actually belong to .M+ . ) ( Take .ξ ∈ D f (b)1/2 . Using (2.7), (2.8), Fubini’s theorem and the properties of spectral integrals, it follows that f

) λ a dν (t) deξ,ξ (λ) (λ) = [0,∞) [0,∞) λ + t ) (f f λ a deξ,ξ (λ) dν (t) = [0,∞) [0,∞) λ + t f / \ = a (a + t1)−1 ξ, ξ dν (t) (f

f

f

.

[0,∞)

a (λ) deξ,ξ

[0,∞)

f ≤

[0,∞)

f =

[0,∞)

/ \ b (b + t1)−1 ξ, ξ dν (t) ||2 || || || b f (λ) deξ,ξ (λ) = ||f (b)1/2 ξ ||

< ∞.

||2 || || || a f (λ) deξ,ξ (λ) ≤ ||f (b)1/2 ξ ||

.

B(H )

) ( Hence, .ξ ∈ D f (a)1/2 and .

|| ||2 || || ||f (a)1/2 ξ ||

B(H )

f = [0,∞)

B(H )

By (i) of Proposition 2.2.24, it may be concluded that .0 ≤ f (a) ≤ f (b) in .Sh (M). u n Corollary 2.2.28 If .0 ≤ a ≤ b in .Sh (M) then .0 ≤ a γ ≤ bγ for all .0 < γ < 1.

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2 Measurable Operators

Proof For .0 < γ < 1 the function .f (λ) = λγ , .λ ≥ 0, has the well-known integral representation λγ =

.

sin π γ π

f 0

∞

λ dt, λ ≥ 0 (λ + t) t 1−γ

and so the result follows at once ( ) from Proposition 2.2.27 applied to the measure .ν given by .dν (t) = π −1 sin π γ t γ −1 dt. u n References: [111, 114, 126, 137].

2.3 The Algebra of τ -Measurable Operators If the von Neumann algebra .M, acting on the Hilbert space H , is equipped with a faithful normal semi-finite trace .τ , then there is an important .∗-subalgebra of .S (M). This is the .∗-algebra of all .τ -measurable operators, denoted by .S (M, τ ), or simply by .S (τ ). This section introduces .τ -measurable operators and develops some of their basic properties. In a von Neumann algebra .M, equipped with a faithful normal semi-finite trace .τ , apart from finite projections, also .τ -finite projections in .P (M) may be considered. A projection .p ∈ P (M) is called .τ -finite if .τ (p) < ∞. If .p ∈ P (M) is a .τ -finite projection, then p is finite. Indeed, suppose that .q ∈ P (M) is such that .q ≤ p and .q ∼ p. Then .τ (q) = τ (p) and .τ (p) = τ (q) + τ (p − q). Since .τ (p) < ∞, this implies that .τ (p − q) = 0, so .p − q = 0 and hence .p = q. In general, finite projections in .M are not .τ -finite. This can be seen, for example, by considering abelian von Neumann algebras, in which all projections are finite but not necessarily .τ -finite. The definition of .τ -measurable operators is similar to the definition of measurable operators, replacing finite projections in .P (M) by .τ -finite projections. Throughout this section, it is assumed that .M is a von Neumann algebra in a Hilbert space H and .τ : M+ → [0, ∞] is a faithful normal semi-finite trace. Definition 2.3.1 A linear subspace D of H is said to be .τ -dense in H( if there exists ) ⊥ < ∞ for a sequence .{pn }∞ in .P (M), such that .pn ↑ 1, .pn (H ) ⊆ D and .τ pn n=1 all n. The following observation is an immediate consequence of the definition. Proposition 2.3.2 If D is a .τ -dense subspace of H , then D is strongly dense in H . ∞ Proof If D is .τ -dense, then there ( ⊥ )exists a sequence .{pn }n=1 in .P (M), such that < ∞ for all n. As observed at the beginning .pn ↑ 1, .pn (H ) ⊆ D and .τ pn ( ) of this section, .τ pn⊥ < ∞ implies that .pn⊥ is a finite projection. Therefore, D is u n strongly dense.

2.3 The Algebra of τ -Measurable Operators

71

Proposition 2.3.3 For a linear subspace D of H the following two statements are equivalent: (i) D is .τ -dense in H . (ii) For ( every ) .ε > 0 there exists a projection .p ∈ P (M) such that .p (H ) ⊆ D and ⊥ ≤ ε. .τ p ⊥ Proof ) D is .τ -dense in H . Since it follows from .pn ↓ 0 and ( ⊥ )First assume( that ⊥ .τ p 1 < ∞ that .τ pn ↓ 0 as .n → ∞, it is clear that (ii) holds. Now, suppose that condition (ii) is For every .n ∈ N, ) ( satisfied. Athere exists .qn ∈ P (M) such that .qn (H ) ⊆ D and .τ qn⊥ ≤ 2−n . Setting .pn = ∞ k=n qk , it is clear that .pn ≤ pn+1 and .pn (H ) ⊆ qn (H ) ⊆ D for all n. Furthermore,

( τ

.

pn⊥

)

( =τ

∞ V

) qn⊥

≤

k=n

∞ ∞ ( ) Σ Σ τ qn⊥ ≤ 2−k = 2−n+1 , k=n

k=n

(A∞ ⊥ ) A∞ ⊥ for all .n = 1, 2, . . .. Therefore, .τ n=1 pn = 0 and so . n=1 pn = 0. Hence, ⊥ .pn ↓ 0, that is, .pn ↑ 1. This shows that D is .τ -dense and so, (ii) implies (i). u n The next result is similar to Proposition 2.2.3. Proposition 2.3.4 If .{Dn }∞ n=1 is a sequence of .τ -dense subspaces of H , then n∞ . D is also . τ -dense. n=1 n Proof Let .ε > 0 be given. ) each .n = 1, 2, . . . there exists ( For A .pn ∈ P (M) such that .pn (H ) ⊆ Dn and .τ pn⊥ ≤ ε2−n−1 . Defining .e = ∞ n=1 pn , it follows that .e (H ) = pn e (H ) ⊆ Dn and ( ) ⊥ =τ .τ e

(

∞ V

n=1

) pn⊥

≤

∞ ( ) Σ τ pn⊥ ≤ ε. n=1

n By Proposition 2.3.3, . ∞ n=1 Dn is .τ -dense in H .

u n

Definition 2.3.5 A linear operator x in H is called .τ -(pre-)measurable whenever x is (pre-)closed, affiliated with .M and its domain .D (x) is .τ -dense in H . The set of all .τ -measurable operators is denoted by .S (M, τ ) or, simply by .S (τ ). Note that if an operator x in H is .τ -pre-measurable, then its closure .x¯ is .τ measurable. Furthermore, a linear operator x in H is .τ -(pre-)measurable if and only if x is (pre-)measurable with respect to .M and its domain .D (x) is .τ -dense. From the definitions, it is evident that .M ⊆ S (τ ) ⊆ S (M). It will be shown in Theorem 2.3.8 that .S (τ ) actually is a .∗-subalgebra of .S (M). It will be convenient to present first several characterizations of .τ -measurable operators.

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2 Measurable Operators

Proposition 2.3.6 Let x be a closed, densely defined operator such that .xηM. The following statements are equivalent: (i) x is .τ -measurable. (ii) .|x| is .τ -measurable. ( ) (iii) There exists a projection .p ∈ P (M) such that .p (H ) ⊆ D (x) and .τ p⊥ < ∞. ( ) (iv) There exists .λ > 0 such that .τ e|x| (λ, ∞) < ∞. ( ) (v) .τ e|x| (λ, ∞) → 0 as .λ → ∞. Proof A closed, densely defined operator x, affiliated with .M, is .τ -measurable if and only if .D (x) is .τ -dense. Since .D (x) = D (|x|), it is immediately clear from Proposition 2.1.4 (vi) that assertions (i) and (ii) are equivalent. It is clear that (i) implies (iii). that (iii) holds and that .p ∈ P (M) is ) ( Suppose such that .p (H ) ⊆ D (x) and .τ p⊥ < ∞. Note that .xp ∈ B (H ). If .λ ∈ R is such that .λ > ||xp||B(H ) , then it follows from Lemma 2.2.9 that .e|x| (λ, ∞) < p ⊥ and ( ) ( ) so, .τ e|x| (λ, ∞) ≤ τ p⊥ < ∞. This shows that (iv) is satisfied. ( ) Assuming that (iv) holds, let .λ0 > 0 be such that .τ e|x| (λ0 , ∞) < ∞. Since ( ) |x| (λ, ∞) ↓ 0 as .λ → ∞, the normality of the trace implies that .τ e|x| (λ, ∞) → .e 0 as .λ → ∞. Therefore, (v) holds. to )show that (v) implies (i), let .ε > 0 be given and take .λ > 0 such that ( Finally, |x| (λ, ∞) ≤ ε. Setting .p = e|x| [0, λ], it is clear that .p (H ) ⊆ D (|x|) = D (x) .τ e ( ) and .τ p⊥ ≤ ε. Since x is assumed to be affiliated with .M, this shows that x is .τ -measurable. u n It follows immediately from (iv) in the above proposition that if .τ (1) < ∞, then any closed densely defined operator which is affiliated with .M is .τ -measurable. Consequently, in this case, .S (τ ) = S (M) is the collection of all closed densely defined operators in H , affiliated with .M (cf. Corollary 2.2.11). The next result is similar to Lemma 2.2.13. Lemma 2.3.7 If x is a .τ -pre-measurable operator in H and D is a .τ -dense subspace of H , then .x −1 (D) is .τ -dense in H . Proof (Given ) .ε(> 0) there exist .p, q ∈ P (M) such that .p (H ) ⊆ D (x), .q( (H ) ⊆) D and .τ p⊥ , .τ q ⊥ ≤ ε/2. Since .xp ∈ M, the null projection .f = n q ⊥ xp of ⊥ ⊥ .q xp belongs to .P (M). Note that .q xpf = 0 and so, .xpf = qxpf . From ( ) ( ) f ⊥ = s q ⊥ xp ∼ r q ⊥ xp ≤ q ⊥ ,

.

( ) ( ) it follows that .τ f ⊥ ≤ τ q ⊥ . Defining .e = f ∧ p, it is clear that .e (H ) ⊆ ( ) ( ) ( ) ( ) p (H ) ⊆ D (x). Furthermore, .τ e⊥ = τ f ⊥ ∨ p⊥ ≤ τ f ⊥ + τ p⊥ ≤ ε. Using the equalities .e = pf e and .xpf = qxpf , it follows that xe = xpf e = qxpf e = qxe.

.

2.3 The Algebra of τ -Measurable Operators

73

Hence .xe (H ) ⊆ D, that is, .e (H ) ⊆ x −1 (D) and consequently, .x −1 (D) is .τ dense. n u Theorem 2.3.8 The set .S (τ ) of all .τ -measurable operators in H is a .∗-subalgebra of .S (M). Moreover, .S (τ ) is an absolutely solid subspace of .S (M), that is, if .x ∈ S (M) and .y ∈ S (τ ) such that .|x| ≤ |y|, then .x ∈ S (τ ). Proof If .x, y ∈ S (τ ), then .x + y ∈ S (M) is the strong sum of the operators x and y. Since .D (x) ∩ D (y) ⊆ D (x + y) and, by Proposition 2.3.4, .D (x) ∩ D (y) is .τ -dense, it follows that .D (x + y) is .τ -dense. Therefore, .x + y ∈ S (τ ). Similarly, if −1 (D (x)) ⊆ .x, y ∈ S (τ ), then .xy ∈ S (M) is the strong product of x and y. Since .y −1 D (xy) and, by Lemma 2.3.7, .y (D (x)) is .τ -dense, the subspace .D (xy) is also .τ -dense. Hence, .xy ∈ S (τ ). Since .D (x) ⊆ D (λx), it is clear that .λx ∈ S (τ ) whenever .x ∈ S (τ ) and .λ ∈ C. This shows that .S (τ ) is a subalgebra of .S (M). To prove that .S (τ ) is a .∗-subalgebra of .S (M), let .x ∈ S (τ ) be given with polar decomposition .x = v |x|. It follows from .x ∗ = |x| v ∗ that .D (x ∗ ) = (v ∗ )−1 (D (x)), which is a .τ -dense subspace of H by Lemma 2.3.7. Hence, .x ∗ ∈ S (τ ). To see that .S (τ ) is absolutely solid in .S (M), assume that .x ∈ S((M), ).y ∈ S (τ ) and .|x| ≤ |y|. By Proposition 2.2.24 (i), this implies that .D |y|1/2 ⊆ ( ) ) ( D |x|1/2 . Since .D (y) = D (|y|) ⊆ D |y|1/2 and .D (y) is .τ -dense, it follows ( 1/2 ) that .D |x| is .τ -dense. Consequently, .|x|1/2 ∈ S (τ ), so .|x| ∈ S (τ ) and hence, by Proposition 2.3.6, .x ∈ S (τ ). u n Remark 2.3.9 Note that it follows from Remark 2.2.18, in combination with the above theorem, that .M is an absolutely solid subspace of .S (τ ). The set of all self-adjoint elements of .S (τ ) is denoted by .Sh (τ ), which is a real linear subspace of .S (τ ). Clearly, .Re (x) , Im (x) ∈ Sh (τ ) whenever .x ∈ S (τ ). The space .Sh (τ ) is a partially ordered vector space with respect to the ordering inherited + + from .Sh (M). Its positive cone is denoted by .S (τ )+ , so( .S (τ ) ) = S (M) ∩ Sh (τ ). + + − ± If .a ∈ Sh (τ ), then .a , .a ∈ S (τ ) , as .D (a) ⊆ D a . Therefore, the positive cone .S (τ )+ is generating in .Sh (τ ). As seen in Proposition 2.2.25 (i), every upward directed system in .Sh (M), which is bounded from above, has a least upper bound in .Sh (M). The next proposition shows that the same holds for .Sh (τ ). { } Proposition 2.3.10 If . aβ is an upward directed net in .Sh (τ ) and if there exists .b ∈ Sh (τ ) such that .aβ ≤ b for all .β, then there exists .a ∈ Sh (τ ) such that .aβ ↑ a. Proof Without loss of generality, it may be assumed that .aβ ≥ 0 for all .β. By Proposition 2.2.25 (i), there exists { } .a ∈ Sh (M) such that .aβ ↑ a in .Sh (M). Since b is also an upper bound of . aβ in .Sh (M), it is clear that .0 ≤ a ≤ b. Since, by Theorem 2.3.8, .S (τ ) is an absolutely solid subspace of .S (M), this implies that .a ∈ Sh (τ ). It is now evident that .aβ ↑ a in .Sh (τ ). u n From the proof of the above proposition, it is also clear that statements (ii) and (iii) of 2.2.25 are valid in .Sh (τ ).

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2 Measurable Operators

For the statement of the next result, some further notation is needed. Recall that s (x) and .r (x) denote the support and range projection, respectively, of an operator x. Define

.

F (τ ) = {x ∈ M : τ (s (x)) < ∞} .

.

(2.9)

Since .s (x) ∼ r (x) for all .x ∈ M, it is clear that .F (τ ) is also given by .F (τ ) = {x ∈ M : τ (r (x)) < ∞}. Since .s (x) = r (x ∗ ), it follows that .x ∈ F (τ ) if and only if .x ∗ ∈ F (τ ). Furthermore, if .x ∈ F (τ ) and .y ∈ M, then .n (x) ≤ n (yx) and so, .s (yx) ≤ s (x) from which it follows that .yx ∈ F (τ ). If .x, y ∈ F (τ ), then .n (x) ∧ n (y) ≤ n (x + y) implies that .s (x + y) ≤ s (x) ∨ s (y) and so, τ (s (x + y)) ≤ τ (s (x)) + τ (s (y)) < ∞,

.

that is, .x + y ∈ F (τ ). Consequently, .F (τ ) is a two-sided ideal in .M. For later reference, some of these observations are collected together in the next lemma. Lemma 2.3.11 The set .F (τ ), defined by (2.9), is a two-sided ideal in .M. Furthermore, .F (τ ) is also given by F (τ ) =

.

U

{eMe : e ∈ P (M) , τ (e) < ∞} .

Proof In view of the above remarks, only the last statement needs to be proved. If .e ∈ P (M) with .τ (e) < ∞ and .x ∈ eMe, then .x = xe, which implies that .s (x) ≤ e. Therefore, .τ (s (x)) < ∞, which shows that .x ∈ F (τ ). Conversely, if .x ∈ F (τ ), then .τ (s (x)) = τ (r (x)) < ∞. Defining .e = s (x) ∨ r (x), it is clear that .τ (e) < ∞ and .x = xe = ex, that is, .x ∈ eMe. The proof is complete. u n Proposition 2.3.12 If .0 ≤ a ∈ Sh (τ ), then there exists an upward directed system { } aβ in .F (τ ) such that .0 ≤ aβ ↑ a holds in .Sh (τ ).

.

Proof First, it will be shown that for any .0 < b ∈ Sh (τ ) there exists .c ∈ F (τ ) such that .0 < c ≤ b. Let .λ ≥ 0 be such that .beb [0, λ] /= 0, in which case .beb [0, λ] ∈ M and .0 < beb [0, λ] ≤ b. Since the trace .τ is semi-finite, there exists an increasing net .{pα } in .P (M) such that .pα ↑α eb [0, λ] and .τ (pα ) < ∞ for all .α. Since b1/2 eb [0, λ] pα eb [0, λ] b1/2 ↑α b1/2 eb [0, λ] b1/2 = beb [0, λ]

.

in .M+ , there exists .α0 such that .c = b1/2 eb [0, λ] pα0 eb [0, λ] b1/2 /= 0. Since 1/2 eb [0, λ] ∈ M, it is now clear that .c ∈ F (τ ) and .0 < c ≤ b. .pα0 ∈ F (τ ) and .b Given .0 ≤ a ∈ Sh (τ ), let .V be the set of all upward directed subsets .V ⊆ (F (τ ))+ satisfying .v ≤ a for all .v ∈ V . Ordering .V by inclusion, it is easy to see that every chain in .V has an upper bound in .V. Consequently, .V contains a maximal element .Vm . According to Proposition 2.3.10, .v0 = sup Vm exists in .Sh (τ ). If .0 ≤ v0 < a, then it follows from the first part of the proof that there exists

2.3 The Algebra of τ -Measurable Operators

75

c ∈ F (τ ) such that .0 < c ≤ a − v0 . Define

.

V1 = {x ∈ F (τ ) : ∃v ∈ Vm such that 0 ≤ x ≤ v + c} .

.

Since the set .Vm is upward directed, it follows readily that the set .V1 is also upward directed. Further, if .x ∈ V1 , then .x ≤ v +c for some .v ∈ Vm and so, .x ≤ v0 +c ≤ a. Hence, .V1 ∈ V. Since .Vm ⊆ V1 , the maximality of .Vm implies that .V1 = Vm . Accordingly, .v + c ≤ v0 for all .v ∈ Vm and so, .v0 + c ≤ v0 , which is absurd as .c > 0. Therefore, .sup Vm = a and the proof is complete. u n Example 2.3.13 Some simple examples of .τ -measurable operators now follow. (a) Let H be a Hilbert space and let .M = B (H ), equipped with the standard trace + .τ : M → [0, ∞]. If .p ∈ P (M), then .τ (p) < ∞ if and only if the range of p is finite dimensional, which is equivalent to saying that p is a finite projection. Consequently, .S (τ ) = S (M) and .S (M) = M, as seen in Example 2.2.19 (a). It should be observed that in this case the ideal .F (τ ), as defined in (2.9), is precisely the ideal of all finite rank operators. Indeed, if .x ∈ M, then .x ∈ F (τ ) if and only if .x = px, where .p ∈ P (M) satisfies .τ (p) < ∞. (b) Let { .(X, Σ, μ) be }a Maharam measure space, .H = L2 (μ) and .M = Mf : f ∈ L∞ (μ) . The mapping .f |−→ Mf is a .∗ -isomorphism from + .L∞ (μ) onto .M. Defining .τ : M → [0, ∞] by ( ) τ Mf =

f

.

X

f dμ, 0 ≤ f ∈ L∞ (μ) ,

it follows that .τ is a faithful normal semi-finite trace on .M. Example 2.2.19 (b) shows that the map .f − | → Mf , .f ∈ L0 (μ), is a .∗-isomorphism from .L0 (μ) onto .S (M). If .Mf ∈ S (M), then it follows from Proposition 2.3.6 (iii) that .Mf ∈ S (τ ) (if and ) only if there exists a projection .(p ∈) P (M) such that .p (H ) ⊆ D Mf (equivalently, .Mf p ∈ M) and .τ p⊥ < ∞. Any projection .p ∈ P (M) is of the form .p = MχA for some .A ∈ Σ and .τ (p) = μ (A). Therefore, if .f ∈ L0 (μ), then .Mf ∈ S (τ ) if and only if there exists .A ∈ Σ such that .μ (X \ A) < ∞ and .f χA ∈ L∞ (μ). Consequently, if the .∗-subalgebra .S (μ) of .L0 (μ) is defined by S (μ) = {f ∈ L0 (μ) : ∃ A ∈ Σ, μ (X \ A) < ∞, f χA ∈ L∞ (μ)} , (2.10)

.

then the map .f |−→ Mf is a .∗-isomorphism from .S (μ) onto .S (τ ). If the measure .μ is finite, then it is clear that .S (μ) = L0 (μ) and so, .S (τ ) = S (M). However, if .μ is an infinite measure, then .S (μ) C L0 (μ). Indeed, in ∞ this case there exists a pairwise Σ∞disjoint sequence .{An }n=1 in .Σ with .μ (An ) ≥ 1 for all n. The function .f = n=1 nχAn belongs to .L0 (μ) but f does not belong to .S (μ). Hence, in this case, the inclusion .S (τ ) ⊆ S (M) is proper.

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(c) The algebra of all .τ -measurable operators depends, in general, on the choice of the trace .τ . This may be illustrated by the following example. Suppose that .(X, Σ, μ) is a measure space, where the measure .μ is .σ -finite but not finite { } and let .M = Mf : f ∈ L∞ (μ) , as in (b). There exists a finite measure .ν on .Σ which is equivalent with .μ (that is, .ν( and).μ have f the same null sets). If + .τ1 : M → [0, ∞] is defined by setting .τ1 Mf = X f dν, .0 ≤ f ∈ L∞ (μ), then .τ1 is a finite normal faithful trace on .M. From Proposition 2.3.6, it is clear that .S (τ1 ) = S (M), whereas .S (τ ) C S (M), since .μ is not a finite measure (see (b)). If .a ∈ S (τ )+ , (then .a)1/2 ∈ S (M)+ (see ) discussion following Remark 2.2.21). ( the Since .D (a) ⊆ D a 1/2 , the domain .D a 1/2 is .τ -dense and so, .a 1/2 ∈ S (τ )+ . The next proposition exhibits a large class of Borel functions f on the spectrum .σ (a) of .a ∈ Sh (τ ), with the property that .f (a) ∈ Sh (τ ). Recall that .Bbc (σ (a)) denotes the space of all complex-valued Borel functions on .σ (a), which are bounded on all compact subsets of .σ (a). Proposition 2.3.14 If .a ∈ Sh (τ ), then .f (a) ∈ S (τ ) for all .f ∈ Bbc (σ (a)). The mapping .f − | → f (a) is a .∗ -homomorphism from .Bbc (σ (a)) into .S (τ ). Proof For any .f ∈ Bbc (σ (a)), it follows from Proposition 2.2.20 that .f (a) ∈ S (M). Since .a ∈ S( (τ ), it follows from Proposition 2.3.6 (iv) that there exists ) |a| (λ, ∞) < ∞ . As in the proof of Proposition 2.2.20, .λ > 0 such that .τ e |f (a)| (ρ, ∞) ≤ e|a| (λ, ∞) (see (2.4)). Hence, there exists .ρ > ( |f (a)| ) 0 such that .e .τ e (ρ, ∞) < ∞, and via Proposition 2.3.6 it may be concluded that .f (a) ∈ S (τ ). The second statement of the proposition follows immediately from the corresponding statement in Proposition 2.2.20. u n Remark 2.3.15 (i) If .a ∈ Sh (τ ), then the condition .f ∈ Bbc (σ (a)) is sufficient for .f (a) to be in .S (τ ) but, not necessary. Indeed, if .τ (1) < ∞, then all closed densely defined operators affiliated with .M are .τ -measurable. Therefore, in | → f (a) is this situation, .f (a) ∈ S (τ ) for all .f ∈ B (σ (a)) and the map .f − a .∗-homomorphism from .B (σ (a)) into .S (τ ) (see also Remark 2.2.21). (ii) Since .S (τ ) is absolutely solid in .S (M), it is clear that the observation made in Remark 2.2.26 concerning the functional calculus in .S (M) is also valid for .S (τ ). References: [93, 127].

2.4 The Algebra of τ -Compact Operators As observed in Proposition 2.3.6, a closed, densely defined operator x, affiliated with the von( Neumann algebra .M, belongs to .S (τ ) if and only if there exists .λ > 0 ( ) ) such that .τ e|x| (λ, ∞) < ∞. In general, if .x ∈ S (τ ), then .τ e|x| (λ, ∞) = ∞

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77

for( certain values of .λ > 0. By way of a simple example, if .τ (1) = ∞, then ) τ e1 (λ, ∞) = τ (1) = ∞ whenever .0 < λ < 1. The set .S0 (τ ), of all .x ∈ S (τ ) ( ) with the property that .τ e|x| (λ, ∞) < ∞ for all .λ > 0, will play an important role in the sequel. Some of the basic properties of the set .S0 (τ ) will now be established.

.

Definition 2.4.1 The collection of all closed, densely defined ( operators ) x in H , affiliated with the von Neumann algebra .M and satisfying .τ e|x| (λ, ∞) < ∞ for all .λ > 0, will be denoted by .S0 (τ ). The elements of .S0 (τ ) are sometimes called .τ -compact operators. Evidently, each .x ∈ S0 (τ ) is .τ -measurable, that is, .S0 (τ ) ⊆ S (τ ). From the definition it is also clear that, if x is a closed, densely defined linear operator in H and .xηM, then .x ∈ S0 (τ ) if and only if .|x| ∈ S0 (τ ). The proposition that follows presents another characterization of the elements of .S0 (τ ). Proposition 2.4.2 If x is a closed, densely defined operator in H , affiliated with the von Neumann algebra .M, then .x ∈ (S0 (τ)) if and only if for every .ε > 0 there exists a projection .p ∈ P (M) such that .τ p⊥ < ∞, .p (H ) ⊆ D (x) and .||xp||B(H ) ≤ ε. Proof Assume that .x ∈ S0 (τ ) and let .ε > 0 be given. Setting .p = e|x| [0, ε] , it is clear that .p ∈ P (M), .p (H ) ⊆ D (|x|) = D (x) and .|||x| p||B(H ) ≤ ε. If .x = v |x| is the polar decomposition of x, then, .

||xp||B(H ) = ||v |x| p||B(H ) ≤ |||x| p||B(H ) ≤ ε.

For the proof of the converse implication, let .λ > 0 be given. By hypothesis, ( ) there exists .p ∈ P (M) such that .τ p⊥ < ∞, .p (H ) ⊆ D (x) and .||xp||B(H ) ≤ ( ) λ. It follows from Lemma 2.2.9 that .e|x| (λ, ∞) < p⊥ and so, .τ e|x| (λ, ∞) ≤ ( ⊥) τ p < ∞. This shows that .x ∈ S0 (τ ). u n Remark 2.4.3 If x is a closed densely defined operator which is affiliated with .M and if .τ (r (x)) < ∞, then .x ∈ S0 (τ ). Indeed, it follows from .r (|x|) ∼ r (x) that ⊥ .τ (r (|x|)) < ∞ and so, the projection .p = n (x) = r (|x|) satisfies .xp = 0 ( ⊥) < ∞. Hence, Proposition 2.4.2 shows that .x ∈ S0 (τ ). In particular, and .τ p .F (τ ) ⊆ S0 (τ ). It is also evident that if .τ (1) < ∞, then .S0 (τ ) is equal to the set of all closed, densely defined operators affiliated with .M. So, in this case, .S0 (τ ) = S (τ ) = S (M). However, if .τ (1) = ∞, then .S0 (τ ) /= S (τ ), as is clear from the discussion preceding Definition 2.4.1. Proposition 2.4.4 The set .S0 (τ ) is an absolutely solid .∗-subalgebra of .S (τ ). Moreover, if .x ∈ S0 (τ ) and .y ∈ S (τ ), then xy, .yx ∈ S0 (τ ). Proof Throughout the proof of this proposition, the characterization of elements of S0 (τ ) given in Proposition 2.4.2 will be used without further reference. It is evident that .λx ∈ S0 (τ ) whenever .x ∈ S0 (τ ) and .λ ∈ C. Take .x1 , x2 ∈ S0 (τ ).(To show that .x1 + x2 ∈ S0 (τ ), let .ε > 0 be given and take .pj ∈ P (M) such ) || || that .τ p⊥ < ∞, .xj pj ∈ M and .||xj pj || ≤ ε, for .j = 1, 2. If .q = p1 ∧ p2 ,

.

j

B(H )

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then ( ) ( ) ( ) ( ) τ q ⊥ = τ p1⊥ ∨ p2⊥ ≤ τ p1⊥ + τ p2⊥ < ∞.

.

Furthermore, .(x1 + x2 ) q = x1 p1 q + x2 p2 q ∈ M and .

||(x1 + x2 ) q||B(H ) ≤ ||x1 p2 q||B(H ) + ||x1 p2 q||B(H ) ≤ 2ε.

This shows that .x1 + x2 ∈ S0 (τ ). It will next be shown that( .x ∈) S0 (τ ) and .y ∈ S (τ ) implies that .xy ∈ S0 (τ ). Let ⊥ .p2 ∈ P (M) be such that .τ p 2 < ∞ and .yp2 ∈ M. If .yp2 = 0, then .(xy) p2 = 0, in which case it is evident that .xy ∈ S0 (τ ). So, it may ( )be assumed that .yp2 /= 0. Given .ε > 0 , there exists .p1 ∈ P (M) such that .τ p1⊥ < ∞ and .xp1 ∈ M with ( ⊥ ) .||xp1 ||B(H ) ≤ ε/ ||yp2 ||B(H ) . Define .e ∈ M by .e = n p y and .q = p2 ∧ e. Since 1 ( ) ( ) ( ) ⊥ = s p ⊥ y ∼ r p ⊥ y and .r p ⊥ y ≤ p ⊥ , it follows that .e⊥ < p ⊥ and so, .e 1( 1 1 1 1 ) ( ⊥) ≤ τ p1⊥ . Hence, .τ e ( ) ( ) ( ) ( ) τ q ⊥ = τ p2⊥ ∨ e⊥ ≤ τ p2⊥ + τ e⊥ < ∞.

.

From the definition of e it is clear that .p1⊥ ye = 0 and so, .ye = p1 ye. Using that .q = eq = p2 q, it follows that xyq = xyeq = xp1 yeq = (xp1 ) (yp2 ) q ∈ M

.

and .||xyq||B(H ) ≤ ||xp1 ||B(H ) ||yp2 ||B(H ) ≤ ε. Consequently, .xy ∈ S0 (τ ) . This shows, in particular, that .S0 (τ ) is a subalgebra of .S (τ ). To show that .S0 (τ ) is a .∗-subalgebra of .S (τ ), let .x ∈ S0 (τ ) have polar decomposition .x = v |x|. By definition, .x ∈ S0 (τ ) implies that .|x| ∈ S0 (τ ). Since ∗ = |x| v ∗ , with .|x| ∈ S (τ ) and .v ∗ ∈ S (τ ), it follows, from what has been .x 0 proved so far, that .x ∗ ∈ S0 (τ ). Hence, .S0 (τ ) is a .∗-subalgebra of .S (τ ). Moreover, if .x ∈ S0 (τ ) and .y ∈ S (τ ), then .x ∗ ∈ S0 (τ ) and .y ∗ ∈ S (τ ), and so .x ∗ y ∗ ∈ S0 (τ ). Therefore, .yx = (x ∗ y ∗ )∗ ∈ S0 (τ ). It remains to show that .S0 (τ ) is absolutely solid in .S (τ ). To this end, first observe that, if .a ∈ S0 (τ ) is self-adjoint and .a ≥ 0, then .a 1/2 ∈ S0 (τ ). Indeed, for .λ > 0, √

e

.

a

(λ, ∞) = ea

( ) ({ }) √ μ ≥ 0 : μ > λ = e a λ2 , ∞

( √ ) and so, .τ e a (λ, ∞) < ∞ for all .λ > 0. Now suppose that .x ∈ S (τ ) and .y ∈ S0 (τ ) such that .|x| ≤ |y|. It follows from Proposition 2.2.24 (ii) that there exists .z ∈ M satisfying .|x|1/2 = z |y|1/2 . Since .|y| ∈ S0 (τ ), it follows from the above observation that .|y|1/2 ∈ S0 (τ ) and so, .|x|1/2 ∈ S0 (τ ). Since .S0 (τ ) is a subalgebra of .S (τ ) , it follows that .|x| ∈ S0 (τ ) and hence, .x ∈ S0 (τ ). u n

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79

The real linear subspace of .S0 (τ ) consisting of all self-adjoint elements in .S0 (τ ) is denoted by .S0,h (τ ). If .a ∈ S0,h (τ ), then .|a| ∈ S0,h (τ ) and .0 ≤ a ± ≤ |a|. Hence, since .S0 (τ ) is absolutely solid in .S (τ ), it is clear that .a + , a − ∈ S0,h (τ ). Consequently, with respect to the ordering inherited from .Sh (τ ), the space .S0,h (τ ) is a partially ordered vector space with generating positive cone .S0 (τ )+ . The next result is concerned with the functional calculus for elements .a ∈ S0,h (τ ). For this purpose, define } { B0 (σ (a)) = f ∈ B (σ (a)) : lim f (λ) = 0 .

.

λ→0

Note that .B0 (σ (a)) is a .∗-algebra with respect to pointwise operations and complex conjugation. Proposition 2.4.5 If .a ∈ S0,h (τ ), then .f (a) ∈ S0 (τ ) for all .f ∈ B0 (σ (a)). The map .f − | → f (a) is a .∗-homomorphism from .B0 (σ (a)) into .S0 (τ ). Proof Let .f ∈ B0 (σ (a)) be given. Proposition 2.1.4 (v) implies that .f (a) ηM. to show that .f (a) ∈ Moreover, .f (a) is closed and densely ( defined. Therefore, ) S0 (τ ), it is sufficient to prove that .τ e|f (a)| (λ, ∞) < ∞ for all .λ > 0. Given .λ > 0, observe that e|f (a)| (λ, ∞) = e|f |(a) (λ, ∞) = ea ({μ ∈ σ (a) : |f (μ)| > λ}) .

.

Since .limμ→0 f (μ) = 0, there exists .α > 0 such that .|f (μ)| ≤ λ for all .μ ∈ σ (a) with .|μ| ≤ α. Consequently, ea ({μ ∈ σ (a) : |f (μ)| > λ}) ≤ ea ({μ ∈ σ (a) : |μ| > α})

.

= e|a| (α, ∞) ( ) ( ) and hence, .τ e|f (a)| (λ, ∞) ≤ τ e|a| (α, ∞) < ∞, as .a ∈ S0 (τ ). The properties of the functional calculus immediately imply that the map .f − | → f (a) is a .∗homomorphism from .B0 (σ (a)) into .S0 (τ ) . u n Remark 2.4.6 (i) If .τ (1) = ∞, then .0 ∈ σ (a) for all .a ∈ S0,h (τ ). Indeed, if .a ∈ S0,h (τ ) and .0 ∈ / σ (a), then there exists .λ > 0 such that .e|a| (λ, ∞) = 1 and so, necessarily .τ (1) < ∞. If .τ (1) < ∞, then .S0 (τ ) = S (τ ) = S (M) (see Remarks 2.4.3) and 2.2.21 shows that .f (b) ∈ S (M) for all .f ∈ B (σ (b)) and all .b ∈ Sh (M) whenever the von Neumann algebra .M is finite. (ii) Given .a ∈ S0,h (τ ) one may also ask for conditions on a function .f ∈ B (σ (a)) implying that .f (a) ∈ S (τ ). An inspection of the proof of Proposition 2.4.5 shows that it is sufficient to require that f is bounded on a neighborhood of the point 0.

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Example 2.4.7 The importance of the algebra .S0 (τ ) may be seen in the following example. Let H be a Hilbert space and let .M = B (H ), equipped with the standard trace .τ . Observe that .S (τ ) = B (H ), as already seen in Example 2.3.13 (a). It will now be shown that .S0 (τ ) = K (H ), the ideal of all compact operators on H . First, let .x ∈ K (H ) be given with polar decomposition .x = v |x|. Since .|x| = v ∗ x, it is clear that also .|x| ∈ K (H ). If x is a finite rank operator, then it follows immediately from Remark 2.4.3 that .x ∈ S0 (τ ). Therefore, it may be assumed that .Ran (x) (equivalently, .Ran (|x|)) is infinite dimensional. By the spectral theorem for compact ∞ self-adjoint .{φn } n=1 in H such that Σ∞operators, there exists an orthonormal system ∞

{λ } φ φ .|x| ξ = λ for all . ξ ∈ H , where . is the sequence of nonn n n n n=1 n=1 zero eigenvalues of .|x| (repeated according to multiplicity) satisfying .λn ↓ 0. Let .ε > 0 be given and .N ∈ N be such that .λN ≤ ε. If p is the projection onto ⊥ is the projection onto .span {φ , . . . , φ } and .span {φn : n ≥ N } ⊕ Ker (|x|), then .p n ( ⊥) Σ 1 φn n=N for all .ξ ∈ H and so, .||xp||B(H ) ≤ ε. This shows that .x ∈ S0 (τ ). Now that .x ∈ S0 (τ ). For every .ε > 0 there exists a projection p such ) ( assume ⊥ < ∞, that is, .p ⊥ is a finite rank projection, and .||xp|| that .τ p B(H ) ≤ ε, that is, || || ⊥ .||x − xp || ≤ ε. Hence, x can be approximated in the operator norm by finite B(H ) rank operators and so, .x ∈ K (H ). References: [94, 113].

2.5 The Measure Topology in S (τ ) This section discusses an important topology, the so-called measure topology, on the algebra .S (τ ) of all .τ -measurable operators, which turns .S (τ ) into a topological .∗ -algebra. As will be seen, this topology is Hausdorff and metrizable and .S (τ ) is complete. Let .M be a semi-finite von Neumann algebra equipped with a fixed faithful normal semi-finite trace .τ . As before, .S (τ ) = S (M, τ ) denotes the .∗-algebra of all .τ -measurable operators (see Definition 2.3.5 and Theorem 2.3.8). For .0 < ε, δ ∈ R, the neighborhood .V (ε, δ) is defined by setting } { ( ) V (ε, δ) = x ∈ S (τ ) : ∃ p ∈ P (M) such that ||xp||B(H ) ≤ ε, τ p⊥ ≤ δ .

.

As will be seen, the sets .{V (ε, δ) : ε, δ > 0} form a neighborhood base at zero for the measure topology (see Proposition 2.5.3). Note already that, if .x ∈ S (τ ), then .x ∈ V (ε, δ) if and only if .|x| ∈ V (ε, δ). Indeed, let .x = v |x| be the polar decomposition of x and suppose ) that .x ∈ V (ε, δ). There exists .p ∈ P (M) ( first such that .||xp||B(H ) ≤ ε and .τ p⊥ ≤ δ. Using that .|x| = v ∗ x, it follows that .

|| || |||x| p||B(H ) = ||v ∗ xp||B(H ) ≤ ||xp||B(H ) ≤ ε

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and so, .|x| ∈ V (ε, δ). The proof of the converse implication is similar and, therefore, omitted. The following alternative description of the sets .V (ε, δ) will be useful. Lemma 2.5.1 If .0 < ε, δ ∈ R, then { ( ) } V (ε, δ) = x ∈ S (τ ) : τ e|x| (ε, ∞) ≤ δ .

.

( ) Proof First assume that .x ∈ S (τ ) is such that .τ e|x| (ε, ∞) ≤ δ. If .p = ( ) e|x| ([0, ε]), then .|||x| p||B(H ) ≤ ε and .τ p⊥ ≤ δ. Hence, .|x| ∈ V (ε, δ) and as observed above, this implies that .x ∈ V (ε, δ). Now assume that .x( ∈ )V (ε, δ). By definition, there exists .p ∈ P (M) such that ⊥ ≤ δ. It follows from Lemma 2.2.9 that .e|x| (ε, ∞) < p ⊥ .||xp||B(H ) ≤ ε and .τ p and so, ( ) ( ) |x| .τ e (ε, ∞) ≤ τ p⊥ ≤ δ, which completes the proof.

u n

The next proposition establishes some basic properties of the sets .V (ε, δ). Proposition 2.5.2 For all .ε, δ, εj , δj > 0 (.j = 1, 2) the following hold: (i) .V (ε1 , δ1 ) ⊆ V (ε2 , δ2 ) whenever .ε1 ≤ ε2 and .δ1 ≤ δ2 . (ii) .λV (ε, δ) = V (|λ| ε, δ) for all .λ ∈ C, .λ /= 0. (iii) The set .V (ε, δ) is balanced, that is, .λV (ε, δ) ⊆ V (ε, δ) for all .λ ∈ C satisfying .|λ| ≤ 1. (iv) The set .V (ε, δ) is absorbing, that is, for each .x ∈ S (τ ) there exists .0 ≤ λ0 ∈ R such that .x ∈ λV (ε, δ) whenever .λ ∈ C and .|λ| ≥ λ0 . (v) If .ε = min (ε1 , ε2 ) and .δ = min (δ1 , δ2 ), then .V (ε, δ) ⊆ V (ε1 , δ1 ) ∩ V (ε2 , δ2 ). (vi) .V (ε1 , δ1 ) + V (ε2 , δ2 ) ⊆ V (ε1 + ε2 , δ1 + δ2 ). (vii) .V (ε1 , δ1 ) · V (ε2 , δ2 ) ⊆ V (ε1 ε2 , δ1 + δ2 ). ∗ (viii) .V n(ε, δ) n = V (ε, δ). (ix) . ε>0 δ>0 V (ε, δ) = {0}. (x) The set .V (ε, δ) is absolutely solid, that is, if .x ∈ V (ε, δ) and .y ∈ S(τ ) satisfies .|y| ≤ |x|, then .y ∈ V (ε, δ). (xi) If .x ∈ V (ε, δ) and .y, z ∈ M, then .yxz ∈ ||y||B(H ) ||z||B(H ) V (ε, δ). Proof Assertions (i) and (ii) follow immediately from the definition of the sets V (ε, δ) and (iii) is an immediate consequence of (i) and (ii). Also, (v) is clear from (i).

.

(iv) Let .x ∈ S)(τ ) be given. It follows from Proposition 2.3.6 (v) that ( |x| .τ e (α, ∞) → 0 as .α → ∞ . Hence, there exists .α0 > 0 such ( ) that .τ e|x| (α0 , ∞) ≤ δ and so, with .p = e|x| ([0, α0 ]), it is clear that

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( ) |||x| p||B(H ) ≤ α0 and .τ p⊥ ≤ δ. This implies that .x ∈ V (α0 , δ) and hence, .(ε/α0 ) x ∈ V (ε, δ). This shows that .x ∈ λ0 V (ε, δ), with .λ0 = α0 /ε. Since the set .V ((ε, δ) )is balanced, it is now clear that .V (ε, δ) || is absorbing. || (vi) Given .xj ∈ V εj , δj , there exists .pj ∈ P (M) such that .||xpj ||B(H ) ≤ εj ( ) and .τ pj⊥ ≤ δj (.j = 1, 2). Defining .p = p1 ∧ p2 , it follows that .

( ) ( ) ( ) ( ) τ p⊥ = τ p1⊥ ∨ p2⊥ ≤ τ p1⊥ + τ p2⊥ ≤ δ1 + δ2

.

and .(x1 + x2 ) p = x1 p + x2 p = x1 p1 p + x2 p2 p, which implies that .

||(x1 + x2 ) p||B(H ) ≤ ||x1 p1 ||B(H ) + ||x2 p2 ||B(H ) ≤ ε1 + ε2 .

Hence, .x1 + x( 2 ∈ V)(ε1 + ε2 , δ1 + δ2 ). || || (vii) Let .xj ∈ V εj , δj and .pj ∈ P (M) be such that .||xpj ||B(H ) ≤ εj and ( ) ) ( ⊥ ≤ δ (.j = 1, 2). If .q = n p ⊥ x , then .p ⊥ x q = 0 and so, .x q = .τ p j 2 j 1 2 1 2 p1 x2 q. Furthermore, ) ( ) ( q ⊥ = s p1⊥ x2 ∼ r p1⊥ x2 ≤ p1⊥

.

( ) ( ) and hence, .τ q ⊥ ≤ τ p1⊥ . Defining .p = q ∧ p2 , it follows that ( ) ( ) ( ) ( ) τ p⊥ = τ q ⊥ ∨ p2⊥ ≤ τ q ⊥ + τ p2⊥ ≤ δ1 + δ2 .

.

Moreover, since x1 x2 p = x1 x2 qp = x1 p1 x2 qp = x1 p1 x2 p2 p ∈ M,

.

it follows that .

||x1 x2 p||B(H ) ≤ ||x1 p1 ||B(H ) ||x2 p2 ||B(H ) ≤ ε1 ε2

and so, .x1 x2 ∈ V (ε1 ε2 , δ1 + δ2 ). (viii) It will be convenient to start with a simple observation. If .v ∈ M is a partial isometry and if .p ∈ P (M) satisfies .p ≤ v ∗ v, then .τ (vpv ∗ ) = τ (p). Indeed, ) ( ) ( ( ) ) ( τ vpv ∗ = τ (vp) (vp)∗ = τ (vp)∗ (vp) = τ pv ∗ vp = τ (p) .

.

( ) Now let .x ∈ V (ε, δ) be given, that is, .τ e|x| (ε, ∞) ≤ δ (see Lemma 2.5.1). If .x = v |x| is the polar decomposition of x, then the spectral measure of the

2.5 The Measure Topology in S (τ )

83

operator .|x ∗ | is given by ( ) ∗ e|x | (B) = ve|x| (B \ {0}) v ∗ + 1 − v ∗ v δ0 (B) ,

.

B ∈ B (R) .

∗ In particular, .e|x | (ε, ∞) = ve|x| (ε, ∞) v ∗ . Since .e|x| (ε, ∞) ≤ observation may be applied to e|x| (0, ∞) = s (|x|) = v ∗ v, the above ( ) ( ) |x| (ε, ∞). This implies that .τ e|x ∗ | (ε, ∞) = τ e|x| (ε, ∞) and .p = e ( ∗ ) hence, .τ e|x | (ε, ∞) ≤ δ. Using Lemma 2.5.1 once again, it follows that ∗ .x ∈ V (ε, δ). (ix) Suppose 2.5.1) this implies ( that .x ∈) V (ε, δ) for all .ε, δ > 0. By Lemma ( that .τ e|x| (ε, ∞) ≤ δ for all .ε, δ > 0. Hence, .τ e|x| (ε, ∞) = 0, and so |x| (ε, ∞) = 0, for all .ε > 0. Therefore, .||x|| .e B(H ) ≤ ε for all .ε > 0, from which it follows that .x = 0. (x) Suppose that .x ∈ V (ε, δ) and .y ∈ S (τ ) satisfies .|y| ≤ |x|. It follows from Proposition 2.2.24 (ii) that there exists .z ∈ M such that .|y|1/2 = z |x|1/2 ) 1/2 (√ ε, ∞ = e|x| (ε, ∞), it follows from and .||z||B(H ) ≤ 1. Since .e|x| ) (√ ) ( √ ε, δ . By the definition of .V ε, δ) , this Lemma 2.5.1 that .|x|1/2 ∈ V ( ) (√ ) 1/2 (√ 1/2 1/2 |y| ε, ∞ ≤ δ. = z |x| ∈V ε, δ and so, .τ e implies that .|y| ( ) Hence, .τ e|y| (ε, ∞) ≤ δ and so .y ∈ V (ε, δ). (xi) From the definition of the sets .V (ε, δ) it is clear that

yx ∈ ||y||B(H ) V (ε, δ) ,

.

x ∈ V (ε, δ) , y ∈ M.

By (viii), this implies that .(yx)∗ ∈ ||y||B(H ) V (ε, δ) and hence, .z∗ (yx)∗ ∈ )∗ ( ||y||B(H ) ||z||B(H ) V (ε, δ). Since .yxz = z∗ (yx)∗ , the desired conclusion follows from (viii). u n Proposition 2.5.3 The collection .V = {V (ε, δ) : ε, δ > 0} is a neighborhood base at 0 for a metrizable Hausdorff vector space topology .Tm on .S (τ ). Moreover, .S (τ ) is a topological .∗-algebra with respect to this topology. Proof Since .0 ∈ V (ε, δ) for all .ε, δ > 0, all sets in .V are non-empty and so it follows from (v) of Proposition 2.5.2 that .V is a filter base. It then follows from (iii), (iv), and (vi) of the same proposition that there exists a unique vector space topology .Tm on .S (τ ) for which .V is a neighborhood base at 0. By (ix), this topology is Hausdorff. Furthermore, (vii) implies that multiplication in .S (τ ) is jointly continuous and from (viii) it is clear that the involution is also continuous. Hence, .S (τ ) is a topological .∗-algebra with respect to the topology .Tm . Finally, it follows from (i) that the countable subcollection .{V (1/n, 1/n)}∞ n=1 is also a neighborhood base for .Tm at 0 and hence, .Tm is a metrizable topology. u n The topology .Tm introduced in the above proposition is called the measure topology on .S (τ ). If a sequence .{xn }∞ n=1 in .S (τ ) is convergent with respect to .Tm

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2 Measurable Operators

Tm to the operator .x ∈ S (τ ), this is denoted by .xn → x, and the sequence .{xn }∞ n=1 is said to converge to x in measure (and the same notation and terminology are used when dealing with nets instead of sequences). It should be pointed out that the measure topology is, in general, not locally convex and does not have bounded neighborhoods of 0 (in the sense of topological vector spaces). The following proposition exhibits one of the fundamental facts concerning the measure topology. In combination with Theorem 2.5.12, this implies that .S (τ ) is the completion of .M with respect to this topology. Proposition 2.5.4 The von Neumann algebra .M is dense in .S (τ ) with respect to the measure topology. Proof Given .x ∈ S (τ ), Proposition 2.3.3 implies that there ( ⊥ )exists a sequence {pn }∞ ∞. n=1 in .P (M) such that .pn (H ) ⊆ D (x) for all n and .τ pn →(0 as .n → ( )) Hence, .xpn ∈ M and .(x − xpn ) pn = 0 for all n. Hence, .x−xpn ∈ V 2−n , τ pn⊥ Tm for all n, which shows that .xpn → x as .n → ∞. Consequently, .M is dense in .S (τ ). u n

.

Remark 2.5.5 As observed in the above proposition, .M is dense in .S (τ ) with respect to the measure topology. If .a ∈ S (τ )+ , then a is actually the limit of an increasing sequence in .M+ . Indeed, defining .an = aea [0, n] for .n ∈ N, it is clear Tm that .an ∈ M+ and that the sequence .{an }∞ n=1 is increasing. To show that .an → a, let .ε, δ > 0 be given. It follows from Proposition 2.3.6 (v) that there exists .N ∈ N such that .τ (ea (n, ∞)) ≤ δ for all .n ≥ N. Since .(a − an ) ea [0, n] = 0 for all n, it Tm is clear that .a − an ∈ V (ε, δ) for all .n ≥ N. This shows that .an → a as .n → ∞. Remark 2.5.6 There are, of course, other possible choices of neighborhood bases for the measure topology at 0. For example, let D be a .τ -dense linear subspace of H and, for .ε, δ > 0, define } ( ) { ⊥ ≤ δ, p (H ) ⊆ D . .VD (ε, δ) = x ∈ S (τ ) : ∃ p ∈ P (M) , ||xp||B(H ) ≤ ε, τ p Observe that ) ( V ε, δ ' ⊆ VD (ε, δ) ⊆ V (ε, δ)

.

(2.11)

whenever .ε > 0 and .0 < δ ' < δ. Indeed, ( it)is obvious that .VD (ε, δ) ⊆ V (ε, δ). Suppose that .0 < δ ' < δ and .x ∈ V ε, δ ' . There exists .p ∈ P (M) such that ( ⊥) ≤ δ ' . Since D is .τ -dense, there exists .q ∈ P (M) .||xp||B(H ) ≤ ε and .τ p ( ⊥) ≤ δ − δ ' . Defining .e = p ∧ q, it follows that such that .q (H ) ⊆ D and .τ q ( ⊥) ( ) .e (H ) = eq (H ) ⊆ D, .||xe||B(H ) = ||xpe||B(H ) ≤ ε and .τ e = τ p⊥ ∨ q ⊥ ≤ ( ) ( ) τ p⊥ + τ q ⊥ ≤ δ. Therefore, .x ∈ VD (ε, δ) and (2.11) holds. It now follows immediately that the sets .VD (ε, δ), .ε, δ > 0, form a neighborhood base at 0 for the measure topology.

2.5 The Measure Topology in S (τ )

85

Another choice of neighborhood base at zero may be found in 2.5.17. Some simple characterizations of convergence in measure are given in the next proposition. Proposition 2.5.7 For a sequence .{xn }∞ n=1 the following statements are equivalent: Tm (i) .xn → 0 as .n → ∞. Tm (ii) .|x(n | → 0 as .n)→ ∞. |x (iii) .τ e n | (ε, ∞) → 0 as .n → ∞ for all .ε > 0. Proof As observed in the remarks preceding Lemma 2.5.1, if .x ∈ S (τ ) and .ε, δ > 0, then .x ∈ V (ε, δ) if and only if .|x| ∈ V (ε, δ). This implies immediately that statements (i) and (ii) are equivalent. For the proof of the equivalence of (i) and (iii), only observe that, given .ε, δ > 0 and .N ∈ N, the statement .xn ∈ V (ε, δ) for all .n ≥ N is equivalent to the assertion ( |x ) | .τ e n (ε, ∞) ≤ δ for all .n ≥ N. u n An immediate and simple consequence of the above proposition is given in the next corollary. Tm Corollary 2.5.8 If .{pn }∞ n=1 is a sequence in .P (M), then .pn → 0 if and only if .τ (pn ) → 0 as .n → ∞. Proof Since the spectral measure of .pn is given by .epn (B) = pn⊥ δ0 (B)+pn δ1 (B), p p .B ∈ B (R), it follows that .e n (ε, ∞) = pn if .0 < ε < 1 and .e n (ε, ∞) = 0 for all .ε ≥ 1. Therefore, the statement of the corollary follows from Proposition 2.5.7 (iii). u n Remark 2.5.9 Proposition 2.5.7 (iii) has other useful consequences. By way of Tm 1/2 Tm + example, if .{an }∞ → 0. n=1 is a sequence in .S (τ ) such that .an → 0, then .an Indeed, ( ) 1/2 ean (ε, 0) = ean ε2 , ∞ ,

.

ε > 0,

from which the result is then evident. Further results of this nature may be found in Sect. 2.8 below. Some simple properties of the measure topology are presented in the next proposition. Proposition 2.5.10 ∞ (i) If .ε, δ > 0 and .{εn }∞ .{δn } n=1 , n n=1 are two sequences in .R satisfying .εn ↓ ε and ∞ .δn ↓ δ as .n → ∞, then . n=1 V (εn , δn ) = V (ε, δ). (ii) For any .ε, δ > 0, the neighborhood .V (ε, δ) is closed with respect to the measure topology. ( ) (iii) The embedding of . M, ||·||B(H ) into .(S (τ ) , Tm ) is continuous.

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2 Measurable Operators

(iv) The unit ball .BM of .M is closed in .S (τ ) with respect to the measure topology. (v) The map .x − | → Rex, .x ∈ S (τ ), is continuous and set .Sh (τ ) is closed with respect to the measure topology. Proof

n (i) From (i) of Proposition 2.5.2 it is clear that .V (ε, δ) ⊆ ∞ n=1 V (εn , δn ). For the proof of the converse inclusion, suppose that .x ∈ V (εn , δn ) for all .n ≥ ( ) 1. It follows from Lemma 2.5.1 that .τ e|x| (εn , ∞) ≤ δn for all n. Since ( ) ( ) |x| (ε , ∞) ↑ e|x| (ε, ∞), and so .τ e|x| (ε , ∞) ↑ τ e|x| (ε, ∞) as .n → ∞, .e n n ( |x| ) this implies that .τ e (ε, ∞) ≤ δ. Hence, .x ∈ V (ε, δ). (ii) Let .{xn }∞ n=1 be a sequence in .V (ε, δ) and suppose that .x ∈ S (τ ) is such Tm that .xn → x as .n(→ ∞. ) By passing to a subsequence, it may be assumed 1 1 that .x − xn ∈ V n , n for all n. For each n , there exists .pn ∈ P (M) ( ) such that .||xn pn ||B(H ) ≤ ε and .τ pn⊥ ≤ δ. Similarly, for each n, there exists ( ⊥) .qn ∈ P (M) such that .||(x − xn ) qn ||B(H ) ≤ 1/n and .τ qn ≤ 1/n. Setting .en = pn ∧ qn , it follows that ( ) ( ) ( ) ( ) τ en⊥ = τ pn⊥ ∨ qn⊥ ≤ τ pn⊥ + τ qn⊥ ≤ δ + 1/n

.

and .

||xn en ||B(H ) = ||xn pn en ||B(H ) ≤ ε,

||(x − xn ) en ||B(H ) = ||(x − xn ) qn en ||B(H ) ≤ 1/n. This implies that .

||xen ||B(H ) ≤ ||xn en ||B(H ) + ||(x − xn ) en ||B(H ) ≤ ε + 1/n

) ( and so, .x ∈ V ε + n1 , δ + n1 for all .n ≥ 1. It follows from (i) that .x ∈ V (ε, δ). (iii) Note that .

{ } n x ∈ M: ||x||B(H ) ≤ ε = V (ε, δ) ,

(2.12)

δ>0

for all .ε > 0. Indeed, if .x ∈ M with .||x||B(H ) ≤ ε, then .e|x| (ε, ∞) = 0 and so, it follows from Lemma 2.5.1 that .x ∈ V (ε, δ) for all( .δ > 0. Conversely, ) if .x ∈ S (τ ) satisfies .x ∈ V (ε, δ) for all .δ > 0, then .τ e|x| (ε, ∞) ≤ δ for ( ) all .δ > 0, which implies that .τ e|x| (ε, ∞) = 0, that is, .e|x| (ε, ∞) = 0. Hence, .|x| ∈ M and .|||x|||B(H ) ≤ ε, equivalently, .x ∈ M and .||x||B(H ) ≤ ε.

2.5 The Measure Topology in S (τ )

87

Therefore, (2.12) holds and this immediately shows that any sequence .{xn }∞ n=1 Tm in .M satisfying .||xn ||B(H ) → 0 as .n n → ∞, also satisfies .xn → 0. (iv) It follows from (2.12) that .BM = δ>0 V (1, δ) and so, by (ii), .BM is closed for the measure topology. (v) Since .Rex = 12 (x + x ∗ ) and the map .x |−→ x ∗ is continuous, it is evident that the map .x |−→ Rex, .x ∈ S (τ ), is continuous with respect to the measure topology. Moreover, since .Sh (τ ) = {x ∈ S (τ ) : x = Rex}, it is now also clear that .Sh (τ ) is closed. u n The next objective is to show that .S (τ ) is a complete metric space with respect to the measure topology. For this purpose, the next lemma will be useful. ( −n −n ) ∞ Lemma 2.5.11 If .{aΣ for n }n=1 is a sequence in .Mh satisfying .an ∈ V 2 , 2 ∞ all n, then the series . n=1 an is convergent in .S (τ ). Proof Defining .pn = ( e|a)n | [0, 2−n ], it follows from Lemma 2.5.1 that −n and .τ p ⊥ ≤ 2−n . Moreover, it is also clear that .a p = p a . .||an pn ||B(H ) ≤ 2 n n n n n Σ∞ The series . n=1 an pn is norm convergent in .M and hence, by Proposition 2.5.10 (iii), Σ it is convergent with respect to the measure topology. It remains to show ⊥ that . ∞ in .S (τ ). For this purpose, define the n=1 an pn is convergent in measure A p projections .qn ∈ P (M) by .qn = ∞ k=n+1 k for all .n = 1, 2, . . .. Observe that .qn ↑ and ( ) ⊥ =τ .τ qn

(

∞ V

) pk⊥

≤

k=n+1

∞ Σ

∞ ( ) Σ τ pk⊥ ≤ 2−k = 2−n .

k=n+1

k=n+1

In particular, .qn ↑ 1. Furthermore, note that .qn pk = pk and so, .qn pk⊥ = 0 for all .k > n. Define the linear operator .b : D → H by D=

∞ U

.

Ran (qn )

n=1

and bξ =

∞ Σ

ak pk⊥ ξ, ξ ∈ D.

(2.13)

ak pk⊥ ξ, ξ ∈ Ran (qn ) ,

(2.14)

.

k=1

Observe that bξ =

n Σ

.

k=1

so .bξ is well defined for all .ξ ∈ D by (2.13). Observe that this operator is .τ -premeasurable. Indeed, from the definition of .D, it is clear that .D is .τ -dense in H . If

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2 Measurable Operators

( ) u ∈ U M' and .ξ ∈ D, then .ξ = qn ξ for some n and so .uξ = uqn ξ = qn uξ , hence .uξ ∈ D. Moreover, using the fact that u commutes with all operators .ak pk⊥ , it follows that{.buξ }∞= ubξ . Hence, b is affiliated with .M. To show that b is closable, suppose that . ξj j =1 is a sequence in .D satisfying .ξj → 0 and .bξj → η as .j → ∞ for some .η ∈ H . Since

.

qn bξj =

∞ Σ

.

qn ak pk⊥ ξj =

k=1

∞ Σ

qn pk⊥ ak pk⊥ ξj =

k=1

n Σ

qn ak pk⊥ ξj

k=1

for all .n = 1, 2, . . ., it is clear that .qn bξj → 0 as .j → ∞. On the other hand, qn bξj → qn η as .j → ∞ and so, .qn η = 0 for all n. Since .qn ↑ 1, this implies that ¯ the closure of b, it is now clear that .η = 0 and hence, b is closable. Denoting by .b Σ ⊥ ¯ ∈ S (τ ). It now follows that .b¯ = ∞ .b a p k k=1 k , where the series converges in the ¯ n = bqn = measure topology. Indeed, for each .n ≥ 1, it follows from (2.14) that .bq Σ n ⊥ and so, a p k=1 k k .

( .

b¯ −

n Σ

) ak pk⊥

qn = 0.

k=1

) ( Σ This implies that .b¯ − nk=1 ak pk⊥ ∈ V 2−n , 2−n for all n, from which the desired conclusion follows. The proof is complete. u n Theorem 2.5.12 The space .S (τ ) is a complete metric space with respect to the measure topology. Proof Assume first that .{xn }∞ n=1 is a sequence in .M, which is Cauchy for the measure topology, that is, for any .ε, δ > 0 there exists .N ∈ N such that .xn − xm ∈ V (ε, δ) for all .n, m ≥ N . From part (iv) of Proposition 2.5.10, it follows that ∞ ∞ .{Rexn } n=1 and .{Imxn }n=1 are both Cauchy sequences. Therefore, for the proof that ∞ the sequence .{xn }n=1 is convergent, it may be assumed that .xn∗ = xn for ( all n. By) passing to a subsequence, it may be assumed further that .xn+1 − xn ∈ V 2−n , 2−n for all .n ≥ 1. Σ Setting .an = xn+1 − xn for all .n = 1, 2, . . ., Lemma 2.5.11 shows that the series . ∞ n=1 an is convergent in the measure topology. Σ∞ This implies that the sequence .{xn }∞ n=1 an ). n=1 is convergent (with limit given by .x1 + Now assume that .{yn }∞ n=1 is an arbitrary Cauchy sequence for the measure topology in .S (τ ). Proposition 2.5.4 implies that .M( is dense)in .S (τ ) and so, for each n, there exists .xn ∈ M such that .yn − xn ∈ V 2−n , 2−n . It is easily verified that .{xn }∞ n=1 is a Cauchy sequence and so, by the first part of the proof, there exists Tm Tm .x ∈ S (τ ) such that .xn → x as .n → ∞. Evidently, .yn → x as .n → ∞ and so, the conclusion may be drawn that .S (τ ) is complete with respect to the measure u n topology. Recall from Definition 2.4.1 that .S0 (τ ) of .S (τ ) consists of all ( the .∗-subalgebra ) operators .x ∈ S (τ ) for which .τ e|x| (λ, ∞) < ∞ for all .λ > 0. Alternatively,

2.5 The Measure Topology in S (τ )

89

by Proposition 2.4.2, the elements .x ∈ S (τ ) belonging to .S0 (τ ) are characterized by the property that for ) .ε > 0 there exists a projection .p ∈ P (M) satisfying ( each ⊥ < ∞. In other words, an element .x ∈ S (τ ) belongs to .||xp||B(H ) ≤ ε and .τ p .S0 (τ ) if and only if for every .ε > 0 there exists .δ > 0 such that .x ∈ V (ε, δ). Proposition 2.5.13 The .∗-subalgebra .S0 (τ ) is closed in .S (τ ) with respect to the measure topology. Tm Proof Suppose that .{xn }∞ n=1 is a sequence in .S0 (τ ) and .x ∈ S (τ ) satisfying .xn → x as .n → ∞. Given .ε > 0, there exists .n ∈ N such that .x − xn ∈ V (ε/2, 1). Moreover, since .xn ∈ S0 (τ ), there exists .δ > 0 such that .xn ∈ V (ε/2, δ). Hence, it follows from Proposition 2.5.2 (vi) that x = (x − xn ) + xn ∈ V (ε/2, 1) + V (ε/2, δ) ⊆ V (ε, 1 + δ) ,

.

which shows that .x ∈ S0 (τ ).

u n

In combination with Theorem 2.5.12, the above proposition implies that .S0 (τ ), equipped with the measure topology, is a complete metric space. Recall the definition of (the two-sided ideal .F (τ ) as given in (2.9). If .x ∈ F (τ ), ) then .xn (x) = 0 and .τ n (x)⊥ = τ (s (x)) < ∞ and so, it is clear from Proposition 2.4.2 that .x ∈ S0 (τ ). Therefore, .F (τ ) ⊆ S0 (τ ) and since .S0 (τ ) is closed in .S (τ ), the closure of .F (τ ) is also contained in .S0 (τ ). In the next proposition it will be shown that this closure is actually equal to .S0 (τ ). Proposition 2.5.14 The closure of .F (τ ) in .S (τ ) with respect to the measure topology is equal to .S0 (τ ). Proof By the preceding discussion, it remains to be shown that .S0 (τ ) is contained in the closure of .F (τ ). First, let .0 ≤ a ∈ S0 (τ ) and define .an ∈ M by setting a .an = ae (1/n, n], .n ∈ N . It follows from .a ∈ S0 (τ ) that ( ) ( ) τ ea (1/n, n] ≤ τ ea (1/n, ∞) < ∞

.

and so, .an ∈ F (τ ) for all n. Since .||aea [0, 1/n]||B(H ) ≤ 1/n, it is clear that Tm Tm a a .ae [0, 1/n] → 0 as .n → ∞ (see Proposition 2.5.10 (iii)). Also, .ae (n, ∞) → 0 (see Remark 2.5.5). Writing a − an = aea [0, 1/n] + aea (n, ∞) ,

.

Tm it is now clear that .an → a. Suppose now that .x ∈ S0 (τ ) is arbitrary and let .x = v |x| be its polar decomposition. By the first part of the proof, there exists a sequence .{an }∞ n=1 in

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2 Measurable Operators

Tm F (τ ) such that .an → |x| as .n → ∞. Consequently, .{van }∞ n=1 is a sequence in Tm .F (τ ) such that .van → v |x| = x. n u .

Two Egorov type results for measure convergence now follow. Lemma 2.5.15 Suppose that the sequence .{xn }∞ in .S (τ ) is convergent in {n=1}∞ measure to .x ∈ S (τ ). There exists a subsequence . xnl l=1 and a sequence .{pk }∞ ( ⊥ ) k=1 ( ) in .P (M) satisfying .pk||((H ) ⊆ D) (x)∩D for all . l ≥ 1, . p ↑ 1 and . τ pk → 0 x nl k || as .k → ∞, such that .|| x − xnl pk ||B(H ) → 0 as .l → ∞ for all .k ≥ 1. n∞ Proof Defining .D = n=1 D (xn ) ∩ D (x), it follows from Proposition 2.3.4 that D is a .τ -dense linear subspace of H . As observed in Remark 2.5.6, the sets .VD (ε, δ), .ε, δ > 0, form a neighborhood base at 0 for the measure topology. { }∞ Therefore, there exists a subsequence . xnl l=1 of .{xn }∞ n=1 which satisfies .x − xnl ∈ ( −l −l−1 ) VD 2 , 2 || for all .l || ≥ 1 . For each l, there exists .ql ∈ P (M) such that ( ) ( ) .ql (H ) ⊆ D, .|| x − xnl ql || ≤ 2−l and .τ ql⊥ ≤ 2−l−1 . Define .pk ∈ P (M) B(H ) (V∞ ⊥ ) ( ) A ≤ by .pk = ∞ q . Evidently, .pk ↑ and, furthermore, .τ pk⊥ = τ l=k ql Σ∞ ( ⊥ )l=k l −k ≤ 2 τ q for all k. In particular, . p ↑ 1. Since . p ≤ q , it is k k k l=k l ||( ) || also clear that .pk (H ) ⊆ D for all .k ≥ 1. Moreover, .|| x − xnl pk ||B(H ) = ||( || ) || x − xn ql pk || ≤ 2−l whenever .l ≥ k. The proof is complete. u n l B(H ) Corollary 2.5.16 If .{xn }∞ n=1 is a sequence in{.S (τ}) which is convergent in measure ∞ to .x ∈ S (τ ), then there exists a subsequence . xnl l=1 and a .τ -dense subspace D of ( ) H such that .D ⊆ D xnl ∩ D (x) for all .l ≥ 1 and .xnl ξ → xξ as .l → ∞ for all .ξ ∈ D. { }∞ Proof Let the subsequence . xnl l=1 and the sequence .{pk }∞ k=1 in .P (M) be as in U∞ Lemma 2.5.15. Defining .D = Ran it is clear that D is .τ -dense and that (p ), k k=1 ( ) .D ⊆ D xnl ∩ D (x) for all .l ≥ 1. Given .ξ ∈ D, there exists .k ≥ 1 such that || ||( || || ) .ξ = pk ξ and so, .||xnl ξ − xξ || = || xnl − x pk ξ ||H → 0 as .l → ∞. u n H As was already observed in Remark 2.5.6, there are several possible choices of neighborhood bases at 0 for the measure topology in .S (τ ). Next, another useful choice of neighborhoods will be discussed. For .ε, δ > 0, define the subsets .U (ε, δ) of .S (τ ) by } { ( ) U (ε, δ) = x ∈ S (τ ) : ∃ p ∈ P (M) such that ||pxp||B(H ) ≤ ε, τ p⊥ ≤ δ .

.

Proposition 2.5.17 The inclusions V (ε, δ) ⊆ U (ε, δ) ⊆ V (ε, 2δ)

.

are valid for all .ε, δ > 0. Consequently, the sets .{U (ε, δ) , ε, δ > 0} are a neighborhood basis for the measure topology at 0.

2.5 The Measure Topology in S (τ )

91

Proof It is clear that .V (ε, δ) ⊆ U (ε,(δ). Let and let .p ∈ ) .x ∈ U (ε, δ)( be given ) P (M) be such that .||pxp||B(H ) ≤ ε, τ p⊥ ≤ δ. If .e = n p ⊥ x , then .p⊥ xe = 0 and so, .xe = pxe. From ( ) ( ) e⊥ = s p ⊥ x ∼ r p ⊥ x ≤ p ⊥

.

( ( ( ) )) ( ) it follows that .τ e⊥ = τ r p⊥ x ≤ τ p⊥ ≤ δ. If .q = p ∧ e, then ( ) ( ) ( ) ( ) τ q ⊥ = τ p⊥ ∨ e⊥ ≤ τ p⊥ + τ e⊥ ≤ 2δ.

.

Furthermore, .xq = xeq = pxeq = pxpq and so, .

||xq||B(H ) = ||pxpq||B(H ) ≤ ||pxp||B(H ) ≤ ε.

Hence, .x ∈ V (ε, 2δ) and so .U (ε, δ) ⊆ V (ε, 2δ).

u n

∞ .{xn } n=1

Corollary 2.5.18 A sequence in .S (τ ) converges in measure to .x ∈ S (τ ) if and only if for any .ε, δ > (0 there ) exist .N ∈ N and .p ∈ P (M) such that ⊥ ≤ δ for all .n ≥ N. .||p (x − xn ) p||B(H ) ≤ ε and .τ p Example 2.5.19 The measure topology may be illustrated by the following simple examples. (a) Let H be a Hilbert space and let .M = B (H ), equipped with the standard trace ( ⊥.τ) . As seen in Example 2.3.13 (a), .S (τ ) = M. Since .p ∈ P (M) and .τ p < 1 imply that .p = 1, it follows that { } V (ε, δ) = x ∈ M : ||x||B(H ) ≤ ε

.

for all .ε > 0 and .0 < δ < 1. Consequently, the measure topology in S (τ ) = M coincides with the operator norm topology in .M = B (H ). As observed in Example 2.4.7, .S0 (τ ) = K (H ), the ideal of all compact operators on H . Furthermore, .F (τ ) coincides with the ideal of finite rank operators (see Example 2.3.13). Consequently, in this case Proposition 2.5.14 is simply the familiar statement that each compact operator in .B (H ) is the norm limit of a sequence of finite rank operators. (b) Let { .(X, Σ, μ) be a}Maharam measure space, let .H = L2 (μ) and let .M = Mf : f ∈(L∞ )(μ) f, as defined in Example 2.1.5 (b), equipped with the trace given by .τ Mf = X f dμ, .0 ≤ f ∈ L∞ (μ). As observed in Example 2.3.13 (b), the mapping .f |−→ Mf is a .∗ -isomorphism from .S (μ) onto .S (τ ), where the .∗-algebra .S (μ) is defined by (2.10). If .{fn }∞ n=1 is a sequence in .S (μ), then Tm .Mfn → 0 in .S (τ ) if and only if for any .ε, δ > 0 there exist .N ∈ N and a || || projection .p = MχA , .A ∈ Σ, such that .||fn χA ||∞ = ||Mfn p||B(H ) ≤ ε and .

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2 Measurable Operators

( ) μ (X \ A) = τ p⊥ ≤ δ for all .n ≥ N, equivalently,

.

μ ({x ∈ X : |fn (x)| > ε}) ≤ δ,

.

n ≥ N.

Tm Consequently, .Mfn → 0 in .S (τ ) if and only if μ ({x ∈ X : |fn (x)| ≥ ε}) → 0 as n → ∞, ∀ ε > 0.

.

This shows that, via the .∗-isomorphism .f − | → Mf , the measure topology in S (τ ) corresponds to the usual topology of convergence in measure in the space .S (μ) with respect to the measure .μ. .

References: [93, 127].

2.6 Measure Topology and Order This section discusses the relation between the order in .Sh (τ ) and the measure topology. Recall from Proposition 2.5.10 (v) that .Sh (τ ) closed in .S (τ ). Proposition 2.6.1 (i) The positive cone .S (τ )+ is closed in .S (τ ) for the measure topology.

Tm (ii) If .{an }∞ n=1 is a sequence in .Sh (τ ) and .a, b ∈ Sh (τ ) are such that .an → a as .n → ∞ and .an ≤ b for all n, then .a ≤ b. (iii) If .{an }∞ n=1 is an increasing sequence in .Sh (τ ) and if .a ∈ Sh (τ ) is such that Tm .an → a as .n → ∞, then .an ↑ a. ∞ (iv) If .{xn }∞ n=1 and .{yn }n=1 are two sequences in .S (τ ) satisfying .|yn | ≤ |xn | for all ∞ n, and if .{xn }n=1 converges to 0 in measure, then .{yn }∞ n=1 also converges to 0 in measure. Proof + (i) Suppose that .a ∈ S (τ ) and that .{an }∞ n=1 is a sequence in .S (τ ) such that Tm .an → a as .n → ∞. Since .Sh (τ ) closed in .S (τ ) , it is clear that .a ∈ Sh (τ ). Defining .e = ea (−∞, 0], it is clear that .a − = −ae = e (−a) e, which implies Tm that .0 ≤ a − ≤ e (an − a) e, as .ean e ≥ 0 for all n. Since .e (an − a) e → 0, it follows from Proposition 2.5.2 (x) that .a − ∈ V (ε, δ) for all .ε, δ > 0. Hence, − = 0, that is, .a ∈ S (τ )+ . .a (ii) Since .an ≤ b is equivalent with .b − an ≥ 0, it follows immediately from (i) that .b − a ≥ 0, that is .a ≤ b. Tm (iii) Suppose that .an ↑ and .an → a as .n → ∞. Since .an ≤ am whenever .m ≥ n, it follows from (ii) that .an ≤ a. This holding for all n shows that a is an upper

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93

∞ bound of .{an }∞ n=1 . Assume that .b ∈ Sh (τ ) is an upper bound of .{an }n=1 , that is, .an ≤ b for all n. Applying (ii) once more, it follows that .a ≤ b. Therefore, a is the least upper bound of .{an }∞ n=1 . (iv) Proposition 2.5.2 (x) shows that, if .x, y ∈ S (τ ) and .|y| ≤ |x|, then .y ∈ V (ε, δ) whenever .x ∈ V (ε, δ). From this observation, the statement is clear. u n

The next result presents an additional relationship between the partial ordering in .Sh (τ ) and the measure topology. The proof requires several observations, which are stated for later reference as a separate lemma. Lemma 2.6.2

{ } (i) Let .ε, δ > 0 be given. Then . a ∈ M+ : τ (a) ≤ εδ ⊆ V (ε, δ). { } (ii) If . aβ is a decreasing net in .M+ and .a ∈ M+ such that .τ (a) < ∞ and Tm .a ≥ aβ ↓ 0 in .M, then .aβ → 0. + (iii) If .a ∈ S (τ ) and .p ∈ P (M), then .a ≤ 2 (pap + (1 − p) a (1 − p)). Proof (i) Take .a ∈ M+ such that .τ (a) ≤ εδ. Since .εea (ε, ∞) ≤ aea (ε, ∞) ≤ a, it is clear that ( ) ετ ea (ε, ∞) ≤ τ (a) ≤ εδ

.

and so, .τ (ea (ε, ∞)) ≤ δ. It follows from Lemma 2.5.1 that .a ∈ V (ε, δ). (ii) If .a ≥ aβ ↓ 0 in .M+ and .τ (a) < ∞, then by the normality of the trace, ( ) Tm .τ aβ ↓ 0 and by (i) this implies that .aβ → 0. (iii) Proposition 2.2.24 (iv) shows that .(p − (1 − p)) a (p − (1 − p)) ≥ 0, which in turn, implies that pa (1 − p) + (1 − p) ap ≤ pap + (1 − p) a (1 − p) .

.

Adding .pap + (1 − p) a (1 − p) to both sides of this inequality yields a = pap + (1 − p) a (1 − p) + pa (1 − p) + (1 − p) ap

.

≤ 2 (pap + (1 − p) a (1 − p)) . { } Theorem 2.6.3 If . aβ is a decreasing net in .S (τ )+ and .a ∈ S0 (τ )+ such that Tm .a ≥ aβ ↓ 0 in .Sh (τ ), then .aβ → 0. Proof Let .ε, δ > 0 be given. It has to be shown that there exists .β0 such that aβ ∈ V (ε, δ) for all .β ≥ β0 . Since a is .τ -measurable, Proposition 2.3.6 (v) implies that .τ (ea (λ, ∞)) → 0 as .λ → ∞ and so, there exists .λ > ε/4 such that .τ (ea (λ, ∞)) ≤ δ/2. Defining .p = ea (ε/4, λ], it is clear that .0 ≤ ap ≤ λea (ε/4, ∞) and so, .ap ∈ M+ . Moreover, .τ (ea (ε/4, ∞)) < ∞, as .a ∈ S0 (τ )+ ,

.

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which implies that .τ (ap) < ∞. It follows from Proposition 2.2.25 (iii) that ap = pap ≥ paβ p ↓ 0

.

Tm in .M. Since .τ (ap) < ∞, it follows from (ii) of Lemma 2.6.2 that .paβ p → 0. Hence, there exists .β0 such that .paβ p ∈ V (ε/4, δ/2) for all .β ≥ β0 . Furthermore, it follows from the definition of p that a (1 − p) ea [0, λ] = aea [0, ε/4]

.

and so, .||a (1 − p) ea [0, λ]||B(H ) ≤ ε/4. Since .τ (ea (λ, ∞)) ≤ δ/2, this shows that .a (1 − p) ∈ V (ε/4, δ/2). Using Proposition 2.5.2 (x), it follows from 0 ≤ (1 − p) aβ (1 − p) ≤ (1 − p) a (1 − p) = a (1 − p)

.

that .(1 − p) aβ (1 − p) ∈ V (ε/4, δ/2) for all .β. Using Proposition 2.5.2 (vi), it follows that paβ p + (1 − p) aβ (1 − p) ∈ V (ε/2, δ)

.

for all .β ≥ β0 . By Lemma 2.6.2 (iii), ( ) 0 ≤ aβ ≤ 2 paβ p + (1 − p) aβ (1 − p)

.

and so, it follows from Proposition 2.5.2 (x) and (ii) that .aβ ∈ V (ε, δ) for all .β ≥ β0 . u n Considering for instance the commutative case (see Example 2.5.19 (b)), it is easy to see that the assumption .a ∈ S0 (τ )+ cannot be omitted in the above theorem. Tm However, if .τ (1) < ∞, then .aβ ↓ 0 in .S (τ ) implies that .aβ → 0, as in this case .S (τ ) = S0 (τ ). An interesting consequence of Theorem 2.6.3 is presented in the next proposition. { } Proposition 2.6.4 Suppose that . aβ is a decreasing net in .S (τ )+ such that .aβ ↓ 0 Tm Tm in .Sh (τ ). If .x ∈ S0 (τ ), then .aβ x → 0 and .xaβ → 0. Proof Without loss of generality, it may be assumed that .0 ≤ aβ ≤ a for all .β for some .a ∈ S (τ )+ . Observe that Proposition 2.4.4 implies that .x ∗ ax ∈ S0 (τ )+ . Since .x ∗ aβ x ↓ 0 and .0 ≤ x ∗ aβ x ≤ x ∗ ax for all .β, it follows from Theorem 2.6.3 Tm that .x ∗ aβ x → 0. Since | ( )∗ ( ) | | 1/2 |2 1/2 1/2 aβ x = |aβ x | , x ∗ aβ x = aβ x

.

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| | | 1/2 |2 Tm this is equivalent to saying that .|aβ x | → 0. As observed in Remark 2.5.9, this | | | 1/2 | Tm 1/2 Tm implies that .|aβ x | → 0, and so, by Proposition 2.5.7, .aβ x → 0. Let .ε, δ > 0 be given. Since .τ (ea (λ, ∞)) → 0 as .λ → ∞, there exists .R > 0 such that .τ (ea (R, ∞)) ≤ δ/2, that is, .a ∈ V (R, δ/2) . Since .0 ≤ aβ ≤ a, it follows from Proposition 2.5.2 (x) that .aβ ∈ V (R, δ/2) for all .β. Observing that 1/2

eaβ (R, ∞) = eaβ

(√

.

) R, ∞ ,

(√ ) 1/2 Tm ∈ V R, δ/2 for all .β. Since .aβ x → 0, there exists ) ( √ 1/2 .β0 such that .a β x ∈ V ε/ R, δ/2 for all .β ≥ β0 . Hence, it follows from Proposition 2.5.2 (vii) that 1/2

this implies that .aβ

1/2 1/2

aβ x = aβ aβ x ∈ V

.

) (√ ) ( √ R, δ/2 V ε/ R, δ/2 ⊆ V (ε, δ)

)∗ ( Tm for all .β ≥ β0 . This shows that .aβ x → 0. Since .xaβ = aβ x ∗ and .x ∗ ∈ S0 (τ ), the second assertion is now also clear. u n As observed in Proposition 2.5.10 (ii), the neighborhoods .V (ε, δ) are closed. Another interesting and useful fact is that they are also closed for taking suprema of increasing nets of positive elements. For the proof of this result, it will be convenient to have the following lemma available. Lemma 2.6.5 Let .a ∈ S (τ ) be positive self-adjoint and let .ε > 0. If .p ∈ P (M) satisfies .p ≤ ea (ε, ∞), then .p < epap (ε, ∞). )( ( ) Proof For convenience, put .b = pap. Since .b = pa 1/2 a 1/2 p = | | ) ( ( 1/2 )∗ ( 1/2 ) a p |a p , it follows that .b1/2 = |a 1/2 p|. In particular, .D b1/2 = ( 1/2 ) (| 1/2 ) D |a p | = D a p . First, it will be shown that .p ∧ eb [0, ε] = 0. Let .q = p ∧ eb [0, ε] and suppose that .ξ ∈ H is such that .qξ = ξ /= 0. This implies that .ξ = eb [0, ε] ξ = ( ) ] ) ( 1/2 [ 0, ε1/2 ξ and so, .ξ ∈ D b1/2 = D a 1/2 p . Since .ξ = pξ , it is now eb ( 1/2 ) . Furthermore, since .p ≤ ea (ε, ∞), it follows that also clear that .ξ ∈ D a ( ) 1/2 a a ε1/2 , ∞ ξ and so, it follows from Lemma 2.2.7 (applied .ξ = e (ε, ∞) ξ = e || || to .a 1/2 ) that .||a 1/2 ξ ||H > ε1/2 ||ξ ||H . This implies that .

|| || || 1/2 || ||b ξ ||

H

||| | || ||| | || = |||a 1/2 p| ξ ||

H

|| || || || = ||a 1/2 pξ ||

H

|| || || || = ||a 1/2 ξ ||

H

> ε1/2 ||ξ ||H .

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2 Measurable Operators

On the other hand, || || || 1/2 || . ||b ξ ||

H

|| [ ] || 1/2 || || 0, ε1/2 ξ || = ||b1/2 eb

H

≤ ε1/2 ||ξ ||H ,

which is a contradiction. Therefore, .p ∧ eb [0, ε] = 0 and this implies that p ∼ p ∨ eb [0, ε] − eb [0, ε] ≤ 1 − eb [0, ε] = eb (ε, ∞) ,

.

that is, .p < eb (ε, ∞).

u n

Using this observation, it may be shown that the neighborhoods .V (ε, δ) are “locally determined” in the following sense. Lemma 2.6.6 Let .ε, δ > 0 be given. If .x ∈ S (τ ), then .x ∈ V (ε, δ) if and only if p |x| p ∈ V (ε, δ) for all .p ∈ P (M) with .τ (p) < ∞.

.

Proof If .x ∈ V (ε, δ), then it follows from Proposition 2.5.2 (xi) that .p |x| p ∈ For the V (ε, δ) for all .p ∈ P (M). ( ) proof of the converse implication, suppose that .x ∈ / V (ε, δ), that is, .τ e|x| (ε, ∞) > δ (see Lemma 2.5.1). Since the trace is semifinite, there exists .p ∈ P (M) such that .p ≤ e|x| (ε, ∞) and .δ( < τ (p) < )∞. It follows from Lemma 2.6.5 that .p < ep|x|p (ε, ∞) and so, .τ ep|x|p (ε, ∞) ≥ / V (ε, δ). u n τ (p) > δ, which shows that .p |x| p ∈ { } Theorem 2.6.7 Suppose that .ε, δ > 0, .a ∈ Sh (τ )+ and that . aβ is a net in + .Sh (τ ) such that .0 ≤ aβ ↑ a in .Sh (τ ). If .aβ ∈ V (ε, δ) for all .β, then .a ∈ V (ε, δ). Proof Take .p ∈ P (M) with .τ (p) < ∞ . It follows from Proposition 2.2.25 (iii) that .0 ≤ paβ p ↑ pap in .Sh (τ ). Since .τ (p) < ∞, Proposition 2.4.4 implies Tm that .pap ∈ S0 (τ )+ , and so, it follows from Theorem 2.6.3 that .paβ p → pap . Since .paβ p ∈ V (ε, δ) for all .β and .V (ε, δ) is closed for the measure topology (see Proposition 2.5.10 (ii)), it follows that .pap ∈ V (ε, δ). This holds for all .p ∈ P (M) with .τ (p) < ∞ and so, Lemma 2.6.6 implies that .a ∈ V (ε, δ) . u n Some properties of subsets which are bounded with respect to the measure topology will now be discussed. The following definition is the usual one in general topological vector spaces, specialized to the present setting. Definition 2.6.8 A non-empty subset W of .S (τ ) is called bounded with respect to the measure topology if for any .ε, δ > 0 there exists .λ > 0 such that .W ⊆ λV (ε, δ). Remark 2.6.9 The notion of boundedness with respect to the measure topology should not be confused with boundedness with respect to a (translation invariant) metric which induces this topology. Consider for example the space .L0 (0, 1) with respect to Lebesgue measure (cf. Example 2.5.19 (b)). The measure topology is

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97

induced by the translation invariant metric d given by f

1

d (f, g) =

.

0

|f (t) − g (t)| dt, f, g ∈ L0 (0, 1) . 1 + |f (t) − g (t)|

Since .d (f, g) ≤ 1 for all .f, g ∈ L0 (0, 1), the space .L0 (0, 1) is bounded with respect d. Evidently, .L0 (0, 1) is not bounded with respect to the measure topology. Since .λV (ε, δ) = V (λε, δ) for all .λ, ε, δ > 0, it is clear that a set .W ⊆ S (τ ) is bounded if and only if for every .δ > 0 there exists .R > 0 such that .W ⊆ V (R, δ). The definition of the neighborhoods .V (R, δ) and Lemma 2.5.1 yield immediately the following result. Lemma 2.6.10 For a non-empty subset W of .S (τ ) the following statements are equivalent: (i) W is bounded with respect to the measure topology.( ) (ii) For every .δ > 0, there exists .R > 0 such that .τ e|x| (R, ∞) ≤ δ for all .x ∈ W . (iii) Given .δ > 0, there exists .R > 0 such that ) every .x ∈ W there exists ( ⊥for ≤ δ. .p ∈ P (M) such that .||xp||B(H ) ≤ R and .τ p As is the case in any topological vector space, if a sequence .{xn }∞ n=1 is convergent, then it is bounded. In addition, the following simple observations should be noted. Proposition 2.6.11 ∞ ∞ (i) If .{xn }∞ n=1 and .{zn }n=1 are bounded sequences in .S (τ ) and .{yn }n=1 is a Tm Tm sequence in .S (τ ) such that .yn → 0, then .xn yn zn → 0. (ii) If .W ⊆ M is norm bounded, then W is bounded with respect to the measure topology.

Proof ∞ (i) Let .ε, δ > 0 be given. Since .{xn }∞ n=1 and .{zn }n=1 are bounded with respect to the measure topology, there exists .R > 0 such that .xn , zn( ∈ V (R, δ/3) for ) all n. Furthermore, there exists .N ∈ N such that .yn ∈ V ε/R 2 , δ/3 for all .n ≥ N. Hence, it follows from Proposition 2.5.2 (vii) that

( ) xn yn zn ∈ V (R, δ/3) V ε/R 2 , δ/3 V (R, δ/3) ⊆ V (ε, δ) ,

.

Tm for all .n ≥ N. This shows that .xn yn zn → 0. (ii) If .M > 0 is such that .||x||B(H ) ≤ M for all .x ∈ W , then .e|x| (M, ∞) = 0 for all .x ∈ W and so, .W ⊆ V (M, δ) for all .δ > 0. Hence, W is bounded for the measure topology. u n

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It will now be shown that any increasing net in .Sh (τ ) which is bounded for the measure topology has a supremum in .Sh (τ ) (see Theorem 2.6.15). The proof of this result is based on a number of observations. Lemma 2.6.12 If .a, b ∈ B (H ) are two commuting self-adjoint operators such that 0 ≤ a ≤ b, then .ea (λ, ∞) ≤ eb (λ, ∞) for all .λ ≥ 0.

.

Proof Note first that, since a and b commute, the spectral measures of a and b commute as well, that is, .ea (δ1 ) eb (δ2 ) = eb (δ2 ) ea (δ1 ) for all Borel sets .δ1 , δ2 ⊆ so R. Moreover, .ea (μ, ∞) → ea (λ, ∞) as .μ ↓ λ, and so it suffices to prove that b a .e [0, λ] e (μ, ∞) = 0 for all .μ > λ. If .0 ≤ λ < μ and .ξ ∈ H is such that b a .ξ = e [0, λ] e (μ, ∞) ξ , then || || ||2 || ||2 ||2 || || || || || || μ ||ξ ||2H ≤ ||a 1/2 ea (μ, ∞) ξ || = ||a 1/2 ξ || ≤ ||b1/2 ξ ||

.

H

|| ||2 || || = ||b1/2 eb [0, λ] ξ || ≤ λ ||ξ ||2H

H

H

H

and so, .ξ = 0. This shows that .eb [0, λ] ea (μ, ∞) = 0 and the proof is complete.

u n

A special case of the desired result now follows. + Lemma 2.6.13 If .{bk }∞ k=1 is an increasing sequence in .M of mutually commuting operators (that is, .bk bl = bl bk for all k and l) which is bounded with respect to the measure topology, then .supk bk exists in .Sh (τ ).

Proof Let .qk be the quadratic form corresponding to the operator .bk , that is, || || || 1/2 ||2 .qk (ξ ) = ||b ξ || for all .ξ ∈ H . Defining .q : D (q) → [0, ∞) by k H

{ } .D (q) = ξ ∈ H : sup qk (ξ ) < ∞ , k

q (ξ ) = sup qk (ξ ) = lim qk (ξ ) , ξ ∈ D (q) , k

k→∞

it follows from Lemmas 1.8.4 and 1.8.5 that .q is a closed quadratic form. Observe first that the domain .D (q) is .τ -dense in H . For this, it is necessary to be show that for ) .δ > 0 there exists a projection .p ∈ P (M) such that .p (H ) ⊆ ( every is bounded with respect D (q) and .τ p⊥ ≤ δ. So, let .δ > 0 be given. Since .{bk }∞ k=1 ( ) to the measure topology, there exists .R > 0 such that .τ ebk (R, ∞) ≤ δ for all b b .k = 1, 2, . . .. It follows from Lemma 2.6.12, that .e k (R, ∞) ≤ e k+1 (R, ∞) for V∞ b k all k and so, the projection .q A = k=1 e (R, ∞) satisfies ( ⊥ ) .τ (q) ≤ δ. Defining ∞ bk .p = 1 − q, it is clear that .p = ≤ δ. If .ξ ∈ p (H ), then k=1 e [0, R] and .τ p

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99

ξ = ebk [0, R] ξ and so,

.

|| || ||2 || || 1/2 || || 1/2 ||2 qk (ξ ) = ||bk ξ || = ||bk ebk [0, R] ξ || ≤ R ||ξ ||2H

.

H

H

for every .k ≥ 1, which implies that .supk qk (ξ ) ≤ R ||ξ ||2H < ∞. This shows that .p (H ) ⊆ D (q) and hence, .D (q) is .τ -dense. Since any .τ -dense subspace of H is in particular norm dense, it follows that .q is a densely defined closed quadratic form and so (see Proposition 1.8.7), there ) ( exists a unique positive self-adjoint operator a in H such that .D (q) = D a 1/2 || 1/2 ||2 and .q (ξ ) = ||a ξ || for all .ξ ∈ D (q). The operator a is the supremum of the H

+ sequence .{bk }∞ k=1 in the set .H of all positive self-adjoint operators in H . This implies in particular that a (and also .a 1/2 ( ) is) affiliated with .M (cf. the proof of Proposition 2.2.25). Moreover, since .D a 1/2 is .τ -dense, the operator .a 1/2 is .τ measurable and hence, a is also .τ -measurable. Consequently, .a = supk bk in .Sh (τ ). u n

For .k = 1, 2, . . ., the function .Yk on .[0, ∞) is defined by setting Yk (λ) =

.

kλ , λ ≥ 0. λ+k

Given .a ∈ S (τ )+ , define f

Yk (λ) dea (λ) = ka (a + k1)−1 .

Yk (a) =

.

[0,∞)

The sequence .{Yk (a)}∞ k=1 is usually called the Yosida approximation of the operator a. The following simple observations are needed in the proof of the next theorem. Lemma 2.6.14 Given .a, b ∈ S (τ )+ , the following statements hold: (i) (ii) (iii) (iv) (v)

Yk (a) ∈ M and .0 ≤ Yk (a) ≤ k1. 0 ≤ Yk (a) ≤ Yk+1 (a) for all .k ≥ 1. .Yk (a) → a in measure as .k → ∞. .Yk (a) ↑k a in .Sh (τ ). If .a ≤ b, then .Yk (a) ≤ Yk (b). . .

Proof (i) Since .0 ≤ Yk (λ) ≤ k for all .λ ≥ 0, it follows from the properties of spectral integrals that .Yk (a) ∈ B (H ) and .0 ≤ Yk (a) ≤ k1. Moreover, .Yk (a) is affiliated with .M and so, it is clear that .Yk (a) ∈ M. (ii) Writing ( .Yk (λ) = λ 1 −

λ λ+k

) ,

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it is clear that .0 ≤ Yk (λ) ≤ Yk+1 (λ) for all .λ ≥ 0. Hence, from the properties of the functional calculus it follows that .0 ≤ Yk (a) ≤ Yk+1 (a) for all .k ≥ 1. (iii) If .k ≥ 1, then a − Yk (a) = a − ka (a + k1)−1 = [a (a + k1) − ka] (a + k1)−1

.

= a 2 (a + k1)−1 . || || Since .||(a + k1)−1 ||B(H ) ≤ k −1 , it is clear that .(a + k1)−1 → 0 in norm in .B (H ) and hence, in measure in .S (τ ) as .k → ∞. Consequently, a 2 (a + k1)−1 → 0 in measure as k → ∞.

.

(iv) Since .0 ≤ Yk (a) ↑k in .Sh (τ ) and .Yk (a) → a in measure as .k → ∞, it follows from Proposition 2.6.1 (iii) that .a = supk Yk (a) in .Sh (τ ). −1 (v) Observing that .0 ≤ a ≤ b implies .(b + k1) ≤ (a + k1)−1 ) (cf. the proof of ( Theorem 2.2.27) and writing .Yk (a) = k 1 − k (a + k1)−1 , it is evident that .Yk (a) ≤ Yk (b). u n { } Theorem 2.6.15 If . aβ is an increasing net in .Sh (τ ) which is bounded with respect to the measure topology, then .supβ aβ exists in .Sh (τ ). { } Proof Without loss of generality, it may be assumed that . aβ is an increasing net { ( )} in .S (τ )+ which is bounded in measure. For each .k ≥ 1, consider the net . Yk aβ β { ( )} in .M. It follows from Lemma 2.6.14 (v) that . Yk aβ β is an increasing net in .M+ . ( ) ( ) Lemma 2.6.14 (i) implies that .0 ≤ Yk aβ ≤ k1 for all .β and so, .bk = supβ Yk aβ { ( )} exists in .M (see Theorem 1.11.2). Moreover, .bk is the strong limit of . Yk aβ β , ( ) that is, .bk ξ = limβ Yk( aβ) ξ for all( .ξ )∈ H . Evidently, .0 ≤ bk ≤ k1. Since, by (ii) of Lemma 2.6.14, .Yk aβ ≤ Yk+1 aβ for all .β, it is also clear that .bk ≤ bk+1 for all k. ( ) so ( ) so Note that .bk bl = {bl bk( for)}all .k, l {≥ 0. .Yk aβ → bk and .Yl aβ → ( Indeed, )} bl . Since both nets . Yk aβ β and . Yl aβ β are norm bounded (by k and l, respectively), this implies that ( ) ( ) so Yk aβ Yl aβ → bk bl .

.

( ) ( ) so ( ) ( ) ( ) ( ) Similarly, .Yl aβ Yk aβ → bl bk . Since .Yk aβ Yl aβ = Yl aβ Yk aβ for all .β, it follows that .bk bl = bl bk . ∞ It will now be shown { } that the sequence .{bk }k=1 is bounded in measure. Let .δ > 0 be given. The net . aβ is bounded in measure ( ) and so, there exists .R > 0 such that .aβ ∈ V (R, δ) for all .β. Since .0 ≤ Yk aβ ( ≤ ) aβ (see (iv) of Lemma 2.6.14), It follows from Proposition 2.5.2 (x) that .Yk aβ ∈ V (R, δ) for all .β and all .k ≥ 1. ( ) Since .Yk aβ ↑β bk , an appeal to Theorem 2.6.7 yields that .bk ∈ V (R, δ) for all k. Hence, .{bk }∞ k=1 is bounded in measure.

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101

( ) According to Lemma 2.6.13, (.a =) supk bk exists in .Sh (τ ). Since .Yk aβ ≤ bk ≤ a for all .β and k, and since .Yk aβ ↑k aβ (by (iv) of Lemma 2.6.14), it is clear that( .aβ) ≤ a for all .β. Furthermore, if .c ∈ Sh (τ ) is such that .aβ ≤ c for all .β, then .Yk aβ ≤ aβ ≤ c for all .β and k, and so, .bk ≤ c for all k. This implies that .a ≤ c and therefore, it may be concluded that .aβ ↑ a in .Sh (τ ). The proof is complete. n u References: [97].

2.7 The Local Measure Topology As before, it will be assumed that .M is a semi-finite von Neumann algebra on a Hilbert space H , equipped with a fixed faithful normal semi-finite trace .τ . Recall that .S (M) and .S (τ ) are the .∗-algebras of all measurable and all .τ measurable operators, respectively. Furthermore, recall that the space .S0 (τ ), defined in Definition 2.4.1, is a two-sided ideal in .S (τ ) by Proposition 2.4.4. Lemma 2.7.1 If .x ∈ S (M) and .p ∈ P (M) satisfies .τ (p) < ∞, then the operators xp, px, and pxp belong to .S0 (τ ). Proof (Clearly, the ) operator xp is measurable. Moreover, .(1 − p) (H ) ⊆ D (xp) and .τ (1 − p)⊥ = τ (p) < ∞. Hence, it follows from Proposition 2.3.6 that .xp ∈ S (τ ). It is now also evident that .pxp ∈ S (τ ). Furthermore, if .x ∈ S (M), then .x ∗ ∈ S (M) and so, by what just has been proved, .x ∗ p ∈ S (τ ) and hence, ∗ ∗ .px = (x p) ∈ S (τ ). Since .p ∈ S0 (τ ) and .S0 (τ ) is a two-sided ideal in .S (τ ), it u n is also clear that .xp, px, pxp ∈ S0 (τ ) . Recall, Sect. 2.5, that for .ε, δ > 0, the set .V (ε, δ) consists of all .x ∈ S((τ ))for which there exists a projection .p ∈ P (M) satisfying .||xp||B(H ) ≤ ε and .τ p⊥ ≤ δ. For .0 < ε, δ ∈ R and .e ∈ P (M) with .τ (e) < ∞, the subset .V (ε, δ, e) of .S (M) is defined by setting V (ε, δ, e) = {x ∈ S (M) : exe ∈ V (ε, δ)} .

.

(2.15)

It will now be shown that these sets form a base at zero for a vector space topology in .S (M), called the local measure topology. The following lemma will be helpful. Lemma 2.7.2 If .x ∈ S (M) is such that .pxp = 0 for all .p ∈ P (M) satisfying τ (p) < ∞, then .x = 0.

.

Proof Since .pxp = 0 implies that .px ∗ p = 0, it follows that also .pRe (x) p = 0 and .pIm (x) p = 0. Therefore, without loss of generality, it may be assumed that ∗ x .x = x . For .n ∈ Z, define .en = e ([n, n + 1)) and set .xn = xen . Fix .n ∈ Z and let .pα ∈ P (M) be such that .pα ↑α en and .τ (pα ) < ∞ for all .α (this is possible as the trace is semi-finite). Since .pα = en pα , it follows that .pα xn pα = pα xen pα = so pα xpα = 0 for all .α. Furthermore, .pα ↑ en implies that .pα → en . Since .xn ∈ M

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and multiplication is jointly so-continuous on norm bounded sets, it follows that so pα xn pα → en xn en = xn . Hence, .xn = 0. This holds for all .n ∈ Z and so .x = 0. u n

.

The next proposition collects some properties of the sets .V (ε, δ, e) (cf. Proposition 2.5.2). ( ) Proposition 2.7.3 For all .ε, εj , δ, δj > 0 and .e, ej ∈ P (M) with .τ (e) , τ ej < ∞ (.j = 1, 2), the following statements hold: (i) .V (ε1 , δ1 , e1 ) ⊆ V (ε2 , δ2 , e2 ) whenever .ε1 ≤ ε2 , .δ1 ≤ δ2 and .e2 ≤ e1 . (ii) If .ε = min (ε1 , ε2 ), .δ = min (δ1 , δ2 ) and .e = e1 ∨ e2 , then .V (ε, δ, e) ⊆ V (ε1 , δ1 , e1 ) ∩ V (ε2 , δ2 , e2 ). (iii) The set .V (ε, δ, e) is balanced and absorbing. (iv) .V (ε1 , δ1 , e) + V (ε2 , δ2 , e) ⊆ V (ε1 + ε2 , δ1 + δ2 , e). (v) .V (ε, δ, e)∗ = V (ε, δ, e). (vi) If .x ∈ V (ε, δ, e) for all .ε, δ > 0 and all .e ∈ P (M) with .τ (e) < ∞, then .x = 0. Proof (i) If .x ∈ V (ε1 , δ1 , e1 ), then .e1 xe1 ∈ V (ε1 , δ1 ) and so it follows from Proposition 2.5.2 (xi) that .e2 xe2 = e2 (e1 xe1 ) e2 ∈ V (ε1 , δ1 ). Proposition 2.5.2 (i) implies that .V (ε1 , δ1 ) ⊆ V (ε2 , δ2 ) and hence, .x ∈ V (ε2 , δ2 , e2 ). (ii) This follows immediately from (i). (iii) The fact that .V (ε, δ, e) is balanced follows immediately from the corresponding property of the set .V (ε, δ); see Proposition 2.5.2 (iii). To show that .V (ε, δ, e) is absorbing, let .x ∈ S (M) be given. Since, by Lemma 2.7.1, .exe ∈ S (τ ) and the set .V (ε, δ) is absorbing in .S (τ ), there exists .λ0 > 0 such that .exe ∈ λ0 V (ε, δ), that is, .x ∈ λ0 V (ε, δ, e). Since .V (ε, δ, e) is balanced, it may be concluded that .V (ε, δ, e) is absorbing. (iv) This follows immediately from Proposition 2.5.2 (vi). (v) If .x ∈ V (ε, δ, e), then .exe ∈ V (ε, δ) and so, it follows from Proposition 2.5.2 (viii) that .ex ∗ e = (exe)∗ ∈ V (ε, δ). Hence, .x ∗ ∈ V (ε, δ, e). This shows that ∗ ∗ .V (ε, δ, e) ⊆ V (ε, δ, e), from which it is clear that .V (ε, δ, e) = V (ε, δ, e). (vi) Suppose .x ∈ V (ε, δ, e) for all .ε, δ > 0 and all .e ∈ P (M) satisfying .τ (e) < ∞. If .e ∈ P (M) and .τ (e) < ∞, then .exe ∈ V (ε, δ) for all .ε, δ > 0 and so, it follows from Proposition 2.5.2 (ix) that .exe = 0. Now Lemma 2.7.2 implies that .x = 0. u n Proposition 2.7.4 The collection Vl = {V (ε, δ, e) : ε, δ > 0, e ∈ P (M) , τ (e) < ∞}

.

is a neighborhood base at 0 for a Hausdorff vector space topology .Tlm in .S (M). The involution .x |−→ x ∗ is continuous with respect to .Tlm .

2.7 The Local Measure Topology

103

Proof Assertions (ii), (iii), and (iv) of Proposition 2.7.3 imply that .Vl is a neighborhood base for a vector space topology .Tlm in .S (M). It follows from (vi) that this topology in Hausdorff and (v) implies that the map .x |−→ x ∗ is continuous with respect to .Tlm . u n The topology .Tlm is called the local measure topology on .S (M). If a net .{xα } in .S (M) is convergent to .x ∈ S (M) with respect to .Tlm , then this is denoted by Tlm .xα → x, and the net .{xα } is said to converge locally in measure to x. Observe that Tlm Tm it follows immediately from the definitions that .xα → x if and only if .exα e → exe for all .e ∈ P (M) with .τ (e) < ∞. The topology on .S (τ ) induced by .Tlm is also referred to as the local measure topology. A neighborhood basis at 0 for the local measure topology in .S (τ ) is given by the sets .Vτ (ε, δ, e) = V (ε, δ, e) ∩ S (τ ), where .ε, δ > 0 and .e ∈ P (M) with .τ (e) < ∞. It follows from Proposition 2.5.2 (xi) that .V (ε, δ) ⊆ Vτ (ε, δ, e) whenever .ε, δ > 0 and .e ∈ P (M) with .τ (e) < ∞. Hence, the local measure topology in .S (τ ) is weaker than the measure topology. It will become clear from the examples that, in general, the local measure topology is not metrizable and that multiplication is not jointly continuous (neither in .S (M), nor in .S (τ )). However, multiplication is continuous in each factor separately, as the next proposition shows. Proposition 2.7.5 If .{xα } is a net in .S (M) which converges to 0 locally in measure, Tlm Tlm then .xα y → 0 and .yxα → 0 for all .y ∈ S (M). Tlm Tm Proof Given .y ∈ S (M), it is shown first that .xα y → 0, that is, .exα ye → 0 for all .e ∈ P (M) with .τ (e) < ∞. To this end, fix .e ∈ P (M) satisfying .τ (e) < ∞. Since ⊥ ≤ n (ye), it follows that .e r (ye) ∼ s (ye) = 1 − n (ye) ≤ e

.

and so, .τ (r (ye)) ≤ τ (e) < ∞. Defining .p = r (ye) ∨ e, it is clear that .τ (p) < ∞ and exα ye = e (pxα p) ye.

.

By hypothesis, .pxα p → 0 with respect to the measure topology. Since, by Tm Tm Lemma 2.7.1, .ye ∈ S (τ ), this implies that .e (pxα p) ye → 0, that is, .exα ye → 0. )∗ ( Tlm Therefore, .xα y → 0. Since .yxα = xα∗ y ∗ and, by Proposition 2.7.4, the mapping .z − | → z∗ is continuous in .S (M) with respect to the local measure topology, it is Tlm Tlm now also clear that .xα → 0 in .S (M) implies that .yxα → 0 for all .y ∈ S (M). n u The relation between the partial ordering in .Sh (M) and the local measure topology will now be considered. Note already that .Sh (M) is .Tlm -closed in .S (M), because the map .x − | → x ∗ is continuous.

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Proposition 2.7.6 (i) The positive cone .S (M)+ is closed in .S (M) for the local measure topology. { } Tlm (ii) If . aβ is a net in .Sh (M) and .a, b ∈ Sh (M) are such that .aβ → a and .aβ ≤ b for all .β, then .a ≤ b. { } Tlm (iii) If . aβ is an increasing net in .Sh (M) and if .a ∈ Sh (M) is such that .aβ → a, then { .a}β ↑ a.{ } (iv) If . aβ and . bβ are two nets in .S (M)+ satisfying .aβ ≤ bβ for all .β, and if Tlm Tlm .bβ → 0, then .aβ → 0. { } Tlm (v) If . aβ is a decreasing net in .Sh (M) such that .aβ ↓ 0 in .Sh (M), then .aβ → 0. Moreover, all of the above statements remain valid if .Sh (M) is replaced by .Sh (τ ). Proof

{ } (i) Suppose that .a ∈ S (M) belongs to the .Tlm -closure of .S (M)+ , and let . aβ be Tlm a net in .S (M)+ such that .aβ → a. Since .Sh (M) is .Tlm -closed in .S (M), this implies that .a ∗ = a. Note that .eaβ e → eae in measure for each .e ∈ P (M) with .τ (e) < ∞, and .eaβ e ≥ 0 for all .β. By Proposition 2.6.1 (i), .S (τ )+ is closed in .Sh (τ ), which implies that .eae ≥ 0. Defining .en = ea (−n, −n + 1] and .an = aen for .n ∈ N, it is clear that .an ∈ M and .an ≤ 0. Fix .n ∈ N and take .pα ∈ P (M) with .τ (pα ) < ∞ such that .pα ↑ en (which is possible as the trace is semi-finite). It follows from .pα = en pα that .pα an pα = pα aen pα = so pα apα ≥ 0 for all .α. Furthermore, .pα ↑ en implies that .pα → en . Since .an ∈ M and multiplication is jointly so-continuous on norm bounded sets, it so follows that .pα an pα → en an en = an . Hence .an ≥ 0, as .M+ is closed in .M with respect to so-convergence. This shows that .an = 0. Since this holds for all .n ∈ N, it follows that .aea (−∞, 0] = 0 and so .a ∈ S (M)+ . The proofs of (ii) and (iii) are now similar to the proofs of the corresponding statements of Proposition 2.6.1. Tlm Tm (iv) If .bβ → 0, then .ebβ e → 0 for all .e ∈ P (M) with .τ (e) < ∞. Since .0 ≤ aβ ≤ bβ implies that .0 ≤ eaβ e ≤ ebβ e , it follows from Proposition 2.6.1 Tlm Tm (iv) that .eaβ e → 0 for all such projections e and consequently, .aβ → 0. (v) If .aβ ↓ 0 in .S (M), then it follows from Proposition 2.2.25 (iii) that .eaβ e ↓ 0 for all .e ∈ P (M) with .τ (e) < ∞. Lemma 2.7.1 implies that .eaβ e ∈ S0 (τ ) Tm for all .β and so, it follows from Theorem 2.6.3 that .eaβ e → 0 for all such Tlm projections e. Hence, .aβ → 0. The final statement of the proposition being easily verified, the proof is complete. u n

2.7 The Local Measure Topology

105

Example 2.7.7 Some examples of the local measure topology now follow, complementing Example 2.5.19. (a) Consider the von Neumann algebra .M = B (H ) equipped with the standard trace .τ . As has been observed before, .S (M) = S (τ ) = B (H ) and the measure topology coincides with the operator norm topology (see Examples 2.2.19, 2.3.13, and 2.5.19). It is interesting to observe that the local measure topology is equal to the weak {operator topology. Indeed,}recall from Example 2.5.19 (a) that .V (ε, δ) = x ∈ B (H ) : ||x||B(H ) ≤ ε whenever .ε > 0 and .0 < δ < 1. Consequently, a base for the local measure topology at 0 is given by the sets { } W (ε, e) = x ∈ B (H ) : ||exe||B(H ) ≤ ε ,

.

where .ε > 0 and .e ∈ P (B (H )) with .τ (e) < ∞. Suppose that U is a neighborhood of 0 for that wo-topology which is of the form |< } >| { U = x ∈ B (H ) : | xξj , ηj | ≤ ε, j = 1, . . . , n ,

.

|| || || || where .0 < ε ∈ R and .ξj , ηj ∈ H (.j = 1, . . . , n) satisfy .||ξj ||H , ||ηj ||H ≤ {1. Let .e ∈ P (B}(H )) be the projection onto the subspace spanned by . ξj , ηj : 1 ≤ j ≤ n . If .x ∈ W (ε, e), then .

|< >| |< >| |< >| | xξj , ηj | = | xeξj , eηj | = | exeξj , ηj | ≤ ||exe||B(H ) ≤ ε

for all .j = 1, . . . , n and so, .x ∈ U . Hence, .W (ε, e) ⊆ U and this shows that the wo-topology is weaker than the local measure topology. To prove the converse, let .ε > 0 and .e ∈ P (B (H )) with .τ (e) < ∞ be given and let .{φ1 , . . . , φn } be an orthonormal basis of .Ran (e). Using the fact that exeξ =

n n Σ Σ

.

< > xφk , φj φj , x ∈ B (H ) , ξ ∈ H,

j =1 k=1

a simple computation shows that |< } >| { U = x ∈ B (H ) : | xφk , φj | ≤ ε/n, j, k = 1, . . . , n ⊆ W (ε, e) .

.

Hence, the local measure topology is weaker than the wo-topology and so, the two topologies coincide. (b) Let { .(X, Σ, μ) be a} Maharam measure space, let .H = L2 (μ) and .M = Mf : f ∈(L∞ )(μ) f, as defined in Example 2.1.5 (b), equipped with the trace given by .τ Mf = X f dμ, .0 ≤ f ∈ L∞ (μ). As observed in Example 2.2.19 (b), the mapping .f |−→ Mf is a .∗ -isomorphism from .L0 (μ) onto .S (M) and this mapping also induces a .∗-isomorphism from .S (μ) onto .S (τ ) (see Example 2.3.13 (b)). Moreover, in Example 2.5.19 (b) it is shown that via this

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∗-isomorphism .f − | → Mf , the measure topology in .S (τ ) corresponds to the usual topology of convergence in measure in the space .S (μ) with respect to the measure .μ. Combining these observations, it is readily verified that for any Tlm sequence .{fn }∞ n=1 in .L0 (μ) the statement .Mfn → 0 is equivalent to saying that

.

μ ({x ∈ A : |fn (x)| ≥ ε}) → 0 as n → ∞

.

for all .ε > 0 and all .A ∈ Σ with .μ (A) < ∞. Consequently, the local measure topology in .S (M) corresponds to the usual topology of local convergence in measure in the space .L0 (μ). The next remark points out the following differences between the measure topology and the local measure topology: (a) multiplication is in general not jointly continuous with respect to the local measure topology; (b) in general there does not exist a neighborhood base of 0 for the local measure topology consisting of absolutely solid sets; (c) the local measure topology is in general not metrizable. Remark 2.7.8 Let H be a separable Hilbert space with orthonormal basis .{φn }∞ n=1 and consider .M = B (H ) equipped with the standard trace .τ . In this case, .S (M) = S (τ ) = B (H ) and the local measure topology coincides with the wo-topology (see Example 2.7.7 (a)). (a) For .n ∈ N define .xn ∈ B (H ) by .xn φj = φj +n for all .j = 1, 2, . . .. The adjoint ∗ ∗ ∗ .xn is given by .xn φj = φj −n whenever .j > n and .xn φj = 0 for .1 ≤ j ≤ n. wo wo ∗ It is easy to see that .xn → 0 and .xn → 0 as .n → ∞. However, .xn∗ xn φj = φj for all .j ≥ 1 and so, .xn∗ xn = 1 for all n. Hence, .xn∗ xn does not converge to zero with respect to the wo-topology. This shows that multiplication is not jointly continuous with respect to the local measure topology (not even on bounded subsets). Tlm (b) Using the same sequence .{xn }∞ n=1 as in (a), observe that .xn → 0 but, since Tlm .|xn | = 1 for all n, .|xn | - 0. This shows that the local measure topology cannot have a neighborhood base at 0 consisting of absolutely solid sets. In particular, the sets .V (ε, δ, e), as defined by (2.15), are not absolutely solid (cf. Proposition 2.5.2 (x)). (c) The local measure topology (that is, the wo-topology) in .B (H ) is not metriz√ able. For .n ∈ N define .an ∈ B (H ) by .an ξ = n φn for all .ξ ∈ H and consider the set .A = {an : n ∈ N}. The claim is that 0 belongs to the wo-closure of A. Indeed, let |< } >| { U = x ∈ B (H ) : | xξj , ηj | ≤ ε, j = 1, . . . , k

.

2.8 Continuity of Operator Functions with Respect to the Measure Topology

107

be of 0, where .ε > 0 and .ξj , ηj ∈ H (.j = 1, . . . , k) with || || || a||wo-neighborhood ||ξj || , ||ηj || ≤ 1. It needs to be shown that there exists n such that .an ∈ U . H H Since

.

.

|< >| >| |< >| √ |< >| √ |< | an ξj , ηj | = n | ξj , φn | | φn , ηj | ≤ n | φn , ηj |

>| √ |< for all j , it suffices to show that there exists n such that . n | φn , ηj | ≤ ε for all .1 ≤ j ≤ k. Observing that ⎡ .

⎢ ⎣

∞ Σ n=1

⎛ ⎝

k Σ

⎞2 ⎤1/2 [∞ ]1/2 k Σ Σ |< |< >| >|2 | | φn , η j |⎠ ⎥ | φn , ηj < ∞, ⎦ ≤

j =1

j =1

n=1

|< >| Σ √ it follows that there exists n such that . kj =1 | φn , ηj | ≤ ε/ n and so, |< >| √ .| φn , ηj | ≤ ε/ n for all .1 ≤ j ≤ k. This establishes the assertion. It will be shown next that no sequence in A converges to zero with respect to the wo{ }∞ wo topology. Indeed, if . ani i=1 is a sequence in A such that .ani → 0 as .i → ∞, { } ∞ then .{ni }∞ i=1 is unbounded and so, . ani i=1 is not norm bounded in .B (H ). { }∞ wo However, by the uniform boundedness principle, .ani → 0 implies that . ani i=1 is norm bounded in .B (H ) and this yields a contradiction. Therefore, 0 belongs to the wo-closure of A but, no sequence in A wo-converges to 0. Hence, the wo-topology is not metrizable. References: [10–12, 45, 47, 48].

2.8 Continuity of Operator Functions with Respect to the Measure Topology As before, let .M be a von Neumann algebra on the Hilbert space .H , equipped with a semi-finite normal faithful trace .τ . The algebra of all .τ -measurable operators is denoted by .S (τ ) and .Sh (τ ) denotes the real subspace of all self-adjoint elements of .S (τ ). Let .Bbc (R) be the algebra of all complex-valued Borel functions on .R which are bounded on compact subsets of .R. For .a ∈ Sh (τ ) and .f ∈ Bbc (R), the normal operator .f (a) is defined via the functional calculus, that is, f f (a) =

.

R

f (λ) dea (λ) .

By Proposition 2.3.14, the map .f − | → f (a) is a .∗-homomorphism from .Bbc (R) into .S (τ ). For fixed .f ∈ Bbc (R), the map .a − | → f (a) from .Sh (τ ) into .S (τ ) is called an operator function. The question now arises as to which conditions ensure that an operator function is continuous with respect to the measure topology.

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The following observation will be used. Lemma 2.8.1 Suppose that .W ⊆ Sh (τ ) is bounded with respect to the measure topology. If .{fn }∞ n=1 is a sequence in .Bbc (R) and .f ∈ Bbc (R) is such that .fn → f as .n → ∞ uniformly on compact subsets of .R, then .fn (a) → f (a) uniformly on W , that is, for every .ε, δ > 0 there exists .N ∈ N such that .f (a) − fn (a) ∈ V (ε, δ) for all .n ≥ N and all .a ∈ W . Proof Since .f (a) − fn (a) = (f − fn ) (a) for all .a ∈ W , it is sufficient to show that .fn (a) → 0 uniformly on W whenever .{fn }∞ n=1 is a sequence in .Bbc (R) such that .fn → 0 uniformly on compact subsets of .R. Let .ε, δ > 0 be given. Since W( is bounded,) it follows from Lemma 2.6.10 that there exists .R > 0 such that |a| (R, ∞) ≤ δ for all .a ∈ W . Since .f → 0 uniformly on .[−R, R], there .τ e n exists .N ∈ N such that .|fn (λ)| ≤ ε for all .λ ∈ [−R, R] and all .n ≥ N. Therefore, if .n ≥ N and .a ∈ W , then e|fn (a)| (ε, ∞) = e|fn |(a) (ε, ∞) = ea ({λ ∈ R : |f (λ)| > ε})

.

≤ ea {λ ∈ R : |λ| > R} = e|a| (R, ∞) ( ) ( ) and so, .τ e|fn (a)| (ε, ∞) ≤ τ e|a| (R, ∞) ≤ δ. This shows that .fn (a) ∈ V (ε, δ) for all .a ∈ W whenever .n ≥ N. n u The following special case of Lemma 2.8.1 deserves explicit mention. Corollary 2.8.2 If .{fn }∞ n=1 is a sequence in .Bbc (R) and .f ∈ Bbc (R) such that Tm .fn → f as .n → ∞ uniformly on compact subsets of .R, then .fn (a) → f (a) for all .a ∈ Sh (τ ). It is now convenient to introduce the set .A of functions on .R by setting } { Tm Tm .A = f ∈ Bbc (R) : an → a in Sh (τ ) =⇒ f (an ) → f (a) in S (τ ) .

(2.16)

Since .S (τ ) is a topological .∗-algebra, it is clear that .A is a conjugate closed subalgebra of .Bbc (R). In Lemma 2.8.3, it will be shown that .A is closed with respect to uniform convergence on compact subsets. Lemma 2.8.3 The algebra .A is closed in .Bbc (R) with respect to uniform convergence on compact subsets of .R. Proof Suppose that .{fn }∞ n=1 is a sequence in .A and .f ∈ Bbc (R) is such that .fn → f as .n → ∞ uniformly on compact subsets of .R. To show that .f ∈ A, let .{ak }∞ k=1 be Tm a sequence in .Sh (τ ) and .a ∈ Sh (τ ) such that .ak → a as .k → ∞ and let .ε, δ > 0 be given. Since .{ak }∞ k=1 is a convergent sequence, it is, in particular, bounded and so, by Lemma 2.8.1, there exists .N1 ∈ N such that .fn (ak ) − f (ak ) ∈ V (ε/3, δ/3) for all Tm .k and all .n ≥ N1 . Furthermore, by Corollary 2.8.2, .fn (a) → f (a) as .n → ∞ and

2.8 Continuity of Operator Functions with Respect to the Measure Topology

109

so, there exists .N2 ∈ N such that .fn (a) − f (a) ∈ V (ε/3, δ/3) for all .n ≥ N2 . Fix Tm .n ≥ max (N1 , N2 ). Since .fn ∈ A implies that .fn (ak ) → fn (a) as .k → ∞, there exists .K ∈ N such that .fn (a) − fn (ak ) ∈ V (ε/3, δ/3) for all .k ≥ K . Therefore, f (a) − f (ak ) = (f (a) − fn (a)) + (fn (a) − fn (ak )) + (fn (ak ) − f (ak ))

.

∈ V (ε/3, δ/3) + V (ε/3, δ/3) + V (ε/3, δ/3) ⊆ V (ε, δ) Tm for all .k ≥ K. This shows that .f (ak ) → f (a) and so, .f ∈ A.

u n

For .ζ ∈ C \ R, the function .rζ : R → C is defined by setting .rζ (λ) = (λ − ζ )−1 . Evidently, .rζ ∈ C0 (R) and so, in particular, .rζ ∈ Bbc (R) for all .ζ ∈ C \ R. Here .C0 (R) denotes the space of all complex-valued continuous functions on .R vanishing at infinity. If .a ∈ Sh (τ ), then .rζ (a) = r (ζ, a) = (a − ζ 1)−1 ∈ M for all .ζ ∈ C \ R (see Proposition 2.1.4 (vii)). Lemma 2.8.4 For all .ζ ∈ C \ R, the function .rζ belongs to .A. Proof Fix .ζ ∈ C \ R. Since .|λ − ζ | ≥ |Im (λ − ζ )| = |Imζ | > 0 for all .λ ∈ R, it follows that .

| | |rζ (λ)| =

1 1 ≤ |λ − ζ | |Imζ |

and so, .

|| || || || ||r (ζ, a)||B(H ) = ||rζ (a)||B(H ) ≤ ||rζ ||∞ ≤ |Imζ |−1

for all .a ∈ Sh (τ ). Let .{an }∞ n=1 be a sequence in .Sh (τ ) and let .a ∈ Sh (τ ) be such Tm that .an → a as .n → ∞. The sequence .{r (ζ, an )}∞ n=1 is bounded in .M and hence, by Proposition 2.6.11 (ii), it is also bounded with respect to the measure topology in .S (τ ). Writing r (ζ, a) − r (ζ, an ) = r (ζ, a) (an − a) r (ζ, an ) ,

.

Tm it follows from Proposition 2.6.11 (i) that .r (ζ, a)−r (ζ, an ) → 0, that is, .rζ (an ) → u n rζ (a) in measure. This shows that .rζ ∈ A. The proof of the next lemma uses the following version of the Stone–Weierstrass theorem, which is stated explicitly for the reader’s convenience. Theorem 2.8.5 (Stone–Weierstrass) Let S be a locally compact Hausdorff space and .C0 (S) be the space of all complex-valued continuous functions on S vanishing at infinity. If .R is a conjugate closed subalgebra of .C0 (S) which separates the points of .S (that is, for all .s /= t in S there exists .f ∈ R such that .f (s) /= f (t) ) and if

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2 Measurable Operators

for every .s ∈ S there exists .f ∈ R such that .f (s) /= 0, then .R is uniformly dense in C0 (S).

.

Lemma 2.8.6 The space .C0 (R) is contained in the algebra .A. Proof Define .R = C0 (R) ∩ A, which is a conjugate closed subalgebra of .C0 (R). Lemma 2.8.4 implies that .rζ ∈ R for all .ζ ∈ C \ R. If .ζ ∈ C \ R, then .rζ (s) /= rζ (t) whenever .s /= t in .R and .rζ (s) /= 0 for all .s ∈ R. Hence, .R satisfies the conditions of the Stone–Weierstrass theorem and so .R is uniformly dense in .C0 (R). Since .A is closed in .Bbc (R) with respect to uniform convergence on compact sets, it is evident that .R is uniformly closed in .C0 (R). Consequently, .R = C0 (R) and hence, .C0 (R) ⊆ A. u n The above lemma immediately leads to the following theorem, where .C (R) denotes the space of all complex-valued continuous functions on .R. Tm Tm Theorem 2.8.7 If .f ∈ C (R) and if .an → a in .Sh (τ ), then .f (an ) → f (a). Proof In view of Lemmas 2.8.3 and 2.8.6, it needs only to be observed that .C0 (R) is dense in .C (R) with respect to uniform convergence on compact sets. To see this, let .{ϕn }∞ n=1 be any sequence in .C (R) such that .ϕn (λ) = 1 for .λ ∈ [−n, n] and .ϕn (λ) = 0 whenever .|λ| ≥ n + 1. If .f ∈ C (R), then .f ϕn ∈ C0 (R) for all n and .f ϕn → f uniformly on compact subsets of .R. u n The following special consequence of the above theorem deserves special mention. Corollary 2.8.8 The map .x |−→ |x| is continuous on .S (τ ) with respect to the Tm Tm measure topology, that is, if .xn → x in .S (τ ), then .|xn | → |x|. Tm Tm Tm Proof It follows from .xn → x in .S (τ ) that .xn∗ → x ∗ and so, .xn∗ xn → x ∗ x, as .S (τ ) is a topological .∗-algebra. Now apply the above theorem with .f ∈ C (R) given by √ .f (λ) = λ if .λ ≥ 0 and .f (λ) = 0 if .λ < 0. u n If the limit operator a is kept fixed, then the class of functions for which Tm Tm .f (an ) → f (a) whenever .an → a in .Sh (τ ) may be extended further, as will be shown in the next theorem. To formulate this result, it is convenient to denote, for any function .f : R → C, the set of points at which f is continuous by .Cf . The proof of this theorem is based on the next lemma, in which the following notation will be used. For any subset .A ⊆ R, define ρA (t) = inf {|t − s| : s ∈ A} ,

.

t ∈R

(where .inf ∅ = +∞). Evidently, .ρA is continuous and .ρA (t) = 0 if and only if t ∈ A.

.

2.8 Continuity of Operator Functions with Respect to the Measure Topology

111

Lemma 2.8.9 Suppose that .f ∈ Bbc (R). If Z is a closed subset of .R such that f (t) = 0 for all .t ∈ Z and .Z ⊆ Cf , then there exists .h ∈ C (R) such that .h (t) = 0 for all .t ∈ Z and .|f (t)| ≤ h (t) for all .t ∈ R.

.

Proof First assume, in addition, for} all .t ∈ R. For .n = 1, 2, . . ., { that .|f (t)| ≤ 1 −n |f ≥ 2 define the sets .An by .An = t ∈ R : . Note that .An ⊆ An+1 and (t)| ( ∞ ) .R = A1 ∪ ∪ A \ A , as a disjoint union (since .|f (t)| ≤ 1 for all .t ∈ R). n−1 n=2 n Moreover, .Z ⊆ Cf implies that .An ∩ Z = ∅ and so, .ρZ (t) + ρAn (t) > 0 for all .t ∈ R and all n. Defining h (t) =

∞ Σ

.

n=1

2−n

ρZ (t) , t ∈ R, ρZ (t) + ρAn (t)

it is clear that .h ∈ C (R) and that .h (t) = 0 for all .t ∈ Z. If .t ∈ A1 , then .ρAn (t) = 0 for all n and so, .h (t) = 1. Hence, .|f (t)| ≤ h (t) for all .t ∈ A1 . If .t ∈ An \ An−1 for some .n ≥ 2, then .ρAk (t) = 0 for all .k ≥ n and so, .h (t) ≥ 2−n+1 . It follows / An−1 that .|f (t)| < 2−n+1 and so, .|f (t)| ≤ h (t). Hence, .|f (t)| ≤ h (t) from .t ∈ for all .t ∈ An \ An−1 . It follows that .|f (t)| ≤ h (t) for all .t ∈ R. Returning to the general case, since .f ∈ Bbc (R), there exists .g ∈ C (R) such that .|f (t)| + 1 ≤ g (t) for all .t ∈ R. Defining .f1 = fg −1 , it is clear that .f1 (t) = 0 for all .t ∈ Z, .Z ⊆ Cf1 = Cf and .|f1 (t)| ≤ 1 for all .t ∈ R. Therefore, by the first part of the proof, there exists .h1 ∈ C (R) such that .h1 (t) = 0 for all .t ∈ Z and .|f1 (t)| ≤ h1 (t) for all .t ∈ R. If .h = h1 g, then h has the required properties. u n Theorem 2.8.10 Suppose .a ∈ Sh (τ ). If .f ∈ Bbc (R) is such that .σ (a) ⊆ Cf , then Tm Tm ∞ .f (an ) → f (a) for any sequence .{an } n=1 in .Sh (τ ) satisfying .an → a. Proof Since .σ (a) is closed and .f | σ (a) is continuous, it is a consequence of Tietze’s extension theorem that there exists .g ∈ C (R) such that .g (λ) = f (λ) for all .λ ∈ σ (a). Applying Lemma 2.8.9 to the set .Z = σ (a) and the function .f − g, it follows that there exists .h ∈ C (R) such that .h (λ) = 0 for all .λ ∈ σ (a) and .|f − g| ≤ h. Since .g (λ) = f (λ) for all .λ ∈ σ (a), it is clear that .g (a) = f (a). Tm It follows from Theorem 2.8.7 that .g (an ) → g (a). Furthermore, .h (λ) = 0 for all Tm .λ ∈ σ (a), so .h (a) = 0. Hence, Theorem 2.8.7 yields that .h (an ) → 0. Observing that .

|f (an ) − g (an )| = |f − g| (an ) ≤ h (an )

Tm for all n, it follows from Proposition 2.6.1 (iv) that .f (an ) − g (an ) → 0 as .n → ∞. Consequently, Tm f (an ) = (f (an ) − g (an )) + g (an ) → g (a) = f (a)

.

and the proof is complete.

u n

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2 Measurable Operators

At the end of this section, it seems to be appropriate to include some observations concerning the uniqueness of the functional calculus. The methods of proof of these results are similar to the ones used above. It will be convenient to define the function .ι : R → R by .ι (λ) = λ, .λ ∈ R. Proposition 2.8.11 Let .a ∈ Sh (τ ) be fixed. If .o : C (R) → S (τ ) is a unital .∗homomorphism satisfying: (i) .o (ι) = a. Tm (ii) .o (fn ) → o (f ) in .S (τ ) whenever .f ∈ C (R) and .{fn }∞ n=1 is a sequence in .C (R) such that .fn → f uniformly on compact subsets of .R, then .o (f ) = f (a) for all .f ∈ C (R). Proof Define the unital .∗-subalgebra .B of .C (R) by setting B = {f ∈ C (R) : o (f ) = f (a)} .

.

It follows from Corollary 2.8.2 and hypothesis (ii) that .B is closed in .C (R) with respect to uniform convergence on compact subsets of .R. Using the notation introduced in the discussion preceding Lemma 2.8.4, a moment’s reflection shows that .rζ ∈ B for all .ζ ∈ C \ R. By the same argument as used in the proof of Lemma 2.8.6, it follows via the Stone–Weierstrass theorem (see Theorem 2.8.5) that .C0 (R) ⊆ B. Since .C0 (R) is dense in .C (R) with respect to uniform convergence on compact subsets of .R (see the proof of Theorem 2.8.7), this implies that .B = C (R). The proof is complete. u n Proposition 2.8.12 Let .a ∈ Sh (τ ) be fixed. If .ψ : Bbc (R) → S (τ ) is a unital ∗-homomorphism satisfying:

.

(i) .ψ (ι) = a. Tm (ii) .ψ (fn ) → ψ (f ) in .S (τ ) whenever .f ∈ Bbc (R) and .{fn }∞ n=1 is a sequence in .Bbc (R) such that .fn → f uniformly on compact subsets of .R. (iii) If .f, fn ∈ Bbc (R) (.n = 1, 2, . . .) are positive functions such that .0 ≤ fn (λ) ↑ f (λ) for all .λ ∈ R, then .ψ (fn ) ↑ ψ (f ) in .Sh (τ ), then .ψ (f ) = f (a) for all .f ∈ Bbc (R). Proof Define the unital .∗-subalgebra .U of .Bbc (R) by setting U = {f ∈ Bbc (R) : ψ (f ) = f (a)} .

.

By Proposition 2.8.11, it is clear that .C (R) ⊆ U. Furthermore, it follows from Remark 2.2.26 and hypothesis (iii) that .U is closed in .Bbc (R) with respect to u n monotone pointwise convergence. This suffices to conclude that .U = Bbc (R). References: [26, 46, 69, 128].

2.9 Trace Preserving ∗-Homomorphisms

113

2.9 Trace Preserving ∗-Homomorphisms In this section some observations concerning subalgebras and .∗-homomorphisms are collected together. As before, .M is a von Neumann algebra on a Hilbert space H , equipped with a normal faithful semi-finite trace .τ . Suppose that .N is a von Neumann subalgebra of .M. It is clear that the restriction of .τ to .N is a trace, which is normal and faithful. In general, however, this restriction need not be semi-finite on .N. Now, it is assumed that the restriction of .τ to .N is semi-finite (and this restriction will simply be denoted by .τ ). In this situation, the .∗-algebras of .τ -measurable operators corresponding to ' ' .M and .N will be denoted by .S (M, τ ) and .S (N, τ ), respectively. Since .M ⊆N , it is clear that every operator in H affiliated with .N is also affiliated with .M. Hence, from the definition of .τ -measurability, it is clear that .S (N, τ ) ⊆ S (M, τ ) and so, .S (N, τ ) is a unital .∗-subalgebra of .S (M, τ ). Note that it follows from Proposition 2.3.6 (equivalence of (i) and (iv)) that S (N, τ ) = {x ∈ S (M, τ ) : xηN} .

.

(2.17)

Denoting the basic neighborhoods at zero for the measure topology in .S (M, τ ) and .S (N, τ ) by .VM (ε, δ) and .VN (ε, δ), respectively, it is immediately clear from Lemma 2.5.1 that VN (ε, δ) = VM (ε, δ) ∩ S (N, τ ) ,

.

ε, δ > 0.

Consequently, the measure topology in .S (N, τ ) is the topology induced by the measure topology in .S (M, τ ). Since .S (N, τ ) is complete for the measure topology, this implies { } in particular that .S (N, τ ) is closed in .S (M, τ ). If . aβ is an increasing net in .S (N, τ )+ and .a ∈ S (N, τ )+ , then .aβ ↑β a in .Sh (N, τ ) if and only if .aβ ↑β a in .Sh (M, τ ) (indeed, in both cases the supremum coincides with the supremum in the sense of the form ordering; { quadratic } cf. Propositions 2.2.25 and 2.3.10). Furthermore, if . aβ is an increasing net in .S (N, τ )+ and if .a ∈ S (M, τ )+ is such that .aβ ↑β a in .Sh (M, τ ), then + .a ∈ S (N, τ ) . Indeed, since .aβ ↑β a with respect to the quadratic from ordering ( ) ∗ and .u aβ u = (aβ )for all .u ∈ U N' , it follows from Corollary 1.8.9 that .u∗ au = a for all .u ∈ U N' and hence, .a ∈ S (N, τ )+ . Now suppose that .M1 and .M2 are two von Neumann algebras on Hilbert spaces .H1 and .H2 , respectively, and let .τ1 and .τ2 be semi-finite normal faithful traces on .M1 and .M2 , respectively. Suppose that .π : S (τ1 ) → S (τ2 ) is a .∗ -homomorphism, that is, .π is an algebra homomorphism satisfying .π (x ∗ ) = π (x)∗ for all .x ∈ S (τ1 ). Evidently, .π (a) ∈ Sh (τ2 ) whenever .a ∈ Sh (τ1 ). Since every .a ∈ S (τ1 )+ has a square root in .S (τ1 )+ (see Proposition 2.3.14), it is also clear that .a ∈ S (τ1 )+ implies that .π (a) ∈ S (τ2 )+ . Hence, .π is an order preserving map from .Sh (τ1 ) into

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2 Measurable Operators

Sh (τ2 ). Furthermore,

.

.

|π (x)| = π (|x|) ,

x ∈ S (τ1 ) .

(2.18)

Indeed, .π (|x|) ≥ 0 and ) ( ( ) π (|x|)2 = π |x|2 = π x ∗ x = π (x)∗ π (x) ,

.

which implies that .π (|x|) = |π (x)|. Also observe that .π (M1 ) ⊆ M2 . In fact, if x ∈ M1 , then .|x| ≤ ||x||B(H ) 1. Since .π is order preserving and .π (1) is a (selfadjoint) projection, this implies that

.

.

|π (x)| = π (|x|) ≤ ||x||B(H ) π (1) ≤ ||x||B(H ) 1

and hence, .π (x) ∈ M2 (as .M2 is absolutely solid in .S (τ2 ); see Remark 2.3.9). Furthermore, it should be noted that .π is injective (that is, .π is a .∗-isomorphism) if and only if .π |M1 is injective. Indeed, suppose that .π |M1 is injective. If .0 /= x ∈ S (τ1 ), then there exists .a ∈ M+ 1 such that .0 < a ≤ |x| and so, 0 < π (a) = π (a) ≤ π (|x|) = |π (x)| ,

.

which shows that .π (x) /= 0. A characterization of continuity of .∗-homomorphisms with respect to the measure topology is given in the following lemma. Lemma 2.9.1 For a unital .∗-homomorphism .π : S (τ1 ) → S (τ2 ), the following statements are equivalent: (i) .π is continuous with respect to the measure topology. (ii) .π |M1 is continuous with respect to the measure topology. (iii) .τ2 (π (pn )) → 0 for every sequence .{pn }∞ n=1 in .P (M1 ) satisfying .τ1 (pn ) → 0 as .n → ∞. Proof Evidently, (i) implies (ii). That (ii) implies (iii) is an immediate consequence of Corollary 2.5.8. Assuming that (iii) holds, let .ε, δ > 0 be given. It follows from (iii) that there exists .δ ' > 0(such)that .τ2 (π (q)) ≤ δ for all .q ∈ P (M1 ) satisfying .τ1 (q) ≤ δ ' . If .x ∈ VM1 ε, δ ' , then there exists .p ∈ P (M1 ) such that .||xp||B(H1 ) ≤ ε and ( ⊥) .τ1 p ≤ δ ' . Since .π |M1 is a contraction with respect to the norms in .M1 and .M2 , this implies that .

||π (x) π (p)||B(H2 ) = ||π (xp)||B(H2 ) ≤ ||xp||B(H1 ) ≤ ε.

2.9 Trace Preserving ∗-Homomorphisms

115

Furthermore, .π (p) ∈ P (M2 ) and ( ) ( ( )) τ2 π (p)⊥ = τ2 π p ⊥ ≤ δ.

.

Hence, .π (x) ∈ VM2 (ε, δ). This shows that ( ( )) π VM1 ε, δ ' ⊆ VM2 (ε, δ)

.

and so, .π is continuous with respect to the measure topology. The proof is complete. u n Next, extensions of .∗-homomorphisms from .M1 into .M2 to .∗-homomorphisms Recall that a .∗-homomorphism .π : M1 → from .S (τ1 ) into .S (τ2 ) will be (discussed. ) + M2 is said to be normal if .π aβ ↑β π (a) in .M+ 2 whenever .aβ ↑β a in .M1 . If .π is a unital normal .∗ -homomorphism, then its range .π (M1 ) is a von Neumann subalgebra of .M2 (and conversely, if .π is a unital .∗-homomorphism and if .π (M1 ) is a von Neumann subalgebra of .M2 , then .π is normal); see Theorem 1.11.10. Proposition 2.9.2 If .π : M1 → M2 is a unital .∗-homomorphism which is continuous with respect to the measure topology, then following statements hold: (i) .π has a unique extension to a .∗-homomorphism .πˆ : S (τ1 ) → S (τ2 ), which is continuous with respect to the measure topology; if .π is an isomorphism, then so is .πˆ . ( ) (ii) If .a ∈ Sh (τ1 ), then .f πˆ a = πˆ (f (a)) for all .f ∈ C (R). Assuming, in addition, that .π is normal, the following statements hold: ( ) (iii) If .a ∈ Sh (τ1 ), then .f πa ˆ = πˆ (f (a)) for all .f ∈ Bbc (R). (iv) If .a ∈ Sh (τ1 ), then the spectral measure of .πˆ (a) is given by ( ) π ea : B − | → π ea (B) ,

.

B ∈ B (R) ;

( ) (v) .πˆ is normal, that is, .πˆ aβ ↑β πˆ (a) in .S (τ2 )+ whenever .aβ ↑β a in .S (τ1 )+ . Proof (i) Since .M1 is dense in .S (τ1 ) (see Proposition 2.5.4) and .S (τ2 ) is complete with respect to the measure topology (by Theorem 2.5.12), it is clear that .π has a unique continuous linear extension .πˆ : S (τ1 ) → S (τ2 ). Since .S (τ1 ) and .S (τ2 ) are topological .∗-algebras, .π ˆ is a .∗-homomorphism. By Lemma 2.9.1 (equivalence of (i) and (ii)), any .∗-homomorphism from .S (τ1 ) into .S (τ2 ) extending .π is continuous with respect to the measure topology and so, the uniqueness of .πˆ is also clear. Furthermore, if .π is a .∗ -isomorphism, then it follows from the remarks preceding Lemma 2.9.1, that .πˆ is a .∗-isomorphism as well.

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2 Measurable Operators

(ii) The map .o : f |−→ πˆ (f (a)) , .f ∈ C (R), satisfies the conditions of Proposition 2.8.11 (with respect to the element .πa ˆ ∈ S (τ2 )) and so, .o (f ) = ( ) f πa ˆ for all .f ∈ C (R). (iii) Now, it is assumed, in addition, that .π is normal. The map .ψ : f − | → πˆ (f (a)), .f ∈ Bbc (R), satisfies the conditions of Proposition 2.8.12 (with ( ) respect to the element .πˆ a ∈ S (τ2 )) and so, .o (f ) = f πˆ a for all .f ∈ Bbc (R). Applying (iii) to .f = χB , .B ∈ B (R), statement (iv) is now also evident. (v) Replacing .aβ and a by .aβ + 1 and .a + 1, respectively, it may be assumed, without loss of generality, that a is invertible (cf. Proposition 2.2.24 (vi)). If .0 ≤ aβ ↑β a in .S (τ1 ), with a invertible, then it follows from Proposition 2.2.25 (iii) that 0 ≤ a −1/2 aβ a −1/2 ↑β 1

.

( ) in .M1 . Since .π a −1/2 = πˆ (a)−1/2 and .π is normal, this implies that ) ( ( ) πˆ (a)−1/2 πˆ aβ πˆ (a)−1/2 = π a −1/2 aβ a −1/2 ↑β 1

.

( ) and, therefore, (once again using Proposition 2.2.25 (iii)), .πˆ aβ ↑β πˆ (a) in .S (τ2 ). The proof is complete. u n Of special importance is the case that the .∗-homomorphism .π : M1 → M2 is trace preserving, that is, .τ2 (π (a)) = τ1 (a) for all .a ∈ M+ 1 . Since .|π (x)| = π (|x|), .x ∈ M1 , it is clear that a trace preserving .∗-homomorphism is injective. Furthermore, Lemma 2.9.1 implies that any unital trace preserving .∗-homomorphism is continuous with respect to the measure topology and so, the assertions of the above proposition hold. Moreover, the following additional result holds in this case. Proposition 2.9.3 Let .π : M1 → M2 be a unital trace preserving normal .∗isomorphism and .πˆ : S (τ1 ) → S (τ2 ) be the unique .∗ -isomorphism extending .π . If the von Neumann subalgebra .N2 of .M2 is defined by setting .N2 = π (M1 ), then .π ˆ (S (τ1 )) = S (N2 , τ2 ) and .πˆ : S (τ1 ) → S (N2 , τ2 ) is a homeomorphism for the measure topology. Proof Since .π normal, its range .N2 = π (M1 ) is a von Neumann subalgebra of M2 . Moreover, .π is a trace preserving .∗-isomorphism from .M1 onto .N2 and so, the restriction of .τ2 to .N2 is semi-finite. Therefore, .S (N2 , τ2 ) is well defined and it is a .∗-subalgebra of .S (M2 , τ2 ) = S (τ2 ) (see the remarks at the beginning of the present section). If .x ∈ S (τ1 ) and .x = v |x| is its polar decomposition, then

.

πˆ (x) = π (v) πˆ (|x|) .

.

2.9 Trace Preserving ∗-Homomorphisms

117

It follows Proposition 2.9.2 (iv) that .eπˆ (|x|) = π e|x| and so, .eπˆ (|x|) (B) ∈ N2 for all .B ∈ B (R). Therefore, .|x| is affiliated with .N2 and hence, .|x| ∈ S (N2 , τ2 ) (see (2.17)). Since .π (v) ∈ N2 , it is now clear that .πˆ (x) ∈ S (N2 , τ2 ). This shows that .πˆ (S (τ1 )) ⊆ S (N2 , τ2 ) . The trace preserving unital .∗-isomorphism −1 : N → M has a unique extension to a .∗ -isomorphism from .S (N , τ ) .π 2 1 2 2 into .S (τ1 ), which is the inverse of .πˆ . Consequently, .S (N2 , τ2 ) = πˆ (S (τ1 )) and .π ˆ −1 : S (N2 , τ2 ) → S (τ1 ) is continuous with respect to the measure topology. The proof is complete. u n For later applications it is illustrative to consider the following special case of the above proposition. Example 2.9.4 Suppose that .M is a von Neumann algebra on a Hilbert space H , equipped with a semi-finite normal faithful trace .τ . As usual, .L∞ [0, 1] is considered as a von Neumann algebra on the Hilbert space .L2 [0, 1], equipped with the Lebesgue integral as its trace. The tensor product .L∞ [0, 1] ⊗M is a von Neumann algebra on the tensor product Hilbert space .L2 [0, 1] ⊗H . Let .τˆ be the unique semi-finite normal faithful trace on .L∞ [0, 1] ⊗M satisfying (f τˆ (f ⊗ x) =

1

.

) f (t) dt τ (x) ,

0

0 ≤ f ∈ L∞ [0, 1] , 0 ≤ x ∈ M.

The map .π : x |−→ 1 ⊗ x, .x ∈ M, is a unital trace preserving normal .∗isomorphism from .M onto the von Neumann subalgebra .C1⊗M of .L∞ ([0,) 1] ⊗M. Consequently, .π extends( uniquely to) a .∗-isomorphism .πˆ : S (τ ) → S τˆ and the ˆ will also be denoted by range of .πˆ is equal to .S C1 ⊗ M, τˆ . If .x ∈ S (τ ), then .πx .1 ⊗ x. Note that it follows from Proposition 2.9.2 (iv) that for any .a ∈ Sh (τ ), the spectral measure of .1 ⊗ a is given by e1⊗a (B) = 1 ⊗ ea (B) ,

.

B ∈ B (R) .

Remark 2.9.5 The notation .1 ⊗ x for the operator .πˆ x, as introduced in the above example, may be justified as follows. Given .x ∈ S (τ ), define the linear operator 1 O x : D (1 O x) → L2 [0, 1] ⊗H

.

in .L2 [0, 1] ⊗H by setting .D (1 O x) = L2 [0, 1] ⊗ D (x) and .

(1 O x) (f ⊗ ξ ) = f ⊗ xξ,

f ∈ L2 [0, 1] , ξ ∈ D (x) .

Since .D (x) is .τ -dense in H , it is easy to see that .L2 [0, 1] ⊗ D (x) is .τˆ -dense in L2 [0, 1] ⊗H and so, in particular, .1Ox is densely defined. Using that .(C1 ⊗ M)' = B (H ) ⊗M' , it is clear that .1 O x, is affiliated with .C1 ⊗ M (and so, in particular, affiliated with .L∞ [0, 1] ⊗M). Furthermore, it is easily verified that .1 O x ∗ ⊆ (1 O x)∗ , which implies that .(1 O x)∗ is densely defined and hence, .1 O x is .τˆ -

.

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2 Measurable Operators

pre-measurable. The( closure of ).1 O x is (momentarily) denoted by .1Ox . It is now clear that .1Ox ∈ S C1 ⊗ M, τˆ . The claim is that .πˆ x = 1Ox. Indeed, since .x( is ).τ -measurable, there exists a ⊥ sequence .{pn }∞ n=1 in .P (M) such that .pn ↑ 1, .τ pn < ∞ and .pn (H ) ⊆ D (x) for all n. Defining D0 =

∞ U

.

) ( (1 ⊗ pn ) L2 [0, 1] ⊗H ,

n=1

it is clear that( .D0) is a .τˆ -dense subspace of .L2 [0, 1] ⊗H and .D0 ⊆ D (1 O x). Furthermore, . πˆ x (1 ⊗ pn ) =( πˆ (xp ) n ) ∈ C1 ⊗ M, which ( ) implies that .1 ⊗ pn maps the space .L2 [0, 1] ⊗H into .D πˆ x . Hence, .D0 ⊆ D πˆ x . It is not difficult to show that .

) ( ) ( πˆ x (1 ⊗ pn ) = 1Ox (1 ⊗ pn ) ,

n ∈ N,

and so, the .τˆ - measurable operators .πˆ x and .1Ox coincide on the .τˆ -dense subspace D0 . Since, by Proposition 2.2.14 (iii), .D0 is a core for both .πx ˆ and .1Ox, it follows that .πˆ x = 1Ox. The closure of the operator .1 O x is sometimes also denoted by .1 ⊗ x. Therefore, the above remarks justify the use of the notation .1 ⊗ x for the operator .πˆ x.

.

It is appropriate to include, at this point, some observations which are somewhat related to the discussions above. In Example 2.3.13 (c) it has been observed that the space of .τ -measurable operators depends, in general, on the choice of the trace .τ . The next proposition characterizes pairs of traces .τ1 , .τ2 for which .S (τ1 ) = S (τ2 ). It turns out that if .S (τ1 ) = S (τ2 ), then also the two measure topologies .Tm (τ1 ) and .Tm (τ2 ), corresponding to .τ1 and .τ2 , respectively, coincide. Actually, the following result holds. Proposition 2.9.6 If .τ1 and .τ2 are two faithful normal semi-finite traces on a von Neumann algebra .M, then the following statements are equivalent: (i) The measure topologies .Tm (τ1 ) and .Tm (τ2 ) coincide on .M. (ii) For every sequence .{pn }∞ n=1 in .P (M), τ1 (pn ) → 0 ⇔ τ2 (pn ) → 0 as n → ∞.

.

(iii) For any .p ∈ P (M), .τ1 (p) < ∞ if and only if .τ2 (p) < ∞. (iv) The .∗-algebras .S (τ1 ) and .S (τ2 ) coincide. (v) .S (τ1 ) = S (τ2 ) and .Tm (τ1 ) = Tm (τ2 ). Proof The equivalence of (i) and (ii) is clear by Lemma 2.9.1. To show that (ii) implies (iii), assume that .p ∈ P (M) is such that .τ1 (p) < ∞ and .τ2 (p) = ∞. Since .τ2 is semi-finite, there exists a sequence .{qn }∞ n=1 of pairwise orthogonal projections in .P (M) satisfying .0 ≤ qn ≤ p and .τ2 (qn ) ≥ 1 for all n. Define

2.9 Trace Preserving ∗-Homomorphisms

119

Σ Σ∞ pn = ∞ k=n qk for .n = 1, 2, . . .. Since . n=1 τ1 (qn ) = Σ τ1 (p1 ) ≤ τ1 (p) < ∞, it follows that .τ1 (pn ) ↓ 0. On the other hand, .τ2 (pn ) = ∞ k=n τ2 (qk ) = ∞ for all n. This contradicts assertion (ii) and it is now clear that (ii) implies (iii). Next, it will be shown that (iii) implies (ii). Suppose that .{pn }∞ n=1 is a sequence in .P (M) such that .τ1 (pn ) → 0 as .n → ∞ and .τ2 (pn ) ≥ ε for all n and some −n for all n. .ε > 0. By passing to a subsequence, it may be assumed that .τ1 (pn ) ≤ 2 V∞ Σ∞ −n+1 Defining .en = k=n pk , it follows that .τ1 (en ) ≤ k=n τ1 (pk ) ≤ 2 and .en ↓ 0. Since .τ1 (e1 ) < ∞, it follows from (iii) that also .τ2 (e1 ) < ∞ and so, .τ2 (en ) → 0 as .n → ∞. On the other hand, .en ≥ pn and hence .τ2 (en ) ≥ τ2 (pn ) ≥ ε > 0 for all n, which is a contradiction. This shows that (iii) implies (ii). Suppose that((iii))is satisfied. If .x ∈ S (τ1 ), then there exists .p ( ∈ )P (M) such that ⊥ < ∞. By hypothesis, this implies that .τ p ⊥ < ∞ and so, it .xp ∈ M and .τ1 p 2 follows from Proposition 2.3.6 (iii) that .x ∈ S (τ2 ). This shows that .S (τ1 ) ⊆ S (τ2 ). Similarly, it follows that .S (τ2 ) ⊆ S (τ1 ) and so, .S (τ1 ) = S (τ2 ). Therefore, (iii) implies (iv). To show that (iv) implies (iii), suppose that .p ∈ P (M) is such that .τ1 (p) < ∞ and .τ2 (p) = ∞. Since .τ2 is semi-finite, there exists a pairwise orthogonal sequence ∞ .{pn } n=1 in .P (M) satisfying .0 ≤ pn ≤ p and .τ2 (pn ) ≥ 1 for all n. Set .p0 = Σ 1− ∞ a be the positive self-adjoint operator with spectral measure n=1 pn and letΣ ∞ given by .ea (B) = n=0 pn δn (B), .B ∈ B (R) (where .δn is the Dirac measure on .R at the point .n ∈ N). Since .ea (B) ∈ M for all .B ∈ B (R), it follows Σ from Proposition 2.1.4 that a is affiliated with .M. Furthermore, .ea (λ, ∞) = n>λ pn for each .λ > 0 and so, .τ1 (ea (λ, ∞)) ≤ τ1 (p) < ∞. By Proposition 2.3.6 (iv), this Σ implies that .a ∈ S (τ1 ). On the other hand, .τ2 (ea (λ, ∞)) = n>λ τ2 (pn ) = ∞ for all .λ > 0, which implies, by the same proposition, that .a ∈ / S (τ2 ). Consequently, if .S (τ1 ) ⊆ S (τ2 ), then .τ2 (p) < ∞ whenever .p ∈ P (M) satisfies .τ1 (p) < ∞. Repeating the argument with the roles of .τ1 and .τ2 interchanged, it follows that (iv) implies (iii). If .S (τ1 ) = S (τ2 ), then, by what already has been proved, assertion (i) holds. Via Lemma 2.9.1, this implies that .Tm (τ1 ) = Tm (τ2 ). Therefore, (iv) implies (v). Evidently, (v) implies (i) and so, the proof is complete. u n .

If .τ1 and .τ2 are both finite, then .S (τ1 ) = S (τ2 ) = S (M) and so, the equivalent conditions in the above proposition hold in this situation. References: [27].

Chapter 3

Singular Value Functions

Abstract This chapter is a detailed study of properties of the generalized singular value function of a .τ -measurable operator. Particular attention is given to the notion of submajorization (in the sense of Hardy–Littlewood–Polya), and several basic submajorization inequalities are obtained. Noncommutative .L1 and .L2 -spaces are introduced, together with fundamental convergence theorems. In particular, noncommutative versions of the dominated convergence theorem and Fatou’s lemma are presented. The chapter concludes with a discussion of contractions in the noncommutative pair .(L1 , L∞ ).

3.1 The Right-Continuous Inverse of a Decreasing Function In this section, the notion of the right-continuous inverse of a decreasing function on the real line will be discussed. The functions that are considered take their values in the extended real line, denoted by .[−∞, ∞] (with the usual convention that .−∞ < t < ∞ for all .t ∈ R). For any non-empty subset .A ⊆ [−∞, ∞], it is clear that .sup A and .inf A exist in .[−∞, ∞]. Set .sup ∅ = −∞ and .inf ∅ = ∞. Suppose that .p : R → [−∞, ∞] is a decreasing function (that is, .p (t1 ) ≥ p (t2 ) whenever .t1 ≤ t2 ). For .t ∈ R, set p (t+) = lim p (s) = sup p (s) ,

.

s↓t

s>t

p (t−) = lim p (s) = inf p (s) . s↑t

s t for all .s ∈ R. Since p is a decreasing function, the set .{s ∈ R : p (s) ≤ t} is an interval of the form .(a, ∞) or .[a, ∞), where .a ∈ [−∞, ∞]. For each .t ∈ R, the end point a of this interval is given by .q (t). .

Remark 3.1.2 Two simple observations will be used frequently: (a) If .q (t) > −∞ and .s ∈ R is such that .s < q (t), then .p (s) > t. (b) If .q (t) < ∞ and .s ∈ R is such that .q (t) < s, then .p (s) ≤ t. In other words, .

(−∞, q (t)) ⊆ {s ∈ R : p (s) > t} ⊆ (−∞, q (t)] .

This implies immediately that an alternative formula for .q (t) is given by q (t) = sup {s ∈ R : p (s) > t} ,

.

t ∈ R.

In the next proposition, some of the properties of the right-continuous inverse are collected. Note that assertions (i), (ii), and (iii) justify the name “right-continuous inverse”. Proposition 3.1.3 If .p : R → [−∞, ∞] is a decreasing function, then the rightcontinuous inverse .q : R → [−∞, ∞] has the following properties: (i) q is decreasing and right-continuous. (ii) If .s ∈ R and .p (s) ∈ R, then q (p (s)) ≤ s ≤ q (p (s) −) .

.

(iii) If .t ∈ R and .q (t) ∈ R, then p (q (t) +) ≤ t ≤ p (q (t) −) .

.

(iv) For all .t ∈ R, q (t−) = inf {s ∈ R : p (s) < t} .

.

(v) For all .t ∈ R, q (t) = sup {s ∈ R : p (s−) > t} = sup {s ∈ R : p (s+) > t} .

.

Proof (i) If .t1 < t2 , then .{s ∈ R : p (s) ≤ t1 } ⊆ {s ∈ R : p (s) ≤ t2 }, from which it is immediately clear that .q (t2 ) ≤ q (t1 ). To show that q is right-continuous,

3.1 The Right-Continuous Inverse of a Decreasing Function

123

suppose that .t ∈ R and let .{tn }∞ n=1 be a sequence in .R such that .tn ↓ t. It has to be shown that .q (tn ) ↑ q (t). If .q (t) = −∞, this is trivial, and so, it may be assumed that .q (t) > −∞. Given any .s ∈ R such that .s < q (t), it follows from Remark 3.1.2 (a) that .p (s) > t, and so, there exists m such that .p (s) > tm . This implies that .s ≤ q (tm ) ≤ q (t), which shows that .q (tn ) ↑ q (t). (ii) Evidently, .q (p (s)) = inf {u ∈ R : p (u) ≤ p (s)} ≤ s. If .t ∈ R such that .t < p (s), then it follows from Remark 3.1.2 (b) that .s ≤ q (t), and so, q (p (s) −) = inf {q (t) : t < p (s)} ≥ s.

.

(iii) If .s > q (t), then .p (s) ≤ t, and hence, p (q (t) +) = sup {p (s) : s > q (t)} ≤ t.

.

If .s < q (t), then .p (s) > t, and so, p (q (t) −) = inf {p (s) : s < q (t)} ≥ t.

.

(iv) For convenience, put .α = inf {s ∈ R : p (s) < t}. First, it will be proved that ∞ .q (t−) ≤ α. Assuming that .α < ∞, let .s ∈ R be such that .p (s) < t. If .{tn } n=1 is any sequence such that .p (s) < tn < t for all n and such that .tn ↑ t, then .s ≥ q (tn ) for all n and hence .s ≥ q (t−). This shows that .q (t−) ≤ α. Next it will be shown that .q (t−) ≥ α. Assuming that .q (t−) < ∞, let .s ∈ R satisfy .s > q (t−). There exists .t1 < t such that .q (t1 ) < s, so .p (s) ≤ t1 < t, and this implies that .s ≥ α. Consequently, .s ≥ α whenever .s > q (t−), and hence, .q (t−) ≥ α. This shows that .q (t−) = α. (v) As observed already in Remark 3.1.2, .q (t) = sup {s ∈ R : p (s) > t}. Since .

{s ∈ R : p (s+) > t} ⊆ {s ∈ R : p (s) > t} ⊆ {s ∈ R : p (s−) > t} ,

it suffices to show that .

sup {s ∈ R : p (s−) > t} ≤ sup {s ∈ R : p (s+) > t} .

Denote the right hand supremum by .β, and suppose that .s ∈ R satisfies p (s−) > t. If .u < s, then .p (u+) ≥ p (s−) > t, and so, .u ≤ β. As this holds for all .u < s, it follows that .s ≤ β. Hence, .β is an upper bound of the set .{s ∈ R : p (s−) > t}. The proof is complete. u n

.

The important special case in which the function p is assumed to be rightcontinuous (that is, .p (s) = p(s+) for all .s ∈ R) will now be considered. Proposition 3.1.4 If .p : R → [−∞, ∞] is a decreasing right-continuous function, then the right-continuous inverse .q : R → [−∞, ∞] has the following properties:

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3 Singular Value Functions

(i) If .s, t ∈ R, then .p (s) ≤ t if and only if .q (t) ≤ s; equivalently, .

{s ∈ R : p (s) > t} = (−∞, q (t)) .

(3.1)

(ii) The right-continuous inverse of q is equal to p, that is, p (s) = inf {t ∈ R : q (t) ≤ s} ,

.

s ∈ R.

Proof (i) From the definition of .q (t), it is immediately clear that .p (s) ≤ t implies that .q (t) ≤ s. Conversely, if .q (t) ≤ s, then .q (t) < s + ε, and so, .p (s + ε) ≤ t for all .ε > 0. Consequently, .p (s) = p (s+) ≤ t. This shows that .

{s ∈ R : p (s) ≤ t} = [q (t) , ∞) ,

from which equality (3.1) follows immediately. (ii) This is an immediate consequence of (i), since .

inf {t ∈ R : q (t) ≤ s} = inf {t ∈ R : p (s) ≤ t} = p (s)

for all .s ∈ R.

u n

The following observations will be useful. Proposition 3.1.5 Suppose that .p : R → [−∞, ∞] is decreasing and rightcontinuous, with right-continuous inverse .q : R → [−∞, ∞]. If .α ∈ R, then .

[p (α) , p (α−)) ⊆ {t ∈ R : q (t) = α} ⊆ [p (α) , p (α−)] .

In particular: (i) If the set .{t ∈ R : q (t) = α} is non-empty, then it is an interval with end points .p (α) and .p (α−) and .q (p (α)) = α. (ii) If .α ∈ R is such that .p (α) < p (α−), then the set .{t ∈ R : q (t) = α} is nonempty. Proof It follows from Propositions 3.1.4 (ii) and 3.1.3 (iv) that .

inf {t ∈ R : q (t) ≤ α} = p (α) ≤ p (α−) = inf {t ∈ R : q (t) < α} .

This implies in particular that .q (t) > α if .t < p (α) and that .q (t) < α whenever t > p (α−). Therefore, .q (t) = α implies that .p (α) ≤ t ≤ p (α−). The above inequality implies further that .q (t) ≤ α if .t > p (α) and that .q (t) ≥ α whenever .t < p (α−). Consequently, .p (α) < t < p (α−) implies that .q (t) = α. This u n suffices for the proof of the proposition. .

3.1 The Right-Continuous Inverse of a Decreasing Function

125

At this point, it is appropriate to include a number of comments and observations. Remark 3.1.6 (i) Given a decreasing function .p : R → [−∞, ∞], the function .p+ : R → [−∞, ∞] is defined by setting .p+ (s) = p (s+) for all .s ∈ R. It is easily verified that .p+ is decreasing and right-continuous. The function .p+ is called the right-continuous regularization of p. It follows from Proposition 3.1.3 (v) that p and .p+ have the same right-continuous inverse. Similarly, the function .p− : R → [−∞, ∞], given by .p− (s) = p (s−), .s ∈ R, is called the leftcontinuous regularization of p. Observe that p and .p− have the same rightcontinuous inverse, as follows from Proposition 3.1.3 (v). (ii) Suppose that the function .p : [0, a) → [−∞, ∞] is decreasing and rightcontinuous, where .a ∈ (0, ∞]. The function p may be extended to a decreasing and right-continuous function .p˜ : R → [−∞, ∞] by setting ⎧ ⎨ ∞ if s < 0 .p ˜ (s) = p (s) if s ∈ [0, a) . ⎩ −∞ if s ≥ a (in case a < ∞) Let .q : R → [−∞, ∞] be the right-continuous inverse of .p. ˜ Note that .0 ≤ q (t) ≤ a for all .t ∈ R. For each .t ∈ R, .

{s ∈ R : p˜ (s) ≤ t} = {s ∈ [0, a) : p (s) ≤ t} ∪ [a, ∞) ,

which implies that q (t) = inf {s ∈ [0, a) : p (s) ≤ t} ,

.

t ∈R

(where .inf ∅ = a). In this setting, q is called the right-continuous inverse of p. If .t ∈ R, then it follows from Proposition 3.1.4 (i) that .{s ∈ R : p˜ (s) > t} = (−∞, q (t)). Since .p˜ (s) = −∞ whenever .s ≥ a and .q (t) ≤ a for all .t ∈ R, this implies that .

{s ∈ [0, a) : p (s) > t} = [0, q (t)) ,

t ∈ R.

As a consequence, it follows that q (t) = m {s ∈ [0, a) : p (s) > t} ,

.

t ∈ R,

(3.2)

where m denotes Lebesgue measure on .R. (iii) Suppose that .p : R → [0, a] is a decreasing right-continuous function, where .a ∈ (0, ∞]. Let .q : R → [−∞, ∞] be the right-continuous inverse of p. If .t < 0, then .q (t) = ∞ and .q (t) = −∞ whenever .t ≥ a (in case .a < ∞). Therefore, it is customary to consider q as a function .q : [0, a) → [−∞, ∞].

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3 Singular Value Functions

Note that p is the right-continuous inverse of q according to the conventions agreed on in (ii). In particular, it follows from (3.2) that p (s) = m {t ∈ [0, a) : q (t) > s} ,

s ∈ R.

.

(3.3)

The above concepts are illustrated by the important example of decreasing rearrangements of functions. Suppose that .(X, Σ, ν) is a finite measure space and let .M (X, ν) denote the collection of all extended real-valued (.Σ-) measurable functions .f : X → [−∞, ∞]. Definition 3.1.7 If .f ∈ M (X, ν), the distribution function .d (f ) : R → [0, ν (X)] is defined by d (s; f ) = ν {x ∈ X : f (x) > s} ,

.

s ∈ R.

Note that the distribution function .d (f ) is also given by f d (s; f ) =

χ(s,∞) (f ) dν,

.

s ∈ R.

X

It is clear that .d (f ) is decreasing, and by the countable additivity of .ν, it follows readily that .d (f ) is right-continuous. Before proceeding, two further definitions are required. ( ' ' ') Definition 3.1.8 Suppose .(X, Σ, ν) and . X , Σ , ν are two finite measure ( that ) ) ( ' ' spaces with .ν (X) = ν X . Two functions .f ∈ M (X, ν) and .g ∈ M X' , ν ' are said to be equimeasurable if .d (f ) = d (g). Of course, it is to be noted that, in the above definition, the distribution function d (f ) is calculated relative to the measure .ν on X, whereas the distribution function ' ' .d (g) is calculated relative to the measure .ν on .X . .

Definition 3.1.9 If .f ∈ M (X, ν), then the decreasing rearrangement .λ (f ) of f is defined to be the right-continuous inverse of the distribution function .d (f ), that is, λ (t; f ) = inf {s ∈ R : d (s; f ) ≤ t} ,

.

t ∈ [0, ν (X)) .

Note that the restriction to the interval .[0, ν (X)) in the above definition is in accordance with Remark 3.1.6 (iii). Furthermore, it should be observed that (3.3) implies that d (s; f ) = m {t ∈ [0, ν (X)) : λ (t; f ) > s} ,

.

s ∈ R.

In other words, .d (f ) = d (λ (f )), that is, the functions f and .λ (f ) are equimeasurable. Consequently, for each .f ∈ M (X, ν), the decreasing rearrangement .λ (f ) is a decreasing right-continuous function on .[0, ν (X)) that is equimeasurable

3.1 The Right-Continuous Inverse of a Decreasing Function

127

with f and .λ (f ) is uniquely determined by this property, as is easily verified. Consequently, two extended real-valued functions f and g,( defined on the finite ( ) ) measure spaces .(X, Σ, ν) and . X' , Σ ' , ν ' with .ν (X) = ν ' X' , respectively, are equimeasurable if and only if .λ (f ) = λ (g). The notion of equimeasurability is sufficiently important to deserve an alternative characterization. In the next proposition, the Borel .σ -algebra in .[−∞, ∞] is denoted by .B ([−∞, ∞]). It is well known that .B ([−∞, ∞]) is the .σ -algebra of subsets of .[−∞, ∞] generated by the semi-ring consisting of all cells .(s1 , s2 ] with .s1 < s2 in .R. ( ) Proposition 3.1.10 Let .(X, Σ, ν) and . X' , Σ ' , ν ' be( finite)measure spaces with ( ) ' ' ' ' .ν (X) = ν X . If .f ∈ M (X, ν) and .g ∈ M X , ν , then f and g are equimeasurable if and only if ( ( ) ) ν f −1 (E) = ν ' g −1 (E) ,

.

E ∈ B ([−∞, ∞]) .

Proof Define the finite measures .νf and .νg' on .B ([−∞, ∞]) by .νf (E) = ( ( ) ) ν f −1 (E) and .νg' (E) = ν ' g −1 (E) for .E ∈ B ([−∞, ∞]), respectively (in other words, .νf and .νg' are the image measures of .ν and .ν ' under f and g, respectively). If the functions f and g are equimeasurable, then .νf and .νg' agree on all intervals of the form .(s, ∞], .s ∈ R, and hence on all intervals .(s1 , s2 ], .s1 < s2 . The uniqueness of measure extension (for finite measures) now implies that .νf = νg' on .B ([−∞, ∞]). The converse implication is trivial as .νf = νg' implies that .d (s; f ) = νf ((s, ∞]) = νg' ((s, ∞]) = d (s; g) for all .s ∈ R. u n ( ' ' ') Corollary 3.1.11 ( ' ) Let .(X, Σ, ν) and . X , Σ ,(ν ' be' ) finite measure spaces with ' .ν (X) = ν X . If .f ∈ M (X, ν), .g ∈ M X , ν are equimeasurable and if .p : [−∞, ∞] → [−∞, ∞] is any Borel function, then .p ◦ f and .p ◦ g are also equimeasurable. Proof If .E ∈ B (R), then it follows from Proposition 3.1.10 that ( ( ( ( ) ( )) )) ν (p ◦ f )−1 (E) = ν f −1 p−1 (E) = ν ' g −1 p−1 (E) ( ) = ν ' (p ◦ g)−1 (E) ,

.

and consequently, .p ◦ f and .p ◦ g are equimeasurable.

u n

If .(X, Σ, ν) is an infinite measure space, then a moment’s reflection shows that the notion of decreasing rearrangement as introduced above is not very useful in general. However, for positive functions (in particular, for the absolute value of a function), this notion still is very useful and important (as will become clear from the subsequent sections). Some of the details will be briefly indicated. Let .(X, Σ, ν) be a Maharam measure space and .f : X → C (or, .f : X → [−∞, ∞]) be a

128

3 Singular Value Functions

measurable function. The distribution function .d (|f |) of .|f | is defined by d (s; |f |) = ν {x ∈ X : |f (x)| > s} ,

.

s ∈ [0, ∞) .

The notion of equimeasurability is defined as above. The right-continuous inverse of .d (|f |), which is denoted by .μ (f ), is called the decreasing rearrangement of the function .|f |. So, .μ (f ) : [0, ∞) → [0, ∞] is given by μ (t; f ) = inf {s ≥ 0 : d (s; |f |) ≤ t} .

.

Also in this situation it follows from (3.3) that d (s; |f |) = m {t ∈ [0, ∞) : μ (t; f ) > s} ,

.

s ≥ 0,

and so, the functions .|f | and .μ (f ) are equimeasurable. In the literature, the decreasing rearrangement of .|f | is frequently denoted by .f ∗ . For obvious reasons, this notation will be avoided in the present text and .μ (f ) is used instead. The following observation is formulated as a lemma for future reference. + Lemma 3.1.12 If .(fn )∞ n=1 is an increasing sequence in .M (X, ν) and if .f ∈ + M (X, ν) is defined by .f (x) = supn fn (x), .x ∈ X, then .μ (t; fn ) ↑n μ (t; f ) for all .t > 0.

Proof Since .g ≤ h in .M (X, ν)+ implies that .μ (t; g) ≤ μ (t; h) for all .t ≥ 0, it is clear that .μ (t; fn ) ↑n and that .supn μ (t; fn ) ≤ μ (t; f ) for all .t ≥ 0. For the proof of the reverse inequality, it may be assumed that .t ≥ 0 is such that .supn μ (t; fn ) < ∞. Let .s ∈ (0, ∞) be such that .supn μ (t; fn ) < s, which implies that .d (s; fn ) ≤ t for all n. Since .

{x ∈ X : fn (x) > s} ↑n {x ∈ X : f (x) > s} ,

it follows that .d (s; fn ) ↑n d (s; f ). Consequently, .d (s; f ) ≤ t, and so, .μ (t; f ) ≤ s. This suffices to show that .μ (t; f ) ≤ supn μ (t; fn ). The proof is complete. u n It should be pointed out that it may occur that .d (s; |f |) = ∞ for all .s ≥ 0 (consider, e.g., the function .f : t − | → t on .[0, ∞)), in which case the distribution function has little utility. Therefore, one restricts its attention in this context usually to the class of functions .f ∈ L0 (ν) (that is, measurable functions that are finite .ν-a.e.) satisfying .d (s; |f |) < ∞ for some .s > 0. The space of functions satisfying these conditions has already been introduced in Example 2.3.13 (b) and is denoted by .S (ν). Observe that if .f ∈ M (X, ν), then .f ∈ S (ν) if and only if .d (s; |f |) → 0 as .s → ∞, if and only if .μ (t; f ) < ∞ for all .t > 0. It should also be pointed out, however, that the notions of decreasing rearrangement and equimeasurability have to be handled with some care in the case of infinite measure spaces, as is shown in the next example (and the reader is advised to keep

3.2 The Singular Value Function

129

this in mind when reading the subsequent sections, where such phenomena will also occur). Example 3.1.13 Let .X = [0, ∞) equipped with Lebesgue measure m and the function .f : [0, ∞) → [0, ∞) be defined by .f (x) = arctan x. The distribution function of f is given by .d (s; f ) = ∞ if .s < π/2 and .d (s; f ) = 0 whenever .s ≥ π/2. Accordingly, .μ (t; f ) = π/2 for all .t ≥ 0. Consequently, the functions f and .μ (f ) are equimeasurable, but the image measures of Lebesgue measure under these two( functions) are quite different. Indeed, the image measure under f is the measure . 1/ cos2 x dx on .[0, π/2), whereas the image measure under .μ (f ) is the point measure concentrated at .π/2 with value .∞. In particular, the analogue of Proposition 3.1.10 fails in this case. References: [8, 76, 85].

3.2 The Singular Value Function Throughout this section, .M is assumed to be a semi-finite von Neumann algebra on a Hilbert space H with a given semi-finite faithful normal trace .τ . Recall that .S (τ ) is the .∗-algebra of all .τ -measurable operators. If .x ∈ S (τ ), then the spectral distribution function .d (|x|) of .|x| is defined by setting ) ( d (s; |x|) = τ e|x| (s, ∞) ,

.

s ≥ 0.

Evidently, the function .d (|x|) : [0, ∞) → [0, ∞] is decreasing.( If .s ∈ [0,)∞) and .sn ↓ s, then .e|x| (sn , ∞) ↑ e|x| (s, ∞) in .M+ , and so, .τ e|x| (sn , ∞) ↑ ( ) τ e|x| (s, ∞) . Hence, .d (|x|) is right-continuous on .[0, ∞). Since .x ∈ S (τ ), it follows from Proposition 2.3.6 (iv) and (v) that there exists .λ > 0 such that .d (λ; |x|) < ∞ and that .d (s; |x|) → 0 as .s → ∞ (but, it may very well happen that .d (s; |x|) = ∞ on an interval .[0, λ0 ) for some .λ0 > 0; see Example 3.2.2 (i) and also Lemma 3.2.3 (ii)). Definition 3.2.1 For .x ∈ S (τ ), the singular value function .μ (x) is defined to be the right-continuous inverse of the spectral distribution function .d (|x|), that is, μ (t; x) = inf {s ≥ 0 : d (s; |x|) ≤ t} ,

.

t ≥ 0.

Consequently, the function .μ (x) : [0, ∞) → [0, ∞] is decreasing and rightcontinuous. Note that, by definition, .μ (x) = μ (|x|) for all .x ∈ S (τ ). Since .d (s; |x|) → 0 as .s → ∞, it is clear that .μ (t; x) < ∞ for all .t > 0 (note that the equality .μ (0; x) = ∞ may occur; see Lemma 3.2.3 (i)). From the properties of

130

3 Singular Value Functions

the right-continuous inverse, it also follows that d (s; |x|) = m {t ≥ 0 : μ (t; x) > s} = d (s; μ (x)) ,

.

s ≥ 0.

(3.4)

Furthermore, if .τ (1) < ∞, then .d (s; |x|) ≤ τ (1) for all .s ≥ 0 and so, .μ (t; x) = 0 for all .t ≥ τ (1). Therefore, in the case that .τ (1) < ∞, one could consider .μ (x) as a function on the interval .[0, τ (1)), but in the present exposition .μ (x) will always be considered as a function on .[0, ∞) (unless explicitly stated otherwise). The following examples illustrate the preceding ideas: Example 3.2.2 (i) Let Σm .(M, τ ) be a semi-finite von Neumann algebra and suppose that .a = j =1 αj pj , where .p1 , . . . , pm ∈ P (M) with .pj pk = 0 whenever .j /= k, and .0 < αj ∈ R (.j = 1, . . . , m) are such that .αj /= αk whenever .j /= k. For the computation of .μΣ (a), it may be assumed that .α1 > α2 > · · · > αm > 0. Setting .pm+1 = 1 − m j =1 pj and .αm+1 = 0, the spectral measure of a is then given by ea =

m+1 Σ

.

pj δαj ,

j =1

where .δαj denotes the Dirac measure at the point .αj . Since ea (λ, ∞) =

Σ

.

pj ,

λ ≥ 0,

αj >λ

the spectral distribution function of a is given by Σ ( ) ( ) d (λ; a) = τ ea (λ, ∞) = τ pj ,

.

λ ≥ 0.

αj >λ

} { ( ) ( ) Defining .k = min 1 ≤ j ≤ m : τ pj = ∞ (if .τ pj < ∞ for all .1 ≤ j ≤ m, set .k = m + 1, in which case .αk = 0), it follows that ⎧ ⎪ 0 if λ ≥ α1 ⎨Σ j .d (λ; a) = τ if αj +1 ≤ λ < αj (j = 1, . . . , k − 1) . (p ) i i=1 ⎪ ⎩∞ if 0 ≤ λ < αk Σj Define .ρj = i=1 τ (pi ) for .j = 1, . . . , m and .ρ0 = 0. It is now easily verified that .μ (a) is given by μ (a) =

k−1 Σ

.

j =1

αj χ[ρj −1 ,ρj ) + αk χ[ρk−1 ,∞)

(3.5)

3.2 The Singular Value Function

131

( ) (note that if .τ pj < ∞ for all .1 ≤ j ≤ m, then .k − 1 = m and the last term is equal to zero in the above formula). Note, in particular, if .a = 1 and .τ (1) = ∞, then .d (λ; 1) = 0 for all .λ ≥ 1 and .d (λ; 1) = ∞ whenever .0 ≤ λ < 1, and so, .μ (1) = χ[0,∞) . It will be convenient to have another form of .μ (a) available. For this purpose, first note that via summation by parts, a may also be written as a=

m+1 Σ

.

βj qj ,

j =1

Σj where .qj = i=1 pi (.1 ≤ j ≤ m + 1), .βj = αj − αj +1 > 0 for .j = 1, . . . , m, and .βm+1 = 0. Note that q1 ≤ q2 ≤ · · · ≤ qm ≤ qm+1 = 1

.

( ) and that .τ qj = ρj , .1 ≤ j ≤ m. Applying a similar summation by parts to equation (3.5), it follows that μ (a) =

k Σ

.

j =1

βj χ[0,τ (qj )) .

( ) Since .μ qj = χ[0,τ (qj )) , this may also be written as μ (a) =

k Σ

.

( ) βj μ qj .

(3.6)

j =1

(ii) Consider the special case that .H = Cn and .M = B (H ) ∼ = Mn (C) equipped with the standard Σ trace .τn . If .a ∈ Mn (C) is positive self-adjoint, then a may be written as .a = m j =1 αj pj , where .α1 > · · · > αm > 0 are the distinct nonzero eigenvalues of a and .pj is the orthogonal projection onto the eigenspace corresponding to .αj . It follows from (3.5) that μ (a) =

m Σ

.

j =1

αj χ[ρj −1 ,ρj ) ,

(3.7)

where the numbers .ρj are defined as in (i) above. For each j , the length of [ ) ( ) the interval . ρj −1 , ρj is .τ pj , which is the dimension of the eigenspace corresponding to .αj . Consequently, (3.7) may be rewritten as μ (a) =

n Σ

.

j =1

λj χ[j −1,j ) ,

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3 Singular Value Functions

where .λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0 is the sequence of eigenvalues of a in which each eigenvalue is repeated according to its multiplicity. If .x ∈ Mn (C) is arbitrary, then .μ (x) = μ (|x|), and the eigenvalues of .|x| are usually called the singular values of x. Applying the above observations to .a = |x|, it follows that μ (x) =

n Σ

.

μj χ[j −1,j ) ,

j =1

where .μ1 ≥ μ2 ≥ · · · ≥ μn ≥ 0 is the sequence of singular values of x, repeated according to multiplicity. (iii) Now suppose that .M = B (H ) (H any Hilbert space), equipped with the standard trace .τ . Suppose that .a ∈ B (H ) is a positive self-adjoint compact operator. From the spectral theorem, it follows that a can be written as a=

Σ

.

αj pj

j

(norm convergent series in .B (H )), where .α1 > α2 > · · · > 0 is the (finite or infinite) sequence of distinct non-zero eigenvalues of a and each .pj is the projection onto the eigenspace corresponding to .αj . Note that ) ( orthogonal .τ pj < ∞ is the dimension of the eigenspace corresponding to .αj . Since the spectral measure of a is given by Σ

ea =

.

( pj δαj + ⎝1 −

j

Σ

⎞ p j ⎠ δ0 ,

j

a calculation analogous to that in (i) above shows that μ (a) =

Σ

.

αj χ[ρj −1 ,ρj ) ,

j

Σj where .ρj = i=1 τ (pi ) for all j and .ρ0 = 0. Since the length of the interval [ ) . ρj −1 , ρj is equal to the multiplicity of the eigenvalue .αj , this may also be written as Σ .μ (a) = λj χ[j −1,j ) , j

where .λ1 ≥ λ2 ≥ · · · > 0 is the sequence of non-zero eigenvalues of a, repeated according to multiplicity. If .x ∈ B (H ) is an arbitrary compact operator, then .|x| is also compact, and the eigenvalues of .|x| are called the

3.2 The Singular Value Function

133

singular values of x. Accordingly, μ (x) =

Σ

.

μj χ[j −1,j ) ,

j

where .μ1 ≥ μ2 ≥ · · · > 0 is the sequence of non-zero singular values of x, repeated according to multiplicity. For later reference, two simple properties of .μ (x) are formulated in the next lemma. Lemma 3.2.3 Suppose that .x ∈ S (τ ): (i) .μ (0; x) < ∞ if and only if .x ∈ M, in which case .μ (0; x) = ||x||B(H ) . (ii) If .λ0 = inf {s ≥ 0 : d (s; |x|) < ∞}, then .limt→∞ μ (t; x) = λ0 . Proof |x| (i) If .x ∈ S (τ ) and .s ≥ 0, then it is clear that = 0 if and}only if .x ∈ M { .e (s, ∞) and .||x||B(H ) ≤ s. Since .μ (0; x) = inf s ≥ 0 : e|x| (s, ∞) = 0 , statement (i) follows immediately. (ii) If .t ≥ 0, then .{s ≥ 0 : d (s; |x|) ≤ t} ⊆ [λ0 , ∞), and so, .μ (t; x) ≥ λ0 for all .t ≥ 0. It follows from .s > λ0 that .d (s; |x|) < ∞, and hence, by Proposition 3.1.3 (ii), .μ (d (s; |x|) ; x) ≤ s. Since .μ (x) is decreasing, this suffices to show that .limt→∞ μ (t; x) = λ0 . u n

It follows in particular from (ii) in the lemma above that if .x ∈ S (τ ), then d (s; |x|) < ∞ for all .s > 0 if and only if .limt→∞ μ (t; x) = 0. Recall from Definition 2.4.1 that the collection of all .x ∈ S (τ ) with the property that .d (s; |x|) < ∞ for all .s > 0 is denoted by .S0 (τ ) (which is a .∗-closed two-sided ideal in .S (τ ); see Proposition 2.4.4). These observations immediately yield the following result.

.

Proposition 3.2.4 The subalgebra .S0 (τ ) of .S (τ ) satisfies } { S0 (τ ) = x ∈ S (τ ) : lim μ (t; x) = 0 .

.

t→∞

There is an intimate relationship between the properties of the neighborhoods for the measure topology and the properties of the singular value function, which will now be discussed and exploited. Recall that a neighborhood base at zero for the measure topology in .S (τ ) is given by the sets } { ( ) V (ε, δ) = x ∈ S (τ ) : ∃ p ∈ P (M) such that ||xp||B(H ) ≤ ε, τ p⊥ ≤ δ ,

.

134

3 Singular Value Functions

where .ε, δ > 0 (see Sect. 2.5). As shown in Lemma 2.5.1, these neighborhoods are also given by { ( ) } V (ε, δ) = x ∈ S (τ ) : τ e|x| (ε, ∞) ≤ δ .

.

In other words, using Proposition 3.1.4 (i), V (s, t) = {x ∈ S (τ ) : d (s; |x|) ≤ t}

.

= {x ∈ S (τ ) : μ (t; x) ≤ s}

(3.8)

for all .s, t > 0. From the definition of .μ (t; x), it is also clear that μ (t; x) = inf {s > 0 : x ∈ V (s, t)}

.

(3.9)

for all .t > 0. Moreover, it follows from .d (μ (t; x) ; |x|) ≤ t that x ∈ V (μ (t; x) , t) ,

.

t > 0, μ (t; x) > 0.

(3.10)

These observations make it possible to transfer properties of the neighborhoods V (s, t) to corresponding properties for singular value functions. This is illustrated in the following propositions.

.

Proposition 3.2.5 Let .x ∈ S (τ ). For all .t ≥ 0, the singular value function .μ (x) admits the characterization } { ( ) ⊥ ≤t . .μ (t; x) = inf ||xp||B(H ) : p ∈ P (M) , p (H ) ⊆ D (x) , τ p (3.11) Proof For .t = 0, (3.11) is precisely (i) of Lemma 3.2.3. If .t > 0, then (3.11) follows immediately from (3.9) and the definition of the sets .V (s, t). u n Remark 3.2.6 (i) Observe that it follows immediately from the above proposition that .μ (t; xe) = 0 whenever .t ≥ τ (e) for all .x ∈ S (τ ) and .e ∈ P (M) with ( .τ (e) )< ∞. Indeed, if .t ≥ τ (e), then, trivially, .xe (1 − e) = 0 and .τ (1 − e)⊥ ≤ t; hence, .μ (t; xe) = 0. In particular, .μ (t; x) = 0 for all .t ≥ τ (s (x)), where .s (x) is the support projection of .x ∈ S (τ ). (ii) It should be observed that τ (s (x)) = inf {t ≥ 0 : μ (t; x) = 0}

.

(with the convention that .inf ∅ = ∞). Indeed, calling the infimum on the right hand side .t0 , it is clear from (i) preceding that .t0 ≤ τ (s (x)). On the other hand, if .t0 < ∞, then .μ (t0 ; x) = 0, which implies that .d (0; x) ≤ t0 , that

3.2 The Singular Value Function

135

( ) is, .τ e|x| (0, ∞) ≤ t0 . Since .s (x) = s (|x|) = e|x| (0, ∞), this shows that .τ (s (x)) ≤ t0 . In the next proposition, some elementary properties of singular value functions are collected. Proposition 3.2.7 For all .x, y ∈ S (τ ), the following hold: (i) (ii) (iii) (iv) (v) (vi)

μ (λx) = |λ| μ (x) for all .λ ∈ C. μ (x ∗ ) = μ (x). .μ (t1 + t2 ; x + y) ≤ μ (t1 ; x) + μ (t2 ; y) for all .t1 , t2 ≥ 0. .μ (t1 + t2 ; xy) ≤ μ (t1 ; x) μ (t2 ; y) for all .t1 , t2 ≥ 0. If .|y| ≤ |x|, then .μ (y) ≤ μ (x). .μ (uxv) ≤ ||u||B(H ) ||v||B(H ) μ (x) for all .u, v ∈ M. . .

Proof First note that by the right-continuity of the singular value functions, it is sufficient to establish the above relations on the interval .(0, ∞). Using (3.9), these properties follow easily from the corresponding properties of the neighborhoods .V (s, t) obtained in Proposition 2.5.2. Indeed, statements (i), (ii), (v), and (vi) are immediate consequences of (ii), (viii), (x), and (xi) of Proposition 2.5.2, respectively. Next (iii) will be proved. Since singular value functions are rightcontinuous, it may be assumed that .t1 , t2 > 0. If .α > μ (t1 ; x) and .β > μ (t2 ; y), then it follows from (3.9) and (vi) of Proposition 2.5.2 that x + y ∈ V (α, t1 ) + V (β, t2 ) ⊆ V (α + β, t1 + t2 ) ,

.

and hence, (3.9) implies that .μ (t1 + t2 ; x + y) ≤ α + β. This suffices for the proof u n of (iii). Statement (iv) follows in the same way from Proposition 2.5.2 (vii). Note that it follows in particular from (vi) in the above proposition that μ (u∗ xu) = μ (x) whenever .x ∈ S (τ ) and .u ∈ U (M) (that is, the singular value function is unitarily invariant).

.

Proposition 3.2.8 Let .φ : [0, ∞) → [0, ∞) be an increasing function that is left-continuous on .(0, ∞) and .a ∈ S (τ )+ . If necessary, .φ (∞) is interpreted as .φ (∞) = limt→∞ φ (t) ∈ [0, ∞]: (i) If .τ (1) = ∞, then μ (φ (a)) = φ ◦ μ (a)

.

holds on .[0, ∞). (ii) If .τ (1) < ∞, then μ (φ (a)) = φ ◦ μ (a) χ[0,τ (1)) .

.

If, in addition, .φ (0) = 0, then .μ (φ (a)) = φ ◦ μ (a) holds on .[0, ∞).

(3.12)

136

3 Singular Value Functions

Proof Given .a ∈ S (τ )+ , it is evident that the function .φ ◦ μ (a) : [0, ∞) → [0, ∞] is decreasing, and it is easily verified that .φ ◦μ (a) is right-continuous. Furthermore, observe that ( ) ( ( )) φ(a) .d (s; φ (a)) = τ e (s, ∞) = τ ea φ −1 (s, ∞) , s ≥ 0. It follows from the stated assumptions on the function .φ that for each .s ≥ φ (0) the set .φ −1 (s, ∞) is either of the form .(α, ∞) for some .α ≥ 0, or .φ −1 (s, ∞) = ∅. Assume first that .φ −1 (s, ∞) = (α, ∞) for some .α ≥ 0. Using (3.4) and the fact that .μ (t; a) > α if and only if .φ (μ (t; a)) > s, it follows that in this case ( ) d (s; φ (a)) = τ ea (α, ∞) = d (α; a) = d (α; μ (a)) = d (s; φ ◦ μ (a)) .

.

If (.φ −1 ( (s, ∞) =))∅ (that is, .φ (t) ≤ s for all .t ≥ 0), then .d (s; φ (a)) = τ ea φ −1 (s, ∞) = 0 and also .d (s; φ ◦ μ (a)) = 0. This shows that d (s; φ (a)) = d (s; φ ◦ μ (a)) ,

s ≥ φ (0) .

.

(3.13)

If .φ (0) = 0, it follows that .d (φ (a)) = d (φ ◦ μ (a)) on .[0, ∞). Since the function φ ◦ μ (a) is decreasing and right-continuous, this implies that

.

μ (φ (a)) = φ ◦ μ (a)

.

holds on .[0, ∞), in case (i) as well as (ii). Now suppose that .φ (0) > 0 and let .0 ≤ s < φ (0). ( ) ( ( )) d (s; φ (a)) = τ eφ(a) (s, ∞) = τ ea φ −1 (s, ∞) ( ) = τ ea [0, ∞) = τ (1) .

.

(3.14)

At the same time, d (s; φ ◦ μ (a)) = m {t ≥ 0 : φ (μ (t; a)) > s} = ∞,

.

(3.15)

since .φ (μ (t; a)) ≥ φ (0) > s for all .t ∈ [0, ∞). Consequently, if .τ (1) = ∞, it follows that d (s; φ (a)) = d (s; φ ◦ μ (a)) ,

.

0 ≤ s < φ (0) .

Combined with the assertion of (3.13), this yields that .d(s; φ(a)) = d(s; φ ◦ μ(a)) holds on .[0, ∞), and, as above, it follows that .μ (φ (a)) = φ ◦ μ (a) on .[0, ∞). By this, (i) is proved. Finally, assume that .τ (1) < ∞. If .0 ≤ t < τ (1) and .s ≥ 0, then it follows from (3.14) and (3.15) that .d (s; φ (a)) ≤ t if and only if .d (s; φ ◦ μ (a)) ≤ t, in

3.2 The Singular Value Function

137

which case .s ≥ φ (0) and so, by (3.13), .d (s; φ (a)) = d (s; φ ◦ μ (a)). Using that φ ◦ μ (a) is decreasing and right-continuous, this implies that

.

μ (t; φ (a)) = inf {s ≥ 0 : d (s; φ (a)) ≤ t}

.

= inf {s ≥ 0 : d (s; φ ◦ μ (a)) ≤ t} = φ (μ (t; a)) whenever .0 ≤ t < τ (1). This completes the proof of assertion (ii).

u n

Remark 3.2.9 In connection with statement (ii) of the above proposition, the following example is illustrative. If .τ (1) < ∞ and the function .φ is given by .φ (t) = 1, .t ∈ [0, ∞), then .φ (a) = 1 for any .a ∈ S (τ )+ . Therefore, .μ (φ (a)) = χ[0,τ (1)) , whereas .φ (μ (t; a)) = 1 for all .t ≥ 1. This shows that the identity .μ (φ (a)) = φ ◦ μ (a) does not hold on the whole interval .[0, ∞), but only on .[0, τ (1)). In the next proposition, some further properties of singular value functions are collected. Proposition 3.2.10 (i) For .x ∈ S (τ ), .μ (x ∗ x) = μ (xx ∗ ). (ii) For positive .x, y ∈ S(τ ), .μ(x 1/2 yx 1/2 ) = μ(y 1/2 xy 1/2 ). (iii) If .x ∈ S(τ ), .0 < α ∈ R, and .e = e|x| (α, ∞), then: (a) .μ (|x| ( e) = )μ (x) χ[0,τ (e)) . (b) .μ t; |x| e⊥ = μ (t + τ (e) ; x) for all .t ≥ 0 whenever .τ (e) < ∞. Proof (i) Since .x ∗ x = |x|2 and .xx ∗ = |x ∗ |2 , it follows from Propositions 3.2.8 and 3.2.7 (ii) that ) ( ( ) ( )2 μ x ∗ x = μ |x|2 = μ (|x|)2 = μ (x)2 = μ x ∗ (| | ) ( ) (| |)2 2 = μ |x ∗ | = μ |x ∗ | = μ xx ∗ .

.

(ii) Writing .x 1/2 yx 1/2 = (y 1/2 x 1/2 )∗ y 1/2 x 1/2 , this follows immediately from (i). (iii) For the proof of (a), note first that .|x| e = φ (|x|), where the function .φ is given by .φ (λ) = λχ(α,∞) (λ), .λ ∈ [0, ∞). The function .φ : [0, ∞) → [0, ∞) is increasing, left-continuous, and .φ (0) = 0, and hence, by Proposition 3.2.8, μ (|x| e) = φ ◦ μ (x) = μ (x) χ(α,∞) (μ (x)) .

.

If .t ≥ 0, then .χ(α,∞) (μ (t; x)) = 1 if and only (if .μ (t; x) > ) α, which by Proposition 3.1.4 is equivalent to .d (α; |x|) = τ e|x| (α, ∞) > t, that is, .0 ≤ t < τ (e). This shows that .χ(α,∞) (μ (x)) = χ[0,τ (e)) , and so, .μ (|x| e) = μ (x) χ[0,τ (e)) .

138

3 Singular Value Functions

Turning to the proof of (b), assume in addition that .τ (e) < ∞. Since .|x| e⊥ = |x| e|x| [0, α] = f (|x|), where .f (λ) = λχ[0,α] (λ), .λ ∈ [0, ∞), it is clear that ) ( ( ( )) d s; |x| e⊥ = τ e|x| f −1 (s, ∞) ,

.

s ≥ 0.

Now observe that .f −1 (s, ∞) = (s, α] if .0 ≤ s < α and .f −1 (s, ∞) = ∅ whenever .s ≥ α. Consequently, ) { τ (e|x| (s, α]) if 0 ≤ s < α, ( ⊥ = .d s; |x| e 0 if s ≥ α. ( ) Since .e|x| (s, α] = e|x| (s, ∞) − e|x| (α, ∞) and .τ e|x| (α, ∞) = τ (e) < ∞, it ( |x| ) follows that .τ e (s, α] = d (s; |x|)−d (α; |x|) (with the convention that .∞−r = ∞ for .r ∈ R). This shows that ) ( d s; |x| e⊥ = (d (s; |x|) − d (α; |x|))+ ,

.

s ≥ 0.

If .t ≥ 0, then the condition .(d (s; |x|) − d (α; |x|))+ ≤ t is equivalent to the condition that .d (s; |x|) ≤ t + d (α; |x|), and so, ) ( { } μ t; |x| e⊥ = inf s ≥ 0 : (d (s; |x|) − d (α; |x|))+ ≤ t

.

= inf {s ≥ 0 : d (s; |x|) ≤ t + d (α; |x|)} = μ (t + d (α; |x|) ; |x|) = μ (t + τ (e) ; |x|) for all .t ≥ 0. This completes the proof of (b).

u n

The following proposition establishes the relation between convergence in measure in .S (τ ) and singular value functions. Note that the notation .lim is used for the limit inferior and .lim for the limit superior throughout this book. Proposition 3.2.11 (i) If .{xn }∞ n=1 is a sequence in .S (τ ), then .xn → 0 for the measure topology if and only if .μ (t; xn ) → 0 as .n → ∞ for all .t > 0. (ii) If .x, xn ∈ S (τ ), .n = 1, 2, . . ., and if .xn → x for the measure topology, then μ (t; x) ≤ limn→∞ μ (t; xn ) ≤ limn→∞ μ (t; xn ) ≤ μ (t−; x)

.

for each .t > 0. Consequently, μ (t; x) = lim μ (t; xn )

.

n→∞

3.2 The Singular Value Function

139

whenever .t > 0 is a point of continuity of .μ (x). In particular, .μ (xn ) → μ (x) a.e. on .[0, ∞). Proof Tm

(i) Suppose that .xn → 0 and let .t > 0 be given. For every .ε > 0, there exists .N ∈ N such that .xn ∈ V (ε, t) for all .n ≥ N and so, by (3.8), .μ (t; xn ) ≤ ε for all .n ≥ N. The converse implication is established similarly. (ii) Let .t > 0 and .ε > 0 be given and observe that Proposition 3.2.7 (iii) implies that μ (t + ε; x) ≤ μ (t; xn ) + μ (ε; x − xn ) ,

.

n ∈ N.

Since .μ (ε; x − xn ) → 0 by (i), it follows that μ (t + ε; x) ≤ limn→∞ μ (t; xn )

.

for all .ε > 0. The right-continuity of .μ (x) at the point t now implies that μ (t; x) ≤ limn→∞ μ (t; xn ) .

.

If .0 < ε < t, then, by Proposition 3.2.7 (iii), μ (t; xn ) ≤ μ (t − ε; x) + μ (ε; xn − x) ,

.

n ∈ N.

Since .μ (ε; xn − x) → 0, it follows that limn→∞ μ (t; xn ) ≤ μ (t − ε; x)

.

for all .ε ∈ (0, t), and consequently, limn→∞ μ (t; xn ) ≤ μ (t−; x) .

.

This suffices to complete the proof of the proposition. u n It will now be shown that the singular value function preserves suprema and infima of directed systems in .S (τ )+ . It will be convenient to observe the following general property of upward directed families of decreasing right-continuous functions on .R+ . Lemma 3.2.12 Suppose that .{fα } is an upward directed family of positive decreasing right-continuous functions on .R+ such that .supα fα (t) < ∞ for all .t > 0. If the function .f : [0, ∞) → [0, ∞], defined by setting f (t) = sup fα (t) ,

.

α

t > 0,

140

3 Singular Value Functions

{ }∞ is right-continuous, then there exists an increasing subsequence . fαn n=1 of .{fα } such that .fαn (t) ↑ f (t) for all .t > 0. Consequently, ( ) the equality .f = supα fα holds in the Dedekind complete vector lattice .L0 R+ . Proof Let .{rk }∞ k=1 be an enumeration of .Q ∩ (0, ∞). Let .α1 be such that .fα1 (r1 ) ≥ f (r1 ) − 1. Next, let .α2 ≥ α1 be such that .fα2 (r1 ) ≥ f (r1 ) − 1/2( and ) .fα2 ((r2 )) ≥ f (r2 ) − 1/2. Continuing this way, let .αn ≥ αn−1 be such that .fα(n r)j ≥ f( rj) − 1/n for .j = 1, . . . , n. The sequence .α1 ≤ α2{≤ (· · ·) satisfies }.fαn rj ↑n f rj for all .j = 1, 2, . . .. If .t > 0, then .f (t) = sup f rj : t ≤ rj , as the function f is right-continuous and decreasing. Hence, ( ) ( ) f (t) = sup f rj = sup sup fαn rj

.

t≤rj

t≤rj

n

( ) = sup sup fαn rj = sup fαn (t) n t≤rj

n

{ }∞ for all .t > 0, which shows that the sequence . fαn n=1 has the desired properties. To justify the final of the lemma, first ( statement ) ( note ) that f evidently is an upper bound of .{fα } in .L0 R+ . Suppose that .g ∈ L0 R+ is also an upper bound of .{fα }. This implies in particular that .fαn ≤ g a.e. on .R+ for all n. Since .fαn (t) ↑ f (t) for all + .t > 0, it follows that .f ≤ g a.e. on .R . Therefore, f is the least upper bound of ( +) ( ) .{fα } in .L0 R , that is, .f = supα fα in .L0 R+ . u n Remark 3.2.13 It is worth noting that if f is a real-valued function, .{fα } is an upward directed system, and if .f (t) = supα fα (t) for every .t ∈ (R+ , )then it is in general not the case that the equality .f = supα fα holds in .L0 R+ . Indeed, if .E denotes the system of all finite subsets of .R+ , partially ordered by inclusion, then .χE (t) ↑E∈E 1 for all( .t ∈) R+ . Evidently, however, it is not the case that + . .supE∈E χE = 1 holds in .L0 R Proposition 3.2.14 { } + + (i) If . aβ is an upward directed )system in .S (τ ) and .a ∈ S (τ ) such that .aβ ↑ a ( holds in .Sh (τ ), then .μ t; aβ ↑β μ (t; a) for all .t ≥ 0. Moreover, there exists { }∞ { } ) ( an increasing subsequence . aβn n=1 of . aβ such that .μ t; aβn ↑n μ (t; a) ( ) for all .t ≥ 0, and consequently, .μ (a) = supβ μ aβ holds in .S (m). { } (ii) If . aβ is a downward directed system in .S (τ )+ such that .aβ ↓ 0 holds in + .Sh (τ ) and if there exists .a ∈ S0 (τ ) such that .0 ≤ aβ ≤ a for all .β, then ) ( .μ t; aβ ↓β 0 for all .t > 0. Moreover, there exists a decreasing subsequence { }∞ { } ( ) . aβn of . aβ such that .μ t; aβn ↓n 0 for all .t > 0, and consequently, ( )n=1 .μ aβ ↓β 0 holds in .S (m). Proof

( ( ) ) (i) It follows from Proposition 3.2.7 (v) that .μ t; aβ (↑β and ) .μ t; aβ ≤ μ (t; a) for all .β and .t ≥ 0, which implies that .supβ μ t; aβ ≤ μ (t; a). For the

3.2 The Singular Value Function

141

) ( proof of the reverse inequality, first take .t > 0 and set .s = supβ μ t; aβ . By (3.8), .aβ ∈ V (s, t) for all .β, and hence, it follows from Theorem 2.6.7 that .a ∈ V (s, t), that is, .μ (t; a) ≤ s. Consequently, ) ( μ (t; a) = sup μ t; aβ ,

t > 0.

.

β

Using the right-continuity of the singular value function at .t = 0, it now follows that ) ( μ (0; a) = sup μ (t; a) = sup sup μ t; aβ

.

t>0

t>0 β

( ) ) = sup sup μ t; aβ = sup μ 0; aβ , β t>0

(

β

and this completes the proof of the first statement of (i). The second statement is now an immediate consequence of Lemma 3.2.12. (ii) If .a ≥ aβ ↓ 0 in .Sh (τ ) with .a ∈ S0 (τ )+ , then it follows from Theorem 2.6.3 Tm

that .aβ → 0. Since the measure topology is metrizable, there exists a decreasing { }∞ Tm subsequence . aβn n=1 such that .aβn → 0. Proposition 3.2.11 (i) implies that ( ( ) ) .μ t; aβn ↓n 0 for all .t > 0. It is now also clear that .μ t; aβ ↓β 0 for all .t > 0, and so, the proof is complete. u n The result of Lemma 2.6.6 may be reformulated in terms of singular value functions. First note that if .x ∈ S (τ ) and .p ∈ P (M), then .μ (pxp) ≤ μ (x), as follows from (vi) of Proposition 3.2.7. Further, if .p ≤ q in .P (M), then .pxp = p (qxq) p, and so, .μ (pxp) ≤ μ (qxq). Since .τ (p ∨ q) ≤ τ (p) + τ (q) for all .p, q ∈ P (M), the set .{p ∈ P (M) : τ (p) < ∞} is upward directed. These observations show that the set .

{μ (pxp) : p ∈ P (M) , τ (p) < ∞}

is an upward directed family of functions on the interval .[0, ∞). Proposition 3.2.15 If .x ∈ S (τ ), then μ (t; x) = sup {μ (t; p |x| p) : p ∈ P (M) , τ (p) < ∞}

.

for all .t ≥ 0. Moreover, .μ (x) is the supremum of the upward directed system {μ (p |x| p) : p ∈ P (M) , τ (p) < ∞} in the space .S (m).

.

Proof Let .t > 0 and set α = sup {μ (t; p |x| p) : p ∈ P (M) , τ (p) < ∞} .

.

142

3 Singular Value Functions

From the remarks made above, it is clear that .α ≤ μ (t; x). For each .p ∈ P (M) with .τ (p) < ∞, it follows from .μ (t; p |x| p) ≤ α that .p |x| p ∈ V (α, t). An application of Lemma 2.6.6 yields that .x ∈ V (α, t), and hence, .μ (t; x) ≤ α. Consequently, .μ (t; x) = α. The case that .t = 0 now follows immediately from the right-continuity of the singular value function (cf. the proof of Proposition 3.2.14). The last statement of the proposition is clear by Lemma 3.2.12. u n Boundedness with respect to the measure topology of subsets of .S (τ ) may also be expressed in terms of the singular value function. By Lemma 2.6.10, a non-empty subset W of .S (τ ) is bounded with respect to the( measure topology if and only if ) for every .δ > 0 there exists .R > 0 such that .τ e|x| (R, ∞) ≤ δ for all .x ∈ W . ) ( Since the condition .d (R; |x|) = τ e|x| (R, ∞) ≤ δ is equivalent to saying that .μ (δ; x) ≤ R, the following result is now clear. Proposition 3.2.16 A non-empty subset W of .S (τ ) is bounded with respect to the measure topology if and only if .

sup μ (t; x) < ∞,

t > 0.

x∈W

{ } + Remark 3.2.17 Suppose that . aβ is an increasing ) .S (τ ) that is bounded for ( net in the measure topology and define .f (t) = supβ μ t; aβ for .t > 0 (so, .f (t) < ∞ for all .t > 0). By Theorem 2.6.15, .a = supβ aβ exists in .Sh (τ ). It follows from ) ( Proposition 3.2.14 (i) that .μ t; aβ ↑β μ (t; a) for all .t > 0, and hence, .f = μ (a). As is well known, the trace of a positive .n×n matrix is the sum of its eigenvalues. In the general case, the trace of a positive element of .M is obtained by integrating the singular value function, as the following formula shows. Proposition 3.2.18 If .a ∈ M+ , then f τ (a) =

∞

μ (t; a) dt.

.

0

Σ Proof Assume first that .a = nj=1 αj pj , where .pj ∈ P (M) (.j = 1, . . . , n) with .pi pj = 0 whenever .i /= j , and .0 < αj ∈ R. Without loss of generality, it may be assumed that .α1 > α2 > · · · > αn > 0. If .τ (pi ) = ∞ for some .1 ≤ i ≤ n, then .a ≥ αi pi , and so, .τ (a) ≥ αi τ (pi ) = ∞, which implies that .τ (a) = ∞. it is clear Furthermore, .μ (a) ≥ μ (αi pi ) = αi μ (pi ) and .μ (pi ) = χ[0,∞)(; hence, f∞ ) that .τ (a) = 0 μ (a) dt = ∞ in this case. Assuming that .τ pj < ∞ for all .1 ≤ j ≤ n, it follows from Example 3.2.2 (i) that μ (a) =

n Σ

.

j =1

αj χ[ρj −1 ,ρj ) ,

3.2 The Singular Value Function

where .ρj = f

Σj

i=1 τ

∞

.

143

(pi ). This implies that

μ (t; a) dt =

0

n Σ

n ( ) Σ ( ) αj ρj − ρj −1 = αj τ pj = τ (a) .

j =1

j =1

Now assume that .0 ≤ a ∈ M is arbitrary. Setting .αn,j = (j/n) ||a||B(H ) for .0 ≤ j ≤ n and all .n ∈ N, define n Σ

an =

.

αn,j −1 ea

(( ]) αn,j −1 , αn,j .

j =1

By the definition of spectral integrals, f 0 ≤ an ↑

.

(0,||a||B(H )

] λde

a

(λ) = a

in .Mh (and also .an → a with respect to the norm .||·||B(H ) ), and hence, the normality of the trace implies that .τ (an ) ↑ τ (a). It follows from the first part of this proof the f

∞

τ (an ) =

μ (t; an ) dt

.

0

for all n. Furthermore, Proposition 3.2.14 (i) shows that .μ (t; an ) ↑ μ (t; a) for all t ≥ 0, and so, the monotone convergence theorem implies that

.

f τ (an ) =

.

∞

f

0

Consequently, .τ (a) =

f∞ 0

∞

μ (t; an ) dt ↑

μ (t; a) dt. 0

u n

μ (t; a) dt.

In the last part of the present section, some alternative formulae for the singular value function will be derived. It will be convenient to introduce the following quantities. Given a self-adjoint operator a in the Hilbert space H and an orthogonal projection e in H , define .αe (a) ∈ [−∞, ∞] by setting } { αe (a) = sup : ||ξ ||H = 1, eξ = ξ, ξ ∈ D (a) .

.

Evidently, if .e ≤ p in .P (M), then( .αe (a) ) ≤ αp (a). If .a ≥ 0, and if the projection e is such that .e (H ) ⊆ D (a) ⊆ D a 1/2 , then it is clear that .αe (a) is also given by ||2 } || { || || αe (a) = sup : ||ξ ||H = 1, eξ = ξ = ||a 1/2 e||

.

B(H )

.

144

3 Singular Value Functions

Assume now that .a ≥ 0 and define .βe (a) ∈ [0, ∞] for any projection e in H by setting {|| ||2 )} ( || || βe (a) = sup ||a 1/2 ξ || : ||ξ ||H = 1, eξ = ξ, ξ ∈ D a 1/2 .

.

H

) ( As above, if e satisfies .e (H ) ⊆ D a 1/2 , then } || {|| || ||2 || || || 1/2 ||2 .βe (a) = sup ||a ξ || : ||ξ ||H = 1, eξ = ξ = ||a 1/2 e|| H

B(H )

.

) ( Since .D (a) ⊆ D a 1/2 , it is evident that .αe (a) ≤ βe (a) for all projections e, and if e is such that .e (H ) ⊆ D (a), then .αe (a) = βe (a). Proposition 3.2.19 If .a ∈ S (τ )+ , then: (i) } { ( ) μ (t; a) = inf αe (a) : e ∈ P (M) , τ e⊥ ≤ t } ( ) { = inf αe (a) : e ∈ P (M) , e (H ) ⊆ D (a) , τ e⊥ ≤ t

.

for all .t ≥ 0. (ii) } { ( ) μ (t; a) = inf βe (a) : e ∈ P (M) , τ e⊥ ≤ t ) ( ) } ( { = inf βe (a) : e ∈ P (M) , e (H ) ⊆ D a 1/2 , τ e⊥ ≤ t

.

for all .t ≥ 0. Proof Let .t ≥ 0 be fixed. Denote the infima in (i) by A and .A' , respectively, and the infima in (ii) by B and .B ' , respectively. From the observations above, it follows that A ≤ B ≤ B ' ≤ A' ,

.

and so, it is sufficient to show that .A' ≤ μ (t; a) and that .μ (t; a) ≤ A. ( ) First it will be proved that .μ (t; a) ≤ A. Suppose that .e ∈ P (M) with .τ e⊥ ≤ t. To show that .μ (t; a) ≤ αe (a), it may be assumed that .αe (a) < ∞. Let .γ ∈ R be such that .αe (a) < γ , and let .q = ea (γ , ∞)∧e. It will be shown that .q = 0. For this purpose, suppose first that .p ∈ P (M) is such that .p (H ) ⊆ D (a) and .p ∧ q /= 0. Take .ξ ∈ H such that .||ξ ||H = 1 and .ξ = pξ = qξ , which implies that .ξ ∈ D (a)

3.2 The Singular Value Function

145

and .ξ = eξ = ea (γ , ∞) ξ . Hence, f .

f

= [0,∞)

a sdeξ,ξ (s) =

(γ ,∞)

a sdeξ,ξ (s) ≥ γ ,

and this contradicts the assumption that . ≤ αe (a) < γ . Consequently, .p ∧ q = 0 for any .p ∈ P (M) with .p (H ) ⊆ D (a). Since ( ⊥ ).D (a) is .τ -dense, there exists a sequence .{pn }∞ n=1 in .P (M) such that .pn ↑ 1, .τ pn ↓ 0, and .pn (H ) ⊆ D (a) for ⊥ all n. Since .pn ∧ 1.14.5 (iii)), it follows ) 0 implies that .q < pn (see Proposition ( q⊥= that .τ (q) ≤ τ pn for all n, and so, .τ (q) = 0, that is, .ea (γ , ∞) ∧ e = q = 0. ( ) Consequently, .ea (γ , ∞) < e⊥ , and hence, .τ (ea (γ , ∞)) ≤ τ e⊥ ≤ t, from which it may be concluded that .μ (t; a) ≤ γ . Since this holds for all .γ > αe (a), it follows that .μ (t; a) ≤ αe (a), and therefore, .μ (t; a) ≤ A. ' Next )will be shown that .A ≤ μ (t; a). If .e ∈ P (M) is such that .e (H ) ⊆ D (a) ( it ⊥ ≤ t, then . ≤ ||ae||B(H ) for all .ξ ∈ H satisfying .ξ = eξ and and .τ e .||ξ ||H = 1. Hence, .αe (a) ≤ ||ae||B(H ) . This implies that } { ( ) A' ≤ inf ||ae||B(H ) : e ∈ P (M) , e (H ) ⊆ D (a) , τ e⊥ ≤ t = μ (t; a) ,

.

by Proposition 3.2.5. The proof of the proposition is complete.

u n

Recall that for each .x ∈ S (τ ), the support projection of x is denoted by .s (x), that is, .s (x) = 1 − n (x). For .t > 0, define Rt (τ ) = {x ∈ S (τ ) : τ (s (x)) ≤ t} .

.

The following proposition presents a geometric interpretation of the singular value function in terms of what might be called generalized approximation numbers. Proposition 3.2.20 If .x ∈ S (τ ), then { } μ (t; x) = inf ||x − y||B(H ) : y ∈ Rt (τ ) , x − y ∈ M

.

for all .t > 0. Proof Let .t > 0 and .x ∈ S (τ ) be fixed. Defining y = xe|x| (μ (t; x) , ∞) ,

.

it follows that .x − y = xe|x| [0, μ (t; x)], and so, .x − y ∈ M with .||x − y||B(H ) ≤ μ (t; x). Moreover, .s (y) ≤ e|x| (μ (t; x) , ∞), which implies that ( ) τ (s (y)) ≤ τ e|x| (μ (t; x) , ∞) = d (μ (t; x) ; |x|) ≤ t.

.

146

3 Singular Value Functions

Consequently, { } inf ||x − y||B(H ) : y ∈ Rt (τ ) , x − y ∈ M ≤ μ (t; x) .

.

To obtain the reverse inequality, suppose that .y ∈ Rt (τ ) is such that .x − y ∈ M. Since .xn (y) = (x − y) n (y), it is clear that .||xn (y)||B(H ) ≤ ||x − y||B(H ) . ) ( Furthermore, .τ n (y)⊥ = τ (s (y)) ≤ t, and so, Proposition 3.2.5 implies that } { ( ) μ (t; x) = inf ||xe||B(H ) : e ∈ P (M) , τ e⊥ ≤ t ≤ ||x − y||B(H ) .

.

This shows that { } μ (t; x) ≤ inf ||x − y||B(H ) : y ∈ Rt (τ ) , x − y ∈ M ,

.

which concludes that proof of the proposition.

u n

References: [52, 55, 62].

3.3 Extension of the Trace to S (τ )+ In this section, it will be shown that the trace on .M+ extends naturally to the positive cone of .S (τ ) so that the basic properties of unitary invariance, linearity, and normality are preserved. Throughout this section, .M is a semi-finite von Neumann algebra equipped with a fixed semi-finite faithful normal trace .τ : M+ → [0, ∞]. As noted in Proposition 3.2.18, the trace on .M+ is obtained by integrating the singular value function. More precisely, if .a ∈ M+ , then the equality f

∞

τ (a) =

μ (t; a) dt

.

0

holds (in the sense that if either side is finite, then so is the other, in which case equality holds). This leads naturally to the following definition that extends the trace to .S (τ )+ . Definition 3.3.1 If .a ∈ S (τ )+ , then the trace .τ (a) is defined by setting f τ (a) =

∞

μ (t; a) dt.

.

0

Note that, f ∞ if .x ∈ S (τ ), then .μ (x) = μ (|x|), and so it is clear that τ (|x|) = 0 μ (t; x) dt. Furthermore, if .a ≤ b in .S (τ )+ , then it follows from Proposition 3.2.7 (v) that .μ (a) ≤ μ (b), and so, it is evident from the definition that .τ (a) ≤ τ (b) (that is, .τ is monotone). .

3.3 Extension of the Trace to S (τ )+

147

Remark 3.3.2 In Sect. 3.4, the set .{x ∈ S (τ ) : τ (|x|) < ∞} will be identified with the noncommutative .L1 -space .L1 (τ ), and it will be shown that .L1 (τ ) is the predual of the von Neumann algebra .M. In the next proposition, the basic properties of the extended trace are collected. Proposition 3.3.3 The functional .τ : S (τ )+ → [0, ∞] possesses the following properties: (i) .τ is additive, positively homogeneous, and unitarily invariant. (ii) .τ is faithful, that is, if .a ∈ S (τ )+ and .τ (a) = 0, then .a = 0. (iii) If .0 < a ∈ S (τ )+ , then there exists .b ∈ M+ such that .0 < b ≤ a and .τ (b) < ∞. ( ) (iv) .τ is normal, that is, if .0 ≤ aβ ↑ a holds in .Sh (τ ), then .τ aβ ↑ τ (a). Proof It will be convenient ( ) to prove statement (iv) first. If .0 ≤ a( β ↑ ) a in .Sh (τ ), then it is clear that .μ t; aβ ↑≤ μ (t; a) (for all) .t ≥ 0, and so, .τ aβ ↑≤ τ (a). It follows from Proposition 3.2.14 (i) that .μ t; aβ ↑β μ (t; a) for all .t ≥ 0, and there { }∞ { } ) ( exists an increasing subsequence . aβn n=1 of . aβ such that .μ t; aβn ↑n μ (t; a) for all .t ≥ 0. The monotone convergence theorem implies that ( ) τ aβn =

f

.

0

∞

) ( μ t; aβn dt ↑

f

∞

μ (t; a) dt = τ (a) ,

0

( ) and hence, .τ (a) ≤ supβ τ aβ . This establishes (iv). (i) Since the singular value function is positively homogeneous and unitarily invariant (see Proposition 3.2.7), the corresponding properties for the trace on + .S (τ ) follow immediately from Definition 3.3.1. For the proof of the additivity of .τ , it should be observed that for any .a ∈ + S (τ )+ , there exists a sequence .{an }∞ n=1 in .M such that .an ↑ a in .Sh (τ ). Indeed, defining the functions .fn on .[0, ∞) by .fn (λ) = min (λ, n), and setting + .an = fn (a) for all .n ∈ N, it is clear that .an ∈ M for all n and that .an ↑ a in .Sh (τ ) (cf. Remark 2.2.26). Given .a, b ∈ S (τ )+ , it follows from the above observation that there exist + ∞ sequences .{an }∞ n=1 and .{bn }n=1 in .M such that .an ↑ a and .bn ↑ b in .Sh (τ ). This implies that .an + bn ↑ a + b in .Sh (τ ). It follows from (iv) that .τ (an ) ↑ τ (a), .τ (bn ) ↑ τ (b), and .τ (an + bn ) ↑ τ (a + b). Furthermore, by the additivity of the trace on .M+ , .τ (an + bn ) = τ (an ) + τ (bn ) for each n. This suffices to conclude that .τ (a +fb) = τ (a) + τ (b). ∞ (ii) If .a ∈ S (τ )+ and .τ (a) = 0, then . 0 μ (t; a) dt = 0. Since .μ (a) is rightcontinuous, this implies that .μ (t; a) = 0 for all .t ≥ 0, and hence, .a = 0. (iii) By the observations in the proof of (i), if .0 < a ∈ S (τ )+ , then there exists a + sequence .{an }∞ n=1 in .M such that .an ↑ a in .Sh (τ ). Hence, there is an m such that .0 < am ≤ a. Since the trace .τ is semi-finite on .M+ , there exists .b ∈ M+ such that .0 < b ≤ am and .τ (b) < ∞. The proof is complete. u n

148

3 Singular Value Functions

For later reference, some further properties of the extended trace are presented in the following proposition. Proposition 3.3.4 (i) If .x ∈ S (τ ), then .τ (x ∗ x) = τ (xx ∗ ) and .τ (|x ∗ |) = τ (|x|). (ii) If .x ∈ S (τ ) and .v, w ∈ M, then .τ (|vxw|) ≤ ||v||B(H ) ||w||B(H ) τ (|x|). Proof These results are immediate consequences of Propositions 3.2.10 (i), Proposition 3.2.7 (ii), (vi) in combination with Definition 3.3.1. u n The following observation is sometimes useful. Proposition 3.3.5 If .a ∈ S (τ )+ , then .

{τ (eae) : e ∈ P (M) , τ (e) < ∞} ↑e τ (a) .

Proof Evidently, .τ (eae) ≤ τ (a) for all .e ∈ P (M) (with .τ (e) < ∞). As has been shown in Proposition 3.2.15, .μ (a) is the supremum in .L0 [0, ∞) of the upward directed system .{μ (eae) : e ∈ P (M) , τ (e) < ∞}. Hence, there exists an increasing sequence .{en }∞ n=1 in .P (M), with .τ (en ) < ∞ for all n, such that .μ (en aen ) ↑n μ (a) a.e. on .[0, ∞) (see also Lemma 3.2.12). It follows from the monotone convergence theorem that f

∞

τ (en aen ) =

.

f

∞

μ (en aen ) dt ↑n

0

μ (a) dt = τ (a) ,

0

and this suffices for the proof of the proposition.

u n

Alternative formulae for the trace may be obtained via the functional calculus. Indeed, the trace is given by integrating against the Lebesgue–Stieltjes measure induced by the trace of the spectral measure. The details follow. Let a be a self-adjoint operator on H , which is affiliated with .M. Denoting by a .B (R) the Borel .σ -algebra of .R, it follows that .e (B) ∈ M for all .B ∈ B (R), and a a hence, the function .τ e : B |−→ τ (e (B)), .B ∈ B (R), is a .σ -additive measure. Note that .τ ea is not .σ -finite in general. For example, if .a = 1, then .τ ea = τ (1) δ1 , and this is not a .σ -finite measure whenever .τ (1) = ∞. Note, furthermore, that the measure .τ ea is supported on the spectrum .σ (a) of a, that is, .(τ ea ) (R \ σ (a)) = 0. For any Borel function .f : R → [0, ∞), the positive self-adjoint operator .f (a) is defined by f f (a) =

.

R

f (λ) dea (λ) .

Note that .f (a) only depends on the values that f takes on .σ (a), and so it may be assumed that .f = 0 on .R \ σ (a). Furthermore, it will be assumed that .f (a) ∈ S (τ )+ , so that .τ (f (a)) is defined. This is the case if .a ∈ S (τ ) and f is bounded on compact subsets of .[0, ∞) (see Proposition 2.3.14), but it may happen that .f (a) ∈

3.3 Extension of the Trace to S (τ )+

149

S (τ )+ under weaker conditions on .f ; for example, if .τ (1) < ∞, then .f (a) ∈ S (τ )+ for all positive Borel functions (see Remark 2.3.15 (i)). The trace of .f (a) is now also given by the formula f τ (f (a)) =

.

R

( ) f (λ) d τ ea (λ) .

(3.16)

Indeed, if g is a step function given by g=

k Σ

.

(3.17)

αj χBj ,

j =1

where .Bj ∈ B (R) and .0 ≤ αj ∈ R for .j = 1, . . . , k, then g (a) =

k Σ

.

( ) αj ea Bj ,

j =1

and hence, τ (g (a)) =

k Σ

.

( )( ) αj τ ea Bj =

j =1

f R

( ) g (λ) d τ ea (λ) .

(3.18)

Next, let .{gn }∞ n=1 be a sequence of step functions, each of which is of the form (3.17), such that .0 ≤ gn (λ) ↑ f (λ) for all .λ ∈ R. From the properties of the functional calculus (see Remark 1.8.10), it follows that .gn (a) ↑ f (a) in the sense of the quadratic form ordering, and hence, .gn (a) ↑ f (a) in .S (τ )+ (see Remark 2.2.26). The normality of the trace implies that .τ (gn (a)) ↑ τ (f (a)). On the other hand, it follows from (3.18) in combination with the monotone convergence theorem that f τ (gn (a)) =

.

R

( ) gn (λ) d τ ea (λ) ↑n

f R

( ) f (λ) d τ ea (λ) ,

from which (3.16) follows. If .a ∈ S (τ )+ , then measure .τ ea also arises as the image of Lebesgue measure m under the function .μ (a). Lemma 3.3.6 (i) Suppose that .τ (1) < ∞ and .a ∈ S (τ )+ . The measure .τ ea on .B [0, ∞) is equal to the image of Lebesgue measure on .(0, τ (1)) under the function .μ (a) : (0, τ (1)) → [0, ∞).

150

3 Singular Value Functions

(ii) Suppose that .τ (1) = ∞ and .a ∈ S0 (τ )+ . The measure .τ ea and the image of Lebesgue measure on .(0, ∞) under the function .μ (a) : (0, ∞) → [0, ∞) have the same restriction to the Borel .σ -algebra .B (0, ∞). Proof (i) Let .ν denote the image of m under the function .μ (a) : (0, τ (1)) → [0, ∞), that is, ν (B) = m {t ∈ (0, τ (1)) : μ (t; a) ∈ B} ,

.

B ∈ B [0, ∞) .

If .0 ≤ α < β, then ν (α, β] = m {t ∈ (0, τ (1)) : α < μ (t; a) ≤ β} .

.

Since .

{t ∈ (0, τ (1)) : μ (t; a) > s} = {t ∈ (0, τ (1)) : d (s; a) > t} = (0, d (s; a))

for all .s ≥ 0, it follows that .

{t ∈ (0, τ (1)) : α < μ (t; a) ≤ β} = [d (β; a) , d (α; a)) ,

and so, ( ) ( ) ν (α, β] = d (α; a) − d (β; a) = τ ea (α, ∞) − τ ea (β, ∞) ( ) ( ) = τ ea (α, β] = τ ea (α, β] .

.

Furthermore, ( ) ν [0, ∞) = m (0, τ (1)) = τ (1) = τ ea [0, ∞) .

.

These observations suffice to conclude that .ν = τ ea on .B [0, ∞). (ii) First note that .a ∈ S0 (τ ) implies that .(τ ea ) (s, ∞) = d (s; a) < ∞ for all .s > 0. Denote the image measure of m on .(0, ∞) under the function .μ (a) : (0, ∞) → [0, ∞) by .ν, that is, ν (B) = m {t > 0 : μ (t; a) ∈ B} ,

.

B ∈ B [0, ∞) .

It has to be shown that .ν (B) = (τ ea ) (B) for every Borel set .B ⊆ (0, ∞). Since .B ∩ (1/n, ∞) ↑ B for such sets B, it is sufficient to show that .ν and .τ ea coincide on .B (ε, ∞) for each .ε > 0. Let .ε > 0 be fixed. If .ε ≤ α < β, then it follows via the same computation as used in the proof of (i), that is, .ν (α, β] = (τ ea ) (α, β]. Since .ν (ε, ∞) =

3.3 Extension of the Trace to S (τ )+

151

d (ε; a) = (τ ea ) (ε, ∞) < ∞, it may be concluded that .ν = τ ea on .B (ε, ∞). The proof is complete. u n Remark 3.3.7 The assumption that .a ∈ S0 (τ ) in (ii) of Lemma 3.3.6 cannot be replaced by .a ∈ S (τ )+ . Indeed, consider on .[0, ∞), equipped with Lebesgue measure, the function a defined by .a (t) = arctan t, .t ≥ 0. Since .μ (t; a) = π/2, .t ≥ 0 (see Example 3.1.13), the image measure .ν of m under the function .μ (a) is given by .ν (B) = ∞ if .π/2 ∈ B and .ν (B) = 0 otherwise, for all Borel sets .B ⊆ [0, ∞). On the other hand, .(τ ea ) (α, β] = tan β − tan α whenever .0 ≤ α < β < π/2. Furthermore, the restriction of the two measures to the open interval .(0, ∞) is essential. Indeed, if .a ≥ 0 is any Lebesgue integrable function on .[0, ∞) for which the two sets .

{t ∈ [0, ∞) : a (t) > 0} ,

{t ∈ [0, ∞) : a (t) = 0}

both have infinite measure, then .a ∈ S0 (m) and .μ (t; a) > 0 for all .t > 0. Hence, the image .ν of m under .μ (a) satisfies .ν ({0}) = 0. On the other hand, however, a .(τ e ) ({0}) = m {t ≥ 0 : a (t) = 0} = ∞. It is worthwhile to point out the following consequence of Lemma 3.3.6. Corollary 3.3.8 If .a ∈ S (τ )+ , then .

{μ (t; a) : t ∈ [0, τ (1))} ⊆ σ (a) ,

(3.19)

σ (a) = {μ (t; a) : t ∈ [0, τ (1))},

(3.20)

and if .a ∈ S0 (τ )+ , then .

where .σ (a) denotes the spectrum of a. / σ (a), then there exists .ε ∈ R such that .0 < ε < λ Proof If .0 < λ ∈ R and .λ ∈ and .(λ − ε, λ + ε) ∩ σ (a) = ∅. Since the spectral measure .ea of a is supported on .σ (a), this implies that the spectral distribution function .d (a) has a constant value, .α say, on the interval .[λ − ε, λ + ε). If .α = ∞, then .μ (t) ≥ λ + ε for all .t ≥ 0. If .α < ∞, then it follows from Proposition 3.1.3 (ii) that μ (α; a) ≤ λ − ε ≤ λ + ε ≤ μ (α−; a) .

.

/ σ (a), then Hence, in both cases, .λ does not belong to that range of .μ (a). If .0 ∈ there exists .ε > 0 such that .[0, ε) ∩ σ (a) = ∅, in which case .d (a) has the constant value .τ (1) on .[0, ε). Consequently, .μ (t; a) ≥ ε whenever .t ∈ [0, τ (1)). This suffices for the proof of (3.19).

152

3 Singular Value Functions

For the proof of (3.20), assume first that .τ (1) < ∞. Since .σ (a) is the support of the spectral measure .ea , it follows from .λ ∈ σ (a) that ea ((λ − ε, λ + ε) ∩ [0, ∞)) /= 0

.

and so, by Lemma 3.3.6 (i), m {t ∈ [0, τ (1)) : μ (t; a) ∈ (λ − ε, λ + ε) ∩ [0, ∞)} /= 0,

.

for all .ε > 0. Consequently, there exists a sequence .{tn }∞ n=1 in .[0, τ (1)) such that .μ (tn ; a) → λ. This completes the proof of (3.20) in the case that .τ (1) < ∞. Now suppose that .τ (1) = ∞ and .a ∈ S0 (τ )+ . As observed in Remark 2.4.6 (i), this implies that .0 ∈ σ (a). Furthermore, .a ∈ S0 (τ )+ implies that .μ (t; a) → 0 as .t → ∞ (see Proposition 3.2.4), and hence, 0 ∈ {μ (t; a) : t ∈ [0, τ (1))}.

.

Having made these two observations, the proof of (3.20) now follows the same lines as in the case that .τ (1) < ∞. The details are left to the reader. u n Proposition 3.3.9 (i) Suppose that .τ (1) < ∞ and .a ∈ S (τ )+ . If .f : [0, ∞) → [0, ∞) is a Borel function, then f

τ (1)

τ (f (a)) =

f (μ (t; a)) dt.

.

0

(ii) Suppose that .τ (1) = ∞ and .a ∈ S0 (τ )+ . If .f : [0, ∞) → [0, ∞) is a Borel function that is bounded on a neighborhood of 0 and .f (0) = 0, then f

∞

τ (f (a)) =

f (μ (t; a)) dt.

.

0

Proof (i) As observed already, .f (a) ∈ S (τ )+ since .τ (1) < ∞. By Lemma 3.3.6 (i), .τ ea is the image measure of m on .(0, τ (1)) under the function .μ (a) : (0, τ (1)) → [0, ∞), and so, it follows from (3.16) in combination with the change of measure formula that f

( ) f (λ) d τ ea (λ) =

τ (f (a)) =

.

[0,∞)

f

τ (1)

f (μ (t; a)) dt. 0

3.3 Extension of the Trace to S (τ )+

153

(ii) Since f is bounded on a neighborhood of 0, it follows from .a ∈ S0 (τ )+ that .f (a) ∈ S (τ )+ (see Remark 2.4.6 (ii)). Since .f (0) = 0, (3.16) and Lemma 3.3.6 (ii), in combination with the the change of measure formula, yield that f f ( a) ( ) .τ (f (a)) = f (λ) d τ e (λ) = f (λ) d τ ea (λ) f

[0,∞)

(0,∞)

∞

=

f (μ (t; a)) dt. 0

The proof is complete. u n The assumptions that .a ∈ S0 (τ ) and .f (0) = 0 cannot be omitted in (ii) of the above proposition, as can be seen from the examples presented in Remark 3.3.7. Next, some remarks concerning trace preserving .∗-isomorphisms. Suppose that .M1 and .M2 are von Neumann algebras on Hilbert spaces .H1 and .H2 , respectively, and let .τ1 and .τ2 be semi-finite normal faithful traces on .M1 and .M2 , respectively. If .π : M1 → M2 is a unital trace preserving .∗-isomorphism, then it follows from Proposition 2.9.2 (i) that there exists a unique .∗-isomorphism .πˆ : S (τ1 ) → S (τ2 ) such that .πˆ |M1 = π . Moreover, .πˆ is the unique extension of .π , which is continuous with respect to the measure topology. The following proposition presents some additional information. Proposition 3.3.10

( ) (i) The .∗-isomorphism .πˆ is (extended) trace preserving, that is, .τ2 πˆ a = τ1 (a) for all .a ∈ S (τ1 )+ . (ii) If .π is assumed, in (addition, to be normal, then .πˆ preserves the singular value ) function, that is, .μ πˆ x = μ (x) for all .x ∈ S (τ1 ). Proof (i) Given .a ∈ S (τ1 )+ , define .an = aea [0, n] for .n ∈ N. The sequence .{an }∞ n=1 Tm

is increasing and .an → a as .n → ∞ (and so, in particular, .0 ≤ an ↑ a Tm

in .Sh (τ1 )). This implies that .0 ≤ πˆ (an ) ↑ and also .πˆ (an ) → πˆ (a), as .πˆ is continuous for the measure topology. Consequently (see Proposition 2.6.1 (iii)), .0 ≤ π ˆ (an ) ↑ πˆ (a) in .Sh (τ2 ). The normality of the (extended) trace implies that ( ) ( ) τ1 (an ) = τ2 (π an ) = τ2 πˆ an ↑n τ2 πˆ a

.

and also τ1 (an ) ↑n τ1 (a) .

.

( ) Hence, .τ2 πˆ a = τ1 (a).

154

3 Singular Value Functions

(ii) Suppose that .x ∈ S (τ1 ). It follows from Proposition | (iv) that the spectral | 2.9.2 measure .eπˆ (|x|) of .πˆ (|x|) is given by .π e|x| . Since .|πˆ (x)| = πˆ (|x|), this implies that ( ( ) ) |) ( | ˆ (x)| = τ2 e|πˆ (x)| (s, ∞) = τ2 eπˆ (|x|) (s, ∞) .d s; |π ) ) ( ( = τ2 π e|x| (s, ∞) = τ1 e|x| (s, ∞) = d (s; |x|) for all .s ≥ 0, from which it is evident that ) ( μ t; πˆ (x) = μ (t; x) ,

.

t ≥ 0.

This completes the proof of the proposition. u n Example 3.3.11 Suppose that .M is a von Neumann algebra on a Hilbert space H , equipped with a semi-finite normal faithful trace .τ . Consider .L∞ [0, 1] as a von Neumann algebra on the Hilbert space .L2 [0, 1], equipped with the Lebesgue integral as its trace. The tensor product von Neumann algebra .L∞ [0, 1] ⊗M on the Hilbert space tensor product .L2 [0, 1] ⊗H is equipped with the corresponding tensor product trace .τˆ (see Example 2.9.4). The map .π : x |−→ 1 ⊗ x, .x ∈ M, is a unital trace preserving normal .∗-isomorphism from .M onto the von Neumann subalgebra .C1 ⊗ M of .L∞ [0, 1] ⊗M. Consequently, .π extends uniquely to a .∗-isomorphism ( ) .π ˆ : x |−→ 1 ⊗ x, from .S (τ ) into .S τˆ , which preserves the singular value function As observed in Proposition 2.9.3, the range of (in particular, .πˆ( is trace preserving). ) .π ˆ is equal to .S C1 ⊗ M, τˆ . References: [45, 55].

3.4 The Space L1 (τ ) As before, .M is a von Neumann algebra on a Hilbert space H , equipped with a fixed semi-finite faithful normal trace .τ . Recall from Theorem 1.15.8 that the set Mτ = {x ∈ M : τ (|x|) < ∞}

.

is a two-sided ideal in .M and is called the domain of definition of .τ . The restriction of .τ to .M+ τ extends to a positive linear functional .τ˙ on .Mτ (and frequently, .τ˙ is also denoted by .τ ). Using the extended trace .τ : S (τ )+ → [0, ∞], the space .L1 (τ ) is defined by L1 (τ ) = {x ∈ S (τ ) : τ (|x|) < ∞} .

.

155

3.4 The Space L1 (τ )

By the definition of the extension of the trace, it follows immediately that { f L1 (τ ) = x ∈ S (τ ) :

∞

.

} μ (t; x) dt < ∞ = {x ∈ S (τ ) : μ (x) ∈ L1 (m)} ,

0

where .L1 (m) denotes the usual Lebesgue space on the half-line .[0, ∞). It is also clear that Mτ = L1 (τ ) ∩ M.

.

Next, observe that .L1 (τ ) is a linear subspace of .S (τ ). Indeed, the fact that .L1 (τ ) is closed under scalar multiplication follows immediately from the identity .μ (αx) = |α| μ (x), so that .τ (|αx|) = |α| τ (|x|), for all .x ∈ S (τ ) and .α ∈ C. To see that .L1 (τ ) is closed under addition, suppose that .x, y ∈ L1 (τ ). It follows from Proposition 3.2.7 (iii) that f

∞

.

f

∞

μ (t; x + y) dt ≤

0

f

∞

μ (t/2; x) dt +

0

μ (t/2; y) dt 0

f

∞

=2 0

f

∞

μ (t; x) dt + 2

μ (t; y) dt < ∞,

0

and so, .x + y ∈ L1 (τ ). Furthermore, it should be noted that if .x ∈ S (τ ), then x ∈ L1 (τ ) if and only if .|x| ∈ L1 (τ ), as .μ (x) = μ (|x|). Also observe that if .x ∈ L1 (τ ), then .μ (t; x) → 0 as .t → ∞, and so, it follows from Proposition 3.2.4 that .x ∈ S0 (τ ). Therefore, .L1 (τ ) ⊆ S0 (τ ). Direct application of Proposition 3.2.7 (v) and Proposition 3.3.4 now yields the following. .

Proposition 3.4.1 (i) (ii) (iii) (iv)

If .x If .x If .x If .x

∈ L1 (τ ), .y ∈ S (τ ), and if .|y| ≤ |x|, then also .y ∈ L1 (τ ). ∈ S (τ ), then .x ∈ L1 (τ ) if and only if .x ∗ ∈ L1 (τ ). ∈ S (τ ), then .x ∗ x ∈ L1 (τ ) if and only if .xx ∗ ∈ L1 (τ ). ∈ L1 (τ ) and .u, v ∈ M, then .uxv ∈ L1 (τ ).

Denoting .L1 (τ )h = {a ∈ L1 (τ ) : a = a ∗ }, it is clear from (ii) in the above proposition that L1 (τ ) = L1 (τ )h ⊕ iL1 (τ )h ,

.

as a direct sum of real linear subspaces. Furthermore, it is clear from (i) that a ± ∈ L1 (τ )h whenever .a ∈ L1 (τ )h and so, .L1 (τ )h = L1 (τ )+ − L1 (τ )+ , where + .L1 (τ ) = L1 (τ ) ∩ S (τ )+ . Consequently, .L1 (τ )h is a partially ordered vector space with generating positive cone, with respect to the partial ordering induced by .Sh (τ ). .

156

3 Singular Value Functions

The restriction of .τ to .L1 (τ )+ is an additive positively homogeneous real-valued functional, and so, by the above remarks, it has a unique extension to a linear functional .τ˙ : L1 (τ ) → C. Usually, this functional .τ˙ will be simply denoted by .τ again. Note that .τ is real-valued on .L1 (τ )h , since .τ takes positive values on + .L1 (τ ) . In the next proposition, some simple properties of the trace functional are collected. Proposition 3.4.2 If .x ∈ L1 (τ ), then: (i) .τ (x ∗ ) = τ (x). (ii) .τ (u∗ xu) = τ (x) for all .u ∈ U (M). (iii) .τ (xy) = τ (yx) for all .y ∈ M. Proof Property (i) is a simple consequence of the above noted fact that .τ is realvalued on .L1 (τ )h . Since the extended trace is unitarily invariant on .S (τ )+ (see (i) of Proposition 3.3.3), (ii) is evident for .x ∈ L1 (τ )+ , and the general case follows by linearity. For the proof of (iii), first note that (ii) implies that .τ (ux) = τ (xu) for all .u ∈ U (M). Since any .y ∈ M is a linear combination of at most four unitary elements of .M, assertion (iii) is now also clear. u n For further analysis of the space .L1 (τ ), it will be convenient to have available the space .L2 (τ ), which is defined by setting { } L2 (τ ) = x ∈ S (τ ) : x ∗ x ∈ L1 (τ ) .

.

( ) Using the definition of the extended trace and the fact that .μ |x|2 = μ (|x|)2 = μ (x)2 for all .x ∈ S (τ ) (see Proposition 3.2.8), it is clear that .L2 (τ ) is also given by f ) } { { ( L2 (τ ) = x ∈ S (τ ) : τ |x|2 < ∞ = x ∈ S (τ ) :

.

∞

} μ (t; x)2 dt < ∞

0

= {x ∈ S (τ ) : μ (x) ∈ L2 (m)} , where m denotes Lebesgue measure on .[0, ∞). It is easy to see that .L2 (τ ) is a linear subspace of .S (τ ). In fact, if .x, y ∈ L2 (τ ), then it follows from Proposition 3.2.7 (iii) that f .

∞

f

∞

μ (t; x + y)2 dt ≤

0

(μ (t/2; x) + μ (t/2; y))2 dt

0

f

∞

=2

(μ (t; x) + μ (t; y))2 dt < ∞,

0

since .μ (x) + μ (y) ∈ L2 (m). Furthermore, it is clear that the assertions of Proposition 3.4.1 are also valid for .L2 (τ ), since they all follow from the corresponding properties of the singular value function.

157

3.4 The Space L1 (τ )

If .x, y ∈ L2 (τ ), then .xy ∈ L1 (τ ). Indeed, it follows from Proposition 3.2.7 (iv) that f ∞ f ∞ . μ (t; xy) dt ≤ μ (t/2; x) μ (t/2; y) dt 0

0

(f

)1/2 (f

∞

≤2

)1/2

∞

2

μ (t; x) dt

2

μ (t; y) dt

0

< ∞.

0

Consequently, an inner product . on .L2 (τ ) may be defined by setting .

( ) = τ y ∗ x ,

x, y ∈ L2 (τ ) .

The corresponding norm is given by .

)1/2 ( ( )1/2 ||x||2 = τ x ∗ x = τ |x|2 ,

x ∈ L2 (τ ) .

) ( Since .μ |x|2 = μ (x)2 , it is clear from the definition of the extended trace that the norm .||·||2 is also given by .||x||2 = ||μ (x)||2 , .x ∈ L2 (τ ). Note that the Cauchy– Schwarz inequality for this inner product states that .

| ( ∗ )| ( ) ( ) |τ y x | ≤ τ x ∗ x 1/2 τ y ∗ y 1/2 ,

x, y ∈ L2 (τ ) .

(3.21)

Furthermore, it should be observed that an inner product .' in .L2 (τ ) may also be defined by setting .' = τ (xy ∗ ) for all .x, y ∈ L2 (τ ). The corresponding norm is then given by .||x||' = τ (xx ∗ )1/2 , .x ∈ L2 (τ ). Since, by Proposition 3.3.4 (i), ' ∗ ∗ .τ (x x) = τ (xx ) for all .x ∈ S (τ ), it follows that .||x|| = ||x||2 , .x ∈ L2 (τ ). Two inner products inducing the same norm necessarily coincide, and hence, .τ (xy ∗ ) = τ (y ∗ x) for all .x, y ∈ L2 (τ ). Replacing in this last identity the element y by .y ∗ yields the following result. Proposition 3.4.3 If .x, y ∈ L2 (τ ), then .xy ∈ L1 (τ ) and .τ (xy) = τ (yx). Note that the above proposition implies that .τ (ab) ≥ 0 whenever .a ∈ L1 (τ )+ and .b ∈ M+ . Indeed, .a 1/2 and .a 1/2 b both belong to .L2 (τ ), and hence, ) ( ) ( τ (ab) = τ a 1/2 a 1/2 b = τ a 1/2 ba 1/2 ≥ 0,

.

since .a 1/2 ba 1/2 ∈ S (τ )+ . The above observations will be used in the proof of an important inequality for the trace functional on .L1 (τ ), which is presented in the next proposition. Its proof is based on an inequality that will also be useful for other purposes and is, therefore, singled out as a separate lemma.

158

3 Singular Value Functions

Lemma 3.4.4 If .x ∈ L1 (τ ) and .y ∈ M, then .

(| | )1/2 ( | ∗ |)1/2 |τ (xy)| ≤ τ |x ∗ | |y| τ |x| |y | .

Proof Let .x = u |x| and .y = v |y| be the polar decompositions of x and y, respectively. Since .xv |y|1/2 ∈ L1 (τ ) and .|y|1/2 ∈ M, it follows from Proposition 3.4.2 (iii) that ) ( ) ( τ (xy) = τ xv |y|1/2 |y|1/2 = τ |y|1/2 xv |y|1/2 ) ( = τ |y|1/2 u |x|1/2 |x|1/2 v |y|1/2 .

.

Since .w = |y|1/2 u |x|1/2 and .z = |x|1/2 v |y|1/2 both belong to .L2 (τ ), the Cauchy– Schwarz inequality (3.21) implies that ( )1/2 ( ∗ )1/2 |τ (xy)| ≤ τ ww ∗ τ z z )1/2 ( )1/2 ( = τ |y|1/2 u |x| u∗ |y|1/2 τ |y|1/2 v ∗ |x| v |y|1/2

.

)1/2 ( )1/2 ( = τ u |x| u∗ |y| τ |x| v |y| v ∗ (| | )1/2 ( | ∗ |)1/2 = τ |x ∗ | |y| τ |x| |y | , using for the last equality that .u |x| u∗ = |x ∗ | and .v |y| v ∗ = |y ∗ |.

u n

Proposition 3.4.5 If .x ∈ L1 (τ ) and .y ∈ M, then .

|τ (xy)| ≤ ||y||B(H ) τ (|x|) .

(3.22)

Proof It follows from .0 ≤ |y| ≤ ||y||B(H ) 1 that .0 ≤ |x ∗ |1/2 |y| |x ∗ |1/2 ≤ ||y||B(H ) |x ∗ |, and so, (| | | |1/2 ) (| |) 1/2 ≤ ||y||B(H ) τ |x ∗ | = ||y||B(H ) τ (|x|) . 0 ≤ τ |x ∗ | |y| |x ∗ |

.

Furthermore, since .|x ∗ |1/2 ∈ L2 (τ ) and .|x ∗ |1/2 |y| ∈ L2 (τ ), it follows from Proposition 3.4.3 that ) (| | | | (| | | |1/2 ) (| | ) 1/2 1/2 1/2 = τ |x ∗ | |x ∗ | |y| = τ |x ∗ | |y| . τ |x ∗ | |y| |x ∗ |

.

Consequently, .0 ≤ τ (|x ∗ | |y|) ≤ ||y||B(H ) τ (|x|). Similarly, .0 ≤ τ (|x| |y ∗ |) ≤ ||y||B(H ) τ (|x|), and hence, (3.22) now follows from Lemma 3.4.4. u n Taking .y = 1 in Proposition 3.4.5, the following important special case is obtained.

159

3.4 The Space L1 (τ )

Corollary 3.4.6 If .x ∈ L1 (τ ), then |τ (x)| ≤ τ (|x|) .

.

(3.23)

Remark 3.4.7 It follows, in particular, from the above corollary that .|τ (xy)| ≤ τ (|xy|) for all .x ∈ L1 (τ ) and .y ∈ M. If .xy = v |xy| is the polar decomposition of xy, then .|xy| = v ∗ xy, and so, it follows from Proposition 3.4.5 that || ) || ( ) ( τ (|xy|) = τ v ∗ xy = τ xyv ∗ ≤ ||yv ∗ ||B(H ) τ (|x|) ≤ ||y||B(H ) τ (|x|) .

.

Consequently, inequality (3.22) may be refined to .

|τ (xy)| ≤ τ (|xy|) ≤ ||y||B(H ) τ (|x|) ,

x ∈ L1 (τ ) , y ∈ M.

It will be convenient to introduce the notation .

||x||1 = τ (|x|) ,

x ∈ L1 (τ ) .

Note that .||x||1 is also given by f .

||x||1 =

0

∞

μ (t; x) dt = ||μ (x)||1 ,

where .||μ (x)||1 denotes the norm of .μ (x) in the Lebesgue space .L1 [0, ∞). Evidently, .||αx||1 = |α| ||x||1 for all .α ∈ C, .x ∈ L1 (τ ), and if .x ∈ L1 (τ ), then .||x||1 = 0 if and only if .x = 0 (cf. Proposition 3.3.3, (ii)). Actually, .||·||1 is a norm on .L1 (τ ), as follows from the alternative description of .||·||1 presented in the next proposition. Proposition 3.4.8 If .x ∈ L1 (τ ), then .

{ } ||x||1 = sup |τ (xy)| : y ∈ M, ||y||B(H ) ≤ 1 .

(3.24)

Proof If .y ∈ M with .||y||B(H ) ≤ 1, then it follows from Proposition 3.4.5 that |τ (xy)| ≤ τ (|x|), and hence,

.

.

{ } sup |τ (xy)| : y ∈ M, ||y||B(H ) ≤ 1 ≤ τ (|x|) = ||x||1 .

On the other hand, if .x = v |x| is the polar decomposition of x, then .|x| = v ∗ x, and so, ) ( ) ( τ (|x|) = τ v ∗ x = τ xv ∗ .

.

Since .||v ∗ ||B(H ) ≤ 1, this shows that the supremum is attained and has value .τ (|x|) = ||x||1 . u n

160

3 Singular Value Functions

From (3.24), it is immediately clear that .||·||1 satisfies the triangle inequality, and hence, this yields the following result. Corollary 3.4.9 The functional .x |−→ ||x||1 is a norm on the space .L1 (τ ). Furthermore, if .x ∈ L1 (τ ) and .y, z ∈ M, then .yxz ∈ L1 (τ ) and .||yxz||1 ≤ ||y||B(H ) ||z||B(H ) ||x||1 . Proof As already pointed out, the first statement is a direct consequence of Proposition 3.4.8. The second assertion is simply a reformulation of Proposition 3.3.4 (ii). u n Remark 3.4.10 It will be useful to observe that τ (|x + y|) ≤ τ (|x|) + τ (|y|) ,

.

x, y ∈ S (τ ) .

(3.25)

Indeed, if .τ (|x|) = ∞ or .τ (|y|) = ∞, then this inequality is trivially satisfied. If τ (|x|) < ∞ and .τ (|y|) < ∞, then .x, y ∈ L1 (τ ) and so .x + y ∈ L1 (τ ), in which case (3.25) is simply the triangle inequality for the norm in .L1 (τ ).

.

Note that (3.23) simply states that the trace functional .τ is bounded with respect to .||·||1 , with .||τ || ≤ 1. Furthermore, it should be noted that .||Rex||1 , ||Imx||1 ≤ ||x||1 for all .x ∈ L1 (τ ), which implies in particular that .L1 (τ )h is closed in .L1 (τ ). Next the properties of the normed space .(L1 (τ ) , ||·||1 ) will be discussed. Proposition 3.4.11 The inclusion of .(L1 (τ ) , ||·||1 ) into .(S (τ ) , Tm ) is continuous. Proof If .x ∈ L1 (τ ) and .ε > 0, then f

ε

εμ (ε; x) ≤

.

f μ (t; x) dt ≤

0

0

∞

μ (t; x) dt = ||x||1 .

Consequently, if .xn → 0 in .L1 (τ ), then .μ (ε; xn ) → 0 for all .ε > 0, and hence, by Tm

Proposition 3.2.11 (i), .xn → 0.

u n

Since, by Proposition 2.6.1 (i), the positive cone .S (τ )+ is closed in .S (τ ) with respect to the measure topology, it is an immediate consequence of Proposition 3.4.11 that .L1 (τ )+ is closed in .L1 (τ ). The next theorem exhibits one of the fundamental properties of the space .L1 (τ ), which is the noncommutative extension of the Beppo–Levi theorem of classical integration theory. { } Theorem 3.4.12 If . aβ is an upward directed system in .L1 (τ )+ || (Beppo–Levi) || such that .supβ ||aβ ||1 < ∞, then there exists .a ∈ L1 (τ )+ such that .0 ≤ aβ ↑β a || || { }∞ and .||a − aβ ||1 ||→ 0. Moreover, there exists an increasing subsequence . aβn n=1 of || { } . aβ such that .||a − aβn || ↓ 0 as .n → ∞ and .0 ≤ aβn ↑n a. 1 { } { } Proof By Proposition 3.4.11, the norm boundedness of . aβ implies that . aβ is bounded in .S (τ ) with respect to the measure topology. Hence, it follows from Theorem 2.6.15 that there exists .a ∈ S (τ )+ such that .0 ≤ aβ ↑ a in .Sh (τ ). The

161

3.4 The Space L1 (τ )

( ) normality of the trace that .τ aβ ||↑ τ (a),|| and || ||(see Proposition 3.3.3, (iv)) implies so, .τ (a) = supβ ||aβ ||1 < ∞, that is, .a ∈ L1 (τ )+ . Consequently, .||a − aβ ||1 = ) ( ) ( τ a − aβ = τ (a) − τ aβ ↓β 0. It is now also clear that there exists an increasing || || { }∞ { } subsequence . aβn n=1 of . aβ such that .||a − aβn ||1 ↓ 0 as .n → ∞. Since .aβn ↑ Tm

and (again using Proposition 3.4.11) .aβn → a, it follows from Proposition 2.6.1 (iii) that .aβn ↑ a holds in .Sh (τ ), and hence, in .L1 (τ )h . The proof is complete. u n Corollary 3.4.13 || ||The norm .||·||1 on .L1 (τ ) is order continuous, that is, if .aβ ↓ 0 in L1 (τ )+ , then .||aβ ||1 ↓ 0.

.

Another important consequence of Theorem 3.4.12 is that the space .L1 (τ ) is complete with respect to .||·||1 , as will be shown in the next proposition. Proposition 3.4.14 The normed linear space .L1 (τ ) is a Banach space. Proof Since .||Rex||1 , ||Imx||1 ≤ ||x||1 for all .x ∈ L1 (τ ), it is clearly enough to Therefore, it is sufficient to prove show that .L1 (τ Σ )h is complete with respect to .||·||1 .Σ ∞ that any series . ∞ a with . a ∈ L and . (τ ) n=1 n n=1 ||an ||1 < ∞ is convergent. || +n|| || 1− || h + − Since .an = an − an and .||an ||1 , .||an ||1 ≤ ||an ||1 for all n, it may be assumed Σn that .an ∈ L1 (τ )+ for all n. Defining .sn = k=1 ak , it is clear that .0 ≤ sn ↑ and .supn ||sn ||1 < ∞. Therefore, it follows from Theorem 3.4.12 that there exists Σ + .a ∈ L1 (τ ) such that .||a − sn ||1 → 0, that is, .a = ∞ n=1 an as a norm convergent u n series in .L1 (τ ), which completes the proof. Remark 3.4.15 If .x ∈ S (τ ), then it follows from Lemma 3.2.3 (i) that .x ∈ M if and only if .μ (x) ∈ L∞ (m), in which case .||x||B(H ) = μ (0; x) = ||μ (x)||∞ . Therefore, in analogy with the spaces .L1 (τ ) and .L2 (τ ), the von Neumann algebra .M is also denoted by .L∞ (τ ), and if .x ∈ L∞ (τ ), then its norm .||x||B(H ) is also denoted by .||x||∞ . Recall that .F (τ ) denotes the two-sided ideal in .M consisting of all those .x ∈ M for which .τ (s (x)) < ∞ (equivalently, .τ (r (x)) < ∞). Since .|x| ≤ ||x||B(H ) s (x), it is clear that .τ (|x|) < ∞ for all .x ∈ F (τ ), that is, .F (τ ) ⊆ L1 (τ ) ∩ M. + Proposition 3.4.16 If .a ∈ L1 (τ )+ , then there exists a sequence .{an }∞ n=1 in .F (τ ) such that .an ↑ a and .||a − an ||1 → 0 as .n → ∞. In particular, .F (τ ) and .L1 (τ ) ∩ M are norm dense in .L1 (τ ).

Proof Given .a ∈ L1 (τ )+ , define .an = aea (1/n, n]. As has been noted before, a a .L1 (τ ) ⊆ S0 (τ ), and so, .τ (e (1/n, n]) ≤ τ (e (1/n, ∞)) < ∞, which implies + that .an ∈ F (τ ) for all n. Observing that .0 ≤ an ↑ a, it follows from Corollary 3.4.13 that .||a − an ||1 → 0 as .n → ∞. The remaining statements of the proposition are now also evident. u n It follows in particular from the above corollary, in combination with Proposition 3.4.14, that .L1 (τ ) is the norm completion of .L1 (τ ) ∩ M with respect to the norm .||·||1 . The next two theorems are noncommutative extensions of well-known results in classical integration theory.

162

3 Singular Value Functions

Theorem 3.4.17 (Fatou’s Lemma) If .{xn }∞ n=1 is a sequence in .L1 (τ ), if .x ∈ S (τ ) and .xn → x in measure, and if .lim.n→∞ ||xn ||1 < ∞, then .x ∈ L1 (τ ) and .

||x||1 ≤ limn→∞ ||xn ||1 .

Proof By Proposition 3.2.11 (ii), .μ (xn ) → μ (x) a.e. on .[0, ∞). Applying the classical version of Fatou’s Lemma for Lebesgue integrals on .[0, ∞), it follows that f .

0

∞

f μ (t; x) dt ≤ limn→∞

∞ 0

μ (t; xn ) dt = limn→∞ ||xn ||1 .

f∞ This shows in particular that . 0 μ (t; x) dt < ∞, and so, .x ∈ L1 (τ ). As the final u n inequality is now evident, the proof is complete. Remark 3.4.18 From the proof of the above theorem, it is clear that τ (|x|) ≤ limn→∞ τ (|xn |)

.

Tm

holds for any sequence .{xn }∞ n=1 in .S (τ ) satisfying .xn → x for some .x ∈ S (τ ). Evidently, Theorem 3.4.17 implies, in particular, that the norm closed unit ball in .L1 (τ ) is closed in .S (τ ) with respect to the measure topology. It is also useful to observe that the closed unit ball in .L1 (τ ) is actually closed in .S (τ ) with respect to the local measure topology. Proposition 3.4.19 Suppose that .{xα } is a net in .L1 (τ ) satisfying .||xα ||1 ≤ 1 for all .α. If .x ∈ S (τ ) and .xα → x locally in measure, then .x ∈ L1 (τ ) and .||x||1 ≤ 1. Proof If .x = v |x| is the polar decomposition of x, then it follows from Tlm

Proposition 2.7.5 that .v ∗ xα → |x|. Consequently, if .e ∈ P (M) is a finite trace { }∞ Tm projection, then .ev ∗ xα e → e |x| e, and so, there exists a subsequence . xαn n=1 Tm

such that .ev ∗ xαn e → e |x| e as .n → ∞. It follows from Proposition 3.3.4 (ii) that |) (| (| |) τ |ev ∗ xαn e| ≤ τ |xαn | ≤ 1,

.

n ∈ N,

and hence, Theorem 3.4.17 implies that .e |x| e ∈ L1 (τ ) and .τ (e |x| e) ≤ 1. Since this holds for all .e ∈ P (M) with .τ (e) < ∞, it follows from Proposition 3.3.5 that .τ (|x|) ≤ 1, that is, .x ∈ L1 (τ ) and .||x||1 ≤ 1. u n Remark 3.4.20 The result of Proposition 3.4.19 may also be formulated as follows. If .{xα } is a net in .S (τ ) and if .x ∈ S (τ ) is such that .xα → x locally in measure, then τ (|x|) ≤ limα τ (|xα |) ,

.

(3.26)

163

3.4 The Space L1 (τ )

where .limα τ (|xα |) is defined by limα τ (|xα |) = sup inf τ (|xα |) .

.

β α≥β

Indeed, suppose that (3.26) does not hold. Without loss of generality, it may be assumed that .

inf τ (|xα |) ≤ 1 − ε,

α≥β

∀β

(3.27)

and that .τ (|x|) ≥ 1 + ε, for some .0 < ε < 1. By Proposition 3.3.5, there exists a finite trace projection .e ∈ P (M) such that .τ (e |x| e) ≥ 1 + ε/2. If .x = v |x| is the Tm

polar decomposition of x, then .v ∗ xα → |x| locally in measure, and so, .ev ∗ xα e → e |x| e. Consequently, there exists a sequence .β1 ≤ β2 ≤ · · · such that .ev ∗ xα e ∈ V (1/n, 1/n) whenever .α ≥ βn . It follows from (3.27) that for each n there exists (| |) Tm ∗ .αn ≥ βn such that .τ |xαn | ≤ 1 − ε/2. It is clear that .ev xαn e → e |x| e as .n → ∞ and also |) (| |) (| τ |ev ∗ xαn e| ≤ τ |xαn | ≤ 1 − ε/2,

.

n ∈ N.

Therefore, by Theorem 3.4.17, .τ (e |x| e) ≤ 1 − ε/2, which yields a contradiction. This shows that (3.26) is valid. Theorem 3.4.21 (Dominated Convergence) If .{xn }∞ n=1 is a sequence in .L1 (τ ), if .x ∈ S (τ ) and .xn → x in measure, and if .μ (xn ) ≤ f a.e. on .[0, ∞) for all n for some .0 ≤ f ∈ L1 (m), then .x ∈ L1 (τ ) and .||x − xn ||1 → 0 (and so, in particular, .τ (xn ) → τ (x)) as .n → ∞. Proof Again by Proposition 3.2.11 (ii), .μ (xn ) → μ (x) a.e., and so, .μ (x) ≤ f holds a.e. on .[0, ∞), which implies that .μ (x) ∈ L1 (m), that is, .x ∈ L1 (τ ). Since .x − xn → 0 for the measure topology, it follows from Proposition 3.2.11 (i) that .μ (t; x − xn ) → 0 for all .t > 0. Now observe, using Proposition 3.2.7 (iii), that μ (t; x − xn ) ≤ μ (t/2; x) + μ (t/2; xn ) ≤ 2f (t/2)

.

for almost all .t ∈ [0, ∞). Since the function .t |−→ 2f (t/2), .t ≥ 0, belongs to L1 (m), an application of the classical dominated convergence theorem of Lebesgue yields that

.

f .

||x − xn ||1 =

∞

μ (t; x − xn ) dt → 0,

n → ∞.

0

Finally, since the trace functional .τ is continuous on .L1 (τ ), it is clear that the last assertion of the proposition holds and the proof is complete. u n

164

3 Singular Value Functions

The discussion that follows will lead to the identification of the space .L1 (τ ) with the pre-dual .M∗ of the von Neumann algebra .M. If .x ∈ L1 (τ ), then .xy ∈ L1 (τ ) for all .y ∈ M, and so, the linear functional .φx on .M may be defined by setting φx (y) = τ (xy) ,

.

y ∈ M.

It follows from Proposition 3.4.5 that .|φx (y)| ≤ τ (|x|) ||y||B(H ) for all .y ∈ M. Consequently, .φx ∈ M∗ and .||φx ||M∗ ≤ τ (|x|) = ||x||1 . Actually, Proposition 3.4.8 states that .

{ } ||φx ||M∗ = sup |τ (xy)| : y ∈ M, ||y||B(H ) ≤ 1 = τ (|x|)

holds for all .x ∈ L1 (τ ). Therefore, the map .o : x |−→ φx is a linear isometry from L1 (τ ) into .M∗ . Note that, since .L1 (τ ) is a Banach space, the range of .o is a closed subspace of .M∗ . To identify the range of .o, the result of the following lemma is needed.

.

Lemma 3.4.22 If .x ∈ L1 (τ )+ , then .φx is a positive normal linear functional on .M. Proof If .y ∈ M+ , then .y 1/2 xy 1/2 ≥ 0, and hence, it follows from Proposition 3.4.2 (iii) that ) ( τ (xy) = τ y 1/2 xy 1/2 ≥ 0,

.

which shows that .φx is positive. To prove that .φx is normal, suppose that .0 ≤ yα ↑ y in .M. Since .x 1/2( and .x 1/2 yα) belong to .L2 (τ ), it follows from Proposition ( )3.4.3 that .τ (xyα ) = τ x 1/2 yα x 1/2 for all .α, and similarly, .τ (xy) = τ x 1/2 yx 1/2 . By Proposition 2.2.25 (iii), 0 ≤ x 1/2 yα x 1/2 ↑α x 1/2 yx 1/2

.

holds in .Sh (τ ), and so, the normality of the trace (see Proposition 3.3.3 (iv)) implies that ) ) ( ( 1/2 .φx (yα ) = τ x yα x 1/2 ↑α τ x 1/2 yx 1/2 = φx (y) , which shows that .φx is normal.

u n

By Theorem 1.11.7, any positive normal functional on .M is ultra-weakly continuous, and hence, the above lemma implies that .φx ∈ M∗ for all .x ∈ L1 (τ )+ . Since any .x ∈ L1 (M) is a linear combination of positive elements of .L1 (τ ) (see the discussion following Proposition 3.4.1), it follows that .φx ∈ M∗ for all .x ∈ L1 (τ ). Consequently, the range of the isometry .o is a closed subspace of .M∗ . Therefore, to show that the range of .o is all of .M∗ , by the Hahn–Banach theorem, it is sufficient to prove that for every .0 /= ψ ∈ (M∗ )∗ there exists .x ∈ L1 (τ ) such that .ψ (φx ) /= 0.

165

3.4 The Space L1 (τ )

The norm dual of .M∗ may be identified with the space .M, in the sense that each ψ ∈ (M∗ )∗ is given by .ψ (φ) = φ (y), .φ ∈ M∗ , for a unique .y ∈ M (see Theorem 1.11.5 (v)). Consequently, it has to be proved that for each .0 /= y ∈ M, there exists .x ∈ L1 (τ ) such that .τ (xy) = φx (y) /= 0. This is achieved in the next lemma.

.

Lemma 3.4.23 If .0 /= y ∈ M, then there exists .x ∈ L1 (τ ) such that .τ (xy) /= 0. Proof First observe that there exists .e ∈ P (M) such that .τ (e) < ∞ and .e |y| e /= 0. Indeed, the trace .τ is semi-finite, and so, there exists an upward directed system so .{eα } in .P (M) such that .0 ≤ eα ↑ 1 and .τ (eα ) < ∞ for all .α. Since .eα → 1 so and this net is uniformly bounded, it follows that .eα |y| eα → |y|, and hence, there exists .α0 such that .eα0 |y| eα0 /= 0. Therefore, the projection .e = eα0 has the desired properties. Let .y = v |y| be the polar decomposition of y and define .x = ev ∗ . Since .τ (e) < ∞, it is clear that .x ∈ L1 (τ ) and ( ) τ (xy) = τ ev ∗ y = τ (e |y| e) > 0,

.

u n

which completes the proof.

Via the preceding lemma, it may now be concluded that the map .o : x |−→ φx is a linear isometry form .L1 (τ ) onto the pre-dual .M∗ of .M. The adjoint of .o defines an isometry .o∗ : (M∗ )∗ → L1 (τ )∗ , and hence, identifying .(M∗ )∗ with .M (as indicated above), .o∗ induces an isometry from .M onto .L1 (τ )∗ . These results will be collected in the next theorem. Theorem 3.4.24 Let .M be a von Neumann algebra equipped with a fixed normal semi-finite faithful trace .τ : (i) Defining, for each .x ∈ L1 (τ ), the linear functional .φx : y − | → τ (xy) on .M, the map .x − | → φx is a linear isometry from .L1 (τ ) onto the pre-dual .M∗ of .M. (ii) Defining, for each .y ∈ M, the linear functional .ψy : x |−→ τ (xy) on .L1 (τ ), the map .y |−→ ψy is a linear isometry from .M onto the Banach dual .L1 (τ )∗ . In particular, .

||y||B(H ) = sup {|τ (xy)| : x ∈ L1 (τ ) , ||x||1 ≤ 1} ,

y ∈ M.

Remark 3.4.25 The duality pairing between .L1 (τ ) and .M is also denoted by ., that is, . = τ (xy) for all .x ∈ L1 (τ ) and .y ∈ M. So, using this notation, the map .x |−→ is a linear isometry from .L1 (τ ) onto .M∗ , and the map .y |−→ is a linear isomorphism from .M onto .L1 (τ )∗ . In other words, the pre-dual ∗ .M∗ may be identified with .L1 (τ ), and the dual .L1 (τ ) may be identified with .M, via the trace duality given by . = τ (xy), .x ∈ L1 (τ ), and .y ∈ M. Remark 3.4.26 Suppose that .M = B (H ), equipped with standard trace .τ . In this case, the space .L1 (τ ) is precisely the ideal .S1 of all trace class operators on H , and Theorem 3.4.24 identifies the dual space of .S1 with .B (H ) via trace duality.

166

3 Singular Value Functions

Furthermore, this theorem shows that the space of all ultra-weakly continuous linear functionals on .B (H ) may be identified with .S1 , via trace duality. In the commutative situation, that is, .M = L∞ (ν), where .(X, Σ, ν) is a Maharam measure space, the above theorem gives the usual identification of the dual space .L1 (ν)∗ with the space .L∞ (ν). This latter statement is actually equivalent to the Radon–Nikodym theorem being valid for the measure space .(X, Σ, ν), which shows that the assumption that .(X, Σ, ν) is a Maharam measure space is crucial in this setting. The next theorem presents an estimate of the trace of the product of two operators in terms of their singular value functions. Because of the length of its proof, two lemmas will be obtained first, which deal with special cases of the theorem to be proved. f∞ Lemma 3.4.27 If .0 ≤ a, b ∈ L1 (τ ) ∩ M, then .τ (ab) ≤ 0 μ (t; a) μ (t; b) dt. Proof Let .0 ≤ a ∈ L1 (τ ) ∩ M be fixed. If .q ∈ P (M) with .τ (q) < ∞, then f

∞

τ (aq) = τ (qaq) =

μ (t; qaq) dt.

.

0

Since .μ (qaq) ≤ μ (a) and, by Remark 3.2.6 (i), .μ (t; qaq) = 0 whenever .t ≥ τ (q), it is clear that .μ (qaq) ≤ μ (a) χ[0,τ (q)) . Noting that .χ[0,τ (q)) = μ (q), this proves the lemma with .b = q. Now suppose that .0 ≤ s ∈ M is a linear combination of finite trace projections. Such an element s may be written as m Σ

s=

.

βj qj ,

j =1

( ) where .0 < βj ∈ R for all j and .q1 ≤ q2 ≤ · · · ≤ qm in .P (M) with .τ qj < ∞. As follows from Example 3.2.2 (i), the singular value function of s is given by m Σ

μ (s) =

.

( ) βj μ qj .

j =1

Using the fact that the lemma has already been proved for the case .b = qj , it follows that τ (as) =

m Σ

.

j =1

m ) Σ ( βj τ aqj ≤ βj j =1

f

∞

( ) μ (a) μ qj dt =

0

which is the statement of the lemma with .b = s.

f

∞

μ (a) μ (s) dt, 0

167

3.4 The Space L1 (τ )

If .0 ≤ b ∈ L1 (τ ) ∩ M is arbitrary, the spectral theorem implies that there exists + an increasing sequence .{sn }∞ n=1 in .M , with each .sn being a linear combination of finite trace projections, such that .0 ≤ sn ↑ b and .||b − sn ||B(H ) → 0 as .n → ∞. For each n, it follows from the above that f

∞

τ (asn ) ≤

.

f

∞

μ (a) μ (sn ) dt ≤

0

μ (a) μ (b) dt.

(3.28)

0

Furthermore, by Proposition 3.4.5, 0 ≤ τ (ab) − τ (asn ) = τ (a (b − sn )) ≤ ||b − sn ||B(H ) τ (a) ,

.

and so, .τ (asn ) ↑n τ (ab). Consequently, the result of lemma now follows from (3.28) by letting .n → ∞. u n f∞ Lemma 3.4.28 If .x, y ∈ M, then .τ (|xy|) ≤ 0 μ (t; x) μ (t; y) dt. Proof First suppose that .x, y ∈ L1 (τ ) ∩ M and let .xy = v |xy| be the polar decomposition of xy, so that .|xy| = v ∗ xy. According to Lemma 3.4.4, | )1/2 (| ∗ | | ∗ |)1/2 ( ) (| τ (|xy|) = τ v ∗ xy ≤ τ |x ∗ v | |y| τ |v x | |y | .

.

(3.29)

It follows from Lemma 3.4.27, with .a = |x ∗ v| and .b = |y|, that (| ∗ | ) .τ |x v | |y| ≤

f

∞

|) (| μ |x ∗ v | μ (|y|) dt ≤

0

f

∞

μ (x) μ (y) dt, 0

since .μ (|x ∗ v|) ≤ μ (x ∗ ) = μ (x). Similarly, | | |) (| τ |v ∗ x | |y ∗ | ≤

f

∞

μ (x) μ (y) dt,

.

0

and so the result of the lemma, in this case, follows from (3.29). Now assume that .x, y ∈ M and that .e ∈ P (M) satisfies .τ (e) < ∞. If .xy = v |xy| is the polar decomposition of xy, then by (3.23) in Corollary 3.4.6, |) ( ) (|( ) τ (e |xy| e) = τ ev ∗ xye ≤ τ | ev ∗ x (ye)| .

.

Since .ev ∗ x and ye both belong to .L1 (τ ) ∩ M, it follows from the first part of the proof that |) (|( ) τ | ev ∗ x (ye)| ≤

f

.

0

∞

( ) μ ev ∗ x μ (ye) dt ≤

f

∞

μ (x) μ (y) dt, 0

168

3 Singular Value Functions

as .μ (ev ∗ x) ≤ μ (x) and .μ (ye) ≤ μ (y). This shows that f τ (e |xy| e) ≤

∞

μ (x) μ (y) dt,

.

e ∈ P (M) , τ (e) < ∞,

0

u n

and now, the result of the lemma follows from Proposition 3.3.5. Theorem 3.4.29 If .x, y ∈ S (τ ), then f τ (|xy|) ≤

∞

(3.30)

μ (t; x) μ (t; y) dt.

.

0

Proof If .x, y ∈ S (τ ), then .D (x) ∩ D (y) is .τ -dense, and so, there exists an ∞ increasing .{pn } n=1 in .P (M) such that .pn (H ) ⊆ D (x) ∩ D (y) for all ) ( sequence ⊥ n and .τ pn → 0 as .n → ∞. This implies that .xpn , ypn ∈ M for all n and Tm

Tm

xpn → x, .ypn → y. It follows from Lemma 3.4.28 that

.

f τ (|(xpn ) (ypn )|) ≤

.

∞

f μ (xpn ) μ (ypn ) dt ≤

0

∞

μ (x) μ (y) dt 0

Tm

holds for all n. Furthermore, since .(xpn ) (ypn ) → xy as .n → ∞, it follows from Fatou’s lemma (see Theorem 3.4.17 and Remark 3.4.18) that f τ (|xy|) ≤ limn→∞ τ (|(xpn ) (ypn )|) ≤

∞

μ (x) μ (y) dt,

.

0

u n

and by this, the proof of the theorem is complete.

For later purposes, some additional properties of the trace are required, which will be discussed next. Proposition 3.4.30 If .x, y ∈ S (τ ) and if .xy, yx ∈ L1 (τ ), then .τ (xy) = τ (yx). Proof For .n ∈ N, define the projections .pn and .qn by ∗ pn = e|x | (1/n, n] ,

.

qn = e|x| (1/n, n] .

If .x = v |x| is the polar decomposition of x, then .|x ∗ | = v |x| v ∗ , from which it follows that the spectral measure of .|x ∗ | is given by ) ( ∗ e|x | (B) = ve|x| (B \ {0}) v ∗ + 1 − vv ∗ δ0 (B) ,

.

B ∈ B (R) .

169

3.4 The Space L1 (τ )

∗ In particular, .e|x | (B) = ve|x| (B) v ∗ for any Borel subset .B ⊆ (0, ∞), and so, ∗ .pn = vqn v for all n. Hence,

pn x = vqn v ∗ x = vqn |x| = v |x| qn = xqn

.

and also .pn x = pn pn x = pn xqn for all n. Observe now that .qn y ∈ L1 (τ ) for all .n ∈ N. Indeed, it follows from the definition of .qn that .qn ≤ n2 |x|2 , and so, .

|qn y|2 = y ∗ qn y ≤ n2 y ∗ x ∗ xy = n2 |xy|2 .

Since the square-root function is operator monotone (see Corollary 2.2.28), this implies that .|qn y| ≤ n |xy|, and hence, .qn y ∈ L1 (τ ) as .xy ∈ L1 (τ ). Furthermore, .xqn = v |x| qn ∈ M. Consequently, using Proposition 3.4.2 (iii), τ (pn xy) = τ (xqn y) = τ ((xqn ) (qn y)) = τ (qn yxqn ) = τ (yxqn ) ,

.

where the last equality follows from the fact that .yxqn ∈ L1 (τ ). Since the functionals .z − | → τ (zxy) and .z |−→ τ (yxz), .z ∈ M, are both linear combinations of positive normal functionals on .M (see Lemma 3.4.22) and since |x ∗ | (0, ∞) = s (x ∗ ) = r (x) and .qn ↑ e|x| (0, ∞) = s (x), it follows that .pn ↑ e τ (xy) = τ (r (x) xy) = lim τ (pn xy)

.

n→∞

= lim τ (yxqn ) = τ (yxs (x)) = τ (yx) . n→∞

u n

The proof is complete.

Remark 3.4.31 The condition in Proposition 3.4.30 that both xy and yx belong to L1 (τ ) cannot be weakened to assuming that only .xy ∈ L1 (τ ), as is shown by the following simple example. Let .H = H1 ⊕ H1 , where .H1 is an infinite dimensional Hilbert space, and let .M = B (H ) with standard trace. If the operators .x, y ∈ B (H ) are defined by setting .x (ξ, η) = (ξ, 0) and .y (ξ, η) = (0, ξ ), .(ξ, η) ∈ H1 ⊕ H1 , then .xy = 0 but .yx = y, which is not even compact.

.

Proposition 3.4.32 If .a, b ∈ S (τ )+ and if .ab ∈ L1 (τ ), then a 1/2 ba 1/2 , b1/2 ab1/2 ∈ L1 (τ )

.

and ) ( ) ( τ (ab) = τ a 1/2 ba 1/2 = τ b1/2 ab1/2 .

.

170

3 Singular Value Functions

( ] Proof For .n ∈ N, let .en = ea 1/n2 , n2 and observe that aen ≤ n2 a 2 en ≤ n2 a 2 .

.

This implies that .

| | ( )∗ ( ) | 1/2 |2 en a 1/2 b = baen b |en a b| = en a 1/2 b ≤ n2 ba 2 b = n2 (ab)∗ (ab) = n2 |ab|2 ,

and so, .

| | | 1/2 | |en a b| ≤ n |ab| .

Since .ab ∈ L1 (τ ), this shows that .en a 1/2 b ∈ L1 (τ ) for all .n ∈ N. In combination with the fact that .en a 1/2 ∈ M, this yields that )( ) (( )) ( en a 1/2 en a 1/2 b = τ en a 1/2 ben a 1/2 ) ( = τ en a 1/2 ba 1/2 en .

τ (en ab) = τ

.

Since .τ (x ∗ x) = τ (xx ∗ ) for all .x ∈ S (τ ), it follows further that ) (( )∗ ( )) ( b1/2 a 1/2 en τ en a 1/2 ba 1/2 en = τ b1/2 a 1/2 en )( )∗ ) (( = τ b1/2 a 1/2 en b1/2 a 1/2 en ) ( )∗ ) (( , = τ b1/2 a 1/2 en b1/2 a 1/2

.

and hence, τ (en ab) = τ

.

((

) ( )∗ ) . b1/2 a 1/2 en b1/2 a 1/2

(3.31)

Since .en ↑ ea (0, ∞) = s (a) implies that .

( ) ( )∗ ( ) ( )∗ b1/2 a 1/2 en b1/2 a 1/2 ↑n b1/2 a 1/2 s (a) b1/2 a 1/2 = b1/2 ab1/2

in .Sh (τ ), the normality of the trace in combination with (3.31) implies that ) ( τ (en ab) ↑n τ b1/2 ab1/2 .

.

171

3.4 The Space L1 (τ )

On the other hand, the functional .x − | → τ (xab), .x ∈ M, is a linear combination of at most four positive normal functionals on .M, and so, .τ (en ab) → τ (s (a) ab) = τ (ab) as .n → ∞. Consequently, ) ( τ b1/2 ab1/2 = τ (ab) ,

.

( ) which shows in particular that .τ b1/2 ab1/2 < ∞, that is, .b1/2 ab1/2 ∈ L1 (τ ). ( 1/2 1/2 )∗ ( 1/2 1/2 ) a b = b1/2 ab1/2 ∈ L1 (τ ), it follows from Finally, since . a b Proposition 3.4.1 (iii) that )( )∗ ( a 1/2 ba 1/2 = a 1/2 b1/2 a 1/2 b1/2 ∈ L1 (τ )

.

) ( ) ( and .τ a 1/2 ba 1/2 = τ b1/2 ab1/2 . By this, the proof of the proposition is complete. u n It is now natural to extend the trace to the set of all positive self-adjoint operators affiliated with the von Neumann algebra .M. Recall that the set of all positive selfadjoint operators in H is denoted by .H+ , which is partially ordered via the quadratic form ordering (see Sect. 1.8). Furthermore, .H+ (M) denotes the set of all .a ∈{ H+} that are affiliated with the von Neumann algebra .M. Observe, in particular, if . aβ is an upward directed system in .H+ (M) and if there exists .b ∈ H+ such that .aβ ≤ b for all .β, then it follows from Proposition 1.8.8 and Corollary 1.8.9 that there exists + + + .a ∈ H (M) such that .aβ ↑β a in .H , and hence in .H (M). Furthermore, it is evident that M+ ⊆ S (τ )+ ⊆ S (M)+ ⊆ H+ (M) .

.

+ + If .a ∈ H+ (M) and .b (∈ S (M) are such .a ∈ S (M) . Indeed, ) ( 1/2 ) that .a ≤ b, (then ) 1/2 1/2 ⊆ D a .a ≤ b implies that .D b , and since .D b is strongly dense, it ( 1/2 ) 1/2 is also strongly dense. Hence, .a ∈ S (M)+ and so .a ∈ follows that .D a + + S (M) . A similar argument shows that if .a ∈ H (M), .b ∈ S (τ )+ , and .a ≤ b, then .a ∈ S (τ )+ . Moreover, it should be observed that for any .a ∈ H+ (M), there exists an + + increasing sequence .{an }∞ n=1 in .M such that .an ↑ a in .H (M). Indeed, defining .fn (t) = min (t, n), .t ≥ 0, for each .n ∈ N, it is clear that .fn (t) ↑n t for all .t ∈ [0, ∞). Setting .an = fn (a), it is clear that .an ∈ M for all n and that .an ↑ a in + .H (M). The extended trace .τ : S (τ )+ → [0, ∞] may now be further extended to + + + .H (M) by setting .τ (a) = ∞ for all .a ∈ H (M) \ S (τ ) . This extension + .τ : H (M) → [0, ∞] is easily seen to be positively homogeneous, monotone (that is, .τ (a) ≤ τ (b) whenever .a ≤ b in .H+ (M)), unitarily invariant, and normal. + For ) proof of the last assertion, suppose that ( the ( ).aβ ↑β a holds in .H (M). Since .τ aβ ≤ τ (a) for all .β, it is clear that .supβ τ aβ ≤ τ (a). To show that the reverse

172

3 Singular Value Functions

( ) inequality holds, it may be assumed that .supβ τ aβ < ∞, in which case it follows + + from the Beppo–Levi ( ) theorem (Theorem 3.4.12) that .a ∈ L1 (τ ) +⊆ S (τ ) , and hence, .supβ τ aβ = τ (a). Furthermore, the restriction .τ : S (M) → [0, ∞] is easily seen to be additive as well. Suppose that a is a self-adjoint operator in H that is affiliated with .M. If .f : R → [0, ∞) is a Borel function, then .f (a) is affiliated with .M, and so, .f (a) ∈ H+ (M). Recall that the measure .τ ea on .B (R) is defined by setting .(τ ea ) (B) = τ (ea (B)), .B ∈ B (R). The following observation is an extension of (3.16). Proposition 3.4.33 Let a be a self-adjoint operator in H that is affiliated with .M: (i) If .f : R → [0, ∞) is a positive Borel function, then f τ (f (a)) =

.

R

( ) f (λ) d τ ea (λ) .

(3.32)

(ii) If .f : R → R is a Borel function, then .f (a) ∈ L1 (τ ) if and only if .f ∈ L1 (τ ea ), and in this case (3.32) is also valid. Proof (i) For .n ∈ N, define the function .fn : R → [0, ∞) by .fn (t) = min (f (t) , n), + .t ≥ 0. Since .fn (a) ∈ M , it follows from (3.16) that f τ (fn (a)) =

.

R

( ) fn (λ) d τ ea (λ)

for each n. Since .fn (t) ↑ f (t), .t ≥ 0, it is clear that .fn (a) ↑ f (a) in .H+ (M), and so, the normality of .τ implies that .τ (fn (a)) ↑ τ (f (a)). In combination with the monotone convergence theorem, this shows that (3.32) holds. (ii) If .f : R → R is an arbitrary Borel function, then .f (a) ∈ L1 (τ ) if and only if .f (a)+ = f + (a) and .f (a)− = f − (a) both belong to .L1 (τ ), and a + − ∈ L (τ ea ). Applying (i) to the functions .f + .f ∈ L1 (τ e ) if and if .f , f 1 and .f − , the claimed result is now clear. u n References: [45, 55, 78, 111].

3.5 The Eigenvalue Function As has been seen in the previous sections, the singular value function is a useful tool in semi-finite von Neumann algebras. In von Neumann algebras with finite trace, it is possible to define an eigenvalue function for self-adjoint elements, which may be exploited in certain problems requiring a more detailed analysis. Therefore, throughout this section, it will be assumed that .M is a von Neumann algebra on a

3.5 The Eigenvalue Function

173

Hilbert space H , which is equipped with a finite faithful normal trace .τ : M+ → [0, ∞). As in the previous section, the symbol .τ also denotes the extended trace on .S (τ )+ , as well as the corresponding trace functional on the space .L1 (τ ). In this case, it should be noted that .M ⊆ L1 (τ ). Furthermore, the assumption that .τ (1) < ∞ implies that .S (τ ) consists of all closed and densely defined operators in H that are affiliated with .M (see Remark 2.4.3). If .a ∈ Sh (τ ), then the spectral distribution function .d (a) of a is defined by setting ( ) d (s; a) = τ ea (s, ∞) ,

.

s ∈ R.

It is clear that the function .d (a) : R → [0, τ (1)] is decreasing, and the normality of the trace implies that .d (a) is right-continuous. Moreover, .lims→−∞ d (s; a) = τ (1) and .lims→∞ d (s; a) = 0. Definition 3.5.1 If .a ∈ Sh (τ ), then the eigenvalue function .λ (a) is defined to be the right-continuous inverse of the spectral distribution function .d (a), that is, λ (t; a) = inf {s ∈ R : d (s; a) ≤ t} ,

.

t ∈ [0, τ (1)) .

Since .lims→∞ d (s; a) = 0, it is clear that .λ (t; a) < ∞ whenever .t ∈ (0, τ (1)) (but, .λ (0; a) = ∞ might occur) and since .lims→−∞ d (s; a) = τ (1), it follows that .λ (t; a) > −∞ for all .t ∈ [0, τ (1)). Therefore, the function .λ (a) : [0, τ (1)) → (−∞, ∞] is decreasing and right-continuous. If .a ∈ S (τ )+ , then it is evident that .λ (t; a) = μ (t; a) for all .t ∈ [0, τ (1)), where, as before, .μ (a) is the singular value function of a. It follows from Remark 3.1.6 (iii), in particular (3.3), that a and the function .λ (a) have the same distribution function, that is, d (s; a) = m {t ∈ [0, τ (1)) : λ (t; a) > s} = d (s; λ (a)) ,

.

s ∈ R.

Note that this equality is equivalent to saying that ( ) τ ea (−∞, s] = m {t ∈ [0, τ (1)) : λ (t; a) ≤ s} .

.

The following proposition exhibits a further relationship between the eigenvalue function and the trace. Recall that, for any .a ∈ Sh (τ ), the measure .τ ea on .B (R) is given by .τ ea (B) = τ (ea (B)), .B ∈ B (R), which is a finite Borel measure as .τ (1) < ∞. The following lemma is an extension of Lemma 3.3.6 (i). Its proof follows along the same lines. Lemma 3.5.2 If .a ∈ Sh (τ ), then the measure .τ ea on .B (R) is the image of Lebesgue measure m on .(0, τ (1)) under the function .λ (a) : (0, τ (1)) → R. Proof If .ν denotes the image measure of m under .λ (a) and if .α < β in .R, then ν (α, β] = m {t ∈ (0, τ (1)) : α < λ (t; a) ≤ β} .

.

174

3 Singular Value Functions

Using that .α < λ (t; a) ≤ β holds if and only if .d (β; a) ≤ t < d (α; a), it follows that ( ) ( ) ν (α, β] = d (α; a) − d (β; a) = τ ea (α, ∞) − τ ea (β, ∞) ( ) ( ) = τ ea (α, β] = τ ea (α, β] ,

.

from which it may be concluded that .ν = τ ea .

u n

Remark 3.5.3 Via the same argument as used in the proof of Corollary 3.3.8, it follows from Lemma 3.5.2 that σ (a) = {λ (t; a) : t ∈ [0, τ (1))}

.

for all .a ∈ Sh (τ ). Remark 3.5.4 Lemma 3.5.2 has another interesting consequence. If .a ∈ Sh (τ ) and if .f : R → R is a Borel function, then d (f (a)) = d (f ◦ λ (a)) .

(3.33)

.

Indeed, using that the spectral measure .ef (a) is given by ( ) ef (a) (B) = ea f −1 (B) ,

B ∈ B (R) ,

.

it follows from Lemma 3.5.2 that ) ( )( ) ( f (a) .d (s; f (a)) = τ e (s, ∞) = τ ea f −1 (s, ∞) } { = m t ∈ (0, τ (1)) : λ (t; a) ∈ f −1 (s, ∞) = m {t ∈ (0, τ (1)) : f (λ (t; a)) ∈ (s, ∞)} = d (s; f ◦ λ (a)) for all .s ≥ 0. If .a ∈ S (τ )+ and .f : R → [0, ∞) is a positive Borel function, then Proposition 3.3.9 (i) states that f τ (f (a)) =

.

0

τ (1)

f f (μ (t; a)) dt =

τ (1)

f (λ (t; a)) dt. 0

The next proposition extends this to operators .a ∈ Sh (τ ) and arbitrary Borel functions .f : R → R.

3.5 The Eigenvalue Function

175

Proposition 3.5.5 Suppose that .a ∈ Sh (τ ): (i) If .f : R → [0, ∞) is a positive Borel function, then f

τ (1)

τ (f (a)) =

f (λ (t; a)) dt.

.

(3.34)

0

(ii) If .f : R → R is a Borel function, then .f (a) ∈ L1 (τ ) if and only if .f ◦ λ (a) ∈ L1 (0, τ (1)), and in this case (3.34) also holds (which is, in particular, the case if f is bounded on the spectrum a). Proof (i) As already observed in Proposition 3.4.33 (i), f τ (f (a)) =

.

R

f (s) dτ ea (s)

(3.35)

holds for any positive Borel function f on .R. Therefore, (3.34) is now an immediate consequence of Lemma 3.5.2 in combination with the change of measure formula. (ii) By Proposition 3.4.33 (ii), .f (a) ∈ L1 (τ ) if and only if .f ∈ L1 (τ ea ) and in this case (3.35) holds. As in the proof of (i), this immediately implies (3.34). u n The following special case of Proposition 3.5.5 is sufficiently important to single out explicitly. Corollary 3.5.6 If .a ∈ Sh (τ ), then f

τ (1)

τ (|a|) =

.

f |λ (t; a)| dt =

0

τ (1)

λ (t; |a|) dt.

0

Furthermore, .a ∈ L1 (τ ) if and only if .λ (a) ∈ L1 (0, τ (1)), and in this case, f

τ (1)

τ (a) =

λ (t; a) dt.

.

(3.36)

0

For further analysis of the eigenvalue function .λ (a), it will be convenient to introduce another closely related function, which will be denoted by .λˇ (a). Given .a ∈ Sh (τ ), the function .t − | → λ (t−; a), .t ∈ (0, τ (1)], is decreasing and leftcontinuous on .(0, τ (1)] (actually, it is the left-continuous regularization of the function .λ (a), in the sense of Remark 3.1.6 (i)). Consequently, the function .λˇ (a), defined by setting λˇ (t; a) = λ ((τ (1) − t) −; a) ,

.

t ∈ [0, τ (1)) ,

176

3 Singular Value Functions

is increasing and right-continuous on .[0, τ (1)). Since the functions .λ (a) and t |−→ λ (t−; a) differ on an at most countable subset of .(0, τ (1)), these two functions are equimeasurable (cf. also Remark 3.1.6 (i)). Furthermore, the map .t |−→ τ (1) − t is measure preserving from .[0, τ (1)) onto .(0, τ (1)], and hence, the functions .λ (a) and .λˇ (a) are equimeasurable. The function .λˇ (a) may be referred to as the increasing eigenvalue function of the operator a. The following is an extension of Proposition 3.2.8 (ii). .

Proposition 3.5.7 Let .a ∈ Sh (τ ) and .φ : R → R be a function that is either increasing or decreasing. If necessary, .φ (∞) and .φ (−∞) are interpreted as .φ (∞) = limt→∞ φ (t) and .φ (−∞) = limt→−∞ φ (t), respectively: (i) If .φ is increasing and left-continuous, then .λ (φ (a)) = φ ◦ λ (a). (ii) If .φ is decreasing and right-continuous, then .λ (φ (a)) = φ ◦ λˇ (a). Proof First note that, in cases (i) and (ii), it follows from the observation in Remark 3.5.4 that d (φ (a)) = d (φ ◦ λ (a)) .

.

(3.37)

In case (i), the assumptions on the function .φ imply that .φ ◦ λ (a) is decreasing and right-continuous, and so, by taking right-continuous inverses in (3.37), it may be concluded that .λ (φ (a)) = φ ◦ λ (a). Now suppose that .φ is decreasing and right-continuous. As has been observed above, the functions .λ (a) and .λˇ (a) are equimeasurable, and hence, by Corollary 3.1.11, the functions .φ ◦ λ (a) and .φ ◦ λˇ (a) are also equimeasurable. Therefore, it follows from (3.37) that ( ) d (φ (a)) = d φ ◦ λˇ (a) .

.

(3.38)

The stated assumptions on the function .φ in case (ii) imply that .φ ◦ λˇ (a) is decreasing and right-continuous, and therefore, taking right-continuous inverses in (3.38) yields that .λ (φ (a)) = φ ◦ λˇ (a). u n For later reference, the next proposition gathers a number of elementary properties of the (increasing) eigenvalue function. Proposition 3.5.8 For .a ∈ Sh (τ ), the following statements hold: ˇ (i) .λ (−a) ( + ) = −λ (a). ( ) (ii) .λ a = λ (a)+ and .λˇ a + = λˇ (a)+ . ( −) ( −) (iii) .λ a = λˇ (a)− and .λˇ a(| = λ|)(a)− . | | (iv) .λ (|a|) = λ (|λ (a)|) = λ |λˇ (a)| . (v) .λ (t; a + β1) = λ (t; a) + β, .t ∈ [0, τ (1)) for all .β ∈ R. Proof Since the function .φ : t |−→ −t, .t ∈ R, is decreasing and continuous, (i) follows immediately from Proposition 3.5.7 (ii). Similarly, the first assertion of

3.5 The Eigenvalue Function

177

(ii) follows by applying Proposition 3.5.7 (i) to the increasing continuous function φ:t − | → t + ,(.t ∈ R. function .φ : t |−→ −t + , .t ∈ R, is decreasing, ) The continuous + + ˇ and hence, .λ −a = −λ (a) . In combination with (i), this yields that

.

( ) ( ) λˇ a + = −λ −a + = λˇ (a)+ ,

.

which is the second assertion of (ii). Statement (iii) now follows from (ii) applied to −a, in combination with (i). For the proof of (iv), observe that (3.33) in Remark 3.5.4 (applied to the function .t |−→ |t|, .t ∈ R) implies that .d (|a|) = d (|λ (a)|). By taking right-continuous inverses, it follows that .λ (|a|) = λ|(|λ (a)|). Since .λˇ (a) is equimeasurable with | | |ˇ .λ (a), and hence, the same holds for .|λ (a)| and .|λ (a)|, it is clear that .λ (|λ (a)|) = |) (| | | λ |λˇ (a)| . Finally, (v) follows from Proposition 3.5.7 (i), applied to the function .φ : t − | → t + β, .t ∈ R. u n ) ( +) ( = λ (a)+ and .λ (a)− = λˇ a − , it is now also clear Remark 3.5.9 Since .λ a that: ( ) ( ) ( ) ( ) (vi) .λ (a) = λ a + − λˇ a − and .|λ (a)| = λ a + + λˇ a − . ( ) ( ) (vii) .λ a + ∧ λˇ a − = 0. .

Example 3.5.10 (i) Let Σm.(M, τ ) be a von Neumann algebra with .τ (1) < ∞ and suppose that .a = j =1 αj pj , where .p1 , . . . , pm ∈ P (M) with .pj pk = 0 whenever .j /= k, and .αj ∈ R (.j = 1, . . . , m) are such that .αj /= αk whenever .j /= k. For the computation of .λ (a),Σ it may be assumed without loss of generality that m .α1 > α2 > · · · > αm and . j =1 pj = 1. The spectral measure is given by ea =

m Σ

.

p j δa j ,

j =1

and hence, ( ) Σ ( ) d (s; a) = τ ea (s, ∞) = τ pj .

.

αj >s

Consequently,

d (s; a) =

.

⎧ ⎪ ⎨ τΣ(1) ⎪ ⎩0

j i=1 τ

if s < αm , (pi ) if αj +1 ≤ s < αj , if s ≥ α1 .

j = 1, . . . , m − 1,

178

3 Singular Value Functions

Defining .ρj = verified that

Σj

i=1 τ

(pi ) for .j = 1, . . . , m and .ρ0 = 0, it is now easily

λ (a) =

m Σ

αj χ[ρj −1 ,ρj ) .

.

j =1

(3.39)

It is now also clear that the increasing eigenvalue function .λˇ (a) is given by ˇ (a) = .λ

m Σ j =1

αj χ[τ (1)−ρj ,τ (1)−ρj −1 ) .

In particular, if .p ∈ P (M), then λˇ (p) = χ[τ (p⊥ ),τ (1)) .

λ (p) = χ[0,τ (p)) ,

.

∼ Mn (C) equipped (ii) Consider the special case that .H = Cn and .M = B (H ) = with theΣstandard trace .τn . If .a ∈ Mn (C) is self-adjoint, then a may be written as .a = m j =1 αj pj , where .α1 > · · · > αm are the distinct eigenvalues of a and .pj is the orthogonal projection onto the eigenspace corresponding to .αj . Note Σ that . m j =1 pj = 1. It follows from (3.39) that λ (a) =

m−1 Σ

.

j =1

αj χ[ρj −1 ,ρj ) ,

where the numbers .ρj are defined as in (i) above. For each j , the length of [ ) ( ) the interval . ρj −1 , ρj is .τ pj , which is the dimension of the eigenspace corresponding to .αj . Therefore, λ (a) =

n Σ

.

λj χ[j −1,j ) ,

j =1

where .λ1 ≥ λ2 ≥ · · · ≥ λn is the sequence of eigenvalues of a in which each eigenvalue is repeated according to its multiplicity. Furthermore, λˇ (a) =

n Σ

.

λn−j +1 χ[j −1,j ) .

j =1

Consequently, in this case, the eigenvalue function .λ (a) may be identified with the eigenvalue sequence of a, taken in decreasing order and repeated according to multiplicity. Moreover, .d (s; a) is just the number of eigenvalues that exceed

3.6 Properties of the Eigenvalue Function

179

s. Similarly, the increasing eigenvalue function .λˇ (a) may be identified with the increasing eigenvalue sequence of a. References: [42, 101].

3.6 Properties of the Eigenvalue Function In this section, some further properties of the eigenvalue function, which will be used in subsequent sections, are discussed in some detail. As in the previous section, .M is a von Neumann algebra on a Hilbert space .H , and .τ is a fixed finite normal faithful trace on .M. First, an alternative description of the eigenvalue function will be derived. Recall from Sect. 3.2 that if .a ∈ Sh (τ ) and .e ∈ P (M), then the quantity .αe (a) is defined by } { αe (a) = sup : ||ξ ||H = 1, eξ = ξ, ξ ∈ D (a) .

.

The next result is similar to Proposition 3.2.19 (i). Proposition 3.6.1 If .a ∈ Sh (τ ), then the eigenvalue function .λ (a) is given by } { ( ) λ (t; a) = inf αe (a) : e ∈ P (M) , τ e⊥ ≤ t

.

for all .t ∈ [0, τ (1)). Proof Let .t ∈ [0, τ (1)) be fixed and denote the right hand infimum by A. If .s ∈ R and .ξ ∈ D (a) are such that .ea (−∞, s] ξ = ξ and .||ξ ||H = 1, then f .

=

f R

a udeξ,ξ

= (−∞,s]

a udeξ,ξ ≤ s,

which shows that .αe (a) ≤ s whenever .e = ea (−∞, s]. Therefore, { ( ) } A ≤ inf αe (a) : e = ea (−∞, s] , s ∈ R, τ ea (s, ∞) ≤ t

.

≤ inf {s ∈ R : d (s; a) ≤ t} = λ (t; a) . The proof of the inequality .λ (t; a) ≤ A is exactly the same as that of the corresponding inequality .μ (t; a) ≤ A in the proof of Proposition 3.2.19 and is, therefore, omitted. u n Remark 3.6.2 By analogy with the expression for the singular value function given in Proposition 3.2.19 (i), it might be expected that the eigenvalue function .λ (t; a)

180

3 Singular Value Functions

would also be given by .

} { ( ) inf αe (a) : e ∈ P (M) , e (H ) ⊆ D (a) , τ e⊥ ≤ t

(3.40)

for all .t ∈ [0, τ (1)). This, however, is false in general, even in the commutative case, as will be illustrated by the following example. Consider the interval .[0, 1] with Lebesgue measure m. Let .M be .L∞ [0, 1], acting via multiplication on the Hilbert space .L2 [0, 1], equipped with the Lebesgue integral as trace .τ . As has been observed in Example 2.3.13 (b), the space .S (τ ) may be identified with .L0 [0, 1], acting via multiplication on .L2 [0, 1]. Define the function .a ∈ L0 [0, 1] by { a (t) =

.

1 t−1/2 t−1

if 0 ≤ t < 1/2, if 1/2 ≤ t < 1,

in which case .a (t) = λ (t; a), .0 ≤ t < 1. It will be shown that .λ (t; a) is not equal to the infimum in (3.40) ) .t = 1/2. Indeed, suppose that .e ∈ P (M) is such that ( ⊥ for ≤ 1/2, that is, .e = χE , where .E ⊆ [0, 1] is a measurable .e (H ) ⊆ D (a) and .τ e set such that .aχE is bounded and .m ([0, 1] \ E) ≤ 1/2. Since .limt↓1 a (t) = −∞, there exists .ε > 0 such that .(1 − ε, 1) ⊆ [0, 1] \ E, and so, the set .F = E ∩ [0, 1/2] has non-zero Lebesgue measure. This implies that {f αe (a) = sup

1

} a |ξ | dt : ξ ∈ L2 [0, 1] , ξ = χE ξ, ||ξ ||2 = 1

1

} a |ξ | dt : ξ ∈ L2 [0, 1] , ξ = χF ξ, ||ξ ||2 = 1 = 1,

.

{f

0

≥ sup 0

2

2

since .a (t) = 1 a.e. on F . This shows that .

} { ( ) inf αe (a) : e ∈ P (M) , e (H ) ⊆ D (a) , τ e⊥ ≤ t ≥ 1,

whereas .λ (1/2; a) = 0. In this connection, it should be noted that, in general, if .a ∈ Sh (τ ) is such that − ∈ M, then it is easy to verify that .λ (t; a) is given by (3.40). .a Proposition 3.6.1 has a number of interesting consequences, which will be discussed next. Proposition 3.6.3 (i) If .a, b ∈ Sh (τ ) and .a ≤ b, then .λ (t; a) ≤ λ (t; b) for all .t ∈ [0, τ (1)). (ii) If .f : R → [0, ∞) is an increasing (respectively, decreasing) Borel function, then the map .a |−→ τ (f (a)), .a ∈ Sh (τ ), is increasing (respectively, decreasing) on .Sh (τ ).

3.6 Properties of the Eigenvalue Function

181

Proof (i) Let .t ∈ [0, τ (1)) be fixed. Since .D (a) and .D (b) are both .τ -dense, it follows from Proposition 2.3.4 that .D (a)∩D (b) is also .τ -dense. ( ⊥ )Therefore, there exists a sequence .{pn }∞ ↓ 0 as .n → ∞ and in .P (M) such that .pn ↑ 1, .τ pn n=1 ( ⊥) ≤ .pn (H ) ⊆ D (a)∩D (b) for all n. Suppose that .e ∈ P (M) is such that .τ e t. It follows from .e ∨ pn − e ∼ pn − e ∧ pn that τ (1 − e ∧ pn ) = τ (1 − pn ) + τ (e ∨ pn − e) ( ) ( ) ( ) ≤ τ pn⊥ + τ e⊥ ≤ t + τ pn⊥ .

.

Since .a ≤ b implies that (and is actually equivalent to) . ≤ for all .ξ ∈ D (a) ∩ D (b) (see the discussion following Theorem 2.2.16), it follows that { } αe∧pn (a) = sup : ||ξ ||H = 1, (e ∧ pn ) ξ = ξ } { ≤ sup : ||ξ ||H = 1, (e ∧ pn ) ξ = ξ

.

= αe∧pn (b) ≤ αe (b) , and hence, by Proposition 3.6.1, ( ( ) ) λ t + τ pn⊥ ; a ≤ αe∧pn (a) ≤ αe (b) .

.

( ) This holds for all .e ∈ P (M) with .τ e⊥ ≤ t, and so, according to ( ( ⊥) ) ( ) Proposition 3.6.1, .λ t + τ pn ; a ≤ λ (t; b) for all .n ∈ N. Since .τ pn⊥ ↓ 0 as .n → ∞ and .λ (a) is right-continuous, it follows that .λ (t; a) ≤ λ (t; b). (ii) This follows immediately from (i) in combination with Proposition 3.5.5 (i). u n Proposition 3.6.4 If .a, b ∈ Sh (τ ), then λ (s + t; a + b) ≤ λ (s; a) + λ (t; b)

.

for all .s, t ≥ 0 satisfying .s + t < τ (1). Proof Evidently, it may be assumed that .s, t > 0. Given (.ε >) 0, it follows ) ( from Proposition 3.6.1 that there exist .p, q ∈ P (M) satisfying .τ p⊥ ≤ s and .τ q ⊥ ≤ t, such that .αp (a) ≤ λ (s; a) + ε and .αq (b) ≤ λ (t; b) + ε. Since .D (a) ( ⊥∩) D (b) is ∞ ↓ 0 and .τ -dense, there exists a sequence .{en } in .P (M) such that .en ↑ 1, .τ en n=1 .en (H ) ⊆ D (a) ∩ D (b) for all n. For each n, it follows that } { αp∧q∧en (a + b) = sup : ξ = (p ∧ q ∧ en ) ξ, ||ξ ||H = 1

.

≤ αp∧q∧en (a) + αp∧q∧en (b) ≤ αp (a) + αq (b) .

182

3 Singular Value Functions

Since ) ( ) ( ) ( ) ( ) ( τ (p ∧ q ∧ en )⊥ ≤ τ p⊥ + τ q ⊥ + τ en⊥ ≤ s + t + τ en⊥ ,

.

it follows from Proposition 3.6.1 that ) ( ( ) λ s + t + τ en⊥ ; a + b ≤ αp (a) + αq (b) ≤ λ (s; a) + λ (t; b) + 2ε

.

( ) for all n. Since .τ en⊥ ↓ 0 and .λ (a + b) is right-continuous, this implies that λ (s + t; a + b) ≤ λ (s; a) + λ (t; b) + 2ε.

.

This holds for all .ε > 0, and so, .λ (s + t; a + b) ≤ λ (s; a) + λ (t; b).

u n

Proposition 3.6.5 Suppose that .a, b ∈ Sh (τ ): (i) If .a − b ∈ M, then .|λ (t; a) − λ (t; b)| ≤ ||a − b||B(H ) for all .t ∈ [0, τ (1)). (ii) If .a, b ∈ L1 (τ ), then f

τ (1)

.

0

|λ (t; a) − λ (t; b)| dt ≤ ||a − b||1 .

Proof (i) Since .a − b ≤ ||a − b||B(H ) 1, it is clear that .a ≤ b + ||a − b||B(H ) 1, and so, it follows from Proposition 3.6.3 (i) in combination with Proposition 3.5.8 (v) that λ (t; a) ≤ λ (t; b) + ||a − b||B(H ) ,

.

t ∈ [0, τ (1)) .

Statement (i) is now clear by interchanging the roles of a and b. (ii) Since .a, b ≤ (a − b)+ + b, it follows from Proposition 3.6.3 (i) that ( ) λ (a) ∨ λ (b) ≤ λ (a − b)+ + b .

.

The same inequality holds with a and b interchanged and so .

|λ (a) − λ (b)| = 2 [λ (a) ∨ λ (b)] − λ (a) − λ (b) ( ) ( ) ≤ λ (a − b)+ + b + λ (b − a)+ + a − λ (a) − λ (b) .

3.6 Properties of the Eigenvalue Function

183

Integrating this last inequality over .[0, τ (1)), and using (3.36) in Corollary 3.5.6, it follows that f

τ (1)

.

( ) ( ) |λ (a) − λ (b)| dt ≤ τ (a − b)+ + b + τ (b − a)+ + a

0

−τ (a) − τ (b) ( ) ( ) = τ (a − b)+ + τ (b − a)+ = τ (|a − b|) = ||a − b||1 . The proof is complete. u n The next theorem complements, in the case that the trace is finite, the result of Theorem 3.4.29. Theorem 3.6.6 Suppose that .a, b ∈ Sh (τ ). If .λ (|a|) λ (|b|) ∈ L1 (0, τ (1)), then ab ∈ L1 (τ ) and

.

f .

τ (1)

λ (a) λˇ (b) dt ≤ τ (ab) ≤

0

f

τ (1)

λ (a) λ (b) dt.

(3.41)

0

Proof First, note that by Proposition 3.5.8 (iv), .μ (λ (a)) = λ (|λ (a)|) = λ (|a|) and μ (λ (b)) = λ (|b|) on .[0, τ (1)). Therefore, it follows from Theorem 3.4.29, applied to the commutative von Neumann algebra .L∞ (0, τ (1)) with Lebesgue integral as the trace, that

.

f .

τ (1)

f |λ (a) λ (b)| dt ≤

0

τ (1)

f

τ (1)

μ (λ (a)) μ (λ (b)) dt =

0

λ (|a|) λ (|b|) dt < ∞.

0

( ) It also follows from Proposition 3.5.8 (iv) that .μ λˇ (b) = μ (λ (b)), and so, by the same argument, f

| | | |λ (a) λˇ (b)| dt < ∞,

τ (1) |

.

0

and this now shows that both integrals in (3.41) are well defined and finite. It is an immediate consequence of Theorem 3.4.29 that f τ (|ab|) ≤

.

τ (1)

λ (|a|) λ (|b|) dt < ∞,

(3.42)

0

and so, .ab ∈ L1 (τ ). For the proof of (3.41), it will be assumed first that .a, b ∈ M+ . Since .τ (ab) = τ (ba) = τ (ab) (keep in mind that .τ (1) < ∞) implies that .τ (ab) ∈ R, it follows

184

3 Singular Value Functions

from Corollary 3.4.6 and (3.42) that f

τ (1)

τ (ab) ≤ |τ (ab)| ≤ τ (|ab|) ≤

(3.43)

λ (a) λ (b) dt.

.

0

From (3.43), applied with b replaced by .||b||B(H ) 1 − b ≥ 0, in combination with Proposition 3.5.8 (i), (v) and (3.36) in Corollary 3.5.6, it follows that f . ||b||B(H ) τ (a) − τ (ab) ≤

f

τ (1) 0 τ (1)

= 0

( ) λ (a) λ ||b||B(H ) 1 − b dt ( ) λ (a) ||b||B(H ) − λˇ (b) dt f

= ||b||B(H ) τ (a) −

τ (1)

λ (a) λˇ (b) dt,

0

and so, f .

τ (1)

λ (a) λˇ (b) dt ≤ τ (ab) .

0

Now suppose that .a, b ∈ S (τ )+ satisfy .λ (a) λ (b) ∈ L1 (0, τ (1)). For each .n ∈ N, define the function .fn on .[0, ∞) by .fn (s) = min (s, n) and set .an = fn (a) and .bn = fn (b). Since .λ (an ) = fn ◦ λ (a), it is clear that .0 ≤ λ (an ) ↑n λ (a) and, similarly, .0 ≤ λ (bn ) ↑n λ (b) on .[0, τ (1)). For all n, it follows from the first part of the proof that f

τ (1)

.

λ (an ) λˇ (bn ) dt ≤ τ (an bn ) ≤

0

f

τ (1)

f λ (an ) λ (bn ) dt ≤

0

τ (1)

λ (a) λ (b) dt. 0

Furthermore, it is clear that f

τ (1)

.

λ (an ) λˇ (bn ) dt ↑

0

f

τ (1)

λ (a) λˇ (b) dt,

n → ∞.

0 Tm

Tm

Tm

It is easily verified (cf. Remark 2.5.5) that .an → a and .bn → b, and so, .an bn → ab. Since μ (t; an bn ) ≤ μ (t/2; an ) μ (t/2; bn ) ≤ μ (t/2; a) μ (t/2; b) ,

.

t ∈ [0, τ (1)) ,

and the function .t − | → μ (t/2; a) μ (t/2; b) is integrable over .[0, τ (1)), it follows from the dominated convergence theorem (Theorem 3.4.21) that τ (an bn ) → τ (ab) ,

.

n → ∞.

3.7 Some Auxiliary Results

185

This implies that the left hand inequality of (3.41) holds for all .a, b ∈ S (τ )+ satisfying .λ (a) λ (b) ∈ L1 (0, τ (1)). f τ (1) Now suppose that .a, b ∈ Sh (τ ) such that . 0 λ (|a|) λ (|b|) dt < ∞. As observed already at the beginning of the present proof, this implies that .ab ∈ L1 (τ ). Write ab = a + b+ + a − b− − a + b− − a − b+ .

.

Using the identities from Proposition 3.5.8, it follows that f

τ (1)

λ (a) λˇ (b)+ dt = +

.

0

f

τ (1)

( ) ( ) ( ) λ a + λˇ b+ dt ≤ τ a + b+

τ (1)

( ) ( ) λ a + λ b+ dt =

0

f ≤

0

f

τ (1)

λ (a)+ λ (b)+ dt.

0

Similarly, it follows that f

τ (1)

) ( λ (a)− λˇ (b)− dt ≤ τ a − b− ≤

τ (1)

( ) λ (a)+ λ (b)− dt ≤ τ a + b− ≤

τ (1)

) ( λ (a)− λ (b)+ dt ≤ τ a − b+ ≤

.

0

f

0

f

0

f f f

τ (1)

λ (a)− λ (b)− dt,

0 τ (1)

λ (a)+ λˇ (b)− dt,

0 τ (1)

λ (a)− λˇ (b)+ dt.

0

Adding the first two above estimates and subtracting the last two finally yield (3.41). u n References: [21, 42, 52, 85].

3.7 Some Auxiliary Results In the present section, some general facts and techniques are discussed, which will be used in the subsequent sections. Suppose that .M is a von Neumann algebra on a Hilbert space H , equipped with a semi-finite normal faithful trace .τ . Recall from Sect. 1.11, if .e ∈ P (M), then eMe = {exe : x ∈ M}

.

is a .∗-subalgebra of .M with unit element e. Note that if .a ∈ M and .b ∈ eMe satisfy 0 ≤ a ≤ b, then .a ∈ eMe. Indeed, .0 ≤ a ≤ b implies that .n (b) ≤ n (a), and so, .s (a) ≤ s (b) ≤ e, from which it follows that .a ∈ eMe. .

186

3 Singular Value Functions

Let .K = e (H ). For any .x ∈ M, the operator .xe ∈ B (K) is defined by xe = (exe) |K . The set .Me = {xe : x ∈ M} is a von Neumann algebra on K (the reduction of .M by e), and the map .x |−→ xe , .x ∈ M, is a linear .∗-preserving surjection from .M onto .Me . Furthermore, the map .φe : x |−→ xe , .x ∈ eMe, is a .∗-isomorphism from .eMe onto .Me . Note that if .a ∈ M is (positive) self-adjoint, then .ae is also (positive) self-adjoint. It will be convenient to identify .Me with .eMe via the map .φe . With this identification, it is clear that .

P (Me ) = {p ∈ P (M) : p ≤ e} .

.

Defining .τe : M+ e → [0, ∞] by setting τe (ae ) = τ (eae) ,

.

a ∈ M+ ,

that is, .τe = τ ◦ φe−1 (or, .τe = τ |eMe when .Me is identified with .eMe via .φe ), it is easy to verify that .τe is a semi-finite normal faithful trace. Note that the trace .τe is finite if and only if .τ (e) < ∞. Next, a description of the .τe -measurable operators will be given. If .x ∈ S (τ ), then the operator .xe in K is defined by .xe = (exe) |K , that is, .D (xe ) = D (exe)∩K ( ) and .xe ξ = (ex) ξ , .ξ ∈ D (xe ). Evidently, .xe is closed and, since .(Me )' = M' e , it is also clear that .xe is affiliated with .Me . Since .x ∈ S (τ ), there a sequence ( ⊥exists ) ∞ .{pn } in .P (M) such that .pn ↑ 1, .pn (H ) ⊆ D (x) and .τ pn for all n. < ∞ n=1 Defining .qn ∈ P (M) by .qn = pn ∧ e, it follows from ( ) τ (e − qn ) = τ (pn ∨ e − pn ) ≤ τ pn⊥

.

that .τ (e − qn ) ↓ 0 as .n → ∞ (and so, in particular, .qn ↑ e). Since qn (K) ⊆ D (x) ∩ K ⊆ D (ex) ∩ K = D (xe ) ,

.

{ }∞ the sequence . (qn )e n=1 is a determining sequence in .P (Me ) for the operator .xe . Consequently, .xe ∈ S (τe ). It should be observed that the above argument shows that the subspace .K ∩ D (x) is .τe -dense in K, and hence, .K ∩ D (x) is a core of the operator .xe . Note that .xf = (xe )f whenever .f ≤ e in .P (M). Defining eS (τ ) e = {exe : x ∈ S (τ )} ,

.

it is clear that .eS (τ ) e is a .∗-subalgebra of .S (τ ) with unit element e. The proof of the next result follows via standard arguments, and therefore, the details are left to the reader. They can be found in the first reference at the end of this section, however. Lemma 3.7.1 The map .φe : x |−→ xe , .x ∈ eS (τ ) e, is a unital .∗-isomorphism from .eS (τ ) e onto .S (τe ), which is a homeomorphism for the measure topology.

3.7 Some Auxiliary Results

187

It should also be observed that .a 1/2 ∈ eS (τ ) e whenever .0 ≤ a ∈ eS (τ ) e. Indeed, ( )2 using that .ea = ae implies that .ea 1/2 = a 1/2 e, it follows that . ea 1/2 e = eae = a. Since .ea 1/2 e ≥ 0, it can be concluded that .ea 1/2 e = a 1/2 , that is, .a 1/2 ∈ eS (τ ) e. An immediate consequence is that .|y| ∈ eS (τ ) e whenever .y ∈ eS (τ ) e. In other words, .|exe| = e |exe| e for all .x ∈ S (τ ). If .a ∈ Sh (τ ), then it is easily verified that the spectral measure of .ae is given by ) ( eae (B) = eeae (B \ {0}) + n (eae) − e⊥ δ0 (B) ,

.

B ∈ B (R) .

If .x ∈ S (τ ), then .|xe | = |exe|e (as the map .φe is a .∗-isomorphism from .eMe onto Me ) and hence,

.

e|xe | (s, ∞) = ee|exe|e (s, ∞) = e|exe| (s, ∞) , 0 ≤ s ∈ R.

.

Consequently, μ (xe ) = μ (exe) ,

.

x ∈ S (τ ) ,

(3.44)

where .μ (xe ) is computed with respect to the reduced von Neumann algebra .Me and the trace .τe . From these observations, it follows, for instance, that .xe ∈ L1 (τe ) and .||xe ||1 ≤ ||x||1 whenever .x ∈ L1 (τ ). It also follows that the extended traces .τ and .τe on .S (τ )+ and .S (τe )+ , respectively, satisfy τe (ae ) = τ (eae) ,

.

a ∈ S (τ )+ .

The following observations will also be used in the subsequent sections. Recall that a non-zero projection .p ∈ P (M) is called minimal if .Mp = Cp. In other words, a projection .0 /= p ∈ P (M) is minimal if and only if .q ∈ P (M) and .q ≤ p imply that .q = p or .q = 0 (that is, p is a non-zero minimal element in the lattice .P (M)). The von Neumann algebra .M is called non-atomic if .M contains no minimal projections. The first observation is that if .M is non-atomic, then .P (M) contains non-zero elements of arbitrary small trace. Lemma 3.7.2 If p is a non-zero element of .P (M) which does not majorize any minimal projection, then for every .ε > 0 there exists .q ∈ P (M) such that .q ≤ p and .0 < τ (q) ≤ ε. In particular, this holds for any non-zero projection in a nonatomic von Neumann algebra. Proof Since .τ is semi-finite, there exists .e ∈ P (M) such that .e ≤ p and .0 < τ (e) < ∞. By hypothesis, e is not minimal, and so, there exists .e1 ∈ P (M) such that .0 < e1 < e. Since .τ (e1 ) + τ (e − e1 ) = τ (e), it follows that either −1 τ (e) or .τ (e − e ) ≤ 2−1 τ (e). Without loss of generality, it may be .τ (e1 ) ≤ 2 1 assumed that .τ (e1 ) ≤ 2−1 τ (e). By repeating this argument, it follows that for each −n . .n ∈ N there exists .en ∈ P (M) such that .en ≤ e ≤ p such that .0 < τ (en ) ≤ 2 u n This clearly suffices for the proof.

188

3 Singular Value Functions

Lemma 3.7.3 Suppose that .M is non-atomic and that .p ≤ q in .P (M). If .θ ∈ R is such that .τ (p) ≤ θ ≤ τ (q), then there exists .e ∈ P (M) such that .p ≤ e ≤ q and .τ (e) = θ . Proof Setting P = {f ∈ P (M) : p ≤ f ≤ q, τ (f ) ≤ θ } ,

.

it is clear that .P = / ∅ and .P is partially ordered by the ordering inherited from P (M). From the normality of the trace, it follows that every chain in .P has an upper bound. By an appeal to Zorn’s lemma, there exists a maximal element .e ∈ P. If .τ (e) < θ , then it follows from Lemma 3.7.2, applied to the .q − e, that ) ( projection there exists .e' ∈ P (M) such that .e' ≤ q − e and .0 < τ e' < θ − τ (e). This implies that .e + e' ∈ P and .e < e + e' , which contradicts the maximality of e. Therefore, .τ (e) = θ and the proof is complete. u n

.

Remark 3.7.4 Without any assumptions on the von Neumann algebra .M, it is clear that the result of the above lemma is not valid. However, if .M is atomic and all minimal projections in .M have equal trace, then an appropriate modification of Lemma 3.7.3 does hold. Indeed, without loss of generality, it may be assumed that the trace of each minimal projection is equal to one. Consequently, if .e ∈ P (M) and .τ (e) < ∞, then .τ (e) ∈ N. Moreover, each projection in .P (M) is the supremum of a mutually orthogonal system of minimal projections. Using these observations, it is not difficult to show that if .p ≤ q in .P (M) and if .θ ∈ N satisfies .τ (p) ≤ θ ≤ τ (q), then there exists .e ∈ P (M) such that .p ≤ e ≤ q and .τ (e) = θ . In some of the proofs, it will be convenient to be able to embed an arbitrary semi-finite von Neumann algebra into a non-atomic von Neumann algebra. One of the ingredients in this construction is the following general observation. Lemma 3.7.5 Suppose that .M is a von Neumann algebra on a Hilbert space H . If (S, Σ, ν) is a non-atomic Maharam measure space, then the von Neumann algebra .L∞ (ν) ⊗M does not contain any minimal projections (that is, .L∞ (ν) ⊗M is nonatomic). .

Proof First note that, since .(S, Σ, ν) is a Maharam measure space, the set P (L∞ (ν)) of all idempotents in .L∞ (ν) is a complete Boolean algebra (in fact, .P (L∞ (ν)) is the measure algebra of .ν). Furthermore, if .a ∈ P (L∞ (ν)), then .a ⊗1 is a central projection in .L∞ (ν) ⊗M. Indeed, .a ⊗ 1 commutes(with all elements )' of the algebraic tensor product .L∞ (ν) ⊗ M, and hence, .a ⊗ 1 ∈ L∞ (ν) ⊗M . ) ( Given .p /= 0 in .P L∞ (ν) ⊗M , define the subset .A of .P (L∞ (ν)) by .

A = {a ∈ P (L∞ (ν)) : p ≤ a ⊗ 1} .

.

Clearly, .A /= ∅ (as .1 ∈ A) and .A is downward directed. Define .e = inf A (in P (L∞ (ν)), which is also the infimum in the Dedekind complete vector lattice .L∞ (ν)). Since the map .g |−→ g ⊗ 1, .g ∈ L∞ (ν), is a normal .∗-isomorphism .

3.7 Some Auxiliary Results

189

from .L∞ (ν) into .L∞ (ν) ⊗M, mapping .L∞ (ν) onto the von Neumann subalgebra L∞ (ν) ⊗ C1, it is clear that

.

a ⊗ 1 ↓a∈A e ⊗ 1

.

holds in .L∞ (ν) ⊗M. Hence, p ≤e⊗1

.

and so, in particular, .e /= 0. Since the measure .ν is non-atomic, the Boolean algebra .P (L∞ (ν)) is non-atomic, and so, there exists .f ∈ P (L∞ (ν)) such that .0 < f < e. Define the projection q in .L∞ (ν) ⊗M by .q = (f ⊗ 1) p = p (f ⊗ 1). Evidently, .q ≤ p. If .q = 0, then .p ≤ (f ⊗ 1)⊥ = f ⊥ ⊗ 1, and by the definition of e, this implies that .e ≤ f ⊥ . Since .f ≤ e, it follows that .f = 0, which is a contradiction. Hence, .q /= 0. If .q = p, then .p ≤ f ⊗ 1, and so, .e ≤ f . Since by assumption .f < e, this is a contradiction. Hence, .q < p. Consequently, for any projection .p /= 0 in .L∞ (ν) ⊗M, there exists a projection q such that .0 < q < p, which shows that p is not minimal. The proof is complete. u n A combination of Lemma 3.7.5 and the observations made in Example 2.9.4 and Example 3.3.11 yields the result of the next proposition. Proposition 3.7.6 Let .M be a von Neumann algebra on a Hilbert space H , equipped with a semi-finite faithful normal trace .τ . Consider the von Neumann algebra .L∞ [0, 1] on .L2 [0, 1] equipped with Lebesgue integration as trace. Let the von Neumann algebra .L∞ [0, 1] ⊗M be equipped with the corresponding tensor product trace .τˆ . The following statements hold: (i) The normal trace preserving .∗-isomorphism .π : x − | → 1 ⊗ x from .M into .L∞ [0, 1] ⊗M extends uniquely to a .∗-isomorphism .π ˆ : x |−→ 1 ⊗ x from ( ) .S (τ ) into .S τˆ , which preserves the singular value function, that is, μ (1 ⊗ x) = μ (x) ,

.

x ∈ S (τ ) .

(ii) If .τ (1) < ∞, then .πˆ preserves the eigenvalue function, that is, λ (1 ⊗ a) = λ (a) ,

.

a ∈ Sh (τ ) .

( ) (iii) The range of .πˆ is .S C1 ⊗ M, τˆ . (iv) The von Neumann algebra .L∞ [0, 1] ⊗M is non-atomic. Proof Only assertion (ii) needs proof. If .a ∈ Sh (τ ), then it follows from Proposition 2.9.2 (iv) that the spectral measure of .1 ⊗ a is given by .e1⊗a (B) =

190

3 Singular Value Functions

1 ⊗ ea (B), .B ∈ B (R). Since .π is trace preserving, this implies that ( ) ( ) d (s; 1 ⊗ a) = τˆ e1⊗a (s, ∞) = τˆ 1 ⊗ ea (s, ∞) ) ( = τ ea (s, ∞) = d (s; a)

.

for all .s ∈ R, from which it is clear that .λ (1 ⊗ a) = λ (a).

u n

The following rather technical results are included for later purposes. If .x ∈ S (τ ), then the number .μ∞ is defined by μ∞ = lim μ (t; x) .

.

t→∞

It follows from Lemma 3.2.3 (ii) that .μ∞ is also given by μ∞ = inf {s ≥ 0 : d (s; |x|) < ∞} .

(3.45)

.

If there exists .t0 ≥ 0 such that .μ (t0 ; x) = μ∞ , then it follows from Proposition 3.1.5 that .

{t ≥ 0 : μ (t; x) = μ∞ } = [d (μ∞ ; |x|) , ∞) .

(3.46)

Lemma 3.7.7 Suppose that the von Neumann algebra .M is non-atomic and .a ∈ S (τ )+ . Let .λ = μ (t; x) for some .t ∈ (0, ∞) satisfying .t ≤ τ (1): (i) If .λ > μ∞ or .λ = 0, then there exists .e ∈ P (M) such that ea (λ, ∞) ≤ e ≤ ea [λ, ∞)

.

and

τ (e) = t.

(ii) If .λ = μ∞ > 0 and .ε ∈ R is such that .0 < ε < μ∞ , then there exists .e ∈ P (M) such that ea (μ∞ , ∞) ≤ e ≤ ea (μ∞ − ε, ∞)

.

and

τ (e) = t.

Proof (i). Suppose first that .λ = 0. In this case, it follows from Remark 3.2.6 (ii) that .τ (s (a)) ≤ t, and hence, there exists .e ∈ P (M) such that .ea (0, ∞) = s (a) ≤ e ≤ 1 = ea [0, ∞) and .τ (e) = t. If .λ > μ∞ , then it follows from Proposition 3.1.3 (iii) that d (λ; a) ≤ t ≤ d (λ−; a) .

.

{ }∞ It follows from (3.45) that there exists a sequence . sj j =1 in .(0, ∞) such that .sj ↑ λ ( ( )) ( ) ) ( and .τ ea sj , ∞ = d sj ; a < ∞ for all j . Since .ea sj , ∞ ↓ ea [λ, ∞), this

3.7 Some Auxiliary Results

191

implies that .d (λ−; a) = τ (ea [λ, ∞)). Consequently, ea (λ, ∞) ≤ ea [λ, ∞) ,

.

( ) ( ) τ ea (λ, ∞) ≤ t ≤ τ ea [λ, ∞) ,

and so, by Lemma 3.7.3, there exists .e ∈ P (M) such that .ea (λ, ∞) ≤ e ≤ ea [λ, ∞) and .τ (e) = t. (ii). If .λ = μ∞ > 0, then d (μ∞ ; a) ≤ t ≤ d (μ∞ −; a) = ∞,

.

and so, { .

ea (μ∞ , ∞) ≤ ea (μ∞ − ε, ∞) , τ (ea (μ∞ , ∞)) ≤ t ≤ τ (ea (μ∞ − ε, ∞)) = ∞.

By Lemma 3.7.3, there exists .e ∈ P (M) such that ea (μ∞ , ∞) ≤ e ≤ ea (μ∞ − ε, ∞)

.

and .τ (e) = t. The proof is complete.

u n

Lemma 3.7.8 Suppose that the von Neumann algebra .M is non-atomic and that + a ∈ S (τ { ) } . Denote .μ∞ = limt→∞ μ (t). If .μ∞ > 0, let .ε be such that .0 < ε < μ∞ . If . tj j is a finite or infinite sequence in .(0, ∞) such that .0 < t1 < t2 < · · · ≤ ( ) { } τ (1) and .λj = μ tj ; a , then there exists a sequence . ej j of projections in .P (M) such that

.

e1 < e2 < e3 < · · ·

.

and

( ) τ ej = tj ∀j,

satisfying: ( [ ) ) (i) .ea λj , ∞ ≤ ej ≤ ea λj , ∞ whenever .λj > μ∞ or .λj = 0. (ii) .ea (μ∞ , ∞) ≤ ej ≤ ea (μ∞ − ε, ∞) whenever .λj = μ∞ > 0. { } Proof The sequence . ej j will be constructed inductively. The existence of a projection .e1 ∈ P (M), satisfying the stated conditions, follows immediately from Lemma 3.7.7. Suppose that .e1 < · · · < ek have been constructed with the stated properties. Assuming that .λk > λk+1 , it follows from Lemma 3.7.7 that there exists .ek+1 ∈ P (M) satisfying either (i) or (ii), and it is easily verified that .ek < ek+1 . Hence, .ek+1 has the desired properties. It remains to consider the case that .λk = λk+1 . If .λk = λk+1 = 0, then a .e (0, ∞) ≤ ek and .τ (ek ) < tk+1 . Since .tk+1 ≤ τ (1), there exists .ek+1 ∈ P (M) such that .ek ≤ ek+1 and .τ (ek+1 ) = tk+1 . Suppose next that .λk = λk+1 > μ∞ . Since .ek ≤ ea [λk+1 , ∞) and .τ (ek ) < tk+1 ≤ τ (ea [λk+1 , ∞)), there exists .ek+1 ∈ P (M) such that .ek ≤ ek+1 ≤

192

3 Singular Value Functions

ea [λk+1 , ∞) and .τ (ek+1 ) = tk+1 . Observing that ea (λk+1 , ∞) = ea (λk , ∞) ≤ ek ≤ ek+1 ≤ ea [λk+1 , ∞) ,

.

it is clear that the projection .ek+1 satisfies (i). Finally, assume that .λk = λk+1 = μ∞ > 0. Since .ek ≤ ea (μ∞ − ε, ∞) and a .τ (e (μ∞ − ε, ∞)) = ∞, there exists .ek+1 ∈ P (M) such that .ek ≤ ek+1 ≤ a e (μ∞ − ε, ∞) and .τ (ek+1 ) = tk+1 . Evidently, .ek+1 satisfies (ii) and so, the proof is complete. u n Remark 3.7.9 Suppose that the von Neumann algebra .M is atomic and all minimal projections in .M have trace equal to one. Using Remark 3.7.4, it is easy to verify that the result of Lemma 3.7.7 still holds under { }the additional assumption that .t ∈ N. Similarly, if it is assumed, in addition, that . tj j is a sequence in .N, then the result of Lemma 3.7.8 is also valid in this case. The details are left to the reader. Lemma 3.7.10 Suppose that .a ∈ S (τ )+ and let .μ∞ = limt→∞ μ (t; a): (i) If .λ ∈ [0, ∞) and .e ∈ P (M) are such that .τ (e) < ∞ and ea (λ, ∞) ≤ e ≤ ea [λ, ∞) ,

.

then .ae = ea = eae and .μ (ea) = μ (a) χ[0,τ (e)) . In particular, f

τ (e)

τ (eae) = τ (ea) =

μ (t; a) dt.

.

0

(ii) If .0 < ε < μ∞ and .e ∈ P (M) is such that .τ (e) < ∞ and ea (μ∞ , ∞) ≤ e ≤ ea (μ∞ − ε, ∞) ,

.

then f

τ (e)

τ (eae) ≥

.

μ (t; a) dt − ετ (e) .

0

Proof (i) First observe that the assumption on e implies that e commutes with all spectral projections of the form .ea (s, ∞), .s ∈ R, and so .eea (B) = ea (B) e for all Borel subsets B of .R. Hence, .ea = ae. It is readily verified that the spectral measure .eea of ea is given by eea (B) = eea (B \ {0}) + (1 − es (a)) δ0 (B) ,

.

B ∈ B (R) ,

3.7 Some Auxiliary Results

193

from which it follows that { e

.

(s, ∞) = ee (s, ∞) =

ea

a

e if 0 ≤ s < λ a e (s, ∞) if s ≥ λ

for all .s ≥ 0. Consequently, { d (s; ea) =

.

τ (e) if 0 ≤ s < λ, d (s; a) if s ≥ λ,

s ≥ 0,

that is, .d (s; ea) = min (d (s; ea) , τ (e)), from which it is immediately clear that .μ (ea) = μ (a) χ[0,τ (e)) . This implies, in particular, that f τ (ea) =

∞

.

f

τ (e)

μ (t; ae) dt =

μ (t; a) dt. 0

0

(ii) For convenience, let .p = ea (μ∞ , ∞) and .q = ea (μ∞ − ε, ∞). Since .μ (pa) = μ (a) χ[0,τ (p)) (cf. Proposition 3.2.10 (iii)), it is clear that f

∞

τ (pap) = τ (pa) =

.

f μ (t; pa) dt =

0

τ (p)

μ (t; a) dt. 0

Since .pa (e − p) = ap (e − p) = 0 and, similarly, .(e − p) ap = 0, it follows that eae = pap + (e − p) a (e − p) .

.

From the definition of q, it is clear that .qaq = aq ≥ (μ∞ − ε) q and .e − p = (e − p)q. Hence, .

(e − p) a (e − p) = (e − p) qaq (e − p) ≥ (μ∞ − ε) (e − p) q (e − p) = (μ∞ − ε) (e − p) .

This shows that τ ((e − p) a (e − p)) ≥ (μ∞ − ε) (τ (e) − τ (p)) .

.

Since .τ (p) = d (μ∞ ; a), it follows from (3.46) that .μ (t; a) = μ∞ for all t ∈ [τ (p) , τ (e)], and hence,

.

τ (eae) = τ (pap) + τ ((e − p)a(e − p)) f τ (p) ≥ μ (t; a) dt + (μ∞ − ε) (τ (e) − τ (p))

.

0

194

3 Singular Value Functions

f =

τ (e)

μ (t; a) dt − ε (τ (e) − τ (p))

0

f ≥

τ (e)

μ (t; a) dt − ετ (e) .

0

The proof is complete. u n References: [27, 42].

3.8 Submajorization for the Eigenvalue Function In this section, a number of inequalities for the eigenvalue function will be discussed, which are centered around the important concept of submajorization in the sense of Hardy, Littlewood and Pólya. In this section, it is assumed that the trace is finite. The next lemma gives the relationship between the eigenvalue function .λ (a) of an element .a ∈ S (τ )h and the eigenvalue function .λ (ae ) of .ae , computed in the reduced von Neumann algebra .Me with respect to the trace .τe . Recall from Proposition 3.6.1 that } { ( ) λ (t; a) = inf αp (a) : p ∈ P (M) , τ p⊥ ≤ t ,

.

t ∈ [0, τ (1)) ,

(3.47)

where } { αp (a) = sup : ξ ∈ H, ||ξ ||H = 1, pξ = ξ ∈ D (a) .

.

If .e ∈ P (M) and .K = e (H ), then .K ∩D (a) is .τe -dense in K, and hence, a core for the operator .ae , as has been noted at the beginning Sect. 3.7. Using this observation, it is not difficult to show that } { αe (a) = αe (ae ) = sup : ξ ∈ D (ae ) , ||ξ ||H = 1 .

.

(3.48)

Moreover, αp (a) = αp (ae ) ,

.

p ∈ P (M) , p ≤ e.

(3.49)

Indeed, if .p ∈ P (M) and .p ≤ e, then it follows from (3.48), applied to the projection p, and the elements a and .ae in the von Neumann algebras .M and .Me ,

3.8 Submajorization for the Eigenvalue Function

195

respectively, that ( ) ( ) αp (a) = αp ap = αp (ae )p = αp (ae ) .

.

Lemma 3.8.1 If .a ∈ S (τ )h and .e ∈ P (M), then ( ( ) ) λ t + τ e⊥ ; a ≤ λ (t; ae ) ≤ λ (t; a) ,

.

t ∈ [0, τ (e)) .

( ) Proof Let .t ∈ [0, τ (e)) be fixed. If .p ∈ P (M) and .τ p⊥ ≤ t, then ( ) ( ) τe (p ∧ e)⊥ = τ (e − p ∧ e) = τ (p ∨ e − p) ≤ τ p⊥ ≤ t,

.

and so, it follows from (3.47), applied in the von Neumann algebra .Me , and from (3.49) that λ (t; ae ) ≤ αp∧e (ae ) = αp∧e (a) { } = sup : ξ ∈ H, ||ξ ||H = 1, (p ∧ e) ξ = ξ ∈ D (a) { } ≤ sup : ξ ∈ H, ||ξ ||H = 1, pξ = ξ ∈ D (a) = αp (a) .

.

Applying (3.47) once again, it follows that .λ (t; ae ) ≤ λ (t; a). ( ) For the proof of the left side inequality, suppose that .p ∈ P (Me ) and .τe p⊥ ≤ t, that is, .p ∈ P (M), .p ≤ e, and .τ (e − p) ≤ t. This implies that ( ) ( ) τ p⊥ = τ (1 − e) + τ (e − p) ≤ t + τ e⊥ ,

.

( ( ) ) and so, .λ t + τ e⊥ ; a ≤ αp (a). On the other hand, it follows from (3.49) that ( ( ⊥) ) ; a ≤ αp (ae ) for all .p ∈ P (Me ) .αp (a) = αp (ae ), which shows that .λ t + τ e ( ⊥) ≤ t. Consequently, it follows from (3.47), applied in .Me , that satisfying .τe p ( ( ⊥) ) ; a ≤ λ (t; ae ). The proof is complete. .λ t + τ e u n Lemma 3.8.2 If .a ∈ S (τ )h , .s ∈ R, and .e ∈ P (M) are such that ea (s, ∞) ≤ e ≤ ea [s, ∞) ,

.

then: (i) .λ (t; ( ae ) )= λ (t; a) for all .t ∈ [0, τ (e)).[ ( )) (ii) .λ t; ae⊥ = λ (t + τ (e) ; a) for all .t ∈ 0, τ e⊥ . Proof (i) From the assumption on e, it follows that .eea (r, ∞) = ea (r, ∞) e for all .r ∈ R, which implies that .eea (B) = ea (B) e for all Borel sets .B ⊆ R. Consequently,

196

3 Singular Value Functions

ea = ae, and the spectral measure .eae of .ae is given by

.

eae (B) = eea (B) = ea (B) e,

.

B ∈ B (R) .

In particular, { eae (r, ∞) =

.

e if r < s, ea (r, ∞) if r ≥ s,

and so, { d (r; ae ) =

.

τ (e) if r < s, d (a; r) if r ≥ s.

(3.50)

If .0 ≤ t < τ (e), then .d (r; ae ) ≤ t if and only if .d (r; a) ≤ t. Indeed, if d (r; ae ) ≤ t, then it follows from (3.50) that .r ≥ s and .d (r; a) ≤ t. On the other hand, since .τ (ea [s, ∞)) ≥ τ (e), it follows from .d (r; a) ≤ t < τ (e) that .r ≥ s, and hence, (3.50) implies that .d (r; ae ) ≤ t. Consequently,

.

λ (t; ae ) = inf {r ∈ R : d (r; ae ) ≤ t}

.

= inf {r ∈ R : d (r; a) ≤ t} = λ (t; a) for all .t ∈ [0, τ (e)). (ii) As observed in the proof of (i), .eea (B) = ea (B) e for all .B ∈ B (R) and .ea = ae. This implies that ea1−e (B) = (1 − e) ea (B) = ea (B) (1 − e) ,

.

B ∈ B (R) ,

and hence, { e

.

a1−e

(r, ∞) = (1 − e) e (r, ∞) = a

ea (r, ∞) − e if r < s, 0 if r ≥ s.

It follows that { d (r; a1−e ) =

.

d (r; a) − τ (e) if r < s, 0 if r ≥ s.

(3.51)

Observe that (3.51) may also be written as .d (r; a1−e ) = (d (r; a) − τ (e))+ . Indeed, if .r < s, then .e ≤ ea [s, ∞) ≤ ea (r, ∞), and so, .d (r; a) − τ (e) ≥

3.8 Submajorization for the Eigenvalue Function

197

0. If .r ≥ s, then .ea (r, ∞)( ≤ )ea (s, ∞) ≤ e, and so, .d (r; a) − τ (e) ≤ 0. Consequently, if .0 ≤ t < τ e⊥ , then λ (t; a1−e ) = inf {r ∈ R : d (r; a1−e ) ≤ t} } { = inf r ∈ R : (d (r; a) − τ (e))+ ≤ t

.

= inf {r ∈ R : d (r; a) − τ (e) ≤ t} = λ (t + τ (e) ; a) . The proof is complete. u n For later reference, the following immediate consequence of Lemmas 3.8.1 and 3.8.2 is stated explicitly. Corollary 3.8.3 Suppose that .a ∈ S (τ )h , .s ∈ R, and .e ∈ P (M) are such that ea (s, ∞) ≤ e ≤ ea [s, ∞) .

.

) ( (i) If .f ∈ P (M) satisfies .f ≥ e, then .λ t; af = λ (t; a) for all .t ∈ [0, τ (e)). ) ) ( ( (ii) If .f ∈ P (M) satisfies .f ≥ e⊥ and .α = τ f − e⊥ , then .λ t + α; af = [ ( ⊥ )) λ (t + τ (e) ; a) for all .t ∈ 0, τ e . Proof (i) It follows from the right side inequality in Lemma 3.8.1 (applied in the von Neumann algebras .Me and .M) that ( ) ( ( ) ) λ (t; ae ) = λ t; af e ≤ λ t; af ≤ λ (t; a)

.

for all .t ∈ [0, τ (e)). Since, by Lemma 3.8.2 (i), .λ (t; ae ) = λ (t; a) for all t ∈ [0, τ (e)), assertion (i) is now clear. (ii) Applying the left side inequality in Lemma 3.8.1 in .Mf to the projection .e⊥ , it follows that [ ( )) ( ) ( ) .λ t + α; af ≤ λ t; ae⊥ , t ∈ 0, τ e⊥ .

) ( ( ) and, by Lemma 3.8.2 (ii), .λ t; ae⊥ = λ (t + τ (e) ; a), .0 ≤ t < τ e⊥ . Consequently, ) ( λ t + α; af ≤ λ (t + τ (e) ; a) ,

.

[ ( )) t ∈ 0, τ e⊥ .

( ) On the other hand, writing .t + τ (e) = t + α + τ f ⊥ , it( follows from ) the left side inequality in Lemma 3.8.1 that .λ (t + τ (e) ; a) ≤ λ t + α; af whenever [ ( ⊥ )) [ ( )) , which is, in particular, the case if .t ∈ 0, τ e⊥ . This .t + α ∈ 0, τ f shows that (ii) holds. u n

198

3 Singular Value Functions

For a first application of Lemma 3.8.2, the following observation is needed. Lemma 3.8.4 Suppose that .M is non-atomic. If .a ∈ S (τ )h and .θ ∈ [0, τ (1)), then there exists .e ∈ P (M) such that .τ (e) = θ and ea (λ (θ ; a) , ∞) ≤ e ≤ ea [λ (θ ; a) , ∞) .

.

Proof For convenience, set .β = λ (θ ; a). It follows from Proposition 3.1.3 (iii) that .d (β; a) ≤ θ ≤ d (β−; a). Let .{βn }∞ n=1 be a sequence in .R such that .βn < β for all n and .βn ↑ β, which implies that .d (βn ; a) ↓ d (β−; a) as .n → ∞. Since .ea (βn , ∞) ↓n ea [β, ∞) in .P (M), the normality of the trace implies that a .τ (e [β, ∞)) = d (β−; a). Hence, ea (β, ∞) ≤ ea [β, ∞) , ( ) ( ) τ ea (β, ∞) ≤ θ ≤ τ ea [β, ∞) , .

and so, it follows from Lemma 3.7.3 that there exists .e ∈ P (M) such that ea (β, ∞) ≤ e ≤ ea [β, ∞) and .τ (e) = θ . The proof is complete. u n

.

Theorem 3.8.5 If .M is non-atomic and .a ∈ L1 (τ )h , then: ft (i) . 0 λ (s; a) ds = sup {τ (ae) : e ∈ P (M) , τ (e) = t}, .t ∈ [0, τ (1)). f τ (1) (ii) . 1−t λ (s; a) ds = inf {τ (ae) : e ∈ P (M) , τ (e) = t}, .t ∈ [0, τ (1)). Proof (i) Let .t ∈ [0, τ (1)) be fixed. If .e ∈ P (M) and .τ (e) = t, then it follows from Theorem 3.6.6 that f τ (ae) ≤

τ (1)

λ (s; a) λ (s; e) ds

.

0

f =

τ (1)

f

t

λ (s; a) χ[0,τ (e)) (s) ds =

λ (s; a) ds,

0

0

which shows that f .

sup {τ (ae) : e ∈ P (M) , τ (e) = t} ≤

t

λ (s; a) ds. 0

For the proof of the converse inequality, set .β = λ (t; a). It follows from Lemma 3.8.4 that there exists .e ∈ P (M) such that .ea (β, ∞) ≤ e ≤ ea [β, ∞) and .τ (e) = t. Note that this implies, in particular, that .eea (B) = ea (B) e, .B ∈ B (R), and so, .ae = ea. Therefore, it follows from Corollary 3.5.6 (in particular, (3.36) applied in the reduced von Neumann algebra .Me ) and

3.8 Submajorization for the Eigenvalue Function

199

Lemma 3.8.2 (i) that f τ (ae) = τ (eae) = τe (ae ) =

τ (e)

λ (s; ae ) ds

.

0

f

t

=

λ (s; a) ds. 0

This shows that (i) holds (note that the supremum is actually a maximum, attained at a projection commuting with a). (ii) Since .λ (−a) = −λˇ (a) (see Proposition 3.5.8 (i)), assertion (ii) follows from (i) applied to .−a. u n Remark 3.8.6 It follows from the proof of the above theorem that f .

t

λ (s; a) ds = max {τ (ae) : e ∈ P (M) , τ (e) = t, ea = ae}

0

for all .t ∈ [0, τ (1)) (and similarly for the infimum in (ii)). The following special (classical) case of the above theorem should be pointed out explicitly. Corollary 3.8.7 If .(X, Σ, ν) is a non-atomic finite measure space and .f ∈ L1 (ν) is real-valued, then f

t

.

{f

}

λ (s; f ) ds = sup

0

f dν : T ∈ Σ, ν (T ) = t T

and f

ν(S)

.

{f λ (s; f ) ds = inf

1−t

} f dν : T ∈ Σ, ν (T ) = t

T

for all .t ∈ [0, ν (T )). Definition 3.8.8 Suppose that .M1 and .M2 are von Neumann algebras on Hilbert spaces .H1 and .H2 , respectively, equipped with finite normal faithful traces .τ1 and .τ2 , respectively, such that .τ1 (1) = τ2 (1). If .a ∈ L1 (τ1 )h and .b ∈ L1 (τ2 )h , then a is said to be .λ-submajorized by b, denoted by .a ≺≺λ b, if f

t

.

0

f

t

λ (s; a) ds ≤

λ (s; b) ds 0

200

3 Singular Value Functions

for all .0 ≤ t < τ1 (1). If, in addition, f

τ1 (1)

.

f λ (s; a) ds =

τ1 (1)

λ (s; b) ds,

0

0

then a is said to be .λ-majorized by b, which is denoted by .a ≺λ b. It should be observed that .a ≺≺λ b is equivalent to saying that .λ (a) ≺≺λ λ (b), where the latter submajorization is with respect to the commutative von Neumann algebra .L∞ [0, τ1 (1)) (equipped with the Lebesgue integral as its trace). Theorem 3.8.9 If .M is a von Neumann algebra equipped with a finite normal faithful trace .τ , then λ (a + b) ≺λ λ (a) + λ (b)

.

for all .a, b ∈ L1 (τ )h . Proof In view of Proposition 3.7.6 (ii), it is sufficient to prove the theorem under the additional assumption may that .M is non-atomic. Let .t ∈ [0, τ (1)) be fixed. If .e ∈ P (M) is such that .τ (e) = t, then it follows from Theorem 3.8.5 (i) that f

t

τ ((a + b) e) = τ (ae) + τ (be) ≤

.

0

f λ (s; a) ds +

t

λ (s; a) ds. 0

Since this holds for all .e ∈ P (M) satisfying .τ (e) = t, the same theorem implies that f t f t . λ (s; a + b) ds ≤ (λ (s; a) + λ (s; a)) ds. 0

0

Finally, it follows from Corollary 3.5.6 that f

τ (1)

.

λ (s; a + b) ds = τ (a + b) = τ (a) + τ (b)

0

f

t

=

(λ (s; a) + λ (s; a)) ds.

0

Observing that .λ (λ (a) + λ (b)) = λ (a) + λ (b), the proof is complete.

u n

For the proof of the next theorem (see Theorem 3.8.14), some further preparations are needed. As before, .M is a von Neumann algebra equipped with a finite normal faithful trace .τ . Lemma 3.8.10 Let .M be non-atomic and suppose that .a, b ∈ Sh (τ ). If .0 ≤ θ1 < θ2 ≤ τ (1), then there exists .f ∈ P (M) such that .τ (f ) = τ (1) − (θ2 − θ1 ) and:

3.8 Submajorization for the Eigenvalue Function

(i) (ii) (iii) (iv)

201

( ) λ (t; af = λ (t; a) for ) all .t ∈ [0, θ1 ). .λ t − (θ2 − θ1 ) ; bf = λ (t; b) for all .t ∈ [θ2 ; τ (1)). ) ( ) ( .λ (t; a) − λ (t; b) ≤ λ t; af − λ t; bf for all .t ∈ [0, θ1 ). ( ) ( ) .λ (t; a) − λ (t; b) ≤ λ t − (θ2 − θ1 ) ; af − λ t − (θ2 − θ1 ) ; bf for all .t ∈ [θ2 ; τ (1)). .

Proof It follows from Lemma 3.8.4 that there exist .e1 , e2 ∈ P (M) such that τ (e1 ) = θ1 , .τ (e2 ) = θ2 , and

.

ea ((λ (θ1 ; a)) , ∞) ≤ e1 ≤ ea [λ (θ1 ; a) , ∞) ,

.

eb ((λ (θ2 ; b)) , ∞) ≤ e2 ≤ eb [λ (θ2 ; b) , ∞) . Since τ (e1 ∨ (1 − e2 )) ≤ τ (e1 ) + τ (1 − e2 ) = τ (1) − (θ2 − θ1 ) ,

.

it follows from Lemma 3.7.3 that there exists a projection .f ∈ P (M) such that f ≥ e1 ∨ (1 − e2 ) ,

.

τ (f ) = τ (1) − (θ2 − θ1 ) .

) ( Since .f ≥ e1 , it follows from Corollary 3.8.3 (i) that .λ t; af = λ (t; a) for all .0 ≤ t ≤ τ (e1 ) = θ1 , which is assertion (i) of the lemma. Since .f ≥ 1 − e2 , it follows from Corollary 3.8.3 (ii) that ) ( λ (t; b) = λ t + α − τ (e2 ) ; bf ,

t ∈ [τ (e2 ) , τ (1)) ,

.

) ( where .α = τ f − e2⊥ = θ1 , that is, ) ( λ (t; b) = λ t − (θ2 − θ1 ) ; bf ,

.

t ∈ [θ2 , τ (1)) .

This proves statement (ii). ) ( It follows from Lemma 3.8.1 that .λ t; bf( ≤ λ) (t; b) for all .t ∈ [0, τ (f )). Since .θ1 ≤ τ (f ), this implies in particular that .λ t; bf ≤ λ (t; b) for all .t ∈ [0, θ1 ), and so, (iii) follows immediately from (i). Similarly, it follows from Lemma 3.8.1 that ) ( ( ) λ (t; a) ≤ λ t − τ f ⊥ ; af ,

.

[ ( ) ) t ∈ τ f ⊥ , τ (1) .

( ) Since .τ f ⊥ ≤ θ2 , this implies, in particular, that ) ( λ (t; a) ≤ λ t − (θ2 − θ1 ) ; af ,

.

t ∈ [θ2 , τ (1)) ,

which, in combination with (ii), yields (iv). The proof is complete.

u n

202

3 Singular Value Functions

For any measurable subset T of .[0, ∞), define the map .σT : T → [0, m (T )) by setting σT (t) = m (T ∩ [0, t)) ,

.

t ∈ T,

where, as before, m denotes Lebesgue measure. If T is a finite disjoint union of cells of the form .[a, b), then .σT is an increasing measure preserving bijection from T onto .[0, m (T )). By way of example, if .0 ≤ θ1 < θ2 ≤ τ (1) and .T = [0, θ1 ) ∪ [θ2 , τ (1)), then .σT is given by .σT (t) = t if .t ∈ [0, θ1 ) and .σT (t) = t − (θ2 − θ1 ) if .t ∈ [θ2 , τ (1)). Consequently, statements (iii) and (iv) of Lemma 3.8.10 may be reformulated as follows. Lemma 3.8.11 Let .M be non-atomic and suppose that .a, b ∈ Sh (τ ). If .0 ≤ θ1 < θ2 ≤ τ (1) and if .T = [0, θ1 ) ∪ [θ2 , τ (1)), then there exists a projection .e ∈ P (M) such that .τ (e) = m (T ) and λ (t; a) − λ (t; b) ≤ λ (σT (t) ; ae ) − λ (σT (t) ; be ) ,

.

t ∈ T.

The following lemma shows that the result of Lemma 3.8.11 continues to hold whenever .T ⊆ [0, τ (1)) is a finite union of cells. Lemma 3.8.12 Let .M be non-atomic and suppose that .a, b ∈ Sh (τ ). If .T ⊆ [0, τ (1)) is a disjoint union of cells, then there exists .e ∈ P (M) such that .τ (e) = m (T ) and λ (t; a) − λ (t; b) ≤ λ (σT (t) ; ae ) − λ (σT (t) ; be ) ,

.

t ∈ T.

(3.52)

Proof Note that the complement .[0, τ (1))\T of T is also a disjoint union of disjoint cells. The proof is by induction on the number n of cells in .[0, τ (1)) \ T . If .n = 0, then there is nothing to be proved (take .e = 1), and the case .n = 1 is precisely Lemma 3.8.11. Suppose that the statement has been proved for some .n ≥ 1 and that .[0, τ (1)) \ T consists of .n + 1 (non-empty) disjoint cells. Let J be any of these cells in .[0, τ (1)) \ T and define .S = T ∪ J , so that .[0, τ (1)) \ S is a disjoint union of n cells. The induction hypothesis asserts that there exists a projection .f ∈ P (M) such that .τ (f ) = m (S) and ( ) ( ) λ (t; a) − λ (t; b) ≤ λ σS (t) ; af − λ σS (t) ; bf ,

.

t ∈ S.

(3.53)

Writing .σS (J ) = [θ1 , θ2 ), define .T1 = [0, θ1 ) ∪ [θ2 , τ (f )) and note that .m (T1 ) = m (T ). It follows from Lemma 3.8.11, applied in the reduced von Neumann algebra .Mf to the operators .af and .bf , that there exists a projection .e ∈ P (M) with .e ≤ f such that .τ (e) = m (T1 ) = m (T ) and ) ( ) ( ) ( ) ( λ t; af − λ t; bf ≤ λ σT1 (t) ; ae − λ σT1 (t) ; ae ,

.

t ∈ T1 .

(3.54)

3.8 Submajorization for the Eigenvalue Function

203

Combination of (3.53) and (3.54) yields that ) ( ) ( λ (t; a) − λ (t; b) ≤ λ σT1 (σS (t)) ; ae − λ σT1 (σS (t)) ; ae

.

whenever .t ∈ S and .σS (t) ∈ T1 . The induction step is now completed by observing that σS−1 (T1 ) = T

.

and that σT (t) = σT1 (σS (t)) ,

.

t ∈ T. u n

The proof is finished. Lemma 3.8.13 If .a, b ∈ L1 (τ )h and T is a measurable subset of .[0, τ (1)), then f

f

m(T )

(λ (t; a) − λ (t; b)) dt ≤

.

λ (t; a − b) dt.

0

T

Proof In view of Proposition 3.7.6, it suffices to prove the lemma under the additional assumption that .M is non-atomic. Suppose first that .T ⊆ [0, τ (1)) is a finite union of disjoint cells. By Lemma 3.8.12, there exists .e ∈ P (M) such that .τ (e) = m (T ) and (3.52) holds. Since the map .σT : T → [0, m (T )) is a measure preserving bijection, it follows that f

f (λ (t; a) − λ (t; b)) dt ≤

.

T

(λ (σT (t) ; ae ) − λ (σT (t) ; be )) dt T

f =

m(T )

(λ (t; ae ) − λ (t; be )) dt

0

f =

0

m(T )

f λ (t; ae ) dt −

m(T )

λ (t; be ) dt. 0

Furthermore, it follows from Corollary 3.5.6 and Theorem 3.6.6 (applied to the reduced von Neumann algebra .Me ) that f

m(T )

.

0

f

m(T )

λ (t; ae ) dt − 0

( ) λ (t; be ) dt = τe (ae ) − τe (be ) = τe (a − b)e = τ (e (a − b) e) = τ ((a − b) e)

204

3 Singular Value Functions

f

τ (1)

≤

f

m(T )

λ (t; a − b) λ (t; e) dt =

0

λ (t; a − b) dt,

0

which proves the lemma for the case that T is a finite union of disjoint cells. If .T ⊆ [0, τ (1)) is an arbitrary measurable set, then there exists a sequence ∞ .{Tn } n=1 , each .Tn a finite union of disjoint cells, such that .m (T ATn ) → 0 as .n → ∞ (where, as usual, .T ATn denotes the symmetric difference of T and .Tn ). By the dominated convergence theorem, f

f (λ (t; a) − λ (t; b)) dt →

.

(λ (t; a) − λ (t; b)) dt,

Tn

T

and similarly, since .m (Tn ) → m (T ), it follows that f

m(Tn )

.

f

m(T )

λ (t; a − b) dt →

λ (t; a − b) dt

0

0

as .n → ∞. Since it follows from the first part of the proof that the assertion of the u n lemma holds for each .Tn , this suffices to complete the proof. The following reformulation of the lemma above is one of the main results in the present section. Theorem 3.8.14 If .M is a von Neumann algebra equipped with a finite normal faithful trace .τ , then λ (a) − λ (b) ≺λ λ (a − b)

.

for all .a, b ∈ L1 (τ )h . Proof If .t ∈ [0, τ (1)), then it follows from the commutative specialization of Theorem 3.8.5 (see Corollary 3.8.7) that f .

t

f λ (s; λ (a) − λ (b)) ds = sup

0

T

(λ (s; a) − λ (s; b)) ds, T

where the supremum is taken over all measurable subsets .T ⊆ [0, τ (1)) satisfying m (T ) = t. Therefore, it follows immediately from Lemma 3.8.13 that

.

f

t

.

0

f

t

λ (s; λ (a) − λ (b)) ds ≤ 0

λ(s; a − b) ds,

t ∈ [0, τ (1)) .

3.9 Submajorization for the Singular Value Function

205

Furthermore, using Corollary 3.5.6 (applied to .M and its commutative specialization), it follows that f .

τ (1)

f

τ (1)

λ (s; λ (a) − λ (b)) ds =

0

f

(λ (s; a) − λ (s; b)) ds = τ (a − b)

0 τ (1)

=

λ (s; a − b) ds,

0

u n

which completes the proof.

Remark 3.8.15 Actually, Theorem 3.8.9 may be obtained as a corollary of Theorem 3.8.14 (in combination with Corollary 3.5.6). Indeed, if .a, b ∈ L1 (τ ), then Theorem 3.8.14 implies that λ (a + b) − λ (a) ≺λ λ (b) ,

.

and so, if .t ∈ [0, τ (1)) and .T ⊆ [0, τ (1)) satisfies .m (T ) = t, then f

f

t

(λ (s; a + b) − λ (s; a)) ds ≤

.

λ (s; b) ds, 0

T

and so, f

f

f

λ (s; a + b) ds ≤

.

T

t

λ (s; a) ds + f

t

≤

λ (s; b) ds 0

T

f

t

λ (s; a) ds +

0

λ (s; b) ds. 0

The result of Theorem 3.8.9 now follows immediately. References: [42, 52, 88].

3.9 Submajorization for the Singular Value Function In this section, a number of inequalities will be discussed, which are all centered around the important concept of submajorization in the sense of Hardy, Littlewood and Pólya. Throughout this section, it will be assumed that .M is a von Neumann algebra on a Hilbert space H , equipped with a fixed semi-finite faithful normal trace .τ . Definition 3.9.1 Suppose that .M1 and .M2 are von Neumann algebras on Hilbert spaces .H1 and .H2 , respectively, equipped with semi-finite normal faithful traces .τ1

206

3 Singular Value Functions

and .τ2 , respectively. If .x ∈ S (τ1 ) and .y ∈ S (τ2 ), then x is said to be submajorized by y, denoted by .x ≺≺ y, if f

t

.

f

t

μ (s; x) ds ≤

(3.55)

μ (s; y) ds

0

0

for all .t ≥ 0. Remark 3.9.2 The notion of submajorization may, of course, in particular be considered for functions on .[0, ∞), that is, with respect to the commutative von Neumann algebra .L∞ (0, ∞) with Lebesgue integral as trace. If .x ∈ S (τ1 ) and .y ∈ S (τ2 ), then it is clear that .x ≺≺ y if and only if .μ (x) ≺≺ μ (y). It should also be observed that, if .τ1 (1) = τ2 (1) < ∞ and if .x ∈ S (τ1 )+ and + .y ∈ S (τ2 ) , then the above introduced notion of submajorization coincides with .λ-submajorization, as was introduced in Definition 3.8.8. Evidently, if .x ∈ S (τ1 ) and .y ∈ S (τ2 ) are such that .μ (x) ≤ μ (y), in particular if .|x| ≤ |y|, then .x ≺≺ y (but the converse does not hold). It is also clear that .x ≺≺ y implies that .αx ≺≺ αy for all .0 ≤ α ∈ R. Remark 3.9.3 It should be observed that, if .x ∈ S (τ1 ) and .y ∈ L1 (τ2 ) satisfy x ≺≺ y, then .x ∈ L1 (τ1 ) and .||x||1 ≤ ||y||1 . Indeed, letting .t → ∞ in (3.55) implies that

.

f τ1 (|x|) =

∞

.

f μ (s; x) ds ≤

0

∞

μ (s; y) ds = τ2 (|y|) .

0

Similarly, if .x ∈ S (τ1 ) and .y ∈ M2 satisfy .x ≺≺ y, then .x ∈ M1 and .||x||B(H1 ) ≤ ||y||B(H2 ) . Indeed, using Lemma 3.2.3 (i), it follows that μ (t; x) ≤

.

1 t

f

t

μ (s; x) ds ≤

0

1 t

f 0

t

μ (s; y) ds ≤ μ (0; y) = ||y||B(H2 )

for all .t > 0, and hence, .μ (0; x) ≤ ||y||B(H2 ) . Consequently, .x ∈ M1 and ||x||B(H1 ) ≤ ||y||B(H2 ) .

.

For a further investigation offthe properties of submajorization, it will be crucial t to have another expression for . 0 μ (s; x) ds available. For functions f on .[0, ∞), this will correspond to the formula f

t

.

0

}

{f

|f | dm : A ⊆ [0, ∞) , m (A) = t ,

μ (s; f ) ds = sup

t ≥ 0,

A

(3.56) where, as before, m denotes Lebesgue measure (and the sets A in this formula are assumed to be Lebesgue measurable). The validity of the above identity depends on the fact that Lebesgue measure on .[0, ∞) is non-atomic (in fact, it is easy to

3.9 Submajorization for the Singular Value Function

207

find examples of measure spaces with atoms for which the analogue of this formula fails). Before embarking on the preparations of the proof of Theorem 3.9.5, observe that the estimate given in the next proposition is valid for any semi-finite von Neumann algebra. Proposition 3.9.4 If .x ∈ S (τ ) and .p ∈ P (M), then f

τ (p)

τ (|pxp|) ≤

μ (s; x) ds

.

0

and f τ (p |x| p) ≤

τ (p)

μ (s; x) ds.

.

0

Proof It follows from Remark 3.2.6 (i) that μ (pxp) = μ (pxp) χ[0,τ (p)) .

.

By Proposition 3.2.7 (vi), .μ (pxp) ≤ μ (x), and hence, f τ (|pxp|) =

∞

.

f

τ (p)

μ (s; pxp) ds =

f

0

0

τ (p)

μ (s; pxp) ds ≤

μ (s; x) ds, 0

which establishes the first inequality. Replacing x by .|x|, the second inequality follows immediately. u n Recall that a von Neumann algebra is called non-atomic if it does not contain any minimal projections (see the discussion preceding Lemma 3.7.2). Theorem 3.9.5 If .M is non-atomic, if .a ∈ S (τ )+ , and if .0 ≤ t ≤ τ (1), then f

t

.

μ (s; a) ds = sup {τ (eae) : e ∈ P (M) , τ (e) = t} .

(3.57)

0

Proof By Proposition 3.9.4, it is clear that f .

t

sup {τ (eae) : e ∈ P (M) , τ (e) = t} ≤

μ (s; a) ds. 0

For the proof of the converse inequality, let .t > 0 be fixed and set .λ = μ (t; a). Furthermore, let .μ∞ = lims→∞ μ (s; a). If .λ = 0 or .λ > μ∞ , then it follows from Lemma 3.7.7 (i) that there exists .e ∈ P (M) such that ea (λ, ∞) ≤ e ≤ ea [λ, ∞)

.

and

τ (e) = t.

208

3 Singular Value Functions

Hence, by Lemma 3.7.10 (i), f τ (eae) =

t

μ (s; a) ds.

.

0

Consequently, (3.57) holds in these cases (and the supremum is actually a maximum that is attained at a projection commuting with a). If .λ = μ∞ > 0 and .0 < ε < μ∞ , then it follows from Lemma 3.7.7 (ii) that there exists .e ∈ P (M) satisfying ea (μ∞ , ∞) ≤ e ≤ ea (μ∞ − ε, ∞)

.

τ (e) = t,

and

and so, by Lemma 3.7.10 (ii), f τ (eae) ≥

.

t

μ (s; a) ds − εt.

0

This holds for all .ε satisfying .0 < ε < μ∞ , which implies that f .

t

sup {τ (eae) : e ∈ P (M) , τ (e) = t} ≥

μ (s; a) ds. 0

By this, the proof of the theorem is complete.

u n

An inspection of the proof of the above proposition immediately yields the following corollary. Corollary 3.9.6 Suppose that .M is non-atomic: (i) If .a ∈ S0 (τ )+ , then f

t

.

μ (s; a) ds = max {τ (eae) : e ∈ P (M) , ae = ea, τ (e) = t}

(3.58)

0

for all .0 ≤ t ≤ τ (1). (ii) If .a ∈ S (τ )+ and .t ≥ 0 is such that .μ (t; a) > lims→∞ μ (s; a), then (3.58) also holds. Proof Both (i) and (ii) imply that either .μ (t; a) = 0 or .α = μ (t; a) > 0 and d (α−; a) < ∞, in which cases the result of the corollary has already been noted in the proof of Theorem 3.9.5. u n

.

Remark 3.9.7 (i) It should be noted that formula (3.56) follows from Theorem 3.9.5 by taking .M = L∞ (0, ∞). In general, the supremum in (3.57) is not a maximum, not even in the commutative setting. Indeed, if the function f on .[0, ∞) is defined

3.9 Submajorization for the Singular Value Function

209

by .f (t) = arctan t, then it is easy to see that the supremum in (3.56) is not attained (note that in this case .μ (s; f ) = π/2 for allf.s ∈ [0, ∞)). t (ii) In view of Corollary 3.9.6, it might be expected that . 0 μ (s; a) ds is equal to sup {τ (eae) : e ∈ P (M) , ae = ea, τ (e) = t}

.

(3.59)

for all .a ∈ S (τ )+ and .0 ≤ t ≤ τ (1), whenever the von Neumann algebra .M is non-atomic. This is, however, false in general as is shown by the following example. Let .H = L2 [0, 1] and consider the non-atomic von Neumann algebra M = L∞ [0, 1] ⊗B (H ) ,

.

equipped with the tensor product trace .τˆ of the Lebesgue integral on .L∞ [0, 1] and the standard trace .τ on .B (H ). Identifying the Hilbert space tensor product .L2 [0, 1] ⊗H with the Bochner space .L2 ([0, 1] , H ), the von Neumann algebra wo .M may be identified with the space .L∞ ([0, 1] , B (H )) of all weak operator measurable .B (H )-valued functions, in which case the trace .τˆ is given by f τˆ (f ) =

1

+ f ∈ Lwo ∞ ([0, 1] , B (H )) .

τ (f (t)) dt,

.

0

( ) Identifying .L2 [0, 1] ⊗H = L2 [0, 1] ⊗L2 [0, 1] with .L2 [0, 1]2 , the von wo Neumann subalgebra ( .L∞2[0, ) 1] ⊗L∞ [0, 1] = L∞ ([0, 1] ,(L∞ [0, )1]) of .M corresponds to .L∞ [0, 1] , acting via multiplication on .L2 [0, 1]2 . ( ) Let .0 < a ∈ L∞ [0, 1]2 be a function such that its spectral sets coincide with the .σ -algebra of all Lebesgue measurable subsets of .[0, 1]2 . If wo .x ∈ L∞ ([0, 1] , B (H )) satisfies .xa = ax, then x commutes with the spectral ( ) ( ) measure of a, and hence with all elements of .L∞ [0, 1]2 . Since .L∞ [0, 1]2 is ( ) maximal abelian, this implies that .x ∈ L∞ [0, 1]2 . In particular, if .e ∈ P (M) is such that .ae = ea, then e is given by a weak operator measurable function .e : [0, 1] → P (L∞ [0, 1]), and its trace is given by f τˆ (e) =

1

τ (e (t)) dt.

.

0

Now observe that .e (t) /= 0 implies that .τ (e (t)) = ∞ (as .e (t) is not even compact). Consequently, .τˆ (e) < ∞ is possible only if .e = 0. Since .0 < ft μ a) ds < ∞, it is now clear that (3.59) does not hold for any .t > 0. (s; 0 Remark 3.9.8 Suppose that the von Neumann algebra .M is atomic and that all minimal projections have trace equal to one. It follows from Remark 3.7.9 that the result of Theorem 3.9.5 is also valid in this case, if the values of t are restricted to

210

3 Singular Value Functions

N, that is,

.

n−1 Σ .

f μ (k) =

n

μ (s; a) ds = sup {τ (eae) : e ∈ P (M) , τ (e) = n} ,

0

k=0

for all .n ∈ N satisfying .n ≤ τ (1). The following theorem follows from Theorem 3.9.5, using the technique discussed in Proposition 3.7.6. Theorem 3.9.9 Let .M be a von Neumann algebra on a Hilbert space H , equipped with the semi-finite faithful normal trace .τ . If .x, y ∈ S (τ ), then .μ (x + y) ≺≺ μ (x) + μ (y), that is, f .

t

f

0

f

t

μ (s; x + y) ds ≤

t

μ (s; x) ds +

0

μ (s; y) ds,

t ≥ 0.

(3.60)

0

Proof In view of Proposition 3.7.6, it is sufficient to prove this theorem under the additional assumption that .M is non-atomic. Given .0 ≤ t ≤ τ (1), suppose that .e ∈ P (M) satisfies .τ (e) = t and let .x + y = v |x + y| be the polar decomposition of .x + y. Using Remark 3.4.10 and Proposition 3.9.4, it follows that |) ( ) (| τ (e |x + y| e) = τ ev ∗ (x + y) e = τ |ev ∗ xe + ev ∗ ye| |) |) (| (| ≤ τ |ev ∗ xe| + τ |ev ∗ ye| f t f t ( ) ( ) ≤ μ s; v ∗ x ds + μ s; v ∗ y ds

.

0

f ≤

t

0

f

t

μ (s; x) ds +

0

μ (s; y) ds. 0

Since this holds for all .e ∈ P (M) with .τ (e) = t, inequality (3.60) is an immediate consequence of Theorem 3.9.5. u n Theorem 3.9.10 If .x, y ∈ S (τ ), then .μ (xy) ≺≺ μ (x) μ (y). Proof As in the proof of Theorem 3.9.9, it suffices to prove the result under the additional assumption that .M is non-atomic. Given .0 ≤ t ≤ τ (1), let .e ∈ P (M) satisfy .τ (e) = t. If .xy = v |xy| is the polar decomposition of xy, then it follows from Theorem 3.4.29 that f ∞ |) (| ∗ ( ) | | ev xye ≤ .τ (e |xy| e) = τ μ s; ev ∗ x μ (s; ye) ds. 0

3.9 Submajorization for the Singular Value Function

211

Since .μ (s; ye) = 0 for all .s ≥ t (see Remark 3.2.6 (i)), .μ (ev ∗ x) ≤ μ (x) and .μ (ye) ≤ μ (y), this implies that f τ (e |xy| e) ≤

t

μ (s; x) μ (s; y) ds,

.

0

and so, the result of the theorem now follows from Theorem 3.9.5.

u n

Further preparation is needed for the next theorem (see Theorem 3.9.14). In the proof of Lemma 3.9.12, the following simple observations will be used. Lemma 3.9.11 If .e ∈ P (M) satisfies .τ (e) < ∞ and if .a ∈ Sh (τ ), then ) ( λ (t; ae ) ≤ μ t; a + ,

.

t ∈ [0, τ (e)) .

( ) If, in addition, there exists .p ∈ P (M) such that .p ≤ s a + and .p ≤ e, then ( ) μ t; pa + p ≤ λ (t; ae ) ,

.

t ∈ [0, τ (p)) .

( ) Proof Since .a ≤ a + implies that .eae ≤ ea + e, it is clear that .ae ≤ a + e . Therefore, it follows from Proposition 3.6.3 (ii), equality (3.44), and Proposition 3.2.7 (vi) that ( ( ) ) ( ) ) ( ( ( ) ) λ (t; ae ) ≤ λ t; a + e = μ t; a + e = μ t; ea + e ≤ μ t; a +

.

for all .t ∈ [0, τ (e)). ( ) Now assume, in addition, that .p ∈ P (M) is such that .p ≤ e and .p ≤ s a + . ( ) Since .pa + p = pap, it is clear that . a + p = ap , and so, it follows from Lemma 3.8.1, applied in the reduced von Neumann algebra .Me , that ( ( ) ) ( ( ) ) μ t; pa + p = λ t; a + p = λ t; ap ≤ λ (t; ae )

.

for all .t ∈ [0, τ (p)).

u n

Lemma 3.9.12 If .a, b ∈ Mh and if .S, T ⊆ [0, ∞) are of finite measure, then f .

S

( ( +) ( )) μ a − μ b+ dt +

f

( ( −) ( )) μ b − μ a − dt

T

f ≤

m(S)+m(T )

( ) μ (a − b)+ dt.

0

( +) > 0( for Proof Without loss ( of− )generality, it may be assumed that .μ t; a ( all )) .t ∈ S and that .μ t; b > 0 for all .t ∈ S, which implies that .m (S) ≤ τ s a +

212

3 Singular Value Functions

( ( )) and .m (T ) ≤ τ s b− (see Remark 3.2.6). Further, it may be assumed that ( ( )) ( ( )) τ s a + + τ s b− ≤ τ (1) .

.

Indeed, if this is not the case, then replace .M by the von Neumann algebra .M ⊗ M2 (C) = M2 (M), equipped with tensor product trace, and replace a and b by [ a˜ =

.

] a0 , 00

[ ] b0 b˜ = , 00

) ( ) ( respectively, and observe that .μ t; a˜ ± = μ t; a ± for all .t ∈ [0, τ (1)). Moreover, in view of Proposition 3.7.6, it may be assumed further ( ) that .M is non-atomic. Suppose that .p ∈ P (M) satisfies .p ≤ s (a + ) and .m (S) ≤ τ (p) < ∞. Similarly, let .q ∈ P (M) be such that .q ≤ s b− and .m (T ) ≤ τ (q) < ∞. Note that such projections exist, as the trace .τ is semi-finite and normal. Since τ (p ∨ q) ≤ τ (p) + τ (q) ≤ τ (s(a + )) + τ (s(b− )) ≤ τ (1) ,

.

Lemma 3.7.3 implies that there exists .e ∈ P (M) such(that .p ∨ )q ≤ e and .τ (e) = + τ (p) + τ (q). It follows from Lemma ( + ) 3.9.11 that .μ t; pa p ≤ λ (t; ae ), .t ∈ [0, τ (p)), and that .λ (t; be ) ≤ μ t; b , .t ∈ [0, τ (e)). Therefore, ) ( ) ( μ t; pa + p − μ t; b+ ≤ λ (t; ae ) − λ (t; be ) ,

.

t ∈ [0, τ (p)) .

(3.61)

The same argument, with a, b, and q replaced by .−b, .−a, and q, respectively, shows that ( ) ( ) μ t; qb− q − μ t; a − ≤ λ (t; −be ) − λ (t; −ae )

.

= λˇ (t; ae ) − λˇ (t; be ) ,

t ∈ [0, τ (q)) ,

(3.62)

where the last equality follows from Proposition 3.5.8 (i), applied in .Me . Recall that, for .c ∈ Me , the function .λˇ (c) is given by λˇ (t; c) = λ ((τ (e) − t) −; c) ,

.

t ∈ [0, τ (e)) .

Writing .τ (e) − T = {τ (e) − t : t ∈ T }, it is clear that .

f ( f ) λˇ (ae ) − λˇ (be ) dt = T

(λ (ae ) − λ (be )) dt. τ (e)−T

(3.63)

3.9 Submajorization for the Singular Value Function

213

Since .S ⊆ [0, τ (p)) and .τ (e) − T ⊆ (τ (p) , τ (e)], the sets S and .τ (e) − T are disjoint, and so, it follows from (3.61), (3.62), and (3.63) that f .

( ( + ) ( )) μ pa p − μ b+ dt +

S

f

( ( − ) ( )) μ qb q − μ a − dt T

f ≤

(λ (ae ) − λ (be )) dt. S∪(τ (e)−T )

Furthermore, Theorem 3.8.14 (or, Lemma 3.8.13), in combination with Lemma 3.9.11, implies that f

f

m(S)+m(T )

(λ (ae ) − λ (be )) dt ≤

.

λ (ae − be ) dt

0

S∪(τ (e)−T )

f =

m(S)+m(T )

) ( λ (a − b)e dt

m(S)+m(T )

) ( μ (a − b)+ dt.

0

f ≤

0

It thus has been proved that f .

S

( ( + ) ( )) μ pa p − μ b+ dt + f

≤

f

( ( − ) ( )) μ qb q − μ a − dt T

m(S)+m(T )

( ) μ (a − b)+ dt

(3.64)

0

( +) for all (projections .p, q ∈ P (M) satisfying .p ≤ s a , .m (S) ≤ τ (p) < ∞ and ) − , .m (T ) ≤ τ (q) < ∞. .q ≤ s b ( +) Fix any .p1 ∈ P (M) satisfying and .m (S) ≤ τ (p1 ) 0. Consequently, .d xˆ + = d xˆ − = d (|x|), and hence, .μ xˆ + = μ xˆ − = μ (x). u n Theorem 3.9.14 Let .M be a von Neumann algebra on a Hilbert space H , equipped with the semi-finite faithful normal trace .τ . If .x, y ∈ S (τ ), then μ (x) − μ (y) ≺≺ μ (x − y) .

.

Proof By the commutative specialization of Theorem 3.9.5, it has to be shown that f

f

m(T )

|μ (x) − μ (y)| dt ≤

.

μ (x − y) dt

(3.66)

0

T

for all measurable subsets .T ⊆ [0, ∞) of finite measure. Let the set T be fixed, and first assume, in addition, that .x, y ∈ M. Define .T1 ⊆ [0, ∞) by setting T1 = {t ∈ T : μ (t; x) > μ (t; y)}

.

and let .T2 = T \T1 . Lemma 3.9.12, applied in the von Neumann algebra .M⊗M2 (C) to the element .xˆ and .yˆ (as defined by (3.65)), implies that f

( ( +) ( )) μ xˆ − μ yˆ + dt +

.

f

T1

( ( −) ( )) μ yˆ − μ xˆ − dt T2

f ≤

m(T )

μ

(( )+ ) dt. xˆ − yˆ

0

Observing that .xˆ − yˆ = (x − y)∧ , it follows from Lemma 3.9.13 that f

f

f

(μ (x) − μ (y)) dt +

.

T1

(μ (y) − μ (x)) dt ≤

m(T )

μ (x − y) dt,

0

T2

which shows that (3.66) holds. Now suppose that .x, y ∈ S (τ ) are arbitrary. Since .D (x) ∩ D (y) is .τ -dense (see ∞ Proposition 2.3.4), there exists an increasing ( ⊥ )sequence .{en }n=1 in .P (M) such that → 0 as .n → ∞. This implies that .en (H ) ⊆ D (x) ∩ D (y) for all n and .τ en Tm

Tm

xen , yen ∈ M for all n and .xen → x, .yen → y as .n → ∞. Proposition 3.2.11 (ii) implies that .μ (xen ) → μ (x) and .μ (yen ) → μ (y) almost everywhere on .[0, ∞). Since .μ ((x − y) en ) ≤ μ (x − y) (see Proposition 3.2.7 (vi)), it follows from the first part of the proof that

.

f

f

m(T )

|μ (xen ) − μ (yen )| dt ≤

.

T

f

μ ((x − y) en ) dt

0 m(T )

≤ 0

μ (x − y) dt

216

3 Singular Value Functions

for all n, and hence, an application of Fatou’s lemma yields (3.66). The proof is complete. n u Remark 3.9.15 It should be observed that Theorem 3.9.9 may be obtained as a consequence of Theorem 3.9.14, in combination with the commutative specialization of Theorem 3.9.5, via the same argument that is used in Remark 3.8.15. ft The next theorem exhibits an important formula for the quantity . 0 μ (x) ds. As was mentioned in Remark 3.4.15, the von Neumann algebra .M is also denoted by .L∞ (τ ) and that .||x||∞ = ||x||B(H ) for all .x ∈ L∞ (τ ). Moreover, .||x||∞ = ||μ (x)||∞ = μ (0; x), .x ∈ L∞ (τ ). Theorem 3.9.16 If .x ∈ S (τ ), then f

t

.

0

{ } μ (s; x) ds = inf ||y||1 + t ||z||∞ : x = y + z, y ∈ L1 (τ ) , z ∈ L∞ (τ )

for all .t > 0. Proof Let .t > 0 be fixed. fThe right hand side infimum is denoted by A (where t inf ∅ = ∞). To prove that . 0 μ (x) ds ≤ A, it may be assumed that .A < ∞. If .y ∈ L1 (τ ), .z ∈ L∞ (τ ), and .x = y + z, then it follows from Proposition 3.2.7 (iii) that .

μ (s; x) ≤ μ (s; y) + μ (0; z) = μ (s; y) + ||z||∞

.

for all .s > 0. Consequently, f

t

.

f μ (s; x) ds ≤

0

0

f ≤

0

t

μ (s; y) ds + t ||z||∞

∞

μ (s; y) ds + t ||z||∞ = ||y||1 + t ||z||∞ ,

ft which shows that . 0 μ (x) ds ≤ A. ft To prove the converse inequality, it may be assumed that . 0 μ (x) ds < ∞. Let .x = v |x| be the polar decomposition of x and define α = μ (t; x) ,

.

y = v (|x| − α1)+ ,

z = x − y.

It follows from Propositions 3.2.7 (vi) and 3.2.8 that ) ( μ (s; y) ≤ μ s; (|x| − α1)+ = (μ (s; x) − α)+ ,

.

s ≥ 0.

217

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

Since .μ (s; x) ≤ α for all .s ≥ t and .μ (s; x) ≥ α whenever .0 < s ≤ t, this implies that f ∞ f t f ∞ + . μ (s; y) ds ≤ μ (s; x) ds − αt, (μ (s; x) − α) ds = 0

0

0

and so, .y ∈ L1 (τ ) and f .

||y||1 ≤

t

μ (s; x) ds − αt.

0

Furthermore, ) ( z = v |x| − (|x| − α1)+ = v (|x| ∧ (α1)) ,

.

where .|x| ∧ (α1) = φ (|x|) with .φ (s) = s ∧ α, .s ∈ [0, ∞). Using Proposition 3.2.8 once again, it follows that μ (s; z) ≤ μ (s; |x| ∧ (α1)) = μ (s; x) ∧ α,

.

s ≥ 0,

and so, .z ∈ L∞ (τ ) and .||z||∞ ≤ α. Consequently, f A ≤ ||y||1 + t ||z||∞ ≤

t

μ (s; x) ds,

.

0

and this suffices to complete the proof of the proposition.

u n

Remark 3.9.17 Anf inspection of the above proof shows that for .x ∈ S (τ ) and t t > 0, the quantity . 0 μ (s; x) ds is also given by

.

f .

0

t

{ } μ (s; x) ds = inf ||y||1 + t ||z||∞ : |x| = y + z, y ∈ L1 (τ )+ , z ∈ L∞ (τ )+ .

The details are left to the reader. References: [42, 55, 65, 81, 88, 95, 100].

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ )) There is an intimate relationship between submajorization of the singular value functions and linear operators that are contractions for both the .L1 -norm and the .L∞ -norm. This important relationship will be discussed in the present section.

218

3 Singular Value Functions

Let .M be a von Neumann algebra on a Hilbert space H , equipped with a fixed semi-finite normal faithful trace .τ . In the present section, the von Neumann algebra .M will be denoted frequently by .L∞ (τ ) (see Remark 3.4.15). Recall, from Proposition 3.4.11, that the Banach space .L1 (τ ) is continuously embedded in .S (τ ) (equipped, as usual, with its measure topology). Similarly, by Proposition 2.5.10 (iii), .L∞ (τ ) = M is also continuously embedded in .S (τ ). Consequently, the pair .(L1 (τ ) , L∞ (τ )) is a Banach couple, in the sense of interpolation theory. Therefore, it is natural to consider the intersection .L1 (τ ) ∩ L∞ (τ ) and the sum .L1 (τ ) + L∞ (τ ) of these Banach spaces. The norm in .L1 (τ ) ∩ L∞ (τ ) is defined by setting .

||x||L1 ∩L∞ = max (||x||1 , ||x||∞ ) ,

x ∈ L1 (τ ) ∩ L∞ (τ ) .

It is easy to see that .L1 (τ ) ∩ L∞ (τ ) is a Banach space with respect to this norm. It should be observed that if .x ∈ S (τ ), then .x ∈ L1 (τ ) ∩ L∞ (τ ) if and only if .μ (x) ∈ L1 ∩ L∞ [0, ∞). Moreover, .

||x||L1 ∩L∞ = ||μ (x)||L1 ∩L∞ ,

x ∈ L1 (τ ) ∩ L∞ (τ ) .

The space .L1 (τ ) ∩ L∞ (τ ) is also denoted by .L1 ∩ L∞ (τ ). The norm on the space .L1 (τ ) + L∞ (τ ) is defined by setting .

{ } ||x||L1 +L∞ = inf ||y||1 + ||z||∞ : x = y + z, y ∈ L1 (τ ) , z ∈ L∞ (τ )

for all .x ∈ L1 (τ ) + L∞ (τ ). A standard argument shows that .L1 (τ ) + L∞ (τ ) is a Banach space with respect to this norm, which is continuously embedded in .S (τ ). It follows f 1 from Theorem 3.9.16 that if .x ∈ S (τ ), then .x ∈ L1 (τ ) + L∞ (τ ) if and only if . 0 μ (t; x) dt < ∞ and that f .

||x||L1 +L∞ =

1

μ (t; x) dt, 0

x ∈ L1 (τ ) + L∞ (τ ) .

(3.67)

In particular, if .x ∈ S (τ ), then .x ∈ L1 (τ ) + L∞ (τ ) if and only if .μ (x) ∈ (L1 + L∞ ) [0, ∞) and .

||x||L1 +L∞ = ||μ (x)||L1 +L∞ ,

x ∈ L1 (τ ) + L∞ (τ ) .

The space .L1 (τ ) + L∞ (τ ) is also denoted by .(L1 + L∞ ) (τ ). For later reference, the following simple observation is formulated as a lemma. Lemma 3.10.1 If .x ∈ S (τ ), then the following statements are equivalent: (i) (ii) (iii) (iv)

xf ∈ L1 (τ ) + L∞ (τ ). a μ x) dt < ∞ for some .a > 0. f0a (t; . 0 μ (t; x) dt < ∞ for all .a > 0. .e |x| e ∈ L1 (τ ) for all .e ∈ P (M) satisfying .τ (e) < ∞. . .

219

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

Proof The equivalence of (i), (ii), and (iii) follows immediately from the discussion preceding, in combination with the fact that .μ (x) is decreasing. Therefore, it suffices to prove that (i) and (iv) are equivalent. Observing that .|x| belongs to .L1 (τ ) + L∞ (τ ) whenever x does, it is clear that (i) implies (iv), since products of elements in .L1 (τ ) with elements in .L∞ (τ ) belong to .L1 (τ ) (see Proposition 3.4.1 (iv)). Conversely, suppose that (iv) holds. If .x ∈ L∞ (τ ), then there is nothing to prove. It may be assumed, therefore, that .x ∈ / L∞ (τ ), that is, .e|x| (s,) ∞) /= 0 for all ( |x| .s ≥ 0. Since .x ∈ S (τ ), there exists .s ≥ 0 such that .τ e (s, ∞) < ∞. Setting |x| (s, ∞), it follows that .0 < τ (e) < ∞, and, by Proposition 3.2.10 (iii), .e = e .μ (e |x| e) = μ (x) χ[0,τ (e)) . Hence, f .

τ (e)

f μ (t; x) dt =

0

∞

μ (t; e |x| e) dt = τ (e |x| e) < ∞,

0

which shows that (iii) and so, (i) holds. The proof is complete.

u n

Recall from Proposition 2.5.14 that the closure of the ideal .F (τ ) in .S (τ ), with respect to the measure topology, is equal to .S0 (τ ). Since .F (τ ) ⊆ L1 (τ )∩L∞ (τ ) ⊆ S0 (τ ), this implies, in particular, that the closure of .L1 (τ ) ∩ L∞ (τ ) in .S (τ ), with respect to the measure topology, is also equal to .S0 (τ ). Proposition 3.10.2 (i) The closure of .L1 (τ )∩L∞ (τ ) in the space .L1 (τ )+L∞ (τ ) is equal to .S0 (τ )∩ (L1 (τ ) + L∞ (τ )). (ii) The space .M = L∞ (τ ) is dense in .L1 (τ ) + L∞ (τ ). (iii) The space .F (τ ) is norm dense in .L1 (τ ) ∩ L∞ (τ ). Proof (i) As observed above, the embedding of .L1 (τ ) + L∞ (τ ) into .S (τ ), with respect to the norm and measure topology, respectively, is continuous. Therefore, if .x ∈ L1 (τ ) + L∞ (τ ) belongs to the closure of .L1 (τ ) ∩ L∞ (τ ), then x belongs to the closure of .L1 (τ ) ∩ L∞ (τ ) in .S (τ ). Since .L1 (τ ) ∩ L∞ (τ ) ⊆ S0 (τ ) (cf. the discussion prior to Proposition 3.4.1) and .S0 (τ ) is closed in .S (τ ) (see Proposition 2.5.13), it follows that .x ∈ S0 (τ ). Conversely, let .0 ≤ a ∈ S0 (τ ) ∩ (L1 (τ ) + L∞ (τ )) be given and define an = aea (1/n, n] ,

.

n = 1, 2, . . . .

Since .τ (ea (1/n, n]) < ∞, it is clear that .an ∈ L1 (τ ) ∩ L∞ (τ ) for all n. Setting .αn = τ (ea (n, ∞)), it follows from Proposition 3.2.10 (iii) that ( ) μ aea (n, ∞) = μ (a) χ[0,αn ) ,

.

220

3 Singular Value Functions

and so, Lemma 3.10.1 implies that .aea (n, ∞) ∈ L1 (τ ) and .

|| a || ||ae (n, ∞)|| = 1

f

αn

μ (t; a) dt. 0

Consequently, .

|| || || || ||a − an ||L1 +L∞ ≤ ||aea (n, ∞)||1 + ||aea [0, 1/n]||∞ f αn 1 ≤ μ (t; a) dt + n 0

for all n. Since .αn → 0 as .n → ∞, this shows that .||a − an ||L1 +L∞ → 0. This suffices for the proof of assertion (i). (ii) Since (i) implies that .L1 (τ ) is contained in the closure of .L∞ (τ ) in .L1 (τ ) + L∞ (τ ), it is evident that .L∞ (τ ) is dense in .L1 (τ ) + L∞ (τ ). (iii) Given .0 ≤ a ∈ L1 (τ ) ∩ L∞ (τ ), define .an = aea (1/n, ∞), .n = 1, 2, . . .. Since .(1/n) ea (1/n, ∞) ≤ a and .τ (a) < ∞, it is clear that .τ (ea (1/n, ∞)) < ∞. Therefore, .an ∈ F (τ ) for all n. Since .a − an = aea [0, 1/n], it is also clear that .||a − an ||∞ → 0 as .n → ∞. Furthermore, .0 ≤ a − an ≤ a, .τ (a) < ∞, and .a − an ↓ 0, which implies that .τ (a − an ) ↓ 0. This suffices for the proof of assertion (iii). u n It will be convenient to introduce the notation G (τ ) = L1 (τ ) + L∞ (τ ) ,

.

H (τ ) = L1 (τ ) ∩ L∞ (τ ) ,

which will be used in the remaining part of this section. If .x ∈ G (τ ) and .y ∈ H (τ ), then it is clear that xy and yx both belong to .L1 (τ ). Note that it follows from Proposition 3.4.30 that .τ (xy) = τ (yx) for all .x ∈ G (τ ) and .y ∈ H (τ ). Consequently, a duality pairing may be defined by setting .

= τ (xy) = τ (yx) ,

x ∈ G (τ ) , y ∈ H (τ ) .

(3.68)

If .y ∈ H (τ ) is such that . = 0 for all .x ∈ G (τ ), then .y = 0 (indeed, τ (|y|) = τ (v ∗ y) = 0, where .y = v |y| is the polar decomposition of y). Similarly, if .x ∈ G (τ ) satisfies . = 0 for all .y ∈ H (τ ), then .x = 0. Indeed, let .x = w |x| be the polar decomposition of x and let .{pα } be a net in .P (M) such that .pα ↑ 1 and .τ (pα ) < ∞ for all .α. Since .pα w ∗ ∈ H (τ ), it follows from Proposition 3.4.32 that ) ( ) < > ( 1/2 .τ |x| pα |x|1/2 = τ (pα |x|) = τ pα w ∗ x = x, pα w ∗ = 0

.

221

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

for all .α. By Proposition 2.2.25 (iii), .|x|1/2 pα |x|1/2 ↑ |x| in .Sh (τ ), and hence, the normality of the trace implies that .τ (|x|) = 0, and so, by Proposition 3.3.3 (ii), .x = 0. It should also be observed that . = τ (ab) ≥ 0 whenever .a ∈( G (τ )+ and ) .b ∈ H (τ )+ . In fact, it follows from Proposition 3.4.32 that .τ (ab) = τ a 1/2 ba 1/2 ≥ 0, as .a 1/2 ba 1/2 ≥ 0. Furthermore, using Corollary 3.4.9, it is easily verified that .

|| ≤ ||x||G(τ ) ||y||H (τ ) ,

x ∈ G (τ ) , y ∈ H (τ ) .

(3.69)

Consequently, via the duality pairing given by (3.68), .H (τ ) may be identified with a point separating subspace of the Banach dual of .G (τ ) and, similarly, .G (τ ) may be identified with a point separating subspace of the Banach dual of .H (τ ). It is now supposed that also .N is a von Neumann algebra on a Hilbert space K, equipped with a semi-finite normal faithful trace .σ . Recall that a linear operator .T : G (σ ) → G (τ ) is called bounded from the Banach couple .(L1 (σ ) , L∞ (σ )) into the Banach couple .(L1 (τ ) , L∞ (τ )) whenever the restriction .T1 of T to .L1 (σ ) is a bounded linear operator from .L1 (σ ) into .L1 (τ ), and the restriction .T∞ of T to .L∞ (σ ) is a bounded linear operator from .L∞ (σ ) into .L∞ (τ ). The linear space of all bounded operators from .(L1 (σ ) , L∞ (σ )) into .(L1 (τ ) , L∞ (τ )) will be denoted by .A (N, M). If .T ∈ A (N, M), then .||T1 ||1 and .||T∞ ||∞ denote the norms of .T1 and .T∞ in the spaces .L (L1 (σ ) , L1 (τ )) and .L (L∞ (σ ) , L∞ (τ )), respectively. The norm in .A (N, M) is defined by setting .

||T ||A(N,M) = max (||T1 ||1 , ||T∞ ||∞ ) ,

T ∈ A (N, M) ,

and it is readily verified that .A (N, M) is a Banach space with respect to this norm. The unit ball of .A (N, M) is denoted by .Σ (N, M), that is, { } Σ (N, M) = T ∈ A (N, M) : ||T ||A(N,M) ≤ 1 .

.

It should be observed that every .T ∈ A (N, M) is a bounded operator from .G (σ ) into .G (τ ) and that .

||T ||L(G(σ ),G(τ )) ≤ ||T ||A(N,M) .

(3.70)

The next proposition presents an alternative characterization of operators in A (N, M).

.

Proposition 3.10.3 If .T : G (σ ) → G (τ ) is a linear operator, then .T ∈ A (N, M) if and only if there exists .0 ≤ c ∈ R such that .T x ≺≺ cx for all .x ∈ G (σ ), in which case .||T ||A(N,M) is precisely the smallest positive number c for which this holds. Proof First, suppose that .T ∈ A (N, M). Let .x ∈ G (σ ) and .t > 0 be given. If .y ∈ L1 (σ ) and .z ∈ L∞ (σ ) are such that .x = y + z, then it follows from

222

3 Singular Value Functions

Theorem 3.9.16 that f t ( ) . μ (s; T x) ds ≤ ||T y||1 + t ||T z||∞ ≤ ||T ||A(N,M) ||y||1 + t ||z||∞ . 0

Applying the same theorem once more, it follows that f .

0

t

f μ (s; T x) ds ≤ ||T ||A(N,M)

t

μ (s; x) ds,

t > 0,

0

that is, .T x ≺≺ ||T ||A(N,M) x. For the proof of the converse implication, suppose that .0 ≤ c ∈ R is such that .T x ≺≺ cx for all .x ∈ G (σ ). If .x ∈ L1 (σ ), then, by Remark 3.9.3, .T x ∈ L1 (τ ) and .||T x||1 ≤ c ||x||1 . Similarly, if .x ∈ N, then .T x ∈ M and .||T x||∞ ≤ c ||x||∞ . Hence, .T ∈ A (N, M) and .||T ||A(N,M) ≤ c. The proof is complete. u n If .T ∈ A (N, M), then .T : G (σ ) → G (τ ) is a bounded linear operator and, by Proposition 3.10.2 (ii), .L∞ (σ ) is dense in .G (σ ). Therefore, the operator T is completely determined by its restriction .T∞ to .L∞ (σ ). In other words, the map .o : T − | → T∞ is a linear injection from .A (N, M) into .L (L∞ (σ ) , L∞ (τ )). The next lemma identifies the image of the map .o. Lemma 3.10.4 An operator .S ∈ L (L∞ (σ ) , L∞ (τ )) belongs to .o (A (N, M)) if and only if there exists a constant .c ≥ 0 such that .

|τ (ySx)| ≤ c ||x||∞ ||y||1 ,

x ∈ L∞ (σ ) , y ∈ L1 (τ ) ,

(3.71)

x ∈ H (σ ) , y ∈ H (τ ) ,

(3.72)

and .

|τ (ySx)| ≤ c ||x||1 ||y||∞ ,

|| || in which case .||o−1 (S)||A(N,M) is equal to the smallest constant .c for which these estimates hold. Proof It should be observed that, by Theorem 3.4.24 (ii), estimate (3.71) simply states that .||Sx||∞ ≤ c ||x||∞ , .x ∈ L∞ (τ ), that is, .||S||∞ ≤ c. First suppose that .S = T∞ = o (T ) for some .T ∈ A (N, M). Since .||T∞ ||∞ ≤ ||T ||A(N,M) , it is evident that (3.71) is satisfied with .c = ||T ||A(N,M) . If .x ∈ H (σ ) and .y ∈ H (τ ), then .

|τ (ySx)| = |τ (yT1 x)| ≤ ||T1 x||1 ||y||∞ ≤ ||T1 ||1 ||x||1 ||y||∞ ,

which implies that (3.72) holds with .c = ||T ||A(N,M) .

223

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

Now assume that .S ∈ L (L∞ (σ ) , L∞ (τ )) satisfies (3.71) and (3.72). It follows from (3.72) that .Sx ∈ L1 (τ ) whenever .x ∈ H (σ ) and .

||Sx||1 ≤ c ||x||1 ,

x ∈ H (σ ) .

(3.73)

Indeed, let .x ∈ H (σ ) be given with polar decomposition .Sx = v |Sx|. Since .τ is semi-finite, there exists a net .{pα } in .P (M) such that .pα ↑α 1 and .τ (pα ) < ∞ for all .α. Using Proposition 3.4.32, it follows from (3.72) that ) ( | ( )| τ |Sx|1/2 pα |Sx|1/2 = τ (pα |Sx|) = |τ pα v ∗ Sx | ≤ c ||x||1

.

for all .α. Since .|Sx|1/2 pα |Sx|1/2 ↑α |Sx| in .M, the claim follows from the normality of the trace. By Proposition 3.4.16, .H (σ ) is dense in .L1 (σ ), and hence, (3.73) implies that .S |H (σ ) has a unique bounded linear extension .S1 : L1 (σ ) → L1 (τ ) satisfying .||S1 ||1 ≤ c. Evidently, the operators S and .S1 coincide on .H (σ ), and so, the linear operator .T : G (σ ) → G (τ ) may be defined by setting T (x + y) = Sx + S1 y,

.

x ∈ L∞ (σ ) , y ∈ L1 (σ ) .

It is clear that .T ∈ A (N, M) with .||T ||A(N,M) = max (||S||∞ , ||S1 ||1 ) ≤ c and that .o (T ) = S. The proof is complete. u n Before proceeding, some notation and facts from Banach space theory should be recalled. Let X and Y be Banach spaces and .Y ∗ be the Banach dual of Y . The space of all bounded linear operators from X into Y is denoted by .L (X, Y ). If .x ∈ X and ∗ ∗ ∗ .y ∈ Y , then the linear functional .x ⊗ y on .L (X, Y ) is defined by setting .

< > < > T , x ⊗ y ∗ = T x, y ∗ ,

T ∈ L (X, Y ) .

If V is a linear subspace of X and W is a linear subspace of .Y ∗ , then the linear subspace of the dual of .L (X, Y ) generated by the functionals .x ⊗ y ∗ , .x ∈ V , .y ∗ ∈ W , is denoted by .V ⊗ W (and this subspace indeed is the algebraic tensor product of the vector spaces V and W ). The locally convex weak topology in .L (X, Y ) determined by .V ⊗ W is denoted by .σ (L (X, Y ) , V ⊗ W ). If .A is a linear subspace of .L (X, Y ), then the restriction of this topology to .A is denoted by .σ (A, V ⊗ W ). In particular, .σ (L (X, Y ∗ ) , X ⊗ Y ) is the weak topology in the space .L (X, Y ∗ ) determined by .X ⊗ Y (considering Y as a subspace of .Y ∗∗ ). The following result is well known. For the reader’s convenience, an indication of the proof is included. Proposition 3.10.5 If X and Y are Banach spaces, then the unit ball of .L (X, Y ∗ ) is compact with respect to .σ (L (X, Y ∗ ) , X ⊗ Y ).

224

3 Singular Value Functions

Proof Let .B (X, Y ) be the space of all continuous bilinear forms on .X×Y , equipped with the norm given by .

{ } ||ϕ||B(X,Y ) = sup |ϕ (x, y)| : x ∈ X, y ∈ Y, ||x||X , ||y||Y ≤ 1 ,

ϕ ∈ B (X, Y ) .

If .x ∈ X and .y ∈ Y , then the linear functional .x ⊗ y on .B (X, Y ) is defined by setting . = ϕ (x, y), .ϕ ∈ B (X, Y ). Let .X ⊗ Y be the linear subspace of the dual of .B (X, Y ), generated by the functionals .x ⊗ y, .x ∈ X, .y ∈ Y . It is a consequence of Tikhonov’s theorem that the unit ball in .B (X, Y ) is compact with respect to .σ (B (X, Y ) , X ⊗ Y ). For any .T ∈ L (X, Y ∗ ), let the bilinear form .ϕT ∈ B (X, Y ) be given by .ϕT (x, y) = , .x ∈ X, .y ∈ Y . It is readily verified that the map .T |−→ ϕT is a linear isometry from .L (X, Y ∗ ) onto .B (X, Y ), which is also a homeomorphism with respect to the topologies .σ (L (X, Y ∗ ) , X ⊗ Y ) and .σ (B (X, Y ) , X ⊗ Y ). u n This suffices for the proof. Recall from Theorem 3.4.24 (ii) that the von Neumann algebra .M = L∞ (τ ) may be identified with the Banach dual of .L1 (τ ), via trace duality. Corollary 3.10.6 The unit ball of .L (L∞ (σ ) , L∞ (τ )) is compact with respect to σ (L (N, M) , N ⊗ L1 (τ )).

.

Identifying .H (τ ) with a subspace of the dual space of .G (τ ), via the duality pairing given by (3.68), the space .A (N, M) may be equipped with the .σ (A (N, M) , G (σ ) ⊗ H (τ )) topology. Proposition 3.10.7 The unit ball .Σ (N, M) of .A (N, M) is compact with respect to .σ (A (N, M) , G (σ ) ⊗ H (τ )). Proof Let .o : A (N, M) → L (N, M) be the restriction map considered in Lemma 3.10.4. It will be shown first that .o (Σ (N, M)) is closed with respect to .σ (L (N, M) , N ⊗ L1 (τ )). Suppose that .{Sα } is a net in .o (Σ (N, M)) and that .S ∈ L (N, M) is such that .Sα → S with respect to .σ (L (N, M) , N ⊗ L1 (τ )). By Lemma 3.10.4, each .Sα satisfies inequalities (3.71) and (3.72) with .c = 1, which clearly implies that the same estimates hold for S. Therefore, it follows from the same lemma that S belongs to .o (Σ (N, M)). By Corollary 3.10.6, the unit ball of .L (N, M) is compact with respect to .σ (L (N, M) , N ⊗ L1 (τ )), and hence, the same holds for .o (Σ (N, M)). Evidently, this implies that .o (Σ (N, M)) is compact with respect to .σ (L (N, M) , N ⊗ H (τ )). It remains, therefore, to be observed that the map .o−1 : o (Σ (N, M)) → Σ (N, M) is continuous for the topologies determined by .N ⊗ H (τ ). This, however, follows by a routine argument, using that .Σ (N, M) is a bounded subset of .L (G (σ ) , G (τ )) (see (3.70)), that .N is dense in .G (σ ) (see Proposition 3.10.2 (ii)) and that the functionals on .G (τ ) induced by .H (τ ) are bounded (see (3.69)). This completes the proof. u n Corollary 3.10.8 For each .x ∈ G (σ ), the subset .{T x : T ∈ Σ (N, M)} of .G (τ ) is σ (G (τ ) , H (τ ))-compact.

.

225

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

Proof It suffices to observe that, for each .x ∈ G (σ ), the map .T |−→ T x, T ∈ Σ (N, M), is continuous with respect to .σ (A (N, M) , G (σ ) ⊗ H (τ )) and .σ (G (τ ) , H (τ )). u n .

To formulate the next result, some further notation is needed. The subset Σ (N, M)+ of .Σ (N, M) is defined by setting

.

Σ (N, M)+ = {T ∈ Σ (N, M) : T x ≥ 0 ∀ 0 ≤ x ∈ G (σ )} .

.

Σn Lemma 3.10.9 Suppose that .a ∈ H (σ ) is given by .a = i=1 αi pi , where .p1 , . . . , pn ∈ P (N) with .pi pj = 0 whenever .i /= j , and .0 < αi ∈ R (.i = 1, . . . , m) are such that .α1 >Σα2 > · · · > αn > 0. Suppose, furthermore, m that .b ∈ H (τ ) is given by .b = j =1 βj qj , where .q1 , . . . , qm ∈ P (M) with .qj qk = 0 whenever .j /= k, and .0 < βj ∈ R (.j = 1, . . . , m) are such that + .β1 > β2 > · · · > βm > 0. There exists .T ∈ Σ (N, M) such that f

∞

.

μ (t; a) μ (t; b) dt = τ (bT a) .

(3.74)

0

Proof It follows from Example 3.2.2 (i) (in particular, formula (3.5)) that the singular value functions .μ (a) and .μ (b) are given by μ (a) =

n Σ

.

αi χ[ξi−1 ,ξi ) ,

μ (b) =

i=1

m Σ j =1

βj χ[ηj −1 ,ηj ) ,

respectively, where .ξ0 = η0 = 0 and ξi =

i Σ

.

σ (pk ) ,

1 ≤ i ≤ n,

τ (qk ) ,

1 ≤ j ≤ m.

k=1

ηj =

j Σ k=1

Define the “conditional expectation type” operator .T : G (σ ) → G (τ ) by setting ) ( n )) ( [ m Σ Σ m [ξi−1 , ξi ) ∩ ηj −1 , ηj ( ) .T x = σ (xpi ) qj , σ (pi ) τ qj j =1 i=1

x ∈ G (σ ) .

It is easy to see that .||T x||1 ≤ ||x||1 for all .x ∈ L1 (σ ) and that .||T x||∞ ≤ ||x||∞ for all .x ∈ L∞ (σ ). It is also evident that .T x ≥ 0 whenever .0 ≤ x ∈ G (σ ). Consequently, .T ∈ Σ (N, M)+ . A direct calculation shows that T satisfies (3.74). The details are left to the reader. u n

226

3 Singular Value Functions

In the proof of the next theorem, the following special case of a classical inequality, due to Hardy, will be used. For the sake of completeness, an indication of the proof is included. Lemma 3.10.10 If .f, g, h : (0, ∞) → [0, ∞) are decreasing functions such that f ≺≺ g, then

.

f

∞

.

f

∞

f (s) h (s) ds ≤

g (s) h (s) ds.

0

0

Proof If h is a step function of the form h=

n Σ

.

j =1

αj χ[0,ρj ) ,

where .0 < ρ1 < · · · < ρn in .[0, ∞) and .0 < αj ∈ R for all j , then the result of the lemma follows immediately from the definition of submajorization. Therefore, it only needs to be observed that any positive decreasing function h on .(0, ∞) is the u n a.e. limit of an increasing sequence of step functions of this type. Proposition 3.10.11 If .0 ≤ a ∈ G (σ ) and .0 ≤ b ∈ H (τ ), then f

∞

.

{ } μ (t; a) μ (t; b) dt = sup τ (bT a) : T ∈ Σ (N, M)+ .

0

Proof If .T ∈ Σ (N, M)+ , then it follows from Proposition 3.10.3 that .T a ≺≺ a, and since the function .μ (b) is decreasing, Lemma 3.10.10 implies that f .

∞

f

∞

μ (t; T a) μ (t; b) dt ≤

μ (t; a) μ (t; b) dt.

0

0

Furthermore, it follows from (3.23) in Corollary 3.4.6 and Theorem 3.4.29 that f τ (bT a) ≤ τ (|bT a|) ≤

∞

μ (t; T a) μ (t; b) dt.

.

0

This shows that {

.

sup τ (bT a) : T ∈ Σ (N, M)

+

}

f

∞

≤

μ (t; a) μ (t; b) dt. 0

For the proof of the converse inequality, first assume, in addition, that .0 ≤ a ∈ H (σ ). It follows from the spectral theorem that there exist sequences .{an }∞ n=1 in ∞ .H (σ ) and .{bn } in .H (τ ), each .an and .bn of the form given in Lemma 3.10.9, n=1 for which .an ↑ a and .bn ↑ b hold in .N and .M, respectively. By Lemma 3.10.9, for

227

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

each n there exists .Tn ∈ Σ (N, M)+ such that f

∞

.

μ (t; an ) μ (t; bn ) dt = τ (bn Tn an ) .

0

Next, observe that .τ (bn Tn an ) ≤ τ (bTn a). Indeed, since .0 ≤ Tn an ≤ Tn a, it follows from Proposition 3.4.32 that ) ( ) ( 1/2 1/2 1/2 1/2 ≤ τ bn (Tn a) bn τ (bn Tn an ) = τ bn (Tn an ) bn ) ( = τ (bn Tn a) = τ (Tn a)1/2 bn (Tn a)1/2 ) ( ≤ τ (Tn a)1/2 b (Tn a)1/2 = τ (bTn a) .

.

Consequently, f .

∞

μ (t; an ) μ (t; bn ) dt ≤ τ (bTn a)

0

for all n. Furthermore, by Proposition 3.2.14 (i), .μ (t; an ) ↑ μ (t; a) and .μ (t; bn ) ↑ μ (t; b) a.e. on .[0, ∞), and so, f

∞

.

f

∞

μ (t; an ) μ (t; bn ) dt ↑n

μ (t; a) μ (t; b) dt.

0

0

This shows that f ∞ } { . μ (t; a) μ (t; b) dt = sup τ (bT a) : T ∈ Σ (N, M)+

(3.75)

0

holds for all .0 ≤ a ∈ H (σ ) and .0 ≤ b ∈ H (τ ). Assume now that .0 ≤ a ∈ G (σ ) is arbitrary. { }Since .F(σ ) ⊆ H (σ ), it follows from Proposition 2.3.12 that there exists a net . aβ in .H (σ ) such that .0 ≤ aβ ↑ a in .Sh (σ ). If .T ∈ Σ (N, M)+ , then it follows from Proposition 3.4.32 that ) ( ) ( ) ( ) ( τ bT aβ = τ b1/2 T aβ b1/2 ≤ τ b1/2 (T a) b1/2

.

≤ τ (bT a) . Using (3.75), this implies that .

} ) } { { ( sup τ (bT a) : T ∈ Σ (N, M)+ ≥ sup sup τ bT aβ : T ∈ Σ (N, M)+ β

f

= sup β

0

∞

) ( μ t; aβ μ (t; b) dt.

228

3 Singular Value Functions

( ) Furthermore, by Proposition 3.2.14 (i), .μ aβ ↑ μ (a) in the space .L0 [0, ∞), and hence, f

∞

.

( ) μ t; aβ μ (t; b) dt ↑β

f

∞

μ (t; a) μ (t; b) dt.

0

0

Consequently, {

.

sup τ (bT a) : T ∈ Σ (N, M)

+

}

f

∞

≥

μ (t; a) μ (t; b) dt, 0

u n

which completes the proof of the theorem. Corollary 3.10.12 If .x ∈ G (σ ) and .y ∈ H (τ ), then f .

∞

μ (t; x) μ (t; y) dt = sup {Reτ (yT x) : T ∈ Σ (N, M)}

0

= sup {|τ (yT x)| : T ∈ Σ (N, M)} = sup {τ (|yT x|) : T ∈ Σ (N, M)} . Proof Let .x ∈ G (σ ) and .y ∈ H (τ ) be fixed, and denote the above suprema by A, B, and C, respectively. Since Reτ (yT x) ≤ |τ (yT x)| ≤ τ (|yT x|)

.

for all .x ∈ G (σ ) and .y ∈ H (τ ), it is clear that .A ≤ B ≤ C. If .T ∈ Σ (N, M), then it follows from Proposition 3.10.3 that .T x ≺≺ x, and so, using Theorem 3.4.29 and Lemma 3.10.10, this implies that f

∞

τ (|yT x|) ≤

.

f

∞

μ (t; T x) μ (t; y) dt ≤

0

μ (t; x) μ (t; y) dt, 0

which shows that f C≤

∞

μ (t; x) μ (t; y) dt.

.

0

Let .x = v |x| and .y = w |y| be the polar decompositions of x and y, respectively. If .T ∈ Σ (N, M)+ , then the operator .T˜ ∈ Σ (N, M), defined by setting ( ) T˜ z = T v ∗ z · w ∗ ,

.

z ∈ G (σ ) ,

satisfies ( ) ) ( ( ) τ y T˜ x = τ yT (|x|) w ∗ = τ w ∗ yT (|x|) = τ (|y| T (|x|)) .

.

229

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

Using Proposition 3.10.11, this implies that } { A ≥ sup τ (|y| T (|x|)) : T ∈ Σ (N, M)+ f ∞ = μ (t; x) μ (t; y) dt,

.

0

u n

which completes the proof. For each .x ∈ S (σ ), the subset .oM (x) of .S (τ ) is defined by oM (x) = {y ∈ S (τ ) : y ≺≺ x} .

.

Note that it follows from Theorem 3.9.9 that the set .oM (x) is absolutely convex. Furthermore, if .x ∈ G (σ ), then it follows from (3.67) that .oM (x) ⊆ G (τ ). The following theorem is one of the main results of the present section. Theorem 3.10.13 If .x ∈ G (σ ), then .oM (x) = {T x : T ∈ Σ (N, M)}. Proof Let .x ∈ G (σ ) be fixed. Defining .K = {T x : T ∈ Σ (N, M)}, it follows from Proposition 3.10.3 that .K ⊆ oM (x), and it is clear that K is absolutely convex. Furthermore, by Corollary 3.10.8, the set K is compact with respect to .σ (G (τ ) , H (τ )). Since .H (τ ) separates the points of .G (τ ) (see the discussion following Proposition 3.10.2), this implies, in particular, that K is .σ (G (τ ) , H (τ ))closed. If there exists an element .z ∈ oM (x) \ K, then it follows from the Hahn–Banach separation theorem that there exists .y ∈ H (τ ) such that τ (yz) > 1 and

.

|τ (yT x)| ≤ 1, T ∈ Σ (N, M) .

However, using Lemma 3.10.10, it follows from Corollaries 3.4.6 and 3.10.12, Theorem 3.4.29 that f ∞ f ∞ .τ (yz) ≤ τ (|yz|) ≤ μ (t; y) μ (t; z) dt ≤ μ (t; y) μ (t; x) dt 0

0

= sup {|τ (yT x)| : T ∈ Σ (N, M)} ≤ 1, which is a contradiction. Consequently, .K = oM (x).

u n

Combination of Corollary 3.10.8 and Theorem 3.10.13 immediately yields the following result. Corollary 3.10.14 If .x ∈ G (σ ), then .oM (x) is an absolutely convex and σ (G (τ ) , H (τ ))-compact subset of .G (τ ).

.

A version of Theorem 3.10.13, which involves positive elements only, will be discussed next. The following simple observation will be useful. Lemma 3.10.15 If .x ∈ G (τ ) satisfies .τ (xy) ≥ 0 for all .y ∈ H (τ )+ , then .x ≥ 0.

230

3 Singular Value Functions

Proof The assumption on x implies, in particular, that .τ (xy) = τ (xy) = τ (yx ∗ ) = τ (x ∗ y) for all .y ∈ H (τ )+ . Since .H (τ ) separates the points of .G (τ ), it follows that .x ∗ = x. Set .e = ex (−∞, 0). If .y ∈ H (τ )+ , then .eye ∈ H (τ )+ , and so, ( ) 0 ≤ τ x − y = −τ (exey) = −τ (xeye) ≤ 0.

.

( ) This shows that .τ x − y = 0 for all .y ∈ H (τ )+ , from which it follows that .x − = 0, that is, .x ≥ 0. n u It follows immediately from Lemma 3.10.15 that the positive cone .G (τ )+ is closed with respect to .σ (G (τ ) , H (τ )), and consequently, .Σ (N, M)+ is closed in .Σ (N, M) with respect to .σ (A (N, M) , G (σ ) ⊗ H (τ )). Therefore, Proposition 3.10.7 implies that .Σ (N, M)+ is also compact with respect to this topology. Consequently, by the same{ argument as was used} to obtain Corollary 3.10.8, for each .x ∈ G (σ )+ the set . T x : T ∈ Σ (N, M)+ is a .σ (G (τ ) , H (τ ))-compact subset of .G (τ )+ . For .x ∈ G (σ )+ , the subset .oM (x)+ of .G (τ )+ is defined by { } oM (x)+ = y ∈ G (τ )+ : y ≺≺ x .

.

Proposition 3.10.16 If .x ∈ G (σ )+ , then: { } (i) .oM (x)+ = T x : T ∈ Σ (N, M)+ . (ii) .oM (x)+ is a convex .σ (G (τ ) , H (τ ))-compact subset of .G (τ )+ . Proof In view{of the above observations, only assertion (i) needs to be proved. } Defining .K = T x : T ∈ Σ (N, M)+ , it is clear that .K ⊆ oM (x)+ . Suppose that there exists an element .z ∈ oM (x)+ \ K. Since K is convex and .σ (G (τ ) , H (τ ))compact, it follows from the Hahn–Banach separation theorem that there exists .y ∈ H (τ ) such that Reτ (yz) > 1

.

and

Reτ (yT x) ≤ 1,

T ∈ Σ (N, M)+ .

Since .τ (yz) = τ (zy ∗ ) = τ (y ∗ z), it is clear that .Reτ (yz) = τ ((Rey) z), and similarly, .Reτ (yT x) = τ ((Rey) T x) for all .T ∈ Σ (N, M)+ . Therefore, replacing y by .Rey, it may be assumed that .y ∗ = y and τ (yz) > 1 and

.

τ (yT x) ≤ 1,

T ∈ Σ (N, M)+ .

(3.76)

( ) Since .y + ≥ y and .z ≥ 0, it is also clear that .τ y + z > 1. Furthermore, if .T ∈ ( ) ( ) Σ (N, M)+ , then .τ y + T x = τ y T˜ x ≤ 1, where .T˜ ∈ Σ (N, M)+ is defined by setting .T˜ w = ey [0, ∞) T w, .w ∈ G (σ ). Consequently, without loss of generality, it may be assumed that (3.76) holds for some .y ∈ H (τ )+ . It now follows from

231

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

Theorem 3.4.29 and Proposition 3.10.11 that f

∞

τ (yz) ≤

.

f μ (t; y) μ (t; z) dt ≤

0

∞

μ (t; y) μ (t; x) dt 0

} { = sup τ (yT x) : T ∈ Σ (N, M)+ ≤ 1, which is a contradiction. Therefore, .K = oM (x)+ and the proof is complete.

u n

The following result is a useful consequence of Theorem 3.10.13 and Proposition 3.10.16 (i). Proposition 3.10.17 If .y ∈ G (τ ) and .x1 , x2 ∈ G (σ ) are such that .y ≺≺ x1 + x2 , then there exist .y1 , y2 ∈ G (τ ) such that .y = y1 + y2 and .yi ≺≺ xi (.i = 1, 2). Moreover, if y is positive, then .y1 and .y2 can be taken to be positive. Proof It follows from Theorem 3.9.9 that .μ (x1 + x2 ) ≺≺ μ (x1 ) + μ (x2 ), and so, y ≺≺ μ (x1 ) + μ (x2 ). Therefore, it follows from Theorem 3.10.13, applied to the von Neumann algebras .L∞ [0, ∞) and .M, that there exists .T ∈ Σ (L∞ [0, ∞) , M) such that

.

T (μ (x1 ) + μ (x2 )) = y.

.

Setting .y1 = T (μ (x1 )) and .y2 = T (μ (x2 )), it is clear that .y = y1 + y2 and yi ≺≺ μ (xi ), that is, .yi ≺≺ xi (.i = 1, 2). If .y ≥ 0, then it follows from Proposition 3.10.16 (i) that the operator T may be taken from .Σ (L∞ [0, ∞) , M)+ , in which case .y1 , y2 ≥ 0. u n

.

It should be observed that via Proposition 3.10.16 (i), the result of Proposition 3.10.11 may also be formulated as f

∞

.

{ } μ (t; x) μ (t; y) dt = sup τ (zy) : z ∈ G (τ )+ , z ≺≺ x

(3.77)

0

for all .x ∈ G (σ )+ and .y ∈ H (τ )+ . Similarly, using Theorem 3.10.13, the result of Corollary 3.10.12 may be formulated as f

∞

.

μ (t; x) μ (t; y) dt = sup {τ (|zy|) : z ∈ G (τ ) , z ≺≺ x}

0

for all .x ∈ G (σ ) and .y ∈ H (τ ). The following proposition extends these results. Proposition 3.10.18 (i) If .x ∈ S (σ ) and .y ∈ G (τ )+ , then f .

0

∞

{ } μ (t; x) μ (t; y) dt = sup τ (zy) : z ∈ H (τ )+ , z ≺≺ x .

(3.78)

232

3 Singular Value Functions

(ii) If .x ∈ S (σ ) and .y ∈ G (τ ), then f

∞

.

μ (t; x) μ (t; y) dt = sup {|τ (zy)| : z ∈ H (τ ) , z ≺≺ x} .

0

(iii) If .x ∈ S (σ ) and .y ∈ S (τ ), then f

∞

.

μ (t; x) μ (t; y) dt = sup {τ (|zy|) : z ∈ H (τ ) , z ≺≺ x} .

0

Proof (i) If .y ∈ G (τ )+ , .x ∈ S (σ ), and .z ∈ H (τ )+ satisfies .z ≺≺ x, then it follows from Remark 3.4.7, Theorem 3.4.29, and Lemma 3.10.10 that f

∞

τ (zy) ≤ τ (|zy|) ≤

.

f μ (t; z) μ (t; y) dt ≤

0

∞

μ (t; x) μ (t; y) dt. 0

Therefore, it is sufficient to prove that f

∞

.

{ } μ (t; x) μ (t; y) dt ≤ sup τ (zy) : z ∈ H (τ )+ , z ≺≺ x

0

holds for all .y ∈ G (τ )+ and .x ∈ S (σ ). Evidently, it may be assumed that + + + + .x ∈ S (σ ) . Assume first that .x ∈ G (σ ) and .y ∈ H (τ ) . If .z ∈ G (τ ) , then it follows from Proposition 2.3.12 that there exists an upward directed system + .{zα } in .H (τ ) such that .0 ≤ zα ↑α z. By Proposition 2.2.25 (iii), this implies 1/2 that .0 ≤ y zα y 1/2 ↑α y 1/2 zy 1/2 , and so it follows from Proposition 3.4.32 and the normality of the trace that ) ) ( ( τ (zα y) = τ y 1/2 zα y 1/2 ↑α τ y 1/2 zy 1/2 = τ (zy) .

.

In combination with (3.77), this shows that .

{ } sup τ (zy) : z ∈ H (τ )+ , z ≺≺ x ≥

f

∞

μ (t; x) μ (t; y) dt, 0

whenever .x ∈ G (σ )+ and .y ∈ H (τ )+ . { } Suppose next that .x ∈ G (σ )+ and .y ∈ G (τ )+ and let . yβ be an upward directed system in .H (τ )+ such that .0 ≤ yβ ↑ y. By Proposition 3.2.14 (i),

233

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

( ) this implies that .μ yβ ↑ μ (y) in .L0 [0, ∞), and so, f

∞

.

f

∞

μ (t; x) μ (t; y) dt = sup

0

β

( ) μ (t; x) μ t; yβ dt

0

{ ( ) } ≤ sup sup τ zyβ : z ∈ H (τ )+ , z ≺≺ x β

{ } ≤ sup τ (zy) : z ∈ H (τ )+ , z ≺≺ x , ( ) where it is used that .0 ≤ τ zyβ ≤ τ (zy) holds for all .z ∈ H (τ )+ and all .β. Finally, if .x ∈ S (σ )+ and .y ∈ G (τ )+ , then there exists a sequence .{xn }∞ n=1 in .G (σ )+ such that .0 ≤ xn ↑ x. Since .μ (xn ) ↑ μ (x) in .L0 [0, ∞), it follows that f ∞ f ∞ . μ (t; x) μ (t; y) dt = sup μ (t; xn ) μ (t; y) dt n

0

0

} { ≤ sup sup τ (zy) : z ∈ H (τ )+ , z ≺≺ xn n

{ } ≤ sup τ (zy) : z ∈ H (τ )+ , z ≺≺ x . This completes the proof of (i). (ii) Suppose that .x ∈ S (σ ) and .y ∈ G (τ ). As in the proof of (i), it is sufficient to show that f ∞ . μ (t; x) μ (t; y) dt ≤ sup {|τ (zy)| : z ∈ H (τ ) , z ≺≺ x} . (3.79) 0

It follows from (3.78) that f

∞

.

{ } μ (t; x) μ (t; y) dt = sup τ (z |y|) : z ∈ H (τ )+ , z ≺≺ x .

0

If .z ∈ H (τ )+ satisfies .z ≺≺ x and .y = v |y| is the polar decomposition of ∗ ∗ ∗ .y ∈ G (τ ), then .τ (z |y|) = τ (zv y) = |τ (zv y)| and .zv ≺≺ x. It is now clear that (3.79) holds. (iii) Suppose now that .x ∈ S (σ ) and .y ∈ S (τ ). Again, it is sufficient to show that f

∞

.

μ (t; x) μ (t; y) dt ≤ sup {τ (|zy|) : z ∈ H (τ ) , z ≺≺ x} ,

0

and so, it may be assumed that .sup {τ (|zy|) : z ∈ H (τ ) , z ≺≺ x} < ∞. This implies, in particular, that .zy ∈ L1 (τ ) for all .z ∈ H (τ ) satisfying .z ≺≺ x. + + Let .{yn }∞ n=1 be a sequence in .G (τ ) such that .0 ≤ yn ↑ |y|. If .z ∈ H (τ ) satisfies .z ≺≺ x, then .z |y| ∈ L1 (τ ), and so, .τ (zyn ) ≤ τ (z |y|) for all n. Since

234

3 Singular Value Functions

μ (yn ) ↑ μ (y), it follows from (3.78) that

.

f

∞

.

f μ (t; x) μ (t; y) dt = sup n

0

∞

μ (t; x) μ (t; yn ) dt 0

{ } = sup sup τ (zyn ) : z ∈ H (τ )+ , z ≺≺ x n

{ } ≤ sup τ (z |y|) : z ∈ H (τ )+ , z ≺≺ x . Furthermore, if .z ∈ H (τ )+ is such that .z ≺≺ x and if .y = v |y| is the polar decomposition of y, then .τ (z |y|) = τ (zv ∗ y) ≤ τ (|y (zv ∗ )|) and .zv ∗ ≺≺ x. This implies that f

∞

.

μ (t; x) μ (t; y) dt ≤ sup {τ (|zy|) : z ∈ H (τ ) , z ≺≺ x} ,

0

which completes the proof of the proposition. u n Remark 3.10.19 An inspection of the proof of Proposition 3.10.18 (i) shows that in formula (3.78) the set .H (τ )+ may be replaced by the smaller set .F (τ )+ . In fact, Proposition 2.3.12 actually implies that for any .z ∈ G (τ )+ there exists an upward directed system .{zα } in .F (τ )+ such that .0 ≤ zα ↑α z. Using such a net .{zα } in the proof of (i) yields the desired strengthening. Consequently, also in parts (ii) and (iii) the space .H (τ ) may be replaced by .F (τ ). For later reference, the latter observation is formulated as a corollary. Corollary 3.10.20 If .x ∈ S (σ ) and .y ∈ S (τ ), then f .

∞

μ (t; x) μ (t; y) dt = sup {τ (|zy|) : z ∈ F (τ ) , z ≺≺ x} .

0

If the von Neumann algebra is non-atomic, then the above results may be sharpened, as will be shown next. Theorem 3.10.21 If the von Neumann algebra .M is non-atomic, then f

∞

.

μ (t; x) μ (t; y) dt = sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x)}

(3.80)

0

for all .x, y ∈ S (τ ). Proof If .z ∈ S (τ ) satisfies .μ (z) ≤ μ (x), then it follows from Theorem 3.4.29 that f

∞

τ (|zy|) ≤

.

0

f μ (t; z) μ (t; y) dt ≤

∞

μ (t; x) μ (t; y) dt, 0

235

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

and so, it is sufficient to show that f

∞

.

μ (t; x) μ (t; y) dt ≤ sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x)} .

(3.81)

0

For this purpose, it may be assumed that .

sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x)} < ∞,

which implies in particular that zy ∈ L1 (τ )

.

for all z ∈ S (τ ) satisfying μ (z) ≤ μ (x) .

(3.82)

Furthermore, both the right and left hand sides of (3.81) depend only on .|x|, and so, it may be assumed, in addition, that .x ∈ S (τ )+ . Inequality (3.81) will now be proved in a number of steps. Suppose first x=

n Σ

.

(3.83)

αj pj ,

j =1

where .0 < αj ∈ R, .1 ≤ j ≤ n, and .p1 < p2 < · · · < pn in .P (M) such that τ (pn ) < ∞. As observed in Example 3.2.2 (i), the singular value function .μ (x) of x is given by

.

μ (x) =

n Σ

.

j =1

αj χ[0,τ (pj )) ,

and hence, f

∞

.

μ (t; x) μ (t; y) dt =

0

n Σ

f τ (pj ) αj

j =1

Given .δ > 0, define .ε = δτ (pn )−1

(Σ n

μ (t; y) dt. 0

)−1

. It follows from Lemmas 3.7.8 ( ) and that there exist projections .e1 < e2 · · · < en in .P (M) such that .τ ej = ( 3.7.10 ) τ pj and f τ (pj ) .

0

j =1 αj

) ( ) ( μ (t; y) dt ≤ τ ej |y| ej + ετ pj ,

1 ≤ j ≤ n.

236

3 Singular Value Functions

Defining .x˜ ∈ S (τ ) by setting x˜ =

n Σ

.

αj ej ,

j =1

( ) ( ) it is clear that .μ (x) ˜ = μ (x). Furthermore, since .μ ej = μ pj ≤ αj−1 μ (x), it ) ) ( ( follows from (3.82) that .ej |y| ∈ L1 (τ ), and so, .τ ej |y| ej = τ ej |y| for all j . Consequently, n Σ .

j =1

f τ (pj ) αj

μ (t; y) dt ≤

0

n Σ

n Σ ) ( ( ) αj τ ej |y| + ε αj τ pj

j =1

j =1

) ≤ τ (x˜ |y|) + δ = τ xv ˜ ∗ y + δ, (

where .y = v |y| is the polar decomposition of y. Since .μ (xv ˜ ∗ ) ≤ μ (x) and ˜ ∗ y|), this shows that τ (xv ˜ ∗ y) ≤ τ (|xv

.

f

∞

.

μ (t; x) μ (t; y) dt ≤ sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x)} + δ.

0

This holds for all .δ > 0, from which it follows that (3.81) holds whenever x is of the form given by (3.83). Suppose next that .0 ≤ x ∈ L1 (τ ) ∩ L∞ (τ ). By the spectral theorem, there exists a sequence .{xn }∞ n=1 such that .0 ≤ xn ↑ x, where each .xn is of the form given by (3.83). Since .0 ≤ xn ↑ x implies that .μ (xn ) ↑ μ (x), it follows from the above that f ∞ f ∞ . μ (t; x) μ (t; y) dt = sup μ (t; xn ) μ (t; y) dt n

0

0

= sup sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (xn )} n

≤ sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x)} . Finally, suppose that .x ∈ S (τ )+ is arbitrary. Let .{xα } be an upward directed system in .L1 (τ ) ∩ L∞ (τ ) such that .0 ≤ xα ↑α x in .S (τ ). By Proposition 3.2.14 (i), .μ (xα ) ↑α μ (x) in .L0 (0, ∞), and hence, f

∞

.

0

f μ (t; x) μ (t; y) dt = sup α

∞

μ (t; xα ) μ (t; y) dt 0

≤ sup sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (xα )} α

≤ sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x)} , which establishes the result of the theorem.

u n

3.10 Contractions for the Couple (L1 (τ ) , L∞ (τ ))

237

Remark 3.10.22 Suppose that the von Neumann algebra is atomic and that all minimal projections have trace equal to one. Using the observations made in Remark 3.7.9, it is not difficult to show that the result of the above theorem is also valid in this situation (and actually, the proof is somewhat simpler, since .S (τ ) = M in this case). The details are left to the reader. Remark 3.10.23 Suppose that the von Neumann algebra .M is either non-atomic or is atomic and all minimal projections have equal trace. If .x, y ∈ S (τ ), then formula (3.80) may be sharpened to f

∞

.

μ (t; x) μ (t; y) dt = sup {τ (|zy|) : z ∈ F (τ ) , μ (z) ≤ μ (x)} .

0

Indeed, denoting the above right hand suprema by A, it is clearly sufficient to show that .

sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x)} ≤ A.

For this purpose, it may be assumed that .A < ∞. Suppose that .z ∈ S (τ ) satisfies μ (z) ≤ μ (x). { }It follows from Proposition 2.3.12 that there exists an upward directed net . aβ in .F (τ )+ such that .0 ≤ aβ ↑ |z|. By Proposition 2.7.6 (v), this implies that .aβ → |z| locally in measure. If .z = v |z| is the polar decomposition of z, then it follows from Proposition 2.7.5 that .vaβ y → zy locally in measure. Furthermore, since .vaβ ∈ F (τ ) and

.

) ( ) ( μ vaβ ≤ μ aβ ≤ μ (z) ≤ μ (x) ,

.

|) (| it is clear that .τ |vaβ y | ≤ A for all .β. Hence, Proposition 3.4.19 implies that .τ (|zy|) ≤ A. This establishes the claimed result. References: [19, 45, 56, 63, 81, 95, 138].

Chapter 4

Symmetric Spaces of τ -Measurable Operators .

Abstract The notion of a bimodule of .τ -measurable operators is introduced and studied in detail. Special attention is given to the duality theory of normed bimodules of .τ -measurable operators. In particular, the Köthe dual of a normed bimodule is introduced and its properties presented. In the final two sections, the discussion is specialized to the case of symmetric spaces of .τ -measurable operators, with particular attention given to the special case that the underlying von Neumann algebra is either non-atomic or atomic with all minimal projections having equal trace.

4.1 Normed M-Bimodules of τ -Measurable Operators Let .M be a von Neumann algebra on a Hilbert space H , equipped with a semi-finite faithful normal trace .τ . As before, the .∗-algebra of all .τ -measurable operators is denoted by .S (τ ). Definition 4.1.1 A linear subspace E of .S (τ ) is called an .M-bimodule of .τ measurable operators (briefly, an .M-bimodule, if no confusion may arise) if .uxv ∈ E whenever .x ∈ E and .u, v ∈ M. If an .M-bimodule E is equipped with a (semi-) norm .‖·‖E , satisfying .

‖uxv‖E ≤ ‖u‖B(H ) ‖v‖B(H ) ‖x‖E ,

x ∈ E, u, v ∈ M,

(4.1)

then E is called a (semi-) normed .M-bimodule of .τ -measurable operators (briefly, a (semi-) normed .M -bimodule). A normed .M-bimodule, which is a Banach space, will also be called a Banach M-bimodule (of .τ -measurable operators).

.

Example 4.1.2 (a) The spaces .L∞ (τ ) = M, .L1 (τ ), .L1 (τ ) ∩ L∞ (τ ) and .L1 (τ ) + L∞ (τ ) (see Sect. 3.10) are all examples of normed .M-bimodules, which are also Banach spaces. Also the space .F (τ ), as defined in Sect. 2.3, is an .M-bimodule. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. G. Dodds et al., Noncommutative Integration and Operator Theory, Progress in Mathematics 349, https://doi.org/10.1007/978-3-031-49654-7_4

239

240

4 Symmetric Spaces of .τ -Measurable Operators

(b) If .(X, ∑, ν) is a Maharam measure space and .M = L∞ (ν), then .E ⊆ S (ν) is an .L∞ (ν)-bimodule if and only if E is an ideal of .ν-measurable functions, that is, E is a linear subspace of .S (ν) with the additional property that .f ∈ E whenever .f ∈ S (ν) satisfies .|f | ≤ |g| for some .g ∈ E. A norm .‖·‖E on E satisfies (4.1) if and only if .‖·‖E is absolutely monotone, that is, .‖f ‖E ≤ ‖g‖E whenever .|f | ≤ |g| in E. In particular, Banach .M-modules are precisely the Banach function spaces E on .(X, ∑, ν) satisfying .E ⊆ S (ν). In the following proposition, some elementary properties of (normed) .Mbimodules are collected. Proposition 4.1.3 If .E ⊆ S (τ ) is an .M-bimodule, then the following assertions hold: (i) (ii) (iii) (iv)

If .x ∈ S (τ ), then .x ∈ E if and only if .|x| ∈ E. If .x ∈ S (τ ), then .x ∈ E if and only if .x ∗ ∈ E. If .x ∈ S (τ ) and .y ∈ E are such that .|x| ≤ |y|, then .x ∈ E. If .x, y ∈ S (τ ), then .xy ∈ E if and only if .|x| y ∈ E.

Moreover, if E is a semi-normed .M-bimodule, then (v) (vi) (vii) (viii)

For all .x ∈ E, .‖|x|‖E = ‖x‖E . For all .x ∈ E, .‖x ∗ ‖E = ‖x‖E . Whenever .|x| ≤ |y| in E, .‖x‖E ≤ ‖y‖E . If .x, y ∈ S (τ ) such that .xy ∈ E, .‖xy‖E = ‖|x| y‖E .

Proof (i) and (v). Let .x = v |x| be the polar decomposition of x. If .x ∈ E, then it follows from .|x| = v ∗ x that .|x| ∈ E, and .‖|x|‖E ≤ ‖x‖E if E is a semi-normed .M-bimodule. Conversely, if .|x| ∈ E, then .x = v |x| implies that .x ∈ E, and .‖x‖E ≤ ‖|x|‖E if E is semi-normed. (ii) and (vi). If .x ∈ E, with polar decomposition .x = v |x|, then it follows from (i) that .|x| ∈ E and so, .x ∗ = |x| v ∗ ∈ E, and .‖x ∗ ‖E ≤ ‖|x|‖E = ‖x‖E in case E is semi-normed. Applying the same argument with x replaced by .x ∗ makes statements (ii) and (vi) clear. (iii) and (vii). In view of (i), it may be assumed that .0 ≤ x ≤ y. It follows from Proposition 2.2.24 (ii) that there exists .z ∈ M such that .x 1/2 = zy 1/2 and 1/2 = y 1/2 z∗ , this implies that .x = zyz∗ . Consequently, .‖z‖B(H ) ≤ 1. Since .x ∗ .x ∈ E, and .‖x‖E ≤ ‖z‖B(H ) ‖z ‖B(H ) ‖y‖E ≤ ‖y‖E in case E is semi-normed. (iv) and (viii). Let .x = v |x| be the polar decomposition of x. If .xy ∈ E, then .|x| y = v ∗ xy ∈ E, and .‖|x| y‖E ≤ ‖v ∗ ‖B(H ) ‖xy‖E ≤ ‖xy‖E in case E is semi-normed. Similarly, if .|x| y ∈ E, then .xy = v |x| y ∈ E, and .‖xy‖E ≤ ‖v‖B(H ) ‖|x| y‖E ≤ ‖|x| y‖E in case E is semi-normed. ⨆ ⨅ Recall that (iii) of the above proposition is exactly the definition of an absolutely solid subspace, and hence this proposition states that any .M-bimodule is an absolutely solid subspace of .S (τ ). Suppose that .E ⊆ S (τ ) is an .M-bimodule. The real linear subspace of E consisting of all self-adjoint elements of E is denoted by .Eh . It is clear from part

4.1 Normed M-Bimodules of τ -Measurable Operators

241

(ii) of the above proposition that .Rex, .Imx ∈ Eh whenever .x ∈ E and so, .E = Eh ⊕ iEh . Furthermore, .E + = E ∩ S (τ )+ is the positive cone of .Eh with respect to the partial ordering induced by .Sh (τ ). If .a ∈ Eh , then .a + = aea [0, ∞) ∈ E + and similarly, .a − = −aea (−∞, 0) ∈ E + . This shows that the positive cone .E + generates .Eh . In particular, every element of E is a linear combination of at most four positive elements. If it is assumed, in addition that E is a normed .M-bimodule, then it is also clear that .‖Rex‖E ≤ ‖x‖E and .‖Imx‖E ≤ ‖x‖E for all .x ∈ E and that .Eh is closed. Furthermore, if .a ∈ Eh , then  + this implies, in particular,  − ‖a‖ ‖a‖ .a  ≤ and .a  ≤ E E. E E  If . aβ is an increasing net in .E + and .a ∈ E + , then .aβ ↑β a in .Eh if and only if .aβ ↑β a in .Sh (τ ). Indeed, it is evident that .aβ ↑β a in .Sh (τ ) implies that .aβ ↑β a in .Eh . Conversely, if .aβ ↑β a in .Eh , then Proposition 2.3.10 implies that there exists + .b ∈ S (τ ) such that .aβ ↑β b in .Sh (τ ). This implies that .0 ≤ b ≤ a and so, it follows from Proposition 4.1.3 (iii) that .b ∈ E + . Hence, .a = b, by which the claim is proved.   Furthermore, if . aβ is an increasing net in .E + and .b ∈ E + is such that + such that .a ↑ a in .E . Indeed, .aβ ≤ b for all .β, then there exists .a ∈ E β β h by Proposition 2.3.10, there exists .a ∈ S (τ )+ such that .aβ ↑β a in .Sh (τ ). Since + .0 ≤ a ≤ b, it follows from Proposition 4.1.3 (iii) that .a ∈ E , which proves the claim. Suppose that .E ⊆ S (τ ) is an .M-bimodule and that .p ∈ P (M).The reduced von Neumann algebra .Mp is equipped with the trace .τp given by .τp ap = τ (pap), + .a ∈ M . Defining pEp = {pxp : x ∈ E} ,

.

    Ep = xp ∈ S τp : x ∈ E ,

it follows immediately from Lemma 3.7.1 that the map .x │−→ xp is a .∗-preserving linear isomorphism from pEp onto .Ep . It is easily verified that .Ep is an .Mp bimodule of .τp -measurable operators. Moreover, if E is a normed .M-bimodule and for .xp ∈ Ep the norm is defined by setting .

  xp 

Ep

= ‖pxp‖E ,

x ∈ E,

then .Ep is a normed .Mp -bimodule and, evidently, the map .x │−→ xp is an isometry from pEp onto .Ep . If .E ⊆ S (τ ) is an .M-bimodule, then the subset .P (E) of .P (M) is defined by P (E) = {p ∈ P (M) : p ∈ E} .

.

Lemma 4.1.4 Given a (semi-normed) .M-bimodule .E ⊆ S (τ ), the following statements hold: (i) If .p ∈ P (E) and .q ∈ P (M) such that .p ∼ q, then .q ∈ P (E) (and .‖q‖E = ‖p‖E ).

242

4 Symmetric Spaces of .τ -Measurable Operators

(ii) If .p ∈ P (E) and .q ∈ P (M) such that .q ≲ p, then .q ∈ P (E) (and .‖q‖E ≤ ‖p‖E ). (iii) If .p, q ∈ P (E), then .p ∨ q ∈ P (E) (and .‖p ∨ q‖E ≤ ‖p‖E + ‖q‖E ). Proof (i) By definition, there exists a partial isometry .v ∈ M such that .p = v ∗ v and ∗ ∗ ∗ .q = vv . This implies that .p = v qv and .q = vpv . Hence, .q ∈ E. If E is semi-normed, then     ‖q‖E = vpv ∗ E ≤ ‖v‖B(H ) v ∗ B(H ) ‖p‖E ≤ ‖p‖E .

.

Similarly, .p = v ∗ qv implies that .‖p‖E ≤ ‖q‖E and so, .‖q‖E = ‖p‖E . (ii) If .q ≲ p, then there exists .q1 ∈ P (M) such that .q1 ≤ p and .q ∼ q1 . It follows from .q1 ≤ p that .q1 = pq1 ∈ P (E). Therefore, (i) implies that .q ∈ P (E). If E is semi-normed, then .‖q‖E = ‖q1 ‖E ≤ ‖p‖E . (iii) Since .p − p ∧ q ≤ p, it is clear that .p − p ∧ q ∈ P (E) (and .‖p − p ∧ q‖E ≤ ‖p‖E ). Using that p ∨ q − q ∼ p − p ∧ q,

.

it follows from (i) that .p ∨ q − q ∈ P (E) (and .‖p ∨ q − q‖E = ‖p − p ∧ q‖E ). Since .p ∨ q = (p ∨ q − q) + q, it is now clear that .p ∨ q ∈ P (E) and .

‖p ∨ q‖E ≤ ‖p ∨ q − q‖E + ‖q‖E ≤ ‖p‖E + ‖q‖E

if E is semi-normed. ⨆ ⨅ Note that it follows from Lemma 4.1.4 (iii) that the set .P (E) is a lattice ideal in .P (M) (that is, .p ∈ P (M), .q ∈ P (E) and .p ≤ q imply .p ∈ P (E), and .p ∨ q ∈ P (E) whenever .p, q ∈ P (E)). Definition 4.1.5 If .E ⊆ S (τ ) is an .M-bimodule, then the carrier projection .cE ∈ P (M) of E is defined by setting cE =

.



{p : p ∈ P (E)} .

Lemma 4.1.6 The carrier projection .cE of an .M-bimodule E is also given by 



{r (x) : x ∈ E}    {s (|x|) : x ∈ E} = s (a) : a ∈ E + . =

cE =

.

{s (x) : x ∈ E} =

4.1 Normed M-Bimodules of τ -Measurable Operators

243

Proof Since .x ∈ E if and only if .x ∗ ∈ E and .s (x) = s (|x|) = r (x ∗ ), it is clear that the four suprema on the right-hand side are equal. If .p ∈ P (E), then .p = s (p) and hence,   s (a) : a ∈ E + . .cE ≤ For the proof of the converse inequality, suppose that .a ∈ E + . If .ε > 0, then εea (ε, ∞) ≤ aea (ε, ∞)

.

and so, .ea (ε, ∞) ∈ P (E). This implies that .ea (ε, ∞) ≤ cE for all .ε > 0. Since ea (1/n, ∞) ↑n ea (0, ∞) = s (a) ,

.

it follows that .s (a) ≤ cE , which suffices to complete the proof.

⨆ ⨅

Corollary 4.1.7 If .E ⊆ S (τ ) is an .M-bimodule, then x = xcE = cE x = cE xcE ,

.

x ∈ E.

Moreover, .cE is a central projection in .M. Proof If .x ∈ E, then Lemma 4.1.6 implies that .x = xs (x) = xs (x) cE = xcE , and similarly, .x = cE x, which proves the first statement. To show that .cE is central, let .y ∈ M be given and suppose that .{pα } ⊆ P (E) so so satisfies .pα ↑α cE . This implies that .pα →α cE and so, .pα y →α cE y. Furthermore, so .pα y ∈ E and so, .pα y = pα ycE for all .α. Consequently, .pα y →α cE ycE and hence, so .cE y = cE ycE . Similarly, .ypα →α ycE and .ypα = cE ypα for all .α, from which it follows that .ycE = cE ycE . This shows that .cE y = ycE . ⨆ ⨅ The next lemma presents another characterization of the carrier projection .cE . Lemma 4.1.8 If .{0} /= E ⊆ S (τ ) is an .M-bimodule, then the carrier projection cE is the largest projection .c ∈ P (M) with the property that for any .p ∈ P (M) satisfying .0 < p ≤ c, there exists .q ∈ P (E) such that .0 < q ≤ p.

.

Proof It will be shown first that .cE has the stated property. To this end, suppose that p ∈ P (M) is such that .0 < p ≤ cE . If .pe = 0 for all .e ∈ P (E), then .e ≤ p⊥ for all .e ∈ P (E) and so, .cE ≤ p⊥ , which is absurd. Hence, there exists .e ∈ P (E) such that .pe /= 0. Setting .q = r (pe), it is clear that .0 < q ≤ p. Furthermore, since .q ∼ s (pe) ≤ e, it follows from Lemma 4.1.4 (ii) that .q ∈ P (E). Suppose now that .c ∈ P (M) is such that for every .p ∈ P (M) satisfying .0 < p ≤ c, there exists .q ∈ P (E) with .0 < q ≤ p. It has to be proved that .c ≤ cE . Defining .c' ∈ P (M) by setting .c' = sup {q ∈ P (E) : q ≤ c}, it is clear from the definition of .cE that .c' ≤ cE . If .c' < c, then it follows from the assumption on c that there exists .q ∈ P (E) such that .0 < q ≤ c − c' . Since .q ≤ c, the definition of .

244

4 Symmetric Spaces of .τ -Measurable Operators

c' implies that .q ≤ c' , which yields a contradiction. Consequently, .c = c' ≤ cE and this completes the proof. ⨆ ⨅

.

Corollary 4.1.9 If .E ⊆ S (τ ) is an .M-bimodule, then the following statements are equivalent: cE = 1. For every .0 /= p ∈ P (M), there exists .q ∈ P (E) such that .0 < q ≤ p. For every .0 /= p ∈ P (M), there exists .x ∈ E such that .pxp /= 0. For every .0 /= p ∈ P (M), there exists a system .{qα } of mutually orthogonal projections .qα ∈ P (E), satisfying .τ (qα ) < ∞ for all .α, such that . α qα = p. (v) For every .0 < a ∈ S (τ )+ , there exists .b ∈ E + such that .0 < b ≤a. (vi) For every .a ∈ S (τ )+ , there exists an upward directed system . aβ in .E + such that .0 ≤ aβ ↑ a.

(i) (ii) (iii) (iv)

.

Proof The equivalence of (i) and (ii) is clear from Lemma 4.1.8 and it is evident that (ii) implies (iii). If condition (iii) is satisfied and .0 /= p ∈ P (M), then there exists .a ∈ E + such that .0 < b = pap ∈ E. Let .ε > 0 be such that .eb (ε, ∞) /= 0. It follows from .0 < εeb (ε, ∞) ≤ beb (ε, ∞) that .eb (ε, ∞) ∈ P (E). Since b .e (ε, ∞) ≤ s (b) ≤ p, this shows that (iii) implies (ii). That (ii) implies (iv) follows from the semi-finiteness of the trace in combination with a standard argument involving Zorn’s lemma. If (iv) is satisfied, then it is clear that . {p : p ∈ P (E)} = 1, that is, .cE = 1. To show that (ii) implies (v), suppose that .0 < a ∈ S (τ )+ and let .ε > 0 be such that .ea (ε, ∞) /= 0. It follows from (ii) that there exists .q ∈ P (E) such that a a .0 < q ≤ e (ε, ∞). Consequently, .εq ∈ E and .0 < εq ≤ εe (ε, ∞) ≤ a. Hence, assertion (v) holds. It will be shown next that (v) implies (vi). Given .0 < a ∈ S (τ ), let .W be the collection of all indexedfamilies .{wα }α∈A in E satisfying   .wα > 0 for all .α and . α∈A wα ≤ a, where . α∈A wα is defined to be .supF α∈F wα , where the supremum is taken  over all finite subsets .F ⊆ A. Ordering the set .W by setting .{wα }α∈A ≺ vβ β∈B whenever .A ⊆ B and .wα = vα for all .α ∈ A, it is easy Hence, by Zorn’s lemma, to see that every chain in .W has a least upper bound.  .W has a maximal element .{wα }α∈A . Defining .w0 = α∈A0 wα , it is clear that 0 + such that .0 ≤ w0 ≤ a. If .w0 < a, then it follows from (v) that there exists .b ∈ E .0 < b ≤ a − w0 . Adding the element b to the family .{wα }α∈A yields an element 0 of .W which is strictly larger than this maximal element. This is a contradiction. + Consequently, .  α∈A0 wα = a. Defining .aβ ∈ E for every finite subset .β ⊆ A0 by setting .aβ = α∈β wα , it is clear that .0 ≤ aβ ↑β a. Hence, assertion (vi) holds. Evidently, (vi) implies (v). Therefore, to complete the proof, it suffices to show (v) implies (ii). Given .0 /= p ∈ P (M), it follows from (v) that there exists .b ∈ E + such that .0 < b ≤ p. Let .ε > 0 be such that .eb (ε, ∞) /= 0. Since .εeb (ε, ∞) ≤ b, it is clear that .eb (ε, ∞) ∈ P (E). Observing that .eb (ε, ∞) ≤ s (b) ≤ p, the proof ⨆ is complete. ⨅ The following observation is an immediate consequence of the equivalence of statements (i) and (ii) in Corollary 4.1.9.

4.1 Normed M-Bimodules of τ -Measurable Operators

245

Corollary 4.1.10 If E and F are two .M-bimodules of .τ -measurable operators satisfying .cE = cF = 1, then .cE∩F = 1. Replacing the von Neumann algebra .M by the reduced von Neumann algebra McE and the .M-bimodule E by .EcE , it is usually assumed that the carrier projection of E equals .1.

.

Example 4.1.11 The .M-bimodules .L∞ (τ ) = M, .L1 (τ ), .L1 (τ ) ∩ L∞ (τ ), L1 (τ ) + L∞ (τ ), and .F (τ ) all have carrier projection equal to .1.

.

Remark 4.1.12 If .E = F (τ ), then .cE = 1 and so, Corollary implies that  4.1.9  for every .a ∈ S (τ )+ there exists an upward directed system . aβ in .F (τ )+ such that .0 ≤ aβ ↑ a. This is precisely the result which has already been obtained in Proposition 2.3.12. In the proof of the next proposition, the following simple observation will be used. Lemma 4.1.13 If .{ek }∞ k=1 is an increasing sequence in .P (M) and if .e ∈ P (M) is such that .ek ↑ e and .τ (e) < ∞, then .ek ∧ q ↑ e ∧ q for all .q ∈ P (M). Proof It follows from e ∧ q − ek ∧ q = e ∧ q − ek ∧ (e ∧ q)

.

∼ ek ∨ (e ∧ q) − ek ≤ e − ek that τ (e ∧ q − ek ∧ q) ≤ τ (e − ek ) → 0,

.

from which it is clear that .ek ∧ q ↑ e ∧ q.

⨆ ⨅

Remark 4.1.14 Note that the condition that .τ (e) < ∞ in Lemma 4.1.13 cannot be omitted. Let us consider the case of a separable Hilbert space H identified with .𝓁2 , of all square summable complex sequences. We shall consider bounded operators on .𝓁2 with respect to standard orthonormal basis in .𝓁2 . Let .ek be the (orthogonal) projection onto the subspace generated by the first k unit vectors. Let q be√ the projection onto the one-dimensional subspace of .𝓁2 generated by √ 2 1 2 1 .( , 2 2 , 4 , 4 , · · · ), that is, ⎛ ⎜ ⎜ ⎜ ⎜ .q = ⎜ ⎜ ⎜ ⎝

⎞ ··· ⎟ ···⎟ ⎟ ···⎟ ⎟. ⎟ ···⎟ ⎠ .. . . . .

√ √ 1 2 2 1 8 4 √4 √2 2 1 2 1 8 √8 4 √4 2 1 2 1 8 √8 16 √4 2 1 2 1 8 8 16 16

.. .

.. .

.. .

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4 Symmetric Spaces of .τ -Measurable Operators

In particular, q is a rank-one projection and .ek ∧ q = 0 or q. Assume that .ek ∧ q = q for some sufficiently large k. Then, .ek ≥ q. We have .ek qek = q. However, a direct verification shows that .ek qek /= q, which implies that .ek ∧q = 0 for all k. Therefore, .ek ∧ q = 0 /= q = q ∧ 1 for all k. Proposition 4.1.15 If .E ⊆ S (τ ) is a normed .M-bimodule, then the embedding of E into .S (τ ) is continuous with respect to the norm topology in E and the local measure topology in .S (τ ), that is, if .{xn }∞ n=1 is a sequence in E satisfying .‖xn ‖E → 0, then .xn → 0 with respect to the local measure topology. Proof Suppose that .{xn }∞ n=1 is a sequence in E such that .‖xn ‖E → 0. It has to be proved that .pxn p → 0 with respect to the measure topology for all .p ∈ P (M) satisfying .τ (p) < ∞. Replacing .M and E by .Mp and .Ep , respectively, it is sufficient to prove the proposition under the additional assumption that .τ (1) < ∞. Tm

that  Assuming  .τ (1) < ∞, it has to be shown that .xn → 0 in .S (τ ), that is, |x | .τ e n (ε, ∞) → 0 for all .ε > 0 (see Proposition 2.5.7). Fixing .ε > 0 and setting |x | .pn = e n (ε, ∞), it follows from 0 ≤ εe|xn | (ε, ∞) ≤ |xn | e|xn | (ε, ∞)

.

that .pn ∈ P (E) for all n and .‖pn ‖E → 0 as .n → ∞. Suppose that .τ (pn )  0 as .n → ∞. Passing to a subsequence, it may be assumed that .‖pn ‖E ≤ 2−n and .τ (pn ) ≥ δ for some .δ > 0 and all n. Define .q ∈ P (M) by setting q=

∞ ∞  

pm .

.

n=1 m=n

  ∞ Since .τ (1) < ∞, .τ δ for all n, and . ∞ m=n pm ≥ m=n pm ↓n q, it follows that ∞ .τ (q) ≥ δ. Furthermore, since .q ≤ p , it is also clear that m m=n  q=

∞ 

.

 pm ∧ q,

n = 1, 2, . . . .

m=n

It follows from Lemma 4.1.13 that  M  .

 pm ∧ q ↑ M q

m=n

and hence, for every .n ∈ N, there exists .Mn ∈ N such that the projection .rn ∈ P (M), defined by rn =

M n

.

m=n

 pm ∧ q,

4.1 Normed M-Bimodules of τ -Measurable Operators

247

satisfies τ (q − rn ) ≤ 2−n−1 .

.

Define .sn ∈ P (M) by setting sn =

∞ 

.

rm ,

n = 1, 2, . . . .

m=n

Since .sn , rm ≤ q for all n and m, it follows that q − sn =

∞ 

.

(q − rm ) .

m=n

Consequently, τ (q − sn ) ≤

∞ 

.

τ (q − rm ) ≤ 2−n

m=n

for all n. This implies, in particular, .sn ↑ q. that n On the other hand, .sn ≤ rn ≤ M p m=n m and hence, it follows from Lemma 4.1.4 (iii) that .sn , rn ∈ P (E) and

.

‖sn ‖E ≤ ‖rn ‖E ≤

Mn  m=n

‖pm ‖E ≤

∞ 

2−m = 2−n+1

m=n

for all .n ∈ N. Since .sn ↑, this clearly implies that .sn = 0 for all n and so, .q = 0. This ⨆ contradicts the fact that .τ (q) ≥ δ. This suffices for the proof of the proposition. ⨅ Corollary 4.1.16 If .E ⊆ S (τ ) is a normed .M-bimodule, then the following statements hold: (i) The positive cone .E + is closed in E with respect to .‖ · ‖E . (ii) If .{an }∞ n=1 is a sequence in .Eh and .a, b ∈ Eh are such that .‖an − a‖E → 0 and .an ≤ b for all n, then .a ≤ b. (iii) If .{an }∞ n=1 is an increasing sequence in .Eh and .a ∈ E such that .‖an − a‖E → 0, then .a ∈ Eh and .an ↑ a. Proof (i) By Proposition 2.7.6 (i), the positive cone .S (τ )+ is closed in .S (τ ) for the local measure topology. Therefore, (i) is an immediate consequence of Proposition 4.1.15. (ii) Since .b − an → b − a in E and .b − an ∈ E + for all .n, it follows from (i) that + .b − a ∈ E , that is, .a ≤ b.

248

4 Symmetric Spaces of .τ -Measurable Operators

(iii) Since .Eh is closed (see the arguments below Proposition 4.1.3), it is clear that .a ∈ Eh . If .m ∈ N, then .am ≤ an for all .n ≥ m and so, it follows from (ii) that .am ≤ a for all m. On the other hand, if .c ∈ Eh is such that .an ≤ c for all n, then (ii) implies that .a ≤ c. Consequently, .a = supn an . ⨆ ⨅ Some criteria for norm completeness for normed .M-bimodules will be discussed next. Lemma 4.1.17 If .E ⊆ S (τ ) is a normed .M-bimodule, then the following conditions are equivalent: (i) E is a Banach space. ∞ + (ii) If .{an }∞ n=1 is a sequence in .E satisfying . n=1 ‖an ‖E < ∞, then ∞  .

an = sup

m 

an

m∈N n=1

n=1

exists in .Eh and ∞  ∞      ‖an ‖E . . an  ≤   n=1

E

n=1

  Proof If (i) is satisfied, then . ∞ series . ∞ a is n=1 ‖an ‖E < ∞ implies that the  ∞ n m n=1 norm convergent in E, with sum .a ∈ E, say. Since the sequence . a n m=1 is n=1  m increasing in m, it follows from Corollary 4.1.16 (iii) that .a = supm∈N n=1 an . ∞ Evidently, .‖a‖E ≤ n=1 ‖an ‖E and so this shows that (i) implies (ii). Suppose now that condition (ii) is satisfied. Since .‖Rex‖E , ‖Imx‖E ≤ ‖x‖E for all .x ∈ E, it is clearly enough to show that .Eh is with respect to complete ∞ .‖·‖E . Therefore, it is sufficient to prove that any series . a , with .an ∈ Eh and n=1 n   ∞ + − a − and .a +  , ‖a ‖ satisfying . < ∞, is convergent. Since . a = a n n E n n n E n=1  − + for all n. It ‖ ‖a .an  ≤ for all n, it may be assumed further that .an ∈ E n E E follows from (ii) that a=

∞ 

.

n=1

exists ∞ in E and that .‖a‖E ≤ . n=m an , it follows that

an = sup

m 

an

m∈N n=1

∞

n=1 ‖an ‖E . Applying the same argument to the series

    ∞ m−1 ∞           ‖an ‖E . a − an  =  an  ≤   n=m  n=m n=1

E

E

4.1 Normed M-Bimodules of τ -Measurable Operators

249

 for all .m ∈ N. Therefore, the series . ∞ n=1 an is norm convergent in E. This shows that (ii) implies (i). ⨆ ⨅ Definition 4.1.18 A normed .M-bimodule m.E ⊆ S (τ ) is said to have the∞Riesz– Fischer property if . ∞ n = supm∈N n=1 a n=1 an exists in E whenever .{an }n=1 is a sequence in .E + satisfying . ∞ n=1 ‖an ‖E < ∞. The following completeness criterion for normed .M-bimodules of .τ -measurable operators is sometimes useful. Theorem 4.1.19 If .E ⊆ S (τ ) is a normed .M-bimodule, then the following statements are equivalent: (i) E is a Banach space. (ii) E has the Riesz–Fischer property. Proof That (i) implies (ii) follows immediately from Lemma 4.1.17. It suffices, therefore, to show that statement (ii) of the present theorem implies statement (ii) + of Lemma 4.1.17. To this end, suppose that .{an }∞ n=1 is  a∞sequence in .E satisfying m ∞ . n=1 ‖an ‖E < ∞. By the Riesz–Fischer property, . n=1 an = supm∈N n=1 an exists in E. It remains to be shown that ∞  ∞      ‖an ‖E . . an  ≤   n=1

E

n=1

If this is not so, then there exists .ε > 0 such that ∞  ∞      ‖an ‖E . . an  > ε +   n=1

n=1

E

Let .n1 < n2 < · · · be in .N such that ∞  .

‖an ‖E ≤ 2−k ,

k = 1, 2, . . .

n=nk +1

and observe that   ∞     nk ∞      ∞     ‖a ‖ ‖an ‖E ≥ ε . a ≥ a − > ε +  n n n E    n=nk +1  n=1 n=1 n=n +1 k E E

for all k. Now arrange the terms of each of the sequences .{an : n > nk }, for .k = 1, 2, . . ., into a single sequence .{wm }∞ m=1 . Since ∞  .

m=1

‖wm ‖E =

∞ ∞   k=1 n=nk +1

‖an ‖E ≤

∞  k=1

2−k < ∞,

250

4 Symmetric Spaces of .τ -Measurable Operators

 the assumption that E has the Riesz–Fischer property implies that . ∞ m=1 wm exists in E. Observing that the term .an occurs at least k times in the sequence .{wm }∞ m=1 whenever .n > nk , it is not difficult to see that ∞  .

m=1

wm ≥ k

∞ 

an ,

k = 1, 2, . . . .

n=nk +1

Consequently,   ∞   ∞          . wm  ≥ k  an   ≥ kε   n=nk +1  m=1 E E

for all k, and this is clearly a contradiction. This suffices to conclude the proof of ⨆ the theorem. ⨅ Corollary 4.1.20 For a normed .M-bimodule .E ⊆ S (τ ), the following statements are equivalent: (i) E is a Banach space. (ii) Every increasing Cauchy sequence in .E + is convergent. (iii) Every increasing Cauchy sequence in .E + is bounded from above in .E + . Proof That (i) implies (ii) is evident and that (ii) implies (iii) follows immediately from Corollary 4.1.16 (iii). To show that (iii).⇒(i), by Theorem 4.1.19, it suffices to prove that E has ∞ + the nthat .{an }n=1 is a sequence in .E such that ∞Riesz–Fischer property. Suppose ∞ . n=1 ‖an ‖E < ∞. Defining .bn = k=1 ak for .n = 1, 2, . . ., it is clear that .{bn }n=1 + is an increasing Cauchy sequence. Hence, by hypothesis, there exists .b ∈ E such that .bn ≤ b for all n. It now follows from Proposition 2.3.10 that there exists + .c ∈ S (τ ) such that .bn ↑n c in .S (τ )+ . Since .0 ≤ c ≤ b, it follows from Proposition 4.1.3 (iii) that .c ∈ E + and hence, .bn ↑n c in .E + . This shows that E has the Riesz–Fischer property. ⨆ ⨅ Definition 4.1.21 A normed .M-bimodule .E ⊆ S (τ ) is said to have the .(σ -) Fatouproperty if for every upward directed net (sequence) . aβ in .E + , satisfying  .supβ aβ  < ∞, there exists .a ∈ E + such that .aβ ↑ a in E and .‖a‖E =  E sup aβ  . β

E

It is clear that the .σ -Fatou property implies the Riesz–Fischer property and hence, the following assertion is an immediate consequence of Theorem 4.1.19. Corollary 4.1.22 If a normed .M-bimodule .E ⊆ S (τ ) has the .σ -Fatou property, then E is a Banach space.

4.1 Normed M-Bimodules of τ -Measurable Operators

251

Example 4.1.23 To illustrate the concepts introduced above, recall that in Sect. 3.4, the space .L2 (τ ) is defined by setting   L2 (τ ) = x ∈ S (τ ) : x ∗ x ∈ L1 (τ ) ,

.

or equivalently,   L2 (τ ) = x ∈ S (τ ) :

∞

.

 μ (t; x)2 dt < ∞ .

(4.2)

0

The inner product .〈·, ·〉 on .L2 (τ ) is defined by setting .〈x, y〉 = τ (y ∗ x), .x, y ∈ L2 (τ ) and the corresponding norm given by  . ‖x‖2 =

1/2

∞

2

μ (t; x) dt

,

x ∈ L2 (τ ) .

(4.3)

0

Using (4.2) and (4.3), in combination with Proposition 3.2.7 (vi), it is easily verified that .L2 (τ ), equipped with the norm .‖·‖2 , is a normed .M-bimodule. It should be observed that .L2 (τ ) has the .σ -Fatou property. Indeed, suppose that .{an }∞ n=1 is an increasing sequence in .L2 (τ )+ such that .supn ‖an ‖2 = M < ∞. Since the function .μ (an ) is decreasing, this implies that .μ (t; an ) ≤ Mt −1/2 for all .t > 0. By Proposition 3.2.11 (i) (or by Proposition 3.2.16), it follows that the sequence .{an }∞ n=1 is bounded in .S (τ ) with respect to the measure topology and hence, Theorem 2.6.15 implies that there exists .a ∈ S (τ )+ such that .0 ≤ an ↑ a in .S (τ ). It follows from Proposition 3.2.14 (i) that .μ (an ) ↑ μ (a) a.e. on .[0, ∞) and so, by (4.3) and the Monotone Convergence Theorem, .‖an ‖2 ↑ ‖a‖2 . Hence, .L2 (τ ) has the .σ -Fatou property. As has been observed in Corollary 4.1.22, this implies, in particular, .L2 (τ ) is a Banach space. It should be observed that .L2 (τ ) in fact has the Fatou property. Remark 4.1.24 Suppose that .(X, ∑, ν) is a Maharam measure space and consider the von Neumann algebra .M = L∞ (ν) with integration as its trace. Let .E ⊆ S (ν) be a normed .L∞ -bimodule, that is, E is a normed ideal of measurable functions contained in .S (ν) (see Example 4.1.2 (b)). Assume that E has the .σ -Fatou property. It should be observed that if .{fn }∞ n=1 is a sequence in E and if .f ∈ S (ν) is such that .fn → f .ν-a.e. on X and .limn→∞ ‖fn ‖E < ∞, then .f ∈ E and .

‖f ‖E ≤ limn→∞ ‖fn ‖E .

(4.4)

Indeed, since .|fn | → |f | .ν-a.e. on X, it is clear that .|f (t)| = limn→∞ |fn (t)|.  Noting that .infk≥n |fk | ≤ fj  for all .j ≥ n, it is clear that      . inf |fk | inf ‖fk ‖E .  ≤ k≥n k≥n E

252

4 Symmetric Spaces of .τ -Measurable Operators

Since .infk≥n |fk | ↑n |f | and E has the .σ -Fatou property, it follows that .f ∈ E and (4.4) holds. It should be observed that if the measure space is .σ -finite and if .E has the .σ Fatou property, then E has the Fatou property. Indeed, if the measure .ν is .σ -finite and if .{fα } isan upward directed system in .S (ν)+ , then there exists an increasing ∞ subsequence . fαn n=1 having the same set of upper bounds as .{fα } (this can be justified like in Theorem 113.2 in [145]). References: [40, 144].

4.2 The Dual of a Normed M-Bimodule In this section, some properties of the dual space of a normed .M-bimodule will be discussed. As before, .M is a von Neumann algebra on a Hilbert space H , equipped with a fixed semi-finite normal faithful trace .τ . Suppose that .E ⊆ S (τ ) is an .M-module and let .E ♯ denote the algebraic dual space of E. For .φ ∈ E ♯ , the conjugate functional .φ¯ ∈ E ♯ is defined by setting φ¯ (x) = φ (x ∗ ),

.

x ∈ E.

A linear functional .φ is called Hermitian if .φ¯ = φ and the real linear subspace of ♯ ♯ .E consisting of all Hermitian linear functionals is denoted by .E . Using the fact h that .E = Eh ⊕ iEh , it is evident that a functional .φ ∈ E ♯ is Hermitian if and only if .φ (a) ∈ R for all .a ∈ Eh . For any .φ ∈ E ♯ , the real and imaginary parts .Reφ and .Imφ are defined by setting Reφ =

.

 1 φ + φ¯ , 2

Imφ =

 1  φ − φ¯ , 2i

respectively. Observing that .φ = Reφ + iImφ, it is clear that ♯

♯

E ♯ = Eh ⊕ iEh .

.

A linear functional .φ ∈ E ♯ is called positive if .φ (a) ≥ 0 for all .a ∈ E + . Since + − E + , it is easy to see that any positive functional is Hermitian. The .Eh = E  + ♯ positive functionals form a cone in .Eh , denoted by . E ♯ . Given .φ ∈ E ♯ and .y ∈ M, the linear functionals .yφ and .φy are defined by setting .

(yφ) (x) = φ (xy) ,

(φy) (x) = φ (yx) ,

x ∈ E,

respectively. If E is a normed .M-bimodule, with Banach dual .E ∗ , and .φ ∈ E ∗ , then it is clear that .yφ and .φy belong to .E ∗ for all .y ∈ M. Moreover, .‖yφ‖E ∗ ≤ ‖y‖B(H ) ‖φ‖E ∗ and .‖φy‖E ∗ ≤ ‖y‖B(H ) ‖φ‖E ∗ .

4.2 The Dual of a Normed M-Bimodule

253

If E is a normed .M-bimodule, with Banach dual .E ∗ , then it is clear that .φ¯ ∈ E ∗   ∗ whenever .φ ∈ E and .φ¯ E ∗ = ‖φ‖E ∗ . Moreover, if .φ ∈ E ∗ , then .Reφ, .Imφ ∈ ♯ Eh∗ = E ∗ ∩ Eh and .‖Reφ‖E ∗ , .‖Imφ‖E ∗ ≤ ‖φ‖E ∗ . In particular, .Eh∗ is a norm closed real subspace of .E ∗ and .E ∗ = Eh∗ ⊕ iEh∗ . Using that .Reφ (x) = φ (Rex) for all ∗ .x ∈ E and .φ ∈ E , it follows easily that h .

‖φ‖E ∗ = sup {|φ (a)| : a ∈ Eh , ‖a‖E ≤ 1} ,

φ ∈ Eh∗ .

(4.5)

Consequently, the space .Eh∗ may be identified with the Banach dual of the real  + normed space .Eh . Furthermore, it is clear that .(E ∗ )+ = E ∗ ∩ E ♯ is a proper closed cone in .Eh∗ , which induces a partial ordering in .Eh∗ by setting .φ ≤ ψ whenever .φ, ψ ∈ Eh∗ satisfy .ψ − φ ∈ (E ∗ )+ . It will be shown next that the dual positive cone .(E ∗ )+ is generating in .Eh∗ , that is, .Eh∗ = (E ∗ )+ − (E ∗ )+ . The following observation will be used. Lemma 4.2.1 The positive cone .E + is 2-normal, that is, if .a ≤ b ≤ c in .Eh , then .‖b‖E ≤ 2 max (‖a‖E , ‖c‖E ). Proof It follows from .b ≤ c that 0 ≤ b+ = eb [0, ∞) beb [0, ∞) ≤ eb [0, ∞) ceb [0, ∞)

.

    and  so, .b+ E ≤ eb [0, ∞) ceb [0, ∞)E ≤ ‖c‖E . Similarly, .−b ≤ −a implies that .b− E ≤ ‖a‖E . Consequently, .‖b‖E ≤ ‖a‖E + ‖c‖E ≤ 2 max (‖a‖E , ‖c‖E ). ⨆ ⨅ Now it follows from a well-known theorem of J. Grosberg and M. Krein that the dual cone .(E ∗ )+ is 2-generating, that is, every .φ ∈ Eh∗ can be written as .φ = φ1 −φ2 , where .φ1 , .φ2 ∈ (E ∗ )+ and .‖φ1 ‖E ∗ + ‖φ2 ‖E ∗ ≤ 2 ‖φ‖E ∗ . For the sake of reference, this result will be recorded in the next proposition. Proposition 4.2.2 If .E ⊆ S (τ ) is a normed .M-bimodule, then the dual positive cone .(E ∗ )+ is generating in .Eh∗ , that is, .Eh∗ = (E ∗ )+ − (E ∗ )+ . To be more precise, for every .φ ∈ Eh∗ , there exist .φ1 , .φ2 ∈ (E ∗ )+ such that .φ = φ1 − φ2 and .‖φ1 ‖E ∗ + ‖φ2 ‖E ∗ ≤ 2 ‖φ‖E ∗ . Applying Proposition 4.2.2 to .Reφ and .Imφ, it follows that each .φ ∈ E ∗ may be written as φ = φ1 − φ2 + i (φ3 − φ4 ) ,

.

 + φj ∈ E ∗ ,

‖φ1 ‖E ∗ + ‖φ2 ‖E ∗ ≤ 2 ‖φ‖E ∗ ,

1 ≤ j ≤ 4,

(4.6)

‖φ3 ‖E ∗ + ‖φ4 ‖E ∗ ≤ 2 ‖φ‖E ∗ .

Corollary 4.2.3 If .E ⊆ S (τ ) is a normed .M-bimodule, then .

for all .x ∈ E.

  ‖x‖E ≤ 4 sup |φ (x)| : 0 ≤ φ ∈ E ∗ , ‖φ‖E ∗ ≤ 1

(4.7)

254

4 Symmetric Spaces of .τ -Measurable Operators

Proof Fix .x ∈ E and let M denote the supremum in right-hand side of (4.7). Note that .|ψ (x)| ≤ M ‖ψ‖E ∗ for all .ψ ∈ (E ∗ )+ . Given .φ ∈ E ∗ with .‖φ‖E ∗ ≤ 1, write .φ as in (4.6), which yields that .

|φ (x)| ≤ M

4 j =1

  φj  ∗ ≤ 4M ‖φ‖E ∗ ≤ 4M. E j

  Consequently, .‖x‖E = sup |φ (x)| : φ ∈ E ∗ , ‖φ‖E ∗ ≤ 1 ≤ 4M.

⨆ ⨅

Another important property of the ordered Banach space .Eh∗ is that the norm is monotone (see Corollary 4.2.9). For the proof, some useful inequalities concerning positive functionals will be obtained first. The proofs of these inequalities are based on the following lemma, where the .M-bimodule .F 2 is defined for any .M-bimodule .F ⊆ S (τ ), by setting F2 =

.

⎧ n ⎨ ⎩

j =1

⎫ ⎬ xj yj : xj , yj ∈ F, j = 1, . . . , n; n ∈ N . ⎭

Lemma 4.2.4 If .E ⊆ S (τ ) is an .M-bimodule, then the set F = x ∈ S (τ ) : |x|2 ∈ E +

!

.

(4.8)

is also an .M-bimodule and satisfies .F 2 = E. Proof It is clear that .λx ∈ F whenever .x ∈ F and .λ ∈ C. If .x, y ∈ F , then (x − y)∗ (x − y) ≥ 0 and so, .x ∗ y + y ∗ x ≤ x ∗ x + y ∗ y. This implies that

.

  0 ≤ (x + y)∗ (x + y) ≤ 2 x ∗ x + y ∗ y .

.

Since .2 (x ∗ x + y ∗ y) ∈ E + , it follows from Proposition 4.1.3 (iii) that .

(x + y)∗ (x + y) ∈ E + ,

and this shows that .x + y ∈ F . If .x ∈ F and .u, v ∈ U (M), then .

  (uxv)∗ (uxv) = v ∗ x ∗ u∗ uxv = v ∗ x ∗ x v ∈ E +

and so, .uxv ∈ F . Since every element of .M is a linear combination of at most four elements of .U (M), it follows that .y1 xy2 ∈ F for all .y1 , y2 ∈ M. Hence, F is an .M-bimodule.  + It will be shown next that . F 2 = E + . If .a ∈ E + , then .a 1/2 ∈ F and so,  1/2 2  2 +  + .a = a ∈ F . Hence, .E + ⊆ F 2 . To prove the converse inclusion, let  2 +  .a ∈ F be given and write .a = nj=1 xj yj with .xj , yj ∈ F . Noting that a is

4.2 The Dual of a Normed M-Bimodule

255

  positive and therefore . nj=1 xj yj = nj=1 yj∗ xj∗ , it is easily verified that n #∗ " # " #∗ " #! 1 " ∗ xj + yj xj∗ + yj − xj∗ − yj xj∗ − yj , 4

a=

.

j =1

from which it follows that #∗ " # 1 " ∗ xj + yj xj∗ + yj . 4 n

0≤a≤

.

j =1

Since .xj∗ + yj ∈ F for all j , it follows from the definition of F that

.

n " #∗ " #  xj∗ + yj xj∗ + yj ∈ E + j =1

 + and so, it follows from Proposition 4.1.3 (iii) that .a ∈ E + . This shows that . F 2 ⊆  + E + and hence, .E + = F 2 . Finally, if .x ∈ S (τ ), then it follows from Proposition 4.1.3 (i) that .x ∈ E if and  + only if .|x| ∈ E + = F 2 , if and only if .x ∈ F 2 (as .F 2 is also an .M-bimodule). ⨆ ⨅ Consequently, .E = F 2 and this completes the proof. Remark 4.2.5 The .M-bimodule F , defined by (4.8), is denoted by .E 1/2 . Lemma 4.2.4 may have some useful applications. By way of example, if .a ∈ E + and .x ∈ M, then .a 1/2 xa 1/2 ∈ E + . Indeed, suppose first, in addition, that .x ∈ M+ . Since .a 1/2 ∈ E 1/2 , it follows from Lemma 4.2.4 that .x 1/2 a 1/2 ∈ E 1/2 , which implies that #∗ " # " #2 " x 1/2 a 1/2 ∈ E 1/2 = E. a 1/2 xa 1/2 = x 1/2 a 1/2

.

If .x ∈ M is arbitrary, then x is a linear combination of at most four elements of .M+ and the result follows. Let .E ⊆ S (τ ) be an .M-bimodule and suppose that .φ : E → C is a positive linear functional. If .x, y ∈ E 1/2 , then it follows from Lemma 4.2.4 that .y ∗ x ∈ E and it is easy to verify that a semi-inner product (that is, an inner product that is not necessarily positive definite) .〈·, ·〉φ on .E 1/2 may be defined by setting .

  〈x, y〉φ = φ y ∗ x ,

x, y ∈ E 1/2 .

256

4 Symmetric Spaces of .τ -Measurable Operators

 2 The Cauchy–Schwartz inequality implies that .〈x, y〉φ  ≤ 〈x, x〉φ 〈y, y〉φ , that is, .

  ∗ 2     φ y x  ≤ φ x ∗ x φ y ∗ y ,

x, y ∈ E 1/2 .

(4.9)

Proposition 4.2.6 Let .E ⊆ S (τ ) be an .M-bimodule. If .φ : E → C is a positive linear functional and .x ∈ E, then .

    |φ (wx)|2 ≤ φ w x ∗  w ∗ φ (|x|)

for all .w ∈ M. In particular, .φ (|x|) = 0 implies that .φ (wx) = 0 for all .w ∈ M. Proof Let .x = v |x| be the polar decomposition of x. Since .wv |x|1/2 and .|x|1/2 both belong to .E 1/2 , it follows from (4.9) that .

 " #2   |φ (wx)|2 = φ wv |x|1/2 |x|1/2  "" #" #∗ # ≤ φ wv |x|1/2 wv |x|1/2 φ (|x|)   = φ wv |x| v ∗ w ∗ φ (|x|) .

Since .v |x| v ∗ = |x ∗ |, this suffices for the proof of the proposition.

⨆ ⨅

Example 4.2.7 In connection with Proposition 4.2.6, it should be pointed out that the inequality .|φ (x)| ≤ φ (|x|), .x ∈ E, does not hold, in general, for positive functionals .φ. Indeed, let .E = M = M2 (C), equipped with the standard trace .τ , and define .0 ≤ a ∈ M2 (C) by setting $ a=

.

% 10 . 00

Define the linear functional  .φ on .M2 (C) by setting .φ (x) = τ (xa), .x ∈ M2 (C). If x ≥ 0, then .φ (x) = τ a 1/2 xa 1/2 ≥ 0 and so, .φ is positive. Let .y ∈ M2 (C) be given by

.

$ y=

.

% η 1 , η2 η

where .0 < η ∈ R. Observing that % η2 η , . |y| = η 1 $

4.2 The Dual of a Normed M-Bimodule

257

it follows that, .φ (y) = η and .φ (|y|) = η2 . This shows that the inequality .|φ (y)| ≤ φ (|y|) does not hold in general (that is, if .0 ≤ a ∈ M2 (C), then the inequality .|τ (xa)| ≤ τ (|x| a), .x ∈ M2 (C), is in general false). Corollary 4.2.8 Suppose that .E ⊆ S (τ ) is a normed .M-bimodule. If .φ ∈ E ♯ is positive, then .

  sup {φ (x) : x ∈ E, ‖x‖E ≤ 1} = sup φ (x) : x ∈ E + , ‖x‖E ≤ 1 .

(4.10)

  In particular, .φ is bounded if and only if .sup φ (x) : x ∈ E + , ‖x‖E ≤ 1 < ∞, in which case .

  ‖φ‖E ∗ = sup φ (x) : x ∈ E + , ‖x‖E ≤ 1 .

(4.11)

Proof Denoting the two suprema in (4.10) by A and B, respectively, it is clear that B ≤ A. On the other hand, if .x ∈ E satisfies .‖x‖E ≤ 1, then .‖x ∗ ‖E ≤ 1 and so, by Proposition 4.2.6,

.

.

  |φ (x)|2 ≤ φ x ∗  φ (|x|) ≤ B 2 .

Hence, .|φ (x)| ≤ B for all .x ∈ E satisfying .‖x‖E ≤ 1, which shows that .A ≤ B. All statements of the corollary are now clear. ⨅ ⨆ An immediate consequence of formula (4.11) is the following result. Corollary 4.2.9 If .E ⊆ S (τ ) is a normed .M-bimodule, then the norm in .Eh∗ is monotone, that is, if .0 ≤ φ ≤ ψ in .Eh∗ , then .‖φ‖E ∗ ≤ ‖ψ‖E ∗ . In combination with estimate (4.7), inequality (4.9) may also be used to derive some inequalities for the norm on E. This will be illustrated by the following lemma, which will be used at a later stage. Lemma 4.2.10 If .E ⊆ S (τ ) is a normed .M-bimodule, then .

1/2 1/2  ‖xy‖E ≤ 4 ‖x‖E y ∗ xy E

for all .0 ≤ x ∈ E and .y ∈ M. Proof Let .xy = v |xy| be the polar decomposition of xy. If .φ ∈ E ∗ is a positive functional satisfying .‖φ‖E ∗ ≤ 1, then it follows from (4.9) that #∗ " ##2   2  ""  x 1/2 y  φ (|xy|)2 = φ v ∗ xy  = φ x 1/2 v "" #∗ " ## "" #∗ " ## x 1/2 y φ x 1/2 v x 1/2 v ≤ φ x 1/2 y           = φ y ∗ xy φ v ∗ xv ≤ y ∗ xy E v ∗ xv E ≤ ‖x‖E y ∗ xy E .

.

258

4 Symmetric Spaces of .τ -Measurable Operators

It follows now from (4.7) that .‖xy‖E ≤ 4 ‖x‖E ‖y ∗ xy‖E complete. 1/2

1/2

and the proof is ⨆ ⨅

Some important concepts will be discussed next, which are analogous to those in Sect. 1.12. Definition 4.2.11 Let .E ⊆ S (τ ) be a normed .M-bimodule. The absolute kernel N (φ) of a positive functional .φ ∈ (E ∗ )+ is defined by setting

.

    N (φ) = x ∈ E : φ (|x|) = φ x ∗  = 0 .

.

Note that it follows from Proposition 4.2.6 that .N (φ) ⊆ Ker (φ) and it is evident that .N (φ)+ = Ker (φ)+ . Lemma 4.2.12 For .φ ∈ (E ∗ )+ , the following statements hold: (i) If .x ∈ E, then .x ∈ N (φ) if and only if .φ (wx) = φ (xw) = 0 for all .w ∈ M. (ii) .N (φ) is a closed linear subspace of E which is .∗-closed, .x ∈ N (φ) implies .|x| ∈ N (φ), and if .x ∈ E and .y ∈ N (φ) satisfy .0 ≤ x ≤ y, then .x ∈ N (φ). Proof (i) Let .x ∈ E be fixed and .x = v |x| be its polar decomposition. If x satisfies ∗ .φ (wx) = φ (xw) = 0 for all .w ∈ M, then it follows from .|x| = v x and ∗ ∗ ∗ .|x | = xv that .φ (|x|) = φ (|x |) = 0, that is, .x ∈ N (φ). For the proof of the converse implication, suppose that .x ∈ N (φ). It follows immediately from Proposition 4.2.6 that .φ (wx) = 0 for all .w ∈ M. Since .φ (xw) = φ (w ∗ x ∗ ), the same proposition, applied to .x ∗ , implies that also .φ (xw) = 0 for all .w ∈ M. (ii) By (i), it is clear that .N (φ) is a closed linear subspace of E. Since .φ is positive, it is also clear that .x ∈ N (φ) whenever .0 ≤ x ≤ y in E and .y ∈ N (φ). Evidently, .N (φ) is .∗-closed and .|x| ∈ N (φ) whenever .x ∈ N (φ). The proof is complete. ⨆ ⨅ Part (ii) of Lemma 4.2.12 motivates the following definition. Definition 4.2.13 A linear subspace J of an .M-bimodule .E ⊆ S (τ ) is called an order ideal if J is .∗-closed, .x ∈ J implies .|x| ∈ J , and if .x ∈ E and .y ∈ J satisfy .0 ≤ x ≤ y, then .x ∈ J . Consequently, by part (ii) of Lemma 4.2.12, the absolute kernel of a positive linear functional .φ ∈ (E ∗ )+ is a closed order ideal in E. Note, furthermore, that any order ideal .J ⊆ E is generated by .J + as a linear subspace. Indeed, if .x ∈ J , then ∗ ∈ J and so, .Rex and .Imx belong to J . Moreover, if .a ∈ J and .a = a ∗ , then .x ± ≤ |a|, which implies that .a ± ∈ J . Consequently, every element .|a| ∈ J and .0 ≤ a of J is a linear combination of positive elements of J . It should also be observed that the intersection of any collection of order ideals is an order ideal.

4.2 The Dual of a Normed M-Bimodule

259

Definition 4.2.14 Let .E ⊆ S (τ ) be an .M-bimodule. An order ideal .J ⊆ E is called order dense if for every .0 < x ∈ E + there exists .y ∈ J + such that .0 < y ≤ x. Note that the intersection of finitely many order dense order ideals is also an order dense order ideal. Indeed, suppose that .J1 and .J2 are two order dense order ideals in E. If .0 < x ∈ E, then there exists .y1 ∈ J1+ such that .0 < y1 ≤ x and then, there exists .y2 ∈ J2+ such that .0 < y2 ≤ y1 . Hence, .y2 ∈ (J1 ∩ J2 )+ and .0 < y2 ≤ x. Like in Lemma 1.12.4, we can characterize order denseness as follows. Lemma 4.2.15 If .E ⊆ S (τ ) is an .M-bimodule and .J ⊆ E is an order ideal, the following statements are equivalent: (i) J is order dense in E.  (ii) For every .0 < x ∈ E + , there exists a family .{xα } in .J + such that . α xα = x. (iii) For every .0 < x ∈ E + , there exists an upward directed system .{xα } in .J + such that .0 ≤ xα ↑α x. (iv) For every .0 < p ∈ P (E), there exists .q ∈ P (M) ∩ J such that .0 < q ≤ p. (v) For every .0 < p ∈ P (E), there exists afamily .{eα } of mutually orthogonal projections in .P (M) ∩ J such that .p = α eα . (vi) For every .0 < p ∈ P (E), there exists an upward directed system .{eα } in .P (M) ∩ J such that .eα ↑ p. Proof The implication (i).⇒(ii) follows from the same argument that was used for the proof of the implication (v).⇒(vi) in Corollary 4.1.9. The implications (ii).⇒(iii) and (iii).⇒(i) are evident. (i).⇒(iv). Given .0 < p ∈ P (E), it follows from the assumption on J that there exists .y ∈ J + such that .0 < y ≤ p. It is easy to see that this implies that y y y .e (0, ∞) = s (y) ≤ p. If .λ > 0 is such that .e [λ, ∞) /= 0, then .λe [λ, ∞) ≤ y. y Since J is an order ideal, this implies that .e [λ, ∞) ∈ J . The implication now follows by observing that .0 < ey [λ, ∞) ≤ ey (0, ∞) ≤ p. The implication (iv).⇒(v) follows by a routine argument involving Zorn’s lemma (cf. the proof of (i).⇒(ii)). Since the implications (v).⇒(vi) and (vi).⇒(i) are evident, ⨆ ⨅ the proof is complete. Recall that a linear functional .φ ∈ M∗ is called normal if .xα ↓α 0 in .M implies that .φ (xα ) → 0. A functional .φ is normal if and only if .φ is ultra-weakly continuous and the collection of all normal functionals is equal to .M∗ , the pre-dual of .M (see Proposition 1.12.7). Every functional .φ ∈ M∗ has a unique decomposition .φ = φn + φs , where .φn is normal and .φs is singular, that is, .φs ∈ M∗s (we denote by ∗ ∗ .Ms the set of all singular functionals in .M ). Recall, furthermore, that a functional ∗ .φ ∈ M is singular if and only if .φ vanishes on some order dense order ideal in .M (see Lemma 1.12.5). These facts motivate the following definition.

260

4 Symmetric Spaces of .τ -Measurable Operators

Definition 4.2.16 If .E ⊆ S (τ ) is a normed .M-bimodule and .φ ∈ E ∗ , then (i) .φ is called normal if .xα ↓α 0 in E implies that .φ (xα ) → 0. (ii) .φ is called completely additive if " " ##  φ x ei = φ (xei ) ,

.

i

i

and

φ

"" i

# #  ei x = φ (ei x) i

for every collection .{ei } of mutually orthogonal projections in .P (M) and all x ∈ E. (iii) .φ is called singular if .φ vanishes on some order dense order ideal in E. .

The collections of all normal, completely additive, and singular functionals on a ∗ , and .E ∗ , respectively. Later normed .M-module .E ⊆ S (τ ) are denoted by .En∗ , .Eca s in the next chapter, it will be shown that, for a large class of normed .M-modules, ∗ ∗ ∗ × of E (to .En = Eca , via trace duality .En may be identified with the Köthe dual .E ∗ be introduced in the next section) and that every .φ ∈ E has a unique decomposition ∗ ∗ .φ = φn + φs with .φn ∈ En and .φs ∈ Es . ∗ ∗ It is evident that .En and .Eca are linear subspaces of .E ∗ . If .φ ∈ En∗ (respectively, ∗ ∗ ) and hence, .Reφ, ¯ ∈ En∗ (respectively, .φ¯ ∈ Eca .φ ∈ Eca ), then it is clear that .φ ∗ ∗ .Imφ ∈ En (respectively, .Reφ, .Imφ ∈ Eca ). If a positive functional .0 ≤ φ ∈ E ∗ vanishes on some order ideal .J ⊆ E, then it is easy to see that .J ⊆ N (φ). Consequently, if .0 ≤ φ ∈ E ∗ , then .φ is singular if and only if its absolute kernel .N (φ) is order dense in E. Since the intersection of finitely many order dense order ideals is again an order dense order ideal, it is also clear that .Es∗ is a linear subspace of .E ∗ . Moreover, since ideals are .∗-closed, it is also clear that .φ ∈ Es∗ implies that .φ¯ ∈ Es∗ and hence, .Reφ and .Imφ also belong to ∗ ∗ ∗ ∗ ∗ .Es . Furthermore, it should be observed that .En ∩ Es = {0}. Indeed, if .φ ∈ En ∩ Es , then there exists an order dense order ideal .J ⊆ E such that .φ (x) = 0 for all + .x ∈ J . If .x ∈ E , then it follows from Lemma 4.2.15 that there exists an upward directed net .{xα } in .J + such that .0 ≤ xα ↑ x. Since .φ (xα ) = 0 for all .α and .φ is normal, it follows that .φ (x) = 0. This shows that .φ = 0. For future reference, these elementary observations are collected in the following lemma. Lemma 4.2.17 If .E ⊆ S (τ ) is a normed .M-bimodule, then .En∗ and .Es∗ are linear subspaces of .E ∗ satisfying .En∗ ∩ Es∗ = {0}. This section ends with some simple, but useful observations. Lemma 4.2.18 If .E ⊆ S (τ ) is a Banach .M-bimodule, then any positive linear functional .φ on E is necessarily bounded. Proof By Corollary 4.2.8, it suffices to show that there exists a constant .M ≥ 0 such that .0 ≤ φ (a) ≤ M for all .a ∈ E + satisfying .‖a‖E ≤ 1. If such M does not exist, then, for each .n ∈ N, there exists .an ∈ E + such that .‖an ‖E ≤ 1 and

4.3 The Köthe Dual of a Normed M-Bimodule

261

φ (an ) ≥ n3 . It follows from the completeness of E that the series

.

b=

∞ 

.

n−2 an

n=1

 is norm convergent in E. Defining .bn = nk=1 k −2 ak , .n ∈ N, it is clear that .0 ≤ bn ↑n and so, it follows from Corollary 4.1.16 (iii) that .0 ≤ bn ↑n b. In particular, −2 a ≤ b and so, .φ (b) ≥ n−2 φ (a ) ≥ n for all n, which is clearly a .0 ≤ n n n contradiction. This proves the lemma. ⨆ ⨅ Lemma 4.2.19 Let .E ⊆ S (τ ) be a normed .M-bimodule. If .{xα } is a downward directed system in .E + and if .φ (xα ) → 0 for all .φ ∈ E ∗ , then .‖xα ‖E ↓α 0. Proof The given assumption implies that 0 belongs to the .σ (E, E ∗ )-closure of the set .{xα }. It follows from the Hahn–Banach separation theorem (or, Mazur’s theorem) that .0 belongs to the norm closed hull of the set .{xα }. Consequently, given  convex n n .ε > 0, there exists a finite set . xαj and .0 ≤ λj ∈ R with . λ = 1 such j =1 j j =1 that      n   . λj xαj    ≤ ε. j =1  E

If .α is such that .0 ≤ xα ≤ xαj (.j = 1, . . . , n), then .0 ≤ xα ≤ .‖xα ‖E ≤ ε. This proves the lemma.

n

j =1 λj xαj

and so, ⨆ ⨅

References: [3, 39, 40, 48, 61, 68].

4.3 The Köthe Dual of a Normed M-Bimodule In this section, the so-called Köthe dual will be identified as an important part of the dual space. Definition 4.3.1 If .E ⊆ S (τ ) is a normed .M-bimodule, then the Köthe dual .E × of E is defined by setting E × = {y ∈ S (τ ) : sup {τ (|xy|) : x ∈ E, ‖x‖E ≤ 1} < ∞} ,

.

and for .y ∈ E × , the quantity .‖y‖E × is defined by setting .

‖y‖E × = sup {τ (|xy|) : x ∈ E, ‖x‖E ≤ 1} .

262

4 Symmetric Spaces of .τ -Measurable Operators

It should be observed that .xy ∈ L1 (τ ) whenever .x ∈ E and .y ∈ E × and that .

|τ (xy)| ≤ τ (|xy|) ≤ ‖x‖E ‖y‖E × ,

x ∈ E, y ∈ E × .

(4.12)

Lemma 4.3.2 Let .E ⊆ S (τ ) be a normed .M-bimodule. (i) If .y ∈ S (τ ), then .y ∈ E × if and only if .xy ∈ L1 (τ ) for all .x ∈ E and .

sup {|τ (xy)| : x ∈ E, ‖x‖E ≤ 1} < ∞.

(4.13)

Moreover, if .y ∈ E × , then .

‖y‖E × = sup {|τ (xy)| : x ∈ E, ‖x‖E ≤ 1} .

(4.14)

(ii) Equipped with .‖·‖E × , the space .E × is a semi-normed .M-module. Proof (i) If .x ∈ E and .y ∈ E × , then it is clear from the definition of × that .xy ∈ L (τ ) and that .|τ (xy)| ≤ τ (|xy|). This shows that .E 1 × .sup {|τ (xy)| : x ∈ E, ‖x‖E ≤ 1} ≤ ‖y‖E × for all .y ∈ E . Suppose now that .y ∈ S (τ ) is such that .xy ∈ L1 (τ ) for all .x ∈ E and that (4.13) is satisfied. Let A denote the supremum in (4.13). If .x ∈ E is such that .‖x‖E ≤ 1 and if ∗ ∗ .xy = v |xy| is the polar decomposition of xy, then .v x ∈ E with .‖v x‖E ≤ 1 and hence,    τ (|xy|) = τ v ∗ xy  ≤ A.

.

This shows that .y ∈ E × and .‖y‖E × ≤ A, which suffices for the proof of (i). (ii) From (i) it follows that .E × is a linear subspace of .S (τ ) and that .‖·‖E × is a semi-norm on .E × . Suppose that .y ∈ E × and v and w are in the closed unit ball of .M. If .x ∈ E is such that .‖x‖E ≤ 1, then also .‖wxv‖E ≤ 1. Furthermore, .xv ∈ E implies that .xvy ∈ L1 (τ ) and so, it follows from Proposition 3.4.2 (iii) and (4.14) that .

|τ (xvyw)| = |τ (wxvy)| ≤ ‖y‖E × .

Using (4.14) once more, it may be concluded that .‖vyw‖E × ≤ ‖y‖E × . This shows that .E × is a semi-normed .M-bimodule with respect to .‖·‖E × . ⨆ ⨅ The following lemma gives a necessary and sufficient condition for .‖·‖E × to be a norm on .E × . Lemma 4.3.3 If E is a normed .M-bimodule, then .‖·‖E × is a norm on .E × if and only if the carrier projection .cE of E is equal to .1.

4.3 The Köthe Dual of a Normed M-Bimodule

263

Proof Suppose that .cE = 1. If .0 /= y ∈ E × , then .s (y) /= 0 and so, by Corollary 4.1.9, there exists .q ∈ P (E) such that .0 < q ≤ s (y). Since  1/2 2 .q |y| q = |y| q  and .|y| q /= 0, it follows that .q |y| q > 0 and hence, .τ (|y| q) = τ (q |y| q) > 0. If .y = v |y| is the polar decomposition of y, then ∗ .qv ∈ E and   τ qv ∗ y = τ (q |y|) = τ (q |y| q) > 0.

.

It is now clear from (4.14) that .‖y‖E × /= 0.   ⊥ = 0 for all .x ∈ E and so, .0 /= c⊥ ∈ E × satisfies .c⊥  If .cE < 1, then .xcE E E E× = 0. This shows that .‖·‖E × is not a norm in this case. The proof is complete. ⨆ ⨅ Remark 4.3.4 Suppose that .E ⊆ S (τ ) is a normed .M-bimodule with .cE = 1. Let F ⊆ E be an .M-bimodule satisfying .cF = 1 and equip F with the norm .‖·‖E . In this case, .F × = E × and .‖y‖E × = ‖y‖F × for all .y ∈ E × . Indeed, it is clear that × ⊆ F × and .‖y‖ × × .E F × ≤ ‖y‖E × for all .y ∈ E . Suppose now that .y ∈ F and that .x ∈ E satisfies .‖x‖E ≤ 1. By Corollary 4.1.9, there exists an upward directed net . aβ in .F + such that .0 ≤ aβ ↑ |x|. If .x = v |x| is the polar decomposition (v) that .vaβ y → xy locally of x, then it follows from Propositions  2.7.6  2.7.5 and  aβ  ≤ ‖x‖E ≤ 1, it is clear that in measure. Since .vaβ ∈ F and .vaβ  ≤ E E   .τ vaβ y  ≤ ‖y‖F × for all .β. By Proposition 3.4.19, this implies that .τ (|xy|) ≤ ‖y‖F × . From the definition of .E × , it is now clear that .y ∈ E × and .‖y‖E × ≤ ‖y‖F × . In particular, if the .M-bimodule .F (τ ) is contained in E, then an element .y ∈ S (τ ) belongs to .E × if and only if .

.

sup {τ (|xy|) : x ∈ F (τ ) , ‖x‖E ≤ 1} < ∞,

and in this case, .‖y‖E × is given by the above supremum. The next lemma collects some further elementary properties of the Köthe dual. Lemma 4.3.5 Let .E ⊆ S (τ ) be a normed .M-bimodule. (i) The Köthe dual .E × is also given by E × = {y ∈ S (τ ) : sup {τ (|yx|) : x ∈ E, ‖x‖E ≤ 1} < ∞}

.

and .

‖y‖E × = sup {τ (|yx|) : x ∈ E, ‖x‖E ≤ 1} ,

y ∈ E×.

(ii) If .y ∈ S (τ ), then .y ∈ E × if and only if .yx ∈ L1 (τ ) for all .x ∈ E and .

sup {|τ (yx)| : x ∈ E, ‖x‖E ≤ 1} < ∞.

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4 Symmetric Spaces of .τ -Measurable Operators

Moreover, if .y ∈ E × , then .

‖y‖E × = sup {|τ (yx)| : x ∈ E, ‖x‖E ≤ 1} .

(iii) If .x ∈ E and .y ∈ E × , then .τ (xy) = τ (yx). (iv) If .0 ≤ x ∈ E and .0 ≤ y ∈ E × , then # " # " τ (xy) = τ x 1/2 yx 1/2 = τ y 1/2 xy 1/2 ≥ 0.

.

Proof (i) It should be observed that .

sup {τ (|xy|) : x ∈ E, ‖x‖E ≤ 1} = sup {τ (|yx|) : x ∈ E, ‖x‖E ≤ 1} (4.15)

for all .y ∈ S (τ ). Indeed, let A and B, respectively, denote the suprema in (4.15). To show that .B ≤ A, it may be assumed that .A < ∞, that is, × and .A = ‖y‖ ∗ × .y ∈ E E × . By Lemma 4.3.2 (ii), this implies that .y ∈ E ∗ and .‖y ‖E × = ‖y‖E × . Consequently, it follows from (4.12) that         τ (|yx|) = τ (yx)∗  = τ x ∗ y ∗  ≤ x ∗ E y ∗ E × ≤ A

.

for all .x ∈ E satisfying .‖x‖E ≤ 1. Hence, .B ≤ A. For the proof of the converse inequality, it may be assumed that .B < ∞, which implies, in particular, that .yx ∈ L1 (τ ) for all .x ∈ E. Let .y = v |y| be the polar decomposition of y. If .x ∈ E satisfies .‖x‖E ≤ 1 and .y ∗ x ∗ = w |y ∗ x ∗ | is the polar decomposition of .y ∗ x ∗ , then .yv ∗ x ∗ ∈ L1 (τ ) and .‖v ∗ x ∗ w ∗ v ∗ ‖E ≤ 1. Hence,       τ (|xy|) = τ y ∗ x ∗  = τ w ∗ v ∗ yv ∗ x ∗ = τ yv ∗ x ∗ w ∗ v ∗   ≤ τ yv ∗ x ∗ w ∗ v ∗  ≤ B,

.

which shows that .A ≤ B. Therefore, (4.15) holds and assertion (i) is now clear. (ii) The proof of this assertion is similar to the proof of part (i) of Lemma 4.3.2 and therefore omitted. If .x ∈ E and .y ∈ E × , then it is clear from the definition of .E × and part (i) of the present lemma that both yx and yx belong to .L1 (τ ). Therefore, statements (iii) and (iv) are immediate consequences of Propositions 3.4.30 and 3.4.32, respectively. ⨆ ⨅

4.3 The Köthe Dual of a Normed M-Bimodule

265

If .y ∈ E × , then it follows from Lemma 4.3.2 (i) that the linear functional .φy : x │−→ τ (xy), .x ∈ E, is bounded and .

  φy  ∗ = ‖y‖E × . E

(4.16)

Evidently, .y − │ → φy is a linear map from the Köthe dual .E × into the Banach dual ∗ .E . Note furthermore that φy = φy ∗ ,

.

y ∈ E×.

(4.17)

Indeed, if .x ∈ E, then   φy (x) = φy (x ∗ ) = τ (x ∗ y) = τ y ∗ x = φy ∗ (x) .

.

It follows from Lemma 4.3.3, in combination with (4.16), that the map .y │−→ φy , y ∈ E × is injective if and only if .cE = 1, in which case it is an isometry from .E × onto a linear subspace of .E ∗ . Identifying .E × with a subspace of .E ∗ , via the map × × is also denoted by .〈·, ·〉, .y │→ φy , .y ∈ E , the trace duality pairing of E and .E that is, .

〈x, y〉 = φy (x) = τ (xy) = τ (yx) ,

.

x ∈ E, y ∈ E × .

It should be observed that Lemma 4.3.2 (i) implies that .

|〈x, y〉| = |τ (xy)| ≤ ‖x‖E ‖y‖E × ,

x ∈ E, y ∈ E × .

Furthermore, if .0 ≤ y ∈ E × , then it follows from Lemma 4.3.5 (iv) above that + .φy is a positive linear functional, that is, .φy (x) ≥ 0 whenever .x ∈ E . Lemma 4.3.6 If .E ⊆ S (τ ) is a normed .M-bimodule and .y ∈ E × , then the following statements hold: (i) The functional .φy is normal, that is, .xα ↓ 0 in E implies that .φy (xα ) →α 0. (ii) If .wα ↑α w in .M, then .φy (wα x) →α φy (wx) and .φy (xwα ) →α φy (xw) for all .x ∈ E. (iii) The functional .φy is completely additive. Proof (i) Since every .y ∈ E × is a linear combination of at most four positive elements of × × .E , it suffices to consider the case that .0 ≤ y ∈ E . Lemma 4.3.5 (iv) implies that # " 1/2 .φy (xα ) = τ y xα y 1/2 ,

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4 Symmetric Spaces of .τ -Measurable Operators

and it follows from Proposition 2.2.25 (iii) that .y 1/2 xα y 1/2 ↓α 0. Since 1/2 x y 1/2 ∈ L (τ ) and the trace is normal, it follows that .φ (x ) ↓ 0. .y α 1 y α α (ii) It should be observed first that .τ (wα z) → τ (wz) whenever .wα ↑α w in .M and .z ∈ L1 (τ ). Indeed, without loss of generality, it may be assumed that .z ≥ 0. In this case, .z1/2 wα z1/2 ↑α z1/2 wz1/2 and so, # # " " τ (wα z) = τ z1/2 wα z1/2 ↑α τ z1/2 wz1/2 = τ (wz) ,

.

which proves the claim. Consequently, if .y ∈ E × and .x ∈ E, then .xy ∈ L1 (τ ) and hence φy (wα x) = τ (wα xy) →α τ (wxy) = φy (wx)

.

whenever .wα ↑α w in .M. Since .φy (xwα ) = φy ∗ (wα x ∗ ), the second assertion of (ii) is now also clear. Statement (iii) follows immediately from (ii) and Definition 4.2.16 (ii). ⨆ ⨅ As observed above, if .y ≥ 0 in .E × , then the functional .φy is also positive. If the carrier projection of E is equal to .1, then the converse implication also holds, as is shown in the next lemma. Lemma 4.3.7 Suppose that .E ⊆ S (τ ) is a normed .M-bimodule satisfying .cE = 1. If .y ∈ E × and .φy ≥ 0, then .y ≥ 0. Proof As observed in the discussion following Lemma 4.3.5, the assumption .cE = 1 implies that the map .z │−→ φz form .E × into .E ∗ is injective. Since .φy ≥ 0 implies, in particular, that .φy = φy , it follows from (4.17) that .φy = φy ∗ and so, .y = y ∗ . If y .p ∈ P (E) satisfies .0 ≤ p ≤ e (−∞, 0), then     0 ≤ φy (p) = τ (py) = τ pey (−∞, 0) y = −τ py − ≤ 0

.

  and so, .τ py − = 0. It follows from Corollary 4.1.9 (iv) that there exists upward   an − y →α .{pα } in .P (E) such that .pα ↑α e (−∞, 0). Since .τ pα y directed system     y − − (cf. the proof of part (ii) of Lemma 4.3.6) and = τ y τ e (−∞, 0) y    − = 0 for all .α, it follows that .τ y − = 0, that is, .y − = 0. Consequently, .τ pα y .y ≥ 0. ⨅ ⨆ The space .E × is always non-trivial, as follows from the next proposition. Proposition 4.3.8 If .E ⊆ S (τ ) is a normed .M-bimodule, then .cE × = 1. Proof Suppose that .p ∈ P (M) satisfies .0 < τ (p) < ∞. Let .BE denote the closed unit ball in E and define the absolutely convex subset K of .L1 (τ ) by setting K = BE ∩ L1 (τ )

.

L1 (τ )

.

4.3 The Köthe Dual of a Normed M-Bimodule

267

It should be observed that .uxv ∈ K whenever .x ∈ K and u and v are in the closed unit ball of .M. It will be shown first that there exists .n ∈ N such that .np ∈ / K. Indeed, if .np ∈ K for all n, then for each n, there exists .xn ∈ BE ∩ L1 (τ ) such that .‖np − xn ‖1 ≤ 1. This implies that .n−1 xn → p in .L1 (τ ) and hence, by Proposition 3.4.11, .n−1 xn → p with respect to the measure topology in .S (τ ). On the other hand, .n−1 xn → 0 in E, and so, by Proposition 4.1.15, .n−1 xn → 0 with respect to the local measure topology in .S (τ ). Since the local measure topology is Hausdorff, it follows that .p = 0, which is a contradiction. Fixing .n0 ∈ N such that .n0 p ∈ / K, it follows from the Hahn–Banach separation theorem that there exists .φ ∈ L1 (τ )∗ such that .

|φ (x)| ≤ 1,

x ∈ K,

n0 φ (p) > 1.

Identifying .L1 (τ )∗ , via trace duality, with the von Neumann algebra .M (see Theorem 3.4.24 (ii)), it follows that there exists .y ∈ M satisfying .

|τ (xy)| ≤ 1,

x ∈ K,

n0 τ (py) > 1.

(4.18)

Furthermore, if .x ∈ K and .xy = w |xy| is the polar decomposition of xy, then w ∗ x ∈ K and hence, .τ (|xy|) = τ (w ∗ xy) ≤ 1. Consequently,

.

τ (|xy|) ≤ 1,

.

x ∈ K.

(4.19)

It will be shown now that .y ∈ E × . If .x ∈ BE , then there exists an upward directed system .{xα } in .L1 (τ ) such that .0 ≤ xα ↑α |x|. Let .x = v |x| be the polar decomposition of x. Since .vxα ∈ K, it follows from (4.19) that .τ (|vxα y|) ≤ 1 for all .α. By Propositions 2.7.5 and 2.7.6 (v), .0 ≤ xα ↑α |x| implies that .vxα y → xy with respect to the local measure topology. Since the norm closed unit ball in .L1 (τ ) is closed with respect to the local measure topology (see Proposition 3.4.19), it follows that .xy ∈ L1 (τ ) and .τ (|xy|) ≤ 1. Therefore, .y ∈ E × and .‖y‖E × ≤ 1. By (4.18), .τ (pyp) = τ (py) > 1/n0 and so, .pyp /= 0. It thus has been shown that, for every .p ∈ P (M) satisfying .0 < τ (p) < ∞, there exists .y ∈ E × such that .pyp /= 0. By Corollary 4.1.9, this suffices to conclude that .cE × = 1. ⨆ ⨅ Corollary 4.3.9 If .E ⊆ S (τ ) is a normed .M-bimodule, then .E × separates the points of E, that is, if .x ∈ E is such that .τ (xy) = 0 for all .y ∈ E × , then .x = 0. Proof If .0 < x ∈ E, then there exists .0 /= p ∈ P (M) and .α > 0 such that αp ≤ x. By Proposition 4.3.8, .cE × = 1 and so, there exists .q ∈ P E × such that .0 < q ≤ p. This implies that .0 < ατ (q) = τ ((αp)q) ≤ τ (xq) (as .φq ≥ 0; cf. Lemma 4.3.5 (iv)). Suppose now  that  .0 /= x ∈ E is arbitrary. By the first part of the proof, there exists .q ∈ P E × such that .τ (|x| q) > 0. If .x = v |x| is the polar decomposition of x, then .τ (|x| q) = τ (v ∗ xq) = τ (xqv ∗ ) and hence, the element .y = qv ∗ ∈ E × satisfies .τ (xy) > 0. The result of the corollary is now clear. ⨆ ⨅ .

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4 Symmetric Spaces of .τ -Measurable Operators

If .E ⊆ S (τ ) is a normed .M-bimodule satisfying .cE = 1, then it follows from Lemma 4.3.2 (ii) and Lemma 4.3.3 that .E × , equipped with the norm .‖·‖E × , is also a normed .M-bimodule. Therefore, the second Köthe dual .E ×× may be defined by  × setting .E ×× = E × , that is,     E ×× = x ∈ S (τ ) : sup τ (|yx|) : y ∈ E × , ‖y‖E × ≤ 1 < ∞

.

and .

  ‖x‖E ×× = sup τ (|yx|) : y ∈ E × , ‖y‖E × ≤ 1   = sup |τ (yx)| : y ∈ E × , ‖y‖E × ≤ 1 ,

x ∈ E ×× .

It is clear that always .E ⊆ E ×× . Note that E is an absolutely solid subspace of ×× (that is, .x ∈ E whenever .x ∈ E ×× and .|x| ≤ |y| for some .y ∈ E). Indeed, by .E Proposition 4.1.3 (iii), E actually is an absolutely solid subspace of .S (τ ). Proposition 4.3.10 If .E ⊆ S (τ ) is a normed .M-bimodule satisfying .cE = 1, then .E ×× , equipped with .‖·‖E ×× , is a normed .M-bimodule such that .cE ×× = 1. Moreover, .E ⊆ E ×× and .‖x‖E ×× ≤ ‖x‖E for all .x ∈ E. Proof It follows from Proposition 4.3.8 that .cE ×× = 1 and so, by Lemma 4.3.3, ‖·‖E ×× is a norm on .E ×× . If .x ∈ E and .y ∈ E × satisfies .‖y‖E × ≤ 1, then it follows from Lemma 4.3.5 (i) that .τ (|yx|) ≤ ‖y‖E × ‖x‖E ≤ ‖x‖E . By the definitions of ×× and .‖·‖ ×× , this implies that .x ∈ E ×× and .‖x‖ ×× ≤ ‖x‖ . .E ⨆ ⨅ E E E .

Remark 4.3.11 If E and F are two normed .M-bimodules of .τ -measurable operators such that .E ⊆ F and .‖x‖F ≤ ‖x‖E for all .x ∈ E, then it follows immediately from the definition that .F × ⊆ E × and .‖y‖E × ≤ ‖y‖F × for all .y ∈ F × . Suppose now that .E ⊆ S (τ ) is a normed .M-bimodule such that .cE = 1. It follows from Proposition 4.3.10 and the above observation that .E ××× ⊆ E × and ××× . On the other hand, Proposition 4.3.10, .‖y‖E × ≤ ‖y‖E ××× for all .y ∈ E × × applied to .E , implies that .E ⊆ E ××× and .‖y‖E ××× ≤ ‖y‖E × for all .y ∈ E × . This shows that .E ××× = E × , with equality of norms. If .E ⊆ S (τ ) is a Banach .M-bimodule, then the definition of the Köthe dual space can be simplified, as the following proposition shows. Proposition 4.3.12 If .E ⊆ S (τ ) is a Banach .M-bimodule, then E × = {y ∈ S (τ ) : xy ∈ L1 (τ ) for all x ∈ E}

.

= {y ∈ S (τ ) : yx ∈ L1 (τ ) for all x ∈ E} . Proof Setting F = {y ∈ S (τ ) : xy ∈ L1 (τ ) ∀ x ∈ E} ,

.

G = {y ∈ S (τ ) : yx ∈ L1 (τ ) ∀ x ∈ E} ,

4.3 The Köthe Dual of a Normed M-Bimodule

269

it is easily verified that both F and G are .M-bimodules. It should be observed first that .F = G. Indeed, if .y ∈ F and .x ∈ E, then .y ∗ ∈ F and .x ∗ ∈ E, which implies that .yx = (x ∗ y ∗ )∗ ∈ L1 (τ ). This shows that .F ⊆ G. The proof of the converse inclusion is similar and hence, .F = G. If .y ∈ F , then .φy : x − │ → τ (xy), .x ∈ E, defines a linear functional on E. Furthermore, if .y ∈ F + , then it follows from Proposition 3.4.32 that # " τ (xy) = τ x 1/2 yx 1/2 ≥ 0,

.

x ∈ E+,

that is, .φy is a positive linear functional. Since it is assumed that E is Banach, Lemma 4.2.18 implies that .φy ∈ E ∗ for all .y ∈ F + . Every element of F is a linear combination of at most four positive elements (see the discussion following Proposition 4.1.3) and hence, .φy ∈ E ∗ for all .y ∈ F . Consequently, if .y ∈ F , then .xy ∈ L1 (τ ) for all .x ∈ E and .

sup {|τ (xy)| : x ∈ E, ‖x‖E ≤ 1} < ∞

and so, by Lemma 4.3.2 (i), .y ∈ E × . This shows that .F ⊆ E × . The converse inclusion is an immediate consequence of Lemma 4.3.2 (i) and so, the proof is complete. ⨆ ⨅ Example 4.3.13 Some examples of Köthe duals now follow. As before, .M is a von Neumann algebra on a Hilbert space H , equipped with a semi-finite normal faithful trace .τ . (a) If .E = L∞ (τ ) = M, then it is easily verified that .E × = L1 (τ ). Moreover, it follows from Proposition 3.4.8 that .‖·‖L∞ (τ )× = ‖·‖1 . Similarly, using that the dual of .L1 (τ ) may be identified with .L∞ (τ ) via trace duality (see Theorem 3.4.24), it is not difficult to see that .L1 (τ )× = L∞ (τ ) and that .‖·‖L (τ )× = ‖·‖∞ = ‖·‖B(H ) . 1 (b) Since .E = L2 (τ ) is a Hilbert space with respect to the inner product given by ∗ .〈x, y〉 = τ (y x), .x, y ∈ L2 (τ ) (see Example 4.1.23), it follows immediately × that .L2 (τ ) = L2 (τ ), with equality of norms. (c) Consider the Banach .M-bimodules .G (τ ) = L1 (τ ) + L∞ (τ ) and .H (τ ) = L1 (τ ) ∩ L∞ (τ ) (see Example 4.1.2 and Sect. 3.10). It will be shown that × .G (τ ) = H (τ ) and .H (τ )× = G (τ ), with equality of norms. Recall that the norms on .G (τ ) and .H (τ ) are given by .

  ‖x‖G(τ ) = inf ‖y‖1 + ‖z‖∞ : x = y + z, y ∈ L1 (τ ) , z ∈ L∞ (τ )  1 = μ (t; x) dt = ‖μ (x)‖L1 +L∞ , x ∈ L1 (τ ) + L∞ (τ ) 0

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4 Symmetric Spaces of .τ -Measurable Operators

and .

‖x‖H (τ ) = max (‖x‖1 , ‖x‖∞ ) = ‖μ (x)‖L1 ∩L∞ ,

x ∈ L1 (τ ) ∩ L∞ (τ ) ,

respectively. Suppose that .y ∈ H (τ ). It is clear that .xy ∈ L1 (τ ) for all .x ∈ G (τ ). If .x ∈ G (τ ) satisfies .‖x‖G(τ ) ≤ 1 and .x = x1 + x2 with .x1 ∈ L1 (τ ) and .x2 ∈ L∞ (τ ), then |τ (xy)| ≤ |τ (x1 y)| + |τ (x2 y)| ≤ ‖x1 ‖1 ‖y‖∞ + ‖x2 ‖∞ ‖y‖1

.

≤ ‖y‖H (τ ) (‖x1 ‖1 + ‖x2 ‖∞ ) . Taking the infimum over all such decompositions .x = x1 + x2 of x, it follows that .

|τ (xy)| ≤ ‖y‖H (τ )

and hence, by Lemma 4.3.2 (i), .y ∈ G (τ )× and .‖y‖G(τ )× ≤ ‖y‖H (τ ) . Conversely, if .y ∈ G (τ )× , then it is clear from (a) above that .y ∈ H (τ ). Moreover, .

‖y‖∞ = sup {τ (|xy|) : x ∈ L1 (τ ) , ‖x‖1 ≤ 1}   ≤ sup τ (|xy|) : x ∈ G (τ ) , ‖x‖G(τ ) ≤ 1 = ‖y‖G(τ )×

and, similarly, .‖y‖1 ≤ ‖y‖G(τ )× . This shows that .‖y‖H (τ ) ≤ ‖y‖G(τ )× and it may be concluded that .G (τ )× = H (τ ), with equal norms. By Proposition 4.3.10, .G (τ ) ⊆ G (τ )×× = H (τ )× and .‖y‖H (τ )× ≤ ‖y‖G(τ ) for all .y ∈ G (τ ). Suppose now that .y ∈ H (τ )× . It follows from Proposition 3.10.18 (iii) that 

1

.

  μ (t; y) dt = sup τ (|zy|) : z ∈ H (τ ) , z ≺≺ χ[0,1) .

0

Observing that .z ≺≺ χ[0,1) implies that .‖z‖1 ≤ 1 and .‖z‖∞ ≤ 1 (see Remark 3.9.3), it is clear that 

1

.

0

  μ (t; y) dt ≤ sup τ (|zy|) : z ∈ H (τ ) , ‖z‖H (τ ) ≤ 1 = ‖y‖H (τ )× .

Hence, .y ∈ G (τ ) and .‖y‖G(τ ) ≤ ‖y‖H (τ )× . Consequently, .H (τ )× = G (τ ) with equality of the norms. References: [40, 45].

4.4 Symmetric Spaces of τ -Measurable Operators

271

4.4 Symmetric Spaces of τ -Measurable Operators Throughout this section, let .M be a von Neumann algebra on a Hilbert space H , equipped with a fixed semi-finite faithful normal trace .τ . Definition 4.4.1 A linear subspace E of .S (τ ), equipped with a norm .‖·‖E , is called symmetrically normed if .x ∈ S (τ ), .y ∈ E, and .μ (x) ≤ μ (y) imply that .x ∈ E and .‖x‖E ≤ ‖y‖E . If, in addition, E is a Banach space, then E is termed a symmetric space (of .τ -measurable operators). It should be pointed out that being symmetrically normed imposes conditions on both the space and the norm. Example 4.4.2 It is readily verified that each of the spaces .L∞ (τ ) = M, .L1 (τ ), L1 (τ ) ∩ L∞ (τ ), and .L1 (τ ) + L∞ (τ ) is symmetrically normed (and also a Banach space; cf. Example 4.1.23). The space .F (τ ), equipped with the norm .‖·‖L1 ∩L∞ , is symmetrically normed but, in general, not complete.

.

Proposition 4.4.3 If .E ⊆ S (τ ) is a symmetrically normed space, then E is a normed .M-bimodule. Proof If .x ∈ E and .u, v ∈ M, then it follows from Proposition 3.2.7 (i), (vi) that   μ (uxv) ≤ ‖u‖B(H ) ‖v‖B(H ) μ (x) = μ ‖u‖B(H ) ‖v‖B(H ) x .

.

Hence, .uxv ∈ E and .‖uxv‖E ≤ ‖u‖B(H ) ‖v‖B(H ) ‖x‖E .

⨆ ⨅

It follows, in particular, from the above proposition that the observations made in Proposition 4.1.3 and Lemma 4.1.4 are valid for symmetrically normed spaces. The following observation will be used frequently. If .p ∈ P (E) and .q ∈ P (M) are such that .τ (q) ≤ τ (p), then .q ∈ P (E) and .‖q‖E ≤ ‖p‖E . Indeed, if .τ (q) ≤ τ (p), then μ (q) = χ[0,τ (q)) ≤ χ[0,τ (p)) = μ (p)

.

and so, the assertion follows immediately from the definition. For symmetrically normed spaces, the result of Proposition 4.1.15 may be improved as follows. Proposition 4.4.4 If .E ⊆ S (τ ) is a symmetrically normed space, then the embedding of E in .S (τ ) is continuous with respect to the norm topology in E and the measure topology in .S (τ ), that is, if .{xn }∞ n=1 is a sequence in E satisfying Tm

‖xn ‖E → 0, then .xn → 0.

.

Proof Let .{xn }∞ n=1 be a sequence in E such that .‖xn ‖E → 0. It has to be shown Tm

that that  |x.xn| → 0 in .S (τ ). By Proposition 2.5.7, this is equivalent to showing |x | .τ e n (ε, ∞) → 0 for all .ε > 0. Fixing .ε > 0 and setting .pn = e n (ε, ∞),

272

4 Symmetric Spaces of .τ -Measurable Operators

it follows from 0 ≤ εe|xn | (ε, ∞) ≤ |xn | e|xn | (ε, ∞)

.

that .pn ∈ P (E) for all n and .‖pn ‖E → 0 as .n → ∞. Suppose that .τ (pn )  0 as .n → ∞. Passing to a subsequence, it may be assumed that .τ (pn ) ≥ δ for all n and some .δ > 0. Let γ = inf {τ (e) : 0 /= e ∈ P (M)} .

.

Assuming first that .γ = 0, there exists .e ∈ P (M) such that .0 < τ (e) ≤ δ. As observed prior to the present proposition, .τ (e) ≤ τ (pn ) implies that .e ∈ P (E) and .0 < ‖e‖E ≤ ‖pn ‖E for all n. This clearly contradicts the fact that .‖pn ‖E → 0 as .n → ∞. Assume now that .γ > 0, in which case each projection .pn dominates a minimal projection .en ∈ P (M) satisfying .τ (en ) ≥ γ (cf. Lemma 3.7.2). It should be observed that a fixed minimal projection can only be dominated by finitely many .pn ’s, as .‖pn ‖E → 0. Therefore, by passing to a subsequence, if necessary, it may be assumed that the minimal projections .{en }∞ n=1  mutually distinct. Defining  are < 2α. Since the minimal .α = infn τ (en ), choose .n0 ∈ N such that .τ en0 projections .en and .en+1 are distinct, it follows that .en ∧ en+1 = 0 and hence, τ (en ∨ en+1 ) = τ (en ) + τ (en+1 ) ≥ 2α,

.

n ∈ N.

Consequently,   τ en0 ≤ 2α ≤ τ (en ∨ en+1 )

.

  and so, .en0 E ≤ ‖en ∨ en+1 ‖E . Using Lemma 4.1.4 (iii), this implies that   0 < en0 E ≤ ‖en ∨ en+1 ‖E ≤ ‖en ‖E + ‖en+1 ‖E ,

.

n ∈ N.

Since .‖en ‖E ≤ ‖pn ‖E → 0 as .n → ∞, this clearly is a contradiction, which completes the proof. ⨅ ⨆ Before formulating the next lemma, it should be recalled that .F (τ ) denotes the M-bimodule consisting of all elements .x ∈ M satisfying .τ (s (x)) < ∞ (see Sect. 2.3).

.

Lemma 4.4.5 Suppose that .E ⊆ S (τ ) is a symmetrically normed space. (i) If the carrier projection .cE of E is equal to .1, then .

{p ∈ P (M) : τ (p) < ∞} ⊆ P (E)

and hence, .F (τ ) ⊆ E.

4.4 Symmetric Spaces of τ -Measurable Operators

273

(ii) If the von Neumann algebra .M is either non-atomic or atomic with minimal projections having equal trace, and if .E /= {0}, then .cE = 1. Proof (i) By Lemma 4.1.4 (iii), the set .P (E) is upward directed and so, the normality of the trace .τ implies that .

sup {τ (p) : p ∈ P (E)} = τ (1) .

(4.20)

Suppose first that .τ (1) = ∞. If .q ∈ P (M) satisfies .τ (q) < ∞, then (4.20) implies that .τ (q) ≤ τ (p) for some .p ∈ P (E) and hence, .q ∈ P (E). This proves the assertion (i) in the case that .τ (1) = ∞. Assuming that .τ (1) < ∞, it has to be shown that .1 ∈ E. It follows   from (4.20) that there exists .p ∈ P (E) such that .τ (p) ≥ 12 τ (1). Since .τ p⊥ ≤ 12 τ (1) ≤ τ (p), it follows that also .p⊥ ∈ P (E) and hence, .1 ∈ P (E). If .x ∈ F (τ ), then the support projection .p = s (x) of x satisfies .τ (p) < ∞ and .|x| ≤ ‖x‖B(H ) p. Since .p ∈ P (E), it follows from Proposition 4.1.3 (iii) that .x ∈ E. The proof of part (i) is complete. (ii) Assume first that .M is non-atomic. Since .E /= {0} there exists .0 /= e0 ∈ P (E). Suppose that .cE < 1. Since .M is non-atomic, it follows from Lemma 3.7.2 ⊥ and .0 < τ (f ) ≤ τ (e ). This that there exists .f ∈ P (M) such that .f ≤ cE 0 implies that .f ∈ P (E) and so .f ≤ cE , which is a contradiction. Hence, it may be concluded that .cE = 1. Suppose now that .M is atomic and all minimal projections have equal trace. Since .E /= {0}, it is clear that .P (E) contains at least one minimal projection of .M. Since all minimal projections have equal trace, .P (E) must contain all the minimal projections of .M and hence, .cE = 1. The proof of the lemma is complete. ⨆ ⨅ As before, in the next theorem the von Neumann algebra .M and its norm .‖·‖B(H ) are denoted by .L∞ (τ ) and .‖·‖∞ , respectively. Theorem 4.4.6 Suppose that the von Neumann algebra .M is non-atomic and let E ⊆ S (τ ), .E /= {0}, be a symmetrically normed space.

.

(i) If .τ (1) < ∞, then .L∞ (τ ) ⊆ E ⊆ L1 (τ ) and .

‖x‖E ≤ ‖1‖E ‖x‖∞ , x ∈ L∞ (τ )

and

‖x‖1 ≤

(ii) If .τ (1) = ∞, then F (τ ) ⊆ E ⊆ L1 (τ ) + L∞ (τ )

.

τ (1) ‖x‖E , x ∈ E. ‖1‖E

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4 Symmetric Spaces of .τ -Measurable Operators

and ⎧ ⎨ ‖x‖E ≤ 2 ‖e‖E ‖x‖L1 ∩L∞ if x ∈ F (τ ) , .

⎩

‖x‖L1 +L∞ ≤ ‖e‖−1 E ‖x‖E if x ∈ E,

(4.21)

where .e ∈ P (E) is any projection satisfying .τ (e) = 1. If, in addition, E is a Banach space, then also .L1 (τ ) ∩ L∞ (τ ) ⊆ E and .‖x‖E ≤ 2 ‖e‖E ‖x‖L ∩L for all .x ∈ L1 (τ ) ∩ L∞ (τ ), where .e ∈ P (E) is any ∞ 1 projection satisfying .τ (e) = 1. Proof (i) Lemma 4.4.5 implies that .1 ∈ E. If .x ∈ L∞ (τ ), then .|x| ≤ ‖x‖∞ 1 and hence, .x ∈ E and .‖x‖E ≤ ‖1‖E ‖x‖∞ . To show that .E ⊆ L1 (τ ), it may be assumed, without loss of generality, that .τ (1) = 1. Moreover, it suffices to show that .a ∈ L1 (τ )+ and .‖a‖1 ≤ ‖a‖E whenever .a ∈ E + . Assume first that .0 ≤ a ∈ E is of the form ‖1‖−1 E n .a = j =1 αj ej , where .ej ∈ P (M) = P (E), .j = 1, . . . , n, are mutually orthogonal, satisfying .τ ej = 1/n, and .0 ≤ αj ∈ R for all j . For .k = 0, 1, . . . , n − 1, define ak =

n 

.

αkj ej ,

j =1

where the numbers .αkj are defined by setting  αkj =

.

αj +k if 1 ≤ j ≤ n − k, αj +k−n if n − k < j ≤ n.

It follows immediately from Example 3.2.2 (i) that .μ (ak ) = μ (a) and so, ‖ak ‖E = ‖a‖E for all .0 ≤ k ≤ n − 1. It is also clear that

.

n−1 

⎛ ⎞ n  ak = ⎝ αj ⎠ 1

k=0

j =1

.

and so, ⎛ .⎝

n  j =1

⎞ αj ⎠ ‖1‖E ≤

n−1  k=0

‖ak ‖E = n ‖a‖E .

(4.22)

4.4 Symmetric Spaces of τ -Measurable Operators

275

On the other hand,

.

n 

‖a‖1 =

j =1

⎞ ⎛ n   1  αj τ ej = ⎝ αj ⎠ . n

(4.23)

j =1

Combining (4.22) and (4.23) yields that .

‖a‖1 ≤

1 ‖a‖E . ‖1‖E

(4.24)

Suppose now that .0 ≤ a ∈ E is arbitrary. Given .n ∈ N, it will be convenient to denote λn,k = μ (k/n; a) ,

.

k = 1, . . . , n.

It follows from Lemma 3.7.8 that for each .n ∈ N,  there  exist projections .0 < fn,1 < · · · < fn,n in .P (M) = P (E) such that .τ fn,k = k/n and  &   ea λn,k , ∞ ≤ fn,k ≤ ea λn,k , ∞ ,

.

1 ≤ k ≤ n.

As observed in Lemma 3.7.10 (i), this implies in particular that .afn,k = fn,k anfor all n and k. For each n, define the mutually orthogonal projections . en,k in .P (E) by setting .en,1 = fn,1 and .en,k = fn,k − fn,k−1 for k=1   .2 ≤ k ≤ n. Observe that .τ en,k = 1/n and &  en,k ≤ ea λn,k , ∞ ,

.

1 ≤ k ≤ n.

For .n ∈ N, define .0 ≤ bn ∈ E by setting bn =

n 

.

λn,k en,k .

k=1

Recalling that .afn,k = fn,k a for all n and k, we have &  λn,k en,k = λn,k en,k ea λn,k , ∞ en,k  & ≤ en,k aea λn,k , ∞ en,k = aen,k

.

for all .1 ≤ k ≤ n and so it is clear that  n   .0 ≤ bn ≤ a en,k ≤ a. k=1

(4.25)

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4 Symmetric Spaces of .τ -Measurable Operators

Consequently, .‖bn ‖E ≤ ‖a‖E for all n. Therefore, it follows from (4.24) that ‖bn ‖1 ≤

.

1 1 ‖a‖E , ‖bn ‖E ≤ ‖1‖E ‖1‖E

n = 1, 2, . . . .

On the other hand, by Example 3.2.2 (i), μ (bn ) =

n 

.

μ (k/n; a) χ' k−1 n

k=1

, nk

#,

which implies that .μ (bn ) ↑ μ (a) a.e. on .[0, ∞). Hence, by the Monotone Convergence Theorem, 

∞

τ (a) =

.

 n

0

= sup ‖bn ‖1 ≤ n

∞

μ (t; a) dt = sup

μ (t; bn ) dt 0

1 ‖a‖E < ∞ ‖1‖E

and so, .0 ≤ a ∈ L1 (τ ) and .‖a‖1 ≤ ‖1‖−1 E ‖a‖E . This completes the proof of assertion (i). (ii) Recall that if .0 ≤ a ∈ S (τ ), then .a ∈ L1 (τ ) + L∞ (τ ) if and only if (1 (1 . 0 μ (t; a) dt < ∞, in which case .‖a‖L1 +L∞ = 0 μ (t; a) dt. Furthermore, since .M is non-atomic, it follows from Theorem 3.9.5 that  .

1

μ (t; a) dt = sup {τ (pap) : p ∈ P (M) , τ (p) = 1} .

0

If .0 ≤ a ∈ E and .p ∈ P (M) is such that .τ (p) = 1, then it follows from (i), applied in the reduced von Neumann algebra .Mp and the symmetrically normed space .Ep , that       τ (pap) = τp ap ≤ ‖p‖−1 Ep ap E

.

=

‖p‖−1 E ‖pap‖Ep

≤

p

‖p‖−1 E ‖a‖E

.

This suffices to prove that .E ⊆ L1 (τ ) + L∞ (τ ) and .‖x‖L1 +L∞ ≤ ‖e‖−1 E ‖x‖E for all .x ∈ E, where .e ∈ P (E) satisfies .τ (e) = 1. Assume now that .0 ≤ a ∈ F (τ ). For convenience, set .μn = μ (n; a), .n = 0, 1, 2, . . .. Note that .μ0 = ‖a‖∞ and so, .ea (μ0 , ∞) = 0. Note, furthermore, that .a ∈ .F (τ ) implies, in particular, that there exists .N ∈ N such that .μ (t; a) = 0 for all .t ≥ N. By Lemma 3.7.8 (i), there exists an increasing sequence .{fn }N n=1

4.4 Symmetric Spaces of τ -Measurable Operators

277

of projections in .P (M) such that .τ (fn ) = n and ea (μn , ∞) ≤ fn ≤ ea [μn , ∞) ,

n = 1, 2, . . . , N.

.

Observe that Lemma 4.4.5 implies that .fn ∈ P (E) for all n. Moreover, by Lemma 3.7.10 (i), .afn = fn a for all n. Define the mutually orthogonal projections .{en }N n=1 in .P (E) by setting .e1 = f1 and .en = fn − fn−1 for .2 ≤ n ≤ N. Observe that .τ (en ) = 1 for all n and hence, .‖en ‖E = ‖em ‖E for all .1 ≤ m, n ≤ N. Furthermore, using that en ≤ ea [μn , μn−1 ] ,

1 ≤ n ≤ N,

.

a computation similar to (4.25) shows that μn en ≤ aen ≤ μn−1 en ,

1 ≤ n ≤ N.

.

(4.26)

This implies, in particular, that N  .

μn en ≤ a

n=1

N 

 en

≤ a.

n=1

Therefore, N  .

 μn = τ

n=1

N 

 μn en

≤ τ (a) = ‖a‖1

n=1

and so, N  .

‖μn−1 en ‖E =

n=1

N 

 μn−1 ‖e1 ‖E ≤ (‖a‖1 + ‖a‖∞ ) ‖e1 ‖E .

n=1

Defining the element .b ∈ E by setting .b = .

N

n=1 μn−1 en ,

it is clear that

‖b‖E ≤ 2 ‖e1 ‖E ‖a‖L1 ∩L∞ .

Furthermore, using (4.26) and the fact that the projections .fn commute with a, it follows that b≥

N 

.

n=1

aen = afN ≥ aea (μN , ∞) = aea (0, ∞) = a.

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4 Symmetric Spaces of .τ -Measurable Operators

Consequently, .

‖a‖E ≤ ‖b‖E ≤ 2 ‖e1 ‖E ‖a‖L1 ∩L∞ .

By this, the first assertion of part (ii) is proved. Assume now, in addition, that E is complete and let .x ∈ L1 (τ ) ∩ L∞ (τ ) be given. Since .F (τ ) is dense in .L1 (τ ) ∩ L∞ (τ ) (see Proposition 3.10.2 (iii)), there exists a sequence .{xn }∞ n=1 in .F(τ ) such that .‖x − xn ‖L1 ∩L∞ → 0 as ∞ .n → ∞. It follows from (4.21) that the sequence .{xn } n=1 is Cauchy in E and so, there exists .y ∈ E such that .‖y − xn ‖E → 0. Since the embeddings of .L1 (τ ) ∩ L∞ (τ ) and E into .S (τ ) are both continuous for the measure topology (see Proposition 4.4.4), it follows that .x = y ∈ E. Since .‖xn ‖E ≤ 2 ‖e‖E ‖xn ‖L1 ∩L∞ for all n, it is also clear that .‖x‖E ≤ 2 ‖e‖E ‖x‖L1 ∩L∞ , where .e ∈ P (M) with .τ (e) = 1. The proof of the theorem is complete. ⨆ ⨅ The case for atomic algebras is different. Theorem 4.4.7 Suppose that the von Neumann algebra .M is atomic and all minimal projections have equal trace. If .E ⊆ S (τ ), .E /= {0}, is a symmetrically normed space, then F (τ ) ⊆ E ⊆ L∞ (τ )

.

and .

‖x‖E ≤ ‖e0 ‖E ‖x‖1 , x ∈ F (τ ) , and ‖x‖∞ ≤ ‖e0 ‖−1 E ‖x‖E , x ∈ E,

(4.27)

where .e0 is any minimal projection in .M. If, in addition, .E is a Banach space, then L1 (τ ) ⊆ E and .‖x‖E ≤ ‖e0 ‖E ‖x‖1 for all .x ∈ L1 (τ ).

.

Proof It should be noted that it has already been observed in Lemma 4.4.5 that F (τ ) ⊆ E. Without loss of generality, it may be assumed that .τ (e) = 1 for all minimal projections .e ∈ P (M). Let .e0 ∈ P (M) be a fixed minimal projection. This implies in particular that .τ (p) ∈ N ∪ {0} for all .p ∈ P (M) satisfying .τ (p) < ∞. Therefore, given .0 ≤ a ∈ E, the function .s │−→ d (s; a) = τ (ea (s, ∞)), .s ∈ [0, ∞), is decreasing with .lims→∞ d (s; a) = 0 and takes its values in .N ∪ {0}. This implies that .

μ0 = μ (0; a) = inf {s ≥ 0 : d (s; a) = 0} < ∞

.

and so, .a ∈ L∞ (τ ) with .‖a‖∞ = μ0 . Consequently, μ0 μ (e0 ) = μ0 χ[0,1) ≤ μ (a)

.

and so, .‖a‖∞ ‖e0 ‖E ≤ ‖a‖E . This suffices for the proof of the second inequality in (4.27).

4.4 Symmetric Spaces of τ -Measurable Operators

279

If .0 ≤ a ∈ F (τ ), then it follows from the spectral theorem that a is given by  n a = nj=1 αj ej , where . ej j =1 are mutually orthogonal minimal projections in .M and .0 ≤ αj ∈ R (.1 ≤ j ≤ n). Therefore,

.

.

‖a‖E ≤

n     αj  ej  = ‖e0 ‖E ‖a‖1 , E j =1

which establishes the first inequality in (4.27). Assume now, in addition, that E is complete. If .x ∈ L1 (τ ), then there exists a sequence .{xn }∞ n=1 in .F (τ ) such that .‖x − xn ‖1 → 0 (see Proposition 3.4.16). It follows from (4.27) that .‖xn − xm ‖E ≤ ‖e0 ‖E ‖xn − xm ‖1 for all m and n and ∞ hence, .{xn }∞ n=1 is a Cauchy sequence in E. Therefore, the sequence .{xn }n=1 is norm convergent in E, with limit equal to x (as follows from the second inequality in (4.27)). It is now also clear that .‖x‖E ≤ ‖e0 ‖E ‖x‖1 and hence, the proof of the theorem is complete. ⨆ ⨅ Remark 4.4.8 The following simple observation is sometimes useful. If the von Neumann algebra .M is atomic and all minimal projections have equal trace, then .pMp is finite dimensional whenever .p ∈ P (M) satisfies .τ (p) < ∞. Indeed, there  n exist mutually orthogonal minimal projections . ej j =1 in .P (M) such that .p = n n j =1 ej . Therefore, if .x ∈ pMp, then .x = i,j =1 ei xej . Let .i, j be fixed such that .ei xej /= 0 and set .y = ei xej . Since .0 < r (y) ≤ ei and .0 < s (y) ≤ ej , it follows that .ei = r (y) and .ej = s (y). Hence, the projections .ei and .ej are equivalent. Let ∗ ∗ .vij be a partial isometry such that .vij v ij = ei and .vij vij = ej . If .y = w |y| is the polar decomposition of y, then w is also a partial isometry satisfying .ww ∗ = ei and .w ∗ w = ej . Using that .ei and .ej are minimal   projections, it follows readily that .w = γij vij for some .γij ∈ C satisfying .γij  = 1. Furthermore, .|y| ej = |y| and so, .|y| = ej |y| ej , which implies that  .|y| = λij ej for some .0 ≤ λij ∈ R. Setting .αij = γij λij , this shows that .x = i,j αij vij , where the summation is taken over all i and j for which .ei and .ej are equivalent. It may be concluded, therefore, that .pMp is finite dimensional. As follows from Lemma 4.4.5, if .E ⊆ S (τ ) is a symmetrically normed space with .cE = 1, then the .M-bimodule .F (τ ) is contained in E. The closure of .F (τ ) in E is important enough to deserve a special notation. Definition 4.4.9 If .E ⊆ S (τ ) is a symmetrically normed space with .cE = 1, then E the closure of .F (τ ) in E is denoted by .E b , that is, .E b = F (τ ) . As is easily verified, .E b is an .M-bimodule in its own right. Furthermore, if E is a Banach space and .F (τ ), equipped with the .L1 ∩ L∞ -norm, is continuously E embedded in E, then .E b = L1 (τ ) ∩ L∞ (τ ) (by the same argument as was used in the last part of the proof of Theorem 4.4.6 (ii)). The following lemma characterizes the elements of the space .E b .

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4 Symmetric Spaces of .τ -Measurable Operators

Lemma 4.4.10 Suppose that .E ⊆ S (τ ) is a symmetrically normed space satisfying cE = 1. If .x ∈ E, then .x ∈ E b if and only if the following three conditions are satisfied:

.

(a) (b) (c)

x ∈ S0 (τ ).  xe|x| (0, s] → 0 as .s ↓ 0. E  |x|  .xe (s, ∞)E → 0 as .s → ∞. . .

Moreover, if .x ∈ S (τ ) and .y ∈ E b satisfy .μ (x) ≤ μ (y), then .x ∈ E b .     Proof It should be noted that .xe|x| (0, s]E = |x| e|x| (0, s]E and similarly  |x|    .xe (s, ∞)E = |x| e|x| (s, ∞)E for all .x ∈ E and .s ≥ 0 (as follows via the polar decomposition of x). It will be shown first that any .x ∈ E b satisfies conditions (a), (b), and (c). Since the space .S0 (τ ) is closed in .S (τ ) with respect to the measure topology (see Proposition 2.5.13) and since the embedding of E into .S (τ ) is continuous with respect to the measure topology (see Proposition 4.4.4), it follows from .F (τ ) ⊆ S0 (τ ) that .E b ⊆ S0 (τ ). Consequently, every .x ∈ E b satisfies condition (a). To show that any .x ∈ E b satisfies condition (b), let .ε > 0 be given and let .y ∈ F (τ ) be such that .‖x − y‖E ≤ ε. Defining .p = r (y), the range projection of y, it follows that .y = py and .τ (p) < ∞. For .0 < s ∈ R, write xe|x| (0, s] = p ⊥ xe|x| (0, s] + pxe|x| (0, s]

.

      and observe that .p⊥ xe|x| (0, s]E ≤ p⊥ x E = p ⊥ (x − y)E ≤ ε. Furthermore,         |x| . pxe (0, s] ≤ ‖p‖E xe|x| (0, s] ≤ s ‖p‖E . E

∞

  This implies that .lims↓0 xe|x| (0, s]E ≤ ε, which suffices to establish that x satisfies (b). To show that every .x ∈ E b satisfies condition (c), it may be assumed, without loss of generality, that .0 ≤ x ∈ E b . Given .ε > 0, let .y ∈ F (τ ) satisfy .‖x − y‖E ≤ ε. Replacing, if necessary, y by .Rey, it may be assumed that .y ∗ = y. Observe that 0 ≤ y − ≤ ey (−∞, 0) (x − y) ey (−∞, 0)

.

    and so, .y − E ≤ ‖x − y‖E . This implies that .x − y + E ≤ 2 ‖x − y‖E . Consequently, it may also be assumed, without loss of generality, that .0 ≤ y ∈ F (τ ). Setting .λ = ‖y‖∞ , let .n ∈ N be such that .n > 2λ. For convenience, set x .e = e (n, ∞) and observe that .

‖xe − eye‖E = ‖e (x − y) e‖E ≤ ε.

4.4 Symmetric Spaces of τ -Measurable Operators

281

Therefore, the element .z = eye satisfies .0 ≤ z ∈ F (τ ), .‖xe − z‖E ≤ ε, and .z ≤ λe. This implies that xe ≥ ne ≥ λe ≥ z ≥ 0

.

and so, 0 ≤ (n − λ) e ≤ xe − z.

.

It follows that .‖e‖E ≤ (n − λ)−1 ε and hence, .

‖z‖E = ‖ze‖E ≤ ‖z‖∞ ‖e‖E ≤ λ (n − λ)−1 ε ≤ ε.

Consequently, .

‖xe‖E ≤ ‖xe − z‖E + ‖z‖E ≤ 2ε,

which suffices to show that x satisfies condition (c). Suppose now that .x ∈ E satisfies conditions (a), (b), and (c). Defining .xn = xe|x| (1/n, n], .n ∈ N, it is clear that .‖x− xn ‖E → 0 as .n →  ∞. Furthermore,  |x| (1/n, ∞) < ∞ and hence, .τ e|x| (1/n, n] < ∞ .x ∈ S0 (τ ) implies that .τ e for all n. Since .‖xn ‖∞ ≤ n, it is now clear that .xn ∈ F (τ ) for all n. Hence, .x ∈ E b . For the proof of the final statement of the lemma, suppose that .y ∈ E b and .x ∈ S(τ ) is such that .μ (x) ≤ μ (y). By the first part of the lemma, the element y satisfies the above conditions (a), (b), and (c). Since .y ∈ S0 (τ ), it is clear that .x ∈ S0 (τ ) (see Proposition 3.2.4). Given .s > 0, let .α = d (s; |x|) and .β = d (s; |y|), and observe that .μ (x) ≤ μ (y) implies, in particular, that .α ≤ β. Therefore, it follows from Proposition 3.2.10 (iii) that " " # # μ |x| e|x| (s, ∞) = μ (x) χ[0,α) ≤ μ (y) χ[0,β) = μ |y| e|y| (s, ∞)

.

    and so, .|x| e|x| (s, ∞)E ≤ |y| e|y| (s, ∞)E . This implies that x satisfies condition (c) above. To show that x also satisfies (b), it should be observed first that if .μ (t; x) = 0 for some .t > 0, then (b) always holds. Indeed, in this case, the support projection .e|x| (0, trace 3.2.6 (ii)) and so,  (see Remark  ∞) has finite   |x| (0, ∞) ∈ E. Therefore, .xe|x| [0, s] ≤ s e|x| (0, ∞) for all .s > 0, which .e E E proves the claim. Consequently, it may be assumed that .μ (t; x) > 0 for all .t > 0. By Proposition 3.2.10 (iii), # " μ t; |y| e|y| [0, s] = μ (t + β; y) ,

.

t ≥ 0.

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4 Symmetric Spaces of .τ -Measurable Operators

Let .0 < s1 < μ (β; y) and set .α1 = d (s1 ; x). It follows from Proposition 3.1.3 that α1 ≥ d (μ (β; y) −; x) ≥ β and hence,

.

" # μ t; |x| e|x| [0, s1 ] = μ (t + α1 ; x) ≤ μ (t + β; x) " # ≤ μ (t + β; y) = μ t; |y| e|y| [0, s] .

.

    This implies that .xe|x| [0, s1 ]E ≤ ye|y| [0, s]E , from which it follows that x satisfies condition (b). Consequently, by the first part of the present lemma, it may be concluded that .x ∈ E b . The proof is complete. ⨆ ⨅ Remark 4.4.11 (a) It is easily seen that conditions (b) and (c) in Lemma 4.4.10 may be replaced by the following single condition: (a’) For every  ⊥  .ε > 0, there exists a finite trace projection .p ∈ P (M) such that .xp  ≤ ε. E

(b) If E is a symmetrically normed space on the interval .(0, ∞), then the result of Lemma 4.4.10 may also  be formulated  as follows.A function .f ∈ E belongs to .E b if and only if .μ (f ) χ(n,∞) E → 0 and .(μ (f ) − n1)+ E → 0 as .n → ∞. The details are left to the reader. For the proof of the next result (see Theorem 4.4.16), some technical preparations are needed. As before, .M is a von Neumann algebra on a Hilbert space H , equipped with a semi-finite normal faithful trace .τ . For sake of simplicity, in the proof of the following lemma, the integral of a function .f ∈ L∞ (0, τ (1)) with respect to Lebesgue measure m is denoted by .m (f ). Lemma 4.4.12 If the von Neumann algebra .M is non-atomic and .τ (1) < ∞, then there exists an abelian non-atomic von Neumann subalgebra .N of .M and a trace preserving normal surjective unital .∗-isomorphism .π : L∞ (0, τ (1)) → N. Proof Without loss of generality, it may be assumed that .τ (1) = 1. For .n = 0, 1, . . . and .k = 0, . . . , 2n − 1, define .∆n,k ∈ P (L∞ (0, 1)) by setting ∆n,k = χ[k2−n ,(k+1)2−n ) .

.

Let .Dn denote the finite Boolean subalgebra of .P (L∞ (0, 1)) generated by  2n −1 . ∆n,k . Observing that .Dn ⊆ Dn+1 for all n, define the Boolean subalgebra .D k=0 ) of .P (L∞ (0, 1)) by setting .D = ∞ n=0 Dn . Since .M is non-atomic, it is not difficult, using Lemma 3.7.3, to construct a collection  .

en,k : k = 0, . . . , 2n − 1, n = 0, 1, . . .



4.4 Symmetric Spaces of τ -Measurable Operators

283

of mutually commuting projections in .P (M) such that:   (a) For each .n = 0, 1, . . ., the collection . en,k : k = 0, . . . , 2n − 1 consists of 2n −1 mutually orthogonal projections satisfying . k=0 en,k = 1. n−1 − 1 and .n = 1, 2, . . .. (b) .en−1,k = e + e for all .0 ≤ k ≤ 2 n,2k n,2k+1   (c) .τ en,k = 2−n for all .n = 0, 1, . . . and .0 ≤ k ≤ 2n − 1. For each .n ≥ 0, let .An denote the Boolean algebra of projections in .P (M)  2n −1 ) generated by . en,k k=0 . Note that .An ⊆ An+1 for all n. Define .A = ∞ n=0 An , which is also a Boolean algebra (with respect to the ordering induced by .P (M)). Let   the Boolean isomorphism .hn : Dn → An be defined by setting .hn ∆n,k = en,k , n .0 ≤ k ≤ 2 − 1. It follows from (b) that .hn+1 |An = hn and hence, the Boolean isomorphism .h : D → A may be defined by setting .h (p) = hn (p) whenever .p ∈ Dn . Since each map .hn is trace preserving, it is clear that h is trace preserving as well. It should be observed that .|p − q| = p∨q −p∧q for all .p, q ∈ D and, similarly, .|e − f | = e∨f −e∧f for all .e, f ∈ A. Consequently, .h (|p − q|) = |h (p) − h (q)| for all .p, q ∈ D and so, .

‖h (p) − h (q)‖1 = τ (|h (p) − h (q)|)

(4.28)

= m (|p − q|) = ‖p − q‖1 for all .p, q ∈ D. If .p ∈ P (L∞ (0, 1)), then the properties of the Lebesgue measure imply that there exists a sequence .{pk }∞ k=1 in .D such that .‖p − pk ‖1 → 0 as .k → ∞. It follows from (4.28) that .‖h (pk ) − h (pl )‖1 → 0 as .k, l → ∞, that is, .{h (pk )}∞ k=1 is a Cauchy sequence in .L1 (τ ). Since .L1 (τ ) is complete, there exists .e ∈ L1 (τ ) Tm

such that .h (pk ) → e in .L1 (τ ). This implies that .h (pk ) → e and so, .e ∈ P (M). It is easily verified that the projection .e ∈ P (M) does not depend on the particular ¯ choice of the sequence .{pk }∞ k=1 in .D satisfying .‖p − pk ‖1 → 0. Setting .h (p) = e, this defines a map .h¯ : P (L∞ (0, 1)) → P (M). Its range .B = h¯ [P (L∞ (0, 1))] is a Boolean algebra (with respect to the ordering induced by .P (M)) and .h¯ : P (L∞ (0, 1)) → B is a trace preserving Boolean isomorphism. Define the .∗-subalgebra .S of .L∞ (0, 1) by S=

.

⎧ n ⎨ ⎩

j =1

⎫ ⎬ αj pj : αj ∈ C, pj ∈ P (L∞ (0, 1)) , j = 1, . . . , n; n ∈ N ⎭

and the map .π0 : S → M by setting ⎛ π0 ⎝

n 

.

j =1

⎞ αj pj ⎠ =

n  j =1

  αj h¯ pj

284

4 Symmetric Spaces of .τ -Measurable Operators

 for all . nj=1 αj pj ∈ S. It is readily verified that .π0 is a unital .∗-isomorphism satisfying .τ (π0 (s)) = m (s) and .‖π0 (s)‖B(H ) = ‖s‖∞ for all .s ∈ S. A routine argument shows that the .∗-isomorphism .π0 : S → M has a unique extension to a unital .∗-isomorphism .π : L∞ (0, 1) → M satisfying .τ (π (f )) = m (f ) and .‖π (f )‖B(H ) = ‖f ‖∞ for all .f ∈ L∞ (0, 1). It should be observed that .π is normal. Indeed, if .fα ↓α 0 in .L∞ (0, 1), then .m (fα ) ↓α 0 and so, since .π is trace preserving and .τ (1) < ∞, it follows that .τ (π (fα )) ↓α 0. This implies that .π (fα ) ↓α 0 in .M. Consequently, .N = π (L∞ (0, 1)) is a von Neumann subalgebra of .M and it is clear that .π : L∞ (0, 1) → N is a trace preserving .∗-isomorphism onto. Evidently, .N is abelian and non-atomic. The proof of the lemma is complete. ⨆ ⨅ If .τ (1) < ∞ and .N is a von Neumann subalgebra of .M, then the .∗-algebra S (N, τ ) of all .τ -measurable operators corresponding to .N is a .∗-subalgebra of .S (τ ), which is closed with respect to the measure topology (see the discussion at the beginning of Sect. 2.9). It should be observed that if .x ∈ S (N, τ ), then the singular value function .μ (x) of x is the same with respect to .(M, τ ) and .(N, τ ). Indeed, .μ (x) depends only on the spectral distribution function of x. If .π : L∞ (0, τ (1)) → N is an .∗-isomorphism satisfying the properties stated in Lemma 4.4.12, then it follows from Propositions 2.9.3 and 3.3.10 (i) that .π has a unique extension to a trace preserving normal unital .∗-isomorphism .π : S (0, τ (1)) → S (τ ) such that .π (S (0, τ (1))) = S (N, τ ). Moreover, .π preserves the singular value function, that is, .μ (πf ) = μ (f ) for all .f ∈ S (0, τ (1)) (see Proposition 3.3.10 (ii)). .

Proposition 4.4.13 Suppose that .M is a non-atomic von Neumann algebra and τ (1) < ∞. Let .E ⊆ S (τ ) be a symmetrically normed space.

.

(i) There exists an abelian non-atomic von Neumann algebra .N of .M such that there exists a surjective normal unital .∗-isomorphism .π : S (0, τ (1)) → S (N, τ ), which preserves the singular value function. (ii) Defining .EN ⊆ S (N, τ ) by setting .EN = E ∩ S (N, τ ) and .‖x‖EN = ‖x‖E , .x ∈ EN , the space .EN is a symmetrically normed subspace of .S (N, τ ), which is closed in E; if E is complete, then .EN is also complete. Furthermore, defining E (0, τ (1)) = π −1 (EN ) ,

.

and .

‖f ‖E(0,τ (1)) = ‖πf ‖E ,

f ∈ E (0, τ (1)) ,

the following statements hold: (iii) .E (0, τ (1)) is a symmetrically normed space on .(0, τ (1)) with respect to the norm .‖·‖E(0,τ (1)) ;

4.4 Symmetric Spaces of τ -Measurable Operators

285

(iv) .E = {x ∈ S (τ ) : μ (x) ∈ E (0, τ (1))} and .‖x‖E = ‖μ (x)‖E(0,τ (1)) for all .x ∈ E; (v) The map .π : E (0, τ (1)) → EN is a surjective isometric isomorphism (in particular, if E is a Banach space, then .E (0, τ (1)) is also a Banach space). Proof (i) The existence of the von Neumann subalgebra .N and the .∗-isomorphism .π : S (0, τ (1)) → S (N, τ ) with the desired properties follows from Lemma 4.4.12 and the discussion preceding the present proposition. (ii) If .x ∈ S (N, τ ), then its singular value functions with respect to .(M, τ ) and .(N, τ ) coincide and so, it is clear that .EN is a symmetrically normed subspace of .S (N, τ ). To show that .EN is closed in E, suppose that .{xn }∞ n=1 is a sequence in .EN and that .x ∈ E satisfies .‖x − xn ‖E → 0. The embedding of E into .S (τ ) Tm

is continuous with respect to the measure topology and hence, .xn → x. Since .S (N, τ ) is closed in .S (τ ) with respect to the measure topology (see Sect. 2.9), it follows that .x ∈ S (N, τ ). Consequently, .x ∈ EN , which shows that .EN is closed in E. Evidently, if E is complete, then .EN is also complete. (iii) From the definition of .E (0, τ (1)), it is clear that .E (0, τ (1)) is a linear subspace of .S (0, τ (1)). By the definition of the norm .‖·‖E(0,τ (1)) , the map .π : E (0, τ (1)) → EN is a surjective isometric isomorphism (which proves (v)). To show that .E (0, τ (1)) is a symmetrically normed space, suppose that .f ∈ S (0, τ (1)) and .g ∈ E (0, τ (1)) satisfy .μ (f ) ≤ μ (g). By the definition of .E (0, τ (1)), .πg ∈ EN . Since .π preserves the singular value function, it follows that .μ (πf ) ≤ μ (πg) and so, .πf ∈ EN . This implies that .f ∈ E (0, τ (1)) and .

‖f ‖E(0,τ (1)) = ‖πf ‖E ≤ ‖πg‖E = ‖g‖E(0,τ (1)) ,

This shows that .E (0, τ (1)) is a symmetrically normed space. (iv) If .x ∈ E, then .μ (x) ∈ S (0, τ (1)) and .y = π (μ (x)) belongs to .S (N, τ ). Since .π preserves the singular value function, it follows that .μ (y) = μ (x) and so, .y ∈ E ∩ S (N, τ ), that is, .y ∈ EN with .‖y‖E = ‖x‖E . Consequently, −1 (y) ∈ E (0, τ (1)) and .μ (x) = π .

‖μ (x)‖E(0,τ (1)) = ‖π (μ (x))‖E = ‖y‖E = ‖x‖E .

Finally, suppose that .x ∈ S (τ ) satisfies .μ (x) ∈ E (0, τ (1)). By definition, the element .y = π (μ (x)) belongs to .EN and .μ (y) = μ (x), as .π preserves the singular value function. Since E is symmetric, this implies that .x ∈ E. The first assertion of (v) has already been noted in the proof of part (iii). The second assertion is an immediate consequence of this, since it has been proven in (ii) that .EN is complete whenever E is complete. The proof of the proposition is finished. ⨆ ⨅

286

4 Symmetric Spaces of .τ -Measurable Operators

Definition 4.4.14 The norm .‖·‖E in normed .M-bimodule .E ⊆ S (τ ) is called a Fatou norm if for any upward directed net .{xα } in .E + and .x ∈ E + , it follows from .0 ≤ xα ↑α x that .‖xα ‖E ↑α ‖x‖E . Remark 4.4.15 Suppose that .(X, ∑, ν) is a Maharam measure space and consider the von Neumann algebra .M = L∞ (ν) with integration as its trace. Let .E ⊆ S (ν) be a normed .L∞ -bimodule, that is, E is a normed ideal of measurable functions contained in .S (ν). Assume that .‖·‖E is a Fatou norm. It should be observed that if ∞ .{fn } n=1 is a sequence in .E and if .f ∈ E is such that .fn → f .ν-a.e. on X, then .

‖f ‖E ≤ limn→∞ ‖fn ‖E .

The proof follows by the same argument as was used in Remark 4.1.24. As is the case for the Fatou property (see Remark 4.1.24), if the measure .ν is ∞ + .σ -finite and if .‖·‖E is a .σ -Fatou norm, that is, for any sequence .{fn } n=1 in .E and + .f ∈ E it follows from .0 ≤ fn ↑ f that .‖fn ‖E ↑ ‖f ‖E , then .‖·‖E is a Fatou norm. Theorem 4.4.16 If .E ⊆ S (τ ) is a symmetrically normed space and .M is either non-atomic or atomic and all minimal projections have equal trace, then the following statements are equivalent: (i) The norm in E is a Fatou norm. (ii) The norm closed unit ball .BE of E is closed in E with respect to the local measure topology. Proof For the proof that (i) implies (ii), assume first, in addition, that .τ (1) < ∞. If .M is atomic and all minimal projections have equal trace, then Remark 4.4.8, in combination with Theorem 4.4.7, forces E to be finite dimensional. Therefore, in this case, it is evident that (ii) follows from (i). Suppose now that .M is non-atomic. Let the abelian von Neumann subalgebra .N of .M, the singular value preserving .∗-isomorphism .π : S (0, τ (1)) → S (N, τ ), and the symmetrically normed space .E (0, τ (1)) be defined as in Proposition 4.4.13. It should be observed that the norm .‖·‖E(0,τ (1)) is a Fatou norm. Indeed, suppose that .{fα } is an upward directed system in .E (0, τ (1))+ and that .f ∈ E (0, τ (1))+ is such that .0 ≤ fα ↑ f . Since the .∗-isomorphism .π is normal, this implies that .π (fα ) ↑α π (f ) in E and hence, .‖π (fα )‖E ↑α ‖π (f )‖E . By the definition of the norm in .E (0, τ (1)), this shows that .‖fα ‖E(0,τ (1)) ↑α ‖f ‖E(0,τ (1)) and hence, .‖·‖E(0,τ (1)) is Fatou. Recall, since .τ (1) < ∞, that the local measure topology coincides with the measure topology. Tm

Let .{xn }∞ n=1 be a sequence in .BE and let .x ∈ E be such that .xn → x as .n → ∞. By Proposition 3.2.11 (ii), this implies in particular that .μ (xn ) → μ (x) a.e. on the interval .(0, τ (1)), and hence it follows from the above observations that .

‖x‖E = ‖μ (x)‖E(0,τ (1)) ≤ limn→∞ ‖μ (xn )‖E(0,τ (1)) = limn→∞ ‖xn ‖E ≤ 1,

that is, .x ∈ BE . This proves that (i) implies (ii) in the case that .τ (1) < ∞.

4.4 Symmetric Spaces of τ -Measurable Operators

287

Turning to the general case, suppose that .{xα } is a net in .BE and that .x ∈ E is such that .xα → x with respect to the local measure topology. If .x = v |x| is the Tlm

polar decomposition of x, then .vxα → |x| and .‖vxα ‖E ≤ 1 for all .α. Therefore, replacing x by .|x| and .xα by .vxα , it may be assumed, without loss of generality, that .x ≥ 0. If .p ∈ P (M) satisfies .τ (p) < ∞, then .pxα p → pxp with respect to the measure topology. Hence, it follows from the special case considered above, applied to the reduced von Neumann algebra .Mp and the symmetrically normed space .Ep , that .‖pxp‖ there exists an upward  E ≤ 1. Since the trace .τ is semi-finite,  directed system . pβ in .P (M) satisfying .τ pβ < ∞ for all .β and .pβ ↑β 1. It     = μ x 1/2 pβ x 1/2 and hence (ii) that .μ pβ xpβ follows from Proposition 3.2.10     that .x 1/2 pβ x 1/2 ∈ E and .x 1/2 pβ x 1/2 E = pβ xpβ E ≤ 1 for all .β. Since .0 ≤ x 1/2 pβ x 1/2 ↑β x and .‖·‖E is a Fatou norm, this implies that .‖x‖E ≤ 1, that is, .x ∈ BE . This shows that (i) implies (ii). Finally, by Proposition 2.7.6 (v), .0 ≤ xα ↑α x in E (and hence, in .S (τ )) implies that .xα → x with respect to the local measure topology and so, since .supα ‖xα ‖E ≤ ‖x‖E < ∞, we have that x lies in the ball of radius .supα ‖xα ‖E and therefore .‖x‖E = sup ‖xα ‖E . This shows that (ii) implies (i). ⨆ ⨅ Remark 4.4.17 It follows from Theorem 4.4.16 that if .M is non-atomic, then the norm in a symmetrically normed space .E ⊆ S (τ ) is a Fatou norm if and only if the sets .{x ∈ E : ‖x‖E ≤ c} are closed with respect to the local measure topology for all .0 ≤ c ∈ R. Equivalently, the function .x │−→ ‖x‖E is lower semi-continuous with respect to the local measure topology. As is well known, this latter statement is also equivalent to requiring that .‖x‖E ≤ limα ‖xα ‖E whenever .xα → x locally in measure in E. It is of some interest to point out that the restriction of any symmetric norm to the space .F (τ ) is always a Fatou norm (assuming that the underlying von Neumann algebra .M is either non-atomic or atomic and all minimal projections have equal trace). The details follow and the proof is divided into several lemmas. The following three lemmas are concerned with symmetrically normed spaces on an interval .[0, a), .0 < a ≤ ∞, equipped with the Lebesgue measure. For any measurable function f on .[0, a) and .τ > 0, the dilation .Dτ f of f is defined by setting .(Dτ f ) (t) = f (t/τ ), .t ∈ [0, a) (where .f (t/τ ) = 0 whenever .t/τ ≥ a). Lemma 4.4.18 Suppose that E is a symmetrically normed space on an interval [0, a) and let .f ∈ E be given by

.

f =

N 

.

j =1

αj χ[aj −1 ,aj ) ,

(4.29)

where .0 = a0 < a1 < · · · < aN ≤ a (and .aN < ∞ if .a = ∞) and .α1 > α2 > · · · > αN > 0. If .1 < n ∈ N, then .

  n−1   ‖f ‖E ≤ D n−1 f  ≤ ‖f ‖E . n E n

(4.30)

288

4 Symmetric Spaces of .τ -Measurable Operators

Proof Since the function f is decreasing, it is clear that .0 ≤ D n−1 f ≤ f , n from which the right-hand side of (4.30) follows. Divide each of the intervals & .Ij = aj −1 , aj , .1 ≤ j ≤ N , into n disjoint intervals .Ij k , .1 ≤ k ≤ n, of equal   length . aj − aj −1 /n. For .1 ≤ k ≤ n, define the functions .fk by setting fk =

N 

.

αj χIj \Ij k .

j =1

 Observing that .μ (fk ) = D n−1 f for all k and that . nk=1 fk = (n − 1) f , it follows n that  n  n          ‖fk ‖E = n D n−1 f  , . (n − 1) ‖f ‖E =  fk  ≤ n   E k=1

E

k=1

⨆ ⨅

which yields the left-hand side estimate of (4.30).

Lemma 4.4.19 Suppose that E is a symmetrically normed space on an interval [0, a) and let .f ∈ E be of the same form as in (4.29). If .{gk }∞ k=1 is a sequence of decreasing functions on .[0, a) such that .0 ≤ gk (t) ↑k f (t), .t ∈ [0, a), then .‖gk ‖E ↑ ‖f ‖E as .k → ∞. .

Proof Given .0 < λ < 1, it follows from Lemma 4.4.18 that there exists .1 < n ∈ N such that .τ = (n − 1) /n satisfies .‖Dτ f ‖E ≥ λ ‖f ‖E and .aj −1 < τ aj < aj , .1 ≤ j ≤ N . Note that Dτ f =

N 

.

j =1

αj χ[τ aj −1 ,τ aj ) .

Since .gk (t) ↑k f (t) for all .t ∈ [0, a), there exists .K ∈ N such that     gK τ aj ≥ λf τ aj = λαj ,

.

1 ≤ j ≤ N.

It should be observed that .gK ≥ λDτ f . Indeed, if .τ aj −1 ≤ t < τ aj , then   gK (t) ≥ gK τ aj ≥ λαj = λ (Dτ f ) (t) .

.

Consequently, .

‖gK ‖E ≥ λ ‖Dτ f ‖E ≥ λ2 ‖f ‖E ,

which shows that .supk ‖gk ‖E ≥ λ2 ‖f ‖E . This holds for all .0 < λ < 1 and so, .supk ‖gk ‖E ≥ ‖f ‖E . The proof is complete. ⨅ ⨆

4.4 Symmetric Spaces of τ -Measurable Operators

289

In the next lemma, .F (0, a) denotes the space of all bounded functions on .[0, a) which are supported on a set of finite measures (that is, .F (0, a) = F (m)). Note that if .a < ∞, then .F (0, a) = L∞ (0, a). It should be recalled from Theorem 4.4.6 that .F (0, a), equipped with the norm .‖·‖L1 ∩L∞ , is continuously embedded in every symmetrically normed space E on .[0, a), that is, there exists a constant .C > 0 such that .

‖f ‖E ≤ C ‖f ‖L1 ∩L∞ , f ∈ F (0, a) .

(4.31)

It should also be recalled (see Remark 4.4.15) that in symmetrically normed spaces on an interval .[0, a) the notions of Fatou norm and .σ -Fatou norm coincide. Lemma 4.4.20 If E is a symmetrically normed space on .[0, a), then the restriction of the norm .‖·‖E to .F (0, a) is a Fatou norm, that is, .‖gk ‖E ↑ ‖f ‖E whenever .0 ≤ gk ↑ f in .F (0, a). Proof Suppose that .0 ≤ f ∈ F (0, a) and that .{gk }∞ k=1 is an increasing sequence in .F (0, a) suchthat .0 ≤ gk ↑ f a.e. on .[0, a). Given .ε > 0, there exists a step   N function .h = j =1 αj χAj , with .m Aj < ∞ for all j , such that .0 ≤ h ≤ f and .‖f − h‖L1 ∩L∞ ≤ ε/C (where C is defined in (4.31)). Hence, .‖f − h‖E ≤ ε. Evidently, .0 ≤ gk ∧h ↑ h a.e. on .[0, a) and so, .μ (t; gk ∧ h) ↑k μ (t; h), .t ∈ [0, ∞). The function .μ (h) is of the form (4.29) and the functions .μ (gk ∧ h) are decreasing on .[0, a). Therefore, it follows from Lemma 4.4.19 that .

‖gk ∧ h‖E = ‖μ (gk ∧ h)‖E ↑k ‖μ (h)‖E = ‖h‖E .

Consequently, .

sup ‖gk ‖E ≥ sup ‖gk ∧ h‖E = ‖h‖E k

k

≥ ‖f ‖E − ‖f − h‖E ≥ ‖f ‖E − ε. This holds for all .ε > 0 and so, .supk ‖gk ‖E ≥ ‖f ‖E . This establishes the lemma.

⨆ ⨅

The above lemma, in combination with Proposition 4.4.13, will now be used to prove the corresponding result for symmetrically normed spaces of .τ -measurable operators. Proposition 4.4.21 Suppose that the von Neumann algebra .M, equipped with a semi-finite normal faithful trace .τ , is either non-atomic or is atomic and all minimal projections have equal trace. If .E ⊆ S (τ ) is a symmetrically normed space, then the restriction of the norm .‖·‖E to the subspace .F (τ ) is a Fatou norm, that is, if .0 ≤ xα ↑α x in .F (τ ), then .‖xα ‖E ↑α ‖x‖E . Proof Suppose that .0 ≤ x ∈ F (τ ) and that .{xα } is an increasing net in .F (τ )+ such that .0 ≤ xα ↑α x. The support projection .p = s (x) of x satisfies .τ (p) < ∞ and

290

4 Symmetric Spaces of .τ -Measurable Operators

x = pxp. Moreover, .xα = pxα p for all .α (see the discussion at the beginning of Sect. 3.7) and .(xα )p ↑α xp in .Ep . Consequently, replacing .M by the reduced von Neumann algebra .Mp and the space E by .Ep , it may be assumed, without loss of generality, that .τ (1) < ∞. If .M is atomic and all minimal projections have equal trace, then it follows from Remark 4.4.8 that E is finite dimensional. In this case, it follows from .0 ≤ xα ↑α x that .‖x − xα ‖E →α 0 and so, it is evident that .‖xα ‖E ↑α ‖x‖E . Assume now that .M is non-atomic. Let the symmetrically normed space .E (0, τ (1)) on the interval .[0, τ (1)) be defined as in Proposition 4.4.13. By Proposition 3.2.14 (i), .0 ≤ xα ↑α x implies that .μ (xα ) ↑α μ (x) in the space .E (0, τ (1)). Furthermore, it is clear that .μ (x) ∈ F (0, τ (1)) = L∞ (0, τ (1)) and so, Lemma 4.4.20 implies that .

.

‖xα ‖E = ‖μ (xα )‖E(0,τ (1)) ↑α ‖μ (x)‖E(0,τ (1)) = ‖x‖E . ⨆ ⨅

This completes the proof of the proposition.

As will be seen in Theorem 4.5.8, the norm .‖·‖E is also Fatou on the subspace E b , the closure of .F (τ ) in E. References: [40, 42, 44, 45, 95–97, 116–118, 140].

.

4.5 The Köthe Dual of a Symmetric Space The Köthe dual of symmetrically normed spaces will be discussed next. As before, M is a von Neumann algebra equipped with a fixed semi-finite faithful normal trace .τ . Recall from Theorem 3.10.21 and Remark 3.10.22 that if the von Neumann algebra .M is either non-atomic or atomic and all minimal projections have equal trace, then

.



∞

.

μ (t; x) μ (t; y) dt = sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x)}

(4.32)

0

for all .x, y ∈ S (τ ). It should be observed that if .E ⊆ S (τ ) is a non-zero symmetrically normed space and if the von Neumann algebra .M is either non-atomic or atomic with all atoms having equal trace, then it follows from Lemma 4.4.5 that the carrier projection .cE of E is equal to .1 and hence, by Lemma 4.3.3, .‖·‖E × is a norm on .E × . Proposition 4.5.1 Suppose that the von Neumann algebra .M is either non-atomic or atomic and all minimal projections have equal trace. Let .{0} /= E ⊆ S (τ ) be a symmetrically normed space. If .y ∈ S (τ ), then .

sup {τ (|xy|) : x ∈ E, ‖x‖E ≤ 1}  = sup 0

∞

 μ (t; x) μ (t; y) dt : x ∈ E, ‖x‖E ≤ 1 .

4.5 The Köthe Dual of a Symmetric Space

291

In particular, .y ∈ E × if and only if  .

∞

sup 0

 μ (t; x) μ (t; y) dt : x ∈ E, ‖x‖E ≤ 1 < ∞,

in which case  .

‖y‖E × = sup

0

∞

 μ (t; x) μ (t; y) dt : x ∈ E, ‖x‖E ≤ 1 .

Proof Using that E is symmetrically normed, it follows immediately from (4.32) that  ∞  . sup μ (t; x) μ (t; y) dt : x ∈ E, ‖x‖E ≤ 1 0

= sup {τ (|zy|) : z ∈ S (τ ) , μ (z) ≤ μ (x) , x ∈ E, ‖x‖E ≤ 1} = sup {τ (|xy|) : x ∈ E, ‖x‖E ≤ 1} . The second statement of the proposition follows immediately from the definition of E × and its norm .‖·‖E × . The proof is complete. ⨆ ⨅

.

Before formulating the next theorem, it will be convenient to introduce some terminology. It should be recalled that if .x, y ∈( S (τ ), then x is ( t said to be t submajorized by y, denoted by .x ≺≺ y, whenever . 0 μ (s; x) ds ≤ 0 μ (s; y) ds for all .t > 0 (see Sect. 3.9). In particular, if .μ (x) ≤ μ (y), then .x ≺≺ y. Definition 4.5.2 A linear subspace E of .S (τ ), equipped with a norm .‖·‖E , is called fully symmetrically normed if .x ∈ S (τ ), .y ∈ E and .x ≺≺ y imply that .x ∈ E, and .‖x‖E ≤ ‖y‖E . If, in addition, E is a Banach space, then .E is called a fully symmetric space (of .τ -measurable operators). Example 4.5.3 From the definitions it is clear that the spaces .L∞ (τ ) = M, .L1 (τ ), L1 (τ ) ∩ L∞ (τ ), and .L1 (τ ) + L∞ (τ ) are fully symmetric spaces.

.

Remark 4.5.4 (a) Evidently, any fully symmetrically normed space E is, in particular, symmetrically normed and hence, by Proposition 4.4.3, E is a normed .M-bimodule of .τ -measurable operators. (b) If .E ⊆ S (τ ) is a fully symmetrically normed space and .E /= {0}, then .cE = 1 (without any additional assumptions on .M or .τ ). In fact, since .E /= {0}, there exists .0 /= p ∈ P (E). If .τ (p) = ∞, then .μ (p) = μ (1) and so, .1 ∈ E and hence, .cE = 1. Suppose that .0 < τ (p) < ∞ and let .q ∈ P (M) be a finite trace projection. Taking .n ∈ N such that .τ (q) ≤ nτ (p), it is easily verified that .q ≺≺ np, and so .q ∈ E. This shows that .P (E) contains all finite trace projections in .P (M), which implies, in particular, that .cE = 1.

292

4 Symmetric Spaces of .τ -Measurable Operators

Theorem 4.5.5 Let .M be von Neumann algebra which is either non-atomic or atomic with all atoms having equal trace. If .{0} /= E ⊆ S (τ ) is a symmetrically normed space, then .E × is a fully symmetric space with the Fatou property. Proof It will be shown first that .E × is fully symmetrically normed. If .y ∈ E × , then it follows from Proposition 4.5.1 that  .

‖y‖E × = sup

∞

0

 μ (t; z) μ (t; y) dt : z ∈ E, ‖z‖E ≤ 1 < ∞.

If .x ∈ S (τ ) satisfies .x ≺≺ y, that is, .μ (x) ≺≺ μ (y), then Lemma 3.10.10 implies that  ∞  ∞ . μ (t; z) μ (t; x) dt ≤ μ (t; z) μ (t; y) dt, z ∈ E, ‖z‖E ≤ 1, 0

0

and hence,  .

∞

sup 0

 μ (t; z) μ (t; x) dt : z ∈ E, ‖z‖E ≤ 1



∞

≤ sup 0

 μ (t; z) μ (t; y) dt : z ∈ E, ‖z‖E ≤ 1 < ∞.

Proposition 4.5.1 shows that .x ∈ E × and .‖x‖E × ≤ ‖y‖E × . Hence, .E × is a fully symmetrically normed space. To show that .E × has the Fatou property, observe that Proposition 4.4.4 implies that the embedding of .E × into .S (τ ) is continuous with respect   to the norm topology in .E × and the measure topology in .S (τ ). Suppose that . aβ is an upward directed    +   system in . E × satisfying .supβ aβ E = M < ∞. This implies that . aβ is bounded with respect to the measure topology in .S (τ ) and hence, it follows from Theorem 2.6.15 that there exists .a ∈ S (τ )+ such that .aβ ↑β a. If .x ∈ E and .‖x‖E ≤ 1, then it follows from Proposition 4.5.1 that  .

0



∞

    μ (t; x) μ t; aβ dt ≤ aβ E × ≤ M



for all .β. Since .μ aβ ↑ μ (a) in .L0 (0, ∞), this implies that  .

0

∞

μ (t; x) μ (t; a) dt ≤ M < ∞,

x ∈ E, ‖x‖E ≤ 1,

× from which it follows (once again using Proposition 4.5.1) that  .a ∈ E and .‖a‖E × ≤ M. On the other hand, since .0 ≤ aβ ≤ a, it is clear that .aβ  × ≤ ‖a‖E × E for all .β and so, .M ≤ ‖a‖E × . This shows that .E × has the Fatou property.

4.5 The Köthe Dual of a Symmetric Space

293

As observed in Corollary 4.1.22, the Fatou property implies that .E × is complete and hence, .E × is a fully symmetric space. ⨆ ⨅ Under the assumptions of Theorem 4.5.5, it follows from this theorem that .E × and .E ×× are also fully symmetric spaces. Moreover, .E ⊆ E ×× and .‖x‖E ×× ≤ ‖x‖E for all .x ∈ E. The next theorem characterizes norms .‖·‖E for which .‖x‖E ×× = ‖x‖E holds for all .x ∈ E. Theorem 4.5.6 Suppose that the von Neumann algebra .M is either non-atomic or atomic with all atoms having equal trace. If .{0} /= E ⊆ S (τ ) is a symmetrically normed space, then the following statements are equivalent: (i) The norm in E is a Fatou norm. (ii) The norm closed unit ball .BE of E is closed in E with respect to the local measure topology. (iii) .‖x‖E ×× = ‖x‖E for all .x ∈ E. Proof It has already been shown in Theorem 4.4.16 that assertions (i) and (ii) are equivalent. It will be proved now that (i) and (ii) imply (iii). Suppose that .‖·‖E is a Fatou norm. Denoting the norm closed unit ball in E by .BE , let K be the closure in .L1 (τ ) of the set .BE ∩ L1 (τ ). Hence, K is a closed absolutely convex subset of .L1 (τ ). It should be observed that .K ∩ E ⊆ BE . Indeed, if .x ∈ K ∩ E, then there exists a sequence .{xn }∞ n=1 in .BE ∩ L1 (τ ) such that .‖x − xn ‖1 → 0 as .n → ∞. This implies, in particular, that .xn → x with respect to the measure topology. Since .BE is closed in E with respect to the local measure topology, it follows that .x ∈ BE , which establishes the claim. Let .z ∈ BE ∩ L1 (τ ) satisfying .‖z‖E = 1 be fixed and define for .ε > 0 the element .zε ∈ E by setting .zε = (1 + ε) z. Keep .ε > 0 fixed for the moment. Since .zε ∈ / BE , it follows from the above that .zε ∈ / K. Therefore, it follows from the Hahn–Banach separation theorem that there exists .y ∈ M (identified with the Banach dual of .L1 (τ )) satisfying .

|τ (xy)| ≤ 1, x ∈ BE ∩ L1 (τ ) and |τ (zε y)| > 1.

Observe that τ (|xy|) ≤ 1,

.

x ∈ BE ∩ L1 (τ ) .

Indeed, if .x ∈ BE ∩ L1 (τ ) and .xy = v |xy| is the polar decomposition of xy, then v ∗ x ∈ BE ∩ L1 (τ ) and so,

.

τ (|xy|) = τ

.

 ∗   v x y ≤ 1.

Suppose that .x ∈ BE . Since the carrier projection of the .M-bimodule .E ∩ L1 (τ ) is equal to .1, there exists an upward directed net .{xα } in .E∩L1 (τ ) such that .0 ≤ xα ↑α |x|. This implies, in particular, that .xα → |x| with respect to the local measure

294

4 Symmetric Spaces of .τ -Measurable Operators Tlm

topology. If .x = v |x| is the polar decomposition of x, then .vxα y → xy. Since .vxα ∈ BE ∩ L1 (τ ) for all .α, it is clear that .τ (|vxα y|) ≤ 1 for all .α and hence, it follows from Proposition 3.4.19 that .τ (|xy|) ≤ 1. This shows that .

sup {τ (|xy|) : x ∈ BE } ≤ 1

and so, .y ∈ E × and .‖y‖E × ≤ 1. Since .|τ (zε y)| > 1, it follows that 1 < |τ (zε y)| ≤ ‖zε ‖E ×× ‖y‖E × ≤ ‖zε ‖E ×× .

.

The definition of .zε implies that .(1 + ε) ‖z‖E ×× > 1 holds for all .ε > 0 and hence, ‖z‖E ×× ≥ 1. It thus has been shown that .‖z‖E ×× ≥ 1 for all .0 ≤ z ∈ E ∩ L1 (τ ) satisfying .‖z‖E = 1. Consequently, .‖z‖E ×× ≥ ‖z‖E for all .z ∈ E ∩ L1 (τ ), that is,

.

.

‖z‖E ×× = ‖z‖E ,

z ∈ E ∩ L1 (τ ) .

  Finally, if .0 ≤ z ∈ E, then there exists an upward directed system . zβ in .E ∩ L1 (τ ) such that .0 ≤ zβ ↑β z. Since .‖·‖E is a Fatou norm, this implies that       .zβ  ↑ ‖z‖E . It follows from the above that .zβ E ×× = zβ E for all .β and so, E β .

    ‖z‖E ×× ≥ sup zβ E ×× = sup zβ E = ‖z‖E . β

β

Since the inequality .‖z‖E ×× ≤ ‖z‖E always holds (see Proposition 4.3.10), this shows that .‖z‖E ×× = ‖z‖E . This suffices to prove that .‖x‖E = ‖x‖E ×× for all .x ∈ E. The proof that (i) (equivalently, (ii)) implies (iii) is complete. Suppose now that the embedding of E into .E ×× is an isometry. As follows from Theorem 4.5.5, the space .E ×× has the Fatou property and so, in particular, .‖·‖E ×× is a Fatou norm. Consequently, if .0 ≤ xα ↑α x holds in E, then .0 ≤ xα ↑α x in ×× and hence, .E .

‖xα ‖E = ‖xα ‖E ×× ↑α ‖x‖E ×× = ‖x‖E .

The proof of the theorem is complete.

⨆ ⨅

Corollary 4.5.7 Suppose that the von Neumann algebra .M is either non-atomic or atomic and all minimal projections have equal trace. Let .E ⊆ S (τ ) be a symmetrically normed space with a Fatou norm. If .x, y ∈ E satisfy .x ≺≺ y, then .‖x‖E ≤ ‖y‖E .

4.5 The Köthe Dual of a Symmetric Space

295

Proof It may be assumed that .E = / {0}. By the previous theorem, the embedding of E into .E ×× is an isometry. Therefore, it follows from Theorem 4.5.5 that .

‖x‖E = ‖x‖E ×× ≤ ‖y‖E ×× = ‖y‖E . ⨆ ⨅

In combination with Proposition 4.4.21, the above results have some interesting consequences. Theorem 4.5.8 Let the von Neumann algebra .M be either non-atomic or atomic with all minimal projections having equal trace. If .{0} /= E ⊆ S (τ ) is a symmetrically normed space, then the following statements hold: (i) (ii) (iii) (iv)

‖x‖E ×× = ‖x‖E for all .x ∈ E b . The restriction of the norm .‖·‖E to the space .E b is a Fatou norm. If .x, y ∈ E b satisfy .x ≺≺ y, then .‖x‖E ≤ ‖y‖E . If E is a Banach space, then .E b is a fully symmetric space.

.

Proof (i) It should be observed first that it follows from Remark 4.3.4 that .F (τ )× = E × isometrically, where .F (τ ) is equipped with the norm .‖·‖E . Consequently, ×× .F (τ ) = E ×× isometrically. By Proposition 4.4.21, the norm .‖·‖E is Fatou on the subspace .F (τ ). Therefore, it follows from Theorem 4.5.6, applied to .F (τ ) equipped with the norm .‖·‖E , that .‖x‖E ×× = ‖x‖E for all .x ∈ F (τ ). If .x ∈ E b , then there exists a sequence .{xn }∞ n=1 in .F (τ ) such that .‖x − xn ‖E → 0 as .n → ∞. In particular, .‖xn ‖E → ‖x‖E . Since .‖x − xn ‖E ×× ≤ ‖x − xn ‖E for all n, it follows that also .‖xn ‖E = ‖xn ‖E ×× → ‖x‖E ×× and hence, .‖x‖E ×× = ‖x‖E . (ii) and (iii) By Theorem 4.5.5, the space .E ×× is fully symmetric and .‖·‖E ×× is a Fatou norm. Therefore, (ii) and (iii) are immediate consequences of part (i). (iv) It is assumed now, in addition, that E is a Banach space. As observed before (see the discussion following Definition 4.4.9), this implies that .E b = E L1 (τ ) ∩ L∞ (τ ) . Suppose that .x ∈ S (τ ) and .y ∈ E b satisfy .x ≺≺ y and let ∞ .{yn } n=1 be a sequence in .L1 (τ ) ∩ L∞ (τ ) such that .‖y − yn ‖E → 0 as .n → ∞. It follows from Theorem 3.10.13 that there exists an .(L1 , L∞ )-contraction T in .L1 (τ ) + L∞ (τ ) such that .T y = x. By Proposition 3.10.3, .T z ≺≺ z for all ×× and .z ∈ L1 (τ ) + L∞ (τ ) and so it follows from Theorem 4.5.5 that .T z ∈ E ×× .‖T z‖E ×× ≤ ‖z‖E ×× for all .z ∈ E . Consequently, .‖x − T yn ‖E ×× → 0 as b .n → ∞ and .T yn ∈ L1 (τ ) ∩ L∞ (τ ) for all n. Since .‖z‖E ×× = ‖z‖E for all .z ∈ E b b and .E is complete with respect to the norm .‖·‖E , this implies that .x ∈ E and .‖x − T yn ‖E → 0. Moreover, .‖T yn ‖E ≤ ‖yn ‖E for all n and hence, .‖x‖E ≤ ‖y‖E . This shows that .E b is a fully symmetric space, which completes the proof of the ⨆ ⨅ theorem. References: [42, 44, 45, 116–118, 140].

Chapter 5

Strongly Symmetric Spaces of τ -Measurable Operators .

Abstract This chapter exhibits the theory of strongly symmetric spaces of .τ measurable operators (without any additional assumptions on the underlying semifinite von Neumann algebra). This includes, in particular, the class of fully symmetric spaces. Duality theory is discussed in detail and a version of the fundamental Hewitt–Yosida decomposition is presented. Several characterizations of order continuity of the norm are given.

5.1 Strongly Symmetric Spaces and Their Köthe Duals As before, .M is a von Neumann algebra on a Hilbert space H , equipped with semifinite normal faithful trace .τ . In Sects. 4.4 and 4.5, most of the important results concerning symmetrically normed spaces E of .τ -measurable operators are obtained under the additional hypothesis that the underlying von Neumann algebra .M is either non-atomic or atomic and all minimal projections have the same trace. In the present chapter, the assumption on the norm of E is slightly strengthened (it is assumed to be strongly symmetric), but it turns out that no additional assumptions on the von Neumann algebra are necessary. Definition 5.1.1 A symmetrically normed space .E ⊆ S (τ ) is called strongly symmetrically normed if its norm .‖·‖E has the additional property that .‖x‖E ≤ ‖y‖E whenever .x, y ∈ E satisfy .x ≺≺ y. Moreover, if E is also a Banach space, then E is called a strongly symmetric space (of .τ -measurable operators). Remark 5.1.2 (a) It is clear that any fully symmetrically normed space (see Definition 4.5.2) is strongly symmetrically normed. (b) By Corollary 4.5.7, if the von Neumann algebra is non-atomic and if .E ⊆ S (τ ) is a symmetrically normed space with a Fatou norm, then E is strongly symmetrically normed.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. G. Dodds et al., Noncommutative Integration and Operator Theory, Progress in Mathematics 349, https://doi.org/10.1007/978-3-031-49654-7_5

297

298

5 Strongly Symmetric Spaces of .τ -Measurable Operators

(c) If the von Neumann algebra .M is non-atomic and if .{0} /= E ⊆ S (τ ) is a symmetrically normed space, then it follows from Theorem 4.5.8 (iii) that .E b , equipped with .‖·‖E , is strongly symmetrically normed. In the proof of the next lemma, it will be used that if .x ∈ S (τ ), then the function 1 .t − │ → t



t

μ (s; x) ds,

t > 0,

0

is decreasing on .(0, ∞). This follows immediately from the fact that .μ (x) is a decreasing function. Indeed, if .0 < t1 < t2 , then 1 . t2



t2

 μ (s; x) ds =

0

1



1

μ (t2 s; x) ds ≤

0

0

1 μ (t1 s; x) ds = t1



t1

μ (s; x) ds, 0

which proves the claim. Lemma 5.1.3 If .E ⊆ S (τ ) is a strongly symmetrically normed space, then the following statements hold: (i) .E ⊆ L1 (τ ) + L∞ (τ ) and the embedding of E into .L1 (τ ) + L∞ (τ ) is continuous. (ii) If .cE = 1, then .F (τ ) ⊆ E and the embedding of .F (τ ) into E is continuous with respect to the norm .‖·‖L1 ∩L∞ in .F (τ ) and the norm in E; moreover, if E is a Banach space, then .L1 (τ ) ∩ L∞ (τ ) ⊆ E with continuous embedding. Proof (i) If / L1 (τ ) + L∞ (τ ), then it follows from Lemma 3.10.1 that  t .x ∈ E, but .x ∈ . μ x) ds = ∞ for all .t > 0. This implies that .nx ≺≺ x and so, .n ‖x‖E ≤ (s; 0 ‖x‖E for all .n ∈ N. Hence .x = 0, which is clearly a contradiction. This shows that .E ⊆ L1 (τ ) + L∞ (τ ). If the embedding of E into .L1 (τ ) + L∞ (τ ) is not continuous, then there exists a sequence .{xn }∞ n=1 in E satisfying .‖xn ‖E → 0 and .‖xn ‖L1 +L∞ = 1 0 μ (s; xn ) ds → ∞ as.n → ∞. Since .μ (xn ) is a decreasing function for t each n, this implies that . 0 μ (s; xn ) ds → ∞ as .n → ∞ for all .t > 0. Let .p ∈ P (E) be fixed such that .0 < τ (p) < ∞ and let .N ∈ N be such that  τ (p) . μ (s; xn ) ds ≥ τ (p) for all .n ≥ N. It should be observed that .p ≺≺ xn 0 whenever .n ≥ N. Indeed, if .0 < t ≤ τ (p), then 1≤

.

1 τ (p)



τ (p)

μ (s; xn ) ds ≤

0

1 t



t

μ (s; xn ) ds 0

t t and so, . 0 μ (s; p) ds = t ≤ 0 μ (s; xn ) ds for all .n ≥ N. If .t ≥ τ (p), then  .

0

t

 μ (s; p) ds = τ (p) ≤ 0

τ (p)

 μ (s; xn ) ds ≤

t

μ (s; xn ) ds 0

5.1 Strongly Symmetric Spaces and Their Köthe Duals

299

for all .n ≥ N, which proves the claim. Since the norm in E is strongly symmetric, it follows from .p ≺≺ xn that .‖p‖E ≤ ‖xn ‖E , .n ≥ N, and this contradicts the assumption that .‖xn ‖E → 0. Consequently, it may be concluded that the embedding of E into .L1 (τ ) + L∞ (τ ) is continuous. (ii) Assuming that .cE = 1, it follows immediately from Lemma 4.4.5 (i) that .F (τ ) ⊆ E. To show that the embedding of .F (τ ) into E is continuous, fix any projection .p ∈ P (E) satisfying .0 < τ (p) < ∞. Suppose that .{xn }∞ n=1 is a sequence in .F (τ ) such that .‖xn ‖L1 ∩L∞ → 0, that is, .‖xn ‖∞ → 0 and .‖xn ‖1 → 0 as .n → ∞. Given .ε > 0, let .N ∈ N be such that .‖xn ‖∞ ≤ ε and .‖xn ‖1 ≤ ετ (p) for all .n ≥ N. If .0 < t ≤ τ (p), then 

t

.

0

 μ (s; xn ) ds ≤ ‖xn ‖∞ t ≤

t

μ (s; εp) ds,

n ≥ N,

0

and if .t > τ (p), then 

t

.

0

 μ (s; xn ) ds ≤ ‖xn ‖1 ≤

t

μ (s; εp) ds,

n ≥ N.

0

Hence, .xn ≺≺ εp and so, .‖xn ‖E ≤ ε ‖p‖E for all .n ≥ N . This shows that xn → 0 in E as .n → ∞. Assuming that E is a Banach space, the proof of the embedding of .L1 (τ ) ∩ L∞ (τ ) into E is now identical to the proof of the last part of Theorem 4.4.6 (ii). ⨆ ⨅

.

Next, the Köthe dual of a strongly symmetrically normed space .E ⊆ S (τ ) will be discussed. It will be assumed that the carrier projection .cE of E is equal to .1, and hence Lemma 5.1.3 implies that F (τ ) ⊆ E ⊆ L1 (τ ) + L∞ (τ ) ,

.

(5.1)

with continuous embeddings. It should be recalled from Corollary 3.10.20 that  .

∞

μ (s; x) μ (s; y) ds = sup {τ (|zy|) : z ∈ F (τ ) , z ≺≺ x}

(5.2)

0

holds for all .x, y ∈ S (τ ). Proposition 5.1.4 If .y ∈ S (τ ), then .

sup {τ (|xy|) : x ∈ E, ‖x‖E ≤ 1}  = sup 0

∞

 μ (t; x) μ (t; y) dt : x ∈ E, ‖x‖E ≤ 1 .

300

5 Strongly Symmetric Spaces of .τ -Measurable Operators

In particular, .y ∈ E × if and only if 

 μ (t; x) μ (t; y) dt : x ∈ E, ‖x‖E ≤ 1 < ∞,

∞

sup

.

0

in which case  .

‖y‖E × = sup

0

∞

 μ (t; x) μ (t; y) dt : x ∈ E, ‖x‖E ≤ 1 .

Proof Using that .F (τ ) ⊆ E, in combination with (5.2), it follows that 

∞

sup

.

0

 μ (t; x) μ (t; y) dt : x ∈ E, ‖x‖E ≤ 1

= sup {τ (|zy|) : z ∈ F (τ ) , z ≺≺ x, x ∈ E, ‖x‖E ≤ 1} ≤ sup {τ (|xy|) : x ∈ F (τ ) , ‖x‖E ≤ 1} ≤ sup {τ (|xy|) : x ∈ E, ‖x‖E ≤ 1} . On the other hand, by Theorem 3.4.29,  τ (|xy|) ≤

∞

μ (t; x) μ (t; y) dt

.

0

for all .x ∈ E satisfying .‖x‖E ≤ 1. This proves the first assertion of the proposition. The remaining statements follow immediately from the definition of the Köthe dual × and the definition of .‖·‖ × . .E ⨆ ⨅ E Theorem 5.1.5 If .E ⊆ S (τ ) is a strongly symmetrically normed space with .cE = 1, then .E × is a fully symmetric space with the Fatou property. Proof Using Proposition 5.1.4 instead of Proposition 4.5.1, the proof of this theorem is exactly the same as the proof of Theorem 4.5.5. ⨆ ⨅ It is of some interest to point out another consequence of Proposition 5.1.4. Proposition 5.1.6 If .E ⊆ S (τ ) is a strongly symmetrically normed space with cE = 1, then there exists a fully symmetric space .F ⊆ S (0, ∞), with the Fatou property, such that:

.

(i) .E × = {y ∈ S (τ ) : μ (y) ∈ F }. (ii) .‖y‖E × = ‖μ (y)‖F for all .y ∈ E × . Proof Let .F ⊆ S (0, ∞) consist of all functions .f ∈ S (0, ∞) for which  .

sup 0

∞

 μ (s; f ) μ (s; x) ds : x ∈ E, ‖x‖E ≤ 1 < ∞

5.1 Strongly Symmetric Spaces and Their Köthe Duals

301

and define  .

‖f ‖F = sup

∞ 0

 μ (s; f ) μ (s; x) ds : x ∈ E, ‖x‖E ≤ 1 ,

f ∈ F.

Suppose that .f, g ∈ F . It follows from Theorem 3.9.9 that .μ (f + g) ≺≺ μ (f ) + μ (g) and hence, Lemma 3.10.10 implies that  .

∞

 μ (s; f + g) μ (s; x) ds ≤

0

∞



∞

μ (s; f ) μ (s; x) ds +

0

μ (s; g) μ (s; x) ds 0

for all .x ∈ E satisfying .‖x‖E ≤ 1. Consequently, .f + g ∈ F and .‖f + g‖F ≤ ‖f ‖F +‖g‖F . This suffices to show that .F is a linear subspace of .S (0, ∞) and .‖·‖F ∞ is a semi-norm on F . If .f ∈ F and .‖f ‖F = 0, then . 0 μ (s; f ) μ (s; x) ds = 0 for all .x ∈ E. In particular, if .p ∈ P (E) satisfies .0 < τ (p) < ∞, then  τ (p) . μ (s; f ) ds = 0, from which it follows that .μ (f ) = 0 and so, .f = 0. 0 Therefore, .‖·‖F is a norm on F . From Proposition 5.1.4, it is clear that an element .y ∈ S (τ ) belongs to .E × if and only if .μ (y) ∈ F and that .‖y‖E × = ‖μ (y)‖F for all .y ∈ E × . To show that F is fully symmetrically normed, suppose that .f ∈ F and that .g ∈ S (0, ∞) satisfies .g ≺≺ f . By Lemma 3.10.10, this implies that 

∞

.



∞

μ (s; g) μ (s; x) ds ≤

0

μ (s; f ) μ (s; x) ds 0

for all .x ∈ E with .‖x‖E ≤ 1. Consequently, .g ∈ F and .‖g‖F ≤ ‖f ‖F . It remains to be proved that F has the Fatou property. Let .{fn }∞ n=1 be an increasing sequence in .F + such that .supn ‖fn ‖F < ∞. This implies, in particular, that .{fn }∞ n=1 is bounded in measure in .S (0, ∞) and hence, there exists .0 ≤ f ∈ S (0, ∞) such that .0 ≤ fn ↑ f a.e. on .(0, ∞). If .x ∈ E satisfies .‖x‖E ≤ 1, then  .

∞

 μ (s; fn ) μ (s; x) ds ↑n

0

∞

μ (s; f ) μ (s; x) ds, 0

from which it follows that  ∞ . sup μ (s; f ) μ (s; x) ds ≤ sup ‖fn ‖F < ∞. ‖x‖E ≤1 0

n

Consequently, .f ∈ F and .‖f ‖F ≤ supn ‖fn ‖F . Finally, by the Fatou property proved above and Corollary 4.1.22, F is a Banach space. This suffices to complete the proof of the proposition. ⨆ ⨅ Corollary 5.1.7 If .E ⊆ S (τ ) is a strongly symmetrically normed space with .cE = 1, then the closed unit ball of .E × is closed in .S (τ ) with respect to the local measure topology.

302

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof Suppose that .{yα } is a net in .BE × and that .y ∈ S (τ ) is such that .yα → y with respect to the local measure topology. If .y = v |y| is the polar decomposition of y, then .v ∗ yα → |y| with respect to the local measure topology and .v ∗ yα ∈ BE × for all .α. Therefore, replacing y by .|y|, it may be assumed, without loss of generality, that .y ≥ 0. If .p ∈ P (M) and .τ (p) < ∞, then .pyα p → pyp in measure and so, there  ∞ exists a subsequence . yαn n=1 such that .pyαn p → pyp in measure as .n → ∞. By  Proposition 3.2.11 (ii) this implies that .μ pyαn p → μ (pyp) a.e. on .(0, ∞). .F ⊆ S (0, ∞), with the Fatou property, be as in Let the fully symmetric

space





Proposition 5.1.6. Since . μ pyαn p

= pyαn p E × ≤ 1 for all n, it is clear that F



.lim.n→∞ μ pyαn p

≤ 1 and hence, by Remark 4.1.24, .μ (pyp) ∈ F and F .



‖μ (pyp)‖F ≤ limn→∞ μ pyαn p F ≤ 1.

This shows that .pyp ∈ E × and .‖pyp‖E × ≤ 1 for all .p ∈ P (M) satisfying .τ (p) < ∞.    Let . pβ be an increasing net in .P (M) such that .pβ ↑β 1 and .τ pβ < ∞  1/2  for all .β. Since .μ pβ ypβ = μ y pβ y 1/2 (see Proposition 3.2.10 (ii)), it follows that .y 1/2 pβ y 1/2 ∈ E × and . y 1/2 pβ y 1/2 E × ≤ 1 for all .β. Furthermore, 1/2 p y 1/2 ↑ y and hence, the Fatou property of .E × implies that .y ∈ E × .0 ≤ y β β and .‖y‖E × ≤ 1, that is, .y ∈ BE × . The proof is complete. ⨆ ⨅ The result of the above corollary will be strengthened in Proposition 5.2.16, where it will be shown that the unit ball of .E × is actually complete for the local measure topology. The embedding of a strongly symmetrically normed space E into its second Köthe dual .E ×× will be discussed next. It should be recalled that always .E ⊆ E ×× and .‖x‖E ×× ≤ ‖x‖E for all .x ∈ E (see Proposition 4.3.10). The next objective is to characterize strongly symmetrically normed spaces for which the embedding of E into .E ×× is an isometry (see Theorem 5.1.9). For this purpose, the following lemma is needed, which may be of interest in its own right. For another application of this lemma, see Proposition 5.1.13. As before, it is assumed that .cE = 1. Recall that this implies that the inclusions of (5.1) hold. Lemma 5.1.8 If .E ⊆ S (τ ) is a strongly symmetrically normed space, then .‖x‖E = ‖x‖E ×× for all .x ∈ E b , the norm closure of .F (τ ) in E. In particular, if E is a strongly symmetric space, then .‖x‖E = ‖x‖E ×× for all .x ∈ L1 (τ ) ∩ L∞ (τ ). Proof Suppose that .0 ≤ a ∈ E is of the form a=

n 

.

j =1

αj ej ,

(5.3)

5.1 Strongly Symmetric Spaces and Their Köthe Duals

303

where  .e1 , . . . , en are mutually orthogonal projections in .P (M) satisfying .0 < τ ej < ∞ and .0 < αj ∈ R, .1 ≤ j ≤ n. Assume first, in addition, that .‖a‖E = 1. For .ε > 0, set .aε = (1 + ε) a. Define the n-dimensional linear subspace W of E by setting .W = span {e1 , . . . , en }. A moment’s reflection shows that every linear functional on W is given by .x │−→ τ (xy), .x ∈ W , where .y = nj=1 ηj ej with .ηj ∈ C, .1 ≤ j ≤ n. As before, .BE denotes the closed unit ball in E. Defining .K = BE ∩W , it is clear that K is absolutely convex and closed in W (with respect to the norm .‖·‖E ). Since .‖aε ‖E = 1 + ε, the element .aε does not belong to K and so, by the Hahn–Banach separation theorem, there exists an element .y = nj=1 ηj ej (.ηj ∈ C, .1 ≤ j ≤ n) such that .

|τ (xy)| ≤ 1, x ∈ K,

and

|τ (aε y)| > 1.

(5.4)

Define the linear operator T from .L1 (τ ) + L∞ (τ ) into itself by setting  n  τ zej  ej , .T z = τ ej j =1

z ∈ L1 (τ ) + L∞ (τ ) ,

and observe that .‖T z‖1 ≤ ‖z‖1 , .z ∈ L1 (τ ), and .‖T z‖∞ ≤ ‖z‖∞ , .z ∈ L∞ (τ ). Consequently, it follows from Proposition 3.10.3 that .T z ≺≺ z for all .z ∈ L1 (τ ) + L∞ (τ ). Since .T z ∈ W ⊆ E for all z, the assumption on the norm .‖·‖E implies that .‖T z‖E ≤ ‖z‖E for all .z ∈ E. In particular, .T z ∈ K whenever .z ∈ BE and so, it follows from (5.4) that .

|τ ((T z) y)| ≤ 1,

z ∈ BE .

Since  n n  n     τ zej  ηk τ ej ek = τ zej ηj = τ (zy) , .τ ((T z) y) = τ ej j =1 k=1 j =1 this implies that .

|τ (zy)| ≤ 1,

z ∈ BE ,

and hence, .‖y‖E × ≤ 1. Using that .|τ (aε y)| > 1, it follows that 1 < (1 + ε) |τ (ay)| ≤ (1 + ε) ‖a‖E ×× ‖y‖E × ≤ (1 + ε) ‖a‖E ×× .

.

This holds for all .ε > 0 and so, .‖a‖E ×× ≥ 1, that is, .‖a‖E ×× = ‖a‖E = 1. Consequently, .‖a‖E ×× = ‖a‖E holds for all elements .0 ≤ a ∈ F (τ ) which are of the form (5.3). If .0 ≤ a ∈ F (τ ) is arbitrary, then it follows from the spectral theorem that there exists an increasing sequence .{ak }∞ k=1 , each .ak of the form (5.3),

304

5 Strongly Symmetric Spaces of .τ -Measurable Operators

such that .0 ≤ ak ↑ a and .‖a − ak ‖L1 ∩L∞ → 0 as .k → ∞. By Lemma 5.1.3 (ii), the embeddings of .F (τ ) (equipped with the norm .‖·‖L1 ∩L∞ ) into E and .E ×× are continuous and so, .‖a − ak ‖E → 0 and .‖a − ak ‖E ×× → 0 as .k → ∞. Since .‖ak ‖E ×× = ‖ak ‖E for all k, this implies that .‖a‖E ×× = ‖a‖E . It may be concluded that .‖x‖E ×× = ‖x‖E for all .x ∈ F (τ ). Finally, if .x ∈ E b , then there exists a sequence .{xk }∞ k=1 such that .‖x − xk ‖E → 0. Using that .‖x − xk ‖E ×× ≤ ‖x − xk ‖E and .‖xk ‖E ×× = ‖xk ‖E , it follows that b .‖x‖E ×× = ‖x‖E . If E is assumed to be complete, then .L1 (τ ) ∩ L∞ (τ ) ⊆ E (cf. the last part of the proof of Theorem 4.4.6 (ii) together with Lemma 5.1.3) and so, the proof of the lemma is complete. ⨆ ⨅ Theorem 5.1.9 If .E ⊆ S (τ ) is a strongly symmetrically normed space with .cE = 1, then the following statements are equivalent: (i) The norm in E is a Fatou norm. (ii) The embedding of E into .E ×× is an isometry. (iii) The norm closed unit ball .BE of E is closed in E with respect to the local measure topology. Proof Suppose that .‖·‖E is a Fatou norm. If .0 ≤ x ∈ E, then there exists an increasing net .{xα } in .F (τ )+ such that .0 ≤ xα ↑α x. By hypothesis, this implies that .‖xα ‖E ↑α ‖x‖E . By Theorem 5.1.5, .E ×× has the Fatou property and so, in particular, .‖·‖E ×× is a Fatou norm. Consequently, .‖xα ‖E ×× ↑α ‖x‖E ×× . Moreover, it follows from Lemma 5.1.8 that .‖xα ‖E = ‖xα ‖E ×× and hence, .‖x‖E = ‖x‖E ×× . This shows that (i) implies (ii). Assuming that assertion (ii) holds, let .{xα } be a net in .BE such that .xα → x with respect to the local measure topology, for some .x ∈ E. By Corollary 5.1.7, the closed unit ball of .E ×× is closed in .S (τ ) with respect to the local measure topology. Since .‖xα ‖E ×× = ‖xα ‖E ≤ 1 for all .α, this implies that .‖x‖E = ‖x‖E ×× ≤ 1. Hence, (ii) implies (iii). Finally, if .0 ≤ xα ↑α x in E (and hence, in .S (τ )), then .xα → x with respect to the local measure topology (see Proposition 2.7.6 (v)) and so, it is clear that (iii) implies (i). ⨅ ⨆ A similar result holds with respect to the Fatou property in strongly symmetrically normed spaces, as is shown in the next theorem. Theorem 5.1.10 If .E ⊆ S (τ ) is a strongly symmetrically normed space with .cE = 1, then the following statements are equivalent: (i) E has the Fatou property. (ii) .E = E ×× and .‖x‖E = ‖x‖E ×× for all .x ∈ E. (iii) The norm closed unit ball .BE of E is closed in .S (τ ) with respect to the local measure topology. Proof (i).⇒(ii) Since the Fatou property implies that .‖·‖E is a Fatou norm, it follows from Theorem 5.1.9 that .‖x‖E = ‖x‖E ×× for all .x ∈ E. If .0 ≤ x ∈ E ×× , then by Corollary 4.1.9, there exists an increasing net .{xα } in .E + such that .0 ≤ xα ↑α x.

5.1 Strongly Symmetric Spaces and Their Köthe Duals

305

Furthermore, by Theorem 5.1.5, .‖·‖E ×× is a Fatou norm. Consequently, .

sup ‖xα ‖E = sup ‖xα ‖E ×× = ‖x‖E ×× < ∞ α

α

and hence, the Fatou property of E implies that .x ∈ E and .‖x‖E = ‖x‖E ×× . (ii).⇒(iii) This implication follows immediately from the fact that, by Corollary 5.1.7, the norm closed unit ball in .E ×× is closed with respect to measure topology in .S (τ ). (iii).⇒(i) Suppose that .{xα } is an increasing net .E + such that .

sup ‖xα ‖E = γ < ∞. α

By Proposition 5.1.4, the embedding of E into .S (τ ) is continuous with respect to the norm topology in E and the measure topology in .S (τ ) and so, the net .{xα } is bounded in .S (τ ) with respect to the measure topology. Hence, it follows from Theorem 2.6.15 that there exists .x ∈ S (τ ) such that .0 ≤ xα ↑α x in .S (τ ). By Proposition 2.7.6 (v), this implies that .xα → x with respect to the local measure topology and so, by hypothesis, .x ∈ γ BE , that is, .x ∈ E and .‖x‖E ≤ supα ‖xα ‖E . This suffices to establish that E has the Fatou property. ⨆ ⨅ Corollary 5.1.11 If .E ⊆ S (τ ) is a strongly symmetrically normed space satisfying cE = 1 with the Fatou property, then E is a fully symmetric space.

.

Proof It follows from Theorem 5.1.10 that .E = E ×× with equality of norms. ⨆ ⨅ Therefore, the result is an immediate consequence of Theorem 5.1.5. Corollary 5.1.12 Suppose that the von Neumann algebra .M is either non-atomic or atomic and all minimal projections have equal trace. If .{0} /= E ⊆ S (τ ) is a symmetrically normed space, then the following statements are equivalent: (i) E has the Fatou property. (ii) .E = E ×× and .‖x‖E ×× = ‖x‖E for all .x ∈ E. (iii) The norm closed unit ball .BE of E is closed in .S (τ ) with respect to the local measure topology. In particular, if E has the Fatou property, then E is a fully symmetric space. Proof (i).⇒(ii) It follows from Corollary 4.5.7 that E is strongly symmetrically normed and so, this implication follows immediately from the corresponding implication in Theorem 5.1.10. The implications (ii).⇒(iii) and (iii).⇒(i) are established in exactly the same way as in Theorem 5.1.10. The final assertion of the corollary is now also clear. ⨆ ⨅ The following proposition presents another application of Lemma 5.1.8. As before, it is assumed that .cE = 1. Proposition 5.1.13 If .E ⊆ S (τ ) is a strongly symmetric space, then .E b is a fully symmetric space (with respect to the norm .‖·‖E ).

306

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof It should be recalled that a strongly symmetric space is, by definition, a Banach space (see Definition 5.1.1). It follows from Lemma 5.1.3 that .L1 (τ ) ∩ L∞ (τ ) ⊆ E and so, E

E b = L1 (τ ) ∩ L∞ (τ ) .

.

To show that .E b is fully symmetric, suppose that .x ∈ S (τ ) and .y ∈ E b are such that .x ≺≺ y. By Lemma 5.1.3 (i), .y ∈ L1 (τ ) + L∞ (τ ) and so, it follows from Theorem 3.10.13 that there exists an .(L1 , L∞ )-contraction T from .L1 (τ ) + L∞ (τ ) into itself such that .T y = x. Since .T z ≺≺ z for all .z ∈ L1 (τ ) + L∞ (τ ) (see Proposition 3.10.3) and, by Theorem 5.1.5, the space .E ×× is fully symmetric, it follows that .T z ∈ E ×× and .‖T z‖E ×× ≤ ‖z‖E ×× for all .z ∈ E ×× . In particular, ×× . .x ∈ E Let .{yn }∞ n=1 be a sequence in .L1 (τ ) ∩ L∞ (τ ) such that .‖y − yn ‖E → 0. Define .xn ∈ L1 (τ ) ∩ L∞ (τ ) by setting .xn = T yn for all n and note that .

‖x − xn ‖E ×× = ‖T (y − yn )‖E ×× ≤ ‖y − yn ‖E ×× ≤ ‖y − yn ‖E

for all n. On the other hand, it follows from Lemma 5.1.8 that .‖z‖E ×× = ‖z‖E for all .z ∈ E b , and hence, .

‖xn − xm ‖E = ‖T (yn − ym )‖E ×× ≤ ‖yn − ym ‖E ×× = ‖yn − ym ‖E

for all .m, n ∈ N. Since E is assumed to be complete, this implies that .x ∈ E and ‖x − xn ‖E → 0. This shows that .x ∈ E b . Observing that .‖xn ‖E = ‖T yn ‖E ×× ≤ ‖yn ‖E ×× = ‖yn ‖E for all n, it is also clear that .‖x‖E ≤ ‖y‖E , which completes the proof. ⨆ ⨅

.

Reference: [45].

5.2 Normal and Singular Functionals Let .M be a von Neumann algebra on a Hilbert space H , equipped with a normal semi-finite faithful trace .τ . The main purpose in this section is to show that in a large class of normed .M-bimodules, every bounded linear functional has a unique decomposition in a normal and a singular part. Moreover, the space of normal functionals will be identified with the Köthe dual space. The proof of these results (see Theorems 5.2.8 and 5.2.9) is divided into a number of lemmas and propositions. Lemma 5.2.1 If for every .e ∈ P (M) with .τ (e) < ∞ there exists .ae ∈ L1 (τ ) such that: (i) .ae = af e whenever .e, f ∈ P (M) are finite trace projections satisfying .e ≤ f . (ii) For all .e ∈ P (M) with .τ (e) < ∞, the operator .eae is self-adjoint.

5.2 Normal and Singular Functionals

307

then there exists a closed, densely defined symmetric operator a on H , which is affiliated with .M, such that .ae = ae for all finite trace projections .e ∈ P (M). Proof Observe first that (i) implies, in particular, that .ae = ae e for all finite trace projections .e ∈ P (M), and so, .D (ae ) = {ξ ∈ H : eξ ∈ D (ae )}, that is, if .ξ ∈ H , then .ξ ∈ D (ae ) if and only if .eξ ∈ D (ae ). Furthermore, if .e, f ∈ P (M) are finite trace projections such that .e ≤ f , then it follows from (i) that    D (ae ) = ξ ∈ H : eξ ∈ D af

.

and so,   D (ae ) ∩ e (H ) = D af ∩ e (H ) ⊆ D af ∩ f (H ) .

.

Since the set .{e ∈ P (M) : τ (e) < ∞} is upward directed, this shows that .

{D (ae ) ∩ e (H ) : e ∈ P (M) , τ (e) < ∞}

(5.5)

is an upward directed system of linear subspaces of H , with respect to inclusion. Therefore,

.D (a0 ) = D (ae ) ∩ e (H ) e∈P (M) τ (e) 0 (see Proposition 2.3.6 (iv)). This implies that there exists a strictly increasing sequence ∞ b .{λn } in .[0, ∞) such that .λn ↑ ∞ and .τ e (λn , λn+1 ] > 1 for all n. Since n=1 the trace .τ is semi-finite, there exists for each n a projection .en ∈ P (M) which satisfies .en ≤ eb (λn , λn+1 ] and .1 ≤ τ (en ) < ∞. It follows from .beb (λn , λn+1 ] ≥ λn eb (λn , λn+1 ] that .en beb (λn , λn+1 ] en ≥ λn en , that is, .en ben ≥ λn en for all n. Consequently, λn τ (en ) ≤ τ (en ben ) = τ (ben ) ≤ Cτ (en )

.

for all n. This is absurd and so, it may be concluded that b is .τ -measurable.

⨆ ⨅

Lemma 5.2.4 Suppose that .E ⊆ S (τ ) is a normed .M-bimodule such that .F (τ ) is continuously embedded in E. If .ψ is a Hermitian bounded linear functional on

5.2 Normal and Singular Functionals

311

E and if a is a closed symmetric operator in H , affiliated with .M, satisfying the conclusion of Lemma 5.2.2, then: (i) .|τ (zae)| ≤ ‖ψ‖E ∗ ‖z‖E for all .z ∈ F (τ )+ and all finite trace projections .e ∈ P (M). (ii) .τ (|zae|) ≤ 4 ‖ψ‖E ∗ ‖z‖E for all .z ∈ F (τ ) and all finite trace projections .e ∈ P (M). (iii) .|a| e ∈ L1 (τ ) and .0 ≤ τ (|a| e) ≤ 4 ‖ψ‖E ∗ ‖e‖E for all .e ∈ P (M) satisfying .τ (e) < ∞. (iv) The operator a is .τ -measurable and self-adjoint. Proof It should be observed first that, since .F (τ ) is embedded continuously in E, the restriction of .ψ to .F (τ ) is bounded (with respect to the norm .‖·‖L1 ∩L∞ ). Consequently, there exists a closed symmetric operator a in H which satisfies the conclusion of Lemma 5.2.2. (i) If .e ∈ P (M) and .τ (e) < ∞, then ψ (ez) = τ (zae) + ψse (z) , z ∈ M,

.

where the functional .ψse is singular on .M. By Lemma 1.12.5 (ii), .ψse vanishes on some order dense order ideal .J ⊆ M. Therefore, if .z ∈ J ∩ F (τ ), then .

|τ (zae)| = |ψ (ez)| ≤ ‖ψ‖E ∗ ‖ez‖E ≤ ‖ψ‖E ∗ ‖z‖E .

(5.8)

Suppose now that .0 ≤ z ∈ F (τ ). Since J is order dense, Lemma 4.2.15 (iii) implies that there exits an upwards directed system .{zα } in J such that .0 ≤ zα ↑α z. Since .0 ≤ z ∈ F (τ ) ⊆ S0 (τ ), it follows from Theorem 2.6.3 that .zα → z with respect to the measure topology. Furthermore, .ae ∈ L1 (τ ) ⊆ S (τ ) and so .zα ae → zae with respect to the measure topology because .S (τ ) is a topological algebra (cf. Proposition 2.5.3). Consequently, there exists a sequence .α1 ≤ α2 ≤ · · · such that .zαn ae → zae as .n → ∞ (as the measure topology is metrizable). It follows via Proposition 3.2.7 that 

μ zαn ae ≤ zαn ∞ μ (ae) ≤ ‖z‖∞ μ (ae) ∈ L1 (m) , n ∈ N,

.

and  so the dominated convergence theorem (see Theorem 3.4.21) yields that τ zαn ae → τ (zae) as .n → ∞. For each .n ∈ N, (5.8) implies that

.

.

 



 τ zα ae  ≤ ‖ψ‖E ∗ zα ≤ ‖ψ‖E ∗ ‖z‖E n n E

and hence, .|τ (zae)|

≤ ‖ψ‖ E ∗ ‖z‖E .

(ii) If .z ∈ F (τ ), then . (Rez)± E ≤ ‖z‖E and . (Imz)± E ≤ ‖z‖E . Therefore, it follows immediately from (i) that .|τ (zae)| ≤ 4 ‖ψ‖E ∗ ‖z‖E . If .zae = v |zae| is the polar decomposition of zae, then .|zae| = v ∗ zae and hence,



 τ (|zae|) = τ v ∗ zae ≤ 4 ‖ψ‖E ∗ v ∗ z E ≤ 4 ‖ψ‖E ∗ ‖z‖E .

.

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

(iii) Let .a = v |a| be the polar decomposition of the operator a (which exists since a is a closed and densely defined operator) and suppose that .e ∈ P (M) satisfies ∗ ∗ .τ (e) < ∞. Since .|a| = v a, it is clear that .|a| e = v ae ∈ L1 (τ ). Since .e |a| e ≥ 0, it is also clear that .τ (|a| e) = τ (e |a| e) ≥ 0. It follows now from the estimate in (ii) that     τ (|a| e) = τ v ∗ ae = τ ev ∗ ae ≤ τ ev ∗ ae



≤ 4 ‖ψ‖E ∗ ev ∗ ≤ 4 ‖ψ‖E ∗ ‖e‖E .

.

E

(iv) Since .F (τ ) is continuously embedded in E, there exists a constant .C > 0 such that .‖x‖E ≤ C ‖x‖F(τ ) for all .x ∈ F (τ ). Therefore, it follows from (iii) that τ (|a| e) ≤ 4 ‖ψ‖E ∗ ‖e‖E ≤ 4C ‖ψ‖E ∗ ‖e‖F(τ ) ,

.

e ∈ P (M) , τ (e) < ∞.

By Lemma 5.2.3, this implies that .|a| ∈ S (τ ). Consequently, .a = v |a| ∈ S (τ ). Since a is symmetric, this yields, in particular, that a is self-adjoint (see Proposition 2.2.12). ⨆ ⨅ Proposition 5.2.5 Let .E ⊆ S (τ ) be a normed .M-bimodule (with .cE = 1). Assume, furthermore, that one of the following conditions is satisfied: (a) E is strongly symmetrically normed. (b) E is symmetrically normed and .M is either non-atomic or .M is atomic and all minimal projections have the same trace. If .ψ ∈ E ∗ is Hermitian, then there exists a unique .a ∈ E × such that for each finite trace projection .e ∈ P (M), there exists a singular functional .ψse ∈ M∗s satisfying ψ (ez) = τ (zae) + ψse (z) ,

.

z ∈ M.

(5.9)

Moreover, the operator a is self-adjoint. Furthermore, if .ψ ≥ 0, then .a ≥ 0 and eψse ≥ 0 for all .e ∈ P (M) satisfying .τ (e) < ∞.

.

Proof First observe that .F (τ ) ⊆ E, with continuous embedding. Indeed, in case (a) this follows from Lemma 5.1.3 (ii), and in case (b), this has been proved in Theorems 4.4.6 and 4.4.7. Consequently, it follows from Lemmas 5.2.2 and 5.2.4 that there exists a self-adjoint operator .a ∈ S (τ ) such that .ae ∈ L1 (τ ) for all .e ∈ P (M) with .τ (e) < ∞ and (5.9) is satisfied. If .e ∈ P (M) and .τ (e) < ∞, then it follows from Lemma 5.2.4 (ii) that .

sup {τ (|zae|) : z ∈ F (τ ) , ‖z‖E ≤ 1} ≤ 4 ‖ψ‖E ∗ < ∞.

As has been observed in Remark 4.3.4, this implies that .ae ∈ E × and .‖ae‖E × ≤ 4 ‖ψ‖E ∗ . If .a = v |a| is the polar decomposition of a, then .e |a| e = ev ∗ ae, which implies that .e |a| e ∈ E × and .‖e |a| e‖E × ≤ 4 ‖ψ‖E ∗ for all finite trace projections

5.2 Normal and Singular Functionals

313

e ∈ P (M). Let .{eα } be an upward directed system in .P (M) such that .eα ↑ 1 and .τ (eα ) < ∞ for all .α (such a system exists, as the trace is semi-finite). It follows from Proposition 3.2.10 (ii) that .μ |a|1/2 eα |a|1/2 = μ (eα |a| eα ) and so, .|a|1/2 eα |a|1/2 ∈ E × and



1/2

. |a| eα |a|1/2 × = ‖eα |a| eα ‖E × ≤ 4 ‖ψ‖E ∗

.

E

for all .α. Furthermore, by Proposition 2.2.25 (iii), .|a|1/2 eα |a|1/2 ↑α |a|. It should be recalled that the space .E × has the Fatou property. Indeed, in case (a) this has been shown in Theorem 5.1.5 and for case (b) in Theorem 4.5.5. Consequently, .|a| ∈ E × and .‖|a|‖E × ≤ 4 ‖ψ‖E ∗ , that is, .a ∈ E × and .‖a‖E × ≤ 4 ‖ψ‖E ∗ . To prove the uniqueness of the operator a, suppose that .b ∈ E × is such that for each .e ∈ P (M) with .τ (e) < ∞, there exists a singular functional .ϕse ∈ M∗s satisfying ψ (ez) = τ (zbe) + ϕse (z) ,

.

z ∈ M.

Since, for each finite trace projection .e ∈ P (M), the functional .z − │ → τ (zbe), for z ∈ M, is normal (note that .be ∈ L1 (τ ) as .F (τ ) ⊆ E), it follows that

.

τ (zbe) = τ (zae) ,

.

z∈M

(and .ϕse = ψse ). Consequently, .be = ae for all .e ∈ P (M) with .τ (e) < ∞. Let .{eα } be a net in .P (M) such that .eα ↑α 1 and .τ (eα ) < ∞ for all .α. This implies that .eα → 1 locally in measure and so, .(b − a) eα → b − a locally in measure. Since .beα = aeα for all .α, it follows that .b − a = 0, that is, .b = a. This proves the uniqueness statement of the proposition. Assume now that .0 ≤ ψ ∈ E ∗ . If .e ∈ P (M) is a finite trace projection, then  ψ (eze) = τ (zeae) + ψse (ze) = τ (zeae) + eψse (z) ,

.

z ∈ M,

and the functional .z │−→ ψ (eze), .z ∈ M, is positive. The functional .z │−→ τ (zeae), .z ∈ M, is normal and .eψse is singular (see Proposition 1.12.2 (ii)). Since the normal and singular parts of a positive functional in .M∗ are both positive, this implies that .τ (zeae) ≥ 0 for all .z ∈ M+ and .eψse ≥ 0. Hence, .eae ≥ 0 in .L1 (τ ) for all .e ∈ P (M) with .τ (e) < ∞ (see Lemma 4.3.7). If .e ∈ P (M) with  − .τ (e) < ∞ is such that .e ≤ s a , then .eae = −ea − e. It follows that .ea − e = 0,  − 1/2  − 1/2 = 0 for all finite trace projections .e ∈ P (M) equivalently, . a  e a satisfying .e ≤ s a − . If .{eα } is an increasing system in .P (M) such that .eα ↑α  1/2  − 1/2  1/2  −  − 1/2  a s a − and .τ (eα ) < ∞, then . a − eα a ↑α a − s a and  − 1/2  −  − 1/2  − −  − − s a = 0, that is, .a = s a a s a = 0. Consequently, a so, . a .a ≥ 0. The proof of the proposition is complete. ⨆ ⨅

314

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proposition 5.2.6 Let .E ⊆ S (τ ) be a normed .M-bimodule (with .cE = 1). Assume, furthermore, that one of the following two conditions is satisfied: (a) E is strongly symmetrically normed. (b) E is symmetrically normed and .M is either non-atomic or atomic and all minimal projections have the same trace. If .0 ≤ ψ ∈ E ∗ , then .ψ admits a unique decomposition ψ = ψn + ψs ,

.

where .ψn ∈ En∗ and .ψs ∈ Es∗ . Moreover, .ψn ≥ 0 and .ψs ≥ 0, and there exists a unique .0 ≤ a ∈ E × such that .ψn (x) = τ (xa), .x ∈ E. Proof Since .ψ ≥ 0 implies, in particular, that .ψ is Hermitian, Proposition 5.2.5 is applicable. Consequently, there exists a unique element .0 ≤ a ∈ E × such that for each finite trace projection .e ∈ P (M), there exists a singular functional .ψse ∈ M∗s satisfying ψ (ez) = τ (zae) + ψse (z) ,

.

z ∈ M.

Define .0 ≤ ψn ∈ E ∗ by setting ψn (x) = τ (xa) ,

.

x ∈ E,

and let .ψs ∈ E ∗ be defined by ψs = ψ − ψn .

.

As observed in Lemma 4.3.6 (i), the functional .ψn is normal. The first objective is to show that .ψs is singular. If .e ∈ P (M) with .τ (e) < ∞ and .z ∈ M, then ψs (ez) = ψ (ez) − τ (eza)

.

(5.10)

= τ (zae) + ψse (z) − τ (eza) = ψse (z) . Consequently,  ψs (eze) = ψse (ze) = eψse (z) ,

.

z ∈ M, e ∈ P (M) , τ (e) < ∞.

By Proposition 5.2.5, this implies that .ψs (eze) ≥ 0 whenever .z ∈ M+ and .e ∈ P (M) with .τ (e) < ∞. Since F+ (τ ) =

.

  eM+ e : e ∈ P (M) , τ (e) < ∞ ,

5.2 Normal and Singular Functionals

315

as follows from Lemma 2.3.11, this implies that .ψs is a positive functional on .F (τ ). Therefore,     J0 = z ∈ F (τ ) : ψs (|z|) = ψs z∗  = 0 ,

.

which is the absolute kernel of .ψs |F(τ ) , is an order ideal in .F (τ ) (and hence, in E). It should be observed next that .J0 is order dense. Indeed, if .e ∈ P (M) satisfies e .0 < τ (e) < ∞, then the functional .eψs is singular on .M and so, there exists  e .f ∈ P (M) satisfying .0 < f ≤ e and . eψs (f ) = 0. Hence,  ψs (f ) = ψs (ef e) = ψse (f e) = eψse (f ) = 0

.

and so, .f ∈ J0 . Therefore, since .τ is semi-finite, it may be concluded, via Lemma 4.2.15, that .J0 is dense order in E. Consequently, .ψs is singular. It will be shown next that .ψs ≥ 0. If .x ∈ E + , then, by Lemma 4.2.15, there exists an upward directed system .{xα } in .J0 such that .0 ≤ xα ↑α x. Since .ψs (xα ) = 0 , it follows that ψs (x) = ψs (x − xα ) = ψ (x − xα ) − ψn (x − xα )

.

≥ −ψn (x − xα ) for all .α. The normality of .ψn implies that .ψn (x − xα ) ↓α 0 and so, .ψs (x) ≥ 0. Finally, by Lemma 4.2.17, .En∗ ∩ Es∗ = {0}, from which the uniqueness of the decomposition is evident. The uniqueness of a follows from the discussion following Lemma 4.3.5. ⨆ ⨅ It might be of some interest to point out the following alternative characterization of positive singular functionals, which is an immediate consequence of Proposition 5.2.6. Corollary 5.2.7 The same assumptions as in Proposition 5.2.6. A functional .0 ≤ ψ ∈ E ∗ is singular if and only if .ψ does not majorize any non-zero positive normal functional. Proof Suppose that .0 ≤ ψ ∈ Es∗ and .0 ≤ φ ∈ En∗ such that .0 ≤ φ ≤ ψ. By definition, .ψ vanishes on some order dense order ideal .J ⊆ E. If .x ∈ J + , then .0 ≤ φ (x) ≤ ψ (x) = 0 and so, .φ (x) = 0. Since J is generated by .J + , it follows that .φ vanishes on J . Hence, .φ is singular. As observed in Lemma 4.2.17, .En∗ ∩ Es∗ = {0}, which implies that .φ = 0. Suppose now that .ψ does not majorize any non-zero positive normal functional. By Proposition 5.2.6, there exist .0 ≤ ψn ∈ En∗ and .0 ≤ ψs ∈ Es∗ such that .ψ = ψn + ψs . Since .0 ≤ ψn ≤ ψ, it follows that .ψn = 0 and so, .ψ = ψs . The proof is ⨆ ⨅ complete.

316

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Theorem 5.2.8 Let .E ⊆ S (τ ) be a normed .M-bimodule (with .cE = 1). Assume, furthermore, that one of the following conditions is satisfied: (a) E is strongly symmetrically normed (b) E is symmetrically normed and .M is either non-atomic or atomic and all minimal projections have the same trace. Every .ψ ∈ E ∗ admits a unique decomposition ψ = ψn + ψs ,

.

where .ψs ∈ Es∗ and .ψn ∈ En∗ is given by .ψn (x) = τ (xy), .x ∈ E, for a unique × ∗ .y ∈ E . Moreover, if .ψ is Hermitian, then .y = y , and if .ψ ≥ 0, then .y ≥ 0 and .ψs ≥ 0. Proof Let .ψ ∈ E ∗ be fixed. It follows from Proposition 4.2.2 that .ψ may be written in the form   (1) .ψ = ψ − ψ (2) + i ψ (3) − ψ (4) , where .0 ≤ ψ (j ) ∈ E ∗ for .1 ≤ j ≤ 4. By Proposition 5.2.6, each .ψ (j ) has a decomposition (j )

(j )

ψ (j ) = ψn + ψs ,

.

 (j ) (j ) (j ) where .0 ≤ ψs ∈ Es∗ and .0 ≤ ψn ∈ En∗ is given by .ψn (x) = τ xaj , .x ∈ E, with .0 ≤ aj ∈ E × (.1 ≤ j ≤ 4). Defining   ψn = ψn(1) − ψn(2) + i ψn(3) − ψn(4)

.

and   ψs = ψs(1) − ψs(2) + i ψs(3) − ψs(4) ,

.

it is clear that .ψ = ψn +ψs is the desired decomposition of .ψ. Since .En∗ ∩Es∗ = {0}, it is clear that this decomposition is unique. The final statements of the theorem have already been observed in Proposition 5.2.6. The uniqueness of y follows from the discussion following Lemma 4.3.5. ⨆ ⨅ The following theorem is an important consequence of the above result. Theorem 5.2.9 Let .E ⊆ S (τ ) be a normed .M-bimodule (with .cE = 1). Assume, furthermore, that one of the following conditions is satisfied: (a) E is strongly symmetrically normed. (b) E is symmetrically normed and .M is either non-atomic or atomic and all minimal projections have the same trace.

5.2 Normal and Singular Functionals

317

For a functional .ψ ∈ E ∗ , the following statements are equivalent: (i) .ψ is normal. (ii) .ψ is completely additive. (iii) .ψ is right completely additive, that is,     ψ x ei = ψ (xei ) , x ∈ E,

.

i

i

(5.11)

for every collection .{ei } of mutually orthogonal projections in .P (M). (iv) .ψ is left completely additive, that is, ψ



.

i

   ei x = ψ (ei x) , x ∈ E, i

for every collection .{ei } of mutually orthogonal projections in .P (M). (v) There exists a unique element .y ∈ E × such that ψ (x) = τ (xy) , x ∈ E.

.

If any of the preceding equivalent assertions are valid, then .ψ is Hermitian if and only if .y = y ∗ , and .ψ is positive if and only if .y ≥ 0. Proof It has already been observed in Lemma 4.3.6 that condition (v) implies (i) and (ii). Evidently, condition (ii) implies (iii) and (iv). Therefore, it is sufficient to show that each of the assertions (i), (iii), and (iv) implies (v). (i).⇒(v) Suppose that .ψ is normal. By Theorem 5.2.8, there exist .y ∈ E × and ∗ ∗ .ψs ∈ Es such that .ψ (x) = τ (xy) + ψs (x), .x ∈ E. This implies that .ψs ∈ En . By ∗ ∗ Lemma 4.2.17, .En ∩ Es = {0} and hence, .ψs = 0. Consequently, .ψ (x) = τ (xy), .x ∈ E. (iii).⇒(v) By Theorem 5.2.8, there exist .y ∈ E × and .ψs ∈ Es∗ such that ψ (x) = τ (xy) + ψs (x) , x ∈ E.

.

It follows from Lemma 4.3.6 that the functional .x │→ τ (xy), for .x ∈ E, satisfies condition (5.11). Consequently, .ψs satisfies also (5.11). The claim is that .ψs = 0. To this end, suppose that .φ ∈ Es∗ satisfies (5.11). Let .J ⊆ E be an order dense order ideal on which .φ vanishes. It will be shown first that .φ (p) = 0 for all .p ∈ P (E). Given .p ∈ P (E), it follows from Lemma 4.2.15 that there exists a mutually orthogonal system .{ei } in .P (M)∩J satisfying .p = i ei . Condition (5.11), applied to .x = p, yields that      φ (p) = φ p ei = φ (pei ) = φ (ei ) = 0.

.

i

i

i

318

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Consequently, .φ (w) = 0 for each .w ∈ E of the form n

w=

.

j =1

(5.12)

λ j pj ,

 where .pj ∈ P (M) with .τ pj < ∞, .λj ∈ C (.1 ≤ j ≤ n), and .n ∈ N. Let .0 ≤ z ∈ F (τ ). The spectral theorem implies that there exists a sequence ∞ .{zk } k=1 , each .zk of the form (5.12), such that .0 ≤ zk ↑k z and .‖z − zk ‖L1 ∩L∞ → 0 as .k → ∞. The inclusion of .F (τ ) (equipped with the norm .‖·‖L1 ∩L∞ ) in E is continuous and so .‖z − zk ‖E → 0 as .k → ∞. Since .φ (zk ) = 0 for all k and .φ is a continuous functional on E, it follows that .φ (z) = 0. Hence, .φ |F(τ ) = 0. Let .0 ≤ x ∈ E be given. Suppose that .e ∈ P (M) is a finite trace projection satisfying .e ≤ ex [n, n + 1) for some .n ∈ N ∪ {0}. Then .xe = xex [n, n + 1) e ∈ F (τ ). Taking maximal systems of mutually orthogonal finite trace projections dominated by the spectral projections .ex [n, n + 1), .n = 0, 1, . . ., it follows that there exists a system .{eα } of mutually orthogonal finite trace projections such that . α eα = 1 (as .cE = 1) and .xeα ∈ F (τ ) for all .α. As has been shown above, .φ |F(τ ) = 0 and so .φ (xeα ) = 0 for all .α. Since .φ satisfies (5.11), this implies that     φ (x) = φ x eα = φ (xeα ) = 0.

.

α

α

By linearity, .φ = 0 and the proof of (iii).⇒(v) is thereby complete. Finally, the proof that condition (iv) implies (v) is similar to the proof of (iii).⇒(v), with the obvious modifications. ⨅ ⨆ Suppose that .E ⊆ S (τ ) and that the assumptions of Theorem 5.2.8 are satisfied. It follows from this theorem that .E ∗ = En∗ ⊕ Es∗ , and by Theorem 5.2.9, the space ∗ × of E, via trace duality. For .ψ ∈ E ∗ , .En may be identified with the Köthe dual .E the decomposition ψ = ψn + ψs ,

.

ψn ∈ En∗ , ψs ∈ Es∗ ,

will be referred to as the Yosida–Hewitt decomposition. Define .Pn : E ∗ → E ∗ by setting .Pn ψ = ψn , .ψ ∈ E ∗ , that is, .Pn is the linear projection onto .En∗ along .Es∗ , and let .Ps = I − Pn . As follows from Proposition 5.2.6, both projections .Pn and .Ps are positive, that is, .Pn ψ ≥ 0 and .Ps ψ ≥ 0 whenever .0 ≤ ψ ∈ E ∗ . Consequently, ∗ .0 ≤ Pn ψ ≤ ψ and .0 ≤ Ps ψ ≤ ψ for all .0 ≤ ψ ∈ E . In particular, it follows from Corollary 4.2.9 that .

‖Pn ψ‖E ∗ ≤ ‖ψ‖E ∗ and ‖Ps ψ‖E ∗ ≤ ‖ψ‖E ∗ ,

Since every .ψ ∈ E ∗ may be written as ψ = ψ1 − ψ2 + i (ψ3 − ψ4 ) ,

.

0 ≤ ψ ∈ E∗.

5.2 Normal and Singular Functionals

319

where .0 ≤ ψj ∈ E ∗ (.1 ≤ j ≤ 4), .‖ψ1 ‖E ∗ + ‖ψ2 ‖E ∗ ≤ 2 ‖ψ‖E ∗ and .‖ψ3 ‖E ∗ + ‖ψ4 ‖E ∗ ≤ 2 ‖ψ‖E ∗ (see (4.6) following Proposition 4.2.2), this implies that the projections .Pn and .Ps are bounded with .‖Pn ‖L(E ∗ ) ≤ 4 and .‖Ps ‖L(E ∗ ) ≤ 4. In particular, .En∗ and .En∗ are norm closed subspaces of .E ∗ . Identifying .En∗ with .E × via trace duality (see Theorem 5.2.9), it follows that ∗ ∗ × may be written as .ψ = .En is positively generated (in fact, every .ψ ∈ En = E





 ∗ + ψ1 − ψ2 + i (ψ3 − ψ4 ), with .ψj ∈ En and . ψj E ∗ ≤ ψj E ∗ (.1 ≤ j ≤ 4). It should be observed that .Es∗ is also positively generated. Actually, the following holds. Proposition 5.2.10 Each .ψ ∈ Es∗ may be written as .ψ = ψ1 − ψ2 + i (ψ3 − ψ4 ), where .0 ≤ ψj ∈ Es∗ (.1 ≤ j ≤ 4) and .‖ψ1 ‖E ∗ + ‖ψ2 ‖E ∗ ≤ 2 ‖ψ‖E ∗ and .‖ψ3 ‖E ∗ + ‖ψ4 ‖E ∗ ≤ 2 ‖ψ‖E ∗ . Proof Writing .ψ = φ1 − φ2 + i (φ3 − φ4 ) with .0 ≤ φj ∈ E ∗ (.1 ≤ j ≤ 4), .‖φ1 ‖E ∗ + ‖φ2 ‖E ∗ ≤ 2 ‖ψ‖E ∗ , and .‖φ3 ‖E ∗ + ‖φ4 ‖E ∗ ≤ 2 ‖ψ‖E ∗ , it follows that ψ = Ps ψ = Ps φ1 − Ps φ2 + i (Ps φ3 − Ps φ4 ) .

.





Since .0 ≤ Ps φj ≤ φj implies that . Ps φj E ∗ ≤ φj E ∗ , setting .ψj = Ps φj (.1 ≤ j ≤ 4) yields the desired decomposition of .ψ. ⨅ ⨆ For later reference, the following consequence of Proposition 5.2.10 is formulated as a lemma. ∗ Lemma 5.2.11 If .{ψn }∞ n=1 is a sequence in .Es , then there exists an order dense order ideal .J ⊆ E such that .ψn (x) = 0 for all .x ∈ J and all .n ∈ N.

Proof Without loss of generality, it may be assumed that .‖ψn ‖E ∗ ≤ 1 for all n. By Proposition 5.2.10, each .ψn may be written as   (n) (n) (n) (n) ψn = φ1 − φ2 + i φ3 − φ4 ,

.



(n)

(n) where .0 ≤ φj ∈ Es∗ and . φj ∗ ≤ 2 (.1 ≤ j ≤ 4). Define .0 ≤ φn ∈ Es∗ by E (n) setting .φn = 4j =1 φj and set φ=

∞ 

.

2−n φn .

n=1

Since .Es∗ is closed, it follows that .0 ≤ φ ∈ Es∗ and so, .φ vanishes on some order (n) dense order ideal .J ⊆ E. If .x ∈ J + , then .0 ≤ φj (x) ≤ φn (x) ≤ 2n φ (x) = 0, (n)

which shows that .φj (x) = 0 for all .n ∈ N and .1 ≤ j ≤ 4. Consequently, + and all n. Hence, all the functionals .ψ , .n ∈ N, vanish .ψn (x) = 0 for all .x ∈ J n on J . ⨆ ⨅

320

5 Strongly Symmetric Spaces of .τ -Measurable Operators

It will be shown next that the subspaces .En∗ and .Es∗ are not only norm closed in but they are actually sequentially .σ (E ∗ , E)-closed as well. This follows from the following theorem. The proof of this theorem uses the so-called Phillips’ lemma, which will be formulated first for future reference.

∗ .E

Lemma 5.2.12 (Phillips’ Lemma) Let .A be a non-empty set, and suppose that A {νn }∞ n=1 is a sequence of bounded finitely additive measures on the .σ -algebra .2 of all subsets of .A. If .νn (A) → 0 as .n → ∞ for every .A ∈ 2A , then

.

lim

.

 α∈A

n→∞

|νn ({α})| = 0.

Theorem 5.2.13 The same conditions as in Theorem 5.2.9. If .{ψk }∞ k=1 is a sequence in .E ∗ and if .ψ ∈ E ∗ is such that .ψk → ψ for the .σ (E ∗ , E)-topology, then .Pn ψk → Pn ψ for the .σ (E ∗ , E)-topology (and hence, also .Ps ψk → Ps ψ for the .σ (E ∗ , E)topology). Proof It suffices to show that .Pn ψk → 0 for the .σ (E ∗ , E)-topology whenever ∗ .ψk → 0 for the .σ (E , E)-topology. It follows from Lemma 5.2.11 that there exists an order dense order ideal .J ⊆ E on which all functionals .Ps ψk vanish. Let .0 ≤ x ∈ E be fixed. It follows from Lemma 4.2.15 that there exists a family .{xα : α ∈ A} in + such that . .J α∈A xα = x. For .k ∈ N, define the finitely additive set function .νk on the .σ -algebra .2A of all subsets of the index set .A by setting  νk (A) = ψk



.

 A ∈ 2A .

xα ,

α∈A

Since .

  sup |νk (A)| : A ∈ 2A ≤ ‖ψk ‖E ∗ ‖x‖E ,

each .νk is of bounded variation. Furthermore, since .ψk → 0 for the .σ (E ∗ , E)topology, it is clear that  .

lim νk (A) = lim ψk

k→∞

k→∞



 = 0,

xα

A ∈ 2A .

α∈A

Consequently, it follows from Phillips’ lemma that .

lim

k→∞



|ψk (xα )| = lim

k→∞

α∈A



|νk ({α})| = 0.

α∈A

Since .(Ps ψk ) (xα ) = 0 for all .k ∈ N and all .α ∈ A, this shows that .

lim

k→∞

 α∈A

|(Pn ψk ) (xα )| = 0.

5.2 Normal and Singular Functionals

321

Using that .Pn ψk is normal, it follows that  .

lim (Pn ψk ) (x) = lim (Pn ψk )

k→∞

k→∞

= lim

k→∞





 xα

α∈A

(Pn ψk ) (xα ) = 0,

α∈A

and this suffices to complete the proof.

⨆ ⨅

Corollary 5.2.14 The space .L1 (τ ) is weakly sequential complete. Proof Suppose that .{xk }∞ k=1 is a weak Cauchy sequence in .L1 (τ ), which implies, in particular, that .{xk }∞ k=1 is uniformly bounded in .L1 (τ ). Identifying the dual ∗ space of .L1 (τ ) with .M, via trace  duality, it follows that there exists .ψ ∈ M ∗ such that .xk → ψ for the .σ M , M -topology, that is, .τ (xk y) → ψ (y), σ (M∗ ,M) −→ Pn ψ. Since .y ∈ M. By Theorem 5.2.13 (applied to .E = M), .Pn xk ∗ .L1 (τ ) = Mn = M∗ (via trace duality), it follows that .Pn xk = xk for all k and hence, .ψ = Pn ψ. Consequently, there exists .x ∈ L1 (τ ) such that .ψ (y) = τ (yx), .y ∈ M. Since .τ (xk y) → τ (xy), .y ∈ M, it is clear that .xk → x weakly. The proof ⨆ ⨅ is complete. If .ψ ∈ En∗ is given by .ψ (x) = τ (xy), .x ∈ E, where .y ∈ E × , then .(wψ) (x) = τ (xwy) and .(ψw) (x) = τ (wxy) = τ (xyw), .x ∈ E, from which it is clear that ∗ × .wψ, .ψw ∈ En (corresponding to wy, .yw ∈ E , respectively). It will be shown next that .wψ, .ψw ∈ Es∗ whenever .ψ ∈ Es∗ and .w ∈ M. Proposition 5.2.15 The same assumptions as in Theorem 5.2.9. If .ψ ∈ Es∗ , then ∗ .wψ, .ψw ∈ Es for all .w ∈ M. Proof As has been observed before (see the discussion following Theorem 5.2.9), Es∗ is positively generated. Therefore, it suffices to prove the statement of the proposition for positive functionals. Let .0 ≤ ψ ∈ Es∗ and .w ∈ M be given. If .x ∈ N (ψ), then .ψ (|x|) = 0 and so, it follows from Proposition 4.2.6 that .(ψw) (x) = ψ (wx) = 0. This shows that .ψw vanishes on the order dense order ideal .N (ψ) and hence, .ψw ∈ Es∗ . Since .(wψ) (x) = (ψw ∗ ) (x ∗ ), .x ∈ E, the same argument shows that .wψ vanishes on .N (ψ) and so, .wψ ∈ Es∗ . The proof is ⨆ ⨅ complete. .

This section ends with another consequence of Theorem 5.2.9. It has been shown that the norm closed unit ball in .E × is closed in .S (τ ) for the local measure topology (see Corollary 5.1.7 and also Theorem 4.5.6 in combination with Theorem 4.5.5). Actually, the norm closed unit ball in .E × is complete with respect to the local measure topology, as is shown in the following proposition. Proposition 5.2.16 The same conditions as in Theorem 5.2.8. The norm closed unit ball in .E × is complete for the local measure topology.

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof Suppose that .{yα } is a Cauchy net for the local measure topology in .BE × . For each finite trace projection .e ∈ P (M), the net .{eyα e} is Cauchy for the measure topology. Since the measure topology is complete, there exists .ze ∈ S (τ ) such Tm

that .eyα e →α ze . Furthermore, since .eyα e ∈ BE × for all .α and .BE × is closed in .S (τ ) with respect to the measure topology, it follows that .ze ∈ BE × . It should be observed that if .e, f ∈ P (M) are finite trace projections such that .e ≤ f , then .eyα e = e (fyα f ) e for all .α, which implies that .ze = ezf e. If .x ∈ F (τ ), then there exists .e ∈ P (M) such that .τ (e) < ∞ and .x = exe. Defining .ψ (x) = τ (xze ), it is easily verified that this definition does not depend on the choice of .e ∈ P (M) satisfying .τ (e) < ∞ and .x = exe and that .ψ is a linear functional on .F (τ ). Furthermore, .ψ is normal. Indeed, if .x ≥ xα ↓α 0 in .F (τ ) and if .e ∈ P (M) satisfies .τ (e) < ∞ and .x = exe, then .xα = exα e for all .α (see the observations made at the beginning of Sect. 3.7). Hence, .ψ (xα ) = τ (xα ze ) →α 0. Observing that .

|ψ (x)| = |τ (xze )| ≤ ‖x‖E ‖ze ‖E × ≤ ‖x‖E

for all .x ∈ F (τ ), it follows from Theorem 5.2.9, applied to .F (τ ) equipped with norm .‖·‖E , that there exists .y ∈ F (τ )× = E × (see Remark 4.3.4) such that .‖y‖E × ≤ 1 and .ψ (x) = τ (xy), .x ∈ F (τ ). Consequently, if .e ∈ P (M) is a finite trace projection and .x ∈ F (τ ), then τ (xeye) = τ (exey) = ψ (exe) = τ (exeze ) = τ (xze ) .

.

Tm

This implies that .eye = ze . Therefore, the element .y ∈ BE × satisfies .eyα e →α eye, that is, .yα →α y with respect to the local measure topology. ⨆ ⨅ References: [39, 48].

5.3 Order Continuity of the Norm As before, it is assumed that .M is a von Neumann algebra on a Hilbert space H , equipped with a semi-finite faithful normal trace .τ . Definition 5.3.1 If .E ⊆ S (τ ) is a normed .M-bimodule, then the norm .‖·‖E is called order continuous if .‖xα ‖E ↓α 0 whenever .{xα } is a downward directed net in + satisfying .x ↓ 0. .E α α An alternative characterization of order continuity is given in the following proposition. Proposition 5.3.2 If .E ⊆ S (τ ) is a normed .M-bimodule, then the norm .‖·‖E is order continuous if and only if .E ∗ = En∗ .

5.3 Order Continuity of the Norm

323

Proof If the norm in E is order continuous, then .xα ↓α 0 in E implies that ‖xα ‖E →α 0 and so, .φ (xα ) →α 0 for all .φ ∈ E ∗ . Hence, .E ∗ = En∗ . Suppose now that .E ∗ = En∗ and that .xα ↓α 0 in E. This implies that .φ (xα ) →α 0 for all .φ ∈ En∗ = E ∗ and so, it follows from Lemma 4.2.19 that .‖xα ‖E ↓α 0. Therefore, the norm in E is order continuous. ⨆ ⨅

.

Another useful observation is presented in the next lemma. Lemma 5.3.3 If .E ⊆ S (τ ) is a normed .M-bimodule, then the following statements are equivalent: (i) E has order continuous norm. + (ii) .‖xn ‖E ↓ 0 for every decreasing sequence .{xn }∞ n=1 in .E satisfying .xn ↓ 0. Proof Since it is clear that (ii) follows from (i), it suffices to show that statement (ii) implies that the norm .‖·‖E is order continuous. For this purpose, suppose that .{xα } is a decreasing net in .E + satisfying .xα ↓α 0. It should be observed that this implies that .{xα } is a Cauchy net for the norm. Indeed, then there

exist  if∞.{xα } is not Cauchy, an .ε > 0 and a decreasing subsequence . xαn n=1 satisfying . xαn − xαn+1 E ≥ ε for all n. It follows from the discussion following Proposition 4.1.3

that there

exists + such that .x



.y ∈ E αn ↓n y. By assertion (ii), this implies that . y − xαn E → 0 as .n → ∞, which is clearly a contradiction. Consequently, .{xα} is a Cauchy ∞ net. This implies that there exists a decreasing subsequence . xαn n=1 such that



. xαn − xα

≤ 1/n for all .α ≥ αn . Let .x ∈ E + be such that .xαn ↓n x. It follows E

from (ii) that . x − xαn E → 0 as .n → ∞ and hence, .‖x − xα ‖E →α 0. This implies that .xα ↓α x (cf. Corollary 4.1.16 (iii)). Hence, .x = 0 and so, .‖xα ‖E →α 0. The proof is complete. ⨆ ⨅ If .E ⊆ S (τ ) is a normed .M-bimodule satisfying .cE = 1, then .E ⊆ E ×× and .‖x‖E ×× ≤ ‖x‖E for all .x ∈ E (see Proposition 4.3.10). If E is strongly symmetrically normed or if E is symmetrically normed and the von Neumann algebra .M is either non-atomic or atomic and all minimal projections have equal trace, then it follows from Theorem 5.2.9, in combination with Proposition 5.3.2, that .‖x‖E ×× = ‖x‖E for all .x ∈ E. Indeed, in this case, it follows that .

  ‖x‖E = sup |φ (x)| : φ ∈ E ∗ , ‖φ‖E ∗ ≤ 1   = sup |τ (xy)| : y ∈ E × , ‖y‖E × ≤ 1 = ‖x‖E ××

for all .x ∈ E. This observation yields, in particular, the following result. Corollary 5.3.4 Suppose that the von Neumann algebra .M is either non-atomic or atomic and all minimal projections have equal trace. If .E ⊆ S (τ ) is a symmetrically normed space with order continuous norm and .cE = 1, then E is strongly symmetrically normed. Proof By the above discussion, the embedding of E into .E ×× is an isometry. Furthermore, it follows from Theorem 4.5.5 that .E ×× is a fully symmetric space and

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

so, in particular, .E ×× is strongly symmetrically normed. Consequently, if .x, y ∈ E satisfy .x ≺≺ y, then .‖x‖E = ‖x‖E ×× ≤ ‖y‖E ×× = ‖y‖E . This proves the corollary. ⨆ ⨅ For the ease of presentation, in the remaining part of the present section it will be assumed that .E ⊆ S (τ ) is a strongly symmetrically normed space and that the carrier projection .cE of E is equal to .1. The following simple observations should go on record. Lemma 5.3.5 If E has order continuous norm, then .F (τ ) is norm dense in E, that is, .E = E b . In particular, E is contained in .S0 (τ ), that is, .limt→∞ μ (t; x) = 0 for all .x ∈ E. Proof It should be recalled from Lemma 4.4.5 (i) that the assumption that .cE = 1 implies that .F (τ ) ⊆ E. If .0 ≤ x ∈ E, then it follows from Proposition 2.3.12 that there exists an upward directed net .{xα } in .F (τ )+ such that .0 ≤ xα ↑α x. Since .x − xα ↓ 0 and the norm in E is order continuous, it follows that .‖x − xα ‖E ↓α 0. The positive cone .E + is generating in E and so, this suffices to show that .E = E b . The final statement of the lemma follows immediately from Lemma 4.4.10. ⨆ ⨅ A combination of Lemma 5.3.5 with the result of Proposition 5.1.13 immediately yields the next corollary. Corollary 5.3.6 If .E ⊆ S (τ ) is a strongly symmetric space with order continuous norm, then E is fully symmetric. In Corollary 5.2.14 it has been observed that the space .L1 (τ ) is weakly sequentially complete. The next objective is to present a characterization of those strongly symmetrically normed spaces which are weakly sequentially complete. For this purpose, the following terminology is introduced. Definition 5.3.7 A strongly symmetrically normed space .E ⊆ S (τ ) is called a KB+ space (or Kantorovich–Banach space) if every increasing sequence .{xn }∞ n=1 in .E satisfying .supn ‖xn ‖E < ∞ is norm convergent in E. It should be observed that KB-spaces are complete. Indeed, it is clear that if E is a KB-space, then increasing Cauchy sequences in .E + are convergent and hence, it follows from Corollary 4.1.20 that E is complete. Some other characterizations of KB-spaces are presented in the next lemma. Lemma 5.3.8 If .E ⊆ S (τ ) is a strongly symmetrically normed space, then the following statements are equivalent: (i) E is a KB-space. (ii) Every increasing net .{xα } in .E + satisfying .supα ‖xα ‖E < ∞ is norm convergent in E. (iii) E has order continuous norm and the Fatou property. Proof That (ii).⇒(iii) and (iii).⇒(i) follows easily from the definitions. To show that (i) implies assertion (ii), suppose that .{xα } is an increasing net in .E + such that

5.3 Order Continuity of the Norm

325

supα ‖xα ‖E < ∞. Since E is complete, it suffices to show that .{xα } is a Cauchy  net. ∞ If .{xα } is not Cauchy, then there exist .ε > 0 and an increasing subsequence . xαn n=1





such that . xαn+1 − xαn E ≥ ε for all n. Since .supn xαn E < ∞, it follows from  ∞ assertion (i) that the sequence . xαn n=1 is convergent, which is a contradiction. ⨅ ⨆

.

Theorem 5.3.9 If .E ⊆ S (τ ) is a strongly symmetric space, then the following statements are equivalent: (i) E is weakly sequentially complete. (ii) E is a KB-space. + Proof (i).⇒(ii) If .{xk }∞ k=1 is an increasing sequence in .E such that .supk ‖xk ‖E < ∞ ∞, then .{xk }k=1 is a weak Cauchy sequence. By hypothesis, there exists .x ∈ E such that .xk → x for the .σ (E, E ∗ )-topology. Since the positive cone .E + is norm closed, it is also .σ (E, E ∗ )-closed, which implies that .x ∈ E + and .xk ↑ x. It now follows from Lemma 4.2.19 that .‖x − xk ‖E → 0. Consequently, E is a KB-space. ∞ (ii).⇒(i) Suppose that .{xk }∞ k=1 is a weak Cauchy sequence in E. Since .{xk }k=1 is norm bounded, by the uniform boundedness principle it follows that there exists .ψ ∈ E ∗∗ such that .xk → ψ for the .σ (E ∗∗ , E ∗ )-topology. As observed in Lemma 5.3.8, E has order continuous norm and so, .E ∗ = En∗ (see Proposition 5.3.2). Therefore, by Theorem 5.2.9, .E ∗ may be identified with the Köthe dual .E × via trace duality. It follows from Theorem 5.2.8 that

 ∗  ∗  ∗ E ∗∗ = E × = E × n ⊕ E × s .

.

 ∗ Identifying . E × n via trace duality with the second Köthe dual space .E ×× , a moment’s reflection shows that the canonical embedding of E into .E ∗∗ corresponds ×× it follows from Theorem that  × 5.2.13 to the ∗ inclusion of E in .E ∗∗. Furthermore, ∗ × ∗ . E is sequentially .σ (E , E )-closed. Consequently, .ψ ∈ E , that is, n n there exists .x ∈ E ×× such that .ψ (y) = τ (xy), .y ∈ E × . Furthermore, by Lemma 5.3.8, E has the Fatou property and so, Theorem 5.1.10 implies that ×× . Consequently, .x ∈ E and .x → x weakly. This shows that E is weakly .E = E k sequentially complete. ⨆ ⨅ A related result is the following characterization of reflexivity of strongly symmetric spaces. Theorem 5.3.10 A strongly symmetric space .E ⊆ S (τ ) is reflexive if and only if the following conditions are satisfied: (a) The space E has the Fatou property. (b) The norms on E and .E × are order continuous. Proof Suppose that E is reflexive. This implies, in particular, that E is weakly sequentially complete and so, by Theorem 5.3.9, E is a KB-space. Hence, by Lemma 5.3.8, E has the Fatou property and the norm on E is order continuous. It follows from Proposition 5.3.2 that the dual space .E ∗ may be identified via trace

326

5 Strongly Symmetric Spaces of .τ -Measurable Operators

duality with the Köthe dual .E × . Consequently, .E × is reflexive and so, .E × is also a KB-space. Therefore, the norm in .E × is order continuous. Suppose now that conditions (a) and (b) are satisfied. It follows from (b) that the Banach duals of E and .E × may be identified, via trace duality, with the Köthe duals .E × and .E ×× , respectively. Furthermore, the Fatou property of E implies that ×× . These observations suffice to conclude that E is reflexive. .E = E ⨆ ⨅ References: [20, 45, 119]

5.4 Elements of Order Continuous Norm In strongly symmetrically normed spaces E in which the norm is not order continuous, the following subset of E is of some interest. As before, in this section it is assumed that .cE = 1. Definition 5.4.1 The subset .E oc of E is defined by setting E oc = {x ∈ E : |x| ≥ xα ↓α 0 ⇒ ‖xα ‖E ↓α 0} .

.

The elements of .E oc are called the elements of order continuous norm in E. Observe already that .E oc is an absolutely solid subset of E in the sense that if oc such that .|x| ≤ |y|, then .x ∈ E oc . Furthermore, it is evident .x ∈ E and .y ∈ E that the norm in E is order continuous if and only if .E = E oc . Remark 5.4.2 By the same argument as was used in the proof of Lemma 5.3.3, it follows that the set .E oc is also given by E oc = {x ∈ E : |x| ≥ xn ↓n 0 ⇒ ‖xn ‖E ↓ 0 as n → ∞} .

.

An alternative characterization of the set .E oc will be given in the next proposition. For this purpose, some further notation is required. If .A ⊆ E and .B ⊆ E ∗ , then the annihilator .A⊥ of A and the inverse annihilator .⊥ B of B are defined by setting   A⊥ = ψ ∈ E ∗ : ψ (x) = 0 ∀x ∈ A ,

.

⊥

B = {x ∈ E : ψ (x) = 0 ∀ψ ∈ B} ,

respectively. Evidently, .A⊥ ⊆ E ∗ and .⊥ B ⊆ E are closed linear subspaces. Proposition 5.4.3 The set .E oc of elements of order continuous norm in .E is also given by E oc =

.

⊥

Es∗ .

5.4 Elements of Order Continuous Norm

327

Proof Suppose that .x ∈ E oc . If .0 ≤ ψ ∈ Es∗ , then there exists an order dense order ideal .J ⊆ E on which .ψ vanishes. By Lemma 4.2.15, there exists an upward directed system .{xα } in .J + such that .0 ≤ xα ↑α |x|. Since .|x| ≥ |x| − xα ↓α 0 and .x ∈ E oc , this implies that .‖|x| − xα ‖E ↓α 0 and so, .ψ (xα ) →α ψ (|x|). Consequently, .ψ (|x|) = 0 and hence, by Proposition 4.2.6, .ψ (x) = 0. Every functional in .Es∗ is a linear combination of positive functionals in .Es∗ and so, it may be concluded that .ψ (x) = 0 for all .ψ ∈ Es∗ , that is, .x ∈ ⊥ Es∗ . This shows that oc ⊆ ⊥ E ∗ . .E s For the proof of the reverse inclusion, suppose that .x ∈ E and .ψ (x) = 0 for all .ψ ∈ Es∗ . Let .x = v |x| be the polar decomposition of x. It follows from Proposition 5.2.15 that .ψv ∈ Es∗ whenever .ψ ∈ Es∗ and so, .ψ (|x|) = (ψv) (x) = 0 for all .ψ ∈ Es∗ . Now suppose that .|x| ≥ xα ↓α 0 and that .0 ≤ φ ∈ E ∗ . Let ∗ .φ = φn + φs be the Yosida–Hewitt decomposition of .φ, where .0 ≤ φn ∈ En and ∗ .0 ≤ φs ∈ Es (see Theorem 5.2.8). Observing that .0 ≤ φs (xα ) ≤ φs (|x|), it is clear that .φs (xα ) = 0 for all .α and hence, .φ (xα ) = φn (xα ) ↓α 0. Since .E ∗ is the linear span of .(E ∗ )+ , this implies that .φ (xα ) →α 0 for all .φ ∈ E ∗ . Consequently, by Lemma 4.2.19, .‖xα ‖E ↓α 0. Therefore .x ∈ E oc , which shows that .⊥ Es∗ ⊆ E oc . The proof is complete. ⨆ ⨅ Corollary 5.4.4 The set .E oc is an .M-bimodule, which is norm closed in E. Proof That .E oc is a norm closed linear subspace of E follows immediately from Proposition 5.4.3. To verify that .E oc is an .M-bimodule, suppose that .x ∈ E oc and ∗ ∗ .v, w ∈ M. By Proposition 5.2.15, .vψw ∈ Es whenever .ψ ∈ Es and so, it follows from Proposition 5.4.3 that ψ (wxv) = (vψw) (x) = 0,

.

ψ ∈ Es∗ .

Using Proposition 5.4.3 once more, this shows that .wxv ∈ E oc .

⨆ ⨅

Remark 5.4.5 The is always contained the norm closure of .F (τ ) in E. Indeed, if .0 ≤ x ∈ E oc , then there exists an upward directed net .{xα } in .F (τ )+ such that .0 ≤ xα ↑α x, that is, .x ≥ x − xα ↓α 0. Hence, .‖x − xα ‖E ↓α 0 and so, b oc is generated by its positive cone (see the discussion following .x ∈ E . Since .E Proposition 4.1.3), this suffices to show that .E oc ⊆ E b . space .E oc

in .E b ,

It may happen that .E oc = {0}. Indeed, this is for example the case if .E = M = L∞ (ν) for some non-atomic measure .ν. In the following proposition some conditions are presented which are equivalent to the statement that the carrier projection .cE oc of .E oc is equal to .1. Proposition 5.4.6 If .E ⊆ S (τ ) is a strongly symmetrically normed space, then the following statements are equivalent: (i) .(E oc )⊥ = Es∗ . (ii) .Es∗ is .σ (E ∗ , E)-closed. (iii) .E oc separates the points of .En∗ .

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

(iv) .cE oc = 1. (v) The map .φ │−→ φ |E oc , .φ ∈ En∗ , is a linear isometry from .En∗ onto .(E oc )∗ . Proof By Proposition 5.4.3, .E oc = .

⊥E∗. s

Consequently,

 oc ⊥ ⊥ ∗ ⊥ σ (E ∗ ,E ) , E = Es = Es∗

(5.13)

where the right-hand side denotes the weak-.∗ closure of .Es∗ in .E ∗ . This shows that statements (i) and (ii) are equivalent. (i).⇒(iii) Suppose that .φ ∈ En∗ is such that .φ (x) = 0 for all .x ∈ E oc , that is, oc ⊥ ∗ ∗ ∗ .φ ∈ (E ) . By statement (i), this implies that .φ ∈ Es . Since .En ∩ Es = {0}, it follows that .φ = 0. Hence, assertion (iii) holds. (iii).⇒(iv) Recall fromProposition 4.3.8 that .cE ×× = 1. Therefore, if .cE oc < 1, then there exists .p ∈ P E ×× such that .0 < p ≤ 1 − cE oc . Defining .φ ∈ E ∗ by setting .φ (x) = τ (xp), .x ∈ E, it is clear that .0 < φ ∈ En∗ (as it is assumed that .cE = 1). If .x ∈ E oc , then .x = xcE oc (see Corollary 4.1.7) and so, .xp = 0. Consequently, .φ (x) = 0 for all .x ∈ E oc . This shows that assertion (iii) does not hold. Consequently, (iii) implies that .cE oc = 1. (iv).⇒(v) If .φ ∈ En∗ , then it follows from Theorem 5.2.9 that .φ is given by × satisfying .‖y‖ .φ (x) = τ (xy), .x ∈ E, for some .y ∈ E E × = ‖φ‖E ∗ . Since .cE oc = 1, it follows from Remark 4.3.4 that .‖y‖(E oc )× = ‖y‖E × . Consequently, .‖φ |E oc ‖(E oc )∗ = ‖φ‖E ∗ . This shows that the map .φ │−→ φ |E oc is a linear isometry from .En∗ into .(E oc )∗ . It remains to be proved that this map is surjective. To this end, let .ψ ∈ (E oc )∗ be given. By the Hahn–Banach theorem, there exists .φ ∈ E ∗ such that .φ |E oc = ψ. If .φ = φn + φs is the Yosida–Hewitt decomposition of .φ, then it follows from Proposition 5.4.3 that .φs (x) = 0 for all .x ∈ E oc and hence, .ψ = φn |E oc . This establishes assertion (v). That (v) implies (iii) is evident. It remains to be shown that (iii) implies (i). Observe that it follows from (5.13) that the inclusion .Es∗ ⊆ (E oc )⊥ always holds. Suppose now that .ψ ∈ (E oc )⊥ and let .ψ = ψn + ψs be its Yosida–Hewitt decomposition. Since .ψs |E oc = 0, it follows that .ψn ∈ (E oc )⊥ , and by assertion (iii), this implies that .ψn = 0. Consequently, .ψ = ψs ∈ Es∗ . This completes the proof of the proposition. ⨆ ⨅ By definition, a functional .ψ ∈ E ∗ is singular if .ψ vanishes on some order dense order ideal .J ⊆ E. It follows, in particular, from Proposition 5.4.6 above that if .cE oc = 1, then a functional .ψ is singular if and only if .ψ vanishes on the fixed order dense order ideal .E oc . Remark 5.4.7 Suppose that .E ⊆ S (τ ) is a strongly symmetric space. Even in the case that .cE oc = 1, the .M-bimodule .E oc need not be symmetric, that is, if .x ∈ E and .y ∈ E oc satisfy .μ (x) ≤ μ (y), then not necessarily .x ∈ E oc . To illustrate this phenomenon, let .E = M = 𝓁∞ and equip .𝓁∞ with the trace .τ given by .τ (x) = ∞ −n oc = c , the space of all 2 ξ , . x = (ξn ) ∈ 𝓁+ n 0 ∞ . It is easily verified that .E n=1 null sequences. Defining the projections .p, q ∈ P (𝓁∞ ) by .p = (0, 1, 1, 1, . . .) and

5.4 Elements of Order Continuous Norm

329

q = (1, 0, 0, . . .), it is clear that .τ (p) = τ (q) and so, .μ (p) = μ (q). However, q ∈ E oc but .p ∈ / E oc . In this connection, it is of some interest to observe that if .E oc /= {0} and .E oc is symmetric, then .F (τ ) ⊆ E oc (and hence, by Remark 5.4.5, .E oc = E b ). Indeed, it is clear that .E oc contains all minimal projections of E. Let p be a non-zero projection ⊥ does not dominate any minimal projection and in .P (E oc ). If .cE oc < 1, then .cE oc ⊥ and .τ (q) ≤ τ (p) (cf. hence, there exists .q ∈ P (M) such that .0 < q ≤ cE oc oc Lemma 3.7.2). Since .E is assumed to be symmetric, this implies that .q ∈ P (E oc ) and so, .q ≤ cE oc , which is a contradiction. Hence, it may be concluded that .cE oc = 1. It now follows from Lemma 4.4.5 (i) that .F (τ ) ⊆ E oc .

. .

If the von Neumann algebra is either non-atomic or atomic with all minimal projections having equal trace, then the above results may be improved, as will be shown in the next propositions. Proposition 5.4.8 For a strongly symmetric space .E ⊆ S (τ ), consider the following statements: (i) .E oc /= {0}. (ii) For every .ε > 0, there exists .δ > 0 such that .‖p‖E ≤ ε whenever .p ∈ P (E) satisfies .τ (p) ≤ δ. (iii) .E oc = E b . (iv) The carrier projection .cE oc of .E oc is equal to .1. The implications (ii).⇒(iii).⇒(iv).⇒(i) are always valid. If the von Neumann algebra M is non-atomic, then also the implication (i).⇒(ii) holds (and hence, all four statements are equivalent).

.

Proof (ii).⇒(iii) By Remark 5.4.5, it is sufficient to show that .F (τ ) ⊆ E oc , that is, oc .q ∈ P (E ) whenever .q ∈ P (M) and .τ (q) < ∞. Fixing .q ∈ P (M) with .τ (q) <   ∞, let . aβ be a decreasing net in .E + such that .q ≥ aβ ↓β 0. It should be recalled that .q ∈ E (see Lemma 5.1.3 (ii)). Given .ε > 0, let .δ > 0 be such that .‖p‖E ≤ ε for all .p ∈ P (E) .τ (p) ≤ δ. Since .τ (q) < ∞, the normality of the trace  satisfying implies that .τ aβ ↓β 0. Observing that .0 ≤ εeaβ (ε, ∞) ≤ aβ , it follows that a a .τ (e β (ε, ∞)) →β 0 and so, there exists .β0 such that .τ (e β (ε, ∞)) ≤ δ whenever .β ≥ β0 . It follows from .0 ≤ aβ ≤ q that aβ = aβ eaβ (ε, ∞) + aβ eaβ [0, ε] ≤ eaβ (ε, ∞) + εq

.



 and hence, . aβ E ≤ ε 1 + ‖q‖E whenever .β ≥ β0 . This shows that .q ∈ E oc , which is sufficient to establish assertion (iii). Evidently, statement (iii) implies (iv) and statement (iv) implies (i). Assume now that the von Neumann algebra .M is non-atomic. (i).⇒(ii) Since .E oc /= {0}, there exists a projection .e ∈ P (E oc ) satisfying .0 < τ (e) < ∞. Furthermore, .M is assumed to be non-atomic and so, there exists a −n for all n (see sequence .{en }∞ n=1 in .P (M) such that .e ≥ en ↓n 0 and .τ (en ) = 2 oc Lemma 3.7.3). Since .e ∈ E , this implies that .‖en ‖E ↓ 0 as .n → ∞. Given .ε > 0,

330

5 Strongly Symmetric Spaces of .τ -Measurable Operators



 there exists .n0 ∈ N such that . en0 E ≤ ε. Let .δ = τ en0 . If .p ∈ P (E) and  .τ (p) ≤ δ, then .μ (p) ≤ χ[0,δ) = μ en0 and hence, .‖p‖E ≤ en0

≤ ε. ⨆ ⨅ E Remark 5.4.9 If .E = M = 𝓁∞ , equipped with the trace .τ considered in Remark 5.4.7, then E satisfies condition (i), but not condition (ii) of Proposition 5.4.8. Indeed, .E oc = co . However, defining .pn = (0, . . . , 0, 1, 1, 1, . . .) (n zeroes), it follows that .τ (pn ) = 2−n for all n, but .‖pn ‖∞ = 1 for all n. Proposition 5.4.10 Suppose that the von Neumann algebra .M is atomic and that all minimal projections have the same trace. If .E ⊆ S (τ ) = M is a strongly symmetric space and .E /= {0}, then .E oc = E b , which is equal to the closed linear span of all minimal projections. Proof First observe that it follows from the assumption on .M that .F (τ ) is the linear span of all minimal projections. Therefore, .E b is equal to the closed linear span of all minimal projections. Furthermore, it is clear that .E oc contains all minimal projections (indeed, if .p ∈ P (M) is minimal, then .pEp = pMp simply consists of all scalar multiples of p). Consequently, .F (τ ) ⊆ E oc and so, Remark 5.4.5 implies that .E oc = E b . The proof is complete. ⨆ ⨅ Corollary 5.4.11 Suppose that the von Neumann algebra .M is atomic and that all minimal projections have the same trace. If .E ⊆ M is a strongly symmetric space, then a functional .ψ ∈ E ∗ is singular if and only if .ψ vanishes on all minimal projections. Furthermore, defining for each .y ∈ E × the linear functional .φy on .E b by setting φy (x) = τ (xy) ,

.

x ∈ Eb,

 ∗ the map .y │−→ φy is a linear isometry from .E × onto . E b . Proof By Proposition 5.4.10, .E oc = E b , which clearly implies that .cE oc = 1. Therefore, condition (iv) of Proposition 5.4.6 is satisfied. The first statement of the present corollary is now an immediate consequence of part (i) of Proposition 5.4.6 and the second statement is a simple reformulation of part (v) of the same proposition. ⨆ ⨅ References: [39, 40].

5.5 Elements of Absolutely Continuous Norm In this section, some important alternative descriptions of elements of order continuous norm will be discussed. For this purpose, the following notion is introduced. As before, E is assumed to be a strongly symmetrically normed space of .τ -measurable operators (always satisfying .cE = 1).

5.5 Elements of Absolutely Continuous Norm

331

Definition 5.5.1 An element .x ∈ E is said to have absolutely continuous norm if ‖en xen ‖E → 0 for every sequence .{en }∞ n=1 in .P (M) satisfying .en ↓n 0. The set of all elements of absolutely continuous norm is denoted by .E an .

.

It is clear from the definition that .E an is a linear subspace of E and that .x ∗ ∈ E an whenever .x ∈ E an . In due course, it will be shown that actually .E an = E oc (see Proposition 5.5.11). The following lemma collects together some of the elementary properties of the space .E an . Lemma 5.5.2 (i) .E oc ⊆ E an . (ii) .E an ⊆ S0 (τ ). (iii) If .x ∈ E an , then .‖en xen ‖E → 0 whenever .{en }∞ n=1 is a sequence in .P (M) satisfying .τ (en ) → 0 as .n → ∞. Proof (i) By Corollary 5.4.4, .E oc is an .M-bimodule and so, each element of .E oc is a linear combination of at most four positive elements (see the discussion following Proposition 4.1.3). Therefore, it is sufficient to show that .(E oc )+ ⊆ E an Suppose that .0 ≤ x ∈ E oc and that .en ↓n 0 in .P (M). Since .μ (en xen ) =  .1/2 μ x en x 1/2 (see Proposition 3.2.10 (ii)) and .x ≥ x 1/2 en x 1/2 ↓n 0 (see Proposition 2.2.25 (iii)), it follows that .





‖en xen ‖E = x 1/2 en x 1/2 ↓n 0. E

This shows that .x ∈ E an . (ii) To show that .x ∈ S0 (τ ) whenever .x ∈ E an , it may be assumed, without loss of generality, that .x ∗ = x. Suppose that .x = x ∗ ∈ E an and that .x ∈ / S0 (τ ). It follows that at least one of .x + and .x − does not belong to .S0 (τ ). For / S0 (τ ), that is, there exists .s > 0 such that definiteness, suppose that .x + ∈  +   x .τ e (s, ∞) = ∞. It should be observed that this implies that .μ x + ≥ s1 (see Lemma 3.2.3 (ii)) and so, .M ⊆ E. Using that the trace is semi-finite, it follows that there exists an increasing sequence .{pn }∞ n=1 in .P (M) such that x + (s, ∞) and .n ≤ τ (p ) < ∞ for all n. Defining .p = sup p .pn ≤ e n n n and .en = p − pn for all n, it is clear that .en ↓n 0, .τ (en ) = ∞ and .en ≤ + ex (s, ∞) = ex (s, ∞) for all n. Since .x ∈ E an , it follows that .‖en xen ‖E → 0 as .n → ∞. On the other hand, .xex (s, ∞) ≥ sex (s, ∞) and so, en xen = en xex (s, ∞) en ≥ sen ex (s, ∞) en = sen ,

.

which implies that .‖en xen ‖E ≥ s ‖en ‖E for all n. Since .τ (en ) = ∞, it is also clear that .μ (en ) = μ (1) and hence, .‖en ‖E = ‖1‖E for all n. Consequently, .‖en xen ‖E ≥ s ‖1‖E for all n, which is clearly a contradiction. Therefore, it may be concluded that .x ∈ S0 (τ ) and this completes the proof of part (ii).

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

(iii) Suppose that .x ∈ E an and that .{en }∞ n=1 is a sequence in .P (M) such that .τ (en ) → 0, but .‖en xen ‖E  0. By passing to a subsequence, it may be assumed that .τ (en ) ≤ 2−n and .‖en xen ‖E ≥ δ for some .0 < δ ∈ R and all n. Defining .pn ∈ P (M) by setting pn =

.

∞ k=n

ek ,

−n+1 . Consequently, it is clear that .pn ↓n and that .τ (pn ) ≤ ∞ k=n τ (ek ) ≤ 2 .pn ↓n 0 and so, the hypothesis on x implies that .‖pn xpn ‖E → 0 as .n → ∞. On the other hand, 0 < δ ≤ ‖en xen ‖E = ‖en pn xpn en ‖E ≤ ‖pn xpn ‖E

.

for all n, which is absurd. This suffices for the proof of part (iii) of the lemma. ⨆ ⨅ For further analysis, the following definition turns out to be useful. Definition 5.5.3 A subset .A of E is said to be of uniformly absolutely continuous norm if .

sup {‖en xen ‖E : x ∈ A} → 0

as .n → ∞ for all sequences .{en }∞ n=1 in .P (M) satisfying .en ↓ 0. It is clear that any set of uniformly absolutely continuous norm is contained in E an . Furthermore, it should be observed that if .0 ≤ y ∈ E an , then the set

.

.

[0, y] = {x ∈ E : 0 ≤ x ≤ y}

is of uniform absolutely continuous norm. Indeed, if .{en }∞ n=1 is a sequence in .P (M) such that .en ↓ 0, then .0 ≤ en xen ≤ en yen for all .x ∈ [0, y] and so, .

sup {‖en xen ‖E : x ∈ [0, y]} ≤ ‖en yen ‖E → ∞, n → ∞.

For the proof of the next proposition, some preparation is needed. In particular, it will be convenient to adapt the following terminology. Definition 5.5.4 A projection .e ∈ P (M) is termed .σ -finite (with respect to .τ ) if there exists a sequence .{pn }∞ n=1 in .P (M) satisfying .τ (pn ) < ∞ for all n such that .pn ↑n e. The von Neumann algebra .M will be called .σ -finite (with respect to .τ ) if the identity is .σ -finite.

5.5 Elements of Absolutely Continuous Norm

333

Several characterizations of .σ -finite projections will now be given. Lemma 5.5.5 For a projection .e ∈ P (M), the following statements are equivalent: (i) e is .σ -finite. (ii) There a sequence .{pn }∞ n=1 of finite trace projections in .P (M) such that exists ∞ .e ≤ p . n n=1 (iii) Every mutually orthogonal system .{qα } of non-zero projections in .P (M) such that .qα ≤ e for all .α is at most countable. (iv) There exists a mutually orthogonal sequence .{qn }∞ n=1 of finite trace projections ∞ in .P (M) such that .e = n=1 qn . Proof (i).⇒(ii) This implication is evident. (ii) Let .{pn }∞ n=1 be a sequence of finite trace projections in .P (M) such that .⇒(iii) ∞ .e ≤ p . Suppose that .{qα }α∈A is a mutually orthogonal system in .P (M) such n=1 n that .0 < qα ≤ e for all .α ∈ A. For each .n ∈ N, define .An = {α ∈ A : pn qα pn /= 0} and observe that .A = ∞ n=1 An . Indeed, suppose that .α ∈ A is such that .pn qα pn = 0 for all n. This implies that .(qα pn )∗ (qα pn ) = 0∞and so, .qα pn = 0 for all n. Hence, .pn ≤ 1 − qα for all n, which implies that .e ≤ n=1 pn ≤ 1 − qα . Since .0 < qα ≤ e, this is impossible. It will be shown next that each of the sets .An is at most countable. Indeed, for each .k ∈ N, set .An,k = {α ∈ An : τ (pn qα pn ) ≥ 1/k}. If .F is a finite subset of .An,k , then    . pn qα pn = pn qα pn ≤ pn , α∈F

α∈F

and so, .

|F|  ≤ τ (pn qα pn ) ≤ τ (pn ) , α∈F k

that  is,  .|F| ≤ kτ (pn ) (where .|·| denotes the cardinality of a set). This shows that An,k  ≤ kτ (pn ) < ∞ for all k, and therefore, .An is at most countable. This suffices for the proof of implication (ii).⇒(iii). (iii).⇒(iv) There exists a maximal mutually orthogonal system .{qα } in .P (M) such that .0 < τ (qα ) < ∞ for all .α and .e = α qα (see Lemma 1.15.6 (ii)). By assumption (iii), the system .{qα } is at most countable. Therefore, (iv) holds. (iv).⇒(i) This is evident (take .pn = nk=1 qk ). ⨆ ⨅

.

Remark 5.5.6 A von Neumann algebra .M is sometimes called countably decomposable if every mutually orthogonal system in .P (M) is at most countable. It follows from Lemma 5.5.5 that this notion (for semi-finite von Neumann algebras) is equivalent to being .σ -finite as defined in Definition 5.5.4.

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

The following corollary is an immediate consequence of Lemma 5.5.5. Corollary 5.5.7 (i) If .f ≤ e in .P (M) and e is .σ -finite, then f is .σ -finite. ∞ (ii) If .{en }∞ n=1 is a sequence in .P (M) and each .en is .σ -finite, then . n=1 en is .σ -finite. The following observations will be needed. Lemma 5.5.8 (i) If .x ∈ S0 (τ ), then the support projection .s (x) and the range projection .r (x) are .σ -finite. Moreover, there is a .σ -finite projection p such that .pxp = x. (ii) If .{xn }∞ n=1 is a sequence in .S0 (τ ), then there exists a .σ -finite projection .p ∈ P (M) such that .xn = pxn p for all n. Proof

 (i) Since .x ∈ S0 (x), it follows that .τ e|x| (1/n, ∞) < ∞ for all .n ∈ N. Observing that e|x| (1/n, ∞) ↑n e|x| (0, ∞) = s (|x|) = s (x) ,

.

it is clear that .s (x) is .σ -finite. Since .x ∗ ∈ S0 (τ ), also the range projection ∗ .r (x) = s (x ) is .σ -finite. Therefore, it follows from Corollary 5.5.7 (ii) that the projection .p = s (x) ∨ r (x) is .σ -finite and .x = r (x) xs (x) = pr (x) xs (x) p = pxp. (ii) If .{xn }∞ n=1 is a sequence in .S0 (τ ), then it follows from the above obserin combination with Corollary 5.5.7 (ii), that the projection .p = vation,  (s (x n ) ∨ r (xn )) is .σ -finite and satisfies n xn = r (xn ) xn s (xn ) = pr (xn ) xn s (xn ) p = pxn p

.

for all n. The proof is complete. ⨆ ⨅ Proposition 5.5.9 Let .E ⊆ S (τ ) be a strongly symmetrically normed space. If an {xn }∞ n=1 is a sequence in .E , then the following statements are equivalent:

.

(i) .‖xn ‖E → 0 as .n → ∞. (ii) .xn → 0 with respect to the measure topology and the set .{xn : n ∈ N} is of uniformly absolutely continuous norm. Proof (i).⇒(ii) By Proposition 4.3.10, the embedding of E into .S (τ ) is continuous Tm

with respect to the measure topology and so, it is clear that .xn → 0 in .S (τ ).

5.5 Elements of Absolutely Continuous Norm

335

To see that .{xn }∞ n=1 has uniformly absolutely continuous norm, suppose that ∞ .{ek } is a sequence in .P (M) satisfying .ek ↓ 0 and let .ε > 0 be given. Choosing k=1 .N ∈ N such that .‖xn ‖E ≤ ε for all .n ≥ N, it follows that .

‖ek xn ek ‖E ≤ ‖xn ‖E ≤ ε,

n ≥ N,

−1 for all .k ∈ N. Since each of the elements .{xn }N n=1 is assumed to have absolutely continuous norm, there exists .K ∈ N such that .‖ek xn ek ‖E ≤ ε for all .1 ≤ n ≤ N −1 and .k ≥ K. Consequently, .

sup {‖ek xn ek ‖E : n ∈ N} ≤ ε

whenever .k ≥ K. This shows that .{xn }∞ n=1 has uniformly absolutely continuous norm. (ii).⇒(i) It is sufficient to show .{xn }∞ n=1 has a subsequence which converges to zero, since this implies that each subsequence has a subsequence converging to zero. Furthermore, without loss of generality, it may be assumed that .xn∗ = xn for all n. It follows from Lemma 5.5.2 (ii) and Lemma 5.5.8 (ii) that there exists a .σ -finite projection .p ∈ P (M) such that .xn = pxn p for all n. Therefore, passing to the reduced von Neumann algebra .Mp and the corresponding strongly symmetrically normed space .Ep , it may be assumed that .1 is .σ -finite. Let .{ek }∞ k=1 be a sequence in .P (M) such that .ek ↑ 1 and .τ (ek ) < ∞ for all k. Tm

Since .xn → 0, it follows from the Egorov type result of Lemma 2.5.15 that, by passing to a subsequence of .{xn }∞ a sequence n=1 , it may  assumed that there exists  be



 ∞ ⊥

. pj in .P (M) satisfying .pj ↑ 1, .τ pj → 0 as .j → ∞ and . xn pj B(H ) → j =1 0 as .n → ∞ for all j . Observe that







⊥ ⊥

⊥

. ‖xn ‖E ≤ ‖xn ek ‖E + ek xn ek + ek xn ek

E E



⊥ ⊥

≤ 2 ‖xn ek ‖E + ek xn ek , E

where it is used that







⊥

. ek xn ek

≤ ‖ek xn ‖E = (ek xn )∗ E = ‖xn ek ‖E . E

Let .ε > 0 be given. Since .ek⊥ ↓ 0, the assumption that .{xn }∞ n=1 is of uniformly absolutely continuous norm implies that there exists a .k ∈ N such that .





sup ek⊥ xn ek⊥ < ε.

n∈N

E

336

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Keeping this k fixed, observe that .













‖xn ek ‖E ≤ xn pj ek E + pj xn pj⊥ ek + pj⊥ xn pj⊥ ek

E E







≤ 2 xn pj B(H ) ‖ek ‖E + pj⊥ xn pj⊥ . E

Since .pj⊥ ↓ 0, again the fact that .{xn }∞ n=1 is of uniformly absolutely continuous norm implies that there exists a .j ∈ N such that .





sup pj⊥ xn pj⊥ < ε.

n∈N

E



Using the fact that . xn pj B(H ) → 0 as .n → ∞, choose .N ∈ N such that



‖ek ‖E < ε for all .n ≥ N . Combining the above estimates, it follows . xn pj

B(H ) that .‖xn ‖E < 7ε for all .n ≥ N and this suffices for the proof of the proposition. ⨅ ⨆ It is worth noting that the implication (ii).⇒(i) of the above proposition remains valid in the case of nets. Corollary 5.5.10 Suppose that .E ⊆ S (τ ) is a strongly symmetrically normed space and that .{xα }α∈A is a net in .E an . If .xα →α 0 for the measure topology and if the set .{xα : α ∈ A} is of uniformly absolutely continuous norm, then .‖xα ‖E →α 0. Proof If .‖xα ‖E α 0, then there exists .0 < ε ∈ R such that the set .B = {α ∈ A : ‖xα ‖E ≥ ε} is cofinal in .A, that is,  for  each .α ∈ A, there exists .β ∈ B such that .α ≤ β. Consequently, the subnet . xβ β∈B of .{xα }α∈A satisfies .xβ →β 0



in measure and . xβ E ≥ ε for all .β ∈ B. Consequently, there exists an increasing sequence .{βn }∞ n=1 in .B such that .xβn ∈ V (1/n, 1/n) for

all

.n ∈ N. Therefore, .xβn →n 0 in measure and hence, by Proposition 5.5.9, . xβn

→ 0 as .n → ∞. E This is a contradiction. ⨅ ⨆ Proposition 5.5.9 has the following consequence. Proposition 5.5.11 If E is a strongly symmetrically normed space, then .E an = E oc . In particular, .E an is an .M-bimodule which is closed in E. Proof In part (i) of Lemma 5.5.2, it has already been observed that .E oc ⊆ E an . For the proof of the reverse inclusion, it should be observed first that .(E an )+ ⊆ (E oc )+ . Indeed, suppose that .0 ≤ x ∈ E an and let .{xn }∞ n=1 be a sequence in E such that .x ≥ xn ↓n 0. By Lemma 5.5.2 (ii), .x ∈ S0 (τ ) and so, it follows from Theorem 2.6.3 that .xn → 0 with respect to the measure topology. Furthermore, the set .{xn }∞ n=1 is of uniformly absolutely continuous norm (see the discussion following Definition 5.5.3). Consequently, it follows from Proposition 5.5.9 that oc .‖xn ‖E → 0 as .n → ∞. By Remark 5.4.2, this suffices to show that .x ∈ E .

5.5 Elements of Absolutely Continuous Norm

+ .x

337

Suppose next that .x ∈ E an and .x ∗ = x. Setting .p = ex (0, ∞), it is clear that = xp. If .{pn }∞ n=1 is a sequence in .P (M) such that .p ≥ pn ↓ 0, then .



pn x + pn = ‖pn xppn ‖E = ‖pn xpn ‖E → 0, E

n → ∞.

This shows that .x + is of absolutely continuous norm in the space .Ep with respect  + ⊆ to the reduced von Neumann algebra .Mp . By the first part of the proof, . Epan  + Epoc and hence, .x + ∈ Epoc . If .{xα } is a net in E satisfying .x + ≥ xα ↓α 0, then .xα = pxα p for all .α and so, .xα ↓α 0 in .Ep . Consequently, .‖xα ‖E ↓ 0, which shows that .x + ∈ E oc . Similarly, .x − = (−x)+ ∈ E oc and hence, .x ∈ E oc . Since .E an is closed under taking adjoints, this suffices to prove that .E an ⊆ E oc . The final assertion of the proposition simply follows from Corollary 5.4.4. ⨆ ⨅ The following theorems collect together a number of useful characterizations of elements of absolutely continuous norm. Theorem 5.5.12 Suppose that .E ⊆ S (τ ) is a strongly symmetrically normed space. If .x ∈ E, then the following statements are equivalent: (i) For all decreasing sequences .{en }∞ n=1 in .P (M), en ↓n 0 =⇒ ‖en xen ‖E →n 0,

.

that is, .x ∈ E an . (ii) For all decreasing systems .{eα } in .P (M), eα ↓α 0 =⇒ ‖eα xeα ‖E →α 0.

.

(iii) For all decreasing sequences .{en }∞ n=1 in .P (M), en ↓n 0 =⇒ ‖xen ‖E →n 0.

.

(iv) For all decreasing systems .{eα } in .P (M), eα ↓α 0 =⇒ ‖xeα ‖E →α 0.

.

(v) For all decreasing systems .{xα } in E, |x| ≥ xα ↓α 0 =⇒ ‖xα ‖E →α 0,

.

that is, .x ∈ E oc . (vi) For all decreasing sequences .{xn }∞ n=1 in E, .

|x| ≥ xn ↓n 0 =⇒ ‖xn ‖E →n 0.

338

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof The equivalence (i).⇔(v) has been obtained in Proposition 5.5.11 and the equivalence (v).⇔(vi) has been observed in Remark 5.4.2. The implications (iv).⇒(iii).⇒(i) and (iv).⇒(ii).⇒(i) are all evident. Therefore, it suffices to prove that (v) implies (iv). For this purpose, suppose that the element .x ∈ E satisfies condition (v) and let .{eα } be a net in .P (M) such that .eα ↓α 0. It follows from Lemma 4.2.10 that .

1/2

1/2

‖|x| eα ‖E ≤ 4 ‖x‖E ‖eα |x| eα ‖E

(5.14)

 for all .α. Furthermore, .μ (eα |x| eα ) = μ |x|1/2 eα |x|1/2 (see Proposition 3.2.10 (ii)), which implies that .|x|1/2 eα |x|1/2 ∈ E and .‖eα |x| eα ‖E =

|x|1/2 eα |x|1/2 for all .α. Since E .

|x| ≥ |x|1/2 eα |x|1/2 ↓α 0,

the assumption on x implies that .‖eα |x| eα ‖E →α 0. Consequently, it follows from (5.14) that .‖|x| eα ‖E →α 0. If .x = v |x| is the polar decomposition of x, then .‖xeα ‖E = ‖v |x| eα ‖E ≤ ‖|x| eα ‖E for all .α and so, .‖xeα ‖E →α 0. The proof is complete. ⨆ ⨅ The following theorem characterizes elements of absolutely continuous norm in terms of sequences of mutually orthogonal projections. Note that E is assumed to be complete in this case. Theorem 5.5.13 Suppose that .E ⊆ S (τ ) is a strongly symmetric space. If .x ∈ E, then the following statements are equivalent: (i) .x ∈ E an . (ii) For all mutually orthogonal sequences .{en }∞ n=1 in .P (M), it follows that .‖xen ‖E → 0 as .n → ∞. (iii) For all mutually orthogonal sequences .{en }∞ n=1 in .P (M), it follows that .‖en xen ‖E → 0 as .n → ∞. Proof (i).⇒(ii) Suppose that .x ∈ E an and let .{en } be a mutually orthogonal e sequence in .P (M). For .n ∈ N, define .pn ∈ P (M) by .pn = ∞ k=n k , in which case .pn ↓n 0. By Theorem 5.5.12, (i) implies condition (iii) of that theorem, and hence .‖xpn ‖E ↓n 0. Since .en = pn en , it follows that .

‖xen ‖E = ‖xpn en ‖E ≤ ‖xpn ‖E ,

which implies that .‖xen ‖E → 0 as .n → ∞. The implication (ii).⇒(iii) is obvious. (iii).⇒(i) Assume first that .x ∈ E + and let .{en }∞ n=1 be a sequence in .P (M) such that .en ↓n 0. Defining .yn = x 1/2 en x 1/2 , it follows that .μ (yn ) = μ (en xen ) (see Proposition 3.2.10 (ii)), which implies that .yn ∈ E and .‖yn ‖E = ‖en xen ‖E for all n. Furthermore, it is clear that .yn ↓n 0. It will now be shown that .{yn }∞ n=1 is a

5.5 Elements of Absolutely Continuous Norm

339

Cauchy sequence E. Indeed, if .{y then there exist .ε > 0 and

n } is not Cauchy,

 in∞ a subsequence . ynk k=1 such that . ynk − ynk+1 E ≥ ε for all .k. Defining .pk = enk − enk+1 , the sequence .{pk }∞ k=1 is mutually orthogonal in .P (M) and .









‖pk xpk ‖E = x 1/2 pk x 1/2 = ynk − ynk+1 E ≥ ε E

for all k, which is contradicting assumption (iii) on x. Therefore, it may be concluded that .{yn }∞ n=1 is a Cauchy sequence in E and hence, .{yn } is norm convergent in E. Since .yn ↓n 0, it follows that .‖yn ‖E → 0 (see Corollary 4.1.16 (iii)), that is, .‖en xen ‖E → 0 as .n → ∞. This shows that .x ∈ (E an )+ . Assume next that .x ∗ = x ∈ E satisfies (iii) and let .p = ex (0, ∞), in which case + .x = px. If .{en }∞ n=1 is a mutually orthogonal sequence in .P (M) satisfying .en ≤ p for all n, then .



en x + en = ‖en xpen ‖E = ‖en xen ‖E → 0, E

n → ∞.

This shows that .x + satisfies condition (iii) in the space .Ep . By what has already + +   and hence, .x + ∈ Epoc (see been proved, this implies that .x + ∈ Epan Proposition 5.5.11). Via the same argument as was used at the end of the proof of Proposition 5.5.11, it follows that .x + ∈ (E oc )+ and so, .x + ∈ (E an )+ . Similarly, − ∈ (E an )+ and hence, .x ∈ E an . .x Suppose now that .x ∈ E is arbitrary, satisfying (iii). This implies that .x ∗ , and hence also .Rex and .Imx satisfy (iii). Consequently, .x = Rex + iImx ∈ E an . This completes the proof of the theorem. ⨆ ⨅ By definition, the norm in E is order continuous if and only if .E = E oc . Therefore, the above theorems yield immediately the following result. Corollary 5.5.14 Let .E ⊆ S (τ ) be a strongly symmetric space. The space E has order continuous norm if and only if each .x ∈ E (equivalently, each .x ∈ E + ) satisfies any (equivalently, all) of the conditions of Theorems 5.5.12 and 5.5.13. The following characterization of elements of absolutely continuous norm will also be useful. As before, the space .L1 (τ ) is identified, via trace duality, with the pre-dual .M∗ of .M (see Theorem 3.4.24 (i)). Theorem 5.5.15 If .E ⊆ S (τ ) is a strongly symmetrically normed space and if x ∈ E, then the following statements are equivalent:

.

(i) .x ∈ E an . (ii) Whenever .{xα } is a bounded net in .M+ satisfying .xα →α 0 with respect to .σ (M, L1 (τ )), it follows that .‖xxα ‖E →α 0. (iii) Whenever .{xα } is a bounded net in .M+ , satisfying .xα →α 0 with respect to .σ (M, L1 (τ )), it follows that .‖xα x‖E →α 0.

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof (i).⇒(ii) Since .E an is an .M-bimodule (see Proposition 5.5.11), each .x ∈ E an is a linear combination of at most four positive elements of .E an (see the discussion following Proposition 4.1.3). Therefore, it is sufficient to consider the   case that .0 ≤ x ∈ E an . In this case, there exists a net . yβ in .F (τ ) such that .0 ≤ yβ ↑β x by Proposition 2.3.12. Since x has order continuous norm, this



implies that . x − yβ E →β 0 (see Proposition 5.5.11). Consequently, for the proof of (i).⇒(ii), it may be assumed that .0 ≤ x ∈ F (τ ). Note that this implies that .0 ≤ x ≤ ‖x‖∞ s (x), with .τ (s (x)) < ∞. Furthermore, it may also be assumed that .0 ≤ xα ≤ 1 for all .α. 1/2 Via the spectral theorem, note that .0 ≤ x 1/2 ≤ ‖x‖∞ s (x). It is now clear that 1/2 .x ∈ L1 (τ ) and hence, using Proposition 3.4.2 (iii), .



 

1/2

x xα x 1/2 = τ x 1/2 xα x 1/2 = τ (xα x) →α 0. 1

Since the embedding of .(L1 (τ ) , ‖·‖1 ) into .(S (τ ) , Tm ) is continuous (see Proposition 3.4.11), this implies that .x 1/2 xα x 1/2 →α 0 with respect to the measure 1/2 x x 1/2 ≤ x for topology. The assumption that .0 ≤ xα ≤ 1 yields that .0 ≤ x  1/2  α an 1/2 all .α and so, since .x ∈ E , it follows that the set . x xα x is of uniformly absolutely continuous norm, by the remarks following Definition 5.5.3. Therefore,



Corollary 5.5.10 implies that . x 1/2 xα x 1/2 E →α 0.    1/2 1/2 (see Proposition 3.2.10 (ii)), it Recalling that .μ x 1/2 xα x 1/2 = μ xα xxα



1/2 1/2

is now clear that also . xα xxα →α 0. Furthermore, since .0 ≤ xα ≤ 1 implies E

1/2

(via the spectral theorem) that .0 ≤ xα ≤ 1 for all .α, it follows that .‖xxα ‖E ≤

1/2

xxα . An application of Lemma 4.2.10 yields that E

.







1/2

1/2

xxα ≤ 4 ‖x‖E xα1/2 xxα1/2 →α 0. E

E

Therefore, it may be concluded that .‖xxα ‖E →α 0, showing that (i).⇒(ii). wo (ii).⇒(i) If .{en }∞ n=1 is a sequence in .P (M) such that .en ↓n 0, then .en →n 0 uwo

and hence .en → n 0 (as the wo-topology coincides with the uwo-topology on norm bounded sets of .B (H )). Since the uwo-topology on .M is equal to .σ (M, L1 (τ )), it follows that .en → 0 as .→ ∞ with respect to .σ (M, L1 (τ )). Therefore, it follows from (i) that .‖xen ‖E →n 0. Via Theorem 5.5.12, this shows that .x ∈ E an . (i).⇒(iii) If .x ∈ E an , then .x ∗ ∈ E an and therefore, if .{xα } is a bounded net in + .M satisfying .xα →α 0 with respect to .σ (M,

L1 (τ )), it follows from the already proved implication (i).⇒(ii) that .‖xα x‖E = (xα x)∗ E = ‖x ∗ xα ‖E →α 0. (iii).⇒(i) If .x ∈ E satisfies (iii), then .x ∗ satisfies (ii) and hence, .x ∗ ∈ E an , which implies that .x ∈ E an . The proof of the theorem is complete. ⨆ ⨅ References: [20, 39, 41, 45].

5.6 Further Characterizations of Order Continuity

341

5.6 Further Characterizations of Order Continuity In this section, some further characterizations of order continuity of the norm will be presented which complement those given in Theorems 5.5.12 and 5.5.13. These characterizations are related to certain Banach space properties: the (well-known) property .(u) of Pełczy´nski, the embedding of isomorphic copies of .𝓁∞ , and norm separability. Throughout this section, .M will be a von Neumann algebra on a Hilbert space H , equipped with a faithful, normal, and semi-finite trace .τ . When considering strongly symmetric spaces .E ⊆ S (τ ), it will always be assumed that the carrier projection .cE of .E satisfies .cE = 1. It should also be pointed out that the results in this section are valid also for symmetric spaces, assuming that .M is either non-atomic or atomic and all minimal projections have the same trace. The details are left to the reader. A Banach space E is said to contain an isomorphic copy of .𝓁∞ if there exist a linear subspace F of E and a surjective linear isomorphism .T : 𝓁∞ → F . The following is a consequence of Theorem 5.5.13. Proposition 5.6.1 If .E ⊆ S (τ ) is a strongly symmetric space and E does not have order continuous norm, then E contains an isomorphic copy of .𝓁∞ . Proof If E does not have order continuous norm, then it follows from Theorem 5.5.13 that there exist .0 < x ∈ E and a sequence .{en }∞ n=1 of mutually orthogonal projections in .P (M) such that .‖xen ‖E  0 as .n → ∞. Passing, if necessary, to a subsequence of .{en }, it may be assumed that there exists .ε > 0 such that .‖xen ‖E ≥ ε for all n. If .(αn ) ∈ 𝓁∞ , then it is not difficult to show that the series . ∞ n=1 αn en is strongly

∞ convergent in .M and that . n=1 αn en B(H ) = ‖(αn )‖∞ . Define the linear map .T : 𝓁∞ → E by setting Tα = x

∞

.

n=1

 αn en ,

α = (αn ) ∈ 𝓁∞ .

It is clear that .

∞



‖T α‖E ≤ ‖x‖E

αn en

n=1

B(H )

= ‖(αn )‖∞ ‖x‖E ,

α = (αn ) ∈ 𝓁∞ ,

and hence, T is bounded with .‖T ‖ ≤ ‖x‖E . For each .n ∈ N and .α ∈ 𝓁∞ , it is clear that .(T α) en = αn xen and so, ε |αn | ≤ ‖αn xen ‖E = ‖(T α) en ‖E ≤ ‖T α‖E .

.

This implies that .ε ‖α‖∞ ≤ ‖T α‖E for all .α ∈ 𝓁∞ and hence, T is an isomorphism onto its range. This completes the proof. ⨆ ⨅ It will be shown that the converse of Proposition 5.6.1 also holds: the norm in a strongly symmetric space containing a copy of .𝓁∞ is not order continuous

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

(see Theorem 5.6.10). An important role in the proof is played by property .(u) of Pełczy´nski (see Definition 5.6.6 and Proposition 5.6.9). Recall that a projection .e ∈ P (M) is said to be .σ -finite if there exists a sequence ∞ .{pn } n=1 of finite trace projections in .P (M) such that .pn ↑n e (see Definition 5.5.4) and that the von Neumann algebra .M is called .σ -finite whenever .1 is .σ -finite. An important consequence of .σ -finiteness is exhibited in Proposition 5.6.3. For the proof, it will be convenient to have the following observations available. Lemma 5.6.2 Let .(M, τ ) be a semi-finite von Neumann algebra and let .x ∈ S (τ )+ be given.   (i) There exists a system . qβ of mutually orthogonal projections in .P (M) such that . β qβ = 1 and .τ qβ xqβ < ∞ for all .β. (ii) If, in addition, .M is .σ -finite, then there exists a sequence .{qn }∞ n=1 of mutually ∞ orthogonal projections in .P (M) such that . n=1 qn = 1 and .τ (qn xqn ) < ∞ for all n. Proof (i) The following claim will be proved first: given .x ∈ S (τ )+ and .0 /= q ∈ P (M), there exists .p ∈ P (M) such that .0 < p ≤ q and .τ (pxp) < ∞. Indeed, if .qxq = 0, then there is nothing to be proved and so it may be assumed that .qxq > 0. Let .m ∈ N be such that .eqxq (0, m) > 0 and observe that qxq (0, m) ≤ q (in fact, .1 − q ≤ n (qxq) and so, .eqxq (0, ∞) = 1 − n (qxq) ≤ .e q). Since the trace is semi-finite, there exists .p ∈ P (M) such that .0 < p ≤ eqxq (0, m) and .τ (p) < ∞. It follows that   τ (pxp) = τ peqxq (0, m) xp ≤ mτ peqxq (0, m) p = mτ (p) < ∞,

.

which proves the claim. + .x ∈ S (τ ) , it follows via Zorn’s lemma that there exists a system  Given  . qβ of non-zero mutually orthogonal projections in .P (M), which is maximal  with respect to the property that .τ qβ xqβ < ∞ for all .β. Set .q = β qβ and suppose that .q /= 1. By the first part of the proof, there exists .p ∈ P (M) such that .0 0 where .μ (e) is continuous (see Proposition 3.2.11 (ii)), that is, for all .0 < t /= τ (e). This implies that .τ (pn ) → τ (e), and therefore, there exists .0 < C ∈ R such that .τ (pn ) ≤ C for all n. Consequently, .μ (pn ) ≤ χ[0,C) for all n. Using the dominated convergence theorem (see Theorem 3.4.21), it may be ⨆ ⨅ concluded that .‖e − pn ‖1 → 0 as .n → ∞. This completes the proof. As follows from the lemma above, a trace .τ is separable if and only if the set {p ∈ P (M) : τ (p) < ∞} is separable with respect to the .‖·‖1 -norm. The following observation is sometimes useful.

.

Lemma 5.6.16 Let .(M, τ ) be a semi-finite von Neumann algebra. The trace .τ is separable if and only if the space .S0 (τ ) is separable with respect to the measure topology. Proof It will be convenient to set .D = P (M) ∩ F (τ ). Since .D ⊆ S0 (τ ), it is clear that if .S0 (τ ) is separable with respect to the measure topology, then .τ is separable. Assuming that .τ is separable, it follows from the .Tm -separability of D Tm that .span (D) ⊆ S (τ ) is separable for the measure topology .Tm (as is the case Tm in any topological vector space). The claim is that .S0 (τ )n ⊆ span (D) . Indeed, let .0 ≤ x ∈ S0 (τ ) be given and observe first that if .a = k=1 αk pk , with .0 < αk ∈ R and .pk ∈ P (M) (.1 ≤ k ≤ n), satisfies .0 ≤ a ≤ x, then .pk ∈ D for all k (in fact, this implies that .pk ∈ S0 (τ ) and so, .τ (pk ) < ∞). Consequently, it follows from the spectral theorem that there exists a sequence .{an }∞ n=1 in .span (D) such that Tm

0 ≤ an ↑n x. Since .x ∈ S0 (τ ), this implies that .an → x (see Theorem 2.6.3). This suffices to prove the claim, from which it follows that .S0 (τ ) is .Tm -separable. ⨆ ⨅

.

348

5 Strongly Symmetric Spaces of .τ -Measurable Operators

The next proposition exhibits an important sufficient condition for separability of the trace. Recall that the underlying Hilbert space of the von Neumann algebra .M is denoted by H . Proposition 5.6.17 If .(M, τ ) is a semi-finite von Neumann algebra and the underlying Hilbert space H is separable, then the trace .τ is separable. Proof By Theorem .M∗ of .M may be represented ∞ 1.11.5 (ii), each .ϕ in the pre-dual ∞ ∞ 〈xξ 〉, {ξ } as .ϕ (x) = , η . x ∈ M, where . n n n n=1 n=1 and .{ηn }n=1 are sequences ∞ ∞ 2 2 in H satisfying

 . n=1 ‖ξn ‖H < ∞ and . n=1 ‖ηn ‖H < ∞. Note that .‖ϕ‖M∗ ≤

(‖ξn ‖H ) ‖ηn ‖H . It is now not difficult to see that .M∗ is separable 2 2 whenever H is separable. Since the pre-dual .M∗ may be identified with .L1 (τ ) via trace duality (see Theorem 3.4.24 (i)), it follows that .L1 (τ ) is norm separable. Using that the embedding of .(L1 (τ ) , ‖·‖1 ) into .(S (τ ) , Tm ) is continuous (see, e.g., Proposition 3.4.11), this implies that .L1 (τ ) is separable for the measure topology. Consequently, .P (M) ∩ F (τ ) is also separable for the measure topology, that is, the trace .τ is separable. ⨆ ⨅ As was observed in Corollary 5.6.12, if a strongly symmetric space .E ⊆ S (τ ) is norm separable, then E has order continuous norm. The converse statement is, in general, not true, as is illustrated by Example 5.6.13. The next objective is to show that, under the assumption that the trace .τ is separable, order continuity of the norm does imply norm separability. For this purpose, some preparation is needed. In particular, a dominated convergence type theorem for strongly symmetric spaces with order continuous norm (see Theorem 5.6.19) will be obtained, which is of interest in its own right. Recall that a strongly symmetric space .E ⊆ S (τ ) is always assumed to satisfy .cE = 1 (and hence .F (τ ) ⊆ E; see Lemma 4.4.5). Lemma 5.6.18 Let .E ⊆ S (τ ) be a strongly symmetric space with order continuous norm. For every .ε > 0, there exists .δ > 0 such that .‖p‖E < ε whenever .p ∈ P (M) satisfies .τ (p) < δ. In particular, if .{pn }∞ n=1 is a sequence of finite trace projections in .P (M) such that .τ (pn ) → 0 as .n → ∞, then .‖pn ‖E → 0. Proof Suppose that .ε > 0 and that such a .δ > 0 does not exist. This implies the −n existence of a sequence .{pn }∞ n=1 in .P (M) satisfying ∞ .τ (pn ) ≤ 2 and .‖pn ‖E ≥ ε for all n. Defining .qn ∈ P (M) by .qn = k=n pk , it follows that .qn ↓n and ∞ −n+1 for all n. Hence, .q ∈ E and .q ↓ 0. By the order .τ (qn ) ≤ n n n k=n τ (pk ) = 2 continuity of the norm, this implies that .‖qn ‖E ↓n 0. Since .pn ≤ qn , it is also clear that .‖pn ‖E ≤ ‖qn ‖E for all n, which yields a contradiction. ⨆ ⨅ In the proof of the theorem which follows, the following notation will be used. For .0 ≤ α ∈ R, define the real functions .fα and .gα by setting .fα (t) = t ∧ α and .gα (t) = (t − α)+ , .t ∈ [0, ∞). For .0 ≤ x ∈ S (τ ), set .x ∧ α1 = fα (x) and + .(x − α1) = gα (x). Note that .x = x ∧ α1 + (x − α1)+ and that .x ∧ α1 ≤ x ∧ β1 and .(x − β1)+ ≤ (x − α1)+ whenever .0 ≤ α ≤ β, as follows from the properties of the functional calculus of x. Similarly, .0 ≤ x ∧ α1 ≤ α1 and .0 ≤ x ∧ α1 ≤ x for

5.6 Further Characterizations of Order Continuity

349

all .α ≥ 0. Furthermore, by Proposition 3.2.8, μ (x ∧ α1) = μ (x) ∧ α1,

.

 μ (x − α1)+ = (μ (x) − α1)+ ,

(5.15)

for all .α ≥ 0. It should also be observed that .(x − αk 1)+ ↓k 0 in .S (τ ) whenever .αk ↑k ∞. This can be seen, for instance, by assuming that .z ∈ S (τ ) satisfies + .0 ≤ z ≤ (x − αk 1) for all k, which implies that  μ (t; z) ≤ μ t; (x − αk 1)+ = (μ (t; x) − αk )+ ↓k 0,

.

t > 0,

and hence, .μ (z) = 0, that is, .z = 0. Similarly, if .αk ↓k 0, then .x ∧ αk 1 ↓k 0 in S (τ ).

.

Theorem 5.6.19 Suppose that .E ⊆ S (τ ) is a strongly symmetric space with order continuous norm. Let .{xn }∞ n=1 be a sequence in E and .y ∈ E be such that .μ (xn ) ≤ μ (y) for all n. If .xn → 0 for the measure topology, then .‖xn ‖E → 0 as .n → ∞. Proof For the proof of the theorem, it may be assumed, without loss of generality, that .y ≥ 0 and .xn ≥ 0 for all n. Assume first, in addition, that .0 ≤ xn ≤ N1 for all n and some .N ∈ N. Given .δ > 0, it follows that 0 ≤ xn = xn exn [0, δ] + xn exn (δ, ∞) ≤ xn exn [0, δ] + Nexn (δ, ∞) .

.

It follows from .xn → 0 in measure that .τ (exn (δ, ∞)) → 0 as .n → ∞ (see Proposition 2.5.7) and so, by Lemma 5.6.18, .‖exn (δ, ∞)‖E → 0 as .n → ∞. Hence .



lim sup ‖xn ‖E ≤ lim sup xn exn [0, δ] E . n→∞

n→∞

Since .μ (xn exn [0, δ]) ≤ δ1 and .μ (xn exn [0, δ]) ≤ μ (xn ) ≤ μ (y), it follows from (5.15) that  μ xn exn [0, δ] ≤ μ (y) ∧ δ1 = μ (y ∧ δ1)

.

and so, .‖xn exn [0, δ]‖E ≤ ‖y ∧ δ1‖E . This shows that .

lim sup ‖xn ‖E ≤ ‖y ∧ δ1‖E ,

δ > 0.

n→∞

As has been observed in the discussion preceding the present theorem, .y ∧ δ1 ↓ 0 in .S (τ ) as .δ ↓ 0. Using once again that the norm is order continuous, this implies that .‖y ∧ δ1‖E ↓ 0 as .δ ↓ 0, from which it follows that .lim supn→∞ ‖xn ‖E = 0, that is, .‖xn ‖E → 0 as .n → ∞. Assume now that .0 ≤ xn ∈ E and .0 ≤ y ∈ E are such that .xn → 0 with respect + to the measure topology and .μ (xn ) ≤ μ (y) for

all n. Since

.(y − N 1) ↓N 0 in E, + the order continuity of the norm implies that . (y − N1) E ↓N 0. Given .ε > 0,

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5 Strongly Symmetric Spaces of .τ -Measurable Operators



let .N ∈ N be such that . (y − N 1)+ E ≤ ε/2. Using (5.15), it follows that   μ (xn − N 1)+ = (μ (xn ) − N1)+ ≤ (μ (y) − N 1)+ = μ (y − N 1)+

.





and so, . (xn − N1)+ E ≤ (y − N1)+ E ≤ ε/2 for all n. Hence, .



‖xn ‖E ≤ ‖xn ∧ N 1‖E + (xn − N 1)+ E ≤ ‖xn ∧ N 1‖E + ε/2

(5.16)

for all n. Since .0 ≤ xn ∧N 1 ≤ xn and .xn → 0 with respect to the measure topology, it follows that .xn ∧ N1 → 0 as .n → ∞ with respect to the measure topology (see Proposition 2.6.1 (iv)). Observing that .0 ≤ xn ∧ N1 ≤ N1 for all n, it follows from the first part of the proof that .‖xn ∧ N 1‖E → 0 as .n → ∞. Therefore, there exists .M ∈ N such that .‖xn ∧ N1‖E ≤ ε/2 for all .n ≥ M. Consequently, by (5.16), this implies that .‖xn ‖E ≤ ε for all .n ≥ M. The proof is complete. ⨆ ⨅ The following lemma is a simple consequence of Theorem 5.6.19. Note that the case in which .e = 0 was already treated in Lemma 5.6.18. Lemma 5.6.20 Let .E ⊆ S (τ ) be a strongly symmetric space with order continuous norm. If .{pn }∞ n=1 and e are finite trace projections in .P (M) such that .pn → e with respect to the measure topology, then .‖e − pn ‖E → 0 as .n → ∞. Proof It follows from Lemma 5.6.15 that there exists .0 < C ∈ R such that τ (pn ) ≤ C for all n. Hence, .τ (e ∨ pn ) ≤ τ (e) + C for all n. Let .q ∈ P (M) be a finite trace projection such that .τ (e ∨ pn ) ≤ τ (q) for all n (such a q always exists; if .τ (1) < ∞, then take .q = 1; if .τ (1) = ∞, then .P (M) contains projections of arbitrarily large finite trace). Since .(e ∨ pn )⊥ ≤ n (e − pn ) (where .n (e − pn ) denotes the null projection of .e − pn ), it follows that .s (e − pn ) ≤ e ∨ pn and so, .τ (s (e − pn )) ≤ τ (q). Hence, .μ (t; e − pn ) = 0 for all .t ≥ τ (q) (see Remark 3.2.6). Since .‖e − pn ‖B(H ) ≤ 2, this implies that .μ (e − pn ) ≤ μ (2q) for ⨆ ⨅ all n. The statement of the lemma now follows from Theorem 5.6.19. .

The following observation is an immediate consequence of the above lemma. Corollary 5.6.21 Let .E ⊆ S (τ ) be a strongly symmetric space with order continuous norm. If the trace .τ is separable, then the set .{p ∈ P (M) : τ (p) < ∞} is separable with respect to the norm topology in E. Proof Since .τ is separable, there exists a countable subset .C of .{p ∈ P (M) : τ (p) < ∞} which is dense with respect to the measure topology. Since the measure topology is metrizable, it follows immediately from Lemma 5.6.20 that .C is dense in .{p ∈ P (M) : τ (p) < ∞} with respect to the norm topology in E. ⨆ ⨅

.

The following result shows that separability of the trace is the appropriate notion for spaces with order continuous norm to be norm separable.

5.6 Further Characterizations of Order Continuity

351

Theorem 5.6.22 Let .E ⊆ S (τ ) be a strongly symmetric space with order continuous norm. The following statements are equivalent: (i) E is norm separable. (ii) The trace .τ is separable. Proof It will be convenient to denote .D = {p ∈ P (M) : τ (p) < ∞}. (i).⇒(ii) Since the embedding of .(E, ‖·‖E ) into .(S (τ ) , Tm ) is continuous (see Proposition 4.4.4), it follows that E is separable with respect to the measure topology. Since .D ⊆ E, it follows that D is separable for the measure topology, that is, .τ is separable. (ii).⇒(i) If .τ is separable, then it follows from Corollary 5.6.21 that the set D is ‖·‖ norm separable and hence, .span (D) E is norm separable. The claim is that .E ⊆ ‖·‖E span (D) . Indeed, if .0 ≤ x ∈ E, then .x ∈ S0 (τ ) (see Lemma 5.3.5) and so, it follows from the spectral theorem that there exists a sequence .{an }∞ n=1 in .span (D) such that .0 ≤ an ↑n x (cf. the proof of Lemma 5.6.16). The order continuity of the ‖·‖ norm in E implies that .‖x − an ‖E → 0 as .n → ∞ and so, .x ∈ span (D) E . This suffices to complete the proof. ⨆ ⨅ Combining the above theorem with the result of Theorem 5.6.10, the next theorem follows. Theorem 5.6.23 If .E ⊆ S (τ ) is a strongly symmetric space, then the following statements are equivalent: (i) E has order continuous norm and the trace .τ is separable. (ii) E is norm separable. Proof (i).⇒(ii) This implication follows immediately from Theorem 5.6.22. (ii).⇒(i) If E is norm separable, then E cannot contain an isomorphic copy of .𝓁∞ and so, by Theorem 5.6.10, E has order continuous norm. Now it follows from ⨆ ⨅ Theorem 5.6.22 that the trace .τ is separable. The following result is now an immediate consequence of Theorem 5.6.23 (and of Proposition 5.6.17). Corollary 5.6.24 Suppose that the trace .τ is separable (which is in particular the case if the underlying Hilbert space H is separable). If .E ⊆ S (τ ) is a strongly symmetric space, then the following statements are equivalent: (i) E has order continuous norm. (ii) E is norm separable. As can been seen in Example 5.6.13, if .(M, τ ) is a semi-finite .σ -finite von Neumann algebra, then the trace .τ need not be separable. However, as is shown in the next proposition, the converse statement holds. Proposition 5.6.25 If .(M, τ ) is a semi-finite von Neumann algebra and the trace .τ is separable, then .M is .σ -finite.

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof Let .{pn }∞ < ∞}, which is n=1 be a countable subset of .{p ∈ P (M) : τ (p) ∞ respect to the measure topology. The claim is that . dense with n=1 pn = 1. Indeed,  ⊥ > 0. Let .e ∈ P (M) be set .q = ∞ p and suppose that . q = / 1, that is, . q n n=1 a finite trace projection such that .0 < e ≤ q ⊥ . Since .e ≤ pn⊥ , it follows that .|e − pn | = e + pn and so, .

‖e − pn ‖1 = τ (|e − pn |) = τ (e) + τ (pn ) ≥ τ (e) > 0,

n ∈ N.

Hence, e does not belong to the .‖·‖1 -closure of the set .{pn }∞ n=1 . By Lemma 5.6.15, this implies that e does not belong to the .Tm -closure of the set .{pn }∞ n=1 . This is a ∞ contradiction. Therefore, it may be concluded that . n=1 pn = 1 and it now follows from Lemma 5.5.5 that .M is .σ -finite. ⨆ ⨅ The following two results relate separability to reflexivity. Theorem 5.6.26 Let .E ⊆ S (τ ) be a strongly symmetric space. If the Banach dual E ∗ is separable and E has the .σ -Fatou property, then E is reflexive.

.

Proof As is well known, separability of .E ∗ implies that of E. Theorem 5.6.23 now implies that E has order continuous norm and that the trace .τ is separable. This implies that the Banach dual .E ∗ may be identified with the Köthe dual .E × via trace duality (see Proposition 5.3.2 in combination with Theorem 5.2.9). Consequently, the space .E × is norm separable and hence, .E × has also order continuous norm (again by Theorem 5.6.23). Furthermore, by Proposition 5.6.25, separability of .τ implies that .M is .σ -finite and hence, by Proposition 5.6.5, E has the Fatou property. Thus, the space E has order continuous norm and the Fatou property and .E × has order continuous norm. Theorem 5.3.10 now implies that E is reflexive. ⨆ ⨅ Theorem 5.6.27 Let .E ⊆ S (τ ) be a strongly symmetric space. If .E ∗∗ is separable, then E is reflexive. Proof The separability of .E ∗∗ implies that both E and .E ∗ are separable. In ∗ = E × (via trace duality). particular, E has order continuous norm  ×and ∗ so, .E × ∗∗ Since .E has the Fatou property and . E = E is separable, it follows from Theorem 5.6.26 that .E × = E ∗ is reflexive. Hence, E is reflexive. ⨆ ⨅ Remark 5.6.28 That Theorem 5.6.27 fails in the setting of general Banach spaces is a well-known result of R.C. James. Another characterization of order continuity of the norm will be discussed next. Some preparation is required. Let .E ⊆ S (τ ) be a normed .M-bimodule (satisfying ∗ .cE = 1, as usual) with Banach dual .E . Recall from the beginning of Sect. 4.2 ∗ that, for each .φ ∈ E , the conjugate functional .φ¯ ∈ E ∗ is given by .φ¯ (x) = φ (x ∗ ), ¯ .φ ∈ E ∗ , is a conjugate linear isometric involution .x ∈ E, and that the map .φ │−→ φ, ∗ ∗∗ ¯ ∈ E ∗∗ by setting in .E . For .Ф ∈ E , define .Ф .

 ¯ (φ) = Ф φ¯ , φ ∈ E ∗ . Ф

5.6 Further Characterizations of Order Continuity

353

¯ is a conjugate linear isometric involution in .E ∗∗ . A functional The map .Ф │−→ Ф ∗∗ ¯ = Ф. The set of all Hermitian elements .Ф ∈ E is termed Hermitian whenever .Ф in .E ∗∗ is a real linear subspace of .E ∗∗ and will be denoted by .(E ∗∗ )h . It is readily verified that a functional .Ф ∈ E ∗∗ is Hermitian if and only if .Ф (φ) ∈ R for all .φ ∈ Eh∗ . For each .Ф ∈ E ∗∗ , let .ReФ, .ImФ ∈ (E ∗∗ )h be defined by setting   1 1 ¯ ¯ .ReФ = 2 Ф + Ф and .ImФ = 2i Ф − Ф , respectively. Observe that .‖ReФ‖E ∗∗ , ∗∗ .‖ImФ‖E ∗∗ ≤ ‖Ф‖E ∗∗ for all .Ф ∈ E , from which it follows, in particular, that ∗∗ ∗∗ ∗∗ .(E )h is closed in .E . Since .Ф = ReФ + iImФ for all .Ф ∈ E , it follows also ∗∗ ∗∗ ∗∗ that .E = (E )h ⊕ i (E )h (as a direct sum of real linear subspaces). Furthermore, it is not difficult to verify that .

  ‖Ф‖E ∗∗ = sup |Ф (φ)| : φ ∈ Eh∗ , ‖φ‖E ∗ ≤ 1

for all .Ф ∈ (E ∗∗ )h (cf. equation (4.5) in Sect. 4.2). This may be used to show that the map .Ф │−→ Ф|Eh∗ , .Ф ∈ (E ∗∗ )h , is a real linear isometry from .(E ∗∗ )h onto  ∗  ∗ the real Banach dual space . Eh∗ . Identifying .(E ∗∗ )h in this way with . Eh∗ , there is no danger of confusion in denoting .(E ∗∗ )h by .Eh∗∗ . Moreover, since .Eh∗ is the real Banach dual of .Eh (as has been observed in Sect. 4.2), it follows also that ∗∗ ∗∗ .E h = (Eh ) . A functional .Ф ∈ E ∗∗ is called positive if .Ф (φ) ≥ 0 for all .φ ∈ (E ∗ )+ . The set of all positive functionals in .E ∗∗ is denoted by .(E ∗∗ )+ . Since the positive cone ∗ + is generating in .E ∗ , it follows that .(E ∗∗ )+ ⊆ E ∗∗ and that .(E ∗∗ )+ is a .(E ) h h proper cone in .Eh∗∗ . The space .Eh∗∗ will be considered equipped with the partial order induced by the cone .(E ∗∗ )+ , that is, if .Ф, Ψ ∈ Eh∗∗ , then .Ф ≤ Ψ if, and only if, .Ψ − Ф ∈ (E ∗∗ )+ . From the general theory of ordered normed linear spaces, it actually follows that .(E ∗∗ )+ is generating, that is, .Eh∗∗ = (E ∗∗ )+ − (E ∗∗ )+ . It should be observed that .(E ∗∗ )+ is .σ (E ∗∗ , E ∗ )-closed in .E ∗∗ . Indeed, suppose that .{Фα } is a net in .(E ∗∗ )+ and .Ф ∈ E ∗∗ is such that .Ф →α Ф with respect to ∗∗ ∗ ∗ + .σ (E , E ). If .φ ∈ (E ) , then .Фα (φ) ≥ 0 for all .α and .Фα (φ) →α Ф (φ), and therefore, .Ф (φ) ≥ 0. Hence, .Ф ∈ (E ∗∗ )+ . For .Ф, Ψ ∈ Eh∗∗ with .Ф ≤ Ψ, define the order interval .[Ф, Ψ] by setting .

  [Ф, Ψ] = 𝚪 ∈ Eh∗∗ : Ф ≤ 𝚪 ≤ Ψ .

  Since .[Ф, Ψ] = Ф + (E ∗∗ )+ ∩ Ψ − (E ∗∗ )+ , it follows from the above observation that .[Ф, Ψ] is .σ (E ∗∗ , E ∗ )-closed in .E ∗∗ . A subset .A ⊆ Eh∗∗ is said to be solid if .[Ф, Ψ] ⊆ A whenever .Ф, Ψ ∈ A with .Ф ≤ Ψ. Note that if A is a linear subspace of .Eh∗∗ , then A is solid if and only if .[0, Ф] ⊆ A for all .Ф ∈ A+ . Let .π : E → E ∗∗ be the canonical isometrical embedding of E into .E ∗∗ , that is, .(π x) (φ) = φ (x) for all .φ ∈ E ∗ and .x ∈ E. Observing that .π x = π x ∗ for all ∗∗ .x ∈ E, it follows that .π x ∈ E h if and only if .x ∈ Eh . This implies, in particular, that the restriction .π|Eh is the canonical embedding of .Eh into its Banach bidual ∗∗ .E . Furthermore, it is easy to see that if .x ∈ E, then .π x ≥ 0 if and only if h ∗ + .x ≥ 0. Indeed, if .x ≥ 0, then .(π x) (φ) = φ (x) ≥ 0 for all .φ ∈ (E ) , so .π x ≥ 0.

354

5 Strongly Symmetric Spaces of .τ -Measurable Operators

 Conversely, if .π x ≥ 0, then, in particular, .(π x) φy ≥ 0 for all .0 ≤ y ∈ E × (where × .φy (z) = τ (zy), .z ∈ E). Therefore, .τ (xy) = (π x) φy ≥ 0 for all .0 ≤ y ∈ E , × which implies that .x ≥ 0 (using, e.g., Lemma 4.3.7, applied in the space .E ). Consequently, the map .π : Eh → π (Eh ) is bi-positive. Theorem 5.6.29 If .E ⊆ S (τ ) is a strongly symmetric space, then the following statements are equivalent: (i) E has order continuous norm. (ii) The space .π (Eh ) is a solid subspace of .Eh∗∗ . (iii) For each .x ∈ E + , the order interval .[0, x] in .Eh is .σ (E, E ∗ )-compact. Proof (i).⇒(ii) Observe first that the order continuity of the norm in E implies that E ∗ may be identified with the Köthe dual .E × via trace duality (see Proposition 5.3.2 and Theorem 5.2.9). To show (Eh ) is solid in .Eh∗∗ , it is sufficient to prove that  that .π ∗∗ + for each .x ∈ E , the set . Ф ∈ Eh : 0 ≤ Ф ≤ π x is contained in .π (Eh ). Given + and .Ф ∈ E ∗∗ such that .0 ≤ Ф ≤ π x, let .{y } be a downward directed .x ∈ E α h system in .E × such that .y ↓α 0. This implies that .

0 ≤ Ф (yα ) ≤ (π x) (yα ) = τ (xyα ) ↓α 0

.

and hence, .Ф (yα ) ↓α 0. This shows that .Ф is a normal functional on .E × and so, by Theorem 5.2.9, there exists a unique .0 ≤ z ∈ E ×× such that .Ф (y) = τ (zy) for all × × .y ∈ E . Observing that .τ ((x − z) y) = (π x − Ф) (y) ≥ 0 for all .0 ≤ y ∈ E , it × ×× follows from Lemma 4.3.7 (applied in .E ) that .x−z ≥ 0 in .E , that is, .0 ≤ z ≤ x. Proposition 4.1.3 (iii) now implies that .z ∈ E and so, .Ф = π z, which suffices for the proof that (i) implies (ii). (ii).⇒(iii) Given .x ∈ E + , observe first that   π ([0, x]) = Ф ∈ Eh∗∗ : 0 ≤ Ф ≤ π x .

.

(5.17)

  Indeed, it is clear that .π ([0, x]) ⊆ Ф ∈ Eh∗∗ : 0 ≤ Ф ≤ π x . Suppose that .Ф ∈ Eh∗∗ is such that .0 ≤ Ф ≤ π x. Since .π (Eh ) is assumed to be solid in .Eh∗∗ , this implies that .Ф = π z for some .z ∈ Eh . As observed before, the map .π : Eh → π (Eh ) is bi-positive and hence, .0 ≤ z ≤ x. Therefore, .Ф ∈ π ([0, x]), which establishes assertion (5.17). Consequently, .π ([0, x]) = [0, π x] (where the latter is the order interval in ∗∗ ∗∗ ∗ .E ) and so, .π ([0, x]) is norm bounded and .σ (E , E )-closed. Therefore, by the h Banach–Bourbaki–Alaoglu theorem, the set .π ([0, x]) is .σ (E ∗∗ , E ∗ )-compact. This implies that .[0, x] is .σ (E, E ∗ )-compact. (iii).⇒(i) Suppose that .{xα } is a downward directed net in E such that .xα ↓α 0 with .0 ≤ xα ≤ x for some .x ∈ E + and such that .‖xα ‖E α 0, in which case there exists .0 < ε ∈ R satisfying .‖xα ‖E ≥ ε for all  .α. Since .[0, x] is assumed to be .σ (E, E ∗ )-compact, there exist a subnet . xα(β) β∈B and an element .y ∈ [0, x] such that .xα(β) →β y with respect to .σ (E, E ∗ ). Observe that .xα(β) ↓β 0 (in fact, if .z ∈ E satisfies .z ≤ xα(β) for all .β ∈ B, then for every .α there exists .β ∈ B such that

5.7 Sets of Uniformly Absolutely Continuous Norm

355

α ≤ α(β) and so .z ≤ xα(β) ≤ xα ). If .β ∈ B, then .0 ≤ xα(γ ) ≤ xα(β)  for all .γ ≥ β. Since . 0, xα(β) is .σ (E, E ∗ )-closed, it follows that .y ∈ 0, xα(β) . Consequently,



∗ .y = 0. This shows that .φ xα(β) → 0 for all .φ ∈ E and hence, . xα(β)

→β 0 E (see Lemma 4.2.19). This contradicts the assumption that .‖xα ‖E ≥ ε for all .α. Therefore, it may be concluded that the norm in E is order continuous, and the proof is complete. ⨆ ⨅ .

For the reader’s convenience, some of the main results in this section are collected together in the following theorem. Theorem 5.6.30 If .E ⊆ S (τ ) is a strongly symmetric space with .cE = 1, then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

The norm in E is order continuous. E does not contain an isomorphic copy of .𝓁∞ . E has property .(u). For each .x ∈ E + , the order interval .[0, x] is .σ (E, E ∗ )-compact. ∗ × via trace duality. .E = E

Moreover, if the trace .τ is separable, then each of the above statements is also equivalent to: (vi) The space E is norm separable. It should be noted that the result of the above theorem is valid also if the space E is symmetric under the assumption that the underlying von Neumann algebra .M is either non-atomic or atomic and all minimal projections have equal trace. References: [38, 41, 43].

5.7 Sets of Uniformly Absolutely Continuous Norm In this section several properties of subsets .A of a strongly symmetric space .E ⊆ S (τ ) which are of uniformly absolutely continuous norm (see Definition 5.5.3) will be discussed in some detail. A principal result is that a bounded subset .A ⊆ E is of absolutely uniformly continuous norm if and only if for every .ε > 0 there exists an = E oc such that .0 ≤ y ∈ E A ⊆ yBM + BM y + εBE

.

(see Theorem 5.7.9), where .BM and .BE denote the closed unit ball in .M and E, respectively. It should be observed that, although it is assumed that the spaces considered are strongly symmetric, the results are also valid for symmetric spaces under the additional assumption that the von Neumann algebra .M is either nonatomic or atomic with all minimal projections having equal trace.

356

5 Strongly Symmetric Spaces of .τ -Measurable Operators

As before, .M is a semi-finite von Neumann algebra equipped with a normal faithful semi-finite trace .τ . Let .E ⊆ S (τ ) be a strongly symmetric space (with .cE = 1, as usual). Recall from Definition 5.5.3 that a subset .A of E is said to be of uniformly absolutely continuous norm if .sup {‖en xen ‖E : x ∈ A} → 0 as ∞ .n → ∞, for all sequences .{en } n=1 in .P (M) satisfying .en ↓ 0. Since .‖en xen ‖E =



(en xen )∗ = ‖en x ∗ en ‖E for all .x ∈ A and .n ∈ N, it is clear that a set .A is of E uniformly absolutely continuous norm if and only if the set .A∗ = {x ∗ : x ∈ A} is of uniformly absolutely continuous norm. The proof of the following lemma is straightforward. Lemma 5.7.1 Let .E ⊆ S (τ ) be a strongly symmetric space. (i) If .A1 , A2 ⊆ E are of uniformly absolutely continuous norm, then so is .A1 + A2 . (ii) If .A ⊆ E and for each .ε > 0 there exists a subset .Aε ⊆ E of uniformly absolutely continuous norm such that .A ⊆ Aε + εBE , then .A is of uniformly absolutely continuous norm. In particular, it follows from Lemma 5.7.1 (i) that a set .A is of absolutely uniformly continuous norm if and only if both sets .ReA = {Rex : x ∈ A} and .ImA = {Imx : x ∈ A} have this property. It should also be observed that if the set .A is of uniformly absolutely continuous norm, then each individual element .x ∈ A has absolutely continuous norm (see Definition 5.5.1), that is, .A ⊆ E an = E oc . In addition, it follows from statement (ii) in the above lemma that if a set .A ⊆ E is uniformly absolutely continuous norm, then so is its norm closure .A. In the next proposition, some further characterizations of sets of uniformly absolutely continuous norm are presented. Proposition 5.7.2 If .E ⊆ S (τ ) is a strongly symmetric space and .A ⊆ E, then the following statements are equivalent: (i) .A is of uniformly absolutely continuous norm. (ii) .supx∈A ‖en xen ‖E → 0 as .n → ∞ whenever .{en }∞ n=1 is a mutually orthogonal sequence in .P (M). (iii) .supx∈A ‖eα xeα ‖E →α 0 for every downward directed system .{eα } in .P (M) satisfying .eα ↓α 0. Proof (i).⇒(ii) Suppose that .{en }∞ n=1 is a mutually orthogonal sequence in .P (M), but that .supx∈A ‖en xen ‖E  0 as .n → ∞. Passing, if necessary, to a subsequence of .{en }∞ n=1 , it may be assumed that there exists .ε > 0 such that .supx∈A ‖en xen ‖E > ∞ ε for all .n ∈ N. This implies that there exists a sequence .{xn }n=1 such that .‖en xn en ‖E > ε for all .n. Setting .pn = m≥n em , .n ∈ N, it is easily

5.7 Sets of Uniformly Absolutely Continuous Norm

357

verified that .pn ↓n 0 in .P (M). Therefore, it follows from assumption (i) that supx∈A ‖pn xpn ‖E → 0 as .n → ∞. On the other hand,

.

0 < ε < ‖en xn en ‖E = ‖en pn xn pn en ‖E

.

≤ ‖pn xn pn ‖E ≤ sup ‖pn xpn ‖E x∈A

for all n, which is a contradiction. Hence, it may be concluded that (i) implies (ii). (ii).⇒(iii) Suppose that (iii) fails for some downward directed system .{eα }α∈A in .P (M) satisfying .eα ↓α 0. Since .supx∈A ‖eα xeα ‖E ↓α , this implies that there exists .0 < ε ∈ R such that .supx∈A ‖eα xeα ‖E > ε for all .α and so, for each .α ∈ A there exists an element .xα ∈ A such that .‖eα xα eα ‖E > ε. Observe that

for each .α ∈ A, there exists .γ ≥ α in .A such that . xα eγ E ≤ ε/4 and . eγ xα E ≤ ε/4. Indeed, an since .xα ∈ E .β1 ≥ α in

, it follows from Theorem 5.5.12 (iv) that there exists ∗ an , there

.A such that . xα eβ

≤ ε/4 for all .β ≥ β1 . Similarly, since also .xα ∈ E E

∗



exists .β2 ≥ α in .A such that . eβ xα E = xα eβ ≤ ε/4 for all .β ≥ β2 . Let now .γ ∈ A be such that .γ ≥ β1 and .γ ≥ β2 . Consequently, there exists an increasing sequence .{αi }∞ i=1 in .A such that .



xα eα ≤ ε/4, i i+1 E



eα

i+1

xαi E ≤ ε/4,

i ∈ N.

Setting .xi = xαi , .ei = eαi , and .pi = ei − ei+1 (.i ∈ N), it follows that .

‖pi xi pi ‖E ≥ ‖ei xi ei ‖E − ‖ei xi ei+1 ‖E − ‖ei+1 xi ei ‖E − ‖ei+1 xi ei+1 ‖E ≥ ‖ei xi ei ‖E − ‖xi ei+1 ‖E − ‖ei+1 xi ‖E − ‖xi ei+1 ‖E > ε/4.

Since the sequence .{pi }∞ i=1 is mutually orthogonal in .P (M), this contradicts assumption (ii). The implication (iii).⇒(i) being obvious, the proof of the proposition is complete. ⨆ ⨅ For the proof of the main result in this section, some further preparation is needed. Definition 5.7.3 Let .E ⊆ S (τ ) be a strongly symmetric space. For .x ∈ E and δ > 0, set

.

ω (x; δ) = sup {‖exe‖E : e ∈ P (M) , τ (e) ≤ δ} .

.

Observe that .ω (x; δ) ≤ ‖x‖E and .ω (x; δ) = ω (x ∗ ; δ) for all .x ∈ E and .δ > 0. Furthermore, .ω (x; δ1 ) ≤ ω (x; δ2 ) whenever .0 < δ1 ≤ δ2 . Lemma 5.7.4 If .A ⊆ E is of uniformly absolutely continuous norm, then .

lim sup ω (x; δ) = 0. δ↓0 x∈A

358

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof Since .ω (x; δ) ↓δ , it is sufficient to show that .

 lim sup ω x; 2−n = 0.

n→∞ x∈A

For each .n ∈ N, there exists a projection .en ∈ P (M) with .τ (en ) ≤ 2−n , and there exists .xn ∈ A satisfying .

 ‖en xn en ‖E ≥ sup ω x; 2−n − 1/n.

(5.18)

x∈A

Set .pn = supm≥n em and observe that .τ (pn ) ≤ 2−n+1 , which implies that .pn ↓n 0 in .P (M). Furthermore, .

‖en xn en ‖E = ‖en pn xn pn en ‖E ≤ ‖pn xn pn ‖E ≤ sup ‖pn xpn ‖E ,

n ∈ N.

x∈A

Since .A is of uniformly absolutely continuous norm, it follows that .supx∈A ‖pn xpn ‖E →n 0 and hence, .‖en xn en ‖E →n 0. Therefore, (5.18) now implies that −n → 0 and the proof is complete. .supx∈A ω x; 2 ⨆ ⨅ n In general, the condition that .limδ↓0 supx∈A ω (x; δ) = 0 is not sufficient for a set .A to be of uniformly absolutely continuous norm. Indeed, consider the interval .(0, ∞) equipped with Lebesgue measure m and let .E = L1 (0, ∞). Defining .A =   χ[n,n+1) : n ∈ N , it is readily verified that .limδ↓0 supx∈A ω (x; δ) = 0, but the set .A is not of uniformly absolutely norm. However, if the trace is finite, then the following holds. Corollary 5.7.5 Suppose that .τ (1) < ∞ and that .E ⊆ S (τ ) is a strongly symmetric space. For a subset .A ⊆ E, the following statements are equivalent: (i) .A is of uniformly absolutely continuous norm. (ii) .limδ↓0 supx∈A ω (x; δ) = 0. Proof The implication (i).⇒(ii) has been obtained in Lemma 5.7.4. To show that (ii) implies (i), suppose that .{en }∞ n=1 is a sequence in .P (M) such that .en ↓n 0. This implies that .τ (en ) ↓n 0, as .τ (1) < ∞. Given .ε > 0, let .δ > 0 be such that .supx∈A ω (x; δ) ≤ ε. There exists .N ∈ N such that .τ (en ) ≤ δ for all .n ≥ N . This implies that .

sup ‖en xen ‖E ≤ sup ω (x; δ) ≤ ε,

x∈A

n ≥ N,

x∈A

which shows that .supx∈A ‖en xen ‖E →n ∞. Hence .A is of uniformly absolutely continuous norm. ⨆ ⨅

5.7 Sets of Uniformly Absolutely Continuous Norm

359

The following simple observation will also be used. Lemma 5.7.6 If .x ∈ S (τ ) and .0 < t ∈ R, then there exists a projection .p ∈ P (M) such that .τ (p) ≤ t and .

‖x (1 − p)‖∞ ≤ μ (t; x) .

Moreover, if .x ∗ = x, then p may be taken such that .px = xp. Proof If .μ (t; x) = 0, then the projection .p = s (x) satisfies .τ (p) ≤ t (see Remark 3.2.6) and .x (1 − p) = 0 (and .x ∗ = x implies that .px = xp). Therefore, preceding it may be assumed that .μ (t; x) > 0. As observed in the discussion  Proposition 3.2.5, .x ∈ V (μ (t; x) , t) and so, by Lemma 2.5.1, .τ e|x| (α, ∞) ≤ t with .α = μ (t; x). Defining .p = e|x| (α, ∞), it follows that .τ (p) ≤ t and .





‖x (1 − p)‖∞ = xe|x| [0, α]

∞

≤ α = μ (t; x) .

Furthermore, if .x ∗ = x, then .px = xp (in fact, .p = ex (−∞, α) + ex (α, ∞)).

⨆ ⨅

Recall that any strongly symmetric space .E ⊆ S (τ ) is continuously embedded in the space .L1 (τ ) + L∞ (τ ) (see Lemma 5.1.3) and so, there is a constant .K > 0 such that .

‖x‖L1 +L∞ ≤ K ‖x‖E ,

x ∈ E.

(5.19)

Lemma 5.7.7 If .E ⊆ S (τ ) is a strongly symmetric space and if the constant .K > 0 satisfies (5.19), then μ (t; x) ≤ K min (t, 1)−1 ‖x‖E

.

for all .x ∈ E and .t > 0. Proof Since .μ (x) is decreasing, it is clear that .μ (t; x) χ[0,t) ≤ μ (x) and hence, 



μ (t; x) χ[0,t) L

.

1 +L∞

≤ ‖μ (x)‖L1 +L∞ =

1

μ (s; x) ds 0

= ‖x‖L1 +L∞ ≤ K ‖x‖E (cf. the discussion at the beginning of Sect. 3.10). Observing that



. χ[0,t)

L1 +L∞

the result of the lemma follows.

 =

1

χ[0,t) (s) ds = min (t, 1) ,

0

⨆ ⨅

360

5 Strongly Symmetric Spaces of .τ -Measurable Operators

As has been observed after Definition 5.5.3, if .0 ≤ y ∈ E an , then the order interval .[0, y] is of uniformly absolutely continuous norm. The next proposition exhibits other important examples of such sets. Note that both .yBM and .BM y are subsets of E, as E is an .M-bimodule. Proposition 5.7.8 Let .E ⊆ S (τ ) be a strongly symmetric space and .y ∈ E an . If ∞ .{en } n=1 is a sequence of projections in .P (M) such that .en ↓n 0, then the following statements hold: (i) .supx∈yBM ‖en x‖E → 0 as .n → ∞. (ii) .supx∈BM y ‖xen ‖E → 0 as .n → ∞. In particular, the sets .yBM and .BM y are of uniformly absolutely continuous norm. Proof (i) Since .y ∈ E an , it follows that .

sup ‖en x‖E = sup ‖en yz‖E ≤ ‖en y‖E →n 0,

x∈yBM

z∈BM

and hence, .supx∈yBM ‖en x‖E → 0 as .n → ∞. The proof of (ii) is similar. ⨆ ⨅ Before proving one of the main results in this section, recall that the set .E an of all elements in E of absolutely continuous norm is an .M-bimodule. Indeed, .E an = E oc (see Proposition 5.5.11) and .E oc is an .M-bimodule (see Corollary 5.4.4). Let an (see Definition 4.1.5). It follows from .cE an ∈ P (M) be the carrier projection of .E Lemma 4.1.8, in combination with the semi-finiteness of the trace, that there exists an upward directed system .{pα } of finite trace projections in .P (E an ) satisfying .pα ↑α cE an . For convenience, the carrier projection .cE an will be denoted by .can . Theorem 5.7.9 Let .E ⊆ S (τ ) be a strongly symmetric space and let .A be a bounded subset of E. The following statements are equivalent: (i) .A is of uniformly absolutely continuous norm. (ii) For each upward directed system .{pα } of finite trace projections in .P (E an ) satisfying .pα ↑α can and every .ε > 0, there exist .αε and a constant .Cε > 0 such that  A ⊆ Cε pαε BM + BM pαε + εBE .

.

(iii) For every .ε > 0, there exists .0 ≤ yε ∈ E an such that A ⊆ yε BM + BM yε + εBE .

.

(5.20)

5.7 Sets of Uniformly Absolutely Continuous Norm

361

Proof (i).⇒(ii) Let .{pα } be an upwards directed system of finite trace projections in .P (E an ) satisfying .pα ↑α can and let .ε > 0 be given. It will be assumed first, in addition, that .x ∗ = x for all .x ∈ A. By Lemma 5.7.4, there exists a .δ > 0 such that .

sup ω (x; δ) ≤ ε/4.

(5.21)

x∈A

Since .A is assumed to be norm bounded in E, Lemma 5.7.7 implies that Cε = 2 sup μ (δ; x) < ∞.

.

x∈A

Furthermore, since .can − pα ↓α 0, it follows from Proposition 5.7.2 that there exists an .αε such that .

 

sup can − pαε x can − pαε E ≤ ε/4.

(5.22)

x∈A

It will be shown that (5.20) holds for this projection .pαε and constant .Cε . Indeed, for convenience, set .p = pαε and let .x ∈ A be given. By Lemma 5.7.6, there exists a projection .e ∈ P (M) such that ex = xe,

.

τ (e) ≤ δ,

‖x (1 − e)‖∞ ≤ μ (δ; x) .

Since .x ∈ E an , it follows that x = can xcan = px (can − p) + pxp + (can − p) xp + (can − p) x (can − p) .

.

Write px (can − p) = pxe (can − p) + px (1 − e) (can − p) ,

.

pxp = pxep + px (1 − e) p (can − p) xp = (can − p) xep + (can − p) x (1 − e) p. Using the fact that .ex = xe = exe, observe that .

‖pxe (can − p)‖E ≤ ‖xe‖E = ‖exe‖E ≤ ω (x; δ) ,

(5.23)

and similarly, .

‖pxep‖E ≤ ω (x; δ) ,

‖(can − p) xep‖E ≤ ω (x; δ) .

(5.24)

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

Since .‖x (1 − e)‖∞ ≤ μ (δ; x), it follows that x (1 − e) (can − p) ∈ μ (δ; x) BM

.

and so, px (1 − e) (can − p) ∈ μ (δ; x) pBM ⊆ (Cε /2) pBM ,

.

px (1 − e) p ∈ μ (δ; x) pBM ⊆ (Cε /2) pBM , (can − p) x (1 − e) p ∈ μ (δ; x) BM p ⊆ (Cε /2) BM p. Consequently, the element z = px (1 − e) (can − p) + px (1 − e) p + (can − p) x (1 − e) p

.

satisfies z ∈ Cε (pBM + BM p) .

.

Furthermore, x − z = pxe (can − p) + pxep + (can − p) xep + (can − p) x (can − p)

.

and so, it follows from (5.21), (5.22), (5.23), and (5.24) that .

‖x − z‖E ≤ 3ω (x; δ) + ε/4 ≤ ε.

This shows that (i) implies (ii) under the additional assumption that .x ∗ = x for all .x ∈ A. However, the general case now follows by noting that if .A ⊆ E is any set of uniformly absolutely continuous norm, then so are sets .ReA and .ImA (see the discussion at the beginning of the present section). It is clear that (iii) follows from (ii) and that (iii) implies (i) follows immediately from Proposition 5.7.8 in combination with Lemma 5.7.1. This completes the proof of the theorem. ⨆ ⨅ The following special case of Theorem 5.7.9 is worth mentioning. Corollary 5.7.10 Suppose that the von Neumann algebra .M is .σ -finite. Let .E ⊆ S (τ ) be a strongly symmetric space and let .A be a bounded subset of E. The following statements are equivalent: (i) .A is of uniformly absolutely continuous norm. (ii) There exists .0 ≤ y ∈ E an ∩ (L1 ∩ L∞ ) (τ ) such that for each .ε > 0 there exists a constant .0 ≤ Kε ∈ R satisfying A ⊆ Kε (yBM + BM y) + εBE .

.

5.7 Sets of Uniformly Absolutely Continuous Norm

363

Proof Only the implication (i).⇒(ii) needs a proof. Since .M is .σ -finite, there exists an a sequence .{pn }∞ n=1 of finite trace projections in .P (E ) such that .pn ↑ can . Indeed, it follows from Lemma 4.1.8, in combination with the semi-finiteness of the trace, that there exists a  mutually orthogonal system .{qα } of finite trace projections in an ) such that . .P (E α qα = can . Since .M is .σ -finite, it follows from n Lemma 5.5.5 that this system is at most countable, .{qn }∞ say. Now take .pn = m=1 qm , .n ∈ N. n=1 Setting .λn = 2−n (τ (pn ) + 1)−1 , .n ∈ N, it follows that .‖λn pn ‖L1 ∩L∞ ≤ 2−n and so, the series y=

.

∞ n=1

λ n pn

is norm convergent in .(L1 ∩ L∞ ) (τ ). Since the embedding of .(L1 ∩ L∞ ) (τ ) into E is continuous, this series is also convergent in E. Since .E an is closed in E (see Proposition 5.5.11), it is now clear that .0 ≤ y ∈ E an ∩ (L1 ∩ L∞ ) (τ ). Given .ε > 0, Theorem 5.7.9 (ii) implies that there exist .n ∈ N and .0 ≤ C ∈ R such that A ⊆ C (pn BM + BM pn ) + εBE .

.

Since .0 ≤ pn ≤ λ−1 n y and .ypn = pn y, there exists .z ∈ BM such that .pn = −1 yz. Consequently, λ−1 zy = λ n n −1 pn BM + BM pn = λ−1 n (yzBM + BM zy) ⊆ λn (yBM + BM y)

.

and so, A ⊆ Kε (yBM + BM y) + εBE ,

.

with .Kε = λ−1 n C. This suffices to complete the proof.

⨆ ⨅

The next characterization of sets of uniformly absolutely continuous norm follows also from Theorem 5.7.9. Theorem 5.7.11 Let .E ⊆ S (τ ) be a strongly symmetric space and .A ⊆ E be a norm bounded subset. The following statements are equivalent: (i) .A is of uniformly absolutely continuous norm.   (ii) For all bounded nets .{yα } and . zβ in .M+ satisfying .yα →α 0 and .zβ →β 0 with respect to .σ (M, L1 (τ )), it follows that .



sup yα xzβ E →(α,β) 0.

x∈A

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof (i).⇒(ii) Without loss of generality, it may be assumed that .yα , .zβ ∈ BM for all .α and .β. Given .ε > 0, Theorem 5.7.9 (iii) yields the existence of a .0 ≤ y ∈ E an such that ε A ⊆ yBM + BM y + BE . 3

.

Consequently, .









ε sup yα xzβ E ≤ sup yα ywzβ E + sup yα wyzβ E + 3 x∈A w∈BM w∈BM



ε ≤ ‖yα y‖E + yzβ E + 3

for all

.α and

.β. By Theorem 5.5.15, there exist .α0 and .β0 such that

.‖yα y‖

E ≤ ε/3 and . yzβ E ≤ ε/3 for all .α ≥ α0 and .β ≥ β0 . Hence, .supx∈A yα xzβ E ≤ ε for all .(α, β) ≥ (α0 , β0 ). (ii).⇒(i) If .{en }∞ n=1 is a sequence in .P (M) such that .en ↓n 0, then .en →n 0 with respect to .σ (M, L1 (τ )) (see, e.g., the proof of implication (ii).⇒(i) in Theorem 5.5.15). Therefore, assumption (ii) implies that .supx∈A ‖en xen ‖E → 0 as .n → ∞. Hence, the set .A is of uniformly absolutely continuous norm. ⨆ ⨅ Considering the definition and characterizations (see Proposition 5.7.2) of sets of uniformly absolutely continuous norm, all conditions are two-sided. It is also useful to consider the corresponding one-sided conditions. Definition 5.7.12 Let .E ⊆ S (τ ) be a strongly symmetrically normed space. A subset .A ⊆ E is said to be of right (respectively, left) uniformly absolutely continuous norm if .

sup ‖xen ‖E → 0,

n → ∞,

sup ‖en x‖E → 0,

n → ∞)

x∈A

respectively, .

x∈A

for all sequences .{en }∞ n=1 of projections in .P (M) satisfying .en ↓n 0. If .A is of both right and left uniformly absolutely continuous norms, then .A is said to be of bi-uniformly absolutely continuous norm. It should be observed that if a set .A is of right or left uniformly absolutely continuous norm, then .A is of uniformly absolutely continuous norm, in particular, an . It is also clear that a set .A is of right (respectively, left) uniformly .A ⊆ E absolutely continuous norm if and only if the set .A∗ = {x ∗ : x ∈ A} is of left (respectively, right) uniformly absolutely continuous norm. It should also be noted that the result of Proposition 5.7.8 may now be formulated by stating that if .y ∈ E an ,

5.7 Sets of Uniformly Absolutely Continuous Norm

365

then the set .yBM (respectively, .BM y) is of left (respectively, right) uniformly absolutely continuous norm. In general, the notions of right and left uniformly absolutely continuous norm are different, as is illustrated in the next example. Example 5.7.13 Let H be a separable infinite-dimensional Hilbert space with orthonormal basis .{ϕn }∞ n=1 . Consider the von Neumann algebra .B (H ) equipped with the canonical trace .τ and let .E ⊆ S (τ ) = B (H ) be any non-zero (strongly) symmetric space (e.g., .E = L1 (τ )). For each .n ∈ N, let .en be the orthogonal projection onto the one-dimensional subspace spanned by .ϕn and let .vn be the partial isometry in H defined by setting vn ξ = 〈ξ, ϕ1 〉 ϕn ,

.

ξ ∈ H.

The initial projection of .vn is .e1 and the final projection is .en (that is, .vn∗ vn = e1 and .vn vn∗ = en ). It should be observed also that .vn = vn e1 = en vn , .|vn | = e1 , and .vn∗  = en for all n. Note that .en ∈ E for all n (see Lemma 4.4.5) and so, in particular, .vn ∈ E for all n. It will be shown first that the set .{vn }∞ n=1 is not of left uniformly absolutely continuous norm in E. In fact, define, for .n ∈ N, the projection .pk = ∞ n=k en , in which case .pk ↓k 0. Observing that pk vk = pk ek vk = ek vk = vk ,

.

k ∈ N,

it follows that .

supn ‖pk vn ‖E ≥ ‖pk vk ‖E = ‖vk ‖E = ‖e1 ‖E > 0

for all k. Hence, the set .{vn }∞ n=1 is not of left uniformly absolutely continuous norm in E. However, the set .{vn }∞ n=1 is of right uniformly absolutely continuous norm in E. Indeed, if .{qn }∞ is a sequence of projections in .P (B (H )) such that .qn ↓n 0, then n=1 .

supn ‖vn qk ‖E = supn ‖vn e1 qk ‖E ≤ ‖e1 qk ‖E .

The minimal projection .e1 belongs to .E oc (see Proposition 5.4.10) and so .e1 ∈ E an . Therefore, it follows from Theorem 5.5.12 that .‖e1 qk ‖E → 0 and so, it may be concluded that the set .{vn }∞ n=1 is of right uniformly absolutely continuous norm in E.  ∞ It is now also clear that the set . vn∗ n=1 is of left uniformly absolutely continuous norm, but not of right  uniformly ∞ absolutely continuous norm. A moment’s reflection shows that the set . vn + vn∗ n=1 is of uniformly absolutely continuous norm, but neither of right nor left uniformly absolutely continuous norm.

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

The following proposition is a refinement of the statement of Proposition 5.5.9. Proposition 5.7.14 If .E ⊆ S (τ ) is a strongly symmetrically normed space and an {xn }∞ n=1 is a sequence in .E , then the following statements are equivalent:

.

(i) .‖xn ‖E → 0 as .n → ∞. (ii) .xn →n 0 with respect to the measure topology and the set .{xn : n ∈ N} is of bi-uniformly absolutely continuous norm. (iii) .xn →n 0 with respect to the measure topology and the set .{xn : n ∈ N} is of left or right uniformly absolutely continuous norm. (iv) .xn →n 0 with respect to the measure topology and the set .{xn : n ∈ N} is of uniformly absolutely continuous norm. Proof The implications (ii).⇒(iii).⇒(iv) are evident and (i).⇒(ii) follows from Proposition 5.5.9. That (iv) implies (ii) follows by an inspection of the proof of implication (i).⇒(ii) in Proposition 5.5.9. ⨆ ⨅ There are some important situations in which all three notions introduced in Definition 5.7.12 coincide with being of uniformly absolutely continuous norm. The first case is when the set .A consists of positive elements. Proposition 5.7.15 Suppose that .E ⊆ S (τ ) is a strongly symmetrically normed space. If .A ⊆ E + is bounded, then the following statements are equivalent: (i) (ii) (iii) (iv)

A is of uniformly absolutely continuous norm. A is of right uniformly absolutely continuous norm. .A is of left uniformly absolutely continuous norm. .A is of bi-uniformly absolutely continuous norm. . .

Proof Since the set .A consists of self-adjoint elements, it is clear that (ii) and (iii) are equivalent. Hence, (ii) implies (iv) and so, statements (ii), (iii), and (iv) are equivalent. It has already been observed that (ii) implies (i). Therefore, it remains to show that (i) implies (ii). (i).⇒(ii) Suppose that .A ⊆ E + is bounded and of uniformly absolutely continuous norm, and let .{en }∞ n=1 be a sequence of projections in .P (M) such that .en ↓n 0. Let .0 ≤ M ∈ R be such that .‖x‖E ≤ M for all .x ∈ A. It follows from Lemma 4.2.10 that .

1/2

1/2

1/2

‖xen ‖E ≤ 4 ‖x‖E ‖en xen ‖E ≤ 4M 1/2 ‖en xen ‖E ,

x ∈ A,

and hence, .

 1/2 sup ‖xen ‖E ≤ 4M 1/2 sup ‖en xen ‖E .

x∈A

x∈A

This implies that .A is of right uniformly absolutely continuous norm. The proof is complete. ⨆ ⨅

5.7 Sets of Uniformly Absolutely Continuous Norm

367

The observations made in Proposition 5.7.15 have the following consequences. For .A ⊆ E, denote .|A| = {|x| : x ∈ A}. Corollary 5.7.16 Let .E ⊆ S (τ ) be a strongly symmetrically normed space. If .A ⊆ E is bounded, then the following statements are equivalent: (i) The set .A is of right (respectively, left) uniformly absolutely continuous norm. (ii) The set .|A| (respectively, .|A∗ |) is of uniformly absolutely continuous norm. Proof (i).⇒(ii) Suppose that .A is of right uniformly absolutely continuous norm and let .{en }∞ n=1 be a sequence in .P (M) such that .en ↓n 0. Recalling (see Proposition 4.1.3 (iv)) that .‖|x| en ‖E = ‖xen ‖E for all .x ∈ E and .n ∈ N, it follows that .

sup ‖en |x| en ‖E ≤ sup ‖|x| en ‖E = sup ‖xen ‖E → 0,

x∈A

x∈A

n → ∞,

x∈A

which shows that .|A| is of uniformly absolutely continuous norm. If .A is of left uniformly absolutely continuous norm, then .A∗ is of right uniformly continuous norm and hence, .|A∗ | is of uniformly absolutely continuous norm. (ii).⇒(i) Suppose that the set .|A| is of uniformly absolutely continuous norm. By Proposition 5.7.15, this implies that .|A| is of right uniformly absolutely continuous norm. Consequently, if .{en }∞ n=1 is a sequence in .P (M) satisfying .en ↓n 0, then .

sup ‖xen ‖E = sup ‖|x| en ‖E → 0, n → ∞.

x∈A

x∈A

Hence, .A is of right uniformly continuous norm. It now follows that if .|A∗ | is of uniformly absolutely continuous norm, then .A∗ is of right uniformly absolutely continuous norm and so, .A is of left uniformly absolutely continuous norm. ⨆ ⨅ It is of some interest to mention the following immediate consequence. an and .x ∈ E is such that Corollary 5.7.17 If .{xn }∞ n=1 is a sequence in .E .‖xn − x‖E → 0 as .n → ∞, then also .‖|xn | − |x|‖E → 0.

Proof This follows from a combination of the results of Proposition 5.7.14 and Tm

Tm

Corollary 5.7.16 with the fact that .xn → x implies that .|xn | → |x| (see Corollary 2.8.8). ⨆ ⨅ A further situation where all three notions introduced in Definition 5.7.12 coincide occurs when the trace .τ is finite. As observed in Proposition 5.7.8, if an , then the set .yB is of left uniformly absolutely continuous norm. The next .y ∈ E M lemma shows that if .τ (1) < ∞, then this set is also of right uniformly absolutely continuous norm. Lemma 5.7.18 Suppose that .E ⊆ S (τ ) is a strongly symmetrically normed space and that .τ (1) < ∞. If .y ∈ E an , then the set .yBM is of right uniformly absolutely continuous norm.

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof Suppose that .{en }∞ n=1 is a sequence in .P (M) such that .en ↓n 0 but ‖yxe ‖ .supx∈B  0. This implies that there exist a sequence .{xn }∞ n E n=1 in .BM M and an .ε > 0 such that .

‖yxn en ‖E ≥ ε > 0,

n ∈ N.

(5.25)

Since .τ (1) < ∞, it follows that .

‖xn en ‖1 ≤ ‖en ‖1 = τ (en ) ↓n 0.

Since the embedding of .(L1 (τ ) , ‖·‖1 ) into .(S (τ ) , Tm ) is continuous, this implies that .xn en → 0 in measure and hence, .yxn en → 0 in measure as .n → ∞. Since .{yxn en }∞ n=1 ⊆ yBM and the set .yBM is of uniformly absolutely continuous norm (see Proposition 5.7.8), it follows that the set .{yxn en }∞ n=1 is of uniformly absolutely continuous norm as well. Therefore, Proposition 4.2.2 now implies that .‖yxn en ‖E → 0 as .n → ∞, which contradicts (5.25). This proves the lemma. ⨆ ⨅ Theorem 5.7.19 Suppose that .τ (1) < ∞ and that .E ⊆ S (τ ) is a strongly symmetrically normed space. If .A is a bounded subset of E, then the following statements are equivalent: (i) .A is of uniformly absolutely continuous norm. (ii) .A is of right uniformly absolutely continuous norm. (iii) .A is of left uniformly absolutely continuous norm. Proof (i).⇒(ii) If .A is of uniformly absolutely continuous norm, then it follows from Theorem 5.7.9 that for every .ε > 0 there exists .0 ≤ y ∈ E an such that A ⊆ yBM + BM y + εBE .

.

By Propositions 5.7.8 and Lemma 5.7.18, both sets .yBM and .BM y are of right uniformly absolutely continuous norm and hence, so is .yBM + BM y. Consequently, for every .ε > 0, there exists a set .Aε ⊆ E of right uniformly absolutely continuous norm satisfying .A ⊆ Aε + εBE . This easily implies that .A is of right uniformly absolutely continuous norm. The implications (ii).⇒(i) and (iii).⇒(i) being evident, it remains to be shown that (i) implies (iii). If the set .A is of uniformly absolutely continuous norm, then so is the set .A∗ and hence, by the already obtained implication (i).⇒(ii), the set .A∗ is of right uniformly absolutely continuous norm. Consequently, .A is of left uniformly absolutely continuous norm. This suffices to complete the proof of the theorem. ⨅ ⨆ Remark 5.7.20 That the condition that .τ (1) < ∞ cannot be omitted in the above theorem is illustrated by Example 5.7.13. The following consequence of Theorem 5.7.19 in combination with Corollary 5.7.16 is worth recording.

5.7 Sets of Uniformly Absolutely Continuous Norm

369

Corollary 5.7.21 Suppose that .τ (1) < ∞ and that .E ⊆ S (τ ) is a strongly symmetrically normed space. If .A ⊆ E is bounded, then the following statements are equivalent: (i) .A is of uniformly absolutely continuous norm. (ii) .|A| is of uniformly absolutely continuous norm. (iii) .|A∗ | is of uniformly absolutely continuous norm. Proof By Theorem 5.7.19, .A is of uniformly absolutely continuous norm if and only if .A is of right uniformly absolutely continuous norm. By Corollary 5.7.16, this is equivalent to the set .|A| being of uniformly absolutely continuous norm. The proof that (i).⇔(iii) is along the same lines. ⨆ ⨅ Remark 5.7.22 The condition that .τ (1) < ∞ cannot be omitted in the above is illustrated by Example 5.7.13. Indeed, in this example, the set corollary ∗ . vn : n ∈ N is of uniformly absolutely continuous norm (as it is of left uniformly    absolutely continuous norm), but the set . vn∗  : n ∈ N is equal to .{en : n ∈ N}, which is not of uniformly absolutely continuous norm. If .τ (1) < ∞ and .E ⊆ S (τ ) is a strongly symmetric space with order continuous norm, then some alternative characterizations of sets of uniformly absolutely continuous norm are available. It will be convenient to introduce the following terminology. Definition 5.7.23 Suppose that .τ (1) < ∞ and let .E ⊆ S (τ ) be a strongly symmetric space. A subset .A ⊆ E is called E-uniformly integrable if .





sup xe|x| (n, ∞) → 0,

x∈A

E

n → ∞.

If .E = L1 (τ ), then such a set is simply called uniformly integrable. Theorem 5.7.24 Suppose that .τ (1) < ∞ and let .E ⊆ S (τ ) be a strongly symmetric space. For a set .A ⊆ E, the following statements are equivalent: (i) .A is bounded and of uniformly absolutely continuous norm. (ii) .A is E-uniformly integrable. (iii) .A is bounded, and for every .ε > 0 there exists a .δ > 0 such that .e ∈ P (M) and .τ (e) ≤ δ imply that .‖xe‖E ≤ ε for all .x ∈ A. Proof It will be shown first that (i) and (iii) are equivalent. (i).⇒(iii) Since .τ (1) < ∞, it follows from Theorem 5.7.19 that .A is of right uniformly absolutely continuous norm. Assuming that (iii) does not hold, there exist −n and .x ∈ A such that .‖x e ‖ ≥ ε for all .ε > 0, .en ∈ P (M) with .τ (en ) ≤ 2 n n n E ∞ .n ∈ N. Defining .pn = e , it follows that . p ↓ and . τ (p ) ≤ 2−n+1 for all k n n n k=1 n and so, .pn ↓n 0. Since .A is of right uniformly absolutely continuous norm, this

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

implies that .‖xn pn ‖E → 0 as .n → ∞. On the other hand, using that .en ≤ pn , it follows that ε ≤ ‖xn en ‖E = ‖xn pn en ‖E ≤ ‖xn pn ‖E ,

.

n ∈ N,

which is a contradiction. This shows that (i) implies (iii). (iii).⇒(i) Suppose that .{en }∞ n=1 is a sequence in .P (M) satisfying .en ↓n 0. By hypothesis, given .ε > 0, there exists .δ > 0 such that if .e ∈ P (M) and .τ (e) ≤ δ, then .‖xe‖E ≤ ε for all .x ∈ A. Since .τ (1) < ∞, it follows that .τ (en ) ↓n 0. Hence, there exists .N ∈ N such that .τ (en ) ≤ δ whenever .n ≥ N. Consequently, .supx∈A ‖xen ‖E ≤ ε for all .n ≥ N . This shows that .A is of right uniformly absolutely continuous norm, which implies that (i) holds. It will be shown next that (ii) and (iii) are equivalent. (iii).⇒(ii) Set .M = supx∈A ‖x‖E . Since .ne|x| (n, ∞) ≤ |x| e|x| (n, ∞), it follows that





|x|



.n e (n, ∞) ≤ |x| e|x| (n, ∞) ≤ ‖x‖E ≤ M, E

E

and hence, .



|x|

e (n, ∞) ≤ M/n, E

x ∈ A,

n ∈ N.

(5.26)

Since .τ (1) < ∞, it follows that .E ⊆ L1 (τ ) with continuous embedding and so, there exists a constant .C > 0 such that .‖y‖1 ≤ C ‖y‖E for all .y ∈ E. Therefore, (5.26) implies that   τ e|x| (n, ∞) ≤ CM/n,

.

x ∈ A,

n ∈ N.

Given .ε > 0, it follows from (iii) that there exists .δ > 0 such that .‖xe‖E ≤ ε whenever .e ∈ P (M) with .τ (e) ≤ δ. Let .n0 ∈ N be such that .CM/n0 ≤ δ, which implies that .τ e|x| (n, ∞) ≤ δ for all .x ∈ A and .n ≥ n0 . By the choice of .δ, it now follows that



|x|

. sup xe (n, ∞) ≤ ε, n ≥ n0 . E

x∈A

This shows that (ii) holds. (ii).⇒(iii) It will be shown first that .A is bounded. Since .τ (1) < ∞, it follows that .L∞ (τ ) ⊆ E and there exists a constant .K > 0 such that .‖y‖E ≤ K ‖y‖∞ for all .y ∈ L∞ (τ ). Since .A is assumed to be E-uniformly integrable, it is possible to choose .n0 ∈ N such that



|x|

. xe (n0 , ∞) ≤ 1, x ∈ A, E

5.7 Sets of Uniformly Absolutely Continuous Norm

371

independently of .x ∈ A. Writing .

|x| = |x| e|x| (n0 , ∞) + |x| e|x| [0, n0 ]





and using that . |x| e|x| (n0 , ∞) E = xe|x| (n0 , ∞) E (see Proposition 4.1.3 (iv)), this implies that .











‖x‖E ≤ xe|x| (n0 , ∞) + |x| e|x| [0, n0 ]

E E





≤ 1 + K |x| e|x| [0, n0 ] ≤ 1 + Kn0 , ∞

for all .x ∈ A. Since .n0 does not depend on x, it is obtained that .A is bounded. For the proof of the second assertion of (iii), let .ε > 0 be given. It follows from (ii) and Theorem 5.7.19 that there exists .n1 ∈ N such that











|x| . |x| e (n1 , ∞) = xe|x| (n1 , ∞) ≤ ε/2, x ∈ A. E

E

By Lemma 5.6.18, there exists .δ > 0 such that .‖e‖E ≤ ε/ (2n1 ) whenever .e ∈ P (M) with .τ (e) ≤ δ. Consequently, if .e ∈ P (M) satisfies .τ (e) ≤ δ, then .











‖xe‖E = ‖|x| e‖E ≤ |x| e|x| (n1 , ∞) e + |x| e|x| [0, n1 ] e

E E





|x| ≤ |x| e (n1 , ∞) + n1 ‖e‖E ≤ ε E

for all .x ∈ A. This shows that the second assertion of (iii) holds as well. The proof of the theorem is complete. ⨆ ⨅ It is of some interest to observe that the “right/left versions” of Theorems 5.7.9 and 5.7.11 are valid as well. The proof of the following theorem follows, with the appropriate modifications, the same lines as the proofs of these two theorems. Note that the “left version” follows immediately from the “right version” applied to the set .A∗ . The details are left to the reader. Theorem 5.7.25 Let .E ⊆ S (τ ) be a strongly symmetric space and let .A be a bounded subset of E. The following statements are equivalent: (i) .A is of right (respectively, left) uniformly absolutely continuous norm. (ii) For each upward directed system .{pα } of finite trace projections in .P (E an ) satisfying .pα ↑α can and every .ε > 0, there exist a .pα and a constant .C > 0 such that A ⊆ CBM pα + εBE

.

(respectively, A ⊆ Cpα BM + εBE ).

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

(iii) For every .ε > 0, there exists .0 ≤ y ∈ E an such that A ⊆ BM y + εBE

.

(respectively, A ⊆ yBM + εBE ).

(iv) For all bounded nets .{xα } in .M+ satisfying .xα →α 0 with respect to .σ (M, L1 (τ )), it follows that .

sup ‖xxα ‖E →α 0 (respectively, sup ‖xα x‖E →α 0).

x∈A

x∈A

For sets of bi-uniformly absolutely continuous norm, there is a similar characterization, which is given in the next theorem. Theorem 5.7.26 Let .E ⊆ S (τ ) be a strongly symmetric space and let .A be a bounded subset of E. The following statements are equivalent: (i) .A is of bi-uniformly absolutely continuous norm. (ii) For each upward directed system .{pα } of finite trace projections in .P (E an ) satisfying .pα ↑α can and every .ε > 0, there exist a .pα and a constant .C > 0 such that A ⊆ Cpα BM pα + εBE .

.

(iii) For every .ε > 0, there exists .0 ≤ y ∈ E an ∩ (L1 ∩ L∞ ) (τ ) such that A ⊆ yBM y + εBE .

.

Proof Only implication (i).⇒(ii) needs a proof. If .A is of bi-uniformly absolutely continuous norm, then a moment’s reflection shows that Theorem 5.7.25 yields the existence of a projection .p = pα and a constant .C > 0 such that A ⊆ CBM p + εBE and A ⊆ CpBM + εBE .

.

(5.27)

The first inclusion of (5.27) implies that .A (1 − p) ⊆ εBE and so, .A ⊆ Ap + εBE . From the second inclusion of (5.27), it follows that .Ap ⊆ CpBE p + εBE . Consequently, A ⊆ Ap + A (1 − p) ⊆ CpBE p + 2εBE .

.

This suffices for the proof of the theorem. References: [41, 103].

⨆ ⨅

5.8 Weakly Compact Subsets

373

5.8 Weakly Compact Subsets In this section, the relation between relative weak compactness and uniform absolute continuity will be discussed in some detail. Some simple observations follow. Lemma 5.8.1 If .E ⊆ S (τ ) is a strongly symmetric space, then the map .x │→ x ∗ , ∗ .x ∈ E, is .σ (E, E )-continuous. Proof For .φ ∈ E ∗ , let .φ¯ ∈ E ∗ be the conjugate functional of .φ, that is, .φ¯ (x) = φ (x ∗ ) for all .x ∈ E (see Sect. 4.2). If .{xα } is a net in E and .x ∈ E is such that ∗ .xα → x with respect to .σ (E, E ), then   φ xα∗ = φ¯ (xα ) →α φ¯ (x) = φ x ∗ ,

.

and hence, .xα∗ → x ∗ with respect to .σ (E, E ∗ ).

φ ∈ E∗, ⨆ ⨅

Consequently, a subset .W ⊆ E is (relatively) weakly compact if and only if the set .W ∗ = {x ∗ : x ∈ W } is (relatively) weakly compact. The following lemma is a well-known result in the Banach space theory. For the reader’s convenience, a proof is presented. Lemma 5.8.2 Let E be a Banach space and .A ⊆ E. If for every .0 < ε ∈ R there exists a weakly compact subset .Aε ⊆ E such that .A ⊆ Aε + εBE , then .A is relatively weakly compact. Proof Let .{xα } be a net in .A. Since .A is norm bounded, it follows from the Banach– Alaoglu theorem that, by passing to a subnet, it may be assumed that .xα →α z with respect to .σ (E ∗∗ , E ∗ ) for some .z ∈ E ∗∗ . It suffices to show that .z ∈ E. Given ε ε ε ε .ε > 0, write .xα = xα + yα with .xα ∈ Aε and .yα ∈ εBE . Since .Aε is weakly   ε compact, there exists a subnet . xα(β) of .{xα } such that .xα(β) →β x ε weakly for ε ε ε some .x ∈ Aε . This implies that .yα(β) = xα(β) − xα(β) →β z − x ε with respect to ∗∗ ∗ ∗∗ ∗ ε .σ (E , E ). Since .εBE ∗∗ is .σ (E , E )-closed, it follows that .‖z − x ‖E ∗∗ ≤ ε. ∗∗ The space E is norm closed in .E and so it may be concluded that .z ∈ E, which ⨆ ⨅ completes the proof. Proposition 5.8.3 Let .E ⊆ S (τ ) be a strongly symmetric space. If .y ∈ E an , then each of the sets .BM y and .yBM is weakly compact. Proof Since .E an is an .M-bimodule, it follows that both .BM y and .yBM are contained in .E an . Therefore, replacing E by .can Ecan , without loss of generality, it may be assumed that .can = 1. Let .{xα } be a net in .BM y and write .xα = zα y, where .{zα } is a net in .BM . The set .BM is .σ (M, L1 (τ ))-compact and so, by passing to a subnet, it may be assumed that .zα →α z with respect to .σ (M, L1 (τ )) for some .z ∈ BM . It will be shown next that .xα →α zy with respect to .σ (E, E ∗ ). Indeed, let ∗ be given. Since .E an = E oc (see Proposition 5.5.11), it follows from .φ ∈ E Proposition 5.4.6 (in combination with Theorem 5.2.9) that there exists a (unique)

374

5 Strongly Symmetric Spaces of .τ -Measurable Operators

w ∈ E × such that .φ (x) = τ (xw), .x ∈ E an . Since .yw ∈ L1 (τ ), this implies that

.

φ (xα ) = τ (zα yw) →α τ (zyw) = φ (zy) .

.

This shows that .xα →α zy with respect to .σ (E, E ∗ ). It may be concluded that the set .BM y is weakly compact in E. Using Lemma 5.8.1, it is now clear that the set .yBM is also weakly compact. ⨆ ⨅ The next theorem is one of the main results in this section. Theorem 5.8.4 If .E ⊆ S (τ ) is a strongly symmetric space and if the bounded set A ⊆ E is of uniformly absolutely continuous norm, then .A is relatively weakly compact.

.

Proof By Theorem 5.7.9, for each .ε > 0 there exists .0 ≤ yε ∈ E an such that A ⊆ yε BM + BM yε + εBE .

.

It follows from Proposition 5.8.3 that the set .yε BM + BM yε is weakly compact and hence, by Lemma 5.8.2, the set .A is relatively weakly compact. The proof is complete. ⨆ ⨅ Remark 5.8.5 The converse of the preceding theorem fails in general, even in the case that .M is commutative. Indeed, if .E = Lp [0, 1], with .1 < p < ∞, then the unit ball .BE is weakly compact, since E is reflexive. However, it is easily verified that .BE is not of uniformly absolutely continuous norm. It is worth pointing out the following consequence of Theorem 5.8.4, which strengthens in some sense the equivalence (i).⇔(iii) of Theorem 5.6.29. Corollary 5.8.6 Let .E ⊆ S (τ ) be a strongly symmetric space. If .0 ≤ x ∈ E, then the following statements are equivalent: (i) The element x has order continuous norm. (ii) The order interval .[0, x] is weakly compact. Proof (i).⇒(ii) If .0 ≤ x ∈ E has order continuous norm, then .0 ≤ x ∈ E an (see Proposition 5.5.11) and so, the set .[0, x] is of uniformly absolutely continuous norm. Theorem 5.8.4 now implies that .[0, x] is relatively weakly compact. Since .[0, x] is weakly closed, it follows that .[0, x] is weakly compact. (ii).⇒(i) The proof of this implication is exactly the same as that of implication (iii).⇒(i) in Theorem 5.6.29. ⨆ ⨅  In a strongly symmetric space .E ⊆ S (τ ), the .σ E, E × -topology may also be considered. Note that .σ E, E × is a locally convex topology in .E which is Hausdorff (as .E × separates the point of E; see Corollary 4.3.9). Since .σ E, E × is weaker than .σ (E, E ∗ ), it follows that any (relatively) .σ (E, E ∗ )-compact subset

5.8 Weakly Compact Subsets

375

 of E is also (relatively) .σ E, E × -compact. In relation to Corollary 5.8.6, the following result is of interest. Proposition 5.8.7 If .E ⊆ S (τ ) is a strongly symmetric space and .0 ≤ x ∈ E, then the order interval .[0, x] is .σ E, E × -compact. Proof As usual, the space .E ××  will ∗ be identified, via trace duality, with a closed subspace of the Banach dual . E × . Given .0 ≤ x ∈ E, it will be shown that the  ∗ order interval .[0, x] (in E) actually is an order interval in . E × . In fact, suppose  × ∗ is such that .0 ≤ φ ≤ x. If .{yα } is a net in .E × satisfying .y ↓α 0, that .φ ∈ E then 0 ≤ φ (yα ) ≤ 〈yα , x〉 = τ (xyα ) ↓α 0,

.

 ∗ which shows that .φ ∈ E × n . Therefore, by Theorem 5.2.9, there exists a unique ×× such that .φ (y) = τ (zy) for all .y ∈ E × (that is, .φ = z ∈ E ×× ). .z ∈ E Since .0 ≤ τ (zy) ≤ τ (xy) for all .0 ≤ y ∈ E × , it follows that .0 ≤ z ≤ x (see Lemma 4.3.7, applied in .E × ), and hence .z ∈ [0, x] (as E is an absolutely solid subspace of .E ×× ).  ∗  ∗ Since order intervals in . E × are norm bounded and .σ E× , E × -closed, it now follows from the Banach–Alaoglu theorem that .[0, x] is .σ E, E × -compact. ⨆ ⨅  Another important class of .σ E, E × -compact sets will now be considered. It will be convenient to set G (τ ) = (L1 + L∞ ) (τ ) ,

.

H (τ ) = (L1 ∩ L∞ ) (τ ) .

Recall that (see Sect. 3.10), for each .x ∈ G (τ ), the set .Ω (x) = ΩM (x) ⊆ G (τ ) is defined by setting Ω (x) = {y ∈ S (τ ) : y ≺≺ x} .

.

By Corollary 3.10.14, the set .Ω (x) is .σ (G (τ ) , H (τ ))-compact and hence, σ (G (τ ) , H (τ ))-closed. This implies, in particular, that if .{0}  /= E ⊆ S (τ ) is a fully symmetric space and .x ∈ E, then the set .Ω (x) is .σ E ×× , E × -closed in ×× (indeed, .E ×× ⊆ G (τ ) and .H (τ ) ⊆ E × ). It should be recalled also that it .E follows from Proposition 5.1.4 that .



∞

.

0

μ (t; x) μ (t; y) dt ≤ ‖x‖E ‖y‖E × ,

x ∈ E, y ∈ E × ,

so, in particular, .μ (x) μ (y) ∈ L1 (m) for all .x ∈ E and .y ∈ E × . Proposition 5.8.8 Let .E ⊆ S (τ ) be a fully symmetric space such that .E × ⊆ × S0 (τ ). For any .x ∈ E, the set .Ω (x) is .σ E, E -compact.

376

5 Strongly Symmetric Spaces of .τ -Measurable Operators

 ∗ Proof Observing that .E ⊆ E ×× and .E ×× ⊆ E × (via trace duality), it  × ∗  ∗ is sufficient to show that .Ω (x) is .σ E × , E × -closed in . E . Indeed, the  Banach–Alaoglu theorem then yields that .Ω (x) is .σ E, E × -compact. To this end, it will be shown first that if .{yα } is a net in .E × satisfying .yα ↓α 0, then .

sup |τ (zyα )| →α 0.

(5.28)

z∈Ω(x)

Indeed, it may be assumed that .0 ≤ yα ≤ y for some .y ∈ E × . Since .y ∈ S0 (τ ), it follows from Proposition 3.2.14 (ii) that .μ (yα ) ↓α 0 in .S (m). Furthermore, Corollary 3.4.6 and Theorem 3.4.29 imply that  .

|τ (zyα )| ≤ 0

∞

 μ (z) μ (yα ) dm ≤

∞

μ (x) μ (yα ) dm, 0

where the second inequality follows from the fact that .z ≺≺ x in combination with Hardy’s lemma (see Lemma 3.10.10). Since .μ (x) μ (y) ≥ μ (x) μ (yα ) ↓α 0 in .L1 (m) and the functional given by the Lebesgue integral is normal, it follows that ∞ . which now implies that (5.28) holds. 0 μ (x) μ (yα ) dm ↓α 0,  ∗   To show that .Ω (x) is .σ E × , E × -closed, suppose that . zβ is a net in .Ω (x)  × ∗  × ∗ × and .φ ∈ E is such that .zβ →β φ with respect to .σ E , E . It will  × ∗ × such that {y } be shown that .φ ∈ E n . Let . α be a downward directed net .E  in    .yα ↓α 0. It follows from (5.28) that there exists an .α0 such that . τ zβ yα  ≤ ε for all .β whenever .α ≥ α0 and so, .

   |φ (yα )| ≤ φ (yα ) − τ zβ yα  + ε,

α ≥ α0 ,

 for all .β. Since .τ zβ yα →β φ (yα ) for each .α, this implies that .|φ (yα )| ≤ ε for  ∗ all .α ≥ α0 . Therefore, it may be concluded that .φ ∈ E × n . By Theorem 5.2.9, there exists .z ∈ E ×× such that .φ = z (via trace duality). Hence, .zβ → z with  respect to .σ E ×× , E × . As observed above, .Ω (x) is .σ E ×× , E × -closed and so,  × ∗ × .z ∈ Ω (x). Consequently, .Ω (x) is .σ E , E . This suffices to finish the proof of the proposition. ⨆ ⨅ Remark 5.8.9 (a) If .E ⊆ S (τ ) is a fully symmetric space with .E × ⊆ S0 (τ ), then, for .x ∈ E, the set .Ω (x) is, in general, not .σ (E, E ∗ )-compact. Indeed, if .E = L∞ (m) on .[0, 1] and .x = 1, then .Ω (x) = BL∞ (m) , which is not weakly compact (but, of course, it is .σ (L∞ , L1 )-compact). (b) The condition that .E × ⊆ S0 (τ ) cannot be omitted 5.8.8. Indeed,  in Proposition ∞ if .E = L1 (m) on .[0, ∞) and .x = χ[0,1) , then . χ[n,n+1) n=1 ⊆ Ω (x), but the  ∞ set . χ[n,n+1) n=1 is not relatively weakly compact.

5.8 Weakly Compact Subsets

377

In relation to Proposition 5.8.8, the following observation is also of interest. Proposition 5.8.10 Suppose that .τ (1) < ∞ and let .E ⊆ S (τ ) be a strongly symmetric space with order continuous norm. If .x ∈ E, the set .Ω (x) is of uniformly absolutely continuous norm. Proof It should be observed first that E is fully symmetric (see Corollary 5.3.6) and so, .Ω (x) ⊆ E for every .x ∈ E. Furthermore, it follows from Lemma 5.3.5 that .L∞ (τ ) is dense in E. Given .x ∈ E and .ε > 0, it will be shown that there exists .0 ≤ Cε ∈ R such that Ω (x) ⊆ Cε BL∞ (τ ) + εBE .

.

(5.29)

Indeed, let .x1 ∈ L∞ (τ ) be such that .‖x − x1 ‖E ≤ ε. If .y ∈ Ω (x), then y ≺≺ x = x1 + (x − x1 )

.

and hence, by Proposition 3.10.17, there exist .y1 , y2 ∈ S (τ ) such that .y = y1 + y2 and .y1 ≺≺ x1 , .y2 ≺≺ x − x1 . This implies that .y1 ∈ L∞ (τ ) with .‖y1 ‖∞ ≤ ‖x1 ‖∞ and .y2 ∈ E with .‖y2 ‖E ≤ ‖x − x1 ‖E ≤ ε. This shows that (5.29) holds with .Cε = ‖x1 ‖∞ . Since E has order continuous norm and .1 ∈ E, it follows from Proposition 5.7.8 that the set .BL∞ (τ ) = BM is of uniformly absolutely continuous norm. Since (5.29) holds for any .ε > 0, it may be concluded that the set .Ω (x) is of uniformly absolutely ⨆ ⨅ continuous norm (see Lemma 5.7.1). The proof is complete. Remark 5.8.11 The condition that .τ (1) < ∞ cannot be omitted in the above proposition. Indeed, if .E = L1 (m) on .[0, ∞) and .x = χ[0,1) , the set .Ω (x) is not of uniformly absolutely continuous norm (cf. Remark 5.8.9 (b)). As observed in Remark 5.8.5, a weakly compact subset of a strongly symmetric space is, in general, not of uniformly absolutely continuous norm. However, for the space .L1 (τ ), the situation is different, as will be shown in the next theorem, which is a special case of a result due to C.A. Akemann (1967) [1], dealing with pre-duals of arbitrary (not necessarily semi-finite) von Neumann algebras. Theorem 5.8.12 If .A ⊆ L1 (τ ) is bounded, then the following statements are equivalent: (i) .A is relatively weakly compact. (ii) For every sequence .{en }∞ n=1 of mutually orthogonal projections in .P (M) it follows that .

sup |τ (xen )| → 0,

n → ∞.

x∈A

(iii) .A is of uniformly absolutely continuous norm.

378

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Proof (i).⇒(ii) Suppose that .A ⊆ L1 (τ ) is relatively weakly compact, but condition (ii) is not satisfied. Then there exists a sequence .{en }∞ n=1 of mutually orthogonal projections in .P (M) such that .supx∈A |τ (xen )| n 0. By passing to a subsequence, if necessary, it may be assumed that there exists a sequence .{xn }∞ n=1 in .A such that .|τ (xn en )| ≥ ε for all .n ∈ N and some .0 < ε ∈ R. Since .A is relatively weakly compact, it follows from the Eberlein–Šmulian theorem that, by passing to a further subsequence, it may be also assumed that there exists .x ∈ L1 (τ ) such that .xn →n x with respect to the weak topology .σ (L1 (τ ) , L∞ (τ )). Since the norm in .L1 (τ ) is order continuous, it follows from Theorem 5.5.13 that .

|τ (xen )| ≤ ‖xen ‖1 → 0,

n → ∞.

Hence, there exists .N ∈ N such that .|τ ((xn − x) en )| ≥ ε/2 for all .n ≥ N . Consequently, replacing .xn by .xn − x, it follows that xn

.

σ (L1 (τ ),L∞ (τ )) →n

0 and

|τ (xn en )| ≥ ε/2,

n ≥ N,

(5.30)

where .{en }∞ n=1 is a sequence of mutually orthogonal projections in .P (M). For each .n ∈ N and .A ⊆ N, set    .νn (A) = τ xn ek . k∈A

Observe that for every .n ∈ N, .νn is a finitely additive measure on the Boolean algebra .2N . Since .



|νn (A)| ≤ ‖xn ‖1

k∈A



ek

∞

≤ ‖xn ‖1 ,

A ∈ 2N ,

it is clear that .νn is bounded. Furthermore, since .xn → 0 with respect to σ (L1 (τ ) , L∞ (τ )), it follows immediately that .νn (A) → 0 as .n → ∞ for each .A ∈ 2N . Therefore, Phillips’ lemma (see Lemma 5.2.12) now implies that ∞ . k=1 |νn ({k})| → 0 as .n → ∞ and so, in particular, .|τ (xn en )| = |νn ({n})| → 0 as .n → ∞. This contradicts (5.30) and establishes that (i) implies (ii). (ii).⇒(iii) Observe first that a set .A ⊆ L1 (τ ) satisfies condition (ii) (respectively, (iii)) if and only if both .ReA and .ImA satisfy (ii) (respectively, (iii)). Therefore, for the proof that (ii) implies (iii), it may be assumed, in addition, that .x ∗ = x for all .x ∈ A. Suppose that the set .A is not of uniformly absolutely continuous norm. It follows from Proposition 5.7.2 that there exists a sequence .{en }∞ n=1 of mutually orthogonal projections in .P (M) such that .supx∈A ‖en xen ‖1  0 as .n → ∞. This implies that there exist an .ε > 0 and a sequence .{xn }∞ n=1 in .A such that .

.

‖en xn en ‖1 ≥ ε,

n ∈ N.

5.8 Weakly Compact Subsets

379

Since the elements .en xn en are self-adjoint, it follows that .

  ‖en xn en ‖1 = τ (|en xn en |) = τ (en xn en )+ + τ (en xn en )−

and so, it may be assumed, without loss of generality, that  τ (en xn en )+ ≥ ε/2,

.

n ∈ N.

Setting .pn = een xn en (0, ∞), it follows that .pn ≤ en (in fact, .pn ≤ s (en xn en ) ≤ en , as .en⊥ ≤ n (en xn en )) and .

(en xn en )+ = pn en xn en pn = pn xn pn ,

n ∈ N.

Therefore,  τ (xn pn ) = τ (pn xn pn ) = τ (en xn en )+ ≥ ε/2,

.

n ∈ N.

Since .pn ≤ en for all n, it is clear that .{pn }∞ n=1 is a sequence of mutual orthogonal projections in .P (M) and so, by hypothesis (ii), .

lim sup |τ (xpn )| = 0,

n→∞ x∈A

in particular, .τ (xn pn ) →n 0. This yields a contradiction and so, it may be concluded that (ii) implies (iii). That (iii) implies (i) is a consequence of Theorem 5.8.4 and hence, the proof is complete. ⨆ ⨅ Some consequences of the above theorem will now be discussed. For the definition of the quantity .ω (x; δ), see Definition 5.7.3. Corollary 5.8.13 If .τ (1) < ∞ and if .A ⊆ L1 (τ ) is bounded, then the following two statements are equivalent: (i) .A is relatively weakly compact. (ii) .limδ↓0 supx∈A ω (x; δ) = 0. Proof This follows immediately from the equivalence of (i) and (iii) in Theorem 5.8.12 in combination with Corollary 5.7.5. ⨆ ⨅ A combination of Theorem 5.8.12 and Theorem 5.7.24 yields the following result. Corollary 5.8.14 If .τ (1) < ∞ and .A ⊆ L1 (τ ), then the following two statements are equivalent: (i) .A is relatively weakly compact. (ii) .A is bounded and is of uniformly absolutely continuous norm.

380

5 Strongly Symmetric Spaces of .τ -Measurable Operators

(iii) .A is uniformly integrable. (iv) .A is bounded, and for every .ε > 0 there exists .δ > 0 such that .e ∈ P (M) and .τ (e) ≤ δ imply that .‖xe‖E ≤ ε for all .x ∈ A. The following observation is a direct consequence of Theorem 5.8.12 in combination with Corollary 5.7.21. Corollary 5.8.15 If .τ (1) < ∞ and .A ⊆ L1 (τ ), then the following statements are equivalent: (i) .A is relatively weakly compact. (ii) .|A| is relatively weakly compact. (iii) .|A∗ | is relatively weakly compact. Remark 5.8.16 If .τ (1) = ∞, then neither of the implications (i).⇒(ii), (iii) of the preceding corollary is valid, in general, as can be seen by an inspection of Example 5.7.13. Corollary 5.8.17 If .A ⊆ L1 (τ ) and if either .|A| or .|A∗ | is relatively weakly compact, then .A is relatively weakly compact. Proof Note that each of the three conditions on .A implies that .A is bounded. If |A| is relatively weakly compact, then it follows from Theorem 5.8.12 that .|A| is of uniformly absolutely continuous norm. Therefore, it follows from Corollary 5.7.16 that .A is of uniformly absolutely continuous norm and hence, .A is relatively weakly compact (once again by Theorem 5.8.12). Similarly, if .|A∗ | is relatively weakly compact, then .A∗ is relatively weakly compact and so, .A is relatively weakly compact (as involution is weakly continuous; ⨆ ⨅ see Lemma 5.8.1).

.

References: [41, 43, 50].

5.9 Embedding Copies of c0 and 𝓁1 Let .M be a semi-finite von Neumann algebra equipped with a fixed semi-finite normal faithful trace .τ . Lemma 5.9.1 Suppose that .{ei }ni=1 are mutually orthogonal projections in .P (M). If .x ∈ S (τ ), then n .

i=1

ei xei ≺≺ x.

Proof For each .ε = (ε1 , . . . , εn ) ∈ {−1, 1}n , define .vε ∈ M by setting vε =

.

n i=1

εi ei .

(5.31)

5.9 Embedding Copies of c0 and 𝓁1

381

Observe that .‖vε ‖B(H ) ≤ 1 for all .ε ∈ {−1, 1}n . A simple computation shows that n .

i=1

ei xei =

1  vε xvε . ε∈{−1,1}n 2n

Using the fact that .μ (y + z) ≺≺ μ (y) + μ (z) for all .y, z ∈ S (τ ) (see Theorem 3.9.9), it follows that μ

 1  μ (vε xvε ) . ei xei ≺≺ n i=1 ε∈{−1,1}n 2

n

.

Since .μ (vε xvε ) ≤ ‖vε ‖2 μ (x) ≤ μ (x) for all .ε ∈ {−1, 1}n (see Proposition 3.2.7 (vi)), it is now clear that (5.31) holds. ⨆ ⨅ Let .E ⊆ S (τ ) be a strongly symmetric space (with .cE = 1, as always). A sequence .{xn }∞ n=1 in E is said to be disjointly supported if there exists a sequence ∞ .{en } of mutually orthogonal projections in .P (M) such that .xn = en xn en for n=1 all n. Let L be any of the sequence spaces .𝓁1 , .c0 , or .𝓁∞ . The space E is said to contain a copy of L if there exist a linear subspace X of E and a surjective linear isomorphism −1 are both positivity preserving, then .T : L → X. If the map T and its inverse .T X is called a positive copy of L. If T maps the unit vector basis of L to a disjointly supported sequence in E, then X is called a disjointly supported copy of L. The following observation will be needed. + Lemma 5.9.2 If .{yn }∞ n=1 is a disjointly supported sequence in .E satisfying:

(i) There exists a .δ > 0 such that .‖yn ‖E ≥ δ for all n.

(ii) There exists a constant .C > 0 such that . nk=1 yk E ≤ C for all n. then the following statements hold: (a) The norm closed linear span X of .{yn : n ∈ N} is a disjointly supported positive copy of .c0 . (b) If, in addition, E has the Fatou property, then E contains a disjointly supported positive copy of .𝓁∞ . Proof Let .{en }∞ n=1 be a sequence of mutually orthogonal projections in .P (M) satisfying .yn = en yn en for all n. (a) Denoting by .c00 the space of all complex sequences .λ = (λn )∞ n=1 which are eventually zero, define the linear map .T0 : c00 → E by setting T0 ξ =

.

∞ n=1

ξn yn ,

ξ = (ξn )∞ n=1 ∈ c00 .

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5 Strongly Symmetric Spaces of .τ -Measurable Operators

Using the fact that the sequence .{yn }∞ n=1 is disjointly supported, a simple calculation shows that .

|T0 ξ | = T0 |ξ | ,

ξ ∈ c00 .

It will be shown first that δ ‖ξ ‖∞ ≤ ‖T0 ξ ‖E ≤ C ‖ξ ‖∞ ,

.

ξ ∈ c00 .

(5.32)

Indeed, suppose that .ξ ∈ c00 and let .k ∈ N be such that .ξn = 0 whenever .n > k. It follows that .

|T0 ξ | = T0 |ξ | =

k n=1

|ξn | yn ≤ ‖ξ ‖∞

k n=1

yn

and hence,



k



. ‖T0 ξ ‖E ≤ ‖ξ ‖∞

n=1 yn ≤ C ‖ξ ‖∞ . E

On the other hand, .

|T0 ξ | = T0 |ξ | ≥ |ξn | yn ,

1 ≤ n ≤ k,

and so, .

‖T0 ξ ‖E ≥ |ξn | ‖yn ‖E ≥ δ |ξn | ,

which implies that .‖T0 ξ ‖E ≥ δ ‖ξ ‖∞ . This proves (5.32). From the right-hand side inequality in (5.32), it follows that for each .ξ = (ξn ) in .c0 , the series . ∞ ξ n=1 n yn is norm convergent in E and hence, the linear map .T : c0 → E may be defined by setting Tξ =

.

∞ n=1

ξn yn ,

ξ ∈ c0 .

Evidently, estimate (5.32) remains valid with .T0 replaced by T and hence, T is an isomorphism from .c0 onto .T (c0 ). This implies, in particular, that .T (c0 ) is equal to the norm closed linear span X of .{yn : n ∈ N}. It is also clear that T is positivity preserving. Furthermore, if .ξ ∈ c0 is such that .T ξ ≥ 0, then −1 .ξn yn = en (T ξ ) en ≥ 0, which implies that .ξn ≥ 0 for all n. Therefore, .T is also positivity preserving. Hence, X is a positive copy of .c0 . Finally, since T maps the unit vectors of .c0 onto the sequence .{yn }∞ n=1 , it follows that X is a disjointly supported copy of .c0 .

5.9 Embedding Copies of c0 and 𝓁1

383

(b) If .0 ≤ ξ = (ξn ) ∈ 𝓁∞ , then it follows from (5.32) that



k



.

ξ y n n ≤ C ‖ξ ‖∞

n=1

for all k. Since .0 ≤

k

n=1 ξn yn

∞ .

n=1

E

↑k and E has the Fatou property, it follows that

ξn yn = supk

k n=1

ξn yn



exists in E and satisfies . ∞ ξn yn E ≤ C ‖ξ ‖∞ . Note that the series n=1  ∞ × . combination n=1 ξn yn is .σ E, E -convergent. Since any .ξ ∈ 𝓁∞ is a linear ofat most four positive elements of .𝓁∞ , it follows that the series . ∞ n=1 ξn yn is × -convergent for each .ξ ∈ 𝓁 . Define the linear map .S : 𝓁 → E by .σ E, E ∞ ∞ setting Sξ =

.

∞ n=1

ξn yn ,

ξ ∈ 𝓁∞ .

The  Fatou property of E implies that the norm closed unit ball of E is σ E, E × -closed. Therefore, it follows from (5.32) that .‖Sξ ‖E ≤ C ‖ξ ‖∞ for all .ξ ∈ 𝓁∞ .  Noting that the right multiplication by elements of .M is .σ E, E × continuous in E, it follows that    k k . (Sξ ) en = ξn yn

.

n=1

n=1

and so, estimate (5.32) implies that



k



δ sup |ξn | ≤

ξ y

n=1 n n

1≤n≤k E

 



k

= (Sξ ) en

≤ ‖Sξ ‖E

.

n=1

E

for all .k ∈ N and .ξ ∈ 𝓁∞ . Hence, .δ ‖ξ ‖∞ ≤ ‖Sξ ‖E . It has thus been shown that δ ‖ξ ‖∞ ≤ ‖Sξ ‖E ≤ C ‖ξ ‖∞ ,

.

ξ ∈ 𝓁∞ ,

and therefore, S is an isomorphism from .𝓁∞ onto its range .Y = S (𝓁∞ ). Therefore, Y is a copy of .𝓁∞ . From the definition of the map S, it is clear that S is positivity preserving. If .ξ ∈ 𝓁∞ is such that .Sξ ≥ 0, then .ξn yn = en (Sξ ) en ≥ 0 and so, .ξn ≥ 0 for all n. This shows that .S −1 is also positivity preserving and so, .Y is a positive

384

5 Strongly Symmetric Spaces of .τ -Measurable Operators

copy of .𝓁∞ . Since S maps the unit vectors in .𝓁∞ onto the sequence .{yn }∞ n=1 , it is evident that Y is a disjointly supported copy of .𝓁∞ . The proof is complete. ⨆ ⨅ It has been shown in Proposition 5.6.1 that if E does not have order continuous norm, then E contains a copy of .𝓁∞ . However, the copy of .𝓁∞ constructed in this proposition is, in general, neither positive nor disjointly supported. Therefore, the result of the following proposition is of some interest. Proposition 5.9.3 If .E ⊆ S (τ ) is a strongly symmetric space which does not have order continuous norm, then the following statements hold: (i) E contains a disjointly supported positive copy of .c0 . (ii) If, in addition, E has the Fatou property, then E contains a disjointly supported positive copy of .𝓁∞ . Proof By hypothesis, there exists an element .0 < x ∈ E such that .x ∈ / E oc = E an . By Theorem 5.5.13, this implies that there exist a .δ > 0 and a sequence .{en }∞ n=1 of mutually orthogonal projections in .P (M) such that .‖en xen ‖E ≥ δ for all n. It follows from Lemma 5.9.1 and from the assumption that E is strongly symmetric that



n

.

ek xek ≤ ‖x‖E , n ∈ N. k=1

E

The result of the proposition now follows by applying Lemma 5.9.2 to the elements yn = en xen . ⨆ ⨅

.

Corollary 5.9.4 If a strongly symmetric space .E ⊆ S (τ ) does not contain a disjointly supported positive copy of .c0 , then E has order continuous norm. The next objective is to characterize those strongly symmetric spaces which do not contain a copy of .c0 . For this purpose, the following lemma is needed. Lemma 5.9.5 Let .E ⊆ S (τ ) be a strongly symmetric space with a Fatou norm. Suppose that .0 ≤ z ∈ E ×× and that .A ⊆ E + satisfies .0 ≤ x ≤ z for all .x ∈ A. If .A is not of uniformly absolutely continuous norm, then E contains a disjointly supported positive copy of .c0 . Proof Since .A is not of uniformly absolutely continuous norm, it follows from Proposition 5.7.2 that there exist a .δ > 0, a sequence .{xn }∞ n=1 in .A, and a sequence ∞ .{en } of mutually orthogonal projections in .P (M) such that .‖en xn en ‖E ≥ δ for n=1 all n. Lemma 5.9.1 implies that 0≤

.

n k=1

en xn en ≤

n k=1

en zen ≺≺ z,

n ∈ N.

(5.33)

5.9 Embedding Copies of c0 and 𝓁1

385

Since E has a Fatou norm, it follows from Theorem 5.1.9 that .‖x‖E = ‖x‖E ×× for all .x ∈ E and so, (5.33) implies that .



n

k=1



n



en xn en =

E

k=1



en xn en

E ××

≤ ‖z‖E ×× ,

n ∈ N.

The result of the lemma now follows from an application of Lemma 5.9.2 (a) to the ⨆ ⨅ elements .yn = en xn en . The following theorem is one of the principal results in the present section. Theorem 5.9.6 If .E ⊆ S (τ ) is a strongly symmetric space, then the following statements are equivalent: (i) (ii) (iii) (iv)

E is a KB-space. E is weakly sequentially complete. E does not contain a copy of .c0 . E does not contain a disjointly supported positive copy of .c0 .

Proof The equivalence of (i) and (ii) has been established in Theorem 5.3.9. That (ii) implies (iii) is clear, as .c0 is not weakly sequentially complete. Implication (iii).⇒(iv) is evident and so, it remains to be shown that (iv) implies (i). + (iv).⇒(i) Let .{xn }∞ n=1 be an increasing norm bounded sequence in .E . It has to ∞ be shown that .{xn }n=1 is norm convergent in E (see Definition 5.3.7). Note that it follows from Corollary 5.9.4 that E has order continuous norm. Therefore it suffices to show that there exists .x ∈ E + such that .xn ↑n x. Since the sequence .{xn }∞ n=1 is norm bounded in .E ×× and .E ×× has the Fatou property, it follows that there exists ×× such that .x ↑ z. Hypothesis (iv), in combination with Lemma 5.9.5, .z ∈ E n n implies that the set .{xn }∞ n=1 is of uniformly absolutely continuous norm and hence, by Theorem 5.8.4, .{xn }∞ compact. The Eberlein–Šmulian n=1 is relatively weakly  ∞ theorem implies that there exist a subsequence . xnk k=1 and .x ∈ E such that     × .xnk →k x. If .0 ≤ y ∈ E , then . xnk , y →k 〈x, y〉 and also . xnk , y ↑k 〈z, y〉. Consequently, .〈x, y〉 = 〈z, y〉 for all .y ∈ E × , which implies that .z = x ∈ E. This suffices to complete the proof. ⨆ ⨅ Copies of .𝓁1 will be considered next. Lemma 5.9.7 Let .E ⊆ S (τ ) be a strongly symmetric space. If .E × does not have order continuous norm, then .E contains a disjointly supported positive copy of .𝓁1 .  oc Proof By hypothesis, there exists an element .0 < y ∈ E × such that .y ∈ / E× =  × an . Without loss of generality, it may be assumed that .‖y‖E × ≤ 1. By E Theorem 5.5.13, there exist .ε > 0 and a sequence .{en }∞ n=1 of mutually orthogonal projections in .P (M) such that .

‖en yen ‖E × ≥ 2ε,

n ∈ N.

386

5 Strongly Symmetric Spaces of .τ -Measurable Operators

Since .en yen ≥ 0, it follows from Corollary 4.2.8 that for each n there exists .0 ≤ xn ∈ E with .‖xn ‖E ≤ 1 such that τ (xn en yen ) ≥ ε.

.

Since .τ (xn en yen ) = τ (en xn en yen ), replacing .xn by .en xn en , it may be assumed, without loss of generality, that .xn = en xn e n for all n. ∞ For each .ξ = (ξn ) ∈ 𝓁1 , it is clear that . n=1 ‖ξn xn ‖E ≤ ‖ξ ‖1 and so, the series ∞ . ξ x is norm convergent in E. Therefore, the bounded linear map .T : 𝓁1 → E n n n=1 may be defined by setting Tξ =

.

∞ n=1

ξn xn ,

ξ = (ξn ) ∈ 𝓁1 .

It will be shown next that .

‖T ξ ‖E ≥ ε ‖ξ ‖1 ,

ξ ∈ 𝓁1 .

(5.34)

For this purpose, let .ξ = (ξn )∞ n=1 ∈ c00 be given and let .k ∈ N be such that .ξn = 0 for .n > k. For each .1 ≤ n ≤ k, let .σn ∈ C be such that .ξn = σn |ξn | and .|σn | = 1. A simple calculation shows that   k  k  k   |σn | en yen = . en yen  n=1 σn en yen  = n=1 n=1 and so, it follows from Lemma 5.9.1 that    k   . σn en yen  ≺≺ y,  n=1

which implies that .



k



σ e ye

n=1 n n n

E×

≤ ‖y‖E × ≤ 1.

It now follows that



k



. ξn xn



≥ n=1

E

       k k τ ξn xn σm em yem   n=1 m=1    k σn ξn τ (xn en yen ) =  n=1

=

k n=1

|ξn | τ (xn en yen ) ≥ ε ‖ξ ‖1 .

5.9 Embedding Copies of c0 and 𝓁1

387

This shows that inequality (5.34) holds for all .ξ ∈ c00 , from which it follows easily that (5.34) holds for all .ξ ∈ 𝓁1 . Consequently, .T : 𝓁1 → X = T (𝓁1 ) is a surjective isomorphism and so, X is a copy of .𝓁1 . It is also clear that T is positivity preserving. Furthermore, if .ξ ∈ 𝓁1 is such that .T ξ ≥ 0, then ξn xn = en (T ξ ) en ≥ 0,

.

n∈N

and so, .ξ ≥ 0. Therefore, .T −1 is also positivity preserving, which shows that X is a positive copy of .𝓁1 . Finally, it is clear that X is disjointly supported. The proof of the lemma is complete. ⨆ ⨅ In the proof of the next theorem, the following consequence of a theorem of Šmulian [62, p. 206, Corollary 2] will be used. Proposition 5.9.8 Let X be a Hausdorff locally convex topological vector space  ' ∞ ' .X . If there exists a sequence .{Kn } of .σ X , X -compact subsets with dual space n=1  of .X' such that . ∞ every relatively .σ X, X' n=1 Kn separates the points of X, then compact subset of X is sequentially relatively .σ X, X' -compact. If E is a Banach space, then a subset .A ⊆ E is said to be weakly sequentially precompact if every sequence in A has a weak Cauchy sequence. Theorem 5.9.9 If .E ⊆ S (τ ) is a strongly symmetric space, then the following statements are equivalent: (i) (ii) (iii) (iv)

Both E and .E × have order continuous norm. The closed unit ball .BE of E is weakly sequentially precompact. E does not contain a copy of .𝓁1 . E has order continuous norm and E does not contain a disjointly supported positive copy of .𝓁1 .

Proof (i).⇒(ii) Observe first that the order continuity of the norm in E and .E × implies that .E × = E ∗ and .E ×× = E ∗∗ , via trace duality. Let .{xn }∞ n=1 be a sequence in .BE . Since E has order continuous norm, it follows from Lemma 5.3.5 that .E ⊆ S0 (τ ). Therefore, by Lemma 5.5.8 (ii), there exists a .σ -finite projection .p ∈ P (M) such that .xn = pxn p for all .n ∈ N. Replacing .M by .Mp = pMp and E by .Ep = pEp (see the observations made preceding Lemma 4.1.4), it may be assumed that .1 is .σ -finite. Let .{pn }∞ n=1 be a sequence of finite trace projections in .P (M) such that .pn ↑n 1. Note that .pn ∈ E and .pn ∈ E × for all n (as .cE = 1). It will be shown next that the set

∞ ∞ .A = [0, kpn ] k=1

n=1

388

5 Strongly Symmetric Spaces of .τ -Measurable Operators

 + is norm dense in . E × . Indeed, let .0 ≤ y ∈ E × and .ε > 0 be given. By the × spectral theorem, there exists a sequence .{yk }∞ k=1 in .E such that .0 ≤ yk ↑k y and × .0 ≤ yk ≤ k1 for all k. Since .E has order continuous norm, there exists a .k ∈ N such that .‖y − yk ‖E × ≤ ε/2. Fixing such a k, it follows that .0 ≤ pn yk pn ≤ kpn , that is, .pn yk pn ∈ [0, kpn ] for all n. Since ‖yk − pn yk pn ‖E × ≤ ‖(1 − pn ) yk pn ‖E × + ‖yk (1 − pn )‖E ×

.

≤ ‖(1 − pn ) yk ‖E × + ‖yk (1 − pn )‖E × and since the order continuity of the norm in .E × implies that .

‖(1 − pn ) yk ‖E × →n 0,

‖yk (1 − pn )‖E × →n 0,

n→∞

(see Theorem 5.5.12), it follows that there exists .N ∈ N such that .

‖yk − pn yk pn ‖E × ≤ ε/2,

n ≥ N,

and hence, .

‖y − pn yk pn ‖E × ≤ ε, n ≥ N.

 + Since .pn yk pn ∈ A for all n, this shows that A is norm dense in . E × . It is now clear that A separates the points of .E ×× . Furthermore, using once again that the norm in .E × is order continuous, it follows from Theorem  × 5.6.29 ×× that each of the sets .[0, kpn ], .k, n ∈ N, is weakly compact (that is, .σ E , E  ××  ×× × compact). Proposition 5.9.8 will now be applied with .X = E , σ E , E , × separates in which case .X' = E × . As has been shown above, the set .A ⊆ E  the points of .E ×× and A is a countable union of .σ E × , E ×× -compact subsets of ∞ × .E . It follows from the Banach–Alaoglu theorem that the bounded set .{xn } n=1 is  ×× × relatively .σ E , E -compact. Therefore, it follows from Proposition 5.9.8 that  ∞  ∞ .{xn } . xnk , which is .σ E ×× , E × -convergent in .E ×× . n=1 has a subsequence k=1  ∞ This implies that . xnk k=1 is a weakly Cauchy sequence in E. This suffices to complete the proof of implication (i).⇒(ii). (ii).⇒(iii) This implication follows immediately from the fact that the space .𝓁1 is weakly sequentially complete, but .𝓁1 is not reflexive. (iii).⇒(iv) Since every separable Banach space embeds isometrically into .𝓁∞ , statement (iii) implies that E does not contain a copy of .𝓁∞ . Therefore, by Theorem 5.6.10, the norm in E is order continuous. It is evident that (iii) implies that E does not contain any disjointly supported positive copy of .𝓁1 . (iv).⇒(i) This implication is an immediate consequence of Lemma 5.9.7. The proof of the theorem is complete. ⨆ ⨅

5.9 Embedding Copies of c0 and 𝓁1

389

Remark 5.9.10 (a) It should be pointed out that the condition that E has order continuous norm cannot be omitted in statement (iv) of the above theorem. By way of example, the space .E = 𝓁∞ contains copies of .𝓁1 (as .𝓁∞ contains isometric copies of any separable Banach space). However, .𝓁∞ does not contain a disjointly supported copy of .𝓁1 . Indeed, the Banach dual .𝓁∗∞ has order continuous norm (in fact, ∗ .𝓁∞ is an abstract L-space) and, by a well-known result in the theory of Banach lattices, this implies that .𝓁∞ does not contain a lattice copy of .𝓁1 . (b) It should be noted that the equivalence (ii).⇔(iii) of the preceding Theorem 5.9.9 is valid in any Banach space. This is Rosenthal’s theorem [28, p. 201]. However, the proof of this implication in the present setting in intrinsic and does not appeal to Rosenthal’s theorem. Theorem 5.9.11 If .E ⊆ S (τ ) is a strongly symmetric space, then the following statements are equivalent: (i) E is reflexive. (ii) E contains no copy of either .c0 or .𝓁1 . (iii) E contains no disjointly supported positive copy of either .c0 or .𝓁1 . Proof The implications (i).⇒(ii).⇒(iii) are evident. Suppose now that (iii) holds. Since E does not contain a disjointly supported positive copy of .c0 , it follows from Theorem 5.9.6 that E is a KB-space and so, E has order continuous norm and has the Fatou property. Since E does not contain a disjointly supported positive copy of × also has order continuous norm. It now follows .𝓁1 , Theorem 5.9.9 implies that .E from Theorem 5.3.10 that E is reflexive. The proof is complete. ⨆ ⨅ Reference: [43].

Chapter 6

Examples

Abstract Several concrete classes of noncommutative spaces are introduced and their basic properties derived. The spaces considered include the noncommutative .Lp -spaces, Lorentz, Marcinkiewicz and Orlicz spaces.

This chapter gives some examples of symmetric spaces and their noncommutative versions, which will be discussed in some detail.

6.1 Construction of Symmetric Spaces In this section, the construction of strongly symmetric spaces .E (τ ) of .τ -measurable operators from strongly symmetric spaces E on .(0, ∞) (equipped with Lebesgue measure m) will be discussed. Furthermore, it will be shown that a number of properties which the space E may possess carry over to the noncommutative space .E (τ ). The interval .(0, ∞) will be equipped with Lebesgue measure m and the algebra of all m-measurable operators (functions) on .(0, ∞) is denoted by .S (0, ∞). Similarly, .S0 (0, ∞) denotes the subspace of .S (0, ∞) consisting of all .f ∈ S (0, ∞) for which .μ (t; f ) → 0 as .t → ∞. If .0 < a < ∞ , then the space .S (0, a) will be identified with the subspace of .S (0, ∞) consisting of all functions in .S (0, ∞) vanishing on .[a, ∞). Throughout this section, .M will be a semi-finite von Neumann algebra equipped with a fixed normal faithful semi-finite trace .τ . Note that .μ (x) ∈ S (0, τ (1)) for all .x ∈ S (τ ). Let E be a strongly symmetrically normed space on .(0, a), where .0 < a ≤ ∞, equipped with the norm .||·||E . Throughout this section it is assumed that .E /= {0}, which implies that the carrier of E is equal to .(0, a). If .τ (1) ≤ a, then the set .E (τ ) ⊆ S (τ ) is defined by setting E (τ ) = {x ∈ S (τ ) : μ (x) ∈ E}

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. G. Dodds et al., Noncommutative Integration and Operator Theory, Progress in Mathematics 349, https://doi.org/10.1007/978-3-031-49654-7_6

391

392

6 Examples

and let .

||x||E(τ ) = ||μ (x)||E ,

x ∈ E (τ ) .

The first objective will be to show that .E (τ ) is a strongly symmetrically normed space (see Theorem 6.1.2). Some preparations are needed. Given .0 < a ≤ ∞, for any measurable function f on .(0, a) and .s > 0, the dilation .Ds f of f is defined by setting { .

(Ds f ) (t) =

f (t/s) if t ∈ (0, a) and t/s ∈ (0, a) , 0 if t ∈ (0, a) and t/s ∈ / (0, a) .

It is well known that if E is a symmetric space on .(0, a), then .Ds f ∈ E whenever f ∈ E and .||Ds f ||E ≤ max (1, s) ||f ||E . This result will at present only be needed in the following special form, which is stated as a lemma for later reference.

.

Lemma 6.1.1 Suppose that .0 < a ≤ ∞ and that E is a symmetrically normed space on .(0, a). If .f ∈ E, then .D2 f ∈ E. Proof Without loss of generality,Σ it may be assumed that .0 ≤ f ∈ E. Assume ∞ first that f is of the form .f = n=1 αn χAn , where .{An } are mutually disjoint measurable subsets of .(0, a) and .0 ≤ αn ∈ R for all n. For each n, let .Bn1 and .Bn2 be disjoint measurable subsets of .An such that .An = Bn1 ∪ Bn2 and .m (Bn1 ) = m (Bn2 ) = 12 m (An ). Defining fk =

Σ∞

.

n=1

αn χBnk ,

k = 1, 2,

it is clear that .f = f1 + f2 . Furthermore, .μ (D2 fk ) ≤ μ (f ), which implies that D2 fk ∈ E for .k = 1, 2 . Consequently, .D2 f = D2 f1 + D2 f2 ∈ E. Now let .0 ≤ f ∈ E be arbitrary. For .n ∈ Z, define the set

.

} { An = x ∈ (0, a) : 2n ≤ f (x) < 2n+1

.

and observe that the sets .{An : n ∈ Z} are mutually disjoint. Define the function s by setting s=

.

Σ∞ n=−∞

2n χAn

and observe that .0 ≤ s ≤ f ≤ 2s. This implies, in particular, that .s ∈ E and so, it follows from the first part of the proof that .D2 s ∈ E. Since .0 ≤ D2 f ≤ 2D2 s, it is clear that .D2 f ∈ E. u n The fundamental result in the present section is the following theorem.

6.1 Construction of Symmetric Spaces

393

Theorem 6.1.2 If E is a strongly symmetrically normed space on .(0, a) and τ (1) ≤ a, then the space .E (τ ), equipped with .||·||E(τ ) , is a strongly symmetrically normed space of .τ -measurable operators.

.

Proof It will be shown first that .E (τ ) is a linear subspace of .S (τ ). Since .μ (λx) = |λ| μ (x), it is clear that .λx ∈ E (τ ) whenever .x ∈ E (τ ) and .λ ∈ C. If .x, y ∈ E (τ ), then it follows from Proposition 3.2.7 (iii) that μ (t; x + y) ≤ μ (t/2; x) + μ (t/2; y)

.

= (D2 μ (x)) (t) + (D2 μ (y)) (t) ,

t > 0.

By Lemma 6.1.1, .D2 μ (x) and .D2 μ (y) both belong to E and hence, .μ (x + y) ∈ E, that is, .x + y ∈ E (τ ). This shows that .E (τ ) is a linear subspace of .S (τ ). It will be shown next that .||·||E(τ ) is a norm on .E (τ ). It is clear that .||λx||E(τ ) = |λ| ||x||E(τ ) for all .x ∈ E (τ ) and .λ ∈ C. If .x, y ∈ E (τ ), then, as just has been shown, .μ (x + y) ∈ E and it follows from Theorem 3.9.9 that .μ (x + y) ≺≺ μ (x)+ μ (y). Since .||·||E is a strongly symmetric norm, this implies that .

||x + y||E(τ ) = ||μ (x + y)||E ≤ ||μ (x) + μ (y)||E ≤ ||μ (x)||E + ||μ (y)||E = ||x||E(τ ) + ||y||E(τ ) .

Hence, .||·||E(τ ) is a norm. It is now also clear that .E (τ ) is a strongly symmetrically normed space with respect to .||·||E(τ ) . u n Recall that a strongly symmetrically normed space which is complete is called a strongly symmetric space. Theorem 6.1.3 If E is a strongly symmetric space on .(0, a) and .τ (1) ≤ a, then E (τ ) is also a strongly symmetric space.

.

Proof It has to be shown that .E (τ ) is a Banach space with respect to .||·||E(τ ) whenever .E is a Banach space. By Theorem 4.1.19, it suffices to prove that .E (τ ) ∞ has the Riesz–Fischer n }n=1 is a sequence in Σ∞property. To this end, suppose that Σ.{a n + .E (τ ) satisfying . n=1 ||an ||E(τ ) < ∞. Setting .sn = k=1 ak , .n ∈ N, it has to be shown that .supn sn exists in .E (τ )+ . Evidently, .{sn }∞ n=1 is a Cauchy sequence in .E (τ ). By Proposition 4.4.4, the embedding of .E (τ ) into .S (τ ) is continuous with respect to the norm topology in .E (τ ) and the measure topology in .S (τ ) and hence, ∞ .{sn } n=1 is a Cauchy sequence for the measure topology. Consequently, there exists Tm

x ∈ S (τ )+ such that .sn → x (see Theorem 2.5.12). Since .sn ↑n , this implies that .0 ≤ sn ↑n x in .S (τ ) and .μ (sn ) ↑n μ (x) on .(0, ∞) (see Proposition 3.2.14). Furthermore, it follows from Theorem 3.9.14 that .

μ (sm ) − μ (sn ) ≺≺ μ (sm − sn )

.

394

6 Examples

and so, .

||μ (sm ) − μ (sn )||E ≤ ||μ (sm − sn )||E = ||sm − sn ||E(τ )

for all m and n. This implies that .{μ (sn )}∞ n=1 is a Cauchy sequence in E. Since E is assumed to be complete, there exists .f ∈ E such that .||μ (sn ) − f ||E → 0. Noting that .μ (sn ) ↑n , it follows that .0 ≤ μ (sn ) ↑n f a.e. on .(0, a). Consequently, it may be concluded that .μ (x) = f and hence, .x ∈ E (τ ). The proof of the theorem is complete. u n The following observation is an immediate consequence of the above theorem. Corollary 6.1.4 If E is a fully symmetric space on .(0, a) and .τ (1) ≤ a, then .E (τ ) is also a fully symmetric space. Proof If .x ∈ S (τ ) and .y ∈ E (τ ) are such that .x ≺≺ y, then, by definition, .μ (x) ≺ ≺ μ (y) and .μ (y) ∈ E. This implies that .μ (x) ∈ E and .||μ (x)||E ≤ ||μ (y)||E , that is, .x ∈ E (τ ) and .||x||E(τ ) ≤ ||y||E(τ ) . u n Remark 6.1.5 Let E be a symmetric function space on .(0, ∞) which is not necessarily strongly symmetric. It is proved in [72] that .E(τ ) is necessarily a symmetric space. For detailed exposition of this result we refer to [80]. If .E = L1 (0, ∞), then E (τ ) = {x ∈ S (τ ) : μ (x) ∈ L1 (0, ∞)} = L1 (τ )

.

and f ||x||E(τ ) = ||μ (x)||1 =

.

0

∞

μ (t; x) dt = τ (|x|) = ||x||1 ,

which is in agreement with the definition of the space .L1 (τ ) as given in Sect. 3.4. Similarly, if .E = L∞ (0, ∞), then E (τ ) = {x ∈ S (τ ) : μ (x) ∈ L∞ (0, ∞)} = L∞ (τ ) = M

.

and .

||x||E(τ ) = ||μ (x)||∞ = ||x||B(H ) = ||x||∞ .

Other examples are the strongly symmetric spaces .E = L1 ∩ L∞ (0, ∞) and E = (L1 + L∞ ) (0, ∞), in which cases the above construction agrees with the definitions of the spaces .L1 ∩ L∞ (τ ) and .(L1 + L∞ ) (τ ), respectively, as given in Sect. 3.10.

.

6.1 Construction of Symmetric Spaces

395

If E is a strongly symmetric space on .(0, a), then it follows from Theorem 4.4.6 that L1 ∩ L∞ (0, a) ⊆ E ⊆ (L1 + L∞ ) (0, a) ,

.

with continuous embeddings. Consequently, L1 ∩ L∞ (τ ) ⊆ E (τ ) ⊆ (L1 + L∞ ) (τ ) ,

.

with continuous embeddings. In particular, the carrier projection .cE(τ ) of .E (τ ) is always equal to .1. Several other examples will be discussed later in this section. Given a strongly symmetric space E on .(0, a), its Köthe dual is denoted by × .E , which is a fully symmetric space on .(0, a) with the Fatou property (see Theorem 4.5.5). For a semi-finite von Neumann algebra .(M, τ ) the following spaces may be considered: .E (τ ), its Köthe dual .E (τ )× , and the space .E × (τ ). The following theorem shows that the last two spaces are actually equal. Theorem 6.1.6 If E is a strongly symmetrically normed space on .(0, a) with τ (1) ≤ a, then .E (τ )× = E × (τ ) with equality of norms.

.

Proof If .y ∈ E × (τ ), that is, .y ∈ S (τ ) and .μ (y) ∈ E × , then it follows from Theorem 3.4.29 and the definition of the norm in .E × that f ∞ .τ (|xy|) ≤ μ (t; x) μ (t; y) dt 0

≤ ||μ (x)||E ||μ (y)||E × = ||x||E(τ ) ||y||E × (τ ) for all .x ∈ E (τ ). Therefore, it follows from Definition 4.3.1 that .y ∈ E (τ )× and .||y||E(τ )× ≤ ||y||E × (τ ) . Suppose now that .y ∈ E (τ )× . If .f ∈ E, then f

a

.

f

∞

|f (t) μ (t; y)| dt ≤

0

μ (t; f ) μ (t; y) dt 0

and it follows from Corollary 3.10.20 that f

∞

.

μ (t; f ) μ (t; y) dt = sup {τ (|zy|) : z ∈ F (τ ) , z ≺≺ f } .

0

If .z ∈ F (τ ) and .z ≺≺ f , then .z ∈ E (τ ) and .||z||E(τ ) = ||μ (z)||E ≤ ||f ||E and hence, τ (|zy|) ≤ ||z||E(τ ) ||y||E(τ )× ≤ ||f ||E ||y||E(τ )× .

.

396

6 Examples

Consequently, f

a

.

0

|f (t) μ (t; y)| dt ≤ ||f ||E ||y||E(τ )× ,

f ∈ E,

and this shows that .μ (y) ∈ E × , that is, .y ∈ E × (τ ), and .

||y||E × (τ ) = ||μ (y)||E × ≤ ||y||E(τ )× .

The proof is complete.

u n

Next we discuss a number of properties that carry over from E to .E (τ ). Theorem 6.1.7 Let E be a strongly symmetrically normed space on .(0, a) and τ (1) ≤ a.

.

(i) If .x ∈ E (τ ) and .μ (x) ∈ E oc , then .x ∈ E (τ )oc . In particular, if E has order continuous norm, then .E (τ ) has order continuous norm. (ii) If E has a Fatou norm, then .E (τ ) has a Fatou norm. (iii) If E has the Fatou property, then .E (τ ) has the Fatou property. (iv) If E is a KB-space, then .E (τ ) is a KB -space. (v) If E is reflexive, then .E (τ ) is reflexive. Proof (i) If .μ (x) ∈ E oc , then .μ (x) ∈ E b (see Remark 5.4.5) and so, by Lemma 4.4.10, .limt→∞ μ (t; x) = 0, that is, .x ∈ S0 (τ ). Suppose that .{xα } is a net in .E (τ ) such that .|x| ≥ xα ↓α 0. It follows from Proposition 3.2.14 (ii) that .μ (x) ≥ μ (xα ) ↓α 0 in E. Since .μ (x) ∈ E oc , this implies that .||μ (xα )||E ↓α 0, that is, .||xα ||E(τ ) ↓α 0. (ii) Suppose that .{xα } is a net in .E (τ )+ and that .x ∈ E (τ )+ such that .0 ≤ xα ↑α x. It follows from Proposition 3.2.14 (i) that .μ (xα ) ↑α μ (x) in E . Since the norm in E is a Fatou norm, this implies that .||μ (xα )||E ↑α ||μ (x)||E , that is, .||xα ||E(τ ) ↑α ||x||E(τ ) . (iii) Suppose that .{xα } is a net in .E (τ )+ such that .0 ≤ xα ↑α and .supα ||xα ||E(τ ) < ∞. Since the embedding of .E (τ ) into .S (τ ) is continuous for the norm topology in .E (τ ) and the measure topology in .S (τ ), it follows that .{xα } is bounded in .S (τ ) with respect to the measure topology. It now follows from Theorem 2.6.15 that there exists .x ∈ S (τ )+ such that .0 ≤ xα ↑α x. By Proposition 3.2.14 (i), this implies that .μ (xα ) ↑α μ (x) in .S (0, a). Since .supα ||μ (xα )||E = supα ||xα ||E(τ ) < ∞ and E is assumed to have the Fatou property, it follows that .μ (x) ∈ E and .||μ (x)||E = supα ||μ (xα )||E . Consequently, .x ∈ E (τ ) and .||x||E(τ ) = supα ||xα ||E(τ ) , which shows that .E (τ ) has the Fatou property. (iv) If E is a KB-space, then it follows from Lemma 5.3.8 that E has order continuous norm and the Fatou property. Consequently, by (i) and (iii), .E (τ )

6.1 Construction of Symmetric Spaces

397

has order continuous norm and the Fatou property. Using Lemma 5.3.8 once again, it follows that .E (τ ) is a KB-space. (v) If E is reflexive, then it follows from Theorem 5.3.10 that .E has the Fatou property and both E and .E × have order continuous norms. By (i) and (iii), the space .E (τ ) has order continuous norm and the Fatou property. Furthermore, by Theorem 6.1.6, .E (τ )× = E × (τ ) and hence, it follows from (i) that also × .E (τ ) has order continuous norm. Using Theorem 5.3.10 once again, it may be concluded that .E (τ ) is reflexive. u n Recall, from Definition 4.4.9, that for any symmetrically normed space E E (satisfying .cE = 1), the subspace .E b is defined by .E b = F (τ ) and that .E b itself is a symmetrically normed space (see Lemma 4.4.10). Consequently, if E is a strongly symmetric space on .(0, a), then .E b is also a strongly symmetric space on b .(0, a) and so, if .τ (1) ≤ a, the corresponding strongly symmetric space .E (τ ) may b be constructed (actually, .E is a fully symmetric space; see Proposition 5.1.13). In the following theorem, this space will be identified with .E (τ )b . Theorem 6.1.8 If E is a strongly symmetric space on .(0, a) and .τ (1) ≤ a, then E (τ )b = E b (τ ). In particular, if .E = E b , then also .E (τ ) = E b (τ ).

.

Proof Given .x ∈ E (τ ), it has to be proved that .x ∈ E (τ )b if and only if .μ (x) ∈ E b . The main ingredient in the proof will be the criterion for membership of .E b exhibited in Lemma 4.4.10. Since the conditions .x ∈ E (τ )b or .μ ((x) ∈ E b both ) imply that .x ∈ S0 (τ ), it may be assumed that .x ∈ S0 (τ ), that is, .τ e|x| (s, ∞) < ∞ for all .s > 0. Given .0 < s ∈ R, it follows from Proposition 3.2.10 (iii) that ( ) ( ) μ xe|x| (s, ∞) = μ |x| e|x| (s, ∞) = μ (x) χ[0,d(s;|x|)) = μ (x) χ[0,d(s;μ(x)))

.

= μ (x) χ{t≥0:μ(t;x)>s} and hence, .

|| || || |x| || ||xe (s, ∞)||

E(τ )

|| || = ||μ (x) χ{t≥0:μ(t;x)>s} ||E .

(6.1)

) ( Since .d (s; |x|) = τ e|x| (s, ∞) < ∞, it also follows from the same proposition that ) ( |x| .μ t; xe [0, s] = μ (t + d (s; |x|) ; x) = μ (t + d (s; μ (x)) ; μ (x)) ( ) = μ t; μ (x) χ{t≥0:μ(t;x)≤s} , t ≥ 0,

398

6 Examples

and hence, .

|| || || |x| || ||xe [0, s]||

E(τ )

|| || = ||μ (x) χ{t≥0:μ(t;x)≤s} ||E .

(6.2)

Hence, it follows from (6.1), (6.2), and Lemma 4.4.10 (applied in .E (τ ) as well as in E) that .x ∈ E (τ )b if and only if .μ (x) ∈ E b . The proof is complete. u n Let E be a strongly symmetric space on .(0, a) and let .E oc be the subspace consisting of all elements of order continuous norm in E. If .E oc /= {0}, then it follows from Proposition 5.4.8 that .E oc = E b . Combining this observation with Theorem 6.1.8, the following result is obtained. Proposition 6.1.9 If E is a strongly symmetric space on .(0, a), .τ (1) ≤ a and E oc /= {0} , then .E (τ )oc = E (τ )b and

.

} { E (τ )oc = E oc (τ ) = x ∈ E (τ ) : μ (x) ∈ E oc .

.

Furthermore, denoting, for each .y ∈ E × (τ ), by .ϕy the functional .ϕy ∈ E (τ )∗ given by .ϕy (x) = τ (xy), .x ∈ E (τ ), the restriction map .y |−→ ϕy |E oc (τ ) , .y ∈ E × (τ ), is an isometric isomorphism from .E × (τ ) onto the Banach dual .E oc (τ )∗ . In particular, × .E (τ ) is the Banach dual space of .E(τ ) in this case. Proof As observed in Remark 5.4.5, the inclusion .E (τ )oc ⊆ E (τ )b always holds. It follows from Theorem 6.1.8 that .E (τ )b = E b (τ ). Since .E b = E oc , it is clear that b oc (τ ). Furthermore, it follows from Theorem 6.1.7 (i) that .E oc (τ ) ⊆ .E (τ ) = E oc E (τ ) . Consequently, E oc (τ ) ⊆ E (τ )oc ⊆ E (τ )b = E b (τ ) = E oc (τ ) ,

.

from which the first statement of the proposition follows. Since .E (τ )oc = E b (τ ), it is also clear that the carrier projection of .E (τ )oc is equal to .1 and hence, all equivalent statements of Proposition 5.4.6 hold for .E (τ ). Combining statement (v) of that proposition with the identification of .E (τ )∗n with .E (τ )× (see Theorem 5.2.9) and using the equality .E (τ )× = E × (τ ) (see Theorem 6.1.6), the second statement of the present proposition follows. u n It is a natural question as to whether each strongly symmetric space .E ⊆ S (τ ) arises as the noncommutative version of some strongly symmetric space on the halfline, via the construction given in Theorem 6.1.2. This is indeed the case if E has the Fatou property and has full carrier, as is shown by the following noncommutative version of the Luxemburg representation theorem. Theorem 6.1.10 If .E ⊆ S (τ ) is a strongly symmetric space with the Fatou property and .cE = 1, then there exists strongly symmetric space .F ⊆ S (m) with the Fatou property such that .E = F (τ ) .

6.2 Lp -spaces

399

Proof It follows from Theorem 5.1.10 that .E = E ×× with equality of norms. Consequently, the result of the theorem follows immediately from an application of Proposition 5.1.6 to the space .E × . u n References: [42, 44, 45, 72, 80, 84, 85, 97, 106, 113, 116–121, 136, 139, 140].

6.2 Lp -spaces For .1 ≤ p ≤ ∞, the .Lp -spaces on .(0, ∞) are given by } { .Lp = Lp (m) = Lp (0, ∞) = f ∈ L0 (0, ∞) : ||f ||p < ∞ , where

{ (f .

||f ||p =

)1/p dm , 1 ≤ p < ∞, ess. supt>0 |f (t)| , p = ∞, ∞ p 0 |f |

f ∈ L0 (0, ∞) .

( ) As is well known, the spaces . Lp , ||·||p are symmetric spaces with the Fatou property for all values .1 ≤ p ≤ ∞ and consequently are fully symmetric (see Corollary 5.1.12). For .1 ≤ p < ∞, the norm .||·||p is order continuous and for .1 < p < ∞, the space .Lp (0, ∞) is reflexive. Since the norm in .Lp is order continuous for .1 ≤ p < ∞, it follows that .L∗p = × L× p = Lq (where .1/p + 1/q = 1). Furthermore, .L∞ = L1 . Suppose that .M is a semi-finite von Neumann algebra equipped with a fixed semi-finite normal faithful trace .τ . For .1 ≤ p ≤ ∞, the corresponding noncommutative .Lp -spaces are defined by setting } { .Lp (τ ) = x ∈ S (τ ) : μ (x) ∈ Lp , ||x||Lp (τ ) = ||μ (x)||p ,

x ∈ Lp (τ ) .

The norm .||·||Lp (τ ) is usually denoted by .||·||p . The spaces .L1 (τ ) and .L2 (τ ) introduced in Sect. 3.4 agree with the above definition. As observed before, .L∞ (τ ) = M and .||·||∞ = ||·||B(H ) . The following result now follows immediately from Theorems 6.1.2, 6.1.3, 6.1.6, 6.1.7 and Corollary 6.1.4. Theorem 6.2.1 Let .1 ≤ p, q ≤ ∞ be such that .1/p + 1/q = 1. ( ) (i) For each .1 ≤ p ≤ ∞, the space . Lp (τ ) , ||·||p is a fully symmetric space of .τ -measurable operators with the Fatou property. (ii) .Lp (τ )× = L× p (0, ∞) = Lq (τ ) for all .1 ≤ p ≤ ∞. (iii) For .1 ≤ p < ∞, the norm in .Lp (τ ) is order continuous. (iv) If .1 ≤ p < ∞, then .Lp (τ )∗ = Lq (τ ) (via trace duality). (v) For .1 < p < ∞ the space .Lp (τ ) is reflexive. (vi) The space .L1 (τ ) is a KB-space.

400

6 Examples

) ( Remark 6.2.2 It follows from Proposition 3.2.8 that .μ (x)p = μ |x|p for all .x ∈ S (τ ) and .1 ≤ p < ∞. This implies that f

∞

.

f μ (t; x) dt = p

0

∞

) ( ) ( μ t; |x|p dt = τ |x|p ,

x ∈ S (τ )

0

(here .τ denotes the extended trace on .S (τ )+ , see Sect. 3.3). Consequently, for .1 ≤ p < ∞ the spaces .Lp (τ ) are also given by ) } { ( Lp (τ ) = x ∈ S (τ ) : τ |x|p < ∞ ,

))1/p ( ( ||x||p = τ |x|p ,

.

x ∈ Lp (τ ) .

Remark 6.2.3 In the case that .M = B(H ), equipped with the standard trace .τ , the Lp (τ ) spaces constructed here are precisely the Schatten ideals

.

Sp := {x ∈ K(H ) :

∞ Σ

.

μ(n; x)p < ∞}.

n=0

This can be pieced together by Examples 2.4.7 and 3.2.2 (iii). We follow the notation of Gokhberg and Krein [60] for these spaces, see also the historical notes at the end of this book. References: [97, 102, 135, 138, 141].

6.3 Lorentz Spaces Let .ψ : [0, ∞) → [0, ∞) be a concave function, continuous on .(0, ∞) and satisfying .ψ (0) = 0. Note that .ψ is necessarily increasing and that .ψ is locally absolutely continuous on .(0, ∞). The derivative .ψ ' (t) exists for all .t ∈ (0, ∞), with the exception of an at most countable set and .ψ may be written as f

t

ψ (t) = ψ (0+) +

.

ψ ' (s) ds,

t > 0,

0

where .ψ (0+) = limt↓0 ψ (t). Note that the function .ψ ' is decreasing and .ψ ' (t) ≥ 0 a.e. on .(0, ∞). For definiteness, define .ψ ' (t) = ψ ' (t+) for all .t ∈ (0, ∞). It will be assumed that .ψ is not the zero function (that is, .ψ (t) > 0 for all .t > 0). Furthermore, denote .ψ (∞) = limt→∞ ψ (t) (with the possible value .∞). For .f ∈ S (0, ∞) define f .

||f ||Aψ = ψ (0+) ||f ||∞ +

∞ 0

μ (t; f ) ψ ' (t) dt.

(6.3)

6.3 Lorentz Spaces

401

Remark 6.3.1 (a) If .ψ (0+) = 0, then the term .ψ (0+) ||f ||∞ in (6.3) is interpreted to be zero for all f . (b) Observe that (6.3) may also be written as f .

||f ||Aψ =

∞

μ (t; f ) dψ (t) , 0

as an improper Riemann–Stieltjes integral. The Lorentz space .Aψ = Aψ (m) = Aψ (0, ∞) is now defined by setting { } Aψ = f ∈ S (0, ∞) : ||f ||Aψ < ∞ .

.

Observe that if .A ⊆ (0, ∞) is measurable and .m (A) < ∞, then f .

||χA ||Aψ = ψ (0+) + f

∞

μ (t; χA ) ψ ' (t) dt

0 m(A)

= ψ (0+) +

ψ ' (t) dt = ψ (m (A)) .

0

In particular, .χA ∈ Aψ .

( ) Theorem 6.3.2 The space . Aψ , ||·||Aψ is a fully symmetric space with the Fatou property. Proof If .f ∈ Aψ and .||f ||Aψ = 0, then .f = 0. Indeed, if .||f ||Aψ = 0, then ψ (0+) ||f ||∞ = 0 and .μ (t; f ) ψ ' (t) = 0 a.e. If .ψ (0+) > 0, then this implies that .||f ||∞ = 0 and so, .f = 0. If .ψ (0+) = 0, then there exists .t0 > 0 such that ' .ψ (t) > 0 on .(0, t0 ) and so, .μ (t; f ) = 0 on .(0, t0 ). This implies that .μ (t; f ) = 0 for all .t > 0 and hence, .f = 0. If .f, g ∈ Aψ , then it is clear that .ψ (0+) ||f + g||∞ ≤ ψ (0+) ||f ||∞ + ψ (0+) ||g||∞ . Furthermore, since .μ (f + g) ≺≺ μ (f )+μ (g) (see Theorem 3.9.9) and .ψ ' is decreasing, it follows from Hardy’s inequality (see Lemma 3.10.10) that .

f

∞

.

0

'

f

∞

μ (t; f + g) ψ (t) dt ≤ 0

'

f

∞

μ (t; f ) ψ (t) dt +

μ (t; g) ψ ' (t) dt.

0

Consequently, .f + g ∈ Aψ and .||f + g||Aψ ≤ ||f ||Aψ + ||g||Aψ . It is now clear that ( ) . Aψ , ||·||A is a symmetrically normed space. ψ

402

6 Examples

Suppose that .f ∈ S (0, ∞) and .g ∈ Aψ are such that .f ≺≺ g. This implies that ||f ||∞ ≤ ||g||∞ and so, .ψ (0+) ||f ||∞ ≤ ψ (0+) ||g||∞ . Furthermore, it follows from Hardy’s inequality that

.

f

∞

.

μ (t; f ) ψ ' (t) dt ≤

0

f

∞

μ (t; g) ψ ' (t) dt

0

and so, .||f ||Aψ ≤ ||g||Aψ . This shows that .Aψ is fully symmetrically normed. It will be shown next that .Aψ has the Fatou property. Since the Lebesgue measure on .(0, ∞) is .σ -finite, it is sufficient to show that .Aψ has the .σ -Fatou property (see Proposition 5.6.5). Suppose that .(fn )∞ n=1 is a sequence in .Aψ such that .0 ≤ fn ↑n and .supn ||fn ||Aψ < ∞. Recall that the space .M + (0, ∞) is the set of all measurable functions on .(0, ∞) with values in .[0, ∞]. Defining .f ∈ M + (0, ∞) by .f (t) = supn fn (t), .t ∈ (0, ∞), it is clear that .||fn ||∞ ↑n ||f ||∞ and so, .ψ (0+) ||fn ||∞ ↑n ψ (0+) ||f ||∞ . Furthermore, it follows from Lemma 3.1.12 that .μ (t; fn ) ↑n μ (t; f ) for all .t > 0 and hence, f

∞

.

μ (t; fn ) ψ ' (t) dt ↑n

0

f

∞

μ (t; f ) ψ ' (t) dt.

0

Consequently, f ψ (0+) ||f ||∞ +

∞

.

0

μ (t; f ) ψ ' (t) dt = sup ||fn ||Aψ < ∞.

(6.4)

n

If .ψ (0+) > 0, then this implies that .||f ||∞ < ∞ and so, in particular, .f ∈ S (0, ∞). If .ψ (0+) = 0, then there exists .t0 > 0 such that .ψ ' (t) > 0 on ' .(0, t0 ). Since (6.4) implies that .μ (t; f ) ψ (t) < ∞ a.e. on .(0, ∞), it follows that .μ (t; f ) < ∞ on .(0, t0 ). Hence, .μ (t; f ) < ∞ for all .t > 0, which shows that .f ∈ S (0, ∞). Therefore, .f ∈ Aψ and (by (6.4)) .||fn ||A ↑n ||f ||Aψ . This shows ψ that .Aψ has the .σ -Fatou property (and hence, the Fatou property). This implies, in particular, that .Aψ is a Banach space (see Corollary 4.1.22). The proof is complete. u n Remark 6.3.3 (a) It should be observed that .ψ (0+) > 0 if and only if .Aψ ⊆ L∞ (0, ∞). Indeed, if .ψ (0+) > 0, then it is clear that .f ∈ L∞ (0, ∞) for all .f ∈ Aψ (and −1 ||f || .||f ||∞ ≤ ψ (0+) Aψ ). Conversely, if .Aψ ⊆ L∞ (0, ∞), then it follows via the Closed Graph Theorem that there exists a constant .C > 0 such that .||f ||∞ ≤ C ||f ||A for all .f ∈ Aψ . Taking .f = χ[0,t] , .t > 0, this implies that ψ −1 > 0. .1 ≤ Cψ (t) and hence, .ψ (0+) ≥ C

6.3 Lorentz Spaces

403

(b) It should also be noted that .ψ (∞) < ∞ if and only if .L∞ (0, ∞) ⊆ Aψ (with continuous embedding). Indeed, .L∞ (0, ∞) ⊆ Aψ is equivalent to .1 ∈ Aψ , that is, f

∞

ψ (∞) = ψ (0+) +

.

ψ ' (t) dt < ∞.

0

Recall (see Definition 4.4.9) that .Abψ denotes the norm closure in .Aψ of the space .F (m) of all functions in .L∞ (0, ∞) supported on sets of finite measure (equivalently, .Abψ is the norm closure of .(L1 ∩ L∞ ) (0, ∞) in .Aψ ). As observed in Theorem 4.5.8 (iv), .Abψ is itself a fully symmetric space (with respect to .||·||Aψ ). The space .Abψ may also be described as follows. Proposition 6.3.4 If .ψ : [0, ∞) → [0, ∞) is a concave function as before, then Abψ = Aψ ∩ S0 (0, ∞) .

.

Proof Since .F (m) ⊆ S0 (0, ∞) and .S0 (0, ∞) is closed in .S (0, ∞) with respect to the measure topology, it is clear that .Abψ ⊆ S0 (0, ∞). Therefore, it only needs to be shown that .Aψ ∩ S0 (0, ∞) ⊆ Abψ . Given .0 ≤ f ∈ Aψ ∩ S0 (0, ∞), define fn = f χ{1/n 0, then .Aψ ⊆ L∞ (0, ∞) and so, .fn = f χ{f >1/n} whenever .n ≥ ||f ||∞ . Defining .gn = f − fn = f χ{f ≤1/n} , it follows that . ||gn ||A ≤ ψ

1 ψ (0+) + n

f

∞

μ (t; gn ) ψ ' (t) dt

0

for all .n ≥ ||f ||∞ . Furthermore, .f ≥ gn ↓n 0 a.e. on .(0, ∞). Since .f ∈ S0 (0, ∞), this implies that .μ (f ) ≥ μ (gn ) ↓n 0 on .(0, ∞) and so, by the dominated convergence theorem, it follows that f .

∞

μ (t; gn ) ψ ' (t) dt ↓ 0,

n → ∞.

0

This shows that .||gn ||Aψ → 0 as .n → ∞ and hence, .f ∈ Abψ .

404

6 Examples

(ii) Assume now that .ψ (0+) = 0. Observe that .0 ≤ fn ↑n f a.e. on .(0, ∞) and .f ≥ f − fn ≥ 0 for all n. Since .f ∈ S0 (0, ∞), this implies that .μ (f ) ≥ μ (f − fn ) ↓n 0 on .(0, ∞) and hence, the dominated convergence theorem implies that f .

||f − fn ||Aψ =

∞

μ (t; f − fn ) ψ ' (t) dt ↓n 0,

n → ∞.

0

This shows that .f ∈ Abψ . The proof is complete.

u n

oc .A ψ

Recall that denotes the closed order ideal (that is, .L∞ -bimodule) of all elements of order continuous norm in .Aψ (see Definition 5.4.1 and Corollary 5.4.4). b oc Since .(0, ∞) is non-atomic, it follows that .Aoc ψ = Aψ whenever .Aψ /= {0} (see Proposition 5.4.8). Proposition 6.3.5 The following two statements hold: (i) If .ψ (0+) > 0, then .Aoc ψ = {0}. (ii) If .ψ (0+) = 0, then .Aoc ψ = Aψ ∩ S0 (0, ∞). Proof (i) If .ψ (0+) > 0, then .||χA ||Aψ ≥ ψ (0+) > 0 for all measurable sets oc does not contain any non-zero .A ⊆ (0, ∞) with .m (A) > 0. Hence, .A ψ oc characteristic functions and so, .Aψ = {0}. (ii) Suppose that .ψ (0+) = 0. If .(fn )∞ n=1 is a sequence in .Aψ such that .χ[0,1] ≥ fn ↓n 0, then .χ[0,1] ≥ μ (fn ) ↓n 0 and so, f .

||fn ||Aψ =

∞

μ (t; fn ) ψ ' (t) dt ↓n 0,

n → ∞,

0

oc by dominated convergence. Hence, .χ[0,1] ∈ Aoc ψ and so .Aψ /= {0}. This implies b that .Aoc ψ = Aψ and the result now follows from Proposition 6.3.4. u n

Theorem 6.3.6 The space .Aψ has order continuous norm if and only if .ψ (0+) = 0 and .ψ (∞) = ∞. Proof If .Aψ has order continuous norm, that is, .Aψ = Aoc ψ , then it follows from Proposition 6.3.5 that .ψ (0+) = 0 and .Aψ ⊆ S0 (0, ∞). The latter inclusion implies / Aψ and hence, .ψ (∞) = ∞ (see Remark 6.3.3 (b)). that .1 ∈ If .ψ (0+) = 0, then it follows from Proposition 6.3.5 that .Aoc ψ = Aψ ∩S0 (0, ∞). Furthermore, if .ψ (∞) = ∞, then .1 ∈ / Aψ and so, .Aψ ⊆ S0 (0, ∞). Consequently, if .ψ (0+) = 0 and .ψ (∞) = ∞, then .Aψ = Aoc ψ , that is, .Aψ has order continuous u n norm.

6.4 Marcinkiewicz Spaces

405

Remark 6.3.7 If .0 < a < ∞, then one may also consider Lorentz spaces .Aψ (0, a) on the interval .(0, a). Note that { } Aψ (0, a) = f χ(0,a) : f ∈ Aψ (0, ∞) .

.

In this situation, the behaviour of the function .ψ at infinity does not play a role for the properties of the space .Aψ (0, a). For example, the space .Aψ (0, a) has order continuous norm if and only if .ψ (0+) = 0. Let .M be a semi-finite von Neumann algebra equipped with a semi-finite faithful normal trace .τ . Given a non-zero concave function .ψ : [0, ∞) → [0, ∞) with .ψ (0) = 0, the corresponding noncommutative Lorentz space is defined by { } Aψ (τ ) = x ∈ S (τ ) : μ (x) ∈ Aψ (0, ∞) ,

.

||x||Aψ (τ ) = ||μ (x)||Aψ ,

x ∈ Aψ (τ ) .

The following theorem follows from the above results in combination with Theorems 6.1.2, 6.1.3, 6.1.7, 6.1.8, Corollary 6.1.4 and Proposition 6.1.9 Theorem 6.3.8

( ) (i) The space . Aψ (τ ) , ||·||Aψ (τ ) is a fully symmetric space of .τ -measurable operators with the Fatou property. (ii) .Aψ (τ )b = Abψ (τ ) = Aψ (τ ) ∩ S0 (τ ). (iii) If .ψ (0+) = 0, then .Aψ (τ )oc = Aoc ψ (τ ) = Aψ (τ ) ∩ S0 (τ ). (iv) If .ψ (0+) = 0 and .ψ (∞) = ∞, then .Aψ (τ ) has order continuous norm (and hence, .Aψ (τ ) is a KB-space). References: [20, 97]

6.4 Marcinkiewicz Spaces Let .ψ : [0, ∞) → [0, ∞) be a non-zero concave function, continuous on .(0, ∞) and satisfying .ψ (0) = 0. For .f ∈ S (0, ∞) define .

||f ||Mψ = sup t>0

1 ψ (t)

f

t

μ (s; f ) ds. 0

The Marcinkiewicz space .Mψ = Mψ (m) = Mψ (0, ∞) is defined by setting { } Mψ = f ∈ S (0, ∞) : ||f ||Mψ < ∞ .

.

406

6 Examples

Observe that for any measurable set .A ⊆ (0, ∞), with .0 < m (A) < ∞, it follows that f t f t 1 1 min (t, m (A)) , t > 0. μ (s; χA ) ds = . χ[0,m(A)) (s) ds = ψ (t) 0 ψ (t) 0 ψ (t) Since .ψ is concave, the function .t − | → ψ (t) /t , .t > 0, is decreasing, which implies that .

||χA ||Mψ =

m (A) . ψ (m (A))

(6.5)

In particular, .χA ∈ Mψ . Theorem 6.4.1 The space .Mψ is a fully symmetric space with the Fatou property. ft Proof If .f ∈ Mψ and .||f ||Mψ = 0, then . 0 μ (s; f ) ds = 0 for all .t > 0 and so, .μ (s; f ) = 0 for all .s > 0. This implies that .f = 0. If .f, g ∈ Mψ , then it follows from .μ (f + g) ≺≺ μ (f ) + μ (g) that 1 . ψ (t)

f 0

t

1 μ (s; f + g) ds ≤ ψ (t)

f

t 0

1 μ (s; f ) ds + ψ (t)

f

t

μ (s; g) ds,

t > 0,

0

which implies that .f + g ∈ Mψ and .||f + g||Mψ ≤ ||f ||Mψ + ||g||Mψ . It is now clear that .||·||Mψ is a norm on .Mψ . If .f ∈ S (0, ∞) and .g ∈ Mψ are such that .f ≺≺ g, then 1 . ψ (t)

f

t

0

1 μ (s; f ) ds ≤ ψ (t)

f

t

μ (s; g) ds,

t > 0,

0

and so, .f ∈ Mψ and .||f ||Mψ ≤ ||g||Mψ . This shows that .Mψ is fully symmetrically normed. To show that .Mψ has the Fatou property, it is sufficient to prove that .Mψ has the .σ -Fatou property. Suppose that .0 ≤ fn ↑n in .Mψ and that .supn ||fn ||Mψ < ∞. Defining .f ∈ M + (0, ∞) by .f (t) = supn fn (t), .t ∈ (0, ∞), it follows from Lemma 3.1.12 that .μ (t; fn ) ↑n μ (t; f ) for all .t > 0. Hence, by the monotone convergence theorem, .

1 ψ (t)

f

t 0

μ (s; f ) ds = sup n

1 ψ (t)

f

t

μ (s; fn ) ds

(6.6)

0

≤ sup ||fn ||Mψ < ∞, t > 0. n

ft This implies, in particular, that . 0 μ (s; f ) ds < ∞ for all .t > 0 and so, .μ (s; f ) < ∞, .s > 0, that is, .f ∈ S (0, ∞). It now follows from (6.6) that .f ∈ Mψ and that

6.4 Marcinkiewicz Spaces

407

||f ||Mψ = supn ||fn ||Mψ . This shows that .Mψ has the Fatou property. In particular, Mψ is a Banach space, due to Corollary 4.1.22. The proof is complete. u n

. .

Remark 6.4.2 It should be noted that in the proof of Theorem 6.4.1 no special properties of .ψ are used, other than .ψ (t) > 0, .t > 0. However, to obtain (6.5) it is used that .ψ is quasiconcave (that is, .ψ : [0, ∞) → [0, ∞) is increasing, .ψ (0) = 0, .ψ (t) > 0 for .t > 0 and the function .t − | → ψ (t) /t, .t > 0, is decreasing). The basic relation between Lorentz and Marcinkiewicz spaces is given in the following theorem. Theorem 6.4.3 Let .ψ : [0, ∞) → [0, ∞) be a non-zero concave function, continuous on .(0, ∞) and satisfying .ψ (0) = 0. The Köthe dual of .Aψ is equal to .Mψ , that is, .A× ψ = Mψ with equality of norms. Proof It will be shown first that .Mψ ⊆ A× ≤ ||g||Mψ for all ψ and that .||g||A× ψ .g ∈ Mψ . To this end, let .g ∈ Mψ be given and suppose that .h ∈ F (m) is a simple function. The decreasing rearrangement .μ (h) of h may be written as μ (h) =

N Σ

.

βk χ[0,tk ) ,

k=1

where .βk ≥ 0 and .0 < t1 < · · · < tN . Hence,

.

||h||Aψ = ψ (0+)

(N Σ

) βk +

k=1

N Σ

f

tk

βk

ψ ' (s) ds =

0

k=1

N Σ

βk ψ (tk ) .

k=1

Consequently, f .

∞

f

∞

|h (t) g (t)| dt ≤

0

μ (t; h) μ (t; g) dt =

0

≤

(N Σ

)

N Σ k=1

f

tk

βk

μ (t; g) dt 0

βk ψ (tk ) ||g||Mψ = ||h||Aψ ||g||Mψ .

k=1

Now let .f ∈ Aψ be arbitrary and let .(hn )∞ n=1 be a sequence of simple functions in .F (m) such that .0 ≤ hn ↑n |f | a.e. on .(0, ∞). It follows from the above that f .

0

∞

hn (t) |g (t)| dt ≤ ||hn ||Aψ ||g||Mψ ≤ ||f ||Aψ ||g||Mψ

408

6 Examples

for all n. The monotone convergence theorem implies that f

∞

.

f

∞

hn (t) |g (t)| dt ↑n

0

|f (t) g (t)| dt

0

and so, f

∞

.

0

|f (t) g (t)| dt ≤ ||f ||Aψ ||g||Mψ .

This shows that .g ∈ A× ψ and .||g||A× ≤ ||g||Mψ . ψ

Suppose now that .g ∈ A× ψ , that is, f

∞

.

0

|f (t) g (t)| dt ≤ ||f ||Aψ ||g||A× ,

f ∈ Aψ .

ψ

Recall that } {f f t |g (s)| ds : A ⊆ (0, ∞) , m (A) = t . μ (s; g) ds = sup 0

(6.7)

A

for all .t > 0 (see Theorem 3.9.5). If .A ⊆ (0, ∞) is measurable with .m (A) = t, then f ∞ f |g (s)| ds = . χA (s) |g (s)| ds ≤ ||χA ||Aψ ||g||A× = ψ (t) ||g||A× . A

ψ

0

ψ

Therefore, it follows from (6.7) that .

1 ψ (t)

f

t 0

μ (s; g) ds ≤ ||g||A× ψ

for all .t > 0. Consequently, .g ∈ Mψ and .||g||Mψ ≤ ||g||A× . The proof is complete. ψ u n Since .Aψ has the Fatou property, it follows from Theorem 5.1.10 that .A×× ψ = Aψ , with equality of norms. Therefore, the following result is an immediate consequence of Theorem 6.4.3. Corollary 6.4.4 The Köthe dual of .Mψ is equal to .Aψ , that is, .Mψ× = Aψ with equality of norms. Next, the spaces .Mψoc and .Mψb will be discussed. It should be observed that it follows from (6.5) that condition (ii) of Proposition 5.4.8 is equivalent to requiring that .limt↓0 t/ψ (t) = 0. Therefore, the following result follows immediately from the commutative specialization of Proposition 5.4.8.

6.4 Marcinkiewicz Spaces

409

Proposition 6.4.5 The following three statements are equivalent. (i) .limt↓0 t/ψ (t) = 0. (ii) .Mψoc = / {0}. oc (iii) .Mψ = Mψb . In the most relevant cases, the following description of .Mψoc is available. Proposition 6.4.6 If .ψ : [0, ∞) → [0, ∞) is a non-zero concave function with ψ (0) = 0, satisfying .limt↓0 t/ψ (t) = 0 and .ψ (∞) = ∞, then

.

oc .Mψ

=

Mψb

{ = f ∈ Mψ : lim

1 t→0,∞ ψ (t)

f

t

} μ (s; f ) ds = 0 .

(6.8)

0

Proof The equality .Mψoc = Mψb follows from Proposition 6.4.5. For sake of convenience, denote the right hand side of (6.8) by .Mψ0 . It will be shown first that 0 .M ψ is a closed fully symmetric subspace of .Mψ . For .f ∈ Mψ , define the function .o (f ) : (0, ∞) → [0, ∞) by setting o (t; f ) =

.

1 ψ (t)

f

t

μ (s; f ) ds,

t > 0.

(6.9)

0

Observing that .o (αf ) = |α| o (f ) and .o (f + g) ≤ o (f ) + o (g) (as .f + g ≺≺ μ (f ) + μ (g)) for all .f, g ∈ Mψ and .α ∈ C, it is clear that .Mψ0 is a linear subspace of .Mψ . Furthermore, if .f, g ∈ Mψ and .f ≺≺ g, then .o (f ) ≤ o (g) and so, .Mψ0 is a fully symmetric subspace of .Mψ . To prove that .Mψ0 is closed in .Mψ , it suffices to observe that if .(fn )∞ n=1 is a sequence in .Mψ and .f ∈ Mψ is such that .||f − fn ||Mψ → 0, then .o (t; fn ) → o (t; f ) uniformly on .(0, ∞) as .n → ∞. Since .μ (f ) − μ (g) ≺≺ f − g (see Theorem 3.9.14) it follows that |f t | f t f t | | | |μ (s; f ) − μ (s; g)| ds . μ (s; g) ds || ≤ | μ (s; f ) ds − 0 0 0 f t ≤ μ (s; μ (f ) − μ (g)) ds 0

f ≤

t

μ (s; f − g) ds

0

and hence, .

|o (t; f ) − o (t; g)| ≤ o (t; f − g) ≤ ||f − g||Mψ ,

for all .f, g ∈ Mψ . Consequently, .Mψ0 is a closed subspace of .Mψ .

t > 0,

410

6 Examples

It will be shown that .(L1 ∩ L∞ ) (0, ∞) ⊆ Mψ0 . If .f ∈ (L1 ∩ L∞ ) (0, ∞), then .

f

1 ψ (t)

t

μ (s; f ) ds ≤

0

t ||f ||∞ → 0, ψ (t)

t ↓0

(as, .limt↓0 t/ψ (t) = 0) and also .

1 ψ (t)

f

t

1 ψ (t)

μ (s; f ) ds ≤

0

f

∞

μ (s; f ) ds =

0

||f ||1 → 0, t → ∞ ψ (t)

(as .limt→∞ ψ (t) = ψ (∞) = ∞), which shows that .f ∈ Mψ0 . This shows that b 0 0 .(L1 ∩ L∞ ) (0, ∞) ⊆ M ψ and hence, as .Mψ is closed in .Mψ , it follows that .Mψ = Mψ

⊆ Mψ0 . (L1 ∩ L∞ ) (0, ∞) Since .Mψoc = Mψb (see Proposition 6.4.5), it remains to be shown that .Mψ0 ⊆ oc Mψ . To this end, first observe that .Mψ0 ⊆ S0 (0, ∞). Indeed, if .f ∈ Mψ0 then, using that the function .t |−→ t/ψ (t), .t > 0, is increasing and that .μ (s; f ) ≥ μ (t; f ) for all .0 < s ≤ t, for .t ≥ 1 it follows that .

1 t 1 μ (t; f ) ≤ μ (t; f ) ≤ ψ (t) ψ (t) ψ (1)

f

t

μ (s; f ) ds. 0

ft Since .ψ (t)−1 0 μ (s; f ) ds → 0 as .t → ∞, this implies that .μ (t; f ) → 0 as ∞ 0 .t → ∞, that is, .f ∈ S0 (0, ∞). Let .f ∈ M be given and suppose that .(fn ) n=1 is a ψ sequence in .Mψ such that .|f | ≥ fn ↓n 0 a.e. Since .f ∈ S0 (0, ∞) this implies that .μ (t; fn ) ↓n 0 for all .t > 0. Via the dominated convergence theorem, it follows that .

1 ψ (t)

f

t

μ (s; fn ) ds ↓n 0,

∀ t > 0.

(6.10)

0

Let .ε > 0 be given. Since .f ∈ Mψ0 , there exists .δ > 0 such that .

1 ψ (t)

f

t

μ (s; f ) ds ≤ ε,

0 < t ≤ δ,

(6.11)

t ≥ R.

(6.12)

0

and there exists .R > δ such that f t 1 . μ (s; f ) ds ≤ ε, ψ (t) 0

By (6.10), the continuous functions .o (fn ) : (0, ∞) → [0, ∞) (see (6.9)) satisfy in particular o (t; fn ) ↓n 0,

.

t ∈ [δ, R] ,

6.4 Marcinkiewicz Spaces

411

and hence, by Dini’s theorem, there exists .N ∈ N such that o (t; fn ) ≤ ε,

.

∀ t ∈ [δ, R] ,

n ≥ N.

(6.13)

It follows from (6.11), (6.12), and (6.13 ) that .

1 ψ (t)

f

t

μ (s; fn ) ds ≤ ε,

n ≥ N,

t > 0,

0

that is, .||fn ||Mψ ≤ ε for all .n ≥ N . This shows that .||fn ||Mψ → 0 as .n → ∞ and it may be concluded that .f ∈ Mψoc . The proof is complete. u n Remark 6.4.7 It should be observed that it is always the||case that||.ψ ' ∈ Mψ (with || ' || .||ψ || = 1 − ψ (0+) /ψ (∞)). If .ψ (0+) = 0, then .||χ[0,a) ψ ' ||M = 1 for all Mψ ψ || || ' / M oc . Similarly, if .ψ (∞) = ∞, then .||χ ' || .a > 0 and so, .ψ ∈ [a,∞) ψ Mψ = 1 ψ for all .a > 0 and so, also in this case .ψ ' ∈ / Mψoc . Consequently, if .Mψ has order continuous norm, then necessarily .ψ (0+) > 0 and .ψ (∞) < ∞, in which case .Mψ = L1 (0, ∞) (with equivalent norms). This shows, in particular, that .Mψ is never reflexive (see Theorem 5.3.10). It should also be noted that the space .Aψ is never reflexive. Recall that .Aψ is reflexive if and only if it has the Fatou property and both .Aψ and .A× ψ have order continuous norm (see Theorem 5.3.10). If .Aψ has order continuous norm, then .ψ (0+) = 0 and .ψ (∞) = ∞ (see Theorem 6.3.6). By the above observations, this implies that .A∗ψ = A× ψ = Mψ does not have order continuous norm. Hence, .Aψ is not reflexive. Remark 6.4.8 (cf. Remark 6.3.7). If .0 < a < ∞, then one may also consider Marcinkiewicz spaces .Mψ (0, a) on the interval .(0, a). Note that { } Mψ (0, a) = f χ(0,a) : f ∈ Mψ (0, ∞) .

.

All of the above results remain valid (such as Theorem 6.4.3 and Corollary 6.4.4). However, in this situation, the behaviour of the function .ψ at infinity does not play any role for the properties of the space .Mψ (0, a). For example (see Proposition 6.4.6), if .limt↓0 t/ψ (t) = 0, then {

oc .Mψ

(0, a) =

Mψb

1 (0, a) = f ∈ Mψ (0, a) : lim t↓0 ψ (t)

f

t

} μ (s; f ) ds = 0 .

0

Let .M be a semi-finite von Neumann algebra equipped with a semi-finite faithful normal trace .τ . Given a non-zero concave function .ψ : [0, ∞) →

412

6 Examples

[0, ∞), continuous on .(0, ∞), with .ψ (0) = 0, the corresponding noncommutative Marcinkiewicz space is defined by { } Mψ (τ ) = x ∈ S (τ ) : μ (x) ∈ Mψ (0, ∞) ,

.

||x||Mψ (τ ) = ||μ (x)||Mψ ,

x ∈ Mψ (τ ) .

The following theorem follows from the above results in combination with Theorems 6.1.2, 6.1.3, 6.1.6, 6.1.7, 6.1.8, Corollary 6.1.4 and Proposition 6.1.9. Theorem 6.4.9

( ) (i) The space . Mψ (τ ) , ||·||Mψ (τ ) is a fully symmetric space of .τ -measurable operators with the Fatou property. (ii) .Aψ (τ )× = A× ψ (τ ) = Mψ (τ ) (with equality of norms). (iii) .Mψ (τ )× = Mψ× (τ ) = Aψ (τ ) (with equality of norms). (iv) If .limt↓0 t/ψ (t) = 0 and .ψ (∞) = ∞, then { Mψ (τ )oc = Mψ (τ )b = x ∈ Mψ (τ ) : lim

1 t→0,∞ ψ (t)

.

f

t

} μ (x; f ) ds = 0 .

0

Remark 6.4.10 If .τ (1) < ∞, then only the properties of the space .Mψ (0, τ (1)) play a role for the properties of .Mψ (τ ). Reference: [97].

6.5 Orlicz Spaces Suppose that .ϕ : [0, ∞] → [0, ∞] is a function satisfying: (a) .ϕ is increasing and left-continuous on .(0, ∞]. (b) .ϕ (0) = 0. (c) .ϕ is not identically equal to 0 or .∞ on .(0, ∞]. Define .s∞ = s∞ (ϕ) = sup {s ≥ 0 : ϕ (s) < ∞}. It follows from (c) that .s∞ > 0. If .0 < s∞ < ∞, then .ϕ (s) < ∞ for all .0 ≤ s < s∞ and .ϕ (s) = ∞ for all .s > s∞ . The value .ϕ (s∞ ) may be finite or .∞. Note that the left-continuity of .ϕ at .∞ implies that .ϕ (∞) = lims↑∞ ϕ (s). Definition 6.5.1 The Young function .o : [0, ∞] → [0, ∞] corresponding to .ϕ is defined by setting f o (t) =

t

ϕ (s) ds,

.

0

t ∈ [0, ∞] .

6.5 Orlicz Spaces

413

The following facts are readily verified. Lemma 6.5.2 The Young function .o satisfies the following properties. (i) .o (0) = 0 and .o is increasing with .o (∞) = limt↑∞ o (t) = ∞ . (ii) .o is convex (that is, .o (αt1 + (1 − α) t2 ) ≤ αo (t1 ) + (1 − α) o (t2 ) for all .0 ≤ α ≤ 1 and .t1 ,.t2 ∈ [0, ∞], with the convention that .0 · ∞ = 0). (iii) .o is left-continuous on .(0, ∞] and continuous on .[0, s∞ ]. Note that .o (t) < ∞ for .0 ≤ t < s∞ and .o (t) = ∞ for .t > s∞ . A useful alternative representation of a Young function .o will be discussed ft next. Suppose that .o (t) = 0 ϕ (s) ds, .t ∈ [0, ∞], where the function .ϕ satisfies conditions (a), (b), and (c) above. Let .θϕ be the Borel measure on .[0, s∞ ) generated by .ϕ (that is, .θϕ ([a,fb)) = ϕ (b) − ϕ (a) for all .0 ≤ a ≤ b < s∞ ). Since .ϕ (s) = θϕ ([0, s)) = [0,s) dθϕ , it follows via Fubini’s theorem that, if .0 ≤ t < s∞ , then f f f f .o (t) = dθϕ (u) ds = dsdθϕ (u) (6.14) f

[0,t) [0,s)

f

[0,t) (u,t)

(t − u) dθϕ (u) =

= [0,t)

(t − u)+ dθϕ (u) .

[0,s∞ )

Since .o is left-continuous, it follows via monotone convergence that (6.14) is also valid for .t = s∞ . This proves the following result. Proposition 6.5.3 For any Young function .o : [0, ∞] → [0, ∞], there exists a Borel measure .θ on .[0, s∞ ) such that f .o (t) = (t − u)+ dθ (u) , 0 ≤ t ≤ s∞ [0,s∞ )

(and .o (t) = ∞ for .t > s∞ ). Consider the interval .(0, ∞) equipped with Lebesgue measure m and suppose that .o : [0, ∞] → [0, ∞] is a Young function. Let .L0 (m) denote the space of all .C-valued m-measurable functions on .(0, ∞) (with the usual identification of functions that are equal m-a.e.). A simple observation first. Lemma 6.5.4 If .f ∈ L0 (m) (or, .f : (0, ∞) → [−∞, ∞] is measurable), then μ (o (|f |)) = o (μ (f )).

.

Proof It suffices to consider the case that .f : (0, ∞) → Σ [0, ∞] is measurable. Suppose first that f is a simple function and write .f = nk=1 αk χAk , where .0 ≤ αk ∈ R (.k = 1, . . . , n) and .A1 , . . . , An are mutually disjoint measurable subsets of .(0, ∞) with .m (Ak ) < ∞ for all k. Without loss of generality, it may be assumed

414

6 Examples

Σ that .α1 ≥ · · · ≥ αn ≥ 0, in which case .μ (f ) = nk=1 αk χ[βk−1 ,βk ) , where .β0 = 0 Σ and .βk = kj =1 αj for .1 ≤ k ≤ n. Observing that o (f ) =

.

Σn k=1

o (μ (f )) =

and

o (αk ) χAk

Σn k=1

o (αk ) χ[βk−1 ,βk ) ,

it is clear that .o (f ) and .o (μ (f )) are equimeasurable and hence, .μ (o (f )) = o (μ (f )) (as .o (μ (f )) is decreasing and right-continuous). Suppose now that .f : (0, ∞) → [0, ∞] is an arbitrary measurable function. There exists an sequence .(fn )∞ n=1 of simple functions such that .0 ≤ fn ↑n f , which implies that .o (fn ) ↑n o (f ) and .μ (fn ) ↑n μ (f ). Consequently, .μ (o (fn )) ↑n μ (o (f )) and .o (μ (fn )) ↑n o (μ (f )). Since, by the first part of the proof, .μ (o (fn )) = o (μ (fn )) for all n, it follows that .μ (o (f )) = o (μ (f )). u n The following consequence of Proposition 6.5.3 will be used. Corollary 6.5.5 If .f, g ∈ L0 (m) satisfy .f ≺≺ g , then .o (|f |) ≺≺ o (|g|). Proof Without loss of generality, it may be assumed that .|g (x)| ≤ s∞ m-a.e. on (0, ∞). Since .f ≺≺ g, it follows that also .|f (x)| ≤ s∞ m-a.e. By Proposition 6.5.3, there exists a Borel measure .θ on .[0, s∞ ) such that f .o (t) = (t − u)+ dθ (u) , 0 ≤ t ≤ s∞ .

.

[0,s∞ )

Using Lemma 6.5.4, for all .t ≥ 0 it follows that f t f t . μ (s; o (|f |)) ds = o (μ (s; f )) ds 0

0

=

f tf f

0

(μ (s; f ) − u)+ dθ (u) ds [0,s∞ )

f

=

t

(μ (s; f ) − u)+ dsdθ (u) .

(6.15)

[0,s∞ ) 0

Let .0 ≤ u < s∞ momentarily be fixed and put .v = d (u; |f |), in which case .

{s ≥ 0 : μ (s; f ) > u} = [0, v) .

Hence, f t f + . (μ (s; f ) − u) ds = 0

f

{0≤s≤t:μ(s;f )>u} min(t,v)

= f

(μ (s; f ) − u) ds

(μ (s; f ) − u) ds

0 min(t,v)

≤ 0

f

t

(μ (s; g) − u) ds ≤ 0

(μ (s; g) − u)+ ds.

6.5 Orlicz Spaces

415

It now follows from (6.15) (applied also to the function g) that f f t f t . μ (s; o (|f |)) ds = (μ (s; f ) − u)+ dsdθ (u) 0

[0,s∞ ) 0

f

f

≤

t

(μ (s; g) − u)+ dsdθ (u)

[0,s∞ ) 0

f =

t

μ (s; o (|g|)) ds. 0

u n

The proof is complete.

Definition 6.5.6 For .f ∈ L0 (m) or .f : (0, ∞) → [−∞, ∞] measurable, define Mo (f ) ∈ [0, ∞] by setting f ∞ .Mo (f ) = o (|f |) dm.

.

0

The function .Mo : L0 (m) → [0, ∞] is called the modular corresponding to .o. Lemma 6.5.7 (i) The function .Mo : L0 (m) → [0, ∞] is convex, that is, Mo (αf1 + (1 − α) f2 ) ≤ αMo (f1 ) + (1 − α) Mo (f2 )

.

(6.16)

for all .0 ≤ α ≤ 1 and .f1 , f2 ∈ L0 (m) (with the convention that .0.∞ = 0). (ii) If .f ∈ L0 (m) and .k ≥ 1, then .Mo (kf ) ≥ kMo (f ). (iii) If .f ∈ L0 (m) and .Mo (kf ) = 0 holds for all .k > 0, then .f = 0. (iv) If .f ∈ L0 (m) and .Mo (kf ) ≤ 1 holds for all .k > 0, then .f = 0. Proof (i) This follows immediately from Lemma 6.5.2. (ii) Taking .f2 = 0 in (6.16), it follows that .Mo (αf ) ≤ αMo (f ) for all .f ∈ L0 (m) and .0 ≤ α ≤ 1. Given .k ≥ 1, taking .α = 1/k and replacing f by kf , this implies that .kMo (f ) ≤ Mo (kf ). (iii) If .f /= 0, then there exists .ε > 0 such that the set A = {x ∈ (0, ∞) : |f (x)| ≥ ε}

.

satisfies .m (A) > 0. Let .t0 > 0 be such that .o (t0 ) > 0 and let .k > 0 be such that .kε ≥ t0 . This implies that .|kf (x)| ≥ t0 and so, .o (|kf (x)|) ≥ o (t0 ), for all .x ∈ A. Hence, f f ∞ .Mo (kf ) = o (|kf |) dm ≥ o (|kf |) dm ≥ o (t0 ) m (A) > 0. 0

This suffices for the proof of (iii).

A

416

6 Examples

(iv) Let .f ∈ L0 (m) be such that .Mo (kf ) ≤ 1 for all .k > 0. It will be shown that .Mo (kf ) = 0 for all .k > 0. Suppose that there exists .k0 > 0 such that .Mo (k0 f ) > 0. It follows from (ii) that Mo (kf ) ≥

.

k Mo (k0 f ) k0

for all .k ≥ k0 . For k large enough, this implies that .Mo (kf ) > 1, which is a contradiction. Consequently, .Mo (kf ) = 0 for all .k > 0 and hence, (iii) implies that .f = 0. u n Some more properties of .Mo are collected together in the following lemma. Lemma 6.5.8 (i) If .f ∈ L0 (m) (or, .f : (0, ∞) → [−∞, ∞] is measurable) and .Mo (f ) < ∞, then .f ∈ S (m). (ii) If .f, fn : (0, ∞) → [0, ∞] (.n ∈ N) are measurable and .fn ↑n f m -a.e., then .Mo (fn ) ↑n Mo (f ). (iii) If .f, g ∈ L0 (m) and .f ≺≺ g, then .Mo (f ) ≤ Mo (g). Proof (i) If .s∞ < ∞, then .m ({x ∈ (0, ∞) : |f (x)| > s∞ }) = 0 and so, .|f (x)| ≤ s∞ m-a.e. Hence, .f ∈ L∞ (m) ⊆ S (m). Suppose now that .s∞ = ∞, in which case .o (t) < ∞ for all .0 ≤ t < ∞. Define .t0 = sup {t ≥ 0 : o (t) = 0}. For .s > t0 , it follows from .|f (x)| > s that .o (|f (x)|) > o (s) > 0 and so, f f ∞ .Mo (f ) = o (|f (x)|) dx ≥ o (|f (x)|) dx {x∈(0,∞):|f (x)|>s}

0

≥ o (s) m ({x ∈ (0, ∞) : |f (x)| > s}) , which implies that d (s; |f |) ≤

.

1 Mo (f ) . o (s)

Consequently, .d (s; |f |) < ∞ for all .s > t0 and .lims→∞ d (s; |f |) = 0. This shows that .f ∈ S (m). (ii) Since .o : [0, ∞] → [0, ∞] is increasing and left-continuous, .0 ≤ fn ↑ f m -a.e. implies that .o (fn ) ↑n o (f ) m-a.e. The monotone convergence theorem implies that .Mo (fn ) ↑n Mo (f ). (iii) This is an immediate consequence of Corollary 6.5.5. u n For .f ∈ L0 (m) define .

( { ) } ||f ||o = inf k > 0 : Mo k −1 f ≤ 1

(with the convention that .inf ∅ = ∞).

(6.17)

6.5 Orlicz Spaces

417

Remark 6.5.9 Suppose the ( )that .f ∈ L0 (m) is such that .||f ||o 0, is decreasing, it follows that .Mo k −1 f ≤ 1 for all .k > ||f ||o . Assume, in addition, that .||f ||o > 0 and let .(kn )∞ n=1 be a sequence −1 |f | ↑ ||f ||−1 |f | || such that .kn > ||f ||o and .kn ↓n ||f . This implies that . k n o ) ( o ) ( (n −1 ) |f | ≤ 1 and hence, .Mo kn−1 |f | ↑n Mo ||f ||−1 . Since .Mo kn |f | for all o ) ( n, it follows that .Mo ||f ||−1 o |f | ≤ 1 . Consequently, if .f ∈ L0 (ν) satisfies .0 < ||f ||o < ∞, then the infimum in (6.17) is actually a minimum. Lemma 6.5.10 The map .||·||o : L0 (m) → [0, ∞] is a function norm, that is, (i) (ii) (iii) (iv)

If .f ∈ L0 (m), then .||f ||o = 0 if and only if .f = 0. If .|f | ≤ |g| in .L0 (m), then .||f ||o ≤ ||g||o . .||λf ||o = |λ| ||f ||o for all .f ∈ L0 (m) and .λ ∈ C. .||f + g||o ≤ ||f ||o + ||g||o for all .f, g ∈ L0 (m).

Proof

( ) (i) If .f = 0, then .Mo k −1 f = 0 for all .k > 0 and hence, .||f ||o = 0. Now suppose that .f ∈ (L0 (m) ) is such that .||f ||o = 0. This implies (see Remark 6.5.9) that .Mo k −1 f ≤ 1 for all .k > 0 and so, it follows from Lemma 6.5.7 (iv) that .f = 0. ) ) ( ( (ii) If .|f | ≤ |g| in .L0 (m), then .Mo k −1 f ≤ Mo k −1 g for all .k > 0, which implies .||f ||o ≤ ||g||o . (iii) If .λ = 0, this is clear. If .λ /= 0, then ( { ) } −1 . ||λf ||o = inf k > 0 : Mo k λf ≤ 1 (( } { )−1 ) f ≤1 = inf k > 0 : Mo k |λ|−1 = inf {|λ| l : l > 0, Mo (lf ) ≤ 1} = |λ| ||f ||o . (iv) It may be assumed that .0 < ||f ||o , ||g||o < ∞. Let .γ = ||f ||o + ||g||o and write .||f ||o = αγ and .||g||o = βγ with .α + β = 1. Using that .Mo is convex (see Lemma 6.5.7 (i)) and using Remark 6.5.9, it follows that ( ( ) ) −1 .Mo γ (f + g) = Mo α (αγ )−1 f + β (βγ )−1 g ( ( ) ) ≤ αMo (αγ )−1 f + βMo (βγ )−1 g ( ( ) ) −1 = αMo ||f ||−1 o f + βMo ||g||o g ≤ α + β = 1. Consequently, .||f + g||o ≤ γ = ||f ||o + ||g||o . The proof is complete. The Orlicz space Lo = Lo (m) = Lo (0, ∞)

.

u n

418

6 Examples

corresponding to the Young function .o is now defined by { } Lo = f ∈ L0 (m) : ||f ||o < ∞ .

.

From Lemma 6.5.10 it is clear that .Lo is an order ideal in .L0 (m) and that .||·||o is a Riesz norm on .Lo . Remark 6.5.11 (a) If .f ∈ L0 (m), then .f ∈ Lo if and only( if there ) exists .λ > 0 such that −1 f ≤ 1 for all .k > ||f || (see .Mo (λf ) < ∞. Indeed, if .f ∈ Lo , then .Mo k o Remark 6.5.9). Suppose now that .f ∈ L0 (m) is such that .Mo (λf ) < ∞ for some .λ > 0. If .0 ≤ α ≤ 1, then .Mo (αλf ) ≤ αMo (λf ) (as .Mo is convex). −1 Therefore, by taking ( −1.0 )< α ≤ 1 such that .αMo (λf ) ≤ 1 and .k = (αλ) , it follows that .Mo k f ≤ 1 and hence, .f ∈ Lo . (b) It follows from Lemma 6.5.8 (i) that .Lo ⊆ S (m). Indeed, if .f ∈ Lo , then there exists .λ > 0 such that .Mo (λf ) < ∞ and so, .λf ∈ S (m). This implies that .f ∈ S (m). (c) If .A ⊆ (0, ∞) is such that .m (A) < ∞, then .χA ∈ Lo . Indeed, let .λ > 0 be such that .o (λ) < ∞. Since .o (λχA ) = o (λ) χA , it follows that .Mo (λχA ) = o (λ) m (A) < ∞. Hence (see (a)), .χA ∈ Lo . This shows, in particular, that the carrier of .Lo is equal to .(0, ∞). Theorem 6.5.12 The space .(Lo , ||·||o ) is a fully symmetric space with the Fatou property. Proof In view of Lemma 6.5.10 it only needs to be proved that .Lo is fully symmetrically normed and has the Fatou property. If .f ∈ S (m) and .g ∈ Lo such that .f ≺≺ g , then it follows immediately from Lemma 6.5.8 (iii) that .f ∈ Lo and .||f ||o ≤ ||g||o . Hence, .Lo is fully symmetrically normed. To show that .Lo has the Fatou property, suppose that .{fn }∞ n=1 is a sequence in .Lo such that .0 ≤ fn ↑n and .supn ||fn ||o < ∞. Define the measurable function .f : such)that (0, ∞) → [0, ∞] by setting .f (x) = (supn f (x), ) .x ∈ (0, ∞). Given .α ∈(R−1 −1 f ≤ 1 for all n. Since .Mo α fn ↑n .supn ||fn ||o < α, it follows that .Mo α n ( ( ) ) Mo α −1 f (see Lemma 6.5.8 (ii)), this implies that .Mo α −1 f ≤ 1. Therefore, .f ∈ Lo and .||f ||o ≤ α. This holds for all .α > supn ||fn ||o , so it follows that .||f ||o ≤ supn ||fn ||o . The reverse inequality being clear, it may be concluded that .||f ||o = supn ||fn ||o . This shows that .Lo has the .σ -Fatou property and hence, the Fatou property (which implies, in particular, that .Lo is a Banach space, due to Corollary 4.1.22). u n The norm .||·||o is called the Luxemburg norm in the Orlicz space .Lo . Remark 6.5.13 Since the carrier of .Lo is equal to .(0, ∞) (see Remark 6.5.11 (c)), it follows that L 1 ∩ L ∞ ⊆ Lo ⊆ L1 + L∞ .

.

6.5 Orlicz Spaces

419

Lemma 6.5.14 Suppose that .f ∈ Lo . (i) If .||f ||o ≤ 1, then .Mo (f ) ≤ ||f ||o ≤ 1. (ii) If .||f ||o > 1, then .Mo (f ) ≥ ||f ||o > 1. In particular, .||f ||o ≤ 1 if and only if .Mo (f ) ≤ 1. Proof (i) If .||f ||o ≤ 1 and .f /= 0, then it follows from Lemma 6.5.7 (ii) and Remark 6.5.9 that .

( ) ||f ||−1 o Mo (f ) ≤ Mo f/ ||f ||o ≤ 1

and so, .Mo (f ) ≤ ||f ||o . ( ) (ii) Suppose that .||f ||o > 1. If .1 < k < ||f ||o , then .Mo k −1 f > 1 and so, ( ) k −1 Mo (f ) ≥ Mo k −1 f > 1,

.

which implies that .Mo (f ) > k. This holds for all .1 < k < ||f ||o , hence Mo (f ) ≥ ||f ||o . u n

.

It follows from the above lemma that the closed unit ball .BLo of .Lo is also given by BLo = {f ∈ L0 (m) : Mo (f ) ≤ 1} .

.

(6.18)

Let .ϕ : [0, ∞] → [0, ∞] be a function satisfying conditions (a), (b), and (c) at the beginning of this section. Definition 6.5.15 The left-continuous inverse .ψ : [0, ∞] → [0, ∞] of .ϕ is defined by setting ψ (v) = inf {u ≥ 0 : ϕ (u) ≥ v} ,

.

v ∈ [0, ∞] .

It should be observed that .ψ (v) < u implies that .ϕ (u) ≥ v. The next lemma, the simple proof of which is left to the reader, shows that .ψ satisfies the same conditions as the function .ϕ. Lemma 6.5.16 (a) .ψ is increasing and left-continuous on .(0, ∞]. (b) .ψ (0) = 0. (c) .ψ is not identically equal to 0 or .∞ on .(0, ∞]. The following observations are sometimes useful. The simple proofs are omitted.

420

6 Examples

Lemma 6.5.17 If .v > 0, then .ψ (v) = ∞ if and only if .ϕ (u) < v for all .0 ≤ u < ∞. Corollary 6.5.18 (i) .s∞ (ψ) = ϕ (∞). (ii) .s∞ (ϕ) = ψ (∞). In particular, if .s∞ (ϕ) < ∞, then .s∞ (ψ) = ∞, and if .s∞ (ψ) < ∞, then .s∞ (ϕ) = ∞. Note that it follows, in particular, from the above corollary that the situation where .s∞ (ϕ) < ∞ and .s∞ (ψ) < ∞ cannot occur. The following lemma shows that the left-continuous inverse of .ψ is equal to .ϕ. Lemma 6.5.19 Let .ϕ : [0, ∞] → [0, ∞] be a function satisfying conditions (a), (b), and (c) at the beginning of this section and let .ψ be the left-continuous inverse of .ϕ. Then ϕ (u) = inf {v ≥ 0 : ψ (v) ≥ u} ,

.

u ∈ [0, ∞] .

Proof For .u = 0 this is clear, so it may be assumed that .u ( > ) 0. Suppose that v (≥ )0 is such that .ψ (v) ≥ u. If .0 ≤ u' < u , then .ϕ u' < v (indeed, if ' ≥ v, then it would follow from the definition of .ψ (v) that .u ≤ ψ (v) ≤ u' , .ϕ u which is a contradiction). Since .ϕ is left-continuous, it now follows that .ϕ (u) ≤ v. Consequently, .

ϕ (u) ≤ inf {v ≥ 0 : ψ (v) ≥ u} .

.

For the proof of the reverse inequality, it may be assumed that .ϕ (u) < ∞. Take any .α such that .ϕ (u) < α. This implies that .ψ (α) ≥ u (indeed, if .ψ (α) < u, then .ϕ (u) ≥ α, which is a contradiction). Therefore, .

inf {v ≥ 0 : ψ (v) ≥ u} ≤ α.

Since this holds for all .α > ϕ (u), it follows that .

The proof is complete.

inf {v ≥ 0 : ψ (v) ≥ u} ≤ ϕ (u) . u n

Let the functions .ϕ and .ψ be as above.fAs before, .o denotes the Young function t corresponding to .ϕ (that is, .o (t) = 0 ϕ (u) du, .t ≥ 0) and let .ψ be the ft Young function corresponding to .ψ (that is, .ψ (t) = 0 ψ (v) dv, .t ≥ 0). In this situation, .ψ is called the Young function complementary to .o. Since .ϕ is the leftcontinuous inverse of .ψ (see Lemma 6.5.19), it follows that .o is the Young function complementary to .ψ.

6.5 Orlicz Spaces

421

The proof of the following important classical inequality is omitted. Proposition 6.5.20 (Young’s Inequality) If .o and .ψ are complementary Young functions, then st ≤ o (s) + ψ (t) ,

.

s, t ≥ 0.

(6.19)

Moreover, equality holds in (6.19) if and only if .s = ψ (t) or .t = ϕ (s). Corollary 6.5.21 If .o and .ψ are complementary Young functions, then f .

∞

|f g| dm ≤ Mo (f ) + Mψ (g) ,

f, g ∈ L0 (m) .

0

As before, consider the interval .(0, ∞) equipped with )Lebesgue measure m and ( × let .o be a Young function. The Köthe dual . Lo , ||·||L× of the Orlicz space .Lo o (equipped with the Luxemburg norm .||·||o ) is itself a fully symmetric space with the × Fatou property (and the carrier of .L× o is equal to .(0, ∞)). The norm in .Lo is given by {f ||g||L× = sup

.

o

0

∞

} |f g| dm : ||f ||o ≤ 1 ,

It follows from Lemma 6.5.14 that also {f ∞ } |f . ||g|| × = sup g| dm : M ≤ 1 , (f ) o L o

0

g ∈ L× o.

g ∈ L× o.

(6.20)

It will be shown next that the space .L× o coincides with the Orlicz space .Lψ and that the norms .||·||ψ and .||·||L× are equivalent. For this purpose, it will be convenient to o prove some lemmas first. As before, .o and .ψ are complementary Young functions (corresponding to the increasing functions .ϕ and .ψ, respectively). ≤ 2 ||g||ψ for all .g ∈ Lψ . Lemma 6.5.22 .Lψ ⊆ L× o and .||g||L× o ( ) Proof If .0 /= g ∈ Lψ , then .Mψ g/ ||g||ψ ≤ 1 (see Remark 6.5.9). If .f ∈ Lo with .||f ||o ≤ 1, then .Mo (f ) ≤ 1 (see Lemma 6.5.14) and so, it follows from Corollary 6.5.21 that .

||g||−1 ψ

f 0

∞

( ) |f g| dm ≤ Mo (f ) + Mψ g/ ||g||ψ ≤ 2.

422

6 Examples

Therefore, f

∞

.

0

|f g| dm ≤ 2 ||g||ψ ,

||f ||o ≤ 1,

which shows that .g ∈ L× o and .||g||L× ≤ 2 ||g||ψ .

u n

o

Lemma 6.5.23 If .g ∈ L× o , then f ∞ |fg| dm ≤ ||g||L× max (Mo (f ) , 1) , .

f ∈ L0 (m) .

o

0

(6.21)

Proof If .Mo (f ) = ∞, then (6.21) is clear and so, it may be assumed that Mo (f ) < ∞. If .Mo (f ) ≤ 1, then .||f ||o ≤ 1 and hence, f ∞ |f g| dm ≤ ||g||L× .

.

o

0

( ) (by the definition of .||g||L× ). If .α = Mo (f ) > 1, then .M α −1 f ≤ α −1 M (f ) = 1 o || || and so, .||α −1 f || ≤ 1. This implies that o

f

∞

.

0

α −1 |f g| dm ≤ ||g||L×

o

f∞ and hence, . 0 |fg| dm ≤ ||g||L× Mo (f ). The proof is complete.

u n

o

Lemma 6.5.24 If .s∞ (ψ) < ∞, then .ϕ (∞) = limu→∞ ϕ (u) = s∞ (ψ) = l < ∞ and .|g| / ||g||L× ≤ l m -a.e. on .(0, ∞) for all .0 /= g ∈ L× o. o

Proof The first statement follows immediately from Corollary 6.5.18. Note that × .o (u) ≤ lu for all .u ≥ 0. Let .0 /= g ∈ L o be given and assume that the } { set .A = x ∈ (0, ∞) : |g (x)| > l ||g||L× satisfies .m (A) > 0. Defining .f = o

(lm (A))−1 χA , it follows that f ∞ ) ( .Mo (f ) = o (f ) dm = o (lm (A))−1 m (A) 0

≤ l (lm (A))−1 m (A) = 1 and so, .||f ||o ≤ 1. On the other hand, f ∞ f |f g| dm = (lm (A))−1 |g| dm . 0

A

> (lm (A)) This is a contradiction.

−1

m (A) l ||g||L× = ||g||L× . o

o

u n

6.5 Orlicz Spaces

423

Theorem 6.5.25 The Köthe dual .L× o of the Orlicz space .Lo (equipped with the Luxemburg norm .||·||o ) is equal to .Lψ (as a set) and .

||g||ψ ≤ ||g||L× ≤ 2 ||g||ψ , o

g ∈ Lψ .

Proof It has already been observed in Lemma 6.5.22 that .Lψ ⊆ L× o and that .||g|| × ≤ 2 ||g||ψ holds for all .g ∈ Lψ . Lo The claim is that ||g||ψ ≤ ||g||L× ,

.

o

g ∈ F (m) .

(6.22)

Note that .F (m) is contained in .L× o as well as in .Lψ (cf. Remark 6.5.11 (c)). Let .0 /= g ∈ F (m) be given and put .k = ||g|| × . For a proof, two cases will be Lo distinguished. ) ( (A) Suppose that .s∞ (ψ) = ∞. Define .f = ψ k −1 |g| and note that .f ∈ F (m) (as .ψ (v) < ∞ for all .v ≥ 0). In this situation, there is equality in Young’s inequality, which implies that .

( ) ( ) |f | k −1 |g| = o (|f |) + ψ k −1 |g|

and, therefore, k −1

f

∞

.

) ( |f g| dm = Mo (f ) + Mψ k −1 |g| .

0

f∞

Since . 0 |fg| dm < ∞, this implies, in particular, that .Mo (f ) < ∞ and ) ( Mψ k −1 |g| < ∞. It follows from Lemma 6.5.23 that

.

k −1

f

.

∞

|f g| dm ≤ max (Mo (f ) , 1)

0

and hence, ) ( Mo (f ) + Mψ k −1 |g| ≤ max (Mo (f ) , 1) .

.

(6.23)

) ( If .Mo (f ) ≥ 1, then (6.23) implies that .Mψ k −1 |g| = 0. If .Mo (f ) < 1, ( −1 ) then it follows from (6.23) that .Mψ k |g| ≤ 1. Consequently, it follows ) ( from (6.23) that .Mψ k −1 |g| ≤ 1, which implies that .||g||ψ ≤ k = ||g||L× . o (B) Suppose now that .s∞ (ψ) < ∞, in which case .limu→∞ ϕ (u) = s∞ (ψ) = l < ∞. It follows from Lemma 6.5.24 that .k −1 |g| ≤ l m-a.e. on .(0, ∞). Since it is possible that .ψ (l) = ∞, some care has to be taken. Let .0 < δ < 1 be fixed for

424

6 Examples

) ( the moment and define .f ∈ F (m) by setting .f = ψ δk −1 |g| . By equality in Young’s inequality, it follows that δk

.

−1

f

∞

) ( |f g| dm = Mo (f ) + Mψ δk −1 |g| .

0

f∞

) ( Since . 0 |fg| dm < ∞, it follows that .Mo (f ) and .Mψ δk −1 |g| are both finite. It follows from Lemma 6.5.23 that f ∞ −1 |f g| dm ≤ δ max (Mo (f ) , 1) .δk 0

and hence, ) ( Mo (f ) + Mψ δk −1 |g| ≤ δ max (Mo (f ) , 1) .

.

This implies, in particular, that Mo (f ) ≤ δ max (Mo (f ) , 1) < max (Mo (f ) , 1)

.

and so, .max (Mo (f ) , 1) = 1. Consequently, ) ( Mψ δk −1 |g| ≤ δ < 1.

.

∞ Taking a sequence .(δn ) n=1 such that .0 < δn < ∞ and .δn ↑n 1, it follows that ) ( −1 |g| ≤ 1 (see Lemma 6.5.8 (ii)). Consequently, .||g||ψ ≤ k = ||g||L× . .Mψ k o

This proves (6.22) in both cases (A) and (B). ∞ Now let .g ∈ L× o (m) be given. There exists a sequence .{gn }n=1 in .F (m) such that .0 ≤ gn ↑n |g|. It follows from (6.22) that .||gn ||ψ ≤ ||gn ||L× ≤ ||g||L× for o o all n. Since .Lψ has the Fatou property, this implies that .|g| ∈ Lψ and .||g||ψ = supn ||gn ||ψ ≤ ||g||L× . The proof is complete. u n o

As follows from Theorem 6.5.25 above, on the Orlicz space .Lψ there are two equivalent norms, the Luxemburg norm .||·||ψ and the norm .||·||L× . The norm .||·||L× o o is called the Orlicz norm on .Lψ and will be denoted by .||·||'ψ . As observed before (see (6.20)), .||·||'ψ is given by ' . ||g||ψ

{f

∞

= sup 0

} |f g| dm : Mo (f ) ≤ 1 ,

g ∈ Lψ .

6.5 Orlicz Spaces

425

Similarly, by interchanging the roles of .o and .ψ, on the Orlicz space .Lo there exist two equivalent norms, the Luxemburg norm .||·||o and the Orlicz norm .||·||'o given by .

||f ||'o = sup

{f

∞

} |f g| dm : Mψ (g) ≤ 1 ,

f ∈ Lo ,

(6.24)

0

and these two norms are equivalent (actually, .||f ||o ≤ ||f ||'o ≤ 2 ||f ||o , .g ∈ Lo ). These norms are related via Köthe duality as follows. ( ) Corollary 6.5.26 The Köthe dual .(Lo , ||·||o )× is equal to . Lψ , ||·||'ψ and the ( )× Köthe dual . Lo , ||·||'o is equal to .(Lψ , ||·||ψ ) (and similarly with .o and .ψ interchanged). Proof The first statement is the result of Theorem 6.5.25. It follows also from Theorem 6.5.25, with .o and .ψ interchanged, that .

( ) Lo , ||·||'o = (Lψ , ||·||ψ )×

and so, .

( )× Lo , ||·||'o = (Lψ , ||·||ψ )×× .

Since the space .(Lψ , ||·||ψ ) has the Fatou property, it follows that .(Lψ , ||·||ψ )×× = u n (Lψ , ||·||ψ ). The proof is complete. Let .o : [0, ∞] → [0, ∞] be a Young function and consider the set Yo = Yo (m) = Yo (0, ∞) = {f ∈ L0 (m) : Mo (f ) < ∞} .

.

Since .Mo : L0 (m) → [0, ∞] is a convex function (see Lemma 6.5.7 (i)), it is clear that .Yo is a convex subset of .L0 (m) (and actually, a subset of .Lo ; see Remark 6.5.11 (a)). However, .Yo is, in general, not a linear subspace of .L0 (m) (.f ∈ Yo does not imply .2f ∈ Yo , in general). Lemma 6.5.27 If .f ∈ Yo implies that .2f ∈ Yo , then .Yo is a linear subspace of L0 (m) and, in this case, .Yo = Lo .

.

Proof For .f ∈ Yo it follows that .2n f ∈ Yo for all .n ∈ N and hence, .λf ∈ Yo for all .λ ∈ C (as .Mo is absolute and monotone). If .f, g ∈ Yo , then . 12 f + 12 g ∈ Yo since .Yo is convex and hence, .f + g ∈ Yo . As observed before, .Yo ⊆ Lo (see Remark 6.5.11 (a)). If .f ∈ Lo , then there u n exists .λ > 0 such that .Mo (λf ) < ∞, that is, .λf ∈ Yo . Hence, .f ∈ Yo . Definition 6.5.28 The Young function .o is said to satisfy a .A2 -condition if there exists a constant .C > 0 such that .o (2u) ≤ Co (u) for all .u ≥ 0. Furthermore, .o

426

6 Examples

is said to satisfy a .A2 -condition at .∞ if there exists a .u0 > 0 and a constant .C > 0 such that .o (2u) ≤ Co (u) < ∞ for all .u ≥ u0 . It is clear that the .A2 -condition implies the .A2 -condition at .∞. If .0 < a < ∞ and m is Lebesgue measure on .(0, a), then the Orlicz space .Lo (m) = Lo (0, a) on .(0, a) may be defined as before, and all of the above results remain valid in this situation. Observe that { } Lo (0, a) = f χ(0,a) : f ∈ Lo (0, ∞) .

.

Lemma 6.5.29 (i) If .o satisfies a .A2 -condition, then .Yo = Lo . (ii) If .a < ∞ and .o satisfies a .A2 -condition at .∞, then .Yo (0, a) = Lo (0, a). Proof (i) If .f ∈ Yo , then f

∞

Mo (2f ) =

.

0

f

≤C

o (2 |f (x)|) dx ∞

o (|f (x)|) dx = CMo (f ) < ∞

0

and hence, .2f ∈ Yo . The result now follows from Lemma 6.5.27. (ii) Let .u0 > 0 be such that .o (2u) ≤ Co (u) < ∞ for all .u ≥ u0 . Given .f ∈ Yo (0, a), define the sets X1 = {x ∈ (0, a) : |f (x)| < u0 } ,

.

X2 = {x ∈ (0, a) : |f (x)| ≥ u0 } .

It follows that f

f o (2 |f (x)|) dx +

Mo (2f ) =

.

X1

f

≤ o (2u0 ) m (X1 ) + C

o (2 |f (x)|) dx X2

o (|f (x)|) dx X2

≤ o (2u0 ) m (X1 ) + CMo (f ) < ∞ and so, .2f ∈ Yo (0, a). Again, the result now follows from Lemma 6.5.27.

u n

Proposition 6.5.30 (i) If .o satisfies a .A2 -condition, then .Lo has order continuous norm. (ii) If .a < ∞ and .o satisfies a .A2 -condition at .∞, then .Lo (0, a) has order continuous norm.

6.5 Orlicz Spaces

427

Proof Both conditions imply that .Lo = Yo (on .(0, ∞) and .(0, a), respectively). Suppose that .{fn }∞ n=1 is a sequence in .Lo such that .fn ↓ 0 and .||fn ||o ≥ δ > 0 for all n. Replacing .fn by .(2/δ) fn , it may be assumed that .||fn ||o > 1 for all n. Since .o (fn ) ↓ 0 and .Mo (f1 ) < ∞, it follows from the dominated convergence theorem that .o (fn ) ↓n 0. On the other hand, it follows from Lemma 6.5.14 (ii) that .Mo (fn ) ≥ ||fn ||o > 1 for all n. This is a contradiction. It may be concluded that .Lo has order continuous norm. u n Corollary 6.5.31 Let m be Lebesgue measure on .(0, ∞) (respectively, on .(0, a) for some .0 < a < ∞) and let .o and .ψ be complementary Young functions. If both .o and .ψ satisfy a .A2 -condition (respectively, a .A2 -condition at .∞), then .Lo is reflexive. Proof In both cases, Proposition 6.5.30 implies that the Luxemburg norm on both Lo and .Lψ is order continuous. Since the Luxemburg norm and the Orlicz norm are ' equivalent on .Lψ , it follows that ) norm .||·||ψ on .Lψ is order continuous ( the Orlicz × ' as well. Since .(Lo , ||·||o ) = Lψ , ||·||ψ (see Corollary 6.5.26) and .Lo has the Fatou property, it now follows from Theorem 5.3.10 that .Lo is reflexive. u n

.

Remark 6.5.32 In the proposition above the space .Lo is considered with the Luxemburg norm .||·||o . However, the same result holds for the Orlicz norm .||·||'o , as these two norms are equivalent. Let .o : [0, ∞] → [0, ∞] be a Young function and define s0 = sup {s ≥ 0 : o (s) = 0} ,

.

s∞ = sup {s ≥ 0 : o (s) < ∞} . Evidently, .0 ≤ s0 ≤ s∞ ≤ ∞, .s0 < ∞ and .s∞ > 0. The function .o : [s0 , s∞ ] → [0, o (s∞ )] is continuous and strictly increasing with .o (s0 ) = 0. Therefore, .o has a continuous and strictly increasing inverse .o−1 : [0, o (s∞ )] → [s0 , s∞ ]. If −1 (t) = s . It is now easy to verify .o (s∞ ) < ∞ and .o (s∞ ) < t ≤ ∞, then set .o ∞ −1 that the function .o : [0, ∞] → [s0 , s∞ ] is continuous and increasing, and that −1 is also given .o (t) > 0 for all .t > 0. Furthermore, it is not difficult to show that .o by o−1 (t) = sup {s ≥ 0 : o (s) ≤ t} ,

.

0 ≤ t < ∞.

(6.25)

The purpose of introducing the function .o−1 is to be able to formulate the following result. Lemma 6.5.33 If .A ⊆ (0, ∞) is measurable and .0 < m (A) < ∞, then .χA ∈ Lo and .

||χA ||o =

1 ( ). o−1 m (A)−1

428

6 Examples

Proof That the assumption that .m (A) < ∞ implies that .χA ∈ Lo has already been observed in Remark 6.5.11 (c). The definition of .||χA ||o implies that .

( ) } { ||χA ||o = inf k > 0 : Mo k −1 χA ≤ 1 ( ) } { = inf k > 0 : Mo k −1 m (A) ≤ 1 ( ) } { = inf k > 0 : Mo k −1 ≤ m (A)−1 ( ) }]−1 [ ( )]−1 [ { = o−1 m (A)−1 , = sup k −1 > 0 : Mo k −1 ≤ m (A)−1

where for the last inequality identity (6.25) is used.

u n

Lemma 6.5.34 The following two statements are equivalent. (i) .o (s) < ∞ for all .0 ≤ s < ∞ (that is, .s∞ = ∞). (ii) For every .ε > 0 there exists .δ > 0 such that .||χA ||o ≤ ε whenever .A ⊆ (0, ∞) with .m (A) ≤ δ. Proof (i).⇒(ii). Since .

lim o−1 (t) = s∞ = ∞,

t→∞

[ ]−1 it follows that .limt→∞ o−1 (t) = 0 . Therefore, given .ε > 0, there exists [ −1 ]−1 .R > 0 such that . o ≤ ε for all .t ≥ R. Define .δ = R −1 . If .A ⊆ (0, ∞) (t) is measurable such that .0 < m (A) ≤ δ, then .m (A)−1 ≥ δ −1 = R and hence, [ −1 ( )]−1 m (A)−1 . o ≤ ε, that is (by Lemma 6.5.33), .||χA ||o ≤ ε. (ii).⇒(i). It will be shown that .limt→∞ o−1 (t) = ∞ (that is, .s∞ = ∞). Given .C > 0, let .ε = 1/C and let .δ > 0 be such that .||χA ||o ≤ ε whenever .A ⊆ (0, ∞) with .m (A) ≤ δ. Let .A ⊆ (0, ∞) be measurable such that .0 < m (A) ≤ δ. By Lemma 6.5.33, this implies that .

o−1

(

1 m (A)−1

) = ||χA ||o ≤ ε,

( ) that is, .o−1 m (A)−1 ≥ 1/ε = C. Since .o−1 is increasing, it follows that −1 (t) ≥ C for all .t ≥ m (A)−1 . This shows that .lim −1 (t) = ∞ and .o t→∞ o the proof is complete. u n Proposition 6.5.35 Consider the Orlicz space .Lo equipped with the Luxemburg norm .||·||o . The following statements hold. b (i) If .o (s) < ∞ for all .0 ≤ s < ∞ (that is, .s∞ = ∞), then .Loc o = Lo . oc (ii) If .s∞ < ∞, then .Lo = {0}.

6.5 Orlicz Spaces

429

Proof (i) If .s∞ = ∞, then it follows from Lemma 6.5.34 that condition (ii) of this lemma b is satisfied. It follows now from Proposition 5.4.8 that .Loc o = Lo . (ii) If .s∞ < ∞, then it follows from Lemma 6.5.34 that condition (ii) of this lemma does not hold. Proposition 5.4.8 implies that .Loc o = {0}. u n Suppose that .M is a semi-finite von Neumann algebra equipped with a semifinite normal faithful trace .τ . Let .o and .ψ be a pair of complementary Young functions. The corresponding noncommutative Orlicz space .Lo (τ ) is now defined by setting Lo (τ ) = {x ∈ S (τ ) : μ (x) ∈ Lo } ,

.

||x||o = ||μ (x)||o ,

x ∈ Lo (τ ) .

Several properties of .(Lo (τ ) , ||·||o ) are collected together in the next theorem. As before, this theorem follows immediately from the above results combined with the results obtained in Sect. 6.1. Theorem 6.5.36 (i) The space .(Lo (τ ) , ||·||o ) is a fully symmetric space of .τ -measurable operators with the Fatou property. (ii) Setting .||x||'o = ||μ (x)||'o , .x ∈ Lo (τ ), it follows that .||·||o and .||·||'o are two equivalent norms on .Lo (τ ) (also referred to as the Luxemburg norm and the Orlicz norm on .Lo (τ ) ( )× ( ), respectively). (iii) .(Lo (τ ) , ||·||o )× = Lψ (τ ) , ||·||'ψ and . Lo (τ ) , ||·||'o = (Lψ (τ ) , ||·||ψ ). (iv) If .o satisfies a .A2 -condition (or, if .τ (1) < ∞, satisfies a .A2 -condition at .∞), then .Lo (τ ) has order continuous norm (and hence, .Lo (τ ) is a KB-space). (v) If both .o and .ψ satisfy a .A2 -condition (or, if .τ (1) < ∞, satisfy a .A2 condition at .∞), then .Lo (τ ) is reflexive. (vi) If .o (s) < ∞ for all .0 ≤ s < ∞, then .Lo (τ )oc = Lo (τ )b = Loc o (τ ) = b Lo (τ ). Remark 6.5.37 Let .o and .ψ be complementary Young functions. Suppose that o (s) < ∞ for all .0 ≤ s < ∞ (but, no condition on .ψ). If .x ∈ S (τ ), then, using the definition of the extended trace .τ (see Sect. 3.3) and Proposition 3.2.8, it follows that f ∞ f ∞ .Mo (μ (x)) = o (μ (t; x)) dt = μ (t; o (|x|)) dt = τ (o (|x|)) .

.

0

0

(6.26) Consequently, in this case, the space .Lo (τ ) is also given by Lo (τ ) = {x ∈ S (τ ) : ∃ λ > 0 such that τ (o (λ |x|)) < ∞}

.

430

6 Examples

and the Luxemburg norm is given by .

{ ( ( )) } ||x||o = inf k > 0 : τ o k −1 |x| ≤ 1 .

Furthermore, it follows from (6.26) in combination with (6.18) that the closed unit ball .BLo (τ ) in .Lo (τ ) is also given by BLo (τ ) = {x ∈ S (τ ) : τ (o (|x|)) ≤ 1} .

.

(6.27)

Assuming that .ψ (s) < ∞ for all .0 ≤ s < ∞ (but, no assumption on .o), it follows from (6.27), applied to .ψ, that the Orlicz norm in .Lo (τ ) is also given by .

||x||'o = sup {τ (|xy|) : y ∈ S (τ ) , τ (ψ (|y|)) ≤ 1} ,

analogous to formula (6.24) in the commutative situation. Reference: [97].

6.6 p-Convexification Let .0 < a ≤ ∞ and consider the interval .(0, a) equipped with Lebesgue measure m. Suppose that .E ⊆ L0 (m) is a normed function space (that is, E is a normed (p) ⊆ L (m) is defined by setting .L∞ (m)-bimodule). For .1 ≤ p < ∞, the set .E 0 { } E (p) = f ∈ L0 (m) : |f |p ∈ E

.

and set .

||1/p || ||f ||E (p) = |||f |p ||E ,

f ∈ E (p) .

( ) It will be shown first that . E (p) , ||·||E (p) is a normed function space. Lemma 6.6.1 .E (p) is an order ideal (that is, an absolutely solid linear subspace) in .L0 (m). Proof It is clear that .αf ∈ E (p) whenever .f ∈ E (p) and .α ∈ C. If .f, g ∈ E (p) , then it follows from the pointwise estimate .

( ) ( ) |f + g|p ≤ (|f | + |g|)p ≤ 2p |f |p ∨ |g|p ≤ 2p |f |p + |g|p

and from the fact that E is an order ideal in .L0 (m), that .f + g ∈ E (p) . Hence, (p) is a linear subspace of .L (m). Furthermore, it is clear that .f ∈ E (p) whenever .E 0 (p) . .f ∈ L0 (m) satisfies .|f | ≤ |g| for some .g ∈ E u n Lemma 6.6.2 Suppose that .1 < p, q, r < ∞ satisfy .1/p +1/q = 1/r. If .f ∈ E (p) and .g ∈ E (q) , then .f g ∈ E (r) and .||f g||E (r) ≤ ||f ||E (p) ||g||E (q) .

6.6 p-Convexification

431

Proof Consider first the case that .r = 1. For a proof, it may be assumed that ||f ||E (p) = ||g||E (q) = 1. Young’s inequality

.

.

|f g| ≤

1 1 |f |p + |g|q , p q

implies that .fg ∈ E and that .

||fg||E ≤

|| || || 1 || |||f |p || + 1 |||g|q || = 1 + 1 = 1. E E p q p q

This proves the case .r = 1. To prove the general case, suppose that .f ∈ E (p) and .g ∈ E (q) . Observing that r .|f | ∈ E (p/r) and .|g|r ∈ E (q/r) , it follows from the first part of the proof that r .|f g| ∈ E and that .

||1/r || ||1/r || ||1/r || ||fg||E (r) = |||f g|r ||E ≤ |||f |r ||E (p/r) |||g|r ||E (q/r) = ||f ||E (p) ||g||E (q) . u n

The proof is complete. Lemma 6.6.3 .||·||E (p) is a function norm on .E (p) .

Proof It is clear that .||αf ||E (p) = |α| ||f ||E (p) for all .f ∈ E (p) and .α ∈ C . For the proof of the triangle inequality it may||be assumed that .p > 1. If .f, g ∈ E (p) , then || p/q p−1 .|f + g| ∈ E (q) with .|||f + g|p−1 ||E (q) = ||f + g||E (p) (where .1/p + 1/q = 1). Therefore, via Lemma 6.6.2 (with .r = 1), it follows that || || || || || p p p−1 || |f |f . ||f + g|| (p) = |||f + g| || = + g| + g| || || E E E || || || || || || p−1 || p−1 || ≤ |||f | |f + g| || + |||g| |f + g| || E

) ( p/q ≤ ||f ||E (p) + ||g||E (p) ||f + g||E (p) ,

E

which implies that .||f + g||E (p) ≤ ||f ||E (p) + ||g||E (p) . If .f, g ∈ E (p) and .|f | ≤ |g|, then .|f |p ≤ |g|p and so, .

||1/p || ||1/p || ||f ||E (p) = |||f |p ||E ≤ |||g|p ||E = ||g||E (p) .

u n Hence, .||·||E (p) is a function norm. ( (p) ) The space . E , ||·||E (p) is called the p-convexification of the normed function space E. It has thus been shown that the following holds. Proposition 6.6.4 If .(E, ||·||E ) is a normed function space, then its pconvexification is also a normed function space.

432

6 Examples

Remark 6.6.5 A normed function space is said to be p-convex, for some .p ≥ 1, if there exists a constant .K ≥ 1 such that ||( )1/p || ) (Σn || Σn || p 1/p p || || ≤ K |f | ||f || . (6.28) k k E || || k=1

k=1

E

(Σ )1/p Σn for all .f1 , . . . , fn ∈ E and all .n ∈ N. Note that . nk=1 |fk |p ≤ k=1 |fk | (Σn )1/p p and so, . k=1 |fk | ∈ E whenever .f1 , . . . , fn ∈ E. If E is p-convex, then the smallest constant .KE for which (6.28) holds is called the p -convexity constant of E. Evidently, a normed function space is always 1 -convex with 1-convexity constant equal to 1. It should be observed that the p-convexification .E (p) of any normed function space E is p-convex with .KE (p) = 1 . Indeed, if .f1 , . . . , fn ∈ E (p) , then ||(Σ )1/p || || || n p || || |fk | . || || k=1

E (p)

||Σn || = ||

k=1

=

(Σn k=1

||1/p (Σn || |fk |p || ≤

k=1

E

p

||fk ||E (p)

)1/p

|| || )1/p |||fk |p || E

.

The next proposition exhibits some properties which carry over from E to .E (p) . Proposition 6.6.6 Let E be a normed function space on .(0, a) and let .p ≥ 1. (i) (ii) (iii) (iv)

If E is a Banach function space, then so is .E (p) . If E has order continuous norm, then .E (p) has order continuous norm. If the norm in E is Fatou, then the norm in .E (p) is also Fatou. If E has the Fatou property, then .E (p) has the Fatou property.

Proof The proofs of (ii), (iii), and (iv) are straightforward and, therefore, omitted. For the proof of (i), it may be assumed that .p > 1. By Corollary 4.1.20, it is ( )+ sufficient to show that every increasing Cauchy sequence in . E (p) is bounded ( )+ ( (p) )+ from above in . E (p) . Let .{fn }∞ n=1 be an increasing Cauchy sequence in . E and set .M = supn ||fn ||E (p) . It follows from the numerical inequality p

p

p−1

0 ≤ fn − fm ≤ pfn

.

(fn − fm ) ,

n ≥ m,

in combination with Lemma 6.6.2, that || || || p || p−1 || p || . ||fn − fm || ≤ p ||f || (q) ||fn − fm ||E (p) n E E

=

p/q p ||fn ||E (p)

||fn − fm ||E (p) ≤ pM p/q ||fn − fm ||E (p) .

{ p }∞ This implies that . fn n=1 is an increasing Cauchy sequence in .E + . Since E is p complete, it follows that there exists .g ∈ E + such that .0 ≤ fn ≤ g for all .n ∈ N.

6.6 p-Convexification

433

)+ ( Consequently, .0 ≤ fn ≤ g 1/p ∈ E (p) for all n. This suffices to complete the proof. n u (p) is also a If E is a Banach function space on .(0, a) and( .p ≥ )∗ 1, then .E (p) Banach function space and hence, the Banach dual . E is a Banach lattice. The following observation is of interest.

Proposition 6.6.7 If E is a Banach function space and .p > 1, then the space ( (p) )∗ E has order continuous norm.

.

Proof By a well-known result in the theory of Banach lattices, it is sufficient to (p) show that any positive disjoint sequence .{un }∞ n=1 in the closed unit ball of .E converges weakly to zero. Suppose that .un - 0 weakly as .n → ∞. (This implies, )∗ by passing to a subsequence if necessary, that there exists .0 ≤ φ ∈ E (p) such that .φ (un ) ≥ ε for all n and some .0 < ε ∈ R. Since .{un }∞ n=1 is a positive disjoint sequence, it follows that (Σn .

) p 1/p

k=1

uk

=

Σn k=1

uk ,

n ∈ N.

As observed in Remark 6.6.5, the space .E (p) is .p-convex with p-convexity constant equal to 1. Consequently, nε = φ

.

≤

)

(Σn k=1

(Σn k=1

uk p

||(Σ ) || || n p 1/p || || || ≤ || u || k=1 k

||uk ||E (p)

)1p

E (p)

||φ||(E (p) )∗

||φ||(E (p) )∗ ≤ n1/p ||φ||(E (p) )∗ ,

which is a contradiction. This suffices to complete the proof.

n ∈ N, u n

Theorem 5.3.10 now yields the following consequence. Corollary 6.6.8 If E is a Banach function space with order continuous norm and the Fatou property, then .E (p) is reflexive for all .p > 1. Some more properties of .E (p) are given in the next proposition. Proposition 6.6.9 Let .E ⊆ S (m) be a normed function space on .(0, a) and .p ≥ 1. (i) If E is strongly symmetrically normed, then .E (p) is strongly symmetrically normed. (ii) If E is fully symmetric, then .E (p) is fully symmetric. Proof (i) Suppose that .f ∈ S (m) and .g ∈ E (p) are such that .μ (f ) ≤ μ (g). This implies that ) ) ( ( μ |f |p = μ (f )p ≤ μ (g)p = μ |g|p

.

434

6 Examples

|| || || || and so, it follows from .|g|p ∈ E that .|f |p ∈ E and .|||f |p ||E ≤ |||g|p ||E , that is, .f ∈ E (p) and .||f ||E (p) ≤ ||g||E (p) . This shows that .E (p) is symmetrically normed. Suppose that .f, g ∈ E (p) satisfy .f ≺≺ g. By Corollary 6.5.5, applied to the p p that function .o (t) = t p , .t ≥ 0, this || .|fp ||| ≺≺ |g| . Since E is strongly || implies || p || || || || symmetric, this implies that . |f | E ≤ |g| E and so, .||f ||E (p) ≤ ||g||E (p) . Consequently, .E (p) is strongly symmetrically normed. (ii) This follows via the same argument used in the second part of the proof of (i). u n Suppose now that .M is a semi-finite von Neumann algebra, equipped with a semi-finite normal faithful trace .τ . Let .E ⊆ S (m) be a strongly symmetrically normed space on .(0, a) with .τ (1) ≤ a, and let .E (τ ) be the corresponding strongly symmetrically normed space .E (τ ) of .τ -measurable operators. For .p ≥ 1, the .pconvexification .E (p) (τ ) of .E (τ ) is defined by setting } { E (p) (τ ) = x ∈ S (τ ) : μ (x) ∈ E (p) { } = x ∈ S (τ ) : |x|p ∈ E (τ )

.

and .

|| ||1/p ||x||E (p) (τ ) = ||μ (x)||E (p) = |||x|p ||E(τ ) ,

x ∈ E (p) (τ ) .

From the observations above, in combination with the results in Sect. 6.1, the following statements are now clear. Theorem 6.6.10 ( ) (i) . E (p) (τ ) , ||·||E (p) (τ ) is a strongly symmetrically normed space. (ii) If E is a strongly (respectively, fully) symmetric space, then .E (p) (τ ) is a strongly (respectively, fully) symmetric space. (iii) If E has the Fatou property, then .E (p) (τ ) has the Fatou property. (iv) If E has order continuous norm, then .E (p) (τ ) has order continuous norm. (v) If .p > 1 and E has order continuous norm and the Fatou property, then (p) (τ ) is reflexive. .E The following noncommutative version of Lemma 6.6.2 is of some interest. Proposition 6.6.11 Let .E ⊆ S (m) be a strongly symmetrically normed space on (0, a) with .τ (1) ≤ a. Suppose that .1 < p, q, r < ∞ satisfy .1/r = 1/p + 1/q. If (p) (τ ) and .y ∈ E (q) (τ ), then .xy ∈ E (r) (τ ) and .x ∈ E .

.

||xy||E (r) (τ ) ≤ ||x||E (p) (τ ) ||y||E (q) (τ ) .

6.7 Lorentz Lp,q -Spaces

435

Proof Given .x ∈ E (p) (τ ) and .y ∈ E (q) (τ ), it will be shown first that .xy ∈ E (r) (τ ). In fact, it follows from Proposition 3.2.7 (iv) that μ (t; xy) ≤ μ (t/2; x) μ (t/2; y) = (D2 μ (x)) (t) (D2 μ (y)) (t) ,

.

t > 0,

that is, .μ (xy) ≤ (D2 μ (x)) (D2 μ (y)). Since .μ (x) ∈ E (p) , it follows from Proposition 6.6.9 (i) and Lemma 6.1.1 that .D2 μ (x) ∈ E (p) . Similarly, .D2 μ (y) ∈ E (q) . It now follows from Lemma 6.6.2 that .(D2 μ (x)) (D2 μ (y)) ∈ E (r) and hence, (r) , that is, .xy ∈ E (r) (τ ). .μ (xy) ∈ E It follows from Theorem 3.9.10 that .μ (xy) ≺≺ μ (x) μ (y). Using that .E (r) is strongly symmetrically normed, Lemma 6.6.2 implies that .

||xy||E (r) (τ ) = ||μ (xy)||E (r) ≤ ||μ (x) μ (y)||E (r) ≤ ||μ (x)||E (p) ||μ (y)||E (q) = ||x||E (p) (τ ) ||y||E (q) (τ ) . u n

The proof is complete.

6.7 Lorentz Lp,q -Spaces Consider the interval .(0, ∞) equipped with Lebesgue measure m. Given .1 ≤ p, q ≤ ∞, for each .f ∈ L0 (m), the quantity .||f ||p,q ∈ [0, ∞] is defined by (f .

||f ||p,q =

∞(

t 1/p μ (t; f )

0

)q dt )1/q , t

1 ≤ q < ∞,

and .

||f ||p,∞ = sup t 1/p μ (t; f ) . t>0

It is clear that .||αf ||p,q = |α| ||f ||p,q for all .f ∈ L0 (m), .α ∈ C, and that .||f ||p,q ≤ ||g||p,q whenever .|f | ≤ |g| in .L0 (m). In general, .||·||p,q is not a function norm (however, it is easily verified that .

( ) ||f + g||p,q ≤ 21/p ||f ||p,q + ||g||p,q ,

f, g ∈ L0 (m) ,

for all values of p and q). The Lorentz .Lp,q -space Lp,q = Lp,q (m) = Lp,q (0, ∞)

.

436

6 Examples

is defined by setting { } Lp,q = f ∈ L0 (m) : ||f ||p,q < ∞ .

.

As is easy to see, .Lp,q is an absolutely solid subspace of .L0 (m). Evidently, .Lp,p = Lp and .||·||p,p = ||·||p for all .1 ≤ p ≤ ∞. It should also be observed that .L∞,q = {0} for all .1 ≤ q < ∞. On the other hand, if .1 ≤ p < ∞ and .1 ≤ q ≤ ∞, then .χA ∈ Lp,q for all measurable .A ⊆ (0, ∞) satisfying .m (A) < ∞. Remark 6.7.1 It should also be noted that .f ∈ Lp,q implies that .μ (t; f ) < ∞ for all .t > 0 and hence, .Lp,q ⊆ S (m). Furthermore, if .1 ≤ p < ∞, then .f ∈ Lp,q implies that .μ (t; f ) → 0 as .t → ∞ and so, .Lp,q ⊆ S0 (m). The next lemma shows that the .Lp,q -spaces are increasing with respect to the second index. Lemma 6.7.2 If .1 ≤ p < ∞ and .1 ≤ q ≤ r ≤ ∞, then there exists a constant C = Cp,q,r > 0 such that

.

.

||f ||p,r ≤ C ||f ||p,q ,

f ∈ L0 (m) ,

and consequently, .Lp,q ⊆ Lp,r . Proof Suppose first that .1 ≤ q < r = ∞. Writing t q/p =

.

q p

f

t

s q/p

0

ds , s

t > 0,

it follows that t

.

1/p

{

μ (t; f ) = t ≤

q/p

μ (t; f )

} q 1/q

{ =

q p

( )1/q q ||f ||p,q p

f

t

s 0

q/p

ds μ (t; f ) s

}1/q

q

for all .t > 0 and so, .

||f ||p,∞ = sup t 1/p μ (t; f ) ≤ t>0

( )1/q q ||f ||p,q p

for all .f ∈ L0 (m). This proves the lemma for .r = ∞.

(6.29)

6.7 Lorentz Lp,q -Spaces

437

Suppose now that .1 ≤ q < r < ∞. If .f ∈ L0 (m), then, using (6.29), it follows that f ∞( )r−q ( )q dt r t 1/p μ (t; f ) t 1/p μ (t; f ) . ||f ||p,r = t 0 )r−q f ∞ ( )q dt ( t 1/p μ (t; f ) ≤ sup t 1/p μ (t; f ) t 0 t>0 ( )(r−q)/q ( )(r−q)/q q q r−q q r−q q ||f ||rp,q , ||f ||p,q ||f ||p,q = = ||f ||p,∞ ||f ||p,q ≤ p p which implies that

.

||f ||p,r

( )(r−q)/qr q ||f ||p,q . ≤ p u n

The proof is complete. It should be observed that, in particular, Lp,1 ⊆ Lp ⊆ Lp,∞

.

for all .1 ≤ p < ∞. The spaces .Lp,∞ are also known as weak .Lp -spaces. Remark 6.7.3 One may also consider the Lorentz .Lp,q -spaces on a finite interval (0, a) for some .0 < a < ∞. In this situation it is worth mentioning that .Lp2 ,∞ (0, a) ⊆ Lp1 ,1 (0, a) whenever .1 ≤ p1 < p2 < ∞. Indeed, if .f ∈ L0 (0, a), then f a f a dt 1/p1 = t 1/p2 μ (t; f ) t 1/p1 −1/p2 −1 dt . ||f ||p ,1 = t μ f (t; ) 1 t 0 0 (f a ) 1/p1 −1/p2 −1 ≤ t dt ||f ||p2 ,∞ .

0

fa

and .

0

t 1/p1 −1/p2 −1 dt < ∞, as .1/p1 − 1/p2 − 1 > −1.

The next proposition exhibits a range of p and q for which .||·||p,q is actually a norm. ( ) Proposition 6.7.4 If .1 ≤ p ≤ q < ∞, then . Lp,q , ||·||p,q is a fully symmetric space with the Fatou property and order continuous norm.

438

6 Examples

Proof Defining .ψ (t) = (p/q) t q/p , .t ≥ 0, it follows from Theorems 6.3.2 and 6.3.6 that the Lorentz space .Aψ is a fully symmetric space with the Fatou property and order continuous norm. Observe that for any .f ∈ Lp,q , we have (f

.

)q dt )1/q t 0 )) (f ∞ ( p q/p 1/q q = μ (t; f ) d t q 0 (f ∞ )1/q q = μ (t; f ) dψ(t) ∞(

||f ||Lp,q =

t 1/p μ (t; f )

0

|| ||1/q = |||f |q ||A , ψ

( ) that is, . Lp,q , ||·||p,q is the q-convexification of .Aψ . The result of the proposition now follows from Propositions 6.6.6 and 6.6.9. u n It will be convenient to have the following notation available. For .f ∈ L0 (m), let .Mf : (0, ∞) → [0, ∞] be defined by setting 1 . (Mf ) (t) = t

f

t

μ (s; f ) ds,

t > 0.

0

Using the fact that the function .μ (f ) is decreasing the following properties of Mf are easily verified. Lemma 6.7.5 (i) Mf is continuous and decreasing. (ii) .μ (t; f ) ≤ (Mf ) (t) for all .t > 0. (iii) .(Mf ) (t) < ∞ for all .t > 0 (equivalently, some .t > 0) if and only if .f ∈ (L1 + L∞ ) (m). (iv) If .0 ≤ fn ↑n f in .L0 (m), then .(Mfn ) (t) ↑n (Mf ) (t) for all .t > 0. (v) If .fn ↓n 0 in .S0 (m) ∩ (L1 + L∞ )(m), then .(Mfn ) (t) ↓n 0 for all .t > 0. For .1 ≤ p, q ≤ ∞ and .f ∈ L0 (m), the quantity .||f ||(p,q) ∈ [0, ∞] is defined by (f .

||f ||(p,q) =

)q dt )1/q t 1/p (Mf ) (t) , t

∞( 0

and .

||f ||(p,∞) = sup t 1/p (Mf ) (t) . t>0

1 ≤ q < ∞,

6.7 Lorentz Lp,q -Spaces

439

The quantities .||·||p,q and .||·||(p,q) are equivalent (see Lemma 6.7.7 ). To see this, the following well-known classical inequality, which is one of the so-called Hardy’s inequalities, will be used. Lemma 6.7.6 If .λ < 1 and .1 ≤ q ≤ ∞, then (f

∞(

tλ

.

0

f

1 t

)q

t

ϕ (s) ds 0

dt t

)1/q

≤ (1 − λ)−1

(f

∞(

0

)q dt t λ ϕ (t) t

)1/q ,

for all measurable functions .ϕ : (0, ∞) → [0, ∞]. In the case that .q = ∞, this inequality should be interpreted as ess. supt>0 t λ

.

1 t

f

t 0

ϕ (s) ds ≤ (1 − λ)−1 ess. supt>0 t λ ϕ (t) .

Lemma 6.7.7 If .1 < p < ∞ and .1 ≤ q ≤ ∞, then .

||f ||p,q ≤ ||f ||(p,q) ≤ p' ||f ||p,q ,

f ∈ L0 (m) ,

where .1/p + 1/p' = 1. Proof The first inequality follows immediately from the fact that .μ (f ) ≤ Mf for all .f ∈ L0 (m). The second inequality follows from and application of Lemma 6.7.6 with .λ = 1/p and .ϕ = μ (f ). u n ( ) Proposition 6.7.8 If .1 < p < ∞ and .1 ≤ q ≤ ∞, then . Lp,q , ||·||(p,q) is a fully symmetric space with the Fatou property and, if .1 ≤ q < ∞, then the norm is order continuous. Proof If .f, g ∈ L0 (m), then .μ (f + g) ≺≺ μ (f ) + μ (g) and so, .M (f + g) ≤ Mf + Mg. This implies that .||·||(p,q) satisfies the triangle inequality and hence, .||·||(p,q) is a norm. The remaining statements of the proposition are now readily u n verified, using Lemma 6.7.5 and Remark 6.7.1 The next proposition identifies the Köthe dual of .L(p,q) . Remark 6.7.9 For .1 < p < ∞ the space .Lp,∞ coincides with the Marcinkiewicz ' space .Mψ , where .ψ (t) = t 1/p , .t ≥ 0, with .1/p + 1/p' = 1. This holds since we have f t 1 1/p μ (s; f ) ds, f ∈ L0 (m) . . ||f ||(p,∞) = sup t (Mf ) (t) = sup 1/p' 0 t>0 t>0 t Proposition 6.7.10 Suppose that .1 < p < ∞ and .1 ≤ q ≤ ∞, and let .1 ≤ p' , q ' ≤ ∞ be such that .1/p + 1/p' = 1 and .1/q + 1/q ' = 1. Considering the space .Lp,q equipped with the norm .||·||(p,q) , it follows that

440

6 Examples

L× p,q = Lp' ,q ' ,

(6.30)

.

with equivalence of norms. Proof If .p = q, then the equality (6.30) corresponds to the fact that .L× p = Lp ' . Therefore, it may be assumed that .p /= q . Suppose first that .1 < p < ∞ and .q = ∞. In this case, by Remark 6.7.9, .Lp,∞ coincides with the Marcinkiewicz ' space .Mψ , where .ψ (t) = t 1/p , .t ≥ 0. Therefore, it follows from Corollary 6.4.4 that × L× p,∞ = Mψ = Aψ

.

(with equality of norms). Observing that f .

||f ||Aψ =

∞

μ (t; f ) ψ ' (t) dt =

0

1 p'

f

∞

μ (t; f ) t 1/p

0

'

1 dt = ' ||f ||p' ,1 t p

for all .f ∈ L0 (m), it follows that .Aψ = Lp' ,1 (with equivalent norms). This implies that .L× p,∞ = Lp' ,1 with equivalence of norms. Furthermore, by Theorem 6.4.3, × .A ψ = Mψ (with equality of norms) and hence, × L× p' ,1 = Aψ = Mψ = Lp,∞ ,

.

with equivalence of norms, which takes also care of the case that .q = 1. It will be shown next that .Lp' ,q ' ⊆ L× p,q whenever .1 < p < ∞ and .1 < q < ∞. To this end, suppose that .g ∈ Lp' ,q ' and that .f ∈ Lp,q satisfies .||f ||(p,q) ≤ 1. Using Hölder’s inequality, it follows that (using Theorem 3.4.29) f .

∞

f

∞

|f g| dt ≤

0

f μ (t; f ) μ (t; g) dt =

0

∞(

t 1/p μ (t; f )

)(

'

t 1/p μ (t; g)

0

(f

∞(

≤ 0

)q dt t 1/p μ (t; f ) t

)1/q (f

∞(

'

t 1/p μ (t; g)

0

) dt t q'

) dt t

)1/q '

= ||f ||p,q ||g||p' ,q ' ≤ ||f ||(p,q) ||g||p' ,q ' ≤ ||g||p' ,q ' . This shows that .g ∈ L× ≤ ||g||p' ,q ' ≤ ||g||(p' ,q ' ) . p,q and .||g||L× p,q × For a proof that .Lp,q ⊆ Lp' ,q ' , suppose that .1 < p < q < ∞. Take first × .g ∈ F (m) (which implies that .g ∈ Lp,q ∩ Lp ' ,q ' ) and define the function f by setting 1 .f (t) = t

f

t 0

'

'

'

s q /p −1 μ (s; g)q −1 ds,

t > 0.

6.7 Lorentz Lp,q -Spaces

441

Observe that .1 < p < q implies .q ' /p' < 1 and so, the function .s |→ ' ' ' s q /p −1 μ (s; g)q −1 , .s > 0, is decreasing. Therefore, f is decreasing and so, .μ (f ) = f . Also note that ) ( f t p' ' ' 1 ' ' q ' /p' −1 s ds μ (t; g)q −1 = ' t q /p −1 μ (t; g)q −1 .f (t) ≥ t 0 q

(6.31)

for all .t > 0. Furthermore, it follows via Lemma 6.7.7 (with .λ = 1/p and .ϕ (s) = ' ' ' s q /p −1 μ (s; g)q −1 ) that (f

.

||f ||p,q

)q dt )1/q t μ (t; f ) = t 0 (f ∞ ( )q )1/q f t 1 dt ' ' ' t 1/p = s q /p −1 μ (s; g)q −1 ds t t 0 0 (f ∞ ( ) )q dt 1/q 1 ' ' ' t 1/p t q /p −1 μ (t; g)q −1 ≤ ' t p 0 (f ∞ ( ) )q ' dt 1/q 1 1 ' q ' /q t 1/p μ (t; g) = ' = ' ||g||p' ,q ' . p t p 0 ∞(

1/p

Using the estimate (6.31) it now follows that q' . ||g|| ' ' p ,q

f = ≤

∞(

t 0 q'

p'

f

t

1/p'

f ∞ )q ' dt ' ' ' = t q /p −1 μ (t; g)q −1 μ (t; g) dt μ (t; g) t 0

f (t) μ (t; g) dt ≤

0

q' ||f ||(p,q) ||g||L×p,q p'

q 'p q 'p q ' /q ||g||p' ,q ' ||g||L×p,q ≤ ' ||f ||p,q ||g||L×p,q ≤ 2 ' p (p ) and hence, .

||g||p' ,q ' ≤

q 'p (p' )2

||g||L×p,q .

This shows that .

with .C = q ' p/p' .

||g||(p' ,q ' ) ≤ C ||g||L×p,q ,

g ∈ F (m) ,

(6.32)

442

6 Examples

∞ Given .0 ≤ g ∈ L× p,q , there exists a sequence .{gn }n=1 in .F (m) such that .0 ≤ gn ↑n g. By (6.32), .||gn ||(p' ,q ' ) ≤ C ||gn ||L×p,q for all n and so, .

sup ||gn ||(p' ,q ' ) ≤ C sup ||gn ||L×p,q ≤ C ||g||L×p,q . n

n

Since the space .Lp' ,q ' has the Fatou property (see Proposition 6.7.8), this implies that .g ∈ Lp' ,q ' with .||g||(p' ,q ' ) ≤ C ||g||L×p,q . This shows that .L× p,q ⊆ Lp' ,q ' and ' ' = L (with equivalent norms), in the case that . 1 < p < q < ∞. hence, .L× p ,q p,q Suppose now that .1 < q < p < ∞, in which case .1 < p' < q ' < ∞. By what just has been proved, it follows that .L× p' q ' = Lp,q (with equivalence of norms). Since ' ' the space .Lp' ,q ' has the Fatou property, .L×× p' q ' = Lp q (with equality of norms) and × ' ' hence, .Lp,q = Lp ,q (with equivalence of norms). This completes the proof of the proposition. u n The next corollary follows immediately from the fact that the norm on .Lp,q is order continuous if .1 < p < ∞ and .1 ≤ q < ∞, and so the Banach dual of the space .Lp,q coincides with the Köthe dual. Corollary ( 6.7.11 If .1) < p < ∞ and .1 ≤ q < ∞, then the Banach dual of the space . Lp,q , ||·||(p,q) may be identified with the space .Lp' ,q ' , with equivalence of norms, via the usual duality pairing. Suppose now that .M is a semi-finite von Neumann algebra, equipped with a semi-finite normal faithful trace .τ . For .1 ≤ p ≤ ∞ and .1 ≤ q ≤ ∞, the noncommutative Lorentz .Lp,q -spaces are defined by setting } { Lp,q (τ ) = x ∈ S (τ ) : μ (x) ∈ Lp,q

.

||x||p,q = ||μ (x)||p,q ,

x ∈ Lp,q (τ ) .

Furthermore, for .1 < p < ∞ and .1 ≤ q ≤ ∞ set .

||x||(p,q) = ||μ (x)||(p,q) ,

x ∈ Lp,q (τ ) .

A combination of the results obtained in this section with those of Sect. 6.1 leads to the following theorem. Theorem 6.7.12 (i) For .1 ≤ p ≤ ∞ and .1 ≤ q ≤ ∞, the space .Lp,q (τ ) is an .M-bimodule of .τ -measurable operators (and, actually, .x ∈ Lp,q (τ ) whenever .x ∈ S (τ ) and .y ∈ Lp,q (τ ) is such that .μ (x) ≤ μ (y)). Furthermore, .

( ) ||x + y||p,q ≤ 21/p ||x||p,q + ||y||p,q ,

x, y ∈ Lp,q (τ ) .

6.8 Sequence Spaces

443

(ii) If .1 ≤ p < ∞ and .1 ≤ q ≤ r ≤ ∞, then .Lp,q (τ ) ⊆ Lp,r (τ ) and there exists a constant .C = Cp,q,r > 0 such that .

||x||p,r ≤ C ||x||p,q ,

x ∈ Lp,q (τ ) .

(iii) If .τ (1) < ∞, then .Lp2 ,∞ (τ () ⊆ Lp1 ,1 (τ ) whenever .1 ≤ p1 < p2 < ∞. ) (iv) If .1 ≤ p ≤ q < ∞, then . Lp,q (τ ) , ||·||p,q is a fully symmetric space of .τ -measurable operators with the Fatou property and order continuous norm. (v) If .1 < p < ∞ and .1 ≤ q ≤ ∞, then .

||x||p,q ≤ ||x||(p,q) ≤ p' ||x||p,q ,

x ∈ Lp,q (τ ) ,

where .1/p + 1/p' = 1. ( ) (vi) If .1 < p < ∞ and .1 ≤ q ≤ ∞, then . Lp,q (τ ) , ||·||(p,q) is a fully symmetric space of .τ -measurable operators with the Fatou property and, if .1 ≤ q < ∞, then the norm is order continuous. (vii) Suppose that .1 < p < ∞ and .1 ≤ q ≤ ∞, and let .1 ≤ p' , q ' ≤ ∞ be such that .1/p + 1/p' = 1 and .1/q + 1/q ' = 1. Considering the space .Lp,q (τ ) equipped with the norm .||·||(p,q) , it follows that Lp,q (τ )× = L× p,q (τ ) = Lp' ,q ' (τ ) ,

.

with equivalence of norms. (viii) If .1 < ( p < ∞ and) .1 ≤ q < ∞, then the Banach dual of the space . Lp,q (τ ) , ||·||(p,q) may be identified with the space .Lp' ,q ' (τ ), with equivalence of norms, via trace duality. Reference: [97].

6.8 Sequence Spaces In Sect. 6.1, strongly symmetric spaces of .τ -measurable operators (associated with a semi-finite von Neumann algebra .(M, τ )) were constructed from strongly symmetric spaces on the positive real line .(0, ∞) equipped with Lebesgue measure. However, if the von Neumann algebra .M is atomic and all minimal projections .p ∈ P (M) have equal trace .τ (p) = 1, then it is also possible to construct strongly symmetric spaces associated with .M corresponding to strongly symmetric sequence spaces on .N0 = N ∪ {0}. It is, of course, possible to repeat the construction once again in this setting. However, it is also possible to reduce the “sequential case” to the situation considered before. This is the direction that will be followed in this section. Consider .N0 = N ∪ {0} = {0, 1, 2, . . .} equipped with counting measure .ν0 . It is easily verified that .S (ν0 ) = l∞ (N0 ) = l∞ , the space of all bounded sequences

444

6 Examples

on .N0 equipped with the .sup-norm. Given .x = {x (n)}∞ n=1 ∈ l∞ , its generalized singular value function .μ (x) is given by μ (x) =

∞ Σ

.

(6.33)

μ (n; x) χ[n,n+1) .

n=0

The sequence .μ (0; x) ≥ μ (1; x) ≥ · · · ≥ 0 is called the decreasing { } rearrangement of the sequence .{|x (n)|} (which is denoted also by . x (n)∗ ). Some simple preliminary observations first. Lemma 6.8.1 If .x, y ∈ l∞ , then .x ≺≺ y if and only if n Σ .

μ (k; x) ≤

k=0

n Σ

μ (k; y) ,

n ∈ N0 .

(6.34)

k=0

ft ft Proof Suppose that .x ≺≺ y, that is, . 0 μ (s; x) ds ≤ 0 μ (s; y) ds for all .t ≥ 0. If .n ∈ N0 , then n Σ .

k=0

f μ (k; x) = 0

n+1

f

n+1

μ (s; x) ds ≤ 0

μ (s; y) ds =

n Σ

μ (k; y) .

k=0

Hence, (6.34) holds. Suppose now fthat (6.34) holds and definefthe functions .ϕ and .ψ on .[0, ∞) by t t setting .ϕ (t) = 0 μ (s; x) ds and .ψ (t) = 0 μ (s; y) ds, .t ≥ 0, respectively. It follows from (6.34) that .ϕ (n) ≤ ψ (n) for all .n ∈ N0 . Since both .ϕ and .ψ are linear on the intervals .[n, n + 1], .n ∈ N0 , it follows that .ϕ (t) ≤ ψ (t) for all .t ≥ 0. The proof is complete. u n A (strongly) symmetrically normed space .F ⊆ l∞ will be called a (strongly) symmetrically normed sequence space. Corollary 6.8.2 If .F ⊆ l∞ is a linear subspace equipped with a norm .||·||F , then (F, ||·||F ) is a strongly symmetrically normed sequence space if and only if

.

(i) If .x ∈ l∞ and .y ∈ F are such that .μ (n; x) ≤ μ (n; y), .n ∈ N0 , then .x ∈ F and .||x||F ≤ ||y||Σ F. Σ (ii) If .x, y ∈ F and . nk=0 μ (n; x) ≤ nk=0 μ (n; y), .n ∈ N0 , then .||x||F ≤ ||y||F . Proof From the definitions, it is clear that (i) and (ii) imply that F is a strongly symmetrically normed sequence space. Suppose now that F is a strongly symmetrically normed sequence space. If .x ∈ l∞ and .y ∈ F satisfy .μ (n; x) ≤ μ (n; y), .n ∈ N0 , then it follows from (6.33) that .μ (t; x) ≤ μ (t; y) for all .tΣ∈ [0, ∞) and hence, Σn .x ∈ F and .||x||F ≤ ||y||F . Furthermore, if .x, y ∈ F and . nk=0 μ (n; x) ≤ k=0 μ (n; y), .n ∈ N0 , then it follows from Lemma 6.8.1 that .x ≺≺ y and hence, .||x||F ≤ ||y||F . u n

6.8 Sequence Spaces

445

It should be observed that if F is a symmetrically normed sequence space and x ∈ l∞ , then .x ∈ F if and only if .{μ (n; x)} ∈ F . Indeed, .μ (x) = μ ({μ (n; x)}). Given a strongly symmetrically normed space .E = E (0, ∞) on .(0, ∞) (equipped with Lebesgue measure m), define

.

E (ν0 ) = {x ∈ l∞ : μ (x) ∈ E} ,

.

||x||E(ν0 ) = ||μ (x)||E ,

x ∈ E (ν0 ) ,

which is a strongly symmetrically normed sequence space (see Theorem 6.1.2). This space will be denoted also by lE = E (ν0 ) .

(6.35)

.

Given a strongly symmetrically normed sequence space .F ⊆ l∞ , a strongly symmetrically normed space .EF on .(0, ∞) will now be constructed satisfying .F = lEF (using the notation introduced in (6.35)). Let .A be the .σ -algebra of subsets of .[0, ∞) generated by the intervals .[n, n + 1), .n ∈ N0 . The conditional expectation operator .EA from .(L1 + L∞ ) (0, ∞) onto .(L1 + L∞ ) (A, m) is given by EA f =

∞ (f Σ

n+1

.

n=0

) f (t) dt χ[n,n+1) ,

n

f ∈ (L1 + L∞ ) (0, ∞) .

Since .EA is an .(L1 , L∞ ) -contraction, it follows that .EA f ≺≺ f for all .f ∈ (L1 + L∞ ) (0, ∞) (see Proposition 3.10.3). Note that the function .EA (μ (f )) is decreasing and right-continuous for each .f ∈ (L1 + L∞ ) (0, ∞) (and hence, .μ (EA (μ (f ))) = EA (μ (f ))). Define the map .ψ : (L1 + L∞ ) (0, ∞) → l∞ by setting ψ (f ) = {(EA μ (f )) (n)}∞ n=1 ,

.

f ∈ (L1 + L∞ ) (0, ∞) .

In the following lemma some simple observations are collected together for later reference. Lemma 6.8.3

Σ (i) If .a ∈ l∞ and .g = ∞ n=0 a (n) χ[n,n+1) , then .μ (a) = μ (g). In particular, if .f ∈ (L1 + L∞ ) (0, ∞) and .x ∈ l∞ is defined by .x (n) = (EA f ) (n), .n ∈ N0 , then .μ (x) = μ (EA f ). (ii) .μ (ψf ) = EA μ (f ) for all .f ∈ (L1 + L∞ ) (0, ∞). + + (iii) If .{fn }∞ n=1 is a sequence in .(L1 + L∞ ) (0, ∞) and .f ∈ (L1 + L∞ ) (0, ∞) is such that .fn ↑n f , then .ψ (fn ) ↑n ψ (f ) in .l∞ . + (iv) If .{fn }∞ n=1 is a sequence in .(L1 + L∞ ) (0, ∞) such that .fn ↓n 0 and .f1 ∈ S0 (0, ∞), then .ψ (fn ) ↓n 0 in .l∞ . (v) If .f, g ∈ (L1 + L∞ ) (0, ∞) and .f ≺≺ g, then .ψ (f ) ≺≺ ψ (g).

446

6 Examples

(vi) If .f, g ∈ (L1 + L∞ ) (0, ∞), then .ψ (f + g) ≺≺ ψ (f ) + ψ (g). (vii) If .f, g ∈ (L1 + L∞ ) (0, ∞), then .ψf − ψg ≺≺ f − g. Proof (i) It is readily verified that the sequence a and the function g have the same f n+1 distribution function and hence, .μ (a) = μ (g). Taking .a (n) = n f (t) dt, .n ∈ N0 , the second statement follows immediately from the first. (ii) This is a direct consequence of (i), observing that .μ (EA μ (f )) = EA μ (f ). (iii) Since .0 ≤ fn ↑n f implies that .μ (fn ) ↑n μ (f ), it follows from the monotone convergence theorem that .0 ≤ EA μ (fn ) ↑n EA μ (f ) on .(0, ∞) and hence, .ψ (fn ) ↑n ψ (f ). (iv) It follows from .fn ↓n 0 and .f1 ∈ S0 (0, ∞) that .μ (fn ) ↓n 0 on .(0, ∞) (see Proposition 3.2.14). Via the dominated convergence theorem, this implies that .ψ (fn ) ↓n 0. (v) If .n ∈ N0 , then (using (ii)), n Σ .

(μ (ψf )) (k) =

k=0

n Σ

(EA μ (f )) (k) =

k=0

f =

n+1

f

=

k=0

k+1

μ (t; f ) dt

k=0 k

μ (t; f ) dt ≤

0 n Σ

n f Σ

n+1

μ (t; g) dt 0

(EA μ (g)) (k) =

n Σ

(μ (ψg)) (k) .

k=0

It follows from Lemma 6.8.1 that .ψ (f ) ≺≺ ψ (g). (vi) Since .f + g ≺≺ μ (f ) + μ (g), it follows from (v) that ψ (f + g) ≺≺ ψ (μ (f ) + μ (g)) = ψ (μ (f ))+ψ (μ (g)) = ψ (f )+ψ (g) .

.

(vii) Observing that (ψf ) (n) − (ψg) (n) = EA (μ (f ) − μ (g)) (n) ,

.

n ∈ N0 ,

it follows from (i) that μ (ψf − ψg) = μ (EA (μ (f ) − μ (g))) ≺≺ μ (f ) − μ (g)

.

(where it is used that .EA h ≺≺ h for all .h ∈ (L1 + L∞ ) (0, ∞)). Since .μ (f )− μ (g) ≺≺ μ (f − g) (see Theorem 3.9.14), the result of (vii) follows. u n

6.8 Sequence Spaces

447

Let .F ⊆ l∞ be a strongly symmetrically normed sequence space. Define .EF ⊆ (L1 + L∞ ) (0, ∞) by setting EF = {f ∈ (L1 + L∞ ) (0, ∞) : ψf ∈ F } .

(6.36)

.

The first objective is to show that .EF is a linear subspace of .L1 + L∞ (see Lemma 6.8.6). In the proof of Lemma 6.8.6 the following observations will be used. Lemma 6.8.4 For .x ∈ l∞ , define the sequence .D2 x ∈ l∞ by .(D2 x) (n) = x (Ln/2|), .n ∈ N0 . If .F ⊆ l∞ is a symmetrically normed sequence space and .x ∈ F , then .D2 x ∈ F . Proof Given .x ∈ F , observe that .D2 x = x1 + x2 , where x1 = (x (0) , 0, x (1) , 0, . . .) ,

.

x2 = (0, x (0) , 0, x (1) , 0, . . .) .

u Since .μ (x1 ) = μ (x2 ) = μ (x), it follows that .x1 , x2 ∈ F and hence, .D2 x ∈ F . n Lemma 6.8.5 If .f ∈ (L1 + L∞ ) (0, ∞), then .ψ (D2 f ) ≤ 2D2 ψ (f ). Proof If .n ∈ N0 , then f ψ (D2 f ) (n) = (EA μ (D2 f )) (n) = (EA D2 μ (f )) (n) =

n+1

μ (t/2; f ) dt

.

f =2

(n+1)/2

f μ (t; f ) dt ≤ 2

n/2

[n/2]+1

n

μ (t; f ) dt = 2D2 ψ (f ) (n) .

[n/2]

Hence, .ψ (D2 f ) ≤ 2D2 ψ (f ).

u n

Lemma 6.8.6 .EF is a linear subspace of .(L1 + L∞ ) (0, ∞). Proof Since .ψ (αf ) = |α| ψ (f ), it is clear that .αf ∈ EF whenever .f ∈ EF and α ∈ C. Suppose that .f, g ∈ EF . If .t ≥ 0, then

.

μ (t; f + g) ≤ μ (t/2; f ) + μ (t/2; g) ,

.

that is, μ (f + g) ≤ D2 μ (f ) + D2 μ (g) = μ (D2 f ) + μ (D2 g) .

.

Hence, EA (μ (f + g)) ≤ EA (μ (D2 f )) + EA (μ (D2 g))

.

and so, ψ (f + g) ≤ ψ (D2 f ) + ψ (D2 g) .

.

(6.37)

448

6 Examples

It follows from Lemma 6.8.5 that .ψ (D2 f ) ≤ 2D2 ψ (f ). Lemma 6.8.4 implies that D2 ψ (f ) ∈ F and hence, .ψ (D2 μ (f )) ∈ F . Similarly, .ψ (D2 g) ∈ F and hence, by (6.37), .ψ (f + g) ∈ F , that is, .f + g ∈ EF . The proof is complete. u n

.

Defining .

||f ||EF = ||ψ (f )||F ,

f ∈ EF ,

(6.38)

the following holds. ( ) Lemma 6.8.7 . EF , ||·||EF is a strongly symmetrically normed space. Proof It will be shown first that .||·||EF is a norm on .EF . Evidently, .||αf ||EF = |α| ||f ||EF for all .f ∈ EF and .α ∈ C. If .f, g ∈ EF , then Lemma 6.8.6 implies that .ψ (f + g) ∈ F and it follows from Lemma 6.8.3 (vi) that .ψ (f + g) ≺≺ ψ (f ) + ψ (g). Since the norm on F is assumed to be strongly symmetric, it follows that .

||ψ (f + g)||F ≤ ||ψ (f ) + ψ (g)||F ≤ ||ψ (f )||F + ||ψ (g)||F ,

that is, .||f + g||EF ≤ ||f ||EF + ||g||EF . Hence, .||·||EF is a norm on .EF . Suppose that .f ∈ S (m) and .g ∈ EF are such that .μ (f ) ≤ μ (g). This implies that .EA μ (f ) ≤ EA μ (g) and hence, .ψ (f ) ≤ ψ (g). Consequently, .ψ (f ) ∈ F , that is, .f ∈ EF , and .||f ||EF ≤ ||g||EF . Hence, .EF is a symmetrically normed space. If .f, g ∈ EF are such that .f ≺≺ g, then it follows from Lemma 6.8.3 (iv) that .ψ (f ) ≺≺ ψ (g). The norm on F is strongly symmetric and hence, .

||f ||EF = ||ψ (f )||F ≤ ||ψ (g)||F = ||g||EF .

The proof of the lemma is complete.

u n

Proposition 6.8.8( Let .F ⊆) l∞ be a strongly symmetrically normed sequence space. The space . EF , ||·||EF , defined by (6.36) and (6.38), is a strongly symmetrically normed space satisfying F = lEF = {x ∈ l∞ : μ (x) ∈ EF }

.

(6.39)

and .||x||F = ||μ (x)||EF for all .x ∈ F . Proof In view of Lemmas 6.8.6 and 6.8.7, only the second statement of the proposition needs to be proved. Observe that .μ (x) = EA (μ (x)) for all .x ∈ l∞ . Using Lemma 6.8.3 (ii), it follows that .μ (ψ (μ (x))) = EA (μ (x)) = μ (x) for all .x ∈ l∞ . Consequently, if .x ∈ l∞ , then .μ (x) ∈ EF if and only if .ψ (μ (x)) ∈ F , if and only if .x ∈ F . Furthermore, if .μ (x) ∈ EF , then .||x||F = ||ψ (μ (x))||F = ||μ (x)||EF . u n

6.8 Sequence Spaces

449

The next lemma deals with the completeness of the space .EF . The situation is the same as in Proposition 6.8.8. Lemma 6.8.9 The space .EF is complete if and only if F is complete. Proof It follows from equality (6.39), in combination with Theorem 6.1.3, that F is complete whenever .EF is complete. Suppose that F is complete. It suffices to show that .EF satisfies the Riesz– ∞ Fischer propertyΣ (see Theorem 4.1.19). For this purpose, Σn let .{fk }k=1 be a sequence ∞ + in .EF such that . n=1 ||fn ||EF < ∞. Defining .sn = k=1 fk , .n ∈ N, it is clear that + such that .{sn } is a Cauchy sequence. It has to be shown that there exists .f ∈ EF .0 ≤ sn ↑n f . The embedding of .EF into .(L1 + L∞ ) (0, ∞) is continuous and hence, .{sn } is a Cauchy sequence in .(L1 + L∞ ) (0, ∞) . Since .(L1 + L∞ ) (0, ∞) is complete, there exists .f ∈ (L1 + L∞ ) (0, ∞) such that .||f − sn ||L1 +L∞ → 0. This implies that .0 ≤ sn ↑n f . It suffices to prove that .f ∈ EF . + Defining the increasing sequence .{xn }∞ n=1 in .F by setting .xn = ψ (sn ), .n ∈ N, it follows from Lemma 6.8.3 (vii) that xn − xm ≺≺ sn − sm ,

.

n, m ∈ N,

and hence, .

||xn − xm ||F = ||μ (xn − xm )||EF ≤ ||sn − sm ||EF ,

n, m ∈ N.

Consequently, .{xn } is a Cauchy sequence in F . Since F is assumed to be complete, there exists .x ∈ F such that .||x − xn ||F → 0 as .n → ∞. Note that .0 ≤ xn ↑n x. By Lemma 6.8.3 (iii), .0 ≤ sn ↑n f implies that also .xn = ψ (sn ) ↑n ψ (f ). Therefore, .ψ (f ) = x ∈ F , that is, .f ∈ EF . The proof is complete. u n It will be shown next that a number of properties that the sequence space F may have transfer to the corresponding space .EF . Proposition 6.8.10 Let F and .EF be as in Proposition 6.8.8. (i) (ii) (iii) (iv) (v)

If F has order continuous norm, then .EF has order continuous norm. If the norm in F is Fatou, then the norm in .EF is a Fatou norm. If the space F has the Fatou property, then .EF has also the Fatou property. If F is a KB-space, then .EF is a KB-space. If F is fully symmetrically normed, then .EF is fully symmetrically normed.

Proof (i) Since F has order continuous norm, it follows that .μ (t; x) → 0 as .t → ∞ for all .x ∈ F , that is, .F ⊆ c0 . Consequently, if .f ∈ EF , then .ψf ∈ c0 and so, f n+1 . μ (t; f ) dt → 0 as .n → ∞, which implies that .μ (t; f ) → 0 as .t → ∞. n This shows that .EF ⊆ S0 (0, ∞). If .{fk }∞ k=1 is a sequence in .EF satisfying

450

6 Examples

fk ↓k 0, then it follows from Lemma 6.8.3 (iv) that .ψfk ↓ 0 in F . Since the norm in F is order continuous, this implies that

.

.

||fk ||EF = ||ψfk ||F ↓k 0.

This shows that .EF has order continuous norm. (ii) Suppose that .f ∈ EF and that .(fk )∞ k=1 is a sequence in .EF such that .0 ≤ fk ↑k f . By Lemma 6.8.3 (iii), this implies that .ψfk ↑k ψf in F . Hence, .

||fk ||EF = ||ψfk ||F ↑k ||ψf ||F = ||f ||EF ,

hence, .||·||EF is a Fatou norm. (iii) Suppose that .(fk )∞ k=1 is a sequence in .EF such that .0 ≤ fk ↑k and .supk ||fk ||E < ∞. This implies that .0 ≤ ψfk ↑k in F and .supk ||ψfk ||F = F supk ||fk ||EF < ∞. Since F has the Fatou property, it follows that there exists .0 ≤ x ∈ F such that .ψfk ↑k x and .||x||F = supk ||ψfk ||F = supk ||fk ||E . F Since the embedding of .EF into .(L1 + L∞ ) (0, ∞) is continuous, it is clear that .supk ||fk ||L1 +L∞ < ∞. The space .(L1 + L∞ ) (0, ∞) has the Fatou property and so, there exists .f ∈ (L1 + L∞ ) (0, ∞) such that .0 ≤ fk ↑k f . It follows from Lemma 6.8.3 (iii) that .ψfk ↑k ψf in F . Consequently, .ψf = x ∈ F and, therefore, .f ∈ EF . Moreover, .||f ||EF = ||x||F = supk ||fk ||EF . (iv) If F is a KB-space, then F has order continuous norm and the Fatou property (see Lemma 5.3.8). It follows from (i) and (iii) that .EF has order continuous norm and the Fatou property. Consequently, .EF is a KB-space. (v) Suppose that .g ∈ EF and that .f ∈ S (0, ∞) is such that .f ≺≺ g (which implies that .f ∈ (L1 + L∞ ) (0, ∞)). It follows from Lemma 6.8.3 (iv) that .ψf ≺≺ ψg. Since F is fully symmetric, this implies that .ψf ∈ F and .||ψf ||F ≤ ||ψg||F . Therefore, .f ∈ EF and .||f ||E ≤ ||g||EF . The proof is F complete. u n Remark 6.8.11 Concerning reflexivity some care has to be taken. Actually, if {0} /= F ⊆ l∞ is a strongly symmetrically normed sequence space, then the space .EF is never reflexive. Indeed, it is easily verified that the subspace .{f ∈ EF : f = 0 a.e. on [1, ∞)} is isomorphic to .L1 (0, 1). This also shows that, in general, the spaces .EF× and .EF × are not equal. Indeed, suppose that .F /= {0} is reflexive. Then its Köthe dual .F × has order continuous norm (see Theorem 5.3.10) and so, .EF × has order continuous norm as well (see Proposition 6.8.10 (i)). Furthermore, F has order continuous norm and the Fatou property and hence, .EF has order continuous norm and the Fatou property (see Proposition 6.8.10, (i) and (iii)). Since .EF is not reflexive, it follows that .EF× does not have order continuous norm (once again by Theorem 5.3.10). Consequently, the spaces .EF× and .EF × are different. .

Let .M be a semi-finite von Neumann algebra equipped with a semi-finite faithful normal trace .τ . Furthermore, assume that .M is atomic and that all minimal

6.8 Sequence Spaces

451

projections .p ∈ P (M) satisfy .τ (p) = 1. For each .x ∈ S (τ ) = M, the generalized singular value function .μ (x) is given by μ (x) =

.

Σ∞ n=0

μ (n; x) χ[n,n+1)

(the numbers .μn (x) = μ (n; x), .n ∈ N0 , are called the singular values of the operator x). Suppose that .F ⊆ l∞ is a strongly symmetrically normed sequence space and define { } F (τ ) = x ∈ M : {μn (x)}∞ n=0 ∈ F , || || || ||x||F (τ ) = ||{μn (x)}∞ x ∈ F (τ ) . n=0 F , .

The important observation is the following lemma. Lemma 6.8.12 Let E be a strongly symmetrically normed space on .(0, ∞) and let the strongly symmetrically normed sequence space .lE be defined by setting lE = {x ∈ l∞ : μ (x) ∈ E}

.

(see (6.35)). Then, lE (τ ) = E (τ ) and ||x||lE (τ ) = ||x||E(τ ) ,

.

x ∈ lE (τ ) .

In particular, if F is a strongly symmetrically normed sequence space and .EF is the strongly symmetrically normed space defined by (6.36) and (6.38) (see also Proposition 6.8.8), then F (τ ) = EF (τ ) and ||x||F (τ ) = ||x||EF (τ ) ,

.

x ∈ F (τ ) .

Proof If .x ∈ M and .{μn (x)}∞ n=0 is its singular value sequence, then μ (x) =

.

Σ∞ n=0

μn (x) χ[n,n+1) = μ ({μn (x)}) .

Consequently, .x ∈ E (τ ) if and only if .μ (x) ∈ E, if and only if .μ ({μn (x)}) ∈ E, if and only if .{μn (x)} ∈ lE , if and only if .x ∈ lE (τ ) . Hence, .lE (τ ) = E (τ ). Furthermore, if .x ∈ lE (τ ), then .

||x||lE (τ ) = ||{μn (x)}||lE = ||μ ({μn (x)})||E = ||μ (x)||E = ||x||E(τ ) .

This proves the first part of the lemma. The second part of the lemma follows now from the fact that .F = lEF (see u n Proposition 6.8.8).

452

6 Examples

The following proposition is now an immediate consequence of Lemma 6.8.12, in combination with Propositions 6.8.8, 6.8.10 and the results in Sect. 6.1. Proposition 6.8.13 Let .(M, τ ) be an atomic semi-finite von Neumann algebra with all minimal projections .p ∈ P (M) satisfying .τ (p) = 1. If .F ⊆ l∞ is a strongly symmetrically normed sequence space, then the following statements hold. ( ) (i) The space . F (τ ) , ||·||F (τ ) is a strongly symmetrically normed space of .τ measurable operators. (ii) If F is complete, then .F (τ ) is complete. (iii) If F has order continuous norm, then .F (τ ) has order continuous norm. (iv) If F has a Fatou norm, then .F (τ ) has a Fatou norm. (v) If F has the Fatou property, then .F (τ ) has the Fatou property. (vi) If F is a KB-space, then .F (τ ) is also a .KB-space. (vii) If F is fully symmetric, then .F (τ ) is fully symmetric. As observed in Remark 6.8.11, some care has to be taken concerning reflexivity. It is still possible, however, to obtain the result of Proposition 6.8.15 below via the following ( argument. Let ) .F( ⊆ l∞ and .EF )be as above. As observed in Lemma 6.8.12, . F (τ ) , ||·||F (τ ) = EF (τ ) , ||·||E(τ ) . Therefore, it follows from the Köthe duality of .EF (τ ) that F (τ )× = EF (τ )× = EF× (τ ) .

.

(6.40)

Let .lE × be the sequence space induced by .EF× (see (6.35)). From the Köthe duality F

of .lEF it follows that .lE × = l× EF . Therefore, using Proposition 6.8.8, it follows that F

F × = l× EF = lE × .

.

F

Consequently, via an appeal to the first part of Lemma 6.8.12, it follows that F × (τ ) = lE × (τ ) = EF× (τ ) .

.

F

(6.41)

A combination of (6.40) and (6.41) yields that .F × (τ ) = F (τ )× . For the sake of later reference this result is formulated in the next proposition. Proposition 6.8.14 If F is a strongly symmetrically normed sequence space, then F (τ )× = F × (τ ) ,

.

with equality of norms. The following proposition is now an immediate consequence of the above. Proposition 6.8.15 Let F be a strongly symmetric sequence space and .(M, τ ) be a semi-finite atomic von Neumann algebra with all minimal projections having trace equal to one. If F is reflexive, then .F (τ ) is reflexive.

6.8 Sequence Spaces

453

Proof If F is reflexive, then F is a KB-space and .F × has order continuous norm. By Proposition 6.8.13, this implies that .F (τ ) is a KB-space. Furthermore, it follows from Proposition 6.8.14 that .F (τ )× = F × (τ ). Since .F × has order continuous norm, it follows from Proposition 6.8.13 (ii) that .F (τ )× has order continuous norm. Hence, it may be concluded that .F (τ ) is reflexive (see Theorem 5.3.10). u n This chapter will now be concluded with an example of this construction which is of considerable importance, the Macaev ideal. This ideal has appeared in Voiculescu’s studies on perturbations of operators and quasi-central approximate units [132, 133]. In turn, Voiculescu’s work and the Macaev ideal and its dual are a crucial instrument in Connes’ noncommutative geometry [22]. The Macaev ideal is constructed by taking F = Aω := {x ∈ l∞ :

∞ Σ

.

n=0

1 μ(n; x) < ∞}, n+1

which is easily to be a strongly symmetric sequence space with norm Σ∞ checked 1 ||x||ω = μ(n; x). Performing the procedure of defining .Aω (τ ) as n=0 n+1 discussed in this section on the von Neumann algebra .M = B(H ) equipped with the standard trace .τ gives the standard definition of the Macaev ideal

.

Sω := Aω (τ ) = {x ∈ K(H ) :

∞ Σ

.

n=0

1 μn (x) < ∞}. n+1

Recall the spaces .Lp (τ ) from Sect. 6.2. In the .B(H ) setting, the construction of Lp (τ ) coincides with the sequence space construction .lp (τ ) as in this section. This space is called the Schatten ideal .Sp := Lp (τ ) = lp (τ ) (see Remark 6.2.3). 1 ∞ Since the sequence .{ n+1 }n=0 is an element of .lq for all .1 < q ≤ ∞, it follows that for any operator .x ∈ Cp , .1 ≤ p < ∞, with . p1 + q1 = 1,

.

||x||Sω =

∞ Σ

.

n=0

||{ 1 }∞ || 1 || || μn (x) ≤ || || ||{μn (x)}∞ n=0 ||p < ∞. n + 1 n=0 q n+1

Hence, the Macaev ideal .Sω contains all Schatten ideals .Sp with .1 ≤ p < ∞. Of equal importance to the Macaev ideal itself is its dual space. It is defined in a similar manner, taking this time Σ 1 μ(n; x) < ∞}, log(n + 2) n

F = AO := {x ∈ l∞ : sup

.

n∈N0

j =0

454

6 Examples

which again is Σ a strongly symmetric sequence space with norm .||x||O = n 1 supn∈N0 log(n+2) j =0 μ(n; x). The dual of the Macaev ideal is then defined as Σ 1 .SO := AO (τ ) = {x ∈ K(H ) : sup μn (x) < ∞}. n∈N0 log(n + 2) n

j =0

The duality of this space to .Sω is implemented by the trace. Namely, any .x ∈ SO defines a linear functional on .Sω by y |→ τ (xy),

.

y ∈ Sω ,

and any linear functional on .Sω is of this form. The proof is omitted here but can be found in [25, Chapter 4]. The space .SO in turn exhibits an interesting relation to the Schatten classes .Sp . Namely, if .x ∈ SO , then ∞

μn (x) ≤

.

log(n + 2) 1 Σ ||x||SO . μn (x) ≤ n+1 n+1 n=0

Hence, .μn (x) ∈ lp for all .1 < p ≤ ∞, and, therefore, .Sp contains .SO for all 1 < p ≤ ∞. This dual space .SO is also commonly referred to as the Dixmier ideal or the Dixmier–Macaev ideal, and denoted by .M1,∞ . The set of traces on this space has a rich structure, which plays a significant role in Connes’ noncommutative geometry. The reader is referred to [80] for a detailed study.

.

References: [57, 58, 60, 79, 80, 89, 94–97, 108, 112, 132, 133].

Chapter 7

Interpolation

Abstract This chapter presents several interpolation theorems in the setting of noncommutative spaces introduced in previous chapters. In particular, several classical interpolation results are obtained in the noncommutative setting, including noncommutative versions of theorems of Marcinkiewicz, Calderón, and Boyd. Applications are given of the complex method to the geometry of noncommutative .Lp -spaces.

7.1 Basic Ideas We gather in this section some basic notions from the theory of interpolation that will be needed in the sequel. Following [76], a Banach couple .X = (X0 , X1 ) is a pair of Banach spaces .X0 and .X1 that are algebraically and topologically embedded in a separated topological linear space. With any Banach couple .X = (X0 , X1 ), we may associate the following Banach spaces: 1. The space .X0 ∩ X1 equipped with the norm ||x||X0 ∩X1 = max(||x||X0 , ||x||X1 ),

.

x ∈ X0 ∩ X1 .

2. The space .X0 + X1 equipped with the norm ||x||X0 +X1 = inf{||y||X0 +||z||X1 : x = y+z, y ∈ X0 , z ∈ X1 },

.

x ∈ X0 +X1 .

Let .X = (X0 , X1 ) and .Y = (Y0 , Y1 ) be Banach couples. A linear map T from the space .X0 + X1 to the space .Y0 + Y1 is called a bounded operator from the couple .X to the couple .Y if and only if T is a bounded operator from .X0 + X1 to .Y0 + Y1 and T maps .X0 into .Y0 , .X1 into .Y1 , respectively. We denote by .A (X, Y) the linear space of all bounded operators from the couple .X to the couple .Y. This is a Banach space in the norm ||T ||A (X,Y) = max(||T ||X0 →Y0 , ||T ||X1 →Y1 ),

.

T ∈ A (X, Y).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. G. Dodds et al., Noncommutative Integration and Operator Theory, Progress in Mathematics 349, https://doi.org/10.1007/978-3-031-49654-7_7

455

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7 Interpolation

A Banach space X is said to be intermediate for the Banach couple .X if and only if .X0 ∩ X1 ⊆ X ⊆ X0 + X1 with continuous inclusions. If .X and .Y are two Banach couples, and X and Y are spaces intermediate for the couple .X and .Y, respectively, then the pair .(X, Y ) is called an interpolation pair for the pair .(X, Y), if and only if every bounded operator from the couple .X to the couple .Y maps X into Y , in which case it is necessarily bounded. If the pair .(X, Y ) is an interpolation pair for .(X, Y), then there exists a constant .c > 0 (interpolation constant) such that ||T ||X→Y ≤ c||T ||A (X,Y) ,

for all T ∈ A (X, Y).

.

An interpolation pair .(X, Y ) for the pair .(X, Y) of Banach couples is called an exact interpolation pair for the pair .(X, Y) if and only if the above inequality holds with .c = 1; it is called an interpolation pair of exponent .θ, 0 ≤ θ ≤ 1, if and only if ||T ||X→Y ≤ c ||T ||1−θ ||T ||θX X →Y

.

0

0

1 →Y1

for all T ∈ A (X, Y),

,

where c is an interpolation constant; it is called an exact interpolation pair of exponent .θ if and only if the last inequality holds with .c = 1. If .X0 = Y0 , .X1 = Y1 , and .X = Y and the above corresponding condition holds, then we say that X is an interpolation space, exact interpolation space, interpolation space of exponent .θ , exact interpolation space of exponent .θ for the couple .X = (X0 , X1 ), respectively. It is not difficult to see [76] that any interpolation space for a Banach couple can be renormed, with an equivalent norm, to become an exact interpolation space. In fact, if X is an interpolation space for the Banach couple .X and if .c > 0 is such that ||T ||X→X ≤ c ||T ||A (X,X) ,

.

for all

T ∈ A (X, X),

then X equipped with the equivalent norm .||.||∗X defined by ||x||∗X =

.

1 c

sup 0/=T ∈A (X,X)

||T x||X , ||T ||A (X,X)

x ∈ X,

(7.1)

is an exact interpolation space for the couple .X.

7.1.1 Conditional Expectations Let .M be a von Neumann algebra equipped with a semi-finite faithful normal trace τ . The von Neumann subalgebra .N ⊆ M is said to be proper if and only if the restriction .τN = τ |N of the trace .τ to .N is again semi-finite. Note that, by the remarks at the beginning of Sect. 2.9, that .S(τN ) is a unital .∗-subalgebra of .S(τ );

.

7.1 Basic Ideas

457

moreover, if .{xα } ⊆ S(τN ) satisfies .0 ≤ xα ↓α 0 in .S(τN ), then .0 ≤ xα ↓α 0 holds in .S(τ ). This remark will be used repeatedly in what follows. Let .N ⊆ M be a proper von Neumann subalgebra. If .x ∈ (L1 + L∞ )(τ ), then, by the above remark, the continuous linear functional y → τ (xy),

.

y ∈ (L1 ∩ L∞ )(τN )

is normal on .(L1 ∩ L∞ )(τN ), and so, by Theorem 5.2.9, there exists a uniquely determined element .EN (x) ∈ (L1 ∩ L∞ )(τN )× = (L1 + L∞ )(τN ) such that τ (EN (x)y) = τ (xy),

.

y ∈ (L1 ∩ L∞ )(τN ).

The mapping EN : (L1 + L∞ )(τ ) → (L1 + L∞ )(τN )

.

is clearly linear and is called the conditional expectation of .(L1 + L∞ )(τ ) on .(L1 + L∞ )(τN ). The following proposition gathers the basic properties of conditional expectations (see [131], and also Proposition 2.1 in [31]). Proposition 7.1.1 Let .N ⊆ M be a von Neumann subalgebra that is proper in the sense that the restriction .τN of the trace .τ to .N is semi-finite. For each .x ∈ L1 (τ ) + M, there exists a uniquely determined element .E(x) ∈ L1 (τN ) + N called the conditional expectation with respect to .N such that τ (xy) = τ (E(x)y),

.

∀y ∈ L1 (τN ) ∩ N.

The conditional expectation operator .E has the following properties: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

E(x ∗ ) = E(x)∗ . .E(x) ≥ 0 if .x ≥ 0. If .x ≥ 0 and .E(x) = 0, then .x = 0. .E(x) = x for all .x ∈ L1 (τN ) + N. ∗ ∗ .E(x) E(x) ≤ E(x x), for all .x ∈ M. .0 ≤ xα ↑α x ⊆ M implies .0 ≤ E(xα ) ↑α E(x) ⊆ N. .E(x) ≺≺ x, for all .x ∈ L1 (τ ) + M. .E(xy) = xE(y), for all .x ∈ L1 (τN ) and .y ∈ M. .E(xy) = E(x)y, for all .x ∈ L1 (τ ) and .y ∈ N. .τ (E(x)) = τ (x) for all .x ∈ L1 (τ ). .

In particular, the above proposition implies that conditional expectations are contractions in .L1 (τ ) and .M and hence are admissible operators for the couple .(L1 (τ ), M). We refer to Sect. 3.10 for the detailed study of the spaces .G(τ ) = (L1 + L∞ )(τ ) and .H (τ ) = (L1 ∩ L∞ )(τ ).

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7 Interpolation

Proposition 7.1.2 Let .N ⊆ M be a proper von Neumann subalgebra, and let .E denote the conditional relative to .N. If .E ⊆ S (τ ) is fully symmetric, ) ( expectation then .E (E) ⊆ E, .E E × ⊆ E × , and τ (E(x)y) = τ (xE(y)),

.

∀x ∈ E, ∀y ∈ E × .

Proof Suppose that .x ∈ E, y ∈ E × . Without loss of generality, it may be assumed that .0 ≤ x, y. There exists a net .(xα ) in .L1 (τN ) ∩ N such that .0 ≤ xα ↑α E (x). Using that for any .0 ≤ w ∈ E × , the linear functional .z |→ τ (wz), for .z ∈ E, is positive and normal (order continuous) (see Lemma 4.3.5 (iv) and Lemma 4.3.6 (i)), and the definition of a conditional expectation, it follows that τ (E(x)y) = sup τ (xα y) = sup τ (xα E(y)) = τ (E(x)E(y)).

.

α

α

A similar argument shows that τ (xE(y)) = τ (E(x)E(y)),

.

and this suffices to conclude the proof of the proposition.

u n

7.1.2 Interpolation Functors A mapping .F from Banach couples to Banach spaces is called an (exact) interpolation functor if for every pair .(X, Y) of Banach couples, the pair .(F(X), F(Y)) is an (exact) interpolation pair for .(X, Y). If .X = (X0 , X1 ) is a Banach couple, set .Σ(X) = X0 + X1 and .a(X) = X0 ∩ X1 . It is easily checked that .Σ and .a are exact interpolation functors. We shall have need of the following theorem (see [76] Theorem I.4.9 and [9] Theorem 2.5.1). Theorem 7.1.3 (Aronszajn–Gagliardo) If X is an (exact) interpolation space for the Banach couple .X, then there exists an exact interpolation functor .F such that F(X) = X

.

with equivalence (equality) of norms. References: [4, 9, 31, 76, 121, 131].

7.2 Interpolation Spaces for the Couple (L1 (τ ), M)

459

7.2 Interpolation Spaces for the Couple (L1 (τ ), M) Now suppose that .M and .N are semi-finite von Neumann algebras equipped with faithful normal semi-finite traces .τ and .σ , respectively. As is usual, for .x ∈ S(τ ) and .y ∈ S(σ ), we write .y ≺≺ x whenever y is submajorized by x (see Definition 3.9.1). It is now appropriate to reformulate Theorem 3.10.13 as an explicit interpolation theorem. Theorem 7.2.1 Let E and F be linear subspaces of .L1 (σ ) + N and .L1 (τ ) + M, respectively: (i) The following two statements are equivalent: (a) .T ∈ A(N, M) =⇒ T (E) ⊆ F. (b) .x ∈ E, y ∈ L1 (τ ) + M and y ≺≺ x =⇒ y ∈ F. (ii) If, in addition, the spaces .E, F are normed spaces, then the following two statements are equivalent: (c) .T ∈ Σ(N, M) =⇒ T (E) ⊆ F and ||T ||E→F ≤ 1. (d) .x ∈ E, y ∈ L1 (τ ) + M and y ≺≺ x =⇒ y ∈ F and ||y||F ≤ ||x||E . Proof (a).⇒(b): Let .x ∈ E, y ∈ L1 (τ ) + M and .y ≺≺ x. By Theorem 3.10.13, there exists .T ∈ Σ(N, M) such that .y = T x. Hence, (a) implies that .y = T x ∈ F . (b).⇒(a): Let .T ∈ A(N, M) and .x ∈ E. Proposition 3.10.3 implies that there exists a number .α ≥ 0 such that .T x ≺≺ αx. Now (b) implies that .T x ∈ F . The proof of (ii) is similar. u n Corollary 7.2.2 Let E, F be Banach spaces that are intermediate for the Banach couples .(L1 (σ ), N), (L1 (τ ), M), respectively. Then: (i) .(E, F ) is an interpolation pair for the pair .((L1 (σ ), N), (L1 (τ ), M)) if and only if x ∈ E, y ∈ L1 (τ ) + M and

.

y ≺≺ x =⇒ y ∈ F.

(ii) .(E, F ) is an exact interpolation pair for the pair .((L1 (σ ), N), (L1 (τ ), M)) if and only if x ∈ E, y ∈ L1 (τ ) + M and y ≺≺ x =⇒ y ∈ F and ||y||F ≤ ||x||E .

.

For the case that .E = F , the preceding Corollary admits the following sharpening. Corollary 7.2.3 Let E be a Banach space that is a linear subspace of .L1 (τ ) + M. Then:

460

7 Interpolation

(i) E is an interpolation space for the Banach couple .(L1 (τ ), M) if and only if x ∈ E, y ∈ L1 (τ ) + M and

.

y ≺≺ x =⇒ y ∈ E.

(ii) E is an exact interpolation space for the Banach couple .(L1 (τ ), M) if and only if x ∈ E, y ∈ L1 (τ ) + M and y ≺≺ x =⇒ y ∈ E and ||y||E ≤ ||x||E .

.

Proof The only additional point to be noted is that either of the stated conditions on E implies that E is intermediate for the couple .(L1 (τ ), M). For (ii), this follows immediately from Lemma 5.1.3. The proof of (i) can be reduced to (ii) by putting an equivalent norm on E that makes E an exact interpolation space, see formula (7.1). u n In the next corollary, we will only deal with exact interpolation spaces. Corollary 7.2.4 If E is an exact interpolation space on .(0, a) for the couple .(L1 (m), L∞ (m)), then .E(τ ) is an exact interpolation space for the couple .(L1 (τ ), M). Of course, part (ii) of Corollary 7.2.3 asserts that the exact interpolation spaces for the couple .(L1 (τ ), M) are precisely those that are fully symmetric. The preceding Corollary 7.2.4 permits ready identification of a wide class of strongly symmetric spaces that are exact interpolation spaces for the couple .(L1 (τ ), M). For example, since the familiar Orlicz, Marcinkiewicz, and Lorentz spaces are fully symmetric, it follows that their noncommutative versions are exact interpolation spaces for the couple .(L1 (τ ), M) for all semi-finite .(M, τ ). It should be noted, however, that there exist strongly symmetric spaces that are not interpolation spaces for the couple .(L1 (τ ), M). One such example is exhibited in [76] Theorem II 5.11 as a closed subspace of a Marcinkiewicz space on .[0, ∞). It will be shown later (see Corollary 7.3.5) that the only interpolation spaces for the couple .(L1 (τ ), M) are those that can be constructed as the noncommutative spaces arising from some fully symmetric function space on .(0, a). References: [44, 76, 117].

7.3 Principal Theorems on Noncommutative Interpolation As before, let .M be a von Neumann algebra equipped with a semi-finite faithful normal trace .τ . Proposition 7.3.1 If .E0 and .E1 are exact interpolation spaces on .[0, a) for the couple .(L1 (m), L∞ (m)), then .(E0 + E1 )(τ ) = E0 (τ ) + E1 (τ ) with equality of norms.

7.3 Principal Theorems on Noncommutative Interpolation

461

Proof If .x ∈ E0 (τ ) + E1 (τ ) and if .x = x0 + x1 with .xi ∈ Ei (τ ), .i = 0, 1, then Proposition 3.10.17 shows that there exist functions .f0 , f1 ∈ L1 (R+ ) + L∞ (R+ ) with μ(x) = f0 + f1

.

and

fi ≺≺ xi ,

i = 0, 1.

Since .E0 , E1 are exact interpolation spaces for the couple .(L1 (m), L∞ (m)), it follows that .fi ∈ Ei , i = 0, 1, and so .x ∈ (E0 + E1 )(τ ). Moreover, ||fi ||Ei ≤ ||μ(xi )||Ei = ||xi ||Ei (τ ) ,

.

i = 0, 1.

Consequently, ||μ(x)||E0 +E1 ≤ ||x||E0 (τ )+E1 (τ ) .

.

Similarly, if .x ∈ (E0 + E1 )(τ ), then .x ∈ E0 (τ ) + E1 (τ ) and ||x||E0 (τ )+E1 (τ ) ≤ ||x||(E0 +E1 )(τ ) ,

.

which completes the proof.

u n

Proposition 7.3.2 Let .E1 , E2 ⊆ L1 (τ )+M be fully symmetric spaces, and suppose that E is a Banach space that is a linear subspace of .L1 (τ ) + M: (i) If E is an exact interpolation space for the couple .(E1 , E2 ), then E is fully symmetric. (ii) If E is an interpolation space for the couple .(E1 , E2 ), then there exists an equivalent norm .|| · ||E ' such that .(E, || · ||E ' ) is fully symmetric. Proof (i) If T is a bounded operator from the couple .(L1 (τ ), M) to itself of norm at most one, then T is a bounded linear operator from the couple .(E1 , E2 ) to itself of norm at most one, since .E1 , E2 are fully symmetric. Since E is an exact interpolation space for the couple .(E1 , E2 ), it follows that T maps E to itself with norm at most one and so E is an exact interpolation space for the pair .(L1 (τ ), M). That E is fully symmetric now from Corollary 7.2.2 (ii). (ii) The same argument as above shows that E is an interpolation space for the couple .(L1 (τ ), M). The assertion of (ii) now follows from the equality (7.1) and u n the statement of (i) preceding. Corollary 7.3.3 If E is an exact interpolation space for the couple .(Lp (0, ∞), Lq (0, ∞)), then E is a fully symmetric function space. If E is an interpolation space for the couple .(Lp (0, ∞), Lq (0, ∞)), then E admits an equivalent norm .||·||E ' such that .(E, || · ||E ' ) is fully symmetric.

462

7 Interpolation

Theorem 7.3.4 Let .E0 , E1 be fully symmetric Banach function spaces on .(0, a). If F is an (exact) interpolation functor, then

.

F(E0 , E1 )(τ ) = F(E0 (τ ), E1 (τ ))

.

with equivalence (equality) of norms. Proof Suppose first that .x ∈ F(E0 , E1 )(τ ) so that, in particular, .μ(x) ∈ F(E0 , E1 ). Since .x ≺≺ μ(x), it follows from Theorem 3.10.13 that there exists .Sx ∈ Σ(L∞ (m), M) such that .Sx (μ(x)) = x. From the assumption that .E0 , E1 are fully symmetric, it also follows that .E0 (τ ), E1 (τ ) are fully symmetric and consequently .Sx maps the couple .(E0 , E1 ) to the couple .(E0 (τ ), E1 (τ )). Since .F is an (exact) interpolation functor, it now follows that .Sx maps .F(E0 , E1 ) continuously (respectively, with norm 1) into .F(E0 (τ ), E1 (τ )). In particular, this implies that x = Sx (μ(x)) ∈ F(E0 (τ ), E1 (τ ))

.

and that ||x||F(E0 (τ ),E1 (τ )) ≤ c||μ(x)||F(E0 ,E1 ) = c||x||F(E0 ,E1 )(τ ) ,

.

with .c = 1 in the case that .F is exact. Conversely, suppose that .x ∈ F(E0 (τ ), E1 (τ )). Since .μ(x) ≺≺ x, it follows again from Theorem 3.10.13 that there exists .Tx ∈ Σ(M, L∞ (m)) such that .Tx (x) = μ(x). Since T maps the couple .(E0 (τ ), E1 (τ )) to the couple .(E0 , E1 ) with norm at most 1, and since .F is an (exact) interpolation functor, it follows that .Tx maps .F(E0 (τ ), E1 (τ )) continuously (respectively, with norm 1) into .F(E0 , E1 ). In particular, this implies that μ(x) = Tx (x) ∈ F(E0 , E1 )

.

and that ||x||F(E0 ,E1 )(τ ) = ||μ(x)||F(E0 ,E1 ) ≤ c||x||F(E0 (τ ),E1 (τ )) ,

.

with .c = 1 in the case that .F is exact. This suffices to complete the proof of the theorem. n u Theorem 7.3.4 combined with the Aronszajn–Gagliardo theorem (Theorem 7.1.3) yields the following interesting consequence.

7.3 Principal Theorems on Noncommutative Interpolation

463

Corollary 7.3.5 If .E0 , E1 are fully symmetric spaces on .(0, a) and if E is an (exact) interpolation space for the couple .(E0 (τ ), E1 (τ )), then there exists a fully symmetric space .F = F ((0, a)), which is an exact interpolation space for the couple .(E0 , E1 ) such that E = F (τ )

.

with equivalence (equality) of norms. Now let .(N, σ ) be a von Neumann algebra equipped with a semi-finite faithful normal trace, like .(M, τ ). Theorem 7.3.6 Let .E0 , E1 , F0 , F1 be fully symmetric spaces on .(0, a): (i) If .(E, F ) is an (exact) interpolation couple for the pair .((E0 , E1 ), (F0 , F1 )), then .(E(σ ), F (τ )) is an (exact) interpolation couple for .((E0 (σ ), E1 (σ )), (F0 (τ ), F1 (τ ))). (ii) If .(E, F ) is an (exact) interpolation couple of exponent .θ , .0 ≤ θ ≤ 1, for the pair .((E0 , E1 ), (F0 , F1 )), then .(E(σ ), F (τ )) is an (exact) interpolation couple of exponent .θ for .((E0 (σ ), E1 (σ )), (F0 (τ ), F1 (τ ))). Proof (i) Let V be a bounded operator from the couple .(E0 (σ ), E1 (σ )) to the couple .(F0 (τ ), F1 (τ ), let .x ∈ E(σ ), and set .y = V (x). By Theorem 3.10.13, there exist operators .Sx ∈ Σ(L∞ (m), N), Ty ∈ Σ(M, L∞ (m)) such that Sx (μ(x)) = x,

.

Ty (y) = μ(y).

Setting V˜ = Ty V Sx ,

.

it follows that .V˜ is a bounded operator from the couple .(E0 , E1 ) to the couple ˜ || ≤ ||V ||, which satisfies .(F0 , F1 ) with .||V V˜ μ(x) = μ(y).

.

Since .(E, F ) is an (exact) interpolation couple for the pair .((E0 , E1 ), (F0 , F1 )) and since .μ(x) ∈ E, it follows that .μ(y) ∈ F , or equivalently, that .y = V (x) ∈ F (τ ). Consequently, ||y||F (τ ) = ||μ(y)||F ≤ c||V ||||μ(x)||E = c||V ||||x||E(τ ) .

.

Here c is an interpolation constant that can be taken to be 1 in the case that .(E, F ) is an exact interpolation couple.

464

7 Interpolation

(ii) If .(E, F ) is an interpolation pair of exponent .θ , then ||V˜ μ(x)||F ≤ c ||V˜ ||1−θ ||V˜ ||θE E →F

.

0

1 →F1

0

||μ(x)||E ,

where c is an interpolation constant. Hence, ||V x||F (τ ) ≤ c ||V ||1−θ E (σ )→F

.

0

||V ||θE

0 (τ )

1 (σ )→F1 (τ )

||x||E(σ ) .

If .(E, F ) is an exact interpolation pair of exponent .θ , then .c = 1.

u n

References: [44, 76].

7.4 The K-Functional If .(X0 , X1 ) is a Banach couple, and if .x ∈ X0 + X1 , define for every .t > 0, K(t, x; X1 , X2 ) ≡ K(t, x) = inf{||x0 ||X0 +t||x1 ||X1 : x = x0 +x1 , xi ∈ Xi , i = 0, 1}.

.

Proposition 7.4.1 Suppose that .X = (X0 , X1 ), Y = (Y0 , Y1 ) are Banach couples. If .T : X0 + X1 → Y0 + Y1 is a bounded linear operator from the couple .X to the couple .Y of norm at most 1, then, for every .x ∈ X0 + X1 and every .t > 0, K(t, T x; Y0 , Y1 ) ≤ K(t, x; X0 , X1 ).

.

Proof If .t > 0 and .x ∈ X0 + X1 , then K(t, T x; Y0 , Y1 ) = inf{||z0 ||Y0 + t||z1 ||Y1 : z0 ∈ Y0 , z1 ∈ Y1 , T x = z0 + z1 }

.

≤ inf{||T x0 ||Y0 + t||T x1 ||Y1 : x0 ∈ X0 , x1 ∈ X1 , x = x0 + x1 } ≤ inf{||x0 ||X0 + t||x1 ||Y1 : x0 ∈ X0 , x1 ∈ X1 , x = x0 + x1 } = K(t, x; X0 , X1 ), u n

as claimed.

Note that Theorem 3.9.16 may be readily formulated in terms of the K-functional for the pair .(L1 (τ ), M) to yield that, for all .x ∈ L1 (τ ) + M, F K(t, x) ≡ K(t, x; L1 (τ ), M) =

t

μ(s; x)ds,

.

t > 0.

(7.2)

0

We now show that the study of the K-functionals for certain pairs of noncommutative fully symmetric spaces may be reduced to the corresponding .K-functionals in the commutative setting.

7.4 The K-Functional

465

Proposition 7.4.2 If .E0 , E1 are fully symmetric spaces on .[0, α), then, for all .x ∈ E0 (τ ) + E1 (τ ) = (E0 + E1 )(τ ), K(t, x; E0 (τ ), E1 (τ )) = K(t, μ(x); E0 , E1 ).

.

Proof By Theorem 3.10.13, there exist operators .Sx ∈ Σ(L∞ (m), M), .Tx ∈ Σ(M, L∞ (m)) such that Sx (μ(x)) = x,

.

Tx (x) = μ(x).

Since .E0 , E1 ⊆ S(m) are fully symmetric, it follows from Corollary 7.2.2 that the pair .(E0 , E0 (τ )) (respectively, .(E1 , E1 (τ ))) is an exact interpolation pair for the pair .((L1 (m), L∞ (m)), (L1 (τ ), M)). Consequently, the linear map Sx : L1 (m) + L∞ (m) → L1 (τ ) + M

.

is a bounded linear operator from the couple .(E0 , E1 ) to the couple .(E0 (τ ), E1 (τ )) of norm at most 1. By Proposition 7.4.1, it follows that K(t, x; E0 (τ ), E1 (τ )) = K(t, Sx (μ(x)); E0 (τ ), E1 (τ ))

.

≤ K(t, μ(x); E0 , E1 ). Since .E0 (τ ), E1 (τ ) are fully symmetric, exactly the same argument as above now shows that the linear map Tx : L1 (τ ) + M → L1 (m) + L∞ (m)

.

is a bounded linear operator from the couple .(E0 (τ ), E1 (τ )) to the couple (E0 (m), E1 (m)) of norm at most 1. Again applying Proposition 7.4.1, it follows that

.

K(t, μ(x); E0 , E1 ) = K(t, Tx (x); E0 , E1 )

.

≤ K(t, x; E0 (τ ), E1 (τ )). u n

This suffices to complete the proof.

We briefly introduce the notation .a(t, p, q) ≈p,q b(t, p, q) for real-valued functions .a, b meaning that there exists a constant .Cp,q > 0 depending only on .p, q such that −1 Cp,q b(t, p, q) ≤ a(t, p, q) ≤ Cp,q b(t, p, q),

.

∀t.

466

7 Interpolation

Similarly, .≈p is used if the constant only depends on p. A simple application of the preceding Proposition 7.4.2 using the K -functional formulae given in [66] yields that, if .1 ≤ p < q ≤ ∞, and if .x ∈ Lp (τ ) + Lq (τ ), then K(t, x; Lp (τ ), Lq (τ )) = K(t, μ(x); Lp (m), Lq (m)) )1/p (F θ t

≈p,q

.

μ(x)p dm

(7.3)

0

(F +t

∞

)1/q μ(x)q dm

q < ∞,

,

tθ

with .θ = (p−1 − q −1 )−1 and K(t, x; Lp (τ ), M) = K(t, μ(x); Lp (m), L∞ (m)) )1/p (F p

.

t

≈p

μ(x)p dm

,

q = ∞.

(7.4)

0

Note that a generalization of this formula will be discussed in Sect. 7.6. References: [8, 9, 66].

7.5 Real Interpolation and the Noncommutative Marcinkiewicz Theorem The effect of Theorems 7.3.4 and 7.3.6 is to reduce certain noncommutative interpolation results to the special case obtained by taking .M to be .L∞ (m). We first illustrate their utility for the real interpolation method. For ease of reference, consider the Lorentz spaces .Lp,q (m) for .1 ≤ p, q ≤ ∞ equipped with the equivalent norm .|| · ||(p,q) (see Lemma 6.7.7). As observed in Theorem)6.1.7 (vi), ( if .1 < p < ∞ and .1 ≤ q ≤ ∞, then the space . L(p,q) (τ ), || · ||(p,q) is a fully symmetric space with the Fatou property and, if .1 ≤ q < ∞, then the norm is order continuous. It is convenient now to recall briefly the K-method of real interpolation due to Peetre [9], Chapter 3. For .0 < θ < 1, 1 ≤ q ≤ ∞ or for .0 ≤ θ ≤ 1, q = ∞, the Banach space .[X0 , X1 ]θ,q;K consists of all elements .x ∈ X0 + X1 for which the functional {(F ||x||θ,q;K =

.

is finite.

) ∞ −θ K(t, x))q dtt q 0 (t supt>0 t −θ K(t; x),

1

, 0 < θ < 1, 1 ≤ q < ∞; 0 ≤ θ ≤ 1, q = ∞

(7.5)

7.5 Real Interpolation and the Noncommutative Marcinkiewicz Theorem

467

The functor .[ . , . ]θ,q;K is an exact interpolation functor of exponent .θ (see [9], Theorem 3.1.2). An immediate consequence of Theorem 7.3.4 now follows. Corollary 7.5.1 If .E0 , E1 are fully symmetric spaces on .[0, a), then [E0 (τ ), E1 (τ )]θ,q:K = [E0 , E1 ]θ,q:K (τ ).

.

Note that, in the case of the Banach couple .(L1 (τ ), M), the K-functional is given by the explicit formula F

t

K(t; x) =

μ(x)dm,

.

x ∈ L1 (τ ) + M.

0

This permits ready identification of the space .[L1 (τ ), M]θ,q;K as a noncommutative Lorentz space (see Theorem 6.3.8). Proposition 7.5.2 (a) If .0 < θ < 1 and .1 ≤ q ≤ ∞, then [L1 (τ ), M]θ,q;K = L(p,q) (τ ),

.

1/p = 1 − θ.

(b) If .θ = 0 and .q = ∞, then [L1 (τ ), M]0,∞;K = L1 (τ ).

.

Proof (a) If .x ∈ L1 (τ ) + M, then, using (7.5), Proposition 7.4.2, and Lemma 6.7.8, (F

∞

||x||θ,q;K =

.

(t −θ K(t; x))q

0

(F

∞

=

∞

=

(t 0

)1

(t −θ K(t; μ(x)))q

0

(F

dt t

1−θ

q

dt t

M(μ(x))(t))

)1 q

q dt

t

= ||μ(x)||L(p,q) = ||x||L(p,q) (τ ) , where .1/p = 1 − θ .

)1 q

468

7 Interpolation

(b) In the case that .θ = 0 and .q = ∞ and again using (7.5), F ||x||0,∞;K = sup K(t; x) = sup t>0

F

∞

= 0

t>0

t

μ(s; x)ds

.

0

μ(s; x)ds = ||x||L1 (τ ) . u n

The proof is thereby complete.

More generally, the theorem that follows identifies the noncommutative interpolation space .[Lp0 ,q0 (τ ), Lp1 ,q1 (τ )]θ,q;K as a noncommutative Lorentz space. Theorem 7.5.3 If .1 ≤ q, pi , qi ≤ ∞, i = 0, 1, set .1/p = (1 − θ )/p0 + θ/p1 . If p0 /= p1 , then

.

[Lp0 ,q0 (τ ), Lp1 ,q1 (τ )]θ,q;K = Lp,q (τ ).

.

(7.6)

The formula also holds in the case that .p0 = p1 = p provided that .1/q = (1 − θ )/q0 + θ/q1 . Proof The theorem follows immediately from Theorem 7.3.4 and its commutative specialization given in [9] Theorem 5.3.1. u n The special case of the usual Marcinkiewicz Theorem for the spaces .Lp (m) (see [9] Theorem 5.3.2) now yields the following noncommutative version of the Marcinkiewicz Theorem. Again, .(M, τ ) and .(N, σ ) are von Neumann algebras with semi-finite faithful normal traces. Theorem 7.5.4 Let .1 ≤ pi , qi , ri , si ≤ ∞, i = 0, 1. Suppose that T : Lpi ,ri (σ ) → Lqi ,si (τ ), i = 0, 1,

.

where .p0 /= p1 , .

q0 /= q1 . If

1 1 1 = (1 − θ ) +θ , p p0 p1

1 1 1 = (1 − θ ) + θ , q q0 q1

with 0 < θ < 1,

then T : Lp,r (σ ) → Lq,r (τ ),

.

1 ≤ r ≤ ∞.

(7.7)

In particular, T : Lp (σ ) → Lq (τ )

.

provided .p ≤ q.

(7.8)

7.6 Holmstedt Formula

469

Proof The assertion (7.7) follows immediately from Theorem 7.3.6 and its commutative specialization given in [9] Theorem 5.3.2. The assertion (7.8) follows from the inclusion .Lq,p ⊆ Lq,q , whenever .p ≤ q. u n A more general consequence of Theorem 7.5.4 via the inclusion .Lq,r ⊆ Lq,s whenever .r ≤ s is that if T : Lpi ,ri (σ ) → Lqi ,si (τ ), i = 0, 1,

.

then T : Lp,r (σ ) → Lq,s (τ )

.

provided .1/p = (1 − θ )/p0 + θ/p1 , 1/q = (1 − θ )/q0 + θ/q1 , p0 /= p1 , q0 /= q1 and .1 ≤ r ≤ s ≤ ∞. A particular case is the following noncommutative extension of an interpolation theorem due to Calderón. Theorem 7.5.5 If T : Lpi ,p (σ ) → Lqi ,∞ (τ ),

.

i = 0, 1,

then T : Lp,r (σ ) → Lq,s (τ )

.

provided .r ≤ s and .pi , qi , p and q satisfy the assumptions of Theorem 7.5.4. References: [8, 9, 36, 79].

7.6 Holmstedt Formula Suppose that .X = (X0 , X1 ) is a Banach couple and that Ei = [X0 , X1 ]θi ,qi ;K ,

0 ≤ θi < 1,

.

1 ≤ qi ≤ ∞,

i = 1, 2.

For all .x ∈ X0 + X1 , the Holmstedt formula expresses the K-functional K(t, x; E0 , E1 ) in terms of the K-functional .K(t, x; X0 , X1 ) via the formula

.

(F

t 1/η

K(t, x; E0 , E1 ). ∼ 0

(F +t

( −θ )q ds s 0 K(s, x; X0 , X1 ) 0 s

∞(

t 1/η

s −θ1 K(s, x; X0 , X1 )

)q1 ds s

)1/q0 )1/q1 ,

(7.9)

470

7 Interpolation

where η = θ1 − θ0 ,

.

0 ≤ θ0 < θ1 < 1,

1 ≤ q0 , q1 < ∞.

Here, .a ∼ b means that there exist constants .0 < c1 , c2 such that .c1 a ≤ b ≤ c2 a, and the right hand side has the natural interpretation in the case that .q0 = ∞ or .q1 = ∞. See [9] Theorem 3.6.1. We shall need the following Hardy type inequality. Lemma 7.6.1 Suppose that .p > 1, and set .1/p' = 1 − 1/p. If .x ∈ L1 (τ ) + M, then ) F ∞ ( ( 1) F s F ∞ ds − 1− p −(1− p1 ) s .t s μ(s; x)ds ≤ t μ(x)dm ' ' s tp tp 0 ' F tp F ∞ 1 t −(1− p1 ) ≤ μ(s; x)ds + μ(s; x)s ds. 1 1 p' 1− p 0 1− p t Proof Using the fact that .s → μ(s; x) is decreasing, the first inequality follows by observing that 1 .μ(s; x) ≤ s

F

s

s > 0.

μ(x)dm, 0

The second inequality follows directly via integration by parts.

u n

Suppose now that .1 < p < ∞ and that .q = 1, and observe that, using Proposition 7.5.2, Lp,1 (τ ) = [L1 (τ ), M]1− 1 ,1;K ,

.

p

and L1 (τ ) = [L1 (τ ), M]0,∞;K .

.

Applying Holmstedt’s formula to the Banach couple .(L1 (τ ), M) with p0 = 0, p1 = p, q0 = ∞, q1 = 1, θ0 = 0, η = θ1 = 1 − 1/p,

.

7.7 Operators of Weak Type and the Calderón Operator

471

and using the Hardy type inequality given in Lemma 7.6.1, one obtains, for all .x ∈ L1 (τ ) + Lp,1 (τ ), K(t, x; L1 (τ ), Lp,1 (τ )) = K(t, x; L0,∞ (τ ), Lp,1 (τ )) = K(t, x, [L1 (τ ), M]0,∞;K , [L1 (τ ), M]1− 1 ,1;K ) p F s ∼ sup μ(s; x)ds 0≤s≤t p

.

+t t

F

t

0

∞(

F

∼

'

p'

F

s −(1−1/p)

μ(x)dm 0

F

p'

)

s

∞

μ(s; x)ds + t

0

t

s

p'

−(1− p1 )

ds s

μ(s; x)ds, (7.10)

where .1/p' = 1 − 1/p. Further, using Proposition 7.5.2, observe that Lp,∞ (τ ) = [L1 (τ ), M]1− 1 ,∞;K .

.

p

Again applying Holmstedt’s formula, we obtain F

tp

K(t, x; L1 (τ ), Lp,∞ (τ )) ∼

.

'

μ(s; x)ds + sup s

0

s≥t p

−(1− p1 )

'

F

s

μ(x)dm. 0

(7.11) References: [8, 9, 36].

7.7 Operators of Weak Type and the Calderón Operator Let .M and .N be a von Neumann algebras with semi-finite faithful normal traces .τ and .σ , respectively. For .1 ≤ p < ∞, the map .T : Lp,1 (σ ) → S(τ ) is said to be of weak type .(p, q), .1 ≤ q ≤ ∞ if T maps .Lp,1 (σ ) continuously into .Lq,∞ (τ ), and there exists a constant M such that μ(t; T x) ≤ Mt −1/q ||x||p,1 ,

.

t >0

(7.12)

for all .x ∈ Lp,1 (σ ). T will be said to be of weak type .(∞, q) if and only if there exists a constant M such that μ(t; T x) ≤ Mt −1/q ||x||L∞ (σ ) ,

.

t > 0,

(7.13)

472

7 Interpolation

for all .x ∈ L∞ (σ ). The least constant M for which equation (7.12) or (7.13) holds is called the weak type .(p, q) norm of T . Remark 7.7.1 Note that, if .p < ∞, and if T is a continuous linear mapping from .Lp (σ ) to .Lq (τ ), then T is of weak type .(p, q) since .Lp,1 (σ ) is continuously embedded in .Lp (σ ) and .Lq (τ ) embeds continuously into .Lq,∞ (τ ). Let .ω be the closed segment in the unit square with end points .( p11 , q11 ) and

( p12 , q12 ), where .1 ≤ p1 < p2 ≤ ∞, 1 ≤ q1 , q2 ≤ ∞, q1 /= q2 . If m denotes the slope of the segment .ω, that is,

.

m=

.

1 q1 1 p1

− −

1 q2 1 p2

(7.14)

,

then the Calderón operator .S(ω) is defined on functions on .(0, ∞) by setting F tm 1 F ∞ 1 ds ds −1 1 −1 1 (S(ω)f )(t) = t q1 s p1 f (s) + t q2 s p2 f (s) , p1 0 s p2 t m s ⎛ 1 ⎞ ⎛ 1 ⎞ (7.15) . F tm F ∞ p1 p2 s s = f (s)d ⎝ 1 ⎠ , f (s)d ⎝ 1 ⎠ + 0 tm t q1 t q2 if .1 ≤ p2 < ∞ and by setting F

tm

(S(ω)f )(t) =

.

⎛ f (s)d ⎝

0

s

1 p1 1

⎞ ⎠

(7.16)

t q1

if .p2 = ∞. Lemma 7.7.2 Let .p1 , p2 , q1 , q2 and m be as above. Suppose that .x ∈ L1 (τ ) + M and that .t > 0: (a) If .1 ≤ p2 < ∞ and if .S(ω)(μ(x))(t) < ∞, then (|x| − μ(t α ; x)1)+ ∈ Lp1 ,1 (τ ),

.

and

|x| ∧ μ(t α ; x)1 ∈ Lp2 ,1 (τ ).

(7.17)

In particular, .x ∈ Lp1 ,1 (τ ) + Lp2 ,1 (τ ). (b) If .p2 = ∞ and if .S(ω)(μ(x))(t) < ∞, then (|x| − μ(t α ; x)1)+ ∈ Lp1 ,1 (τ ),

.

|x| ∧ μ(t α ; x)1 ∈ M.

(7.18)

In particular, .x ∈ Lp1 ,1 (τ ) + N. F ∞ 1 −1 (c) If .1 < p2 < ∞ and . t s p2 μ(s; x)ds < ∞, then (|x| − μ(t α ; x)1)+ ∈ L1 (τ ),

.

and

In particular, .x ∈ L1 (τ ) + Lp2 ,1 (τ ).

|x| ∧ μ(t α ; x)1 ∈ Lp2 ,1 (τ ).

(7.19)

7.7 Operators of Weak Type and the Calderón Operator

473

Proof (a) If .1 ≤ p2 ∞, and observe that t

.

− q1

1

1 p1

+t

F

∞(

μ(s; x) − μ(t α ; x)

0

− q1 2

1 p2 F

=t

− q1 1

1 p1

=t

− q1

1 p1

=t

− q1

1 p1

1

1

F

∞

)+

ds s

1

s p1

1

μ(s; x) ∧ μ(t α ; x)s p2

0

ds s

F tα 1 ds ( ) 1 ds − q1 1 α p 1 2 +t μ(s; x) − μ(t ; x) s μ(t α ; x)s p2 s p2 0 s 0 F ∞ 1 ds −1 1 μ(s; x)s p2 + t q2 p2 t α s ) ( F tα 1 ds α α −1 −1 + μ(t α ; x) t p2 q2 − t p1 q1 μ(s; x)s p1 s 0 F ∞ 1 ds −1 1 + t q2 μ(s; x)s p2 p2 t α s F tα F ∞ 1 ds 1 ds −1 1 μ(s; x)s p1 μ(s; x)s p2 + t q2 s p2 t α s 0 tα

= S(ω)(μ(x))(t) < ∞. Using Proposition 3.2.8, observe that μ((|x| − μ(t α ; x)1)+ ) = (μ(x) − μ(t α ; x))+ ,

.

μ(|x| ∧ t α 1) = μ(x) ∧ t α ,

and from this, it follows that the assertion (7.17) holds. Finally, using the spectral theorem, F

∞

|x| =

.

λde|x| (λ) =

0

F

∞

((λ − μ(t α ; x)+ + λ ∧ μ(t α ; x))de|x| (λ)

0

= (|x| − μ(t α ; x)1)+ + |x| ∧ t α 1 ∈ Lp1 ,1 (τ ) + Lp2 ,1 (τ ). That .x ∈ Lp1 ,1 (τ ) + Lp2 ,1 (τ ) follows immediately from the polar decomposition. (b) If .p2 = ∞, then t

.

− q1

1

1 p1

F

∞(

μ(s; x) − μ(t α ; x)

0

)+

1

s p1

ds −1 1 ≤ t q1 s p1

F

tα

1

μ(s; x)s p1 0

ds s

= S(ω)(μ(x))(t) < ∞. This implies that .(|x| − μ(t α ; x)1)+ ∈ Lp1 ,1 (τ ), and since it is clear that .|x| ∧ μ(t α ; x)1 ∈ M, it follows that .x ∈ Lp1 ,1 (τ ) + M.

474

7 Interpolation

F ∞ 1 −1 (c) Now suppose that . t s p2 μ(s; x)ds < ∞. The same calculation as in (a) with .p1 = 1 yields t

.

F

− q1

∞(

μ(s; x) − μ(t ; x)

1

α

)+

0

=t

F

− q1 1

tα

μ(s; x)ds + t

− q1 2

0

1 p2

ds + t

F

− q1

2

1 p2

∞

μ(s; x)s

1 p2

tα

F

∞

1

μ(s; x) ∧ μ(t α ; x)s p2

0

ds s

ds < ∞, s

F tα using the fact that . 0 μ(s; x)ds < ∞ since .x ∈ L1 (τ ) + M. Consequently, (|x| − μ(t m ; x)1)+ ∈ L1 (τ )

.

and

|x| ∧ μ(t α ; x)1 ∈ Lp2 ,1 (τ ),

and it follows as above that .x ∈ L1 (τ ) + Lp2 ,1 (τ ).

u n

If .D ⊆ S(σ ) is a linear subspace, the continuous mapping .T : D → S(τ ) will be called sublinear if, for all .x, y ∈ D and scalars .λ, there exist partial isometries .v, w ∈ N such that |T (x + y)| ≤ v ∗ |T x|v + w ∗ |T y|w,

.

|T (λx)| ≤ |λ||T x|.

(7.20)

If .D ⊆ S(σ ) is a convex subset, then the map .T : D → S(τ ) is called midpoint subconvex if there exist partial isometries .v, w ∈ M such that | ( )| | | |. T 1 x + 1 y | ≤ 1 v ∗ |T x|v + 1 w ∗ |T y|w | 2 | 2 2 2

(7.21)

for all .x, y ∈ D. As follows from [55] Lemma 4.3, for any .x1 , x2 ∈ S(τ ), there exist partial isometries .v, w ∈ M such that |x1 + x1 | ≤ v ∗ |x1 |v + w ∗ |x2 |w,

.

and from this, it is clear that any (sub)linear map is midpoint subconvex. Theorem 7.7.3 Suppose that .1 ≤ p1 < p2 ≤ ∞, 1 ≤ q1 , q2 ≤ ∞ and that q1 /= q2 . If .T : Lp1 ,1 (σ ) + Lp2 ,1 (σ ) −→ S(τ ) is a midpoint subconvex map that is simultaneously of weak types .(pi , qi ), i = 1, 2 with corresponding weak type norms .Mi , i = 1, 2, and if .t > 0 satisfies .

S(ω)(μ(x))(t) < ∞,

.

then μ(t; T x) ≤ c(pi , qi ) max {M1 , M2 } S(ω)(μ(x))(t)

.

7.7 Operators of Weak Type and the Calderón Operator

475

whenever .x ∈ Lp1 ,1 (σ ) + Lp2 ,1 (σ ) in the case that .p2 < ∞, and .x ∈ Lp1 ,1 (σ ) + N in the case that .p2 = ∞. Proof It will suffice to prove the theorem only in the case that .p2 < ∞ as the proof in the case that .p2 = ∞ is similar. Let .x ∈ Lp1 ,1 (σ )+Lp2 ,1 (σ ), and let .β = μt α (x), where .α is the slope of the segment .ω. Let .x = u|x| be the polar decomposition of x and write .|x| = (|x| − β1)+ + |x| ∧ β1. Suppose now that .p1 < p2 . If .t > 0 satisfies .S(ω)(μ(x))(t) < ∞, then it follows from Lemma 7.7.2(a) that + ∈L .(|x| − β1) p1 ,1 (σ ), and .|x| ∧ β1 ∈ Lp2 ,1 (σ ). Now observe that μ(t; T x) = μ(t; T (u(|x| − β1)+ + u(|x| ∧ β1))) ( ( )) 1 1 + = μ t; T 2u(|x| − β1) + 2u|x| ∧ β1 2 2

.

) 1 ( ∗ μ t; v |T (2u(|x| − β1)+ )|v + w ∗ |T (2u|x| ∧ β1)|w 2 1 t 1 t ≤ μ( ; T (2u(|x| − β1)+ )) + μ( ; T (2u(|x| ∧ β1))) 2 2 2 2

≤

1

≤ max {Mi 2 qi } i=1,2

× (t

− q1

1

||μ(u(|x| − β1)+ )||p1 ,1 + t

− q1

2

||μ(u(|x| ∧ β1))||p2 ,1 )

1

≤ max {pi 2 qi Mi } i=1,2

1 − q1 1 − q1 t 1 ||(μ(x) − β)+ ||p1 ,1 + t 2 ||μ(x) ∧ β||p2 ,1 ) p1 p2 ( F tα 1 s 1/p1 = max {pi 2 qi Mi } (μ(x) − μ(t α ; x))d( 1/q ) i=1,2 t 1 0 ) F ∞ F α t s 1/p2 s 1/p2 α + μ(x)d( 1/q ) + μ(t ; x) d( 1/q ) t 2 t 2 0 tα (F α ) F ∞ t 1/p1 1/p2 1 s s μ(x)d( 1/q ) + μ(x)d( 1/q ) = max {pi 2 qi Mi } i=1,2 t 1 t 2 0 tα ( ) 1 + max {pi 2 qi Mi } μ(t α ; x)[t α/p2 −1/q2 − t α/p1 −1/q1 ] ×(

i=1,2

1

= max {pi 2 qi Mi }S(ω)(μ(x))(t), i=1,2

using the fact that .α/p2 − 1/q2 = α/p1 − 1/q1 . References: [8, 19, 36, 42, 55].

u n

476

7 Interpolation

7.8 Noncommutative Marcinkiewicz Theorem for Sublinear Mappings In what follows, the following Hardy inequality (see [8, Chapter 3, Lemma 3.9]), which complements that given in Lemma 6.7.6, will be needed. Lemma 7.8.1 If .λ < 1 and .1 ≤ q ≤ ∞, then (F

∞(

t

.

F

∞

1−λ

0

t

ds ϕ(s) s

)q

dt t

) q1

1 ≤ 1−λ

(F

∞

(t

1−λ

ϕ(t))

q dt

0

t

)1 q

,

(7.22)

where .ϕ is a non-negative measurable function on .(0, ∞). In the case that .q = ∞, this inequality should be interpreted as F ess. sup t 1−λ

∞

ϕ(s)

.

t

t>0

ds ≤ ess. sup t 1−λ ϕ(t). s

(7.23)

It is now possible to present the following noncommutative extension of the Marcinkiewicz interpolation theorem in the formulation given by Calderón. Theorem 7.8.2 Suppose that .1 ≤ p1 < p2 < ∞ and .1 ≤ q1 , q2 ≤ ∞ with q1 /= q2 . Let .0 < θ < 1 and define .p, q by setting

.

.

1−θ 1 θ = + , p p1 p2

1−θ 1 θ = + . q q1 q2

(7.24)

Let .T : Lp1 ,1 (σ ) + Lp2 ,1 (σ ) −→ S(τ ) be a sublinear map that is simultaneously of weak types .(pi , qi ), i = 1, 2, with corresponding weak type constants .Mi , i = 1, 2. If .1 ≤ r ≤ ∞, then T : Lp,r (σ ) → Lq,r (τ ),

.

(7.25)

and there exists a constant .c = c(p1 , q1 ; p2 , q2 ; θ ; r) such that ||T x||Lq,r (τ ) ≤

.

c max{M1 , M2 }||x||Lp,r (τ ) . θ (1 − θ )

(7.26)

Proof Since .(1/p, 1/q) is an interior point of the segment .σ , it follows that ] [ 1 1 1 1 1 = − − . , m q q1 p p1

] [ 1 1 1 1 1 = − − . m q q2 p p2

(7.27)

7.8 Noncommutative Marcinkiewicz Theorem for Sublinear Mappings

477

Suppose first that .r < ∞ and suppose that .x ∈ Lp1 ,1 (σ ) + Lp2 ,1 (σ ). By 1 q

Theorem 7.7.3, setting .M = maxi=1,2 {pi i in .Lq,r (m), it follows that {F ||T x||Lq,r (τ ) ≤ M

∞

.

Mi

} and using the Minkowski inequality

dt t [

1

}1 r

[t q S(ω)μ(t; x)]r

0

{F ]r } 1r } F tm 1 ∞ 1 1 1 1 dt ds − ≤ max , s p1 μ(s; x) t q q1 M( p1 p2 t s 0 0 {

{F

∞

+

[

F

1 1 q − q2

t

tm

s

1 p2

0

0

ds μ(s; x) s

]r

dt t

} 1r ).

The substitution .u = t m in each of the integrals and the equalities in (7.27) yield {F

∞

[ t

.

1 1 q − q1

F

tm

s

1 p1

0

0 1

= |m|− r

{F

ds μ(s; x) s

∞[

1

up

− p1

∞

[ t

1 1 q − q2

F

= |m|

− 1r

tm

s {F

1 p2

} 1r

1

s p1 μ(s; x)

ds μ(s; x) s

∞[

u 0

dt t

0

0

0

u

1

0

{F

F

]r

1 1 p − p2

F

]r

u

s 0

1 p2

dt t

ds s

]r

du u

} 1r ,

} 1r

ds μ(s; x) s

]r

du u

} 1r .

Applying the Hardy inequalities given in Lemmas 6.7.7 and 7.8.1, we obtain the estimate ||T x||Lq,r (τ )

.

)1 ( (F ∞ 1 1 1 du r r −1/r p ≤ max{ , }|m| M c1 [u μ(u; x) ] u p1 p2 0 )1 ) (F ∞ r 1 du + c2 [u p μ(u; x) ]r u 0 = max{

1 1 , }|m|−1/r M(c1 + c2 )||x||Lp,r (τ ) , p1 p2

478

7 Interpolation

where 1 1 1 = − =θ . c1 p1 p

(

1 1 − p1 p2

) ,

) ( 1 1 1 1 1 . = − = (1 − θ ) − c2 p p2 p1 p2

This suffices to complete the proof in the case that .r < ∞. The case that .r = ∞ is similar. n u The noncommutative Marcinkiewicz theorem in the special case of noncommutative .Lp -spaces now follows. Corollary 7.8.3 Suppose that .1 ≤ p1 < p2 < ∞ and .1 ≤ q1 , q2 ≤ ∞ with q1 /= q2 . Suppose, in addition, that .pi ≤ qi , i = 1, 2. Let .0 < θ < 1 and define .p, q by setting .

.

1−θ 1 θ = + , p p1 p2

1−θ 1 θ = + . q q1 q2

(7.28)

If .T : Lp1 ,1 (σ ) + Lp2 ,1 (σ ) −→ S(τ ) is a sublinear map that is simultaneously of weak types .(pi , qi ), i = 1, 2, with corresponding weak type constants .Mi , i = 1, 2, then T : Lp (σ ) → Lq (τ ),

.

and there exists a constant .c = c(p1 , p2 , q1 , q2 ) such that ||T x||Lq (τ ) ≤

.

c max{M1 , M2 }||x||Lp (σ ) . θ (1 − θ )

Proof Since .pi ≤ qi , i = 1, 2, it follows that .p ≤ q. Setting .r = p in the assertion (7.25), it follows that T : Lp,p (σ ) = Lp (σ ) → Lq,p (τ ) ⊆ Lq,q (τ ) = Lq (τ ).

.

u n References: [8, 36, 76].

7.9 The Boyd Interpolation Theorem Recall that, for any measurable function f on the interval .(0, a) with .a ∈ (0, ∞], the dilation .Ds f of f is defined by setting { f (t/s) if t ∈ (0, a) and t/s ∈ (0, a) , . (Ds f ) (t) = 0 if t ∈ (0, a) and t/s ∈ / (0, a) .

7.9 The Boyd Interpolation Theorem

479

It is well known that, if E is a symmetric space on .(0, a), then .Ds defines a bounded linear operator on E with norm .||Ds || ≤ max(1, s). Suppose now that E is a symmetric space on .(0, a). Define the lower Boyd index .pE of E by setting } { 1 pE = sup p > 0 : ∃c > 0 ∀s ≥ 1 ||Ds f ||E ≤ cs p ||f ||E ,

.

and the upper Boyd index .qE of E by setting } { 1 qE = inf q > 0 : ∃c > 0 ∀0 < s ≤ 1 ||Ds f ||E ≤ cs q ||f ||E .

.

The Boyd indices admit the following equivalent formulation. Proposition 7.9.1 log s log s = sup s→∞ log ||Ds || s>1 log ||Ds ||

pE = lim

.

and qE = lim

.

s→0+

log s log s = inf . log ||Ds || 0 0, 0 ≤ f ∈ S(m),

480

7 Interpolation

To proceed further, again suppose that .1 ≤ p < q < ∞ and define the operators Pp , Qq on .S(m) by setting

.

Pp (f )(t) = t

.

F

− p1

t

0

Qq f (t) = t

.

− q1

F

∞

t

ds = s f (s) s 1 p

F

1

−1

1

−1

up

f (ut)du;

0

F

ds = s f (s) s 1 q

1

∞

uq

f (ut)du,

1

and note that Sp,q = Pp + Qq .

(7.29)

.

The theorem that follows is due to Boyd [15]. Theorem 7.9.2 Let .E ⊆ S(m) be a symmetric space and suppose that .1 ≤ p < q < ∞: (i) The operator .Pp is bounded on E if and only if .p < pE . (ii) The operator .Qq is bounded on E if and only if .qE < q. Proof (Montgomery-Smith [91]) (i) Assume first that .1 ≤ p < pE . Note that, if .0 ≤ f ∈ E, then it follows from [76] II.2.18 that 0 ≤ Pp (f ) ≤ Pp (μ(f )).

.

Now observe that Pp (μ(f ))(t) = t

− p1

F

t

1

s p μ(s; f ) 0

.

≤p

0 Σ

ds =p s

μ(t2 ; f )2 n

n/p

F

=p

n=−∞

1

1

μ(tu; f )du p 0

(7.30)

0 Σ

n/p

D2−n μ(f )(t)2

.

n=−∞

Since .p < pE , it follows from Proposition 7.9.1 that there exist .c > 0 and .p < p' < pE such that '

||D2−n μ(f )||E ≤ c2−n/p ||μ(f )||E ,

.

n ≥ 0.

Consequently, p

0 Σ

.

n=−∞

2n/p ||D2−n μ(f )||E ≤ pc

0 Σ n=−∞

'

2n/p 2−n/p ||μ(f )||E ≤ c' ||f ||E < ∞.

7.9 The Boyd Interpolation Theorem

481

Since E is a Banach space, the Riesz–Fischer theorem implies that ∞ Σ

p

D2−n μ(f )2n/p ∈ E.

.

n=−∞

From the inequality (7.30), it follows further that .Pp (μ(f )) ∈ E and ||Pp μ(f )||E ≤ c' ||f ||E .

.

To see that the boundedness of .Pp on E implies that .1 ≤ p < pE , suppose that, for all .f ∈ E, .||Pp (f )||E ≤ C||f ||E for some .C > 1. It will be shown that 1 ≤ p < p/(1 − 1/C) ≤ pE .

(7.31)

.

To this end, it will suffice to show that there is a number .0 < k < 1 such that, for all numbers .s = k −n , n = 1, 2, . . . , it follows that ||Ds f ||E ≤ cs (1−1/C)/p ||f ||E .

(7.32)

.

By induction and a straightforward use of Fubini’s theorem (see, for example, [8] p.151, equation (5.42)), it follows that F n+1 .Pp (f )(t)

=

1

0

−1

(log s p )n 1/p ds . D1/s f (t) n! 1 −p )n

Since the function .s → D1/s μ(f ) (log sn!

is decreasing, it follows that −1

n+1 .Pp (μ(f ))

(log s p )n 1/p s , ≥ D1/s μ(f ) n!

0 < s < 1.

Consequently, ( ||D1/s μ(f )||E ≤ C

.

)

n!

n+1 1

s p (log s

− p1 n )

||f ||E .

Now set .k = e−pC . Using the estimate .n! ≤ ce−n nn , straightforward to see that, if .s = k n , k = 1, 2, . . . , then ||D1/s μ(f )||E ≤ Ccs −(1−1/C)/p ||f ||E ,

.

and this is just the estimate (7.32), and (7.31) now follows.

n = 1, 2, . . . , it is

482

7 Interpolation

This establishes the assertion of (i). The proof of (ii) is similar and the details are omitted. u n Corollary 7.9.3 Let .E ⊆ S(m) be a symmetric space on .(0, a), with .a ∈ (0, ∞]: (i) If .1 ≤ p < q < ∞, then .Sp,q is bounded on E if and only if 1 ≤ p < pE ≤ qE < q.

.

(ii) If .1 ≤ p < ∞, then .Sp,∞ is bounded on .E if and only if .p < pE . Corollary 7.9.3 together with Lemma 7.7.2 now yields the following consequence. Corollary 7.9.4 Let .E ⊆ S(m) be a strongly symmetric space and suppose that x ∈ E(τ ):

.

(i) If .1 ≤ p < pE ≤ qE < q < ∞, then .Sp,q (μ(x)) < ∞ a.e. and .E(τ ) ⊆ Lp,1 (τ ) + Lq,1 (τ ). (ii) If .1 ≤ p < pE , then .Sp,∞ (μ(x)) < ∞ a.e. and .E(τ ) ⊆ Lp,1 (τ ) + M. We now prove the noncommutative Boyd Theorem. Theorem 7.9.5 (Noncommutative Boyd Theorem) Let .E ⊆ S(m) be strongly symmetric. Suppose that .1 ≤ p < q ≤ ∞ and that .T : Lp,1 (σ ) + Lq,1 (σ ) → S(τ ) is a midpoint subconvex map that is simultaneously of weak types .(p, p) and .(q, q), that is, 1

μ(t; T x) ≤ Cr t − r ||x||Lr,1 (σ ) ,

.

t > 0,

x ∈ Lr,1 (σ ), r = p, q.

(7.33)

If .1 ≤ p < pE ≤ qE < q < ∞, then .E(σ ) ⊆ Lp,1 (σ ) + Lq,1 (σ ), and for all x ∈ E(σ ),

.

||T (x)||E(τ ) ≤ c(p, q) max{Cp , Cq }||Sp,q ||E→E ||x||E(σ ) .

.

Proof Suppose that .x ∈ E(σ ), or equivalently, .μ(x) ∈ E. Since .1 ≤ p < pE ≤ qE < ∞, it follows from Corollary 7.9.4 that .Sp,q (μ(x)) < ∞ a.e. and that .E(σ ) ⊆ Lp,1 (σ ) + Lq,1 (σ ). By Lemma 7.7.2 and Theorem 7.7.3, it follows that μ(T x) ≤ c(p, q) max{Cp , Cq }Sp,q (μ(x))

.

a.e.

Since .Sp,q (μ(x)) ∈ E, it follows that .μ(T x) ∈ E, and so ||T x||E(τ ) = ||μ(T x)||E ≤ c(p, q) max{Cp , Cq }||Sp,q ||E→E ||μ(x)||E

.

= c(p, q) max{Cp , Cq }||Sp,q ||E→E ||x||E(σ ) . u n

7.10 Weak Type (p, p) and Strong Type (∞, ∞) Interpolation

483

Via Remark 7.7.1, Theorem 7.9.5 has the following consequence. Corollary 7.9.6 Let .E ⊆ S(m) be a strongly symmetric space on .(0, a) . If 1 ≤ p < pE (≤ qE < q < ∞, then the pair .(E(σ ),)E(τ )) is an interpolation pair for the pair . (Lp (σ ), Lq (σ )), (Lp,∞ (τ ), Lq,∞ (τ )) , with interpolation constant depending only on .p, q, E.

.

Proof It needs to be only shown that .E(τ ) is intermediate for the Banach couple (Lp (τ ), Lq (τ )). From Theorem 7.9.5, it follows that .E(τ ) ⊆ Lp (τ ) + Lq (τ ), and so it remains to be shown that .Lp (τ ) ∩ Lq (τ ) ⊆ E(τ ). This follows immediately from its commutative specialization given in [79] Proposition 2.b.3 and the fact that .Lp (τ ) ∩ Lq (τ ) = (Lp ∩ Lq )(τ ) as follows from [44] Theorem 3.2. The same arguments apply verbatim with .E(τ ) replaced by .E(σ ). u n .

References: [4, 15, 30, 36, 42, 44, 76, 79, 91].

7.10 Weak Type (p, p) and Strong Type (∞, ∞) Interpolation Theorem 7.10.1 Let E ⊆ S(m) be strongly symmetric. Suppose that 1 ≤ p < ∞ and that T : Lp,1 (σ ) + N → S(τ ) is a midpoint subconvex map that is simultaneously of weak type (p, p) and strong type (∞, ∞), that is, ||T x||M ≤ C∞ ||x||N ,

.

||T y||Lp,∞ (τ ) ≤ Cp ||y||Lp,1 (σ ) ,

for all x ∈ Lp,1 (σ ), y ∈ N, respectively. If 1 ≤ p < pE , then E(σ ) ⊆ Lp,1 (σ )+N, and for all x ∈ E(σ ), ||T (x)||E(τ ) ≤ c(p) max{Cp , C∞ }||Sp,∞ ||E→E ||x||E(σ ) .

.

Proof The proof follows by exactly the same argument given in the proof of Theorem 7.9.5, using Corollary 7.9.3 (ii), Lemma 7.7.2 (b), and Theorem 7.7.3. n u Again using Remark 7.7.1, we obtain the following consequence. Corollary 7.10.2 Let E ⊆ S(m) be a strongly symmetric space. If 1 ≤ (p < pE , then the pair) (E(σ ), E(τ )) is an interpolation pair for the pair (Lp (σ ), N), (Lp (τ ), M) , with interpolation constant depending only on p and E. Proof As in the proof of Corollary 7.9.6, it suffices to show that E(τ ) is intermediate for the pair (Lp (τ ), M). From Theorem 7.10.1, it follows that E(τ ) ⊆ Lp (τ ) + M. That Lp (τ ) ∩ M ⊆ E(τ ) is proved in [29] Lemma 5.15 by adapting the relevant part of the proof of [79] Proposition 2.b.3. u n References: [29, 30, 36, 79].

484

7 Interpolation

7.11 Strong Type (1, 1) and Weak Type (q, q) Interpolation, 1 0.

7.11 Strong Type (1, 1) and Weak Type (q, q) Interpolation, 1 < q < ∞

485

Using now the Holmstedt formulae given in (7.11), (7.10) yields the estimate F

tq

'

μ(s; T x)ds ≤ C ' K(t, T x; L1 (τ ), Lq,∞ (τ ))

.

0

≤ C2 K(t, x; L1 (σ ), Lq,1 (σ )) (F q ' F t

≤ C2' 1

Replacing t by .t q ' = t F

t

.

1− q1

)

∞

μ(s; x)ds + t

tq

0

s

'

1 q −1

μ(s; x)ds .

now gives

(F

t

μ(s; T x)ds ≤ C2

0

μ(s; x)ds + t

1− q1

F

∞

s

0

1 q −1

) μ(s; x)ds ,

(7.34)

t

for all .x ∈ L1 (σ ) + L1,q (σ ), t > 0. Using Fubini’s theorem, observe that F

t

F

t

Qq (μ(x))(s)ds =

0

− q1

(F

∞

s

u F

0

u

t

=

1

sq 0

−1 ∞

+ (F

(F

s

s

1 q −1

) μ(s; x)ds du

u

) du μ(s; x)ds

(F

t

− q1

0

F

.

= q'

1 q −1

− q1

u

) du μ(s; x)ds

0

t t

μ(s; x)ds + t

0

1− q1

F

∞

1

sq

−1

) μ(s; x)ds ,

t > 0.

t

(7.35) Comparing this with (7.34), it follows that there exists a constant .C3 such that T x ≺≺

.

C3 Qq (μ(x)). q'

(7.36)

It is now possible to prove the following complement to Theorem 7.10.1. It is a noncommutative extension of the first statement of [6] Theorem 3. Theorem 7.11.1 Let .E ⊆ S(m) be strongly symmetric. Suppose that .1 < q < ∞ and that .T : L1 (σ ) + Lq,1 (σ ) → S(τ ) be a midpoint subconvex map that is simultaneously of strong type .(1, 1) and weak type .(q, q), that is, ||T x||L1 (τ ) ≤ C1 ||x||L1 (σ ) ,

.

||T y||Lq,∞ (τ ) ≤ Cq ||y||Lq,1 (σ )

486

7 Interpolation

for all .x ∈ L1 (σ ) and .y ∈ Lq,1 (σ ). If .1 ≤ qE < q < ∞, then .E(σ ) ⊆ L1 (σ ) + Lq,1 (σ ), and if E is fully symmetric, then .T (E(σ )) ⊆ E(τ ), and there exists a constant .C(q) such that ||T x||E(τ ) ≤ C(q)||x||E(σ ) .

.

Proof Suppose that .x ∈ E(σ ) or, equivalently, that .μ(x) ∈ E, and suppose that .1 ≤ qE < q < ∞. By Theorem 7.9.2, .Qq (μ(x)) ∈ E . In particular, .Qq (μ(x))(t) < ∞almost everywhere. It is clear that .Qq (μ(x)) is decreasing and so F

∞

1

sq

.

−1

μ(s; x)ds = Qq (1) < ∞.

1

By Lemma 7.7.2 (c), it follows that .x ∈ L1 (σ ) + Lq,1 (σ ), and so .E(σ ) ⊆ L1 (σ ) + Lq,1 (σ ). The submajorization inequality (7.36) and the assumption that E is fully symmetric now imply that .T x ∈ E(τ ) and ||T x||E(τ ) ≤

.

C4 ||Qq ||||x||E(σ ) . q' u n

This completes the proof of the theorem.

The lemma that follows is well known. However, we include a proof for lack of a convenient reference. Lemma 7.11.2 If .0 ≤ φ, ψ are increasing concave functions on .[0, 1) with .φ(0) = ψ(0) = 0 for which 0 ≤ φ(t) ≤ ψ(t),

.

0 ≤ t ≤ 1,

(7.37)

then F .

0

1

F

1

μ(t; x)dφ(t) ≤

μ(t; x)dψ(t),

x ∈ S[0, 1).

(7.38)

0

Σn Proof Observe first that, if .s = Σ k=1 αk χAk is a simple function on .[0, 1), with decreasing rearrangement .μ(s) = nk=1 αk χ[0,tk ) with .0 < αk ∈ R and .0 < t1 < · · · < tn , then it follows directly from (7.37) that F

1

.

μ(t; s)dφ(t) = φ(0+)

0

n Σ

αk +

k=1

≤ ψ(0+)

n Σ k=1

n Σ

αk φ(tk )

k=1

αk +

n Σ k=1

F

1

αk ψ(tk ) =

μ(t; s)dψ(t). 0

7.11 Strong Type (1, 1) and Weak Type (q, q) Interpolation, 1 < q < ∞

487

Now suppose that .x ∈ L∞ . For each .n = 1, 2, . . . , there exists a simple function .sn such that .||x − sn ||∞ < 1/n. Using the commutative specialization of Theorem 3.9.14 (which is a classical result of Lorentz–Shimogaki), it follows that .||μ(x) − μ(sn )||∞ < 1/n, n = 1, 2, . . . , and it follows by a simple argument that the inequality (7.38) holds whenever .x ∈ L∞ . Now suppose that .x ∈ S[0, 1). It is clear that we may suppose that F

1

μ(t; x)dψ(t) < ∞

.

0

or else there is nothing to be proved. Observing that .μ(x) ∧ n ↑n μ(x), and that μ(x) ∧ n ∈ L∞ , n ≥ 1, it follows from the monotone convergence theorem that

.

F

1

.

F

1

μ(t; x)dφ(t) = sup n

0

F

1

≤ sup n

μ(t; x) ∧ ndφ(t)

0

F

1

μ(t; x) ∧ ndψ(t) =

0

μ(t; x)dψ(t), 0

u n

and this suffices to prove the lemma.

Before proceeding, some further comments are necessary. Suppose that .ψ : [0, ∞) → [0, ∞) is an increasing concave function such that .ψ(0) = 0. Recall that the Lorentz space .Aψ is defined to be the collection of all .f ∈ S(m) such that F ||f ||Aψ =

.

∞

μ(t; f )dψ(t) < ∞,

0

see Sect. 6.3. Note that the improper Stieltjes integral preceding may be written as F ||f ||Aψ = ψ(0+) +

∞

.

μ(t; f )ψ ' (t)dt.

0

We recall that, if E is a symmetric space on .[0, ∞), then the fundamental function .ϕE is defined by setting ϕE (t) = ||χ[0,t) ||E ,

.

t ≥ 0.

If .ψE denotes the least concave majorant (see [76, Chapter 2]) of the quasiconcave function .ϕE , then it is shown in [105] Proposition 12.1.3 that the Lorentz space b embeds continuously with norm one into E and that, in general, the space .A ψE b cannot be replaced by .A . Recall that .Ab is the closure in .A .A ψE ψE of .L1 ∩ L∞ ψE ψE with respect to .|| · ||AψE (as in Sect. 6.3 and Definition 4.4.9). However, in the case that E is a symmetric space on .[0, 1), then it may be shown [76] Lemma II.5.1

488

7 Interpolation

that .AbψE = AψE . Consequently, in this case, we have the continuous norm one embedding AψE ⊆ E.

(7.39)

.

In what follows, if E is a symmetric space on .[0, ∞), then .E[0, 1) will denote the symmetric space on .[0, 1) defined by setting E[0, 1) = {x ∈ S[0, 1) : μ(x) ∈ E},

||x||E[0,1) = ||x||E .

.

We shall need the following embedding that is stated in [6]. We thank Sergei Astashkin for the proof and his permission to include it here. Lemma 7.11.3 If E is a symmetric space on .[0, ∞), then .L1 ∩Lq,∞ ⊆ E whenever 1 ≤ qE < q.

.

Proof Let .ϕE denote the fundamental function of E. Since .1 ≤ qE < q, it follows that, for every r such that .qE < r < q, there is a constant .C > 0 such that, for all .0 < t ≤ 1, we have ϕE (t) = ||χ[0,t) ||E = ||D 1 χ[0,1) ||E ≤ C1 ||D 1 || ≤ Ct 1/r ,

.

t

t

and consequently, ψE (t) ≤ Ct 1/r ,

0 ≤ t ≤ 1,

.

(7.40)

where .ψ is the least concave majorant of .ϕ. In particular, .ψE (0) = 0. Using Lemma 7.11.2, it follows from the estimate (7.40) and the fact that .r < q that F

1

t

.

− q1

F

=

t

.

− q1

− q1

t

− q1

1

dt r

0

0

Consequently, .t

1

dψE (t) ≤ C C r

F

1

t −1/q+1/r−1 dt < ∞.

0

χ[0,1) ∈ AψE [0, 1). By the embedding (7.39), it now follows that

χ[0,1) ∈ E and ||t

.

− q1

χ[0,1) ||E ≤ ||t

− q1

χ[0,1) ||AψE .

(7.41)

Now suppose that .0 ≤ x ∈ L1 ∩ Lq,∞ . We set μ(x) = μ(x)χ[0,1) + μ(x)χ[1,∞) .

.

(7.42)

7.11 Strong Type (1, 1) and Weak Type (q, q) Interpolation, 1 < q < ∞

489

Observe first that we have the pointwise estimate ( μ(t; x)χ(0,1) (t) ≤

.

) sup μ(s; x)s

t −1/q = ||x||Lq,∞ t −1/q ,

1/q

0 0.

t

If .x, y ∈ L1 (τ ), .y ≺≺tl x, and .||x||1 = ||y||1 , then we write .y ≺tl x. Note that .x ≺tl y if and only if .y ≺hd x. We frequently consider in this section an operator .T : S(0, ∞) → S(0, ∞) and discuss its restriction on .Lp (0, ∞), .1 ≤ p ≤ ∞, and the notation .||T ||Lp →Lp stands for the norm of this restriction. Now, we present the main theorems of this section. The first two theorems provide necessary conditions for a fully symmetric space .E(τ ) to be an exact interpolation space for the couple .(Lp (τ ), Lq (τ )), whereas the third main result of this section, Theorem 7.12.6, provides sufficient conditions for that. The proofs of Theorems 7.12.4, 7.12.5, and 7.12.6 will be presented further below.

7.12 Interpolation for the Pair (Lp (τ ), Lq (τ )), 1 ≤ p < q ≤ ∞

491

Remark 7.12.3 Recall that whenever E is an interpolation space for the couple (Lp (0, ∞), Lq (0, ∞)), it follows that it admits an equivalent fully symmetric norm due to Corollary 7.3.3. This means that the space .E(τ ) is well defined with respect to that norm. Below, in the statements of Theorems 7.12.4–7.12.6, we refer to the space .E(τ ) defined through this procedure.

.

Theorem 7.12.4 Let .(M, τ ) be a semi-finite von Neumann algebra equipped with a semi-finite faithful normal trace .τ . Let E be an interpolation space for the couple p ≺≺ p .(Lp (0, ∞), Lq (0, ∞)), .1 ≤ p < q ≤ ∞. If .|y| hd |x| with .x ∈ E(τ ), then .y ∈ E(τ ) and .||y||E ≤ cp,E ||x||E , where .cp,E is a constant depending on .E(τ ) and p only. Theorem 7.12.5 Let .(M, τ ) be a semi-finite von Neumann algebra equipped with a semi-finite faithful normal trace .τ . Let E be an interpolation space for the couple q ≺ |x|q with .x ∈ E(τ ), then .(Lp (0, ∞), Lq (0, ∞)), .1 ≤ p < q < ∞. If .|y| tl .y ∈ E(τ ) and .||y||E ≤ cq,E ||x||E , where .cq,E is a constant depending on .E(τ ) and q only. Theorem 7.12.6 Let E be a fully symmetric function space and let .1 ≤ p < q ≤ ∞. We have that .E(τ ) is an interpolation space for the couple .(Lp (τ ), Lq (τ )) provided one of the following conditions holds: (i) .q = / ∞, and there exist two positive constants .cp,E and .cq,E such that: (a) For any .f ∈ E and .g ∈ (Lp + L∞ )(0, ∞), if .|g|p ≺≺hd |f |p , then .g ∈ E and .||g||E ≤ cp,E ||f ||E . (b) For any .f ∈ E and .g ∈ (L1 + Lq )(0, ∞), if .|g|q ≺≺tl |f |q , then .g ∈ E and .||g||E ≤ cq,E ||f ||E . (ii) .q = ∞, and there exists a positive constant .cp,E such that: (a) For any .f ∈ E and .g ∈ (Lp + L∞ )(0, ∞), if .|g|p ≺≺hd |f |p , then .g ∈ E and .||g||E ≤ cp,E ||f ||E . The following corollary is a combination of Theorems 7.12.4, 7.12.5, and 7.12.6. Corollary 7.12.7 Let .1 ≤ p < q < ∞. Let E be a fully symmetric space. Then the following assertions are equivalent: (i) E is an interpolation space for the couple .(Lp (0, ∞), Lq (0, ∞)). (ii) .E(τ ) is an interpolation space for the couple .(Lp (τ ), Lq (τ )) for an arbitrary semi-finite von Neumann algebra .M equipped with a semi-finite faithful normal trace .τ . (iii) .E(τ ) is an interpolation space for the couple .(Lp (τ ), Lq (τ )) for some nonatomic semi-finite von Neumann algebra .M equipped with a semi-finite faithful normal trace .τ such that .τ (1) = ∞. (iv) There exist two positive constants .cp,E and .cq,E such that: (a) For any .f ∈ E and .g ∈ (Lp + L∞ )(0, ∞), if .|g|p ≺≺hd |f |p , then .g ∈ E and .||g||E ≤ cp,E ||f ||E .

492

7 Interpolation

(b) For any .f ∈ E and .g ∈ (L1 + Lq )(0, ∞), if .|g|q ≺≺tl |f |q , then .g ∈ E and .||g||E ≤ cq,E ||f ||E . Proof The implications .(i) ⇒ (ii), (iii) follow from Theorem 7.3.6. The implications .(ii) ⇒ (i), (iii) are trivial. The implication .(i) ⇒ (iv) follows from Theorems 7.12.4 and 7.12.5. The implication .(iv) ⇒ (i) follows from Theorem 7.12.6. It suffices to prove the implication .(iii) ⇒ (i). Let .M be a non-atomic von Neumann algebra equipped with a semi-finite faithful normal trace .τ such that .τ (1) = ∞. Let x be a positive element in .E(τ ) ∩ S0 (τ ) such that .τ (s(x)) = ∞. Therefore, for each .n ≥ 1, en := ex ([n, n + 1)) ∨ ex ([

.

1 1 , )) n+1 n

are .τ -finite projections. By Theorem 4.4.12, there exist an abelian non-atomic von Neumann subalgebra .Nn of .en Men and a trace preserving normal surjective unital .∗-isomorphism .nn : L∞ (0, τ (en )) → Nn . We obtain that ∞ o

( ) ∞ nn : L∞ (0, ∞) → ⊕∞ n=1 Nn , ⊕n=1 τ (en · en ) ,

.

n=1

o o∞ Σ Σn defined by setting .( ∞ ), is a n=1 nn )(x) := n=1 nn (xχ( n−1 k=1 τ (ek ),( k=1 τ (ek )] trace preserving normal surjective unital .∗-isomorphism. Therefore, .L∞ (0, ∞) is (trace preserving) .∗-isomorphic to a von Neumann subalgebra of .s(x)Ms(x). For convenience, we write that .L∞ (0, ∞) is a von Neumann subalgebra of .s(x)Ms(x). Moreover, the trace .τ is semi-finite on .L∞ (0, ∞). Let .E be the conditional expectation from .s(x)Ms(x) onto .L∞ (0, ∞) (see Proposition 7.1.1). Observe that .E extends to .(L1 +L∞ )(s(x)Ms(x)) with .E(y) = y for any .y ∈ (L1 + L∞ )(0, ∞). Let T : (Lp + Lq )(0, ∞) → (Lp + Lq )(0, ∞)

.

with .||T ||Lp →Lp , ||T ||Lq →Lq < ∞. Note that .T ◦E : (Lp +Lq )(τ ) → (Lp +Lq )(τ ) with .||T ◦ E||Lp (τ )→Lp (τ ) , ||T ◦ E||Lq (τ )→Lq (τ ) < ∞. It follows from assertion (iii) that .||T ◦ E||E(τ )→E(τ ) < ∞. In particular, .T ◦ E is a bounded mapping from E into .E(τ ). Also, since .E(y) = y for any .y ∈ E and T is a bounded mapping from .(Lp + Lq )(0, ∞) into .(Lp + Lq )(0, ∞), it follows that .T ◦ E is a bounded mapping from E into .(Lp +Lq )(0, ∞). Therefore, .T ◦E maps E into .E(τ )∩(Lp +Lq )(0, ∞) (equipped with either .||·||E or .||·||Lp +Lq ). Note that .T ◦ E(y) = T (y) for any .y ∈ E(0, ∞). It suffices to note that E(τ ) ∩ (Lp + Lq )(0, ∞) = E.

.

7.12 Interpolation for the Pair (Lp (τ ), Lq (τ )), 1 ≤ p < q ≤ ∞

493

Indeed, since .E(τ ) is an interpolation space for the couple .(Lp (τ ), Lq (τ )), it follows that .E(τ ) ⊂ (Lp + Lq )(τ ). In particular, .E ⊂ (Lp + Lq )(0, ∞). Therefore, every element in E is an element in .(Lp + Lq )(0, ∞) and .E(τ ), i.e., .E ⊂ E(τ ) ∩ (Lp + Lq )(0, ∞). On the other hand, let .x ∈ E(τ ) ∩ (Lp + Lq )(0, ∞). We have .x ∈ (Lp + Lq )(0, ∞) with μ(x) ∈ E ∩ (Lp + Lq )(0, ∞) = E.

.

Hence, .x ∈ E. Since .x ∈ E(τ ) ∩ (Lp + Lq )(0, ∞) is arbitrarily taken, it follows u n that .E(τ ) ∩ (Lp + Lq )(0, ∞) ⊂ E, which completes the proof. A similar argument also yields the following result, the commutative specialization of which is due to Lorentz and Shimogaki. Corollary 7.12.8 Let .1 ≤ p < ∞. Let E be a fully symmetric space. Then, the following assertions are equivalent: (i) E is an interpolation space for the couple .(Lp (0, ∞), L∞ (0, ∞)). (ii) .E(τ ) is an interpolation space for the couple .(Lp (τ ), L∞ (τ )) for an arbitrary semi-finite von Neumann algebra .M equipped with a semi-finite faithful normal trace .τ . (iii) .E(τ ) is an interpolation space for the couple .(Lp (τ ), L∞ (τ )) for some nonatomic semi-finite von Neumann algebra .M equipped with a semi-finite faithful normal trace .τ such that .τ (1) = ∞. (iv) There exists a positive constant .cp,E such that: For any .f ∈ E(0, ∞) and .g ∈ (Lp + L∞ )(0, ∞), if .|g|p ≺≺hd |f |p , then .g ∈ E(0, ∞) and .||g||E ≤ cp,E ||f ||E . Theorem 7.12.6 with .p = 1, q = ∞ and Corollary 7.12.8 with .p = 1 imply the following. Corollary 7.12.9 The following statements are equivalent: (i) .E(τ ) is an interpolation space for the couple .(L1 (τ ), L∞ (τ )) for all semi-finite .(M, τ ). (ii) There exists a constant .cE such that, for any .f ∈ E(0, ∞) and .g ∈ (L1 + L∞ )(0, ∞), if .|g| ≺≺hd |f |, then .g ∈ E(0, ∞) and .||g||E ≤ cE ||f ||E . Using Corollary 7.12.7 (i) in the case .p = 1, 1 < q < ∞ and Corollary 7.12.9 yields the following complement to Corollary 7.12.8 (which is probably implicit but does not seem to be explicitly stated in the literature). Corollary 7.12.10 If .1 < q < ∞, then the following statements are equivalent: (i) E is an interpolation space for the pair .(L1 , Lq )(0, ∞). (ii) E is an interpolation space for the pair .(L1 , L∞ )(0, ∞), and there exists a constant .cq,E such that, for any .f ∈ E and .g ∈ (L1 + Lq )(0, ∞), if .|g|q ≺≺tl |f |q , then .g ∈ E and .||g||E ≤ cq,E ||f ||E .

494

7 Interpolation

(iii) .E(τ ) is an interpolation space for the couple .(L1 (τ ), Lq (τ )), for all semi-finite .(M, τ ). Corollary 7.12.11 If .1 ≤ p < q < ∞, then the following statements are equivalent: (i) E is an interpolation space for each of the pairs .(L1 , Lq )(0, ∞), (Lp , L∞ ). (ii) .E(τ ) is an interpolation space for the pair .(Lp (τ ), Lq (τ )) for all semi-finite .(M, τ ). Proof (i).⇒(ii) Since E is an exact interpolation space for the pair .(L1 , Lq )(0, ∞), it follows from the commutative specialization of Corollary 7.12.10 (ii) that there exists a constant .cq,E such that, for any .f ∈ E and .g ∈ (L1 + Lq )(0, ∞), if q q .|g| ≺≺tl |f | , then .g ∈ E and .||g||E ≤ cq,E ||f ||E . Since E is an exact interpolation space for the pair .(Lp , L∞ )(0, ∞), it follows from Corollary 7.12.8 (iv) that there exists a constant .cp,E such that, for any .f ∈ E(0, ∞) and .g ∈ (Lp + L∞ )(0, ∞), if .|g|p ≺≺hd |f |p , then .g ∈ E(0, ∞) and .||g||E ≤ cp,E ||f ||E . The assertion of (ii) now follows from the implication (iv).⇒(ii) of Corollary 7.12.7. (i).⇒(ii) By Theorem 7.3.6, it suffices to observe that if E is an exact interpolation space for the pair .(Lp , Lq )(0, ∞), then E is an exact interpolation space for each of the pairs .(L1 , Lq )(0, ∞), .(Lp , L∞ ), and .(L1 , L∞ ). u n The commutative specialization of the previous equivalence is due to Arazy and Cwikel. The rest of this section is devoted to the proofs of the main results, Theorems 7.12.4, 7.12.5, and 7.12.6. The following lemma (the first partition lemma) is an important ingredient in the proof (see also [67, 122] for similar results). Lemma 7.12.12 Let .f, g ∈ L1 (0, ∞) be positive decreasing step functions. Assume that .g ≺hd f. There exists a partition .{Ik , Jk }k≥0 of .(0, ∞) such that: (i) (ii) (iii) (iv)

For every .k ≥ 0, .Ik and .Jk are disjoint intervals of finite length. (Ik ∪ Jk ) ∩ (Il ∪ Jl ) = ∅ for .k /= l. f and g are constant on .Ik and on .Jk . .g|Ik ∪Jk ≺hd f |Ik ∪Jk for every .k ≥ 0. .

Proof We choose a countable collection of pairwise disjoint intervals .{(ai , bi )}i∈I such that both f and g are constant on each .(ai , bi ) (those constants are denoted by .fi and .gi , respectively) and such that u .

(ai , bi ) = (0, ∞)

i∈I

modulo a set of measure .0. We introduce a natural order on .I by setting .i1 < i2 if the interval .(ai1 , bi1 ) is strictly on the left to the interval .(ai2 , bi2 ).

7.12 Interpolation for the Pair (Lp (τ ), Lq (τ )), 1 ≤ p < q ≤ ∞

495

There exists a subset .J ⊂ I such that {t : f (t) < g(t)} =

.

u (ai , bi ). i∈J

Since .g ≺ f, it follows that F

t

.

0

F (f − g)+ (s)ds −

t

0

F (g − f )+ (s)ds =

t

F

t

f (s)ds −

0

g(s)ds ≥ 0.

0

For each .i ∈ J, denote by .ci the minimal .t > 0 such that F

t

.

0

F (f − g)+ (s)ds =

bi 0

(g − f )+ (s)ds.

Since .f < g on .(ai , bi ), .i ∈ J, it follows that, for every .i ∈ J, we have F

ai

.

0

F (f − g)+ (s)ds =

bi 0

F (f − g)+ (s)ds ≥

bi 0

(g − f )+ (s)ds.

Hence, .ci ≤ ai for .i ∈ J. For each .i ∈ J, the set u

(ci−1 , ci ) ∩ {t : f (t) > g(t)} =

.

Ii,j ,

j ∈I\J

where Ii,j = (ci−1 , ci ) ∩ (aj , bj ),

.

i ∈ J,

j ∈ I\J.

Some of those intervals may be empty. By the definition of .ci , we have F

bi

.

ai

F (g − f )+ (s)ds =

ci ci−1

(f − g)+ (s)ds =

Σ F j ∈I\J Ii,j

(f − g)+ (s)ds.

Set .K = J × (I\J). For .k = (i, j ) ∈ K, set .Ik = Ii,j and ) ( ' , Jk = Ji,j = ai + (gi − fi )−1 ci,j , ai + (gi − fi )−1 ci,j

.

496

7 Interpolation

where ci,j =

ΣF

.

l∈I\J Ii,l l 0. There exist ξ, η ∈ Lp' (τ ) such that .||(ξ, η)||l (2) (L ' (τ )) ≤ 1 and

.

p'

||T (x, y)||l (2) (L

.

p

p

p (τ ))

− ε ≤ ||.

(7.60)

516

7 Interpolation

Now observe that || = ||

.

≤ ||T ∗ ||l (2) (L p'

(2) p' (τ ))→lp (Lp (τ ))

≤ 21/p ||(x, y)||l (2) (L p'

p (τ ))

||(x, y)||l (2) (L p'

p (τ ))

,

where the final inequality follows from (7.51). Combining this with (7.60) yields, for all .x, y ∈ Lp (τ ), ||T (x, y)||l (2) (L

.

p

p (τ ))

− ε ≤ 21/p ||(x, y)||l (2) (L p'

p (τ ))

,

and from this, the inequality (7.58) follows immediately.

u n

A direct consequence of the Clarkson inequalities above is that, if .1 < p < ∞, then .Lp (τ ) has the (so-called) Kadec–Klee property, that is, weak and norm convergences coincide for sequences in the unit sphere of .Lp (τ ). Corollary 7.14.2 Suppose that .1 < p < ∞. If .x, xn ∈ Lp (τ ), n = 1, 2, . . . , if ||xn ||Lp (τ ) →n ||x||Lp (τ ) , and if .xn →n x for the weak topology .σ (Lp (τ ), Lp' (τ )), where .1 = 1/p + 1/p' , then .||xn − x||Lp (τ ) →n 0.

.

Proof Since the norm on any Banach space is lower semi-continuous for sequential weak convergence, and since the sequence .{xn }n≥1 converges weakly to x, it follows that 2||x||Lp (τ ) ≤ lim inf ||xn + x||Lp (τ ) ≤ lim sup ||xn + x||Lp (τ )

.

n→∞

n→∞

≤ lim sup(||xn ||Lp (τ ) + ||x||Lp (τ ) ) = 2||x||Lp (τ ) . n→∞

In particular, ||xn + x||Lp (τ ) →n 2||x||Lp (τ ) .

.

If it is not the case that .||xn − x||Lp (τ ) →n 0, then, by passing to a subsequence if necessary, it may be assumed that there exists .ε > 0 such that ||xn − x||Lp (τ ) ≥ ε,

.

n ≥ 1.

If .1 < p ≤ 2, it follows from (7.51) that ) p' ( ' p p' p p − ||xn + x||Lp (τ ) →n 0, εp ≤ 2 ||x||Lp (τ ) + ||xn ||Lp (τ )

.

7.14 Applications of Complex Method to the Geometry of Noncommutative. . .

517

while if .2 ≤ p < ∞, then it follows from (7.49) that ( ) p p p εp ≤ 2p−1 ||x||Lp (τ ) + ||xn ||Lp (τ ) − ||x + xn ||Lp (τ ) →n 0.

.

In each case, this yields a contradiction and consequently .||xn − x||Lp (τ ) →n 0 for all .1 < p < ∞. n u It is worth noting the following consequence of (7.50). Corollary 7.14.3 Suppose that .1 ≤ p ≤ 2. If .x, y ∈ Lp (τ ) are self-adjoint, then p

p

p

||x + iy||Lp (τ ) ≤ ||x||Lp (τ ) + ||y||Lp (τ ) .

.

(7.61)

Proof Replacing .x, y in (7.50) by .x + iy, x − iy, respectively, yields p

p

p

p

||x + iy||Lp (τ ) + ||x − iy||Lp (τ ) ≤ 2(||x||Lp (τ ) + ||y||Lp (τ ) ).

.

Since .x = x ∗ , y = y ∗ , it follows that ||x − iy||Lp (τ ) = ||(x + iy)∗ ||Lp (τ ) = ||x + iy||Lp (τ ) , p

.

p

p

and the Corollary now follows immediately.

u n

7.14.2 Uniform Convexity and Uniform Smoothness Let X be a Banach space with .dim X ≥ 2: (i) The modulus of convexity .δX (ε), 0 < ε ≤ 2, of X is defined by δX (ε) = inf{1 − ||x + y||/2 : x, y ∈ X, ||x|| = ||y|| = 1, ||x − y|| = ε}.

.

(ii) The modulus of smoothness .ρX (t), t > 0, of X is defined by ρX (t) = sup{(||x + ty|| + ||x − ty||) /2 − 1 : x, y ∈ X, ||x|| = ||y|| = 1}.

.

(iii) The Banach space X is said to be uniformly convex if .δX (ε) > 0 for every .ε > 0 and uniformly smooth if .limt→0 ρX (t)/t = 0. (iv) A uniformly convex (respectively, smooth) Banach space X is said to be of power type p (respectively, power type q) if, for some .0 < K < ∞, δX (ε) ≥ Kεp ,

.

(respectively, ρX (t) ≤ Kt p ).

518

7 Interpolation

The Clarkson inequalities given in Theorem 7.14.1 readily imply that .Lp (τ ) is uniformly convex and uniformly smooth for all .1 < p < ∞. Theorem 7.14.4 Suppose that .1 < p < ∞ and .1 = 1/p + 1/p' . If .0 < ε < 2 and .t > 0, then: (i) If .1 < p ≤ 2, then '

εp .δLp (τ ) (ε) ≥ ' p' 2p

and

ρLp (τ ) (t) ≤

tp . p

(ii) If .2 < p < ∞, then εp .δLp (τ ) (ε) ≥ p2p

'

and

tp ρLp (τ ) (t) ≤ ' . p

Proof (i) Suppose that .1 < p ≤ 2 and let .ε > 0 be given. If .x, y ∈ Lp (τ ) satisfy ||x||Lp (τ ) = ||y||Lp (τ ) = 1,

||x − y||Lp (τ ) = ε,

.

then it follows from (7.51) that '

' εp x + y p' 1+ p −p' ||Lp (τ ) + ' ≤ 2 p .|| = 1. 2 2

Consequently, '

||

.

εp x + y p' ||Lp (τ ) ≤ 1 − p' , 2 2

so that ( ) 1' ' ' εp p εp x+y ||Lp (τ ) ≤ 1 − p' ≤ 1 − ' p' . .|| 2 2 p2 This clearly implies the first assertion of (i). The estimate for the modulus of uniform smoothness follows in a similar manner. Indeed, if .x, y ∈ Lp (τ ), if .t > 0, and if .||x||Lp (τ ) = ||y||Lp (τ ) = 1, and again appealing to (7.51), it follows that p'

p'

p'

||x + ty||Lp (τ ) + ||x − ty||Lp (τ ) ≤ 2(1 + t p ) p .

.

7.14 Applications of Complex Method to the Geometry of Noncommutative. . .

519

1

Using the concavity of the function .(·) p' , this yields

.

||x + ty||Lp (τ ) + ||x − ty||Lp (τ ) 2

⎛ ≤⎝

p'

p'

||x + ty||Lp (τ ) + ||x − ty||Lp (τ ) 2

⎞ 1' p

⎠

1

≤ (1 + t p ) p . Consequently, using the fact that .1 < p ≤ 2, it follows that .

||x + ty||Lp (τ ) + ||x − ty||Lp (τ ) 2

1

− 1 ≤ (1 + t p ) p − 1 ≤

and this yields the stated modulus of uniform smoothness. The proof of (ii) is similar and is omitted.

tp , p u n

Corollary 7.14.5 If .1 < p < ∞, then .Lp (τ ) is uniformly convex and uniformly smooth. The estimate of the modulus of convexity given in Theorem 7.14.4 is best possible only in the case that .2 < p < ∞. To obtain the optimal order for the case that .1 < p < 2, it is necessary to prove a sharper version of the inequalities of Theorem 7.14.4. For this, it will be necessary to use the following variant of the interpolation formula (7.48). If X is a Banach space, if .1 ≤ p ≤ ∞, and if .w > 0, then the Banach space (2) .lp,w (X) is the set of all pairs .(x, y) ∈ X × X equipped with the norm ||(x, y)||l (2)

.

p,w (X)

( p p )1/p = ||x||X + w||y||X .

Again referring to Calderón (see [18], section 13.6; see also [9] Theorem 5.5.3), if X0 , X1 are Banach spaces, if .1 ≤ p0 , p1 ≤ ∞, if .w0 , w1 > 0 and if at least one of (2) (2) the spaces .lp0 ,w0 (X0 ), lp1 ,w1 (X1 ) is reflexive, then

.

[ .

lp(2) (X0 ), lp(2) (X1 ) 0 ,w0 1 ,w1

] θ

(2) = lp,w ([X0 , X1 ]θ )) , p(1−θ)/p0

where .1/p = (1 − θ )/p0 + θ/p1 and .w = w0

pθ/p1

w1

(7.62)

.

The following generalized Hölder inequality will be needed. We include a short proof below (see also [24], or [64]). Lemma 7.14.6 Suppose that .x, y ∈ S(τ ) and that .p, p0 , p1 satisfy .1/p = 1/p0 + 1/p1 . If .x ∈ Lp0 (τ ), y ∈ Lp1 (τ ), then .xy ∈ Lp (τ ), in which case ||xy||Lp (τ ) ≤ ||x||Lp0 (τ ) ||y||Lp1 (τ ) .

.

520

7 Interpolation

Proof By the classical Hölder inequality, we have ||μ(x)μ(y)||Lp (m) ≤ ||μ(x)||Lp0 (τ ) ||μ(y)||Lp1 (m) .

.

By Theorem 3.9.10, we have .μ(xy) ≺≺ μ(x)μ(y). Since .Lp (m) is a fully symmetric space, it follows that ||μ(xy)||Lp (m) ≤ ||μ(x)μ(y)||Lp (m) .

.

Therefore, ||μ(xy)||Lp (τ ) = ||μ(xy)||Lp (m) ≤ ||μ(x)||Lp0 (m) ||μ(y)||Lp1 (m)

.

= ||x||Lp0 (τ ) ||y||Lp1 (τ ) . The proof is thereby complete.

u n

Theorem 7.14.7 (i) If .2 ≤ p < ∞, then there exists a positive constant .Cp depending only on p and which satisfies .Cp ≤ 2p − 1, such that, for all .x, y ∈ Lp (τ ), [ ( )]1/p ( )1/2 1 p p ||x + y||Lp (τ ) + ||x − y||Lp (τ ) ≤ ||x||2Lp (τ ) + Cp ||y||2Lp (τ ) . 2 (7.63)

.

(ii) If .1 < p ≤ 2, then there exists a constant .cp depending only on p and which satisfies .cp ≥ (p − 1)/(p + 1) such that, for all .x, y ∈ Lp (τ ), .

( )1/2 [ 1 ( )]1/p p p ||x + y||Lp (τ ) + ||x − y||Lp (τ ) ||x||2Lp (τ ) + cp ||y||2Lp (τ ) ≤ , 2 (7.64)

or equivalently, ( )1/2 [ ( )]1/p p p 2 2 p−1 ||x|| . ||x + y||L (τ ) + cp ||x − y||L (τ ) ≤ 2 + ||y|| . (τ ) (τ ) L L p p p p (7.65) Proof (i) It will be shown first that, if the inequality (7.63) holds for some .p ∈ [2, ∞), then it holds also for 2p. Suppose then that the inequality (7.63) holds for some fixed .p ∈ [2, ∞). Let .x, y ∈ L2p (τ ) and set .a = x ∗ x + y ∗ y, b = x ∗ y + y ∗ x. Observe that .a, b ∈ Lp (τ ) and .

) ) 1( 1( p p 2p 2p ||a + b||Lp (τ ) + ||a − b||Lp (τ ) . ||x + y||L2p + ||x − y||L2p = 2 2

7.14 Applications of Complex Method to the Geometry of Noncommutative. . .

521

By the assumption that the inequality (7.63) holds for p, and using the Hölder inequality Lemma 7.14.6, it follows that .

) 1( p p ||a + b||Lp (τ ) + ||a − b||Lp (τ ) 2 )p/2 ( ≤ ||aL2 p (τ ) + Cp ||b||2Lp (τ ) )p/2 ( = ||x ∗ x + y ∗ y||2Lp (τ ) + Cp ||y ∗ x + x ∗ y||2Lp (τ ) ]p/2 [ ≤ (||x||2L2p (τ ) + ||y||2L2p (τ ) )2 + 4Cp ||x||2L2p (τ ) ||y||2L2p (τ ) ]p/2 [ = (||x||2L2p (τ ) )2 + 2(2Cp + 1)||x||2L2p (τ ) ||y||2L2p (τ ) + (||x||2L2p (τ ) )2 )p ( ≤ ||x||2L2p (τ ) + (2Cp + 1)||y||2L2p (τ ) .

Consequently, .

)1/2 )1 ( 1( 2p 2p 2p ||x + y||L2p (τ ) + ||x − y||L2p (τ ) ≤ ||x||2L2p (τ ) + (2Cp + 1)||y||2L2p (τ ) 2

and so (7.63) holds with p replaced by 2p and .C2p ≤ 2Cp +1. Via the parallelogram law, observe that .C2 = 1. By induction, it follows that .C2n ≤ 2n − 1, n ≥ 1. Now suppose that .2n < p < 2n+1 and define .θ by setting 1/p = (1 − θ )/2n + θ/2n−1 .

.

Observe via (7.62) that [ .

] (2) (2) (2) l2,2n −1 (L2n (τ )), l2,2n+1 −1 (L2n+1 (τ )) = l2,Cp (Lp (τ )) θ

with .Cp = (2n − 1)1−θ (2n+1 − 1)θ and that [ .

] (2) (2) l2n (L2n (τ )), L2n+1 (L2n+1 (τ )) = lp(2) (Lp (τ )). θ

Now using the same interpolation argument applied to the mapping .(x, y) → (x + y, x−y), as in the proof of equation (7.51), Theorem 7.14.1 (b), yields the inequality (7.63) with Cp = (2n − 1)1−θ (2n+1 − 1)θ ≤ 2n+1 − 1 ≤ 2p − 1,

.

and this suffices to complete the proof of (i).

522

7 Interpolation

(ii) Using an argument similar to that used in the proof of Theorem 7.14.1, and noting that (2)

(2) lp,w (Lq (τ )) = lp' ,1/w (Lq ' (τ )),

.

1 < p, q < ∞,

part (ii) follows from (i) by duality. One obtains cp =

.

p−1 1 1 = . ≥ Cp ' 2p' − 1 p+1

Finally, the inequality (7.65) follows from (7.65) by replacing .x, y by .x + y, x − y, respectively. u n It is to be remarked that the preceding proof follows the outline given in [102]. Optimal estimates for the modulus of uniform convexity .δLp (τ ) (ε) in the case that .1 < p < 2 and on the modulus of uniform smoothness .ρLp (τ ) in the case that .2 < p < ∞ now follow. Corollary 7.14.8 (i) If .2 ≤ p < ∞, then ρLp (τ ) (t) ≤

.

Cp 2 t , 2

t > 0.

(ii) If .1 < p ≤ 2, then δLp (τ ) (ε) ≥

.

cp 2 ε . 8

Proof Only the estimate given in (ii) will be proved; the proof of (i) is similar and will be omitted. Suppose then .1 < p ≤ 2, that .ε > 0 and that ||x||Lp (τ ) = 1 = ||y||Lp (τ ) ,

.

||x − y||Lp (τ ) = ε.

Using the inequality (7.65), it follows that .

( )1/2 ||x + y||2Lp (τ ) + cp ε2 ≤2

so that ) ( cp ε2 ≤ 4 1 − ||(x + y)/2||2Lp (τ ) ( )( ) ≤ 4 1 − ||(x + y)/2||Lp (τ ) 1 + ||(x + y)/2||Lp (τ ) ) ( ≤ 8 1 − ||(x + y)/2||Lp (τ )

.

and the assertion of (ii) follows.

u n

7.14 Applications of Complex Method to the Geometry of Noncommutative. . .

523

Combining Theorem 7.14.4 with Corollary 7.14.8 now yields the following. Corollary 7.14.9 If .1 < p < ∞, then .Lp (τ ) is uniformly convex of power type max{2, p} and uniformly smooth of power type .min{2, p}.

.

Remark 7.14.10 As is clear from the preceding sections, the power type estimates for the moduli of convexity and concavity have been based on the associated Clarkson type inequalities. It is interesting to note the following characterizations of these power type estimates in terms of Clarkson type inequalities as has been observed by Ball, Carlen and Lieb [7]. Proposition 7.14.11 If X is a Banach space, and if .2 ≤ r < ∞, then the following statements are equivalent: (i) There exists a constant .K > 0 such that, for all .x, y ∈ X, .

|| || || || || x + y ||r || x − y ||r ||x||r + ||y||r || + || || || . || 2 || || 2K || ≤ 2

(7.66)

(ii) .δX (ε) ≥ (ε/C)r for some constant .C > 0. Proposition 7.14.12 If X is a Banach space and .1 < r ≤ 2, then the following statements are equivalent: (i) There exists a constant .K > 0 such that, for all .x, y ∈ X, .

||x + y||r + ||x − y||r ≤ ||x||r + ||Ky||r . 2

(7.67)

(ii) .ρX (t) ≤ (Ct)r for some constant .C > 0. To conclude this section, an interesting application will be given to a noncommutative extension of a classical theorem of Orlicz concerning unconditionally convergent series based on the following theorem of Kadec. Theorem 7.14.13 (Kadec) Let XΣ be a uniformly convex Banach space with modulus of uniform convexity .δX (·). If . ∞ j =1 xj is an unconditionally convergent series Σ θ x converges in X for all choices of signs .θn = ±1, n ≥ 1, in X, that is, . ∞ j j j =1 Σ δ (||x|| ) < ∞. then . ∞ X j =1 X The preceding Theorem 7.14.13 together with Corollary 7.14.8 (i) and Theorem 7.14.4 (ii) now yield the following consequence. Σ Corollary 7.14.14 If . ∞ j =1 xj is an unconditionally convergent series in .Lp (τ ), 1 < p < ∞, then ∞ Σ .

j =1

max{p,2}

||xj ||Lp (τ )

< ∞.

524

7 Interpolation

7.14.3 Type and Cotype of the Spaces Lp (τ ), 1 ≤ p < ∞ Denote by .{rn } the usual Rademacher sequence on .[0, 1], that is, rn (t) = sgn sin(2n n t),

.

t ∈ [0, 1], n ≥ 0.

Definition 7.14.15 The Banach space X is said to be of type p for some .1 < p ≤ 2 if there exists a constant .0 < C < ∞ such that, for every finite sequence .(xj )nj=1 ⊆ X, it follows that F

1

.

||

0

n Σ

⎛ rj (t)xj ||X dt ≤ C ⎝

j =1

n Σ

⎞1/p ||xj ||X ⎠ p

,

j =1

and X is said to be of cotype q for some .q ≥ 2 if there exists a constant .0 < C < ∞ such that, for every finite sequence .(xj )nj=1 ⊆ X, it follows that F .

1

||

⎛

n Σ

0

rj (t)xj ||X dt ≥ C −1 ⎝

n Σ

j =1

⎞1/q ||xj ||X ⎠ q

.

j =1

It should be noted that every Banach space X is simultaneously of type 1 and cotype .∞, that is, has trivial type and cotype. Indeed, if .(xj )nj=1 ⊆ X is a finite sequence, then it is clear that F

1

.

||

0

n Σ

rj (t)xj ||X dt ≤

j =1

n Σ

||xj ||X ;

j =1

and F

1

||xi ||X = ||

ri (t)[

.

0

F

1

≤ 0

||

n Σ

xj rk (t)]dt||X

j =1 n Σ

rj (t)xj ||X dt,

i = 1, 2, . . . ,

j =1

which implies that F n .||(||xj ||X )j =1 ||∞

1

= max ||xi ||X ≤ 1≤i≤n

0

||

n Σ j =1

rj (t)xj ||X dt.

7.14 Applications of Complex Method to the Geometry of Noncommutative. . .

525

To see the reasons for the restrictions on .p, q above, it will be first necessary to recall the classical Khintchine Inequality (see [79], I.2.b.3). If .1 ≤ p < ∞, there exist scalars .0 < Ap , Bp such that ⎛ Ap ⎝

n Σ

.

⎞1/2 |an |2 ⎠

⎛ ≤⎝

F 0

j =1

|p ⎞1/p ⎛ ⎞1/2 | n Σ | | aj rj (t)|| dt ⎠ ≤ Bq ⎝ |an |2 ⎠ | | |j =1 j =1 |

| n 1 |Σ

(7.68) for all finite sequences of scalars .(aj )nj=1 . It follows immediately from the Khintchine inequality that the spaces .R and .C are simultaneously of type 2 and cotype 2. It is a further consequence of the Khintchine inequality that no Banach space .X /= 0 can be of type .p > 2 or of cotype .q < 2. Indeed, if .x ∈ X with .||x||X = 1 and .x1 = x2 = · · · = xn = n1 x, then F 1 Σn F 1 1 Σn j =1 rj (t)|dt j =1 xj rj (t)||X dt 0 || 0 n| Σn = ≈ n1/2−1/r . . r ( j =1 ||xj ||X )1/r n1/2−1 From this, it follows that estimation from above is possible for any .n ∈ N only if r ≤ 2 and estimation from below only if .r ≥ 2. Before proceeding, it is to be noted that the .L1 -averages that appear in the definition of type and cotype can be replaced more generally with .Lr -averages for .1 ≤ r < ∞. This is a consequence of the following inequalities due to Kahane.

.

Theorem 7.14.16 (Kahane’s Inequalities) For every .1 < r < ∞, there exists a constant .0 < Kr < ∞ such that, for any Banach space X and every finite sequence n .{xj } j =1 ⊆ X, ||

|| || ||r ⎞1/r ⎛ || || F 1 ||Σ || || n || || || dt ≤ ⎝ || || dt ⎠ r (t)x r (t)x j j j j || || || || 0 ||j =1 ||j =1 || || X X || || || || F 1 Σ || n || || ≤ Kr rj (t)xj || || || dt. 0 ||j =1 ||

|| n 1 ||Σ

F .

0

(7.69)

X

Suppose that H is a Hilbert space and that .{xj }nj=1 ⊆ H is a finite sequence. Observe that F

1

.

0

||

n Σ j =1

xj rj (t)||2H dt =

n Σ i,j =1

F

1

0

ri (t)rj (t)dt =

n Σ

||xj ||2H .

j =1

Consequently, from Kahane’s inequalities, it follows that H is simultaneously of type 2 and cotype 2; in particular, .L2 (τ ) has type 2 and cotype 2. Conversely, it

526

7 Interpolation

has been proved by Kwapie´n (1972) that a Banach space is isomorphic to a Hilbert space if and only if it has type 2 and cotype 2. The theorem that now follows exhibits the type and cotype of the space .Lp (τ ) in the case that .1 < p < ∞. Theorem 7.14.17 If .1 < p < ∞, then .Lp (τ ) is of type .min{p, 2} and cotype max{p, 2}. In particular, let .{xj }nj=1 ⊆ Lp (τ ) be a finite sequence. If .1 < p ≤ 2, then

.

√ .

⎛ n Σ cp ⎝ ||xj ||2L

⎞1/2 ⎠

p (τ )

j =1

⎞1/2 ⎛ ⎞1/p ⎛ F 1 Σ n n Σ p ≤ ⎝ || rj (t)xj ||2Lp (τ )⎠ ≤ ⎝ ||xj ||Lp (τ ) ⎠ , 0

j =1

j =1

(7.70) and if .2 ≤ p < ∞, then ⎛ n Σ p .⎝ ||xj ||

⎞1/p ⎠

Lp (τ )

j =1

⎛ ⎛ ⎞1/2 ⎞1/2 F 1 Σ n n Σ √ ≤ ⎝ || rj (t)xj ||2Lp (τ )⎠ ≤ C p ⎝ ||xj ||2Lp (τ )⎠ . 0

j =1

j =1

(7.71) Here, the constants .cp , Cp are those given in Theorem 7.14.7. Proof Suppose that .1 < p ≤ 2. Using Theorem 7.14.1 (7.51), observe that F

1

.

0

||

n Σ

p'

rj (t)xj ||Lp (τ ) dt

j =1

1 = 2 F ≤ 0

⎛

F

1

⎝||

0

⎛ 1

⎝||

n−1 Σ j =1

n−1 Σ j =1

p'

rj (t)xj + xn ||Lp (τ ) + ||

n−1 Σ j =1

⎞ p'

rj (t)xj − xn ||Lp (τ ) ⎠ dt

⎞p' /p rj (t)xj dt||Lp (τ ) + ||xn ||Lp (τ ) ⎠ p

p

dt

⎧⎛ ⎫p' /p ⎞p/p' ⎪ )p/p' ⎪ (F 1 n−1 ⎨ F 1 Σ ⎬ p' p' ≤ ⎝ || rj (t)xj ||Lp (τ ) dt ⎠ + ||xn ||Lp (τ ) ⎪ ⎪ 0 ⎩ 0 j =1 ⎭ ⎫p' /p ⎧⎛ ⎞p/p' ⎪ ⎪ n−1 ⎨ F 1 Σ ⎬ ' p p = ⎝ || rj (t)xj ||Lp (τ ) dt ⎠ + ||xn ||Lp (τ ) , ⎪ ⎪ ⎩ 0 j =1 ⎭

7.14 Applications of Complex Method to the Geometry of Noncommutative. . .

527

where the penultimate step follows from the fact that .p' /p ≥ 1 and application of the Minkowski inequality in the space .Lp' /p [0, 1]. Repeating the argument .(n − 1) times yields the inequality ⎛ F ⎝ .

1

||

0

n Σ j =1

⎞1/p'

⎛

p' rj (t)xj ||Lp (τ ) dt ⎠

≤⎝

n Σ j =1

⎞1/p p ||xj ||Lp (τ ) ⎠

.

Since .p' ≥ 2, it follows that ⎛ .

⎝

F

1

||

0

n Σ j =1

⎞1/2 rj (t)xj ||2Lp (τ ) dt ⎠

⎛ ≤⎝

F

1

||

0

n Σ j =1

⎞1/p' p'

rj (t)xj ||Lp (τ ) dt ⎠

,

and the right hand inequality of (7.70) now follows. The left hand inequality of (7.70) follows in a similar fashion using instead Theorem 7.14.7(7.64). Indeed, F

1

.

0

||

n Σ

p

rj (t)xj ||Lp (τ ) dt

j =1

⎛

F

1 = 2

1

⎝||

0

j =1

⎛

F

1

≥

⎝||

0

n−1 Σ

n−1 Σ j =1

p rj (t)xj + xn ||Lp (τ )

+ ||

n−1 Σ j =1

⎞ p rj (t)xj − xn ||Lp (τ ) ⎠ dt

⎞p/2 rj (t)xj dt||2Lp (τ ) + cp ||xn ||2Lp (τ ) ⎠

dt

⎧⎛ ⎫p/2 ⎞2/p ⎪ )2/p ⎪ (F 1 n−1 ⎨ F 1 Σ ⎬ p p/2 p ≥ ⎝ || rj (t)xj ||Lp (τ ) dt ⎠ + cp ||xn ||Lp (τ ) ⎪ ⎪ 0 ⎩ 0 j =1 ⎭ ⎧⎛ ⎫p/2 ⎞2/p ⎪ ⎪ n−1 ⎨ F 1 Σ ⎬ p 2 ⎝ ⎠ = || rj (t)xj ||Lp (τ ) dt + cp ||xn ||Lp (τ ) , ⎪ ⎪ ⎩ 0 j =1 ⎭ again using fact that .p/2 ≤ 1 and the (reverse) Minkowski inequality in the space Lp/2 [0, 1]. Repeating the argument .(n − 1) times yields the inequality

.

⎛ F ⎝ .

1 0

||

n Σ j =1

⎞1/p p rj (t)xj ||Lp (τ ) dt ⎠

⎛ ⎞1/2 n √ ⎝Σ ≥ cp ||xj ||2Lp (τ ) ⎠ . j =1

528

7 Interpolation

Since .p ≤ 2, it follows that ⎛ F .⎝

1 0

||

n Σ j =1

⎞1/2 rj (t)xj ||2Lp (τ ) dt ⎠

⎛ F ≥⎝

1

||

0

n Σ j =1

⎞1/p rj (t)xj ||Lp (τ ) dt ⎠ p

,

and the left hand inequality of (7.70) now follows. The proof of the inequalities (7.71) in the case that .2 ≤ p < ∞ is similar using the Clarkson inequalities given in Theorem 7.14.1(7.52) and Theorem 7.14.7(7.63). The details are omitted. u n Remark 7.14.18 The above approach that derives the type and cotype inequalities for the spaces .Lp (τ ) in the case that .1 < p < ∞ based on the Clarkson inequalities may be found in [73], which shows the equivalence in the Banach space setting of type and cotype inequalities with inequalities of Clarkson type. See also [135]. Remark 7.14.19 It should be noted that the type and cotype of the spaces .Lp (τ ) in the case that .1 < p < ∞ also follow from the moduli for uniform smoothness and convexity given in Corollary 7.14.9. Indeed, it is well known (see [79] Theorem 1.e.16) that a Banach space X that has modulus of convexity of power type q for some .q ≥ 2 is also of cotype q; and a Banach space X that has modulus of smoothness of power type p, for some .1 < p ≤ 2, is also of type p. The case for .p = 1 now follows. Theorem 7.14.20 The space .L1 (τ ) has cotype 2. Proof Let .{xj }nj=1 ⊆ L1 (τ ) be a finite sequence. It will be shown that ⎛ ⎞1/2 F n √ −1 Σ 2 ⎝ ⎠ ||xj ||L1 (τ ) ≤ .(2 e)

1

||

0

j =1

n Σ

xj rj (t)dt||L1 (τ ) dt.

(7.72)

j =1

To this end, it will suffice to assume that .xj = xj∗ , 1 ≤ j ≤ n and to prove the inequality ⎛ ⎞1/2 F n Σ −1 ⎝ 2 ⎠ ||xj ||L1 (τ ) ≤ .( e) √

0

j =1

1

||

n Σ

xj rj (t)dt||L1 (τ ) dt.

j =1

Indeed, setting xj' = Re xj ,

.

xj'' = Im xj ,

1 ≤ n,

noting that ||Re xj ||L1 (τ ) , ||Im xj ||L1 (τ ) ≤ ||xj ||L1 (τ )

.

(7.73)

7.14 Applications of Complex Method to the Geometry of Noncommutative. . .

529

and assuming that the inequality (7.73) holds for each of .{xj' }nj=1 , {xj'' }nj=1 , observe that ⎛ ⎞1/2 n √ −1 Σ ⎝ ||xj ||2L1 (τ ) ⎠ .(2 e) j =1

⎛ ⎞1/2 n √ −1 Σ ≤ (2 e) ⎝ (||xj' ||L1 (τ ) + ||xj'' ||L1 (τ ) )2 ⎠ ⎛

j =1

⎞ n n Σ √ −1 Σ ||xj' ||2L1 (τ ) )1/2 + ( ||xj'' ||2L1 (τ ) )1/2 ⎠ ≤ (2 e) ⎝( j =1

≤

1 2

F

F ≤ 0

1

||

0 1

||

n Σ

j =1

xj' rj (t)||L1 (τ ) dt +

j =1 n Σ

1 2

F

1

||

0

n Σ

xj'' rj (t)||L1 (τ ) dt

j =1

xj rj (t)||L1 (τ ) dt.

j =1

Assume then that .xj = xj∗ , 1 ≤ j ≤ n. To prove the inequality (7.73), it may be clearly assumed further that ⎛ ∞ Σ .⎝ ||xj ||2L j =1

⎞1/2

1 (τ )

⎠

= 1. (n)

(n)

Using the fact that the Banach dual of the space .l2 (L1 (τ )) is the space .l2 (M) with respect to the dual pairing =

n Σ

.

τ (xj yj ),

xj ∈ L1 (τ ), yj ∈ M, 1 ≤ j ≤ n

j =1

there exist .yj ∈ M, 1 ≤ j ≤ n, which may be taken to be self-adjoint, such that ||{yj }nj=1 ||l (n) (M) = (

n Σ

.

2

j =1

||yj ||2M )1/2 = 1

and n Σ .

j =1

τ (xj yj ) = ||{xj }nj=1 ||l (n) (L 2

1 (τ ))

= 1.

530

7 Interpolation

Now set n ||

o(t) = −i

.

(1 + iyj rj (t)) ∈ M,

t ∈ [0, 1],

j =1

and observe that, for each .t ∈ [0, 1], ||o(t)||M ≤

n ||

.

(1 + ||yj ||2M )1/2

j =1

⎤ ⎡ n 1 ⎣Σ 2 ⎦ log(1 + ||yj ||M ) = exp 2 j =1

⎤ n √ 1 ⎣Σ ≤ exp ||yj ||2M ⎦ = e. 2 ⎡

j =1

Now, as is easily seen by expanding .o and using the independence of the equidistributed random variables .rj (·), 1 ≤ j ≤ n, F

1

.

τ (xj rj (t))o(t)dt = τ (xj yj ),

1 ≤ j ≤ n, t ∈ [0, 1],

0

and so 1=

Σ

.

⎛

F

1

τ (xj yj ) =

τ ⎝o(t)

0

F ≤ sup ||o(t)||M t∈[0,1]

≤

√ e

F

1 0

||

n Σ

0

n Σ

⎞ xj rj (t)⎠ dt

j =1 1

||

n Σ

xj rj (t)||L1 (τ ) dt

j =1

xj rj (t)||L1 (τ ) dt.

j =1

This establishes the inequality (7.73), and this completes the proof of the Theorem. u n Remark 7.14.21 The preceding theorem and its proof are due to TomczakJaegermann [130] where essentially the same proof shows that the predual of any von Neumann algebra has cotype 2. References: [9, 18, 20, 24, 54, 64, 73, 75, 79, 102, 120, 129, 130, 135, 141, 146].

7.15 The Calderón Family for a Banach Couple of Symmetric Spaces

531

7.15 The Calderón Family for a Banach Couple of Symmetric Spaces Suppose now that .E0 , E1 ⊆ S(ν) are Banach function spaces on the Maharam measure space .(o, Σ, ν). Following [76], if .0 < θ < 1, E will denote the class of all complex .ν-measurable functions x on .o such that |x| ≤ λ|x0 |1−θ |x1 |θ

(7.74)

.

for some .λ > 0, .x0 ∈ E0 , and .x1 ∈ E0 with .||x0 ||E0 ≤ 1 and .||x||E1 ≤ 1. Setting ||x||E = inf{λ},

.

x ∈ E,

(7.75)

where the infimum is taken over all .λ satisfying (7.74), the space .(E, || · ||E ) is a normed order ideal. Indeed, at the outset, it is clear that, if .x ∈ E and .y ∈ S(ν) satisfy .|y| ≤ |x|, then .y ∈ E and .||y||E ≤ ||x||E , and if .α is any scalar, then .αx ∈ E and .||αx||E = |α|||x||E . If .||x|| = 0, then there exists a sequence .λn ↓n 0 and sequences .x0,n ∈ E0 , x1,n ∈ E1 with .||x0,n ||E0 ≤ 1, ||x1,n ||E1 ≤ 1, n ≥ 1 such that |x| ≤ (λn |x0,n |)1−θ (λn |x1,n |)θ

.

n ≥ 1.

Now .λn |x0,n |, λn |x1,n | →n 0 for the respective norm topologies and hence for the measure topology in .S(ν) . Consequently, passing to a suitable subsequence, (λn |x0,n |)1−θ (λn |x1,n |)θ →n 0

.

ν-almost everywhere and it follows that .x = 0. Suppose now that .x, y ∈ E and that .ε > 0. If .x0 , y0 ∈ E0 , x1 , y1 ∈ E1 , and .λ, μ satisfy .

||x0 ||E0 , ||y0 ||E0 ≤ 1,

||x1 ||E1 , ||y1 ||E1 ≤ 1

|x| ≤ λ|x0 |1−θ |x1 |θ ,

|y| ≤ μ|y0 |1−θ |y1 |θ ,

.

and .

and 0 < λ ≤ ||x||E + ε/2,

.

0 < μ ≤ ||y||E + ε/2.

532

7 Interpolation

Then, using Hölder’s inequality |x + y| ≤ |x| + |y|

.

≤ λ1−θ |x0 |1−θ λθ |x1 |θ + μ1−θ |y0 |1−θ μθ |y1 |θ ≤ (λ|x0 | + μ|y0 |)1−θ (λ|x1 | + μ|y1 |)θ ≤ (λ + μ)|z0 |1−θ |z1 |θ , where z0 =

.

λ|x0 | + μ|y0 | ∈ E0 , λ+μ

z1 =

λ|x1 | + μ|y1 | ∈ E1 λ+μ

satisfy that .||z0 ||E0 ≤ 1 and .||z1 ||E1 ≤ 1. Consequently, .x + y ∈ E and ||x + y||E ≤ λ + μ ≤ ||x||E + ||y||E + ε,

.

and this yields the triangle inequality. To see that E is a Banach space, suppose that .{xn }∞ n=1 ⊆ E+ satisfies Σ∞ . ||x || < ∞. Let . ε > 0 be given. There exist . λ > 0, x0n ∈ E0 , x1n ∈ E1 n E n n=1 with .||x0n ||E0 ≤ 1, ||x1n ||E1 ≤ 1 for all .n ∈ N such that 0 < λn ≤ ||xn ||E + ε/2n , and |xn | ≤ λn |x0n |1−θ |x1n |θ , n ∈ N.

.

Σ In particular, . ∞ n=1 λn < ∞. Using Hölder’s inequality, it follows that ∞ Σ .

|xn | ≤

n=1

∞ Σ

1−θ λn |x0n |

θ |x1n ≤

n=1

(∞ Σ

λn |x0n |

)1−θ ( ∞ Σ

n=1

)θ λn |x1n |

n=1

= λ|x0 |1−θ |x1 |θ , where λ=

∞ Σ

.

n=1

Σ∞

∞ ∞ Σ Σ λn , x0 = (λn /λ)|x0n |, x1 = (λn /λ)|x1n |. n=1

n=1

Since . n=1 λn /λ = 1 and .||x0n ||X0 ≤ 1, ||x1n ||X1 ≤ 1 for all .n ∈ N, it follows from the Riesz–Fischer property in .E0 , E1 respectivelyΣthat .x0 ∈ E0 , x1 ∈ E1 and ∞ .||x0 ||E0 ≤ 1, ||x1 ||E1 ≤ 1, respectively. Consequently, . n=1 |xn | ∈ E, and so E is complete, by the Riesz–Fischer theorem. The Banach function space E constructed above is denoted by .E01−θ E1θ and will be referred to as the Calderón product. The family of spaces .E01−θ E1θ , 0 < θ < 1 is known as the Calderón family connecting .E0 and .E1 . The importance of the

7.15 The Calderón Family for a Banach Couple of Symmetric Spaces

533

Calderón family is shown in the following result, due to Calderón [18], in the case that at least one of the spaces .E0 , E1 is reflexive. Theorem 7.15.1 If at least one of the spaces .E0 , E1 has order continuous norm, then .E01−θ E1θ coincides isometrically with the space .[E0 , E1 ]θ . The preceding theorem is proved in [76] Theorem IV.1.14 under the assumption that .E01−θ E1θ has order continuous norm. The theorem as stated now follows as in [76] from the following observation. Proposition 7.15.2 If at least one of .E0 , E1 has order continuous norm, then the space .E = E01−θ E1θ has order continuous norm. Proof Suppose that the norm on .E0 is order continuous, that .0 ≤ x ∈ E01−θ E1θ , and that .{en }∞ n=1 ⊆ Σ satisfy .en ↓n φ. With .λ > 0 as in (7.74), it follows that |χen x| ≤ λ|χen x0 |1−θ |χen x1 |θ , | |1−θ | | χ x en 0 | 1−θ | ≤ λ||χen x0 ||E0 | |x1 |θ . | | ||χen x0 ||1−θ | E0

.

Since .χen x0 ↓n 0 ⊆ E0 and since the norm on .E0 is order continuous, it follows that ||χen x||E ≤ λ||χen x0 ||1−θ E0 →n 0,

.

and this suffices to complete the proof of the Proposition.

u n

Remark 7.15.3 What has been used in the above Proposition is the equality .E an = E oc . While this assertion has been shown in Proposition 5.5.11 in the setting of general noncommutative strongly symmetric spaces, an inspection of the proof shows that it remains valid in the setting of Banach function spaces E on some Maharam measure space. In this setting, one need only note that the key result on norm convergence given in Proposition 5.5.9 is a well-known theorem of W.A.J. Luxemburg. Corollary 7.15.4 If .E0 , E1 ⊆ S(ν) are fully symmetric spaces, at least one of which has order continuous norm, then .E = E01−θ E1θ is fully symmetric and has order continuous norm. Proof Since .E0 , E1 are fully symmetric, it follows from Theorem 7.15.1 and Proposition 7.3.2 that E is fully symmetric. That E has order continuous norm follows from Proposition 7.15.2. u n Corollary 7.15.5 If at least one of the spaces .E0 , E1 has order continuous norm, then .[E0 (τ ), E1 (τ )]θ coincides isometrically with the space .(E01−θ E1θ )(τ ).

534

7 Interpolation

Proof By Theorem 7.3.4, we have [E0 , E1 ]θ (τ ) = [E0 (τ ), E1 (τ )]θ ,

.

and the claim of corollary follows now from Theorem 7.15.1.

u n

An immediate consequence of Theorem 7.15.1 and the Riesz–Thorin interpolation equality (7.47) is the following identification. Corollary 7.15.6 If .1 ≤ p0 , p1 ≤ ∞, and if .E0 = Lp0 (ν), E1 = Lp1 (ν), then E01−θ E1θ coincides isometrically with .Lp (ν), where .1/p = (1 − θ )/p0 + θ/p1 .

.

It should be noted that the expression for the norm in the space .E = E01−θ E1θ given by (7.74) may be reformulated as follows: θ 1−θ ||x||E = inf{||x0 ||1−θ |x1 |θ , E0 ||x1 ||E1 : |x| = |x0 |

.

x0 ∈ E0 , x1 ∈ E1 }.

(7.76)

Indeed, for the moment, denote by .||x||' the expression on the right hand side of (7.76). If .|x| = |x0 |1−θ |x1 |θ for some .x0 ∈ E0 , x1 ∈ E1 , then θ 1−θ x = ||x0 ||1−θ (x1 /||x1 ||E1 )θ . E0 ||x1 ||E1 (x0 /||x0 ||E0 )

.

θ The definition of .|| · ||E implies that .||x||E ≤ ||x0 ||1−θ E0 ||x1 ||E1 , and it follows that ' .||x||E ≤ ||x|| . On the other hand, suppose that .λ > 0, x0 ∈ E0 , x1 ∈ E1 satisfy 1−θ |x |θ . Observe that .||x0 ||E0 ≤ 1, ||x1 ||E1 ≤ 1 and .|x| ≤ λ|x0 | 1 1

|x| = (λϕ 1−θ |x0 |)1−θ (λ|x1 |)θ ,

.

where .ϕ is defined to be .|x|(λ|x0 |1−θ |x1 |θ )−1 on the support of .λ|x0 |1−θ |x1 |θ and zero elsewhere. Since .|ϕ| ≤ 1, it follows that 1

θ ||x||' ≤ ||λϕ 1−θ |x0 |||1−θ E0 ||λ|x1 |||E1 ≤ λ,

.

and this implies that .||x||' ≤ ||x||E and establishes (7.76). References: [18, 76, 118].

7.16 Köthe Duality and the Calderón–Lozanovskii Construction The following fundamental theorem that identifies the Köthe dual of a Calderón product is due to Lozanovskii [83].

7.16 Köthe Duality and the Calderón–Lozanovskii Construction

535

Theorem 7.16.1 (Lozanovskii Duality Theorem) If .E0 , E1 ⊆ S(ν) are Banach function spaces and if .0 < θ < 1, then ( .

E01−θ E1θ

)×

( )1−θ ( × )θ E1 . = E0×

(7.77)

A proof of this basic result may be found in the paper of Reisner [104]. In this section, noncommutative extensions of several theorems due to Lozanovskii will be given based on Theorem 7.3.4. In the sequel, a is a number satisfying .0 < a ≤ ∞. Theorem 7.16.2 Let .E0 , E1 ⊆ S(m) be fully symmetric spaces on .[0, a) and let 0 < θ < 1. If at least one of the spaces .E0 , E1 has order continuous norm, and if at least one of the spaces .E0× , E1× has order continuous norm, then

.

.

[ × ] × [E0 (τ ), E1 (τ )]∗θ = [E0 (τ ), E1 (τ )]× θ = E0 (τ ), E1 (τ ) θ .

Proof In view of Theorem 7.15.1 and Corollary 7.15.4, it suffices only to prove the second equality. To this end, using Theorems 7.3.4, 7.15.1, and 7.16.1 in succession, observe that .

× [E0 (τ ), E1 (τ )]× θ = [E0 , E1 ]θ (τ ) )× ( = E01−θ E1θ (τ ) ( ) = (E0× )1−θ (E1× )θ (τ ) [ ] = E0× , E1× θ (τ ) ] [ = E0 (τ )× , E1 (τ )× θ .

u n

This proves the assertion.

The theorem that follows is again due to Lozanovskii and is an extension of the reflexivity criterion for Banach lattices due to Ogasawara. Theorem 7.16.3 Suppose that .E0 , E1 ⊆ S(ν) are Banach function spaces and let 0 < θ < 1. If at least one of .E0 , E1 is a KB-space and if at least one of .E0× , E1× is a KB-space, then .E = E01−θ E1θ is reflexive.

.

Proof Without loss of generality, it may be assumed that .E0 , E0× are KB-spaces. In particular, the norms on .E0 , E0× are order continuous. By Proposition 7.15.2, the norm on E is order continuous and so E ∗ = E × = (E0× )1−θ (E1× )θ .

.

536

7 Interpolation

Again, since the norm on .E0× is order continuous, it follows that the norm on .E × is order continuous and so E ∗∗ = (E × )∗ = E ×× = (E0×× )1−θ (E1×× )θ = (E0 )1−θ (E1×× )θ

.

using the fact that .E0×× = E0 , by the assumption that .E0 is a KB-space. Using again the order continuity of the norm on .E0 , it follows that the norm on .E ×× is order continuous. Consequently, E ∗∗∗ = E ××∗ = E ××× = (E0× )1−θ (E1××× )θ

.

= (E0× )1−θ (E1× )θ = E × = E ∗ , where the equality .E1××× = E1× follows from the fact that .E1× has the Fatou property. Consequently, .E ∗ is reflexive and so also E is reflexive. u n A noncommutative counterpart now follows. Theorem 7.16.4 Suppose that .E0 , E1 ⊆ S(m) are fully symmetric function spaces on .[0, a). If at least one of .E0 , E1 is a KB-space, and if at least one of .E0× , E1× is a KB-space, and if .0 < θ < 1, then .[E0 (τ ), E1 (τ )]θ is a reflexive Banach space. Proof By Theorem 7.16.3, the space .E = E01−θ E1θ is reflexive and consequently, so is the noncommutative space ) ( E(τ ) = E01−θ E1θ (τ ) = [E0 , E1 ]θ (τ ) = [E0 (τ ), E1 (τ )]θ

.

u n

by Theorem 5.9.6 (v).

In the case that .E0 = E1 , the preceding proposition reduces to a special case of the noncommutative extension of Theorem 5.3.10. A further comment may be made on the special case obtained that .E0 = L∞ (m). If .1 < p < ∞ and if E is any symmetric space on .[0, a), then, setting .θ = 1/p, it is easily seen that E θ L∞ (m)1−θ = E (p) ,

.

where .E (p) is the p-convexification of E (see Sect. 6.6), that is, space of those functions .f ∈ S(m) for which .|f |p ∈ E with norm given by setting .

||f ||

E (p)

|| || 1 = |||f |p || p . E

Theorem 7.16.5 If E is a fully symmetric KB-space on .[0, a) and if .1 < p < ∞, then .E (p) (τ ) is reflexive. Conversely, if .E is a strongly symmetric space on (p) (τ ) is reflexive, then .E(τ ) is a .KB-space, that is, .E(τ ) has order .[0, a), and if .E continuous norm and has the Fatou property.

7.16 Köthe Duality and the Calderón–Lozanovskii Construction

537

Proof The first assertion is an immediate consequence of Theorem 7.16.4 and the fact that .L∞ (m)× = L1 (m). Suppose then that .E (p) (τ ) is reflexive. This implies that the norm on .E (p) (τ ) is order continuous, and consequently, .limt→∞ μt (x) = 0 1

1

for every .x ∈ E (p) (τ ). Since .μ(x) p = μ(|x| p ) holds for each .x ∈ S(τ ), it follows also that .limt→∞ μt (x) = 0 holds for every .x ∈ E(τ ). Suppose now that .{xα } ⊆ E(τ ) and that .0 ≤ xα ↓α 0 holds on .E(τ ). It follows that .μ(xα ) ↓α 0 holds in 1

S0 (m). Since the map .t → t p , t ≥ 0, is operator monotone (see Corollary 2.2.28),

.

1

1

1

it follows that .0 ≤ xαp ↓α holds in .E (p) (τ ), and since .0 ≤ μ(xαp ) = μ(xα ) p ↓α 0, 1

it follows that .xαp ↓α 0 holds also in .E (p) (τ ). Consequently, by order continuity of the norm on .E (p) (τ ), it follows that 1

1

p p ||xα ||E(τ ) = ||xα ||E (p) (τ ) ↓α 0,

.

so that .E(τ ) has order continuous norm. To show that .E(τ ) has the Fatou property, it suffices to show that .E(τ ) = E ×× (τ ). Now, from the reflexivity of .E (p) (τ ), it follows that E (p) (τ ) = E (p) (τ )××

.

= [L∞ (τ ), E(τ )]×× 1 p

[ ] = L∞ (τ ), E ×× (τ ) 1

p

)(p) ( = E ×× (τ ), by an appeal to Theorem 7.16.2 and Theorem 5.3.10. Thus, if .0 ≤ x ∈ E ×× (τ ), 1

then .x p ∈ (E ×× )(p) (τ ) = E (p) (τ ). This implies that .x ∈ E(τ ) and the proof is n u complete. The preceding Proposition reduces to [83] Theorem 4, in the commutative setting. A further consequence of the Lozanovskii duality theorem is the following factorization theorem for .L2 (ν). Theorem 7.16.6 (Lozanovskii [83] Theorem 5) If .E ⊆ S(ν) is a Banach function space and if the norm on E is a Fatou norm, then E 1/2 (E × )1/2 = L2 (ν),

.

where the equality holds pointwise and isometrically. The following facts will be needed for the proof. Lemma 7.16.7 Suppose that .E ⊆ S(ν) is a Banach function space. If the norm on E is a Fatou norm, and if .E × is reflexive, then E is reflexive.

538

7 Interpolation

Proof Since the norm on E is a Fatou norm, the natural embedding of E in .E ×× is an isometry and so E is a closed subspace of .E ×× . Since .E × is reflexive, it follows that the norm on .E × is order continuous and so .(E × )∗ = E ×× . Since .(E × )∗ is reflexive, it follows that E is a closed subspace of a reflexive space and is therefore reflexive. u n Remark 7.16.8 The preceding lemma is stated (without proof) in [83] Lemma 21 without the assumption that the norm on E is a Fatou norm. However, an example due to Anton Schep (private communication) shows that this is false. The following lemma can be found in Theorem B, part 2 in [16] (see also [83]). Lemma 7.16.9 If .E ⊆ S(ν) is a Banach function space, and if the norm on E is a Fatou norm, then so is the norm on .E 1/2 (E × )1/2 . Proof of Theorem 7.16.6 Setting .F = E 1/2 (E × )1/2 , and successively applying the Lozanovskii Duality Theorem 7.16.2, observe that F × = (E × )1/2 (E ×× )1/2

.

and F ×× = (E ×× )1/2 (E ××× )1/2 = (E ×× )1/2 (E × )1/2 = F × ,

.

where the equality is pointwise and isometric. If .z ∈ F × , it follows immediately that .|z| |z| ∈ L1 (ν), i.e., that .z ∈ L2 (ν) and that 1/2

1/2

||z||L2 (ν) ≤ ||z||F × ||z||F ×× = ||z||F × .

.

In particular, .F × ⊆ L2 (ν). This implies that L2 (ν) = L2 (ν)× ⊆ (F × )× = F ×

.

with the embedding of norm at most one, as is easily seen. Consequently, the equality .F × = L2 (ν) holds and so .F × is reflexive. Since the norm on E is a Fatou norm, it follows from Lemma 7.16.9 that the norm on F is a Fatou norm. It follows that F is isometrically embedded in .F ×× = (F × )∗ , which is reflexive. Consequently, F is reflexive and so F = F ×× = L2 (ν)× = L2 (ν).

.

u n Proposition 7.16.10 Let E be a fully symmetric Banach function space on .[0, a). If either E or .E × has order continuous norm, then .

] [ E(τ ), E × (τ ) 1 = L2 (τ ). 2

7.17 Schmidt Decomposition and Lozanovskii Factorization

539

Proof The proposition follows immediately from its commutative specialization, which is a special case of [83] Theorem 5, and Theorem 3.2. u n References: [16, 83, 104].

7.17 Schmidt Decomposition and Lozanovskii Factorization If .(X, Σ, ν) is a .σ -finite measure space, if .E ⊆ S(ν) is a Banach function space, if f ∈ L1 (m), and if .ε > 0, then there exist .g ∈ E and .h ∈ E × such that

.

f = gh,

||g||E ||h||E × ≤ (1 + ε)||f ||1 .

.

If E has the Fatou property, then .ε may be taken to be 0, in which case ||g||E ||h||E × ≤ ||f ||1 .

.

This is known as the Lozanovskii Factorization Theorem, see the references at the end of this section. The aim of the final section of this chapter is to show that this theorem may be readily extended to the noncommutative setting. In contrast to the companion theorem of Lozanovskii given in Theorem 7.16.6, the proof does not depend on interpolation, but rather on the following Schmidt decomposition due to Ovcinnikov. Recall first that the von Neumann subalgebra .N ⊆ M is said to be .proper if the restriction of the trace to .N is again semi-finite. Theorem 7.17.1 If .0 ≤ x ∈ S(τ ), there exists a proper von Neumann subalgebra Mx ⊆ L∞ (0, ∞) with .μ(x) ∈ S(m|Mx ), a proper commutative subalgebra .Mx ⊆ M, and a positive rearrangement preserving algebra isomorphism .Jx of .S(m|Mx ) onto .S(τ |Mx ) whose restriction to the projections of .Mx is a Boolean algebra isomorphism onto the projections of .Mx and for which .

μ(Jx (μ(x))) = μ(x).

.

Proof Define .μ∞ , t0 by setting μ∞ = lim μ(t; x)

.

and

t→∞

t0 = inf{t > 0 : μ(t; x) = μ∞ }.

Set F x0 = e (μ∞ , ∞)x =

.

x

R

λdex0 (λ),

540

7 Interpolation

where ex0 (−∞, λ] = ex (−∞, λ]ex (μ∞ , ∞) + ex (−∞, μ∞ ]

.

if .λ ≥ 0 and zero otherwise. If d(λ; x0 ) = τ (ex0 (λ, ∞)),

.

λ > 0,

then the set {λ > 0 : μ(x0 ) ◦ d(x0 )(λ) /= λ}

.

is of measure zero for the spectral measure induced on .(0, ∞) by the spectral distribution .ex0 (−∞, λ], .λ ∈ R, and consequently, F

F

x0 =

μ(x0 ) ◦ d(x0 )(λ)dex0 (λ) =

.

(0,∞)

R

μ(t; x0 )d e(t), ˜

where F e(−∞, ˜ t] =

χ(−∞,t] ◦ d(x0 )(λ)dex0 (λ)

.

(0,∞)

⎧ x ⎪ ⎪ ⎨e (μ∞ , ∞) = ex (μ∞ , ∞)ex [μ(t; x), ∞) ⎪ ⎪ ⎩0

if t ≥ t0 ; if 0 < t < t0 ; if t ≤ 0.

This yields, in particular, that F x0 =

.

R

μ(t; x)d e(t). ˜

Now define the function .φ0 by setting φ0 (t) = τ (e(−∞, ˜ t]),

.

t ∈ (−∞, ∞).

The function .φ0 is non-decreasing, right-continuous on .(−∞, ∞), .φ0 (t) = 0 if t ≤ 0, .φ0 is constant within intervals of constancy of .μ(x0 ), and the equality

.

φ0 (t−) = t

.

holds whenever .t ≥ 0 is not an interior point of some interval of constancy of .μ(x0 ).

7.17 Schmidt Decomposition and Lozanovskii Factorization

541

Define projections .ei , .i = 0, 1, 2, . . . as follows. If .μ∞ = 0 or if .μ∞ > 0 and t0 = +∞, we set

.

ei = ex (μ∞ , ∞),

i = 0, 1, 2, . . .

.

If .μ∞ > 0 and .t0 < ∞, then t0 = τ (ex (μ∞ , ∞)) < ∞,

.

and there exists .ε > 0 such that τ (ex (μ∞ − ε, ∞)) = ∞.

.

By semi-finiteness, there exist projections .ei , i = 0, 1, 2, . . . such that ex (μ∞ , ∞) = e0 ≤ e1 ≤ · · · ≤ ex (μ∞ − ε, ∞),

.

τ (ei ) < ∞ and

.

lim τ (ei ) = +∞.

i→∞

Set ∞ φ1 (t) = Σi=1 τ (ei − e0 )χ [τ (ei−1 ),τ (ei )) (t),

.

t ∈ (−∞, ∞).

Denote by I the union of the intervals of constancy of .μ(x0 ) and the intervals [τ (ei ), τ (ei+1 )), i = 0, 1, 2, · · · . Let .Mx be the von Neumann subalgebra of .L∞ (m) generated by the members of I and the Lebesgue measurable subsets of .(0, ∞)\I . It is clear that .Mx is a proper subalgebra of .L∞ (m). Denote by .Ex the conditional expectation of .(L1 + L∞ )(m) onto .(L1 + L∞ )(m|Mx ). If we set

.

φ(t) = φ0 (t) + φ1 (t),

.

it is not difficult to see that F F . Ex (f )(t)dt = [0,∞)

[0,∞)

t ∈ (−∞, ∞),

Ex (f )(t)dφ(t),

f ∈ (L1 + L∞ )([0, ∞)).

Denote by .Mx the commutative subalgebra of .M generated by the spectral projections .{e˜t , t ∈ (−∞, ∞)} and the projections .ei , i = 0, 1, 2, . . . and note that .Mx is proper in the sense that the restriction of .τ to .Mx is again normal, faithful, and semi-finite. Now define the map Jx : S(m|Mx ) −→ S(τ |Mx )

.

542

7 Interpolation

by setting F Jx (f ) =

.

[0,∞)

˜˜ f (t)d e(t),

f ∈ S(m|Mx ),

where ∞ Σ (ei − e0 )χ[τ (ei−1 ),τ (ei )) (t),

˜˜ e(−∞, t] = e(−∞, ˜ t] +

.

t ∈ (−∞, ∞).

i=1

Note that ˜˜ e(−∞, t] = e(−∞, ˜ t] + e(∞, t],

t ∈ (−∞, ∞),

.

where Σ

e(∞, t] =

.

(ei+1 − ei ),

t ∈ (−∞, ∞),

{i:τ (ei )≤t}

and that the equality ˜˜ τ (e(−∞, t]) = φ(t)

.

holds for each .t ∈ (−∞, ∞). If .f ∈ S(m|Mx ) and if .λ > 0, set Fλ = {s ≥ 0 : |f |(s) > λ}

.

and observe that (F τ (χ(λ,∞) (|Jx f |)) = τ

.

[0,∞)

F = F =

[0,∞)

[0,∞)

˜˜ χFλ (t)d e(t)

)

˜˜ χFλ (t)dτ (e(∞, t]) χFλ (t)dt.

From this, it now follows that the map .Jx is singular value preserving in the sense that μ(Jx f ) = μ(f ),

.

f ∈ S(m|Mx ),

and from this, it follows, in particular, that μ(Jx (μ(x))) = μ(x).

.

7.17 Schmidt Decomposition and Lozanovskii Factorization

543

The remaining assertions of the theorem now follow, for example, from the appropriate specialization of [37] Corollary 6.3. u n It is to be remarked that if .0 ≤ x ∈ S(τ ) and if .limt→∞ μ(t; x) = 0, then the first part of the proof of the preceding theorem shows that F x=

.

[0,∞)

μ(t; x)d e(t), ˜

and this is just the Schmidt decomposition given by Ovcinnikov. Based on the Schmidt decomposition, it is now possible to extend the Lozanovskii factorization theorem given at the beginning of this section to the noncommutative setting. Proposition 7.17.2 Let E be a strongly symmetric space on .[0, a). If .0 ≤ x ∈ L1 (τ ) and if .ε > 0 is given, then there exist .0 ≤ y ∈ E(τ ), .0 ≤ z ∈ E × (τ ) = E(τ )× with .yz = zy such that .x = yz and such that .||y||E(τ ) ||z||E × (τ ) ≤ (1 + ε)||x||1 . If E has the Fatou property, then .ε may be taken to be 0. Proof Since .0 ≤ x ∈ L1 (τ ), it follows that .μ(t; x) → 0 as .t → ∞. Using the remark following Theorem 7.17.1, let F x = Jx (μ(x)) =

.

[0,∞)

μ(t; x)d e(t) ˜

be the Schmidt decomposition of x. Adhering to the notation of Theorem 7.17.1 and applying Lozanovskii’s factorization theorem in the commutative setting to the Köthe space .E(m|Mx ) to obtain the existence of functions .0 ≤ f ∈ E(m|Mx ), × .0 ≤ g ∈ E(m|Mx ) such that μ(x) = f g,

.

(1 + ε)||μ(x)||1 ≤ ||f ||E(m|Mx ) ||g||E(m|Mx )× .

.

We set F y = Jx (f ) =

.

[0,∞)

f (t)d e˜t ,

F z = Jx (g) =

.

[0,∞)

g(t)d e˜t ,

544

7 Interpolation

and observe that F x = Jx (μ(x)) =

.

μ(t)d e˜t

) (F

(F =

[0,∞)

[0,∞)

f (t)d e˜t

[0,∞)

) g(t)d e˜t

= yz.

Since .f, g ∈ S(Mx ), it follows from Theorem 7.17.1 that μ(y) = μ(Jx f ) = μ(f )

.

and μ(z) = μ(Jx g) = μ(g),

.

consequently, (1 + ε)||x||1 ≥ ||y||E(τ ) ||z||E(τ )× ,

.

and the proof is complete. References: [44, 59, 83, 94].

u n

Brief Historical Notes

The following is a review of the history and structure of the theory of noncommutative integration theory and the corresponding Banach space geometry. While we do not pretend to be historians by any measure, we do wish to highlight the contributions of many mathematicians to the subject. Noncommutative Measure Theory The modern theory of noncommutative measure and integration finds its roots in the seminal papers of J. Dixmier [34] and I. E. Segal [111]. Their key insight was that the study of rings of operators is not merely analogous to classical integration, as was understood by Murray and von Neumann in their original works, but that the subject was a true generalization and that this could be proved concretely. The fundamental paper of Segal [111] develops, in analogy with the classical theory of integration over abstract measure spaces, a calculus for “measurable” operators (these being unbounded operators that play the role of measurable functions), treats the notions of “integrable” and “square-integrable” operators, and obtains extensions of such basic results as the Riesz–Fischer, Radon–Nikodym, Lebesgue monotone convergence, and Fubini theorems. His motivations were drawn from specific analytical situations arising in the theory of operator algebras, quantum field theory, and harmonic analysis on groups. A central role is played by the work of von Neumann and Murray on rings of operators, and other important influences were the papers of H. A. Dye [51] on the extension of the Radon–Nikodym theorem to finite rings of operators and that of Ambrose [2] on standard rings. Gage Spaces The approach of Segal is very measure-theoretic and is based on the notion of a “gage space”, which is a kind of noncommutative analogue of the concept of “measure space”. More precisely, a gage space .r is a system .(H, U, m), composed of a Hilbert space .H, a ring .U of operators in .H, and a non-negative realvalued function on the lattice of (orthogonal) projections in .U, which is completely additive, unitarily invariant, and is such that every projection in .U is the least upper bound of projections on which m is finite. Here, the term “ring of operators” means © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. G. Dodds et al., Noncommutative Integration and Operator Theory, Progress in Mathematics 349, https://doi.org/10.1007/978-3-031-49654-7

545

546

Brief Historical Notes

an algebra of bounded operators that is self-adjoint (i.e., closed under the adjoint operation), closed in the weak operator topology and contains the identity operator I . It may be assumed further, without loss of generality, that m has the property that the only projection with gage 0 is the projection .P = 0. Gage spaces whose rings are commutative are essentially the same as abstract measure spaces. Indeed, let .(o, Σ, μ) be a measure space that is localizable in the sense of Segal [110]. Such measure spaces are precisely those measure algebras that are locally finite and Dedekind complete, or equivalently, are precisely those measure spaces for which the classical Radon–Nikodym theorem is valid. Further, such spaces are also characterized the validity of the Riesz representation theorem, that is, that .L∞ (μ) is the Banach dual of the space .L1 (μ) . Suppose that .H is the Hilbert space of all complex-valued functions on .o with the usual inner product, and let .U be the algebra consisting of all multiplication operators of the form .Mh (f ) = hf, f ∈ H, for some bounded measurable function on .o. For every projection P in .U, there exists a measurable set S (unique, modulo null sets) such that P is the operation of multiplication by the indicator function of S, and one defines .m(p) = μ(S). It is easily verified that .r = (H, U, m) is a gage space. Conversely, a gage space that is commutative in the sense that the underlying ring of operators is commutative is algebraically equivalent to a gage space built on a measure space. Here, two gage spaces .(Hi , Ui , mi ), i = 1, 2) are said to be algebraically equivalent if there exists an adjoint preserving algebraic isomorphism of .U1 onto .U2 that carries .m1 onto .m2 . Measurable Operators To proceed further, Segal introduces the notion of a measurable operator. A closed linear operator T in .H is said to be affiliated with .U if T commutes with every unitary operator in the commutant .U' of .U. Of course, if T is bounded and everywhere defined, then the bi-commutant theorem of von Neumann implies that T is affiliated with .U if and only if .T ∈ U. The closed operator T in .H is said to be measurable with respect to .U if T is affiliated with .U, and the domain .D(T ) of T is strongly dense in .H. This means that there exists an increasing sequence of closed subspaces .Kn ⊆ D(T ), each invariant under every unitary operator in the commutant .U' such that .K⊥ n is algebraically finite ↓ {0}. Equipped with the (in the sense of Murray and von Neumann) and .K⊥ n algebraic operations of strong sum and strong product (notions that go back to von Neumann and Murray in the case of type .I I1 factors), the collection of .Umeasurable operators are then shown to have the structure of a .∗-algebra. In the case that the gage space .r = (H, U, m) is that given by a localizable measure space .(o, Σ, μ), the operator T on .H is measurable with respect to .U if and only if it has the form .Tf = kf for some .Σ-measurable function k on .o. Here, the domain of T consists of all f such that .kf ∈ L2 (μ). Segal then introduces a notion of almost everywhere convergence, which he calls “convergence nearly everywhere” (convergence n.e.) for sequences of measurable operators, which coincides with the usual notion of almost everywhere convergence in the case of localizable measure spaces, and shows that the usual properties of continuity of the algebraic operations continue to hold.

Brief Historical Notes

547

Introduction of the Integral As is standard in integration theory, the integral is now defined for a class of relatively simple operators and then extended to its natural full domain. The definition of the gage m is first extended to the two-sided ideal .S of operators .T ∈ U of finite metric rank, that is, those operators .T ∈ U for which the gage m of the projection on the closure of the range of T is finite. Such operators are called elementary operators. The extension of m to .S, again denoted by m, is a positive linear functional and has the property that .m(ST ) = m(T S) whenever T is an elementary operator and .S ∈ U and is called the integral, or trace, with respect to the gage space. In order to complete the elementary operators to obtain the integrable operators, a norm analogous to the .L1 -norm in measure theory is now introduced on the ideal .S by setting ||T ||1 = sup{|m(ST )| : S ∈ U, ||S||∞ ≤ 1},

.

T ∈ S,

where .|| · ||∞ denotes the operator norm. The functional .|| · ||1 is indeed a norm on .S and has the further properties that, for all .T ∈ S and .V ∈ U, .||T ||1 = ||T ∗ ||1 = m(|T |), where .|T | is given by the polar decomposition .T = U |T |, and .||V T ||1 , ||T V ||1 ≤ ||V ||∞ ||T ||1 . Integrable Operators A measurable operator T with respect to the gage space r is said to be integrable if it is the limit n.e. of a sequence .{Tn } of elementary operators that is a Cauchy sequence for the norm .|| · ||1 . Since .|m(Tn ) − m(Tm )| ≤ ||Tn − Tm ||1 for all .n, m, it follows that .limn→∞ m(Tn ) exists. This limit depends only on the measurable operator T and not on the approximating sequence .{Tn }. Accordingly, the integral of T , again denoted by .m(T ), is defined by setting .m(T ) = limn→∞ m(Tn ), is well defined, and is a linear functional on the linear set of all integrable operators. If T is integrable and if .S ∈ U, then .ST , T S and .|T | are again integrable (here the products are taken in the strong sense). The .L1 -norm of an integrable operator T is now defined exactly as in the case of an elementary operator, is indeed a norm on the linear space .L1 (r) of integrable operators, and the above properties of the norm for elementary operators continue to hold for integrable operators. Further, the integral of a positive integrable operator is again positive, and if T is a positive integrable operator, then there exists a sequence .{Tn } of positive integrable operators that is Cauchy in the .L1 -norm that converges to T n.e. The integral may now be extended to the class of all non-negative measurable operators as follows. If T is non-negative and measurable, then .m(T ) ∈ [0, ∞] is defined by setting

.

m(T ) = sup{m(S) : 0 ≤ S ≤ T , S elementary}

.

in the sense that if either side exists and is finite, then the other side exists and has the same value. The extended integral is additive and positively homogeneous and has the property that, if T is any measurable operator, then .m(T ∗ T ) = m(T T ∗ ). Square-Integrable Operators A measurable operator T is said to be squareintegrable with respect to the gage space .r if and only if the squares of its real

548

Brief Historical Notes

and imaginary parts are integrable. For any two square-integrable operators .X, Y , the (strong) product .X∗ Y is integrable and the Cauchy–Schwartz inequality |m(X∗ Y )|2 ≤ m(X∗ X)m(Y ∗ Y )

.

is valid. Moreover, a measurable operator T is square-integrable if and only if T ∗ T is integrable. The .L2 -norm of a measurable operator T is then defined to be ∗ 1/2 and denoted by .||T || . The .L -norm is indeed a norm on the linear .(m(T T )) 2 2 space .L2 (r) of square-integrable operators, and the elementary operators are dense in .L2 (r). Moreover, if T is square-integrable and if .V ∈ U, then .V T , T V are square-integrable and .||V T ||2 , ||T V ||2 ≤ ||V ||∞ ||T ||2 . .

Riesz–Fischer Theorem For any gage space .r, the space .L1 (r) (.L2 (r)) is complete for the .L1 (.L2 )-norms, respectively, and this of course is an extension of the classical theorem of Riesz–Fischer in the case of commutative gage spaces. An extension of the classical Lebesgue monotone convergence theorem for measurable operators continues in the following sense: if .{Tn } is a monotone increasing sequence of positive measurable operators converging nearly everywhere to a measurable operator T , then T is the minimal positive measurable operator bounding the sequence .{Tn } and .m(Tn ) → mn (T ). Radon–Nikodym Theorem An extension of the classical Radon–Nikodym theorem to the setting of finite von Neumann algebras algebras was given by H. A. Dye [51] for positive linear functionals that are countably additive on projections. Segal treats the case of a more restricted class of functionals, while also generalizing to the semi-finite setting. The extended indefinite integral of an operator T over a gage space .r is the functional n on .U defined by setting .n(X) = m(T X), X ∈ U. It is then the case that a positive linear functional on the ring of a gage space is the extended indefinite integral of a positive integral operator if and only if its restriction to the unit sphere of .U is continuous for the strong operator topology. If .φ is a continuous linear functional on .L1 (r), then there exists a unique element .T ∈ U such that .φ(x) = m(XT ), X ∈ L1 (r). This yields the identification of .U with the Banach dual of .L1 (r) and extends a classical measure-theoretic theorem of F. Riesz. A special case of this theorem had been observed earlier by J. Dixmier [32] in the case of the unique gage space over the ring of all bounded operators on some Hilbert space. Noncommutative .Lp -Spaces Segal’s fundamental work was subsequently developed, extended, and simplified by his students R. A. Kunze [77], W. F. Stinespring [114], and E. J. Nelson [93]. General .Lp -spaces, .1 ≤ p ≤ ∞ with respect to a gage space .r, are defined in Kunze [77]. If T is measurable with respect to .r, one sets .||T ||p = m(|T |p )1/p if .1 ≤ p < ∞ and defines .||T ||∞ to be the operator norm if .T ∈ U. The space .Lp (r) is then the collection of all measurable operators for which the .Lp -norm is finite. The spaces .Lp (r), 1 ≤ p ≤ ∞ are Banach spaces, and the classical Hölder inequality is valid in the sense that, if .T ∈ Lp (r) and .S ∈ Lp' (r), where .1/p + 1/p' = 1, then T S is integrable and .||T S||1 ≤ ||T ||p ||S||p' . Some of

Brief Historical Notes

549

the basic properties of noncommutative .Lp -spaces had been earlier established by J.Dixmier [34]. However, the .Lp -spaces as introduced by Dixmier are defined as abstract completions, whereas in the approach of Kunze, points in the completion are identified as measurable operators, just as a general element of .Lp in classical measure theory is a measurable function. It is worth noting that both these approaches are used extensively in the modern study of noncommutative analysis and that both provide substantial insight. Noncommutative Riesz–Thorin Interpolation Using the framework of noncommutative measure theory, Kunze showed that the Hausdorff–Young theorem, which asserts that the .Lp -norm of a function in .L1 and .Lp , 1 < p ≤ 2 of a locally compact abelian group is greater than or equal to the .Lp' -norm, .1/p + 1/p' = 1, of its Fourier transform, may be extended to the setting of more general locally compact unimodular groups. Here, the Fourier transform in .Lp (1 ≤ p ≤ 2) of a unimodular group G is defined to be the operator of left convolution by the function acting on the Hilbert space .L2 (G) with respect to Haar measure. This operator is shown to be measurable with respect to the von Neumann algebra generated by the left translations. The method of proof of the more general Hausdorff–Young inequality is by establishing the interpolation theorem of Riesz–Thorin theorem in the setting of gage spaces. Measure Convergence The paper of W.F. Stinespring [114] is (essentially) a continuation of that of Segal and develops further aspects of the theory of noncommutative integration which Segal did not have occasion to develop in [111]. The new developments in [114] again have clearly in mind the subsequent application to harmonic analysis and the duality of unimodular groups. A key new idea introduced by Stinespring is that of convergence in measure for a sequence of measurable operators. In the commutative setting, this notion coincides with the classical definition of convergence in measure and, for general gage spaces, is stronger than that of n.e. convergence in the sense that any sequence converging in measure contains a subsequence converging n.e. A dominated convergence theorem is proved, together with an extension of Fatou’s Lemma. An extension of Segal’s monotone convergence theorem is given by introducing .[0, ∞]-valued quadratic functionals on the underlying Hilbert space to extend to unbounded selfadjoint operators the usual quadratic form ordering for bounded operators. This more “pointwise” version of the monotone convergence theorem proves useful in Stinespring’s discussion of the Fubini Theorem for tensor products of gage spaces. The idea of convergence in measure introduced by Stinespring [114] inspired the subsequent work of Nelson [93], which presented a new approach to the theory of noncommutative integration based on the stronger notion of measurability with respect to a trace. The approach of Nelson requires less by way of knowledge of von Neumann algebra techniques and considerably simplifies the original treatments due to Segal [111] and Kunze[77]. The work of Nelson as presented by M.Terp [127] will provide the basis of our approach in the present monograph.

550

Brief Historical Notes

A Functional Analytic Approach While the viewpoint adopted by Segal and his students is explicitly measuretheoretic with the aim of creating a framework for application, among others, to the harmonic analysis of unimodular groups, the approach of Dixmier [32, 34] is very much more from the viewpoint of functional analysis and the theory of Banach spaces with a view to application to the further study of rings of operators. The Commutative Setting In some sense, the difference in viewpoints taken by Segal and Dixmier is indicated in the difference of their approaches to the study of commutative von Neumann algebras [33, 110]. A compact Hausdorff space .o is said to be Stonian if the space .C(o) of all complex-valued continuous functions on .o is Dedekind complete, that is, every upward filtering subset of real-valued elements of .C(o) that is bounded from above with respect to the pointwise partial order on .C(o) has a least upper bound with respect to that partial order (which is not necessarily the pointwise supremum): and hyperstonian if .o is Stonian and the normal measures on .o separate the points of .o. Dixmier [33] shows first that, if .o is a hyperstonian space, then there exists a locally compact space R and a Radon measure .μ on R such that .o is the spectrum of the commutative .C ∗ -algebra .L∞ (R, μ). Conversely, if .(R, μ) is a Radon measure space, then the spectrum .o of .L∞ (R, μ) is hyperstonian,f with the normal measures on .o being given by the linear forms of the type .f → fgdμ, g ∈ L1 (R, μ). A particular consequence is that if .o is hyperstonian, then the space .C(o) is the dual of the space of normal measures on .o. On the other hand, if .o is a hyperstonian space, then there exists a Hilbert space .H and a commutative von Neumann algebra .U such that .o is the spectrum of .U. Conversely, if .U is a commutative von Neumann algebra in some Hilbert space H , then the spectrum .o of .U is hyperstonian. Further, it is noted in [33] that it follows from a result of R. Pallu de la Barrière [98] that the normal measures on .Ω correspond to the normal linear functionals on .U of the form .A → for each pair of fixed elements .x, y ∈ H . It is worth pausing for a moment to consider how deeply unnatural it is to consider such a completion (that is, passing from a locally compact Hausdorff space to a measurable space) without the additional impetus of the theory of .C ∗ - and .W ∗ algebras. Only in passing back from the noncommutative to the commutative does the full extent of the depth and richness of the subject begin to unfold. The Dual and Bidual of the Space of Compact Operators First, it is shown that the Banach dual .I' of the Banach space .I of compact operators may be identified with the Banach ideal of trace class operators with respect to trace duality. Here, a compact operator A is said to be of trace class if and only if the eigenvalue Σ∞ sequence .{λk (|A|)} is summable and the trace norm .||A||1 is given by .||A||1 = k=1 λk (|A|). Further, the Banach bidual .I'' may be identified with .B. Equivalently, .B is the Banach dual of the trace class operators equipped with the trace norm. These results had been obtained earlier by Schatten [107] and von Neumann and Schatten [109], and this is noted in [32]. One consequence is that the spaces .I, I' , I'' = B are reflexive if and only if the underlying Hilbert space is finite dimensional. Of

Brief Historical Notes

551

course, it would later turn out that the existence of a pre-dual is fundamental to both understanding the measure structure of von Neumann algebras and their abstract characterization as .W ∗ -algebras. A Decomposition Theorem Dixmier considered next the Banach dual .B' of .B. If .θ is any element of the dual .B' , and if .ϕ is the restriction of .θ to .I, then the difference .θ − ϕ vanishes on .I. This yields the unique decomposition .θ = ϕ + ψ, with .ϕ ∈ I' and .ψ ∈ I⊥ . Here, .I⊥ ⊆ B' is the set of those uniformly continuous linear functionals that vanish on .I. One consequence of this decomposition is that a linear functional .ϕ on .B is given by an element of .I' if and only if .ϕ is continuous for the ultrastrong topology introduced by von Neumann. Further, this decomposition is an .l1 -type decomposition in the sense that .||θ || = ||ϕ|| + ||ψ||. Moreover, if .θ ≥ 0, then .ϕ ≥ 0 and .ψ ≥ 0. If it is noted that the Banach dual space of the sequence space .l∞ is the space of all finitely additive measures on the Boolean algebra of all subsets of the set of natural numbers .N, then this may be viewed as a a noncommutative counterpart to a special case of a decomposition theorem, published at about the same time, by K. Yosida and E. Hewitt [142] that each bounded additive measure can be uniquely represented as the sum of a countably additive measure and a purely finitely additive measure. This decomposition theorem for elements in the Banach dual of .B was to be subsequently extended to general von Neumann algebras by Takesaki [123, 124] in the study of the conjugate space of an operator algebra. An Analogy What emerges from the above discussion, as noted explicitly by Dixmier, is the remarkable analogy between the classical Banach spaces ' ' ' .c0 , l1 , l∞ , (l∞ ) and the Banach spaces .I, I , B, B . Normal Linear Functionals Motivated further by analogies between classical integration theory and the theory of rings of operators, and developing further the ideas of [32], Dixmier [34] showed that any von Neumann algebra .M is the Banach dual space of the linear subspace .M∗ ⊆ M∗ of all ultraweakly continuous functionals on .M, that is, those linear functionals of the form A→

.

∞ Σ , i=1

where .{xi }, {yi } are fixed sequences in the underlying Hilbert space such that ∞ Σ .

i=1

||xi || < ∞, 2

∞ Σ

||yi ||2 < ∞.

i=1

Each element .ϕ ∈ M∗ is a linear combination of positive elements of .M∗ , and a positive linear functional .ϕ on .M is an element of .M∗ if and only if .ϕ is normal, that is, whenever .Aα ↓α 0 ⊆ M is a downward filtering system with greatest lower bound 0 with respect to the quadratic form ordering, then it follows that

552

Brief Historical Notes

infα ϕ(Aα ) = 0 . Moreover, a linear functional on .M is ultraweakly continuous if and only if it is continuous for the ultrastrong topology.

.

The Space .L1 (τ ) The space .L1 (τ ), 1 ≤ p ≤ ∞ is now introduced in the case that .M is a semi-finite von Neumann algebra equipped with a faithful normal semifinite trace .τ . The space .L∞ (τ ) is taken to be .M with the operator norm. If one sets .m = {A ∈ M : τ (|A|) < ∞}, then .m is a two-sided ideal in .M, called the domain of definition of .τ , and the functional .|| · ||1 : A → τ (|A|) is shown to be a norm on .m. Indeed, one has that ||A||1 = τ (|A|) = sup{|τ (BA)| : B ∈ M, ||B|| ≤ 1}

.

whenever .A ∈ m. In contrast to the definition of Segal, the space .L1 (τ ) is now defined to be the completion of the ideal .m with respect to the norm .|| · ||1 . It is then shown that .L∞ (τ ) = M is the Banach dual of the space .L1 (τ ). The paper by Dixmier [34] played an important role in recognizing the identification of this space as the noncommutative analogue of the classical .L1 -space. The Spaces .Lp (τ ), 1 < p < ∞ For .p ≥ 1, set .m1/p = {A ∈ M : |A|p ∈ m}, and define .||A||p = [τ (|A|p ]1/p if .A ∈ m1/p . By establishing the noncommutative counterpart to the classical Hölder inequality ||AB||1 ≤ ||A||p ||B||q , A ∈ m1/p , B ∈ m1/q , p−1 + q −1 = 1,

.

Dixmier shows that, if .A ∈ m1/p , 1 ≤ p < ∞, then ||A||p = sup{|τ (AB)| : B ∈ m1/q , ||B||q ≤ 1},

.

(7.78)

where .1/p + 1/q = 1. Being the least upper bound of a family of semi-norms, it follows immediately using the faithfulness of .τ that .|| · ||p is a norm, and this, of course, is just the noncommutative counterpart to the classical Minkowski inequality. The space .Lp (τ ), 1 < p < ∞ is now defined to be the completion of the two-sided ideal .m1/p equipped with the norm .|| · ||p . It is then shown that if .1 < p < ∞, then Banach dual of .Lp (τ ) is .Lq (τ ), where .1/p + 1/q = 1. The method of proof depends on first showing that, if .2 ≤ q < ∞, then .Lq (τ ) is uniformly convex, and hence reflexive. It follows from equation 7.78 that .Lp (τ ) is a closed subspace of the Banach dual .(Lq (τ ))∗ if .1 < p < 2 and consequently is also reflexive. That the Banach dual of .Lp (τ ) is .Lq (τ ), where .1/p + 1/q = 1 now follows from a simple density argument again using equation 7.78. Uniform convexity of .Lq (τ ), 2 ≤ q < ∞ is achieved by showing the noncommutative Clarkson inequality: If .q ≥ 2 and if .A, B ∈ Lq (τ ), then [ q q q q] ||A + B||q + ||A − B||q ≤ 2q/p ||A||q + ||B||q .

.

Brief Historical Notes

553

It is to be remarked that the noncommutative Clarkson inequalities in the case .1 < p < 2, and consequently, the uniform convexity of .Lp (τ ) if .1 < p < ∞, were subsequently established by L. Zsido [146]. In the special case that .M is the von Neumann algebra of all bounded operators on some Hilbert space, the Clarkson inequalities in the case .1 < p < 2 are due to C. A. McCarthy [89]. See also [112]. Symmetric Norms and Trace Ideals Symmetric Norms on .Rn The study of .Lp -spaces lies at the heart of classical functional analysis, and much of the discussion so far has been directed toward the development of the theory of their noncommutative counterparts. However, the seeds of a somewhat wider theory embracing a broader class of function spaces were already sown by von Neumann in [134]. The setting here is the finite-dimensional von Neumann algebra of all .n × n complex matrices equipped with the matrix trace and acting in the finite -dimensional Hilbert space .ln2 . Drawing on well-known ideas in the theory of convex bodies, von Neumann introduced the notion of a symmetric norm .ϕ (called by him a symmetric gage function on the space .Rn ), that is, a norm that is symmetric in the sense that it is invariant under permutations and coordinatewise multiplication by .±1. The conjugate norm .ψ defined by setting ψ(v1 , . . . , vn ) = max{

n Σ

.

ui vi : ϕ(u1 , . . . , un ) = 1}

i=1

is again symmetric if .ϕ is symmetric and the conjugate of .ψ is again .ϕ. A very special example of a symmetric norm is given by setting ϕp (u1 , . . . , un ) =

(∞ Σ

.

)1/p |ui |

p

,

1 ≤ p < ∞,

i=1

and, in the case that .p = ∞, ϕ∞ (u1 , . . . , un ) = max{|ui |}.

.

i

Each .ϕp is a symmetric norm, and the conjugate of .ϕp is .ϕq , 1/p + 1/q ≤ 1. The Matrix Metrics of von Neumann Via a symmetric norm .ϕ on .Rn , von Neumann now introduces a corresponding metric .| · |ϕ on the space of all complex .n × n matrices .Mn (C) by setting |X|ϕ = ϕ(s1 (X), . . . , sn (X)),

.

where .s1 (X), . . . , sn (X) are the singular values of X, that is, the eigenvalues of |X| taken in non-increasing order and repeated according to multiplicity. If .ψ is the conjugate to .ϕ, and if .| · |ψ is the corresponding matrix metric, then von Neumann

.

554

Brief Historical Notes

showed that |A|ψ = max{|Tr(AX)| : |X|ϕ = 1},

.

where .Tr denotes the usual matrix trace. The proof given by von Neumann is based on showing that, for all .A, B ∈ Mn (C), .

max{ReTr(AU BV ) : U, V unitary} =

n Σ

si (A)si (B).

i=1

This yields, in particular, the Hölder inequality |Tr(AB)| ≤ |A|ϕ |B|ψ

.

and as well the Minkowski inequality |A + B|ϕ ≤ |A|ϕ + |B|ϕ .

.

It might now be remarked that there is a striking parallel here with the definition and properties of Orlicz spaces [143] and with the theory of function norms and their associate norms in the setting of normed Köthe spaces [144]. This so-called parallel will be given subsequently a much more precise formulation. Symmetrically Normed Ideals The ideas of von Neumann outlined above were subsequently extended to the setting of ideals of compact operators in a separable Hilbert space by R. Schatten [108] and developed further in the monograph of I.C.Gohberg and S.G. Krein [60], in which essential use is made of the material presented in Schatten’s book. The scope of the material presented in [60] is quite extensive, and we will be concerned here only with the basic foundations of the theory of symmetrically normed ideals. In contrast to the methods of von Neumann and Schatten, the approach of Gohberg and Krein is based on the systematic use of properties of the singular values of compact operators. A key idea in this approach is the observation that a symmetric norming function .o (the obvious counterpart to the symmetric gage function of von Neumann with the additional normalizing condition .o(1, 0, . . . ) = 1) on the space .c00 of all finitely non-zero real sequences has the property that .o(ξ ) ≤ o(η) whenever .ξ = (ξ1 , . . . , ξn , 0, . . . ), η = (η1 , . . . , ηn , 0, . . . ) ∈ c00 satisfy ξ1 ≥ ξ2 ≥ · · · ≥ ξn ≥ 0;

.

η1 ≥ η2 ≥ · · · ≥ η n ≥ 0

and k Σ .

j =1

ξj ≤

k Σ j =1

ηj ,

1 ≤ k ≤ n.

Brief Historical Notes

555

Given a symmetric norming function .o, a functional .|·|o is then defined on the twosided ideal .F of finite rank operators (in some Hilbert space H ) by setting .|X|o = o(s(X)), X ∈ F, where .s(X) = (s1 (X), s2 (X), . . . ) is √ the singular value sequence of X, that is, the sequence of eigenvalues of .|X| = X∗ X arranged in decreasing order and repeated according to multiplicity. The preceding observation shows that the functional .| · |o is monotone with respect to submajorization, that is, .o(X) ≤ o(Y ) whenever k Σ .

sj (X) ≤

j =1

k Σ

X, Y ∈ F, 1 ≤ k.

sj (Y ),

j =1

Since .o is a norm on .c00 , it now follows readily from the well-known submajorization inequalities k Σ .

n Σ

sj (X + Y ) ≤

j =1

j =1

sj (X) +

n Σ

sj (Y ),

X, Y ∈ F, 1 ≤ k,

j =1

that is, for all .X, Y ∈ F, |X + Y |o ≤ |X|o + |Y |o .

.

It might be pointed out that von Neumann’s proof of the Minkowski inequality in the matrix setting is based directly on the symmetry property of the symmetric norming function and not on ideas involving submajorization. It now follows readily that .| · |o is a norm on the two-sided ideal .F of the ring .R of all bounded linear operators in the Hilbert space H with the further property that |AXB|o ≤ |A| |X|o |B|,

.

X ∈ F.A, B ∈ R.

Here .| · | denotes the operator norm. Any such norm is called a symmetric norm, and a two-sided ideal in .R equipped with a symmetric norm is called a symmetrically normed ideal. A symmetric norm on .F is unitarily invariant, and conversely, any unitarily invariant norm on .F is symmetric. Moreover, if .||·||S is a unitarily invariant norm on .F, then the equality |A|S = o(s(A)),

.

A∈F

defines a symmetric norming function .o on .c00 via the mapping ξ → ||

Σ

.

ξj (·, φj )φj |o , ξ = {ξj } ∈ c00 ,

j

where .{φj } is some fixed orthonormal basis in the underlying Hilbert space.

556

Brief Historical Notes

Symmetrically Normed Ideals Generated by a Symmetric Norming Function Following [60], suppose now that .o is a symmetric norming function on .c00 . Let .ξ = {ξk }k≥1 be an arbitrary sequence of real numbers, set ξ (n) = (ξ1 , . . . , ξn , 0, 0, . . . ), n ∈ N,

.

and define co = {ξ ∈ c0 : sup o(ξ (n) ) < ∞}.

.

n

The domain of .o is now extended to .cφ by setting o(ξ ) = lim o(ξ (n) ).

.

n→∞

The properties of a symmetric norming function are preserved under this extension. Denoting by .S∞ the ideal of all compact operators in H , we now set So = {X ∈ S∞ : s(X) ∈ co }

.

and define |X|o = o(s(X)) :

.

X ∈ So .

The functional .| · |o is a norm on .So and equipped with this norm, and .So is a symmetrically normed ideal that is a Banach space. The symmetrically normed ideal .So has the additional property that it is stable under submajorization, that is, if .A ∈ So and .B ∈ S∞ satisfy k Σ .

j =1

sj (B) ≤

k Σ

sj (A),

k = 1, 2, . . . ,

j =1

then .B ∈ So and .|B|o ≤ |A|o . A further very special property enjoyed by symmetrically normed ideals is the following counterpart to Fatou’s Lemma. Suppose that .A ∈ R is the weak operator limit of a sequence of operators .{An }∞ n=1 from .So for which .supn |An |o < ∞. If .So /= S∞ , then .A ∈ Sφ and .|A|o ≤ supn |An |o . Separable Symmetrically Normed Ideals The subspace .S(0) o is now defined to be the closure with respect to the norm .|·|o of the set .R of all finite rank operators. The (0) subspace .So is a separable symmetrically normed ideal in the ring .R, and every (0) separable symmetrically normed ideal coincides with some ideal .So . Further, .So

Brief Historical Notes

557 (0)

is separable if and only if .So = So , and in turn, this is equivalent to the condition that .

lim o(ξn+1 , ξn+2 , . . . ) = 0,

n→∞

ξ = {ξn } ∈ co .

A symmetric norming function .o that satisfies this condition is said to be mononormalizing. The Symmetrically Normed Ideals .Sp , 1 ≤ p ≤ ∞ Of particular importance are the symmetrically normed ideals .Sp , 1 ≤ p ≤ ∞, which correspond to the symmetric norming functions op (ξ ) =

(∞ Σ

.

)1/p |ξk |

p

,

1 ≤ p < ∞,

ξ ∈ c00 ,

k=1

and .o∞ (ξ ) = maxk |ξk | in the case that .p = ∞. The collection .Sp consists of all compact operators A for which the singular value sequence .{sk (A)}∞ k=1 is an element of the classical sequence space .lp , if .1 ≤ p < ∞, and .c0 if .p = ∞, with corresponding norms given by .|A|p = ||{sk (A)}∞ k=1 ||p , where .|| · ||p denotes the usual sequence norm. As is clear from the foregoing discussion, the spaces .Sp , 1 ≤ p ≤ ∞ are separable symmetrically normed ideals. They are precisely the specializations of the more general .Lp -spaces considered by Segal, Kunze and Dixmier to the case that the underlying von Neumann algebra is the algebra of all bounded linear operators in a separable Hilbert space, equipped with the standard trace. Functions Associated to Symmetric Norming Functions If .o is a symmetric norming function on .c00 , and if .ξ = (ξ1 , ξ2 , . . . ) ∈ c00 , the associate function .ψ is defined by setting ψ(ξ ) = max

Σ

.

ξj ηj : o(η) ≤ 1, η = (η1 , η2 , . . . ) ∈ c00 .

j

The terminology here is adopted from Schatten [108], though, as indicated earlier, the basic idea goes back to von Neumann. The term adjoint is used in [60] where an equivalent formulation is given. The function .ψ associated to a symmetric norming function .o is itself a symmetric norming function, and the function associated to .ψ is .o. If .ψ = ψp , 1 ≤ p ≤ ∞, then the associate .ψ is just .oq , 1/p + 1/q = 1. The notion of the associate symmetric norming function permits identification of the Banach space dual to a separable symmetrically normed ideal with a symmetrically normed ideal. If .o is a symmetric norming function with associate function .o, and if .So does not coincide elementwise with .S1 ( which is the case if (0) .o is not equivalent to .o1 ), then the Banach dual of the separable space .S o may be identified with the associate space .Sψ with respect to trace duality. It follows, in

558

Brief Historical Notes

particular, that if both .o and its associate .ψ are mononormalizing, that is, if both So and .Sψ are separable, then .S is reflexive.

.

(0)

The Symmetrically Normed Ideals .S|| , S|| , Sπ While attention had been previously focussed on the noncommutative counterparts to the classical .Lp -spaces, the monograph [60] of Gohberg and Krein introduced new concrete examples of nonseparable symmetrically normed ideals, which deserve special mention. Let .|| = (π1 , π2 , . . . ) be a decreasing sequence of non-negative real numbers for which ∞ Σ .

πj = ∞ and

lim πn = 0.

n→∞

k=1

Such a sequence is referred to by [60] as binormalizing. Associated with a binormalizing sequence are the functions o|| (ξ ) = sup

[ n Σ

.

n

ξj∗ /

k=1

n Σ

] πj ,

ξ ∈ c00

k=1

and oπ (ξ ) =

Σ

.

πk ξk∗ ,

ξ ∈ c00 .

k

Here the sequence .ξ ∗ = (ξ1∗ , ξ2∗ , . . . ) is the decreasing rearrangement of the sequence .ξ , that is, the sequence .|ξ | = (|ξ1 |, |ξ2 |, . . . ) rearranged in decreasing order. Each of the functions .o|| , oπ is an s.n. function. The function .oπ is the function associated to the function .o|| , and so the symmetrically normed ideals .So|| = S|| and .Soπ = Sπ are mutually associated. The s.n. ideal .S|| consists of all compact operators A for which ⎡ .

sup ⎣ n

n Σ

sj (A)/

j =1

n Σ

⎤ πj ⎦ < ∞

j =1

and is a nonseparable s.n ideal with the norm ⎡ |A|S|| = sup ⎣

n Σ

.

n

j =1

sj (A)/

n Σ j =1

⎤ πj ⎦ .

Brief Historical Notes

559 (0)

It is an example of a noncommutative Marcinkiewicz space. Its separable part .S|| consists of all compact operators .A ∈ S|| for which ⎡ .

lim ⎣

n→∞

n Σ

sj (A)/

j =1

n Σ

⎤ πj ⎦ = 0.

j =1

The s.n. ideal .Sπ consists of all compact operators A for which ∞ Σ .

πj sj (A) < ∞

j =1

and is a separable s.n. ideal with the norm |A|Sπ =

∞ Σ

.

πj sj (A).

j =1

It is an example of a noncommutative Lorentz space. For the triple of spaces S(0) || , Sπ , S|| ,

.

each is the Banach space dual of the preceding one. It is of interest to note that these examples were independently discovered by Garling [58], based on his earlier study [57] of symmetric Köthe sequence spaces. The Macaev Ideal .Sω As pointed out by Gohberg and Krein [60], their study of the ideals .S(0) || , Sπ , S|| , was inspired by the introduction by V. I. Macaev [87] of the s.n. ideal .Sω , where .o is the binormalizing sequence .{1/(2n − 1)}∞ n=1 . This ideal is known as the Macaev ideal and arose in the study of various questions in the theory of non-self-adjoint operators. The Macaev ideal .Sω has a number of remarkable properties. For example, it contains all of the spaces .Sp , 1 ≤ p < ∞, and the nonseparable space .So dual to the Macaev ideal lies in the gap between the spaces .S1 and .Sp , 1 < p ≤ ∞, that is, S1 ⊆ So ⊆

n

.

Sp .

1 0. Here .{Eλ } is the spectral resolution of the measurable operator .|A|. The generalized singular value function .sA (·) (or, in the language of Grothendieck, the decreasing rearrangement) of A is then defined to be the left-continuous inverse to the distribution function .nA given by the formula sA (α) = inf{λ : nA (λ) < α}.

.

There are a number of striking parallels with the two-sided ideal of compact operators in .H. The definition implies that .sA (α) → 0 as .α → ∞ whenever .A ∈ C0 (r). Further, Ovchinnikov establishes a Courant–Fischer minimax characterization of the function .sA , and a further geometric characterization of the singular value .sA (α) as the distance of A to the set of all operators of metric rank less than .α. From this, follow pointwise estimates for the sum and product: If .A, B ∈ C0 (r), then .A, B ∈ C0 (r) and sA+B (α + β) ≤ sA (α) + sB (β),

.

sAB (αβ) ≤ sA (α)sB (β).

Consequently, .C0 (r) forms a subring with involution in the ring of measurable operators, and .C0 (r) ∩ U is a norm closed, two-sided ideal in .U, which is the norm closure of the set of elements in .U of finite metric rank. Each positive operator A in .U may be expressed in terms of its singular value function by the formula f A=

∞

.

-α , sA (α)d E

-α = I − EsA (α)+0 , E

0

and for a positive compact operator, this is just the classical Schmidt decomposition. This has the consequence that, if .A ∈ C0 (r), then A is integrable if and only if .sA is summable with respect to Lebesgue measure on .[0, ∞), in which case f ||A||L1 (r) =

∞

sA (α)dα.

.

0

Under the assumption that .r contains no minimal projections, it is then shown that, if .A ∈ C0 (r), then, for every .t > 0, f .

0

t

sA (α)dα = sup{|m(BA)| : B ∈ U, r(B) ≤ t, ||B||∞ ≤ 1},

562

Brief Historical Notes

where .r(·) denotes metric rank. The submajorization inequalities f

t

.

f

t

sA+B (α)dα ≤

0

f

t

sA (α)dα +

sB (α)dα,

0

A, B ∈ C0 (r),

t > 0,

0

now follow directly, and the assumption that .U should have no atoms may be removed via the isomorphism .U → U ⊗ 1 of .U into the non-atomic von Neumann tensor product .U⊗L∞ [0, 1], which preserves the generalized singular value function. Finally, if .ψ is a non-decreasing concave function on .[0, ∞), then f

∞

.

f

∞

sAB (α)dψ(α) ≤

0

sA (α)sB (α)dψ(α),

A, B ∈ C0 (r),

0

which readily implies the submajorization inequalities f .

t

f

t

sAB (α)dα ≤

0

sA (α)sB (α)dα,

A, B ∈ C0 (r),

t > 0.

0

A self-contained and comprehensive exposition of generalized s-numbers and their properties was given subsequently by T. Fack and H. Kosaki [55]. Noncommutative Symmetric Spaces Ovchinnikov [95, 96] now applies his earlier results to introduce and study a wide class of noncommutative symmetric spaces that contained as special cases the noncommutative .Lp -spaces of Segal, Kunze and Dixmier as well as the symmetrically normed ideals of Schatten and Gohberg and Krein. In the terminology of [95], a normed linear space E of measurable operators is called symmetric if, for any .A ∈ E and any measurable operator B, the inequality .sB (·) ≤ sA (·) implies that .B ∈ E and .||B||E ≤ ||A||E . Such a space is continuously embedded in the space of measurable operators in the sense that each null sequence contains a subsequence converging to zero almost everywhere. Construction of Noncommutative Symmetric Banach Spaces A general method of constructing noncommutative spaces is now given by Ovˇcinnikov as follows. Let .ψt (·), t ∈ T , be a family of non-decreasing concave functions on .[0, ∞) that take value 0 at 0. If .ET (r) is the collection of all measurable operators for which f ||A||ET (r) = sup

∞

.

t∈T

sA (α)dψt (α) < ∞,

0

then the space .ET (r), equipped with the norm .|| · ||ET (r) is a symmetric Banach space. One sees immediately that if the family .{ψt , t ∈ T } consists of a single element, then the space .ET (r) is the familiar Lorentz space .Aψ (see [76]) if .r is given by .L∞ [0, ∞) acting by multiplication on .L2 [0, ∞) and m taken to be integration with respect to Lebesgue measure. In the case of a general semifinite von Neumann algebra, the space .Eψ (r) := ET (r) can be viewed as the noncommutative Lorentz space consisting of all .A ∈ C0 (r) for which .sA (·) ∈ Aψ

Brief Historical Notes

563

with norm given by .||A||ψ = ||sA (·)||ψ . A wealth of further concrete examples are available by an appeal to the basic theory of rearrangement-invariant spaces and their associate spaces on the positive semi-axis. Indeed, if E is a rearrangementinvariant Banach space on .[0, ∞), then the associate space .E × consists of those measurable g for which f

∞

||g||E × := sup{

.

|f g|(s)ds : f ∈ E, ||f ||E ≤ 1} < ∞.

0

The space .E × is a rearrangement-invariant Banach space, and if E has the (socalled) Fatou property, that is, whenever .0 ≤ fn ↑n ⊆ E satisfies .supn ||fn ||E < ∞, it follows that .supn fn exists in E and .||fn ||E ↑n ||f ||E , then .f ∈ E if and only if f .

sup{

∞

|f g|(s)ds : g ∈ E × , ||g||E × ≤ 1} < ∞

0

in which case f ||f ||E = sup{

∞

.

|f g|(s)ds : g ∈ E × , ||g||E × ≤ 1}

0

f = sup{

∞

sf (s)sg (s)ds : g ∈ E × , ||g||E × ≤ 1}.

0

Here, for the sake of consistency, we have used the notation .sf to denote the classical decreasing rearrangement, rather than the traditional notation .f ∗ . Now suppose that E is a rearrangement-invariant Banach space on the semi-axis with the Fatou fα property. If .T = {g ∈ E × : ||g||E × ≤ 1} and if .ψg (α) = 0 sg (s)ds, α ∈ [0, ∞) whenever .g ∈ T , then .ET (r) is just the space consisting of all .A ∈ C0 (r) such that .s(A) ∈ E equipped with the norm .A → ||A||ET (r) = ||s(A)||E . As is well known, many classical rearrangement-invariant spaces, including the .Lp -spaces, Orlicz, Lorentz, and Marcinkiewicz spaces have the Fatou property, and so the above construction yields noncommutative counterparts to these spaces. Noncommutative Interpolation It is noted first in [95] that each noncommutative symmetric Banach E is an intermediate space (in the sense of interpolation theory) for the noncommutative pair .(L1 (r), M) in the sense that L1 (r) ∩ M ⊆ E ⊆ L1 (r) ⊆ L1 (r) + M

.

with continuous embeddings, where f ||A||L1 (r)+U = inf{||A1 ||L1 (r) + ||A2 ||∞ } =

1

sA (α)dα,

.

0

564

Brief Historical Notes

where the infimum is taken over all representations .A = A1 + A2 with A1 ∈ L1 (r), A2 ∈ U. It is then shown that various interpolation theorems for rearrangement-invariant Banach function spaces generalize to the setting of symmetric spaces of measurable operators. In particular, a rearrangementinvariant space E of measurable operators is an interpolation space for the couple .(L1 (r), U ∩ C0 (r)) if and only if E is stable under submajorization, that is, whenever .A ∈ E and f t f t . sB (α)dα ≤ sA (α)dα, ∀t > 0, .

0

0

it follows that .B ∈ E. This theorem specializes to a theorem of A. Calderón [19] in the case of commutative measure spaces and to a a theorem of Russu [106] in the setting of symmetrically normed ideals. This criterion immediately implies that each of the spaces .ET (r) constructed above is an interpolation space for the couple .(L1 (r), U ∩ C0 (r)). Also indicated in [95] is a general method of reducing the study of interpolation theorems in noncommutative spaces to the corresponding theorems in the commutative setting, a particular example being the noncommutative Riesz– Thorin interpolation theorem of Kunze. Related contributions to noncommutative interpolation dating from about this period are due to J. Peetre and G. Sparr [99, 100] and J. Arazy [4]. Further Contributions Independently of the work of Ovchinnikov, parallel investigations were made by F. J. Yeadon [136, 138, 140] dating from his PhD thesis [139]. Motivated by the work of Dixmier, Segal and Stinespring on noncommutative .Lp -spaces in the setting of gage spaces .r = (H, U, m), and the subsequent contributions of Garling [58] on ideals of operators in Hilbert spaces, F. J. Yeadon studies noncommutative .Lp -spaces [138] and introduces symmetric operator norms [140], with a view to applications in noncommutative ergodic theory [140]. Yeadon first extends the notion of decreasing rearrangement to the class .K = K(U, m) of operators, which he calls totally measurable consisting of those (locally) measurable A for which the distribution function .λ → nA (λ) is finite for at least one .λ > 0. It is shown that the space .Lp (r), 1 ≤ p ≤ ∞ as defined by Segal and Kunze is precisely the space of all .A ∈ K for which the .sA (·) ∈ fLp [0, ∞) in which case ∞ .||A||Lp (r) = ||sA (·)||Lp [0,∞) . Further, if .A, B ∈ K and if . 0 sA (α)sB (α)dα < ∞, f∞ then .AB ∈ L1 (r) and .||AB||L1 (r) ≤ 0 sA (α)sB (α)dα. From this follows easily a Hölder inequality that shows that .Lp (r) and .Lq (r), .1/p + 1/q = 1 are normed spaces in duality. A local version of the Radon–Nikodym theorem previously established by Segal is then proved for ultraweakly continuous linear functionals, and this permits identification of .Lq (r) with the Banach dual of .Lp (r) if .1 ≤ p < ∞ and .1/p + 1/q = 1. As has been noted earlier, this duality had already been proved by Dixmier [34] based (in part) on first establishing the uniform convexity of .Lp (r) for .2 < p < ∞. By contrast, the approach of [138] based on

Brief Historical Notes

565

the systematic use of the decreasing rearrangement is more direct and does not use uniform convexity arguments. Noncommutative Function Norms Motivated by the theory of function norms as developed by W. A. J. Luxemburg and A. C. Zaanen [86] (see also [84, 85, 144]), Yeadon [140] now introduces the notion of an operator norm on the class .L of locally measurable operators. This class of operators is an algebra of operators affiliated with the semi-finite von Neumann algebra .U. It is, in general, larger than the algebra of measurable operators in the sense of Segal and had been studied earlier by Yeadon [137]. An operator norm .γ : L → [0, ∞] is an extended norm that satisfies γ (XAY ) ≤ ||X||∞ ||Y ||∞ γ (A),

.

A ∈ L.

The corresponding normed operator space .Lγ is then defined by Lγ = {A ∈ L : γ (A) < ∞}

.

so that .Lγ is a .U-bimodule and .γ is a norm on .Lγ . an operator norm for which Lγ ⊆ L1 (r) + U is said to be (a) symmetric if .γ (A) ≤ γ (B) whenever .sA (·) ≤ sB (·) and (b) fully symmetric if .γ is symmetric and the norm is monotone with respect to submajorization. Following the development by Luxemburg and Zaanen of the theory of function norms in the commutative setting [86], Yeadon introduces in [136] the notion of the associate .γ × of an operator norm .γ by setting

.

γ × (B) = sup{|m(AB)| : γ (A) ≤ 1}

.

and establishes there that a necessary and sufficient condition that .γ should coincide with the bi-associate norm .γ ×× is that .γ should have the Fatou property in the sense that .γ (supα Aα ) = supα (Aα ) for every increasing net .{Aα } of locally measurable operators. Criteria for completeness and reflexivity for normed operator spaces analogous to those given by Luxemburg and Zaanen [86] are also given in [136]. To any symmetric function space .Lγ ⊆ (L1 + L∞ ) on .[0, ∞) is associated the class of operators Lγ (U) = {A ∈ L1 + U : γ (s(A)) < ∞}

.

equipped with the norm .γ (A) = γ (s(A). Provided the function norm .γ has the (sequential) Fatou property in the sense that .γ (supn fn ) = supn γ (fn ) for every increasing sequence .{fn } of positive measurable functions, then .Lγ (U) is a fully symmetric operator norm. It is shown that if .γ is an operator norm defined on the space .F of elementary operators (in the sense of Segal), and if .γ (En ) →n 0 whenever .En is a sequence of projections in .F decreasing to 0, then the dual of the normed space .(F, γ ) may be identified with the normed operator space .Lγ ×

566

Brief Historical Notes

via trace duality, and the completion of .(F, γ ) may be identified with the closure of .F in the normed operator space .Lγ ×× . The condition here is easily seen to be satisfied if the operator norm .γ is derived from a function norm that satisfies the Lebesgue monotone convergence theorem, that is, .fn ↓n 0 ⊆ Lγ implies .||fn ||γ ↓ 0. This class of spaces consists of those symmetric spaces on .[0, ∞) that are separable. Fully symmetric operator spaces are characterized as those that are exact interpolation spaces for the noncommutative pair .(L1 , U), which, again, is the exact extension of the Calderón characterization in the commutative setting. Finally, a mean ergodic theorem is given in certain spaces .Lγ . An exact extension of the pointwise ergodic theorem in .L1 based on a weak type estimate had been given earlier [139].

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Index

Symbols ∗ -algebra, 26 cone of positive elements, 28 product of family, 30 subalgebra, 26 K-functional, 464 Holmstedt formula, 469 .∗-Algebra homomorphism, 25 isomorphism, 25 subalgebra, 25 unital, 25 .λ-submajorization, 199 .M-bimodule, 239 Banach –, 239 Riesz–Fischer property, 249 (semi-)normed, 239 (.σ )-Fatou property, 250, 343 .C

A Absolutely solid subspace, 52, 61, 240 Annihilator, 326 inverse, 326 Aronszajn–Gagliardo theorem, 458

B Banach algebra, 1, 26 Banach .∗-algebra, 25, 26 Banach couple, 218, 455 bounded operator, 221, 455 intermediate Banach space, 456 interpolation pair, 456 K-functional, 464

Banach space containing isomorphic copy of .l∞ , 341 modulus of convexity, 517 modulus of smoothness, 517 of power type p, 517 property .(u), 344 of type/cotype p, 524 uniformly convex, 517 uniformly smooth, 517 weakly sequentially precompact subset, 387 Beppo–Levi theorem, 160 Boyd interpolation theorem, 480

C Calderón family, 532 Calderón operator, 472 Calderón product, 532, 534 Clarkson inequalities, 511 Commutant, 30 Complex interpolation method, 509 Conditional expectation operator, 457 Copy, 381 disjointly supported, 381 isomorphic copy of .l∞ , 341 positive, 381

D Decreasing rearrangement of a measurable function, 126, 128 Dilation, 287, 392 Distribution function of a measurable function, 126, 128

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. G. Dodds et al., Noncommutative Integration and Operator Theory, Progress in Mathematics 349, https://doi.org/10.1007/978-3-031-49654-7

573

574 Dominated convergence theorem, 163 Double commutant theorem, 31

E Eigenvalue function, 173, 179 increasing, 176 Element of absolutely continuous norm, 331 Element of order continuous norm, 326 Equimeasurable function, 126

F Fatou’s lemma, 162 Form, 19 Cauchy–Schwartz inequality, 19 closed, 20 core, 20 corresponding to operator, 22 densely defined, 21 domain, 19 inner product, 20 ordering, 20 polarization identity, 19 positive, 19 quadratic, 19 sesquilinear, 19 symmetric, 19 Functional calculus Borel, normal operators, 14, 28 continuous, .C ∗ -algebras, 27 self-adjoint measurable operators, 62 self-adjoint .τ -compact operators, 79 self-adjoint .τ -measurable operators, 76, 112 trace formula operators affiliated with .M, 172 trace formula .τ -measurable operators, 148, 152, 175

H Hardy’s inequality, 226, 439, 470, 476 Head majorization, 490 Hilbert space, 1 direct sum, 38 Hölder inequality, 519 Holmstedt formula, 469

I Ideal of measurable functions, 240 Interpolation functor, 458 exact, 458

Index Interpolation pair, 456 exact, 456 of exponent .θ, 456 interpolation constant, 456 Interpolation space, 456 exact, 456 of exponent .θ, 456 K Kadec–Klee property, 516 Kahane’s inequalities, 525 Kantorovich–Banach space, 324 Khintchine inequality, 525 Köthe dual, 261, 290, 299, 534 second, 268 L Lattice, 4 of closed linear subspaces of H , 4 complete, 4 of projections, 4, 39 Linear functional absolute kernel, 258 completely additive, 33, 260 conjugate functional, 252 Hermitian, 252, 353 normal, 33, 259, 260 positive, 32, 252, 353 real/imaginary part, 252 singular, 259, 260 Yosida–Hewitt decomposition, 318 Linear subspace .τ -dense, 70 affiliated, 49 determining sequence, 53 strongly dense, 53 Lorentz .Lp,q space, 435 Lorentz space, 401 Lozanovskii duality theorem, 535 Lozanovskii factorization, 539 Luxemburg representation theorem, 398 M Maharam measure space, 27, 51 Marcinkiewicz space, 405 Marcinkiewicz theorem, 468, 476 Measure separable, 346 Measure topology, 80, 83, 118, 133, 138 bounded set, 96, 142 local –, 101, 246 Midpoint subconvex map, 474

Index N Net decreasing, 4 increasing, 4 Noncommutative Boyd theorem, 482 Noncommutative Lorentz .Lp,q -space, 442 Noncommutative Lorentz space, 405 Noncommutative .Lp -space, 399 Noncommutative Marcinkiewicz space, 412 Noncommutative Orlicz space, 429 Norm Fatou, 286 Hilbert space, 1 on .L1 (τ ), 160 Luxemburg, 418 operator, 1 order continuous, 322, 339, 341 Orlicz, 424 supremum, 10 weak type .(p, q), 472 Normed function space p-convex, 432 p-convexification, 430

O Operator, 1 .τ -compact, 77 .τ -measurable, 70, 71 absolute value, 17, 18 adjoint, 1, 7 affiliated, 49 Borel functional calculus, 14 central support, 39 closable, 6 closed, 6 core, 6 densely defined, 7 determining sequence, 55 direct sum, 9 domain, 5 extension, 5 final projection, 18 graph, 6 Hermitian, 1 initial projection, 18 left/right support projection, 8 measurable, 55 normal, 7 null projection, 8, 15 normal, 1 partial isometry, 18 polar decomposition, 18

575 positive, 2, 7, 15, 61 positive/negative part, 16 quadratic form, 22 quadratic form ordering, 23, 67 range projection, 8, 30, 39 resolvent, 8 resolvent commuting, 17, 64 restriction, 5 self-adjoint, 1, 7, 15 spectral measure, 13 spectrum, 8 strong sum/product, 12, 60 support projection, 8, 15, 30, 39, 145 symmetric, 15 unitary, 2 weak type .(p, q), 471 Yosida approximation, 99 Operator function, 107 Operator topology strong, 2 ultra-strong, 3 ultra-weak, 3 weak, 2 Order ideal, 258 order dense, 259 Order isomorphism, 4 Orlicz space, 417

P Phillips’ lemma, 320 Projection, 2 .∗-algebra, 25 .σ -finite, 41, 332 .τ -finite, 70 Abelian, 42 carrier, 242, 262 central, 39 complement, 2 countable type, 41 countably decomposable, 41 equivalent, 40 finite/infinite, 41 majorization, 40 minimal, 187 properly infinite, 41 von Neumann algebra, 39

R Riesz–Thorin theorem, 511 Right-continuous inverse, 121

576 S Schmidt decomposition, 539 Sequence disjointly supported, 381 property .(u), 344 weak Cauchy, 344 Series unconditionally Cauchy, 344 Singular value function, 129 generalized, 444, 451 Solid subset, 353 So-topology, see Operator topology, strong Spectral distribution function of a positive .τ -measurable operator, 129 of a self-adjoint .τ -measurable operator, 173 Spectral measure, 9 change of measure formula, 13 finitely additive, 9 integration, 10 null set, 9 support, 14 Spectral theorem, 13 Spectrum operator, 8 unital Banach algebra, 27 Sublinear map, 474 Submajorization, 206, 291, 490 Symmetrically normed sequence space, 444 strongly –, 444 Symmetrically normed space, 271, 290 p-convexification, 434 fully –, 291 Kantorovich–Banach space, 324 strongly –, 297 subset of left/right/bi-uniformly absolutely continuous norm, 364 subset of uniformly absolutely continuous norm, 332, 356 Symmetric space, 271 E-uniformly integrable subset, 369 fully –, 291 lower/upper Boyd index, 479 (relatively) weakly compact subset, 373 strongly –, 297 uniformly integrable subset, 369

T Tail majorization, 490 Trace .τ -measurable operators, 146 on all operators affiliated with .M, 171 center-valued, 44 faithful, 45

Index finite, 45 ideal of definition, 46, 154 normal, 45 on .L1 (τ ), 156 semi-finite, 45 separable, 347 von Neumann algebra, 45 Trace duality between .L1 (τ ) and .M, 165 U uso-topology, see Operator topology, ultra-strong uwo-topology, see Operator topology, ultra-weak

V von Neumann algebra, 30 .σ -finite, 43, 332 affiliated linear subspace, 49 affiliated operator, 49 center, 31 central projection, 39 central support, 39 continuous, 43 countably decomposable, 43, 333 discrete, 43 factor, 31, 34, 43 finite, 43, 44, 46 induced, 34 (non-)atomic, 187 normal positive linear functional, 33 normal positive linear map, 33 positive linear functional, 32 positive linear map, 33 pre-dual, 32 product of family, 39 projection, 39 proper subalgebra, 456 properly infinite, 43 purely infinite, 43 reduced, 34 semi-finite, 43, 46 strongly dense linear subspace, 53 trace, 45 trace preserving .∗-homomorphism, 116 type decomposition, 43 type I, 42 type I.∞ , 43 type I.f in , 43 type I.n , 42 type II, 42 type II.1 , 42

Index type II.∞ , 42 type III, 42 weight, 45 W Weak .Lp -space, 437 wo-topology, see Operator topology, weak

577 Y Yosida–Hewitt decomposition, 318 Young function, 412 .A2 -condition, 425 .A2 -condition at .∞, 426 complementary, 420 modular, 415 Young’s inequality, 421