Universitext Editors F.W. Gehring P.R. Halmos Universitext Editors: F.W. Gehring, P.R. Halmos Booss/Bleecker: Topology and Analysis Charlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics Cohn: A Classical Invitation to Algebraic Numbers and Class Fields Curtis: Matrix Groups, 2nd. ed. van Dalen: Logic and Structure Devlin: Fundamentals of Contemporary Set Theory Edwards: A Formal Background to Mathematics I alb Edwards: A Formal Background to Higher Mathematics II alb Endler: Valuation Theory Frauenthal: Mathematical Modeling in Epidemiology Gardiner: A First Course in Group Theory Godbillon: Dynamical Systems on Surfaces Greub: Multilinear Algebra Hermes: Introduction to Mathematical Logic Hurwitz/Kritikos: Lectures on Number Theory Kelly/Matthews: The Non-Euclidean, The Hyperbolic Plane Kostrikin: Introduction to Algebra Luecking/Rubel: Complex Analysis: A Functional Analysis Approach Lu: Singularity Theory and an Introduction to Catastrophe Theory Marcus: Number Fields McCarthy: Introduction to Arithmetical Functions Meyer: Essential Mathematics for Applied Fields Moise: Introductory Problem Course in Analysis and Topology 0ksendal: Stochastic Differential Equations Porter/Woods: Extensions of Hausdorff Spaces Rees: Notes on Geometry Reisel: Elementary Theory of Metric Spaces Rey: Introduction to Robust and Quasi-Robust Statistical Methods Rickart: Natural Function Algebras Schreiber: Differential Forms Smorynski: Self-Reference and Modal Logic Stanisic: The Mathematical Theory of Turbulence Stroock: An Introduction to the Theory of Large Deviations Sunder: An Invitation to von Neumann Algebras Tolle: Optimization Methods V.S. Sunder An Invitation to von Neumann Algebras Springer-Verlag New York Berlin Heidelberg London Paris TokYo V. S. Sunder Indian Statistical Institute New Delhi-I 100 16 India AMS Classification: 46-01 Library of Congress Cataloging in Publication Data Sunder, V. S. An invitation to von Neumann algebras. (Universitext) Bibliography: p. Includes index. I. von Neumann algebras. L Title. QA326.S86 1986 512'.55 86-10058 © 1987 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. 9 8 7 6 5 432 I ISBN-13: 978-0-387-96356-3 e-ISBN-13 978-1-4613-8669-8 DOl: 10.1007/978-1-4613-8669-8 PREFACE Why This Book: The theory of von Neumann algebras has been growing in leaps and bounds in the last 20 years. It has always had strong connections with ergodic theory and mathematical physics. It is now beginning to make contact with other areas such as differential geometry and K-Theory. There seems to be a strong case for putting together a book which (a) introduces a reader to some of the basic theory needed to appreciate the recent advances, without getting bogged down by too much technical detail; (b) makes minimal assumptions on the reader's background; and (c) is small enough in size to not test the stamina and patience of the reader. This book tries to meet these requirements. In any case, it is just what its title proclaims it to be -- an invitation to the exciting world of von Neumann algebras. It is hoped that after perusing this book, the reader might be tempted to fill in the numerous (and technically, capacious) gaps in this exposition, and to delve further into the depths of the theory. For the expert, it suffices to mention here that after some preliminaries, the book commences with the Murray - von Neumann classification of factors, proceeds through the basic modular theory to the III). classification of Connes, and concludes with a discussion of crossed-products, Krieger's ratio set, examples of factors, and Takesaki's duality theorem. Although the material is standard, some of the treatment (particularly in Sections 4.1 - 4.3) may be new. Shortcuts taken: In order to accommodate all the above-mentioned material in a volume this size, it was necessary to take some shortcuts: vi Preface (i) Some theorems, though stated in full generality, are only proved under additional (sometimes very severe) simplifying assumptions -- typically, to the effect that some operator is bounded. Some other results suffer a sorrier fate -- they are not even graced with an apology for a proof. (ii) Arguments of a purely set-topological nature often receive step-motherly treatment; where the argument is painless, it has been included; where it is not, the reader is entreated to accept, in good faith, the validity of the relevant statement. (iii) The exercises are an integral part of the book. Several "lemmas" have been relegated to the exercises; any exercise, which is even slightly non-obvious, is furnished with "hints", which are often more in the nature of outlines of solutions. The exercises, rather than being compiled at ends of sections, punctuate the text at junctures where they seem to fit in most naturally. (iv) Both exercises and unproved results are treated just like properly established theorems, in that they are unabashedly used in subsequent portions of the text. The prospective reader: This book is aimed at two classes of readers: graduate students with a reasonably firm background in analysis, as well as mature mathematicians working in other areas of mathematics. As a matter of fact, this book grew out of a course of (twelve) lectures given by the author while visiting the Indian Statistical Institute at Calcutta in the summer of 1984. It was largely due to the positive response of that audience -- consisting entirely of members of the second category mentioned above -- that the author embarked on this venture. The reader is assumed to be familiar with elementary aspects of: (a) measure theory -- monotone convergence, Fubini's Theorem, absolute continuity, LP spaces for p = 1,2,""; (b) analytic functions of one complex variable -- sparseness of zero-sets, contour integration, theorems of Cauchy, Morera, and Liouville; (c) functional analysis -- the "three principles", weak and weak* topologies; (d) Hilbert spaces and operators -- orthonormal basis, subspaces and projections, bounded operators, self-adjoint operators. (The necessary background material from Hilbert space theory is rapidly surveyed in Section 0.1.) In the latter part of the book, a nodding acquaintance with abstract harmonic analysis will be helpful, although it is not essential. For the reader who has been denied such a pleasure, a Preface vii brief appendix (on topological groups) should serve to perform the necessary introduction, which should precede the furtherance of that acquaintance in Sections 3.2 and 3.3. An attempt has also been made, in Section 3.2, to compile the necessary results from the theory before proceeding to use them. Trappings: This volume is equipped with some of the standard fittings, such as a list of symbols, an index of terms used, some notes of a bibliographical nature, and a bibliography. The bibliographical notes are somewhat terse; for more details, the reader may consult [Tak 4]. The terseness also extends to the bibliography, which lists only those books and papers that bear directly on the treatment here; for an extensive bibliography, the reader might consult [Dix]. If the reader ,spots some inaccuracy in the notes or the references, or anywhere else in the text for that matter, the author would appreciate being informed of such an error. The title: The author would like to take this opportunity to thank Professor Arveson for kindly permitting the use of a title that is highly reminiscent of his delightful little book on C*-algebras. If this volume manages to capture even a miniscule fraction of the charm displayed in that volume, it would have accomplished all that the author could have hoped for. ACKNOWLEDGMENTS I would like to thank the following people for the roles they have played in the production of this book: Professor A. K. Roy, for having invited me to spend six wonderful weeks at Calcutta; the en tire audience for the course of lectures I ga ve at Calcu tta, for their enthusiasm and positive response; Professor M. G. Nadkarni, for some discussions concerning Krieger's ratio set; Krishna, for having faithfully and enthusiastically attended all those seminars I organized, whereby I learnt the theory of von Neumann algebras; Shobha Madan, for painstakingly reading large portions of the manuscript and picking out several errors; Professor W. Arveson for a very encouraging letter which boosted my sagging morale at a crucial stage; Shri V. P. Sharma, for an extremely efficient job of typing, cheerfully performed in an amazingly short period of time; and finally, Vyjayanthi, for reasons too uncountable to enumerate, and to whom this book is fondly dedicated. CONTENTS Preface v Acknowledgments ix List of Symbols xiii Chapter 0 Introduction 1 0.1 Basic operator theory 1 0.2 The predual :f(lf). 5 0.3 Three locally convex topologies on :f(lf) 8 0.4 The double commutant theorem 11 Chapter 1 The Murray - von Neumann Classification of Factors 19 1.1 The relation ... - ... (reI M) 19 1.2 Finite projections 22 1.3 The dimension function 27 Chapter 2 The Tomita - Takcsaki Theory 36 2.1 Noncommutative integration 37 2.2 The GNS construction 39 2.3 The Tomita-Takesaki theorem (for states) 45 2.4 Weights and generalized Hilbert algebras 52 2.5 The KMS boundary condition 63 2.6 The Radon-Nikodym theorem and condi tiona I expectations 73 Chapter 3 The Connes Classification of Type m Factors 84 3.1 The unitary cocycle theorem 85 3.2 The Arveson spectrum of an action 93 3.3 The Connes spectrum of an action 102 3.4 Alternative descriptions of reM) 108 xii Contents Chaptcr 4 Crossed-Products 114 4.1 Discrete crossed-products 115 4.2 The modular operator for a discrete crossed-prod uct 122 4.3 Examples of factors 132 4.4 Continuous crossed-products and Takesaki's duality theorem 148 4.5 The structure of properly infinite von Neumann algebras 155 Appendix: Topological Groups 161 Notcs 164 Bibliography 167 Index 169