Encyclopaedia of Mathematical Sciences Volume 122

Operator and Non-Commutative Geometry III Subseries Editors: Joachim Cuntz Vaughan F.R. Jones B. Blackadar Operator Algebras Theory of C∗-Algebras and von Neumann Algebras

ABC Bruce Blackadar Department of Mathematics University of Nevada, Reno Reno, NV 89557 USA e-mail: [email protected]

Founding editor of the Encyclopedia of Mathematical Sciences: R. V. Gamkrelidze

Library of Congress Control Number: 2005934456

Mathematics Subject Classification (2000): 46L05, 46L06, 46L07, 46L08, 46L09, 46L10, 46L30, 46L35, 46L40, 46L45, 46L51, 46L55, 46L80, 46L85, 46L87, 46L89, 19K14, 19K33

ISSN 0938-0396 ISBN-10 3-540-28486-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-28486-4 Springer Berlin Heidelberg New York

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Typesetting: by the author and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11543510 46/TechBooks 543210 to my parents and in memory of Gert K. Pedersen Preface to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry

The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930’s and 1940’s. A von Neumann is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann’s bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. A factor is a with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, II and III. C∗-algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satisfying an algebraic relation connecting the norm and the involution. They also obtained the fundamental result that a commutative unital C∗-algebra is isomorphic to the algebra of complex valued continuous functions on a compact space – its spectrum. Since then the subject of operator algebras has evolved into a huge math- ematical endeavour interacting with almost every branch of mathematics and several areas of theoretical physics. Up into the sixties much of the work on C∗-algebras was centered around representation theory and the study of C∗-algebras of type I (these algebras are characterized by the fact that they have a well behaved representation theory). Finite dimensional C∗-algebras are easily seen to be just direct sums of matrix algebras. However, by taking algebras which are closures in norm of finite dimensional algebras one obtains already a rich class of C∗-algebras – the so-called AF-algebras – which are not of type I. The idea of taking the closure of an inductive limit of finite-dimensional algebras had already appeared in the work of Murray-von Neumann who used it to construct a fundamental example of a factor of type II – the ”hyperfinite” (nowadays also called approximately finite dimensional) factor. VIII Preface to the Subseries

One key to an understanding of the class of AF-algebras turned out to be K-theory. The techniques of K-theory, along with its dual, Ext-theory, also found immediate applications in the study of many new examples of C∗-algebras that arose in the end of the seventies. These examples include for instance ”the noncommutative tori” or other crossed products of abelian C∗-algebras by groups of homeomorphisms and abstract C∗-algebras gener- ated by isometries with certain relations, now known as the algebras On.At the same time, examples of algebras were increasingly studied that codify data from differential geometry or from topological dynamical systems. On the other hand, a little earlier in the seventies, the theory of von Neu- mann algebras underwent a vigorous growth after the discovery of a natural infinite family of pairwise nonisomorphic factors of type III and the advent of Tomita-Takesaki theory. This development culminated in Connes’ great classification theorems for approximately finite dimensional (“injective”) von Neumann algebras. Perhaps the most significant area in which operator algebras have been used is mathematical physics, especially in quantum statistical mechanics and in the foundations of quantum field theory. Von Neumann explicitly mentioned quantum theory as one of his motivations for developing the theory of rings of operators and his foresight was confirmed in the algebraic quantum field theory proposed by Haag and Kastler. In this theory a von Neumann algebra is associated with each region of space-time, obeying certain axioms. The inductive limit of these von Neumann algebras is a C∗-algebra which contains a lot of information on the quantum field theory in question. This point of view was particularly successful in the analysis of superselection sectors. In 1980 the subject of operator algebras was entirely covered in a single big three weeks meeting in Kingston Ontario. This meeting served as a re- view of the classification theorems for von Neumann algebras and the suc- cess of K-theory as a tool in C∗-algebras. But the meeting also contained a preview of what was to be an explosive growth in the field. The study of the von Neumann algebra of a foliation was being developed in the far more precise C∗-framework which would lead to index theorems for foliations incor- porating techniques and ideas from many branches of mathematics hitherto unconnected with operator algebras. Many of the new developments began in the decade following the Kingston meeting. On the C∗-side was Kasparov’s KK-theory – the bivariant form of K-theory for which operator algebraic methods are absolutely essential. Cyclic cohomology was discovered through an analysis of the fine structure of exten- sions of C∗-algebras These ideas and many others were integrated into Connes’ vast Noncommutative Geometry program. In cyclic theory and in connection with many other aspects of noncommutative geometry, the need for going be- yond the class of C∗-algebras became apparent. Thanks to recent progress, both on the cyclic homology side as well as on the K-theory side, there is now a well developed bivariant K-theory and cyclic theory for a natural class of topological algebras as well as a bivariant character taking K-theory to Preface to the Subseries IX cyclic theory. The 1990’s also saw huge progress in the classification theory of nuclear C∗-algebras in terms of K-theoretic invariants, based on new insight into the structure of exact C∗-algebras. On the von Neumann algebra side, the study of subfactors began in 1982 with the definition of the index of a subfactor in terms of the Murray-von Neu- mann theory and a result showing that the index was surprisingly restricted in its possible values. A rich theory was developed refining and clarifying the index. Surprising connections with knot theory, statistical mechanics and quantum field theory have been found. The superselection theory mentioned above turned out to have fascinating links to subfactor theory. The subfactors themselves were constructed in the representation theory of loop groups. Beginning in the early 1980’s Voiculescu initiated the theory of free prob- ability and showed how to understand the free von Neumann algebras in terms of random matrices, leading to the extraordinary result that the von Neumann algebra M of the free group on infinitelymany generators has full fundamental group, i.e. pMp is isomorphic to M for every non-zero projec- tion p ∈ M. The subsequent introduction of free entropy led to the solution of more old problems in von Neumann algebras such as the lack of a Cartan subalgebra in the free group von Neumann algebras. Many of the topics mentioned in the (obviously incomplete) list above have become large industries in their own right. So it is clear that a conference like the one in Kingston is no longer possible. Nevertheless the subject does retain a certain unity and sense of identity so we felt it appropriate and useful to create a series of encylopaedia volumes documenting the fundamentals of the theory and defining the current state of the subject. In particular, our series will include volumes treating the essential techni- cal results of C∗-algebra theory and von Neumann algebra theory including sections on noncommutative dynamical systems, entropy and derivations. It will include an account of K-theory and bivariant K-theory with applications and in particular the index theorem for foliations. Another volume will be devoted to cyclic homology and bivariant K-theory for topological algebras with applications to index theorems. On the von Neumann algebra side, we plan volumes on the structure of subfactors and on free probability and free entropy. Another volume shall be dedicated to the connections between oper- ator algebras and quantum field theory.

October 2001 subseries editors: Joachim Cuntz Vaughan Jones Preface

This volume attempts to give a comprehensive discussion of the theory of operator algebras (C*-algebras and von Neumann algebras.) The volume is intended to serve two purposes: to record the standard theory in the Encyclo- pedia of Mathematics, and to serve as an introduction and standard reference for the specialized volumes in the series on current research topics in the subject. Since there are already numerous excellent treatises on various aspects of the subject, how does this volume make a significant addition to the literature, and how does it differ from the other books in the subject? In short, why another book on operator algebras? The answer lies partly in the first paragraph above. More importantly, no other single reference covers all or even almost all of the material in this volume. I have tried to cover all of the main aspects of “standard” or “classi- cal” theory; the goal has been to be, well, encyclopedic. Of course, in a subject as vast as this one, authors must make highly subjective judgments as to what to include and what to omit, as well as what level of detail to include, and I have been guided as much by my own interests and prejudices as by the needs of the authors of the more specialized volumes. A treatment of such a large body of material cannot be done at the detail level of a textbook in a reasonably-sized work, and this volume would not be suitable as a text and certainly does not replace the more detailed treatments of the subject. But neither is this volume simply a survey of the subject (a fine survey-level book is already available [Fil96].) My philosophy has been to not only state what is true, but explain why: while many proofs are merely outlined or even omitted, I have attempted to include enough detail and ex- planation to at least make all results plausible and to give the reader a sense of what material and level of difficulty is involved in each result. Where an ar- gument can be given or summarized in just a few lines, it is usually included; longer arguments are usually omitted or only outlined. More detail has been included where results are particularly important or frequently used in the sequel, where the results or proofs are not found in standard references, and XII Preface in the few cases where new arguments have been found. Nonetheless, through- out the volume the reader should expect to have to fill out compactly written arguments, or consult references giving expanded expositions. I have concentrated on trying to give a clean and efficient exposition of the details of the theory, and have for the most part avoided general discussions of the nature of the subject, its importance, and its connections and applica- tions in other parts of mathematics (and physics); these matters have been amply treated in the introductory article to this series. See the introduction to [Con94] for another excellent overview of the subject of operator algebras. There is very little in this volume that is truly new, mainly some simplified proofs. I have tried to combine the best features of existing expositions and arguments, with a few modifications of my own here and there. In preparing this volume, I have had the pleasure of repeatedly reflecting on the outstanding talents of the many mathematicians who have brought this subject to its present advanced state, and the theory presented here is a monument to their collective skills and efforts. Besides the unwitting assistance of the numerous authors who originally developed the theory and gave previous expositions, I have benefited from comments, suggestions, and technical assistance from a number of other spe- cialists, notably G. Pedersen, N. C. Phillips, and S. Echterhoff, my colleagues B.-J. Kahng, V. Deaconu, and A. Kumjian, and many others who set me straight on various points either in person or by email. I am especially grate- ful to Marc Rieffel, who, in addition to giving me detailed comments on the manuscript, wrote an entire draft of Section II.10 on group C*-algebras and crossed products. Although I heavily modified his draft to bring it in line with the rest of the manuscript stylistically, elements of Marc’s vision and refreshing writing style still show through. Of course, any errors or misstate- ments in the final version of this (and every other) section are entirely my responsibility. Speaking of errors and misstatements, as far-fetched as the possibility may seem that any still remain after all the ones I have already fixed, I would appreciate hearing about them from readers, and I plan to post whatever I find out about on my website. No book can start from scratch, and this book presupposes a level of knowledge roughly equivalent to a standard graduate course in functional analysis (plus its usual prerequisites in analysis, topology, and algebra.) In particular, the reader is assumed to know such standard theorems as the Hahn- Banach Theorem, the Krein-Milman Theorem, and the Riesz Representation Theorem (the Open Mapping Theorem, the Closed Graph Theorem, and the Uniform Boundedness Principle also fall into this category but are explicitly stated in the text sinced they are more directly connected with operator theory and operator algebras.) Most of the likely readers will have this background, or far more, and indeed it would be difficult to understand and appreciate the material without this much knowledge. Beginning with a quick treatment of the basics of Hilbert space and operator theory would seem to be the proper Preface XIII point of departure for a book of this sort, and the early sections will be useful even to specialists to set the stage for the work and establish notation and terminology.

September 2005 Bruce Blackadar Contents

I Operators on Hilbert Space ...... 1 I.1 HilbertSpace...... 1 I.1.1 Inner Products ...... 1 I.1.2 Orthogonality ...... 2 I.1.3 Dual Spaces and Weak Topology ...... 3 I.1.4 Standard Constructions ...... 4 I.1.5 Real Hilbert Spaces ...... 5 I.2 Bounded Operators ...... 5 I.2.1 Bounded Operators on Normed Spaces ...... 5 I.2.2 Sesquilinear Forms ...... 6 I.2.3 Adjoint ...... 7 I.2.4 Self-Adjoint, Unitary, and Normal Operators . . . . 8 I.2.5 Amplifications and Commutants ...... 9 I.2.6 Invertibility and Spectrum ...... 10 I.3 Other Topologies on L(H)...... 13 I.3.1 Strong and Weak Topologies ...... 13 I.3.2 Properties of the Topologies ...... 14 I.4 Functional Calculus ...... 17 I.4.1 Functional Calculus for Continuous Functions. . . . 18 I.4.2 Square Roots of Positive Operators ...... 19 I.4.3 Functional Calculus for Borel Functions ...... 19 I.5 Projections ...... 19 I.5.1 Definitions and Basic Properties ...... 20 I.5.2 Support Projections and Polar Decomposition . . . 21 I.6 TheSpectralTheorem...... 23 I.6.1 Spectral Theorem for Bounded Self-Adjoint Operators ...... 23 I.6.2 Spectral Theorem for Normal Operators ...... 25 I.7 Unbounded Operators...... 27 I.7.1 Densely Defined Operators ...... 27 I.7.2 Closed Operators and Adjoints ...... 29 XVI Contents

I.7.3 Self-Adjoint Operators ...... 30 I.7.4 The Spectral Theorem and Functional Calculus for Unbounded Self-Adjoint Operators ...... 32 I.8 Compact Operators ...... 36 I.8.1 Definitions and Basic Properties ...... 36 I.8.2 The Calkin Algebra ...... 37 I.8.3 Fredholm Theory ...... 37 I.8.4 Spectral Properties of Compact Operators ...... 40 I.8.5 Trace-Class and Hilbert-Schmidt Operators ...... 41 I.8.6 Duals and Preduals, σ-Topologies ...... 43 I.8.7 Ideals of L(H) ...... 44 I.9 Algebras of Operators ...... 47 I.9.1 Commutant and Bicommutant ...... 47 I.9.2 Other Properties ...... 48

II C*-Algebras ...... 51 II.1 Definitions and Elementary Facts ...... 51 II.1.1 Basic Definitions ...... 51 II.1.2 Unitization ...... 53 II.1.3 Power series, Inverses, and Holomorphic Functions 54 II.1.4 Spectrum ...... 54 II.1.5 Holomorphic Functional Calculus ...... 55 II.1.6 Norm and Spectrum ...... 57 II.2 Commutative C*-Algebras and Continuous Functional Calculus...... 59 II.2.1 Spectrum of a Commutative Banach Algebra . . . . 59 II.2.2 Gelfand Transform ...... 60 II.2.3 Continuous Functional Calculus ...... 61 II.3 Positivity, Order, and Comparison Theory ...... 63 II.3.1 Positive Elements ...... 63 II.3.2 Polar Decomposition ...... 67 II.3.3 Comparison Theory for Projections ...... 72 II.3.4 Hereditary C*-Subalgebras and General Comparison Theory ...... 75 II.4 Approximate Units ...... 79 II.4.1 General Approximate Units ...... 79 II.4.2 Strictly Positive Elements and σ-Unital C*-Algebras...... 81 II.4.3 Quasicentral Approximate Units ...... 82 II.5 Ideals, Quotients, and Homomorphisms ...... 82 II.5.1 Closed Ideals ...... 83 II.5.2 Nonclosed Ideals ...... 85 II.5.3 Left Ideals and Hereditary Subalgebras ...... 89 II.5.4 Prime and Simple C*-Algebras ...... 93 II.5.5 Homomorphisms and Automorphisms ...... 95 Contents XVII

II.6 States and Representations ...... 100 II.6.1 Representations ...... 101 II.6.2 Positive Linear Functionals and States ...... 103 II.6.3 Extension and Existence of States ...... 106 II.6.4 The GNS Construction ...... 107 II.6.5 Primitive Ideal Space and Spectrum ...... 111 II.6.6 Matrix Algebras and Stable Algebras ...... 116 II.6.7 Weights ...... 118 II.6.8 Traces and Dimension Functions ...... 121 II.6.9 Completely Positive Maps ...... 124 II.6.10 Conditional Expectations ...... 132 II.7 Hilbert Modules, Multiplier Algebras, and Morita Equivalence137 II.7.1 Hilbert Modules ...... 137 II.7.2 Operators ...... 141 II.7.3 Multiplier Algebras ...... 144 II.7.4 Tensor Products of Hilbert Modules ...... 147 II.7.5 The Generalized Stinespring Theorem ...... 149 II.7.6 Morita Equivalence ...... 150 II.8 Examples and Constructions ...... 154 II.8.1 Direct Sums, Products, and Ultraproducts ...... 154 II.8.2 Inductive Limits ...... 156 II.8.3 Universal C*-Algebras and Free Products ...... 158 II.8.4 Extensions and Pullbacks ...... 167 II.8.5 C*-Algebras with Prescribed Properties ...... 176 II.9 Tensor Products and Nuclearity ...... 179 II.9.1 Algebraic and Spatial Tensor Products ...... 180 II.9.2 The Maximal Tensor Product ...... 180 II.9.3 States on Tensor Products...... 182 II.9.4 Nuclear C*-Algebras ...... 184 II.9.5 Minimality of the Spatial Norm ...... 186 II.9.6 Homomorphisms and Ideals ...... 187 II.9.7 Tensor Products of Completely Positive Maps . . . 190 II.9.8 Infinite Tensor Products ...... 191 II.10 Group C*-Algebras and Crossed Products ...... 192 II.10.1 Locally Compact Groups ...... 193 II.10.2 Group C*-Algebras ...... 197 II.10.3 Crossed products ...... 199 II.10.4 Transformation Group C*-Algebras ...... 205 II.10.5 Takai Duality ...... 211 II.10.6 Structure of Crossed Products ...... 212 II.10.7 Generalizations of Crossed Product Algebras . . . . 212 II.10.8 Duality and Quantum Groups ...... 214 XVIII Contents

IIIVonNeumannAlgebras ...... 221 III.1 Projections and Type Classification ...... 222 III.1.1 Projections and Equivalence ...... 222 III.1.2 Cyclic and Countably Decomposable Projections . 225 III.1.3 Finite, Infinite, and Abelian Projections ...... 227 III.1.4 Type Classification ...... 231 III.1.5 Tensor Products and Type I von Neumann Algebras...... 232 III.1.6 Direct Integral Decompositions ...... 237 III.1.7 Dimension Functions and Comparison Theory . . . 240 III.1.8 Algebraic Versions ...... 243 III.2 Normal Linear Functionals and Spatial Theory ...... 244 III.2.1 Normal and Completely Additive States...... 245 III.2.2 Normal Maps and Isomorphisms ofvonNeumannAlgebras...... 248 III.2.3 Polar Decomposition for Normal Linear Functionals and the Radon-Nikodym Theorem . . . 257 III.2.4 Uniqueness of the Predual and Characterizations ofW*-Algebras...... 259 III.2.5 Traces on von Neumann Algebras ...... 260 III.2.6 Spatial Isomorphisms and Standard Forms ...... 269 III.3 Examples and Constructions of Factors...... 275 III.3.1 Infinite Tensor Products ...... 275 III.3.2 Crossed Products and the Group Measure Space Construction ...... 280 III.3.3 Regular Representations of Discrete Groups . . . . . 288 III.3.4 Uniqueness of the Hyperfinite II1 Factor ...... 291 III.4 ModularTheory...... 293 III.4.1 Notation and Basic Constructions ...... 293 III.4.2 Approach using Bounded Operators ...... 295 III.4.3 The Main Theorem ...... 295 III.4.4 Left Hilbert Algebras ...... 296 III.4.5 Corollaries of the Main Theorems ...... 299 III.4.6 The Canonical Group of Outer Automorphisms and Connes’ Invariants ...... 302 III.4.7 The KMS Condition and the Radon-Nikodym TheoremforWeights...... 306 III.4.8 The Continuous and Discrete Decompositions ofavonNeumannAlgebra...... 310 III.4.8.1 The Flow of Weights...... 312 III.5 Applications to Representation Theory of C*-Algebras . . . . . 313 III.5.1 Decomposition Theory for Representations ...... 313 III.5.2 The Universal Representation and Second Dual . . 318 Contents XIX

IV Further Structure ...... 323 IV.1 TypeIC*-Algebras...... 323 IV.1.1 First Definitions ...... 323 IV.1.2 Elementary C*-Algebras ...... 326 IV.1.3 Liminal and Postliminal C*-Algebras ...... 327 IV.1.4 Continuous Trace, Homogeneous, and Subhomogeneous C*-Algebras...... 329 IV.1.5 Characterization of Type I C*-Algebras ...... 337 IV.1.6 Continuous Fields of C*-Algebras ...... 340 IV.1.7 Structure of Continuous Trace C*-Algebras ...... 344 IV.2 Classification of Injective Factors ...... 350 IV.2.1 Injective C*-Algebras ...... 352 IV.2.2 Injective von Neumann Algebras ...... 353 IV.2.3 Normal Cross Norms...... 360 IV.2.4 Semidiscrete Factors ...... 362 IV.2.5 Amenable von Neumann Algebras ...... 365 IV.2.6 Approximate Finite Dimensionality ...... 367 IV.2.7 Invariants and the Classification of Injective Factors ...... 367 IV.3 Nuclear and Exact C*-Algebras ...... 368 IV.3.1 Nuclear C*-Algebras ...... 368 IV.3.2 Completely Positive Liftings ...... 374 IV.3.3 Amenability for C*-Algebras...... 378 IV.3.4 Exactness and Subnuclearity...... 383 IV.3.5 Group C*-Algebras and Crossed Products ...... 391

V K-Theory and Finiteness ...... 395 V.1 K-TheoryforC*-Algebras...... 395 V.1.1 K0-Theory...... 396 V.1.2 K1-Theory and Exact Sequences ...... 402 V.1.3 Further Topics ...... 408 V.1.4 Bivariant Theories ...... 411 V.1.5 Axiomatic K-Theory and the Universal CoefficientTheorem...... 413 V.2 Finiteness ...... 418 V.2.1 Finite and Properly Infinite Unital C*-Algebras . . 418 V.2.2 Nonunital C*-Algebras ...... 423 V.2.3 Finiteness in Simple C*-Algebras ...... 430 V.2.4 Ordered K-Theory...... 434 V.3 StableRankandRealRank ...... 444 V.3.1 Stable Rank ...... 445 V.3.2 Real Rank ...... 452 V.4 Quasidiagonality ...... 457 V.4.1 Quasidiagonal Sets of Operators ...... 457 V.4.2 Quasidiagonal C*-Algebras ...... 460 XX Contents

V.4.3 Generalized Inductive Limits ...... 464

References ...... 479

Index ...... 505