Operator Algebras (eBook)

Preface to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry 7
Preface 11
Contents 15
I Operators on Hilbert Space 21
I.1 Hilbert Space 21
I.1.1 Inner Products 21
I.1.2 Orthogonality 22
I.1.3 Dual Spaces and Weak Topology 23
I.1.4 Standard Constructions 24
I.1.5 Real Hilbert Spaces 25
I.2 Bounded Operators 25
I.2.1 Bounded Operators on Normed Spaces 25
I.2.2 Sesquilinear Forms 26
I.2.3 Adjoint 27
I.2.4 Self-Adjoint, Unitary, and Normal Operators 28
I.2.5 Amplifications and Commutants 29
I.2.6 Invertibility and Spectrum 30
I.3 Other Topologies on L(H) 33
I.3.1 Strong and Weak Topologies 33
I.3.2 Properties of the Topologies 34
I.4 Functional Calculus 37
I.4.1 Functional Calculus for Continuous Functions 38
I.4.2 Square Roots of Positive Operators 39
I.4.3 Functional Calculus for Borel Functions 39
I.5 Projections 39
I.5.1 Definitions and Basic Properties 40
I.5.2 Support Projections and Polar Decomposition 41
I.6 The Spectral Theorem 43
I.6.1 Spectral Theorem for Bounded Self-Adjoint Operators 43
I.6.2 Spectral Theorem for Normal Operators 45
I.7 Unbounded Operators 47
I.7.1 Densely De.ned Operators 47
I.7.2 Closed Operators and Adjoints 49
I.7.3 Self-Adjoint Operators 50
I.7.4 The Spectral Theorem and Functional Calculus for Unbounded Self-Adjoint Operators 52
I.8 Compact Operators 56
I.8.1 Definitions and Basic Properties 56
I.8.2 The Calkin Algebra 57
I.8.3 Fredholm Theory 57
I.8.4 Spectral Properties of Compact Operators 60
I.8.5 Trace-Class and Hilbert-Schmidt Operators 61
I.8.6 Duals and Preduals, s-Topologies 63
I.8.7 Ideals of L (H) 64
I.9 Algebras of Operators 67
I.9.1 Commutant and Bicommutant 67
I.9.2 Other Properties 68
II C*-Algebras 71
II.1 Definitions and Elementary Facts 71
II.1.1 Basic De.nitions 71
II.1.2 Unitization 73
II.1.3 Power series, Inverses, and Holomorphic Functions 74
II.1.4 Spectrum 74
II.1.5 Holomorphic Functional Calculus 75
II.1.6 Norm and Spectrum 77
II.2 Commutative C*-Algebras and Continuous Functional Calculus 79
II.2.1 Spectrum of a Commutative Banach Algebra 79
II.2.2 Gelfand Transform 80
II.2.3 Continuous Functional Calculus 81
II.3 Positivity, Order, and Comparison Theory 83
II.3.1 Positive Elements 83
II.3.2 Polar Decomposition 87
II.3.3 Comparison Theory for Projections 92
II.3.4 Hereditary C*-Subalgebras and General Comparison Theory 95
II.4 Approximate Units 99
II.4.1 General Approximate Units 99
II.4.2 Strictly Positive Elements and s-Unital C*-Algebras 101
II.4.3 Quasicentral Approximate Units 102
II.5 Ideals, Quotients, and Homomorphisms 102
II.5.1 Closed Ideals 103
II.5.2 Nonclosed Ideals 105
II.5.3 Left Ideals and Hereditary Subalgebras 109
II.5.4 Prime and Simple C*-Algebras 113
II.5.5 Homomorphisms and Automorphisms 115
II.6 States and Representations 120
II.6.1 Representations 121
II.6.2 Positive Linear Functionals and States 123
II.6.3 Extension and Existence of States 126
II.6.4 The GNS Construction 127
II.6.5 Primitive Ideal Space and Spectrum 131
II.6.6 Matrix Algebras and Stable Algebras 136
II.6.7 Weights 138
II.6.8 Traces and Dimension Functions 141
II.6.9 Completely Positive Maps 144
II.6.10 Conditional Expectations 152
II.7 Hilbert Modules, Multiplier Algebras, and Morita Equivalence 157
II.7.1 Hilbert Modules 157
II.7.2 Operators 161
II.7.3 Multiplier Algebras 164
II.7.4 Tensor Products of Hilbert Modules 167
II.7.5 The Generalized Stinespring Theorem 169
II.7.6 Morita Equivalence 170
II.8 Examples and Constructions 174
II.8.1 Direct Sums, Products, and Ultraproducts 174
II.8.2 Inductive Limits 176
II.8.3 Universal C*-Algebras and Free Products 178
II.8.4 Extensions and Pullbacks 187
II.8.5 C*-Algebras with Prescribed Properties 196
II.9 Tensor Products and Nuclearity 199
II.9.1 Algebraic and Spatial Tensor Products 200
II.9.2 The Maximal Tensor Product 200
II.9.3 States on Tensor Products 202
II.9.4 Nuclear C*-Algebras 204
II.9.5 Minimality of the Spatial Norm 206
II.9.6 Homomorphisms and Ideals 207
II.9.7 Tensor Products of Completely Positive Maps 210
II.9.8 Infinite Tensor Products 211
II.10 Group C*-Algebras and Crossed Products 212
II.10.1 Locally Compact Groups 213
II.10.2 Group C*-Algebras 217
II.10.3 Crossed products 219
II.10.4 Transformation Group C*-Algebras 225
II.10.5 Takai Duality 231
II.10.6 Structure of Crossed Products 232
II.10.7 Generalizations of Crossed Product Algebras 232
II.10.8 Duality and Quantum Groups 234
III Von Neumann Algebras 241
III.1 Projections and Type Classi.cation 242
III.1.1 Projections and Equivalence 242
III.1.2 Cyclic and Countably Decomposable Projections 245
III.1.3 Finite, In.nite, and Abelian Projections 247
III.1.4 Type Classi.cation 251
III.1.5 Tensor Products and Type I von Neumann Algebras 252
III.1.6 Direct Integral Decompositions 257
III.1.7 Dimension Functions and Comparison Theory 260
III.1.8 Algebraic Versions 263
III.2 Normal Linear Functionals and Spatial Theory 264
III.2.1 Normal and Completely Additive States 265
III.2.2 Normal Maps and Isomorphisms of von Neumann Algebras 268
III.2.3 Polar Decomposition for Normal Linear Functionals 277
III.2.4 Uniqueness of the Predual and Characterizations of 279
III.2.5 Traces on von Neumann Algebras 280
III.2.6 Spatial Isomorphisms and Standard Forms 289
III.3 Examples and Constructions of Factors 295
III.3.1 Infinite Tensor Products 295
III.3.2 Crossed Products and the Group Measure 300
III.3.3 Regular Representations of Discrete Groups 308
III.3.4 Uniqueness of the Hyperfinite II1 Factor 311
III.4 Modular Theory 313
III.4.1 Notation and Basic Constructions 313
III.4.2 Approach using Bounded Operators 315
III.4.3 The Main Theorem 315
III.4.4 Left Hilbert Algebras 316
III.4.5 Corollaries of the Main Theorems 319
III.4.6 The Canonical Group of Outer Automorphisms and 322
III.4.7 The KMS Condition and the Radon-Nikodym Theorem 326
III.4.8 The Continuous and Discrete Decompositions 330
III.5 Applications to Representation Theory of C*-Algebras 333
III.5.1 Decomposition Theory for Representations 333
III.5.2 The Universal Representation and Second Dual 338
IV Further Structure 343
IV.1 Type I C*-Algebras 343
IV.1.1 First Definitions 343
IV.1.2 Elementary C*-Algebras 346
IV.1.3 Liminal and Postliminal C*-Algebras 347
IV.1.4 Continuous Trace, Homogeneous, 349
IV.1.5 Characterization of Type I C*-Algebras 357
IV.1.6 Continuous Fields of C*-Algebras 360
IV.1.7 Structure of Continuous Trace C*-Algebras 364
IV.2 Classification of Injective Factors 370
IV.2.1 Injective C*-Algebras 372
IV.2.2 Injective von Neumann Algebras 373
IV.2.3 Normal Cross Norms. 380
IV.2.4 Semidiscrete Factors 382
IV.2.5 Amenable von Neumann Algebras 385
IV.2.6 Approximate Finite Dimensionality 387
IV.2.7 Invariants and the Classification of Injective Factors 387
IV.3 Nuclear and Exact C*-Algebras 388
IV.3.1 Nuclear C*-Algebras 388
IV.3.2 Completely Positive Liftings 394
IV.3.3 Amenability for C*-Algebras 398
IV.3.4 Exactness and Subnuclearity 403
IV.3.5 Group C*-Algebras and Crossed Products 411
V K-Theory and Finiteness 415
V.1 K-Theory for C*-Algebras 415
V.1.1 K0-Theory 416
V.1.2 K1-Theory and Exact Sequences 422
V.1.3 Further Topics 428
V.1.4 Bivariant Theories 431
V.1.5 Axiomatic K-Theory and the Universal Coe.cient Theorem 433
V.2 Finiteness 438
V.2.1 Finite and Properly In.nite Unital C*-Algebras 438
V.2.2 Nonunital C*-Algebras 443
V.2.3 Finiteness in Simple C*-Algebras 450
V.2.4 Ordered K-Theory 454
V.3 Stable Rank and Real Rank 464
V.3.1 Stable Rank 465
V.3.2 Real Rank 472
V.4 Quasidiagonality 477
V.4.1 Quasidiagonal Sets of Operators 477
V.4.2 Quasidiagonal C*-Algebras 480
V.4.3 Generalized Inductive Limits 484
V.4.3.36 One might hope that the conclusion of V.4.3.35 could be strengthened 494
References 499
Index 525

Preface (P. 11)

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 Encyclopedia 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 signi.cant 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 "classical" operator algebra 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 .ne 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 explanation to at least make all results plausible and to give the reader a sense of what material and level of diffculty is involved in each result. Where an argument 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 in the few cases where new arguments have been found. Nonetheless, throughout 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 applications 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.