Operator Algebras: Theory of C*-Algebras and Von Neumann Algebras

Contents......Page 12
I.1.1 Inner Products......Page 18
I.1.2 Orthogonality......Page 19
I.1.3 Dual Spaces and Weak Topology......Page 20
I.1.4 Standard Constructions......Page 21
I.2.1 Bounded Operators on Normed Spaces......Page 22
I.2.2 Sesquilinear Forms......Page 23
I.2.3 Adjoint......Page 24
I.2.4 Self-Adjoint, Unitary, and Normal Operators......Page 25
I.2.5 Amplifications and Commutants......Page 26
I.2.6 Invertibility and Spectrum......Page 27
I.3.1 Strong and Weak Topologies......Page 30
I.3.2 Properties of the Topologies......Page 31
I.4 Functional Calculus......Page 34
I.4.1 Functional Calculus for Continuous Functions......Page 35
I.5 Projections......Page 36
I.5.1 Definitions and Basic Properties......Page 37
I.5.2 Support Projections and Polar Decomposition......Page 38
I.6.1 Spectral Theorem for Bounded Self-Adjoint Operators......Page 40
I.6.2 Spectral Theorem for Normal Operators......Page 42
I.7.1 Densely Defined Operators......Page 44
I.7.2 Closed Operators and Adjoints......Page 46
I.7.3 Self-Adjoint Operators......Page 47
I.7.4 The Spectral Theorem and Functional Calculus for Unbounded Self-Adjoint Operators......Page 49
I.8.1 Definitions and Basic Properties......Page 53
I.8.3 Fredholm Theory......Page 54
I.8.4 Spectral Properties of Compact Operators......Page 57
I.8.5 Trace-Class and Hilbert-Schmidt Operators......Page 58
I.8.6 Duals and Preduals, σ-Topologies......Page 60
I.8.7 Ideals of L(H)......Page 61
I.9.1 Commutant and Bicommutant......Page 64
I.9.2 Other Properties......Page 65
II.1.1 Basic Definitions......Page 67
II.1.2 Unitization......Page 69
II.1.4 Spectrum......Page 70
II.1.5 Holomorphic Functional Calculus......Page 71
II.1.6 Norm and Spectrum......Page 73
II.2.1 Spectrum of a Commutative Banach Algebra......Page 75
II.2.2 Gelfand Transform......Page 76
II.2.3 Continuous Functional Calculus......Page 77
II.3.1 Positive Elements......Page 79
II.3.2 Polar Decomposition......Page 83
II.3.3 Comparison Theory for Projections......Page 88
II.3.4 Hereditary C*-Subalgebras and General Comparison Theory......Page 91
II.4.1 General Approximate Units......Page 95
II.4.2 Strictly Positive Elements and σ-Unital C*-Algebras......Page 97
II.5 Ideals, Quotients, and Homomorphisms......Page 98
II.5.1 Closed Ideals......Page 99
II.5.2 Nonclosed Ideals......Page 101
II.5.3 Left Ideals and Hereditary Subalgebras......Page 105
II.5.4 Prime and Simple C*-Algebras......Page 109
II.5.5 Homomorphisms and Automorphisms......Page 111
II.6 States and Representations......Page 116
II.6.1 Representations......Page 117
II.6.2 Positive Linear Functionals and States......Page 119
II.6.3 Extension and Existence of States......Page 122
II.6.4 The GNS Construction......Page 123
II.6.5 Primitive Ideal Space and Spectrum......Page 127
II.6.6 Matrix Algebras and Stable Algebras......Page 132
II.6.7 Weights......Page 134
II.6.8 Traces and Dimension Functions......Page 137
II.6.9 Completely Positive Maps......Page 140
II.6.10 Conditional Expectations......Page 148
II.7.1 Hilbert Modules......Page 153
II.7.2 Operators......Page 157
II.7.3 Multiplier Algebras......Page 160
II.7.4 Tensor Products of Hilbert Modules......Page 163
II.7.5 The Generalized Stinespring Theorem......Page 165
II.7.6 Morita Equivalence......Page 166
II.8.1 Direct Sums, Products, and Ultraproducts......Page 170
II.8.2 Inductive Limits......Page 172
II.8.3 Universal C*-Algebras and Free Products......Page 174
II.8.4 Extensions and Pullbacks......Page 183
II.8.5 C*-Algebras with Prescribed Properties......Page 192
II.9 Tensor Products and Nuclearity......Page 195
II.9.2 The Maximal Tensor Product......Page 196
II.9.3 States on Tensor Products......Page 198
II.9.4 Nuclear C*-Algebras......Page 200
II.9.5 Minimality of the Spatial Norm......Page 202
II.9.6 Homomorphisms and Ideals......Page 203
II.9.7 Tensor Products of Completely Positive Maps......Page 206
II.9.8 Infinite Tensor Products......Page 207
II.10 Group C*-Algebras and Crossed Products......Page 208
II.10.1 Locally Compact Groups......Page 209
II.10.2 Group C*-Algebras......Page 213
II.10.3 Crossed products......Page 215
II.10.4 Transformation Group C*-Algebras......Page 221
II.10.5 Takai Duality......Page 227
II.10.7 Generalizations of Crossed Product Algebras......Page 228
II.10.8 Duality and Quantum Groups......Page 230
III Von Neumann Algebras......Page 236
III.1.1 Projections and Equivalence......Page 237
III.1.2 Cyclic and Countably Decomposable Projections......Page 240
III.1.3 Finite, Infinite, and Abelian Projections......Page 242
III.1.4 Type Classification......Page 246
III.1.5 Tensor Products and Type I von Neumann Algebras......Page 247
III.1.6 Direct Integral Decompositions......Page 252
III.1.7 Dimension Functions and Comparison Theory......Page 255
III.1.8 Algebraic Versions......Page 258
III.2 Normal Linear Functionals and Spatial Theory......Page 259
III.2.1 Normal and Completely Additive States......Page 260
III.2.2 Normal Maps and Isomorphisms of von Neumann Algebras......Page 263
III.2.3 Polar Decomposition for Normal Linear Functionals and the Radon-Nikodym Theorem......Page 272
III.2.4 Uniqueness of the Predual and Characterizations of W*-Algebras......Page 274
III.2.5 Traces on von Neumann Algebras......Page 275
III.2.6 Spatial Isomorphisms and Standard Forms......Page 284
III.3.1 Infinite Tensor Products......Page 290
III.3.2 Crossed Products and the Group Measure Space Construction......Page 295
III.3.3 Regular Representations of Discrete Groups......Page 303
III.3.4 Uniqueness of the Hyperfinite II[sub(1)] Factor......Page 306
III.4.1 Notation and Basic Constructions......Page 308
III.4.3 The Main Theorem......Page 310
III.4.4 Left Hilbert Algebras......Page 311
III.4.5 Corollaries of the Main Theorems......Page 314
III.4.6 The Canonical Group of Outer Automorphisms and Connes\' Invariants......Page 317
III.4.7 The KMS Condition and the Radon-Nikodym Theorem for Weights......Page 321
III.4.8 The Continuous and Discrete Decompositions of a von Neumann Algebra......Page 325
III.5.1 Decomposition Theory for Representations......Page 328
III.5.2 The Universal Representation and Second Dual......Page 333
IV.1.1 First Definitions......Page 338
IV.1.2 Elementary C*-Algebras......Page 341
IV.1.3 Liminal and Postliminal C*-Algebras......Page 342
IV.1.4 Continuous Trace, Homogeneous, and Subhomogeneous C*-Algebras......Page 344
IV.1.5 Characterization of Type I C*-Algebras......Page 352
IV.1.6 Continuous Fields of C*-Algebras......Page 355
IV.1.7 Structure of Continuous Trace C*-Algebras......Page 359
IV.2 Classification of Injective Factors......Page 365
IV.2.1 Injective C*-Algebras......Page 367
IV.2.2 Injective von Neumann Algebras......Page 368
IV.2.3 Normal Cross Norms......Page 375
IV.2.4 Semidiscrete Factors......Page 377
IV.2.5 Amenable von Neumann Algebras......Page 380
IV.2.7 Invariants and the Classification of Injective Factors......Page 382
IV.3.1 Nuclear C*-Algebras......Page 383
IV.3.2 Completely Positive Liftings......Page 389
IV.3.3 Amenability for C*-Algebras......Page 393
IV.3.4 Exactness and Subnuclearity......Page 398
IV.3.5 Group C*-Algebras and Crossed Products......Page 406
V.1 K-Theory for C*-Algebras......Page 410
V.1.1 K[sub(0)]-Theory......Page 411
V.1.2 K[sub(1)]-Theory and Exact Sequences......Page 417
V.1.3 Further Topics......Page 423
V.1.4 Bivariant Theories......Page 426
V.1.5 Axiomatic K-Theory and the Universal Coefficient Theorem......Page 428
V.2.1 Finite and Properly Infinite Unital C*-Algebras......Page 433
V.2.2 Nonunital C*-Algebras......Page 438
V.2.3 Finiteness in Simple C*-Algebras......Page 445
V.2.4 Ordered K-Theory......Page 449
V.3 Stable Rank and Real Rank......Page 459
V.3.1 Stable Rank......Page 460
V.3.2 Real Rank......Page 467
V.4.1 Quasidiagonal Sets of Operators......Page 472
V.4.2 Quasidiagonal C*-Algebras......Page 475
V.4.3 Generalized Inductive Limits......Page 479
References......Page 493
Index......Page 518
A......Page 520
C......Page 521
F......Page 523
I......Page 524
N......Page 525
P......Page 526
S......Page 527
T......Page 528
W......Page 529