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

I Operators on Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I.1 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 The Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

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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

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III Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

of von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . 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

of W*-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 Modular Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

Theorem for Weights . . . . . . . . . . . . . . . . . . . . . . . . . 306

III.4.8 The Continuous and Discrete Decompositions

of a von Neumann Algebra . . . . . . . . . . . . . . . . . . . . 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 Type I C*-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-Theory for C*-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

Coefficient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Stable Rank and Real Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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