Introduction to Operator Algebras

INTRODUCTION TO OPERATOR ALGEBRAS......Page 1
Half-title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Introduction......Page 6
Contents......Page 12
1.1. Banach spaces of operators on a Hilbert space......Page 18
1.2. Locally convex topologies in B(H)......Page 25
1.3. Von Neumann's double commutation theorem......Page 32
1.4. Tensor products of Von Neumann algebras......Page 37
1.5. Comparison of projections and central cover......Page 47
1.6. Kaplansky's density theorem......Page 52
1.7. Ideals in Von Neumann algebras......Page 55
1.8. Normal positive linear functionals......Page 57
1.9. Polar decomposition and orthogonal decomposition......Page 63
1.10. Radon–Nikodyn theorems......Page 67
1.11. The equivalence of the topologies s* and τ in a bounded ball......Page 72
1.12. Normal * homomorphisms......Page 77
1.13. Comparison of cyclic projections and spatial * isomorphic theorem......Page 80
1.14. σ-Finite Von Neumann algebras......Page 83
2.1. Definition and basic properties of C*-algebras......Page 88
2.2. Positive cones of C*-algebras......Page 92
2.3. States and the Gelfand–Naimark–Segal construction......Page 96
2.4. Approximate identities and quotient C*-algebras......Page 106
2.5. Extreme points of the unit ball and the existence of an identity......Page 111
2.6. Transitivity theorem and irreducible * representations......Page 115
2.7. Pure states and regular maximal left ideals......Page 120
2.8. Ideals and quotient C*-algebras......Page 124
2.9. Hereditary C*-subalgebras......Page 129
2.10. Comparison, disjunction and quasi-equivalence of * representations......Page 133
2.11. The enveloping Von Neumann algebra......Page 136
2.12. The multiplier algebra......Page 140
2.13. Finite dimensional C*-algebras......Page 146
2.14. The axioms for C*-algebras......Page 147
2.15. Real C*-algebras......Page 165
3.1. Tensor products of Banach spaces and cross-norms......Page 180
3.2. Tensor products of C*-algebras and the spatial C*-norm......Page 182
3.3. The maximal C*-norm......Page 188
3.4. States on algebraic tensor product......Page 192
3.5. The inequality λ(·) ≤ α 0 (·) ≤ α(·) ≤ γ(·)......Page 196
3.6. Completely positive maps......Page 204
3.7. The inductive limit of C*-algebras......Page 213
3.8. Infinite tensor products of C*-algebras......Page 217
3.9 Nuclear C*-algebras......Page 222
4.1. Projections of norm one......Page 232
4.2. W*-algebras and their * representations......Page 236
4.3. Tensor products of W*-algebras......Page 240
4.4. Completely additive functionals and singular functionals......Page 243
4.5. The characterizations of weakly compact subsets in predual......Page 250
5.1. Measure theory on locally compact Hausdorff spaces......Page 254
5.2. Stonean spaces......Page 259
5.3. Abelian W*-algebras......Page 265
5.4. * Representations of abelian C*-algebras......Page 274
6.1. The classification of Von Neumann algebras......Page 284
6.2. An ergodic type theorem for Von Neumann algebras......Page 286
6.3. Finite Von Neumann algebras......Page 292
6.4. Properly infinite Von Neumann algebras......Page 302
6.5. Semi-finite Von Neumann algebras......Page 305
6.6. Purely infinite Von Neumann algebras......Page 315
6.7. Discrete (type (I)) Von Neumann algebras......Page 319
6.8. Continuous Von Neumann algebras and type (II) Von Neumann algebras......Page 323
6.9. The types of tensor products of Von Neumann algebras......Page 325
7.1. Dimension functions......Page 330
7.2. Hyperfinite type (II 1) factors......Page 334
7.3. Construction of factors of type (II) and type (III)......Page 344
7.4 The existences of non-hyperfinite type (II 1) factors and non-nuclear C*-algebras......Page 355
8.1. The KMS condition......Page 364
8.2. Tomita–Takesaki theory......Page 371
8.3. The modular automorphism group of a σ-finite W*-algebra......Page 377
9.1. Preliminaries......Page 386
9.2. The Arveson spectrum......Page 391
9.3. The Connes spectrum......Page 402
9.4. The Connes classification of type (III) factors (σ-finite case)......Page 406
9.5. Examples of type (III λ) factors......Page 410
10.1. Polish spaces......Page 422
10.2. Borel subsets and Sousline subsets......Page 427
10.3. Borel maps and standard Borel spaces......Page 431
10.4. Borel cross sections......Page 437
11.1. The standard Borel structure of W(X*)......Page 443
11.2. Sequences of Borel choice functions......Page 447
11.3. The Borel spaces of Von Neumann algebras......Page 452
11.4. Borel subsets of factorial Borel space......Page 455
12.1. Measurable fields of Hilbert spaces......Page 465
12.2. Measurable fields of operators......Page 472
12.3. Measurable fields of Von Neumann algebras......Page 476
12.4. Decomposition of a Hilbert space into a direct integral......Page 481
12.5. The relations between a decomposable Von Neumann algebra and its components......Page 487
12.6. The constant fields of operators and Von Neumann algebras......Page 491
12.7. Borel subsets of the Borel space of Von Neumann algebras......Page 494
12.8. Borel subsets of the state space of a separable C*-algebra......Page 504
13.1. The spectrum of a C*-algebra......Page 507
13.2. Elementary C*-algebras and CCR (liminary) algebras......Page 518
13.3. GCR (postliminary) algebras and NGCR (antiliminary) algebras......Page 523
13.4. The existence of type (III) factorial * representations of a NGCR algebra......Page 529
13.5. Type I C*-algebras......Page 545
13.6. Separable type I C*-algebras......Page 550
14.1. Choquet theory of boundary integrals on compact convex sets......Page 555
14.2. The C-measure and C-isomorphism of a state......Page 559
14.3. Extremal decomposition and central decomposition......Page 570
14.4 Ergodic decomposition and tracial decomposition......Page 576
15.1. The definition of (AF)-algebras......Page 587
15.2. Dimensions and isomorphic theorem......Page 596
15.3. The Bratteli diagrams of (AF)-algebras......Page 602
15.4. Ideals of (AF)-algebras......Page 607
15.5. Dimension groups......Page 611
15.6. Scaled dimension groups and stably isomorphic theorem......Page 619
15.7. The tracial state space on an (AF)-algebra......Page 623
16.1. W*-crossed products......Page 630
16.2. Takesaki's duality theorem......Page 641
16.3. Group algebras and Group C*-algebras......Page 647
16.4. C*-crossed products......Page 657
16.5. Takai's duality theorem......Page 676
16.6. Some examples of crossed products......Page 686
17.1. The coupling constant......Page 699
17.2. Index for subfactors......Page 710
17.3. The fundamental construction......Page 715
17.4. The values of index for subfactors......Page 724
Appendix. Weak Topology and Weak* Topology......Page 731
References......Page 736
Notation Index......Page 746
Subject Index......Page 750