Fundamentals of the Theory of Operator Algebras. V2 (eBook)

Front Cover 1
Fundamentals of the Theory of Operator Algebras 2
Copyringt Page 5
Contents 6
Preface 10
Contents of Volume I 14
Chapter 6. Comparison Theory of Projection 16
6.1. Polar decomposition and equivalence 16
6.2. Ordering 22
6.3. Finite and infinite projections 28
6.4. Abelian projections 36
6.5. Type decomposition 39
6.6. Type I algebras 43
6.7. Examples 50
6.8. Ideals 58
6.9. Exercises 62
Chapter 7. Normal States and Unitary Equivalence of von Neumann Algebras 71
7.1. Completely additive states 71
7.2. Vector states and unitary implementation 82
7.3. A second approach to normal states 90
7.4. The predual 98
7.5. Normal weights on von Neumann algebras 102
7.6. Exercises 107
Chapter 8. The Trace 121
8.1. Traces 121
8.2. The trace in finite algebras 127
8.3. The Dixmier approximation theorem 137
8.4. The dimension function 147
8.5. Tracial weights on factors 154
8.6. Further examples of factors 163
8.7. Exercises 182
Chapter 9. Algebra and Commutant 201
9.1. The type of the commutant 202
9.2. Modular theory 208
9.3. Unitary equivalence of type I algebras 277
9.4. Abelian von Neumann algebras 282
9.5. Spectral multiplicity 287
9.6. Exercises 306
Chapter 10. Special Representation of C*- Algebras 328
10.1. The universal representation 328
10.2. Irreducible representations 344
10.3. Disjoint representations 351
10.4. Examples 361
10.5. Exercises 383
Chapter 11. Tensor Products 417
11.1. Tensor products of represented C*-algebras 417
11.2. Tensor products of von Neumann algebras 429
11.3. Tensor products of abstract C*-algebras 463
11.4. Infinite tensor products of C*-algebras 480
11.5. Exercises 494
Chapter 12. Approximation by Matrix Algebras 506
12.1. Isomorphism of uniformly matricial algebras 507
12.2. The finite matricial factor 512
12.3. States and representations of matricial C*-algebras 521
12.4. Exercises 537
Chapter 13. Crossed Products 553
13.1. Discrete crossed products 554
13.2. Continuous crossed products 574
13.3. Crossed products by modular automorphism groups 591
13.4. Exercises 606
Chapter 14. Direct Integrals and Decompositions 615
14.1. Direct integrals 616
14.2. Decompositions relative to abelian algebras 642
14.3. Appendix—Borel mappings and analytic sets 648
14.4. Exercises 658
Bibliography 666
Index of Notation 672
Index 678