Von Neumann algebras

VON NEUMANN ALGEBRAS......Page 1
Half-title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
1. Modular theory......Page 12
2. Connes\' classification of type III factors......Page 16
3. Structure theory for type III factors......Page 19
4. Examples......Page 24
5. Classification of injective factors......Page 30
Preface to the Second Edition......Page 36
Introduction......Page 38
Part I: Global Theory......Page 40
1. Commutant and bicommutant......Page 42
2. Hermitian operators in a von Neumann algebra......Page 44
4. Cyclic projections in a von Neumann algebra......Page 45
5. Homomorphisms......Page 48
6. Ideals in von Neumann algebras......Page 50
7. Maximal abelian von Neumann subalgebras......Page 53
1. Induced and reduced von Neumann algebras......Page 58
2. Product of Iron Neumann algebras......Page 61
3. Operators in a tensor product of Hilbert spaces......Page 63
4. Tensor products of von Neumann algebras......Page 66
1. Topologies on L(H)......Page 74
2. The above topologies compared......Page 78
3. Linear forms on L(H)......Page 79
4. The von Neumann density theorem......Page 85
5. Kaplansky\'s density theorem......Page 88
1. Positive linear forms on a *-algebra of operators......Page 94
2. Normal positive linear forms on a von Neumann algebra......Page 97
3. Normal positive linear mappings......Page 100
4. Structure of normal homomorphisms......Page 102
5. Application: Isomorphisms of tensor products......Page 103
6. Support of a normal positive linear form......Page 104
7. Polar decomposition of a linear form......Page 106
8. Decomposition of a hermitian form into positive and negative parts......Page 112
1. Definition of Hilbert algebras......Page 118
2. The commutation theorem......Page 120
3. Bounded elements in Hilbert algebras......Page 122
4. Central elements in Hilbert algebras......Page 124
5. Elementary operations on Hilbert algebras......Page 126
1. Definition of traces......Page 134
2. Traces and Hilbert algebras......Page 138
3. Trace-elements......Page 142
4. An ordering in the set of traces......Page 143
5. An application: isomorphisms of standard von Neumann algebras......Page 145
6. Normal traces on L(H)......Page 146
7. A first classification of von Neumann algebras......Page 150
8. Classification and elementary operations......Page 155
9. The commutant of the tensor product of two semi-finite von Neumann algebras......Page 157
10. The space L[sup(1)] defined by a trace......Page 158
11. Trace and determinant......Page 160
1. Basic measures......Page 168
2. Existence of basic measures......Page 170
3. Structure of abelian von Neumann algebras......Page 173
1. A second classification of von Neumann algebras......Page 178
2. Abelian projections......Page 179
4. Definition of types......Page 182
5. Complete Hilbert algebras and type I factors......Page 183
1. A lemma......Page 188
2. Construction of certain von Neumann algebras......Page 189
3. Examples taken from measure theory......Page 192
4. Existence of different types of factors......Page 194
Part II: Reduction of von Neumann Algebras......Page 200
1. Borel spaces, measures......Page 202
2. Fields of vectors......Page 203
3. Measurable fields of Hilbert spaces......Page 205
4. Basic properties of measurable fields of Hilbert spaces......Page 207
5. Square-integrable vector fields......Page 209
6. Basic properties of direct integrals......Page 211
7. Measurable fields of subspaces......Page 214
8. Measurable fields of tensor products......Page 215
1. Measurable fields of linear mappings......Page 220
2. Examples......Page 221
3. Decomposable linear mappings......Page 222
4. Diagonalisable operators......Page 226
5. Characterisation of decomposable mappings......Page 228
6. Constant fields of operators......Page 229
1. A preliminary theorem......Page 236
2. Measurable fields of von Neumann algebras......Page 237
3. Relations between a decomposable von Neumann algebra and its components......Page 239
4. Constant fields of von Neumann algebras......Page 242
5. Reduction of discrete or continuous von Neumann algebras......Page 245
6. Measurable  fields of homomorphisms......Page 248
1. Measurable fields of Hilbert algebras......Page 252
2. Decomposable Hilbert algebras......Page 253
3. Involution and von Neumann algebras associated with U......Page 254
4. Elements bounded relative to U......Page 256
5. Central elements relative to U......Page 258
6. Uniqueness and existence of the decomposition......Page 259
1. Measurable fields of traces......Page 264
2. Decomposition of traces......Page 266
3. Uniqueness of the decomposition......Page 269
4. Reduction of properly infinite, purely infinite, finite and semi-finite von Neumann algebras......Page 271
2. Existence theorems......Page 274
3. Uniqueness theorems......Page 278
Part III: Further Topics......Page 282
1. Comparison of projections......Page 284
2. A theorem on comparability......Page 286
3. Cyclic projections of A and cyclic projections of A\'......Page 289
4. Applications: I. Properties of cyclic and separating elements......Page 290
5. Applications: II. Characterisation of standard von Neumann algebras......Page 293
1. Definitions......Page 300
3. Finite projections......Page 301
4. Semi-finite projections......Page 303
5. Properly infinite projections......Page 304
6. Purely infinite projections......Page 305
7. Comparison of projections and dimension......Page 306
1. Structure of discrete von Neumann algebras......Page 310
2. Isomorphisms of discrete von Neumann algebras......Page 311
1. Definition......Page 316
2. Traces on Ẑ[sup(+)]......Page 317
3. Relations between scalar traces and operator traces......Page 318
4. Existence and uniqueness theorems for operator traces......Page 321
1. The approximation theorem......Page 326
2. An application: two-sided ideals of A and ideals of Z......Page 329
3. Characters of finite von Neumann algebras......Page 332
1. The coupling operator......Page 338
2. Properties of the coupling operator......Page 340
3. Applications: I. Comparing the strong and ultra-strong, weak and ultra-weak topologies......Page 342
4. Applications: II. Conditions for an isomorphism to be spatial......Page 345
1. Factors contained in a finite von Neumann algebra......Page 348
2. Existence and uniqueness of continuous hyperfinite factors......Page 349
3. Some inequalities......Page 351
4. A new definition of hyperfinite factors......Page 354
5. Hyperfinite factors and elementary operations......Page 359
6. Further examples of finite factors......Page 360
7. Existence of finite, non-hyperfinite, factors......Page 362
1. Statement of the theorem......Page 370
2. Fundamental projections......Page 371
3. Weights on the set of fundamental projections......Page 372
4. Construction of a trace to within ε......Page 375
5. The proof of the theorem concluded......Page 378
6. Consequences of the theorem......Page 379
7. More on tensor products......Page 384
1. Derivations of algebras......Page 390
2. Derivations of C*-algebras: continuity, extension......Page 391
3. Derivations of von Neumann algebras......Page 393
4. Automorphisms of von Neumann algebras......Page 396
1. Spectrum......Page 400
3. Extension of the Gelfand isomorphism......Page 401
Appendix 2......Page 404
2. Partial isometries......Page 406
3. Polar decomposition of an operator......Page 407
Appendix 4......Page 408
2. Polish spaces......Page 412
3. Souslin sets......Page 413
4. Measurability of Souslin sets......Page 414
5. Existence of measurable mappings......Page 415
Topology......Page 418
Topological vector spaces......Page 419
Index of Notation......Page 422
Index of Terminology......Page 424
Journal Articles......Page 428
General Texts......Page 475
Books, Monographs and Conference Proceedings on Operator Algebra......Page 476