Fundamentals of the theory of operator algebras. Volume 2, Advanced theory

This work and Fundamentals of the Theory of Operator Algebras. Volume I, Elementary Theory present an introduction to functional analysis and the initial fundamentals of $C^\*$- and von Neumann algebra theory in a form suitable for both intermediate graduate courses and self-study. The authors provide a clear account of the introductory portions of this important and technically difficult subject. Major concepts are sometimes presented from several points of view; the account is leisurely when brevity would compromise clarity. An unusual feature in a text at this level is the extent to which it is self-contained; for example, it introduces all the elementary functional analysis needed. The emphasis is on teaching. Well supplied with exercises, the text assumes only basic measure theory and topology. The book presents the possibility for the design of numerous courses aimed at different audiences. Contents ......Page 7 Preface ......Page 11 6.1 Polar decomposition and equivalence ......Page 17 6.2 Ordering ......Page 23 6.3 Finite and infinite projections ......Page 29 6.4 Abelian projections ......Page 37 6.5 Type decomposition ......Page 40 6.6 Type I algebras ......Page 44 6.7 Examples ......Page 51 6.8 Ideals ......Page 59 6.9 Exercises ......Page 63 7.1 Completely additive states ......Page 72 7.2 Vector states and unitary implementation ......Page 83 7.3. A second approach to normal states ......Page 91 7.4. The predual ......Page 99 7.5. Normal weights on von Neumann algebras ......Page 103 7.6 Exercises ......Page 108 8.1 Traces ......Page 122 8.2 The trace in finite algebras ......Page 128 8.3. The Dixmier approximation theorem ......Page 138 8.4 The dimension function ......Page 148 8.5. Tracial weights on factors ......Page 155 8.6 Further examples of factors ......Page 164 8.7 Exercises ......Page 183 CHAPTER 9: ALGEBRA AND COMMUTANT ......Page 202 9.1 The type of the commutant ......Page 203 9.2. Modular theory ......Page 209 9.3 Unitary equivalence of type I algebras ......Page 278 9.4 Abelian von Neumann algebras ......Page 283 9.5 Spectral multiplicity ......Page 288 9.6 Exercises ......Page 307 10.1 The universal representation ......Page 329 10.2 Irreducible representations ......Page 345 10.3 Disjoint representations ......Page 352 10.4 Examples ......Page 362 10.5 Exercises ......Page 384 11.1 Tensor products of represented CY-algebras ......Page 418 11.2 Tensor products of von Neumann algebras ......Page 430 11.3 Tensor products of abstract C*-algebras ......Page 464 11.4 Infinite tensor products of C*-algebras ......Page 481 11.5 Exercises ......Page 495 CHAPTER 12: APPROXIMATION BY MATRIX ALGEBRAS ......Page 507 12.1 Isomorphism of uniformly matricial algebras ......Page 508 12.2 The finite matricial factor ......Page 513 12.3 States and representations of matricial C*-algebras ......Page 522 12.4 Exercises ......Page 538 CHAPTER 13: CROSSED PRODUCTS ......Page 554 13.1 Discrete crossed products ......Page 555 13.2 Continuous crossed products ......Page 575 13.3 Crossed products by modular automorphism groups ......Page 592 13.4 Exercises ......Page 607 CHAPTER 14: DIRECT INTEGRALS AND DECOMPOSITIONS ......Page 616 14.1 Direct integrals ......Page 617 14.2 Decompositions relative to abelian algebras ......Page 643 14.4 Exercises ......Page 659 Bibliography ......Page 667 Index of Notation ......Page 673 Index ......Page 679