Operator algebras. Theory of C-star-algebras and von Neumann algebras (bad p480)

This book offers a comprehensive introduction to the general theory of C\*-algebras and von Neumann algebras. Beginning with the basics, the theory is developed through such topics as tensor products, nuclearity and exactness, crossed products, K-theory, and quasidiagonality. The presentation carefully and precisely explains the main features of each part of the theory of operator algebras; most important arguments are at least outlined and many are presented in full detail. Contents......Page 12 I.1.1 Inner Products......Page 18 I.1.2 Orthogonality......Page 19 I.1.3 Dual Spaces and Weak Topology......Page 20 I.1.4 Standard Constructions......Page 21 I.2.1 Bounded Operators on Normed Spaces......Page 22 I.2.2 Sesquilinear Forms......Page 23 I.2.3 Adjoint......Page 24 I.2.4 Self-Adjoint, Unitary, and Normal Operators......Page 25 I.2.5 Amplifications and Commutants......Page 26 I.2.6 Invertibility and Spectrum......Page 27 I.3.1 Strong and Weak Topologies......Page 30 I.3.2 Properties of the Topologies......Page 31 I.4 Functional Calculus......Page 34 I.4.1 Functional Calculus for Continuous Functions......Page 35 I.5 Projections......Page 36 I.5.1 Definitions and Basic Properties......Page 37 I.5.2 Support Projections and Polar Decomposition......Page 38 I.6.1 Spectral Theorem for Bounded Self-Adjoint Operators......Page 40 I.6.2 Spectral Theorem for Normal Operators......Page 42 I.7.1 Densely Defined Operators......Page 44 I.7.2 Closed Operators and Adjoints......Page 46 I.7.3 Self-Adjoint Operators......Page 47 I.7.4 The Spectral Theorem and Functional Calculus for Unbounded Self-Adjoint Operators......Page 49 I.8.1 Definitions and Basic Properties......Page 53 I.8.3 Fredholm Theory......Page 54 I.8.4 Spectral Properties of Compact Operators......Page 57 I.8.5 Trace-Class and Hilbert-Schmidt Operators......Page 58 I.8.6 Duals and Preduals, σ-Topologies......Page 60 I.8.7 Ideals of L(H)......Page 61 I.9.1 Commutant and Bicommutant......Page 64 I.9.2 Other Properties......Page 65 II.1.1 Basic Definitions......Page 67 II.1.2 Unitization......Page 69 II.1.4 Spectrum......Page 70 II.1.5 Holomorphic Functional Calculus......Page 71 II.1.6 Norm and Spectrum......Page 73 II.2.1 Spectrum of a Commutative Banach Algebra......Page 75 II.2.2 Gelfand Transform......Page 76 II.2.3 Continuous Functional Calculus......Page 77 II.3.1 Positive Elements......Page 79 II.3.2 Polar Decomposition......Page 83 II.3.3 Comparison Theory for Projections......Page 88 II.3.4 Hereditary C*-Subalgebras and General Comparison Theory......Page 91 II.4.1 General Approximate Units......Page 95 II.4.2 Strictly Positive Elements and σ-Unital C*-Algebras......Page 97 II.5 Ideals, Quotients, and Homomorphisms......Page 98 II.5.1 Closed Ideals......Page 99 II.5.2 Nonclosed Ideals......Page 101 II.5.3 Left Ideals and Hereditary Subalgebras......Page 105 II.5.4 Prime and Simple C*-Algebras......Page 109 II.5.5 Homomorphisms and Automorphisms......Page 111 II.6 States and Representations......Page 116 II.6.1 Representations......Page 117 II.6.2 Positive Linear Functionals and States......Page 119 II.6.3 Extension and Existence of States......Page 122 II.6.4 The GNS Construction......Page 123 II.6.5 Primitive Ideal Space and Spectrum......Page 127 II.6.6 Matrix Algebras and Stable Algebras......Page 132 II.6.7 Weights......Page 134 II.6.8 Traces and Dimension Functions......Page 137 II.6.9 Completely Positive Maps......Page 140 II.6.10 Conditional Expectations......Page 148 II.7.1 Hilbert Modules......Page 153 II.7.2 Operators......Page 157 II.7.3 Multiplier Algebras......Page 160 II.7.4 Tensor Products of Hilbert Modules......Page 163 II.7.5 The Generalized Stinespring Theorem......Page 165 II.7.6 Morita Equivalence......Page 166 II.8.1 Direct Sums, Products, and Ultraproducts......Page 170 II.8.2 Inductive Limits......Page 172 II.8.3 Universal C*-Algebras and Free Products......Page 174 II.8.4 Extensions and Pullbacks......Page 183 II.8.5 C*-Algebras with Prescribed Properties......Page 192 II.9 Tensor Products and Nuclearity......Page 195 II.9.2 The Maximal Tensor Product......Page 196 II.9.3 States on Tensor Products......Page 198 II.9.4 Nuclear C*-Algebras......Page 200 II.9.5 Minimality of the Spatial Norm......Page 202 II.9.6 Homomorphisms and Ideals......Page 203 II.9.7 Tensor Products of Completely Positive Maps......Page 206 II.9.8 Infinite Tensor Products......Page 207 II.10 Group C*-Algebras and Crossed Products......Page 208 II.10.1 Locally Compact Groups......Page 209 II.10.2 Group C*-Algebras......Page 213 II.10.3 Crossed products......Page 215 II.10.4 Transformation Group C*-Algebras......Page 221 II.10.5 Takai Duality......Page 227 II.10.7 Generalizations of Crossed Product Algebras......Page 228 II.10.8 Duality and Quantum Groups......Page 230 III Von Neumann Algebras......Page 236 III.1.1 Projections and Equivalence......Page 237 III.1.2 Cyclic and Countably Decomposable Projections......Page 240 III.1.3 Finite, Infinite, and Abelian Projections......Page 242 III.1.4 Type Classification......Page 246 III.1.5 Tensor Products and Type I von Neumann Algebras......Page 247 III.1.6 Direct Integral Decompositions......Page 252 III.1.7 Dimension Functions and Comparison Theory......Page 255 III.1.8 Algebraic Versions......Page 258 III.2 Normal Linear Functionals and Spatial Theory......Page 259 III.2.1 Normal and Completely Additive States......Page 260 III.2.2 Normal Maps and Isomorphisms of von Neumann Algebras......Page 263 III.2.3 Polar Decomposition for Normal Linear Functionals and the Radon-Nikodym Theorem......Page 272 III.2.4 Uniqueness of the Predual and Characterizations of W*-Algebras......Page 274 III.2.5 Traces on von Neumann Algebras......Page 275 III.2.6 Spatial Isomorphisms and Standard Forms......Page 284 III.3.1 Infinite Tensor Products......Page 290 III.3.2 Crossed Products and the Group Measure Space Construction......Page 295 III.3.3 Regular Representations of Discrete Groups......Page 303 III.3.4 Uniqueness of the Hyperfinite II[sub(1)] Factor......Page 306 III.4.1 Notation and Basic Constructions......Page 308 III.4.3 The Main Theorem......Page 310 III.4.4 Left Hilbert Algebras......Page 311 III.4.5 Corollaries of the Main Theorems......Page 314 III.4.6 The Canonical Group of Outer Automorphisms and Connes' Invariants......Page 317 III.4.7 The KMS Condition and the Radon-Nikodym Theorem for Weights......Page 321 III.4.8 The Continuous and Discrete Decompositions of a von Neumann Algebra......Page 325 III.5.1 Decomposition Theory for Representations......Page 328 III.5.2 The Universal Representation and Second Dual......Page 333 IV.1.1 First Definitions......Page 338 IV.1.2 Elementary C*-Algebras......Page 341 IV.1.3 Liminal and Postliminal C*-Algebras......Page 342 IV.1.4 Continuous Trace, Homogeneous, and Subhomogeneous C*-Algebras......Page 344 IV.1.5 Characterization of Type I C*-Algebras......Page 352 IV.1.6 Continuous Fields of C*-Algebras......Page 355 IV.1.7 Structure of Continuous Trace C*-Algebras......Page 359 IV.2 Classification of Injective Factors......Page 365 IV.2.1 Injective C*-Algebras......Page 367 IV.2.2 Injective von Neumann Algebras......Page 368 IV.2.3 Normal Cross Norms......Page 375 IV.2.4 Semidiscrete Factors......Page 377 IV.2.5 Amenable von Neumann Algebras......Page 380 IV.2.7 Invariants and the Classification of Injective Factors......Page 382 IV.3.1 Nuclear C*-Algebras......Page 383 IV.3.2 Completely Positive Liftings......Page 389 IV.3.3 Amenability for C*-Algebras......Page 393 IV.3.4 Exactness and Subnuclearity......Page 398 IV.3.5 Group C*-Algebras and Crossed Products......Page 406 V.1 K-Theory for C*-Algebras......Page 410 V.1.1 K[sub(0)]-Theory......Page 411 V.1.2 K[sub(1)]-Theory and Exact Sequences......Page 417 V.1.3 Further Topics......Page 423 V.1.4 Bivariant Theories......Page 426 V.1.5 Axiomatic K-Theory and the Universal Coefficient Theorem......Page 428 V.2.1 Finite and Properly Infinite Unital C*-Algebras......Page 433 V.2.2 Nonunital C*-Algebras......Page 438 V.2.3 Finiteness in Simple C*-Algebras......Page 445 V.2.4 Ordered K-Theory......Page 449 V.3 Stable Rank and Real Rank......Page 459 V.3.1 Stable Rank......Page 460 V.3.2 Real Rank......Page 467 V.4.1 Quasidiagonal Sets of Operators......Page 472 V.4.2 Quasidiagonal C*-Algebras......Page 475 V.4.3 Generalized Inductive Limits......Page 479 References......Page 493 Index......Page 518 A......Page 520 C......Page 521 F......Page 523 I......Page 524 N......Page 525 P......Page 526 S......Page 527 T......Page 528 W......Page 529 This volume attempts to give a comprehensive discussion of the theory of operator algebras (C•-algebras and von Neumann algebras.) The volume is intended to serve two purposes: to record the standard theory in the Encyc- pedia of Mathematics, and to serve as an introduction and standard reference for the specialized volumes in the series on current research topics in the subject. Since there are already numerous excellent treatises on various aspects of thesubject,howdoesthisvolumemakeasigni?cantadditiontotheliterature, and how does it di?er from the other books in the subject? In short, why another book on operator algebras? The answer lies partly in the?rst paragraph above. More importantly, no other single reference covers all or even almost all of the material in this volume. I have tried to cover all of the main aspects of “standard” or “clas- cal” operator algebra theory; the goal has been to be, well, encyclopedic. Of course, in a subject as vast as this one, authors must make highly subjective judgments as to what to include and what to omit, as well as what level of detail to include, and I have been guided as much by my own interests and prejudices as by the needs of the authors of the more specialized volumes. This Book Is The Most Comprehensive Treatment Available Of The General Theory Of C*-algebras And Von Neumann Algebras. Beginning With The Basics, The Theory Is Developed Through Such Topics As Tensor Products, Nuclearity And Exactness, Crossed Products, Classification Of Injective Factors, K-theory, Finiteness, Stable Rank, And Quasidiagonality.--book Jacket. I. Operators On Hilbert Space -- Ii. C*-algebras -- Iii. Von Neumann Algebras -- Iv. Further Structure -- V. K-theory And Finiteness. B. Blackadar. Includes Bibliographical References (pages 479-503) And Index.