Blackadar Operator Algebras

Theory of C*-Algebras and von Neumann Algebras
2006
ISBN: 978-3-540-28517-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 122, 528 Seiten

Reihe: Encyclopaedia of Mathematical Sciences

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.

Autoren/Hrsg.

Blackadar, Bruce

Inhaltsverzeichnis

1;Preface to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry;7
2;Preface;11
3;Contents;15
4;I Operators on Hilbert Space;21
4.1;I.1 Hilbert Space;21
4.1.1;I.1.1 Inner Products;21
4.1.2;I.1.2 Orthogonality;22
4.1.3;I.1.3 Dual Spaces and Weak Topology;23
4.1.4;I.1.4 Standard Constructions;24
4.1.5;I.1.5 Real Hilbert Spaces;25
4.2;I.2 Bounded Operators;25
4.2.1;I.2.1 Bounded Operators on Normed Spaces;25
4.2.2;I.2.2 Sesquilinear Forms;26
4.2.3;I.2.3 Adjoint;27
4.2.4;I.2.4 Self-Adjoint, Unitary, and Normal Operators;28
4.2.5;I.2.5 Amplifications and Commutants;29
4.2.6;I.2.6 Invertibility and Spectrum;30
4.3;I.3 Other Topologies on L(H);33
4.3.1;I.3.1 Strong and Weak Topologies;33
4.3.2;I.3.2 Properties of the Topologies;34
4.4;I.4 Functional Calculus;37
4.4.1;I.4.1 Functional Calculus for Continuous Functions;38
4.4.2;I.4.2 Square Roots of Positive Operators;39
4.4.3;I.4.3 Functional Calculus for Borel Functions;39
4.5;I.5 Projections;39
4.5.1;I.5.1 Definitions and Basic Properties;40
4.5.2;I.5.2 Support Projections and Polar Decomposition;41
4.6;I.6 The Spectral Theorem;43
4.6.1;I.6.1 Spectral Theorem for Bounded Self-Adjoint Operators;43
4.6.2;I.6.2 Spectral Theorem for Normal Operators;45
4.7;I.7 Unbounded Operators;47
4.7.1;I.7.1 Densely De.ned Operators;47
4.7.2;I.7.2 Closed Operators and Adjoints;49
4.7.3;I.7.3 Self-Adjoint Operators;50
4.7.4;I.7.4 The Spectral Theorem and Functional Calculus for Unbounded Self-Adjoint Operators;52
4.8;I.8 Compact Operators;56
4.8.1;I.8.1 Definitions and Basic Properties;56
4.8.2;I.8.2 The Calkin Algebra;57
4.8.3;I.8.3 Fredholm Theory;57
4.8.4;I.8.4 Spectral Properties of Compact Operators;60
4.8.5;I.8.5 Trace-Class and Hilbert-Schmidt Operators;61
4.8.6;I.8.6 Duals and Preduals, s-Topologies;63
4.8.7;I.8.7 Ideals of L (H);64
4.9;I.9 Algebras of Operators;67
4.9.1;I.9.1 Commutant and Bicommutant;67
4.9.2;I.9.2 Other Properties;68
5;II C*-Algebras;71
5.1;II.1 Definitions and Elementary Facts;71
5.1.1;II.1.1 Basic De.nitions;71
5.1.2;II.1.2 Unitization;73
5.1.3;II.1.3 Power series, Inverses, and Holomorphic Functions;74
5.1.4;II.1.4 Spectrum;74
5.1.5;II.1.5 Holomorphic Functional Calculus;75
5.1.6;II.1.6 Norm and Spectrum;77
5.2;II.2 Commutative C*-Algebras and Continuous Functional Calculus;79
5.2.1;II.2.1 Spectrum of a Commutative Banach Algebra;79
5.2.2;II.2.2 Gelfand Transform;80
5.2.3;II.2.3 Continuous Functional Calculus;81
5.3;II.3 Positivity, Order, and Comparison Theory;83
5.3.1;II.3.1 Positive Elements;83
5.3.2;II.3.2 Polar Decomposition;87
5.3.3;II.3.3 Comparison Theory for Projections;92
5.3.4;II.3.4 Hereditary C*-Subalgebras and General Comparison Theory;95
5.4;II.4 Approximate Units;99
5.4.1;II.4.1 General Approximate Units;99
5.4.2;II.4.2 Strictly Positive Elements and s-Unital C*-Algebras;101
5.4.3;II.4.3 Quasicentral Approximate Units;102
5.5;II.5 Ideals, Quotients, and Homomorphisms;102
5.5.1;II.5.1 Closed Ideals;103
5.5.2;II.5.2 Nonclosed Ideals;105
5.5.3;II.5.3 Left Ideals and Hereditary Subalgebras;109
5.5.4;II.5.4 Prime and Simple C*-Algebras;113
5.5.5;II.5.5 Homomorphisms and Automorphisms;115
5.6;II.6 States and Representations;120
5.6.1;II.6.1 Representations;121
5.6.2;II.6.2 Positive Linear Functionals and States;123
5.6.3;II.6.3 Extension and Existence of States;126
5.6.4;II.6.4 The GNS Construction;127
5.6.5;II.6.5 Primitive Ideal Space and Spectrum;131
5.6.6;II.6.6 Matrix Algebras and Stable Algebras;136
5.6.7;II.6.7 Weights;138
5.6.8;II.6.8 Traces and Dimension Functions;141
5.6.9;II.6.9 Completely Positive Maps;144
5.6.10;II.6.10 Conditional Expectations;152
5.7;II.7 Hilbert Modules, Multiplier Algebras, and Morita Equivalence;157
5.7.1;II.7.1 Hilbert Modules;157
5.7.2;II.7.2 Operators;161
5.7.3;II.7.3 Multiplier Algebras;164
5.7.4;II.7.4 Tensor Products of Hilbert Modules;167
5.7.5;II.7.5 The Generalized Stinespring Theorem;169
5.7.6;II.7.6 Morita Equivalence;170
5.8;II.8 Examples and Constructions;174
5.8.1;II.8.1 Direct Sums, Products, and Ultraproducts;174
5.8.2;II.8.2 Inductive Limits;176
5.8.3;II.8.3 Universal C*-Algebras and Free Products;178
5.8.4;II.8.4 Extensions and Pullbacks;187
5.8.5;II.8.5 C*-Algebras with Prescribed Properties;196
5.9;II.9 Tensor Products and Nuclearity;199
5.9.1;II.9.1 Algebraic and Spatial Tensor Products;200
5.9.2;II.9.2 The Maximal Tensor Product;200
5.9.3;II.9.3 States on Tensor Products;202
5.9.4;II.9.4 Nuclear C*-Algebras;204
5.9.5;II.9.5 Minimality of the Spatial Norm;206
5.9.6;II.9.6 Homomorphisms and Ideals;207
5.9.7;II.9.7 Tensor Products of Completely Positive Maps;210
5.9.8;II.9.8 Infinite Tensor Products;211
5.10;II.10 Group C*-Algebras and Crossed Products;212
5.10.1;II.10.1 Locally Compact Groups;213
5.10.2;II.10.2 Group C*-Algebras;217
5.10.3;II.10.3 Crossed products;219
5.10.4;II.10.4 Transformation Group C*-Algebras;225
5.10.5;II.10.5 Takai Duality;231
5.10.6;II.10.6 Structure of Crossed Products;232
5.10.7;II.10.7 Generalizations of Crossed Product Algebras;232
5.10.8;II.10.8 Duality and Quantum Groups;234
6;III Von Neumann Algebras;241
6.1;III.1 Projections and Type Classi.cation;242
6.1.1;III.1.1 Projections and Equivalence;242
6.1.2;III.1.2 Cyclic and Countably Decomposable Projections;245
6.1.3;III.1.3 Finite, In.nite, and Abelian Projections;247
6.1.4;III.1.4 Type Classi.cation;251
6.1.5;III.1.5 Tensor Products and Type I von Neumann Algebras;252
6.1.6;III.1.6 Direct Integral Decompositions;257
6.1.7;III.1.7 Dimension Functions and Comparison Theory;260
6.1.8;III.1.8 Algebraic Versions;263
6.2;III.2 Normal Linear Functionals and Spatial Theory;264
6.2.1;III.2.1 Normal and Completely Additive States;265
6.2.2;III.2.2 Normal Maps and Isomorphisms of von Neumann Algebras;268
6.2.3;III.2.3 Polar Decomposition for Normal Linear Functionals;277
6.2.4;III.2.4 Uniqueness of the Predual and Characterizations of;279
6.2.5;III.2.5 Traces on von Neumann Algebras;280
6.2.6;III.2.6 Spatial Isomorphisms and Standard Forms;289
6.3;III.3 Examples and Constructions of Factors;295
6.3.1;III.3.1 Infinite Tensor Products;295
6.3.2;III.3.2 Crossed Products and the Group Measure;300
6.3.3;III.3.3 Regular Representations of Discrete Groups;308
6.3.4;III.3.4 Uniqueness of the Hyperfinite II1 Factor;311
6.4;III.4 Modular Theory;313
6.4.1;III.4.1 Notation and Basic Constructions;313
6.4.2;III.4.2 Approach using Bounded Operators;315
6.4.3;III.4.3 The Main Theorem;315
6.4.4;III.4.4 Left Hilbert Algebras;316
6.4.5;III.4.5 Corollaries of the Main Theorems;319
6.4.6;III.4.6 The Canonical Group of Outer Automorphisms and;322
6.4.7;III.4.7 The KMS Condition and the Radon-Nikodym Theorem;326
6.4.8;III.4.8 The Continuous and Discrete Decompositions;330
6.5;III.5 Applications to Representation Theory of C*-Algebras;333
6.5.1;III.5.1 Decomposition Theory for Representations;333
6.5.2;III.5.2 The Universal Representation and Second Dual;338
7;IV Further Structure;343
7.1;IV.1 Type I C*-Algebras;343
7.1.1;IV.1.1 First Definitions;343
7.1.2;IV.1.2 Elementary C*-Algebras;346
7.1.3;IV.1.3 Liminal and Postliminal C*-Algebras;347
7.1.4;IV.1.4 Continuous Trace, Homogeneous,;349
7.1.5;IV.1.5 Characterization of Type I C*-Algebras;357
7.1.6;IV.1.6 Continuous Fields of C*-Algebras;360
7.1.7;IV.1.7 Structure of Continuous Trace C*-Algebras;364
7.2;IV.2 Classification of Injective Factors;370
7.2.1;IV.2.1 Injective C*-Algebras;372
7.2.2;IV.2.2 Injective von Neumann Algebras;373
7.2.3;IV.2.3 Normal Cross Norms.;380
7.2.4;IV.2.4 Semidiscrete Factors;382
7.2.5;IV.2.5 Amenable von Neumann Algebras;385
7.2.6;IV.2.6 Approximate Finite Dimensionality;387
7.2.7;IV.2.7 Invariants and the Classification of Injective Factors;387
7.3;IV.3 Nuclear and Exact C*-Algebras;388
7.3.1;IV.3.1 Nuclear C*-Algebras;388
7.3.2;IV.3.2 Completely Positive Liftings;394
7.3.3;IV.3.3 Amenability for C*-Algebras;398
7.3.4;IV.3.4 Exactness and Subnuclearity;403
7.3.5;IV.3.5 Group C*-Algebras and Crossed Products;411
8;V K-Theory and Finiteness;415
8.1;V.1 K-Theory for C*-Algebras;415
8.1.1;V.1.1 K0-Theory;416
8.1.2;V.1.2 K1-Theory and Exact Sequences;422
8.1.3;V.1.3 Further Topics;428
8.1.4;V.1.4 Bivariant Theories;431
8.1.5;V.1.5 Axiomatic K-Theory and the Universal Coe.cient Theorem;433
8.2;V.2 Finiteness;438
8.2.1;V.2.1 Finite and Properly In.nite Unital C*-Algebras;438
8.2.2;V.2.2 Nonunital C*-Algebras;443
8.2.3;V.2.3 Finiteness in Simple C*-Algebras;450
8.2.4;V.2.4 Ordered K-Theory;454
8.3;V.3 Stable Rank and Real Rank;464
8.3.1;V.3.1 Stable Rank;465
8.3.2;V.3.2 Real Rank;472
8.4;V.4 Quasidiagonality;477
8.4.1;V.4.1 Quasidiagonal Sets of Operators;477
8.4.2;V.4.2 Quasidiagonal C*-Algebras;480
8.4.3;V.4.3 Generalized Inductive Limits;484
8.4.4;V.4.3.36 One might hope that the conclusion of V.4.3.35 could be strengthened;494
9;References;499
10;Index;525

Leseproben

Preface (P. 11)

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 Encyclopedia 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 the subject, how does this volume make a signi.cant addition to the literature, and how does it differ from the other books in the subject? In short, why another book on operator algebras?

The answer lies partly in the first 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 "classical" 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.

A treatment of such a large body of material cannot be done at the detail level of a textbook in a reasonably-sized work, and this volume would not be suitable as a text and certainly does not replace the more detailed treatments of the subject. But neither is this volume simply a survey of the subject (a .ne survey-level book is already available [Fil96].)

My philosophy has been to not only state what is true, but explain why: while many proofs are merely outlined or even omitted, I have attempted to include enough detail and explanation to at least make all results plausible and to give the reader a sense of what material and level of diffculty is involved in each result. Where an argument can be given or summarized in just a few lines, it is usually included, longer arguments are usually omitted or only outlined.

More detail has been included where results are particularly important or frequently used in the sequel, where the results or proofs are not found in standard references, and in the few cases where new arguments have been found. Nonetheless, throughout the volume the reader should expect to have to fill out compactly written arguments, or consult references giving expanded expositions.

I have concentrated on trying to give a clean and efficient exposition of the details of the theory, and have for the most part avoided general discussions of the nature of the subject, its importance, and its connections and applications in other parts of mathematics (and physics), these matters have been amply treated in the introductory article to this series. See the introduction to [Con94] for another excellent overview of the subject of operator algebras. There is very little in this volume that is truly new, mainly some simplified proofs.