Fundamentals of the Theory of Operator Algebras. V2

Preface

Contents of Volume I

Chapter 1 Linear Spaces

Chapter 2 Basics of Hilbert Space and Linear Operators

Chapter 3 Banach Algebras

Chapter 4 Elementary C*-Algebra Theory

Chapter 5 Elementary von Neumann Algebra Theory

Chapter 6: Comparison Theory of Projections

6.1 Polar decomposition and equivalence

6.2 Ordering

6.3 Finite and infinite projections

6.4 Abelian projections

6.5 Type decomposition

6.6 Type I algebras

6.7 Example

6.7.1 Lemma

6.7.2 Theorem

6.7.3 Remark

6.7.4 Proposition

6.7.5 Theorem

6.7.6 Example

6.7.7 Example

6.7.8 Theorem

6.7.9 Remark

6.7.10 Theorem

6.8 Ideals

6.9 Exercises

Chapter 7: Normal States and Unitary Equivalence of Von Neumann Algebras

7.1 Completely additive states

7.2 Vector states and unitary implementation

7.3 A second approach to normal states

7.4 The predual

7.5 Normal weights on von Neumann algebras

7.6 Exercises

Chapter 8: The Trace

8.1 Traces

8.2 The trace in finite algebras

8.3 The Dixmier approximation Theorem

8.4 The dimension function

8.5 Tracial weights on factors

8.6 Further examples of factors

8.7 Exercises

Chapter 9: Algebra and Commutant

9.1 The type of the commutant

9.2 Modular theory

9.3 Unitary equivalence of type I algebras

9.4 Abelian von Neumann algebras

9.5 Spectral multiplicity

9.6 Exercises

Chapter 10: Special Representations of C*-Algebras

10.1 The universal representation

10.2 Irreducible representations

10.3 Disjoint representations

10.4 Examples

10.5 Exercises

Chapter 11: Tensor Products

11.1 Tensor products of represented C*-algebras

11.2 Tensor products of von Neumann algebras

11.3 Tensor products of abstract C*-algebras

11.4 Infinite tensor products of C*-algebras

11.5 Exercises

Chapter 12: Approximation by Matrix Algebras

12.1 Isomorphism of uniformly matricial algebras

12.2 The finite matricial factor

12.3 States and representations of matricial C*-algebras

12.4 Exercises

Chapter 13: Crossed Products

13.1 Discrete crossed products

13.2 Continuous crossed products

13.3 Crossed products by modular automorphism groups

13.4 Exercises

Chapter 14: Direct Integrals and Decompositions

14.1 Direct integrals

14.2 Decompositions relative to abelian algebras

14.3 Appendix—Borel mappings and analytic sets

14.4 Exercises

Bibliography

Index of Notation

Algebras and related matters

Direct sums and integrals

Equivalences and orderings

Inner products and norms

Linear operators

Linear spaces

Linear topological spaces, Banach spaces, Hilbert spaces

Modular theory

Multiplicity theory

Sets and mappings

Special Banach spaces

States and weights

Tensor products and crossed products

Index