0 ：Advanced Theory ( Graduate Studies in Mathematics )

Publication subTitle ：Advanced Theory

Publication series ：Graduate Studies in Mathematics

Author： Richard V. Kadison;John R. Ringrose  

Publisher： American Mathematical Society‎

Publication year： 1997

E-ISBN: 9781470420734

P-ISBN(Paperback): 9780821808207

Subject： O177 functional analysis

Keyword： Analysis

Language： ENG

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From the reviews for Volumes I and II: … these two volumes represent a magnificent achievement. They will be an essential item on every operator algebraist's bookshelves and will surely become the primary source of instruction for research students in von Neumann algebra theory. —Bulletin of the London Mathematical Society This book is extremely clear and well written and ideally suited for an introductory course on the subject or for a student who wishes to learn the fundamentals of the classical theory of operator algebras. —Zentralblatt MATH This work and Fundamentals of the Theory of Operator Algebras. Volume I, Elementary Theory (Graduate Studies in Mathematics, Volume 15) 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

Chapter

Cover

Title

Copyright

Contents

Preface

Contents of Volume I

Chapter 6. Comparison Theory of Projection

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. Examples

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

An operator-theoretic construction

Measure-theoretic examples

8.7. Exercises

Chapter 9. Algebra and Commutant

9.1. The type of the commutant

9.2 Modular theory

A first approach to modular theory

Tomitd's theorem—a second approach

A further extension of 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 Representation of C*-Algebras

10.1. The universal representation

10.2. Irreducible representations

10.3. Disjoint representations

10.4. Examples

Abelian C*-algebras

Compact operators

B(H) and the Calkin algebra

Uniformly matricial algebras

10.5. Exercises

Chapter 11. Tensor Products

11.1 Tensor products of represented C*-algebras

11.2. Tensor products of von Neumann algebras

Elementary properties

The commutation theorem

The type of tensor products

Tensor products of unbounded operators

11.3. Tensor products of abstract C*-algebras

The spatial tensor product

C*-norms on U [omitted] B

Nuclear 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 A. Exercises

Bibliography

Index of Notation

Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

Back Cover

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