Elementary Theory - 1st Edition - ISBN: 9780123933041, 9781483214092

Elementary Theory

1st Edition

Fundamentals of the Theory of Operator Algebras

Authors: Richard V. Kadison John R. Ringrose
Editors: Samuel Eilenberg Hyman Bass
eBook ISBN: 9781483214092
Imprint: Academic Press
Published Date: 17th March 1994
Page Count: 416
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Description

Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory provides information pertinent to the fundamental aspects of the theory of operator algebras. This book discusses the finite-dimensional linear algebra. Organized into five chapters, this volume begins with an overview of the fundamental aspects of linear functional analysis that are needed in the study of operator algebras. This text then discusses the continuous linear operators, continuous linear functionals, weak topologies, and convexity in the context of linear topological spaces. Other chapters consider the elementary geometry of Hilbertspaces and the simplest properties of Hilbert space operators. This book discusses as well algebras that have a Banach-space structure relative to which the multiplication is continuous. The final chapter deals with those C*-algebras that are strong-operator closed in their action on some Hilbert space, which play a fundamental role in the subject.

This book is a valuable resource for mathematicians.

Table of Contents


Preface

Contents of Volume II

Chapter 1. Linear Spaces

1.1. Algebraic Results

1.2. Linear Topological Spaces

1.3. Weak Topologies

1.4. Extreme Points

1.5. Normed Spaces

1.6. Linear Functionals on Normed Spaces

1.7. Some Examples of Banach Spaces

1.8. Linear Operators Acting on Banach Spaces

1.9. Exercises

Chapter 2. Basics of Hilbert Space and Linear Operators

2.1. Inner Products on Linear Spaces

2.2. Orthogonality

2.3. The Weak Topology

2.4. Linear Operators

General Theory

Classes of Operators

2.5. The Lattice of Projections

2.6. Constructions with Hilbert Spaces

Subspaces

Direct Sums

Tensor Products and the Hilbert-Schmidt Class

Matrix Representations

2.7. Unbounded Linear Operators

2.8. Exercises

Chapter 3. Banach Algebras

3.1. Basics

3.2. The Spectrum

The Banach Algebra L1(R) and Fourier Analysis

3.3. The Holomorphic Function Calculus

Holomorphic Functions

The Holomorphic Function Calculus

3.4. The Banach Algebra C(X)

3.5. Exercises

Chapter 4. Elementary C*-Algebra Theory

4.1. Basics

4.2. Order Structure

4.3. Positive Linear Functionals

4.4. Abelian Algebras

4.5. States and Representations

4.6. Exercises

Chapter 5. Elementary von Neumann Algebra Theory

5.1. The Weak- and Strong-Operator Topologies

5.2. Spectral Theory for Bounded Operators

5.3. Two Fundamental Approximation Theorems

5.4. Irreducible Algebras—An Application

5.5. Projection Techniques and Constructs

Central Carriers

Some Constructions

Cyclicity, Separation, and Countable Decomposability

5.6. Unbounded Operators and Abelian Von Neumann Algebras

5.7. Exercises

Bibliography

Index of Notation

Index

Details

No. of pages:
416
Language:
English
Copyright:
© Academic Press 1983
Published:
Imprint:
Academic Press
eBook ISBN:
9781483214092

About the Author

Richard V. Kadison

Affiliations and Expertise

Department of Mathematics, University of Pennsylvania,Philadelphia, Pennsylvania

John R. Ringrose

Affiliations and Expertise

School of Marhematics, University of Newcastle, Newcastle upon Tyne, England

About the Editor

Samuel Eilenberg

Affiliations and Expertise

Columbia University

Hyman Bass

Affiliations and Expertise

Department of Mathematics, Columbia University, New York, New York