Elementary Theory

Elementary Theory

Fundamentals of the Theory of Operator Algebras

1st Edition - March 17, 1994

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  • Authors: Richard V. Kadison, John R. Ringrose
  • eBook ISBN: 9781483214092

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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


    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


    Index of Notation


Product details

  • No. of pages: 416
  • Language: English
  • Copyright: © Academic Press 1994
  • Published: March 17, 1994
  • Imprint: Academic Press
  • eBook ISBN: 9781483214092

About the Authors

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 Editors

Samuel Eilenberg

Affiliations and Expertise

Columbia University

Hyman Bass

Affiliations and Expertise

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

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