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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.
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
Chapter 2. Basics of Hilbert Space and Linear Operators
2.1. Inner Products on Linear Spaces
2.3. The Weak Topology
2.4. Linear Operators
Classes of Operators
2.5. The Lattice of Projections
2.6. Constructions with Hilbert Spaces
Tensor Products and the Hilbert-Schmidt Class
2.7. Unbounded Linear Operators
Chapter 3. Banach Algebras
3.2. The Spectrum
The Banach Algebra L1(R) and Fourier Analysis
3.3. The Holomorphic Function Calculus
The Holomorphic Function Calculus
3.4. The Banach Algebra C(X)
Chapter 4. Elementary C*-Algebra Theory
4.2. Order Structure
4.3. Positive Linear Functionals
4.4. Abelian Algebras
4.5. States and Representations
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
Cyclicity, Separation, and Countable Decomposability
5.6. Unbounded Operators and Abelian Von Neumann Algebras
Index of Notation
- No. of pages:
- © Academic Press 1983
- 17th March 1994
- Academic Press
- eBook ISBN:
Department of Mathematics, University of Pennsylvania,Philadelphia, Pennsylvania
School of Marhematics, University of Newcastle, Newcastle upon Tyne, England
Department of Mathematics, Columbia University, New York, New York
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