Theory of operator spaces
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Bibliographic Information
Theory of operator spaces
（AMS chelsea publishing, 386）
American Mathematical Society, c2022
- :pbk.
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Originally published as Operator Spaces by The Clarendon Press; New York, Oxford University Press. © 2000
Reprinted with corrections by the American Mathematical Society, 2022
Includes bibliographical references and index
Description and Table of Contents
Description
This book provides the main results and ideas in the theories of completely bounded maps, operator spaces, and operator algebras, along with some of their main applications. It requires only a basic background in functional analysis to read through the book. The descriptions and discussions of the topics are self-explained. It is appropriate for graduate students new to the subject and the field.
The book starts with the basic representation theorems for abstract operator spaces and their mappings, followed by a discussion of tensor products and the analogue of Grothendieck's approximation property. Next, the operator space analogues of the nuclear, integral, and absolutely summing mappings are discussed. In what is perhaps the deepest part of the book, the authors present the remarkable ""non-classical"" phenomena that occur when one considers local reflexivity and exactness for operator spaces. This is an area of great beauty and depth, and it represents one of the triumphs of the subject. In the final part of the book, the authors consider applications to non-commutative harmonic analysis and non-self-adjoint operator algebra theory.
Operator space theory provides a synthesis of Banach space theory with the non-commuting variables of operator algebra theory, and it has led to exciting new approaches in both disciplines. This book is an indispensable introduction to the theory of operator spaces.
Table of Contents
Matrix and operator conventions
Examples and three basic theorems: The representation theorem
Constructions and examples
The extension theorem
Operator systems and decompositions
Injectivity
Tensor products: The projective tensor product
The injective tensor product
The Haagerup tensor product
Infinite matrices and asymptotic constructions
The Grothendieck programme: The approximation property
Mapping spaces
Absolutely summing mappings
Local theory and integrality: Local reflexivity, exactness, and nuclearity
Local reflexivity and exact integrality
Some algebraic applications: Non-commutative harmonic analysis
An abstract characterization for non-self-adjoint operator algebras
Preliminaries
Bibliography
Index of notation
Index
by "Nielsen BookData"