Quantum Functional Analysis Non coordinate Approach 1st Edition by Helemskii ISBN 082185254X 978-0821852545
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Product details:
ISBN 10: 082185254X
ISBN 13: 978-0821852545
Author: A. Y. Helemskii
This book contains a systematic presentation of quantum functional analysis, a mathematical subject also known as operator space theory. Created in the 1980s, it nowadays is one of the most prominent areas of functional analysis, both as a field of active research and as a source of numerous important applications. The approach taken in this book differs significantly from the standard approach used in studying operator space theory. Instead of viewing “quantized coefficients” as matrices in a fixed basis, in this book they are interpreted as finite rank operators in a fixed Hilbert space. This allows the author to replace matrix computations with algebraic techniques of module theory and tensor products, thus achieving a more invariant approach to the subject. The book can be used by graduate students and research mathematicians interested in functional analysis and related areas of mathematics and mathematical physics. Prerequisites include standard courses in abstract algebra and functional analysis.
Table of contents:
Part I. The beginning: Spaces and operators
Chapter 1. Preparing the stage
1.1. Operators on normed spaces
1.2. Operators on Hilbert spaces
1.3. The diamond multiplication
Bimodules
1.5. Amplifications of linear spaces
1.6. Amplifications of linear and bilinear operators
1.7. Spatial tensor products of operator spaces
1.8. Involutive algebras and C*-algebras
1.9. A technical lemma
Chapter 2. Abstract operator (= quantum) spaces
2.1. Semi-normed bimodules
2.2. Protoquantum and abstract operator (= quantum) spaces. General properties
2.3. First examples. Concrete quantizations
Chapter 3. Completely bounded operators
3.1. Principal definitions and counterexamples
3.2. Conditions of automatic complete boundedness, and applications
3.3. The repeated quantization
3.4. The complete boundedness and spatial tensor products
Chapter 4. The completion of abstract operator spaces
Part II. Bilinear operators, tensor products and duality
Chapter 5. Strongly and weakly completely bounded bilinear operators
5.1. General definitions and properties
5.2. Examples and counterexamples
Chapter 6. New preparations: Classical tensor products
6.1. Tensor products of normed spaces
6.2. Tensor products of normed modules
Chapter 7. Quantum tensor products
7.0. The general universal property
7.1. The Haagerup tensor product
7.2. The operator-projective tensor product
7.3. The operator-injective tensor product
7.4. Column and row Hilbertian spaces as tensor factors
7.5. Functorial properties of quantum tensor products
7.6. Algebraic properties of quantum tensor multiplications
Chapter 8. Quantum duality
8.1. Quantization of spaces in duality
8.2. Quantum dual and quantum predual space
8.3. Examples
8.4. The self-dual Hilbertian space of Pisier
8.5. Duality and quantum tensor products
8.6. Quantization of spaces, set in vector duality
8.7. Quantization of the space of completely bounded operators
8.8. Quantum adjoint associativity
Part III. Principal theorems, revisited in earnest
Chapter 9. Extreme flatness and the Extension Theorem
9.0. New preparations: More about module tensor products
9.1. One-sided Ruan modules
9.2. Extreme flatness and extreme injectivity
9.3. Extreme flatness of certain modules
9.4. The Arveson-Wittstock Theorem
Chapter 10. Representation Theorem and its gifts
10.1. The Ruan Theorem
10.2. The fulfillment of earlier promises
Chapter 11. Decomposition Theorem
11.1. Complete positivity and the Stinespring Theorem
11.2. Complete positivity and complete boundedness: An interplay
11.3. Paulsen trick and the Decomposition Theorem
Chapter 12. Returning to the Haagerup tensor product
12.1. Alternative approach to the Haagerup tensor product
12.2. Decomposition of multilinear operators
12.3. Self-duality of the Haagerup tensor product
Chapter 13. Miscellany: More examples, facts and applications
13.1. CAR operator space
13.2. Further examples
13.3. Schur and Herz Schur multipliers
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