Quantum functional analysis : non-coordinate approach /
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...
| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I. :
American Mathematical Society,
c2010
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| Series: | University lecture series (Providence, R.I.) ;
56 |
| Subjects: |
Table of Contents:
- ch. 0 Three basic definitions and three principal theorems
- ch. 1 Preparing the stage
- 1.1 Operators on normed spaces
- 1.2. Operators on Hilbert spaces
- 1.3. diamond multiplication
- 1.4. 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. technical lemma
- ch. 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
- ch. 3 Completely bounded operators
- 3.1. Principal definitions and counterexamples
- 3.2. Conditions of automatic complete boundedness, and applications
- 3.3. repeated quantization
- 3.4. complete boundedness and spatial tensor products
- ch. 4 completion of abstract operator spaces
- ch. 5 Strongly and weakly completely bounded bilinear operators
- 5.1. General definitions and properties
- 5.2. Examples and counterexamples
- ch. 6 New preparations: Classical tensor products
- 6.1. Tensor products of normed spaces
- 6.2. Tensor products of normed modules
- ch. 7 Quantum tensor products
- 7.0. general universal property
- 7.1. Haagerup tensor product
- 7.2. operator-projective tensor product
- 7.3. operator-injective tensor product
- 7.1. Column and row Hilbertian spaces as tensor factors
- 7.5. Functorial properties of quantum tensor products
- 7.6. Algebraic properties of quantum tensor multiplications
- ch. 8 Quantum duality
- 8.1. Quantization of spaces in duality
- 8.2. Quantum dual and quantum predual space
- 8.3. Examples
- 8.4. 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
- ch. 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. Arveson-Wittstock Theorem
- ch. 10 Representation Theorem and its gifts
- 10.1. Ruan Theorem
- 10.2. fulfillment of earlier promises
- ch. 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
- ch. 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
- ch. 13 Miscellany: More examples, facts and applications
- 13.1. CAR operator space
- 13.2. Further examples
- 13.3. Schur and Herz-Schur multipliers.