Hilbert C*-Modules and Quantum Markov Semigroups

✍ Scribed by Lunchuan Zhang

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 222
• Category: Library

✦ Synopsis

This book explains the basic theory of Hilbert C-module in detail, covering a wide range of applications from generalized index to module framework. At the center of the book, the Beurling-Deny criterion is characterized between operator valued Dirichlet forms and quantum Markov semigroups, hence opening a new field of quantum probability research. The general scope of the book includes: basic theory of Hilbert C-modules; generalized indices and module frames; operator valued Dirichlet forms; and quantum Markov semigroups.
This book will be of value to scholars and graduate students in the fields of operator algebra, quantum probability and quantum information.

✦ Table of Contents

Preface
Contents
1 Basic Theory of Hilbert C-Modules
1.1 Hilbert C-Modules and Bounded Module Mappings
1.1.1 Hilbert C-Modules
1.1.2 Bounded Module Mappings
1.1.3 Multiplier Theorem
1.2 Polar Decomposition and Wold Decomposition
1.2.1 Unitary Equivalence Between Hilbert C-Modules
1.2.2 Polar Decomposition and Wold Decomposition
1.2.3 Topics on Module Operator Equations
1.3 Tensor Products of Hilbert C-Modules
1.3.1 Exterior Tensor Product of Hilbert C-Modules
1.3.2 Interior Tensor Product of Hilbert C-Modules
1.4 The KSGNS Construction
1.4.1 GNS Construction and Stinespring Representation Theorem
1.4.2 KSGNS Representation Theorem
References
2 Kasprove's Stabilization and Fredholm Generalized Index Theory
2.1 Kasprove's Stabilization Theorem
2.1.1 σ-unital C-Algebras
2.1.2 Kasprove's Stabilization Theorem
2.2 Morita Equivalence and C-Correspondences
2.2.1 Morita Equivalence Theorem
2.2.2 C-Correspondences and Cuntz-Pimsner Algebras
2.3 Generalized Index Theory of Fredholm Module Operators
2.3.1 A Brief Introduction to Fredholm Operators Theory on Hilbert Spaces
2.3.2 A Brief Approach to K0-Group of C-Algebras
2.3.3 Fredholm Modules Mappings and Their Generalized Index
2.4 Introduction to Modules Frames Theory
2.4.1 Existence of Module Frame and Reconstruction Formula
2.4.2 Module Frames and Unitary Equivalence Between Hilbert C-Modules
2.4.3 Unitary Equivalence of Closed Submodules and Stable Isomorphism Between HereditaryC- Subalgebras
References
3 Quantum Markov Semigroups Based on Hilbert C-Modules
3.1 Module Operator Semigroups
3.1.1 Background Knowledge
3.1.2 Module Operator Semigroups and Related Concepts
3.1.3 Resolvents and Laplace Transforms
3.1.4 Hille-Yosida Type Theorem
3.2 Abstract Cauchy Problem Based on Hilbert A-Modules
3.2.1 The Relationship Between Classical Solutions of Cauchy Problem and Strongly Continuous Module Operator Semigroups
3.2.2 The Relationship Between Mild Solutions of Cauchy Problem and Strongly Continuous Module Operator Semigroups
3.2.3 Characterizations of Strongly Continuous Module Operator Groups
3.3 Quantum Stone Theorem and Its Application
3.3.1 Quantum Stone Theorem
3.3.2 Spectral Decomposition of a Class of Stationary Quantum Processes
3.4 A Class of Markov Module Operator Semigroups and Operator-Valued Dirichlet Forms
3.4.1 Characterization of Operator-Valued Quadratic Forms
3.4.2 A Class of Markov Module Operator Semigroups
3.4.3 Operator-Valued Dirichlet Forms
3.5 Hypercontractivity and Logarithmic Sobolev Inequality in Probability Gage Space
3.5.1 Hypercontractivity and Logarithmic Sobolev Inequality
3.5.2 The Spectral Gap and Weak Spectral Gap Properties
References
A Fundamentals of C-Algebras and von Neumann Algebras
A.1 Basic Concepts of C-Algebras
A.2 Gelfand Representation of Commutative C-Algebras and Functional Calculus of Normal Elements of C-Algebras
A.2.1 Gelfand Representation of Commutative C-Algebras
A.2.2 Function Calculus of Normal Elements of C-Algebras
A.3 Positive Cone and Ordered Structure in C*-Algebra
A.4 Positive Linear Functionals and GNS Construction
A.5 Basic Knowledge of Completely Positive Mappings
A.6 Fundamentals of von Neumann Algebras
A.6.1 Basic Concepts of von Neumann Algebra
A.6.2 Types of Factors
A.6.3 Characteristics of Type II1 Factor
References
Index