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Operator Algebras and Quantum Statistical Mechanics: Volume 1: C*- and W*- Algebras. Symmetry Groups. Decomposition of States - Softcover
Synopsis
Notes and Remarks.- C*-Algebras and von Newmann Algebras.- 2.1. C*-Algebras.- 2.1.1. Basic Definitions and Structure.- 2.2. Functional and Spectral Analysis.- 2.2.1. Resolvents, Spectra, and Spectral Radius.- 2.2.2. Positive Elements.- 2.2.3. Approximate Identities and Quotient Algebras.- 2.3. Representations and States.- 2.3.1. Representations.- 2.3.2. States.- 2.3.3. Construction of Representations.- 2.3.4. Existence of Representations.- 2.3.5. Commutative C*-Algebras.- 2.4. von Neumann Algebras.- 2.4.1. Topologies on ?(H).- 2.4.2. Definition and Elementary Properties of von Neumann Algebras.- 2.4.3. Normal States and the Predual.- 2.4.4. Quasi-Equivalence of Representations.- 2.5. Tomita-Takesaki Modular Theory and Standard Forms of von Neumann Algebras.- 2.5.1. ?-Finite von Neumann Algebras.- 2.5.2. The Modular Group.- 2.5.3. Integration and Analytic Elements for One-Parameter Groups of Isometries on Banach Spaces.- 2.5.4. Self-Dual Cones and Standard Forms.- 2.6. Quasi-Local Algebras.- 2.6.1. Cluster Properties.- 2.6.2. Topological Properties.- 2.6.3. Algebraic Properties.- 2.7. Miscellaneous Results and Structure.- 2.7.1. Dynamical Systems and Crossed Products.- 2.7.2. Tensor Products of Operator Algebras.- 2.7.3. Weights on Operator Algebras; Self-Dual Cones of General von Neumann Algebras; Duality and Classification of Factors; Classification of C*-Algebras.- Notes and Remarks.- Groups, Semigroups, and Generators.- 3.1. Banach Space Theory.- 3.1.1. Uniform Continuity.- 3.1.2. Strong, Weak, and Weak* Continuity.- 3.1.3. Convergence Properties.- 3.1.4. Perturbation Theory.- 3.1.5. Approximation Theory.- 3.2. Algebraic Theory.- 3.2.1. Positive Linear Maps and Jordan Morphisms.- 3.2.2. General Properties of Derivations.- 3.2.3. Spectral Theory and Bounded Derivations.- 3.2.4. Derivations and Automorphism Groups.- 3.2.5. Spatial Derivations and Invariant States.- 3.2.6. Approximation Theory for Automorphism Groups.- Notes and Remarks.- Decomposition Theory.- 4.1. General Theory.- 4.1.1. Introduction.- 4.1.2. Barycentric Decompositions.- 4.1.3. Orthogonal Measures.- 4.1.4. Borel Structure of States.- 4.2. Extremal, Central, and Subcentral Decompositions.- 4.2.1. Extremal Decompositions.- 4.2.2. Central and Subcentral Decompositions.- 4.3. Invariant States.- 4.3.1. Ergodic Decompositions.- 4.3.2. Ergodic States.- 4.3.3. Locally Compact Abelian Groups.- 4.3.4. Broken Symmetry.- 4.4. Spatial Decomposition.- 4.4.1. General Theory.- 4.4.2. Spatial Decomposition and Decomposition of States.- Notes and Remarks.- References.- Books and Monographs.- Articles.- List of Symbols.
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