In this book we describe the elementary theory of operator algebras and parts of the advanced theory which are of relevance, or potentially of relevance, to mathematical physics. Subsequently we describe various applications to quantum statistical mechanics. At the outset of this project we intended to cover this material in one volume but in the course of develop­ ment it was realized that this would entail the omission of various interesting topics or details. Consequently the book was split into two volumes, the first devoted to the general theory of operator algebras and the second to the applications. This splitting into theory and applications is conventional but somewhat arbitrary. In the last 15-20 years mathematical physicists have realized the importance of operator algebras and their states and automorphisms for problems offield theory and statistical mechanics. But the theory of 20 years ago was largely developed for the analysis of group representations and it was inadequate for many physical applications. Thus after a short honey­ moon period in which the new found tools of the extant theory were applied to the most amenable problems a longer and more interesting period ensued in which mathematical physicists were forced to redevelop the theory in relevant directions. New concepts were introduced, e. g. asymptotic abelian­ ness and KMS states, new techniques applied, e. g. the Choquet theory of barycentric decomposition for states, and new structural results obtained, e. g. the existence of a continuum of nonisomorphic type-three factors.

Inhalt

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.