Operator Algebras and Quantum Statistical Mechanics [electronic resource] : C*- and W*-Algebras Symmetry Groups Decomposition of States / by Ola Bratteli, Derek W. Robinson
Imprint
Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1979
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 developยญ 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 honeyยญ 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 abelianยญ 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
CONTENT
Notes and Remarks -- C*-Algebras and von Newmann Algebras -- 2.1. C*-Algebras -- 2.2. Functional and Spectral Analysis -- 2.3. Representations and States -- 2.4. von Neumann Algebras -- 2.5. TomitaโTakesaki Modular Theory and Standard Forms of von Neumann Algebras -- 2.6. Quasi-Local Algebras -- 2.7. Miscellaneous Results and Structure -- Notes and Remarks -- Groups, Semigroups, and Generators -- 3.1. Banach Space Theory -- 3.2. Algebraic Theory -- Notes and Remarks -- Decomposition Theory -- 4.1. General Theory -- 4.2. Extremal, Central, and Subcentral Decompositions -- 4.3. Invariant States -- 4.4. Spatial Decomposition -- Notes and Remarks -- References -- Books and Monographs -- Articles -- List of Symbols
