Operator algebras in statistical mechanics and noncommutative probability theory (Q1089591)

The paper is a survey of various results concerning some applications of the theory of operator algebras, especially in quantum mechanics and noncommutative probability theory. It is designed to be in a sense complementary to \textit{O. Bratteli}'s and \textit{D. W. Robinson}'s book ''Operator Algebras and Quantum Statistical Mechanics I, II'', (1979; Zbl 0421.46048) and (1981; Zbl 0463.46052), and includes mainly topics not dealt with in the book. Among these are: a description of the algebra of quasi-local observables for a quantum spin system, some properties of KMS-states, a connection between invariant states and first integrals, and first integrals of the multidimensional isotropic Heisenberg model. All this constitutes the more physically oriented part of the article. The other part is devoted to noncommutative probability theory on von Neumann algebras and presents noncommutative analogues of two classical theorems: the central limit theorem for random variables satisfying Rosenblatt's condition and the individual ergodic theorem. Somewhere between these two parts is placed a central limit theorem for the distribution of eigenvalues of the multi-particle Schrödinger operator. Most of the results in the paper have appeared before, however, some of them are presented with complete proofs and for some others explanatory comments and remarks are provided.