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Causal nets of operator algebras. Mathematical aspects of algebraic quantum field theory. (English) Zbl 0749.46038
Mathematische Lehrbücher und Monographien. II. Abteilung: Mathematische Monographien. 80. Berlin: Akademie Verlag. xviii, 460 p. (1992).

This book meets the request of people working in Mathematical Physics, at least those who are concerned with functional-analytic methods.

Part I (Fundamentals of operator algebras, C * -algebras, von Neumann algebras, decomposition theory) provides basic material that is presumably widely known; thus it is more or less to be regarded as an introduction.

Part II as well as Part IV are related to Quantum field theory and Algebraic statistical mechanics ( nets of C * -algebras and von Neumann algebras, quasi-local algebras, net automorphisms, causality, cluster properties, CAR and CCR algebras).

Part V deals with modular theory and (local) type III, or III-1, algebras, and nuclearity.

Part III yields a very elaborate presentation of superselection theory ( representations, intertwiners, endomorphisms), Hilbert C * -systems, inverse superselection theory, Mackey theory.

All in all, this is a very comprehensive book.

Reviewer: A.Wehrl (Wien)
MSC:
46L60Applications of selfadjoint operator algebras to physics
46LxxSelfadjoint operator algebras (C * -algebras, W * -algebras, etc.)
46N50Applications of functional analysis in quantum physics
46-02Research monographs (functional analysis)
81T05Axiomatic quantum field theory; operator algebras
82B10Quantum equilibrium statistical mechanics (general)