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Zbl 0749.46038
Baumgärtel, Hellmut; Wollenberg, Manfred
Causal nets of operator algebras. Mathematical aspects of algebraic quantum field theory. (English)
[B] Mathematische Lehrbücher und Monographien. II. Abteilung: Mathematische Monographien. 80. Berlin: Akademie Verlag. xviii, 460 p. (1992). ISBN 3-05-501421-9/hbk

This book meets the request of people working in Mathematical Physics, at least those who are concerned with functional-analytic methods.\par Part I (Fundamentals of operator algebras, $C\sp*$-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.\par Part II as well as Part IV are related to Quantum field theory and Algebraic statistical mechanics ( nets of $C\sp*$-algebras and von Neumann algebras, quasi-local algebras, net automorphisms, causality, cluster properties, CAR and CCR algebras).\par Part V deals with modular theory and (local) type III, or III-1, algebras, and nuclearity.\par Part III yields a very elaborate presentation of superselection theory ( representations, intertwiners, endomorphisms), Hilbert $C\sp*$-systems, inverse superselection theory, Mackey theory.\par All in all, this is a very comprehensive book.
[A.Wehrl (Wien)]

MSC 2000:: *46L60 Appl. of selfadjoint operator algebras to physics
46Lxx Selfadjoint operator algebras
46N50 Appl. of functional analysis in quantum physics
46-02 Research monographs (functional analysis)
81T05 Axiomatic quantum field theory
82B10 Quantum equilibrium statistical mechanics (general)

Keywords: decomposition theory; Quantum field theory; Algebraic statistical mechanics; nets of $C\sp*$-algebras and von Neumann algebras; quasi-local algebras; net automorphisms; causality; cluster properties; CAR and CCR algebras; modular theory; nuclearity; superselection theory; representations; intertwiners; endomorphisms; Hilbert $C\sp*$-systems; inverse superselection theory

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Zentralblatt MATH Berlin [Germany]