Baumgärtel, Hellmut; Wollenberg, Manfred 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) Cited in 33 Documents MSC: 46L60 Applications of selfadjoint operator algebras to physics 46Lxx Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.) 46N50 Applications of functional analysis in quantum physics 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 81T05 Axiomatic quantum field theory; operator algebras 82B10 Quantum equilibrium statistical mechanics (general) Keywords:decomposition theory; Quantum field theory; Algebraic statistical mechanics; nets of \(C^*\)-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 \(C^*\)-systems; inverse superselection theory PDF BibTeX XML Cite \textit{H. Baumgärtel} and \textit{M. Wollenberg}, Causal nets of operator algebras. Mathematical aspects of algebraic quantum field theory. Berlin: Akademie Verlag (1992; Zbl 0749.46038)