# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 46L60 Applications of selfadjoint operator algebras to physics 46Lxx Selfadjoint operator algebras (${C}^{*}$-algebras, ${W}^{*}$-algebras, etc.) 46N50 Applications of functional analysis in quantum physics 46-02 Research monographs (functional analysis) 81T05 Axiomatic quantum field theory; operator algebras 82B10 Quantum equilibrium statistical mechanics (general)