Classification of \(JBW^ *\)-triples of type I
type: Item
instance of: scholarly article
title: Classification of \(JBW^ *\)-triples of type I
zbMATH Open document ID: 0615.46045
DOI: 10.1007/BF01163661
author: Günter Horn
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173881
review text: JB\({}^ *\)-triples from a category of normed Jordan triple systems generalizing that of Jordan \(C^ *\)-algebras and, hence, that of \(C^ *\)-algebras. A standard example of a \(JB^ *\)-triple is a closed subspace of the space of all bounded operators on a complex Hilbert space which is also closed under the Jordan triple product \(\{xy^ *z\}:=(xy^ *z+zy^ *x)\). The category of \(JB^ *\)-triples is equivalent to the category of bounded symmetric domains with base point in complex Banach spaces as was shown by W. Kaup (1984). \(JB^ *\)- triples also occur naturally in the solution of the contractive projection problem for \(C^ *\)-algebras (Y. Friedman, B. Russo (1985)). \(JB^ *\)-triples which are Banach dual spaces are called \(JBW^ *\)- triples. In analogy to the classification of \(W^ *\)-algebras a \(JBW^ *\)-triple is said to be of type I if it is generated by an abelian tripotent (where a tripotent is an element satisfying \(\{ee^ *e\}=e)\). Cartan factors are certain well-known, explicitly described, indecomposable \(JBW^ *\)-triples of type I. In this paper we show that a JBW\({}^ *\)-triple of type I is isometrically isomorphic to an \(\ell ^{\infty}\)-sum of tensor products of abelian von Neumann algebras and Cartan factors. As a corollary we obtain that an indecomposable \(JBW^ *\)-triple of type I is a Cartan factor.
Mathematics Subject Classification ID: 46H70
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 17C65
zbMATH DE Number: 3996409
zbMATH Keywords: JB\({}^ *\)-triples
zbMATH Keywords: category of normed Jordan triple systems
zbMATH Keywords: Jordan triple product
zbMATH Keywords: Banach dual spaces
zbMATH Keywords: Cartan factors
zbMATH Keywords: \(JBW^ *\)-triple of type I
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3996409