Classification of \(JBW^ *\)-triples of type I
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.
category of normed Jordan triple systems
scientific article; zbMATH DE number 3996409
scientific article