Classification of continuous $JBW^ *$-triples

Abstract:

We show that every $JB{W^{\ast }}$-triple without a direct summand of type I is isometrically isomorphic to an ${l^\infty }$-sum $\mathcal {R}{ \otimes ^\infty }H(A, \alpha )$ where $\mathcal {R}$ is a ${w^{\ast }}$-closed right ideal in a ${W^{\ast }}$-algebra $B$ and $H(A, \alpha )$ are the elements of a ${W^{\ast }}$-algebra $A$ which are symmetric under a C-linear involution $\alpha$ of $A$. Both $A$ and $B$ do not have a direct (${W^{\ast }}$-algebra) summand of type I. In order to refine the decomposition $\mathcal {R}{ \otimes ^\infty }H(A, \alpha )$ we define and characterize types of $JB{W^{\ast }}$-triples.