Weak\(^ *\)-continuity of Jordan triple products and its applications.

*(English)*Zbl 0621.46044A \(JB^*\)-triple is a complex Banach space with a certain ternary algebraic structure. These spaces include all \(C^*\)-algebras, \(JC^*\)- algebras, and the ranges of contractive projections on such spaces. The main result of this paper is that a dual Banach space which is a \(JB^*\)-triple has a unique predual and a separately weak* continuous triple product. Secondary results include the characterization of all closed ideals as M-ideals and an analogue of the Gelfand-Naimark-Segal construction of representations. Subsequent work by the authors and others has shown the main result to have a foundational role in the spectral theory, representation theory, and classification theory of \(JB^*\)-triples.

Some preprints of this article were circulated under the title ”On biduals, preduals, and ideals of \(JB^*\)-triples.”

Some preprints of this article were circulated under the title ”On biduals, preduals, and ideals of \(JB^*\)-triples.”

##### MSC:

46H70 | Nonassociative topological algebras |

47L50 | Dual spaces of operator algebras |

17C65 | Jordan structures on Banach spaces and algebras |