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Weak$$^ *$$-continuity of Jordan triple products and its applications. (English) Zbl 0621.46044
A $$JB^*$$-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.”

##### MSC:
 46H70 Nonassociative topological algebras 47L50 Dual spaces of operator algebras 17C65 Jordan structures on Banach spaces and algebras
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