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Closed Tripotents and Weak Compactness in the Dual Space of a Jb*‐Triple

We revise the concept of compact tripotent in the bidual space of a JB*‐triple. This concept was introduced by Edwards and Rüttimann generalizing the ideas developed by Akemann for compact projections in the bidual of a C*‐algebra. We also obtain some characterizations of weak compactness in the dual space of a JC*‐triple, showing that a bounded subset in the dual space of a JC*‐triple is relatively weakly compact if and only if its restriction to any abelian maximal subtriple C is relatively weakly compact in the dual of C. This generalizes a very useful result by Pfitzner in the setting of C*‐algebras. As a consequence we obtain a Dieudonné theorem for JC*‐triples which generalizes the one obtained by Brooks, Saitô and Wright for C*‐algebras.