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

We revise the concept of compact tripotent in the bidual spaceof a JB*-triple. This concept was introduced by Edwards andRüttimann generalizing the ideas developed by Akemann forcompact projections in the bidual of a C*-algebra. We also obtainsome characterizations of weak compactness in the dual spaceof a JC*-triple, showing that a bounded subset in the dual spaceof a JC*-triple is relatively weakly compact if and only ifits restriction to any abelian maximal subtriple C is relativelyweakly compact in the dual of C. This generalizes a very usefulresult by Pfitzner in the setting of C*-algebras. As a consequencewe obtain a Dieudonné theorem for JC*-triples which generalizesthe one obtained by Brooks, Saitô and Wright for C*-algebras.