CLOSED TRIPOTENTS AND WEAK COMPACTNESS IN THE DUAL SPACE OF A JB*-TRIPLE

Abstract

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.