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 Ruttimann 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 Dieudonnet heorem for JC*-triples which generalizes the one obtained by Brooks, Saito and Wright for C*-algebras. One of the most celebrated and useful results characterizing weakly compact subsets in the dual space of a C*-algebra is due to Pfitzner, who established that weak com- pactness in the dual space of a C*-algebra is commutatively determined (see (30)). More concretely, Pfitzner shows, in a 'tour de force', that if K is a bounded subset in the dual space of a C*-algebra A ,t henK is relatively weakly compact if and only if the restriction of K to each maximal abelian subalgebra of A is relatively weakly compact. This result has many important consequences, one of the most interesting being that every C*-algebra satisfies property (V) of Pelczynski. Pfitzner's result is the latest advance in the study of weak compactness in the dual space of a C*-algebra developed by Takesaki (33), Akemann (1), Akemann, Dodds and Gamlen (3), Saito( 32) and Jarchow (22, 23). In the more general setting of dual spaces of JB*-triples the study of weak compactness has been developed by Chu and Iochum (12) and by Peralta and Rodr´ iguez-Palacios (28, 29). However, all the results concerning weak compactness in the dual space of a JB*-triple give characterizations in terms of the abelian subtriples of its bidual instead of the abelian subtriples of the JB*-triple itself. The question clearly is whether a bounded subset in the dual space of a JB*-triple, E, is relatively weakly compact whenever its restriction to any abelian subtriple of E is. In the main result of this paper we show that weak compactness in the dual space of a JC*-triple is commutatively determined, by showing that a bounded subset K in the dual space of a JC*-triple E is relatively weakly compact if and only if the restriction of K to each separable abelian subtriple of E also is relatively weakly compact (see Theorem 3.5).