Weak compactness in the dual space of a JB*-triple is commutatively determined

Abstract

We prove the following criterium of weak compactness in the dual of a JB*-triple: a bounded set $K$ in the dual of a JB*-triple $E$ is not relatively weakly compact if and only if there exist a sequence of pairwise orthogonal elements $(a_n)$ in the closed unit ball of $E$, a sequence $(\varphi_{n} )$ in $K$, and $\vartheta >0$ satisfying that $|\varphi_{n}(a_{n})|>\vartheta$ for all $n \in {\mathsf N}$. This solves a question stimulated by the main result in [11] and posed in [9].