Weak compactness in the dual space of a JB*-triple is commutatively determined - Fernández-Polo, Francisco J.; Peralta, Antonio M. (MATHEMATICA SCANDINAVICA, 2009)
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 $(φ_{n} )$ in $K$, and $ϑ >0$ satisfying that $|φ_{n}(a_{n})|>ϑ$ for all $n \in {\mathsf N}$. This solves a question stimulated by the main result in [11] and posed in [9].