MATH. SCAND. 105 (2009), 307–319 WEAK COMPACTNESS IN THE DUAL SPACE OF A JB*-TRIPLE IS COMMUTATIVELY DETERMINED

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 (an) in the closed unit ball of E, a sequence (ϕn) in K, and ϑ> 0 satisfying that |ϕn(an) |> ϑ for all n ∈ N. This solves a question stimulated by the main result in [11] and posed in [9]