Weak-local triple derivations on C*-algebras and JB*-triples

Abstract

We prove that every weak-local triple derivation on a JB${}^{\ast }$-triple $E$ (i.e. a linear map $T:E\to E$ such that for each $\varphi \in E^{\ast }$ and each $a\in E$, there exists a triple derivation ${\delta }_{a,\varphi }:E\to E$, depending on $\varphi$ and $a$, such that $\varphi T(a)=\varphi {\delta }_{a,\varphi }(a)$) is a (continuous) triple derivation.