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

We prove that every weak-local triple derivation on a JB⁎-triple E   (i.e. a linear map T:E→ET:E→E such that for each ϕ∈E⁎ϕ∈E⁎ and each a∈Ea∈E, there exists a triple derivation δa,ϕ:E→Eδa,ϕ:E→E, depending on ϕ and a  , such that ϕT(a)=ϕδa,ϕ(a)ϕT(a)=ϕδa,ϕ(a)) is a (continuous) triple derivation. We also prove that conditions(h1)(h1){a,T(b),c}=0{a,T(b),c}=0 for every a, b, c in E   with a,c⊥ba,c⊥b;(h2)(h2)P2(e)T(a)=−Q(e)T(a)P2(e)T(a)=−Q(e)T(a) for every norm-one element a in E, and every tripotent e   in E⁎⁎E⁎⁎ such that e≤s(a)e≤s(a) in E2⁎⁎(e), where s(a)s(a) is the support tripotent of a   in E⁎⁎E⁎⁎, are necessary and sufficient to show that a linear map T on a JB⁎-triple E is a triple derivation.