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

TL;DR

This paper proves that every weak-local triple derivation on JB*-triples is actually a continuous triple derivation, establishing a significant structural property of these maps.

Contribution

It demonstrates that weak-local triple derivations on JB*-triples are necessarily continuous triple derivations, extending understanding of their structure.

Findings

Weak-local triple derivations are continuous triple derivations.

The result applies to JB*-triples, a broad class of Jordan Banach structures.

The proof confirms the structural rigidity of weak-local derivations.

Abstract

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