Local triple derivations on C*-algebras and JB*-triples

TL;DR

This paper proves that all continuous local triple derivations on JB*-triples are actual triple derivations and that such derivations on C*-algebras are automatically continuous, establishing key structural properties.

Contribution

It demonstrates that local triple derivations on JB*-triples are genuine derivations and establishes their automatic continuity, extending to C*-algebras.

Findings

Every continuous local triple derivation on a JB*-triple is a triple derivation.

Local triple derivations on C*-algebras are automatically continuous.

Connections between local triple derivations and generalized Jordan derivations are explored.

Abstract

In a first result we prove that every continuous local triple derivation on a JB$^*$-triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple derivations on a JB$^*$-triple are continuous even if not assumed a priori to be so. In particular every local triple derivation on a C$^*$-algebra is a triple derivation. We also explore the connections between (bounded local) triple derivations and generalised (Jordan) derivations on a C$^*$-algebra.