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Local triple derivations on C∗‐algebras and JB∗‐triples

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. These results provide positive answers to the conjectures posed by Mackey (Bull. London Math. Soc. 45 (2013) 811–824). 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 generalized (Jordan) derivations on a C∗‐algebra.