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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.

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