Triple derivations on C ∗-algebras and JB ∗-triples had been extensively studied in the literature. In this paper, we characterize the structure of triple derivations on semisimple complex Banach ∗-algebras. In particular, we show that every triple derivation on a semisimple complex Banach ∗-algebra is automatically continuous and is a special kind of generalized derivations. Our theorems improve and generalize some known results for C ∗-algebras obtained in Barton and Friedman (Bounded derivations of JB ∗-triples, Quart. J. Math. Oxford Ser.41 (1990), 255-268), Burgos et al. (Local triple derivations on C ∗-algebras and JB ∗-triples, Bull. Lond. Math. Soc.46 (2014), 709-724) and Burgos et al. (Local triple derivations on C∗-algebras, Comm. Algebra42 (2014), 1276-1286). The analogous result for standard operator ∗-algebras on Hilbert spaces is also described.
All Science Journal Classification (ASJC) codes