The Structure of Triple Derivations on Semisimple Banach *-Algebras

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. Algebra 42 (2014), 1276-1286). The analogous result for standard operator *-algebras on Hilbert spaces is also described.