Automatic continuity of derivations on C⁎-algebras and JB⁎-triples

We introduce the notion of Banach Jordan triple modules and determine the precise conditions under which every derivation from a JB⁎-triple E into a Banach (Jordan) triple E-module is continuous. In particular, every derivation from a real or complex JB⁎-triple into its dual space is automatically continuous, motivating the study (which we have carried out elsewhere) of weakly amenable JB⁎-triples. Specializing to C⁎-algebras leads to a unified treatment of derivations and Jordan derivations into modules, shedding light on a celebrated theorem of Barry Johnson.