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. © 2013 Elsevier Inc.

We prove that every (not necessarily linear nor continuous) 2-local triple derivation on a von Neumann algebra M is a triple derivation, equivalently, the set Dert(M), of all triple derivations on M, is algebraically 2-reflexive in the set M(M) =MM of all mappings from M into M.

We prove a Jordan version of Dorofeev's boundedness theorem for completely additive measures and use it to show that every (not necessarily linear nor continuous) 2-local triple derivation on a continuous JBW*-triple is a triple derivation. 2-local triple derivations are well understood on von Neumann algebras. JBW*-triples, which are properly defined in Section I, are intimately related to infinite dimensional holomorphy and include von Neumann algebras as special cases. In particular, continuous JBW*-triples can be realized as subspaces of continuous von Neumann algebras which are stable for the triple product xy*z + zy*x and closed in the weak operator topology.