Ternary weakly amenable C*-algebras and JB*-triples

A well-known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan-Banach triples, focussing on commutative JB*-triples and some Cartan factors. © 2012 2012. Published by Oxford University Press. All rights reserved