Zero jordan product determined Banach algebras

Abstract

A Banach algebra A is said to be a zero Jordan product determined Banach algebra if, for every Banach space X, every bilinear map phi : A x A -> X satisfying phi(a, b) = 0 whenever a, b is an element of A are such that ab + ba = 0, is of the form phi(a, b) = sigma(ab + ba) for some continuous linear map sigma. We show that all C*-algebras and all group algebras L-1(G) of amenable locally compact groups have this property and also discuss some applications.