Zero Lie product determined Banach algebras

A Banach algebra A is said to be zero Lie product determined if every continuous bilinear functional rho : A x A -> C with rho (a, b) = 0 whenever a and b commute is of the form rho(a; b) - T( ab - ba) for some T is an element of A*. In the first part of the paper we give some general remarks on this class of algebras. In the second part we consider amenable Banach algebras and show that all group algebras L-1(G) with G an amenable locally compact group are zero Lie product determined.