The purpose of this article is to develop the notions of amenability
for vector valued group algebras. We prove that L1(G, A) is approximately
weakly amenable where A is a unital separable Banach algebra. We give
necessary and sufficient conditions for the existence of a left invariant mean
on L∞(G, A∗), LUC(G, A∗), WAP(G, A∗) and C0(G, A∗).