Homomorphism weak amenability of certain Banach algebras

In this paper we introduce the notion of phi-commutativity for a Banach algebra A, where phi is a continuous homomorphism on A and study the concept of phi-weak amenability for phi-commutative Banach algebras. We give an example to show that the class of phi-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We characterize phi-weak amenability of phi- commutative Banach algebras and prove some hereditary properties. Moreover we verify some of the previous available results about commutative weakly amenable Banach algebras, for phi-commutative phi-weakly amenable Banach algebras.