Homomorphism weak amenability of certain Banach algebras

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