Homomorphism Weak amenability of certain Banach algebras

Abstract

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