- - - Department of Mathematics, Faculty of Science, University of Isfahan, Isfahan, Iran
- - - Department of Mathematics, Faculty of Science, University of Isfahan, Isfahan, Iran

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.

Banach algebra, varphi-commutative, varphi-derivation, varphi-weakly amenability