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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.

For a Banach algebra A, A00 is (􀀀 1)-weakly amenable if A0 is a Banach A00-bimodule and H1(A00; A0) = f0g. In this paper we prove some important properties of this notion, for instance if A00 is (􀀀 1)-weakly amenable then A is essential and there is no non-zero point derivation on A. We also give some examples, namely, the second dual of every C-algebras is (􀀀 1)-weakly amenable. Finally, we study the relationships between the (􀀀 1)-weakly amenability of A00 and the weak amenability of A00 or A.

The main purpose of this paper is to investigate weak amenability of semigroup algebras. We relate this to a new notion of weak amenability modulo an ideal of Banach algebras. As an important result, we show that l1(S) is weakly amenable modulo I , where I is the corresponding ideal of the group congruence  .