A GENERALIZATION OF THE WEAK AMENABILITY OF BANACH ALGEBRAS

Abstract. Let A be a Banach algebra and let ϕ and ψ be continuous homo-morphisms on A. We consider the following module actions on A, a · x = ϕ(a)x, x · a = xψ(a) (a, x ∈ A). We denote by A(ϕ,ψ) the above A-module. We call the Banach algebra A, (ϕ,ψ)-weakly amenable if every derivation from A into (A(ϕ,ψ)) ∗ is inner. In this paper among many other things we investigate the relations between weak amenability and (ϕ,ψ)-weak amenability ofA. Some conditions can be imposed on A such that the (ϕ′′, ψ′′)-weak amenability of A∗ ∗ implies the (ϕ,ψ)-weak amenability of A. 1. Introduction an