Module derivations into iterated duals of triangular Banach algebras

Abstract

Let A be a Banach algebra, A and B be Banach A-module with compatible actions and X be a Banach left A-A-module and Banach right B-A-module. Then the corresponding triangular Banach algebra Tri(A,X, B) is a Banach A-module with compatible actions. In this paper, we study n-weak module amenability of module extension Banach algebras to provide necessary and sufficient conditions for n-weak module amenability (as an A-module) of Tri(A,X, B), when A and B are not necessarily unital and not have bounded approximate identity. This not only fixes the gaps in some known results in the literature but also extends that results and gives a direct proof for them. Furthermore, we characterize n-weak module amenability of triangular matrix algebras related to inverse semigroups and some triangular Banach algebra related to locally compact groups.