Module derivations into iterated duals of triangular Banach algebras

Abstract

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 of triangular Banach algebra $\mathrm{Tri}(A,X,B)$, 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.