n-Weak Module Amenability of Triangular Banach Algebras

Abstract

Let $\mathcal{A}$, $\mathcal{B}$ be Banach $\mathfrak{A}$-modules with compatible actions and $\mathcal{M}$ be a left Banach $\mathcal{A}$-$\mathfrak{A}$-module and a right Banach $\mathcal{B}$-$\mathfrak{A}$-module. In the current paper, we study module amenability, $n$-weak module amenability and module Arens regularity of the triangular Banach algebra $\mathcal T=[ {cc} \mathcal A & \mathcal M & \mathcal B ]$ (as an $\mathfrak T:=\Big{[ {cc} α& & α ] | α\in\mathfrak A\Big}$-module). We employ these results to prove that for an inverse semigroup $S$ with subsemigroup $E$ of idempotents, the triangular Banach algebra $\mathcal T_0=[ {cc} \ell^1(S)& \ell^1(S) & \ell^1(S) ]$ is permanently weakly module amenable (as an $\mathfrak T_0=[ {cc} \ell^1(E)& & \ell^1(E) ]$-module). As an example, we show that ${\mathcal{T}}_0$ is ${\mathfrak{T}}_0$-module Arens regular if and only if the maximal group homomorphic image $G_S$ of $S$ is finite.