n-Weak Module Amenability of Triangular Banach Algebras

TL;DR

This paper investigates the properties of triangular Banach algebras, focusing on module amenability, $n$-weak module amenability, and Arens regularity, with applications to inverse semigroups and their associated algebras.

Contribution

It introduces new results on $n$-weak module amenability and Arens regularity of triangular Banach algebras, especially in the context of inverse semigroups.

Findings

Triangular Banach algebra $ T_0$ is permanently weakly module amenable.

$ T_0$ is $T_0$-module Arens regular iff the maximal group image of $S$ is finite.

Provides conditions linking algebraic properties to the structure of inverse semigroups.

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 $\mathfrak T:=\Big{[ {cc} \alpha & & \alpha ] | \alpha\in\mathfrak A\Big}$-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.