Generalized notions of module character amenability

TL;DR

This paper explores advanced concepts in Banach algebra theory, introducing new notions of module character amenability and contractibility, and characterizes these properties in terms of diagonals and approximate amenability, especially for inverse semigroup algebras.

Contribution

It defines module character contractibility and introduces module approximately character amenable Banach algebras, providing characterizations and conditions for inverse semigroup algebras.

Findings

Characterizations of module character contractible Banach algebras.

Necessary and sufficient conditions for $ ext{l}^1(S)$ to be module approximately character amenable.

Results on hereditary properties of module $( ext{phi}, ext{varphi})$-amenability.

Abstract

In this paper, we study the hereditary properties of module $(\phi,\varphi)$-amenability on Banach algebras. We also define the concept of module character contractibility for Banach algebras and obtain characterizations of module character contractible Banach algebras in terms of the existence of module $(\phi,\varphi)$-diagonals. We introduce module approximately character amenable Banach algebras. Finally, for every inverse semigroup $S$ with subsemigroup $E$ of idempotents, we find necessary and sufficient conditions for the $\ell^1(S)$ and its second dual to be module approximate character amenable (as a $\ell^1(E)$-module).