Module Pseudo-amenability of Banach algebras

TL;DR

This paper introduces the concepts of module pseudo-amenability and module pseudo-contractibility for Banach algebras, providing characterizations and conditions for specific algebraic structures like inverse semigroups.

Contribution

It defines new notions of module pseudo-amenability and characterizes them for Banach algebras, especially for inverse semigroup algebras.

Findings

Module pseudo-amenability coincides with module approximate amenability for Banach algebras with bounded approximate identity.

Complete characterization of module pseudo-amenability for Banach algebras is provided.

Necessary and sufficient conditions for $ ext{l}^1(S)$ and its second dual to be $ ext{l}^1(E)$-module pseudo-amenable are established.

Abstract

The notions of module pseudo-amenable and module pseudo-contractible Banach algebras are introduced. For a Banach algebra with bounded approximate identity, module pseudo-amenability and module approximate amenability are the same properties. It is given a complete characterization of module pseudo-amenability for a Banach algebra. For every inverse semigroup $S$ with subsemigroup $E$ of idempotents, necessary and sufficient conditions are obtained for the $\ell^1(S)$ and its second dual to be $\ell^1(E)$-module pseudo-amenable.