Module amenability of the second dual and module topological center of semigroup algebras

TL;DR

This paper investigates the module topological center of the second dual of semigroup algebras and establishes conditions for their module amenability, linking algebraic structure to finiteness properties.

Contribution

It introduces the concept of the module topological center for second duals of Banach algebras and characterizes module amenability for semigroup algebras in terms of group homomorphic images.

Findings

The module topological center of $ell^{1}(S)^{**}$ is explicitly determined.

$ell^{1}(S)^{**}$ is $ell^{1}(E)$-module amenable iff a certain group homomorphic image of $S$ is finite.

Provides a connection between algebraic properties of semigroups and module amenability.

Abstract

Let $S$ be an inverse semigroup with an upward directed set of idempotents $E$. In this paper we define the module topological center of second dual of a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find it for $ \ell ^{1}(S)^{**}$ (as an $\ell^{1}(E)$-module). We also prove that $ \ell ^{1}(S)^{**}$ is $\ell^{1}(E)$-module amenable if and only if an appropriate group homomorphic image of $S$ is finite.