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

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 ${\ell }^1(S{)}^{\ast \ast }$ (as an ${\ell }^1(E)$-module). We also prove that ${\ell }^1(S{)}^{\ast \ast }$ is ${\ell }^1(E)$-module amenable if and only if an appropriate group homomorphic image of $S$ is finite.

9 pages