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.
Categories: math.FA