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In this paper we define the module topological center of the second dual  of a Banach algebra  which is a Banach -module with compatible actions on another Banach algebra . We calculate the module topological center of ℓ 1(S)**, as an ℓ 1(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that ℓ 1(S)** is ℓ 1(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.

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