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

In this paper we define the module topological center of the second dual A** of a Banach algebra A which is a Banach u-module with compatible actions on another Banach algebra A. 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.