Super module amenability of inverse semigroup algebras

Abstract

In this paper we compare the notions of super amenability and super module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. We find conditions for the two notions to be equivalent. In particular, we study arbitrary module actions of l1(ES) on l1(S) for an inverse semigroup S with the set of idempotents ES and show that under certain conditions, l1(S) is super module amenable if and only if S is finite. We also study the super module amenability of l1(S)∗∗ and module biprojectivity of l1(S), for arbitrary actions.