(2n+1)-weak module amenability of triangular Banach algebras on Inverse semigroup algebras

Abstract

Let be a commutative (not necessary unital) inverse semigroup with the set of idempotents E ‎then l^1 (S) is a commutative Banach‎ l^1 (E)-module with canonical actions‎. ‎Recently‎, ‎it is shown that the triangular Banach algebra T=Tri(l^1(S), M, l^1(S)) is (n) -weakly l^1 (E)-module amenable‎, ‎provided that M=l^1 (S) and is unital or E satisfies condition D_k for some k∈N ‎. ‎In this paper‎, ‎we show that T is (2n+1)-weakly l^1 (E)-module amenable‎, ‎without any additional conditions on S and E, ‎if M is a certain quotient space of l^1 (S).