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

Abstract

Let "S" be a commutative (not necessary unital) inverse semigroup with the set of idempotents E‎then l^1 ("S") is a commutative Banach‎l^1 ("E")-module with canonical actions‎. ‎Recently‎, ‎it is shown that the triangular Banach algebra ‎T=[■(l^1 ("S" )&M@&〖 l〗^1 ("S" ))] is ("n)" -weakly l^1 ("E")-module amenable‎, ‎provided thatM=l^1 ("S") and "S" is unital or Esatisfies condition D_kfor 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" andE,‎if M is a certain quotient space of l^1 ("S").