Second module cohomology group of inverse semigroup algebras (Q711610)

Let \(S\) be a commutative inverse semigroup and let \(E\) be its semigroup of idempotents. \textit{M. Amini} and \textit{D. E. Bagha} in [Semigroup Forum 71, No. 1, 18--26 (2005; Zbl 1086.43001)] studied the first \(l^1(E)\)-module cohomology group of the semigroup algebra \(l^1(S)\) with coefficients in the dual space \((l^1(S))'=l^\infty(S)\) and showed that \(l^1(S)\) is always weak \(l^1(E)\)-module amenable, that is, \({\mathcal H}^1_{l^1(E)}( l^1(S), l^\infty(S))=0\). In the paper under review the authors show that the second \(l^1(E)\)-module cohomology group of \(l^1(S)\) with coefficients in the \(n\)-th dual space \(l^1(S)^{(n)}\) is a Banach space for every odd \(n\in {\mathbb N}\).