Module cohomology group of inverse semigroup algebras (Q1953958)

This paper studies the module cohomology groups of the semigroup algebra \(\ell^1(S)\) of an inverse semigroup \(S\) with coefficients in the \(n\)th conjugate space \(\ell^1(S)^{(n)}\), for odd \(n\), where \(\ell^1(S)\) is considered as a Banach module over the semigroup algebra \(\ell^1(E)\) on the subsemigroup \(E\) of the idempotents in \(S\). The difference with the classical case is that here the derivations are assumed to be module homomorphisms as well. The authors show that the first module cohomology group is zero when \(S\) is commutative, and the second module cohomology group is a Banach space when \(S\) is a Clifford semigroup (that is, a completely regular inverse semigroup). The condition that \(S\) is commutative is used in the proof of the first statement (main result of Section 2), but is not stated in the abstract of the paper. Also, the definition of \(n\)-weak module amenability given in Section 1 (and used in the commutative case in Section 2) is only valid for commutative Banach algebras.