Module cohomology group of inverse semigroup algebras

Let S be an inverse semigroup and let E be its sub- semigroup of idempotents. In this paper we dene the n-th mod- ule cohomology group of Banach algebras and show that the rst module cohomology group H1 ℓ1(E)(ℓ1(S); ℓ1(S)(n)) is zero, for ev- ery odd n 2 N. Next, for a Cli ord semigroup S we show that H2 ℓ1(E)(ℓ1(S); ℓ1(S)(n)) is a Banach space, for every odd n 2 N.