Module and Hochschild cohomology of certain semigroup algebras

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We study the relation between module and Hochschild cohomology groups of Banach algebras with a compatible module structure. More precisely, we show that for every commutative Banach A - A-bimodule X and every k ∈ N, the seminormed spaces Hᵏₐ (A,X*) and Hᵏ (A/J, X*) are isomorphic, where J is the closed ideal of A generated by the elements of the form a (α· b)-(a· α)b with a,b ∈ A and α∈ A. As an example, we calculate the module cohomologies of inverse semigroup algebras with coefficients in some related function algebras. In particular, we show that for an inverse semigroup S with the set of idempotents E, when ℓ¹(E) acts on ℓ¹(S) by multiplication from right and trivially from left, the first module cohomology H¹ℓ₁₍ₑ₎ (ℓ¹(S), ℓ¹(Gₛ)⁽²ⁿ⁺¹⁾) is trivial for each n ∈ N. As a consequence we conclude that the second module cohomology H²ℓ₁₍ₑ₎ (ℓ¹(S),ℓ¹(Gₛ)⁽²ⁿ⁺¹⁾) is a Banach space, where Gₛ is the maximal group homomorphic image of S.