First order cohomology of Banach semigroup algebras (Q1375915)

Denoting by \(A\) a (complex) Banach algebra and by \(X\) a Banach \(A\)-bimodule, the authors show here that when \(S\) is a semigroup and \(A\) is the convolution Banach semigroup algebra \(\ell^1(S)\), it is often possible to compute the first-order cohomology group \(H^1(A, X)\), that is the space of \(X\)-valued bounded derivations modulo the inner derivations. Namely, the cases which are most interesting in this paper are \(X= A^*\) and \(X= A\) with their natural bimodule products and two other important bimodules given by \(X= (A\otimes A)^*\) and \(X= A\otimes A\), as well as bimodules associated with semigroups action on a set.