Thompson’s semigroup and the first Hochschild cohomology

Abstract In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson’s group F. We introduce the notion of unique factorization semigroup which contains Thompson’s semigroup S and the free semigroup Fₙ on n (2) generators. Let B (S) and B (Fₙ) be the Banach algebras generated by the left regular representations of S and Fₙ, respectively. We prove that all derivations on B (S) and B (Fₙ) are automatically continuous, and every derivation on B (S) is induced by a bounded linear operator in L (S), the weak-operator closed Banach algebra consisting of all bounded left convolution operators on l² (S). Moreover, we prove that the first continuous Hochschild cohomology group of B (S) with coefficients in L (S) vanishes. These conclusions provide positive indications for the left amenability of Thompson’s semigroup.