The first cohomology of the Banach algebra associated with Thompsons semigroup

In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompsons group $mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompsons semigroup $mathcal{S}$ and the free semigroup $mathcal{F}_n$ on $n$ generators ($geq2$). Let $mathfrak{B}(mathcal{S})$ and $mathfrak{B}(mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $mathcal{S}$ and $mathcal{F}_n$, respectively. It is proved that all derivations on $mathfrak{B}(mathcal{S})$ and $mathfrak{B}(mathcal{F}_n)$ are automatically continuous, and every derivation on $mathfrak{B}(mathcal{S})$ is induced by a bounded linear operator in $mathcal{L}(mathcal{S})$, the weak closed Banach algebra consisting of all bounded left convolution operators on $l^2(mathcal{S})$. Moreover, we show that the first continuous Hochschild cohomology group of $mathfrak{B}(mathcal{S})$ with coefficients in $mathcal{L}(mathcal{S})$ vanishes. These conclusions provide positive indications for the left amenability of Thompsons semigroup.