Thompson's semigroup and the first Hochschild cohomology

In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson's group $\mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompson's 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 Thompson's semigroup.