  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.
