Thompson's semigroup and the first Hochschild cohomology

TL;DR

This paper uses algebraic cohomology to analyze the structure and amenability of Thompson's semigroup, showing all derivations are continuous and the first Hochschild cohomology group vanishes, indicating potential amenability.

Contribution

It introduces the notion of unique factorization semigroup containing Thompson's semigroup and proves properties of derivations and cohomology related to its Banach algebra.

Findings

All derivations on the associated Banach algebras are automatically continuous.

Every derivation on the semigroup's Banach algebra is induced by a bounded linear operator.

The first Hochschild cohomology group of the Banach algebra vanishes.

Abstract

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.