Thompson's semigroup and the first Hochschild cohomology

arXiv2104.10556MaRDI QIDQ6365816

Linzhe Huang

Publication date: 21 April 2021

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

Mathematics Subject Classification ID

Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) (47B37) Means on groups, semigroups, etc.; amenable groups (43A07) Algebras of operators on Banach spaces and other topological linear spaces (47L10)

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