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Thompson’s semigroup and the first Hochschild cohomology
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 07 March 2024, pp. 1-21
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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 (
$\geq 2$) generators. 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. We prove 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-operator closed Banach algebra consisting of all bounded left convolution operators on 
$l^2(\mathcal {S})$. Moreover, we prove 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.
