Hochschild cohomology for free semigroup algebras

TL;DR

This paper investigates the Hochschild cohomology of free semigroup operator algebras, establishing automatic continuity of derivations and computing the vanishing of certain cohomology groups, thus advancing the understanding of their algebraic structure.

Contribution

It introduces a new computational method for Hochschild cohomology and proves the triviality of first and higher cohomology groups for key free semigroup algebras.

Findings

All derivations of these algebras are automatically continuous.

The first Hochschild cohomology group of _{\u03bb} with coefficients in _{\u03bbl} is zero.

Higher cohomology groups of non-commutative disc algebras vanish when |bb|<init.

Abstract

This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set $\{z_{\alpha}\}_{\alpha\in\Lambda}$, with the left regular free semigroup algebra $\mathfrak{L}_{\Lambda}$ and the non-commutative disc algebra $\mathfrak{A}_{\Lambda}$ serving as two typical examples. We establish that all derivations of these algebras are automatically continuous. By introducing a novel computational approach, we demonstrate that the first Hochschild cohomology group of $\mathfrak{A}_{\Lambda}$ with coefficients in $\mathfrak{L}_{\Lambda}$ is zero. Utilizing the Ces\`aro operators and conditional expectations, we show that the first normal cohomology group of $\mathfrak{L}_{\Lambda}$ is trivial. Finally, we prove that the higher cohomology groups of the non-commutative disc algebras with coefficients in the complex field vanish when $|\Lambda|<\infty$. These methods extend to compute the cohomology groups of a specific class of operator algebras generated by the left regular representations of cancellative semigroups, which notably include Thompson's semigroup.