1. Hochschild cohomology for free semigroup algebras
- Author
-
Huang, Linzhe, Ma, Minghui, and Wei, Xiaomin
- Subjects
Mathematics - Operator Algebras ,Mathematics - Functional Analysis ,47L10, 46K50, 47B47 - 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., Comment: 19 pages
- Published
- 2024