Hochschild cohomology for free semigroup algebras
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911963569717248 |
|---|---|
| author | Huang, Linzhe Ma, Minghui Wei, Xiaomin |
| author_facet | Huang, Linzhe Ma, Minghui Wei, Xiaomin |
| contents | This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set $\{z_α\}_{α\inΛ}$, with the left regular free semigroup algebra $\mathfrak{L}_Λ$ and the non-commutative disc algebra $\mathfrak{A}_Λ$ 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 $\mathfrak{A}_Λ$ with coefficients in $\mathfrak{L}_Λ$ is zero. Utilizing the Cesàro operators and conditional expectations, we show that the first normal cohomology group of $\mathfrak{L}_Λ$ is trivial. Finally, we prove that the higher cohomology groups of the non-commutative disc algebras with coefficients in the complex field vanish when $|Λ|<\infty$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14729 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hochschild cohomology for free semigroup algebras Huang, Linzhe Ma, Minghui Wei, Xiaomin Operator Algebras Functional Analysis 47L10, 46K50, 47B47 This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set $\{z_α\}_{α\inΛ}$, with the left regular free semigroup algebra $\mathfrak{L}_Λ$ and the non-commutative disc algebra $\mathfrak{A}_Λ$ 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 $\mathfrak{A}_Λ$ with coefficients in $\mathfrak{L}_Λ$ is zero. Utilizing the Cesàro operators and conditional expectations, we show that the first normal cohomology group of $\mathfrak{L}_Λ$ is trivial. Finally, we prove that the higher cohomology groups of the non-commutative disc algebras with coefficients in the complex field vanish when $|Λ|<\infty$. 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. |
| title | Hochschild cohomology for free semigroup algebras |
| topic | Operator Algebras Functional Analysis 47L10, 46K50, 47B47 |
| url | https://arxiv.org/abs/2407.14729 |