Simplicity of action-based $C^{*}$-algebras from hyperbolic actions
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914523247542272 |
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| author | Lou, Tianyi |
| author_facet | Lou, Tianyi |
| contents | We study the simplicity of $C^{*}$-algebras built from group actions. For a faithful isometric action of a group $G$ on a countable metric space $X$, we use the associated action representation on $\ell^2(X)$ to define the action-based $C^{*}$-algebra $C^{*}_{X}G$. We define generalized versions of the properties $P_{\text{naive}}$ and $P_{\text{analytic}}$ relative to the action and show that the naive form implies the analytic form. We also prove that the properties $P_{\text{analytic}}$ associated with a continuous action ensure the simplicity of the action-based $C^*$-algebra. As an application, we deduce that big mapping class groups satisfy the property $P_{\text{naive}}^{\mathbb{X}}$ and the associated action-based $C^*$-algebra is simple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12520 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simplicity of action-based $C^{*}$-algebras from hyperbolic actions Lou, Tianyi Operator Algebras Group Theory 46L05, 20F65, 57K20 We study the simplicity of $C^{*}$-algebras built from group actions. For a faithful isometric action of a group $G$ on a countable metric space $X$, we use the associated action representation on $\ell^2(X)$ to define the action-based $C^{*}$-algebra $C^{*}_{X}G$. We define generalized versions of the properties $P_{\text{naive}}$ and $P_{\text{analytic}}$ relative to the action and show that the naive form implies the analytic form. We also prove that the properties $P_{\text{analytic}}$ associated with a continuous action ensure the simplicity of the action-based $C^*$-algebra. As an application, we deduce that big mapping class groups satisfy the property $P_{\text{naive}}^{\mathbb{X}}$ and the associated action-based $C^*$-algebra is simple. |
| title | Simplicity of action-based $C^{*}$-algebras from hyperbolic actions |
| topic | Operator Algebras Group Theory 46L05, 20F65, 57K20 |
| url | https://arxiv.org/abs/2604.12520 |