Year: 2026
Type: preprint
Abstract: 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 $P_{\mathrm{naive}}$ and $P_{\mathrm{analytic}}$ relative to this action and show that the naive form implies the analytic form. We also prove that $P_{\mathrm{analytic}}$ associated with a coefficient-separating action representation ensures the simplicity of the action-based $C^*$-algebra. As an application, for every connected orientable boundaryless infinite type surface containing a connected nondisplaceable finite type subsurface, we deduce that its mapping class group satisfies the properties $P_{\mathrm{naive}}^{\mathbb{X}}$ and $P_{\mathrm{analytic}}^{\mathbb{X}}$ for one fixed hyperbolic space $\mathbb X$.
Source: arXiv (Cornell University)
Author: Lou, Tianyi
Institutions: Institut Fourier, Université Grenoble Alpes
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7154482282
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