Simplicity of action-based $C^{*}$-algebras from hyperbolic actions

TL;DR

This paper investigates the conditions under which action-based $C^{*}$-algebras derived from group actions are simple, establishing links between properties of the actions and algebraic simplicity, with applications to big mapping class groups.

Contribution

It introduces generalized properties $P_{naive}$ and $P_{analytic}$ for group actions and proves that $P_{analytic}$ ensures the simplicity of the associated $C^{*}$-algebra, with applications to big mapping class groups.

Findings

Naive property implies analytic property for group actions.

$P_{analytic}$ property guarantees simplicity of the $C^{*}$-algebra.

Big mapping class groups satisfy $P_{naive}^{ ext{X}}$ and produce simple $C^{*}$-algebras.

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 $P_{\text{naive}}$ 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.