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

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.