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

Abstract

We study the simplicity of $C^{\ast }$-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 $C^{\ast }$-algebra $C_X^{\ast }G$. 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 $C^{\ast }$-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 $\mathbb{X}$.

arXiv comment25 pages. We rewrite the proof of Theorem 5.3 and add Proposition 5.8 to show that the action-based $C^{\ast }$-algebra of the mapping class group of a boundaryless infinite type surface containing a connected nondisplaceable finite type subsurface is not simple