On Intermediate $C^{\ast }$-subalgebras of $C^{\ast }$-simple Group Actions

Abstract

We show that for a large class of actions $\Gamma \curvearrowright \mathcal{A}$ of $C^{\ast }$-simple groups $\Gamma$ on unital $C^{\ast }$-algebras $\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^{\ast }$-subalgebra $\mathcal{B}$, $C_{\lambda }^{\ast }(\Gamma )\subseteq \mathcal{B}\subseteq \mathcal{A}{⋊}_r\Gamma$, is of the form ${\mathcal{A}}_1{⋊}_r\Gamma$, where ${\mathcal{A}}_1$ is a unital $\Gamma$-$C^{\ast }$-subalgebra of $\mathcal{A}$.

We have added more examples of plump subgroups, in particular showed that every non-elementary subgroup of a non-torsion convergence group is plump. We have also proved a similar result in von Neumann algebraic setting. This version also includes an appendix with Yongle Jiang containing an example of a faithful action of a $C^{\ast }$-simple group, for which every intermediate $C^{\ast }$-algebra splits

References in corpus (2)

• On Virtual Conjugacy of Generalized Odometers
• A new look at C*-simplicity and the unique trace property of a group