On Intermediate $C^*$-subalgebras of $C^*$-simple Group Actions

Tattwamasi Amrutam

On Intermediate $C^$-subalgebras of $C^$-simple Group Actions

Operator Algebras 2019-02-08 v2

Authors: Tattwamasi Amrutam

Abstract

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

Keywords

c*-simple groups group actions intermediate c*-subalgebras hyperbolic groups crossed products amenable radical

Cite

@article{arxiv.1811.11381,
  title  = {On Intermediate $C^*$-subalgebras of $C^*$-simple Group Actions},
  author = {Tattwamasi Amrutam},
  journal= {arXiv preprint arXiv:1811.11381},
  year   = {2019}
}

Comments

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 $C^*$-simple group, for which every intermediate $C^*$-algebra splits