{
  "id": "1704.02595",
  "version": "v1",
  "published": "2017-04-09T12:45:25.000Z",
  "updated": "2017-04-09T12:45:25.000Z",
  "title": "Uniformly recurrent subgroups and simple $C^*$-algebras",
  "authors": [
    "Gabor Elek"
  ],
  "comment": "The paper subsumes and expands our note titled \"On uniformly recurrent subgroups of finitely generated groups.\" (arXiv:1702.01631)",
  "categories": [
    "math.DS",
    "math.OA"
  ],
  "abstract": "We study uniformly recurrent subgroups (URS) introduced by Glasner and Weiss \\cite{GW}. Answering their query we show that any URS $Z$ of a finitely generated group is the stability system of a minimal $Z$-proper action. We also show that for any sofic $URS$ $Z$ there is a $Z$-proper action admitting an invariant measure. We prove that for a $URS$ $Z$ all $Z$-proper actions admits an invariant measure if and only if $Z$ is coamenable. In the second part of the paper we study the $\\C^*$-algebras associated to URS's. We prove that if an URS is generic then its $\\C^*$-algebra is simple. We give an example of URS $Z$ for which the associated simple $C^*$-algebra is not locally reflexive, in particular, it admits both a uniformly amenable trace and a nonuniformly amenable trace.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-09T12:45:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "20E99",
      "46L05"
    ],
    "keywords": [
      "invariant measure",
      "study uniformly recurrent subgroups",
      "proper actions admits",
      "amenable trace",
      "stability system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}