{
  "id": "2108.08832",
  "version": "v1",
  "published": "2021-08-19T17:52:40.000Z",
  "updated": "2021-08-19T17:52:40.000Z",
  "title": "Inclusions of $C^*$-algebras arising from fixed-point algebras",
  "authors": [
    "Siegfried Echterhoff",
    "Mikael Rørdam"
  ],
  "comment": "14 pages",
  "categories": [
    "math.OA"
  ],
  "abstract": "We examine inclusions of $C^*$-algebras of the form $A^H \\subseteq A \\rtimes_{r} G$, where $G$ and $H$ are groups acting on a unital simple $C^*$-algebra $A$ by outer automorphisms and when $H$ is finite. We show that $A^H \\subseteq A$ is $C^*$-irreducible, in the sense that all intermediate $C^*$-algebras are simple, if $H$ moreover is abelian. We further show that $A^H \\subseteq A \\rtimes_{r} G$ is $C^*$-irreducible when $H$ is abelian, if the two actions of $G$ and $H$ on $A$ commute, and the combined action of $G \\times H$ on $A$ is outer. We illustrate these results with examples of outer group actions on the irrational rotation $C^*$-algebras. We exhibit, among other examples, $C^*$-irreducible inclusions of AF-algebras that have intermediate $C^*$-algerbras that are not AF-algebras, in fact, the irrational rotation $C^*$-algebra appears as an intermediate $C^*$-algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-19T17:52:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L05",
      "46L35",
      "46L55"
    ],
    "keywords": [
      "fixed-point algebras",
      "algebras arising",
      "inclusions",
      "irrational rotation",
      "intermediate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}