{
  "id": "2312.06189",
  "version": "v1",
  "published": "2023-12-11T08:12:36.000Z",
  "updated": "2023-12-11T08:12:36.000Z",
  "title": "$K$-stability of $C^*$-algebras generated by isometries and unitaries with twisted commutation relations",
  "authors": [
    "Shreema Subhash Bhatt",
    "Bipul Saurabh"
  ],
  "categories": [
    "math.OA",
    "math.KT"
  ],
  "abstract": "In this article, we prove $K$-stability for a family of $C^*$-algebras, which are generated by a finite set of unitaries and isometries satisfying twisted commutation relations. This family includes the $C^*$-algebra of doubly non-commuting isometries and free twist of isometries. Next, we consider the $C^*$-algebra $A_{\\mathcal{V}}$ generated by an $n$-tuple of $\\mathcal{U}$-twisted isometries $\\mathcal{V}$ with respect to a fixed $n\\choose 2$-tuple $\\mathcal{U}=\\{U_{ij}:1\\leq i<j \\leq n\\}$ of commuting unitaries (see \\cite{NarJaySur-2022aa}). Under the assumption that the spectrum of the commutative $C^*$-algebra generated by $(\\{U_{ij}:1\\leq i<j \\leq n\\})$ does not contain any element of finite order in the torus group $\\bbbt^{n\\choose 2}$, we show that $A_{\\mathcal{V}}$ is $K$-stable. Finally, we prove the same result for the $C^*$-algebra generated by a tuple of free $\\mathcal{U}$-twisted isometries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-11T08:12:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58B32",
      "58B34",
      "46L89",
      "47L55"
    ],
    "keywords": [
      "isometries satisfying twisted commutation relations",
      "twisted isometries",
      "free twist",
      "finite set",
      "finite order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}