{
  "id": "2404.06959",
  "version": "v1",
  "published": "2024-04-10T12:18:22.000Z",
  "updated": "2024-04-10T12:18:22.000Z",
  "title": "Regular inclusions of simple unital $C^*$-algebras",
  "authors": [
    "Keshab Chandra Bakshi",
    "Ved Prakash Gupta"
  ],
  "comment": "16 pages",
  "categories": [
    "math.OA",
    "math.FA"
  ],
  "abstract": "We prove that an inclusion $\\mathcal{B} \\subset \\mathcal{A}$ of simple unital $C^*$-algebras with a finite-index conditional expectation is regular if and only if there exists a finite group $G$ that admits a cocycle action $(\\alpha,\\sigma)$ on the intermediate $C^*$-subalgebra $\\mathcal{C}$ generated by $\\mathcal{B}$ and its centralizer $\\mathcal{C}_\\mathcal{A}(\\mathcal{B})$ such that $\\mathcal{B}$ is outerly $\\alpha$-invariant and $(\\mathcal{B} \\subset \\mathcal{A}) \\cong ( \\mathcal{B} \\subset \\mathcal{C}\\rtimes^r_{\\alpha, \\sigma} G)$. Prior to this characterization, we prove the existence of two-sided and unitary quasi-bases for the minimal conditional expectation of any such inclusion, and also show that such an inclusion has integer Watatani index and depth at most $2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-10T12:18:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple unital",
      "regular inclusions",
      "integer watatani index",
      "minimal conditional expectation",
      "finite-index conditional expectation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}