{
  "id": "0710.3521",
  "version": "v1",
  "published": "2007-10-18T14:30:30.000Z",
  "updated": "2007-10-18T14:30:30.000Z",
  "title": "$C^*-$crossed product of groupoid actions on categories",
  "authors": [
    "Han Li"
  ],
  "comment": "14 pages",
  "categories": [
    "math.OA",
    "math.CT"
  ],
  "abstract": "Suppose that $G$ is a groupoid acting on a small category $H$ in the sense of \\cite[Definition 4]{NOT} and $H\\times_\\alpha G$ is the resulting semi-direct product category (as in \\cite[Proposition 8]{NOT}). We show that there exists a subcategory $H_r \\subseteq H$ satisfying some nice property called ``regularity'' such that $H_r \\times_\\alpha G = H\\times_\\alpha G$. Moreover, we show that there exists a so-called ``quasi action'' (see Definition \\ref{quasi}) $\\beta$ of $G$ on $C^*(H_r)$ (where $C^*(H_r)$ is the semigroupoid $C^*$-algebra as defined in \\cite{EXE}) such that $C^*(H_r\\times_\\alpha G) = C^*(H_r)\\times_\\beta G$ (where the crossed product for $\\beta$ is as defined in Definition \\ref{cross}).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-18T14:30:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L05",
      "18B40"
    ],
    "keywords": [
      "crossed product",
      "groupoid actions",
      "resulting semi-direct product category",
      "quasi action",
      "nice property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.3521L"
    }
  }
}