{
  "id": "0708.4045",
  "version": "v1",
  "published": "2007-08-30T00:32:37.000Z",
  "updated": "2007-08-30T00:32:37.000Z",
  "title": "Stable rank for inclusions of C*-algebras",
  "authors": [
    "Hiroyuki Osaka"
  ],
  "comment": "9 pages",
  "categories": [
    "math.OA"
  ],
  "abstract": "When a unital \\ca $A$ has topological stable rank one (write $\\tsr(A) = 1$), we know that $\\tsr(pAp) \\leq 1$ for a non-zero projection $p \\in A$. When, however, $\\tsr(A) \\geq 2$, it is generally faluse. We prove that if a unital C*-algebra $A$ has a simple unital C*-subalgebra $D$ of $A$ with common unit such that $D$ has \\PSP and $\\sup_{p\\in P(D)}\\tsr(pAp) < \\infty$, then $\\tsr(A) \\leq 2.$ As an application let $A$ be a simple unital \\ca with $\\tsr(A) = 1$ and \\PSP, $\\{G_k\\}_{k=1}^n$ finite groups, $\\af_k$ actions from $G_k$ to ${\\rm Aut}((...((A\\times_{\\af_1}G_1)\\times_{\\af_2} G_2)...)\\times_{\\af_{k-1}}G_{k-1}).$ $(G_0 = \\{1\\})$ Then $$ \\tsr((... ((A\\times_{\\af_1}G_1)\\times_{\\af_2} G_2)...)\\times_{\\af_n}G_n) \\leq 2. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-08-30T00:32:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L05"
    ],
    "keywords": [
      "inclusions",
      "simple unital",
      "non-zero projection",
      "common unit",
      "finite groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.4045O"
    }
  }
}