{
  "id": "1105.1914",
  "version": "v1",
  "published": "2011-05-10T11:13:11.000Z",
  "updated": "2011-05-10T11:13:11.000Z",
  "title": "Fixed points of normal completely positive maps on B(H)",
  "authors": [
    "Bojan Magajna"
  ],
  "comment": "17 pages",
  "categories": [
    "math.OA",
    "math-ph",
    "math.FA",
    "math.MP"
  ],
  "abstract": "Given a sequence of bounded operators $a_j$ on a Hilbert space $H$ with $\\sum a_j^*a_j=1=\\sum a_ja_j^*$, we study the map $\\Psi$ defined on $B(H)$ by $\\Psi(x)=\\sum a_j^*xa_j$ and its restriction $\\Phi$ to the Hilbert-Schmidt class $C^2(H)$. In the case when the sum $\\sum a_j^*a_j$ is norm-convergent we show in particular that the operator $\\Phi-1$ is not invertible if and only if the C$^*$-algebra $A$ generated by $(a_j)$ has an amenable trace. This is used to show that $\\Psi$ may have fixed points in $B(H)$ which are not in the commutant $A'$ of $A$ even in the case when the weak* closure of $A$ is injective. However, if $A$ is abelian, then all fixed points of $\\Psi$ are in $A'$ even if the operators $a_j$ are not positive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-10T11:13:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L07",
      "47N50",
      "81R15"
    ],
    "keywords": [
      "fixed points",
      "positive maps",
      "hilbert-schmidt class",
      "hilbert space",
      "bounded operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.1914M"
    }
  }
}