{
  "id": "math/0310421",
  "version": "v2",
  "published": "2003-10-27T21:07:43.000Z",
  "updated": "2004-05-04T19:45:50.000Z",
  "title": "Representations of Group Algebras in Spaces of Completely Bounded Maps",
  "authors": [
    "Roger R. Smith",
    "Nico Spronk"
  ],
  "comment": "29 pages. Accepted in Indiana Univ. math J",
  "journal": "Indiana Univ. Math. J. 54 (3):873-896, 2005.",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded maps on B(H). We study the range of the map Gamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\\otimes pi(s)^*dmu(s) where we identify CB^sigma(B(H)) with the extended Haagerup tensor product B(H)\\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated by integrating pi to L^1(G) is unital exactly when pi is norm continuous to show that Gamma_pi(L^1(G))\\subset B(H)\\otimes^{eh}B(H) exactly when pi is norm continuous. For the case that G is abelian, we study Gamma_pi(M(G)) as a subset of the Varopoulos algebra. We also characterise positive definite elements of the Varopoulos algebra in terms of completely positive operators.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-05-04T19:45:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L07",
      "22D20",
      "22D10",
      "22D25",
      "22B05"
    ],
    "keywords": [
      "group algebra",
      "bounded maps",
      "varopoulos algebra",
      "extended haagerup tensor product",
      "characterise positive definite elements"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....10421S"
    }
  }
}