{
  "id": "math/0303171",
  "version": "v4",
  "published": "2003-03-13T18:49:16.000Z",
  "updated": "2003-09-08T16:32:20.000Z",
  "title": "Operator space structure and amenability for Figà-Talamanca-Herz algebras",
  "authors": [
    "Anselm Lambert",
    "Matthias Neufang",
    "Volker Runde"
  ],
  "comment": "25 pages; some minor, hopefully clarifying revisions",
  "journal": "J. Funct. Anal. 211 (2004), 245-269",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "Column and row operator spaces - which we denote by COL and ROW, respectively - over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group $G$ and $p,p' \\in (1,\\infty)$ with $\\frac{1}{p} + \\frac{1}{p'} = 1$, we use the operator space structure on $CB(COL(L^{p'}(G)))$ to equip the Figa-Talamanca-Herz algebra $A_p(G)$ with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for $p \\leq q \\leq 2$ or $2 \\leq q \\leq p$ and amenable $G$, the canonical inclusion $A_q(G) \\subset A_p(G)$ is completely bounded (with cb-norm at most $K_G^2$, where $K_G$ is Grothendieck's constant). As an application, we show that $G$ is amenable if and only if $A_p(G)$ is operator amenable for all - and equivalently for one - $p \\in (1,\\infty)$; this extends a theorem by Z.-J. Ruan.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2003-09-08T16:32:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A15",
      "43A30",
      "46B70",
      "46J99",
      "46L07",
      "47L25",
      "47L50"
    ],
    "keywords": [
      "operator space structure",
      "figà-talamanca-herz algebras",
      "amenability",
      "row operator spaces",
      "arbitrary banach spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......3171L"
    }
  }
}