{
  "id": "1511.03423",
  "version": "v1",
  "published": "2015-11-11T09:11:46.000Z",
  "updated": "2015-11-11T09:11:46.000Z",
  "title": "Similarity degree of Fourier algebras",
  "authors": [
    "Hun Hee Lee",
    "Ebrahim Samei",
    "Nico Spronk"
  ],
  "comment": "14 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We show that for a locally compact group $G$, amongst a class which contains amenable and small invariant neighbourhood groups, that its Fourier algebra $A(G)$ satisfies a completely bounded version Pisier's similarity property with similarity degree at most $2$. Specifically, any completely bounded homomorphism $\\pi: A(G)\\to B(H)$ admits an invertible $S$ in $B(H)$ for which $\\|S\\|\\|S^{-1}\\|\\leq ||\\pi||_{cb}^2$ and $S^{-1}\\pi(\\cdot)S$ extends to a $*$-representation of the $C^*$-algebra $C_0(G)$. This significantly improves some results due to Brannan and Samei (J. Funct. Anal. 259, 2010) and Brannan, Daws and Samei (M\\\"{u}nster J. Math 6, 2013). We also note that $A(G)$ has completely bounded similarity degree $1$ if and only if it is completely isomorphic to an operator algebra if and only if $G$ is finite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-11T09:11:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46K10",
      "43A30",
      "46L07",
      "43A07"
    ],
    "keywords": [
      "fourier algebra",
      "small invariant neighbourhood groups",
      "bounded version pisiers similarity property",
      "bounded similarity degree",
      "compact group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151103423L"
    }
  }
}