{
  "id": "1703.00882",
  "version": "v1",
  "published": "2017-03-02T18:17:22.000Z",
  "updated": "2017-03-02T18:17:22.000Z",
  "title": "Extending representations of Banach algebras to their biduals",
  "authors": [
    "Eusebio Gardella",
    "Hannes Thiel"
  ],
  "comment": "16 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We show that a representation of a Banach algebra $A$ on a Banach space $X$ can be extended to a canonical representation of $A^{**}$ on $X$ if and only if certain orbit maps $A\\to X$ are weakly compact. We apply this to study when the essential space of a representation is complemented. This provides a tool to disregard the difference between degenerate and nondegenerate representations on Banach spaces. As an application we show that a C*-algebra $A$ has an isometric representation on an $L^p$-space, for $p\\in[1,\\infty)\\setminus\\{2\\}$, if and only if $A$ is commutative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-02T18:17:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47L10",
      "43A65",
      "46E30"
    ],
    "keywords": [
      "banach algebra",
      "extending representations",
      "banach space",
      "nondegenerate representations",
      "essential space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}