{
  "id": "1708.09635",
  "version": "v1",
  "published": "2017-08-31T09:32:40.000Z",
  "updated": "2017-08-31T09:32:40.000Z",
  "title": "The radical of the bidual of a Beurling algebra",
  "authors": [
    "Jared T. White"
  ],
  "comment": "17 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove that the bidual of a Beurling algebra on $\\mathbb{Z}$, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that ${\\rm rad\\,}(\\ell^{\\, 1}(\\oplus_{i=1}^\\infty \\mathbb{Z})\")$ contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight $\\omega$ on $\\mathbb{Z}$ such that the bidual of $\\ell^{\\, 1}(\\mathbb{Z}, \\omega)$ contains a radical element which is not nilpotent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-31T09:32:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A20",
      "46H10",
      "16N20"
    ],
    "keywords": [
      "beurling algebra",
      "first arens product",
      "contains nilpotent elements",
      "results settles",
      "banach algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}