{
  "id": "2006.00851",
  "version": "v1",
  "published": "2020-06-01T10:52:42.000Z",
  "updated": "2020-06-01T10:52:42.000Z",
  "title": "Orthogonal $\\ell_1$-sets and extreme non-Arens regularity of preduals of von Neumann algebras",
  "authors": [
    "Mahmoud Filali",
    "Jorge Galindo"
  ],
  "comment": "22 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We propose a new definition for a Banach algebra $\\mathfrak{A}$ to be extremely non-Arens regular, namely that the quotient $\\mathfrak{A}^\\ast/\\mathscr{WAP}(\\mathfrak{A})$ of $\\mathfrak{A}^\\ast$ with the space of its weakly almost periodic elements contains an isomorphic copy of $\\mathfrak{A}^\\ast.$ This definition is simpler and formally stronger than the original one introduced by Granirer in the nineties. We then identify sufficient conditions for the predual $\\mathfrak{V}_\\ast$ of a von Neumann algebra $\\mathfrak{V}$ to be extremely non-Arens regular in this new sense. These conditions are obtained with the help of orthogonal $\\ell_1$-sets of $\\mathfrak{V}_\\ast.$ We show that some of the main algebras in Harmonic Analysis satisfy these conditions. Among them,there is ${\\small \\bullet}$ the weighted semigroup algebra of any weakly cancellative discrete semigroup, for any diagonally bounded weight, ${\\small \\bullet}$ the weighted group algebra of any non-discrete locally compact infinite group and for any weight, ${\\small \\bullet}$ the weighted measure algebra of any locally compact infinite group, for any diagonally bounded weight, ${\\small \\bullet}$ the Fourier algebra of any locally compact infinite group having its local weight greater or equal than its compact covering number, ${\\small \\bullet}$ the Fourier algebra of any countable discrete group containing an infinite amenable subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-01T10:52:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47D35",
      "22D15",
      "43A46",
      "43A60",
      "47C15"
    ],
    "keywords": [
      "von neumann algebra",
      "extreme non-arens regularity",
      "orthogonal",
      "extremely non-arens regular",
      "non-discrete locally compact infinite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}