{
  "id": "1811.02393",
  "version": "v1",
  "published": "2018-11-06T15:03:55.000Z",
  "updated": "2018-11-06T15:03:55.000Z",
  "title": "Left ideals of Banach algebras and dual Banach algebras",
  "authors": [
    "Jared T. White"
  ],
  "comment": "20 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We define what it means for a Banach algebra to be topologically left Noetherian. We show that if $G$ is a compact group, then $L^{\\, 1}(G)$ is topologically left Noetherian if and only if $G$ is metrisable. We prove that given a Banach space $E$ such that $E'$ has BAP, the algebra of compact operators $\\mathcal{K}(E)$ is topologically left Noetherian if and only if $E'$ is separable; it is topologically right Noetherian if and only if $E$ is separable. We then define what it means for a dual Banach algebra to be weak*-topologically left Noetherian, and give some examples for which this condition holds. Finally we give a unified approach to classifying the weak*-closed left ideals of certain dual Banach algebras that are also multiplier algebras, with applications to $M(G)$ for $G$ a compact group, and $\\mathcal{B}(E)$ for $E$ a reflexive Banach space with AP.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-06T15:03:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16P40",
      "46H10",
      "43A10",
      "43A20",
      "47L10"
    ],
    "keywords": [
      "dual banach algebra",
      "left ideals",
      "topologically left noetherian",
      "compact group",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}