{
  "id": "2309.15570",
  "version": "v1",
  "published": "2023-09-27T10:51:33.000Z",
  "updated": "2023-09-27T10:51:33.000Z",
  "title": "Weak*-Simplicity of Convolution Algebras on Discrete Groups",
  "authors": [
    "Jared T. White"
  ],
  "comment": "18 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We prove that, given a discrete group $G$, and $1 \\leq p < \\infty$, the algebra of $p$-convolution operators $CV_p(G)$ is weak*-simple, in the sense of having no non-trivial weak*-closed ideals, if and only if $G$ is an ICC group. This generalises the basic fact that $vN(G)$ is a factor if and only if $G$ is ICC. When $p=1$, $CV_p(G) = \\ell^1(G)$. In this case we give a more detailed analysis of the weak*-closed ideals, showing that they can be described in terms of the weak*-closed ideals of $\\ell^1(FC(G))$; when $FC(G)$ is finite, this leads to a classification of the weak*-closed ideals of $\\ell^1(G)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-27T10:51:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A15",
      "43A20",
      "47L10",
      "46H10"
    ],
    "keywords": [
      "discrete group",
      "convolution algebras",
      "basic fact",
      "icc group",
      "convolution operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}