{
  "id": "2202.03111",
  "version": "v1",
  "published": "2022-02-07T12:46:21.000Z",
  "updated": "2022-02-07T12:46:21.000Z",
  "title": "Symmetry and inverse-closedness of some $p$-Beurling algebras",
  "authors": [
    "Prakash A. Dabhi",
    "Karishman B. Solanki"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $(G,d)$ be a metric space with the counting measure $\\mu$ satisfying some growth conditions. Let $\\omega(x,y)=(1+d(x,y))^\\delta$ for some $0<\\delta\\leq1$. Let $0<p\\leq1$. Let $\\mathcal A_{p\\omega}$ be the collection of kernels $K$ on $G\\times G$ satisfying $\\max\\{\\sup_x\\sum_y |K(x,y)|^p\\omega(x,y)^p, \\sup_y\\sum_x |K(x,y)|^p\\omega(x,y)^p\\}<\\infty$. Each $K \\in \\mathcal A_{p\\omega}$ defines a bounded linear operator on $\\ell^2(G)$. If in addition, $\\omega$ satisfies the weak growth condition, then we show that $\\mathcal A_{p\\omega}$ is inverse closed in $B(\\ell^2(G))$. We shall also discuss inverse-closedness of $p$-Banach algebra of infinite matrices over $\\mathbb Z^d$ and the $p$-Banach algebra of weighted $p$-summable sequences over $\\mathbb Z^{2d}$ with the twisted convolution. In order to show these results, we prove Hulanicki's lemma and Barnes' lemma for $p$-Banach algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-07T12:46:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B37",
      "43A15",
      "46K05",
      "47G10"
    ],
    "keywords": [
      "beurling algebras",
      "banach algebra",
      "inverse-closedness",
      "weak growth condition",
      "metric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}