{
  "id": "2208.06241",
  "version": "v1",
  "published": "2022-08-11T11:01:50.000Z",
  "updated": "2022-08-11T11:01:50.000Z",
  "title": "Variable Lebesgue algebra on a Locally Compact group",
  "authors": [
    "Parthapratim Saha",
    "Bipan Hazarika"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For a locally compact group $H$ with a left Haar measure, we study variable Lebesgue algebra $\\mathcal{L}^{p(\\cdot)}(H)$ with respect to a convolution. We show that if $\\mathcal{L}^{p(\\cdot)}(H)$ has bounded exponent, then it contains a left approximate identity. We also prove a necessary and sufficient condition for $\\mathcal{L}^{p(\\cdot)}(H)$ to have an identity. We observe that a closed linear subspace of $\\mathcal{L}^{p(\\cdot)}(H)$ is a left ideal if and only if it is left translation invariant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-11T11:01:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A10",
      "43A15",
      "43A75",
      "43A77"
    ],
    "keywords": [
      "locally compact group",
      "study variable lebesgue algebra",
      "left haar measure",
      "left approximate identity",
      "left translation invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}