{
  "id": "2103.14478",
  "version": "v1",
  "published": "2021-03-26T14:06:42.000Z",
  "updated": "2021-03-26T14:06:42.000Z",
  "title": "The Operator Norm on Weighted Discrete Semigroup Algebras $\\ell^1(S, ω)$",
  "authors": [
    "H. V. Dedania",
    "J. G. Patel"
  ],
  "comment": "7 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\omega$ be a weight on a right cancellative semigroup $S$. Let $\\|\\cdot\\|_{\\omega}$ be the weighted norm on the weighted discrete semigroup algebra $\\ell^1(S, \\omega)$. In this paper, we prove that the weight $\\omega$ satisfies F-property if and only if the operator norm $\\| \\cdot \\|_{\\omega op}$ of $\\| \\cdot \\|_{\\omega}$ is exactly equal to another weighted norm $\\| \\cdot \\|_{\\widetilde{\\omega}_1}$ [Theorem 2.5 ($iii$)]. Though its proof is elementary, the result is unexpectedly surprising. In particular, $\\| \\cdot \\|_{1 op}$ is same as $\\| \\cdot \\|_1$ on $\\ell^1(S)$. Moreover, various examples are discussed to understand the relating among $\\| \\cdot \\|_{\\omega op}$, $\\| \\cdot \\|_{\\omega}$, and $\\ell^1(S, \\omega)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-26T14:06:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46H05",
      "43A20"
    ],
    "keywords": [
      "weighted discrete semigroup algebra",
      "operator norm",
      "weighted norm",
      "right cancellative semigroup",
      "satisfies f-property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}