{
  "id": "1205.2802",
  "version": "v3",
  "published": "2012-05-12T18:02:18.000Z",
  "updated": "2013-12-20T13:56:46.000Z",
  "title": "The Banach algebra of continuous bounded functions with separable support",
  "authors": [
    "M. R. Koushesh"
  ],
  "comment": "9 pages. arXiv admin note: text overlap with arXiv:1204.6660",
  "journal": "Studia Math. 210 (2012), 227-237",
  "categories": [
    "math.FA",
    "math.GN"
  ],
  "abstract": "We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$, which we explicitly construct as a subspace of the Stone--\\v{C}ech compactification of $X$, is countably compact, and if $X$ is non-separable, is moreover non-normal; in addition $C_0(Y)=C_{00}(Y)$. When the underlying field of scalars is the complex numbers, the space $Y$ coincides with the spectrum of the ${C}^*$-algebra $C_s(X)$. Further, we find the dimension of the algebra $C_s(X)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-12-20T13:56:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46J10",
      "46J25",
      "46E25",
      "46E15",
      "54C35",
      "54D35",
      "46H05",
      "16S60"
    ],
    "keywords": [
      "banach algebra",
      "continuous bounded functions",
      "separable support",
      "commutative gelfand-naimark type theorem",
      "locally compact hausdorff space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.2802K"
    }
  }
}