{
  "id": "1308.6555",
  "version": "v3",
  "published": "2013-08-29T18:52:44.000Z",
  "updated": "2013-10-29T18:12:32.000Z",
  "title": "On embeddings of $C_0(K)$ spaces into $C_0(L,X)$ spaces",
  "authors": [
    "Leandro Candido"
  ],
  "comment": "This is a reorganization of the previous manuscript. Some results have been removed, some improved",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $C_0(K, X)$ denote the space of all continuous $X$-valued functions defined on the locally compact Hausdorff space $K$ which vanish at infinity, provided with the supremum norm. If $X$ is the scalar field, we denote $C_0(K, X)$ by simply $C_0(K)$. In this paper we prove that for locally compact Hausdorff spaces $K$ and $L$ and for Banach space $X$ containing no copy of $c_0$, if there is a isomorphic embedding of $C_0(K)$ into $C_0(L,X)$ where either $X$ is separable or $X^*$ has the Radon-Nikod\\'ym property, then either $K$ is finite or $|K|\\leq |L|$. As a consequence of this result, if there is a isomorphic embedding of $C_0(K)$ into $C_0(L,X)$ where $X$ contains no copy of $c_0$ and $L$ is scattered, then $K$ must be scattered.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-10-29T18:12:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E40",
      "46B25"
    ],
    "keywords": [
      "locally compact hausdorff space",
      "supremum norm",
      "scalar field",
      "banach space",
      "isomorphic embedding"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.6555C"
    }
  }
}