{
  "id": "1412.8481",
  "version": "v1",
  "published": "2014-12-29T21:01:46.000Z",
  "updated": "2014-12-29T21:01:46.000Z",
  "title": "An application of virtual degeneracy to two-valued subsets of $L_{p}$-spaces",
  "authors": [
    "Anthony Weston"
  ],
  "comment": "3 page note",
  "categories": [
    "math.FA"
  ],
  "abstract": "Suppose $0 < p < 2$ and that $(\\Omega, \\mu)$ is a measure space for which $L_{p}(\\Omega, \\mu)$ is at least two-dimensional. Kelleher, Miller, Osborn and Weston have shown that if a subset $B$ of $L_{p}(\\Omega, \\mu)$ does not have strict $p$-negative type, then $B$ is affinely dependent (when $L_{p}(\\Omega, \\mu)$ is considered as a real vector space). Examples show that the converse of this statement is not true in general. In this note we describe a class of subsets of $L_{p}(\\Omega, \\mu)$ for which the converse statement holds. We prove that if a two-valued set $B \\subset L_{p}(\\Omega, \\mu)$ is affinely dependent (when $L_{p}(\\Omega, \\mu)$ is considered as a real vector space), then $B$ does not have strict $p$-negative type. This result is peculiar to two-valued subsets of $L_{p}(\\Omega, \\mu)$ and generalizes an elegant theorem of Murugan. It follows, moreover, that of certain types of isometry with range in $L_{p}(\\Omega, \\mu)$ cannot exist.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-29T21:01:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B04",
      "46B85"
    ],
    "keywords": [
      "two-valued subsets",
      "virtual degeneracy",
      "real vector space",
      "application",
      "affinely dependent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.8481W"
    }
  }
}