{
  "id": "2107.04662",
  "version": "v1",
  "published": "2021-07-09T20:30:05.000Z",
  "updated": "2021-07-09T20:30:05.000Z",
  "title": "On linear continuous operators between distinguished spaces $C_p(X)$",
  "authors": [
    "Jerzy Kakol",
    "Arkady Leiderman"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $\\Delta$-space. If there exists a linear continuous surjective mapping $T:C_p(X) \\to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) \\to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,\\alpha])$, where $\\alpha$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,\\alpha]) \\to C_p(Y)$ is given. We also observe that for every countable ordinal $\\alpha$ all closed linear subspaces of $C_p([1,\\alpha])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $\\Delta$-spaces we prove that a linear continuous surjection $T:C_p(X) \\to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X \\subset \\mathbb{R}^n$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-09T20:30:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "46A03",
      "46A20"
    ],
    "keywords": [
      "linear continuous operators",
      "linear continuous surjective mapping",
      "infinite compact",
      "tychonoff space",
      "special attention"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}