{
  "id": "2204.01557",
  "version": "v1",
  "published": "2022-04-04T15:04:45.000Z",
  "updated": "2022-04-04T15:04:45.000Z",
  "title": "The Josefson--Nissenzweig theorem and filters on $ω$",
  "authors": [
    "Witold Marciszewski",
    "Damian Sobota"
  ],
  "comment": "46 pages, comments are welcome!",
  "categories": [
    "math.FA",
    "math.GN",
    "math.LO"
  ],
  "abstract": "For a free filter $F$ on $\\omega$, endow the space $N_F=\\omega\\cup\\{p_F\\}$, where $p_F\\not\\in\\omega$, with the topology in which every element of $\\omega$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\\cup\\{p_F\\}$ for $A\\in F$. Spaces of the form $N_F$ constitute the class of the simplest non-discrete Tychonoff spaces. In this paper we study them in the context of the celebrated Josefson--Nissenzweig theorem from Banach space theory, e.g., we completely describe those filters $F$ for which the spaces $N_F$ carry sequences $\\langle\\mu_n\\colon n\\in\\omega\\rangle$ of finitely supported signed measures satisfying the following two conditions: $\\|\\mu_n\\|=1$ for every $n\\in\\omega$, and $\\mu_n(f)\\to 0$ for every bounded continuous real-valued function $f$ on $N_F$. As a consequence, we obtain a description of a wide class of filters $F$ having the following properties: (1) if $X$ is a Tychonoff space and $N_F$ is homeomorphic to a subspace of $X$, then the space $C_p^*(X)$ of bounded continuous real-valued functions on $X$ contains a complemented copy of the space $c_0$ endowed with the pointwise topology, (2) if $K$ is a compact Hausdorff space and $N_F$ is homeomorphic to a subspace of $K$, then the Banach space $C(K)$ of continuous real-valued functions on $K$ is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space $K$ contains a non-trivial convergent sequence, then the space $C(K)$ is not Grothendieck.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-04T15:04:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "josefson-nissenzweig theorem",
      "compact hausdorff space",
      "bounded continuous real-valued function",
      "supported signed measures satisfying",
      "simplest non-discrete tychonoff spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}