{
  "id": "2305.08543",
  "version": "v1",
  "published": "2023-05-15T11:13:17.000Z",
  "updated": "2023-05-15T11:13:17.000Z",
  "title": "On projectional skeletons and the Plichko property in Lipschitz-free Banach spaces",
  "authors": [
    "Antonio J. Guirao",
    "Vicente Montesinos",
    "Andrés Quilis"
  ],
  "comment": "16 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space of all $\\mathbb{R}$-trees has the Plichko property witnessed by molecules, and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of $\\ell_\\infty$ containing $c_0$ is an $r$-Lipschitz retract for $r < 2$, which implies in particular that $\\mathcal{F}(\\ell_\\infty)$ is not $r$-Plichko for $r < 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-15T11:13:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "46B08",
      "46B26"
    ],
    "keywords": [
      "plichko property",
      "lipschitz-free banach spaces",
      "metric space",
      "study projectional skeletons",
      "dirac measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}