{
  "id": "1611.02133",
  "version": "v1",
  "published": "2016-11-07T15:53:30.000Z",
  "updated": "2016-11-07T15:53:30.000Z",
  "title": "Stability constants of the weak$^*$ fixed point property for the space $\\ell_1$",
  "authors": [
    "Emanuele Casini",
    "Enrico Miglierina",
    "Łukasz Piasecki",
    "Roxana Popescu"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "The main aim of the paper is to study some quantitative aspects of the stability of the weak$^*$ fixed point property for nonexpansive maps in $\\ell_1$ (shortly, $w^*$-fpp). We focus on two complementary approaches to this topic. First, given a predual $X$ of $\\ell_1$ such that the $\\sigma(\\ell_1,X)$-fpp holds, we precisely establish how far, with respect to the Banach-Mazur distance, we can move from $X$ without losing the $w^*$-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in $\\ell_1$ containing all $\\sigma(\\ell_1,X)$-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the $w^*$-fpp in the restricted framework of preduals of $\\ell_1$. Namely, we show that every predual $X$ of $\\ell_1$ with a distance from $c_0$ strictly less than $3$, induces a weak$^*$ topology on $\\ell_1$ such that the $\\sigma(\\ell_1,X)$-fpp holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-07T15:53:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47H10",
      "46B45",
      "46B25"
    ],
    "keywords": [
      "fixed point property",
      "stability constants",
      "fpp holds",
      "banach-mazur distance",
      "complementary approaches"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}