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Weak$^*$ fixed point property in $\ell _1$ and polyhedrality in Lindenstrauss spaces

Volume 241 / 2018

Emanuele Casini, Enrico Miglierina, Łukasz Piasecki, Roxana Popescu Studia Mathematica 241 (2018), 159-172 MSC: 47H10, 46B45, 46B25. DOI: 10.4064/sm8697-5-2017 Published online: 6 November 2017

Abstract

The aim of this paper is to study the $w^*$-fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of $w^*$-closed subsets of the dual sphere is equivalent to the $w^*$-fixed point property. Then, our main result shows the equivalence between another, stronger geometrical property of the dual ball and the stable $w^*$-fixed point property. This last property was introduced by Fonf and Veselý as a strengthening of polyhedrality. In the last section we show that also the first geometrical assumption that we introduce can be related to a polyhedrality concept for the predual space. Indeed, we give a hierarchical structure of various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we rectify an old result about norm-preserving compact extension of compact operators.

Authors

  • Emanuele CasiniDipartimento di Scienza e Alta Tecnologia
    Università dell’Insubria
    via Valleggio 11
    22100 Como, Italy
    e-mail
  • Enrico MiglierinaDipartimento di Discipline Matematiche
    Finanza Matematica ed Econometria
    Università Cattolica del Sacro Cuore
    via Necchi 9
    20123 Milano, Italy
    e-mail
  • Łukasz PiaseckiInstytut Matematyki
    Uniwersytet Marii Curie-Skłodowskiej
    Pl. Marii Curie-Skłodowskiej 1
    20-031 Lublin, Poland
    e-mail
  • Roxana PopescuDepartment of Mathematics
    University of Pittsburgh
    Pittsburgh, PA 15260, U.S.A.
    e-mail

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