Weak∗ fixed point property in ℓ1 and polyhedrality in Lindenstrauss spaces

Enrico Miglierina, Lukasz Piasecki, Emanuele Casini, Łukasz Piasecki, Roxana Popescu

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)159-172
Number of pages14
JournalStudia Mathematica
Volume241
DOIs
Publication statusPublished - 2018

Keywords

  • Extension of compact operators
  • Lindenstrauss spaces
  • Mathematics (all)
  • Nonexpansive mappings
  • Polyhedral spaces
  • Stability of the w∗-fixed point property
  • W∗-fixed point property
  • ℓ1 space

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