Weak$^*$ Fixed Point Property in $\ell_1$ and Polyhedrality in Lindenstrauss Spaces

TL;DR

This paper investigates the $w^*$-fixed point property in duals of Lindenstrauss spaces, linking it to geometric properties like polyhedrality, and establishes hierarchical relations among these notions.

Contribution

It introduces new geometric conditions equivalent to the $w^*$-fixed point property and relates them to polyhedrality in Lindenstrauss spaces, advancing the understanding of fixed point theory.

Findings

Equivalence between certain $w^*$-closed subsets and the fixed point property.

A new geometric property of the dual ball characterizes the stable $w^*$-fixed point property.

Hierarchical structure among polyhedrality notions in Lindenstrauss spaces.

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, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable $w^*$-fixed point property. The last geometrical notion was introduced by Fonf and Vesel\'{y} as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators.