Weak${}^{\ast }$ Fixed Point Property in ${\ell }_1$ and Polyhedrality in Lindenstrauss Spaces

Abstract

The aim of this paper is to study the $w^{\ast }$-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 $w^{\ast }$-closed subsets of the dual sphere is equivalent to the $w^{\ast }$-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 $w^{\ast }$-fixed point property. The last geometrical notion was introduced by Fonf and Veselý 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.