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

Emanuele Casini, Enrico Miglierina, Łukasz Piasecki

Published 2016-04-26Version 1

The main aim of this paper is to study the $w^*$-fixed point property for nonexpansive mappings in the duals of separable Lindestrauss 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} in 2004 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 seen as a polyhedral concept for the predual space. Finally, we prove several relationships between various polyhedrality notions in the framework of Lindenstrauss spaces and we provide some examples showing that none of the considered implications can be reverted.