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