Weak∗ fixed point property in ℓ1 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, 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 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.