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

Emanuele Casini; Enrico Miglierina; Łukasz Piasecki; Roxana Popescu

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arXiv · 1604.07587

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

Emanuele Casini·Enrico Miglierina·Łukasz Piasecki·Roxana Popescu

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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.

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Emanuele Casini, Enrico Miglierina, Łukasz Piasecki, Roxana Popescu. 2016-04-26. Weak$^*$ Fixed Point Property in $\ell_1$ and Polyhedrality in Lindenstrauss Spaces. https://arxiv.org/abs/1604.07587

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