Separable Lindenstrauss spaces whose duals lack the weak$^*$ fixed point property for nonexpansive mappings

TL;DR

This paper investigates conditions under which dual spaces of certain Banach spaces lack the weak* fixed point property for nonexpansive mappings, providing characterizations for separable Lindenstrauss spaces.

Contribution

It characterizes when duals of separable Lindenstrauss spaces lack the weak* fixed point property, especially in relation to containing an isometric copy of c.

Findings

Dual space $X^*$ lacks $w^*$-fixed point property if $X$ contains an isometric copy of c.

Characterizations of weak-star topologies that fail the fixed point property in $ ext{l}_1$ space.

Complete characterization of separable Lindenstrauss spaces with failing $w^*$-fixed point property.

Abstract

In this paper we study the $w^*$-fixed point property for nonexpansive mappings. First we show that the dual space $X^*$ lacks the $w^*$-fixed point property whenever $X$ contains an isometric copy of the space $c$. Then, the main result of our paper provides several characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in $\ell_1$ space. This result allows us to obtain a characterization of all separable Lindenstrauss spaces $X$ inducing the failure of $w^*$-fixed point property in $X^*$.