Stability constants of the weak${}^{\ast }$ fixed point property for the space ${\ell }_1$

Abstract

The main aim of the paper is to study some quantitative aspects of the stability of the weak${}^{\ast }$ fixed point property for nonexpansive maps in ${\ell }_1$ (shortly, $w^{\ast }$-fpp). We focus on two complementary approaches to this topic. First, given a predual $X$ of ${\ell }_1$ such that the $\sigma ({\ell }_1,X)$-fpp holds, we precisely establish how far, with respect to the Banach-Mazur distance, we can move from $X$ without losing the $w^{\ast }$-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in ${\ell }_1$ containing all $\sigma ({\ell }_1,X)$-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the $w^{\ast }$-fpp in the restricted framework of preduals of ${\ell }_1$. Namely, we show that every predual $X$ of ${\ell }_1$ with a distance from $c_0$ strictly less than $3$, induces a weak${}^{\ast }$ topology on ${\ell }_1$ such that the $\sigma ({\ell }_1,X)$-fpp holds.