Stability constants of the weak$^*$ fixed point property for the space $\ell_1$

TL;DR

This paper investigates the stability of the weak* fixed point property in the space 3b, providing quantitative bounds on how much the predual space can vary while maintaining this property.

Contribution

It establishes precise quantitative estimates for the stability of the weak* fixed point property in 3b, depending on the geometry of the predual space and its distance from c0.

Findings

The stability radius depends only on the smallest radius containing all cluster points of extreme points.

Preduals of 3b within a distance less than 3 from c0 preserve the weak* fixed point property.

Explicit bounds are provided for the stability of the property under Banach-Mazur perturbations.

Abstract

The main aim of the paper is to study some quantitative aspects of the stability of the weak$^*$ fixed point property for nonexpansive maps in $\ell_1$ (shortly, $w^*$-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 $w^*$-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 $w^*$-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$^*$ topology on $\ell_1$ such that the $\sigma(\ell_1,X)$-fpp holds.