Stability constants of the weak⁎ fixed point property for the space ℓ1

Abstract

The main aim of the paper is to study some quantitative aspects of the stability of the weak⁎ fixed point property for nonexpansive mappings in ℓ1 (shortly, w⁎-fpp). We focus on two complementary approaches to this topic. First, given a predual X of ℓ1 such that the σ(ℓ1,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 ℓ1 containing all σ(ℓ1,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 ℓ1. Namely, we show that every predual X of ℓ1 with a distance from c0 strictly less than 3, induces a weak⁎ topology on ℓ1 such that the σ(ℓ1,X)-fpp holds.