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

Enrico Miglierina, Lukasz Piasecki, Emanuele Casini, Łukasz Piasecki, Roxana Popescu

Research output: Contribution to journalArticle

6 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)673-684
Number of pages12
JournalJournal of Mathematical Analysis and Applications
Volume452
DOIs
Publication statusPublished - 2017

Keywords

  • Analysis
  • Applied Mathematics
  • Lindenstrauss spaces
  • Renorming
  • Stability of weak⁎ fixed point property
  • Weak⁎ fixed point property
  • ℓ1 space

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