On stability of metric spaces and Kalton's property $Q$

TL;DR

This paper explores the relationship between property Q and upper stability in metric spaces, demonstrating that property Q implies upper stability and providing new insights into the structure of reflexive Banach spaces.

Contribution

It establishes that property Q implies upper stability and offers a direct proof linking upper stability to the asymptotic structure of reflexive Banach spaces.

Findings

Property Q implies upper stability in metric spaces

Reflexive Banach spaces are upper stable due to their asymptotic structure

Provides a new perspective on invariants distinguishing Banach space classes

Abstract

The first named author introduced the notion of upper stability for metric spaces as a relaxation of stability. The motivation was a search for a new invariant to distinguish the class of reflexive Banach spaces from stable metric spaces in the coarse and uniform category. In this paper we show that property $Q$ does in fact imply upper stability. We also provide a direct proof of the fact that reflexive spaces are upper stable by relating the latter notion to the asymptotic structure of Banach spaces.