Property $(\beta)$ and uniform quotient maps

TL;DR

This paper investigates the geometric property $(eta)$ to understand when Banach spaces are uniform quotients of $ ext{ell}_p$, providing positive answers for $1<p<2$ and conditions for $c_0$ quotients.

Contribution

It applies the geometric property $(eta)$ to characterize uniform quotient maps from $ ext{ell}_p$, advancing understanding of Banach space quotients.

Findings

Positive answer for $1<p<2$ regarding uniform quotients of $ ext{ell}_p$

Necessary condition for a Banach space to have $c_0$ as a uniform quotient

Application of property $(eta)$ to quotient map analysis

Abstract

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of $\ell_p$, $1 < p \neq 2 < \infty$, must be isomorphic to a linear quotient of $\ell_p$. We apply the geometric property $(\beta)$ of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case $1<p<2$. We also give a necessary condition for a Banach space to have $c_0$ as a uniform quotient.