A new coarsely rigid class of Banach spaces

TL;DR

This paper establishes that reflexive asymptotic-$c_0$ Banach spaces are coarsely rigid, providing a metric characterization via concentration inequalities, and explores the limits of this characterization with quasi-reflexive examples.

Contribution

It introduces a metric characterization of reflexive asymptotic-$c_0$ spaces and proves their coarse rigidity, expanding understanding of coarse embeddings in Banach space theory.

Findings

Reflexive asymptotic-$c_0$ spaces are coarsely rigid.

A metric concentration inequality characterizes this class.

The inequality is not equivalent to non-embeddability of Hamming graphs.

Abstract

We prove that the class of reflexive asymptotic-$c_0$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_0$ space $Y$, then $X$ is also reflexive and asymptotic-$c_0$. In order to achieve this result we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$c_0$ space, we show that this concentration inequality is not equivalent to the non equi-coarse embeddability of the Hamming graphs.