On uniform and coarse rigidity of $L^p([0,1])$

TL;DR

This paper investigates the rigidity properties of $L^p$ spaces, showing conditions under which they admit embeddings into Banach spaces that are finitely representable in $L^2$, highlighting uniform and coarse embedding behaviors.

Contribution

It establishes new rigidity results for $L^p$ spaces with amenable isometry groups, connecting uniform and coarse embeddings to finite representability in $L^2$ spaces.

Findings

$L^p([0,1])$ spaces admit specific embeddings under certain conditions.

Embeddings into spaces finitely representable in $L^2$ are characterized.

Rigidity phenomena are linked to the structure of isometry groups.

Abstract

If $X$ is an almost transitive Banach space with amenable isometry group (for example, if $X=L^p([0,1])$ with $1\leqslant p<\infty$) and $X$ admits a uniformly continuous map $X\overset\phi\longrightarrow E$ into a Banach space $E$ satisfying $$\inf_{\|x-y\|=r} \| \phi(x)-\phi(y)\|>0 $$ for some $r>0$, then $X$ admits a simultaneously uniform and coarse embedding into a Banach space $V$ that is finitely representable in $L^2(E)$.