On uniform and coarse rigidity of $L^p([0,1])$
Volume 268 / 2023
Studia Mathematica 268 (2023), 235-240
MSC: Primary 46B80; Secondary 46B20.
DOI: 10.4064/sm220603-6-8
Published online: 20 September 2022
Abstract
If $X$ is an almost transitive Banach space with amenable isometry group (for example, if $X=L^p([0,1])$ with $1\leq p \lt \infty $) and $X$ admits a uniformly continuous map $X\overset \phi \longrightarrow E$ into a Banach space $E$ satisfying $$ \inf _{\|x-y\|=r}\|\phi (x)-\phi (y)\| \gt 0 $$ for some $r \gt 0$ (that is, $\phi $ is almost uncollapsed), then $X$ admits a simultaneously uniform and coarse embedding into a Banach space $V$ that is finitely representable in $L^2(E)$.
