Coarse embeddings into superstable spaces

Abstract

Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of ${\ell }_p$, for some $p\in [1,\infty )$. In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then $X$ must contain an isomorphic copy of ${\ell }_p$, for some $p\in [1,\infty )$. In these notes, we show that if a Banach space coarsely embeds into a superstable Banach space, then $X$ has a spreading model isomorphic to ${\ell }_p$, for some $p\in [1,\infty )$. In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any superstable Banach space.

arXiv comment29 pages