# Coarse embeddings into superstable spaces

## TL;DR

This paper explores the coarse embeddability of Banach spaces into superstable spaces, revealing that such embeddings imply the existence of specific spreading models and identifying reflexive spaces that cannot embed into superstable spaces.

## Contribution

It establishes that coarse embeddings into superstable spaces imply the existence of p spreading models, extending previous results on uniform embeddings.

## Key findings

- Coarse embeddings imply p spreading models.
- Existence of reflexive Banach spaces not coarsely embeddable into superstable spaces.
- Extension of embedding results from uniform to coarse embeddings.

## 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.