# Coarse embeddings into $c_0(\Gamma)$

## TL;DR

This paper investigates the conditions under which large Banach spaces can be coarsely embedded into $c_0( ext{cardinality})$, revealing limitations related to cotype and isomorphism properties.

## Contribution

It establishes that large Banach spaces admitting coarse embeddings into $c_0( ext{cardinality})$ lack nontrivial cotype and are linearly isomorphic to $c_0( ext{density})$ if they have a symmetric basis.

## Key findings

- Spaces with large density fail to have nontrivial cotype.
- Such spaces contain $ ext{ extlangle} ext{infty}^n ext{ extrangle}$ uniformly.
- Spaces with symmetric bases are linearly isomorphic to $c_0( ext{density})$.

## Abstract

Let $\lambda$ be a large enough cardinal number (assuming GCH it suffices to let $\lambda=\aleph_\omega$). If $X$ is a Banach space with $\text{dens}(X)\ge\lambda$, which admits a coarse (or uniform) embedding into any $c_0(\Gamma)$, then $X$ fails to have nontrivial cotype, i.e. $X$ contains $\ell_\infty^n$ $C$-uniformly for every $C>1$. In the special case when $X$ has a symmetric basis, we may even conclude that it is linearly isomorphic with $c_0(\text{dens}X)$.