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arXiv · 1703.01891

Coarse embeddings into $c_0(\Gamma)$

Petr Hajek·Thomas Schlumprecht
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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 $c_0(\text{dens}X)$.

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Petr Hajek, Thomas Schlumprecht. 2017-03-06. Coarse embeddings into $c_0(\Gamma)$. https://arxiv.org/abs/1703.01891

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