Coarse embeddings of metric spaces into Banach spaces

Abstract

There are several characterizations of coarse embeddability of a discrete metric space into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces $L_p(\mu )$, we get their coarse embeddability into a Hilbert space for $0<p<2$. This together with a theorem by Banach and Mazur yields that coarse embeddability into ${\ell }_2$ and into $L_p(0,1)$ are equivalent when $1\le p<2$. A theorem by G.Yu and the above allow to extend to $L_p(\mu )$, $0<p<2$, the range of spaces, coarse embedding into which guarantees for a finitely generated group $\Gamma$ to satisfy the Novikov Conjecture.

8 pages