Coarse Embeddings of Metric Spaces into Banach Spaces

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 Lp(μ), 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 l2 and into Lp(0, 1) are equivalent when 1 ≤ p < 2. A theorem by G.Yu and the above allow to extend to Lp(μ), 0 < p < 2, the range of spaces, coarse embedding into which guarantees for a finitely generated group Γ to satisfy the Novikov Conjecture.