Let \(X\) and \(E\) be Banach spaces. The metrics on these spaces induce both uniform and coarse structures, and we can ask whether the following two statements are equivalent to each other:
- \(X\) uniformly embeds into \(E\).
- \(X\) coarsely embeds into \(E\).
Let us first write down what it means for a map \(\phi \colon X \to E\) to be a uniform, resp. a coarse embedding:
- \(\phi\) is a uniform embedding if for all sequences of pairs \(((x_n,y_n))_{n \in \mathbb{N}}\) in \(X \times X\) we have \[\lim_{n \to \infty} \|x_n – y_n\| = 0 \Leftrightarrow \lim_{n \to \infty} \|\phi(x_n) – \phi(y_n)\| = 0\,.\]
- \(\phi\) is a coarse embedding if for all sequences of pairs \(((x_n,y_n))_{n \in \mathbb{N}}\) in \(X \times X\) we have \[\lim_{n \to \infty} \|x_n – y_n\| = \infty \Leftrightarrow \lim_{n \to \infty} \|\phi(x_n) – \phi(y_n)\| = \infty\,.\]
Note that \(\phi\) is not assumed to be linear! This means that for the above question we completely discard the linear structures of \(X\) and \(E\).
One probably asks oneself now why somebody would be interested in investigating the above question, and why it actually should have any positive answer at all (since at first glance a uniform and a coarse embedding do not have much in common)?
Let us first answer the second part: If \(E\) is a Hilbert space, then the answer is indeed yes! That is to say, a Banach space coarsely embeds into a Hilbert space if and only if it uniformly embeds into a Hilbert space. (This follows from a combination of the main result of arXiv:math/0411269 by Randrianarivony together with a result by Aharoni-Maurey-Mityagin cited as [AMM] therein.) Further, Kalton showed that the same holds true for embeddings into \(\ell^\infty\) (Theorem 5.3 in link).
To answer why the question is `interesting’ we need to mention first the following results: Yu proved that a (separable) metric space of bounded geometry satisfies the coarse Baum-Connes conjecture if the space coarsely embeds into a Hilbert space; and he showed together with Kasparov that the metric space at least still satisfies the coarse Novikov conjecture if it admits a coarse embedding into a uniformly convex Banach space. Now if uniformly convex Banach spaces would always admit coarse embeddings into Hilbert spaces, the latter result would be a special case of the former one. Hence the general question about coarse embeddability between Banach spaces came up.
Now what does this have to do with uniform embeddings? When constructing the first example of a separable metric space (but not with bounded geometry) which does not coarsely embed into a Hilbert space, Dranishnikov-Gong-Lafforgue-Yu (link) used ideas of Enflo who answered negatively the question whether every separable metric space uniformly embeds into a Hilbert space (link). This hinted at a connection between these two types of embeddings between Banach spaces, and by now, having several positive results for the main question of this post (i.e. under different assumptions on the space \(E\)), we know that there is indeed such a connection.