Coarse and uniform embeddings

Abstract

In these notes, we study the relation between uniform and coarse embeddings between Banach spaces. In order to understand this relation better, we also look at the problem of when a coarse embedding can be assumed to be topological. Among other results, we show that if a Banach space $X$ uniformly embeds into a minimal Banach space $Y$, then $X$ simultaneously coarsely and uniformly embeds into $Y$, and if a Banach space $X$ coarsely embeds into a minimal Banach space $Y$, then $X$ simultaneously coarsely and homeomorphically embeds into $Y$ by a map with uniformly continuous inverse.Comment: in Journal of Functional Analysis (2017

Equivariant Geometry of Banach Spaces and Topological Groups

“…We should mention here that B. Braga [12] has been able to use our construction above coupled with a result of E. Odell and T. Schlumprecht [48] to show that ℓ 2 admits a simultaneously uniform and coarse embedding into every Banach space with an unconditional basis and finite cotype.…”

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

“…Furthermore, it was proven in [13, Theorem 2.1] that Z p,X is p-AUSable and that Z * p,X is p -AUSable, where p is the conjugate of p, i.e., 1/ p + 1/ p = 1. 9 In particular, if X is infinite dimensional, those spaces are not quasi-reflexive. Therefore, it is natural to look at those spaces when looking for counterexamples for the existence of concentration inequalities.…”

“…Since ℓ 1 coarse Lipschitzly embeds into a reflexive space (see [AL85], Theorem 1), by Theorem 1.4 of [Bra17b], ℓ 1 coarse Lipschitzly embeds into a reflexive space by a continuous map. As ℓ 1 is a Schur space, i.e., every weakly convergent sequence converges in norm, it follows that ℓ 1 weakly sequentially continuously coarse Lipschitzly embeds into a reflexive space.…”

Coarse embeddings into superstable spaces

“…Although this will not be needed for the main result in these notes, Corollary 3.2 can actually be improved to show the existence of a coarsely equivalent translation-invariant stable metric on X. Indeed, it has been shown by the first named author (see [Br,Theorem 1.6]) that if X and Y are Banach spaces and f : X Ñ Y is a coarse embedding, then there is a coarse embedding f : X Ñ ℓ 1 pY q with uniformly continuous inverse (meaning ρ f ptq ą 0 whenever t ą 0). Thus, the same proof as in Corollary 3.2 with ℓ 1 pY q replacing Y and f replacing f will yield that Id : pX, }¨}q Ñ pX, dq is a coarse embedding with uniformly continuous inverse.…”