Coarse and uniform embeddings

TL;DR

This paper explores the relationship between uniform and coarse embeddings of Banach spaces, establishing conditions under which these embeddings coincide or can be strengthened to topological or homeomorphic embeddings.

Contribution

It demonstrates that uniform embeddings into minimal Banach spaces can be strengthened to coarse and uniform embeddings, and coarse embeddings can be made homeomorphic with uniformly continuous inverses.

Findings

Uniform embeddings into minimal Banach spaces imply simultaneous coarse and uniform embeddings.

Coarse embeddings into minimal Banach spaces can be upgraded to homeomorphic embeddings.

Results clarify the interplay between coarse, uniform, and topological embeddings in Banach space theory.

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.