Coarse quotients of metric spaces and embeddings of uniform Roe algebras

TL;DR

This paper explores the relationship between embeddings of uniform Roe algebras with large range and coarse quotients of metric spaces, revealing how the large-scale geometry of one space influences another.

Contribution

It establishes a connection between embeddings of uniform Roe algebras with large range and the existence of coarse quotients, linking algebraic properties to geometric structures.

Findings

If $Y$ has property A and admits a large range embedding, then there is a bijective coarse quotient from $X$ to $Y.

Large scale geometry of $Y$ is controlled by $X$ in such embeddings.

Finite asymptotic dimension of $X$ implies the same for $Y$.

Abstract

We study embeddings of uniform Roe algebras which have "large range" in their codomain and the relation of those with coarse quotients between metric spaces. Among other results, we show that if $Y$ has property A and there is an embedding $\Phi:\mathrm{C}^*_u(X)\to \mathrm{C}^*_u(Y)$ with "large range" and so that $\Phi(\ell_\infty(X))$ is a Cartan subalgebra of $\mathrm{C}^*_u(Y)$, then there is a bijective coarse quotient $X\to Y$. This shows that the large scale geometry of $Y$ is, in some sense, controlled by the one of $X$. For instance, if $X$ has finite asymptotic dimension, so does $Y$.