Bijective rigidity of uniform Roe algebras and injectivity of the comparison map

TL;DR

This paper establishes a deep connection between the algebraic structure of uniform Roe algebras and the coarse geometric properties of the underlying spaces, proving injectivity of a key map and its implications for bijective coarse equivalences.

Contribution

It proves that isomorphic uniform Roe algebras imply bijective coarse equivalences if and only if a certain comparison map is injective, and shows this map is always injective.

Findings

Injectivity of the 0th comparison map is equivalent to bijective coarse equivalence.

The 0th comparison map is unconditionally injective.

If the space is coarsely connected, the map is split-injective.

Abstract

We show that, for uniformly locally finite metric spaces $X$ and $Y$ with isomorphic uniform Roe algebras $C^*_u(X)$ and $C^*_u(Y)$, the existence of a bijective coarse equivalence $f \colon X \to Y$ is equivalent to the injectivity of the $0$th comparison map appearing in the HK conjecture for coarse groupoids. We further prove that the $0$th comparison map is injective unconditionally. Moreover, if the underlying space is coarsely connected, this map is in fact split-injective.