# Low dimensional properties of uniform Roe algebras

## TL;DR

This paper investigates the relationship between low dimensional properties of uniform Roe algebras and the asymptotic dimension of the underlying space, providing characterizations and $K$-theoretic insights.

## Contribution

It offers new characterizations of low dimensionality in uniform Roe algebras and answers open questions about $K_0$-groups for non-amenable groups.

## Key findings

- Characterization of stable rank one and other properties in low dimensional cases
- $K_0$-classes originate from the canonical Cartan in low dimensions
- Counterexamples for non-amenable groups' $K_0$-groups

## Abstract

The goal of this paper is to study when uniform Roe algebras have certain $C^*$-algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as low dimensional, and connect these to low dimensionality of the underlying space in the sense of the asymptotic dimension of Gromov. Some of these results (for example, on stable rank one, cancellation, strong quasidiagonality, and finite decomposition rank) give definitive characterizations, while others (on real rank zero) are only partial and leave a lot open. We also establish results about $K$-theory, showing that all $K_0$-classes come from the inclusion of the canonical Cartan in low dimensional cases, but not in general; in particular, our $K$-theoretic results answer a question of Elliott and Sierakowski about vanishing of $K_0$ groups for uniform Roe algebras of non-amenable groups. Along the way, we extend some results about paradoxicality, proper infiniteness of projections in uniform Roe algebras, and supramenability from groups to general metric spaces. These are ingredients needed for our $K$-theoretic computations, but we also use them to give new characterizations of supramenability for metric spaces.