Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^n$

TL;DR

This paper investigates the homotopy type of the unitary group of the uniform Roe algebra on integer lattices, revealing its structure and equivalences for certain dimensions.

Contribution

It establishes the homotopy equivalence of the stabilizing map and determines the homotopy type for dimensions one and two, linking it to classical unitary groups and Eilenberg--MacLane spaces.

Findings

Stabilizing map is a homotopy equivalence.

Homotopy type for n=1,2 involves classical unitary groups.

Results connect uniform Roe algebra unitary groups to known topological spaces.

Abstract

We study the homotopy type of the space of the unitary group $\U_1(C^\ast_u(|\mathbb{Z}^n|))$ of the uniform Roe algebra $C^\ast_u(|\mathbb{Z}^n|)$ of $\mathbb{Z}^n$. We show that the stabilizing map $\U_1(C^\ast_u(|\mathbb{Z}^n|))\to\U_\infty(C^\ast_u(|\mathbb{Z}^n|))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of $\U_1(C^\ast_u(|\mathbb{Z}^n|))$, which is the product of the unitary group $\U_1(C^\ast(|\mathbb{Z}^n|))$ (having the homotopy type of $\U_\infty(\mathbb{C})$ or $\mathbb{Z}\times B\U_\infty(\mathbb{C})$ depending on the parity of $n$) of the Roe algebra $C^\ast(|\mathbb{Z}^n|)$ and rational Eilenberg--MacLane spaces.