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

We study the homotopy type of the space of the unitary group ${\text{U}}_1(C_u^{\ast }(\mathrm{\mid }{\mathbb{Z}}^n\mathrm{\mid }))$ of the uniform Roe algebra $C_u^{\ast }(\mathrm{\mid }{\mathbb{Z}}^n\mathrm{\mid })$ of ${\mathbb{Z}}^n$. We show that the stabilizing map ${\text{U}}_1(C_u^{\ast }(\mathrm{\mid }{\mathbb{Z}}^n\mathrm{\mid }))\to {\text{U}}_{\mathrm{\infty }}(C_u^{\ast }(\mathrm{\mid }{\mathbb{Z}}^n\mathrm{\mid }))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of ${\text{U}}_1(C_u^{\ast }(\mathrm{\mid }{\mathbb{Z}}^n\mathrm{\mid }))$, which is the product of the unitary group ${\text{U}}_1(C^{\ast }(\mathrm{\mid }{\mathbb{Z}}^n\mathrm{\mid }))$ (having the homotopy type of ${\text{U}}_{\mathrm{\infty }}(\mathbb{C})$ or $\mathbb{Z}\times B{\text{U}}_{\mathrm{\infty }}(\mathbb{C})$ depending on the parity of $n$) of the Roe algebra $C^{\ast }(\mathrm{\mid }{\mathbb{Z}}^n\mathrm{\mid })$ and rational Eilenberg–MacLane spaces.