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