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# Homotopy type of the unitary group of the uniform Roe algebra on $ℤn$
We study the homotopy type of the space of the unitary group $U1(Cu∗(|ℤn|))$ of the uniform Roe algebra $Cu∗(|ℤn|)$ of $ℤn$. We show that the stabilizing map $U1(Cu∗(|ℤn|))→U∞(Cu∗(|ℤn|))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of $U1(Cu∗(|ℤn|))$, which is the product of the unitary group $U1(C∗(|ℤn|))$ (having the homotopy type of $U∞(ℂ)$ or $ℤ×BU∞(ℂ)$ depending on the parity of $n$) of the Roe algebra $C∗(|ℤn|)$ and rational Eilenberg–MacLane spaces.