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Homotopy type of the unitary group of the uniform Roe algebra on ℤn
Journal of Topology and Analysis  (IF0.457),  Pub Date : 2021-04-26, DOI: 10.1142/s1793525321500357
Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya

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