Homotopy type of the unitary group of the uniform Roe algebra on Z(n)

We study the homotopy type of the space of the unitary group U-1(C-u*(|Z(n)|)) of the uniform Roe algebra C-u*(|Z(n)|) of Z(n). We show that the stabilizing map U-1(C-u*(|Z(n)|)) -> U-infinity(C-u*(|Z(n)|)) is a homotopy equivalence. Moreover, when n = 1, 2, we determine the homotopy type of U-1(C-u*(|Z(n)|)), which is the product of the unitary group U-1(C-u*(|Z(n)|)) (having the homotopy type of U-infinity(C) or ZxB U-infinity(C) depending on the parity of n) of the Roe algebra C *(|Z(n)|) and rational Eilenberg-MacLane spaces.