Abstract
We study the homotopy type of the space of the unitary group [Formula: see text] of the uniform Roe algebra [Formula: see text] of [Formula: see text]. We show that the stabilizing map [Formula: see text] is a homotopy equivalence. Moreover, when [Formula: see text], we determine the homotopy type of [Formula: see text], which is the product of the unitary group [Formula: see text] (having the homotopy type of [Formula: see text] or [Formula: see text] depending on the parity of [Formula: see text]) of the Roe algebra [Formula: see text] and rational Eilenberg–MacLane spaces.
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