In this paper, we characterize when the uniform Roe algebra of a metric space with bounded geometry is (stably) finite and when it is properly infinite in standard form for . Moreover, we show that the uniform Roe algebra is a (non-sequential) spatial AF algebra in the sense of Phillips and Viola if and only if the underlying metric space has asymptotic dimension zero.
We also consider the ordered groups of uniform Roe algebras for metric spaces with low asymptotic dimension, showing that (1) the ordered group is trivial when the metric space is non-amenable and has asymptotic dimension at most one, and (2) when the metric space is a countable locally finite group, the (ordered) group is a complete invariant for the (bijective) coarse equivalence class of the underlying locally finite group. It happens that in both cases the ordered group does not depend on .
Cite this article
Yeong Chyuan Chung, Kang Li, Structure and -theory of uniform Roe algebras. J. Noncommut. Geom. 15 (2021), no. 2, pp. 581–614DOI 10.4171/JNCG/405