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Noncommutative dimension theories of uniform Roe algebras Li, Kang


I will report on recent developments in noncommutative dimension theories of uniform Roe algebras associated to metric spaces with bounded geometry.\\ In a joint work with Rufus Willett, we show that for uniform Roe algebras, being AF, having stable rank one, having cancellation, and having finite decomposition rank, are all equivalent to the underlying metric space having asymptotic dimension zero. A countable group has asymptotic dimension zero if and only if it is locally finite. In a joint work with Hung-Chang Liao, we show that uniform Roe algebras of locally finite countable groups can be completely classified by $K_0$ groups together with units. To our best knowledge, this is the first classification result for non-separable and non-simple $C^*$-algebras. As a contrast, if the metric space $X$ is non-amenable and has asymptotic dimension one, then the $K_0$ group of the uniform Roe algebra over $X$ is always zero. Finally, we answer negatively to a question of Elliott and Sierakowski about the vanishing of $K_0$ of the uniform Roe algebras of non-amenable groups with high asymptotic dimension.\\ If time permits, we will discuss the relation between nuclear dimension of uniform Roe algebra and asymptotic dimension of its underlying metric space.

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