Low-dimensional properties of uniform Roe algebras

Li K, Willett R (2018)

Publication Type: Journal article

Publication year: 2018

Journal

Journal of the London Mathematical Society-Second Series London Mathematical Society

Book Volume: 97

Pages Range: 98-124

Journal Issue: 1

DOI: 10.1112/jlms.12100

Abstract

The goal of this paper is to study when uniform Roe algebras have certain Cz.ast; - -algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as low dimensional, and connect these to low dimensionality of the underlying space in the sense of the asymptotic dimension of Gromov. Some of these results (for example, on stable rank one, cancellation, strong quasidiagonality, and finite decomposition rank) give definitive characterizations, while others (on real rank zero) are only partial and leave a lot open. We also establish results about K-theory, showing that all K0-classes come from the inclusion of the canonical Cartan in low-dimensional cases, but not in general; in particular, our K-theoretic results answer a question of Elliott and Sierakowski about vanishing of K0 groups for uniform Roe algebras of non-amenable groups. Along the way, we extend some results about paradoxicality, proper infiniteness of projections in uniform Roe algebras, and supramenability from groups to general metric spaces. These are ingredients needed for our K-theoretic computations, but we also use them to give new characterizations of supramenability for metric spaces.

Authors with CRIS profile

Kang Li

Involved external institutions

Westfälische Wilhelms-Universität (WWU) Münster DE Germany (DE) University of Hawaiʻi at Mānoa US United States (USA) (US)

How to cite

APA:

Li, K., & Willett, R. (2018). Low-dimensional properties of uniform Roe algebras. Journal of the London Mathematical Society-Second Series, 97(1), 98-124. https://dx.doi.org/10.1112/jlms.12100

MLA:

Li, Kang, and Rufus Willett. "Low-dimensional properties of uniform Roe algebras." Journal of the London Mathematical Society-Second Series 97.1 (2018): 98-124.

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