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# Boundary quotient C*-algebras of products of odometers

Published:2017-12-03

• Hui Li,
Department of Mathematics and Statistics, University of Windsor, Windsor ON N9B 3P4
• Dilian Yang,
Department of Mathematics and Statistics, University of Windsor, Windsor ON N9B 3P4
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## Abstract

In this paper, we study the boundary quotient C*-algebras associated to products of odometers. One of our main results shows that the boundary quotient C*-algebra of the standard product of $k$ odometers over $n_i$-letter alphabets ($1\le i\le k$) is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_i: 1\le i\le k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph C*-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz-Pimsner C*-algebra is isomorphic to the boundary quotient C*-algebra. Some relations between the boundary quotient C*-algebra and the C*-algebra $\mathrm{Q}_\mathbb{N}$ introduced by Cuntz are also investigated.
 Keywords: C*-algebra, semigroup, odometer, topological $k$-graph, product system, Zappa-Szép product
 MSC Classifications: 46L05 - General theory of $C^*$-algebras

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