Boundary Quotient $\text{C}^{\ast }$-algebras of Products of Odometers. (7th January 2019)

• Record Type: Journal Article
• Authors: Li, Hui; Yang, Dilian
• Is Part Of: Canadian journal of mathematics. Volume 71:Number 1(2019)
• Page Start: 183
• Page End: 212
• Publication Date: 2019-01-07
• Subjects: 46L05; C∗ -algebra, -- semigroup, -- odometer, -- topological k-graph, -- product system, -- Zappa–Szép product; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2017-034-5
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: In this paper, we study the boundary quotient $\text{C}^{\ast }$ -algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{\ast }$ -algebra of the standard product of $k$ odometers over $n_{i}$ -letter alphabets $(1\leqslant i\leqslant k)$ is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_{i}:1\leqslant i\leqslant k\}$ is rationally independent, if and only if the associated single-vertex $k$ -graph $\text{C}^{\ast }$ -algebra is simple. To achieve this, one of our main steps is to construct a topological $k$ -graph such that its associated Cuntz–Pimsner $\text{C}^{\ast }$ -algebra is isomorphic to the boundary quotient $\text{C}^{\ast }$ -algebra. Some relations between the boundary quotient $\text{C}^{\ast }$ -algebra and the $\text{C}^{\ast }$ -algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.