Boundary Quotient $\text{C}^{* }$-algebras of Products of Odometers. (7th January 2019) - Li, Hui; Yang, Dilian
Abstract: In this paper, we study the boundary quotient $\text{C}^{* }$ -algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{* }$ -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 $\text{C}^{* }$ -algebra is simple. To achieve this, one of our main steps is to construct a topological $k$ -graph such that its associated Cuntz–Pimsner $\text{C}^{* }$ -algebra is isomorphic to the boundary quotient $\text{C}^{* }$ -algebra. Some relations between the boundary quotient $\text{C}^{* }$ -algebra and the $\text{C}^{* }$ -algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.