Boundary Quotient C*-algebras of Products of Odometers

TL;DR

This paper analyzes boundary quotient C*-algebras from products of odometers, establishing their nuclearity, conditions for being Kirchberg algebras, and connecting them to topological k-graphs and Cuntz's algebra _N.

Contribution

It constructs a topological k-graph model for these C*-algebras and characterizes when they are simple, nuclear, and Kirchberg algebras, advancing understanding of their structure.

Findings

Boundary quotient C*-algebra is always nuclear.

It is a UCT Kirchberg algebra iff ng n_i are rationally independent.

The results settle a previously constructed boundary quotient C*-algebra.

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 $\Q_\bN$ introduced by Cuntz are also investigated. As an easy consequence of our main results, it settles a boundary quotient C*-algebra constructed by Brownlowe-Ramagge-Robertson-Whittaker.