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Boundary Quotient $\text{C}^{\ast }$ -algebras of Products of Odometers

  • Hui Li (a1) and Dilian Yang (a2)

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.

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Author H. L. was partially supported by Research Center for Operator Algebras of East China Normal University and was partially supported by Science and Technology Commission of Shanghai Municipality (STCSM), grant No. 13dz2260400. Author D. Y. was partially supported by an NSERC Discovery Grant 808235.

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