On the $K$ -theory of $C^{\ast }$ -algebras arising from integral dynamics

We investigate the $K$ -theory of unital UCT Kirchberg algebras ${\mathcal{Q}}_{S}$ arising from families $S$ of relatively prime numbers. It is shown that $K_{\ast }({\mathcal{Q}}_{S})$ is the direct sum of a free abelian group and a torsion group, each of which is realized by another distinct $C^{\ast }$ -algebra naturally associated to $S$ . The $C^{\ast }$ -algebra representing the torsion part is identified with a natural subalgebra ${\mathcal{A}}_{S}$ of ${\mathcal{Q}}_{S}$ . For the $K$ -theory of ${\mathcal{Q}}_{S}$ , the cardinality of $S$ determines the free part and is also relevant for the torsion part, for which the greatest common divisor $g_{S}$ of $\{p-1:p\in S\}$ plays a central role as well. In the case where $|S|\leq 2$ or $g_{S}=1$ we obtain a complete classification for ${\mathcal{Q}}_{S}$ . Our results support the conjecture that ${\mathcal{A}}_{S}$ coincides with $\otimes _{p\in S}{\mathcal{O}}_{p}$ . This would lead to a complete classification of ${\mathcal{Q}}_{S}$ , and is related to a conjecture about $k$ -graphs.