On the K-theory of C*-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 $\mid S\mid \le 2$ 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.