On the -theory of -algebras arising from integral dynamics

Abstract

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.