A note on crossed products of rotation algebras

Abstract

We compute the $K$-theory of crossed products of rotation algebras ${\mathcal{A}}_{\theta }$, for any real angle $\theta$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of ${\mathcal{A}}_{\theta }$ into the crossed products is injective at the level of $K_0$-groups. We then give an explicit set of generators for the $K_0$-groups and compute the tracial ranges concretely.

11 pages