Symmetrized non-commutative tori revisited

Abstract

For the flip action of ${\mathbb{Z}}_2$ on an $n$-dimensional noncommutative torus $A_{\theta }$, using an exact sequence by Natsume, we compute the K-theory groups of $A_{\theta }⋊{\mathbb{Z}}_2$. The novelty of our method is that it also provides an explicit basis of $K_0(A_{\theta }⋊{\mathbb{Z}}_2)$, for any $\theta$. As an application, for a simple $n$-dimensional torus $A_{\theta }$, using classification techniques, we determine the isomorphism class of $A_{\theta }⋊{\mathbb{Z}}_2$ in terms of the isomorphism class of $A_{\theta }$.