Symmetrized non-commutative tori revisited

Abstract

For the flip action of $\mathbb{Z}_2$ on an $n$-dimensional noncommutative torus $A_θ,$ using an exact sequence by Natsume, we compute the K-theory groups of $A_θ\rtimes \mathbb{Z}_2$. The novelty of our method is that it also provides an explicit basis of $\mathrm{K}_{0}(A_θ\rtimes \mathbb{Z}_2),$ for any $θ.$ As an application, for a simple $n$-dimensional torus $A_θ,$ using classification techniques, we determine the isomorphism class of $A_θ\rtimes \mathbb{Z}_2$ in terms of the isomorphism class of $A_θ.$