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 $A_\theta \rtimes \mathbb{Z}2$. The novelty of our method is that it also provides an explicit basis of $\mathrm{K}{0}(A_{\theta}\rtimes \mathbb{Z}2),$ for any $\theta.$ As an application, for a simple $n$-dimensional torus $A{\theta},$ using classification techniques, we determine the isomorphism class of $A_{\theta}\rtimes \mathbb{Z}2$ in terms of the isomorphism class of $A{\theta}.$