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 \operatorname{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} .