Symmetrized non-commutative tori revisited

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 $$\mathrm{K}${0}(A{\theta}\rtimes $\mathbb{Z}$2),$forany$\theta.$Asanapplication,forasimple$n$-dimensionaltorus$A{\theta},$ using classification techniques, we determine the isomorphism class of $A_{\theta }⋊{\mathbb{Z}}_2$ in terms of the isomorphism class of $A_{\theta }\mathrm{.}$