Isomorphism and Morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of $S{L}_2(\mathbb{Z})$

Let $\theta ,{\theta }^{\mathrm{\prime }}$ be irrational numbers and $A,B$ be matrices in $S{L}_2(\mathbb{Z})$ of infinite order. We compute the $K$-theory of the crossed product ${\mathcal{A}}_{\theta }{⋊}_A\mathbb{Z}$ and show that ${\mathcal{A}}_{\theta }{⋊}_A\mathbb{Z}$ and $$\mathcal{A}${\theta’} \rtimes_B $\mathbb{Z}$$are$*$-isomorphicifandonlyif$\theta = \pm\theta’ \pmod{$\mathbb{Z}$}$and$I-A^{-1}$ismatrixequivalentto$I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that ${\mathcal{A}}_{\theta }{⋊}_A\mathbb{Z}$ and $$\mathcal{A}${\theta’} \rtimes_B $\mathbb{Z}$$areMoritaequivalentifandonlyif$\theta$and$\theta’$ are in the same $G{L}_2(\mathbb{Z})$ orbit and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Finally, we determine the Morita equivalence class of ${\mathcal{A}}_{\theta }⋊F$ for any finite subgroup $F$ of $S{L}_2(\mathbb{Z})$.