Isomorphism and Morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of SL₂(Z)

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Let θ, θ' be irrational numbers and A, B be matrices in SL₂(Z) of infinite order. We compute the K-theory of the crossed product Aθₐ Z and show that Aθ ₐZ and Aθ' Z are *-isomorphic if and only if θ= ±θ' Z and I-A⁻¹ is matrix equivalent to I-B⁻¹. Combining this result and an explicit construction of equivariant bimodules, we show that Aθ ₐZ and Aθ' Z are Morita equivalent if and only if θ and θ' are in the same GL₂(Z) orbit and I-A⁻¹ is matrix equivalent to I-B⁻¹. Finally, we determine the Morita equivalence class of Aθ F for any finite subgroup F of SL₂(Z).