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

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 Aθ⋊AZ and show that Aθ⋊AZ and Aθ′⋊BZ are ⁎-isomorphic if and only if θ=±θ′(modZ) 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′⋊BZ 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).