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

2017/11/14 by Bönicke, Christian, Chakraborty, Sayan, He, Zhuofeng +1 · 1 citation
#46L35 #46L55 #46L80 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1711.05055

Abstract

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 A_θ\rtimes_A ℤ and show that A_θ \rtimes_Aℤ and A_θ' \rtimes_B ℤ are *-isomorphic if and only if θ= ±θ' \pmodℤ and I-A^-1 is matrix equivalent to I-B^-1. Combining this result and an explicit construction of equivariant bimodules, we show that A_θ \rtimes_Aℤ and A_θ' \rtimes_B 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 A_θ \rtimes F for any finite subgroup F of SL₂(ℤ).

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• Morita equivalence classes for crossed product of rational rotation algebras

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