Isomorphism and Morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of $SL_2(Z)$

Zhuofeng He 贺卓丰  (The University of Tokyo)

15:20-16:20, Dec 17, 2019   Science Building A510

Abstract:

Let $\theta, \theta'$ be irrational numbers and $A, B$ be matrices in $SL_2(\Z)$ of infinite order. We compute the $K$-theory of the crossed product $\mathcal{A}_\theta\rtimes_A \Z$ and show that $\mathcal{A}_{\theta} \rtimes_A\Z$ and $\mathcal{A}_{\theta'} \rtimes_B \Z$ are $*$-isomorphic if and only if $\theta = \pm\theta' \pmod \Z$ and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that $\mathcal{A}_{\theta} \rtimes_A\Z$ and $\mathcal{A}_{\theta'} \rtimes_B \Z$ are Morita equivalent if and only if $\theta$ and $\theta'$ are in the same $GL_2(\Z)$ orbit and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Finally, we determine the Morita equivalence class of $\mathcal{A}_{\theta} \rtimes F$ for any finite subgroup $F$ of $SL_2(\Z)$.

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