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

Abstract

Let $\theta ,{\theta }^{\prime }$ 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 ${\mathcal{A}}_{{\theta }^{\prime }}{⋊}_B\mathbb{Z}$ are $\ast$-isomorphic if and only if $\theta =\pm {\theta }^{\prime }(\mathrm{mod}\mathbb{Z})$ and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that ${\mathcal{A}}_{\theta }{⋊}_A\mathbb{Z}$ and ${\mathcal{A}}_{{\theta }^{\prime }}{⋊}_B\mathbb{Z}$ are Morita equivalent if and only if $\theta$ and ${\theta }^{\prime }$ 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 $S{L}_2(\mathbb{Z})$.

32 pages; an alternative proof of Theorem 5.3 was added; other minor corrections

References in corpus (1)

• Every simple higher dimensional noncommutative torus is an AT algebra