The structure of crossed products of irrational rotation algebras by finite subgroups of SL_2 (Z)

Abstract

Let F be a finite subgroup of SL_2 (Z) (necessarily isomorphic to one of Z/2Z, Z/3Z, Z/4Z, or Z/6Z), and let F act on the irrational rotational algebra A_{\theta} via the restriction of the canonical action of SL_2 (Z). Then the crossed product of A_{\theta} by F, and the fixed point algebra for the action of F on A_{\theta}, are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z/2Z on any simple d-dimensional noncommutative torus A_{\Theta}. Along the way, we prove a number of general results which should have useful applications in other situations.