The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(ℤ)

Abstract

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