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

TL;DR

This paper investigates the structure of crossed products and fixed point algebras of irrational rotation algebras under finite subgroup actions of SL_2(Z), showing they are AF algebras and establishing general results with broader applications.

Contribution

It demonstrates that crossed products and fixed point algebras of irrational rotation algebras by finite SL_2(Z) subgroups are AF algebras, and extends results to noncommutative tori with flip actions.

Findings

Crossed products are AF algebras

Fixed point algebras are AF

Results applicable to noncommutative tori

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.