Furstenberg transformations on irrational rotation algebras

Abstract

We introduce a general class of automorphisms of rotation algebras, the non-commutative Furstenberg transformations. We prove that fully irrational non-commutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by traces (Blackadar's Second Fundamental Comparability Question). We also prove that several classes of simple quotients of the $C$*-algebras of discrete subgroups of five-dimensional nilpotent Lie groups, considered by Milnes and Walters, are crossed products of simple $C$*-algebras ($C$*-algebras of minimal ordinary Furstenberg transformations) by automorphisms which have the tracial Rokhlin property. It follows that these algebras also have stable rank one, real rank zero, and order on projections determined by traces.