Furstenberg transformations on irrational rotation algebras

TL;DR

This paper introduces noncommutative Furstenberg transformations on rotation algebras, proves they have the tracial Rokhlin property, and shows their crossed products have desirable structural properties like stable rank one and real rank zero.

Contribution

It establishes the tracial Rokhlin property for a broad class of noncommutative Furstenberg transformations and applies this to analyze the structure of related crossed product C*-algebras.

Findings

Fully irrational transformations have the tracial Rokhlin property.

Crossed products have stable rank one and real rank zero.

Certain simple quotients of nilpotent Lie group C*-algebras are crossed products with these properties.

Abstract

We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative 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.