Z-actions on AH algebras and Z^2-actions on AF algebras

TL;DR

This paper studies automorphisms and Z^2-actions on specific classes of C*-algebras, establishing conditions for the Rohlin property and classifying actions up to cocycle conjugacy.

Contribution

It proves that uniform outerness implies the Rohlin property for Z-actions on certain AH algebras and classifies Z^2-actions on AF algebras using the OrderExt group.

Findings

Uniform outerness implies the Rohlin property under technical assumptions.

Z-actions with the Rohlin property are cocycle conjugate if asymptotically unitarily equivalent.

Classifies Z^2-actions on AF algebras using the OrderExt group.

Abstract

We consider Z-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two Z-actions with the Rohlin property on such a C^*-algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer Z^2-actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as classification invariants.