Z-stability of crossed products by strongly outer actions II

TL;DR

This paper proves that certain crossed product C*-algebras retain Z-stability under specific conditions and classifies actions of the Klein bottle group on Z, advancing understanding of group actions on finite C*-algebras.

Contribution

It establishes Z-stability for crossed products by strongly outer actions of amenable groups on Z-stable C*-algebras and classifies Klein bottle group actions on Z.

Findings

Crossed products are Z-stable under given conditions.

All strongly outer Klein bottle group actions on Z are cocycle conjugate.

First classification of non-abelian infinite group actions on stably finite C*-algebras.

Abstract

We consider a crossed product of a unital simple separable nuclear stably finite Z-stable C*-algebra A by a strongly outer cocycle action of a discrete countable amenable group \Gamma. Under the assumption that A has finitely many extremal tracial states and \Gamma is elementary amenable, we show that the twisted crossed product C*-algebra is Z-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Z are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite C*-algebras.