Ergodic automorphisms on Kirchberg algebras

TL;DR

This paper demonstrates that any countable infinite discrete group can act ergodically on unital Kirchberg algebras, using Pimsner constructions and extension theory, with special results for amenable groups.

Contribution

It introduces a method to realize unital subalgebras as fixed point algebras via Pimsner construction and extends ergodic actions to all countable infinite groups.

Findings

Every countable infinite discrete group admits an ergodic action on any unital Kirchberg algebra.

For amenable groups, every point-wise outer action can be perturbed to an ergodic action.

The paper provides new constructions linking group actions and Kirchberg algebra substructures.

Abstract

Combining the theory of extensions of C*-algebras and the Pimsner construction, we show that every countable infinite discrete group admits an ergodic action on arbitrary unital Kirchberg algebra. In the proof, we give a Pimsner construction realizing many unital subalgebras of a given unital Kirchberg algebra as the fixed point algebras of single automorphisms. Furthermore, for amenable infinite discrete groups, we show that every point-wise outer action on arbitrary unital Kirchberg algebra has an ergodic cocycle perturbation with the help of Gabe--Szab\'{o}'s theorem and Baum--Connes' conjecture.