Ergodic automorphisms on Kirchberg algebras

TL;DR

The paper develops a unified framework to realize ergodic actions of countable infinite groups on unital Kirchberg algebras by combining extension theory with the Pimsner construction. It shows that every such group $G$ admits an ergodic action on any unital Kirchberg algebra $A$, and for amenable $G$ any point-wise outer $G$-action can be perturbed to ergodic by a cocycle, leveraging the Gabe–Szabó theorem and Baum–Connes. The approach centralizes a KK-theoretic realization of fixed-point algebras as corners of Toeplitz–Pimsner algebras and uses equivariant extensions to produce cocycle perturbations, all within the Kirchberg–Phillips classification framework. Collectively, this provides a constructive method to realize ergodic dynamics and fixed-point algebras in the Kirchberg setting, broadening the landscape of ergodic actions in noncommutative topology.

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--Szabó's theorem and Baum--Connes' conjecture.

Theorems & Definitions (47)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Corollary 1.6
• Corollary 1.7
• Remark 1.8
• Lemma 2.1
• proof