# On pathological properties of fixed point algebras in Kirchberg algebras

## TL;DR

This paper explores how fixed point algebras in Kirchberg algebras can exhibit pathological properties, including differences from the original algebra and failure of approximation properties, through constructed group actions.

## Contribution

It constructs specific outer group actions on Kirchberg algebras with fixed point algebras displaying unusual and pathological properties.

## Key findings

- Fixed point algebra can differ significantly from the original algebra.
- Existence of outer actions with fixed point algebra almost equal to reduced group C*-algebra.
- Fixed point algebras can fail the completely bounded approximation property.

## Abstract

We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group $\Gamma$ and any infinite group $\Lambda$, we construct an outer action of $\Lambda$ on the Cuntz algebra $\mathcal{O}_2$ whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}^\ast_{\rm r}(\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.