On pathological properties of fixed point algebras in Kirchberg algebras

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 ${\mathrm{C}}_{\mathrm{r}}^{\ast }(\Gamma )$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

9 pages, some explanations expanded, to appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics