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.

arXiv comment9 pages, some explanations expanded, to appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics

References in corpus (10)

• Non commutative Lp spaces without the completely bounded approximation property
• Examples of groups which are not weakly amenable
• Quasitraces on exact C*-algebras are traces
• Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems
• Equivariant Kirchberg-Phillips-type absorption for amenable group actions
• Group C*-algebras as decreasing intersection of nuclear C*-algebras
• There is no separable universal II_1-factor
• Poly-$\mathbb{Z}$ group actions on Kirchberg algebras II
• Homogeneity of the pure state space of a separable C*-algebra
• Poly-$\mathbb{Z}$ group actions on Kirchberg algebras I

Cited by in corpus (2)

• Equivariant ${\mathcal{O}}_2$-absorption theorem for exact groups
• Non-amenable tight squeezes by Kirchberg algebras