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On pathological properties of fixed point algebras in Kirchberg algebras. (English) Zbl 1470.46095

Summary: 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 \(C_r^*(\Gamma)\). Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

MSC:

46L55 Noncommutative dynamical systems
46L05 General theory of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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