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On pathological properties of fixed point algebras in Kirchberg algebras

Part of: Selfadjoint operator algebras Locally compact groups and their algebras

Published online by Cambridge University Press:  20 September 2019

Yuhei Suzuki
Yuhei Suzuki*
Affiliation:
Graduate school of mathematics, Nagoya University, Chikusaku, Nagoya, 464-8602, Japan ([email protected])
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Abstract

We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

Keywords

Fixed point algebrasnuclearitynon-commutative dynamical systems

MSC classification

Primary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations 46L55: Noncommutative dynamical systems
Secondary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3087 - 3096
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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