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Actions of certain torsion-free elementary amenable groups on strongly self-absorbing \(\mathrm {C}^*\)-algebras

  • Gábor SzabóEmail author
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Abstract

In this paper we consider a bootstrap class \(\mathfrak {C}\) of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on \(\mathrm {C}^*\)-algebras. This class includes all torsion-free abelian groups, poly-\(\mathbb {Z}\)-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group \(\Gamma \in \mathfrak {C}\) and any strongly self-absorbing \(\mathrm {C}^*\)-algebra \(\mathcal {D}\):
  1. (1)

    There is a unique strongly outer \(\Gamma \)-action on \(\mathcal {D}\) up to (very strong) cocycle conjugacy.

     
  2. (2)

    If \(\alpha : \Gamma \curvearrowright A\) is a strongly outer action on a separable, unital, nuclear, simple, \(\mathcal {D}\)-stable \(\mathrm {C}^*\)-algebra with at most one trace, then it absorbs every \(\Gamma \)-action on \(\mathcal {D}\) up to (very strong) cocycle conjugacy.

     
In fact we establish more general relative versions of these two results for actions of amenable groups that have a predetermined quotient in the class \(\mathfrak {C}\). For the monotracial case, the proof comprises an application of Matui–Sato’s equivariant property (SI) as a key method.

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsKU LeuvenLeuvenBelgium

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