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Ƶ-Stability of Crossed Products by Strongly Outer Actions II
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 6, December 2014
- pp. 1441-1496
- 10.1353/ajm.2014.0043
- Article
- Additional Information
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We consider a crossed product of a unital simple separable nuclear stably
finite $\cal Z$-stable $C^*$-algebra $A$ by a strongly outer cocycle
action of a discrete countable amenable group $\Gamma$. Under the
assumption that $A$ has finitely many extremal tracial states and $\Gamma$
is elementary amenable, we show that the twisted crossed product
$C^*$-algebra is $\cal Z$-stable. As an application, we also prove that
all strongly outer cocycle actions of the Klein bottle group on $\cal Z$
are cocycle conjugate to each other. This is the first classification
result for actions of non-abelian infinite groups on stably finite
$C^*$-algebras.