# Strongly outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras

@article{Gardella2018StronglyOA, title={Strongly outer actions of amenable groups on \$\mathcal\{Z\}\$-stable \$C^*\$-algebras}, author={Eusebio Gardella and Ilan Hirshberg}, journal={arXiv: Operator Algebras}, year={2018} }

Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a countable amenable group. If the trace space $T(A)$ is a Bauer simplex and $\partial_eT(A)$ has finite $G$-orbits, we show that $\alpha$ is strongly outer if and only if $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is finite, then these conditions are also equivalent to $\alpha\otimes\mathrm{id}_{\mathcal{Z…

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