论文标题
动力比较和$ \ MATHCAL {Z} $ - 简单$ C^*$ - 代数的交叉产品的稳定性
Dynamical comparison and $\mathcal{Z}$-stability for crossed products of simple $C^*$-algebras
论文作者
论文摘要
我们建立了$ \ MATHCAL {Z} $ - 在$ \ Mathcal {z} $ - 稳定的$ C^*$ - 在温和的技术假设下，我们称McDuff属性相对于不变痕迹，在$ \ MATHCAL {z} $上的外部动作的交叉产品稳定性。我们使用弱的动力比较形式获得了这种结果，我们以极大的一般性来验证。我们通过证明在许多有兴趣的情况下对不变痕迹的麦克粉丝是自动的，从而补充了我们的结果。例如，对于一个可分类的$ c^*$ - 代数$ a $的每项动作，这种情况就是这种情况。 If $G = \mathbb{Z}^d$ and the action $G\curvearrowright \partial_eT(A)$ is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus $\mathcal{Z}$-stability of the corresponding crossed product, also in case $\partial_e T(A)$ has infinite covering dimension.
We establish $\mathcal{Z}$-stability for crossed products of outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group $G$ on a classifiable $C^*$-algebra $A$ whose trace space $T(A)$ is a Bauer simplex with finite dimensional boundary $\partial_e T(A)$, and such that the induced action $G\curvearrowright \partial_eT(A)$ is free. If $G = \mathbb{Z}^d$ and the action $G\curvearrowright \partial_eT(A)$ is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus $\mathcal{Z}$-stability of the corresponding crossed product, also in case $\partial_e T(A)$ has infinite covering dimension.
