论文标题
非交换性弗斯滕伯格边界
Noncommutative Furstenberg boundary
论文作者
论文摘要
我们介绍并研究了边界作用的概念以及离散量子组的Furstenberg边界的概念。至于经典组，边界动作的属性被证明编码了与所讨论的离散量子组相关的操作员代数的重要属性；例如，我们证明，如果对Furstenberg边界上的动作是忠实的，那么量子组C*-Algebra最多承认缩放自动形态群体。为了获得这些结果，我们开发了Hamana的量子群动作的注射信封理论的版本，并建立了有关量子亚组相对舒适性的几个事实。然后，我们表明，在我们的意义上，VAES和Vergnioux研究的自由正交量子群的Gromov边界动作也是边界动作。我们通过证明这些行动承认独特的固定状态来获得这一点。此外，我们证明了这些行动是忠实的，因此在一般情况下结束了新的独特的KMS国家财产，并在仅限于单模型的情况下进行了独特的痕量财产的新证明。我们证明了给定离散量子组的所有边界动作的交叉产物的简单性，并使用它来获得新的简单性结果，以实现自由正交量子组的Gromov边界作用的交叉产物。
We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.
