|XXXIX Workshop on Geometric Methods in Physics||27.06-3.07.2021|
|IX School on Geometry and Physics||21-25.06.2021|
Noncommutative Furstenberg boundary
For a discrete quantum group $G$, I will introduce the concept of a $G$-boundary action on a $C^*$-algebra. It turns out that there exists a largest $G$-boundary, which we call the noncommutative Furstenberg boundary of $G$. For a discrete quantum group $G$ of Kac type, I will show that the unique trace property of $C^*(G)$ follows from the the faithfulness of the $G$-action on the noncommutative Furstenberg boundary of $G$. Then, I will discuss the Gromov boundary of a generic free orthogonal quantum group, showing that this is indeed a boundary in our sense, and that the corresponding action is faithful. Joint work in progress with Kalantar, Skalski and Vergnioux.
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