Vol. 15, No. 3, 2022

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Noncommutative Furstenberg boundary

Mehrdad Kalantar, Paweł Kasprzak, Adam Skalski and Roland Vergnioux

Vol. 15 (2022), No. 3, 795–842

We introduce and study the notions of boundary actions and 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, and 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.

discrete quantum group, quantum group action, noncommutative boundary, free orthogonal quantum group
Mathematical Subject Classification
Primary: 46L55
Secondary: 20G42, 46L05, 46L65
Received: 2 April 2020
Revised: 15 September 2020
Accepted: 28 October 2020
Published: 10 June 2022
Mehrdad Kalantar
Department of Mathematics
University of Houston
Houston, TX
United States
Paweł Kasprzak
Department of Mathematical Methods in Physics
Faculty of Physics
University of Warsaw
Adam Skalski
Institute of Mathematics of the Polish Academy of Sciences
Roland Vergnioux
Normandie Université, UNICAEN, CNRS, LMNO