Realization of quantum group Poisson boundaries as crossed products

TL;DR

This paper demonstrates that the Poisson boundary of a quantum group action can be represented as a von Neumann crossed product, extending classical results to the quantum setting and unifying previous findings.

Contribution

It proves that the Poisson boundary for a quantum group action is isomorphic to a von Neumann crossed product, generalizing and unifying earlier results in the quantum and classical cases.

Findings

Poisson boundary realized as von Neumann crossed product

Extension of classical results to quantum groups

Unification of previous partial results

Abstract

For a locally compact quantum group $\mathbb{G}$, consider the convolution action of a quantum probability measure $\mu$ on $L_\infty(\mathbb{G})$. As shown by Junge--Neufang--Ruan, this action has a natural extension to a Markov map on $\mathcal{B}(L_2(\mathbb{G}))$. We prove that the Poisson boundary of the latter can be realized concretely as the von Neumann crossed product of the Poisson boundary associated with $\mu$ under the action of $\mathbb{G}$ induced by the coproduct. This yields an affirmative answer, for general locally compact quantum groups, to a problem raised by Izumi (2004) in the commutative situation, in which he settled the discrete case, and unifies earlier results of Jaworski, Neufang and Runde.