Realization of quantum group Poisson boundaries as crossed products
  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. 
Mehrdad Kalantar, Matthias Neufang and Zhong-Jin Ruan
math.OA