For a locally compact quantum group G, consider the convolution action of a quantum probability measure μ on L∞ (G). As shown by Junge-Neufang-Ruan, this action has a natural extension to a Markov map on B (L2(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 μ under the action of G induced by the coproduct. This yields an affirmative answer, for general locally compact quantum groups, to a problem raised by Izumi in the commutative situation, in which he settled the discrete case, and unifies earlier results of Jaworski, Neufang and Runde.
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