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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Journal Bulletin of the London Mathematical Society
Kalantar, M. (Mehrdad), Neufang, M, & Ruan, Z.-J. (Zhong-Jin). (2014). Realization of quantum group Poisson boundaries as crossed products. Bulletin of the London Mathematical Society, 46(6), 1267–1275. doi:10.1112/blms/bdu081