Poisson Boundaries over Locally Compact Quantum Groups

Abstract. We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups G. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, answering a conjecture by Furstenberg, admits a non-commutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SUq(2) arising from measures on its spectrum. We further show that the Poisson boundary of the natural Markov operator extension of the convolution action of a quantum probability measure µ on L∞(G) to B(L2(G)), as introduced and studied – for general completely bounded multipliers on L1(G) – by M. Junge, M. Neufang and Z.-J. Ruan, can be identified precisely with the crossed product of the Poisson boundary of µ under the coaction of G induced by the coproduct. This yields an affirmative answer, for general locally compact quantum groups, to a problem raised by M. Izumi (2004) in the commutative situation, in which he settled the discrete case, and unifies earlier results of W. Jaworski, M. Neufang and V. Runde. 1