Poisson boundary of the dual of SUq(n)

We prove that for any non-trivial product-type action of SUq(n) (0<q<1) on an ITPFI factor N, the relative commutant of the fixed point algebra in N is isomorphic to the algebra of bounded measurable functions on the quantum flag manifold. This is equivalent to the computation of the Poisson boundary of the dual discrete quantum group. The proof relies on a connection between the Poisson integral and the Berezin transform. Our main technical result says that a sequence of Berezin transforms defined by a random walk on the dominant weights of SU(n) converges to the identity on the quantum flag manifold. This is a q-analogue of some known results on quantization of coadjoint orbits of Lie groups.