Poisson Boundary of the Dual of SU q (n)

We prove that for any non-trivial product-type action α of SU_q(n) (0<q<1) on an ITPFI factor N, the relative commutant is isomorphic to the algebra L^∞() 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. Supported by JSPS. Partially supported by the Norwegian Research Council. Supported by the SUP-program of the Norwegian Research Council.