  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.
