NON-COMMUTATIVE POISSON BOUNDARIES AND COMPACT QUANTUM GROUP ACTIONS

MASAKI IZUMI

Abstract. In contrast to the ordinary compact group case, the infinite tensor product (abbreviated as ITP) action of a compact quantum group on a factor may allow non-trivial relative commutant of the fixed point subalgebra. We give a probabilistic description of the relative commutant in terms of a non-commutative Markov operator. In view of the ordinary Poisson boundary theory of random walks on discrete groups, the quantum homogeneous space T\SU_{q}(2) may be regarded as the “Poisson boundary” of a non-commutative random walk on the dual object of SU_{q}(2). An analogy of the Poisson integral formula is also given.