Non-commutative Poisson Boundaries and Compact Quantum Group Actions
Masaki Izumi
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 ${\bf T}\backslash SU_q(2)$ 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.