Non-commutative Poisson Boundaries and Compact Quantum Group Actions

We discuss some relationships between two different fields, a non-commutative version of the Poisson boundary theory of random walks and the infinite tensor product (ITP) actions of compact quantum groups on von Neumann algebras. In contrast to the ordinary compact group case, the 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 particular, we show that the following three objects can be naturally identified in the case of the quantum group SUq(2): (1) the relative commutant of the fixed point algebra under the action, (2) the space of harmonic elements for some non-commutative Markov operator on the dual quantum group of SUq(2), and (3) the weak closure L∞(T\SUq(2)) of one of the Podles quantum spheres. In view of the ordinary Poisson boundary theory of random walks on discrete groups, it shows that symbolically the quantum homogeneous space T\SUq(2) may be regarded as the “Poisson boundary” of a non-commutative random walk on the dual object of SUq(2). An analogy of the Poisson integral formula is also given.