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Non-commutative Poisson boundaries and compact quantum group actions. (English) Zbl 1037.46056

Author’s abstract: “We discuss some relationships between two different fields, a noncommutative 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 a nontrivial relative commutant of the fixed point subalgebra. We give a probabilistic description of the relative commutant in terms of a noncommutative Markov operator. In particular, we show that the following three objects can be naturally identified in the case of the quantum group \(\text{SU}_q(2)\): (1) the relative commutant of the fixed point algebra under the action, (2) the space of harmonic elements for some noncommutative Markov operator on the dual quantum group of \(\text{SU}_q (2)\), and (3) the weak closure \(L^\infty ({\mathbf T}\setminus \text{SU}_q (2))\) 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 \({\mathbf T}; \text{SU}_q (2)\) may be regarded as the “Poisson boundary” of a noncommutative random walk on the dual object of \(\text{SU}_q (2)\). An analogy of the Poisson integral formula is also given.”

MSC:

46L65 Quantizations, deformations for selfadjoint operator algebras
60J99 Markov processes
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