In [6], Bichon, De Rijdt and Vaes introduced the notion of monoidally equivalent compact quantum groups. In this paper we prove that there is a natural bijective correspondence between actions of monoidally equivalent quantum groups on unital C * -algebras or on von Neumann algebras. We apply this correspondence to study the behavior of Poisson and Martin boundaries under monoidal equivalence of quantum groups.
The Poisson and Martin boundaries for invariant random walks on the dual of the orthogonal quantum groups Ao(F ), are identified with higher dimensional Podleś spheres that we describe in terms of generators and relations. This provides the first such identification for random walks on non-amenable discrete quantum groups.
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