The Martin Boundary of a Discrete Quantum Group

We consider the Markov operator Pφ on a discrete quantum group given by convolution with a q-tracial state φ. In the study of harmonic elements x, Pφ(x) = x, we define the Martin boundary Aφ. It is a separable C ∗-algebra carrying canonical actions of the quantum group and its dual. We establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on Aφ. The C ∗-algebraAφ has a natural time evolution, and the unit can always be represented by a KMS state. Any such state gives rise to a u.c.p. map from the von Neumann closure of Aφ in its GNS representation to the von Neumann algebra of bounded harmonic elements, which is an analogue of the Poisson integral. Under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual. Next we apply these results to identify the Martin boundary of the dual of SUq(2) with the quantum homogeneous sphere of Podleś. This result extends and unifies previous results by Ph. Biane and M. Izumi.