The Martin boundary of a discrete quantum group

TL;DR

This paper studies the Martin boundary of a discrete quantum group, establishing a representation theorem linking harmonic elements to positive functionals on a C*-algebra, and identifies the boundary for the dual of SU_q(2).

Contribution

It introduces a new framework for understanding harmonic elements and the Martin boundary in the setting of discrete quantum groups, extending classical results to the quantum case.

Findings

The Martin boundary A_phi is a separable C*-algebra with canonical quantum group actions.

Positive harmonic elements correspond to positive linear functionals on A_phi.

The Martin boundary of the dual of SU_q(2) is identified with the quantum homogeneous sphere of Podles.

Abstract

We consider the Markov operator P_phi on a discrete quantum group given by convolution with a q-tracial state phi. In the study of harmonic elements x, P_phi(x)=x, we define the Martin boundary A_phi. 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 A_phi. The C*-algebra A_phi 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 A_phi 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 SU_q(2) with the quantum homogeneous sphere of Podles. This result extends and unifies previous results by Ph. Biane and M. Izumi.