Poisson boundaries over locally compact quantum groups

TL;DR

This paper extends classical harmonic analysis results to the setting of locally compact quantum groups, establishing non-commutative analogues of Poisson boundaries and harmonic function theorems, with applications to quantum groups like SU_q(2).

Contribution

It introduces non-commutative versions of classical theorems on harmonic functions and Poisson boundaries for locally compact quantum groups, including the Choquet--Deny theorem and characterizations of amenability.

Findings

Choquet--Deny theorem holds for compact quantum groups

Non-commutative analogue of Kaimanovich--Vershik and Rosenblatt's result

Concrete realization of Poisson boundaries for SU_q(2)

Abstract

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet--Deny theorem holds for compact quantum groups; also, the result of Kaimanovich--Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a non-commutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group $SU_{q}(2)$ arising from measures on its spectrum.