In this paper we study the harmonic elements of (convolution) Markov maps associated to (ergodic) actions of locally compact quantum groups on (σ-finite) von Neumann algebras. We give several equivalent conditions under which the harmonic elements are trivial.

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete Kac quantum groups, we show that the existence of unique stationary SAT states entails rigidity results concerning amenable extensions of quantum group von Neumann algebras.

No abstract