On irreducibility of Koopman representations corresponding to measure contracting actions

Abstract

We introduce a notion of measure contracting actions and show that Koopman representations corresponding to ergodic measure contracting actions are irreducible. We also show that the actions of Higman–Thompson groups on intervals equipped with Lebesgue measure and the actions of weakly branch groups on the boundaries of rooted trees equipped with non-uniform Bernoulli measures are measure contracting. This gives a new point of view on irreducibility of the corresponding Koopman representations.