Faithful actions on generalized Furstenberg boundary

Let G be a countable discrete group that act minimally on a compact Hausdorff space X by homeomorphisms. Our goal is to establish the equivalence between the faithfulness of the action of G on the generalized Furstenberg boundary ∂F(G, X) and a weakened version of the generalized Powers' averaging property. This result provides valuable insights into the state space of the crossed product C(X)ᵣG.