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 ${\mathrm{\partial }}_F(G,X)$ and a weakened version of the generalized Powers’ averaging property. This result provides valuable insights into the state space of the crossed product $C(X){⋊}_{r}G$.