Faithful actions on generalized Furstenberg boundary

TL;DR

This paper explores the relationship between group actions on boundaries and averaging properties, revealing how faithfulness on the generalized Furstenberg boundary relates to the structure of the associated crossed product.

Contribution

It establishes an equivalence between faithfulness of group actions on the boundary and a weakened form of Powers' averaging property, advancing understanding of crossed product state spaces.

Findings

Faithfulness on the boundary correlates with a weakened Powers' averaging property.

Provides new insights into the structure of crossed product C*-algebras.

Connects boundary actions with operator algebra properties.

Abstract

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 $\partial_F(G, X)$ and a weakened version of the generalized Powers' averaging property. This result provides valuable insights into the state space of the crossed product $C(X)\rtimes_{r}G$.