For a countable discrete group $$\Gamma $$ Γ and a minimal $$\Gamma $$ Γ -space X , we study the notion of $$(\Gamma , X)$$ ( Γ , X ) -boundary, which is a natural generalization of the notion of topological $$\Gamma $$ Γ -boundary in the sense of Furstenberg. We give characterizations of the $$(\Gamma , X)$$ ( Γ , X ) -boundary in terms of essential or proximal extensions. The characterization is used to answer a problem of Hadwin and Paulsen in negative. As an application, we find necessary and sufficient condition for the corresponding reduced crossed product to be exact.

中文翻译：

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Furstenberg 最小动作边界

对于可数离散群 $$\Gamma $$ Γ 和最小 $$\Gamma $$ Γ -space X ，我们研究 $$(\Gamma , X)$$ ( Γ , X ) -boundary 的概念，其中是 Furstenberg 意义上的拓扑 $$\Gamma $$ Γ 边界概念的自然推广。我们根据基本或近端扩展给出了 $$(\Gamma , X)$$ ( Γ , X ) -边界的特征。该表征用于以否定形式回答 Hadwin 和 Paulsen 的问题。作为应用，我们发现相应的约减交叉积是精确的充要条件。