AI 中文总结
本文将拓扑Halmos-von Neumann定理及其范畴等价性推广到局部紧基空间上的开群丛框架，建立了两类阿贝尔群丛间的庞特里亚金型对偶性，拓展了经典阿贝尔群对偶理论。
AI 中文摘要
拓扑Halmos-von Neumann定理是拓扑动力学的基础结果，它完全分类了具有离散谱且为遍历的阿贝尔拓扑群的连续作用，该定理基于此类作用（在选定特殊点的意义下）、作用群的紧化及其对偶群子群之间的对应关系。我们将经典结论及 underlying 范畴等价性推广到局部紧基空间上的开群丛框架中，作为副产品，建立了Étale Hausdorff阿贝尔群丛与开真Hausdorff阿贝尔群丛之间的庞特里亚金型对偶性，推广了离散与紧阿贝尔群之间的经典对偶性。
英文摘要
The topological Halmos--von Neumann theorem is a fundamental result of topological dynamics that completely classifies continuous actions of abelian topological groups which are ergodic and have discrete spectrum. It rests on a correspondence between such actions (up to the choice of a distinguished point), compactifications of the acting group and subgroups of its dual group. We generalize both the classical statement and the underlying categorical equivalences to the framework of open group bundles over locally compact base spaces. As a byproduct, we establish a Pontryagin-type duality between étale Hausdorff abelian group bundles and open proper Hausdorff abelian group bundles, extending the classical duality between discrete and compact abelian groups.
