Compactifications of Abelian Group Bundles and a Topological Halmos--von Neumann-type Theorem
Abstract
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.
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BibTeX
@misc{anglewitz2026compactifications,
title = {{Compactifications of Abelian Group Bundles and a Topological Halmos--von Neumann-type Theorem}},
author = {Dustin Anglewitz and Patrick Hermle and Henrik Kreidler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.28335},
eprint = {2608.28335},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28335},
}
RIS
TY - UNPB
TI - Compactifications of Abelian Group Bundles and a Topological Halmos--von Neumann-type Theorem
AU - Dustin Anglewitz
AU - Patrick Hermle
AU - Henrik Kreidler
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2608.28335
UR - https://arxiv.org/abs/2608.28335
AB - 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.
ER -