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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterization of distal and point-distal minimal transformation groups


Author: Joseph F. Kent
Journal: Proc. Amer. Math. Soc. 32 (1972), 304-308
MSC: Primary 54.80
DOI: https://doi.org/10.1090/S0002-9939-1972-0288744-9
MathSciNet review: 0288744
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Abstract: If G is a locally compact topological group, let $ BC(G)$ denote the set of real-valued, bounded, uniformly continuous functions on G with the compact-open topology. Using the fact that the distal (weakly distal) functions are the elements of $ BC(G)$ whose orbit closures are compact distal (point-distal) minimal sets, we can characterize compact distal and point-distal minimal transformation groups. Let $ (X,G,\phi )$ be a right transformation group where X is compact Hausdorff and minimal under G. Then X is a compact distal (point-distal) minimal set if and only if there is a point $ x \in X$ such that for any homomorphism $ h:X \to BC(G),h(x)$ is a right distal (weakly distal) function.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0288744-9
Keywords: Distal transformation group, point-distal transformation group, distal function, weakly distal function, von Neumann almost periodic function, functions coming from a transformation group
Article copyright: © Copyright 1972 American Mathematical Society