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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniformity and uniformly continuous functions for locally compact groups


Author: Paul Milnes
Journal: Proc. Amer. Math. Soc. 109 (1990), 567-570
MSC: Primary 22D05
MathSciNet review: 1023345
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Abstract: We show that a locally compact group $ G$ has equivalent right and left uniform structures if (and only if) the sets of bounded, complex-valued, right and left uniformly continuous functions on $ G$ coincide. Along the way it is seen that $ G$ has equivalent right and left uniform structures if (and only if) each $ \sigma $-compact subgroup of $ G$ has equivalent right and left uniform structures. We also note that a bounded function $ f:G \to \mathbb{C}$ is right uniformly continuous if (and only if) $ f{\vert _H}$ is right uniformly continuous for each $ \sigma $-compact subgroup $ H$ of $ G$ . $ \sigma $-compactness cannot be weakened to compact generation for these last results; a $ \sigma $-compact group is exhibited which has inequivalent right and left uniform structures, and for which each compactly generated subgroup has equivalent right and left uniform structures.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1023345-7
PII: S 0002-9939(1990)1023345-7
Keywords: Locally compact group, uniform structure, uniformly continuous function
Article copyright: © Copyright 1990 American Mathematical Society