Uniformity and uniformly continuous functions for locally compact groups
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- by Paul Milnes PDF
- Proc. Amer. Math. Soc. 109 (1990), 567-570 Request permission
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) $f{|_H}$ 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.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 567-570
- MSC: Primary 22D05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1023345-7
- MathSciNet review: 1023345