Uniformity and uniformly continuous functions for locally compact groups

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.