Uniformity and uniformly continuous functions for locally compact groups

Abstract

We show that a locally compact group G 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 G coincide. Along the way it is seen that G G has equivalent right and left uniform structures if (and only if) each σ \sigma -compact subgroup of G G has equivalent right and left uniform structures. We also note that a bounded function f : G → C f:G \to \mathbb {C} is right uniformly continuous if (and only if) f | H f{|_H} is right uniformly continuous for each σ \sigma -compact subgroup H H of G G . σ \sigma -compactness cannot be weakened to compact generation for these last results; a σ \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.