Uniformity and uniformly continuous functions for locally compact groups
Abstract: We show that a locally compact group has equivalent right and left uniform structures if (and only if) the sets of bounded, complex-valued, right and left uniformly continuous functions on coincide. Along the way it is seen that has equivalent right and left uniform structures if (and only if) each -compact subgroup of has equivalent right and left uniform structures. We also note that a bounded function is right uniformly continuous if (and only if) is right uniformly continuous for each -compact subgroup of . -compactness cannot be weakened to compact generation for these last results; a -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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Keywords: Locally compact group, uniform structure, uniformly continuous function
Article copyright: © Copyright 1990 American Mathematical Society