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The dual space of the C*-algebra of bounded uniformly continuous functions ona uniform space carries several natural topologies. One of these is thetopology of uniform convergence on bounded uniformly equicontinuous sets, orthe UEB topology for short. In the particular case of a topological group andits right uniformity, the UEB topology plays a significant role in thecontinuity of convolution. In this paper we derive a useful characterisation ofbounded uniformly equicontinuous sets on locally compact groups. Then wedemonstrate that for every locally compact group G the UEB topology on thespace of finite Radon measures on G coincides with the right multipliertopology. In this sense the UEB topology is a generalisation to arbitrarytopological groups of the multiplier topology for locally compact groups. Inthe final section we prove results about UEB continuity of convolution.

Uniformly equicontinuous sets, right multiplier topology, and continuity of convolution