Uniform structure in topological groups

Abstract

A characterization is given of those locally compact groups having equivalent right and left uniform structures. It is shown that α \alpha -compact locally compact groups have equivalent uniformities iff for each right uniformly discrete set B B such that card ( B ) ≤ α (B) \leq \alpha and each neighborhood U U of e , ∩ x ∈ B x U x − 1 e,{ \cap _{x \in B}}xU{x^{ - 1}} is a neighborhood of e e . It is also shown that a group is not unimodular iff it contains an open σ \sigma -compact group which is not unimodular.