Suppose there are two locally
compact group topologies on a group and that the sets of irreducible unitary
representations of the group continuous with respect to each of the topologies
coincide. Then the topologies are equal if they are comparable or there is a normal
subgroup open and σ-compact in one of the topological groups. This is a result of
Klaus Bichteler’s, but the work presented here represents a much shorter method
than that used by Bichteler, using little representation theory, but using results
involving compatible topologies: Topologies containing in common a Hausdorff
topology.
