Topological groups with finite von Neumann algebras of type I

A complete description of all topological groups admitting a separating system of continuous irreducible finite-dimensional unitary representations of uniformly bounded dimensions is obtained. A similar description is obtained for the formally more general class of all unitarily representable topological groups whose von Neumann algebra is a sum of homogeneous summands of finite and uniformly bounded degrees. Related results are obtained; in particular, a description is given of all locally bounded topological groups all of whose irreducible unitary representations are finite-dimensional.