  We study the relationship between operator algebras, $C^*$ and von Neumann,
acting on a Hilbert space and unitary representations of topological groups on
the same space. We obtain certain correspondences between both these families
of objects. In particular, we prove that the study of Abelian subalgebras of a
norm-separable $C^*$-algebra can be, in many aspects, reduced to the study of
unitary representations of Polish groups. It is shown that our techniques have
applications to the construction and classification of Abelian subalgebras of
operator $C^*$-algebras on separable Hilbert spaces.
