Operator algebras and unitary representations of Polish groups

Abstract: 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.