Topological properties of unbounded bicommutants

Abstract

We consider different possible definitions of unbounded commutants and unbounded bicommutants of a set or an algebra of unbounded operators. We investigate their behavior with respect to various topologies. In particular we give sufficient conditions in order that bicommutants be the closure of the original set of operators with respect to some of those topologies. We investigate some special classes of algebras (symmetric, self-adjoint, regular, V* algebras) for which several or all of the bicommutants coincide and are the closure of the algebra with respect to some or all of the considered topologies.