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.

No abstract

A method of direct integral decomposition for very general classes of unbounded closed operators is developed. This method is applied to the reduction theory of partial *-algebras and Op*-algebras.As a consequence, representations of partial ^-algebras are decomposed into irreducible ones and some states are decomposed into extremal states.

Non-self-adjoint representations of *-algebras in a Hilbert space give rise by an extension and transposition procedure to representations in larger spaces, such as distribution spaces. Those new representations provide examples of operators of nested Hilbert spaces which would be very singular operators when considered in the Hilbert space only.