This paper is concerned with
the structure of abelian algebras.𝒜 of operators on Hilbert space ℋ such that
𝒜x = ℋ for some vector x in H. It is shown that if a transitive algebra 𝒯−
contains such an algebra then F is dense in the weak topology on ℒ(ℋ). It is
also shown that when an algebra of this type is semi-simple then it is a
reflexive operator algebra. The algebras investigated have the property that
every densely defined linear trans-formation commuting with the algebra is
bounded.