The Interpolation In CSL Subalgebras Of Von Neumann Algebras

The transitivity theorem was given by Kadison on the C* algebras, and then using it we need no longer distinguish topological and algebraic irreducibility for C* algebras [6]. Af-ter that, Lance provided a necessary and sufficient condition for the transitivity on the nest algebra[19]. Hopenwasser expanded the Lance theorem to the CSL algebras[10], and then Katsoulis had arrived at similar results on nest subalgebras of von Neumann algebras [15]. In this paper, we discuss the transitivity on CSL subalgebras of von Neumann algebras, that is, given any CSL subalgebra of von Neumann algebras (?) and vectoers f, g, we give necessary and sufficent conditions for the existence of an operator A e (?) so that A f=g. Further more the necessary and sufficient conditions for variable interpolation are described. As a special case of this paper, the related results of Lance, Hopenwasser and Katoulis are covered.