Remarks on the Interpolation and the Similarity Problem for Nest Subalgebras of von Neumann Algebras

Given a nest subalgebra of a von Neumann algebra A and vectors f, g we give necessary and sufficient conditions for the existence of an operator A in A so that Af = g. This generalizes a well-known theorem of E. C. Lance and the main result in [Anoussis (1992); Katsoulis et al. (1993)]. We also use the generalized Ringrose criterion to show that the similarity problem for continuous nests in certain crossed product factors of type III has a negative answer.