Reflexivity of non-commutative Hardy algebras

We obtain some partial results in the general case and we turn to the case of a correspondence over a factor. Under some additional assumptions on the representation π:M→B(H) we show that ρπ(H∞(E)) is reflexive. Then we apply these results to analytic crossed products ρπ(H∞(Mα)) and obtain their reflexivity for any automorphism α∈Aut(M) whenever M is a factor. Finally, we show also the reflexivity of the compression of the Hardy algebra to a suitable coinvariant subspace M, which may be thought of as a generalized symmetric Fock space.