Reflexivity of non-commutative Hardy algebras

Abstract

Let H∞(E) be a non-commutative Hardy algebra associated with a W⁎-correspondence E. These algebras were introduced in 2004 by Muhly and Solel, and generalize the classical Hardy algebra of the unit disc H∞(D). As a special case one obtains also the algebra Fd∞ of Popescu, which is H∞(Cd) in our setting.In this paper we view the algebra H∞(E) as acting on a Hilbert space via an induced representation. We write it ρπ(H∞(E)) and we study the reflexivity of ρπ(H∞(E)). This question was studied by Arias and Popescu in the context of the algebra Fd∞, and by other authors in several other special cases. As it will be clear from our work, the extension to the case of a general W⁎-correspondence E over a general W⁎-algebra M requires new techniques and approach.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.