2014/10/07 by Leonid Helmer, Helmer, Leonid · 1 citation
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #math.OA
paper · pdf · doi:10.48550/arxiv.1410.1788
39 pages
arxiv created 2014/10/07 · arxiv updated 2014/10/08
Let H∞(E) be a non commutative Hardy algebra, associated with a W^*-correspondence E. These algebras were introduced in 2004, ~\citeMuS3, by P. Muhly and B. Solel, and generalize the classical Hardy algebra of the unit disc H∞(\mathbbD). As a special case one obtains also the algebra F∞ of Popescu, which is H∞(ℂn) in our setting. In this paper we view the algebra H^∞(E) as acting on a Hilbert space via an induced representation ρ(H∞(E)), and we study the reflexivity of ρ(H∞(E)). This question was studied by A. Arias and G. Popescu in the context of the algebra F∞, 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 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 ρ(H∞( αM)) 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 \mathfrakM, which may be thought of as a generalized symmetric Fock space.