Reflexivity of non-commutative Hardy algebras

Leonid Helmer

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish
Pages (from-to)2752-2794
Number of pages43
JournalJournal of Functional Analysis
Issue number7
StatePublished - 1 Apr 2017


  • Nonselfadjoint algebras
  • Operator algebra
  • Reflexivity
  • W-correspondence

ASJC Scopus subject areas

  • Analysis


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