## 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 F_{d}^{∞} of Popescu, which is H^{∞}(C^{d}) 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 ρ_{π}(H^{∞}(E)) and we study the reflexivity of ρ_{π}(H^{∞}(E)). This question was studied by Arias and Popescu in the context of the algebra F_{d}^{∞}, 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 ρ_{π}(H^{∞}(E)) is reflexive. Then we apply these results to analytic crossed products ρ_{π}(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 language | English |
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Pages (from-to) | 2752-2794 |

Number of pages | 43 |

Journal | Journal of Functional Analysis |

Volume | 272 |

Issue number | 7 |

DOIs | |

State | Published - 1 Apr 2017 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis