Reflexivity of non commutative Hardy Algebras

Abstract

Let $H^{\infty }(E)$ be a non commutative Hardy algebra, associated with a $W^{\ast }$-correspondence $E$. These algebras were introduced in 2004, ~\cite{MuS3}, by P. Muhly and B. Solel, and generalize the classical Hardy algebra of the unit disc $H^{\infty }(\mathbb{D})$. As a special case one obtains also the algebra ${\mathcal{F}}^{\infty }$ of Popescu, which is $H^{\infty }({\mathbb{C}}^n)$ in our setting. In this paper we view the algebra $H^{\infty }(E)$ as acting on a Hilbert space via an induced representation $\rho (H^{\infty }(E))$, and we study the reflexivity of $\rho (H^{\infty }(E))$. This question was studied by A. Arias and G. Popescu in the context of the algebra ${\mathcal{F}}^{\infty }$, 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 $W^{\ast }$-correspondence $E$ over a general $W^{\ast }$-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 $\pi :M\to B(H)$ we show that ${\rho }_{\pi }(H^{\infty }(E))$ is reflexive. Then we apply these results to analytic crossed products $\rho (H^{\infty }({}_{\alpha }M))$ and obtain their reflexivity for any automorphism $\alpha \in Aut(M)$ whenever $M$ is a factor. Finally, we show also the reflexivity of the compression of the Hardy algebra to a suitable coinvariant subspace $\mathfrak{M}$, which may be thought of as a generalized symmetric Fock space.

arXiv comment39 pages