Reflexivity of non commutative Hardy Algebras

TL;DR

This paper investigates the reflexivity of non-commutative Hardy algebras associated with W*-correspondences, extending previous results and introducing new techniques for general cases, especially over factors.

Contribution

It extends the study of algebra reflexivity to general W*-correspondences over W*-algebras, providing new partial results and applying them to crossed products and coinvariant subspaces.

Findings

Reflexivity established for certain representations over factors.

Applied results to analytic crossed products with automorphisms.

Proved reflexivity of algebra compressions to coinvariant subspaces.

Abstract

Let $H^{\infty}(E)$ be a non commutative Hardy algebra, associated with a $W^*$-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 $W^*$-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 $\pi:M\rightarrow B(H)$ 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.