Representations of Hardy Algebras: Absolute Continuity, Intertwiners and Superharmonic Operators

TL;DR

This paper studies how to extend certain algebraic representations from tensor algebras to Hardy algebras, generalizing classical and recent noncommutative operator theory results.

Contribution

It extends the theory of extending representations from tensor algebras to Hardy algebras, generalizing classical and noncommutative cases.

Findings

Established conditions for extending representations to ultra-weakly continuous forms

Connected classical Sz.-Nagy-Foiaș calculus with noncommutative algebra representations

Extended the representation theory of noncommutative disc algebras

Abstract

Suppose $\mathcal{T}_{+}(E)$ is the tensor algebra of a $W^{*}$-correspondence $E$ and $H^{\infty}(E)$ is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of $\mathcal{T}_{+}(E)$ on a Hilbert space to ultra-weakly continuous completely contractive representations of $H^{\infty}(E)$ on the same Hilbert space. Our work extends the classical Sz.-Nagy - Foia\c{s} functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu's noncommutative disc algebra.