Progress in noncommutative function theory

TL;DR

This paper explores noncommutative Hardy algebras as operator-valued function spaces, demonstrating their properties resemble bounded analytic functions and establishing a framework for noncommutative function theory.

Contribution

It provides a representation-theoretic framework for non-self-adjoint operator algebras as noncommutative function spaces, linking Hardy algebras to Schur class functions.

Findings

Representation spaces are parameterized by unit balls of W*-correspondences.

Elements behave similarly to bounded analytic functions.

Supports viewing noncommutative Hardy algebras as noncommutative function theory.

Abstract

In this expository paper we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of representations. We will show that these spaces of representations can be parameterized as unit balls of certain $W^{*}$-correspondences and the functions can be viewed as Schur class operator functions on these balls. We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory.