The Poisson Kernel for Hardy Algebras

TL;DR

This paper introduces a generalized Poisson kernel for Hardy algebras, extending classical harmonic analysis concepts to noncommutative operator algebra settings and relating it to model theory and curvature.

Contribution

It defines a new Poisson kernel for Hardy algebras, generalizing classical formulas and connecting to characteristic functions and model theory in noncommutative analysis.

Findings

Generalized Poisson kernel reproduces function values on noncommutative domains.

Links Poisson kernel to characteristic operator functions and model spaces.

Connects the kernel to point evaluations and curvature concepts.

Abstract

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras $H^{\infty}(E)$, which we call Hardy algebras, and which are noncommutative generalizations of classical $H^{\infty}$, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which ``reproduces'' the values, on $\mathbb{D}((E^{\sigma})^*)$, of the ``functions'' coming from $H^{\infty}(E)$. We present results that are natural generalizations of the Poisson integral formuala. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the ``model space'' for the canonical model that can be attached to a point in the disc $\mathbb{D}((E^{\sigma})^*)$. We also connect our Poission kernel to various "point evaluations" and to the idea of curvature.