The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras $H^{\mathrm{\infty }}(E)$, which we call Hardy algebras, and which are noncommutative generalizations of classical $H^{\mathrm{\infty }}$, 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 $H^{\mathrm{\infty }}(E)$. 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 $\mathbb{D}((E^{\sigma }{)}^{\ast })$. We also connect our Poission kernel to various “point evaluations” and to the idea of curvature.