# The Poisson Kernel for Hardy Algebras - Mathematics > Operator Algebras

Abstract: This note contributes to a circle of ideas that we have been developingrecently in which we view certain abstract operator algebras $H^{\infty}E$,which we call Hardy algebras, and which are noncommutative generalizations ofclassical $H^{\infty}$, as spaces of functions defined on their spaces ofrepresentations. We define a generalization of the Poisson kernel, whichreproduces- the values, on $\mathbb{D}E^{\sigma}^*$, of thefunctions- coming from $H^{\infty}E$. We present results that are naturalgeneralizations of the Poisson integral formuala. They also are easily seen tobe generalizations of formulas that Popescu developed. We relate our Poissonkernel to the idea of a characteristic operator function and show how thePoisson kernel identifies the model space- for the canonical model that canbe attached to a point in the disc $\mathbb{D}E^{\sigma}^*$. We alsoconnect our Poission kernel to various -point evaluations- and to the idea ofcurvature.

Author: Paul S. Muhly, Baruch Solel

Source: https://arxiv.org/