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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, which``reproduces- the values, on $\mathbb{D}E^{\sigma}^*$, of the``functions- 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


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