Representation of Free Herglotz Functions

A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right half plane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.