Representation of free Herglotz functions

TL;DR

This paper develops a free analytic analogue of the classical Herglotz representation, extending it to noncommutative settings and analyzing the topological properties of the resulting function set.

Contribution

It introduces a free Herglotz representation theorem and explores the structure and closure properties of noncommutative Herglotz functions.

Findings

Established a free analogue of the classical Herglotz representation.

Characterized the specialization to free probabilistic cases.

Proved the set of noncommutative Herglotz functions from conditional expectations is topologically closed.

Abstract

A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. 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.