Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not

TL;DR

This paper explores the relationship between Cauchy transforms and free Pick functions, demonstrating that while the classical correspondence holds for bounded cases, it fails generally for unbounded cases, but can be restored in homomorphic free probability settings.

Contribution

The paper establishes that Cauchy transforms from homomorphic conditional expectations uniquely parametrize free Pick functions in a specific operator-valued free probability context.

Findings

Classical parametrization fails for unbounded cases.

Homomorphic conditional expectations restore the parametrization.

Provides insights into free probability and operator-valued functions.

Abstract

Nevanlinna showed that Cauchy transforms of probability measures parametrize all functions from the upper half plane into itself satisfying a certain asymptotic condition at infinity. We show that the correspondence fails in general for the unbounded case for somewhat trivial reasons; however, we show that in a setting of "homomorphic" operator valued free probability that Cauchy transforms of homomorphic conditional expectations parametrize free Pick functions.