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

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.