Cauchy transforms arising from homomorphic conditional expectations   parametrize free Pick functions but those arising from conditional   expectations do not
  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. 
J. E. Pascoe, Ryan Tully-Doyle
math.FA math.OA