Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not
Functional Analysis
2016-07-25 v1 Operator Algebras
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.
Cite
@article{arxiv.1607.06737,
title = {Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not},
author = {J. E. Pascoe and Ryan Tully-Doyle},
journal= {arXiv preprint arXiv:1607.06737},
year = {2016}
}
Comments
12 pages