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

J. E Pascoe; Ryan Tully-Doyle

doi:10.48550/arxiv.1607.06737

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

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

J. E Pascoe and Ryan Tully-Doyle

arXiv (Cornell University)

22 Jul 2016

DOI: https://doi.org/10.48550/arxiv.1607.06737

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Abstract

Mathematics - Functional Analysis

Mathematics - Operator Algebras

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.

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Title: Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not
Creators: J. E Pascoe
Ryan Tully-Doyle
Publication Details: arXiv (Cornell University)
Resource Type: Preprint
Language: English
Academic Unit: Mathematics
Other Identifier: 991021879757004721

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

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Abstract

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