Upgrading subordination properties in free probability theory

TL;DR

This paper extends the understanding of subordination functions in free probability, demonstrating their properties in non-tracial operator-valued C*-probability spaces using matrix constructions and analytic methods.

Contribution

It introduces a modified form of subordination for conditional expectations, broadening the applicability of subordination functions in free probability theory.

Findings

Existence of subordination functions in non-tracial settings confirmed

Matrix construction shows subordination functions satisfy modified conditions

Analytic methods establish properties of subordination functions in new context

Abstract

The existence of Voiculescu's subordination functions in the context of non-tracial operator-valued C*-probability spaces has been established using analytic function theory methods. We use a matrix construction to show that the subordination functions thus obtained also satisfy an appropriately modified form of subordination for conditional expectations.