A Characterization of Freeness by a Factorization Property of R-transform

TL;DR

This paper characterizes freeness in free probability through a factorization property of R-transforms, linking B-valued and D-valued transforms, with applications to random matrices.

Contribution

It introduces a new factorization criterion for freeness with amalgamation over D based on R-transform properties.

Findings

Freeness characterized by R-transform factorization

Application to random matrix models

Bridges B-valued and D-valued free probability

Abstract

Let M be a B-probability space. Assume that B itself is a D-probability space; then M can be viewed as a D-probability space as well. Let X be in M. We characterize freeness of X from B with amalgamation over D in terms of a certain factorization condition linking the B-valued and D-valued R-transforms of X. We give an application to random matrices.