Matricial R-transform

TL;DR

This paper introduces the matricial R-transform, a unified noncommutative analogue of the logarithm of the Fourier transform, for various types of noncommutative independence, generalizing free probability concepts.

Contribution

It derives a linearization formula for the matricial R-transform using Toeplitz operators, unifying multiple noncommutative independence types.

Findings

Derived a linearization formula for the matricial R-transform.

Unified various noncommutative independence types under a single framework.

Established the matricial R-transform as a noncommutative logarithm of the convolution.

Abstract

We study the addditon problem for strongly matricially free random variables which generalize free random variables. Using operators of Toeplitz type, we derive a linearization formula for the `matricial R-transform' related to the associated convolution. It is a linear combination of Voiculescu's R-transforms in free probability with coefficients given by internal units of the considered array of subalgebras. This allows us to view this formula as the `matricial linearization property' of the R-transform. Since strong matricial freeness unifies the main types of noncommutative independence, the matricial R-transform plays the role of a unified noncommutative analog of the logarithm of the Fourier transform for free, boolean, monotone, orthogonal, s-free and c-free independence.