Matricial R-transform
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.
Cite
@article{arxiv.1101.4389,
title = {Matricial R-transform},
author = {Romuald Lenczewski},
journal= {arXiv preprint arXiv:1101.4389},
year = {2015}
}
Comments
This is an improved version of the original paper (this concerns mainly Sections 6,7 and 8; in particular, the proof of Lemma 7.1 is more detailed and Corollary 8.1 is new)