Free Decompositions of R-Diagonal Random Variables
Abstract
Basic notions for *-noncommutative probability spaces and B-valued
*-noncommutative probability spaces, including Voiculescu's free independence
and Speicher's cumulants, are recalled.
In both the scalar and more generally, the algebra-valued setting, R-diagonal
random variables are defined and we recall some results regarding their
decompositions into products of a Haar unitary and a self adjoint element that
are *-free from one another.
Various classes of B-valued Haar unitaries are compared and contrasted
with several distinguishing examples.
Decompositions of the particular case of C^2-valued circular elements are
investigated more thoroughly with computational methods, resulting in a proof
that every tracial C^2-valued circular having a free decomposition with a
Haar unitary must have a free even decomposition with a normalizing Haar
unitary.
Subject
free probabilityfree decomposition
polar decomposition
even decomposition
R-diagonal
circular, Haar unitary
cumulant
Citation
Griffin, John (2021). Free Decompositions of R-Diagonal Random Variables. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /195255.