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.