Eta-diagonal distributions and infinite divisibility for R-diagonals

TL;DR

This paper explores the properties and infinite divisibility of R-diagonal distributions in free probability, introducing eta-diagonal distributions as Boolean counterparts and providing a parametrization of infinitely divisible cases.

Contribution

It introduces eta-diagonal distributions as Boolean analogs of R-diagonal distributions and establishes a parametrization of infinitely divisible R-diagonals via probability measures.

Findings

Parametrization of infinitely divisible R-diagonal distributions by probability measures.

Proof that the set of infinitely divisible R-diagonals is closed under free multiplicative convolution.

Establishment of properties of eta-diagonal distributions and their relation to R-diagonals.

Abstract

The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an eta-diagonal distribution that is the Boolean counterpart of an R-diagonal distribution. We establish a number of properties of eta-diagonal distributions, then we examine the canonical bijection relating eta-diagonal distributions to infinitely divisible R-diagonal ones. The overall result is a parametrization of an arbitrary $\boxplus$-infinitely divisible R-diagonal distribution that can arise in a C*-probability space, by a pair of compactly supported Borel probability measures on $[ 0, \infty )$. Among the applications of this parametrization, we prove that the set of $\boxplus$-infinitely divisible R-diagonal distributions is closed under the operation $\boxtimes$ of free multiplicative convolution.