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arXiv · 1608.03515

Eta-diagonal distributions and infinite divisibility for R-diagonals

Hari Bercovici·Alexandru Nica·Michael Noyes·Kamil Szpojankowski
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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.

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Hari Bercovici, Alexandru Nica, Michael Noyes, Kamil Szpojankowski. 2016-08-11. Eta-diagonal distributions and infinite divisibility for R-diagonals. https://arxiv.org/abs/1608.03515

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