Eta-diagonal distributions and infinite divisibility for 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 ⊞ 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 η-diagonal distribution that is the Boolean counterpart of an $R$-diagonal distribution. We establish a number of properties of η-diagonal distributions, then we examine the canonical bijection relating η-diagonal distributions to infinitely divisible R R -diagonal ones. The overall result is a parametrization of an arbitrary ⊞-infinitely divisible $R$-diagonal distribution that can arise in a $C^*$-probability space by a pair of compactly supported Borel probability measures on [0,∞) [ 0 , ∞ ) . Among the applications of this parametrization, we prove that the set of ⊞-infinitely divisible $R$-diagonal distributions is closed under the operation ⊞ of free multiplicative convolution.