Eta-diagonal distributions and infinite divisibility for R-diagonals

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 boxed plus 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 boxed plus-infinitely divisible R-diagonal distribution that can arise in a C*-probability space by a pair of compactly supported Borel probability measures on [0, infinity). Among the applications of this parametrization, we prove that the set of boxed plus-infinitely divisible R-diagonal distributions is closed under the operation boxed times of free multiplicative convolution.