Abstract

Let Dc(k) be the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a C∗-probability space. On Dc(k) one has an operation ⊞ of free additive convolution, and one can consider the subspace Dinf-div c (k) of distributions which are infinitely divisible with respect to this operation. The linearizing transform for ⊞ is the R-transform (one has R µ⊞ν = Rµ + Rν, ∀ µ, ν ∈ Dc(k)). We prove the equality {Rµ | µ ∈ D inf-div c (k)} = {ηµ | µ ∈ Dc(k)}, (I) where for µ ∈ Dc(k) we denote by ηµ the η-series associated to µ. (The series ηµ is the counterpart of Rµ in the theory of Boolean convolution. A quick way to define it is by the formula ηµ = Mµ/(1 + Mµ), with Mµ the moment seris of µ.) As a consequence of (I), one can define a bijection B: Dc(k) → Dinf-div c (k) via the formula R B(µ) = ηµ, ∀ µ ∈ Dc(k). We show that B is a multi-variable analogue of a bijection studied by Bercovici and Pata for k = 1, and we prove a theorem about convergence in moments which parallels the Bercovici-Pata result. On the other hand we prove the formula B(µ ⊠ ν) = B(µ) ⊠ B(ν), (II) (III) with µ, ν considered in a space Dalg(k) ⊇ Dc(k) where the operation of free multiplicative convolution ⊠ always makes sense. An equivalent reformulation of (III) is that η µ⊠ν = ηµ ⋆ ην, ∀ µ, ν ∈ Dalg(k). (IV) This shows that, in a certain sense, η-series behave in the same way as R-transforms in connection to the operation of multiplication of free k-tuples of non-commutative random variables.

Citations