η-series and a Boolean Bercovici–Pata bijection for bounded k-tuples

Let Dc(k)Dc(k) be the space of (non-commutative) distributions of k  -tuples of selfadjoint elements in a C∗C∗-probability space. On Dc(k)Dc(k) one has an operation ⊞ of free additive convolution, and one can consider the subspace Dcinf-div(k) of distributions which are infinitely divisible with respect to this operation. The linearizing transform for ⊞ is the R  -transform (one has Rμ⊞ν=Rμ+RνRμ⊞ν=Rμ+Rν, ∀μ,ν∈Dc(k)∀μ,ν∈Dc(k)). We prove that the set of R  -transforms {Rμ|μ∈Dcinf-div(k)} can also be described as {ημ|μ∈Dc(k)}{ημ|μ∈Dc(k)}, where for μ∈Dc(k)μ∈Dc(k) we denote ημ=Mμ/(1+Mμ)ημ=Mμ/(1+Mμ), with MμMμ the moment series of μ  . (The series ημημ is the counterpart of RμRμ in the theory of Boolean convolution.) As a consequence, one can define a bijection B:Dc(k)→Dcinf-div(k) via the formulaequation(I)RB(μ)=ημ,∀μ∈Dc(k). We show that BB is a multi-variable analogue of a bijection studied by Bercovici and Pata for k=1k=1, and we prove a theorem about convergence in moments which parallels the Bercovici–Pata result. On the other hand we prove the formulaequation(II)B(μ⊠ν)=B(μ)⊠B(ν),B(μ⊠ν)=B(μ)⊠B(ν), with μ,νμ,ν considered in a space Dalg(k)⊇Dc(k)Dalg(k)⊇Dc(k) where the operation of free multiplicative convolution ⊠ always makes sense. An equivalent reformulation of (II) is thatequation(III)ημ⊠ν=ημην,∀μ,ν∈Dalg(k), where is an operation on series previously studied by Nica and Speicher, and which describes the multiplication of free k-tuples in terms of their R-transforms. Formula (III) 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.