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

AbstractLet 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 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+Rν, ∀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)}, where for μ∈Dc(k) we denote ημ=Mμ/(1+Mμ), with Mμ the moment series of μ. (The series ημ is the counterpart of Rμ in the theory of Boolean convolution.) As a consequence, one can define a bijection B:Dc(k)→Dcinf-div(k) via the formula(I)RB(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(II)B(μ⊠ν)=B(μ)⊠B(ν), 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 (II) is that(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