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

Abstract

Let D c ( k ) be the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a C ∗ -probability space. On D c ( k ) one has an operation ⊞ of free additive convolution, and one can consider the subspace D c inf - div ( 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 = R μ + R ν , ∀ μ , ν ∈ D c ( k ) ). We prove that the set of R-transforms { R μ | μ ∈ D c inf - div ( k ) } can also be described as { η μ | μ ∈ D c ( k ) } , where for μ ∈ D c ( k ) we denote η μ = M μ / ( 1 + M μ ) , with M μ the moment series of μ. (The series η μ is the counterpart of R μ in the theory of Boolean convolution.) As a consequence, one can define a bijection B : D c ( k ) → D c inf - div ( k ) via the formula (I) R B ( μ ) = η μ , ∀ μ ∈ D c ( 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 D alg ( k ) ⊇ D c ( k ) where the operation of free multiplicative convolution ⊠ always makes sense. An equivalent reformulation of (II) is that [Display omitted] 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.