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

Let D-c(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 El 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 boxed plus is the R-transform (one has R-mu boxed plus nu = R-mu + R-nu, for all(mu), nu is an element of D-c(k)). We prove that the set of R-transforms {R-mu vertical bar mu is an element of D-c(inf-div)(k)} can also be described as {eta (mu) vertical bar mu is an element of D-c(k)}, where for mu is an element of D-c(k) we denote eta(mu) = M-mu/(1 + M mu), with M-mu, the moment series of g. (The series n. is the counterpart of RA 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 R-B(mu) = eta(mu), for all(mu) is an element of D-c(k). (I) 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(mu boxed times nu) = B(mu) boxed times B(nu), (II) with mu, nu considered in a space D-alg(k) superset of D-c(k) where the operation of free multiplicative convolution boxed times always makes sense. An equivalent reformulation of (II) is that 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, eta-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. (c) 2007 Elsevier Inc. All rights reserved.