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

Abstract

On the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a $C^*$-probability space $D_c(k)$, one has an operation $\freeplus$ of free additive convolution, and one can consider the subspace $D_c^{inf-div}$ of distributions which are infinitely divisible with respect to this operation. The linearizing transform for free additive convolution is the R-transform. Thus, one has $R_{\mu\freeplus\nu}=R_{\mu}+R_{\nu}$. The eta-series $\eta_{\mu}$ is the counterpart of $R_{\mu}$ in the theory of Boolean convolution. We prove that the space of eta-series of distributions belonging to $D_c(k)$ coincides with the space of R-transforms of distributions which are infinitely divisible with respect to free additive convolution. As a consequence of this fact, one can define a bijection $B : D_c(k) \to D_c^{inf-div}$ via the formula $R_{B(\mu)} = \eta_{\mu}$, for all distributions $\mu$ in $D_c(k)$. 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 $B(\mu\freetimes\nu) = B(\mu) \freetimes B(\nu),$ with $\mu,\nu$ considered in a space $D^{alg}(k)$ containing $D_c (k)$ where the operation of free multiplicative convolution $\freetimes$ always makes sense. An equivalent reformulation for this equality is that $\eta_{\mu\freetimes\nu}=\eta_{\mu} \freestar \eta_{\nu},$ for all $\mu,\nu\in D^{alg}(k)$. This 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.