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 $C^{\ast }$-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 $R_{μ\freeplusν}=R_μ+R_ν$. 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 $B(μ\freetimesν) = B(μ) \freetimes B(ν),$ 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 $η_{μ\freetimesν}=η_μ \freestar η_ν,$ 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.

LaTeX, 41 pages. Minor changes and corrections, added references

Cited by in corpus (2)

• Appell polynomials and their relatives II. Boolean theory
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