Eta-series and a Boolean Bercovici-Pata bijection for bounded k-tuples
| dc.creator | Belinschi, Serban T. | |
| dc.creator | Nica, Alexandru | |
| dc.date | 2006-08-25 | |
| dc.date | 2007-06-25 | |
| dc.date.accessioned | 2026-07-07T08:12:00Z | |
| dc.date.available | 2026-07-07T08:12:00Z | |
| dc.description | 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 $R_{μ\freeplusν}=R_μ+R_ν$. The eta-series $η_μ$ is the counterpart of $R_μ$ 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 $R_{B(μ)} = η_μ$, for all distributions $μ$ 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 $μ,ν$ 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 $μ,ν\in D^{alg}(k)$. 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. | |
| dc.description | LaTeX, 41 pages. Minor changes and corrections, added references | |
| dc.identifier | https://arxiv.org/abs/math/0608622 | |
| dc.identifier | http://arxiv.org/abs/math/0608622 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132305 | |
| dc.subject | Operator Algebras | |
| dc.subject | Combinatorics | |
| dc.subject | 46L54 | |
| dc.title | Eta-series and a Boolean Bercovici-Pata bijection for bounded k-tuples | |
| dc.type | text |