IUMJ

Title: On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

Authors: Serban T. Belinschi and Alexandru Nica

Issue: Volume 57 (2008), Issue 4, 1679-1714

Abstract:

Let $mathcal{M}$ denote the space of Borel probability measures on $\mathbb{R}$. For every $t\geq0$ we consider the transformation $\mathbb{B}_t:\mathcal{M}\to\mathcal{M}$ defined by \[ mathbb{B}_t(\mu)=\big(\mu^{\boxplus(1+t)}\big)^{\uplus(1/(1+t))},\quad\mu\in\mathcal{M}, \] where $\boxplus$ and $\uplus$ are the operations of free additive convolution and respectively of Boolean convolution on $\mathcal{M}$, and where the convolution powers with respect to $\boxplus$ and $\uplus$ are defined in the natural way. We show that $\mathbb{B}_s\circ\mathbb{B}_t=\mathbb{B}_{s+t}$, $\forall\,s$, $t\geq0$ and that, quite surprisingly, every $\mathbb{B}_t$ is a homomorphism for the operation of free \textit{multiplicative} convolution $\boxtimes$ (that is, $\mathbb{B}_t(\mu\boxtimes\nu)=\mathbb{B}_t(\mu)\boxtimes\mathbb{B}_t(\nu)$ for all $\mu$, $\nu\in\mathcal{M}$ such that at least one of $\mu$, $\nu$ is supported on $[0,\infty)$). We prove that for $t=1$ the transformation $\mathbb{B}_1$ coincides with the canonical bijection $\mathbb{B}:\mathcal{M}\to\mathcal{M}_{\infdiv}$ discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here $\mathcal{M}_{\infdiv}$ stands for the set of probability distributions in $\mathcal{M}$ which are infinitely divisible with respect to the operation $\boxplus$. As a consequence, we have that $\mathbb{B}_t(\mu)$ is $\boxplus$-infinitely divisible for every $\mu\in\mathcal{M}$ and every $t\geq1$. On the other hand we put into evidence a relation between the transformations $\mathbb{B}_t$ and the free Brownian motion; indeed, Theorem 1.6 of the paper gives an interpretation of the transformations $\mathbb{B}_t$ as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to $\boxtimes$, and always reaches $\boxplus$-infinite divisibility by the time $t=1$.