Limits Laws for Geometric Means of Free Random Variables

Let $\{T_{k}\}_{k=1}^{\infty}$ be a family of *--free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the free central limit Theorem. More precisely, let $B_{n}=T_{1}^{*}T_{2}^{*}... T_{n}^{*}T_{n}... T_{2}T_{1}$ then $B_{n}$ is a positive operator and $B_{n}^{1/2n}$ converges in distribution to an operator $\Lambda$. We completely determine the probability distribution $\nu$ of $\Lambda$ from the distribution $\mu$ of $|T|^{2}$. This gives us a natural map $\mathcal{G}:\mathcal{M_{+}}\to \mathcal{M_{+}}$ with $\mu\mapsto \mathcal{G}(\mu)=\nu.$ We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution $\nu$ and the distribution of the Lyapunov exponents for the sequence $\{T_{k}\}_{k=1}^{\infty}$ introduced in \cite{LyaV}.