2008/02/28 by Gabriel H. Tucci, Tucci, Gabriel H.
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #math.OA #math.PR
paper · pdf · doi:10.48550/arxiv.0802.4226
Published in Indiana Journal of Mathematics, vol. 59, no. 1, pp. 1-13, 2010
arxiv created 2010/10/02 · arxiv updated 2010/10/05
Let \Tk\k=1∞ 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 Bn=T1*T2*... Tn*Tn... T2T1 then Bn is a positive operator and Bn1/2n converges in distribution to an operator Λ. We completely determine the probability distribution ν of Λ from the distribution μ of |T|2. This gives us a natural map G:\mathcalM+→ \mathcalM+ with μ↦ G(μ)=ν. 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 ν and the distribution of the Lyapunov exponents for the sequence \Tk\k=1∞ introduced in \citeLyaV.