PIRSA:10070016  ( MP4 Medium Res , MP3 , PDF ) Which Format?
Some limit theorems in operator-valued noncommutative probability
Speaker(s): Serban Belinschi
Abstract: A famous result in classical probability - Hinv{c}in's Theorem - establishes a bijection between infinitely divisible probability distributions and limits of infinitesimal triangular arrays of independent random variables. Analogues of this result have been proved by Bercovici and Pata for scalar-valued {em free probability}. However, very little is known for the case of operator-valued distributions, when the field of scalars is replaced by a $C^*$-algebra; essentially the only result known in full generality that we are aware of is Voiculescu's operator-valued central limit theorem. In this talk we will use a recent breakthrough in the description of infinite divisibility of operator-valued distributions achieved by Popa and Vinnikov to prove a Hinv{c}in-type theorem for operator-valued free random variables and to formulate a free - to - conditionally free Bercovici-Pata bijection. Time permitting, we will discuss in more detail relaations between the operator-valued free, Boolean and monotone central limits. This is joint work with Mihai V. Popa and Victor Vinnikov.
Date: 05/07/2010 - 10:30 am
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