Some limit theorems in operator-valued noncommutative probability

          @misc{ pirsa_PIRSA:10070016,
            doi = {10.48660/10070016},
            url = {https://pirsa.org/10070016},
            author = {Belinschi, Serban},
            keywords = {},
            language = {en},
            title = {Some limit theorems in operator-valued noncommutative probability},
            publisher = {Perimeter Institute},
            year = {2010},
            month = {jul},
            note = {PIRSA:10070016 see, \url{https://pirsa.org}}
          }
          

Abstract

A famous result in classical probability - Hin\v{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 Hin\v{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.