Abstract

Distributional symmetries and invariance principles provide deep structural results in classical probability. For example, the de Finetti theorem characterizes an infinite sequence of random variables to be conditional independent and identically distributed if and only if its joint distribution is invariant under permuting these random variables. Recently significant progress was made in transferring de Finetti type results to an operator algebraic setting of noncommutative probability. Here commuting squares provide a noncommutative generalization of conditional independence in classical probability.   I will briefly overview some of these newer developments, in particular when applied to subfactors. My talk will focus on ongoing research on distributional symmetries in the context of Jones-Temperley-Lieb algebras.  It is based on joint work with Gwion Evans, Rolf Gohm, Arundhathi Krishnan, and Stephen Wills.