A simplicial approach to noncommutative distributional symmetries

Gwion Evans (Aberystwyth University)

Tuesday 22nd March, 2016 16:00-17:00 Maths 522


Distributional symmetries (or invaricance principles) play an important role in noncommutative proabability theory.  It was shown by Gohm and Köstler that in the setting of von Neumann algebras and the corresponding noncommutative proabability spaces there is a noncommutative version of the classical de Finetti theorem for spreadable sequences, which states that a spreadable sequence of random variables is braidable and therefore exchangeable and thus conditionally independent and identically distributed in a noncommutative sense.  Braidability and exchangeability are distributional symmetries arising from actions of the braid group and symmetric group, respectively.  In this talk I will show that spreadability (the invariance of a distribution when one passes from a sequence of random variables to a subsequence) has a homological flavour arising from the (semi-)simplicial category.  After reviewing some basic constructions from semi-simplicial category theory, I will show how one can make an explicit connection with spreadable sequences, illustrated by examples in the von Neumann setting, which may provide some insight into constructing further examples that distinguish between spreadable and braidable sequences.  This is joint work with Rolf Gohm and Claus Köstler.

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