Extended de Finetti theorems for boolean independence and monotone independence

Liu, Weihua

doi:10.1090/tran/7034

Extended de Finetti theorems for boolean independence and monotone independence

Author: Weihua Liu
Journal: Trans. Amer. Math. Soc. 370 (2018), 1959-2003
MSC (2010): Primary 46L53; Secondary 46L54, 60G09
DOI: https://doi.org/10.1090/tran/7034
Published electronically: October 24, 2017
MathSciNet review: 3739198
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Abstract: We construct several new spaces of quantum sequences and their quantum families of maps in the sense of Sołtan. The noncommutative distributional symmetries associated with these quantum maps are noncommutative versions of spreadability and partial exchangeability. Then, we study simple relations between these symmetries. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by integers, rather than unilateral sequences. At the end of the paper, we will show Ryll-Nardzewski type theorems for monotone independence and boolean independence: Roughly speaking, an infinite bilateral sequence of random variables is monotonically (boolean) spreadable if and only if the variables are identically distributed and monotone (boolean) with respect to the conditional expectation onto its tail algebra. For an infinite sequence of noncommutative random variables, boolean spreadability is equivalent to boolean exchangeability.

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