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Abstract
The extended de Finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of RyllNardzewski, exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and Kümmerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors.
Original language  English 

Pages (fromto)  10731120 
Number of pages  48 
Journal  Journal of Functional Analysis 
Volume  258 
Issue number  4 
DOIs  
Publication status  Published  15 Feb 2010 
Keywords
 Noncommutative de Finetti theorem
 Distributional symmetries
 Exchangeability
 Spreadability
 Noncommutative conditional independence
 Mean ergodic theorem
 Noncommutative Kolmogorov zero–one law
 Noncommutative Bernoulli shifts
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 1 Finished

Quantum Control : Approach Based on Scattering Theory for Noncommutative Markov Chains
Engineering and Physical Sciences Research Council
01 Jun 2009 → 31 May 2012
Project: Externally funded research