A noncommutative extended de Finetti theorem

2008/06/23 by Claus Köstler, Köstler, Claus · 2 citations
Mathematics · #46L53 #47A53 #60G09 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.0806.3621

openalex publication_date 2008/06/23 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/30

Abstract

The extended de Finetti theorem characterizes exchangeable infinite random sequences 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 Ryll-Nadzewski, exchangeability turns out to be stronger than spreadability for infinite noncommutative random sequences. 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.

Citations

• A quantum-mechanical central limit theorem

Cited by

• Full classification of de Finetti type theorems for *-random variables in classical and free probability
• On the Ryll-Nardzewski Theorem for Quantum Stochastic Processes

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