Communications in Mathematical Physics

, Volume 291, Issue 2, pp 473–490 | Cite as

A Noncommutative de Finetti Theorem: Invariance under Quantum Permutations is Equivalent to Freeness with Amalgamation

  • Claus Köstler
  • Roland SpeicherEmail author


We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen “exchangeability” (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables \({(x_i)_{i\in\mathbb{N}}}\) , we prove that invariance of the joint distribution of the x i ’s under quantum permutations is equivalent to the fact that the x i ’s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the x i ’s.


Hopf Algebra Conditional Expectation Factorization Property Noncommutative Analogue Classical Permutation 
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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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