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

By Claus Köstler, Roland Speicher. 2008

95 citations recorded by the source corpus.

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 xi’s under quantum permutations is equivalent to the fact that the xi’s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the xi’s.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f315c39e7f0c683c9001d15aabfa7b141c92f98a

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Citations in the corpus

Free Probability and Random Matrices (2014, background): “So we are, for example, not addressing such exciting topics as free stochastic and Malliavin calculus [40, 108, 114], or the rectangular version of free probability [28], or the strong version of asymptotic freeness [48, 58, 89], or free monotone transport [83], or the relation with representation t”
Easy quantum groups and quantum subgroups of a semi-direct product quantum group (2013, methodology): “After the work of Köstler and Speicher in [KS09], de Finetti theorems became a central piece of the theory of easy quantum groups.”
On the classification of easy quantum groups (2012, unknown): “s of some of the easy quantum groups have been studied in [VV07], [B12a], [B12b], [F12]. The link between easy quantum groups and free probability theory has been discovered by Ko¨stler and Speicher, [KS09]. Their noncommutative version of the De Finetti theorem hints at the deep connection between ”
Classification results for easy quantum groups (2009, background): “that might fall one day into an extension of our formalism. Here is a list of topics, waiting to be developed: (1) De Finetti theorems. These are available for Sn,On from the book [24], for S+ n from [25] then [19], and for O+ n from [20]. We intend to develop a global approach to the problem, by us”
The Full Classification of Orthogonal Easy Quantum Groups (2013, methodology): “[CuSp11a] S. Curran and R. Speicher.”
De Finetti theorems for easy quantum groups (2009, background): “These are compact quantum groups in the sense of Woronowicz [35].”
Unitary Easy Quantum Groups: the free case and the group case (2015, background): “The most important applications are de Finetti theorems, [KS09] and [BCS12b].”