Quantum exchangeable sequences of algebras

Curran, Stephen

doi:10.1512/iumj.2009.58.3939

Cited by 23 publications

(52 citation statements)

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“…Köstler and Speicher discovered in [42] that a free de Finetti theorem holds, with S N replaced by S + N . Curran found a bit later a more advanced proof, and generalizations, using the Weingarten formula [30], [31]. These results, along with [50], suggested a whole new approach to probabilistic invariance questions, by axiomatizing and classifying the compact quantum groups having an "elementary" Tannakian dual, and then by studying the actions of such quantum groups on random variables.…”

Section: Introductionmentioning

confidence: 99%

A duality principle for noncommutative cubes and spheres

Banica¹

2016

J. Noncommut. Geom.

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Section: Introductionmentioning

confidence: 99%

A duality principle for noncommutative cubes and spheres

Banica¹

2016

J. Noncommut. Geom.

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“…A similar extension of the noncommutative de Finetti theorem was obtained by Curran [2]. He considered quantum exchangeability of * -representations of a *algebra and proved that, for an infinite sequence of such, freeness with amalgamation over a tail algebra holds.…”

Section: Introductionmentioning

confidence: 59%

“…However, he did assume faithfulness of the state in the W * -noncommutative probability space. Although our results up to this point overlap with and are similar in spirit to Curran's, we have included our proof because (a) it is different from that contained in [2] (and closer to the proof of [8]) and (b) we must allow the states on the relevant W * -algebras to be non-faithful.…”

Section: Introductionmentioning

confidence: 62%

Quantum symmetric states on free product $C^*$-algebras

Dykema

Köstler

Williams

2016

Trans. Amer. Math. Soc.

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Abstract. We introduce symmetric states and quantum symmetric states on universal unital free product C * -algebras of the form A = * ∞ 1 A for an arbitrary unital C * -algebra A, as a generalization of the notions of exchangeable and quantum exchangeable random variables. We prove existence of conditional expectations onto tail algebras in various settings and we define a natural C * -subalgebra of the tail algebra, called the tail C * -algebra. Extending and building on the proof of the noncommutative de Finetti theorem of Köstler and Speicher, we prove a de Finetti type theorem that characterizes quantum symmetric states in terms of amalgamated free products over the tail C * -algebra, and we provide a convenient description of the set of all quantum symmetric states on A in terms of C * -algebras generated by homomorphic images of A and the tail C * -algebra. This description allows a characterization of the extreme quantum symmetric states. Similar results are proved for the subset of tracial quantum symmetric states, though in terms of von Neumann algebras and normal conditional expectations. The central quantum symmetric states are those for which the tail algebra is in the center of the von Neumann algebra, and we show that the central quantum symmetric states form a Choquet simplex whose extreme points are the free product states, while the tracial central quantum symmetric states form a Choquet simplex whose extreme points are the free product traces.

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“…Usually, such actions are extended to the von Neumann algebra generated by M in the GNS representation of ϕ. However, we cannot apply [4,Thm 3.3] to do this here, because the action β N is not multiplicative. This is the reason why we will have to deal only with the algebra generated by the variables all along.…”

Section: Quantum Invariance and Bi-freeness For Families Of Pairsmentioning

confidence: 99%

“…Let ρ j be the unique homomorphism from the noncommutative polynomial algebra C⟨X ℓ , X r ⟩ to M such that ρ j (X ℓ ) = x ℓ j and ρ j (X r ) = x r j . Then, classical bi-exchangeability is equivalent to the exchangeability of the sequence (ρ j ) j∈N in the sense of [4,Def 4.3].…”

Section: 2mentioning

confidence: 99%

On bi-free De Finetti theorems

Freslon¹,

Weber²

2016

Annales Mathématiques Blaise Pascal

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Abstract. We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bifreeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting. We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of n-freeness.

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Quantum exchangeable sequences of algebras

Cited by 23 publications

References 0 publications

A duality principle for noncommutative cubes and spheres

A duality principle for noncommutative cubes and spheres

Quantum symmetric states on free product $C^*$-algebras

On bi-free De Finetti theorems

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