Traffic Distributions and Independence II: Universal Constructions for Traffic Spaces

Abstract : We investigate questions related to the notion of \traffics introduced by the third author as a non-commutative probability space with additional operations and equipped with the notion of traffic independence. We prove that any sequence of unitarily invariant random matrices that converges in non-commutative distribution converges as well in traffic distribution whenever it fulfils some factorisation property. We provide an explicit description of the limit which allows to recover and extend some applications (a result by Mingo and Popa on the asymptotic freeness from the transposed ensembles, and of Accardi, Lenczewski and Salapata on the freeness of infinite transitive graphs). We also improve the theory of traffic spaces by considering a positivity axiom related to the notion of state in non-commutative probability. We construct the free product of traffic spaces and prove that it preserves the positivity condition. This analysis leads to our main result stating that every non-commutative probability space endowed with a tracial state can be enlarged and equipped with a structure of traffic space.
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Pré-publication, Document de travail
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https://hal.archives-ouvertes.fr/hal-01249890
Contributeur : Camille Male <>
Soumis le : jeudi 5 décembre 2019 - 19:02:31
Dernière modification le : vendredi 10 janvier 2020 - 21:09:02

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UCTS-Final-Arxiv.pdf
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  • HAL Id : hal-01249890, version 2

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Guillaume Cébron, Antoine Dahlqvist, Camille Male. Traffic Distributions and Independence II: Universal Constructions for Traffic Spaces. 2016. ⟨hal-01249890v2⟩

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