# Universal constructions for spaces of traffics

Abstract : We investigate questions related to the notion of traffics introduced in [8] as a noncommu-tative probability space with numerous additional operations and equipped with the notion of traffic independence. We prove that any sequence of unitarily invariant random matrices that converges in noncommutative distribution converges in distribution of traffics whenever it fulfills some factorization property. We provide an explicit description of the limit which allows to recover and extend some applications (on the freeness from the transposed ensembles [12] and the freeness of infinite transitive graphs [1]). We also improve the theory of traffic spaces by considering a positivity axiom related to the notion of state in noncom-mutative probability. We construct the free product of spaces of traffics and prove that it preserves the positivity condition. This analysis leads to our main result stating that every noncommutative probability space endowed with a tracial state can be enlarged and equipped with a structure of space of traffics.
Keywords :
Type de document :
Pré-publication, Document de travail
2016
Domaine :

https://hal.archives-ouvertes.fr/hal-01249890
Contributeur : Camille Male <>
Soumis le : lundi 4 janvier 2016 - 08:42:04
Dernière modification le : mardi 11 octobre 2016 - 13:42:19

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UCST_Arxiv.pdf
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• HAL Id : hal-01249890, version 1

### Citation

Guillaume Cébron, Antoine Dahlqvist, Camille Male. Universal constructions for spaces of traffics. 2016. <hal-01249890>

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