Universal constructions for spaces of traffics

Guillaume Cébron (1) , Antoine Dahlqvist (2) , Camille Male (3)

1 Saarland University [Saarbrücken]
2 CAM - University of Cambridge [UK]
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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.

Mots clés: free probability; random matrices

Domaines: Probabilités [math.PR]; Algèbres d'opérateurs [math.OA]

Guillaume Cébron, Antoine Dahlqvist, Camille Male. Universal constructions for spaces of traffics. 2016. ⟨hal-01249890v1⟩