Traffic Distributions and Independence II: Universal Constructions for Traffic Spaces

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

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 University of Sussex
3 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: free probability; random matrices

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

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