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 factorization property and we provide an explicit description of the limit. 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.