Traffic Distributions and Independence II: Universal Constructions for Traffic Spaces

TL;DR

This paper advances the theory of traffic spaces in noncommutative probability, establishing conditions for convergence of traffics, describing their limits, and constructing free products that preserve positivity, thereby broadening the framework's applicability.

Contribution

It introduces a positivity axiom for traffic spaces, provides explicit descriptions of limits of traffics, and constructs free products that maintain positivity, extending the theory's scope.

Findings

Convergence of unitarily invariant matrices in traffics under factorization.

Explicit description of traffic space limits enabling application extensions.

Construction of free products preserving positivity in traffic spaces.

Abstract

We investigate questions related to the notion of traffics introduced by the author C. Male as a noncommutative 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 by Mingo and Popa and the freeness of infinite transitive graphs by Accardi, Lenczewski and Salapata). We also improve the theory of traffic spaces by considering a positivity axiom related to the notion of state in noncommutative 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.