Rigid structures in the universal enveloping traffic space

Seminar | April 18 | 3:10-4 p.m. | 1011 Evans Hall

 Benson Au, U.C. Berkeley

 Department of Statistics

For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, C\'{e}bron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ that extends the trace $\varphi$. The CDM construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure. We show that $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{A}, \varphi)$. If $(\mathcal{A}, \varphi)$ is itself a free product, then we show how this structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$. Here, we find a duality between classical independence and free independence. We apply our results to prove the asymptotic freeness of a large class of dependent random matrices, generalizing results of Bryc, Dembo, and Jiang and of Mingo and Popa. The talk will be accessible to non-specialists in non-commutative probability. This is joint work with Camille Male.