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Rigid structures in the universal enveloping traffic spaceSeminar: Probability Seminar  April 18  3:104 p.m.  1011 Evans Hall Benson Au, U.C. Berkeley 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 nonspecialists in noncommutative probability. This is joint work with Camille Male. 5106422781 

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