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Canonical structures in traffic spaces: with a view toward random matrices

Banff International Research Station for Mathematical Innovation and Discovery

For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, Cébron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau)$ that extends the trace $\varphi$ [CDM16]. This construction comes equipped with some canonical independence structure: in a joint work in progress with Male, we show that $(\mathcal{G}(\mathcal{A}), \tau)$ can be realized as the free product of three natural subalgebras (in the sense of Voiculescu), and that there exists a canonical homomorphic conditional expectation $P$ onto a subalgebra intermediate to $\mathcal{A}$ and $\mathcal{G}(\mathcal{A})$. Combining this with the coherent convergence properties of $(\mathcal{G}(\mathcal{A}), \tau)$ proved in [CDM16], we show that free independence describes the asymptotic behaviour of a large class of dependent random matrices (in particular, we recover and explain a result of Bryc, Dembo, and Jiang on random Markov matrices [BDJ06]).

Mathematics; Functional analysis; Probability theory and stochastic processes; Operator theory/algebras

video/mp4

Author affiliation: University of California, Berkeley

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61628

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/