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arxiv: 0712.2904 · v2 · submitted 2007-12-18 · 🧮 math.OA

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Random matrices, free probability, planar algebras and subfactors

A. Guionnet, D. Shlyakhtenko, V.F.R. Jones

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keywords algebraplanaralgebrasassociatedfamilygivenprobabilityrandom
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Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated von Neumann algebras are II$_{1}$ factors whose inclusions realize the given planar algebra as a system of higher relative commutants. We thus give an alternative proof to a result of Popa that every planar algebra can be realized by a subfactor.

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  1. Gauging the Categorical Connes' $\tilde{\chi}(M)$

    math.OA 2026-04 unverdicted novelty 7.0

    For any finite group G there exists a McDuff II1 factor M such that its categorical Connes tilde-chi(M) is braided equivalent to Rep(G), providing the first non-modular braided fusion category realized this way.