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arXiv · 0712.2904

Random matrices, free probability, planar algebras and subfactors

A. Guionnet·V. F. R. Jones·D. Shlyakhtenko
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Abstract

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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A. Guionnet, V. F. R. Jones, D. Shlyakhtenko. 2008-07-08. Random matrices, free probability, planar algebras and subfactors. https://arxiv.org/abs/0712.2904

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