Random matrices, free probability, planar algebras and subfactors

By Alice Guionnet, Vaughan F. R. Jones, D. Shlyakhtenko. 2007

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Author(s): Guionnet, A; Jones, VFR; Shlyakhtenko, D | 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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e12c33010047621e082af2f74c40c3e3427100c4