×

Random matrices, free probability, planar algebras and subfactors. (English) Zbl 1219.46057

Blanchard, Etienne (ed.) et al., Quanta of maths. Conference on non commutative geometry in honor of Alain Connes, Paris, France, March 29–April 6, 2007. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-5203-3/pbk). Clay Mathematics Proceedings 11, 201-239 (2010).
Summary: 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.
For the entire collection see [Zbl 1206.00042].

MSC:

46L54 Free probability and free operator algebras
15B52 Random matrices (algebraic aspects)
46L37 Subfactors and their classification
PDFBibTeX XMLCite
Full Text: arXiv