Celebratio Mathematica

A. Guionnet, V. F. R. Jones, and D. Shlyakhtenko: “Random matrices, free probability, planar algebras and subfactors,” pp. 201–239 in Quanta of maths: Conference on non commutative geometry in honor of Alain Connes (Paris, 29 March–6 April 2007). Edited by E. Blanchard, D. Ellwood, M. Khalkhali, M. Marcolli, H. Moscovici, and S. Popa. Clay Mathematics Proceedings 11. American Mathematical Society (Providence, RI), 2010. MR 2732052 Zbl 1219.46057 ArXiv 0712.2904 incollection

Sorin Teodor Popa	Related
Étienne Blanchard	Related
Alain Connes	Related
Masoud Khalkhali	Related
Dimitri Y. Shlyakhtenko	Related
Alice Guionnet	Related
David Alexandre Ellwood	Related
Matilde Marcolli	Related
Henri Moscovici	Related

A. Guionnet, V. Jones, and D. Shlyakhtenko: “A semi-finite algebra associated to a subfactor planar algebra,” J. Funct. Anal. 261 : 5 (September 2011), pp. 1345–1360. MR 2807103 Zbl 1230.46054 article

We canonically associate to any planar algebra two type \( \mathrm{II}_{\infty} \) factors \( \mathfrak{M}_{\pm} \). The subfactors constructed previously by the authors in [Guionnet et al. 2010] are isomorphic to compressions of \( \mathfrak{M}_{\pm} \) to finite projections. We show that each \( \mathfrak{M}_{\pm} \) is isomorphic to an amalgamated free product of type \( \mathrm{I} \) von Neumann algebras with amalgamation over a fixed discrete type \( \mathrm{I} \) von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that \[ \mathfrak{M}_{+} \cong \mathfrak{M}_{-} \] is the amplification a free group factor on a finite number of generators. As an application, we show that the factors \( M_j \) constructed in [Guionnet et al. 2010] are isomorphic to interpolated free group factors \[ L(\mathbb{F}(r_j)),\quad r_j = 1 + 2\delta^{-2j}(\delta - 1)I , \] where \( \delta^2 \) is the index of the planar algebra and \( I \) is its global index. Other applications include computations of laws of Jones–Wenzl projections.

Dimitri Y. Shlyakhtenko	Related
Alice Guionnet	Related

A. Guionnet, V. F. R. Jones, D. Shlyakhtenko, and P. Zinn-Justin: “Loop models, random matrices and planar algebras,” Comm. Math. Phys. 316 : 1 (2012), pp. 45–97. MR 2989453 Zbl 1277.82013 ArXiv 1012.0609 article

Jean Zinn-Justin	Related
Dimitri Y. Shlyakhtenko	Related
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Vaughan F. R. Jones

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