2009/11/25 by Alice Guionnet, A. Guionnet, Vaughan F. R. Jones +6 · 2 citations
Mathematics · #46L37 #46L54 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46L37 #msc:46L54
paper · pdf · doi:10.48550/arxiv.0911.4728
arxiv created 2009/11/25 · openalex publication_date 2009/11/25 · arxiv updated 2009/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We canonically associate to any planar algebra two type II∞ factors M+ and M-. The subfactors constructed previously by the authors in a previous paper are isomorphic to compressions of M+ and M- to finite projections. We show that each \mathfrakM± is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M+ ≅ M- is the amplification a free group factor on a finite number of generators. As an application, we show that the factors Mj constructed in our previous paper are isomorphic to interpolated free group factors L(\mathbbF(rj)), rj=1+2δ-2j(δ-1)I, where δ2 is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.