A semi-finite algebra associated to a planar algebra
Abstract
We canonically associate to any planar algebra two type II_{\infty} 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 \mathfrak{M}_{\pm} 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_{+} \cong 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 our previous paper are isomorphic to interpolated free group factors L(\mathbb{F}(r_{j})), 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2009
- DOI:
- 10.48550/arXiv.0911.4728
- arXiv:
- arXiv:0911.4728
- Bibcode:
- 2009arXiv0911.4728G
- Keywords:
-
- Mathematics - Operator Algebras;
- 46L37;
- 46L54