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+2δ^{-2j}(δ-1)I, where δ^{2} is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.