A semi-finite algebra associated to a subfactor planar algebra

We canonically associate to any planar algebra two type II∞ factors M±. The subfactors constructed previously by the authors in Guionnet et al. (2010) [6], are isomorphic to compressions of M± to finite projections. We show that each M± 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 Guionnet et al. (in press) [6] are isomorphic to interpolated free group factors L(F(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.