  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.
