In Jones's paper "Planar Algebras I", Theorem 4.2.1 establishes that an extremal finite index subfactor admits a spherical C*-planar algebra structure, and Theorem 4.3.1 establishes that spherical C*-planar algebra arise from subfactors. It seems unlikely, due to the key ingredient of the proof of Theorem 4.3.1, that these processes are inverses of one another.

Question: Is there a class of subfactors for which one can associate planar algebras in a reversible way? (I.e. for which there is a known inverse for the way one passes from the subfactor to the planar algebra, and vice versa?)

(In particular I am wondering if this is now possible in light of the work of Jones, Shlyakhtenko and Guionnet.)