Planar algebras and the Ocneanu-Szymanski theorem

We give a very simple `planar algebra' proof of the part of the Ocneanu-Szymanski theorem which asserts that for a finite index, depth two, irreducible $II_1$-subfactor $N \subset M$, the relative commutants $N^\prime \cap M_1$ and $M^\prime \cap M_2$ admit mutually dual Kac algebra structures. In the hyperfinite case, the same techniques also prove the other part, which asserts that $N^\prime \cap M_1$ acts on $M$ with invariants $N$.