Planar algebras and the ocneanu-szymanski theorem

Abstract

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 II₁-subfactor N⊂M, the relative commutants N'∩M₁ and M'∩M₂ admit mutually dual Kac algebra structures. In the hyperfinite case, the same techniques also prove the other part, which asserts that N'∩M₁ acts on M with invariants N.