Abstract

 
 
 We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II1 factor, and vice versa.
 
 
Highlights
At first glance there is nothing planar about a subfactor
We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions
The most surprising result of [J1] was that [M : N ] is “quantized” — to be precise, if [M : N ] < 4 there is an integer n ≥ 3 with [M : N ] = 4 cos2 π/n. This led to a surge of interest in subfactors and the major theorems of Pimsner, Popa and Ocneanu ([PP],[Po1],[O1]). These results turn around a “standard invariant” for finite index subfactors, known variously as the “tower of relative commutants”, the “paragroup”, or the “λ-lattice”
Summary
At first glance there is nothing planar about a subfactor. A factor M is a unital ∗-algebra of bounded linear operators on a Hilbert space, with trivial center and closed in the topology of pointwise convergence. The following tangle: defines a rotation of period k (2k boundary points) so it is a consequence of the planar algebra structure that the rotation x1 ⊗x2 ⊗· · ·⊗xk → x2 ⊗x3 ⊗· · ·⊗xk ⊗x1, which makes no sense on Mk−1, is well defined on N -central vectors and has period k This result is an essential technical ingredient of the proof of Theorem 4.2.1. By enlarging the class of tangles in the planar operad, say so as to include oriented edges and boundary points, or discs with an odd number of boundary points, one would obtain a notion of planar algebra applicable to more examples. Deborah Craig for her patience and first-rate typing, and Tsukasa Yashiro for the pictures
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