Let N ⊆ M be II1 factors with [M : N ] < ∞. There is a “standard invariant” for N ⊆ M , which we shall describe using the planar algebra formalism of [19]. The vector spaces Pk of N −N invariant vectors in the N −N bimodule ⊗M admit an action of the operad of planar tangles as in [19] and [21]. In more usual notation the vector space Pk is the relative commutant N ′ ∩ Mk−1 in the tower Mk of [16]. The conditional expectation EN from M to N is in P2 and generates a planar subalgebra called the… CONTINUE READING