A planar algebraic description of conditional expectations

Let N subset of M be a unital inclusion of arbitrary von Neumann algebras. We give a 2-C*- categorical/planar algebraic description of normal faithful conditional expectations E : M -> N subset of M with finite index and their duals E' : N' -> M' subset of N' by means of the solutions of the conjugate equations for the inclusion morphism iota : N subset of Mand its conjugate morphism (iota) over bar : M -> N. In particular, the theory of index for conditional expectations admits a 2-C *- categorical formulation in full generality. Moreover, we show that a pair (N subset of M, E) as above can be described by a Q-system, and vice versa. These results are due to Longo in the subfactor/simple tensor unit case [ R. Longo, Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial, Comm. Math. Phys. 130 (1990) 285-309; A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159 (1994) 133-150]