A planar algebraic description of conditional expectations

Abstract

Let [Formula: see text] be a unital inclusion of arbitrary von Neumann algebras. We give a 2-[Formula: see text]-categorical/planar algebraic description of normal faithful conditional expectations [Formula: see text] with finite index and their duals [Formula: see text] by means of the solutions of the conjugate equations for the inclusion morphism [Formula: see text] and its conjugate morphism [Formula: see text]. In particular, the theory of index for conditional expectations admits a 2-[Formula: see text]-categorical formulation in full generality. Moreover, we show that a pair [Formula: see text] 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].