A planar algebraic description of conditional expectations

Abstract

Let $\mathcal{N}\subset \mathcal{M}$ be a unital inclusion of arbitrary von Neumann algebras. We give a 2-{$C^{\ast }$}-categorical/planar algebraic description of normal faithful conditional expectations $E:\mathcal{M}\to \mathcal{N}\subset \mathcal{M}$ with finite index and their duals $E^{\prime }:{\mathcal{N}}^{\prime }\to {\mathcal{M}}^{\prime }\subset {\mathcal{N}}^{\prime }$ by means of the solutions of the conjugate equations for the inclusion morphism $\iota :\mathcal{N}\to \mathcal{M}$ and its conjugate morphism $\overline{\iota }:\mathcal{M}\to \mathcal{N}$. In particular, the theory of index for conditional expectations admits a 2-{$C^{\ast }$}-categorical formulation in full generality. Moreover, we show that a pair $(\mathcal{N}\subset \mathcal{M},E)$ 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 [Lon90, Thm.\ 5.2], [Lon94, Thm.\ 5.1].

20 pages

References in corpus (2)

• Q-system completion for C* 2-categories
• Galois Correspondence and Fourier Analysis on Local Discrete Subfactors