A planar algebraic description of conditional expectations

TL;DR

This paper provides a 2-{$C^*$}-categorical and planar algebraic framework for describing conditional expectations with finite index in von Neumann algebras, extending Longo's subfactor results to a more general setting.

Contribution

It introduces a novel 2-{$C^*$}-categorical approach to index theory of conditional expectations, connecting them with Q-systems and conjugate equations.

Findings

Provides a categorical formulation of index theory for conditional expectations.

Shows the equivalence between pairs of inclusions with expectations and Q-systems.

Extends Longo's subfactor results to arbitrary von Neumann algebra inclusions.

Abstract

Let $\mathcal{N}\subset\mathcal{M}$ be a unital inclusion of arbitrary von Neumann algebras. We give a 2-{$C^*$}-categorical/planar algebraic description of normal faithful conditional expectations $E:\mathcal{M}\to\mathcal{N}\subset\mathcal{M}$ with finite index and their duals $E':\mathcal{N}'\to\mathcal{M}'\subset\mathcal{N}'$ 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-{$C^*$}-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].