# A planar algebraic description of conditional expectations

November 08, 2021

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].

Keywords:

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