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].

arXiv comment20 pages

References in corpus (15)

• Nets of Subfactors
• Operator algebras in rigid C*-tensor categories
• Skein theory for the D_{2n} planar algebras
• Local fields in boundary conformal QFT
• Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)
• Rigid C^*-tensor categories of bimodules over interpolated free group factors
• Q-system completion for C* 2-categories
• Minimal index and dimension for 2-$C^{\ast }$-categories with finite-dimensional centers
• On Landauer's principle and bound for infinite systems
• Phase boundaries in algebraic conformal QFT
• Infinite index extensions of local nets and defects
• Compact Hypergroups from Discrete Subfactors
• Galois Correspondence and Fourier Analysis on Local Discrete Subfactors
• Realization of rigid C${}^{\ast }$-tensor categories via Tomita bimodules
• Minimal index and dimension for inclusions of von Neumann algebras with finite-dimensional centers