Let N subset of 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 : M -> N subset of M with finite index and their duals E' : N' -> M' subset of N' by means of the solutions of the conjugate equations for the inclusion morphism iota : N subset of Mand its conjugate morphism (iota) over bar : M -> 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 (N subset of 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 [ 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].

Giorgetti, L. (2022). A planar algebraic description of conditional expectations. INTERNATIONAL JOURNAL OF MATHEMATICS, 33(5) [10.1142/S0129167X22500379].

A planar algebraic description of conditional expectations

Giorgetti, L
2022

Abstract

Let N subset of 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 : M -> N subset of M with finite index and their duals E' : N' -> M' subset of N' by means of the solutions of the conjugate equations for the inclusion morphism iota : N subset of Mand its conjugate morphism (iota) over bar : M -> 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 (N subset of 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 [ 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].
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05
English
Jones-Kosaki index
conditional expectation
conjugate equations
Q-system
Giorgetti, L. (2022). A planar algebraic description of conditional expectations. INTERNATIONAL JOURNAL OF MATHEMATICS, 33(5) [10.1142/S0129167X22500379].
Giorgetti, L
Articolo su rivista
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/303683
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact