Classification of amenable subfactors of type II
type: Item
instance of: scholarly article
title: Classification of amenable subfactors of type II
zbMATH Open document ID: 0853.46059
DOI: 10.1007/BF02392646
published in: Acta Mathematica
publication date: 1997-01-05T00:00:00Z
review text: A central problem arising in the theory of subfactors is the classification of subfactors \(N\subset M\) of finite index of the hyperfinite (or approximately finite dimensional) factors \(M\). The physically relevant invariant for such a subfactor is the lattice of its higher relative commutants \(\{M'_i\cap M_j\}_{i,j}\) in the Jones tower \(N\subset M\subset M_1\subset\dots\); this is called the standard invariant of \(N\subset M\), denoted \(\mathcal G_{N,M}\). The inclusions between these higher relative commutants are described by Jones' principal graph \(\Gamma_{N,M}\). An abstract characterization of the invariant \(\mathcal G_{N,M}\) is given in [\textit{A. Ocneanu}, Quantized groups, string algebras, and Galois theory for von Neumann algebras, London Math. Soc. Lecture Notes 136, 119-172 (1988; Zbl 0696.46048] as the paragroup of all irreducible correspondences (bimodules) generated under Connes' composition (or fusion) rule by \(N\subset M\), and \(\Gamma_{N,M}\) as its ``fusion rule matrix'' (a Cayley-type graph). The main result of the paper under review is that for a large class of subfactors, called strongly amenable, the standard invariant \(\mathcal G_{N,M}\) is the complete invariant. It should be noted that all the examples of subfactors coming out this far for quantum field theories and polynomial invariants for knots are strongly amenable. Also, the classical problems of classifying actions of amenable discrete groups and compact Lie groups on hyperfinite factors were given equivalent formulations in terms of classification of certain subfactors that are strongly amenable, in an earlier work by \textit{A. Wasserman} and the author [Actions of compact Lie groups on von Neumann algebras, C. R. Acad. Sci. Paris Sér. I Math. 315, No. 4, 421-426 (1992; Zbl 0794.46050)]. The result is in some sense the best that can be obtained, as it is shown that strongly amenable subfactors give the largest class of subfactors which can be reconstructed in a direct …
Mathematics Subject Classification ID: 46L37
zbMATH DE Number: 764288
zbMATH Keywords: operator algebras
zbMATH Keywords: factors
zbMATH Keywords: Jones tower
zbMATH Keywords: amenable subfactors
author: Sorin Popa
reviewed by: Miroslav Engliš
MaRDI profile type: Publication
cites work: Q5666451
cites work: Q3805292
cites work: Entropy of groups and subfactors
cites work: On the Existence of Central Sequences in Subfactors
cites work: A note on intermediate subfactors
cites work: Q4299841
cites work: Une classification des facteurs de type ${\rm III}$
cites work: Outer conjugacy classes of automorphisms of factors
cites work: Classification of injective factors. Cases \(\mathrm{II}_1\), \(\mathrm{II}_\infty\), \(\mathrm{III}_\lambda\), \(\lambda\neq 1\)
cites work: Q4161729
cites work: Compact metric spaces, Fredholm modules, and hyperfiniteness
cites work: Q3803660
cites work: Entropy for automorphisms of \(II_1\) von Neumann algebras
cites work: The flow of weights on factors of type III
cites work: Q3261641
cites work: Amenability and virtual diagonals for von Neumann algebras
cites work: Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras. I
cites work: Superselection sectors with braid group statistics and exchange algebras
cites work: Q4693152
cites work: Minimizing indices of conditional expectations onto a subfactor
cites work: Application of fusion rules to classification of subfactors
cites work: Classification of subfactors with the principal graph \(D^{(1)}_ n\)
cites work: Index for subfactors
cites work: Hecke algebra representations of braid groups and link polynomials
cites work: Normalcy in operator algebras
cites work: Random walks on discrete groups: Boundary and entropy
cites work: Symmetric Random Walks on Groups
cites work: Extension of Jones' theory on index of arbitrary factors
cites work: Index of subfactors and statistics of quantum fields. I
cites work: Index of subfactors and statistics of quantum fields. II: Correspondences, braid group statistics and Jones polynomial
cites work: Minimal index and braided subfactors
cites work: On rings of operators. IV
cites work: Action of discrete amenable groups on von Neumann algebras
cites work: Q3472677
cites work: Some extremal functions in Fourier analysis
cites work: Q4048325
cites work: Entropy and index for subfactors
cites work: Iterating the Basic Construction
cites work: Finite dimensional approximation of pairs of algebras and obstructions for the index
cites work: On a problem of R.V. Kadison on maximal Abelian *-subalgebras in factors
cites work: Singular maximal Abelian *-subalgebras in continuous von Neumann algebras
cites work: Maximal injective subalgebras in factors associated with free groups
cites work: Classification of subfactors: The reduction to commuting squares
cites work: Q4734327
cites work: Q3488019
cites work: Relative dimension, towers of projections and commuting squares of subfactors
cites work: Q4014846
cites work: Q4022202
cites work: Free states of the canonical anticommutation relations
cites work: Finite subalgebras of a von Neumann algebra
cites work: Duality for crossed products and the structure of von Neumann algebras of type III
cites work: Conditional expectations in von Neumann algebras
cites work: Ergodic actions of compact groups on operator algebras. III: Classification for SU(2)
cites work: Q3977381
cites work: Hecke algebras of type \(A_ n\) and subfactors
cites work: Quantum groups and subfactors of type B, C, and D
description: scientific article; zbMATH DE number 764288