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**Classification of subfactors: The reduction to commuting squares.**
*(English)*
Zbl 0757.46054

This article contains the proof for a difficult and fundamental result in the theory of subfactors in the sense of V. Jones. It is shown that any inclusion \(M\supset N\) of hyperfinite factors with finite index and finite depth is canonically obtained (as an inductive limit over the iterated “basic construction”) from a so-called commuting square of inclusions of finite-dimensional algebras. It is also shown that, in the finite depth case, the commuting square can be chosen canonically and that two hyperfinite inclusions are isomorphic iff their associated commuting squares are isomorphic. Therefore the problem of classifying finite index depth inclusions of hyperfinite factors is reduced to the classification of finite dimensional commuting squares. The classification of these, in turn, is a combinatorial (though still difficult) problem. In particular, one obtains a complete classification of subfactors with index \(<4\) in terms of commuting squares. A version of the main theorem had first been stated and a proof had been announced, but never published, by Ocneanu.

Reviewer: J.Cuntz (Heidelberg)

### MSC:

46L37 | Subfactors and their classification |

46L35 | Classifications of \(C^*\)-algebras |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

### Keywords:

basic construction; subfactors; commuting square of inclusions of finite- dimensional algebras### References:

[1] | [C] Connes, A.: Classification of injective factors. Ann. Math.104, 73-115 (1976) · Zbl 0343.46042 |

[2] | [J1] Jones, V.F.R.: Index for subfactors. Invent. Math.72, 1-25 (1983) · Zbl 0508.46040 |

[3] | [J2] Jones, V.F.R.: Subfactors and related topics, Operator Algebras and Applications, Vol. 2, London Math. Soc. Lect. Note Series136, 103-118 (1988) |

[4] | [J3] Jones, V.F.R.: Actions of finite groups on the hyperfinite II1 factor. Mem. Am. Math. Soc.28, No. 237, (1980) |

[5] | [JPP] Jones, V., Pimsner, M., Popa, S.: Private correspondence, 1982-1984 |

[6] | [GHJ] Goodman, F., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras. MSRI Publications 14, Springer Verlag, 1989 · Zbl 0698.46050 |

[7] | [MvN] Murray, F., von Neumann, J.: Rings of operators IV. Ann. Math.44, 716-808 (1943) · Zbl 0060.26903 |

[8] | [Oc] Ocneanu, A.: Quantized groups string algebras and Galois theory for algebras. Operator Algebras and Applications, Vol. 2, London Math. Soc. Lect. Note Series136, 119-172 (1988) · Zbl 0696.46048 |

[9] | [PiPo1] Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Sci. Ec Norm. Super., IV. Ser.19, 57-106 (1986) · Zbl 0646.46057 |

[10] | [PiPo2] Pimsner, M., Popa, S.: Iterating the basic construction. Trans. Am. Math. Soc.,310, 127-134 (1988) · Zbl 0706.46047 |

[11] | [PiPo3] Pimsner, M. Popa, S.: Finite dimensional approximation of pairs of algebras and obstructions for the index. Preprint 1988 |

[12] | [Po1] Popa, S.: Maximal injective subalgebras in factors associated with free groups. Adv. Math.50, 27-48 (1983) · Zbl 0545.46041 |

[13] | [Po2] Popa, S.: Relative dimension, towers of projections and commuting squares of subfactors. Pac. J. Math.137, 95-122 (1989) · Zbl 0699.46042 |

[14] | [Po3] Popa, S.: Matrices entiéres et algébres envelopantes associées aux sousfacteurs. Preprint 1990 |

[15] | [Po4] Popa, S.: On a problem of R. V. Kadison on maximal abelian *-subalgebras in factors. Invent. Math. 65 269-281 (1981) · Zbl 0481.46028 |

[16] | [Wa] Wassermann, A.: Coactions and Yang-Baxter equations for ergodic actions and subfactors. Operator Algebras and Applications Vol. 2, London Math. Soc. Lect. Note Series136, 203-236 (1988) · Zbl 0809.46079 |

[17] | [W] Wenzl, H.: Hecke algebras of typeA n and subfactors. Invent. Math.92, 349-383 (1988) · Zbl 0663.46055 |

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