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.


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
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