##
**Subfactors of type \(II_ 1\) factors and related topics.**
*(English)*
Zbl 0674.46034

Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 2, 939-947 (1987).

[For the entire collection see Zbl 0657.00005.]

The paper begins with an attractive review of the (Jones) index with references to the Murray-von Neumann paper, and the index 2 decomposition of M. Goldman. This bit of history provides us with a hint that the result by the author on the values of the index is unexpected. While the great variety of applications of the Jones index have been discussed elsewhere, the present paper focuses on (i) an elegant proof of Jones’ theorem from “Index for subfactors”, Invent. Math. 72, 1-25 (1983; Zbl 0508.46040), and (ii) a discussion of braids and links.

The index is defined for \(N\subset M\), \(II_ 1\)-factors as the dimension of \(L^ 2(M,tr)\) over N, and the aforementioned results provides a dichotomany for the values [M:N] of the index. If \([M:N]<4\), it must be one of the Jones number 4 \(\cos^ 2(\pi /n)\), and every \(r\geq 4\) is attained by [M:N]. It is possible to choose both factors hyperfinite, as follows from the proof which is included. The construction of M, N from a prescribed value r (as specified) uses the Hecke algebra generators, when \(r<4\) is one of the Jones numbers. If \(r\geq 4\), a pair M, N is obtained by general von Neumann algebra theory. The enlightening and readable discussion of link invariants is based on work of the author, and of P. Freyd, D. Yetter, J. Hoste, W. Lickorish, K. Millet, A Oceanu, among others.

The paper begins with an attractive review of the (Jones) index with references to the Murray-von Neumann paper, and the index 2 decomposition of M. Goldman. This bit of history provides us with a hint that the result by the author on the values of the index is unexpected. While the great variety of applications of the Jones index have been discussed elsewhere, the present paper focuses on (i) an elegant proof of Jones’ theorem from “Index for subfactors”, Invent. Math. 72, 1-25 (1983; Zbl 0508.46040), and (ii) a discussion of braids and links.

The index is defined for \(N\subset M\), \(II_ 1\)-factors as the dimension of \(L^ 2(M,tr)\) over N, and the aforementioned results provides a dichotomany for the values [M:N] of the index. If \([M:N]<4\), it must be one of the Jones number 4 \(\cos^ 2(\pi /n)\), and every \(r\geq 4\) is attained by [M:N]. It is possible to choose both factors hyperfinite, as follows from the proof which is included. The construction of M, N from a prescribed value r (as specified) uses the Hecke algebra generators, when \(r<4\) is one of the Jones numbers. If \(r\geq 4\), a pair M, N is obtained by general von Neumann algebra theory. The enlightening and readable discussion of link invariants is based on work of the author, and of P. Freyd, D. Yetter, J. Hoste, W. Lickorish, K. Millet, A Oceanu, among others.

Reviewer: P.E.T.Jörgensen

### MSC:

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

46L51 | Noncommutative measure and integration |

46L53 | Noncommutative probability and statistics |

46L54 | Free probability and free operator algebras |

46L10 | General theory of von Neumann algebras |

46L60 | Applications of selfadjoint operator algebras to physics |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |