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Braid groups, Hecke algebras and type \(II_ 1\) factors. (English) Zbl 0659.46054

Geometric methods in operator algebras, Proc. US-Jap. Semin., Kyoto/Jap. 1983, Pitman Res. Notes Math. Ser. 123, 242-273 (1986).
[For the entire collection see Zbl 0632.00012.]
This very interesting paper deals with the index \[ [M:N]=\lim_{i\to \infty}rank_ N(M_{i+1)}/rank_ N(M_ i) \] where \(N\subset M\) are \(II_ 1\) factors, \(M_ 0=N\), \(M_ 1=M\), \(M_{i+1}=End_{M_{n- 1}}(M_ i)\) for \(i\geq 2\), and “rank” means the rank of a maximal free N-submodule. The algebra generated by elements \(e_ i\in End_{M_{i- 1}}(M_ i)\) (the “conditional expectations” from \(M_ i\) onto \(M_{i-1})\) and relations \(e^ 2_ i=e_ i\), \(e_ ie_{i+1}e_ i=[M:N]^{-1}e_ i\) and \(e_ ie_ j=e_ je_ i\) if \(| i-j| \geq 2\), are in some sense related to representations of the braid groups. The index [M:N] is calculated, when \([M:N]<4:\) in fact \([M:N]=4 \cos^ 2\pi /m\) for some \(m=3,4,5,..\). [see the author, Invent Math. 72, 1-25 (1983; Zbl 0508.46040)]. Two beautiful accounts of the theory have appeared: by A. Connes [Sémin. Bourbaki 1984-85, No.647, Astérisque 133-134, 289-308 (1986; Zbl 0597.57005)] and V. Jones [Proc. Intern. Congress, Math. Berkeley 1986, 939-947 (1987)].
Reviewer: G.Corach

MSC:

46L35 Classifications of \(C^*\)-algebras
20F36 Braid groups; Artin groups
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)