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Iterating the basic construction. (English) Zbl 0706.46047

Summary: Let \(N\subset M\) be a pair of type \(II_ 1\) factors with finite Jones’ index and \(N\subset M\subset M_ 1\subset M_ 2\subset...\subset M_ n\subset...\subset M_{2n+1}\) be the associated tower of type \(II_ 1\) factors obtained by iterating Jones’ basic construction. We give an explicit formula of a projection in \(M_{2n+1}\) which implements the conditional expectation of \(M_ n\) onto N, thus showing that \(M_{2n+1}\) comes naturally from the basic construction associated to the pair \(N\subset M_ n\). From this we deduce several properties of the relative commutant \(N'\cap M_ n\).

MSC:

46L35 Classifications of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
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References:

[1] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1 – 25. · Zbl 0508.46040 · doi:10.1007/BF01389127
[2] A. Ocneanu, in preparation.
[3] Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57 – 106. · Zbl 0646.46057
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