Iterating the basic construction. (English) Zbl 0706.46047

Pimsner, Mihai; Popa, Sorin

Trans. Am. Math. Soc. 310, No. 1, 127-133 (1988).

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

Keywords:

finite Jones’ index; associated tower of type \(II_ 1\) factors; iterating Jones’ basic construction

References:

[1]	V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1 – 25. · Zbl 0508.46040
[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