Entropy and index for subfactors. (English) Zbl 0646.46057

Pimsner, Mihai; Popa, Sorin

Ann. Sci. Éc. Norm. Supér. (4) 19, No. 1, 57-106 (1986).

This paper is related to that of V. Jones [Invent. Math. 72, 1–25 (1983; Zbl 0508.46040)] where the index \([M:N]\) is defined for finite von Neumann factors \(M\), \(N\) with \(N\subset M\). The authors prove several interesting results.
For example: a characterization of the case “\([M:N]\) is finite” is given; \([M:N]^{-1}=\inf \| E_ N(f)\|\) where \(f\) runs through all nonzero projections of \(M\) \((E_N\) denotes the trace preserving conditional expection onto \(N)\); a relation between \([M:N]\) and the so-called relative entropy \(H\), \(H\le \log [M:N]\), is established [see A. Connes and E. Störmer, Acta Math. 134, 289–306 (1975; Zbl 0326.46032)]; computation of \(H\) for \(\text{II}_1\)-factors, in particular: if \(N'\cap M={\mathbb{C}}\) then \(H=\log [M:N]\) and if \(4<[M:N]<3+2\sqrt{2}\) then the converse is also true; applications to Jones’ pair of subfactors; computation of \([M:N]\) and \(H\) in the finite-dimensional case.

Reviewer: Hellmut Baumgärtel (Berlin)

MSC:

46L35

Classifications of \(C^*\)-algebras

Keywords:

Jones index; finite von Neumann factors; trace preserving conditional expection; relative entropy; Jones pair of subfactors

Citations:

Zbl 0508.46040; Zbl 0326.46032

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