Extension of Jones' theory on index of arbitrary factors
The author extends Jones' index theory of \(II_ 1\)-factors [Invent. Math. 72, 1-25 (1983; Zbl 0508.46040)] to arbitrary factors. Let M be a factor on a Hilbert space H with a subfactor N. The index is defined as the one with respect to a normal conditional expectation \(E: M\to N\) assuming its existence which is assured in the \(II_ 1\)-case as the canonical conditional expectation for the trace. Based on the works of \textit{A. Connes} [J. Funct. Anal. 35, 153-164 (1980; Zbl 0443.46042)] and \textit{U. Haagerup} [ibid. 32, 175-206 (1979; Zbl 0426.46046), ibid. 33, 339-361 (1979; Zbl 0426.46047)], the author associates E canonically to the operator valued weight \(E^{-1}\) of N' to M' satisfying \(d(\phi.E)/d\psi =d\phi /d(\psi.E^{-1})\) where \(\phi\) and \(\psi\) are normal faithful semifinite weights on N and M' respectively. The index of E is then defined as the (extended) number \(E^{-1}(1)\). It is shown (Theorem 2.2) that the number \(E^{-1}(1)\) does not depend on the space H. The author shows how the results of Jones, notably basic construction, in his paper can be transformed into this situation and finally proves (Theorem 5.4) the fact that Index E belongs to the set \(\{\) 4 cos\({}^ 2(\pi /n)\); \(n=3,4,...\}\cup [4,\infty]\).
scientific article; zbMATH DE number 3981832