Extension of Jones' theory on index of arbitrary factors
type: Item
instance of: scholarly article
title: Extension of Jones' theory on index of arbitrary factors
zbMATH Open document ID: 0607.46034
DOI: 10.1016/0022-1236(86)90085-6
author: Hideki Kosaki
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: 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\({}^ 2(\pi /n)\); \(n=3,4,...\}\cup [4,\infty]\).
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 3981832
zbMATH Keywords: index of subfactors
zbMATH Keywords: Jones' index theory
zbMATH Keywords: normal conditional expectation
zbMATH Keywords: operator valued weight
zbMATH Keywords: normal faithful semifinite weights
reviewed by: Jun Tomiyama
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90085-6
OpenAlex ID: W1992635693
cites work: On the spatial theory of von Neumann algebras
cites work: Operator valued weights in von Neumann algebras. II
cites work: Index for subfactors
cites work: Entropy and index for subfactors
cites work: Conditional expectations in von Neumann algebras
cites work: Q3759407
description: scientific article; zbMATH DE number 3981832