Index for von Neumann algebras with finite dimensional centers
type: Item
instance of: scholarly article
title: Index for von Neumann algebras with finite dimensional centers
zbMATH Open document ID: 0826.46051
author: Tamotsu Teruya
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1995-11-27T00:00:00Z
review text: Let \(M\) and \(N\) be von Neumann algebras and let \(E\in {\mathcal E}(M, N)\), the set of normal faithful conditional expectations from \(M\) onto \(N\). \textit{H. Kosaki} [J. Funct. Anal. 66, 123-140 (1986; Zbl 0607.46034)] extended the Jones' index theory to pairs of arbitrary factors \(N\subset M\) defining Index \(E\), for \(E\in {\mathcal E}(M, N)\), to be \(E^{-1}(1)\), where the operator-valued weight \(E^{- 1}\) [\textit{U. Haagerup}, J. Funct. Anal. 32, 175-206 (1979; Zbl 0426.46046)] is defined using \textit{A. Connes'} spatial Radon-Nikodym derivatives [J. Funct. Anal. 35, 153-164 (1980; Zbl 0443.46042)]. The author allows \(M\) and \(N\) to be von Neumann algebras with finite-dimensional centres. Kosaki's Index \(E\) is now a positive operator affiliated to \(Z(M)\), the centre of \(M\). When there exists an \(E\in {\mathcal E}(M, N)\) such that Index \(E\) is bounded, \(N\) is said to be of finite index in \(M\). When \(N\) is not of finite index the author sets the `index' \([M; N]_0\) to be \(\infty\) (see below). \textit{H. Araki} [Publ. Res. Ind. Math. Sci., Kyoto Univ. 11, 809-833 (1976; Zbl 0326.46031)] using the modular operator of \textit{M. Takesaki} and Tomita [`Tomita's theory of modular Hilbert algebras and its applications', Lect. Notes Math. 128 (1970; Zbl 0193.425)] extended the relative entropy for density matrices to faithful normal states of von Neumann algebras. (Intuitively the relative entropy is the logarithm of the number of intermediate states.) Supposing that there exists a faithful normal state \(\varphi\) of \(M\) for which there exists an \(E\in {\mathcal E}(M, N)\) with \(\varphi\circ E= \varphi\), the author adapts Araki to define the entropy \(K_\varphi(M|N)\) of \(M\) relative to \(\varphi\) and \(N\). \(K_E\) denotes the supremum of the \(K_\varphi\) such that \(\varphi\circ E= \varphi\). Assume henceforth that \(Z(M)\cap Z(N)= \mathbb{C}\) and \(E\in {\mathcal E}(M, N)\). If \(N\) is of finite index in \(M\) then the…
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L37
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 597440
zbMATH Keywords: Connes' spatial Radon-Nikodym derivatives
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: normal faithful conditional expectations
zbMATH Keywords: Jones' index theory to pairs of arbitrary factors
zbMATH Keywords: operator-valued weight
zbMATH Keywords: density matrices
zbMATH Keywords: faithful normal states of von Neumann algebras
zbMATH Keywords: relative entropy
zbMATH Keywords: tensor products
zbMATH Keywords: crossed products by finite groups
reviewed by: Aubrey Wulfsohn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195168432
OpenAlex ID: W1982217704
cites work: Relative entropy of states of von Neumann algebras
cites work: Relative entropy for states of von Neumann algebras. II
cites work: On the spatial theory of von Neumann algebras
cites work: Q4693152
cites work: Operator valued weights in von Neumann algebras. I
cites work: Operator valued weights in von Neumann algebras. II
cites work: Minimizing indices of conditional expectations onto a subfactor
cites work: Index for pairs of finite von Neumann algebras
cites work: Index for subfactors
cites work: Some remarks on index and entropy for von Neumann subalgebras
cites work: Actions of finite groups on finite von Neumann algebras and the relative entropy
cites work: Q3825555
cites work: Extension of Jones' theory on index of arbitrary factors
cites work: Index of subfactors and statistics of quantum fields. I
cites work: Entropy and index for subfactors
cites work: Index for 𝐶*-subalgebras
DOI: 10.2977/PRIMS/1195168432
Recommended article: Q6134285
Recommended article: Index for pairs of finite von Neumann algebras
Recommended article: Index theory in von Neumann algebras
Recommended article: Index semigroup and indexing on von Neumann algebras
Recommended article: Index for 𝐶*-subalgebras
Recommended article: Q4712476
Recommended article: Perturbations of von Neumann subalgebras with finite index
Recommended article: On the dimension theory of von Neumann algebras
Recommended article: Q4256733
Recommended article: Indice d'esperances conditionnelles et algebres de von Neumann finies.
description: scientific article; zbMATH DE number 597440