Index for pairs of finite von Neumann algebras
type: Item
instance of: scholarly article
title: Index for pairs of finite von Neumann algebras
zbMATH Open document ID: 0663.46052
author: S. H. Smith
published in: Pacific Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The Jones' index of a pair (N,M) of finite von Neumann algebras with finite dimensional centres has been given two definitions: one ring- theoretic, and one using Markov traces. We extend the second definition to the case of finite, countably decomposable von Neumann algebras and we show that the two definitions agree when the algebras are direct sums of finite factors. We also study Markov traces on such pairs. More precisely: When N is of finite index in M, i.e. if M acts on some Hilbert space H in such a way that the commutant \(N'_ H\) of N is finite, and that the coupling operators \(c_ M(H)^{\pm 1}\) and \(c_ N(H)^{\pm 1}\) are bounded, then we define two bounded, linear, normal maps \(C^ M_ N\) in \(L_*(Z(N))\) and \(D^ M_ N\) in \(L_*(Z(M))\) in terms of the coupling operators and the centre-valued traces, which do not depend on the chosen representation and which have the same spectral radius. The index of N in M, denoted by [M:N], is then the common spectral radius of \(C^ M_ N\) and of \(D^ M_ N\).
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 4084358
zbMATH Keywords: Jones' index
zbMATH Keywords: finite von Neumann algebras with finite dimensional centres
zbMATH Keywords: Markov traces
zbMATH Keywords: finite, countably decomposable von Neumann algebras
zbMATH Keywords: direct sums of finite factors
zbMATH Keywords: coupling operators
zbMATH Keywords: centre- valued traces
zbMATH Keywords: common spectral radius
MaRDI profile type: Publication
full work available at URL: pjm.1990.146.43
OpenAlex ID: W2015072658
DOI: 10.2140/PJM.1990.146.43
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description: scientific article; zbMATH DE number 4084358