Non-commutative central limits
type: Item
instance of: scholarly article
title: Non-commutative central limits
zbMATH Open document ID: 0695.46023
DOI: 10.1007/BF00341282
author: D. Goderis
author: P. Vets
author: A. Verbeure
publication date: 1989-01-01T00:00:00Z
review text: For the non-commutative or quantum system \((M,\rho)\), where M is a von Neumann algebra of operators on a Hilbert space and \(\rho\) is a vector state on M, a CCR-\(C^*\)-algebra of normal fluctuations \(\overline{W(H,\sigma_{\rho})}\) of the system (M,\(\rho)\) and quasi-free state \(\omega_ s\) of \(\overline{W(H,\sigma_{\rho})}\) are defined and investigated (here \(H=M_{sa}\) and \(\sigma_{\rho}(x,y)=-i\rho ([x,y])\), \(x,y\in M_{sa})\). Using \(\overline{W(H,\sigma_{\rho})}\) and \(\omega_ s\) the central limit theorem for the operators from M is formulated and stability of this theorem under small perturbations is studied. In particular, two systems (M,\(\rho)\) and \((M,\rho '_ n)\), where \(\rho '_ n\) is a weakly perturbed state with respect to \(\rho\), are considered and it is proved that the central limit of the relative entropies of \(\rho\) and \(\rho '_ n\) yields the relative entropy of the limit quasi-free representations.
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4138552
zbMATH Keywords: von Neumann algebra of operators
zbMATH Keywords: \(CCR\)-\(C^*\)-algebra of normal fluctuations
zbMATH Keywords: quasi-free state
zbMATH Keywords: central limit theorem
zbMATH Keywords: relative entropies
reviewed by: Vladimir Chilin
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00341282
OpenAlex ID: W2093534346
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4138552