Absolute continuity for linear forms on \(B^*\)-algebras and a Radon- Nikodým type theorem (quadratic version)
type: Item
instance of: scholarly article
title: Absolute continuity for linear forms on \(B^*\)-algebras and a Radon- Nikodým type theorem (quadratic version)
zbMATH Open document ID: 0534.46044
DOI: 10.1007/BF02848539
author: Gerd Niestegge
publication date: 1983-01-01T00:00:00Z
review text: In this paper a Radon-Nikodym type theorem for positive linear functionals on \(C^*\)-algebras is presented which is based on the following notion of absolute continuity. Let A be a \(C^*\)-algebra with positive cone \(A_+\), closed unit ball S, and let f be a positive functional on A. A bounded linear functional g on A is said to be absolutely continuous with respect to f if, for each \(\epsilon>0\), there exists \(\delta>0\) such that \(| g(x)|<\epsilon\) for all \(x\in A_+\cap S\) with \(f(x)<\delta\). If \(y\in A\), define \(f^ y\) by \(f^ y(x)=f(y^*xy)\) for \(x\in A.\) The main result of the paper is the following theorem: If A is a \(C^*\)- algebra and g, f are positive functionals on A such that g is absolutely continuous with respect to f, then there exists a sequence \(\{y_ n\}\) in A such that \(\lim_{n\to \infty}\| g-f^{y_ n}\| =0.\) Applications are made to vector states on von Neumann algebras.
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 3848037
zbMATH Keywords: Radon-Nikodym type theorem for positive linear functionals on \(C^*\)- algebras
zbMATH Keywords: vector states on von Neumann algebras
MaRDI profile type: Publication
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 3848037