Norm additivity conditions for normal linear functionals on von Neumann algebras
type: Item
instance of: scholarly article
title: Norm additivity conditions for normal linear functionals on von Neumann algebras
zbMATH Open document ID: 0853.46058
author: Masaharu Kusuda
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1997-01-05T00:00:00Z
review text: Let \(M\) be a von Neumann algebra and let \(\varphi\) and \(\psi\) be bounded linear functionals on \(M\). We then have the norm inequality \(|\varphi+ \psi|\leq |\varphi|+ |\psi|\). On the other hand, it is well-known that if \(\varphi\) and \(\psi\) are positive, then \(|\varphi+ \psi|= |\varphi|+ |\psi|\). In general, however, such an equality does not necessarily hold if both \(\varphi\) and \(\psi\) are not positive. The purpose of this paper is to investigate when the norm equality \(|\varphi+ \psi|= |\varphi|+ |\psi|\) holds for given normal linear functionals \(\varphi\) and \(\psi\). Then the fact to play an essential role is the following: Let \(M\) be a von Neumann algebra and let \(\varphi\) be a normal linear functional on \(M\). Then we have \[ \varphi(\cdot)= |\varphi |(\nu\cdot),\;|\varphi|(\cdot)= \varphi(\nu^*\cdot),\text{ and } |\varphi|= |\;|\varphi|\;| \] for all partial isometries \(\nu^*\) in \(M\) satisfying that \(\varphi(\nu^*)= |\varphi|\), where \(|\varphi|\) denotes the absolute value of \(\varphi\). In connection with this fact, we may expect that the set of those elements \(x\), in the unit ball of \(M\), with \(\varphi(x)= |\varphi |\) has nice information on norms and absolute values of normal linear functionals on \(M\). In fact, by employing such a set, we will give necessary and sufficient conditions for a pair of normal linear functionals \(\varphi\) and \(\psi\) to satisfy that \(|\varphi+ \psi|= |\varphi|+ |\psi|\) or that \(|\varphi+ \psi|= |\varphi|+ |\psi|\).
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 878497
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: normal linear functional
zbMATH Keywords: partial isometries
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195163922
OpenAlex ID: W1986651924
DOI: 10.2977/PRIMS/1195163922
description: scientific article; zbMATH DE number 878497