On a positive linear map preserving absolute values
type: Item
instance of: scholarly article
title: On a positive linear map preserving absolute values
zbMATH Open document ID: 0878.15008
DOI: 10.1016/S0024-3795(97)80015-8
author: Woo Young Lee
author: Leroy B. Beasley
author: An Hyun Kim
published in: Linear Algebra and its Applications
publication date: 1997-12-14T00:00:00Z
review text: Suppose \(A\) and \(B\) are unital \(C^*\)-algebras and suppose that \(A^+= \{a^*a\); \(a\in A\}\) is a closed convex cone of positive elements in \(A\). If \(a\in A\), then \(|a|= (a^*a)^{1/2}\) is called the absolute value of \(a\). A linear map \(\varphi:A\to B\) is called positive if \(\varphi(A^+)\subset B^+\) and is called disjoint if \(xy=0\) in \(A\) implies \(\varphi(x).\varphi(y)=0\) in \(B\). \textit{L. T. Gardner} [Proc. Am. Math. Soc. 76, 271-278 (1979; Zbl 0425.46042)] proposed the following problem: Does a positive disjoint linear map of \(C^*\)-algebras preserve absolute values? In this note the authors give an affirmative answer for the case of an \(n\times n\) matrix algebra, \(n\geq 2\), and also a necessary and sufficient condition for a linear map to preserve absolute values.
reviewed by: K.Burian
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1040499
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: cone
zbMATH Keywords: positive elements
zbMATH Keywords: absolute value
MaRDI profile type: Publication
cites work: Linear Maps of C ∗ -Algebras Preserving the Absolute Value
cites work: Affine mappings of invertible operators
cites work: Isometries of operator algebras
cites work: Q3997462
cites work: On the Jordan Structure of C ∗ -Algebras
full work available at URL: s0024-3795(97)80015-8
OpenAlex ID: W2086304494
description: scientific article; zbMATH DE number 1040499