Two generalizations of the Gleason-Kahane-Żelazko theorem
type: Item
instance of: scholarly article
title: Two generalizations of the Gleason-Kahane-Żelazko theorem
zbMATH Open document ID: 0868.46038
author: Erik Christensen
published in: Pacific Journal of Mathematics
publication date: 1996-12-15T00:00:00Z
review text: We consider a unital Banach algebra \({\mathfrak A}\), and a continuous unital linear mapping \(\varphi\) of \({\mathfrak A}\) into \(M_n (\mathbb{C})\) -- the \(n \times n\) matrices over \(\mathbb{C}\). The first generalization states that if \(\varphi\) sends invertible elements to invertible elements, then the kernel of \(\varphi\) is contained in a proper two sided closed ideal of finite codimension. The second result characterizes this property for \(\varphi\) in saying that \(\varphi ({\mathfrak A}_{\text{inv}})\) is contained in \(\text{GL}_n (\mathbb{C})\) if and only if for each \(a\) in \({\mathfrak A}\) and each natural number \(k\): \[ \text{trace} (\varphi (a^k))= \text{trace} (\varphi (a)^k). \]
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46H10
zbMATH DE Number: 950184
zbMATH Keywords: trace
zbMATH Keywords: determinant
zbMATH Keywords: Banach algebra
zbMATH Keywords: invertible elements
zbMATH Keywords: ideal
MaRDI profile type: Publication
full work available at URL: pjm.1997.177.27
OpenAlex ID: W2057780279
DOI: 10.2140/PJM.1997.177.27
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description: scientific article; zbMATH DE number 950184