\(JB^{\ast}\)-algebras of topological stable rank 1
type: Item
instance of: scholarly article
title: \(JB^{\ast}\)-algebras of topological stable rank 1
zbMATH Open document ID: 1161.46041
DOI: 10.1155/2007/37186
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/55384
review text: \(C^*\)-algebras of topological stable rank 1 were introduced by \textit{M. A. Rieffel} [Proc. Lond. Math. Soc. (3) 46, 301--333 (1983; Zbl 0533.46046)]. Rieffel also proved that a \(C^*\)-algebra \(A\) is of topological stable rank 1 (tsr 1) if and only if its invertible elements are norm dense. The paper under review considers this characterisation as starting point to extend the notion of topological stable rank 1 to the more general setting of \(JB^*\)-algebras. The author proves, among other results, that a complex spin factor and a finite-dimensional \(JB^*\)-algebra have stable rank one, providing an example of a special \(JBW^*\)-algebra of tsr 1 whose enveloping von Neumann algebra is not trs 1. The author also proves some interesting characterisations of invertible extreme points of the closed unit ball of a \(JB^*\)-algebra. Concretely, an extreme point of the closed unit ball of a unital \(JB^*\)-algebra is unitary if and only if its distance to the invertible elements is strictly less than 1. Consequently, given a \(JB^*\)-algebra of tsr 1 \(A\), the extreme points of the closed unit ball of \(A\) and unitary elements in \(A\) coincide. This characterisation extends an analogous results obtained by \textit{G. K. Pedersen} in [J.~Oper. Theory 26, No. 2, 345--381 (1991; Zbl 0784.46043)].
reviewed by: Antonio M. Peralta
Mathematics Subject Classification ID: 46L70
zbMATH DE Number: 5282473
zbMATH Keywords: JB*-algebra
zbMATH Keywords: topological stable rank 1
author: Akhlaq A. Siddiqui
MaRDI profile type: Publication
OpenAlex ID: W1991368537
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description: scientific article; zbMATH DE number 5282473