Von Neumann regularity and quadratic conorms in JB\(^*\)-triples and C\(^*\)-algebras
type: Item
instance of: scholarly article
title: Von Neumann regularity and quadratic conorms in JB\(^*\)-triples and C\(^*\)-algebras
zbMATH Open document ID: 1157.46042
author: A. Morales-Campoy
author: Antonio M. Peralta
author: Maribel Ramírez
author: María Burgos
author: Amin Kaidi
publication date: 2008-05-26T00:00:00Z
review text: For an element \(a\) in a \(\text{JB}^*\)-triple \(A\), the quadratic operator \(Q(a)\) is defined, for each element \(b\) in \(A\), by \[ Q(a)(b)= \{a b a\}, \] and \(a\) is said to be von Neumann regular if there exists an element \(b\) in \(A\) such that \(Q(a)b\) and \(a\) coincide. The authors show that an element \(a\) in \(A\) is von Neumann regular if and only if the conjugate linear operator \(Q(a)\) has norm-closed range, and discuss the notion of von Neumann regularity in terms of the the spectral properties of elements of \(A\). The authors go on to define the quadratic co-norm \(\gamma^q(a)\) of an element \(a\) of \(A\) by \[ \gamma^a(a)= \text{inf}\{\| Q(a)(b)\|: b\in A,\,\text{dist}(b,\ker(Q(a)))\geq 1\}. \] It is shown that \(\gamma^q(a)\) coincides with the infimum of the squares of the elements in the triple spectrum of \(a\). This leads to the description of non-zero tripotents in \(A\) as those elements the norm and quadratic co-norm of which are both one. These results allow the authors to show that when \(A\) and \(B\) are \(\text{JBW}^*\)-triples, which are, of course, well-supplied with tripotents, a contractive linear bijection from \(A\) onto \(B\) is a triple isomorphism if and only if it preserves quadratic co-norms. In the final two sections of this interesting paper, the authors study the norm-continuity of the quadratic co-norm in a \(\text{JB}^*\)-triple \(A\) and then apply all their results to the case when \(A\) is a \(C^*\)-algebra, thereby producing some new findings in this area.
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 17C10
Mathematics Subject Classification ID: 47B48
zbMATH DE Number: 5278754
zbMATH Keywords: von Neumann regularity
zbMATH Keywords: quadratic conorm
zbMATH Keywords: C\(^*\)-algebra
zbMATH Keywords: JB\(^*\)-triple
zbMATH Keywords: triple spectrum
reviewed by: C. M. Edwards
MaRDI profile type: Publication
full work available at URL: s10114-007-0985-x
OpenAlex ID: W2075198186
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DOI: 10.1007/S10114-007-0985-X
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 5278754