Central kernels of subspaces of JB\(^{*}\)-triples
type: Item
instance of: scholarly article
title: Central kernels of subspaces of JB\(^{*}\)-triples
zbMATH Open document ID: 1128.46027
author: Christopher S. Hoskin
author: C. M. Edwards
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-06-06T00:00:00Z
review text: For any closed subspace \(L\) of a complex Banach space \(A\), there exists a greatest M-ideal \(k_n(L)\) of \(A\) contained in \(L\). The space \(k_n (L)\) is called the norm central kernel of \(L\) in \(A\). When \(A\) is a dual space and \(L\) is a weak*-closed subspace, there exists a greatest M-summand \(k(L)\) of \(A\) contained in \(A\). \(k(L)\) is termed the central kernel of \(L\) in \(A\). The definition of these two central kernels is given by purely geometric statements. The results of the paper under review describe the norm central kernel of an arbitrary closed subspace \(L\) of a JB*-triple \(A\) and the central kernel of a weak*-closed subspace \(M\) of a JBW*-triple \(W\). The main results of the paper establish a purely algebraic description of these two central kernels based on the JB*-triple structure existing in \(E\) and \(W\). It is the interplay between the geometric, holomorphic, and algebraic structure of JB*-triples which makes the latter an appropriate structure to describe central kernels of subspaces. The final section considers some specialisations to the settings of C*-algebras and von Neumann algebras. We should also add that the notion of the central kernel of a weak*-closed subspace \(L\) of a JBW*-triple \(W\) was already introduced and studied in [\textit{C.\,M.\thinspace Edwards} and \textit{G.\,T.\,Rüttimann}, J.~Algebra 250, No.\,1, 90--114 (2002; Zbl 1019.17010)].
reviewed by: Antonio M. Peralta
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 17C65
Mathematics Subject Classification ID: 46B04
zbMATH DE Number: 5161749
zbMATH Keywords: M-summand
zbMATH Keywords: M-ideal
zbMATH Keywords: central kernel
zbMATH Keywords: norm central kernel
zbMATH Keywords: JB*-triple
zbMATH Keywords: JBW*-triple
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.060
OpenAlex ID: W1986276840
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DOI: 10.1016/J.JMAA.2006.10.060
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description: scientific article; zbMATH DE number 5161749