Weak-local triple derivations on \(\mathrm{C}^{\ast}\)-algebras and \(\mathrm{JB}^{\ast}\)-triples
type: Item
instance of: scholarly article
title: Weak-local triple derivations on \(\mathrm{C}^{\ast}\)-algebras and \(\mathrm{JB}^{\ast}\)-triples
zbMATH Open document ID: 1372.47044
author: María Burgos
author: J. C. Cabello
author: Antonio M. Peralta
published in: Linear Algebra and its Applications
publication date: 2016-08-17T00:00:00Z
review text: A triple derivation on a \(\mathrm{JB}^\ast\)-triple \((E,\{\cdot,\cdot,\cdot\})\) is a linear mapping \(\delta: E\to E\) satisfying the Leibniz rule \(\delta\{a,b,c\} = \{\delta(a), b,c\} +\{a,\delta(b),c\}+\{a,b,\delta(c)\}\) for every \(a,b,c\in E\). A \(\mathrm{JB}^\ast\)-triple \(E\) is a weak-local triple derivation if, for each \(\phi \in E^\ast\) and each \(a\in E\), there exists a triple derivation \(\delta_{a,\phi} : E\to E\), depending on \(\phi\) and \(a\), such that \(\phi T(a) = \phi \delta_{a,\phi} (a)\). In the paper under review, the authors prove that every weak-local triple derivation on a \(\mathrm{JB}^\ast\)-triple \(E\) is a (continuous) triple derivation. In addition, they give some sufficient and necessary conditions for a linear map \(T\) on a \(\mathrm{JB}^\ast\)-triple \(E\) to be a triple derivation.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 46L57
Mathematics Subject Classification ID: 17C65
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 6617607
zbMATH Keywords: \(\mathrm{C}^{\ast}\)-algebra
zbMATH Keywords: \(\mathrm{JB}^{\ast}\)-triple
zbMATH Keywords: triple derivation
zbMATH Keywords: weak-local triple derivation
zbMATH Keywords: weak-local derivation
MaRDI profile type: Publication
full work available at URL: j.laa.2016.06.042
OpenAlex ID: W2964028687
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DOI: 10.1016/J.LAA.2016.06.042
description: scientific article; zbMATH DE number 6617607