Jordan, Jordan right and Jordan left derivations on convolution algebras
type: Item
instance of: scholarly article
title: Jordan, Jordan right and Jordan left derivations on convolution algebras
zbMATH Open document ID: 1414.43004
author: Mohammad Hossein Ahmadi Gandomani
author: M. J. Mehdipour
published in: Bulletin of the Iranian Mathematical Society
publication date: 2019-03-22T00:00:00Z
review text: A Jordan derivation on a ring $R$ is an additive mapping $d$ that satisfies \[ d(x^2) = d(x) x + x d(x) \] for all $x \in R$; $d$ is said to be a Jordan left derivation if \[ d(x^2) = 2xd(x) \] for all $x \in R$. Jordan right derivations are defined similarly. Previous work, which is reviewed in the introduction of the paper, for Jordan derivations, Jordan left derivations and Jordan right derivations has been done for semiprime rings and semisimple Banach algebras. Now let $\mathcal{G}$ be a locally compact group and denote by $L_0^{\infty}(\mathcal{G})$ the subspace of $L^{\infty}({\mathcal{G}})$ consisting of all functions that vanish at infinity. With respect to the first Arens product, $L_0^{\infty}(\mathcal{G})^{\ast}$ is a noncommutative Banach algebra that is not semiprime and not semisimple. The goal of the paper under review is to see what results concerning Jordan derivations, Jordan left derivations and Jordan right derivations for semiprime rings or semisimple Banach algebras are true in the $L_0^{\infty}(\mathcal{G})^{\ast}$ setting. Results are given for Jordan derivations, Jordan left derivations and Jordan right derivations on $L_0^{\infty}(\mathcal{G})^{\ast}$ when $\mathcal{G}$ is a locally compact abelian group. An example of such a result is that when $\mathcal{G}$ is abelian, $d$ is a Jordan right derivation on $L_0^{\infty}(\mathcal{G})^{\ast}$ if and only if it is a right derivation on $L_0^{\infty}(\mathcal{G})^{\ast}$.
reviewed by: Michael J. Puls
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 16W25
zbMATH DE Number: 7040769
zbMATH Keywords: locally compact group
zbMATH Keywords: Jordan derivation
zbMATH Keywords: Jordan right derivation
zbMATH Keywords: Jordan left derivation
zbMATH Keywords: $k$-centralizing mapping
MaRDI profile type: Publication
full work available at URL: s41980-018-0125-7
OpenAlex ID: W2801946797
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Wikidata QID: Q129539254
DOI: 10.1007/S41980-018-0125-7
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description: scientific article; zbMATH DE number 7040769