On approximately higher ring derivations
type: Item
instance of: scholarly article
title: On approximately higher ring derivations
zbMATH Open document ID: 1144.39024
author: Yong-Soo Jung
author: Ick-Soon Chang
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-05-15T00:00:00Z
review text: Let \({\mathcal A}\) be an algebra and \(n_0 \in \{0, 1, \dots, \}\cup \{\infty\}\). A sequence \((d_j)_{j=1}^{n_0}\) of mappings on \({\mathcal A}\) is called a higher ring derivation of rank \(n_0\) if for each \(0 \leq j \leq n_0\), \[ d_j(ab) = \sum_{\ell=0}^j d_\ell(a)d_{j-\ell}(b) \qquad (a, b \in {\mathcal A}). \] It is obvious that \(d_0\) is a homomorphism and \(d_1\) is a \(d_0\)-derivation in the sense of \textit{M. Mirzavaziri} and \textit{M. S. Moslehian} [Proc. Am. Math. Soc. 134, No. 11, 3319--3327 (2006; Zbl 1116.46061)]. In this paper the authors use ideas of \textit{R. Badora} [J. Math. Anal. Appl. 276, No. 2, 589--597 (2002; Zbl 1014.39020)] and \textit{T. Miura, G. Hirasawa} and \textit{S.-E. Takahasi} [J. Math. Anal. Appl. 319, No. 2, 522--530 (2006; Zbl 1104.39025)] to establish the stability of higher derivations on Banach algebras as well as superstability of such mappings under the surjectivity of \(d_0\).
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 5275648
zbMATH Keywords: derivation
zbMATH Keywords: higher derivation
zbMATH Keywords: Banach algebra
zbMATH Keywords: Hyers-Ulam-Rassias stability
zbMATH Keywords: superstability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.01.083
OpenAlex ID: W2027015973
cites work: On approximate ring homomorphisms
cites work: On approximate derivations
cites work: Approximately isometric and multiplicative transformations on continuous function rings
cites work: On stability of additive mappings
cites work: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings
cites work: Ring derivations on semi-simple commutative Banach algebras
cites work: On the Stability of the Linear Functional Equation
cites work: On the Hyers-Ulam stability of \(\psi\)-additive mappings
cites work: Continuity of module and higher derivations
cites work: Continuity of Derivations on Commutative Algebras
cites work: Continuity of Derivations and a Problem of Kaplansky
cites work: Q4714196
cites work: A perturbation of ring derivations on Banach algebras
cites work: On the Stability of the Linear Mapping in Banach Spaces
cites work: The functional equation of multiplicative derivation is superstable on standard operator algebras
cites work: Derivations on commutative normed algebras
cites work: The image of a derivation is contained in the radical
cites work: Q3259049
DOI: 10.1016/J.JMAA.2008.01.083
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description: scientific article; zbMATH DE number 5275648