Isomorphisms between \(C^{*}\)-ternary algebras
type: Item
instance of: scholarly article
title: Isomorphisms between \(C^{*}\)-ternary algebras
zbMATH Open document ID: 1113.39033
author: Choonkil Park
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: Following \textit{H.~Zettl} [Adv.\ Math.\ 48, 117--143 (1983; Zbl 0517.46049)], a \(C^*\)-ternary ring is defined to be a Banach space \({\mathcal A}\) with a ternary product \((x, y, z)\mapsto [x, y, z]\) from \({\mathcal A}\) into \({\mathcal A}\) which is linear in the outer variables, conjugate linear in the middle variable, and associative in the sense that \([x, y, [z, t, s]]=[x, [t, z, y], s]=[[x, y, z], t, s]\), and satisfies \(\| [x, y, z]\| \leq\| x\| \| y\| \| z\| \) and \(\| [x, x, x]\| =\| x\| ^{3}\). A linear mapping \(\varphi: {\mathcal A} \to {\mathcal B}\) between \(C^*\)-ternary rings is called a homomorphism if \(\varphi([x, y, z])=[\varphi(x), \varphi(y), \varphi(z)]\) (\(x, y, z\in {\mathcal A}\)). A linear mapping \(\delta: {\mathcal A} \to {\mathcal A}\) is called a derivation if \(\delta([x, y, z])= [\delta(x), y, z]+[x, \delta(y), z]+[x, y, \delta(z)]\) (\((x, y, z\in {\mathcal A}\)). The stability of \(C^*\)-ternary homomorphisms and \(C^*\)-ternary derivations was proved by the reviewer in [Approximate \(C^*\)-ternary ring homomorphisms, Bull.\ Braz.\ Math.\ Soc.\ (to appear) and Almost derivations on \(C^*\)-ternary rings, Bull.\ Belg.\ Math.\ Soc.--Simon Stevin (to appear)]. The author of the present paper investigates isomorphisms between \(C^*\)-ternary rings and derivations associated to three Cauchy--Jensen type additive functional equations. Some other results in this area may be also found in his paper [J.~Math.\ Phys.\ 47, No.~10, 103512 (2006; Zbl 1112.39023)].
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 5083262
zbMATH Keywords: Cauchy-Jensen functional equation
zbMATH Keywords: \(C^*\)-ternary ring isomorphism
zbMATH Keywords: Hyers-Ulam-Rassias stability
zbMATH Keywords: \(C^{*}\)-ternary derivation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.04.010
OpenAlex ID: W2013848316
cites work: Hypersymmetry: A ℤ3-graded generalization of supersymmetry
cites work: Approximate homomorphisms of ternary semigroups
cites work: Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces
cites work: On the stability of \(J^*\)-homomorphisms
cites work: Q4549797
cites work: Leibniz and Lie algebra structures for Nambu algebra
cites work: On stability of additive mappings
cites work: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings
cites work: On the Stability of the Linear Functional Equation
cites work: On the asymptoticity aspect of Hyers-Ulam stability of mappings
cites work: Stability of \(\Psi\)-additive mappings: Applications to nonlinear analysis
cites work: On the Hyers-Ulam-Rassias stability of approximately additive mappings
cites work: The cubic chessboard
cites work: Almost derivations on \(C^*\)-ternary rings
cites work: Lie \(\ast\)-homomorphisms between Lie \(C^*\)-algebras and Lie \(\ast\)-derivations on Lie \(C^*\)-algebras
cites work: Q3023624
cites work: Homomorphisms between Poisson JC*-algebras
cites work: Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between \(C^*\)-algebras
cites work: Hyers–Ulam–Rassias stability of a generalized Apollonius–Jensen type additive mapping and isomorphisms between C *‐algebras
cites work: Q3733629
cites work: Solution of a problem of Ulam
cites work: On the Stability of the Linear Mapping in Banach Spaces
cites work: The problem of S. M. Ulam for approximately multiplicative mappings
cites work: On the stability of functional equations in Banach spaces
cites work: On the stability of functional equations and a problem of Ulam
cites work: Q4820379
cites work: On the Behavior of Mappings which do not Satisfy Hyers-Ulam Stability
cites work: Proprieta’ locali e approssimazione di operatori
cites work: Q3259049
cites work: On special classes of n-algebras
cites work: A characterization of ternary rings of operators
DOI: 10.1016/J.JMAA.2006.04.010
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description: scientific article; zbMATH DE number 5083262