Isomorphisms between \(C^{*}\)-ternary algebras
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)].
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