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Permuting triderivations and permuting trihomomorphisms in Banach algebras. (English) Zbl 1452.39006
Summary: Using the direct method, we prove the Hyers-Ulam stability of permuting triderivations and permuting trihomomorphisms in Banach algebras and unital \(C^\ast\)-algebras associated with the triadditive \(s\)-functional inequality \begin{multline*} \| f (x+y, z-w, a+b)+f(x-y, z+w, a-b)-2f (x, z, a)+2f(x, w, b)-2f(y, z, b)+2f(y, w, a) \| \\ \leq \| s (2f((x+y) / 2, z-w, a+b)+2f((x-y) / 2, z+w, a-b) \\ -2f(x, z, a)+ 2f(x, w, b)-2f (y, z, b)+2f(y, w, a)) \|, \end{multline*} where \(s\) is a fixed nonzero complex number with \(|s|< 1\).
MSC:
39B52 Functional equations for functions with more general domains and/or ranges
39B62 Functional inequalities, including subadditivity, convexity, etc.
39B82 Stability, separation, extension, and related topics for functional equations
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
47B47 Commutators, derivations, elementary operators, etc.
17A40 Ternary compositions
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