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The pexiderized Apollonius-Jensen type additive mapping and isomorphisms between \(C^*\)-algebras. (English) Zbl 1144.39027
Let \(X, Y\) be Banach modules over a \(C^*\)-algebra. The authors prove the Hyers-Ulam-Rassias stability of the following pexiderized Apollonius-Jensen functional equation in Banach modules over a unital \(C^*\)-algebra:
\[ \begin{split} F\left( \left(\sum_{i=1}^n z_i\right) - \left(\sum_{i=1}^n x_i\right) \right) + G\left( \left(\sum_{i=1}^n z_i\right) - \left(\sum_{i=1}^n y_i\right) \right)\\= 2H\left( \left(\sum_{i=1}^n z_i\right) - \frac{\left(\sum_{i=1}^n x_i\right) + \left(\sum_{i=1}^n y_i\right)}{2} \right) . \end{split} \] It is shown that the mappings \(F, G, H : X \rightarrow Y\) satisfy the functional equation above and \(F(0)=G(0)\) if and only if the mappings \(F, G, H \) are additive and \(F=G=H\). The paper can be regarded as a continuation of the work by C. Park [Math. Nachr. 281, No. 3, 402–411 (2008; Zbl 1142.39023)].

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46L05 General theory of \(C^*\)-algebras
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