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Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping. (English) Zbl 1114.39011

The authors prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital \(C^*\)-algebras involving a product of powers of norms. \[ \sum_{i=1}^nr_if\Big(\sum_{j=1}^nr_j(x_i-x_j)\Big) + \Big(\sum_{i=1}^nr_i\Big) f\Big(\sum_{i=1}^nr_ix_i\Big)=\Big(\sum_{i=1}^nr_i\Big)\sum_{i=1}^nr_if(x_i), \]
where \(r_1, \dots, r_n \in(0, \infty)\). They also investigate the isomorphisms between \(C^*\)-algebras.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
46L05 General theory of \(C^*\)-algebras
47B48 Linear operators on Banach algebras
39B52 Functional equations for functions with more general domains and/or ranges
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