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Approximation of a generalized Euler-Lagrange type additive mapping on Lie \(C^\ast\)-algebras. (English) Zbl 1361.39016
Summary: Using fixed point method, we prove some new stability results for Lie \((\alpha,\beta,\gamma)\)-derivations and Lie \(C^\ast\)-algebra homomorphisms on Lie \(C^\ast\)-algebras associated with the Euler-Lagrange type additive functional equation
\[ \sum\limits_{j=1}^n f\left(-r_jx_j+\sum\limits_{1\leq i\leq n, i\neq j} r_ix_i\right)+2\sum\limits_{i=1}^n r_if(x_i)=nf\left(\sum\limits_{i=1}^n r_ix_i\right) \]
where \(r_1,\dots,r_n\in\mathbb{R}\) are given and \(r_i,r_j\neq 0\) for some \(1\leq i<j\leq n\).
MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras
16W25 Derivations, actions of Lie algebras
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