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Stability of (\(\alpha\), \(\beta\), \(\gamma\))-derivations on Lie \( C^*\)-algebra associated to a periderized quadratic type functional equation. (English) Zbl 1360.39021
A \(C^{\star}\)-algebra \(A\) endowed with the Lie product \([x,y] = xy-yx\) is called a Lie \(C^{\star}\)-algebra. A \(\mathbb{C}\)-linear mapping \(D : A \to A\) is called a Lie derivation of the Lie \(C^{\star}\)-algebra \(A\) if \(D\) satisfies \(D[x,y] = [D(x),y] + [x, D(y)]\) for all \(x, y \in A\). The mapping \(D\) is called a \((\alpha, \beta, \gamma )\)-derivation if \(\alpha D[x,y] =\beta [D(x),y)] + \gamma [x,D(y)]\) for all \(x, y \in A\). In this paper, the authors prove some stability results for the \((\alpha, \beta, \gamma )\)-derivation \(D: A \to A\) and the homomorphism associated to the quadratic functional equation \(f(kx+y)+f(kx+\sigma (y)) = 2k g(x) +2 g(y)\) for all \(x, y \in A\).
MSC:
39B82 Stability, separation, extension, and related topics for functional equations
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
39B52 Functional equations for functions with more general domains and/or ranges
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