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Stability of \((\alpha,\beta,\gamma)\)-derivatives on Lie \(C^*\)-algebras. (English) Zbl 1217.39034
The authors investigate the stability of \((\alpha, \beta, \gamma)\)-derivations on a Lie C*-algebra \(A\) associated to the following functional equation
\[ f\left(\frac{x_2-x_1}{3}\right)+f\left( \frac{x_1-3x_3}{3}\right) +f\left( \frac{3x_1+3x_3-x_2}{3}\right)=f(x_1).\tag{1} \]
An \((\alpha, \beta, \gamma)\)-derivation of \(A\) is a linear map \(d:A \to A\) such that for certain \(\alpha, \beta, \gamma \in {\mathbb C}\) the identity
\[ \alpha d[x,y]= \beta [d(x),y]+ \gamma [x, d(y)], \]
for every \(x,y\in A\) (\([\;, \;]\) denotes the Lie bracket of \(A\)). They show that if \(f\) is an approximate solution of (1) (in the sense of the norm of \(A\)), then there exists a unique \((\alpha, \beta, \gamma)\)-derivation \(d\) of \(A\) which approximates \(f\).

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46K70 Nonassociative topological algebras with an involution
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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