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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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##### References:
 [1] DOI: 10.1063/1.522727 · Zbl 0308.17005 [2] DOI: 10.1063/1.3093269 · Zbl 1214.46034 [3] DOI: 10.1016/j.geomphys.2009.11.004 · Zbl 1188.39029 [4] DOI: 10.1016/j.geomphys.2010.01.012 · Zbl 1192.39020 [5] DOI: 10.1155/S016117129100056X · Zbl 0739.39013 [6] DOI: 10.1006/jabr.2001.8880 · Zbl 0998.17010 [7] DOI: 10.1073/pnas.27.4.222 · Zbl 0061.26403 [8] DOI: 10.1088/0305-4470/24/3/012 · Zbl 0731.17001 [9] DOI: 10.1016/j.geomphys.2007.10.005 · Zbl 1162.17014 [10] DOI: 10.1088/0305-4470/36/26/309 · Zbl 1040.17021 [11] DOI: 10.1016/j.jmaa.2003.10.051 · Zbl 1051.46052 [12] Park C., J. Math. Phys. 50 (2009) [13] Park C., J. Lie Theory 15 pp 393– (2005) [14] DOI: 10.1007/s10114-005-0629-y · Zbl 1121.39030 [15] Park C., Rev. Bull. Calcutta Math. Soc. 11 pp 83– (2003) [16] Park C., Acta Math. Sci. Ser. B Engl. Ed. 25 pp 449– (2005) [17] Park C., Int. J. Nonlinear Anal. Appl. 1 (2) pp 1– (2010) [18] DOI: 10.1016/0024-3795(88)90210-8 · Zbl 0668.17004 [19] DOI: 10.1016/0021-9045(89)90041-5 · Zbl 0672.41027 [20] Rassias J. M., Discuss. Math. 7 pp 193– (1985) [21] Rassias J. M., Bull. Sci. Math. 108 (2) pp 445– (1984) [22] DOI: 10.1016/0022-1236(82)90048-9 · Zbl 0482.47033 [23] DOI: 10.1090/S0002-9939-1978-0507327-1 [24] DOI: 10.1006/jmaa.2000.6788 · Zbl 0958.46022 [25] DOI: 10.1023/A:1006499223572 · Zbl 0981.39014 [26] DOI: 10.1088/0305-4470/38/12/011 · Zbl 1063.22023
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