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On the generalized Hyers–Ulam–Rassias stability in Banach modules over a \(C^{*}\)-algebra. (English) Zbl 1053.46028
In 1940, S. M. Ulam [“Problems in modern mathematics. First published under the title ‘A collection of mathematical problems’ ” (Science Editions. New York: John Wiley and Sons, Inc.) (1964; Zbl 0137.24201)] posed the following problem concerning the stability of group homomorphisms: Under what condition does there exist an additive mapping near an approximately additive mapping? In 1941, D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] gave a partial solution of Ulam’s problem in the context of Banach spaces. In 1978, Th. M. Rassias [Proc. Am. Math. Soc. 72, 297-300 (1978; Zbl 0398.47040)] generalized Hyers’ result. Over the last decades, several stability problems for functional equations have been investigated in the spirit of Hyers–Ulam–Rassias; cf. S. Czerwik (ed.) [“Stability of Functional Equations of Ulam–Hyers–Rassias Type” (Hadronic Press, ISBN 1-57485-057-1) (2003)]. A generalized Hyers–Ulam–Rassias stability of the quadratic functional equation \[ f(\sum_ {i=1}^ nx_ i)+\sum_ {1\leq i<j\leq n} f(x_ i-x_ j)=n\sum_ {i=1}^ n f(x_ i) \] was investigated by J.-H. Bae and K.-W. Jun [J. Math. Anal. Appl. 258, 183-193 (2001; Zbl 0983.39013)]. In the present paper, the authors extend the generalized Hyers–Ulam–Rassias stability of the above \(n\)-dimensional quadratic functional equation to Banach modules over a \(C^{*}\)-algebra.
C. G. Park [J. Math. Anal. Appl. 291, 214-223 (2004; Zbl 1046.39023)] recently treated the stability of generalized quadratic mappings in Banach modules.

MSC:
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
39B82 Stability, separation, extension, and related topics for functional equations
39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
46L05 General theory of \(C^*\)-algebras
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