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On approximately higher ring derivations. (English) Zbl 1144.39024

Let 𝒜 be an algebra and n 0 {0,1,,}{}. A sequence (d j ) j=1 n 0 of mappings on 𝒜 is called a higher ring derivation of rank n 0 if for each 0jn 0 ,

d j (ab)= =0 j d (a)d j- (b)(a,b𝒜)·

It is obvious that d 0 is a homomorphism and d 1 is a d 0 -derivation in the sense of M. Mirzavaziri and M. S. Moslehian [Proc. Am. Math. Soc. 134, No. 11, 3319–3327 (2006; Zbl 1116.46061)]. In this paper the authors use ideas of R. Badora [J. Math. Anal. Appl. 276, No. 2, 589–597 (2002; Zbl 1014.39020)] and T. Miura, G. Hirasawa and S.-E. Takahasi [J. Math. Anal. Appl. 319, No. 2, 522–530 (2006; Zbl 1104.39025)] to establish the stability of higher derivations on Banach algebras as well as superstability of such mappings under the surjectivity of d 0 .

MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
47B47Commutators, derivations, elementary operators, etc.
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