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On approximately higher ring derivations. (English) Zbl 1144.39024

Let $𝒜$ be an algebra and ${n}_{0}\in \left\{0,1,\cdots ,\right\}\cup \left\{\infty \right\}$. A sequence ${\left({d}_{j}\right)}_{j=1}^{{n}_{0}}$ of mappings on $𝒜$ is called a higher ring derivation of rank ${n}_{0}$ if for each $0\le j\le {n}_{0}$,

${d}_{j}\left(ab\right)=\sum _{\ell =0}^{j}{d}_{\ell }\left(a\right){d}_{j-\ell }\left(b\right)\phantom{\rule{2.em}{0ex}}\left(a,b\in 𝒜\right)·$

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:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges 47B47 Commutators, derivations, elementary operators, etc.
References:
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