On approximately higher ring derivations
Let \({\mathcal A}\) be an algebra and \(n_0 \in \{0, 1, \dots, \}\cup \{\infty\}\). A sequence \((d_j)_{j=1}^{n_0}\) of mappings on \({\mathcal A}\) is called a higher ring derivation of rank \(n_0\) if for each \(0 \leq j \leq n_0\), \[ d_j(ab) = \sum_{\ell=0}^j d_\ell(a)d_{j-\ell}(b) \qquad (a, b \in {\mathcal A}). \] It is obvious that \(d_0\) is a homomorphism and \(d_1\) is a \(d_0\)-derivation in the sense of \textit{M. Mirzavaziri} and \textit{M. S. Moslehian} [Proc. Am. Math. Soc. 134, No. 11, 3319--3327 (2006; Zbl 1116.46061)]. In this paper the authors use ideas of \textit{R. Badora} [J. Math. Anal. Appl. 276, No. 2, 589--597 (2002; Zbl 1014.39020)] and \textit{T. Miura, G. Hirasawa} and \textit{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 \(d_0\).
scientific article; zbMATH DE number 5275648
scientific article