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On approximate derivations. (English) Zbl 1093.39024
The author of the present pleasant paper establishes that if \(A\) is a subalgebra of a Banach algebra \(B\) and \(f: A \to B\) satisfies \(\| f(x+y) - f(x) - f(y)\| \leq \delta\) and \(\| f(xy) - xf(y) - f(x)y\| \leq \varepsilon\), for all \(x, y \in A\) and for some \(\delta, \varepsilon \geq 0\), then there exists a unique additive derivation \(d:A \to B\) such that \(\| f(x) - d(x)\| \leq \delta \quad (x \in A)\), and \(x\left(f(y) - d(y)\right) = 0 \quad (x, y \in A)\). The result and its proof are still true for a more general case if we consider a normed algebra \(A\) and replace \(B\) by a Banach \(A\)-bimodule \(X\).
He also proves that if \(B\) is a normed algebra with an identity belonging to \(A\), then every mapping \(f: A \to B\) satisfying \(\| f(xy) - xf(y) - f(x)y\| \leq \varepsilon \quad (x, y \in A)\) must fulfil \(f(xy) = xf(y) - f(x)y \quad (x, y \in A)\). This superstability result is nice, since there is not assumed any (approximately) additive condition on \(f\). Some similar results in which one considers generalized derivations can be found in the reviewer’s paper [“Hyers-Ulam-Rassias stability of generalized derivations”, Int. J. Math. Math. Sci. (to appear)].

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