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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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