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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 $\parallel f\left(x+y\right)-f\left(x\right)-f\left(y\right)\parallel \le \delta$ and $\parallel f\left(xy\right)-xf\left(y\right)-f\left(x\right)y\parallel \le \epsilon$, for all $x,y\in A$ and for some $\delta ,\epsilon \ge 0$, then there exists a unique additive derivation $d:A\to B$ such that $\parallel f\left(x\right)-d\left(x\right)\parallel \le \delta \phantom{\rule{1.em}{0ex}}\left(x\in A\right)$, and $x\left(f\left(y\right)-d\left(y\right)\right)=0\phantom{\rule{1.em}{0ex}}\left(x,y\in A\right)$. 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 $\parallel f\left(xy\right)-xf\left(y\right)-f\left(x\right)y\parallel \le \epsilon \phantom{\rule{1.em}{0ex}}\left(x,y\in A\right)$ must fulfil $f\left(xy\right)=xf\left(y\right)-f\left(x\right)y\phantom{\rule{1.em}{0ex}}\left(x,y\in A\right)$. 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 39B52 Functional equations for functions with more general domains and/or ranges 47B47 Commutators, derivations, elementary operators, etc.