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Multiplicative derivations on $$C(X)$$. (English) Zbl 0843.46018
Summary: Let $$X$$ be a completely regular topological space satisfying the first axiom of countability with no isolated points, and let $$C(X)$$ be the algebra of all continuous functions on $$X$$. A mapping $$d: C(X)\to C(X)$$ is called a multiplicative derivation if $$d(fg)= fd(g)+ gd(f)$$ for every pair of functions $$f,g\in C(X)$$ (no linearity or continuity of $$d$$ is assumed). We obtain a complete description of such mappings and give examples to show that the above assumptions on the space $$X$$ are essential.

##### MSC:
 46E25 Rings and algebras of continuous, differentiable or analytic functions 47B47 Commutators, derivations, elementary operators, etc.
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##### References:
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