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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.

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