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Some preserver problems on algebras of smooth functions. (English) Zbl 1211.46022
This paper is an important contribution to the increasingly popular area of preserver problems on function spaces. Interested readers can see the monograph [L. Molnár, “Selected preserver problems on algebraic structures of linear operators and on function spaces” (Lecture Notes in Mathematics 1895; Berlin: Springer) (2007; Zbl 1119.47001)].
Let $$X$$ be a Hausdorff topological space, locally homeomorphic to a fixed Banach space of dimension at least one (admitting a $$C^k$$ bump function). Let $$T: C^k(Y) \rightarrow C^k(X)$$ be a bijection preserving order (pointwise) in both directions, then $$Tf(x)=t(x,f(\tau(x)))$$ for all $$f \in C^k(Y)$$ and $$x \in X$$, where $$\tau: X \rightarrow Y$$ is a homeomorphism and $$t: X \times \mathbb R \rightarrow \mathbb R$$ is given by $$t(x,c)=Tc(x)$$. Thus linearity (or additive) of $$T$$ is determined by its action on the set of constant functions on $$Y$$.
It is also shown that multiplicative bijections are induced by composition with a $$C^k$$ diffeomorphism and hence any such map is linear.

MSC:
 4.6e+11 Topological linear spaces of continuous, differentiable or analytic functions
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References:
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