Multiplicative bijections of 𝒞(𝒳,ℐ)

Marovt, Janko

doi:10.1090/S0002-9939-05-08069-X

Multiplicative bijections of $\mathcal {C(X},I\mathcal {)}$
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by Janko Marovt PDF

Proc. Amer. Math. Soc. 134 (2006), 1065-1075 Request permission

Abstract:

Let $\mathcal {X}$ be a compact Hausdorff space which satisfies the first axiom of countability, let $I=\left [ 0,1\right ]$ and let $\mathcal {C}(\mathcal {X}$,$I)$ be the set of all continuous functions from $\mathcal {X}$ to $I.$ If $\varphi :\mathcal {C}(\mathcal {X}$,$I)\rightarrow \mathcal {C}(\mathcal {X}$,$I)$ is a bijective multiplicative map, then there exist a homeomorphism $\mu : \mathcal {X\rightarrow X}$ and a continuous map $k:\mathcal {X} \rightarrow \left ( 0,\infty \right ) ,$ such that $\varphi (f)(x)=f(\mu (x))^{k(x)}$ for all $x\in \mathcal {X}$ and for all $f\in \mathcal {C}(\mathcal {X},I).$

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