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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Multiplicative bijections of $\mathcal{C(X},I\mathcal{)}$


Author: Janko Marovt
Journal: Proc. Amer. Math. Soc. 134 (2006), 1065-1075
MSC (2000): Primary 46J10; Secondary 46E05
Published electronically: July 20, 2005
MathSciNet review: 2196040
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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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Additional Information

Janko Marovt
Affiliation: EPF-University of Maribor, Razlagova 14, 2000 Maribor, Slovenia
Email: janko.marovt@uni-mb.si

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08069-X
PII: S 0002-9939(05)08069-X
Keywords: Preserver, multiplicative map
Received by editor(s): September 10, 2004
Received by editor(s) in revised form: October 27, 2004
Published electronically: July 20, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.