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Multiplicatively spectrum-preserving maps of function algebras

Authors: N. V. Rao and A. K. Roy
Journal: Proc. Amer. Math. Soc. 133 (2005), 1135-1142
MSC (2000): Primary 46J10, 46J20
Published electronically: October 15, 2004
MathSciNet review: 2117215
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Abstract: Let $X$ be a compact Hausdorff space and $\mathcal{A}\subset C(X)$ a function algebra. Assume that $X$ is the maximal ideal space of $\mathcal A$. Denoting by $\sigma (f)$ the spectrum of an $f\in \mathcal{A}$, which in this case coincides with the range of $f$, a result of Molnár is generalized by our Main Theorem: If $\Phi :\mathcal{A} \rightarrow \mathcal{A}$ is a surjective map with the property $\sigma (fg)=\sigma (\Phi (f)\Phi (g))$ for every pair of functions $f,g\in \mathcal{A}$, then there exists a homeomorphism $\Lambda :X\rightarrow X$ such that

\begin{displaymath}\Phi (f)(\Lambda (x))=\tau (x)f(x) \end{displaymath}

for every $x\in X$ and every $f\in \mathcal{A}$ with $\tau =\Phi (1)$.

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Additional Information

N. V. Rao
Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606

A. K. Roy
Affiliation: Indian Statistical Institute-Calcutta, Statistics and Mathematics Unit, 203 B.T. Road, Calcutta 700 108, India

Keywords: Automorphism, function algebra, spectrum, boundaries
Received by editor(s): January 21, 2003
Received by editor(s) in revised form: April 2, 2003, and December 2, 2003
Published electronically: October 15, 2004
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society

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