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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Multiplicatively spectrum-preserving maps of function algebras
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by N. V. Rao and A. K. Roy
Proc. Amer. Math. Soc. 133 (2005), 1135-1142
DOI: https://doi.org/10.1090/S0002-9939-04-07615-4
Published electronically: October 15, 2004

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 \[ \Phi (f)(\Lambda (x))=\tau (x)f(x) \] for every $x\in X$ and every $f\in \mathcal {A}$ with $\tau =\Phi (1)$.
References
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Bibliographic Information
  • N. V. Rao
  • Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606
  • Email: rnagise@math.utoledo.edu
  • A. K. Roy
  • Affiliation: Indian Statistical Institute-Calcutta, Statistics and Mathematics Unit, 203 B.T. Road, Calcutta 700 108, India
  • Email: ashoke@isical.ac.in
  • 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
  • © Copyright 2004 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 1135-1142
  • MSC (2000): Primary 46J10, 46J20
  • DOI: https://doi.org/10.1090/S0002-9939-04-07615-4
  • MathSciNet review: 2117215