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
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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 \[ \Phi (f)(\Lambda (x))=\tau (x)f(x) \] for every $x\in X$ and every $f\in \mathcal {A}$ with $\tau =\Phi (1)$.References
- Lajos Molnár, Some characterizations of the automorphisms of $B(H)$ and $C(X)$, Proc. Amer. Math. Soc. 130 (2002), no. 1, 111–120. MR 1855627, DOI 10.1090/S0002-9939-01-06172-X
- Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
- Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
- Errett Bishop and Karel de Leeuw, The representations of linear functionals by measures on sets of extreme points, Ann. Inst. Fourier (Grenoble) 9 (1959), 305–331. MR 114118, DOI 10.5802/aif.95
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