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

DOI:
https://doi.org/10.1090/S0002-9939-04-07615-4

Published electronically:
October 15, 2004

MathSciNet review:
2117215

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Abstract | References | Similar Articles | Additional Information

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 $\tau =\Phi (1)$.

- 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 https://doi.org/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**

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Additional 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

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