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

 

Uniform algebra isomorphisms and peripheral multiplicativity


Authors: Aaron Luttman and Thomas Tonev
Journal: Proc. Amer. Math. Soc. 135 (2007), 3589-3598
MSC (2000): Primary 46J10, 46J20; Secondary 46H40
Published electronically: June 22, 2007
MathSciNet review: 2336574
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Abstract: Let $ \varphi\colon A\to B$ be a surjective operator between two uniform algebras with $ \varphi(1)=1$. We show that if $ \varphi$ satisfies the peripheral multiplicativity condition $ \sigma_\pi\big(\varphi(f)\,\varphi(g)\big)=\sigma_\pi(fg)$ for all $ f,g\in A$, where $ \sigma_\pi(f)$ is the peripheral spectrum of $ f$, then $ \varphi$ is an isometric algebra isomorphism from $ A$ onto $ B$. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.


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Additional Information

Aaron Luttman
Affiliation: Division of Science and Mathematics, Bethany Lutheran College, Mankato, Minnesota 56001
Email: luttman@blc.edu

Thomas Tonev
Affiliation: Department of Mathematical Sciences, The University of Montana/Missoula, Montana 59812-1032
Email: tonevtv@mso.umt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08881-8
PII: S 0002-9939(07)08881-8
Keywords: Uniform algebra, peaking function, peak set, generalized peak point, Choquet boundary, Shilov boundary, homeomorphism, spectrum of an element, peripheral spectrum, peripheral range, peripherally multiplicative operator, algebra isomorphism
Received by editor(s): November 23, 2005
Received by editor(s) in revised form: August 14, 2006
Published electronically: June 22, 2007
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society