Uniform algebra isomorphisms and peripheral multiplicativity
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- by Aaron Luttman and Thomas Tonev PDF
- Proc. Amer. Math. Soc. 135 (2007), 3589-3598 Request permission
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.References
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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
- 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
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 3589-3598
- MSC (2000): Primary 46J10, 46J20; Secondary 46H40
- DOI: https://doi.org/10.1090/S0002-9939-07-08881-8
- MathSciNet review: 2336574