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June 2016 Maps Which Preserve a Certain Norm Condition between the Exponential Groups of Uniform Algebras
Tatsuya NOGAWA
Tokyo J. Math. 39(1): 39-44 (June 2016). DOI: 10.3836/tjm/1471873311

Abstract

Let $\A_j$ be a uniform algebra with a Choquet boundary $Ch\A_j$, $j = 1, 2$. In this paper we prove that if $\phi : \exp\A_1 \to \exp\A_2$ is a surjection and satisfies the equality \begin{equation*} \max \left\{ \left\| \frac{\phi (f)}{\phi (g)} -1 \right\|_\infty, \left\| \frac{\phi (g)}{\phi (f)} -1 \right\|_\infty \right \} =\max \left\{ \left\| \frac{f}{g} -1 \right\|_\infty, \left\| \frac{g}{f} -1 \right\|_\infty \right \} \end{equation*} for any $f, g \in \exp\A_1$, then $\phi$ is of the form \begin{equation*} \phi(f)(y) = \left\{ \begin{array}{ll} \phi(1)(y) f(\varphi(y))^{\kappa(y)} & \text{for}~ y \in K, \\ \phi(1)(y) \overline{f(\varphi(y))}^{\kappa(y)} & \text{for}~ y \in Ch\A_2 \setminus K \\ \end{array} \right. \end{equation*} for any $f \in \exp\A_1$, where $\kappa$ is a continuous function from $Ch\A_2$ into $\{ 1, -1 \}$, $\varphi$ is a homeomorphism from $Ch\A_2$ onto $Ch\A_1$ and $K$ is a clopen subset of $Ch\A_2$.

Citation

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Tatsuya NOGAWA. "Maps Which Preserve a Certain Norm Condition between the Exponential Groups of Uniform Algebras." Tokyo J. Math. 39 (1) 39 - 44, June 2016. https://doi.org/10.3836/tjm/1471873311

Information

Published: June 2016
First available in Project Euclid: 22 August 2016

zbMATH: 1360.46045
MathSciNet: MR3543130
Digital Object Identifier: 10.3836/tjm/1471873311

Subjects:
Primary: 46J10
Secondary: 47B48

Rights: Copyright © 2016 Publication Committee for the Tokyo Journal of Mathematics

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Vol.39 • No. 1 • June 2016
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