Tokyo Journal of Mathematics

Maps Which Preserve a Certain Norm Condition between the Exponential Groups of Uniform Algebras

Tatsuya NOGAWA

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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 $Ch\A_2$.

Article information

Source
Tokyo J. Math., Volume 39, Number 1 (2016), 39-44.

Dates
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1471873311

Digital Object Identifier
doi:10.3836/tjm/1471873311

Mathematical Reviews number (MathSciNet)
MR3543130

Zentralblatt MATH identifier
1360.46045

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 47B48: Operators on Banach algebras

Citation

NOGAWA, Tatsuya. Maps Which Preserve a Certain Norm Condition between the Exponential Groups of Uniform Algebras. Tokyo J. Math. 39 (2016), no. 1, 39--44. doi:10.3836/tjm/1471873311. https://projecteuclid.org/euclid.tjm/1471873311


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