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

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 $Ch\A_2$.