Extremely Strong Boundary Points and Real-linear Isometries

Abstract

Let $X,Y$ be locally compact Hausdorff spaces, $A$ be a complex subspace of $C_0(X)$ and $T: A \longrightarrow C_0(Y)$ be a real-linear isometry, whose range is not assumed to be a complex subspace of $C_0(Y)$. In this paper, using the set $\Theta(A)$ and $\tau(A)$ consisting of all extremely strong boundary points and strong boundary points of $A$, respectively we introduce appropriate subsets $Y_0$ and $Y_1$ of $Y$ and give a description of $T$ on these sets. More precisely, we show that there exist continuous functions $\Phi:Y_0\longrightarrow \Theta(A)$, $\alpha:Y_0\longrightarrow [-1,1]$ and $w:Y_0\longrightarrow \mathbf{T}$, where $\mathbf{T}$ is the unit circle, such that \[Tf(y)=w(y) \cdot (\mathrm{Re}(f(\Phi(y)))+\alpha(y) i \, \mathrm{Im}(f(\Phi(y))) \] for all $f\in A$ and $y\in Y_0$. The result is improved in the case where either i) $T(A)$ is a complex subspace of $C_0(Y)$ and $\Theta(A)= \mathrm{ch}(A)$, where $\mathrm{ch}(A)$ is the Choquet boundary of $A$ or ii) $T(A)$ satisfies a certain separating property.

In the first case we show that there exists a clopen subset $K$ of $Y_0$ such that \[ (Tf)(y)= w(y)\left\lbrace \begin{array}{@{}cl} f( \Phi(y)) & \,\, y\in K,\\ \overline{f(\Phi(y))} & \,\, y\notin K, \end{array} \right. \] for each $f\in A$ and $y\in Y_0$. In the second case we obtain similar results for $\tau(A)\cap \mathrm{ch}(A)$ and $Y_1$ instead of $\Theta(A)$ and $Y_0$.