Let $X$ be a compact Hausdorff space and $C(X)$ the Banach space of continuous $\mathbb K$-valued functions equipped with the supremum norm. We denote the dual space of $C(X)$ by $C(X)^*$.
A well-known theorem, due to Arens & Kelley (1947), states that the extreme points of the unit ball of $C(X)^*$ are precisely the functionals of the form$$\alpha= \theta\cdot\delta_x,$$ where $\theta$ is an unitary scalar and $\delta_x$ is the functional "evaluation at $x$", that is, $ \delta_x(f)=f(x), $ for every $f\in C(X)$.
I wonder if there is a version of this theorem for $C_0(X)$ spaces, that is, the Banach space of continuous $\mathbb K$-valued functions which vanish at infinity for some locally compact (not necessarily compact) Hausdorff space $X$.
Actually, I'm pretty sure I have seen people using this theorem for the $C_0(X)$ spaces, however I still fail to find any proof of this. I was able to prove, by myself, a version of this theorem for $C_0(X)$ spaces in the case $\mathbb K=\mathbb R$. I still have no answer for the case $\mathbb K=\mathbb C$.