Gelfand-Kolmogorov-type theorem for $C_0(X)$? I am looking for good references for Gelfand-Kolmogorov-type theorems for different function spaces—the space of vanishing functions, in particular. Explicitly, I am after a proof of the following fact:

Let $\mathfrak{A} = C_0(X)$ be the C*-algebra of vanishing functions on $X$ a locally compact and Hausdorff space. Then $X$ is homeomorphic with the set of characters $\Omega(\mathfrak{A})$ endowed with the weak-star topology.

 A: Update, 8/30. Landsman's (2017) Foundations of Quantum Theory gives a different, elegant proof using the one-point compactification of $X$, cf. Theorem C.45.
I'm including my go at a proof, below. I hobbled this together from strategies in Haskell Curry's answer on this thread and Nustruev's (2000) Smooth Manifolds and Observables, 3.16. I am not sure if this is standard, though.
Proof
Consider the map
\begin{equation*}
    \theta : X \to \Omega(C_0 (X) ) : x \mapsto ( f  \mapsto f(x)).
\end{equation*}
It is clear that $\theta$ is injective: each pair of points $x\neq y \in X$ specify characters $\theta(x) \neq \theta(y)  $ (as they will differ on some $f$). We now endeavor to show that $\theta$ is surjective. We do so by exploiting the fact that $X$ is locally compact and Hausdorff, which conditions allow us to exploit Riesz representations.
Explicitly, consider any $\chi \in \Omega (C_0 (X) ) $. By $\chi : C_0 (\Omega) \to \mathbb{C}$ a non-zero *-homomorphism, it is also a continuous linear functional on $C_0(X)$. Thus, by the Riesz-Markov theorem, there is a unique, regular, countably additive, complex-valued Borel measure $\mu$ on $X$ such that
\begin{equation*}
    \forall f \in C_0(X), \quad \chi (f) = \int_X d\mu(x) \, f(x).
\end{equation*}
Now suppose towards a contradiction that $\chi$ does not correspond to any point in $X$. That is, for every $y \in X$, there is some $f_y \in C_0 (X)$ such that $\chi(f_y) \neq f_y (y)$. By definition, $\chi$ is non-zero, so pick some $f$ such that $\chi(f) = \lambda \neq 0$. Since $f$ vanishes at infinity, all of its level sets (for non-zero complex $\lambda$) must be compact. In particular, the set of points $L = f^{-1} (\lambda)$ is compact. Now take each $y\in L$. By $f_y$ continuous, the set of all $x$ such that $f_y(x) \neq \chi(f_y) $ is open (it is the preimage of $\mathbb{C}\setminus \chi (f_y) $ , which is open). So the sets
\begin{equation*}
    U_y = \{ x\in X  \,|\, f_y(x) \neq \chi(f_y) \}, \quad y\in L
\end{equation*}
form an open cover of $L$. By $L$ compact, we can form an open sub-cover $\{U_{y_1}, ..., U_{y_m}\} $. Now consider the function
\begin{equation*}
    g = (f - \chi(f) )^2 + \sum^m_{i=1} (f_{y_i} - \chi(f_{y_i}) )^2. 
\end{equation*}
This function is an element of $C_0(X)$ because we have only used the algebraic operations in the above. However, $g$ is non-vanishing. And by construction,
\begin{equation*}
    \chi(g) = (\chi(f) - \chi(f) )^2 + \sum^m_{i=1} (\chi(f_{y_i}) - \chi(f_{y_i}) )^2 = 0.
\end{equation*}
The only way to square these two facts about $g$ with the existence of the Riesz representation of $\chi$ above is to suppose that $\mu$ vanishes everywhere. But this contradicts our assumption that $\chi$ is non-zero. So $\theta$ is a bijection.
All that remains is to show that $\theta$ is continuous. Recall that the weak-* topology on $\Omega(C_0 (X) )$ is the weakest topology that renders continuous all functions of the form $\tilde{f} : \Omega(C_0 (X) ) \to \mathbb{C} :: \chi \mapsto \chi(f)$ for some $f\in C_0 (X) $. Thus, open sets of the form $\tilde{f}^{-1} (V)$ for $V$ open in $\mathbb{C}$ form an open basis for $\Omega(C_0 (X) )$. Note, then, that $\theta^{-1} (\tilde{f}^{-1} (V)) $ is the set of points $x \in X$ for which $f(x) \in V $, and this set is open in $X$ by virtue of $f \in C_0 (X)$. So $\theta$ is a homeomorphism.
