Let $\mathcal{A}$ be an abelian C${}^{\ast}$-Algebra with unit. We know that $\mathcal{A}\cong C(\Omega(\mathcal{A}))$, where $\Omega(\mathcal{A})\subseteq\mathcal{A}'_{\geq 0}$. Note that for $\varphi\in\mathcal{A}'$ we define $\varphi\geq 0$ exactly in case $\langle f,\varphi\rangle$ for all $f\in\mathcal{A}$ with $f\geq 0$ (whereby the order relation is the usual one on C${}^{\ast}$-Algebras).
Consider now the linear hull $\langle\Omega(\mathcal{A})\rangle$. Question 1. Is this dense in $\mathcal{A}'$ under the weak topology, that is, is it $\sigma(\mathcal{A}',\mathcal{A}'')$-weakly dense?
Application. Suppose that $\xi,\eta\in\mathcal{A}''$ satisfy $\xi\leq\eta$ on $\Omega(\mathcal{A})$, that is $\langle\xi,\varphi\rangle\leq\langle\eta,\varphi\rangle$ for all $\varphi\in\Omega(\mathcal{A})$. If the above is true, then it holds that $\xi\leq\eta$, that is, that $\langle\xi,\varphi\rangle\leq\langle\eta,\varphi\rangle$ for all $\varphi\in\mathcal{A}'$ with $\varphi\geq 0$. It is trivial to show this for $\xi,\eta\in\mathcal{A}$ (viewed as a subspace of $\mathcal{A}''$ in the canonical fashion), but I cannot seem to show this without the above result. Perhaps there is a counterexample, so that the desired application fails and the above result is necessarily false.
Further observations. Due to the Riesz-Representation theorem, and since $\Omega(\mathcal{A})$ is compact Hausdorff for abelian C${}^{\ast}$-Algebras with a unit, we have $\mathcal{A}'\cong C(\Omega(\mathcal{A}))'=\langle\{T_{\mu}\mid\mu~\text{(reg.) prob. meas. on $\Omega(\mathcal{A})$}\}\rangle$, where $T_{\mu}:f\in C(\Omega(\mathcal{A}))\mapsto\int f~\mathrm{d}\mu$. So it is necessary and sufficient to show that all probability measures on $\Omega(\mathcal{A})$ can be weakly approximated by linear combinations of characters. Here we replace/identify each character $\tau\in\Omega(\mathcal{A})$ with $\hat{\tau}:\hat{a}\in C(\Omega(\mathcal{A}))\mapsto \tau(a)=\hat{a}(\tau)$, where $a\in\mathcal{A}\mapsto\hat{a}\in C(\Omega(\mathcal{A}))$ is the canonical Gelfand-C${}^{\ast}$-isomorphism. That is, each character is the identified with the point measure $\hat{\tau}=\delta_{\tau}$. Now, the convex hull of the set of point measures can be easily shown to be $w^{\ast}$-dense in the set of probability measures identified as a subspace of $C(K)'$, where $K=\Omega(\mathcal{A})$. Is it also $w$-dense?
Hence the reduced problem:
Question 2. Is the convex hull of the set of point measures over a compact Hausdorff space weakly dense in the set of probability measures?
No. The span of $\Omega(\mathcal{A})$ is not dense in $\mathcal{A}'$ in the norm topology (for instance, if $\Omega(\mathcal{A})=[0,1]$, it is easy to see that Lebesgue measure cannot be approximated in norm by finite-support measures; see also the answers to this question on MO in the case $\Omega(\mathcal{A})=\beta\mathbb{N}$). Letting $E$ be the norm-closure of the span of $\Omega(\mathcal{A})$ in $\mathcal{A}'$, then $E$ is a closed proper subspace of $\mathcal{A}'$. By Hahn-Banach, there are then continuous functional on $\mathcal{A}'$ which vanish on $E$, and these functionals show that the span of $\Omega(A)$ is not weakly dense.