A nonstandard-analytic proof of a theorem regarding noncommutative ergodic optimizations

In a previous article, we extended the notion of ergodic optimization to the setting of C*-dynamical systems of countable discrete groups. Among the key results of that paper was that given an action $G \stackrel{\Xi}{\curvearrowright} \mathfrak{M}$ of a countable discrete amenable group $G$ on a W*-probability space $(\mathfrak{M}, \rho)$ by $\rho$-preserving $*$-automorphisms of $\mathfrak{M}$, a positive element $x \in \mathfrak{M}$, and a right F{\o}lner sequence $\mathcal{F} = (F_k)_{k \in \mathbb{N} }$ for $G$, the sequence $$\left( \left\| \frac{1}{|F_k|} \sum_{g \in F_k} \Xi_g x \right\| \right)_{ k \in \mathbb{N} }$$ converges to a value $\Gamma(x)$ which can be described in the language of ergodic optimization. We provide here an alternate, more direct proof of that theorem using the tools of nonstandard analysis.