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

Abstract

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 $\ast$-automorphisms of $\mathfrak{M}$, a positive element $x\in \mathfrak{M}$, and a right Følner sequence $\mathcal{F}=(F_k{)}_{k\in \mathbb{N}}$ for $G$, the sequence $${\left(\Vert \frac{1}{|F_k|}\sum _{g\in F_k}{\Xi }_{g}x\Vert \right)}_{k\in \mathbb{N}}$$ 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.