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

TL;DR

This paper offers a nonstandard analysis-based proof of a theorem on ergodic optimization in noncommutative dynamical systems, extending previous results to a more direct and alternative approach.

Contribution

It introduces a nonstandard analysis method to prove a key theorem in noncommutative ergodic optimization, providing a more direct proof compared to prior approaches.

Findings

Established convergence of ergodic averages in noncommutative setting

Provided a nonstandard analysis proof technique for ergodic theorems

Extended previous results to a more accessible proof framework

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 $*$-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.