Noncommutative Ergodic Optimization

TL;DR

This paper extends ergodic optimization theory to non-commutative C*-dynamical systems, linking optimization to convergence of ergodic averages and employing nonstandard analysis for proofs.

Contribution

It introduces a non-commutative framework for ergodic optimization and provides new convergence results and alternative proofs using nonstandard analysis.

Findings

Established a link between ergodic optimization and convergence of ergodic averages in C*-dynamical systems.

Provided new proofs of existing results using nonstandard analysis techniques.

Extended classical ergodic optimization theory to the non-commutative setting.

Abstract

We extend the theory of ergodic optimization and maximizing measures to the non-commutative field of C*-dynamical systems. We then provide a result linking the ergodic optimizations of elements of a C*-dynamical system to the convergence of certain ergodic averages in a suitable seminorm. We also provide alternate proofs of several results in this article using the tools of nonstandard analysis.