Disjointness and unique ergodicity of C*-dynamical systems

TL;DR

This paper investigates the concept of disjointness in noncommutative C*-dynamical systems, establishing an ergodic theorem and exploring unique ergodicity, with applications to various group actions and examples.

Contribution

It introduces a noncommutative ergodic theorem for disjoint C*-dynamical systems and relates unique ergodicity to disjointness, extending classical concepts to the noncommutative setting.

Findings

Established an ergodic theorem for disjoint C*-dynamical systems.

Connected unique ergodicity with disjointness in noncommutative systems.

Provided examples involving actions of groups beyond integers.

Abstract

We study an ergodic theorem for disjoint C*-dynamical systems, where disjointness here is a noncommutative version of the concept introduced by Furstenberg for classical dynamical systems. This is applied to W*-dynamical systems. We also consider specific examples of disjoint C*-dynamical and W*-dynamical systems, including for actions of other groups than $\mathbb{Z}$. Unique ergodicity and unique ergodicity relative to the fixed point algebra are closely related to disjointness, and are used to give examples of disjoint C*-dynamical systems.