A note on noncommutative unique ergodicity and weighted means

TL;DR

This paper explores noncommutative unique ergodicity in $C^*$-dynamical systems using Riesz means, establishing convergence criteria and constructing examples of uniquely ergodic yet non-ergodic operators.

Contribution

It introduces Riesz summation as a weaker convergence method for unique ergodicity and constructs a non-ergodic, uniquely ergodic entangled Markov operator.

Findings

Riesz means characterize unique ergodicity in $C^*$-dynamical systems.

Convergence of Riesz means to a fixed point projection is equivalent to unique ergodicity.

Existence of a non-ergodic, uniquely ergodic entangled Markov operator.

Abstract

In this paper we study unique ergodicity of $C^*$-dynamical system $(\ga,T)$, consisting of a unital $C^*$-algebra $\ga$ and a Markov operator $T:\ga\mapsto\ga$, relative to its fixed point subspace, in terms of Riesz summation which is weaker than Cesaro one. Namely, it is proven that $(\ga,T)$ is uniquely ergodic relative to its fixed point subspace if and only if its Riesz means {equation*} \frac{1}{p_1+...+p_n}\sum_{k=1}^{n}p_kT^kx {equation*} converge to $E_T(x)$ in $\ga$ for any $x\in\ga$, as $n\to\infty$, here $E_T$ is an projection of $\ga$ to the fixed point subspace of $T$. It is also constructed a uniquely ergodic entangled Markov operator relative to its fixed point subspace, which is not ergodic.