A note on noncommutative unique ergodicity and weighted means
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2008
- DOI:
- 10.48550/arXiv.0803.0073
- arXiv:
- arXiv:0803.0073
- Bibcode:
- 2008arXiv0803.0073A
- Keywords:
-
- Mathematics - Operator Algebras;
- Mathematics - Dynamical Systems;
- 47A35;
- 46L35;
- 46L55
- E-Print:
- 11 pages. submitted. Linear Alg. Applications (to appear)