  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.
