On the uniform convergence of ergodic averages for $C^*$-dynamical systems

We investigate some ergodic and spectral properties of general (discrete) $C^*$-dynamical systems $({mathfrak A},Phi)$ made of a unital $C^*$-algebra and a multiplicative, identity-preserving $*$-map $Phi:{mathfrak A}to{mathfrak A}$, particularising the situation when $({mathfrak A},Phi)$ enjoys the property of unique ergodicity with respect to the fixed-point subalgebra. For $C^*$-dynamical systems enjoying or not the strong ergodic property mentioned above, we provide conditions on $lambda$ in the unit circle ${zin{mathbb C}mid |z|=1}$ and the corresponding eigenspace ${mathfrak A}_lambdasubset{mathfrak A}$ for which the sequence of Cesaro averages $left(frac1{n}sum_{k=0}^{n-1}lambda^{-k}Phi^kright)_{n>0}$, converges point-wise in norm. We also describe some pivotal examples coming from quantum probability, to which the obtained results can be applied.