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

We investigate some ergodic and spectral properties of general (discrete) $C^*$-dynamical systems $({\mathfrak A},Φ)$ made of a unital $C^*$-algebra and a multiplicative, identity-preserving $*$-map $Φ:{\mathfrak A}\to{\mathfrak A}$, particularising the situation when $({\mathfrak A},Φ)$ 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 $λ$ in the unit circle $\{z\in{\mathbb C}\mid |z|=1\}$ and the corresponding eigenspace ${\mathfrak A}_λ\subset{\mathfrak A}$ for which the sequence of Cesaro averages $\left(\frac1{n}\sum_{k=0}^{n-1}λ^{-k}Φ^k\right)_{n>0}$, converges point-wise in norm. We also describe some pivotal examples coming from quantum probability, to which the obtained results can be applied.