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 \(\Phi :{\mathfrak A}\rightarrow {\mathfrak A}\), 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 \(\{z\in {\mathbb {C}}\mid |z|=1\}\) and the corresponding eigenspace \({\mathfrak A}_\lambda \subset {\mathfrak A}\) for which the sequence of Cesaro averages \(\left( \frac{1}{n}\sum _{k=0}^{n-1}\lambda ^{-k}\Phi ^k\right) _{n>0}\), converges point-wise in norm. We also describe some pivotal examples coming from quantum probability, to which the obtained results can be applied.

中文翻译：

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关于动力系统$$ C ^ * $$ C ∗-动力学系统的遍历平均值的一致收敛性

我们研究了由单个\（C ^ * \）-代数构成的一般（离散）\（C ^ * \）-动力系统\（（{\ mathfrak A}，\ Phi）\）的遍历和谱图性质一个可乘的，保留身份的\（* \）- map \（\ Phi：{\ mathfrak A} \ rightarrow {\ mathfrak A} \），特别说明了\（（{{mathfrak A}，\ Phi）\ ）就定点子代数而言具有独特的遍历性。对于不具有上述强遍历特性的\（C ^ * \）动力系统，我们在\\\\ zb中的\\\\ lambda \）中提供条件（\ {\ mathbb {C}} \ | z | = 1 \} \中）以及对应的特征空间\（{\ mathfrak A} _ \ lambda \ subset {\ mathfrak A} \），Cesaro序列的平均空间为\（\ left（\ frac {1} {n} \ sum _ {k = 0 } ^ {n-1} \ lambda ^ {-k} \ Phi ^ k \ right）_ {n> 0} \），按规范逐点收敛。我们还描述了一些来自量子概率的关键例子，可以将获得的结果应用到这些例子中。