Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.

中文翻译：

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根据量子伯努利噪声的量子费勒半群

量子伯努利噪声 (QBN) 是作用于伯努利泛函的湮灭和创造算子家族，它们同时满足典型的反对易关系。在本文中，我们旨在根据 QBN 研究量子 Feller 半群。我们首先研究由恒等算子和 QBN 生成的代数的局部结构。然后，我们使用我们在这里获得的新结果从 QBN 构造一类具有 Feller 属性的量子马尔可夫半群。作为我们结果的应用，我们研究了一个与 QBN 相关的特殊量子 Feller 半群，当它简化为某个阿贝尔子代数时，该半群产生由 Ornstein-Uhlenbeck 算子生成的半群。最后，我们找到了存在对半群对角的忠实不变状态的充分条件。