In this paper, we construct, for a certain class of semigroup dynamical systems, two operator algebras that are universal with respect to their corresponding covariance conditions: one being selfadjoint, and another being nonselfadjoint. We prove that the C*-envelope of the nonselfadjoint operator algebra is precisely the selfadjoint one. This result leads to a number of new examples of operator algebras and their C*-envelopes, with many from number fields and commutative rings. We further establish the functoriality of these operator algebras along with their applications.

中文翻译：

---

半群动力系统的C *-包络和扩张理论

在本文中，对于一类半群动力系统，我们构造了两个算子代数，它们在其对应的协方差条件方面是通用的：一个是自伴的，另一个是非自伴的。我们证明了非自伴算子的C *包络恰好是自伴算子的C *包络。这一结果导致了许多新的算子代数及其C *包络的例子，其中许多来自数域和交换环。我们进一步建立了这些算子代数及其应用的功能。