标题： $K$- 代数和有理群理论 $C^*$- 通过广义绕数的代数 显示英文标题

标题： $K$-Theory of Adelic and Rational Group $C^*$-algebras via Generalized Winding Numbers

摘要： 我们采用以下方法来分析周期性阿代尔函数中的同伦等价性。 首先，我们引入预周期函数的概念，并通过构造广义绕数来定义其同伦不变量。 随后，我们建立周期性阿代尔函数与预周期函数之间的基本对应关系。 通过将广义绕数扩展到周期性阿代尔函数，我们证明该不变量完全表征了周期性阿代尔函数空间内的同伦等价类。 在此分类基础上，我们获得了有理群的$K_{1}$-群的显式描述，$C^\ast$-代数，$K_{1}(C^{*}(\mathbb{Q}))$。 最后，我们采用类似策略来确定$K_1(C^{\ast}(\mathbb{A}))$的结构。 显示英文摘要

摘要： We take the following approach to analyze homotopy equivalence in periodic adelic functions. First, we introduce the concept of pre-periodic functions and define their homotopy invariant through the construction of a generalized winding number. Subsequently, we establish a fundamental correspondence between periodic adelic functions and pre-periodic functions. By extending the generalized winding number to periodic adelic functions, we demonstrate that this invariant completely characterizes homotopy equivalence classes within the space of periodic adelic functions. Building on this classification, we obtain an explicit description of the $K_{1}$-group of the rational group $C^\ast$-algebra, $K_{1}(C^{*}(\mathbb{Q}))$. Finally, we employ a similar strategy to determine the structure of $K_1(C^{\ast}(\mathbb{A}))$.