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

Abstract: 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 $K_1(C^{\ast}(\mathbb{A}))$.