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

2025/08/30 by Wenqing Wu, Hang Wang, Wu, Wenqing +1
Computer Science · Mathematics · #46L80 (K-theory and operator algebras #58B34 (Noncommutative geometry #Algebraic and Geometric Analysis #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Polynomial and algebraic computation #\`a la Connes) #advanced mathematical theories #including cyclic theory)

paper · pdf · doi:10.48550/arxiv.2509.00390

openalex publication_date 2025/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 K₁-group of the rational group C^∗-algebra, K₁(C^*(ℚ)). Finally, we employ a similar strategy to determine the structure of K₁(C^∗(\mathbbA)).

Citations

• A K-theoretic Selberg trace formula
• Cuntz-Li algebras from a-adic numbers
• C*-algebras associated with integral domains and crossed products by actions on adele spaces
• The Regular C*-algebra of an Integral Domain

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