On K-theoretic invariants of semigroup C*-algebras attached to number fields, Part II

TL;DR

This paper demonstrates how to recover the zeta function and ideal class group of a number field from the K-theoretic invariants of associated semigroup C*-algebras, linking algebraic and dynamical perspectives.

Contribution

It shows that the semigroup C*-algebra and its canonical subalgebra encode key number-theoretic invariants, providing a new approach to their reconstruction.

Findings

Reconstruction of the zeta function from C*-algebra data

Recovery of the ideal class group as a group from the algebraic structure

Alternative dynamical systems interpretation of the invariants

Abstract

This paper continues the study of K-theoretic invariants for semigroup C*-algebras attached to ax+b-semigroups over rings of algebraic integers in number fields. We show that from the semigroup C*-algebra together with its canonical commutative subalgebra, it is possible to reconstruct the zeta function of the underlying number field as well as its ideal class group (as a group). In addition, we give an alternative interpretation of this result in terms of dynamical systems.