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

TL;DR

This paper demonstrates that semigroup C*-algebras associated with rings of integers can uniquely determine number fields up to arithmetic equivalence, especially for finite Galois extensions, under certain conditions.

Contribution

It establishes a link between semigroup C*-algebras and the classification of number fields, showing that these algebras encode key arithmetic information.

Findings

Semigroup C*-algebras determine number fields up to arithmetic equivalence.

For finite Galois extensions, isomorphic C*-algebras imply isomorphic number fields.

The results depend on the number fields having the same number of roots of unity.

Abstract

We show that semigroup C*-algebras attached to ax+b-semigroups over rings of integers determine number fields up to arithmetic equivalence, under the assumption that the number fields have the same number of roots of unity. For finite Galois extensions, this means that the semigroup C*-algebras are isomorphic if and only if the number fields are isomorphic.