On the C*-algebra associated with the full adele ring of a number field

TL;DR

This paper characterizes the primitive ideal space of a C*-algebra linked to a number field's adele ring and shows that isomorphisms of these algebras reflect isomorphisms of the underlying number fields.

Contribution

It provides an explicit description of the primitive ideal space and uses K-theoretic invariants to distinguish places, linking algebraic and number-theoretic structures.

Findings

Primitive ideal space explicitly described

K-theoretic invariants distinguish places

Isomorphisms of C*-algebras imply number field isomorphisms

Abstract

The multiplicative group of a number field acts by multiplication on the full adele ring of the field. Generalising a theorem of Laca and Raeburn, we explicitly describe the primitive ideal space of the crossed product C*-algebra associated with this action. We then distinguish real, complex, and finite places of the number field using K-theoretic invariants. Combining these results with a recent rigidity theorem of the authors implies that any *-isomorphism between two such C*-algebras gives rise to an isomorphism of the underlying number fields that is constructed from the *-isomorphism.