C*-algebras associated with integral domains and crossed products by actions on adele spaces

TL;DR

This paper computes the K-theory of C*-algebras linked to rings of integers in number fields, using duality theorems to relate finite and infinite adele spaces, advancing understanding of their algebraic structure.

Contribution

It introduces a duality theorem for global fields that connects crossed products over finite and infinite adele spaces, enabling K-theory computations.

Findings

K-theory of C*-algebras associated with rings of integers is computed.

A duality theorem for global fields is established.

Crossed products over finite and infinite adeles are identified.

Abstract

We compute the K-theory for C*-algebras naturally associated with rings of integers in number fields. The main ingredient is a duality theorem for arbitrary global fields. It allows us to identify the crossed product arising from affine transformations on the finite adeles with the analogous crossed product algebra over the infinite adele space.