Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras

TL;DR

This paper demonstrates that *-isomorphisms between certain crossed product C*-algebras uniquely determine the underlying number fields, linking operator algebra isomorphisms to number field isomorphisms and characterizations.

Contribution

It establishes a rigidity result connecting *-isomorphisms of specific crossed product C*-algebras to explicit isomorphisms of the underlying number fields, and extends Neukirch--Uchida type theorems.

Findings

Constructed explicit number field isomorphisms from *-isomorphisms of crossed product C*-algebras.

Proved an analogue of the Neukirch--Uchida theorem using topological full groups.

Characterized number fields via associated discrete groups.

Abstract

We prove that the class of crossed product C*-algebras associated with the action of the multiplicative group of a number field on its ring of finite adeles is rigid in the following explicit sense: Given any *-isomorphism between two such C*-algebras, we construct an isomorphism between the underlying number fields. As an application, we prove an analogue of the Neukirch--Uchida theorem using topological full groups, which gives a new class of discrete groups associated with number fields whose abstract isomorphism class completely characterises the number field.