Constructing number field isomorphisms from \(*\)-isomorphisms of certain crossed product \(\mathrm{C}^*\)-algebras. (English) Zbl 1543.46039
Summary: We prove that the class of crossed product \(\mathrm{C}^*\)-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 \(\mathrm{C}^*\)-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.
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 11R56 | Adèle rings and groups |
| 11M55 | Relations with noncommutative geometry |
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