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

Chris Bruce, Takuya Takeishi

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.

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

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.
Paper Structure (43 sections, 57 theorems, 99 equations)

This paper contains 43 sections, 57 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Main result
  4. Outline of the proof
  5. Structure of the paper
  6. Preliminaries
  7. Notation and terminology
  8. Ext-groups and KK-theory
  9. Number-theoretic background
  10. The C*-algebra associated with the action on the ring of finite adeles
  11. Crossed products modulo a finite group
  12. General observations
  13. Translation actions
  14. Trivial actions
  15. Tensor product decompositions
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $K$ and $L$ be number fields. Then, the following are equivalent: In particular, $K\cong L$ if and only if $\mathfrak A_K\cong \mathfrak A_L$.

Theorems & Definitions (124)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5: Neu and Hasse
  • Lemma 2.6
  • ...and 114 more