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On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids

Chris Bruce, Xin Li

Abstract

We study semigroup C*-algebras of semigroups associated with number fields and initial data arising naturally from class field theory. Using K-theoretic invariants, we investigate how much information about the initial number-theoretic data is encoded in our semigroup C*-algebras.

On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids

Abstract

We study semigroup C*-algebras of semigroups associated with number fields and initial data arising naturally from class field theory. Using K-theoretic invariants, we investigate how much information about the initial number-theoretic data is encoded in our semigroup C*-algebras.

Paper Structure

This paper contains 18 sections, 40 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semigroup C*-algebras
  4. Congruence monoids and semigroup C*-algebras associated with their actions
  5. Class fields associated with congruence monoids
  6. Reconstruction of initial data in number-theoretic context
  7. Pure infiniteness
  8. K-theory
  9. K-theory for our semigroup C*-algebras
  10. The case of the rational number field
  11. On the summands in the K-theory formula for our semigroup C*-algebras
  12. The case of real quadratic fields and totally positive elements
  13. K-theory for boundary quotients
  14. Duality Theorems
  15. Rational K-theory computations for boundary quotients
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that $K$ and $L$ are number fields with rings of algebraic integers $R$ and $S$. Let $\mathfrak{m}$ and $\mathfrak{n}$ be moduli for $K$ and $L$, and let $\Gamma$ and $\Lambda$ be subgroups of $(R/\mathfrak{m})^*$ and $(S/\mathfrak{n})^*$, respectively. Suppose that there is an isomorphism $

Theorems & Definitions (79)

  • Theorem : see Theorem \ref{['thm:reconstruction']}
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 69 more