K-theory for ring C*-algebras - the case of number fields with higher roots of unity

TL;DR

The paper computes the K-theory of ring C*-algebras $\mathfrak A[R]$ attached to rings of integers in number fields in the presence of higher roots of unity. Building on Morita equivalences to adele-crossed products, a duality between infinite and finite adeles, and La-Lück's K-theory for group C*-algebras, the authors reduce the problem to an inductive limit controlled by endomorphisms $\eta_c$ and then apply the Pimsner–Voiculescu sequence. They obtain explicit formulas: for $|\mu|>2$, $K_*(\mathfrak A[R]) \cong K_0(C^*(\mu)) \otimes_{\mathbb{Z}} \Lambda^*(\Gamma)$ as a $\mathbb{Z}/2\mathbb{Z}$-graded group, where $K^\times = \mu \times \Gamma$, and in general parity of real places fixes whether $\Lambda^*(\Gamma)$ or $K_0(C^*(\mu)) \otimes_{\mathbb{Z}} \Lambda^*(\Gamma)$ appears. Moreover, the classification results imply that ring C*-algebras for different rings of integers are isomorphic. The work combines adelic duality, inductive limit analysis, and equivariant K-theory to yield a complete invariant for these algebras and demonstrates the power of these techniques in noncommutative geometry.

Abstract

We compute K-theory for ring C*-algebras in the case of higher roots of unity and thereby completely determine the K-theory for ring C*-algebras attached to rings of integers in arbitrary number fields.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 3.1: Langer-Lück
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Proposition 4.3