Primitive ideals and K-theoretic approach to Bost-Connes systems

TL;DR

This paper demonstrates that the Dedekind zeta function is an invariant of Bost-Connes $C^*$-algebras and explores their primitive ideals and $K$-theoretic properties.

Contribution

It establishes the Dedekind zeta function as an invariant of Bost-Connes $C^*$-algebras and analyzes their primitive ideals using $K$-theory.

Findings

Dedekind zeta function is an invariant of Bost-Connes $C^*$-algebras

Characterization of second maximal primitive ideals

Application of $K$-theory to algebra quotients

Abstract

By KMS-classification theorem, the Dedekind zeta function is an invariant of Bost-Connes systems. In this paper, we show that it is in fact an invariant of Bost-Connes $C^*$-algebras. We examine second maximal primitive ideals of Bost-Connes $C^*$-algebras, and apply $K$-theory to some quotients.