Irreducible Representations of Bost-Connes systems

Abstract

The classification problem of Bost-Connes systems was studied by Cornellissen and Marcolli partially, but still remains unsolved. In this paper, we will give a representation-theoretic approach to this problem. We generalize the result of Laca and Raeburn, which concerns with the primitive ideal space on the Bost-Connes system for $\mathbb{Q}$. As a consequence, the Bost-Connes $C^{\ast }$-algebra for a number field $K$ has $h_K^1$-dimensional irreducible representations and does not have finite-dimensional irreducible representations for the other dimensions, where $h_K^1$ is the narrow class number of $K$. In particular, the narrow class number is an invariant of Bost-Connes $C^{\ast }$-algebras.

15 pages. I rewrote Introduction, and I changed section numbers. I corrected several mistakes and did some minor changes

References in corpus (2)

• Quantum Statistical Mechanics, L-series and Anabelian Geometry
• Bost-Connes systems, Hecke algebras, and induction

Cited by in corpus (2)

• Primitive ideals and K-theoretic approach to Bost-Connes systems
• Bost-Connes systems for local fields of characteristic zero