Irreducible representations of Bost–Connes systems

Abstract

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 Q. As a consequence, the Bost-Connes C *

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

“…The homomorphism of the bottom line is injective by [7,Lemma 3.12]. This implies that Φ is injective.…”

“…By [7,Lemma 3.12], P S is contained in ker π S,γ = P S,γ for any γ. Let B = B(π S,γ (1 YK )ℓ 2 (J K /Γ S )).…”

“…By [7,Lemma 3.12], there exists a sequence {w n } n∈N ⊂ J K such that w n x converges to y in X K . For any n ∈ N, letπ n be the representation…”

“…Primitive ideals revisited. In [7,Section 3.3], the primitive ideal structure of A K is formally determined. PrimA K has a bundle structure over 2 PK with torus fibers.…”

“…This seems to be optimal because we use the class field theory to construct the C * -algebra A K , although the partition function does not reflect such a construction so much. So we can expect that even the C * -algebra A K itself may have plenty of information of the original field K. Indeed, the narrow class number h 1 K is an invariant of A K by [7], looking at maximal primitive ideals of K.…”