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 Q. As a consequence, the Bost-Connes C *algebra for a number field K has h 1 K -dimensional irreducible representations and does not have finite-dimensional irreducible representations for the other dimensions, where h 1 K is the narrow class number of K. In particular, the narrow class number is an invariant of Bost-Connes C * -algebras.

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

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

“…Then A K = 1 YK ÃK 1 YK and 1 YK is a full projection. This presentation is crucial to determine the primitive ideal space of A K in [7].…”

“…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 )).…”

“…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.…”

Bost-Connes Systems for Local Fields of Characteristic Zero

“…The primitive ideal space. The primitive ideal space can be determined in the same way as in [9], [8] by applying Williams's theorem [10] to ÃK . Since the action of Z on X K is free, any primitive ideal is of the form of I x = ker π x for x ∈ Y K , where π x is the representation of A K on B(ℓ 2 N) determined by It is interesting to compare it with the case of number fields (cf.…”