Primitive ideals and \(K\)-theoretic approach to Bost-Connes systems (Q317345)

As the main classification theorem obtained, it is shown that, if the Bost-Connes \(C^*\)-algebras associated to two number fields are isomorphic, then the Dedekind zeta functions associated to them coincide.NEWLINENEWLINEThe Bost-Connes \(C^*\)-algebra of a number field \(K\) is defined to be the semigroup crossed product \(C^*\)-algebra of the \(C^*\)-algebra of all continuous functions on the quotient space of the product space of the profinite completion of the integer ring of \(K\) and the Galois group corresponding to the extension of \(K\) by abstract or concrete roots of unity by a product action of multiplication and Artin reciprocity map of the group of invertibles of the profinite completion, by the action of the semigroup of integral ideals of \(K\).NEWLINENEWLINEThe Bost-Connes \(C^*\)-algebra can be written as a full corner of a non-unital, group crossed product \(C^*\)-algebra of the \(C^*\)-algebra of all continuous functions vanishing at infinity on the quotient space of the product space of the finite adéle ring of \(K\) and the Galois group by the same product action, by the action of the group of fractional ideals of \(K\).NEWLINENEWLINEThe primitive ideal space of a Bost-Connes \(C^*\)-algebra is formally determined in another paper by Takeishi. In this paper under review, second maximal (or pre-maximal in a sense) primitive ideals of a \(C^*\)-algebra are introduced as a notion and the space of these ideals is examined, and determined in the case of Bost-Connes \(C^*\)-algebras.NEWLINENEWLINEAs a consequence, it is shown that the \(C^*\)-algebra accociated to the sets of maximal primitive ideals and second maximal primitive ideals of a Bost-Connes \(C^*\)-algebra is a simple \(C^*\)-algebra with a unique unbounded trace up to scalar multiplication, and the evaluation of the \(K_0\)-group of the \(C^*\)-algebra under the trace is computed.NEWLINENEWLINEBy using these results as well as the main theorem of \textit{D. Stuart} and \textit{R. Perlis} [J. Number Theory 53, No. 2, 300--308 (1995; Zbl 0863.11082)], the proof of the main result is completed.