Reconstructing the Bost–Connes semigroup actions from K-theory

Abstract

We complete the classification of Bost-Connes systems. We show that two Bost-Connes C*-algebras for number fields are isomorphic if and only if the original semigroups actions are conjugate. Together with recent reconstruction results in number theory by Cornelissen-de Smit-Li-Marcolli-Smit, we conclude that two Bost-Connes C*-algebras are isomorphic if and only if the original number fields are isomorphic.

𝐶*-algebras of right LCM monoids and their equilibrium states

“…A general construction of full and reduced C * -algebras for left cancellative monoids was introduced by Li, [Li12], and soon fundamental questions about the interplay between nuclearity of the C * -algebra and amenability of the monoid or its left inverse hull were raised, see [Li13,Nor14]. Subsequent work on semigroup C *algebras that centered on computing K-theory revealed impressively rich connections to number theory, see [Li14,KT], and geometric group theory, see [ELR16].…”

Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras

“…This construction of C*-algebras from number fields is one of the easiest and most natural. Our main result says that the rigidity phenomenon from [38,Corollary 1.2] holds even in this setting: Theorem 1.1. Let K and L be number fields.…”

“…The Hecke C*algebras constructed from totally positive ax + b-groups in [42] are less ad hoc. These are full corners in the Bost-Connes C*-algebra by [41], so the results of the second-named author and Kubota in [38] yield a rigidity theorem for such Hecke C*-algebras. However, from the perspective of rigidity for C*-algebras constructed from number fields, there are other C*-algebras that are more natural from a number-theoretic or dynamical view point.…”

“…Inspired by Li's rigidity results for ax + b-semigroup C*-algebras, the problem of rigidity phenomena for the Bost-Connes C*-algebras-without any additional data such as time evolutions-was considered by the second-named author [78,79], where it was proven that the Bost-Connes C*-algebra remembers both the Dedekind zeta function and the narrow class number of the underlying number field; as in the case of semigroup C*-algebras, this implies rigidity if one of the number fields is Galois. A complete solution to the rigidity problem was later obtained by the second-named author and Kubota [38]: Any *-isomorphism between the Bost-Connes C*-algebras associated with two number fields gives rise to a conjugacy between the Bost-Connes semigroup dynamical systems. Moreover, this conjugacy is constructed from K-theoretic invariants of the C*-algebras.…”

Ground states of groupoid C∗-algebras, phase transitions and arithmetic subalgebras for Hecke algebras

“…Recently, Takeishi used K-theoretical techniques inspired by [23] to prove that the C * -algebra in the Bost-Connes system associated to a number field K recovers the Dedekind zeta-function of K, [29], thus refining the findings of [18] that showed that the entire Bost-Connes system determines the Dedekind zetafunction. Finally, the full reconstruction of number fields up to isomorphism from the C * -algebras of the respective Bost-Connes systems has been recently achieved by Kubota and Takeishi [12], see also [6].…”