C*-algebras of Toeplitz type associated with algebraic number fields

Abstract

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra $\cT[R]$. It is an extension of the ring C*-algebra $\cA[R]$ studied previously by the first named author in collaboration with X.Li. In contrast to $\cA[R]$, it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$-semigroup $R\rtimes R^\times$ on $\ell^2 (R\rtimes R^\times)$. The algebra $\cT[R]$ carries a natural one-parameter automorphism group $(\sigma_t)_{t\in\Rz}$. We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The "partition functions" are partial Dedekind $\zeta$-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\zeta$-functions, which we then use to show uniqueness of the $\beta$-KMS state for each inverse temperature $\beta\in(1,2]$.