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

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra ${mathfrak T }[R]$ . It is an extension of the ring C*-algebra ${mathfrak A }[R]$ studied previously by the first named author in collaboration with X. Li. In contrast to ${mathfrak A }[R]$ , it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$ -semigroup $Rrtimes R^times $ on $ell ^2 (Rrtimes R^times )$ . The algebra ${mathfrak T }[R]$ carries a natural one-parameter automorphism group $(sigma _t)_{tin {mathbb R }}$ . 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 $beta in (1,2]$ .