$$C^*$$ -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 $${\mathfrak T }[R]$$ . 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 $$R\rtimes R^\times $$ on $$\ell ^2 (R\rtimes R^\times )$$ . The algebra $${\mathfrak T }[R]$$ carries a natural one-parameter automorphism group $$(\sigma _t)_{t\in {\mathbb R }}$$ . 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 $$\beta \in (1,2]$$ .