C*-algebras from actions of congruence monoids on rings of algebraic integers
Chris Bruce · Transactions of the American Mathematical Society · 2019
Let K K be a number field with ring of integers R R . Given a modulus m \mathfrak {m} for K K and a group Γ \Gamma of residues modulo m \mathfrak {m} , we consider the semidirect product R ⋊ R m , Γ R\rtimes R_{\mathfrak {m},\Gamma } obtained by restricting the multiplicative part of the full a x + b ax+b -semigroup over R R to those algebraic integers whose residue modulo m \mathfrak {m} lies in Γ \Gamma , and we study the left regular C*-algebra of this semigroup. We give two presentations of this C*-algebra and realize it as a full corner in a crossed product C*-algebra. We also establish a faithfulness criterion for representations in terms of projections associated with ideal classes in a quotient of the ray class group modulo m \mathfrak {m} , and we explicitly describe the primitive ideals using relations only involving the range projections of the generating isometries; this leads to an explicit description of the boundary quotient. Our results generalize and strengthen those of Cuntz, Deninger, and Laca and of Echterhoff and Laca for the C*-algebra of the full a x + b ax+b -semigroup. We conclude by showing that our construction is functorial in the appropriate sense; in particular, we prove that the left regular C*-algebra of R ⋊