C*-algebras from actions of congruence monoids on rings of algebraic integers

Abstract

Let K be a number field with ring of integers R. Given a modulus m for K and a group Γ of residues modulo m, we consider the semi-direct product R Rm,Γ obtained by restricting the multiplicative part of the full ax + b-semigroup over R to those algebraic integers whose residue modulo m lies in Γ, 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, 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 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 Rm,Γ embeds canonically into the left regular C*-algebra of the full ax + b-semigroup. Our methods rely heavily on Li's theory of semigroup C*-algebras.

Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

“…The map a → [a] m extends uniquely to a surjective group homomorphism from the group of quotients R −1 m R m of R m onto (R/m) * , see [Bru,Lemma 2.1]. By [Bru,Lemma 2.2…”

“…Then R m,Γ is a multiplicative subsemigroup of R × ; such semigroups are called congruence monoids in the literature on semigroups, see, for example, [G-HK]. By [Bru,Proposition 3.1], the group of quotients…”

“…Preliminaries for the proof. The semigroup R R m,Γ canonically embeds into the [Bru,Proposition 3.2], the semigroup R R m,Γ is left Ore and its group of left quotients coincides with (R −1…”

“…A groupoid model. The material below on adeles and a groupoid model for C * λ (P m,Γ ) is from [Bru,Section 5]. It was motivated by similar results from [C-D-L, Section 5] for the special case where m and Γ are trivial.…”

“…By [Bru,Theorem 7.1], the C*-algebra C * λ (R R m,Γ ) has a unique maximal ideal I P m K . The boundary quotient of C * λ (R R m,Γ ), as defined in [Li2,Section 7] (see also Chapter 5.7]), is the quotient C * λ (R R m,Γ )/I P m K .…”