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

## TL;DR

This paper constructs and analyzes C*-algebras arising from actions of congruence monoids on rings of algebraic integers, generalizing previous work on $ax+b$-semigroups and providing explicit descriptions of their structure and primitive ideals.

## Contribution

It introduces new presentations of these C*-algebras, establishes faithfulness criteria, describes primitive ideals explicitly, and demonstrates functorial properties and embeddings into larger semigroup C*-algebras.

## Key findings

- Explicit descriptions of primitive ideals using relations involving isometries.
- Faithfulness criteria for representations based on ideal class projections.
- Embedding of the constructed C*-algebra into the full $ax+b$-semigroup C*-algebra.

## Abstract

Let $K$ be a number field with ring of integers $R$. Given a modulus $\mathfrak{m}$ for $K$ and a group $\Gamma$ of residues modulo $\mathfrak{m}$, we consider the semi-direct product $R\rtimes R_{\mathfrak{m},\Gamma}$ obtained by restricting the multiplicative part of the full $ax+b$-semigroup over $R$ to those algebraic integers whose residue modulo $\mathfrak{m}$ lies in $\Gamma$, 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 $\mathfrak{m}$, 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 $ax+b$-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 $R\rtimes R_{\mathfrak{m},\Gamma}$ 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.