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 $\mathfrak{m}$ for $K$ and a group $\Gamma$ of residues modulo $\mathfrak{m}$, we consider the semi-direct product $R⋊R_{\mathfrak{m},\Gamma }$ 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 $R⋊R_{\mathfrak{m},\Gamma }$ 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.

final version, accepted for publication in Trans. Amer. Math. Soc.; updated introduction and bibliography; 29 pages

References in corpus (1)

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

Cited by in corpus (7)

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• Partition functions as C*-dynamical invariants and actions of congruence monoids
• Algebraic actions I. C*-algebras and groupoids
• Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras