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

Bruce, C. (2020) C*-algebras from actions of congruence monoids on rings of algebraic integers. Transactions of the American Mathematical Society, 373(1), pp. 699-726. (doi: 10.1090/tran/7966)

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 RoRm,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 o 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bruce, Dr Chris
• Authors: Bruce, C.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Transactions of the American Mathematical Society
• Publisher: American Mathematical Society
• ISSN: 0002-9947
• ISSN (Online): 1088-6850
• Published Online: 01 October 2019