The Regular C*-algebra of an Integral Domain

2008/07/09 by Joachim Cuntz, Xin Li, Cuntz, Joachim +1 · 3 citations
Mathematics · #11R56 (Secondary) #46L05 (Primary) 11R04 #58B34 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #math.OA #msc:11R04 #msc:11R56 #msc:46L05 #msc:58B34

paper · pdf · doi:10.48550/arxiv.0807.1407

30 pages

arxiv created 2008/07/09 · openalex publication_date 2008/07/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To each integral domain R with finite quotients we associate a purely infinite simple C*-algebra in a very natural way. Its stabilization can be identified with the crossed product of the algebra of continuous functions on the "finite adele space" corresponding to R by the action of the ax+b-group over the quotient field Q(R). We study the relationship to generalized Bost-Connes systems and deduce for them a description as universal C*-algebras with the help of our construction.

Citations

• On Bost-Connes type systems for number fields

Cited by

• K-Theory of Adelic and Rational Group C^*-algebras via Generalized Winding Numbers
• C*-algebras associated with integral domains and crossed products by actions on adele spaces
• Quantum Statistical Mechanics of ℚ-lattices and noncommutative geometry

Related

• C^*-algebras associated with the ax+b-semigroup over \mathbb N
• C*-algebras associated with integral domains and crossed products by actions on adele spaces
• A crossed-product approach to the Cuntz-Li algebras
• Partial crossed product description of the C*-algebras associated with integral domains
• The ideal structure of the Hecke C*-algebra of Bost and Connes