The Regular C*-algebra of an Integral Domain

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.

30 pages

Cited by in corpus (19)

• C*-algebras associated with endomorphisms and polymorphisms of compact abelian groups
• C*-Algebras of algebraic dynamical systems and right LCM semigroups
• Nuclearity of semigroup C*-algebras and the connection to amenability
• The 2-adic ring $C^{\ast }$-algebra of the integers and its representations
• C*-algebras of Toeplitz type associated with algebraic number fields
• The inner structure of boundary quotients of right LCM semigroups
• Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras
• Cuntz-Li relations, Inverse semigroups and Groupoids
• Cuntz-Li algebras from a-adic numbers
• On the K-theory of crossed products by automorphic semigroup actions
• Partial crossed product description of the C*-algebras associated with integral domains
• C*-algebras associated with integral domains and crossed products by actions on adele spaces
• Operator algebras associated to integral domains
• Cuntz's $ax+b$-semigroup C*-algebra over $\mathbb{N}$ and product system C*-algebras
• Inverse semigroups and the Cuntz-Li algebras
• $C^{\ast }$-Envelope and Dilation Theory of Semigroup Dynamical Systems
• A crossed-product approach to the Cuntz-Li algebras
• KMS states of a generalized Toeplitz algebras
• Quantum Statistical Mechanics of $\mathbb{Q}$-lattices and noncommutative geometry