The 2-adic ring $C^{\ast }$-algebra of the integers and its representations

Abstract

We study the 2-adic version of the ring $C^{\ast }$-algebra of the integers. First, we work out the precise relation between the Cuntz algebra $\cO_2$ and our 2-adic ring $C^{\ast }$-algebra in terms of representations. Secondly, we prove a 2-adic duality theorem identifying the crossed product arising from 2-adic affine transformations on the 2-adic numbers with the analogous crossed product algebra over the real numbers. And finally, as an outcome of this duality result, we construct an explicit imprimitivity bimodule and prove that it transports one canonical representation into the other.

Changes to Remark 3.2. New Remark 6.5 and added paragraph after Proposition 6.3. Corrected typos

References in corpus (3)

• The Regular C*-algebra of an Integral Domain
• A class of $C^{\ast }$-algebras generalizing both graph algebras and homeomorphism $C^{\ast }$-algebras IV, pure infiniteness
• Dilations of semigroup crossed products as crossed products of dilations

Cited by in corpus (8)

• K-theory for ring C*-algebras - the case of number fields with higher roots of unity
• A look at the inner structure of the $2$-adic ring $C^{\ast }$-algebra and its automorphism groups
• The inner structure of boundary quotients of right LCM semigroups
• Diagonal automorphisms of the $2$-adic ring $C^{\ast }$-algebra
• Permutative representations of the $2$-adic ring $C^{\ast }$-algebra
• Phase transition on Exel crossed products assocaited to dilation matrices
• On the entropy and index of the winding endomorphisms of p-adic ring C${}^{\ast }$-algebras
• A crossed-product approach to the Cuntz-Li algebras