Dilating Covariant Representations of a Semigroup Dynamical System Arising from Number Theory

Abstract

In Cuntz et al. (Math Ann 355(4):1383–1423, 2013. doi:10.1007/s00208-012-0826-9), studied the ${C^*}$-algebra ${\mathfrak {T}[R]}$ generated by the left-regular representation of the ${ax + b}$-semigroup of a number ring R on ${\ell^2(R \rtimes R^\times)}$. They were able to describe it as a universal ${C^*}$-algebra defined by generators and relations, and show that it has an interesting KMS-structure and that it is functorial for injective ring homomorphisms. In this paper we show that ${\mathfrak {T}[R]}$ can be realized as the ${C^*}$-envelope of the isometric semicrossed product of a certain semigroup dynamical system ${(\mathcal {A}_R, \alpha, R^\times)}$. We do this by proving that a representation of ${\mathcal {A}_R \times_\alpha^{\rm is}R^\times}$ is maximal if it is also a representation of ${\mathfrak {T}[R]}$. We also show how to explicitly dilate any representation of ${\mathcal {A}_R \times_\alpha^{\rm is}R^\times}$ to a representation of ${\mathfrak {T}[R]}$.