Dilating covariant representations of a semigroup dynamical system arising from number theory (Q255347)

Inspired by a paper by \textit{J. Cuntz} et al. [Math. Ann. 355, No. 4, 1383--1423 (2013; Zbl 1273.22008)], who studied the \(C^\ast\)-algebra \(\mathfrak{I}(R)\) generated by the left-regular representation of the \(ax+ b\)-semigroup of a number ring \(R\) on \(\ell^2(R\rtimes R^\times)\), the author of the paper under review is able to prove that \(\mathfrak{I}(R)\) can be realized as the \(C^\ast\)-envelope of the isometric semicrossed product of a certain semigroup dynamical system \((\mathcal{A}_R,\alpha,R^\times)\). He also shows how to explicitly dilate any representation of \(\mathcal{A}_R \times^{\text{is}}_\alpha R^\times\) to a representation of \(\mathfrak{I}(R)\).