Crossed products as compact quantum metric spaces

Abstract By employing the external Kasparov product, in [18], Hawkins, Skalski, White, and Zacharias constructed spectral triples on crossed product C $^\ast $ -algebras by equicontinuous actions of discrete groups. They further raised the question for whether their construction turns the respective crossed product into a compact quantum metric space in the sense of Rieffel. By introducing the concept of groups separated with respect to a given length function, we give an affirmative answer in the case of virtually Abelian groups equipped with certain orbit metric length functions. We further complement our results with a discussion of natural examples such as generalized Bunce-Deddens algebras and higher-dimensional noncommutative tori.