Crossed product C*-algebras associated with non-minimal actions on the circle

We examine crossed product C*-algebras associated with free non-minimal actions of countably infinite discrete abelian groups on the circle, extending the work of Putnam, Schmidt, and Skau.

We obtain a large class of unital separable nuclear and non-simple C*-algebras that are quasidiagonal, have stable rank one, and admit a unique tracial state.

We determine their ideal structure and establish an improved uniform upper bound for their nuclear dimension.

Finally, in the case $G = \mathbb{Z}^d$, we compute the K-theory and trace pairing map.