$C^{\ast }$-algebras from Anzai flows and their $K$-groups

We study the $C^{\ast }$-algebra ${\mathcal{A}}_{n,\theta }$ generated by the Anzai flow on the $n$-dimensional torus ${\mathbb{T}}^n$. It is proved that this algebra is a simple quotient of the group $C^{\ast }$-algebra of a lattice subgroup ${\mathfrak{D}}_n$ of a $(n+2)$-dimensional connected simply connected nilpotent Lie group $F_n$ whose corresponding Lie algebra is the generic filiform Lie algebra ${\mathfrak{f}}_n$. Other simple infinite dimensional quotients of $C^{\ast }({\mathfrak{D}}_n)$ are also characterized and represented as matrix algebras over simple affine Furstenberg transformation group $C^{\ast }$-algebras of the lower dimensional tori. The $K$-groups of the ${\mathcal{A}}_{n,\theta }$ and other simple quotients of $C^{\ast }({\mathfrak{D}}_n)$ are studied, the Pimsner-Voiculescu 6-term exact sequence being a useful tool. The rank of the $K$-groups of ${\mathcal{A}}_{n,\theta }$ is studied as explicitly as possible, and is proved to be the same as for more general transformation group $C^{\ast }$-algebras of ${\mathbb{T}}^n$ including the Furstenberg transformation group $C^{\ast }\text{-algebras}$ $A_{F_{f,\theta }}$. An error (about these $K$-groups) in the literature is addressed.