On the structure of the reduced semigroup $C^*$-algebra for the semigroup $Z\rtimes Z^{\times}$

We study the structure of the reduced semigroup $C^*$-algebra for the semigroup that is a semidirect product of the additive group $Z$ of all integers and the multiplicative semigroup of integers without zero $Z^{\times}:=Z\setminus\{0\}$. In the report we discuss various ways for the representation of this algebra as cross products with the infinite dihedral group $D$ and the multiplicative semigroup of natural numbers $\mathbb{N}$.