On a representation of a semigroup $C^*$-algebra as a crossed product

We construct the semidirect product $\mathbb{Z}\rtimes_{\varphi}\mathbb{Z}^{\times}$ of the additive group $\mathbb{Z}$ of all integers and the multiplicative semigroup $\mathbb{Z}^{\times}$ of integers without zero relative to a semigroup homomorphism $\varphi$ from $\mathbb{Z}^{\times}$ to the endomorphism semigroup of $\mathbb{Z}$. It is shown that this semidirect product is a normal extension of the semigroup $\mathbb{Z}\times \mathbb{N}$ by the residue class group modulo two, where $\mathbb{N}$ is the multiplicative semigroup of all natural numbers. We study the structures of the reduced semigroup $C^*$-algebras for the semigroups $\mathbb{Z}\rtimes_{\varphi}\mathbb{Z}^{\times}$ and $\mathbb{Z}\times \mathbb{N}$. We introduce a dynamical system for the semigroup $C^*$-algebra of the semigroup $\mathbb{Z}\times \mathbb{N}$ and its covariant representation. The semigroup $C^*$-algebra of the semigroup $\mathbb{Z}\rtimes_{\varphi}\mathbb{Z}^{\times}$ is represented as a crossed product of the $C^*$-algebra of the semigroup $\mathbb{Z}\times \mathbb{N}$ by the residue class group modulo two.