Abstract
The paper deals with the reduced semigroup \(C^{*}\) -algebra for the semidirect product of semigroups \(S\) and \(P\) , where \(P\) acts on \(S\) by automorphisms. We represent this \(C^{*}\) -algebra as the reduced crossed product of the reduced semigroup \(C^{*}\) -algebra for \(S\) by the semigroup \(P\) which acts by automorphisms. The purpose of the paper is to demonstrate that the semicrossed product \(C^{*}\) -algebras and the semidirect products of semigroups are closely related. We show that the reduced semigroup \(C^{*}\) -algebra for a semidirect product \(S\rtimes_{\beta}^{a}P\) is isomorphic to the reduced semicrossed product \(C^{*}\) -algebra \(C^{*}_{r}(S)\rtimes_{\alpha,r}^{a}P\) . We apply this result to the study of the structure of the reduced semigroup \(C^{*}\) -algebra for the semidirect product \(\mathbb{Z}\rtimes\mathbb{Z}^{\times}\) of the additive group \(\mathbb{Z}\) of all integers and the multiplicative semigroup \(\mathbb{Z}^{\times}\) of integers without zero.
