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.
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This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program (‘‘PRIORITY-2030’’).
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Lipacheva, E.V. A Semigroup \(\boldsymbol{C}^{\mathbf{*}}\)-algebra for a Semidirect Product of Semigroups. Lobachevskii J Math 45, 452–461 (2024). https://doi.org/10.1134/S1995080224010347
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DOI: https://doi.org/10.1134/S1995080224010347