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Published:
May 2, 2026DOI:
10.4314/mejs.v18i1.4Keywords:
Article Details
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Main Article Content
Operator Algebras Associated with the Multiplicative Semigroup of Natural Numbers and Integer Groups Z
Kibrom Gebrehiwot Gebremeskel
Abstract
This paper investigates operator algebras arising from the left regular representations of the multiplicative semigroup of natural numbers and the integer group. It is shown that the Banach algebra generated by it contains neither non-trivial projections nor compact operators. Additionally, it is proven that the -algebra, generated by a family of bounded operators in, does not contain any non-zero compact operators. The study further examines the von Neumann algebras generated by in. Moreover, it is shown that the, where is the C∗-algebra generated by in and represents the space of continuous functions on the unit circle. The results connect naturally with number theory through arithmetic semigroup actions and with quantum mechanics as models for noncommutative observables and symmetries. These links suggest that the structural rigidity established in this work may have implications for both arithmetic and quantum spectral analysis.
