The study of $C^*$-algebras arising from number-theoretic data has its roots in the seminal construction of the Bost-Connes system, a landmark bridge between quantum statistical mechanics and the distribution of prime numbers. This program was later formalized and expanded by Cuntz, Li, Marcolli, Consani, Exel, among others, who developed a general framework to associate universal operator algebras with rings and integral domains through the representation of affine semigroups. In this talk, we investigate the $C^*$-algebraic structures associated with a specific category of commutative associative semifields. By relaxing the classical requirement for additive inverses, we uncover a novel class of algebras. We prove that these constructions naturally recover and generalize the classical Toeplitz algebra $\mathcal{T}$, identifying the ideal of compact operators as a fundamental geometric residue of the arithmetic space. Finally, we compute the $K$-theory of the associated $C^*$-algebras.

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Bost-Connes system,semifield,Toeplitz algebra,$K$-theory.