Krein-Space Representations of Arithmetic Functions Determined by Primes

In this paper, we study representations of the algebra \(\mathcal {A}\) generated by all arithmetic functions, determined by fixed primes (or prime numbers). The main purposes of this paper are (i) to establish nice representational models of \(\mathcal {A}\) under primes, (ii) to study fundamental properties of such representations, (iii) to investigate how \( \mathcal {A}\) is acting as operators in representations, and (iv) to consider new free probability models on Krein-space operator algebras.