Free W*-Dynamical Systems From p-Adic Number Fields and the Euler Totient Function

Abstract

In this paper, we study relations between free probability on crossed product W * -algebras with a von Neumann algebra over p-adic number fields ℚp (for primes p), and free probability on the subalgebra Φ, generated by the Euler totient function ϕ, of the arithmetic algebra A , consisting of all arithmetic functions. In particular, we apply such free probability to consider operator-theoretic and operator-algebraic properties of W * -dynamical systems induced by ℚp under free-probabilistic (and hence, spectral-theoretic) techniques.

Keywords

• p-Adic number fields ℚp
• p-Adic von neumann algebras ℳp
•  dynamical systems induced by ℚp
•  arithmetic functions
•  the arithmetic algebra A
•  the euler totient function ϕ