In this paper, we study how a p-adic number field acts on an arbitrarily fixed W^*-algebra, and how it affects the original free-probabilistic information on the W^*-algebra, for each prime p. In particular, by understanding the σ-algebra of as a semigroup equipped with the setintersection, we act on a unital tracial W^*-probability space (M,tr), creating the corresponding semigroup W^*-dynamical system. From such a dynamical system, construct the crossed product W^*-algebra equipped with a suitable linear functional. We study free probability on such W^*-dynamical operator-algebraic structures determined by primes, and those on corresponding free products of such structures over primes. As application, we study cases where given W^*-probability spaces are generated by countable discrete groups.

Free Probability, Free Probability Spaces, p-Adic Number Fields, Von Neumann Algebras, W^*-Dynamical Systems, Crossed Product W^*-Algebras.