In this paper, we study asymptotic semicircular laws induced both by arbitrarily fixed C * -probability spaces, and p-adic number fields {Q p } p∈P , as p → ∞ in the set P of all primes.In this paper, we study asymptotic-semicircular laws over "both" primes and unital C * -probability spaces. Since we generalize the asymptotic semicircularity of [25] up to C * -algebra-tensor, the patterns and results of this paper would be similar to those of [25], but generalize-or-universalize them. OverviewIn Section 2, fundamental concepts and backgrounds are introduced. In Sections 3-6, suitable free-probabilistic models are considered, where they contain p-adic number-theoretic information, for our purposes.In Section 7, we establish-and-study C * -probability spaces containing both analytic data from Q p , and free-probabilistic information of fixed unital C * -probability spaces. Then, our free-probabilistic structure LS A , a free product Banach * -probability space, is constructed, and the free probability on LS A is investigated in Section 8.

In this paper, we establish free-probabilistic models (H(Gp), ψp) on Hecke algebras H(Gp), and construct Hilbert-space representations of H (Gp), preserving free-probabilistic information from (H(Gp), ψp) , for primes p. From such free-probabilistic structures with representations, we study spectral properties of operators in C *-algebras generated by {H(Gp)} p:primes , via their free distributions.