Semicircular Elements Induced by p-Adic Number Fields ℚp

In this chapter, we introduce the frontier motivation for this monograph. Not only in the modern number theory, but also in quantum physics, the p-acdic numbers Q_p , where p is a prime, play important roles, especially, they provide non-Eucledean geometric, analytic, and topological models. Starting from naturally determined mutually orthogonal projections on Q_p , we construct weighted-semicircular elements preserving number-theoretic data, as Banach-space operators. By wisely handling such number-theoretic data, expressed by the weights of those weighted-semicircular elements, the corresponding semicircular elements are constructed. It shows that the mutually-orthogonal projections on Q_p (which are Hilbert-space operators) induce semicircular elements (as Banach-space operators).