Semicircular elements induced by p-adic number fields

In this paper, we study semicircular-like elements, and semicircular elements induced by \(p\)-adic analysis, for each prime \(p\). Starting from a \(p\)-adic number field \(\mathbb{Q}_{p}\), we construct a Banach \(*\)-algebra \(\mathfrak{LS}_{p}\), for a fixed prime \(p\), and show the generating elements \(Q_{p,j}\) of \(\mathfrak{LS}_{p}\) form weighted-semicircular elements, and the corresponding scalar-multiples \(\Theta_{p,j}\) of \(Q_{p,j}\) become semicircular elements, for all \(j\in\mathbb{Z}\). The main result of this paper is the very construction of suitable linear functionals \(\tau_{p,j}^{0}\) on \(\mathfrak{LS}_{p}\), making \(Q_{p,j}\) be weighted-semicircular, for all \(j\in\mathbb{Z}\)