Semicircular elements induced by p-adic number fields

Abstract

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 $j\in\mathbb{Z}$.