Opuscula Math. 37, no. 5 (), 665-703
http://dx.doi.org/10.7494/OpMath.2017.37.5.665
Opuscula Mathematica

# 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 $$j\in\mathbb{Z}$$.
Keywords: free probability, primes, $$p$$-adic number fields $$\mathbb{Q}_{p}$$, Hilbert-space representations, $$C^{*}$$-algebras, wighted-semicircular elements, semicircular elements.
Mathematics Subject Classification: 05E15, 11R47, 11R56, 46L10, 46L40, 47L15, 47L30, 47L55.