Opuscula Math. 37, no. 5 (2017), 665-703
http://dx.doi.org/10.7494/OpMath.2017.37.5.665

Opuscula Mathematica

Semicircular elements induced by p-adic number fields

Ilwoo Cho
Palle E. T. Jorgensen

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.

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• Ilwoo Cho
• St. Ambrose University, Department of Mathematics and Statistics, 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, USA
• Palle E. T. Jorgensen
• The University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa City, IA 52242-1419, USA
• Communicated by P.A. Cojuhari.
• Accepted: 2016-12-04.
• Published online: 2017-07-05.