Opuscula Math. 37, no. 5 (2017), 665-703

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
  • Received: 2016-10-10.
  • Accepted: 2016-12-04.
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Cite this article as:
Ilwoo Cho, Palle E. T. Jorgensen, Semicircular elements induced by p-adic number fields, Opuscula Math. 37, no. 5 (2017), 665-703, http://dx.doi.org/10.7494/OpMath.2017.37.5.665

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