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.

Full text (pdf)

  1. S. Albeverio, P.E.T. Jorgensen, A.M. Paolucci, Multiresolution wavelet analysis of integer scale Bessel functions, J. Math. Phys. 48 (2007) 7, 073516, 24.
  2. S. Albeverio, P.E.T. Jorgensen, A.M. Paolucci, On fractional Brownian motion and wavelets, Complex Anal. Oper. Theory 6 (2012) 1, 33-63.
  3. D. Alpay, P.E.T. Jorgensen, Spectral theory for Gaussian processes: reproducing kernels, boundaries, and \(L_2\)-wavelet generators with fractional scales, Numer. Funct. Anal. Optim. 36 (2015) 10, 1239-1285.
  4. D. Alpay, P.E.T. Jorgensen, D. Kimsey, Moment problems in an infinite number of variables, Infin. Dimens. Anal. Quantum Probab. Relat. Top. Prob. 18 (2015) 4, 1550024.
  5. D. Alpay, P.E.T. Jorgensen, D. Levanony, On the equivalence of probability spaces, J. Theo. Prob. (2016), to appear.
  6. D. Alpay, P.E.T. Jorgensen, G. Salomon, On free stochastic processes and their derivatives, Stochastic Process. Appl. 124 (2014) 10, 3392-3411.
  7. I. Cho, Free distributional data of arithmetic functions and corresponding generating functions, Complex Anal. Oper. Theory 8 (2014) 2, 537-570.
  8. I. Cho, Dynamical systems on arithmetic functions determined by prims, Banach J. Math. Anal. 9 (2015) 1, 173-215.
  9. I. Cho, Free product \(C^*\)-algebras induced by \(*\)-algebras over \(p\)-adic number fields (2016), submitted.
  10. I. Cho, T. Gillespie, Free probability on the Hecke algebra, Complex Anal. Oper. Theory 9 (2015), 1491-1531.
  11. I. Cho, P.E.T. Jorgensen, Krein-Space Operators Induced by Dirichlet Characters, Special Issues: Contemp. Math.: Commutative and Noncommutative Harmonic Analysis and Applications, Amer. Math. Soc. (2014), 3-33.
  12. A. Connes, Noncommutative Geometry, Academic Press, San Diego, CA, 1994.
  13. A. Connes, Hecke algebras, type III-factors, and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (New Series) 1 (1995) 3, 411-457.
  14. A. Connes, Trace formula in noncommutative geometry and the zeroes of the Riemann zeta functions, arXiv:math/9811068 [math.NT] (1998).
  15. T. Gillespie, Superposition of zeroes of automorphic \(L\)-functions and functoriality, University of Iowa, PhD Thesis (2010).
  16. T. Gillespie, Prime number theorems for Rankin-Selberg \(L\)-functions over number fields, Sci. China Math. 54 (2011) 1, 35-46.
  17. P.E.T. Jorgensen, Operators and Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics, 2nd ed., Dover Publications, 2008.
  18. P.E.T. Jorgensen, A.M. Paolucci, Wavelets in mathematical physics: \(q\)-oscillators, J. Phys. A. 36 (2003) 23, 6483-6494.
  19. P.E.T. Jorgensen, A.M. Paolucci, States on the Cuntz algebras and \(p\)-adic random walks, J. Aust. Math. Soc. 90 (2011) 2, 197-211.
  20. P.E.T. Jorgensen, A.M. Paolucci, \(q\)-frames and Bessel functions, Numer. Funct. Anal. Optim. 33 (2012) 7-9, 1063-1069.
  21. P.E.T. Jorgensen, A.M. Paolucci, Markov measures and extended zeta functions, J. Appl. Math. Comput. 38 (2012) 1-2, 305-323.
  22. F. Radulescu, Random matrices, amalgamated free products and subfactors of the \(C^*\)-algebra of a free group of nonsingular index, Invent. Math. 115 (1994), 347-389.
  23. F. Radulescu, Conditional expectations, traces, angles between spaces and representations of the Hecke algebras, Lib. Math. 33 (2013) 2, 65-95.
  24. F. Radulescu, Free group factors and Hecke operators, notes taken by N. Ozawa, Proceedings of the 24th Conference in Operator Theory, Theta Advanced Series in Math., Theta Foundation, 2014.
  25. R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994), 611-628.
  26. R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Amer. Math. Soc. Mem. 132 (1998) 627.
  27. R. Speicher, A conceptual proof of a basic result in the combinatorial approach to freeness, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 213-222.
  28. R. Speicher, P. Neu, Physical Applications of Freeness, XII-th International Congress of Math. Phy. (ICMP '97), International Press, 1999, 261-266.
  29. V.S. Vladimirov, \(p\)-adic quantum mechanics, Comm. Math. Phys. 123 (1989) 4, 659-676.
  30. V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics, Ser. Soviet & East European Math., vol. 1, World Scientific, 1994.
  31. D. Voiculescu, Free probability and the von Neumann algebras of free groups, Rep. Math. Phys. 55 (2005) 1, 127-133.
  32. D. Voiculescu, Symmetries arising from free probability theory, Frontiers in Number Theory, Physics and Geometry (2006), 231-243.
  33. D. Voiculescu, Aspects of free analysis, Jpn. J. Math. 3 (2008) 2, 163-183.
  34. D. Voiculescu, K. Dykemma, A. Nica, Free Random Variables, CRM Monograph Series, vol. 1, 1992.
  • 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.
  • Received: 2016-10-10.
  • Accepted: 2016-12-04.
  • Published online: 2017-07-05.
Opuscula Mathematica - cover

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

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.