Opuscula Math. 35, no. 4 (2015), 445-484

Opuscula Mathematica

On dynamical systems induced by p-adic number fields

Ilwoo Cho

Abstract. In this paper, we construct dynamical systems induced by \(p\)-adic number fields \(\mathbb{Q}_{p}\). We study the corresponding crossed product operator algebras induced by such dynamical systems. In particular, we are interested in structure theorems, and free distributional data of elements in the operator algebras.

Keywords: prime fields, \(p\)-adic number fields, the Adele ring, \(p\)-adic von Neumann algebras, \(p\)-adic dynamical systems.

Mathematics Subject Classification: 05E15, 11R47, 46L54, 47L15, 47L55.

Full text (pdf)

  1. W. Arveson, Four Lectures on Noncommutative Dynamics, arXiv:math.OA/0207278v1, (2002), Preprint.
  2. W. Arveson, Noncommutative Dynamics and \(E\)-Semigroups, Springer Monographs in Math., Springer, 2003.
  3. D. Bump, Automorphic Forms and Representations, Cambridge Studies in Adv. Math., vol. 55, Cambridge Univ. Press, 1996.
  4. I. Cho, Operators induced by prime numbers, Methods Appl. Math. 19 (2013) 4, 313-340.
  5. I. Cho, \(p\)-adic Banach space operators and adelic Banach space operators, Opuscula Math. 34 (2014) 1, 29-65.
  6. I. Cho, Classification on arithmetic functions and corresponding free-moment \(L\)-functions, Bulletin Korean Math. Soc., to appear.
  7. I. Cho, Free distributional data of arithmetic functions and corresponding generating functions, Complex Anal. Oper. Theory, DOI: 10.1007/s11785-013-0311-9, (2013). http://dx.doi.org/10.1007/s11785-013-0311-9
  8. I. Cho, Histories distorted by partial isometries, J. Phy. Math. 3 (2011), article ID: P110301.
  9. I. Cho, Frames, fractals and radial operators in Hilbert space, J. Math. Sci.: Adv. Appl. 5 (2010) 2, 333-393.
  10. I. Cho, Direct producted \(W^{*}\)-probability spaces and corresponding amalgamated free stochastic integration, Bull. Korean Math. Soc. 44 (2007) 1, 131-150.
  11. I. Cho, T. Gillespie, Arithmetic functions and corresponding free probability determined by primes, submitted.
  12. I. Cho, P.E.T. Jorgensen, Krein-space operators induced by Dirichlet characters, Commutative and Noncommutative Harmonic Analysis and Applications, Contemp. Math. Amer. Math. Soc. 603 (2013), 3-34.
  13. I. Cho, P.E.T. Jorgensen, Krein-space representations of arithmetic functions determined by primes, submitted.
  14. T. Gillespie, Superposition of Zeroes of Automorphic \(L\)-functions and Functoriality, Univ. of Iowa, PhD Thesis, 2010.
  15. T. Gillespie, Prime number theorems for Rankin-Selberg \(L\)-functions over number fields, Sci. China Math. 54 (2011) 1, 35-46.
  16. 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.
  17. R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Amer. Math. Soc. Mem. 132, no. 627, (1998).
  18. 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.
  19. D. Voiculescu, K. Dykemma, A. Nica, Free Random Variables, CRM Monograph Series, vol. 1, 1992.
  • Ilwoo Cho
  • St. Ambrose University, Department of Mathematics, 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, USA
  • Communicated by P.A. Cojuhari.
  • Received: 2013-11-15.
  • Revised: 2014-10-29.
  • Accepted: 2014-11-07.
  • Published online: 2015-02-06.
Opuscula Mathematica - cover

Cite this article as:
Ilwoo Cho, On dynamical systems induced by p-adic number fields, Opuscula Math. 35, no. 4 (2015), 445-484, http://dx.doi.org/10.7494/OpMath.2015.35.4.445

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.