Opuscula Math. 38, no. 4 (2018), 501-535

Opuscula Mathematica

Banach *-algebras generated by semicircular elements induced by certain orthogonal projections

Ilwoo Cho
Palle E. T. Jorgensen

Abstract. The main purpose of this paper is to study structure theorems of Banach \(*\)-algebras generated by semicircular elements. In particular, we are interested in the cases where given semicircular elements are induced by orthogonal projections in a \(C^{*}\)-probability space.

Keywords: free probability, orthogonal projections, weighted-semicircular elements, semicircular elements.

Mathematics Subject Classification: 46L10, 46L54, 47L15, 47L30, 47L55.

Full text (pdf)

  1. M. Brannan, K. Kirkpatrick, Quantum groups and generalized circular elements, Pacific J. Math. 282 (2016) 1, 35-61.
  2. I. Cho, On dynamical systems induced by \(p\)-adic number fields, Opuscula Math. 35 (2015) 4, 445-484.
  3. I. Cho, \(p\)-adic free stochastic integrals for \(p\)-adic weighted-semicircular motions determined by primes \(p\), Libertas Math. (New Series) 36 (2016) 2, 65-110.
  4. I. Cho, Free semicircular families in free product Banach \(\ast\)-algebras induced by \(p\)-adic number fields, Complex Anal. Oper. Theory 11 (2017) 3, 507-565.
  5. I. Cho, P.E.T. Jorgensen, Semicircular elements induced by \(p\)-adic number fields, Opuscula Math. 37 (2017) 5, 665-703.
  6. A. Deya, R. Schott, On the rough-paths approach to noncommutative stochastic calculus, J. Funct. Anal. 265 (2013) 4, 594-628.
  7. T. Gillespie, Prime number theorems for Rankin-Selberg \(L\)-functions over number fields, Sci. China Math. 54 (2011) 1, 35-46.
  8. I. Gomez, M. Caetagnino, A quantum version of spectral decomposition theorem of dynamical systems, Quantum Chaos Hierarchy: Ergodic, Mixing and Exact, Chaos Solitons Fractals 70 (2015), 99-116.
  9. N.A. Ivanov, T. Omland, \(C^{*}\)-simplicity of free products with amalgamation and radical classes of groups, J. Funct. Anal. 272 (2017) 9, 3712-3741.
  10. Y. Liu, Y. Yu, On an extension of Stevic-Sharma operator from the general space to weighted-type spaces on the unit ball, Complex Anal. Oper. Theory 11 (2017) 2, 261-288.
  11. M.A. Navarro, S. Villegas, Semi-stable radial solutions of \(p\)-Laplace equations in \(\mathbb{R}^N\), Nonlinear Anal. 149 (2017), 111-116.
  12. A. Nica, Free probability aspect of irreducible meandric systems, and some related observations about meanders, Infinite Dimes. Anal. Quantum Probab. Relat. Top. 19 (2016) 2, 1650011, 22 pp.
  13. 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.
  14. R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Amer. Math. Soc. Mem. 132 (1998) 627.
  15. 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.
  16. D. Voiculescu, K. Dykemma, A. Nica, Free random variables, CRM Monograph Series, vol. 1, Amer. Math. Soc., Providence, 1992.
  17. Y. Yin, Z. Bai, J. Hu, On the semicircular law of large-dimensional random quaternion matrices, J. Theor. Probab. 29 (2016) 3, 1100-1120.
  • Ilwoo Cho
  • Saint 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, Iowa City, IA 52242-1419, USA
  • Communicated by P.A. Cojuhari.
  • Received: 2017-06-12.
  • Accepted: 2017-08-21.
  • Published online: 2018-04-11.
Opuscula Mathematica - cover

Cite this article as:
Ilwoo Cho, Palle E. T. Jorgensen, Banach *-algebras generated by semicircular elements induced by certain orthogonal projections, Opuscula Math. 38, no. 4 (2018), 501-535, https://doi.org/10.7494/OpMath.2018.38.4.501

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.