Difference between Semicircular-like Laws Induced by p-Adic Number Fields and the Semicircular Law

Ilwoo Cho^1,

¹Department of Mathematics & Statistics, Saint Ambrose University, Davenport, Iowa, U. S. A.

Cite this paper:
Ilwoo Cho. Difference between Semicircular-like Laws Induced by p-Adic Number Fields and the Semicircular Law. Turkish Journal of Analysis and Number Theory. 2017; 5(5):165-190. doi: 10.12691/tjant-5-5-4

Abstract

In this paper, we study "semicircular-like" elements in free product Banach *-algebras induced by Haar-measurable functions over p-adic number fields , for primes p. And we investigate how the free distributions of operators generated by our mutually-free weighted-semicircular elements are close enough to (or far from) those of free reduced words generated by arbitrary mutually-free semicircular elements.

Keywords:
free probability p-Adic number fields Hilbert-space representations weighted-semicircular elements semicircular elements weight ratios

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References:

[1]	I. Cho, Representations and Corresponding Operators Induced by Hecke Algebras, Compl. Anal. Oper. Theo., 10, no. 7, (2016). 1453-1499.
[2]	I. Cho, Free W*-Dynamical Systems from p-Adic Number Fields and the Euler Totient Functions, Mathematics, 3, (2015). 1095-1138.
[3]	I. Cho, On Dynamical Systems Induced by p-Adic Number Fields, Opuscula Math., 35, no. 4, (2015). 445-484.
[4]	I. Cho, Free Semicircular Families in Free Product Banach *-Algebras Induced by p-Adic Number Fields over Prims p, Compl. Anal. Oper. Theo., 11, no. 3, (2017). 507-565.
[5]	I. Cho, Free Distributional Data of Arithmetic Functions and Corresponding Generating Functions, Compl. Anal. Oper. Theo., vol. 8, issue:2, (2014). 537-570.
[6]	I. Cho, Dynamical Systems on Arithmetic Functions Determined by Prims, Banach J. Math. Anal., 9, no. 1, (2015). 173-215.
[7]	I. Cho, and T. Gillespie, Free Probability on the Hecke Algebra, Compl. Anal.Oper. Theo., vol. 9, issue 7, (2015). 1491-1531.
[8]	I. Cho, and P. E. T. Jorgensen, Krein-Space Operators Induced by Dirichlet Char-acters, Special Issues: Contemp. Math.: Commutative and Noncommutative Harmonic Analysis and Applications, Amer. Math. Soc., (2014). 3-33.
[9]	T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, PhD Thesis. (2010).
[10]	T. Gillespie, Prime Number Theorems for Rankin-Selberg L-Functions over Number Fields, Sci. China Math., 54, no. 1, (2011). 35-46.
[11]	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.
[12]	R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Amer. Math. Soc. Mem., vol 132, no. 627, (1998).
[13]	V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Math-ematical Physics, Ser. Soviet & East European Math., vol 1, (1994). World Scientific.
[14]	D. Voiculescu, K. Dykemma, and A. Nica, Free Random Variables, CRM Monograph Series, vol 1., (1992). Published by Amer. Math. Soc..