Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
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Turkish Journal of Analysis and Number Theory. 2017, 5(2), 31-56
DOI: 10.12691/tjant-5-2-2
Open AccessArticle

p-Adic Number Fields Acting On W*-Probability Spaces

Ilwoo Cho1,

1Department of Mathematics, 421 Ambrose Hall, Saint Ambrose University, 518 W. Locust St., Davenport, Iowa, 52803, U. S. A.

Pub. Date: February 15, 2017

Cite this paper:
Ilwoo Cho. p-Adic Number Fields Acting On W*-Probability Spaces. Turkish Journal of Analysis and Number Theory. 2017; 5(2):31-56. doi: 10.12691/tjant-5-2-2

Abstract

In this paper, we study how a p-adic number field acts on an arbitrarily fixed W*-algebra, and how it affects the original free-probabilistic information on the W*-algebra, for each prime p. In particular, by understanding the σ-algebra of as a semigroup equipped with the setintersection, we act on a unital tracial W*-probability space (M,tr), creating the corresponding semigroup W*-dynamical system. From such a dynamical system, construct the crossed product W*-algebra equipped with a suitable linear functional. We study free probability on such W*-dynamical operator-algebraic structures determined by primes, and those on corresponding free products of such structures over primes. As application, we study cases where given W*-probability spaces are generated by countable discrete groups.

Keywords:
Free Probability Free Probability Spaces p-Adic Number Fields Von Neumann Algebras W*-Dynamical Systems Crossed Product W*-Algebras.

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References:

[1]  I. Cho, Free Product C*-Algebras Induced by *-Algebras over p-Adic Number Fields for Primes p, (2016). Submitted to J. Numb. Theo. Anal. Acad.
 
[2]  I. Cho, Semicircular Elements Induced by p-Adic Number Fields, (2016). Submitted to Adv. Oper. Theo.
 
[3]  I. Cho, On Dynamical Systems Induced by p-Adic Number Fields, Opuscula Math., 35, no. 4, (2015). 445-484.
 
[4]  I. Cho, Free-Distributional Data of Arithmetic Functions and Corresponding Generating Functions Determined by Gaps of Primes, Compl. Anal. Oper. Theo., vol 8, issue 2, (2014). 537-570.
 
[5]  I. Cho, Classification of Arithmetic Functions and Corresponding Free-Moment L-Functions, Bull. Korean Math. Soc., 52, no. 3, (2015). 717-734.
 
[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., (2014).
 
[8]  I. Cho, and 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.
 
[9]  T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, (2010). PhD Thesis.
 
[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 W*-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 Mathematical 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).