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Turkish Journal of Analysis and Number Theory

**ISSN (Print):**
2333-1100

**ISSN (Online):**
2333-1232

**Frequency:**
bimonthly

**Editor-in-Chief:**
Mehmet Acikgoz, Feng Qi, Cenap ozel

**Website:**
http://www.sciepub.com/journal/TJANT

### Article

*p*-Adic Number Fields Acting On*W*^{*}-Probability Spaces^{1}Department of Mathematics, 421 Ambrose Hall, Saint Ambrose University, 518 W. Locust St., Davenport, Iowa, 52803, U. S. A.

*Turkish Journal of Analysis and Number Theory*. 2017, 5(2), 31-56

doi: 10.12691/tjant-5-2-2

Copyright © 2017 Science and Education Publishing

**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.

Correspondence to: Ilwoo Cho, Department of Mathematics, 421 Ambrose Hall, Saint Ambrose University, 518 W. Locust St., Davenport, Iowa, 52803, U. S. A.. Email: choilwoo@sau.edu

### Abstract

*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

^{*}-Dynamical Systems, Crossed Product W

^{*}-Algebras.

### 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. | ||

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### Article

**Fixed Point Theorems for Expansive Mappings in G-metric Spaces**

^{1}Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan

*Turkish Journal of Analysis and Number Theory*. 2017, 5(2), 57-62

doi: 10.12691/tjant-5-2-3

Copyright © 2017 Science and Education Publishing

**Cite this paper:**

Rahim Shah, Akbar Zada. Fixed Point Theorems for Expansive Mappings in G-metric Spaces.

*Turkish Journal of Analysis and Number Theory*. 2017; 5(2):57-62. doi: 10.12691/tjant-5-2-3.

Correspondence to: Rahim Shah, Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan. Email: safeer_rahim@yahoo.com, rahimshahstd@upesh.edu.pk

### Abstract

### Keywords

### References

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[18] | R. Shah, A. Zada and I. Khan, Some fixed point theorems of integral type contraction in cone b-metric spaces, Turkish. J. Ana. Num. Theor., vol. 3(6), 2105, 165-169. | ||

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[20] | K. P. R. Rao, L. Bhanu, Z. Mustafa, Fixed and related fixed point theorems for three maps in G-metric space, J. Adv. Stud. Topol., vol. 3(4), 2012, pp. 12-19. | ||

[21] | W. Shatanawi, Z. Mustafa, On coupled random fixed point results in partially ordered metric spaces, Mat. Vesn., vol. 64, 2012, pp. 139-146. | ||

[22] | A. Zada, R. Shah, T. Li, Integral Type Contraction and Coupled Coincidence Fixed Point Theorems for Two Pairs in G-metric Spaces, Hacet. J. Math. Stat., vol. 45(5), 2016, pp. 1475-1484. | ||

### Article

**Some New Integral Inequalities for -Times Differentiable - Convex Functions in the First Sense**

^{1}Department of Mathematics, Faculty of Sciences and Arts, Giresun University-Giresun-TÜRKİYE

^{2}Institute of Science, Ordu University-Ordu-TÜRKİYE

*Turkish Journal of Analysis and Number Theory*. 2017, 5(2), 63-68

doi: 10.12691/tjant-5-2-4

Copyright © 2017 Science and Education Publishing

**Cite this paper:**

Mahir Kadakal, Huriye Kadakal, İmdat İşcan. Some New Integral Inequalities for -Times Differentiable - Convex Functions in the First Sense.

*Turkish Journal of Analysis and Number Theory*. 2017; 5(2):63-68. doi: 10.12691/tjant-5-2-4.

Correspondence to: Mahir Kadakal, Department of Mathematics, Faculty of Sciences and Arts, Giresun University-Giresun-TÜRKİYE. Email: mahirkadakal@gmail.com

### Abstract

*n*-time differentiable -convex functions in the first sense.

### Keywords

### References

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[20] | S. Maden, H. Kadakal, M. Kadakal and İ. İşcan, “Some new integral inequalities for n-times differentiable convex and concave functions”. https://www.researchgate.net/publication/312529563, (Submitted) | ||

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