June  2020, 28(2): 739-776. doi: 10.3934/era.2020038

Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$

421 Ambrose Hall, Saint Ambrose Univ., Dept. of Math. & Stat., 518 W. Locust St., Davenport, Iowa, 52803, USA

Received  February 2020 Revised  March 2020 Published  May 2020

In this paper, we study the Banach $ * $-probability space $ (A\otimes_{\Bbb{C}}\Bbb{LS}, $ $ \tau_{A}^{0}) $ generated by a fixed unital $ C^{*} $-probability space $ (A, $ $ \varphi_{A}), $ and the semicircular elements $ \Theta_{p,j} $ induced by $ p $-adic number fields $ \Bbb{Q}_{p}, $ for all $ p $ $ \in $ $ \mathcal{P}, $ $ j $ $ \in $ $ \Bbb{Z}, $ where $ \mathcal{P} $ is the set of all primes, and $ \Bbb{Z} $ is the set of all integers. In particular, from the order-preserving shifts $ g\times h_{\pm } $ on $ \mathcal{P} $ $ \times $ $ \Bbb{Z}, $ and $ * $-homomorphisms $ \theta $ on $ A, $ we define the corresponding $ * $-homomorphisms $ \sigma_{(\pm ,1)}^{1:\theta } $ on $ A\otimes_{\Bbb{C}}\Bbb{LS}, $ and consider free-distributional data affected by them.

Citation: Ilwoo Cho. Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$. Electronic Research Archive, 2020, 28 (2) : 739-776. doi: 10.3934/era.2020038
References:
[1]

S. AlbeverioP. E. T. Jorgensen and A. M. Paolucci, On fractional Brownian motion and wavelets, Compl. Anal. Oper. Theo., 6 (2012), 33-63.  doi: 10.1007/s11785-010-0077-2.  Google Scholar

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D. Alpay and P. E. T. Jorgensen, Spectral theory for Gaussian processes: Reproducing kernels, Random Functions and Operator Theory, 83 (2015), 211-229.   Google Scholar

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D. Alpay and P. Jorgensen, Spectral theory for Gaussian processes: Reproducing kernels, boundaries, & $L_{2}$ -wavelet generators with fractional scales, Numb. Funct. Anal, Optim., 36 (2015), 1239-1285.  doi: 10.1080/01630563.2015.1062777.  Google Scholar

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D. AlpayP. Jorgensen and G. Salomon, On free stochastic processes and their derivatives, Stochastic Process. Appl., 124 (2014), 3392-3411.  doi: 10.1016/j.spa.2014.05.007.  Google Scholar

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I. Cho, Free semicircular families in free product Banach $\ast$-algebras induced by $p$-adic number fields over primes $p$, Compl. Anal. Oper. Theo., 11 (2017), 507-565.  doi: 10.1007/s11785-016-0625-5.  Google Scholar

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I. Cho, Adelic analysis and functional analysis on the finite adele ring, Opuscula Math., 38 (2017), 139-185.  doi: 10.7494/OpMath.2018.38.2.139.  Google Scholar

[10]

I. Cho, Semicircular-like and semicircular laws on Banach $*$-probability spaces induced by dynamical systems of the finite adele ring, Adv. Oper. Theo., 4 (2019), 24-70.  doi: 10.15352/aot.1802-1317.  Google Scholar

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I. Cho, $p$-adic free stochastic integrals for $p$-adic weighted-semicircular motions determined by primes $p$, Libertas Math. (New S.), 36 (2016), 65-110.   Google Scholar

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I. Cho and P. E. T. Jorgensen, Semicircular elements induced by $p$-adic number fields, Opuscula Math., 37 (2017), 665-703.  doi: 10.7494/OpMath.2017.37.5.665.  Google Scholar

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J.-B. Bost and A. Connes, Hecke algebras, type $III$-factors, and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.), 1 (1995), 411-457.  doi: 10.1007/BF01589495.  Google Scholar

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A. Connes, Trace formula in noncommutative Geometry and the zeroes of the Riemann zeta functions, Available at: http://www.alainconnes.org/en/download.php. Google Scholar

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B. DragovichYa. Radyno and A. Khennikov, Distributions on adéles, J. Math. Sci., 142 (2007), 2105-2112.  doi: 10.1007/s10958-007-0120-7.  Google Scholar

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B. DragovichA. Khennikov and D. Mihajiović, Linear fractional $p$-adic and adelic dynamical systems, Rep. Math. Phy., 60 (2007), 55-68.  doi: 10.1016/S0034-4877(07)80098-X.  Google Scholar

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T. L. Gillespie, Superposition of Zeroes of Automorphic $L$-Functions and Functoriality, Thesis (Ph.D.)–The University of Iowa, 2011, 75 pp.  Google Scholar

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T. Gillespie and G. H. Ji, Prime number theorems for Rankin-Selberg $L$-functions over number fields, Sci. China Math., 54 (2011), 35-46.  doi: 10.1007/s11425-010-4137-x.  Google Scholar

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U. Haagerup and F. Larsen, Brown's spectrial distribution measure for $R$-diagonal elements in finite von Neumann algebras, J. Funct. Anal., 176 (2000), 331-367.  doi: 10.1006/jfan.2000.3610.  Google Scholar

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P. E. T. Jorgensen and A. M. Paolucci, Markov measures and extended zeta functions, J. Appl. Math. Comput., 38 (2012), 305-323.  doi: 10.1007/s12190-011-0480-5.  Google Scholar

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P. E. T. Jorgensen and A. M. Paolucci, States on the Cuntz algebras and $p$-adic random walks, J. Aust. Math. Soc., 90 (2011), 197-211.  doi: 10.1017/S1446788711001212.  Google Scholar

[23]

I. KaygorodovI. Shestakov and U. Umirbaev, Free generic Poisson fields and algebras, Comm. Alg., 46 (2018), 1799-1812.  doi: 10.1080/00927872.2017.1358269.  Google Scholar

[24]

T. Kemp and R. Speicher, Strong Haagerup inequalities for free $R$-diagonal elements, J. Funct. Anal., 251 (2007), 141-173.  doi: 10.1016/j.jfa.2007.03.011.  Google Scholar

[25]

L. Makar-Limanov and I. Shestakov, Polynomials and Poisson dependence in free Poisson algebras and free Poisson fields, J. Alg., 349 (2012), 372-379.  doi: 10.1016/j.jalgebra.2011.08.008.  Google Scholar

[26]

A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, 335. Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511735127.  Google Scholar

[27]

F. Rǎdulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group of nonsingular index, Invent. Math., 115 (1994), 347-389.  doi: 10.1007/BF01231764.  Google Scholar

[28]

F. Radulescu, Free group factors and Hecke operators, The Varied Landscape of Operator Theory, Theta Ser. Adv. Math., Theta, Bucharest, 17 (2014), 241-257.   Google Scholar

[29]

R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc., 132 (1998). doi: 10.1090/memo/0627.  Google Scholar

[30]

R. Speicher, A conceptual proof of a basic result in the combinatorial approach to freeness, Infinit. Dimention. Anal. Quant. Prob. Relat. Topics, 3 (2000), 213-222.  doi: 10.1142/S0219025700000157.  Google Scholar

[31]

V. S. Vladimirov and I. V. Volovich, $p$-adic quantum mechanics, Comm. Math. Phy., 123 (1989), 659-676.  doi: 10.1007/BF01218590.  Google Scholar

[32]

V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet and East European Mathematics, 1. World Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/1581.  Google Scholar

[33]

D.-V. Voiculescu, Aspects of free analysis, Jpn. J. Math., 3 (2008), 163-183.  doi: 10.1007/s11537-008-0753-4.  Google Scholar

[34]

D. Voiculescu, Free probability and the von Neumann algebras of free groups, Rep. Math. Phy., 55 (2005), 127-133.  doi: 10.1016/S0034-4877(05)80008-4.  Google Scholar

[35]

D. V. Voiculescu, K. J. Dykemma and A. Nica, Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups, CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992.  Google Scholar

show all references

References:
[1]

S. AlbeverioP. E. T. Jorgensen and A. M. Paolucci, On fractional Brownian motion and wavelets, Compl. Anal. Oper. Theo., 6 (2012), 33-63.  doi: 10.1007/s11785-010-0077-2.  Google Scholar

[2]

S. Albeverio, P. E. T. Jorgensen and A. M. Paolucci, Multiresolution wavelet analysis of integer scale Bessel functions, J. Math. Phy., 48 (2007), 073516, 24 pp. doi: 10.1063/1.2750291.  Google Scholar

[3]

D. AlpayP. E. T. Jorgensen and D. Levanony, On the equivalence of probability spaces, J. Theo. Prob., 30 (2017), 813-841.  doi: 10.1007/s10959-016-0667-7.  Google Scholar

[4]

D. Alpay, P. E. T. Jorgensen and D. P. Kimsey, Moment problems in an infinite number of variables, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 18 (2015), 1550024, 14 pp. doi: 10.1142/S0219025715500241.  Google Scholar

[5]

D. Alpay and P. E. T. Jorgensen, Spectral theory for Gaussian processes: Reproducing kernels, Random Functions and Operator Theory, 83 (2015), 211-229.   Google Scholar

[6]

D. Alpay and P. Jorgensen, Spectral theory for Gaussian processes: Reproducing kernels, boundaries, & $L_{2}$ -wavelet generators with fractional scales, Numb. Funct. Anal, Optim., 36 (2015), 1239-1285.  doi: 10.1080/01630563.2015.1062777.  Google Scholar

[7]

D. AlpayP. Jorgensen and G. Salomon, On free stochastic processes and their derivatives, Stochastic Process. Appl., 124 (2014), 3392-3411.  doi: 10.1016/j.spa.2014.05.007.  Google Scholar

[8]

I. Cho, Free semicircular families in free product Banach $\ast$-algebras induced by $p$-adic number fields over primes $p$, Compl. Anal. Oper. Theo., 11 (2017), 507-565.  doi: 10.1007/s11785-016-0625-5.  Google Scholar

[9]

I. Cho, Adelic analysis and functional analysis on the finite adele ring, Opuscula Math., 38 (2017), 139-185.  doi: 10.7494/OpMath.2018.38.2.139.  Google Scholar

[10]

I. Cho, Semicircular-like and semicircular laws on Banach $*$-probability spaces induced by dynamical systems of the finite adele ring, Adv. Oper. Theo., 4 (2019), 24-70.  doi: 10.15352/aot.1802-1317.  Google Scholar

[11]

I. Cho, $p$-adic free stochastic integrals for $p$-adic weighted-semicircular motions determined by primes $p$, Libertas Math. (New S.), 36 (2016), 65-110.   Google Scholar

[12]

I. Cho and P. E. T. Jorgensen, Semicircular elements induced by $p$-adic number fields, Opuscula Math., 37 (2017), 665-703.  doi: 10.7494/OpMath.2017.37.5.665.  Google Scholar

[13] A. Connes, Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994.   Google Scholar
[14]

J.-B. Bost and A. Connes, Hecke algebras, type $III$-factors, and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.), 1 (1995), 411-457.  doi: 10.1007/BF01589495.  Google Scholar

[15]

A. Connes, Trace formula in noncommutative Geometry and the zeroes of the Riemann zeta functions, Available at: http://www.alainconnes.org/en/download.php. Google Scholar

[16]

B. DragovichYa. Radyno and A. Khennikov, Distributions on adéles, J. Math. Sci., 142 (2007), 2105-2112.  doi: 10.1007/s10958-007-0120-7.  Google Scholar

[17]

B. DragovichA. Khennikov and D. Mihajiović, Linear fractional $p$-adic and adelic dynamical systems, Rep. Math. Phy., 60 (2007), 55-68.  doi: 10.1016/S0034-4877(07)80098-X.  Google Scholar

[18]

T. L. Gillespie, Superposition of Zeroes of Automorphic $L$-Functions and Functoriality, Thesis (Ph.D.)–The University of Iowa, 2011, 75 pp.  Google Scholar

[19]

T. Gillespie and G. H. Ji, Prime number theorems for Rankin-Selberg $L$-functions over number fields, Sci. China Math., 54 (2011), 35-46.  doi: 10.1007/s11425-010-4137-x.  Google Scholar

[20]

U. Haagerup and F. Larsen, Brown's spectrial distribution measure for $R$-diagonal elements in finite von Neumann algebras, J. Funct. Anal., 176 (2000), 331-367.  doi: 10.1006/jfan.2000.3610.  Google Scholar

[21]

P. E. T. Jorgensen and A. M. Paolucci, Markov measures and extended zeta functions, J. Appl. Math. Comput., 38 (2012), 305-323.  doi: 10.1007/s12190-011-0480-5.  Google Scholar

[22]

P. E. T. Jorgensen and A. M. Paolucci, States on the Cuntz algebras and $p$-adic random walks, J. Aust. Math. Soc., 90 (2011), 197-211.  doi: 10.1017/S1446788711001212.  Google Scholar

[23]

I. KaygorodovI. Shestakov and U. Umirbaev, Free generic Poisson fields and algebras, Comm. Alg., 46 (2018), 1799-1812.  doi: 10.1080/00927872.2017.1358269.  Google Scholar

[24]

T. Kemp and R. Speicher, Strong Haagerup inequalities for free $R$-diagonal elements, J. Funct. Anal., 251 (2007), 141-173.  doi: 10.1016/j.jfa.2007.03.011.  Google Scholar

[25]

L. Makar-Limanov and I. Shestakov, Polynomials and Poisson dependence in free Poisson algebras and free Poisson fields, J. Alg., 349 (2012), 372-379.  doi: 10.1016/j.jalgebra.2011.08.008.  Google Scholar

[26]

A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, 335. Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511735127.  Google Scholar

[27]

F. Rǎdulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group of nonsingular index, Invent. Math., 115 (1994), 347-389.  doi: 10.1007/BF01231764.  Google Scholar

[28]

F. Radulescu, Free group factors and Hecke operators, The Varied Landscape of Operator Theory, Theta Ser. Adv. Math., Theta, Bucharest, 17 (2014), 241-257.   Google Scholar

[29]

R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc., 132 (1998). doi: 10.1090/memo/0627.  Google Scholar

[30]

R. Speicher, A conceptual proof of a basic result in the combinatorial approach to freeness, Infinit. Dimention. Anal. Quant. Prob. Relat. Topics, 3 (2000), 213-222.  doi: 10.1142/S0219025700000157.  Google Scholar

[31]

V. S. Vladimirov and I. V. Volovich, $p$-adic quantum mechanics, Comm. Math. Phy., 123 (1989), 659-676.  doi: 10.1007/BF01218590.  Google Scholar

[32]

V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet and East European Mathematics, 1. World Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/1581.  Google Scholar

[33]

D.-V. Voiculescu, Aspects of free analysis, Jpn. J. Math., 3 (2008), 163-183.  doi: 10.1007/s11537-008-0753-4.  Google Scholar

[34]

D. Voiculescu, Free probability and the von Neumann algebras of free groups, Rep. Math. Phy., 55 (2005), 127-133.  doi: 10.1016/S0034-4877(05)80008-4.  Google Scholar

[35]

D. V. Voiculescu, K. J. Dykemma and A. Nica, Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups, CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992.  Google Scholar

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