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December  2019, 12(8): 2221-2252. doi: 10.3934/dcdss.2019143

Free probability on $ C^{*}$-algebras induced by hecke algebras over primes

Ilwoo Cho 1,, and  Palle Jorgense 2,

1. 

St. Ambrose Univ., Dept. of Math. & Stat., 421 Ambrose Hall, 518 W. Locust St., Davenport, IA 52803, USA
2. 

Univ. of Iowa, Dept. of Math., 14 McLean Hall, Iowa City, IA 52242, USA

* Corresponding author: Ilwoo Cho

Received  August 2016 Revised  March 2017 Published  January 2019

In this paper, we establish free-probabilistic models $ \left( \mathcal{H}(G_{p}),\text{ }\psi _{p}\right)$ on Hecke algebras $ \mathcal{H}(G_{p})$, and construct Hilbert-space representations of $ \mathcal{H} (G_{p}),$ preserving free-probabilistic information from $ \left( \mathcal{H}(G_{p}),\text{ }\psi _{p}\right) ,$ for primes $ p.$ From such free-probabilistic structures with representations, we study spectral properties of operators in $ C^{*}$-algebras generated by $ \left\{ \mathcal{H}(G_{p})\right\}_{p:primes}$, via their free distributions.

Keywords: Free probability, free product, primes, Hecke algebras, representations.
Mathematics Subject Classification: Primary: 46L10, 46L53, 46L54, 47L55; Secondary: 05E15, 11G15, 11R47, 11R56.
Citation: Ilwoo Cho, Palle Jorgense. Free probability on $ C^{*}$-algebras induced by hecke algebras over primes. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2221-2252. doi: 10.3934/dcdss.2019143
References:
[1]

K. Abu-Ghanem, D. Alpay, F. Colombo and I. Sabadini, Gleason's problem and schur multipliers in the multivariable quaternionic setting, J. Math. Anal. Appl., 425 (2015), 1083-1096.  doi: 10.1016/j.jmaa.2015.01.022.  Google Scholar

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D. Alpay, F. Colombo and I. Sabadini, Inner product spaces and krein spaces in the quaternionic setting, recent adv. inverse scattering, schur anal. stochastic process, Opre. Theo., Adv. Appl., 224 (2015), 33-65.  doi: 10.1007/978-3-319-10335-8_4.  Google Scholar

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M. Aschbacher, Finite Group Theory, Second edition. Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9781139175319.  Google Scholar

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

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D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511609572.  Google Scholar

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I. Cho, p-adic Banach-space operators and adelic Banach-space operators, Opuscula Math., 34 (2014), 29-65.  doi: 10.7494/OpMath.2014.34.1.29.  Google Scholar

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I. Cho, Operators induced by prime numbers, Methods Appl. Math., 19 (2012), 313-339.  doi: 10.4310/MAA.2012.v19.n4.a1.  Google Scholar

[9]

I. Cho, Certain group dynamical systems induced by hecke algebras, Opuscula Math., 36 (2016), 337-373.  doi: 10.7494/OpMath.2016.36.3.337.  Google Scholar

[10]

I. Cho, Free probability on hecke algebras and certain group algebras induced by Hecke algebras, Opuscula Math., 36 (2016), 153-187.  doi: 10.7494/OpMath.2016.36.2.153.  Google Scholar

[11]

I. Cho, Free probability on $ W^{*}$-dynamical systems determined by general linear group $ GL_{2}(\Bbb{Q}_{p})$, Bollettino dell'Unione Math. Italiana, 10 (2017), 725-764.  doi: 10.1007/s40574-016-0111-z.  Google Scholar

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I. Cho, Representations and corresponding operators induced by Hecke algebras, Compl. Anal. Oper. Theo., 10 (2016), 437-477.  doi: 10.1007/s11785-014-0418-7.  Google Scholar

[13]

I. Cho and T. Gillespie, Free probability on the Hecke algebra, Compl. Anal. Oper. Theo., 9 (2015), 1491-1531.  doi: 10.1007/s11785-014-0403-1.  Google Scholar

[14]

I. Cho and P. E. T. Jorgensen, Krein-space operators induced by Dirichlet characters, Special Issues: Contemp. Math. Amer. Math. Soc., 603 (2013), 3-33.  doi: 10.1090/conm/603/12046.  Google Scholar

[15]

C. W. Curtis, Note on the structure constants of Hecke algebras of induced representations of finite Chevalley groups, Pacific J. Math., 279 (2015), 181-202.  doi: 10.2140/pjm.2015.279.181.  Google Scholar

[16]

T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, 2010, PhD Thesis. Google Scholar

[17]

T. Gillespie, 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

[18]

D. Hilbert, Mathematical problems, Bull. Ame. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3.  Google Scholar

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G. Johnson, A note on the double affine Hecke algebra of type $ GL_{2}$, Comm. Alg., 44 (2016), 1018-1032.  doi: 10.1080/00927872.2014.999924.  Google Scholar

[20]

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.   Google Scholar

[21]

R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Amer. Math. Soc. Mem. 132 (1998), x+88 pp. doi: 10.1090/memo/0627.  Google Scholar

[22]

V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, P-adic analysis and mathematical physics, Ser. Soviet & East European Math., 1 (1994), xx+319 pp. doi: 10.1142/1581.  Google Scholar

[23]

D. Voiculescu, K. Dykemma and A. Nica, Free Random Variables, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[24]

C. Zhong, On the formal affine Hecke algebra, J. Inst. Math. Jussieu, 14 (2015), 837-855.  doi: 10.1017/S1474748014000188.  Google Scholar

show all references

References:
[1]

K. Abu-Ghanem, D. Alpay, F. Colombo and I. Sabadini, Gleason's problem and schur multipliers in the multivariable quaternionic setting, J. Math. Anal. Appl., 425 (2015), 1083-1096.  doi: 10.1016/j.jmaa.2015.01.022.  Google Scholar

[2]

D. Alpay, F. Colombo and I. Sabadini, Inner product spaces and krein spaces in the quaternionic setting, recent adv. inverse scattering, schur anal. stochastic process, Opre. Theo., Adv. Appl., 224 (2015), 33-65.  doi: 10.1007/978-3-319-10335-8_4.  Google Scholar

[3]

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

[4]

M. Aschbacher, Finite Group Theory, Second edition. Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9781139175319.  Google Scholar

[5]

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

[6]

D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511609572.  Google Scholar

[7]

I. Cho, p-adic Banach-space operators and adelic Banach-space operators, Opuscula Math., 34 (2014), 29-65.  doi: 10.7494/OpMath.2014.34.1.29.  Google Scholar

[8]

I. Cho, Operators induced by prime numbers, Methods Appl. Math., 19 (2012), 313-339.  doi: 10.4310/MAA.2012.v19.n4.a1.  Google Scholar

[9]

I. Cho, Certain group dynamical systems induced by hecke algebras, Opuscula Math., 36 (2016), 337-373.  doi: 10.7494/OpMath.2016.36.3.337.  Google Scholar

[10]

I. Cho, Free probability on hecke algebras and certain group algebras induced by Hecke algebras, Opuscula Math., 36 (2016), 153-187.  doi: 10.7494/OpMath.2016.36.2.153.  Google Scholar

[11]

I. Cho, Free probability on $ W^{*}$-dynamical systems determined by general linear group $ GL_{2}(\Bbb{Q}_{p})$, Bollettino dell'Unione Math. Italiana, 10 (2017), 725-764.  doi: 10.1007/s40574-016-0111-z.  Google Scholar

[12]

I. Cho, Representations and corresponding operators induced by Hecke algebras, Compl. Anal. Oper. Theo., 10 (2016), 437-477.  doi: 10.1007/s11785-014-0418-7.  Google Scholar

[13]

I. Cho and T. Gillespie, Free probability on the Hecke algebra, Compl. Anal. Oper. Theo., 9 (2015), 1491-1531.  doi: 10.1007/s11785-014-0403-1.  Google Scholar

[14]

I. Cho and P. E. T. Jorgensen, Krein-space operators induced by Dirichlet characters, Special Issues: Contemp. Math. Amer. Math. Soc., 603 (2013), 3-33.  doi: 10.1090/conm/603/12046.  Google Scholar

[15]

C. W. Curtis, Note on the structure constants of Hecke algebras of induced representations of finite Chevalley groups, Pacific J. Math., 279 (2015), 181-202.  doi: 10.2140/pjm.2015.279.181.  Google Scholar

[16]

T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, 2010, PhD Thesis. Google Scholar

[17]

T. Gillespie, 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

[18]

D. Hilbert, Mathematical problems, Bull. Ame. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3.  Google Scholar

[19]

G. Johnson, A note on the double affine Hecke algebra of type $ GL_{2}$, Comm. Alg., 44 (2016), 1018-1032.  doi: 10.1080/00927872.2014.999924.  Google Scholar

[20]

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.   Google Scholar

[21]

R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Amer. Math. Soc. Mem. 132 (1998), x+88 pp. doi: 10.1090/memo/0627.  Google Scholar

[22]

V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, P-adic analysis and mathematical physics, Ser. Soviet & East European Math., 1 (1994), xx+319 pp. doi: 10.1142/1581.  Google Scholar

[23]

D. Voiculescu, K. Dykemma and A. Nica, Free Random Variables, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[24]

C. Zhong, On the formal affine Hecke algebra, J. Inst. Math. Jussieu, 14 (2015), 837-855.  doi: 10.1017/S1474748014000188.  Google Scholar

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