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

Ilwoo Cho; Palle Jorgense

doi:10.3934/dcdss.2019143

Article Contents

2019, Volume 12, Issue 8: 2221-2252. Doi: 10.3934/dcdss.2019143

This issue Previous Article Shadowing is generic on dendrites Next Article Orbit portraits in non-autonomous iteration

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
* Corresponding author: Ilwoo Cho

Received: August 2016

Revised: March 2017

Published: December 2019

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 46L10, 46L53, 46L54, 47L55; Secondary: 05E15, 11G15, 11R47, 11R56.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[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.
[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.
[4]	M. Aschbacher, Finite Group Theory, Second edition. Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9781139175319.
[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.
[6]	D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511609572.
[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.
[8]	I. Cho, Operators induced by prime numbers, Methods Appl. Math., 19 (2012), 313-339. doi: 10.4310/MAA.2012.v19.n4.a1.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, 2010, PhD Thesis.
[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.
[18]	D. Hilbert, Mathematical problems, Bull. Ame. Math. Soc., 8 (1902), 437-479. doi: 10.1090/S0002-9904-1902-00923-3.
[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.
[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.
[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.
[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.
[23]	D. Voiculescu, K. Dykemma and A. Nica, Free Random Variables, American Mathematical Society, Providence, RI, 1992.
[24]	C. Zhong, On the formal affine Hecke algebra, J. Inst. Math. Jussieu, 14 (2015), 837-855. doi: 10.1017/S1474748014000188.