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Free probability on $ C^{*}$-algebras induced by hecke algebras over primes
| 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 |
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.
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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.
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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.
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D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997.
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I. Cho,
p-adic Banach-space operators and adelic Banach-space operators, Opuscula Math., 34 (2014), 29-65.
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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. |
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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. |
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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. |
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T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, 2010, PhD Thesis. |
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T. Gillespie,
Prime number theorems for Rankin-Selberg L-functions over number fields, Sci. China Math., 54 (2011), 35-46.
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G. Johnson,
A note on the double affine Hecke algebra of type $ GL_{2}$, Comm. Alg., 44 (2016), 1018-1032.
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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.
|
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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. |
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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. |
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D. Voiculescu, K. Dykemma and A. Nica, Free Random Variables, American Mathematical Society, Providence, RI, 1992. |
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C. Zhong,
On the formal affine Hecke algebra, J. Inst. Math. Jussieu, 14 (2015), 837-855.
doi: 10.1017/S1474748014000188. |
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. |
| [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. |
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