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

Ilwoo Cho; Palle Jorgense

doi:10.3934/dcdss.2019143

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

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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: July 31, 2016

Revised: February 28, 2017

Published: December 2019

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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.

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

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