Opuscula Math. 36, no. 2 (2016), 153-187
http://dx.doi.org/10.7494/OpMath.2016.36.2.153

Opuscula Mathematica

# Free probability on Hecke algebras and certain group C*-algebras induced by Hecke algebras

Ilwoo Cho

Abstract. In this paper, by establishing free-probabilistic models on the Hecke algebras $$\mathcal{H}\left(GL_{2}(\mathbb{Q}_{p})\right)$$ induced by $$p$$-adic number fields $$\mathbb{Q}_{p}$$, we construct free probability spaces for all primes $$p$$. Hilbert-space representations are induced by such free-probabilistic structures. We study $$C^{*}$$-algebras induced by certain partial isometries realized under the representations.

Keywords: free probability, free moments, free cumulants, Hecke algebras, normal Hecke subalgebras, representations, groups, group $$C^{*}$$-algebras.

Mathematics Subject Classification: 05E15, 11R47, 46L54, 47L15, 47L55.

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• Ilwoo Cho
• St. Ambrose University, Department of Mathematics, 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, USA
• Communicated by P.A. Cojuhari.
• Revised: 2015-05-19.
• Accepted: 2015-07-06.
• Published online: 2015-12-18.