Opuscula Math. 36, no. 3 (2016), 337-373
http://dx.doi.org/10.7494/OpMath.2016.36.3.337

Opuscula Mathematica

# Certain group dynamical systems induced by Hecke algebras

Ilwoo Cho

Abstract. In this paper, we study dynamical systems induced by a certain group $$\mathfrak{T}_{N}^{K}$$ embedded in the Hecke algebra $$\mathcal{H}(G_{p})$$ induced by the generalized linear group $$G_{p} = GL_{2}(\mathbb{Q}_{p})$$ over the $$p$$-adic number fields $$\mathbb{Q}_{p}$$ for a fixed prime $$p$$. We study fundamental properties of such dynamical systems and the corresponding crossed product algebras in terms of free probability on the Hecke algebra $$\mathcal{H}(G_{p})$$.

Keywords: free probability, free moments, free cumulants, Hecke algebra, normal Hecke subalgebra, free probability spaces, representations, operators, Hilbert spaces, dynamical systems, crossed product algebras.

Mathematics Subject Classification: 05E15, 11R47, 46L10, 47L30, 47L55.

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• Ilwoo Cho
• St. Ambrose University, Department of Mathematics and Statistics, 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: 2016-02-21.