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

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.