In this paper, by establishing free-probabilistic models on the Hecke algebras (mathcal{H}left(GL_{2}(mathbb{Q}_{p})ight)) 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.
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