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Classical Hecke operators on Maass forms are unitarely equivalent, up to acommuting phase, to completely positive maps on II$_1$ factors, associated to apair of isomorphic subfactors, and an intertwining unitary. This representationis obtained through a quantized representation of the Hecke operators. in thisrepresentation, the Hecke operators act on the Berezin's quantization,deformation algebra of the fundamental domain of $\PSL(2,\Z)$ in the upperhalfplane. The Hecke operators are inheriting from the ambient, non-commutativealgebra on which they act, a rich structure of matrix inequalities. Using thisconstruction we obtain that, for every prime $p$, the essential spectrum of theclassical Hecke operator $T_p$ is contained in the interval $[-2\sqrt p, 2\sqrtp]$, predicted by the Ramanujan Petersson conjectures. In particular, given anopen interval containing $[-2\sqrt p, 2\sqrt p]$, there are at most a finitenumber of possible exceptional eigenvalues lying outside this interval. Themain tool for obtaining this representation of the Hecke operators (unitarelyequivalent to the classical representation, up to commuting phase) is a Schurrtype, positive "square root" of the state on $\PGL(2,\Q)$, measuring thedisplacement of fundamental domain by translations in $\PGL(2,\Q)$. The "squareroot" is obtained from the matrix coefficients of the discrete seriesrepresentations of $\PSL(2,\R)$ restricted to $\PGL(2, \Q)$. The methods inthis paper may also be applied to any finite index, modular subgroup$\Gamma_0(p^n)$, $n\geq 1$, of $\PSL(2,\Z)$. In this case the essential norm ofthe Hecke operator is equal to the norm of the corresponding convolutionoperator on the cosets Hilbert space $\ell^2((\Gamma_0(p^n))\backslash\PSL(2,\Z[1/p])$.

Type II$_1$ von Neumann algebra representations of Hecke operators on Maass forms and the Ramanujan-Petersson conjectures