Type Ii1 Von Neumann Representations for Hecke Operators on Maass Forms and Ramanujan-peterson Conjecture

We prove that classical Hecke operators on Maass forms are a special case of completely positive maps on II1 factors, associated to a pair of isomorphic subfactors. This representation induces several matrix inequalities on the eigenvalues of the Hecke operators Maass forms. In particular the family of eigenvalues corresponding to an eigenvector is a completely bounded multiplier of the Hecke algebra. Moreover it follows that the Ramanujan-Peterson conjecture holds true, with the possible exception of a finite number of eigenvectors. Given a type II1 von Neumann algebra M , with faithfull trace τ , and two subfactors P0, P1 of M of equal index and a von Neumann algebra isomorphism θ : P0 → P1, every unitary U on the GNS space of M , L2(M, τ), that implements θ, (i.e. UpU∗ = θ(p) for p in P0) implements a completely positive map ΨU on M ′ with values in M ′ by the formula ΨU(x) = E P ′ 1 M ′(UxU ∗), x ∈ M ′. Here M ′ ⊆ P ′ 1 ⊆ B(L2(M, τ)) are the commutants, and E P ′ 1 M ′ is the canonical expectation from M ′ onto P ′ 1. Note that UP0U ∗ = P1 and hence UP ′ 0U ∗ = P ′ 1 so that UM ′U∗ is contained in P ′ 1. Given θ, one can always construct such a unitary U , which is unique up to left multiplication with unitary phase u in P ′ 1 (that is U1 = uU ). The distance from the set of the completely positive maps ΨU to the automorphisms of M ′, measures how far is the automorphism θ from being extended to an automorphism of M . If we let M be the factor generated by the image of PSL2(Z) through the discrete series representation π13 of PSL2(R), then as proven by Jones ([GHJ]), M is unitarely equivalent to the factor L(PSL2(Z)) associated to the left regular representation of the discrete group PSL2(Z).