A universal, non-commutative C∗-algebra associated to the Hecke algebra of double cosets

Let G be a discrete group and Γ an almost normal subgroup. The operation of cosets concatanation extended by linearity gives rise to an operator system that is embeddable in a natural C* algebra. The Hecke algebra naturally embeds as a diagonal of the tensor product of this algebra with its opposite. When represented on the l2 space of the group, by left and right convolution operators, this representation gives rise to abstract Hecke operators that in the modular group case, are unitarily equivalent to the classical operators on Maass wave forms