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Conditional expectations, traces, angles between spaces and Representations of the Hecke algebras

In this paper we extend the results in [Ra] on the representation of the Hecke algebra, determined by the matrix coefficients of a projective, unitary representation, in the discrete series of representations of the ambient group, to a more general, vector valued case. This method is used to analyze the traces of the Hecke operators. We construct representations of the Hecke algebra of a group $G$, relative to an almost normal subgroup $Γ$, into the von Neumann algebra of the group $G$, tensor matrices. The representations we obtain are a lifting of the Hecke operators to this larger algebra. By summing up the coefficients of the terms in the representation one obtains the classical Hecke operators. These representations were used in the scalar case in [Ra], to find an alternative representation of the Hecke operators on Maass forms, and hence to reformulate the Ramanujan Petersson conjectures as a problem on the angle (see e.g. A. Connes's paper [Co] on the generalization of CKM matrix) between two subalgebras of the von Neumann algebra of the group $G$: the image of the representation of the Hecke algebra and the algebra of the almost normal subgroup.