On the Γ-equivariant Form of the Berezin's Quantization of the Upper Half Plane

Introduction Let Γ be a fuchsian subgroup of P SL(2, R). In this paper we consider the Γ-equivariant form of the Berezin's quantization of the upper half plane which will correspond to a deformation quantization of the (singular) space H/Γ. Our main result is that the von Neumann algebra associated to the Γ− equivariant form of the quantization is stable isomorphic with the von Neumann algebra associated to Γ. Moreover the dimension of each algebra in the deformation quantization, as a left module over the group von Neumann algebra L(Γ), is a linear function of the deformation parameter (the " Planck constant "). Recall that the Muray-von Neumann construction, for the dimension of projec-tive, left modules over type II von Neumann algebras with trivial center, allows all positive real numbers as possible value for the dimension. Consequently the above isomorphism is meaningful for all values of the deformation parameter. This will be particularly interesting when Γ is the group Γ = P SL(2, Z). We use the terminology (introduced in [KD], [FR], see also [DV]) of von Neumann algebras L(F t), t > 1 corresponding to free groups with a (possible) fractional " number t of generators " (even if the group itself may not make sense). In this case the von Neumann algebras associated to the equivariant form of the Berezin quantization will be free groups von Neumann algebras where the " number of generators " is a bijective function of the deformation parameter. 2 The difference between the Berezin quantization of the upper half plane and its Γ-equivariant form is easy to establish. In the classical case the von Neumann algebras associated to the deformation are simply isomorphic to B(H), the algebra of all bounded operators on a Hilbert space. In the equivariant case these algebras are type II 1 factors. This is a consequence of the formulae for the traces in this algebras: classically the trace is an integral over H of the restriction to the diagonal of the reproducing kernel, while in the Γ-equivariant case the trace is the integral over a fundamental domain of Γ in H. This last fact is in a particular a generalization of the computation in [GHJ] of the type II 1 factor trace of a product of two Toeplitz operators having automorphic forms as symbols. There exists a remarkable analogy, at least at the formal level, between Rieffel's ([Ri]) construction …