THE GAMMA-EQUIVARIANT FORM OF THE BEREZIN QUANTIZATION OF THE UPPER HALF-PLANE

In this paper we define the Gamma equivariant form of Berezin quantiza tion, where Gamma is a discrete lattice in PSL(2, R). The Gamma equiva riant form of the quantization corresponds to a deformation of the spa ce H/Gamma (H being the upper halfplane). The von Neumann algebras in the deformation (obtained via the Gelfand-Naimark-Segal construction f rom the trace) are type II1 factors. When Gamma is PSL(2, Z), this fac tors correspond tin the setting considered by K. Dykema and independen tly by the author, based on the random matrix model of D. Voiculescu) to free groups von Neumann algebras with ''fractional number of genera tors''. The number of generators turns out to be a function of the Pla nck's ''deformation'' constant. The Connes' cyclic 2-cohomology asocia ted to the deformation is analyzed and turns out to be (by using an au tomorphic forms construction) the coboundary of an (unbounded) cycle.