On the $Γ$-equivariant form of the Berezin's quantization of the upper half plane

Abstract

Let $Γ$ be a fuchsian subgroup of \pslr. 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 $\Bbb H/Γ$. 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 $Γ$.