On the $\mathrm{\Gamma }$-equivariant form of the Berezin's quantization of the upper half plane

Let $\mathrm{\Gamma }$ be a fuchsian subgroup of \pslr. In this paper we consider the $\mathrm{\Gamma }$-equivariant form of the Berezin’s quantization of the upper half plane which will correspond to a deformation quantization of the (singular) space $\mathbb{H}\mathrm{/}\mathrm{\Gamma }$. Our main result is that the von Neumann algebra associated to the $\mathrm{\Gamma }-$ equivariant form of the quantization is stable isomorphic with the von Neumann algebra associated to $\mathrm{\Gamma }$.