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

TL;DR

This paper explores a $ ext{Gamma}$-equivariant version of Berezin's quantization on the upper half plane, linking it to deformation quantization of the quotient space and analyzing the associated von Neumann algebra's properties.

Contribution

It introduces a $ ext{Gamma}$-equivariant form of Berezin's quantization for the upper half plane and establishes its von Neumann algebra's stable isomorphism with that of $ ext{Gamma}$.

Findings

The $ ext{Gamma}$-equivariant quantization corresponds to a deformation quantization of $ ext{H}/ ext{Gamma}$.

The associated von Neumann algebra is stable isomorphic to the algebra linked to $ ext{Gamma}$.

The work connects geometric quantization with operator algebra structures in a new setting.

Abstract

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