Some estimates for the Banach space norms in the von Neumann algebras associated with the Berezin's quantization of compact Riemann

Let $\text{G}$ be any cocompact, discrete subgroup of $\text{pslr}$. In this paper we find estimates for the predual and the uniform Banach space norms in the von Neumann algebras associated with the Berezin’ s quantization of a compact Riemann surface $\mathbb{D}\mathrm{/}\text{G}$. As a corollary, for large values of the deformation parameter $1\mathrm{/}h$, these von Neumann algebras are isomorphic. Using the results in [AS], [AC], [GHJ] on the von Neumann dimension of the Hilbert spaces in the discrete series of unitary representations of $PSL(2,\mathbb{R})$, as left modules over $\mathrm{\Gamma }$ we deduce that the fundamental group ([MvN]) of the von Neumann $\text{Cal}L(\mathrm{\Gamma })$ contains the positive rational numbers. Equivalently, this proves that the algebras $\Cal L(\Gamma)\otimes M_n(\Bbb C)$,areisomorphicforall$n$.