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

Abstract

Let $\G$ be any cocompact, discrete subgroup of $\pslr$. 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 $\Bbb D/\G$. As a corollary, for large values of the deformation parameter $1/h$, 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 $PSL(2,\Bbb R)$, as left modules over $\Gamma$ we deduce that the fundamental group ([MvN]) of the von Neumann $\Cal L(\Gamma)$ contains the positive rational numbers. Equivalently, this proves that the algebras $\Cal L(\Gamma)\otimes M_n(\Bbb C)$, are isomorphic for all $n$.