Some estimates for the Banach space norms in the von Neumann algebras associated with the Berezin's quantization of compact Riemann
Abstract
Description
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 $Γ$ we deduce that the fundamental group ([MvN]) of the von Neumann $\Cal L(Γ)$ contains the positive rational numbers. Equivalently, this proves that the algebras $\Cal L(Γ)\otimes M_n(\Bbb C)$, are isomorphic for all $n$.
AMSTEX, 31 pages. This paper has been circulated (in hard copy version) under the Title : The fundamental group of the von Neumann algebra of any cocompact subgroup of $PSL(2,R)$ contains the positive rationals. This is a revised version- Some (rather minor) errors have been corrected and the Title has been changed to something which looked more appropriate with the method used. One of the references ([Ra1]) may be retrived from this archive as funct-an/9502001 or from the author
AMSTEX, 31 pages. This paper has been circulated (in hard copy version) under the Title : The fundamental group of the von Neumann algebra of any cocompact subgroup of $PSL(2,R)$ contains the positive rationals. This is a revised version- Some (rather minor) errors have been corrected and the Title has been changed to something which looked more appropriate with the method used. One of the references ([Ra1]) may be retrived from this archive as funct-an/9502001 or from the author