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

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

1995-11-08

arxiv.org/abs/funct-an/9511001v2

Mathematics

Operator Algebras

AMSTEX, 31 pages. This paper has been circulated (in hard copy version) under the Title : The fundamental group of the von Neu

Scientific paper

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 $n$.

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