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

TL;DR

This paper estimates Banach space norms in von Neumann algebras linked to Berezin's quantization of compact Riemann surfaces, revealing algebraic isomorphisms and properties of the fundamental group for large deformation parameters.

Contribution

It provides new estimates for norms in von Neumann algebras associated with Berezin's quantization and establishes algebraic isomorphisms for large deformation parameters.

Findings

Von Neumann algebras are isomorphic for large deformation parameter 1/h.

The fundamental group of the von Neumann algebra contains positive rationals.

Algebras tensoring with matrix algebras are isomorphic for all sizes.

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 $n$.