1 Introduction Berezin-Töplitz Quantization

The general idea of quantization is to find a way to pass from the classical setting to the quantum one. In the classical situation, we have a symplectic manifold (M,ω) standing for the space of “states” of some physical system. The topological data of the manifold is contained in the structure of the function algebra C(M). The smooth structure on M gives rise to a distinguished subalgebra C∞ 0 (M) of smooth functions vanishing at ∞, and the symplectic structure gives rise to a Poisson structure on C∞ c (M) the subalgebra consisting of the compactly supported smooth functions. One of the natural generalizations of the idea of a function algebra on a manifold (as well as on more general topological spaces) is that of a C∗algebra, which will now be defined. Perhaps now is a good place to state that from now on (unless stated otherwise) all scalars are complex.