Toeplitz-Berezin quantization and non-commutative differential geometry

Abstract

In this survey article we describe how the recent work in quantization in multi-variable complex geometry (domains of holomorphy, symmetric domains, tube domains, etc.) leads to interesting results and problems in C-algebras which can be viewed as examples of the “non-commutative geometry” in the sense of A. Connes. At the same time, one obtains new functional calculi (of pseudodifferential type) with possible applications to partial differential equations and group representations.