DEFORMATION QUANTIZATION FOR ACTIONS OF KÄHLERIAN LIE GROUPS. VOLUME: 236

Let ?? be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action a of ?? on a Fréchet algebra ?. Denote by ?8 the associated Fréchet algebra of smooth vectors for this action. In the Abelian case ??=R2n and a isometric, Marc Rieffel proved that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Fréchet algebra structures {?a?}??R on ?8. When ? is a C*-algebra, every deformed Fréchet algebra (?8,?a?) admits a compatible pre-C*-structure, hence yielding a deformation theory at the level of C*-algebras too.

In this memoir, the authors prove both analogous statements for general negatively curved Kählerian groups. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderón-Vaillancourt Theorem. In particular, the authors give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.

Authors
Pierre Bieliavsky, Université Catholique de Louvain, Louvain le Neuve, Belgium.
Victor Gayral, Laboratoire de Mathématiques, Reims, France

Table of Contents
Introduction
Notations and conventions
Oscillatory integrals
Tempered pairs for Kählerian Lie groups
Non-formal star-products
Deformation of Fréchet algebras
Quantization of polarized symplectic symmetric spaces
Quantization of Kählerian Lie groups
Deformation of C*-algebras
Bibliography