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Deformation Quantization for Actions of KĂ€hlerian Lie Groups
About this Title
Pierre Bieliavsky, University of Louvain, Belgium and Victor Gayral, University of Reims, France
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 236, Number 1115
ISBNs: 978-1-4704-1491-7 (print); 978-1-4704-2281-3 (online)
DOI: https://doi.org/10.1090/memo/1115
Published electronically: December 18, 2014
Keywords: Strict deformation quantization,
Symmetric spaces,
Representation theory of Lie groups,
Deformation of $C^*$-algebras,
Symplectic Lie groups,
Coherent states,
Noncommutative harmonic analysis,
Noncommutative geometry
MSC: Primary 22E30, 46L87, 81R60, 58B34, 81R30, 53C35, 32M15, 53D55
Table of Contents
Chapters
- 1. Introduction
- Notations and conventions
- 2. Oscillatory integrals
- 3. Tempered pairs for KĂ€hlerian Lie groups
- 4. Non-formal star-products
- 5. Deformation of Fréchet algebras
- 6. Quantization of polarized symplectic symmetric spaces
- 7. Quantization of KĂ€hlerian Lie groups
- 8. Deformation of $C^*$-algebras
Abstract
Let $\mathbb {B}$ be a Lie group admitting a left-invariant negatively curved KĂ€hlerian structure. Consider a strongly continuous action $\alpha$ of $\mathbb {B}$ on a FrĂ©chet algebra $\mathcal {A}$. Denote by $\mathcal {A}^\infty$ the associated FrĂ©chet algebra of smooth vectors for this action. In the Abelian case $\mathbb {B}=\mathbb {R}^{2n}$ and $\alpha$ isometric, Marc Rieffel proved that Weylâs operator symbol composition formula (the so called Moyal product) yields a deformation through FrĂ©chet algebra structures $\{\star _{\theta }^\alpha \}_{\theta \in \mathbb {R}}$ on $\mathcal {A}^\infty$. When $\mathcal {A}$ is a $C^*$-algebra, every deformed FrĂ©chet algebra $(\mathcal {A}^\infty ,\star ^\alpha _\theta )$ admits a compatible pre-$C^*$-structure, hence yielding a deformation theory at the level of $C^*$-algebras too. In this memoir, we prove both analogous statements for general negatively curved KĂ€hlerian 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 CalderĂłn-Vaillancourt Theorem. In particular, we give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.- P. Bieliavsky, Espaces symĂ©triques symplectiques. Ph.D. thesis, UniversitĂ© Libre de Bruxelles (1995); math.DG/0703358.
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