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# memo_has_moved_text();Deformation Quantization for Actions of Kählerian Lie Groups

Pierre Bieliavsky and Victor Gayral

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: http://dx.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

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Chapters

• Chapter 1. Introduction
• Notations and conventions
• Chapter 2. Oscillatory integrals
• Chapter 3. Tempered pairs for Kählerian Lie groups
• Chapter 4. Non-formal star-products
• Chapter 5. Deformation of Fréchet algebras
• Chapter 6. Quantization of polarized symplectic symmetric spaces
• Chapter 7. Quantization of Kählerian Lie groups
• Chapter 8. Deformation of $C^*$-algebras

### Abstract

Let $\mathbb {B}$ be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action $\alpha$ of $\mathbb {B}$ on a Fréchet algebra $\mathcal {A}$. Denote by $\mathcal {A}^\infty$ the associated Fréchet algebra of smooth vectors for this action. In the Abelian case $\mathbb {B}=\mathbb {R}^{2n}$ and $\alpha$ isometric, Marc Rieffel proved that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Fréchet algebra structures $\{\star _{\theta }^\alpha \}_{\theta \in \mathbb {R}}$ on $\mathcal {A}^\infty$. When $\mathcal {A}$ is a $C^*$-algebra, every deformed Fréchet 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 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, we give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.