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Deformation Quantization for Actions of Kählerian Lie Groups

Pierre Bieliavsky Université Catholique de Louvain, Louvain le Neuve, Belgium
Victor Gayral Laboratoire de Mathématiques, Reims, France
Available Formats:
Electronic ISBN: 978-1-4704-2281-3
Product Code: MEMO/236/1115.E
154 pp
List Price: $86.00 MAA Member Price:$77.40
AMS Member Price: $51.60 Click above image for expanded view Deformation Quantization for Actions of Kählerian Lie Groups Pierre Bieliavsky Université Catholique de Louvain, Louvain le Neuve, Belgium Victor Gayral Laboratoire de Mathématiques, Reims, France Available Formats:  Electronic ISBN: 978-1-4704-2281-3 Product Code: MEMO/236/1115.E 154 pp  List Price:$86.00 MAA Member Price: $77.40 AMS Member Price:$51.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2362015
MSC: Primary 22; 46; 81; 58; 53; 32;

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, 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.

• Chapters
• 1. Introduction
• Notations and conventions
• 2. Oscillatory integrals
• 3. Tempered pairs for Kählerian Lie groups
• 4. Non-formal star-products
• 5. Deformation of Fréchet algebras
• 6. Quantization of polarized symplectic symmetric spaces
• 7. Quantization of Kählerian Lie groups
• 8. Deformation of $C^*$-algebras
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Volume: 2362015
MSC: Primary 22; 46; 81; 58; 53; 32;

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, 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.

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