Electronic ISBN:  9781470422813 
Product Code:  MEMO/236/1115.E 
154 pp 
List Price:  $86.00 
MAA Member Price:  $77.40 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 236; 2015MSC: Primary 22; 46; 81; 58; 53; 32;
Let \(\mathbb{B}\) be a Lie group admitting a leftinvariant 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 nonAbelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKBquantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a nonAbelian version of the CalderónVaillancourt Theorem. In particular, the authors give an oscillating kernel formula for WKBstar products on symplectic symmetric spaces that fiber over an exponential Lie group. 
Table of Contents

Chapters

1. Introduction

Notations and conventions

2. Oscillatory integrals

3. Tempered pairs for Kählerian Lie groups

4. Nonformal starproducts

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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Let \(\mathbb{B}\) be a Lie group admitting a leftinvariant 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 nonAbelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKBquantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a nonAbelian version of the CalderónVaillancourt Theorem. In particular, the authors give an oscillating kernel formula for WKBstar 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. Nonformal starproducts

5. Deformation of Fréchet algebras

6. Quantization of polarized symplectic symmetric spaces

7. Quantization of Kählerian Lie groups

8. Deformation of $C^*$algebras