# Deformation Quantization for Actions of Kählerian Lie Groups

Share this page
*Pierre Bieliavsky; Victor Gayral*

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.

#### Table of Contents

# Table of Contents

## Deformation Quantization for Actions of Kahlerian Lie Groups

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Notations and conventions 714 free
- Chapter 2. Oscillatory integrals 1118
- Chapter 3. Tempered pairs for Kählerian Lie groups 4350
- Chapter 4. Non-formal star-products 5966
- Chapter 5. Deformation of Fréchet algebras 6774
- Chapter 6. Quantization of polarized symplectic symmetric spaces 7986
- 6.1. Polarized symplectic symmetric spaces 8188
- 6.2. Unitary representations of symmetric spaces 8693
- 6.3. Locality and the one-point phase 9097
- 6.4. Unitarity and midpoints for elementary spaces 9198
- 6.5. The ⋆-product as the composition law of symbols 96103
- 6.6. The three-point kernel 98105
- 6.7. Extensions of polarization quadruples 101108

- Chapter 7. Quantization of Kählerian Lie groups 105112
- Chapter 8. Deformation of 𝐶*-algebras 119126
- 8.1. Wavelet analysis 119126
- 8.2. A tempered pair from the one-point phase 126133
- 8.3. Extension of the oscillatory integral 128135
- 8.4. A Calderón-Vaillancourt type estimate 130137
- 8.5. The deformed 𝐶*-norm 136143
- 8.6. Functorial properties of the deformation 141148
- 8.7. Invariance of the 𝐾-theory 142149

- Bibliography 153160
- Back Cover Back Cover1166