Deformation quantization for actions of Kählerian Lie groups
Author(s)
Bibliographic Information
Deformation quantization for actions of Kählerian Lie groups
（Memoirs of the American Mathematical Society, no. 1115）
American Mathematical Society, 2015, c2014
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Note
"Volume 236, number 1115 (fifth of 6 numbers), July 2015"
Includes bibliographical references (p. 153-154)
Description and Table of Contents
Description
Let $\mathbb{B}$ be a Lie group admitting a left-invariant negatively curved Kahlerian structure. Consider a strongly continuous action $\alpha$ of $\mathbb{B}$ on a Frechet algebra $\mathcal{A}$. Denote by $\mathcal{A}^\infty$ the associated Frechet 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 Frechet algebra structures $\{\star_{\theta}^\alpha\}_{\theta\in\mathbb{R}}$ on $\mathcal{A}^\infty$. When $\mathcal{A}$ is a $C^*$-algebra, every deformed Frechet 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 Kahlerian 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 Calderon-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
Introduction
Notations and conventions
Oscillatory integrals
Tempered pairs for Kahlerian Lie groups
Non-formal star-products
Deformation of Frechet algebras
Quantization of polarized symplectic symmetric spaces
Quantization of Kahlerian Lie groups
Deformation of $C^*$-algebras
Bibliography
by "Nielsen BookData"
