Deformation Quantization for Actions of Kahlerian Lie Groups Part I: Fréchet Algebras

Details Deformation Quantization for Actions of Kahlerian Lie Groups Part I: Fréchet Algebras Deformation Quantization for Actions of Kahlerian Lie Groups Part I: Fréchet Algebras

2011-09-15

arxiv.org/abs/1109.3419v3

Mathematics

Operator Algebras

Version 2, one reference added

Scientific paper

Let $\B$ be a Lie group admitting a left-invariant negatively curved Kahlerian structure. Consider any tempered action $\alpha$ of $\B$ on a Fr\'echet algebra $(\CA,\mu)$. Denote by $\CA^\infty$ its associated Fr\'echet algebra of smooth vectors for the action $\alpha$. In the Abelian case $\B=\R^{2n}$ and $\alpha$ isometrical, Marc Rieffel proved in \cite{Ri} that Weyl's operator symbol composition formula yields a deformation of $\mu$ through Fr\'echet algebra structures $\{\mu_{\theta}\}_{\theta\in\R}$ on $\CA^\infty$. In this paper, we prove the analogous statement in the general negatively curved Kahlerian group and tempered action case. The construction relies on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geometrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces.

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