Deformation Quantization for Actions of Kahlerian Lie Groups

In this memoir, the authors prove both analogous statements for general negatively curved Kahlerian groups. 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, andin establishing a non-Abelian version of the Calderon-Vaillancourt Theorem.

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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.