Deformation Quantization for Actions of Kählerian Lie Groups

Abstract

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 the action $\alpha$. In the Abelian case $\mathbb{B}={\mathbb{R}}^{2n}$ and $\alpha$ isometric, Marc Rieffel proved that Weyl's operator symbol composition formula yields a deformation through Fréchet algebra structures ${{\star }_{\theta }^{\alpha }}_{\theta \in \mathbb{R}}$ on ${\mathcal{A}}^{\infty }$. When $\mathcal{A}$ is a $C^{\star }$-algebra, every deformed algebra $({\mathcal{A}}^{\infty },{\star }_{\theta }^{\alpha })$ admits a compatible pre-$C^{\star }$-structure. In this paper, we prove both analogous statements in the general negatively curved Kählerian group and (non-isometric) "tempered" action case. The construction relies on the one hand 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, and, on the second hand, in establishing a non-Abelian version of the Calderòn-Vaillancourt Theorem. In particular, we give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.

In this version, we have removed section 7.7 (section 8.7 in the published version). Proposition 7.47 contained a mistake which invalidates our proof of the invariance of the $K$-theory under deformation (Theorem 7.50)

References in corpus (1)

• Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces

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