Abstract

Let B be a Lie group admitting a left-invariant negatively curved K¨ahlerian

structure. Consider a strongly continuous action α of B on a Fr´ echet algebra A.

Denote by A∞ the associated Fr´ echet algebra of smooth vectors for this action. In

the Abelian case B = R2n and α isometric, Marc Rieffel proved that Weyl’s opera-

tor symbol composition formula (the so called Moyal product) yields a deformation

through Fr´ echet algebra structures {

α}θ∈R

θ

on

A∞.

When A is a

C∗-algebra,

ev-

ery deformed Fr´ echet algebra

(A∞, α)

θ

admits a compatible

pre-C∗-structure,

hence

yielding a deformation theory at the level of

C∗-algebras

too. In this memoir, we

prove both analogous statements for general negatively curved K¨ ahlerian 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 Calder´on-Vaillancourt

Theorem. In particular, we give an oscillating kernel formula for WKB-star prod-

ucts on symplectic symmetric spaces that fiber over an exponential Lie group.

Received by the editor August 24, 2012 and, in revised form, June 26, 2013.

Article electronically published on December 18, 2014.

DOI: http://dx.doi.org/10.1090/memo/1115

2010 Mathematics Subject Classification. Primary 22E30, 46L87, 81R60, 58B34, 81R30,

53C35, 32M15, 53D55.

Key words and phrases. Strict deformation quantization, Symmetric spaces, Representation

theory of Lie groups, Deformation of

C∗-algebras,

Symplectic Lie groups, Coherent states, Non-

commutative harmonic analysis, Noncommutative geometry.

c

2014 American Mathematical Society

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