4 1. INTRODUCTION

sequence of weights μ. This explains why the non-Abelian framework forces to

consider such symbol spaces. Applying the results of chapter 4 to this situation,

we get a new associative product on

A∞

defined by the formula

a

α

θ

b :=

(

α(a)

θ

α(b)

)

(e) .

Then, the main result of this chapter, stated as Theorem 5.8 in the text, is the

following fact:

Universal Deformation Formula for Actions of K¨ ahlerian Lie Groups on

Fr´ echet Algebras:

Let (A,α, B) be a Fr´ echet algebra endowed with a tempered action of a normal

j-group. Then, (A∞, α)

θ

is an associative Fr´ echet algebra, (abusively) called the

Fr´ echet deformation of A.

Following the terminology introduced in [17], the word “universal” refers to the

fact that our deformation procedure applies to any Fr´ echet algebra the K¨ahlerian

Lie group acts on. The word “universal” does not refer to the possibility that our

construction might yield all such deformation procedures valid for a given K¨ahlerian

Lie group. However, regarding the last sentence, the following remark can never-

theless be made. In order to get rid of technicalities let us consider, for a short

moment, the purely formal framework of formal Drinfel’d twists based on the bi-

algebra underlying the enveloping algebra U(b) of the Lie group B. Given such a

Drinfel’d twist F ∈ U(b) ⊗U(b)[[ν]], one defines the internal symmetry of the twist

as the group G(F ), consisting of diffeomorphisms that preserve the twist:

G(F ) := {ϕ : B → B | ϕ

˜

F =

˜}

F ,

where

˜

F denotes the left-invariant (formal sequence of) operator(s) associated to

the twist F . Within the present work we treat (all) the situations where, in the

elementary normal case, the internal symmetry equals the automorphism group of

a symmetric symplectic space structure on B whose underlying aﬃne connection

consists in the canonical torsion free invariant connection on the group B (see [3]).

The rest of the memoir is devoted to the construction of a pre-C∗-structure on

(A∞, α),

θ

in the case A is a C∗-algebra. The method we use is a generalization of

Unterberger’s Fuchs calculus [33] and fits within the general framework of “Moyal

quantizer” as defined in [12] (see also [18, Section 3.5]).

In chapter 6, we define a special class of symplectic symmetric spaces which

naturally give rise to explicit WKB-quantizations (i.e. invariant star-products rep-

resentable through oscillatory kernels) that underlie pseudo-differential operator

calculus. Roughly speaking, an elementary symplectic symmetric space is a sym-

plectic symmetric space that consists of the total space of a fibration in flat fibers

over a Lie group Q of exponential type. In that case, a variant of Kirillov’s orbit

method yields a unitary and self-adjoint representation on an Hilbert-space H, of

the symmetric space M:

Ω : M → Usa(H) ,