1. INTRODUCTION 5

with associated “quantization rule”:

Ω :

L1(M)

→ B(H) , F → Ω(F ) :=

M

F (x)Ω(x) dx .

Weighting the above mapping by the multiplication by a (growth controlled) func-

tion m defined on the base Q yields a pair of adjoint maps:

Ωm :

L2(M)

→

L2(H)

and σm :

L2(H)

→

L2(M)

,

where L2(H) denotes the Hilbert space of Hilbert-Schmidt operators on H. Both

of the above maps are equivariant under the whole automorphism group of M.

Note that this last feature very much contrasts with the usual notion of coherent-

state quantization for groups (as opposed to symmetric spaces). The corresponding

“Berezin transform” Bm := σm ◦ Ωm is explicitly controlled. In particular, when

invertible, the associated star-product F1 F2 := Bm1σm(Ωm(F1)Ωm(F2)) − is of os-

cillatory (WKB) type and its associated kernel is explicitly determined. Note that,

because entirely explicit, this chapter yields a proof of Weinstein’s conjectural form

for star-product WKB-kernels on symmetric spaces [36] in the situation consid-

ered here. The chapter ends with considerations on extending the construction to

semi-direct products.

Chapter 7 is entirely devoted to applying the construction of chapter 6 to

the particular case of K¨ ahlerian Lie groups with negative curvature. Such a Lie

group is always a normal j-group is the sense of Pyatetskii-Shapiro. Each of its

elementary factors admits the structure of an elementary symplectic symmetric

space. Accordingly to [3], the obtained non-formal star products coincide with the

one described in chapters 2-4.

Chapter 8 deals with the deformation theory for

C∗-algebras.

We eventually

prove the following statement:

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

C∗-Algebras:

Let (A, α, B) be a C∗-algebra endowed with a strongly continuous and isometric

action of a normal j-group. Then, there exists a canonical C∗-norm on the Fr´echet

algebra (A∞, α).

θ

Its C∗-closure is (abusively) called the C∗-deformation of A.

The above statement follows from a non-Abelian generalization of the Calder´on-

Vaillancourt Theorem in the context of the usual Weyl pseudo-differential calculus

on R2n (see Theorem 8.20). Our result asserts that the element Ωm(F ) associated

to a function F in

B1(B,A)

naturally consists of an element of the spatial tensor

product of A by B(H). Its proof relies on a combination of a resolution of the

identity obtained from wavelet analysis considerations (see section 8.1) and further

properties of our oscillatory integral defined in chapter 2. We finally prove that the

K-theory is an invariant of the deformation.

Acknowledgments We warmly thank the referee for his positive criticism.

His comments, remarks and suggestions have greatly improved this work.