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 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 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.
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