2 1. INTRODUCTION

associated to a translation invariant symplectic structure

ω0

on

R2n.

The associated

star-product therefore corresponds here to Moyal’s product. In the special case

where the Fr´ echet algebra A is a

C∗-algebra,

Rieffel also constructed a deformed

C∗-structure,

so that

(A∞, α)

θ

becomes a

pre-C∗-algebra,

which in turn yields

a deformation theory at the level of

C∗-algebras

too. Many further results have

been proved then (for example continuity of the field of deformed C∗-algebras [26],

invariance of the K-theory [27]...), and many applications have been found (for

instance in locally compact quantum groups [28], quantum fields theory [9, 10],

spectral triples [15]...).

In this memoir, we investigate the deformation theory of C∗-algebras endowed

with an isometric action of a negatively curved K¨ ahlerian Lie group. Most of

the results we present here are of a pure analytical nature. Indeed, once a family

{Kθ}θ∈R of associative (i.e. such that the associated deformed product (1) is at least

formally associative) two-point functions has been found, in order to give a precise

meaning to the associated multiplication rule, there is no doubt that the integrals

in (1) need to be interpreted in a suitable (here oscillatory) sense. Indeed, there is

no reason to expect the two-point function Kθ to be integrable: it is typically not

even bounded in the non-Abelian case! Thus, already in the case of an isometric

action on a

C∗-algebra,

we have to face a serious analytical diﬃculty. We stress

that contrarily to the case of

R2n,

in the situation of a non-Abelian group action,

this is an highly non-trivial feature of our deformation theory.

The memoir is organized as follows.

In chapter 2, we start by introducing non-Abelian and unbounded versions of

Fr´ echet-valued symbol spaces on a Lie group G, with Lie algebra g:

Bμ(G,

E) := f ∈

C∞(G,

E) : ∀X ∈ U(g), ∀j ∈ N, ∃C 0 : Xf

j

≤ C μj ,

where E is a Fr´ echet space, μ := {μj}j∈N is a countable family of specific posi-

tive functions on G, called weights (see Definition 2.1) aﬃliated to a countable set

of semi-norms {. j}j∈N defining the Fr´ echet topology on E and where X is the

left-invariant differential operator on G associated to an element X ∈ U(g). For

example,

B1(G,

C) consists of the smooth vectors for the right regular representa-

tion of G on the space of bounded right-uniformly continuous functions on G (the

uniform structures on G are generally not balanced in our non-Abelian situation)

and it coincides with Laurent Schwartz’s space B when G =

Rn.

We shall also

mention that function spaces on Lie groups of a similar type are considered in [30]

and in [20] in the context of actions of

Rd

on locally convex algebras.

We then define a notion of oscillatory integrals on Lie groups G that are en-

dowed with a specific type of smooth function S ∈ C∞(G, R) (see Definitions 2.17,

2.22 and 2.24). We call such a pair (G, S) an admissible tempered pair. The main

result of this chapter is that associated to an admissible tempered pair (G, S), and

given a growth-controlled function m, the oscillatory integral

D(G, E) → E , F →

G

m

eiS

F ,

canonically extends from D(G, E), the space of smooth compactly supported func-

tions, to our symbol space

Bμ(G,

E). This construction is explained in Definition