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 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 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
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