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 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 ) := : 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) ,
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