Jean Ludwig

A class of symmetric and a class of Wiener group algebras

Abstract It is shown that connected groups of polynomial growth and compact extensions of nilpotent group have symmetric group algebras and that the group algebras of discrete solvable groups have the Wiener property.

Polynomial growth and ideals in group algebras

Let G be a locally compact group with polynomial growth and symmetric group algebra L1 (G). To every closed subset C of Prim* (L1(G)), there exists a smallest twosided closed ideal j (C) in L1(G), whose hull is equal to C. If H is a closed normal subgroup of G, then H1 is a set of synthesis in Prim* (L1(G)).

La convexité de l'application moment d'un groupe de Lie

Let π be a unitary representation of a Lie group G. The moment mapping Ψπ of π assigns to every C∞ vector ξ in the Hilbert space H of π the linear functional Ψπ(ξ) of the Lie algebra g of G by the rule ψπ(ξ)(X)=1i〈dπ(X)ξ, ξ〉H, X ϵ g In this paper, we study the moment set Iπ of π, i.e., the closure of the image of Ψπ. It is shown that for solvable G, Iπ is always convex and that if furthermore π is irreducible, then Iπ is the closure (in g∗) of the convex hull of the Kirillov-Pukanszky orbit of π. If G is compact and if π is irreducible, then we show that Iπ is the convex hull of the orbit of the highest weight Λ of π, if and only if the number Πi = 1n 〈2Λ − αi, αi〉 is different from 0. Here α1, …, αn denote the simple roots of g.

Estimate of the Lp-Fourier transform norm on nilpotent Lie groups

Abstract Let 1 1 p + 1 q =1 . It is well known that the norm of the Lp-Fourier transform of the additive group R n is || F p ( R n )||=A p n , where A p = p 1 p q 1 q 1 2 . For a nilpotent Lie group G, we obtain the estimate || F p (G)||⩽A p 2 dim G−m 2 , where m is the maximal dimension of the coadjoint orbits. Such a result was known only for some particular cases.

Lie groups whose coadjoint orbits are of dimension smaller or equal to two

We give a complete classification of the class of connected, simply connected Lie groups whose coadjoint orbits are of dimension smaller or equal to two.

On the behaviour of sequences in the dual of a nilpotent Lie group.

Let A be a separable C*-algebra and let C he a subset of the dual space ,4 of A. Denote by A + the cone of the positive elements in A. We say that x e A + is an element with bounded trace on C, if the function n -~ tr it(x) is bounded on C. Put BT~:= (xeA+/x with bounded trace on (7} and let BTc be the two-sided selfadjoint ideal in A generated by BT + . Let us set hc for the hull of BTc in ,4, i.e.

On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transformℰ was introduced by D. Arnal and J. C. Cortet,ℰ:L(G) →C∞(V,L(∝2d)), whereL(G) is the Schwartz space ofG andV × ∝2k is aG-invariant Zariski open set ing* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.

On the Hilbert-Schmidt semi-norms ofL1 of a nilpotent Lie group

In (2) appears the convolution product o f f * with f on the (non-abelian) group G combined with the abelian Fourier-transform on r A consequence of this mixture of non-abelian with abelian operations is the fact that the integrand ( f * f*) o exp) need not be nonnegative. This prohibits the extension of (2) to general Ll-functions. It makes it also difficult to give estimations of IH(f)la.s. in terms o f / I [7]. It would be much better if one could replace (2) by

Représentations irréductibles bornées des groupes de Lie exponentiels

Let G be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations (T,U) of G on a Banach space U by giving a G-orbit in n∗ (n being the nilradical of g), a topologically irreducible representation of L1(Rn, ω), for a certain weight ω and a certain n ∈ N, and a topologically simple extension norm. If G is not symmetric, i.e., if the weight ω is exponential, we get a new type of representations which are fundamentally different from the induced representations. Recu par les editeurs 12 novembre, 1999. Etude effectuee dans le cadre du projet de recherche MEN/CUL/98/007. Classification (AMS) par sujet: 43A20. Mots cles: groupe de Lie resoluble exponentiel, representation bornee topologiquement irreductible, orbite, norme d’extension, sous-espace invariant, ideal premier, ideal primitif. c ©Societe Mathematique du Canada 2001. 944

Déformations covariantes sur les orbites polarisées d'un groupe de Lie

Abstract We prove the existence of covariant star-products on the orbits of the coadjoint representation of a Lie group which admits polarisations.