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Let Mθ=(R⋉C2,·θ)(θan irrational number), be the Mautner group. We describe the C∗-algebra of Mθas a subalgebra of C0(C2,B(L2(R)))

Abstract.We continue the description of the C*-algebras of low-dimensional nilpotent Lie groups as algebras of operator fields defined over their spectra. We show here that the C*-algebras of the three 6-dimensional nilpotent Lie groups having the group G5,2as quotient group modulo the centre have norm controlled dual limits.

Abstract.Soit un groupe de Lie nilpotent d'algèbre de Lie . Soit le dual unitaire de Get prenons une représentation dont l'orbite coadjointe associée se note . Comme la réciprocité de Frobenius le suggère, il existe une sorte de dualité entre l'induction et la restriction de représentations, spécialement dans les formules de désintégration en irréductibles. Soient Kun sous-groupe analytique de Get la restriction de à K. Il est bien connu que, selon la géométrie de et la structure de K, la restriction est à multiplicités soit uniformément finies soit uniformément égales à l'infini. Dans ce papier, nous écrivons les formules de Plancherel et de Penney–Plancherel pour la restriction supposée être à multiplicités finies. La méthode des orbites nous est alors très utile pour accomplir cette étude. Let be a nilpotent Lie group with Lie algebra . Let be the unitary dual of Gand associated to a coadjoint orbit . As suggested by the Frobenius reciprocity, there is a sort of duality between the induction and the restriction of unitary representations, namely in the disintegration formulae into irreducibles. Let Kbe an analytic subgroup of Gand be the restriction of to K. It is well known, depending upon the geometry of and the structure of K, that the multiplicities of are either uniformly finite or uniformly infinite. In this paper, we write down the Plancherel and the Penney–Plancherel formulae for supposed to be of finite multiplicities. The orbit method turns out to be very useful for our study.

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transformE was introduced by D. Arnal and J. C. Cortet,E:L(G) ?C8(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.

We say that a locally compact groupG hasT1 primitive ideal space if the groupC*-algebra,C*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC*(G), denoted by PrimG. This means of course that every primitive ideal inC*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT1 primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT1 primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC*(G) are always maximal which implies of course that PrimG isT1.

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)).

For a closed normal subgroupN of a locally compact groupG view a closed subset of Prim*L1(G/N) as a subsetE of Prim*L1(G) in the canonical way and writeN⊥ for Prim*L1(G/N) as a subset of Prim*L1(G); then the injection theorem says: IfE is spectral (i.e. of synthesis), then is so; and if andN⊥ are spectral, thenE is too. In case of a group of polynomial growth with symmetricL1-algebra, where smallest idealsj (E) with given hulls exist, it is known thatN⊥ is always spectral. For a closed,G-invariant subsetF of Prim*L1(N) define a closed subsetE of Prim*L1(G) by . Denote by e (I') the ideal generated byC00(G)*I', where theG-invariant idealI' ofL1(N) is viewed as a subset of measures onG, then the projection theorem states: IfE is spectral, thenF is so, and ifF is spectral withe (j (F))=j (E) thenE is spectral. All assumptions are fulfilled for instance, ifG andN are of polynomial growth with symmetricL1-algebra and eitherSIN-groups or solvable.

Nous donnons dans ce papier une nouvelle caractérisation des convoluteurs de Schwartz pour un groupe de Lie nilpotentGd'algèbre de Lie g. Nous indiquons ensuite une condition suffisante pour qu'une fonctionϕ∈C∞(g*) soit la transformée de Fourier d'un convoluteur et nous exprimons une condition plus précise dans le cas du groupe de Heisenberg.