Exhausting families of representations and spectra of pseudodifferential operators

Families of representations of suitable Banach algebras provide a powerful tool in the study of the spectral theory of (pseudo)differential operators and of their Fredholmness. We introduce the new concept of an exhausting family of representations of a C*-algebra A. An {\em exhausting family} of representations of a C*-algebra A is a set F of representations of A with the property that every irreducible representation of A is weakly contained in some \phi \in F. An exhausting family F of representations of A has the property that `"a \in A is invertible if, and if, \phi(a) is invertible for any \phi \in F." Consequently, the spectrum of a is given by \Spec(a) = \cup_{\phi \in F} \Spec(\phi(a)). In other words, every exhausting family of representations is invertibility sufficient, a concept introduced by Roch in 2003. We prove several properties of exhausting families and we provide necessary and sufficient conditions for a family of representations to be exhausting. Using results of Ionescu and Williams (2009), we show that the regular representations of amenable, second countable, locally compact groupoids with a Haar system form an exhausting family of representations. If $A$ is a separable C*-algebra, we show that a family F of representations of $A$ is exhausting if, and only if, it is invertibility sufficient. However, this result is not true, in general, for non-separable C*-algebras. With an eye towards applications, we extend our results to the case of unbounded operators. A typical application of our results is to parametric families of differential operators arising in the analysis on manifolds with corners, in which case we recover the fact that a parametric operator F is invertible if, and only if, its Mellin transform is invertible. In view of possible applications, we have tried to make this paper accessible to non-specialists in C*-algebras.Comment: Additional reference

Exhaustive families of representations of C∗C^*-algebras associated to NN-body Hamiltonians with asymptotically homogeneous interactions

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study $N$-body type Hamiltonians with interactions. More precisely, let $Y$ be a linear subspace of a finite dimensional Euclidean space $X$, and $v_Y$ be a continuous function on $X/Y$ that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form $H = - \Delta + \sum_{Y \in S} v_Y$, where the subspaces $Y$ belong to some given family S of subspaces. We prove results on the spectral theory of the Hamiltonian when $S$ is any family of subspaces and extend those results to other operators affiliated to a larger algebra of pseudo-differential operators associated to the action of $X$ introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.Comment: 5 page

Exhausting families of representations and spectra of pseudodifferential operators

A powerful tool in the spectral theory and the study of Fred-holm conditions for (pseudo)differential operators is provided by families of representations of a naturally associated algebra of bounded operators. Motivated by this approach, we define the concept of an strictly norming family of representations of a C *-algebra A. Let F be a strictly norming family of representations of A. We have then that an abstract differential operator D affiliated to A is invertible if, and only if, φ(D) is invertible for all φ ∈ F. This property characterizes strictly norming families of representations. We provide necessary and sufficient conditions for a family of representations to be strictly norming. If A is a separable C *-algebra, we show that a family F of representations is strictly norming if, and only if, every irreducible representation of A is weakly contained in a representation φ ∈ F. However, this result is not true, in general, for non-separable C *-algebras. A typical application of our results is to parametric families of differential operators arising in the analysis on manifolds with corners, in which case we recover the fact that a parametric operator P is invertible if, and only if, its Mellin transform P (τ) is invertible, for all τ ∈ R n. The paper is written to be accessible to non-specialists in C *-algebras

C*-algèbres de Sp(n,1) et K-théorie

Cette thèse est consacrée à la K-théorie des C*-algèbres de groupes, maximales et réduites. Nous nous intéressons plus particulièrement aux groupes d'isométries d'un espace hyperbolique quaternionien, Sp(n,1). Nous donnons une description explicite de la K-théorie de la C*-algèbre maximale de ces groupes en fonction de certaines de leurs représentations unitaires irréductibles, dites séries isolées. Ces résultats servent ensuite à calculer l'image de l'application d'assemblage de Baum-Connes, qui à chaque représentation d'un sous-groupe compact maximal associe (dans notre cas) l'indice en K-théorie d'un opérateur de Dirac sur l'espace hyperbolique. Ce calcul, en utilisant alors des propriétés d'universalité de l'opérateur de Dirac, nous permet de calculer l'indice d'un opérateur défini par Wong, et lié à la construction géométrique (induction cohomologique) des séries isolées. Nous déterminons également la structure de la C*-algèbre maximale des groupes Sp(n,1).This thesis is devoted to K-theory for groups C*-algebras, maximal and reduced. We are interested in isomerty groups of quaternionic hyperbolic spaces, Sp(n,1). We describe explicitely the K-theory of the maximal C*-algebra of these groups in terms of some of their unitary irreducible representations, called isolated series. These results are then used to compute the range of the Baum-Connes assembly map, which in this case associates to each representation of a maximal compact subgroup an element of the K-theory namely the index of a Dirac operator acting on the hyperbolic space. Using universality property of these operators, we are then able to compute the index of another operator defined by Wong that is related to the geometric construction by cohomological induction of isolated series. We also completely describe the structure of the maximal C*-algebra of the groups Sp(n,1).STRASBOURG-Sc. et Techniques (674822102) / SudocSudocFranceF

Weed dynamic in Conservation Agriculture: experiences from the Isite-BFC regional network of farmers and cropping system experiments on agroecology in France.

ISBN: 978-84-09-37744-2International audienceConservation Agriculture (CA) relies on three fundamental pillars: diversified crop rotation, permanent soil coverand no soil disturbance. Weed control relies on few tools because pre-sowing tillage, pre-emergence herbicidespraying and in-crop mechanical weeding are not possible. This could lead to drastic changes in weed communitiesand quickly after the transition to CA, with fewer annual species (weed seeds remain on the soil surface, a conditiondeemed to be unfavourable to weed germination) and higher perennial species. However, the implementation ofCA principles could be transcribed into a wide array of cropping systems because the objectives of farmers differ,and/or because systems are implemented in different production situations (e.g., associated or not to livestock, soiltype, irrigation). Therefore, the Isite-BFC regional network gathers CA farmers and experimenters from cooperatives and research institute (INRAE) to share their experiences, detailed practices and weed surveys initiated since2007 in some sites. Weed diversity was high in all systems compared to what is known from tillage-based agriculture. Weed community changes over time depending on the diversity of crop rotation tested and initial weedingpressure. Since CA is challenged by potential glyphosate ban in Europe, the application of glyphosate was stoppedin 2018 in some sites and thus, cropping systems were redesigned accordingly to ensure weed management overthe long run, economic profitability and multiperformance