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.. .. ,

. .. C-*--algebra,

. .. Elliptic-operator, 42 exhaustive family of morphisms, p.43

. .. Fredholm-operator, , p.22

G. .. Fredholm, , vol.63, p.95

. .. Lie,

. .. Hilbert-module, 24 induction of representations for C * -algebras

. .. Jacobson-topology, 47 representation of a C * -algebra, vol.63, p.95

. .. Weighted-sobolev-spaces, , p.11

, Analyse sur les espaces singuliers et théorie de l'indice

, Mots-clefs : analyse globale, opérateurs elliptiques, algèbres d'opérateurs, théorie de l'indice

, bien connue dans le cadre lisse, à des domaines dits singuliers. Les méthodes utilisées reposent d'une part sur l'emploi d'algèbres d'opérateurs et d'outils issus de la géométrie non commutative, d'autre part sur l'introduction de calculs pseudodifférentiels adaptés à la géométrie du domaine, souvent via un groupoïde qui résout les singularités. La première partie de la thèse s'intéresse à l'étude d'une classe particulière de ces groupoïdes, dits Fredholm

, Je conclus cette partie avec la résolution d'un problème aux limites pour un domaine à singularité de type cusp rotationnel. La seconde partie s'intéresse aux opérateurs équivariants sur des variétés compactes, sous l'action d'un groupe fini. On répond à la question suivante : étant donnée une représentation irréductible du groupe, à quelle condition un opérateur différentiel est-il Fredholm entre les composantes isotypiques correspondantes des espaces de Sobolev ?, Dans un travail commun avec A. Baldare, M. Lesch et V. Nistor, nous définissons une notion correspondante d'ellipticité associée à une représentation irréductible fixée et montrons qu'elle caractérise les opérateurs de Fredholm