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Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras

Manuscripts available from http://iecl.univ-lorraine.fr/ Victor.Nistor.
Carvalho was partially supported by Fundação para a Ciência e a Tecnologia (Portugal) UID/MAT/04721/2013.
Nistor has been partially supported by ANR-14-CE25-0012-01 (SINGSTAR)..

Qiao was partially supported by NSF of China (11301317,11571211).
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  • We characterize the groupoids for which an operator is Fredholm if and only if its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called Fredholm. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean or asymptotically hyperbolic, on products of such manifolds, and on many other non-compact manifolds. Moreover, we show that the desingularization of groupoids preserves the class of Fredholm groupoids.

    Mathematics Subject Classification: 46L05(primary), 45B05, 47L80, 58J40, 46N20.


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