Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras

Catarina Carvalho; Victor Nistor; Yu Qiao

doi:10.3934/era.2017.24.008

Article Contents

2017, Volume 24: 68-77. Doi: 10.3934/era.2017.24.008

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Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras

1.
Dep. Matemática, Instituto Superior Técnico, University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
2.
Université de Lorraine, UFR MIM, Ile du Saulcy, CS 50128,57045 METZ, France
3.
Pennsylvania State University, Math. Dept., University Park, PA 16802, USA
4.
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, China

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)..

Received: October 19, 2016

Revised: July 28, 2017

Published: August 2017

Qiao was partially supported by NSF of China (11301317,11571211).

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Abstract

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.

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

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