# American Institute of Mathematical Sciences

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

## 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

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

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.

Citation: Catarina Carvalho, Victor Nistor, Yu Qiao. Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras. Electronic Research Announcements, 2017, 24: 68-77. doi: 10.3934/era.2017.24.008
 [1] B. Ammann, A. D. Ionescu and V. Nistor, Sobolev spaces on Lie manifolds and regularity for polyhedral domains, Doc. Math., 11 (2006), 161-206 (electronic).   Google Scholar [2] B. Ammann, R. Lauter and V. Nistor, Pseudodifferential operators on manifolds with a Lie structure at infinity, Ann. of Math.(2), 165 (2007), 717-747.  doi: 10.4007/annals.2007.165.717.  Google Scholar [3] I. Androulidakis and G. Skandalis, Pseudodifferential calculus on a singular foliation, J. Noncommut. Geom., 5 (2011), 125-152.  doi: 10.4171/JNCG/72.  Google Scholar [4] C. Carvalho, V. Nistor and Yu Qiao, Fredholm conditions on non-compact manifolds: Theory and examples, ArXiv and Hal preprint 2017, submitted. Google Scholar [5] S. Echterhoff, The primitive ideal space of twisted covariant systems with continuously varying stabilizers, Math. Ann., 292 (1992), 59-84.  doi: 10.1007/BF01444609.  Google Scholar [6] R. Exel, Invertibility in groupoid $C^*$-algebras, in Operator Theory, Operator Algebras and Applications, Oper. Theory Adv. Appl., 242, Birkhäuser/Springer, Basel, 2014,173–183. doi: 10.1007/978-3-0348-0816-3_9.  Google Scholar [7] E. Gootman and J. Rosenberg, The structure of crossed product $C^{*}$ -algebras: a proof of the generalized Effros-Hahn conjecture, Invent. Math., 52 (1979), 283-298.  doi: 10.1007/BF01389885.  Google Scholar [8] N. Groẞe and C. Schneider, Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces, Math. Nachr., 286 (2013), 1586-1613.  doi: 10.1002/mana.201300007.  Google Scholar [9] M. Ionescu and D. Williams, The generalized Effros-Hahn conjecture for groupoids, Indiana Univ. Math. J., 58 (2009), 2489-2508.  doi: 10.1512/iumj.2009.58.3746.  Google Scholar [10] M. Ionescu and D. Williams, Irreducible representations of groupoid $C^*$ -algebras, Proc. Amer. Math. Soc., 137 (2009), 1323-1332.  doi: 10.1090/S0002-9939-08-09782-7.  Google Scholar [11] M. Khoshkam and G. Skandalis, Regular representation of groupoid $C^*$ -algebras and applications to inverse semigroups, J. Reine Angew. Math., 546 (2002), 47-72.  doi: 10.1515/crll.2002.045.  Google Scholar [12] R. Lauter, B. Monthubert and V. Nistor, Pseudodifferential analysis on continuous family groupoids, Doc. Math., 5 (2000), 625-655 (electronic).   Google Scholar [13] R. Lauter and V. Nistor, Analysis of geometric operators on open manifolds: A groupoid approach, in Quantization of Singular Symplectic Quotients, Progr. Math., 198, Birkhäuser, Basel, 2001,181–229.  Google Scholar [14] K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, volume 213 of LMS Lect. Note Series, Cambridge U. Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar [15] I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567-593.  doi: 10.1353/ajm.2002.0019.  Google Scholar [16] B. Monthubert, Pseudodifferential calculus on manifolds with corners and groupoids, Proc. Amer. Math. Soc., 127 (1999), 2871-2881.  doi: 10.1090/S0002-9939-99-04850-9.  Google Scholar [17] P. S. Muhly, J. Renault and D. Williams, Continuous-trace groupoid $C^*$ -algebras. Ⅲ, Trans. Amer. Math. Soc., 348 (1996), 3621-3641.  doi: 10.1090/S0002-9947-96-01610-8.  Google Scholar [18] V. Nistor, Desingularization of Lie groupoids and pseudodifferential operators on singular spaces, to appear in Communications in Analysis and Geometry, arXiv: 1512.08613 [math. DG]. Google Scholar [19] V. Nistor and N. Prudhon, Exhausting families of representations and spectra of pseudodifferential operators, to appear in J. Oper. Theory, arXiv: 1411.7921 [math. OA]. Google Scholar [20] V. Nistor, A. Weinstein and P. Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math., 189 (1999), 117-152.  doi: 10.2140/pjm.1999.189.117.  Google Scholar [21] J. Renault, A Groupoid Approach to $C^{*}$ -Algebras Lecture Notes in Mathematics, 793, Springer, Berlin, 1980.  Google Scholar [22] J. Renault, Représentation des produits croisés d'algébres de groupoïdes, J. Operator Theory, 18 (1987), 67-97.   Google Scholar [23] J. Renault, The ideal structure of groupoid crossed product $C^*$-$algebras, J. Operator Theory, 25 (1991), 3-36. Google Scholar [24] J. Renault, Topological amenability is a Borel property, Math. Scand., 117 (2015), 5-30. doi: 10.7146/math.scand.a-22235. Google Scholar [25] S. Roch, Algebras of approximation sequences: structure of fractal algebras, in Singular Integral Operators, Factorization and Applications, Oper. Theory Adv. Appl., 142, Birkhäuser, Basel, 2003,287–310. Google Scholar [26] A. Sims and D. Williams, Amenability for Fell bundles over groupoids, Illinois J. Math., 57 (2013), 429-444. Google Scholar [27] E. Van Erp and R. Yuncken, A groupoid approach to pseudodifferential operators, arXiv: 1511.01041 [math. DG], 2015. Google Scholar [28] D. Williams, Crossed Products of$C{^*}$-Algebras, Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/134. Google Scholar show all references ##### References:  [1] B. Ammann, A. D. Ionescu and V. Nistor, Sobolev spaces on Lie manifolds and regularity for polyhedral domains, Doc. Math., 11 (2006), 161-206 (electronic). Google Scholar [2] B. Ammann, R. Lauter and V. Nistor, Pseudodifferential operators on manifolds with a Lie structure at infinity, Ann. of Math.(2), 165 (2007), 717-747. doi: 10.4007/annals.2007.165.717. Google Scholar [3] I. Androulidakis and G. Skandalis, Pseudodifferential calculus on a singular foliation, J. Noncommut. Geom., 5 (2011), 125-152. doi: 10.4171/JNCG/72. Google Scholar [4] C. Carvalho, V. Nistor and Yu Qiao, Fredholm conditions on non-compact manifolds: Theory and examples, ArXiv and Hal preprint 2017, submitted. Google Scholar [5] S. Echterhoff, The primitive ideal space of twisted covariant systems with continuously varying stabilizers, Math. Ann., 292 (1992), 59-84. doi: 10.1007/BF01444609. Google Scholar [6] R. Exel, Invertibility in groupoid$C^*$-algebras, in Operator Theory, Operator Algebras and Applications, Oper. Theory Adv. Appl., 242, Birkhäuser/Springer, Basel, 2014,173–183. doi: 10.1007/978-3-0348-0816-3_9. Google Scholar [7] E. Gootman and J. Rosenberg, The structure of crossed product$C^{*} $-algebras: a proof of the generalized Effros-Hahn conjecture, Invent. Math., 52 (1979), 283-298. doi: 10.1007/BF01389885. Google Scholar [8] N. Groẞe and C. Schneider, Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces, Math. Nachr., 286 (2013), 1586-1613. doi: 10.1002/mana.201300007. Google Scholar [9] M. Ionescu and D. Williams, The generalized Effros-Hahn conjecture for groupoids, Indiana Univ. Math. J., 58 (2009), 2489-2508. doi: 10.1512/iumj.2009.58.3746. Google Scholar [10] M. Ionescu and D. Williams, Irreducible representations of groupoid$C^*$-algebras, Proc. Amer. Math. Soc., 137 (2009), 1323-1332. doi: 10.1090/S0002-9939-08-09782-7. Google Scholar [11] M. Khoshkam and G. Skandalis, Regular representation of groupoid$C^*$-algebras and applications to inverse semigroups, J. Reine Angew. Math., 546 (2002), 47-72. doi: 10.1515/crll.2002.045. Google Scholar [12] R. Lauter, B. Monthubert and V. Nistor, Pseudodifferential analysis on continuous family groupoids, Doc. Math., 5 (2000), 625-655 (electronic). Google Scholar [13] R. Lauter and V. Nistor, Analysis of geometric operators on open manifolds: A groupoid approach, in Quantization of Singular Symplectic Quotients, Progr. Math., 198, Birkhäuser, Basel, 2001,181–229. Google Scholar [14] K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, volume 213 of LMS Lect. Note Series, Cambridge U. Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883. Google Scholar [15] I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567-593. doi: 10.1353/ajm.2002.0019. Google Scholar [16] B. Monthubert, Pseudodifferential calculus on manifolds with corners and groupoids, Proc. Amer. Math. Soc., 127 (1999), 2871-2881. doi: 10.1090/S0002-9939-99-04850-9. Google Scholar [17] P. S. Muhly, J. Renault and D. Williams, Continuous-trace groupoid$C^*$-algebras. Ⅲ, Trans. Amer. Math. Soc., 348 (1996), 3621-3641. doi: 10.1090/S0002-9947-96-01610-8. Google Scholar [18] V. Nistor, Desingularization of Lie groupoids and pseudodifferential operators on singular spaces, to appear in Communications in Analysis and Geometry, arXiv: 1512.08613 [math. DG]. Google Scholar [19] V. Nistor and N. Prudhon, Exhausting families of representations and spectra of pseudodifferential operators, to appear in J. Oper. Theory, arXiv: 1411.7921 [math. OA]. Google Scholar [20] V. Nistor, A. Weinstein and P. Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math., 189 (1999), 117-152. doi: 10.2140/pjm.1999.189.117. Google Scholar [21] J. Renault, A Groupoid Approach to$C^{*} $-Algebras Lecture Notes in Mathematics, 793, Springer, Berlin, 1980. Google Scholar [22] J. Renault, Représentation des produits croisés d'algébres de groupoïdes, J. Operator Theory, 18 (1987), 67-97. Google Scholar [23] J. Renault, The ideal structure of groupoid crossed product$C^*$-$ algebras, J. Operator Theory, 25 (1991), 3-36.   Google Scholar [24] J. Renault, Topological amenability is a Borel property, Math. Scand., 117 (2015), 5-30.  doi: 10.7146/math.scand.a-22235.  Google Scholar [25] S. Roch, Algebras of approximation sequences: structure of fractal algebras, in Singular Integral Operators, Factorization and Applications, Oper. Theory Adv. Appl., 142, Birkhäuser, Basel, 2003,287–310.  Google Scholar [26] A. Sims and D. Williams, Amenability for Fell bundles over groupoids, Illinois J. Math., 57 (2013), 429-444.   Google Scholar [27] E. Van Erp and R. Yuncken, A groupoid approach to pseudodifferential operators, arXiv: 1511.01041 [math. DG], 2015. Google Scholar [28] D. Williams, Crossed Products of $C{^*}$ -Algebras, Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/134.  Google Scholar
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