The Fredholm property for groupoids is a local property. (English) Zbl 1430.58020
Fredholm Lie groupoids were introduced as a tool for the study of partial differential equations on open manifolds. This article extends the definition to the setting of locally compact groupoids and proves that “the Fredholm property is local” in the following sense: let \(\mathcal{G}\) be a topological groupoid with unit space \(X\), and \(\{Ui\}_{i\in I}\) an open cover of \(X\). The author shows that \(\mathcal{G}\) is a Fredholm groupoid if, and only if, its reductions \(\mathcal{G}_{U_i}^{U_i}\) are Fredholm groupoids for all \(i\in I\).
Reviewer: Marta Macho Stadler (Leioa)
MSC:
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 58H05 | Pseudogroups and differentiable groupoids |
| 46L05 | General theory of \(C^*\)-algebras |
| 47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |
Keywords:
Fredholm operator; Fredholm groupoid; \( C^*\)-algebra; pseudodifferential operator; primitive spectrum; locally compact groupoidReferences:
| [1] | Ammann, B.; Lauter, R.; Nistor, V., On the geometry of Riemannian manifolds with a Lie structure at infinity, Int. J. Math. Math. Sci., 1-4, 161-193 (2004) · Zbl 1071.53020 · doi:10.1155/S0161171204212108 |
| [2] | Atiyah, MF; Patodi, VK; Singer, IM, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Camb. Philos. Soc., 79, 71-99 (1976) · Zbl 0325.58015 · doi:10.1017/S0305004100052105 |
| [3] | Atiyah, MF; Singer, IM, The index of elliptic operators. I, Ann. Math. Second Ser., 87, 484-530 (1968) · Zbl 0164.24001 · doi:10.2307/1970715 |
| [4] | Bigonnet, B.; Pradines, J., Graphe d’un feuilletage singulier, C. R. Acad. Sci. Paris Sér. I Math., 300, 439-442 (1985) · Zbl 0581.57013 |
| [5] | Bohlen, K.; Schrohe, E., Getzler rescaling via adiabatic deformation and a renormalized index formula, J. Math. Pures Appl., 9, 220-252 (2018) · Zbl 1403.58008 · doi:10.1016/j.matpur.2017.07.016 |
| [6] | Bohlen, K., Schulz, R.: Quantization on manifolds with an embedded submanifold (2017). arXiv e-prints, arXiv:1710.02294 |
| [7] | Carvalho, C., Côme, R., Qiao, Y.: Gluing action groupoids: Fredholm conditions and layer potentials (2018). arXiv e-prints, arXiv:1811.07699v1 · Zbl 1449.35467 |
| [8] | Carvalho, Catarina; Nistor, Victor; Qiao, Yu, Fredholm Conditions on Non-compact Manifolds: Theory and Examples, 79-122 (2018), Cham · Zbl 1446.58011 · doi:10.1007/978-3-319-72449-2_4 |
| [9] | Carvalho, C.; Qiao, Y., Layer potentials \(C^*\)-algebras of domains with conical points, Cent. Eur. J. Math., 11, 27-54 (2013) · Zbl 1275.46054 · doi:10.2478/s11533-012-0066-y |
| [10] | Connes, A.: A survey of foliations and operator algebras. In: Operator Algebras and Applications, Part I (Kingston, Ont., 1980), Proceedings of Symposia in Pure Mathematics, vol. 38, pp. 521-628. American Mathematical Society, Providence (1982) · Zbl 0531.57023 |
| [11] | Crainic, M.: Cyclic cohomology of étale groupoids: the general case. \(K\)-Theory 17(4), 319-362 (1999) · Zbl 0937.19007 · doi:10.1023/A:1007756702025 |
| [12] | Crainic, M.; Fernandes, RL, Integrability of Lie brackets, Ann. Math. (2), 157, 575-620 (2003) · Zbl 1037.22003 · doi:10.4007/annals.2003.157.575 |
| [13] | Dauge, M.: Elliptic boundary value problems on corner domains. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988). (Smoothness and asymptotics of solutions) · Zbl 0668.35001 |
| [14] | Debord, C., Holonomy groupoids of singular foliations, J. Differ. Geom., 58, 467-500 (2001) · Zbl 1034.58017 · doi:10.4310/jdg/1090348356 |
| [15] | Debord, C.; Lescure, J-M; Rochon, F., Pseudodifferential operators on manifolds with fibred corners, Université de Grenoble. Annales de l’Institut Fourier, 65, 1799-1880 (2015) · Zbl 1377.58025 · doi:10.5802/aif.2974 |
| [16] | Debord, C.; Skandalis, G., Adiabatic groupoid, crossed product by \(\mathbb{R}_+^{\ast }\) and pseudodifferential calculus, Adv. Math., 257, 66-91 (2014) · Zbl 1300.58007 · doi:10.1016/j.aim.2014.02.012 |
| [17] | Debord, C., Skandalis, G.: Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus (2017). arXiv e-prints, arXiv:1705.09588v2 · Zbl 1460.58013 |
| [18] | Dixmier, J.: \(C^*\)-algebras. North-Holland Publishing Co., Amsterdam (1977). Translated from the French by Francis Jellett, North-Holland Mathematical Library, vol. 15 (1977) · Zbl 0372.46058 |
| [19] | Georgescu, V.; Iftimovici, A., Localizations at infinity and essential spectrum of quantum Hamiltonians. I. General theory, Rev. Math. Phys., 18, 417-483 (2006) · Zbl 1109.47004 · doi:10.1142/S0129055X06002693 |
| [20] | Gilkey, P.B.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11. Publish or Perish, Inc., Wilmington (1984) · Zbl 0565.58035 |
| [21] | Gualtieri, M.; Li, S., Symplectic groupoids of log symplectic manifolds, Int. Math. Res. Not. IMRN, 11, 3022-3074 (2014) · Zbl 1305.22004 · doi:10.1093/imrn/rnt024 |
| [22] | Hebey, Emmanuel; Robert, Frédéric, Sobolev spaces on manifolds, 375-415 (2008) · Zbl 1236.58021 · doi:10.1016/B978-044452833-9.50008-5 |
| [23] | Hilsum, M.; Skandalis, G., Morphismes \(K\)-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. École Norm. Sup. (4), 20, 325-390 (1987) · Zbl 0656.57015 · doi:10.24033/asens.1537 |
| [24] | Hörmander, L.: The analysis of linear partial differential operators. III. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274. Springer, Berlin (1985). (Pseudodifferential operators) (1985) · Zbl 0601.35001 |
| [25] | Ionescu, M.; Williams, D., The generalized Effros-Hahn conjecture for groupoids, Indiana Univ. Math. J., 58, 2489-2508 (2009) · Zbl 1213.46065 · doi:10.1512/iumj.2009.58.3746 |
| [26] | Joyce, D.: On manifolds with corners. In: Advances in Geometric Analysis, Advanced Lectures in Mathematics (ALM), vol. 21, pp. 225-258. Int. Press, Somerville (2012) · Zbl 1317.58001 |
| [27] | Khoshkam, M.; Skandalis, G., Regular representation of groupoid \(C^*\)-algebras and applications to inverse semigroups, J. Reine Angew. Math., 546, 47-72 (2002) · Zbl 1029.46082 |
| [28] | Kondrat’ ev, VA, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč, 16, 209-292 (1967) · Zbl 0162.16301 |
| [29] | Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, vol. 52. American Mathematical Society, Providence (1997) · Zbl 0947.35004 |
| [30] | Lauter, R., Pseudodifferential analysis on conformally compact spaces, Mem. Am. Math. Soci., 163, xvi+92 (2003) · Zbl 1024.58009 |
| [31] | Lauter, R.; Monthubert, B.; Nistor, V., Pseudodifferential analysis on continuous family groupoids, Doc. Math., 5, 625-655 (2000) · Zbl 0961.22005 |
| [32] | Mackenzie, K.: Lie groupoids and Lie algebroids in differential geometry. London Mathematical Society Lecture Note Series, vol. 124. Cambridge University Press, Cambridge (1987) · Zbl 0683.53029 |
| [33] | Mazzeo, R., Elliptic theory of differential edge operators. I, Commun. Partial Differ. Equ., 16, 1615-1664 (1991) · Zbl 0745.58045 · doi:10.1080/03605309108820815 |
| [34] | Melrose, R.B.: The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics, vol. 4 . A K Peters, Ltd., Wellesley (1993) · Zbl 0796.58050 · doi:10.1201/9781439864609 |
| [35] | Melrose, R.B.: Geometric scattering theory. Stanford Lectures. Cambridge University Press, Cambridge (1995) · Zbl 0849.58071 |
| [36] | Mougel, J.; Prudhon, N., Exhaustive families of representations of \(C^\ast \)-algebras associated with \(N\)-body Hamiltonians with asymptotically homogeneous interactions, C. R. Math. Acad. Sci. Paris, 357, 200-204 (2019) · Zbl 1409.81046 · doi:10.1016/j.crma.2019.01.010 |
| [37] | Măntoiu, M.: C*-algebraic spectral sets, twisted groupoids and operators (2018). arXiv e-prints, arXiv:1809.03347v2 · Zbl 1524.22011 |
| [38] | Muhly, PS; Renault, JN; Williams, DP, Equivalence and isomorphism for groupoid \(C^\ast \)-algebras, J. Oper. Theory, 17, 3-22 (1987) · Zbl 0645.46040 |
| [39] | Muhly, PS; Renault, JN; Williams, DP, Continuous-trace groupoid \(C^\ast \)-algebras. III, Trans. Am. Math. Soc., 348, 3621-3641 (1996) · Zbl 0859.46039 · doi:10.1090/S0002-9947-96-01610-8 |
| [40] | Măntoiu, M., \(C^\ast \)-algebras, dynamical systems at infinity and the essential spectrum of generalized Schrödinger operators, J. Reine Angew. Math., 550, 211-229 (2002) · Zbl 1036.46052 |
| [41] | Măntoiu, M.; Purice, R.; Richard, S., Spectral and propagation results for magnetic Schrödinger operators; a \(C^*\)-algebraic framework, J. Funct. Anal., 250, 42-67 (2007) · Zbl 1173.46048 · doi:10.1016/j.jfa.2007.05.020 |
| [42] | Nistor, V.; Prudhon, N., Exhaustive families of representations and spectra of pseudodifferential operators, J. Oper. Theory, 78, 247-279 (2017) · Zbl 1424.46073 |
| [43] | Nistor, V.; Weinstein, A.; Xu, P., Pseudodifferential operators on differential groupoids, Pacific J. Math., 189, 117-152 (1999) · Zbl 0940.58014 · doi:10.2140/pjm.1999.189.117 |
| [44] | Paterson, ALT, Continuous family groupoids, Homol. Homotopy Appl., 2, 89-104 (2000) · Zbl 0992.22001 · doi:10.4310/HHA.2000.v2.n1.a6 |
| [45] | Paterson, A.L.T.: The analytic index for proper, Lie groupoid actions. Groupoids in Analysis. Geometry, and Physics (Boulder, CO, 1999), Contemporary Mathematics, vol. 282, pp. 115-135. American Mathematical Society, Providence (2001) · Zbl 0995.19005 |
| [46] | Rabinovich, V.; Schulze, B-W; Tarkhanov, N., Boundary value problems in oscillating cuspidal wedges, Rocky Mt. J. Math., 34, 1399-1471 (2004) · Zbl 1083.35151 · doi:10.1216/rmjm/1181069808 |
| [47] | Raeburn, I., Williams, D.P.: Morita Equivalence and Continuous-trace \(C^*\)-Algebras, Mathematical Surveys and Monographs, vol. 60. American Mathematical Society, Providence (1998) · Zbl 0922.46050 |
| [48] | Renault, J.: A groupoid approach to \(C^{\ast }\)-algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980) · Zbl 0433.46049 |
| [49] | Renault, J.: Induced representations and hypergroupoids. SIGMA Symmetry Integr. Geom. Methods Appl. 10:Paper 057,18 (2014) · Zbl 1309.22007 |
| [50] | Rieffel, MA, Induced representations of \(C^{\ast }\)-algebras, Adv. Math., 13, 176-257 (1974) · Zbl 0284.46040 · doi:10.1016/0001-8708(74)90068-1 |
| [51] | Schulze, B.-W.: Pseudo-Differential Operators on Manifolds with Singularities, Studies in Mathematics and its Applications, vol. 24. North-Holland Publishing Co., Amsterdam (1991) · Zbl 0747.58003 |
| [52] | Tu, J-L, Non-Hausdorff groupoids, proper actions and \(K\)-theory, Doc. Math., 9, 565-597 (2004) · Zbl 1058.22005 |
| [53] | Van Erp, E., Yuncken, R.: A groupoid approach to pseudodifferential operators (2015). arXiv e-prints, arXiv:1511.01041 · Zbl 1385.58010 |
| [54] | Erp, E.; Yuncken, R., On the tangent groupoid of a filtered manifold, Bull. Lond. Math. Soc., 49, 1000-1012 (2017) · Zbl 1385.58010 · doi:10.1112/blms.12096 |
| [55] | Vasy, A., Propagation of singularities in many-body scattering, Ann. Sci. École Norm. Sup. (4), 34, 313-402 (2001) · Zbl 0987.81114 · doi:10.1016/S0012-9593(01)01066-7 |
| [56] | Williams, D.P.: Crossed Products of \(C^\ast \)-algebras, Mathematical Surveys and Monographs, vol. 134. American Mathematical Society, Providence (2007) · Zbl 1119.46002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
