Coarse geometry and Callias quantisation

Guo, Hao; Hochs, Peter; Mathai, Varghese

doi:10.1090/tran/8202

Coarse geometry and Callias quantisation

Authors: Hao Guo, Peter Hochs and Varghese Mathai
Journal: Trans. Amer. Math. Soc. 374 (2021), 2479-2520
MSC (2020): Primary 19K56; Secondary 46L08, 53D50, 46L80
DOI: https://doi.org/10.1090/tran/8202
Published electronically: January 26, 2021
MathSciNet review: 4223023
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Consider a proper, isometric action by a unimodular, locally compact group $G$ on a complete Riemannian manifold $M$. For equivariant elliptic operators that are invertible outside a cocompact subset of $M$, we show that a localised index in the $K$-theory of the maximal group $C^*$-algebra of $G$ is well-defined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions.

By using the maximal group $C^*$-algebra instead of its reduced counterpart, we can apply the trace given by integration over $G$ to recover an index defined earlier by the last two authors, and developed further by Braverman, in terms of sections invariant under the group action. This leads to refinements of index-theoretic obstructions to Riemannian metrics of positive scalar curvature on noncompact manifolds, and also on orbifolds and other singular quotients of proper group actions. As a motivating application in another direction, we prove a version of Guillemin and Sternberg’s quantisation commutes with reduction principle for equivariant indices of $\mathrm {Spin}^{c}$ Callias-type operators.

References

Herbert Abels, Parallelizability of proper actions, global $K$-slices and maximal compact subgroups, Math. Ann. 212 (1974/75), 1–19. MR 375264, DOI https://doi.org/10.1007/BF01343976
Nicolae Anghel, An abstract index theorem on noncompact Riemannian manifolds, Houston J. Math. 19 (1993), no. 2, 223–237. MR 1225459
N. Anghel, On the index of Callias-type operators, Geom. Funct. Anal. 3 (1993), no. 5, 431–438. MR 1233861, DOI https://doi.org/10.1007/BF01896237
M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) Soc. Math. France, Paris, 1976, pp. 43–72. Astérisque, No. 32-33. MR 0420729
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI https://doi.org/10.1017/S0305004100049410
Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and $K$-theory of group $C^\ast $-algebras, $C^\ast $-algebras: 1943–1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR 1292018, DOI https://doi.org/10.1090/conm/167/1292018
N. Bourbaki, Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1306, Hermann, Paris, 1963 (French). MR 0179291
Maxim Braverman, Index theorem for equivariant Dirac operators on noncompact manifolds, $K$-Theory 27 (2002), no. 1, 61–101. MR 1936585, DOI https://doi.org/10.1023/A%3A1020842205711
Maxim Braverman, The index theory on non-compact manifolds with proper group action, J. Geom. Phys. 98 (2015), 275–284. MR 3414957, DOI https://doi.org/10.1016/j.geomphys.2015.08.014
Maxim Braverman and Simone Cecchini, Callias-type operators in von Neumann algebras, J. Geom. Anal. 28 (2018), no. 1, 546–586. MR 3745871, DOI https://doi.org/10.1007/s12220-017-9832-1
Ulrich Bunke, A $K$-theoretic relative index theorem and Callias-type Dirac operators, Math. Ann. 303 (1995), no. 2, 241–279. MR 1348799, DOI https://doi.org/10.1007/BF01460989
Constantine Callias, Axial anomalies and index theorems on open spaces, Comm. Math. Phys. 62 (1978), no. 3, 213–234. MR 507780
Simone Cecchini, Callias-type operators in $C^*$-algebras and positive scalar curvature on noncompact manifolds, J. Topol. Anal. 12 (2020), no. 4, 897–939. MR 4146567, DOI https://doi.org/10.1142/S1793525319500687

Xiaoman Chen, Jinming Wang, Zhizhang Xie, and Guoliang Yu, Delocalized eta invariants, cyclic cohomology and higher rho-invariants, arXiv:1901.02378, 2019.

Johannes Ebert, Elliptic regularity for Dirac operators on families of noncompact manifolds, arXiv:1905.12299, 2019.

Guihua Gong, Qin Wang, and Guoliang Yu, Geometrization of the strong Novikov conjecture for residually finite groups, J. Reine Angew. Math. 621 (2008), 159–189. MR 2431253, DOI https://doi.org/10.1515/CRELLE.2008.061

Mikhael Gromov and H. Blaine Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196 (1984). MR 720933

V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI https://doi.org/10.1007/BF01398934

Hao Guo, Index of equivariant Callias-type operators and invariant metrics of positive scalar curvature, arXiv:1803.05558, 2018.

Hao Guo, Peter Hochs, and Varghese Mathai, Equivariant Callias index theory via coarse geometry, arXiv:1902.07391, 2019.

Hao Guo, Zhizhang Xie, and Guoliang Yu, A Lichnerowicz vanishing theorem for the maximal Roe algebra, arXiv:1905.12299, 2019.

Uffe Haagerup and Agata Przybyszewska, Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces, arXiv:math/0606794, 2006.

Bernhard Hanke, Daniel Pape, and Thomas Schick, Codimension two index obstructions to positive scalar curvature, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 6, 2681–2710 (English, with English and French summaries). MR 3449594

Nigel Higson and John Roe, Analytic $K$-homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. Oxford Science Publications. MR 1817560

Peter Hochs and Varghese Mathai, Geometric quantization and families of inner products, Adv. Math. 282 (2015), 362–426. MR 3374530, DOI https://doi.org/10.1016/j.aim.2015.07.004

Peter Hochs and Varghese Mathai, Quantising proper actions on Spin$^c$-manifolds, Asian J. Math. 21 (2017), no. 4, 631–685. MR 3691850, DOI https://doi.org/10.4310/AJM.2017.v21.n4.a2

Peter Hochs and Yanli Song, An equivariant index for proper actions III: The invariant and discrete series indices, Differential Geom. Appl. 49 (2016), 1–22. MR 3573821, DOI https://doi.org/10.1016/j.difgeo.2016.07.003

Peter Hochs and Yanli Song, An equivariant index for proper actions I, J. Funct. Anal. 272 (2017), no. 2, 661–704. MR 3571905, DOI https://doi.org/10.1016/j.jfa.2016.08.024

Peter Hochs and Yanli Song, An equivariant index for proper actions II: properties and applications, J. Noncommut. Geom. 12 (2018), no. 1, 157–193. MR 3782056, DOI https://doi.org/10.4171/JNCG/273

Peter Hochs and Hang Wang, An equivariant orbifold index for proper actions, J. Geom. Phys. 154 (2020), 103710, 11. MR 4099480, DOI https://doi.org/10.1016/j.geomphys.2020.103710

Jens Kaad and Matthias Lesch, A local global principle for regular operators in Hilbert $C^*$-modules, J. Funct. Anal. 262 (2012), no. 10, 4540–4569. MR 2900477, DOI https://doi.org/10.1016/j.jfa.2012.03.002

Dan Kucerovsky, A short proof of an index theorem, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3729–3736. MR 1860509, DOI https://doi.org/10.1090/S0002-9939-01-06164-0

Dan Kucerovsky, Functional calculus and representations of $C_0(\Bbb C)$ on a Hilbert module, Q. J. Math. 53 (2002), no. 4, 467–477. MR 1949157, DOI https://doi.org/10.1093/qjmath/53.4.467

E. C. Lance, Hilbert $C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992

Yiannis Loizides, Quasi-polynomials and the singular $[Q,R]=0$ theorem, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 090, 15. MR 4032363, DOI https://doi.org/10.3842/SIGMA.2019.090

Varghese Mathai and Weiping Zhang, Geometric quantization for proper actions, Adv. Math. 225 (2010), no. 3, 1224–1247. With an appendix by Ulrich Bunke. MR 2673729, DOI https://doi.org/10.1016/j.aim.2010.03.023

Eckhard Meinrenken, Symplectic surgery and the ${\rm Spin}^c$-Dirac operator, Adv. Math. 134 (1998), no. 2, 240–277. MR 1617809, DOI https://doi.org/10.1006/aima.1997.1701

Eckhard Meinrenken and Reyer Sjamaar, Singular reduction and quantization, Topology 38 (1999), no. 4, 699–762. MR 1679797, DOI https://doi.org/10.1016/S0040-9383%2898%2900012-3

Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, DOI https://doi.org/10.2307/1970335

Paul-Emile Paradan, Localization of the Riemann-Roch character, J. Funct. Anal. 187 (2001), no. 2, 442–509. MR 1875155, DOI https://doi.org/10.1006/jfan.2001.3825

Paul-Emile Paradan and Michèle Vergne, Equivariant Dirac operators and differentiable geometric invariant theory, Acta Math. 218 (2017), no. 1, 137–199. MR 3710795, DOI https://doi.org/10.4310/ACTA.2017.v218.n1.a3

John Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (1993), no. 497, x+90. MR 1147350, DOI https://doi.org/10.1090/memo/0497

John Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, vol. 90, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. MR 1399087

John Roe, Positive curvature, partial vanishing theorems and coarse indices, Proc. Edinb. Math. Soc. (2) 59 (2016), no. 1, 223–233. MR 3439130, DOI https://doi.org/10.1017/S0013091514000236

Thomas Schick, The topology of positive scalar curvature, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1285–1307. MR 3728662

Youliang Tian and Weiping Zhang, An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math. 132 (1998), no. 2, 229–259. MR 1621428, DOI https://doi.org/10.1007/s002220050223

Ján Špakula and Rufus Willett, Maximal and reduced Roe algebras of coarsely embeddable spaces, J. Reine Angew. Math. 678 (2013), 35–68. MR 3056102, DOI https://doi.org/10.1515/crelle.2012.019

Zhizhang Xie and Guoliang Yu, Delocalized eta invariants, algebraicity, and $K$-theory of group $C^*$-algebras, arXiv:1805.07617, 2018.

Guoliang Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2) 147 (1998), no. 2, 325–355. MR 1626745, DOI https://doi.org/10.2307/121011

Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR 1728880, DOI https://doi.org/10.1007/s002229900032

Guoliang Yu, A characterization of the image of the Baum-Connes map, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 649–657. MR 2732068