Coarse geometry and Callias quantisation

Guo, Hao; Hochs, Peter; Mathai, Varghese

doi:10.1090/tran/8202

Coarse geometry and Callias quantisation
HTML articles powered by AMS MathViewer

by Hao Guo, Peter Hochs and Varghese Mathai PDF

Trans. Amer. Math. Soc. 374 (2021), 2479-2520 Request permission

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 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 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) Astérisque, No. 32-33, Soc. Math. France, Paris, 1976, pp. 43–72. 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 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 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 10.1023/A:1020842205711
Maxim Braverman, The index theory on non-compact manifolds with proper group action, J. Geom. Phys. 98 (2015), 275–284. MR 3414957, DOI 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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, DOI 10.1017/CBO9780511526206
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 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 10.1016/j.aim.2010.03.023
Eckhard Meinrenken, Symplectic surgery and the $\textrm {Spin}^c$-Dirac operator, Adv. Math. 134 (1998), no. 2, 240–277. MR 1617809, DOI 10.1006/aima.1997.1701
Eckhard Meinrenken and Reyer Sjamaar, Singular reduction and quantization, Topology 38 (1999), no. 4, 699–762. MR 1679797, DOI 10.1016/S0040-9383(98)00012-3
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 10.2307/1970335
Paul-Emile Paradan, Localization of the Riemann-Roch character, J. Funct. Anal. 187 (2001), no. 2, 442–509. MR 1875155, DOI 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 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 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, DOI 10.1090/cbms/090
John Roe, Positive curvature, partial vanishing theorems and coarse indices, Proc. Edinb. Math. Soc. (2) 59 (2016), no. 1, 223–233. MR 3439130, DOI 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 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 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 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 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