A short proof of an index theorem

Kucerovsky, Dan

doi:10.1090/S0002-9939-01-06164-0

A short proof of an index theorem
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Proc. Amer. Math. Soc. 129 (2001), 3729-3736 Request permission

Abstract:

We give a $KK$-theoretical proof of an index theorem for Dirac-Schrödinger operators on a noncompact manifold.

References

N. Anghel, $L^2$-index theorems for perturbed Dirac operators, thesis, Ohio State University, 1989.
N. Anghel, $L^2$-index formulae for perturbed Dirac operators, Comm. Math. Phys. 128 (1990), no. 1, 77–97. MR 1042444
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, Global theory of elliptic operators, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969) Univ. Tokyo Press, Tokyo, 1970, pp. 21–30. MR 0266247
P. Lorenzen, Die Definition durch vollständige Induktion, Monatsh. Math. Phys. 47 (1939), 356–358. MR 38, DOI 10.1007/BF01695507
Saad Baaj and Pierre Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les $C^{\ast }$-modules hilbertiens, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 21, 875–878 (French, with English summary). MR 715325
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
Paul Baum, Ronald G. Douglas, and Michael E. Taylor, Cycles and relative cycles in analytic $K$-homology, J. Differential Geom. 30 (1989), no. 3, 761–804. MR 1021372
Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
Constantine Callias, Axial anomalies and index theorems on open spaces, Comm. Math. Phys. 62 (1978), no. 3, 213–234. MR 507780
J. Brüning and H. Moscovici, $L^2$-index for certain Dirac-Schrödinger operators, Max Planck Institut für Math., report, (1991, June), 1–30.
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
Jeff Cheeger, On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 91–146. MR 573430
Paul R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973), 401–414. MR 0369890, DOI 10.1016/0022-1236(73)90003-7
Jeff Fox, Peter Haskell, and Iain Raeburn, Kasparov products, $KK$-equivalence, and proper actions of connected reductive Lie groups, J. Operator Theory 22 (1989), no. 1, 3–29. MR 1026072
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
Nigel Higson, $C^*$-algebra extension theory and duality, J. Funct. Anal. 129 (1995), no. 2, 349–363. MR 1327182, DOI 10.1006/jfan.1995.1054
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
Matthias Lesch, Deficiency indices for symmetric Dirac operators on manifolds with conic singularities, Topology 32 (1993), no. 3, 611–623. MR 1231967, DOI 10.1016/0040-9383(93)90012-K
G. G. Kasparov, The operator $K$-functor and extensions of $C^*$-algebras, Tr. Math. USSR Izvestiya 16 (1981), 513–636.
Dan Kucerovsky, The $KK$-product of unbounded modules, $K$-Theory 11 (1997), no. 1, 17–34. MR 1435704, DOI 10.1023/A:1007751017966
D. Kucerovsky, Kasparov products in $KK$-theory and unbounded operators with applications to index theory, thesis, University of Oxford (Magd.), 1995.
J. A. Mingo and W. J. Phillips, Equivariant triviality theorems for Hilbert $C^{\ast }$-modules, Proc. Amer. Math. Soc. 91 (1984), no. 2, 225–230. MR 740176, DOI 10.1090/S0002-9939-1984-0740176-0
William L. Paschke, $K$-theory for commutants in the Calkin algebra, Pacific J. Math. 95 (1981), no. 2, 427–434. MR 632196
M. A. Shubin, Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Translated from the Russian by Stig I. Andersson. MR 883081, DOI 10.1007/978-3-642-96854-9
Georges Skandalis, Une notion de nucléarité en $K$-théorie (d’après J. Cuntz), $K$-Theory 1 (1988), no. 6, 549–573 (French, with English summary). MR 953916, DOI 10.1007/BF00533786
Alain Valette, A remark on the Kasparov groups $\textrm {Ext}(A,\,B)$, Pacific J. Math. 109 (1983), no. 1, 247–255. MR 716300