Abstract
Let \({\mathcal{A}}\) be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then \({\mathcal{A}}^{-1}\rtimes G \) is KK-theoretically Poincaré dual to \(\big(\mathcal A {\hat {\otimes}_{C_0(X)}} C_\tau (X)\big) \rtimes G\) , where \({\mathcal{A}}^{-1}\) is the inverse of \({\mathcal{A}}\) in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov’s duality theorem. As applications we obtain a version of the above Poincaré duality with X replaced by a compact G-manifold M and Poincaré dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum- Connes conjecture.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anantharaman Delaroche, C., Renault, J.: Amenable groupoids. With a foreword by Georges Skandalis and Appendix B by E. Germain. Monographies de L’Enseignement Mathématique 36. L’Enseignement Mathématique, Geneva, 2000. 196 pp
Atiyah M. and Segal G. (2004). Twisted K-theory. Ukr. Mat. Visn. 1(3): 287–330
Baum P., Connes A. and Higson N. (1994). Classifying space for proper actions and K-theory of group C*-algebras. Contemp. Math. 167: 241–291
Brodzki, J., Mathai, V., Rosenberg, J., Szabo, R.J.: D-branes, RR-fields and duality on noncommutative manifolds. Preprint: arXiv:hep-th/0607020 v1, 4 July (2006)
Crocker D., Kumjian A., Raeburn I. and Williams D.P. (1997). An equivariant Brauer group and actions of groups on C*-algebras. J. Funct. Anal. 146: 151–184
Echterhoff S. and Williams D.P. (2001). Locally inner actions on C 0(X)-algebras. J. Oper. Theory 45(1): 131–160
Emerson H. (2003). Noncommutative Poincaré duality for boundary actions of hyperbolic groups. J. Reine Angew. Math. 564: 1–33
Emerson H. (2003). The Baum-Connes conjecture, noncommutative Poincaré duality and the boundary of the free group. Int. J. Math. Math. Sci. 38: 2425–2445
Emerson H. and Meyer R. (2007). A descent principle for the Dirac dual Dirac method. Topology 46: 185–209
Higson N. and Kasparov G. (2001). E-theory and KK-theory for groups which act properly and isometrically on Hilbert space. Invent. Math. 144: 23–74
Husemoller, D.: Fibre bundles, third edition. Graduate Texts in Mathematics, 20, xx+353 pp. Springer, New York (1994)
Kaminker J. and Putnam I. (1997). K-theoretic duality of shifts of finite type. Commun. Math. Phys. 187(3): 509–522
Kasparov G. (1988). Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91: 147–201
Kasparov G. and Skandalis G. (2003). Groups acting properly on “bolic” spaces and the Novikov conjecture. Ann. Math. 158: 165–206
Kobayashi S. and Nomizu K. (1996). Foundations of Differential Geometry, Vol. II. Wiley, New York
Meyer R. (2001). Generalized fixed-point algebras and square integrable representations. J. Funct. Anal. 186: 167–195
Müller, C., Wockel, C.: Equivalences of smooth and continuous principal bundles with infinite-dimensional structure groups. Preprint, arXiv:math.DG/0604142v1, 6 April (2006)
Packer J. and Raeburn I. (1989). Twisted crossed products of C*-algebras. Proc. Camb. Philos. Soc. 106(2): 293–311
Pflaum, M.J.: Analytic and geometric study of stratified spaces. Lecture Notes in Mathematics 1768, Springer, Heidelberg (2000)
Raeburn I. and Williams D.-P. (1985). Pull-backs of C*-algebras and crossed products by certain diagonal actions. Trans. Am. Math. Soc. 287(2): 755–777
Tu J.-L. (1999). La conjecture de Novikov pour les fouilletages hyperboliques. K-Theory 16(2): 129–184
Tu J.-L. (1999). La conjecture de Baum-Connes pour les feuilletages moyennables. K-Theory 17(3): 215–264
Tu, J.-L.: Twisted K-theory and Poincaré duality. Preprint arXiv:math.KT/0609556v1, 20 Sep (2006)
Williams D.P. (1989). The structure of crossed products by smooth actions. J. Austral. Math. Soc. Ser. A 47(2): 226–235
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the EU-Network Quantum Spaces and Noncommutative Geometry (Contract HPRN-CT-2002-00280) and the Deutsche Forschungsgemeinschaft (SFB 478) and by the National Science and Engineering Research Council of Canada Discovery Grant program.
Rights and permissions
About this article
Cite this article
Echterhoff, S., Emerson, H. & Kim, H.J. KK-theoretic duality for proper twisted actions. Math. Ann. 340, 839–873 (2008). https://doi.org/10.1007/s00208-007-0171-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-007-0171-6