Skip to main content Accessibility help


  • IAIN FORSYTH (a1) (a2) (a3) and ADAM RENNIE (a4)


We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$ -deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).


Corresponding author


Hide All
[1] Atiyah, M. and Hirzebruch, F., ‘Spin-manifolds and group actions’, in: Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham) (Springer, Berlin, 1970), 1828.
[2] Baaj, S. and Julg, P., ‘Théorie bivariante de Kasparov et opérateurs non bornées dans les C -modules hilbertiens’, C. R. Acad. Sci. Paris 296 (1983), 875878.
[3] Berline, N., Getzler, E. and Vergne, M., Heat Kernels and Dirac Operators (Springer, Berlin, 2004).
[4] Brain, S., Mesland, B. and van Suijlekom, W. D., ‘Gauge theory for spectral triples and the unbounded Kasparov product’, J. Noncommut. Geom. 10(1) (2016), 135206.
[5] Carey, A. L., Gayral, V., Rennie, A. and Sukochev, F. A., ‘Index theory for locally compact noncommutative geometries’, Mem. Amer. Math. Soc. 231(1085) (2014).
[6] Carey, A. L., Neshveyev, S., Nest, R. and Rennie, A., ‘Twisted cyclic theory, equivariant KK-theory and KMS states’, J. reine angew. Math. 650 (2011), 161191.
[7] Chamseddine, A., Connes, A. and van Suijlekom, W., ‘Inner fluctuations in noncommutative geometry without the first order condition’, J. Geom. Phys. 73 (2013), 222234.
[8] Connes, A., Noncommutative Geometry (Academic Press, San Diego, CA, 1994).
[9] Connes, A. and Landi, G., ‘Noncommutative manifolds: the instanton algebra and isospectral deformations’, Commun. Math. Phys. 221 (2001), 141159.
[10] Dabrowski, L. and Sitarz, A., ‘Noncommutative circle bundles and new Dirac operators’, Commun. Math. Phys. 318 (2013), 111130.
[11] Dabrowski, L., Sitarz, A. and Zucca, A., ‘Dirac operators on noncommutative principal circle bundles’, Int. J. Geom. Methods Mod. Phys. 11 (2014).
[12] Gracia-Bondía, J. M., Várilly, J. C. and Figueroa, H., Elements of Noncommutative Geometry (Birkhäuser, Boston, 2001).
[13] Higson, N. and Roe, J., Analytic K-Homology (Oxford University Press, Oxford, 2000).
[14] Kaad, J. and Lesch, M., ‘Spectral flow and the unbounded Kasparov product’, Adv. Math. 248 (2013), 495530.
[15] Kasparov, G. G., ‘The operator K-functor and extensions of C -algebras’, Math. USSR Izv. 16 (1981), 513572.
[16] Kucerovsky, D., ‘The KK-product of unbounded modules’, J. K-Theory 11 (1997), 1734.
[17] Kucerovsky, D., ‘A lifting theorem giving an isomorphism of KK-products in bounded and unbounded KK-theory’, J. Operator Theory 44 (2000), 255275.
[18] Lance, E. C., Hilbert C -Modules (Cambridge University Press, Cambridge, 1995).
[19] Lawson, H. B. and Michelsohn, M.-L., Spin Geometry (Princeton University Press, Princeton, NJ, 1989).
[20] Mesland, B., ‘Unbounded bivariant K-theory and correspondences in noncommutative geometry’, J. Reine Angew. Math. 691 (2014), 101172.
[21] Mesland, B. and Rennie, A., ‘Nonunital spectral triples and metric completeness in unbounded KK-theory’, J. Funct. Anal. 271(9) (2016), 24602538.
[22] Pask, D. and Rennie, A., ‘The noncommutative geometry of graph C -algebras I: the index theorem’, J. Funct. Anal. 233 (2006), 92134.
[23] Phillips, N. C., Equivariant K-Theory and Freeness of Group Actions on C -Algebras (Springer, Berlin, 1987).
[24] Raeburn, I. and Williams, D. P., Morita Equivalence and Continuous-Trace C -algebras (American Mathematical Society, Providence, RI, 1998).
[25] Reinhart, B. L., ‘Foliated manifolds with bundle-like metrics’, Ann. of Math. (2) 69 (1959), 119132.
[26] Rieffel, M. A., ‘Deformation quantization for actions of ℝ d ’, Mem. Amer. Math. Soc. 106(506) (1993).
[27] Schwieger, K. and Wagner, S., ‘Free actions of compact groups on C -algebras, part I’, Adv. Math. 317 (2017), 224266.
[28] Slebarski, S., ‘Dirac operators on a compact Lie group’, Bull. Lond. Math. Soc. 17 (1985), 579583.
[29] Várilly, J. C., An Introduction to Noncommutative Geometry (European Mathematical Society, Zurich, 2006).
[30] Zucca, A., Dirac Operators on Quantum Principal G-Bundles (Digital Library, Scuola Internazionale Superiore di Studi Avanzati, Trieste, 2013).
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed