6 A. L. CAREY, V. GAYRAL, A. RENNIE, and F. A. SUKOCHEV The applications to the index theory for Dirac-type operators on manifolds of bounded geometry are contained in Chapter 4. Also in Chapter 4 is a version of the Atiyah L2-index Theorem that applies to covering spaces of noncompact manifolds of bounded geometry. In Chapter 5 we make a start on noncommutative examples, looking at torus actions on C∗-algebras and at the Moyal plane. Any further treatment of noncommutative examples would add considerably to the length of this article, and is best left for another place. Acknowledgements. This research was supported by the Australian Re- search Council, the Max Planck Institute for Mathematics (Bonn) and the Banff International Research Station. A. Carey also thanks the Alexander von Humboldt Stiftung and colleagues in the University of unster and V. Gayral also thanks the CNRS and the University of Metz. We would like to thank our colleagues John Phillips and Magda Georgescu for discussions on nonunital spectral flow. Special thanks are given to Steven Lord, Roger Senior, Anna Tomskova and Dmitriy Zanin for careful readings of this manuscript at various stages. We also thank Emmanuel Pedon for discussions on the Kato inequality, Raimar Wulkenhaar for discussions on index computations for the Moyal plane, and Gilles Carron, Thierry Coulhon, Batu uneysu and Yuri Korduykov for discussions related to heat-kernels on non- compact manifolds. Finally, it is a pleasure to thank the referees: their exceptional efforts have greatly improved this work.
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