A reconstruction theorem for almost-commutative spectral triples

Creators: Ćaćić, Branimir

Abstract

We propose an expansion of the definition of almost-commutative spectral triple that accommodates non-trivial fibrations and is stable under inner fluctuation of the metric, and then prove a reconstruction theorem for almost-commutative spectral triples under this definition as a simple consequence of Connes's reconstruction theorem for commutative spectral triples. Along the way, we weaken the orientability hypothesis in the reconstruction theorem for commutative spectral triples, and following Chakraborty and Mathai, prove a number of results concerning the stability of properties of spectral triples under suitable perturbation of the Dirac operator.

Additional Information

© 2011 Springer. Received: 8 March 2011; Revised: 6 September 2011; Accepted: 23 September 2011; Published online: 26 October 2011. The author would like to thank his advisor, Matilde Marcolli, for her extensive comments and for her advice, support and patience, Partha Sarathi Chakraborty and Nigel Higson for technical advice on some of the stability results, and Nikolaĭ Ivankov, Helge Krüger, Bram Mesland, Kevin Teh, Rafael Torres and Dapeng Zhang for helpful comments and conversations. The author also gratefully acknowledges the financial and administrative support of the Department of Mathematics of the California Institute of Technology, as well as the hospitality and support of the Max Planck Institute for Mathematics.

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